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Brane Effective Actions, Kappa-Symmetry and Applications

  • Joan SimónEmail author
Open Access
Review Article

Abstract

This is a review on brane effective actions, their symmetries and some of their applications. Its first part covers the Green-Schwarz formulation of single M- and D-brane effective actions focusing on kinematical aspects: the identification of their degrees of freedom, the importance of world volume diffeomorphisms and kappa symmetry to achieve manifest spacetime covariance and supersymmetry, and the explicit construction of such actions in arbitrary on-shell supergravity backgrounds.

Its second part deals with applications. First, the use of kappa symmetry to determine supersymmetric world volume solitons. This includes their explicit construction in flat and curved backgrounds, their interpretation as Bogomol’nyi-Prasad-Sommerfield (BPS) states carrying (topological) charges in the supersymmetry algebra and the connection between supersymmetry and Hamiltonian BPS bounds. When available, I emphasise the use of these solitons as constituents in microscopic models of black holes. Second, the use of probe approximations to infer about the non-trivial dynamics of strongly-coupled gauge theories using the anti de Sitter/conformal field theory (AdS/CFT) correspondence. This includes expectation values of Wilson loop operators, spectrum information and the general use of D-brane probes to approximate the dynamics of systems with small number of degrees of freedom interacting with larger systems allowing a dual gravitational description.

Its final part briefly discusses effective actions for N D-branes and M2-branes. This includes both Super-Yang-Mills theories, their higher-order corrections and partial results in covariantising these couplings to curved backgrounds, and the more recent supersymmetric Chern-Simons matter theories describing M2-branes using field theory, brane constructions and 3-algebra considerations.

1 Introduction

Branes have played a fundamental role in the main string theory developments of the last twenty years:
  1. 1.

    The unification of the different perturbative string theories using duality symmetries [312, 495] relied strongly on the existence of non-perturbative supersymmetric states carrying Ramond-Ramond (RR) charge for their first tests.

     
  2. 2.

    The discovery of D-branes as being such non-perturbative states, but still allowing a perturbative description in terms of open strings [423].

     
  3. 3.

    The existence of decoupling limits in string theory providing non-perturbative formulations in different backgrounds. This gave rise to Matrix theory [48] and the anti de Sitter/conformal field theory (AdS/CFT) correspondence [366]. The former provides a non-perturbative formulation of string theory in Minkowski spacetime and the latter in AdS × M spacetimes.

     
At a conceptual level, these developments can be phrased as follows:
  1. 1.

    Dualities guarantee that fundamental strings are no more fundamental than other dynamical extended objects in the theory, called branes.

     
  2. 2.

    D-branes, a subset of the latter, are non-perturbative states1 defined as dynamical hyper-surfaces where open strings can end. Their weakly-coupled dynamics is controlled by the microscopic conformal field theory description of open strings satisfying Dirichlet boundary conditions. Their spectrum contains massless gauge fields. Thus, D-branes provide a window into non-perturbative string theory that, at low energies, is governed by supersymmetric gauge theories in different dimensions.

     
  3. 3.

    On the other hand, any source of energy interacts with gravity. Thus, if the number of branes is large enough, one expects a closed string description of the same system. The crucial realisations in [48] and [366] are the existence of kinematical and dynamical regimes in which the full string theory is governed by either of these descriptions: the open or the closed string ones.

     
The purpose of this review is to describe the kinematical properties characterising the super-symmetric gauge theories emerging as brane effective field theories in string and M-theory, and some of their important applications. In particular, I will focus on D-branes, M2-branes and M5-branes. For a schematic representation of the review’s content, see Figure 1.
Figure 1

Layout of the main relations covered in this review.

These effective theories depend on the number of branes in the system and the geometry they probe. When a single brane is involved in the dynamics, these theories are abelian and there exists a spacetime covariant and manifestly supersymmetric formulation, extending the Green-Schwarz worldsheet one for the superstring. The main concepts I want to stress in this part are
  1. a)

    the identification of their dynamical degrees of freedom, providing a geometrical interpretation when available,

     
  2. b)

    the discussion of the world volume gauge symmetries required to achieve spacetime covariance and supersymmetry. These will include world volume diffeomorphisms and kappa symmetry,

     
  3. c)

    the description of the couplings governing the interactions in these effective actions, their global symmetries and their interpretation in spacetime,

     
  4. d)

    the connection between spacetime and world volume supersymmetry through gauge fixing,

     
  5. e)

    the description of the regime of validity of these effective actions.

     
For multiple coincident branes, these theories are supersymmetric non-abelian gauge field theories. The second main difference from the abelian set-up is the current absence of a spacetime covariant and supersymmetric formulation, i.e., there is no known world volume diffeomorphic and kappa invariant formulation for them. As a consequence, we do not know how to couple these degrees of freedom to arbitrary (supersymmetric) curved backgrounds, as in the abelian case, and we must study these on an individual background case.
The covariant abelian brane actions provide a generalisation of the standard charged particle effective actions describing geodesic motion to branes propagating on arbitrary on-shell super-gravity backgrounds. Thus, they offer powerful tools to study the dynamics of string/M-theory in regimes that will be precisely described. In the second part of this review, I describe some of their important applications. These will be split into two categories: supersymmetric world volume solitons and dynamical aspects of the brane probe approximation. Solitons will allow me to
  1. a)

    stress the technical importance of kappa symmetry in determining these configurations, linking Hamiltonian methods with supersymmetry algebra considerations,

     
  2. b)

    prove the existence of string theory Bogomol’nyi-Prasad-Sommerfield (BPS) states carrying different (topological) charges,

     
  3. c)

    briefly mention microscopic constituent models for certain black holes.

     
Regarding the dynamical applications, the intention is to provide some dynamical interpretation to specific probe calculations appealing to the AdS/CFT correspondence [13] in two main situations
  1. a)

    classical on-shell probe action calculations providing a window to strongly coupled dynamics, spectrum and thermodynamics of non-abelian gauge theories by working with appropriate backgrounds with suitable boundary conditions,

     
  2. b)

    probes approximating the dynamics of small systems interacting among themselves and with larger systems, when the latter can be reliably replaced by supergravity backgrounds.

     

Content of the review: I start with a very brief review of the Green-Schwarz formulation of the superstring in Section 2. This is an attempt at presenting the main features of this formulation since they are universal in brane effective actions. This is supposed to be a reminder for those readers having a standard textbook knowledge of string theory, or simply as a brief motivation for newcomers, but it is not intended to be self-contained. It also helps to set up the notation for the rest of this review.

Section 3 is fully devoted to the kinematic construction of brane effective actions. After describing the general string theory set-up where these considerations apply, it continues in Section 3.1 with the identification of the relevant dynamical degrees of freedom. This is done using open string considerations, constraints from world volume supersymmetry in p + 1 dimensions and the analysis of Goldstone mode in supergravity. A second goal in Section 3.1 is to convey the idea that spacetime covariance and manifest supersymmetry will require these effective actions to be both diffeomorphic and kappa symmetry invariant, where at this stage the latter symmetry is just conjectured, based on our previous world sheet considerations and counting of on-shell degrees of freedom. As a warm-up exercise, in Section 3.2, the bosonic truncations of these effective actions are constructed, focusing on diffeomorphism invariance, spacetime covariance, physical considerations and a set of non-trivial string theory duality checks that are carried in Section 3.3. Then, I proceed to discuss the explicit construction of supersymmetric brane effective actions propagating in a fixed Minkowski spacetime in Section 3.4. This has the virtue of being explicit and provides a bridge towards the more technical and abstract, but also more geometrical, superspace formalism, which provides the appropriate venue to covariantise the results in this particular background to couple the brane degrees of freedom to arbitrary curved backgrounds in Section 3.5. The main result of the latter is that kappa symmetry invariance is achieved whenever the background is an on-shell supergravity background. After introducing the effective actions, I discuss both their global bosonic and fermionic symmetries in Section 3.6, emphasising the difference between space-time and world volume (super)symmetry algebras, before and after gauge fixing world volume diffeomorphisms and kappa symmetry. Last, but not least, I include a discussion on the regime of validity of these effective theories in Section 3.7.

Section 4 develops the general formalism to study supersymmetric bosonic world volume solitons. It is proven in Section 4.1 that any such configuration must satisfy the kappa symmetry preserving condition (215). Reviewing the Hamiltonian formulation of these brane effective actions in 4.2, allows me to establish a link between supersymmetry, kappa symmetry, supersymmetry algebra bounds and their field theory realisations in terms of Hamiltonian BPS bounds in the space of bosonic configurations of these theories. The section finishes connecting these physical concepts to the mathematical notion of calibrations, and their generalisation, in Section 4.3.

In Section 5, I apply the previous formalism in many different examples, starting with vacuum infinite branes, and ranging from BIon configurations, branes within branes, giant gravitons, baryon vertex configurations and supertubes. As an outcome of these results, I emphasise the importance of some of these in constituent models of black holes.

In Section 6, more dynamical applications of brane effective actions are considered. Here, the reader will be briefly exposed to the reinterpretation of certain on-shell classical brane action calculations in specific curved backgrounds and with appropriate boundary conditions, as holographic duals of strongly-coupled gauge theory observables, the existence and properties of the spectrum of these theories, both in the vacuum or in a thermal state, and including their non-relativistic limits. This is intended to be an illustration of the power of the probe approximation technique, rather than a self-contained review of these applications, which lies beyond the scope of these notes. I provide relevant references to excellent reviews covering the material highlighted here in a more exhaustive and pedagogical way.

In Section 7, I summarise the main kinematical facts regarding the non-abelian description of N D-branes and M2-branes. Regarding D-branes, this includes an introduction to super-Yang-Mills theories in p + 1 dimensions, a summary of statements regarding higher-order corrections in these effective actions and the more relevant results and difficulties regarding the attempts to covariantise these couplings to arbitrary curved backgrounds. Regarding M2-branes, I briefly review the more recent supersymmetric Chern-Simons matter theories describing their low energy dynamics, using field theory, 3-algebra and brane construction considerations. The latter provides an explicit example of the geometrisation of supersymmetric field theories provided by brane physics.

The review closes with a brief discussion on some of the topics not covered in this review in Section 8. This includes brief descriptions and references to the superembedding approach to brane effective actions, the description of NS5-branes and KK-monopoles, non-relatistivistic kappa symmetry invariant brane actions, blackfolds or the prospects to achieve a formulation for multiple M5-branes.

In appendices, I provide a brief but self-contained introduction to the superspace formulation of the relevant supergravity theories discussed in this review, describing the explicit constraints required to match the on-shell standard component formulation of these theories. I also include some useful tools to discuss the supersymmetry of AdS spaces and spheres, by embedding them as surfaces in higher-dimensional flat spaces. I establish a one-to-one map between the geometrical Killing spinors in AdS and spheres and the covariantly-constant Killing spinors in their embedding flat spaces.

2 The Green-Schwarz Superstring: A Brief Motivation

The purpose of this section is to briefly review the Green-Schwarz (GS) formulation of the superstring. This is not done in a self-contained way, but rather as a very swift presentation of the features that will turn out to be universal in the formulation of brane effective actions. There exist two distinct formulations for the (super)string:
  1. 1.

    The worldsheet supersymmetry formulation, called the Ramond-Neveu-Schwarz (RNS) formulation2, where supersymmetry in 1 + 1 dimensions is manifest [432, 404].

     
  2. 2.

    The GS formulation, where spacetime supersymmetry is manifest [256, 257, 258].

     

The RNS formulation describes a 1 + 1 dimensional supersymmetric field theory with degrees of freedom transforming under certain representations of some internal symmetry group. After quantisation, its spectrum turns out to be arranged into supersymmetry multiplets of the internal manifold, which is identified with spacetime itself. This formulation has two main disadvantages: the symmetry in the spectrum is not manifest and its extension to curved spacetime backgrounds is not obvious due to the lack of spacetime covariance.

The GS formulation is based on spacetime supersymmetry as its guiding symmetry principle. It allows a covariant extension to curved backgrounds through the existence of an extra fermionic gauge symmetry, kappa symmetry, that is universally linked to spacetime covariance and supersymmetry, as I will review below and in Sections 3 and 4. Unfortunately, its quantisation is much more challenging. The first volume of the Green, Schwarz and Witten book [260] provides an excellent presentation of both these formulations. Below, I just review its bosonic truncation, construct its supersymmetric extension in Minkowski spacetime, and conclude with an extension to curved backgrounds.

Bosonic string: The bosonic GS string action is an extension of the covariant particle action describing geodesic propagation in a fixed curved spacetime with metric gmn
$${S_{{\rm{particle}}}} = - m\int d \tau \,\sqrt {- {{\dot X}^m}{{\dot X}^n}{g_{mn}}(X)} .$$
(1)
The latter is a one-dimensional diffeomorphic invariant action equaling the physical length of the particle trajectory times its mass m. Its degrees of freedom Xm (τ) are the set of maps describing the embedding of the trajectory with affine parameter τ into spacetime, i.e., the local coordinates xm of the spacetime manifold become dynamical fields Xm (τ) on the world line. Diffeomorphisms correspond to the physical freedom in reparameterising the trajectory.
The bosonic string action equals its tension Tf times its area
$${S_{{\rm{string}}}} = - {T_f}\int {{d^2}} \sigma \,\sqrt {- \det {\mathcal G}} .$$
(2)
This is the Nambu-Goto (NG) action [402, 249]: a 1 + 1 dimensional field theory with coordinates σμ μ = 0,1 describing the propagation of a Lorentzian worldsheet, through the set of embeddings Xm (σ) m = 0,1 …d − 1, in a fixed d-dimensional Lorentzian spacetime with metric gmn (X). Notice, this is achieved by computing the determinant of the pullback of the spacetime metric into the worldsheet
$${{\mathcal G}_{\mu \nu}} = {\partial _\mu}{X^m}{\partial _\nu}{X^n}\,{g_{mn}}(X).$$
(3)
Thus, it is a nonlinear interacting theory in 1 + 1 dimensions. Furthermore, it is spacetime covari-ant, invariant under two-dimensional diffeomorphisms and its degrees of freedom {Xm} are scalars in two dimensions, but transform as a vector in d-dimensions.
Just as point particles can be charged under gauge fields, strings can be charged under 2-forms. The coupling to this extra field is minimal, as corresponds to an electrically-charged object, and is described by a Wess-Zumino (WZ) term
$$S = {Q_f}\int {{{\mathcal B}_{(2)}}} ,$$
(4)
where the charge density Qf was introduced and \({\mathcal B}\) stands for the pullback of the d-dimensional bulk 2-form B(2), i.e.,
$${{\mathcal B}_{(2)}} = {1 \over 2}{\partial _\mu}{X^m}{\partial _\nu}{X^n}\,{B_{mn}}(X)\,d{\sigma ^\mu} \wedge d{\sigma ^\nu}.$$
(5)
Thus, the total bosonic action is:
$${S_{{\rm{string}}}} = - {T_f}\int {{d^2}} \sigma \,\sqrt {- \det {\mathcal G}} + {Q_f}\int {{{\mathcal B}_{(2)}}} .$$
(6)
Notice the extra coupling preserves worldsheet diffeomorphism invariance and spacetime covariance. In the string theory context, this effective action describes the propagation of a bosonic string in a closed string background made of a condensate of massless modes (gravitons and Neveu-Schwarz Neveu-Schwarz (NS-NS) 2-form B2(X)). In that case,
$${T_f} = {Q_f} = {1 \over {2\pi \alpha \prime}} = {1 \over {2\pi \ell _s^2}},$$
(7)
where s stands for the length of the fundamental string.
For completeness, let me stress that at the classical level, the dynamics of the background fields (couplings) is not specified. Quantum mechanically, the consistency of the interacting theory defined in Eq. (6) requires the vanishing of the beta functions of the general nonlinear sigma models obtained by expanding the action around a classical configuration when dealing with the quantum path integral. The vanishing of these beta functions requires the background to solve a set of equations that are equivalent to Einstein’s equations coupled to an antisymmetric tensor3. This is illustrated in Figure 2.
Figure 2

Different superstring formulations require curved backgrounds to be on-shell.

Supersymmetric extension: The addition of extra internal degrees of freedom to overcome the existence of a tachyon and the absence of fermions in the bosonic string spectrum leads to super-symmetry. Thus, besides the spacetime vector {Xm}, a set of 1 + 1 scalars fields θα transforming as a spinor under the bulk (internal) Lorentz symmetry SO(1, d − 1) is included.

Instead of providing the answer directly, it is instructive to go over the explicit construction, following [260]. Motivated by the structure appearing in supersymmetric field theories, one looks for an action invariant under the supersymmetry transformations
$$\delta {\theta ^A} = {\epsilon ^A}\,,\quad \quad \delta {X^m} = {\bar \epsilon ^A}{\Gamma ^m}{\theta ^A}\,,$$
(8)
where ϵA is a constant spacetime spinor, \({{\bar\epsilon} ^A} = \epsilon{^{At}}C\) with C the charge conjugation matrix and the label A counts the amount of independent supersymmetries \(A = 1,2, \ldots {\mathcal N}\). It is important to stress that both the dimension d of the spacetime and the spinor representation are arbitrary at this stage.
In analogy with the covariant superparticle [118], consider the action
$${S_1} = - {{{T_f}} \over 2}\int {{d^2}} \sigma \,\sqrt h \,{h^{\mu \nu}}{\Pi _\mu}\cdot{\Pi _\nu}.$$
(9)
This uses the Polyakov form of the action4 involving an auxiliary two-dimensional metric hμν. Πμν stands for the components of the supersymmetric invariant 1-forms
$${\Pi ^m} = d{X^m} + {\bar \theta ^A}{\Gamma ^m}d{\theta ^A},$$
(10)
whereas \({\Pi _\mu} \cdot {\Pi _\nu} \equiv \Pi _\mu ^m\Pi _\nu ^n{\eta _{mn}}\).

Even though the constructed action is supersymmetric and 2d diffeomorphic invariant, the number of on-shell bosonic and fermionic degrees of freedom does not generically match. To reproduce the supersymmetry in the spectrum derived from the quantisation of the RNS formulation, one must achieve such matching.

The current standard resolution to this situation is the addition of an extra term to the action while still preserving supersymmetry. This extra term can be viewed as an extension of the bosonic WZ coupling (4), a point I shall return to when geometrically reinterpreting the action so obtained [294]. Following [260], it turns out the extra term is
$${S_2} = {T_f}\int {{d^2}} \sigma \left({- {\epsilon ^{\mu \nu}}{\partial _\mu}{X^m}\left({{{\bar \theta}^1}{\Gamma _m}{\partial _\nu}{\theta ^1} - {{\bar \theta}^2}{\Gamma _m}{\partial _\nu}{\theta ^2}} \right) + {\epsilon ^{\mu \nu}}{{\bar \theta}^1}{\Gamma ^m}{\partial _\mu}{\theta ^1}{{\bar \theta}^2}{\Gamma _m}{\partial _\nu}{\theta ^2}} \right).$$
(11)
Invariance under global supersymmetry requires, up to total derivatives, the identity
$${\delta _\epsilon}{S_2} = 0\quad \Leftrightarrow \quad 2\bar \epsilon {\Gamma _m}{\psi _{\left[ 1 \right.}}{\bar \psi _2}{\Gamma ^m}{\psi _{\left. 3 \right]}} = 0\,,$$
(12)
for (Ψ1, Ψ2, Ψ3) = (θ, θ′ = ∂θ/∂σ1, θ = dθ/∂σ0). This condition restricts the number of spacetime dimensions d and the spinor representation to be
  • d = 3 and θ is Majorana;

  • d = 4 and θ is Majorana or Weyl;

  • d = 6 and θ is Weyl;

  • d = 10 and θ is Majorana-Weyl.

Let us focus on the last case, which is well known to match the superspace formulation of \({\mathcal N} = 2\) type IIA/B5 Despite having matched the spacetime dimension and the spinor representation by the requirement of spacetime supersymmetry under the addition of the extra action term (11), the number of on-shell bosonic and fermionic degrees of freedom remains unequal. Indeed, Majorana-Weyl fermions in d =10 have 16 real components, which get reduced to 8 on-shell components by Dirac’s equation. The extra \({\mathcal N} = 2\) gives rise to a total of 16 on-shell fermionic degrees of freedom, differing from the 8 bosonic ones coming from the 10-dimensional vector representation after gauge-fixing worldsheet reparameterisations.

The missing ingredient in the above discussion is the existence of an additional fermionic gauge symmetry, kappa symmetry, responsible for the removal of half of the fermionic degrees of freedom.6 This feature fixes the fermionic nature of the local parameter κ(σ) and requires θ to transform by some projector operator
$${\delta _\kappa}\theta = (1 + {\Gamma _\kappa})\,\kappa ,\qquad {\rm{with}}\qquad \Gamma _\kappa ^2 = 1.$$
(13)
Here is a Clifford-valued matrix depending non-trivially on {Xm, θ}. The existence of such transformation is proven in [260].

The purpose of going over this explicit construction is to reinterpret the final action in terms of a more geometrical structure that will be playing an important role in Section 3.1. In more modern language, one interprets as the action describing a superstring propagating in super-Poincaré [259]. The latter is an example of a supermanifold with local coordinates ZM = {Xm, θα}. It uses the analogue of the superfield formalism in global supersymmetric field theories but in supergravity, i.e., with local supersymmetry. The superstring couples to two of these superfields, the supervielbein \(E_M^A(z)\) and the NS-NS 2-form superfield Bac, where the index M stands for curved superspace indices, i.e., M = {m, α}, and the index A for tangent flat superspace indices, i.e., A = {a, α}7.

In the case of super-Poincaré, the components \(E_M^A\) are explicitly given by
$$E_m^a = \delta _m^a\,,\quad \quad E_\alpha ^{\underline{\alpha}} = \delta _\alpha ^{\underline{\alpha}}\,,\quad \quad E_m^{\underline{\alpha}} = 0\,,\quad \quad E_\alpha ^a = {\left({\bar \theta {\Gamma ^a}} \right)_{\underline{\alpha}}}\delta _\alpha ^{\underline{\alpha}}.$$
(14)
These objects allow us to reinterpret the action S1 + S2 in terms of the pullbacks of these bulk objects into the worldsheet extending the bosonic construction
$$\begin{array}{*{20}c} {{{\mathcal G}_{\mu \nu}} = {\Pi _\mu}\cdot{\Pi _\nu} = {\partial _\mu}{Z^M}E_M^a(Z){\partial _\nu}{Z^N}E_N^b(Z){\eta _{ab}}\,,}\\ {{{\mathcal B}_{\mu \nu}} = {\partial _\mu}{Z^M}E_M^A(Z){\partial _\nu}{Z^N}E_N^C(Z)\,{B_{AC}}(Z).\quad \quad}\\ \end{array}$$
(15)
Notice this allows us to write both Eqs. (9) and (11) in terms of the couplings defined in Eq. (15). This geometric reinterpretation is reassuring. If we work in standard supergravity components, Minkowski is an on-shell solution with metric gmn = ηmn, constant dilaton and vanishing gauge potentials, dilatino and gravitino. If we work in superspace, super-Poincaré is a solution to the superspace constraints having non-trivial fermionic components. The ones appearing in the NS-NS 2-form gauge potential are the ones responsible for the WZ term, as it should for an object, the superstring, that is minimally coupled to this bulk massless field.

It is also remarkable to point out that contrary to the bosonic string, where there was no a priori reason why the string tension Tf should be equal to the charge density Qf, its supersymmetric and kappa invariant extension fixes the relation Tf = Qf. This will turn out to be a general feature in supersymmetric effective actions describing the dynamics of supersymmetric states in string theory.

Curved background extension: One of the spins of the superspace reinterpretation in Eq. (15) is that it allows its formal extension to any \({\mathcal N} = 2\) type IIA/B curved background [263]
$$S = - {1 \over {2\pi \alpha \prime}}\int {{d^2}} \sigma \sqrt {- \det {{\mathcal G}_{\mu \nu}}} + {1 \over {2\pi \alpha \prime}}\int {{{\mathcal B}_{(2)}}} .$$
(16)
The dependence on the background is encoded both in the superfields \(E_M^A\) and Bac.

The counting of degrees of freedom is not different from the one done for super-Poincaré. Thus, the GS superstring (16) still requires to be kappa symmetry invariant to have an on-shell matching of bosonic and fermionic degrees of freedom. It was shown in [89] that the effective action (16) is kappa invariant only when the \({\mathcal N} = 2d = 10\) type IIA/B background is on-shell8. In other words, superstrings can only propagate in properly on-shell backgrounds in the same theory.

It is important to stress that in the GS formulation, kappa symmetry invariance requires the background fields to be on-shell, whereas in the RNS formulation, it is quantum Weyl invariance that ensures this self-consistency condition, as illustrated in Figure 2.

The purpose of Section 3.1 is to explain how these ideas and necessary symmetry structures to achieve a manifestly spacetime covariant and supersymmetric invariant formulation extend to different half-BPS branes in string theory. More precisely, to M2-branes, M5-branes and D-branes.

3 Brane Effective Actions

This review is concerned with the dynamics of low energy string theory, or M-theory, in the presence of brane degrees of freedom in a regime in which the full string (M-) theory effective action9 reduces to
$$S \approx {S_{{\rm{SUGRA}}}} + {S_{{\rm{brane}}}}.$$
(17)
The first term in the effective action describes the gravitational sector. It corresponds to \({\mathcal N} = 2d = 10\) type IIA/IIB supergravity or \({\mathcal N} = 1 \, d = 11\) supergravity, for the systems discussed in this review. The second term describes both the brane excitations and their interactions with gravity.
More specifically, I will be concerned with the kinematical properties characterising Sbrane when the latter describes a single brane, though in Section 7, the extension to many branes will also be briefly discussed. From the perspective of full string theory, it is important to establish the regime in which the full dynamics is governed by Sbrane. This requires one to freeze the gravitational sector to its classical on-shell description and to neglect its backreaction into spacetime. Thus, one requires
$$\vert T_{mn}^{{\rm{background}}}\vert \gg \vert T_{mn}^{{\rm{brane}}}\vert ,$$
(18)
where Tmn stands for the energy-momentum tensor. This is a generalisation of the argument used in particle physics by which one decouples gravity, treating Newton’s constant as effectively zero.

Condition (18) is definitely necessary, but not sufficient, to guarantee the reliability of Sbrane. I will postpone a more thorough discussion of this important point till Section 3.7, once the explicit details on the effective actions and the assumptions made for their derivations have been spelled out in Sections 3.13.6.

Below, I focus on the identification of the degrees of freedom and symmetries to describe brane physics. The distinction between world volume and spacetime symmetries and the preservation of spacetime covariance and supersymmetry will lead us, once again, to the necessity and existence of kappa symmetry.

3.1 Degrees of freedom and world volume supersymmetry

In this section, I focus on the identification of the physical degrees of freedom describing a single brane, the constraints derived from world volume symmetries to describe their interactions and the necessity to introduce extra world volume gauge symmetries to achieve spacetime supersymmetry and covariance. I will first discuss these for Dp-branes, which allow a perturbative quantum open string description, and continue with M2 and M5-branes, applying the lessons learnt from strings and D-branes.

Dp-branes: Dp-branes are p + 1 dimensional hypersurfaces Σp +1 where open strings can end. One of the greatest developments in string theory came from the realisation that these objects are dynamical, carry Ramond-Ramond (RR) charge and allow a perturbative worldsheet description in terms of open strings satisfying Dirichlet boundary conditions in p + 1 dimensions [423].The quantisation of open strings with such boundary conditions propagating in 10-dimensional ℝ1,9 Minkowski spacetime gives rise to a perturbative spectrum containing a set of massless states that fit into an abelian vector supermultiplet of the super-Poincaré group in p + 1 dimensions [425, 426]. Thus, any physical process involving open strings at low enough energy, \(E\sqrt {{\alpha {\prime}}} \ll 1\), and at weak coupling, gs ≪ 1, should be captured by an effective supersymmetric abelian gauge theory in p + 1 dimensions.

Such vector supermultiplets are described in terms of U(1) gauge theories to achieve a manifestly ISO(1,p) invariance, as is customary in gauge theories. In other words, the formulation includes additional polarisations, which are non-physical and can be gauged away. Notice the full ISO(1, 9) of the vacuum is broken by the presence of the Dp-brane itself. This is manifestly reflected in the spectrum. Any attempt to achieve a spacetime supersymmetric covariant action invariant under the full ISO(1, 9) will require the introduction of both extra degrees of freedom and gauge symmetries. This is the final goal of the GS formulation of these effective actions.

To argue this, analyse the field content of these vector supermultiplets. These include a set of 9−p scalar fields XI and a gauge field V1 in p+ 1 dimensions, describing p −1 physical polarisations. Thus, the total number of massless bosonic degrees of freedom is
$${\rm{D}}p{\rm{- brane:}}\quad 10 - (p + 1) + (p - 1) = 8\,.$$
Notice the number of world volume scalars XI matches the number of transverse translations broken by the Dp-brane and transform as a vector under the transverse Lorentz subgroup SO(9−p), which becomes an internal symmetry group. Geometrically, these modes XI (σ) describe the transverse excitations of the brane. This phenomena is rather universal in brane physics and constitutes the essence in the geometrisation of field theories provided by branes in string theory.

Since Dp-branes propagate in 10 dimensions, any covariant formalism must involve a set of 10 scalar fields Xm (σ), transforming like a vector under the full Lorentz group SO(1, 9). This is the same situation we encountered for the superstring. As such, it should be clear the extra bosonic gauge symmetries required to remove these extra scalar fields are p+ 1 dimensional diffeomorphisms describing the freedom in embedding Σp +1 in ℝ1,9. Physically, the Dirichlet boundary conditions used in the open string description did fix these diffeomorphisms, since they encode the brane location in ℝ1,9.

What about the fermionic sector? The discussion here is entirely analogous to the superstring one. This is because spacetime supersymmetry forces us to work with two copies of Majorana-Weyl spinors in 10 dimensions. Thus, matching the eight on-shell bosonic degrees of freedom requires the effective action to be invariant under a new fermionic gauge symmetry. I will refer to this as kappa symmetry, since it will share all the characteristics of the latter for the superstring.

M-branes: M-branes do not have a perturbative quantum formulation. Thus, one must appeal to alternative arguments to identify the relevant degrees of freedom governing their effective actions at low energies. In this subsection, I will appeal to the constraints derived from the existence of supermultiplets in p +1 dimensions satisfying the geometrical property that their number of scalar fields matches the number of transverse dimensions to the M-brane, extending the notion already discussed for the superstring and Dp-branes. Later, I shall review more stringy arguments to check the conclusions obtained below, such as consistency with string/M theory dualities.

Let us start with the more geometrical case of an M2-brane. This is a 2+1 surface propagating in d =1 + 10 dimensions. One expects the massless fields to include 8 scalar fields in the bosonic sector describing the M2-brane transverse excitations. Interestingly, this is precisely the bosonic content of a scalar supermultiplet in d =1 + 2 dimensions. Since the GS formulation also fits into a scalar supermultiplet in d = 1 + 1 dimensions for a long string, it is natural to expect this is the right supermultiplet for an M2-brane. To achieve spacetime covariance, one must increase the number of scalar fields to eleven Xm (σ), transforming as a vector under SO(1,10) by considering a d =1 + 2 dimensional diffeomorphic invariant action. If this holds, how do fermions work out?

First, target space covariance requires the background to allow a superspace formulation in d = 1 + 10 dimensions10. Such formulation involves a single copy of d = 11 Majorana fermions, which gives rise to a pair of d =10 Majorana-Weyl fermions, matching the superspace formulation for the superstring described in Section 2. d =11 Majorana spinors have 2[11/2] = 32 real components, which are further reduced to 16 due to the Dirac equation. Thus, a further gauge symmetry is required to remove half of these fermionic degrees of freedom, matching the eight bosonic on-shell ones. Once again, kappa symmetry will be required to achieve this goal.

What about the M5-brane? The fermionic discussion is equivalent to the M2-brane one. The bosonic one must contain a new ingredient. Indeed, geometrically, there are only five scalars describing the transverse M5-brane excitations. These do not match the eight on-shell fermionic degrees of freedom. This is reassuring because there is no scalar supermultiplet in d =6 dimensions with such number of scalars. Interestingly, there exists a tensor supermultiplet in d = 6 dimensions whose field content involves five scalars and a two-form gauge potential V2 with self-dual field strength. The latter involves 6-2 choose 2 physical polarisations, with self-duality reducing these to three on-shell degrees of freedom. To keep covariance and describe the right number of polarisations, the d =1 + 5 theory must be invariant under U(1) gauge transformations for the 2-form gauge potential. I will later discuss how to keep covariance while satisfying the self-duality constraint.

Brane scan: World volume supersymmetry generically constrains the low energy dynamics of supersymmetric branes. Even though our arguments were concerned with M2, M5 and D-branes, they clearly are of a more general applicability. This gave rise to the brane scan programme [3, 196, 193, 191]. The main idea was to classify the set of supersymmetric branes in different dimensions by matching the number of their transverse dimensions with the number of scalar fields appearing in the list of existent supermultiplets. For an exhaustive classification of all unitary representations of supersymmetry with maximum spin 2, see [468]. Given the importance of scalar, vector and tensor supermultiplets, I list below the allowed multiplets of these kinds in different dimensions indicating the number of scalar fields in each of them [73].

Let me start with scalar supermultiplets containing X scalars in d = p +1 dimensions, the results being summarised in Table 1. Notice, we recover the field content of the M2-brane in d =3 and X = 8 and of the superstring in d =2 and X = 8.
Table 1

Scalar multiplets with X scalars in p + 1 worldvolume dimensions.

p + 1

X

X

X

X

1

1

2

4

8

2

1

2

4

8

3

1

2

4

8

4

 

2

4

 

5

  

4

 

6

  

4

 
Concerning vector supermultiplets with X scalars in d = p + 1 dimensions, the results are summarised in Table 2. Note that the last column describes the field content of all Dp-branes, starting from the D0-brane (p = 0) and finishing with the D9 brane (p = 9) filling in all spacetime. Thus, the field content of all Dp-branes matches with the one corresponding to the different vector supermultiplets in d = p + 1 dimensions. This point agrees with the open string conformal field theory description of D branes.
Table 2

Vector multiplets with X scalar degrees of freedom in p + 1 worldvolume dimensions.

p + 1

X

X

X

X

1

2

3

5

9

2

1

2

4

8

3

0

1

3

7

4

 

0

2

6

5

  

1

5

6

  

0

4

7

   

3

8

   

2

9

   

1

10

   

0

Finally, there is just one interesting tensor multiplet with X = 5 scalars in six dimensions, corresponding to the aforementioned M5 brane, among the six-dimensional tensor supermultiplets listed in Table 3.
Table 3

Tensor multiplets with X scalar degrees of freedom in p + 1 world volume dimensions.

p + 1

X

X

6

1

5

Summary: All half-BPS Dp-branes, M2-branes and M5-branes are described at low energies by effective actions written in terms of supermultiplets in the corresponding world-volume dimension. The number of on-shell bosonic degrees of freedom is 8. Thus, the fermionic content in these multiplets satisfies
$$8 = {1 \over 4}M\,{\mathcal N}\,,$$
(19)
where M is the number of real components for a minimal spinor representation in D spacetime dimensions and \({\mathcal N}\) the number of spacetime supersymmetry copies.

These considerations identified an \({\mathcal N} = 8\) = 8 supersymmetric field theory in d = 3 dimensions (M2 brane), \({\mathcal N} = (2,0)\) supersymmetric gauge field theory in d = 6 (M5 brane) and an \({\mathcal N} = 4\) supersymmetric gauge field theory in d = 4 (D3 brane), as the low energy effective field theories describing their dynamics11. The addition of interactions must be consistent with such d dimensional supersymmetries.

By construction, an effective action written in terms of these on-shell degrees of freedom can neither be spacetime covariant nor ISO(1,D − 1) invariant (in the particular case when branes propagate in Minkowski, as I have assumed so far). Effective actions satisfying these two symmetry requirements involve the addition of both extra, non-physical, bosonic and fermionic degrees of freedom. To preserve their non-physical nature, these supersymmetric brane effective actions must be invariant under additional gauge symmetries
  • world volume diffeomorphisms, to gauge away the extra scalars,

  • kappa symmetry, to gauge away the extra fermions.

3.1.1 Supergravity Goldstone modes

Branes carry energy, consequently, they gravitate. Thus, one expects to find gravitational configurations (solitons) carrying the same charges as branes solving the classical equations of motion capturing the effective dynamics of the gravitational sector of the theory. The latter is the effective description provided by type IIA/B supergravity theories, describing the low energy and weak coupling regime of closed strings, and \({\mathcal N} = 1 \, d = 11\) supergravity. The purpose of this section is to argue the existence of the same world-volume degrees of freedom and symmetries from the analysis of massless fluctuations of these solitons, applying collective coordinate techniques that are a well-known notion for solitons in standard, non-gravitational, gauge theories.

In field theory, given a soliton solving its classical equations of motion, there exists a notion of effective action for its small excitations. At low energies, the latter will be controlled by massless excitations, whose number matches the number of broken symmetries by the background soliton [243] 12. These symmetries are global, whereas all brane solitons are on-shell configurations in supergravity, whose relevant symmetries are local. To get some intuition for the mechanism operating in our case, it is convenient to review the study of the moduli space of monopoles or instantons in abelian gauge theories. The collective coordinates describing their small excitations include not only the location of the monopole/instanton, which would match the notion of transverse excitation in our discussion given the pointlike nature of these gauge theory solitons, but also a fourth degree of freedom associated with the breaking of the gauge group [431, 288]. The reason the latter is particularly relevant to us is because, whereas the first set of massless modes are indeed related to the breaking of Poincaré invariance, a global symmetry in these gauge theories, the latter has its origin on a large U(1) gauge transformation.

This last observation points out that the notion of collective coordinates can generically be associated with large gauge transformations, and not simply with global symmetries. It is precisely in this sense how it can be applied to gravity theories and their soliton solutions. In the string theory context, the first work where these ideas were applied was [127] in the particular set-up of 5-brane solitons in heterotic and type II strings. It was later extended to M2-branes and M5-branes in [332]. In this section, I follow the general discussion in [6] for the M2, M5 and D3-branes. These brane configurations are the ones interpolating between Minkowski, at infinity, and AdS times a sphere, near their horizons. Precisely for these cases, it was shown in [236] that the world volume theory on these branes is a supersingleton field theory on the corresponding AdS space.

Before discussing the general strategy, let me introduce the on-shell bosonic configurations to be analysed below. All of them are described by a non-trivial metric and a gauge field carrying the appropriate brane charge. The multiple M2-brane solution, first found in [198], is
$$\begin{array}{*{20}c} {d{s^2} = {U^{- {2 \over 3}}}{\eta _{\mu \nu}}d{x^\mu}d{x^\nu} + {U^{{1 \over 3}}}{\delta _{pq}}d{y^p}d{y^q}\,,} \\ {{A_3} = \pm {1 \over {3!}}\,{U^{- 1}}{\varepsilon _{\mu \nu \rho}}d{x^\mu} \wedge d{x^\nu} \wedge d{x^\rho}\,.\;\;} \\ \end{array}$$
(20)
Here, and in the following examples, describe the longitudinal brane directions, i.e., μ = 0,1,2 for the M2-brane, whereas the transverse Cartesian coordinates are denoted by \({y^p}, p = 3, \ldots 10\). The solution is invariant under ISO(1, 2) × SO(8) and is characterised by a single harmonic function U in ℝ8
$$U = 1 + {\left({{R \over r}} \right)^6},\qquad \quad {r^2} = {\delta _{pq}}{y^p}{y^q}.$$
(21)
The structure for the M5-brane, first found in [273], is analogous but differs in the dimensionality of the tangential and transverse subspaces to the brane and in the nature of its charge, electric for the M2-brane and magnetic for the M5-brane below
$$\begin{array}{*{20}c} {d{s^2} = {U^{- {1 \over 3}}}{\eta _{\mu \nu}}d{x^\mu}d{x^\nu} + {U^{{2 \over 3}}}{\delta _{mn}}d{y^m}d{y^n},\quad} \\ {{R_4} = d{A_3} = \pm {1 \over {4!}}\,{\delta ^{mn}}{\partial _m}U{\varepsilon _{npqsu}}d{y^p} \wedge d{y^q} \wedge d{y^s} \wedge d{y^s}.} \\ \end{array}$$
(22)
In this case, μ = 0,1…, 5 and p = 6, …, 10. The isometry group is ISO(1, 5) × SO(5) and again it is characterised by a single harmonic function U in ℝ5
$$U = 1 + {\left({{R \over r}} \right)^3}\,,\qquad \quad {r^2} = {\delta _{pq}}{y^p}{y^q}\,.$$
(23)
The D3-brane, first found in [195], similarly has a non-trivial metric and self-dual five form RR field strength
$$\begin{array}{*{20}c} {d{s^2} = {U^{- {1 \over 2}}}{\eta _{\mu \nu}}d{x^\mu}d{x^\nu} + {U^{{1 \over 2}}}{\delta _{mn}}d{y^m}d{y^n},\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{F_5} = \pm {1 \over {5!}}({\delta ^{mn}}{\partial _m}U{\varepsilon _{npqstu}}d{y^p} \wedge d{y^q} \wedge d{y^s} \wedge d{y^t} \wedge d{y^u}\quad \quad \quad} \\ {+ 5{\partial _m}{U^{- 1}}{\varepsilon _{\mu \nu \rho \sigma}}d{y^m} \wedge d{x^\mu} \wedge d{x^\nu} \wedge d{x^\rho} \wedge d{x^\sigma}),} \\ \end{array}$$
(24)
with isometry group ISO(1, 3) × SO(6). It is characterised by a single harmonic function U in ℝ6
$$U = 1 + {\left({{R \over r}} \right)^4},\qquad \quad {r^2} = {\delta _{pq}}{y^p}{y^q}.$$
(25)
All these brane configurations are half-BPS supersymmetric. The subset of sixteen supercharges being preserved in each case is correlated with the choice of sign in the gauge potentials fixing their charges. I shall reproduce this correlation in the effective brane action in Section 3.5.
Let me first sketch the argument behind the generation of massless modes in supergravity theories, where all relevant symmetries are gauge, before discussing the specific details below. Consider a background solution with field content \(\varphi _i^{(0)}\), where i labels the field, including its tensor character, having an isometry group G ′. Assume the configuration has some fixed asymptotics with isometry group G, so that G ′ ⊂ G. The relevant large gauge transformations ξi (yP) in our discussion are those that act non-trivially at infinity, matching a broken global transformation asymptotically ϵi, but differing otherwise in the bulk of the background geometry
$$\underset{r \rightarrow \infty}{\lim} {\xi _i}(y) = {\epsilon _i}\,.$$
(26)
In this way, one manages to associate a gauge transformation with a global one, only asymptotically. The idea is then to perturb the configuration \(\varphi _i^{(0)}\) by such pure gauge, δξiφi and finally introduce some world volume dependence on the parameter ϵi, i.e., ϵi (xμ). At that point, the transformation δξiφi is no longer pure gauge. Plugging the transformation in the initial action and expanding, one can compute the first order correction to the equations of motion fixing some of the ambiguities in the transformation by requiring the perturbed equation to correspond to a massless normalisable mode.

In the following, I explain the origin of the different bosonic and fermionic massless modes in the world volume supermultiplets discussed in Section 3.1 by analysing large gauge diffeomorphisms, supersymmetry and abelian tensor gauge transformations.

Scalar modes: These are the most intuitive geometrically. They correspond to the breaking of translations along the transverse directions to the brane. The relevant gauge symmetry is clearly a diffeomorphism. Due to the required asymptotic behaviour, it is natural to consider \({\epsilon ^p} = {U^s}{{\bar \phi}^p}\), where \({{\bar \phi}^p}\) is some constant parameter. Notice the dependence on the harmonic function guarantees the appropriate behaviour at infinity, for any s. Dynamical fields transform under diffeomorphisms through Lie derivatives. For instance, the metric would give rise to the pure gauge transformation
$${h_{mn}} = {{\mathcal L}_\epsilon}g_{mn}^{(0)}\,.$$
(27)
If we allow \({{\bar \phi}^p}\) to arbitrarily depend on the world volume coordinates xμ\({{\bar \phi}^p} \rightarrow {\phi ^p}({x^\mu})\), the perturbation hmn will no longer be pure gauge. If one computes the first-order correction to Einstein’s equations in supergravity, including the perturbative analysis of the energy momentum tensor, one discovers the lowest-order equation of motion satisfied by ϕp is
$${\partial ^\mu}{\partial _\mu}{\phi ^p} = 0\,,$$
(28)
for s = −1. This corresponds to a massless mode and guarantees its normalisability when integrating the action in the directions transverse to the brane. Later, we will see that the lowest-order contribution (in number of derivatives) to the gauge-fixed world-volume action of M2, M5 and D3-branes in flat space is indeed described by the Klein-Gordon equation.
Fermionic modes: These must correspond to the breaking of supersymmetry. Consider the supersymmetry transformation of the 11-dimensional gravitino ψm
$$\delta {\Psi _m} = {\tilde D_m}\zeta \,,$$
(29)
where \({\tilde D}\) is some non-trivial connection involving the standard spin connection and some contribution from the gauge field strength. The search for massless fermionic modes leads us to consider the transformation \(\zeta = {U^s}\bar \lambda\) for some constant spinor \({\bar \lambda}\). First, one needs to ensure that such transformation matches, asymptotically, with the supercharges preserved by the brane. Consider the M5-brane, as an example. The preserved supersymmetries are those satisfying δ ψm = 0. This forces \(s = - {7 \over {12}}\) and fixes the six-dimensional chirality of \({\bar \lambda}\) to be positive, i.e., \({{\bar \lambda}_ +}\). Allowing the latter to become an arbitrary function of the world volume coordinates λ +(xμ), δ ψm becomes non-pure gauge. Plugging the latter into the original Rarita-Schwinger equation, the linearised equation for the perturbation reduces to
$${\Gamma ^\mu}{\partial _\mu}{\lambda _ +} = 0\,.$$
(30)
The latter is indeed the massless Dirac equation for a chiral six-dimensional fermion. A similar analysis holds for the M2 and D3-branes. The resulting perturbations are summarised in Table 4.
Table 4

Summary of supergravity Goldstone modes.

Symmetry

 

M2

M5

D3

Reparametrisations:

εm =

\({U^{- 1}}{{\bar \phi}^m}\)

\({U^{- 1}}{{\bar \phi}^m}\)

\({U^{- 1}}{{\bar \phi}^m}\)

Local supersymmetry:

ζ =

\({U^{- 2/3}}{{\bar \lambda}_ -}\)

\({U^{- 7/12}}{{\bar \lambda}_ +}\)

\({U^{- 5/8}}{{\bar \lambda}_ +}\)

Tensor gauge symmetry:

Λ =

 

\(_{(\ast\bar H = \bar H)}^{{U^{- 1}}{{\bar V}_{(2)}}}\)

\(_{(i\ast\bar F = \bar F)}^{{U^{- 1}}{{\bar V}_{(1)}}}\)

Vector modes: The spectrum of open strings with Dirichlet boundary conditions includes a vector field. Since the origin of such massless degrees of freedom must be the breaking of some abelian supergravity gauge symmetry, it must be the case that the degree form of the gauge parameter must coincide with the one-form nature of the gauge field. Since this must hold for any D-brane, the natural candidate is the abelian gauge symmetry associated with the NS-NS two-form
$$\delta {B_2} = d{\Lambda _1}\,.$$
(31)
Proceeding as before, one considers a transformation with \({\Lambda _1} = {U^k}{{\bar V}_1}\) for some number k and constant one-form \({{\bar V}_1}\) When \({{\bar V}_1}\) is allowed to depend on the world volume coordinates, the perturbation
$$\delta {B_2} = d{U^k} \wedge {V_1}({x^\mu})\,,$$
(32)
becomes physical. Plugging this into the NS-NS two-form equation of motion, one derives dF = 0 where F = dV1 for both of the four-dimensional duality components, for either k = ±1. Clearly, only k = −1 is allowed by the normalisability requirement.
Tensor modes: The presence of five transverse scalars to the M5-brane and the requirement of world volume supersymmetry in six dimensions allowed us to identify the presence of a two-form potential with self-dual field strength. This must have its supergravity origin in the breaking of the abelian gauge transformation
$$\delta {A_3} = d{\Lambda _2}\,,$$
(33)
where indeed the gauge parameter is a two-form. Consider then \({\Lambda _2} = {U^k}{{\bar V}_2}\) for some number k and constant two form \({{\bar V}_2}\). When \({{\bar V}_2}\) is allowed to depend on the world volume coordinates, the perturbation
$$\delta {A_3} = d{U^k} \wedge {V_2}({x^\mu})\,,$$
(34)
becomes physical. Plugging this into the A3 equation of motion, we learn that each world volume duality component *xF3 = ±F3 with F3 = dV2 satisfies the bulk equation of motion if dF3 = 0 for a specific choice of k. More precisely, self-dual components require k =1, whereas anti-self-dual ones require k = −1. Normalisability would fix k = −1. Thus, this is the origin of the extra three bosonic degrees of freedom forming the tensor supermultiplet in six dimensions.

The matching between supergravity Goldstone modes and the physical content of world volume supersymmetry multiplets is illustrated in Figure 5. Below, a table presents the summary of supergravity Goldstone modes where ± indices stand for the chirality of the fermionic zero modes. In particular, for the M2 brane it describes negative eight dimensional chirality of the 11-dimensional spinor λ while for the M5 and D3 branes, it describes positive six-dimensional and four-dimensional chirality.

Thus, using purely effective field theory techniques, one is able to derive the spectrum of massless excitations of brane supergravity solutions. This method only provides the lowest order contributions to their equations of motion. The approach followed in this review is to use other perturbative and non-perturbative symmetry considerations in string theory to determine some of the higher-order corrections to these effective actions. Our current conclusion, from a different perspective, is that the physical content of these theories must be describable in terms of the massless fields in this section.

3.2 Bosonic actions

After the identification of the relevant degrees of freedom and gauge symmetries governing brane effective actions, I focus on the construction of their bosonic truncations, postponing their super-symmetric extensions to Sections 3.4 and 3.5. The main goal below will be to couple brane degrees of freedom to arbitrary curved backgrounds in a world volume diffeomorphic invariant way.

I shall proceed in order of increasing complexity, starting with the M2-brane effective action, which is purely geometric, continuing with D-branes and their one form gauge potentials and finishing with M5-branes including their self-dual three form field strength13.

Bosonic M2-brane: In the absence of world volume gauge field excitations, all brane effective actions must satisfy two physical requirements
  1. 1.

    Geometrically, branes are p + 1 hypersurfaces Σp +1 propagating in a fixed background with metric gmn. Thus, their effective actions should account for their world volumes.

     
  2. 2.

    Physically, all branes are electrically charged under some appropriate spacetime p + 1 gauge form Cp+1. Thus, their effective actions should contain a minimal coupling accounting for the brane charges.

     
Both requirements extend the existent effective action describing either a charged particle (p = 0) or a string (p = 1). Thus, the universal description of the purely scalar field Xm brane degrees of freedom must be of the form
$${S_p} = - {T_p}\,\int\nolimits_{{\Sigma _{p + 1}}} {{d^{p + 1}}} \sigma \,\sqrt {- \det {\mathcal G}} + {Q_p}\,\int\nolimits_{{\Sigma _{p + 1}}} {{{\mathcal C}_{p + 1}}} \,,$$
(35)
where Tp and Qp stand for the brane tension and charge density14. The first term computes the brane world volume from the induced metric \({{\mathcal G}_{\mu \nu}}\)
$${{\mathcal G}_{\mu \nu}} = {\partial _\mu}{X^m}{\partial _\nu}{X^n}{g_{mn}}(X),$$
(36)
whereas the second WZ term Cp+1 describes the pullback of the target space p + 1 gauge field Cp+1(X) under which the brane is charged
$${{\mathcal C}_{(p + 1)}} = {1 \over {(p + 1)!}}{\epsilon ^{{\mu _1} \ldots {\mu _{p + 1}}}}{\partial _{{\mu _1}}}{X^{{m_1}}} \ldots {\partial _{{\mu _{p + 1}}}}{X^{{m_{p + 1}}}}\,{C_{{m_1} \ldots {m_{p + 1}}}}(X).$$
(37)
At this stage, one assumes all branes propagate in a background with Lorentzian metric gmn (X) coupled to other matter fields, such as Cp+1(X), whose dynamics are neglected in this approximation. In string theory, these background fields correspond to the bosonic truncation of the supergravity multiplet and their dynamics at low energy is governed by supergravity theories. More precisely, M2 and M5-branes propagate in d =11 supergravity backgrounds, i.e., m,n = 0,1,… 10, and they are electrically charged under the gauge potential A3(X) and its six-form dual potential A6, respectively (see Appendix A for conventions). D-branes propagate in d =10 type IIA/B backgrounds and the set {Cp+1(X)} correspond to the set of RR gauge potentials in these theories, see Eq. (523).
The relevance of the minimal charge coupling can be understood by considering the full effective action involving both brane and gravitational degrees of freedom (17). Restricting ourselves to the kinetic term for the target space gauge field, i.e., R = dCp+1, the combined action can be written as
$$\int\nolimits_{{{\mathcal M}_D}} {\left({\,{1 \over 2}R \wedge \star R + {Q_p}\hat n \wedge {C_{p + 1}}\,} \right){.}}$$
(38)
Here \({{\mathcal M}_D}\) stands for the D-dimensional spacetime, whereas \({\hat n}\) is a (Dp − 1)-form whose components are those of an epsilon tensor normal to the brane having a δ-function support on the world volume15. Thus, the bulk equation of motion for the gauge potential Cp+1 acquires a source term whenever a brane exists. Since the brane charge is computed as the integral of *R over any topological (Dp − 2)-sphere surrounding it, one obtains
$$\int\nolimits_{{\Sigma _{D - p - 2}}} \star R = \int\nolimits_{{B_{D - p - 1}}} d \star R = \int\nolimits_{{B_{D - p - 1}}} {{Q_p}} \hat n = {Q_p},$$
(39)
where the equation of motion was used in the last step. Thus, minimal WZ couplings do capture the brane physical charge.
Since M2-branes do not involve any gauge field degree of freedom, the above discussion covers all its bosonic degrees of freedom. Thus, one expects its bosonic effective action to be
$${S_{M2}} = - {T_{{\rm{M2}}}}\int {{d^3}} \sigma \,\sqrt {- \det {\mathcal G}} + {Q_{{\rm{M2}}}}\int {{{\mathcal A}_3}} \,,$$
(40)
in analogy with the bosonic worldsheet string action. If Eq. (40) is viewed as the bosonic truncation of a supersymmetric M2-brane, then |Qm2| = T m2. Besides its manifest spacetime covariance and its invariance under world-volume diffeomorphisms infinitesimally generated by
$${\delta _\xi}{X^m} = {{\mathcal L}_\xi}{X^m} = {\xi ^\mu}{\partial _\mu}{X^m},$$
(41)
this action is also quasi-invariant (invariant up to total derivatives) under the target space gauge transformation δΛA3 = d Λ2 leaving \({\mathcal N} = 1 \, d = 11\) supergravity invariant, as reviewed in Eq. (553) of Appendix A.2. This is reassuring given that the full string theory effective action (17) describing both gravity and brane degrees of freedom involves both actions.
Bosonic D-branes: Due to the perturbative description in terms of open strings [423], D-brane effective actions can, in principle, be determined by explicit calculation of appropriate open string disk amplitudes. Let me first discuss the dependence on gauge fields in these actions. Early bosonic open string calculations in background gauge fields [1], allowed to determine the effective action for the gauge field, with purely Dirichlet boundary conditions [214] or with mixed boundary conditions [354], gave rise to a non-linear generalisation of Maxwell’s electromagnetism originally proposed by Born and Infeld in [108]:
$$- \int\nolimits_{{\Sigma _{p + 1}}} {{d^{p + 1}}} \sigma \,\sqrt {- \det ({\eta _{\mu \nu}} + 2\pi \alpha \prime {F_{\mu \nu}})} .$$
(42)
I will refer to this non-linear action as the Dirac-Born-Infeld (DBI) action. Notice, this is an exceptional situation in string theory in which an infinite sum of different α ′ contributions is analytically computable. This effective action ignores any contribution from the derivatives of the field strength F, i.e., ∂μF terms or higher derivative operators. Importantly, it was shown in [1] that the first such corrections, for the bosonic open string, are compatible with the DBI structure.
Having identified the non-linear gauge field dependence, one is in a position to include the dependence on the embedding scalar fields Xm (σ) and the coupling with non-trivial background closed string fields. Since in the absence of world-volume gauge-field excitations, D-brane actions should reduce to Eq. (35), it is natural to infer the right answer should involve
$$\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + 2\pi \alpha \prime {F_{\mu \nu}})} ,$$
(43)
using the general arguments of the preceding paragraphs. Notice, this action does not include any contribution from acceleration and higher derivative operators involving scalar fields, i.e., μ νXm terms and/or higher derivative terms.16 This proposal has nice properties under T-duality [24, 77, 16, 75], which I will explore in detail in Section 3.3.2 as a non-trivial check on Eq. (43). In particular, it will be checked that absence of acceleration terms is compatible with T-duality.
The DBI action is a natural extension of the NG action for branes, but it does not capture all the relevant physics, even in the absence of acceleration terms, since it misses important background couplings, responsible for the WZ terms appearing for strings and M2-branes. Let me stress the two main issues separately:
  1. 1.
    The functional dependence on the gauge field V1 in a general closed string background. D-branes are hypersurfaces where open strings can end. Thus, open strings do have endpoints. This means that the WZ term describing such open strings is not invariant under the target space gauge transformation δB2 = d Λ1
    $$\delta \int\nolimits_{{\Sigma _2}} b = \int\nolimits_{{\Sigma _2}} d \Lambda = \int\nolimits_{\partial {\Sigma _2}} \Lambda ,$$
    (44)
    due to the presence of boundaries. These are the D-branes themselves, which see these endpoints as charge point sources. The latter has a minimal coupling of the form \(\int\nolimits_{\partial {\Sigma _2}} {{V_1}}\), whose variation cancels Eq. (44) if the gauge field transforms as δV1 = dXm (σm under the bulk gauge transformation. Since D-brane effective actions must be invariant under these target space gauge symmetries, this physical argument determines that all the dependence on the gauge field V1 should be through the gauge invariant combination \({\mathcal F} = 2\pi {\alpha {\prime}}d{V_1} - {\mathcal B}\).
     
  2. 2.
    The coupling to the dilaton. The D-brane effective action is an open string tree level action, i.e., the self-interactions of open strings and their couplings to closed string fields come from conformal field theory disk amplitudes. Thus, the brane tension should include a \(g_s^{- 1}\) factor coming from the expectation value of the closed string dilaton eϕ. Both these considerations lead us to consider the DBI action
    $${S_{{\rm{DBI}}}} = - {T_{{{\rm{D}}_{\rm{p}}}}}\int {{d^{p + 1}}} \sigma \,{e^{- \phi}}\sqrt {- \det ({\mathcal G} + {\mathcal F})} ,$$
    (45)
    where \({T_{{{\rm{D}}_{\rm{p}}}}}\) tension.
     
  3. 3.
    The WZ couplings. Dp-branes are charged under the RR potential Cp +1. Thus, their effective actions should include a minimal coupling to the pullback of such form. Such coupling would not be invariant under the target space gauge transformations (527). To achieve this invariance in a way compatible with the bulk Bianchi identities (525), the D-brane WZ action must be of the form
    $$\int\nolimits_{{\Sigma _{p + 1}}} {\mathcal C} \wedge {e^{\mathcal F}},$$
    (46)
    where \({\mathcal C}\) stands for the corresponding pullbacks of the target space RR potentials Cr to the world volume, according to the definition given in Eq. (523). Notice this involves more terms than the mere minimal coupling to the bulk RR potential Cp +1. An important physical consequence of this fact will be that turning on non-trivial gauge fluxes on the brane can induce non-trivial lower-dimensional D-brane charges, extending the argument given above for the minimal coupling [185]. This property will be discussed in more detail in the second part of this review. For a discussion on how to extend these couplings to massive type IIA supergravity, see [255].
     
Putting together all previous arguments, one concludes the final form of the bosonic D-brane action is:17
$${S_{{\rm{Dp}}}} = - {T_{{\rm{Dp}}}}\int\nolimits_{{\Sigma _{p + 1}}} {{d^{p + 1}}} \sigma \,{e^{- \phi}}\sqrt {- \det ({\mathcal G} + {\mathcal F})} + {Q_{{\rm{Dp}}}}\int\nolimits_{{\Sigma _{p + 1}}} {\mathcal C} \wedge {e^{\mathcal F}}.$$
(47)
If one views this action as the bosonic truncation of a supersymmetric D-brane, the D-brane charge density equals its tension in absolute value, i.e., \(\vert {Q_{{{\rm{D}}_{\rm{p}}}}}\vert \, = {T_{{{\rm{D}}_{\rm{p}}}}}\). The latter can be determined from first principles to be [423, 24]
$${T_{{\rm{Dp}}}} = {1 \over {{g_s}\sqrt {\alpha {\prime}}}}{1 \over {{{(2\pi \sqrt {\alpha {\prime}})}^p}}}.$$
(48)
Bosonic covariant M5-brane: The bosonic M5-brane degrees of freedom involve scalar fields and a world volume 2-form with self-dual field strength. The former are expected to be described by similar arguments to the ones presented above. The situation with the latter is more problematic given the tension between Lorentz covariance and the self-duality constraint. This problem has a fairly long history, starting with electromagnetic duality and the Dirac monopole problem in Maxwell theory, see [105] and references therein, and more recently, in connection with the formulation of supergravity theories such as type IIB, with the self-duality of the field strength of the RR 4-form gauge potential. There are several solutions in the literature based on different formalisms:
  1. 1.

    One natural option is to give-up Lorentz covariance and work with non-manifestly Lorentz invariant actions. This was the approach followed in [420] for the M5-brane, building on previous work [213, 295, 441].

     
  2. 2.

    One can introduce an infinite number of auxiliary (non-dynamical) fields to achieve a covariant formulation. This is the approach followed in [384, 502, 375, 177, 66, 98, 99, 100].

     
  3. 3.

    One can follow the covariant approach due to Pasti, Sorokin and Tonin (PST-formalism) [416, 418], in which a single auxiliary field is introduced in the action with a non-trivial non-polynomial dependence on it. The resulting action has extra gauge symmetries. These allow one to recover the structure in [420] as a gauge fixed version of the PST formalism.

     
  4. 4.

    Another option is to work with a Lagrangian that does not imply the self-duality condition but allows it, leaving the implementation of this condition to the path integral. This is the approach followed by Witten [497], which was extended to include non-linear interactions in [140]. The latter work includes kappa symmetry and a proof that their formalism is equivalent to the PST one.

     
In this review, I follow the PST formalism. This assigns the following bosonic action to the M5-brane [417]
$$\begin{array}{*{20}c} {{S_{{\rm{M5}}}} = - {T_{{\rm{M5}}}}\int {{d^6}} \sigma \left({\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\tilde H}_{\mu \nu}})} - \sqrt {- \det {\mathcal G}} {1 \over {4{\partial _\mu}a{\partial ^\mu}a}}{\partial _\delta}a(\sigma){{\mathcal H}^{{\ast}\mu \nu \delta}}{{\mathcal H}_{\mu \nu \rho}}{\partial ^\rho}a(\sigma)} \right)} \\ {+ {T_{{\rm{M5}}}}\int {\left({{{\mathcal A}_6} + {1 \over 2}{{\mathcal H}_3} \wedge {{\mathcal A}_3}} \right)} .\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(49)
As in previous effective actions, all the dependence on the scalar fields Xm is through the bulk fields and their pullbacks to the six-dimensional world volume. As in D-brane physics, all the dependence on the world volume gauge potential V2 is not just simply through its field strength 2, but through the gauge invariant 3-form
$${{\mathcal H}_3} = d{V_2} - {{\mathcal A}_3}\,.$$
(50)
The physics behind this is analogous. \({\mathcal F}\) describes the ability of open strings to end on D-branes, whereas describes the possibility of M2-branes to end on M5-branes [469, 479]18. Its world volume Hodge dual and the tensor \({{\tilde H}_{\mu \nu}}\) are then defined as
$${{\mathcal H}^{{\ast}\mu \nu \rho}} = {1 \over {6\sqrt {- \det {\mathcal G}}}}{\varepsilon ^{\mu \nu \rho {\alpha _1}{\alpha _2}{\alpha _3}}}{{\mathcal H}_{{\alpha _1}{\alpha _2}{\alpha _3}}}\,,$$
(51)
$${\tilde H_{\mu \nu}} = {1 \over {\sqrt {\vert {{(\partial a)}^2}\vert}}}{\mathcal H}_{\mu \nu \rho}^{\ast}{\partial ^\rho}a(\sigma)\,.$$
(52)
The latter involves an auxiliary field a (σ) responsible for keeping covariance and implementing the self-duality constraint through the second term in the action (49). Its auxiliary nature was proven in [418, 416], where it was shown that its equation of motion is not independent from the generalised self-duality condition. The full action also includes a DBI-like term, involving the induced world volume metric \({{\mathcal G}_{\mu \nu}} = {\partial _\mu}{X^m}{\partial _\nu}{X^n}{g_{mn}}(X)\), and a WZ term, involving the pullbacks \({{\mathcal A}_3}\) and \({{\mathcal A}_6}\) of the 3-form gauge potential and its Hodge dual in \({\mathcal N} = 1 \, d = 11\) supergravity [11].
Besides being manifestly invariant under six-dimensional world volume diffeomorphisms and ordinary abelian gauge transformations δV2 = d Λ1, the action (49) is also invariant under the transformation
$$\delta a(\sigma) = \Lambda (\sigma)\,,\quad \quad \delta {V_{\mu \nu}} = {{\Lambda (\sigma)} \over {\sqrt {\vert {{(\partial a)}^2}\vert}}}\left({2{{\delta {{\mathcal L}_{{\rm{DBI}}}}} \over {\delta {{\tilde H}^{\mu \nu}}}} - {{(d{V_2})}_{\mu \nu \rho}}{{{\partial ^\rho}a} \over {\sqrt {\vert {{(\partial a)}^2}\vert}}}} \right)\,.$$
(53)
Given the non-dynamical nature of a (σ), one can always fully remove it from the classical action by gauge fixing the symmetry (53). It was shown in [417] that for an M5-brane propagating in Minkowski, the non-manifest Lorentz invariant formulation in [420] emerges after gauge fixing (53). This was achieved by working in the gauge \({\partial _\mu}a(\sigma) = \delta _\mu ^5\) and \({V_{\mu 5}} = 0\). Since μa is a world volume vector, six-dimensional Lorentz transformations do not preserve this gauge slice. One must use a compensating gauge transformation (53), which also acts on Vμν. The overall gauge fixed action is invariant under the full six-dimensional Lorentz group but in a non-linear non-manifestly Lorentz covariant way as discussed in [420].

As a final remark, notice the charge density QM5 of the bosonic M5-brane has already been set equal to its tension \({T_{{\rm{M5}}}} = 1/{(2\pi)^5}\ell _p^6\).

3.3 Consistency checks

The purpose of this section is to check the consistency of the kinematic structures governing classical bosonic brane effective actions with string dualities [312, 495]. At the level of supergravity, these dualities are responsible for the existence of a non-trivial web of relations among their classical Lagrangians. Here, I describe the realisation of some of these dualities on classical bosonic brane actions. This will allow us to check the consistency of all brane couplings. Alternatively, one can also view the discussions below as independent ways of deriving the latter.

The specific dualities I will be appealing to are the strong coupling limit of type IIA string theory, its relation to M-theory and the action of T-duality on type II string theories and D-branes. Figure 3 summarises the set of relations between the brane tensions discussed in this review under these symmetries.
Figure 3

Set of half-BPS branes discussed in this review, their tensions and some of their connections under T-duality and the strongly-coupled limit of type IIA.

M-theory as the strong coupling limit of type IIA: From the spectrum of 1/2-BPS states in string theory and M-theory, an M2/M5-brane in ℝ1,9 × S1 has a weakly-coupled description in type IIA
  • either as a long string or a D4-brane, if the M2/M5-brane wraps the M-theory circle, respectively

  • or as a D2-brane/NS5 brane, if the M-theory circle is transverse to the M2/M5-brane world volume.

The question to ask is: how do these statements manifest in the classical effective action? The answer is by now well known. They involve a double or a direct dimensional reduction, respectively. The idea is simple. The bosonic effective action describes the coupling of a given brane with a fixed supergravity background. If the latter involves a circle and one is interested in a description of the physics nonsensitive to this dimension, one is entitled to replace the d-dimensional supergravity description by a d-1 one using a Kaluza-Klein (KK) reduction (see [197] for a review on KK compactifications). In the case at hand, this involves using the relation between d = 11 bosonic supergravity fields and the type IIA bosonic ones summarised below [409]
$$\begin{array}{*{20}c} {ds_{11}^2 = {e^{- {2 \over 3}\phi}}\,ds_{10}^2 + {e^{{4 \over 3}\phi}}{{(dy + {C_1})}^2}\,,} \\ {{A_3} = {C_3} + dy \wedge {B_2}\,,\quad \quad \quad \quad} \\ \end{array}$$
(54)
where the left-hand side 11-dimensional fields are rewritten in terms of type IIA fields. The above reduction involves a low energy limit in which one only keeps the zero mode in a Fourier expansion of all background fields on the bulk S1. In terms of the parameters of the theory, the relation between the M-theory circle R and the 11-dimensional Planck scale p with the type IIA string coupling gs and string length s is
$$R = {g_s}{\ell _s},\quad \quad \ell _p^3 = {g_s}\ell _s^3.$$
(55)
The same principle should hold for the brane degrees of freedom {ΦA}. The distinction between a double and a direct dimensional reduction comes from the physical choice on whether the brane wraps the internal circle or not:
  • If it does, one partially fixes the world volume diffeomorphisms by identifying the bulk circle direction y with one of the world volume directions σp, i.e., Y (σ) = σp, and keeps the zero mode in a Fourier expansion of all the remaining brane fields, i.e., ΦA = ΦA (σ ′) where {σ} = {σ′, σp}. This procedure is denoted as a double dimensional reduction [192], since both the bulk and the world volume get their dimensions reduced by one.

  • If it does not, there is no need to break the world volume diffeomorphisms and one simply truncates the fields to their bulk zero modes. This procedure is denoted as a direct reduction since the brane dimension remains unchanged while the bulk one gets reduced.

T-duality on closed and open strings: From the quantisation of open strings satisfying Dirichlet boundary conditions, all D-brane dynamics are described by a massless vector supermultiplet, whose number of scalar fields depends on the number of transverse dimensions to the D-brane. Since D-brane states are mapped among themselves under T-duality [160, 424], one expects the existence of a transformation mapping their classical effective actions under this duality. The question is how such transformation acts on the action. This involves two parts: the transformation of the background and the one of the brane degrees of freedom.

Let me focus on the bulk transformation. T-duality is a perturbative string theory duality [241]. It says that type IIA string theory on a circle of radius R and string coupling gs is equivalent to type IIB on a dual circle of radius R ′ and string coupling gs related as [121, 122, 240]
$$R\prime = {{\alpha \prime} \over R}\,,\qquad g{\prime _s} = {g_s}{{\sqrt {\alpha \prime}} \over R}\,,$$
(56)
when momentum and winding modes are exchanged in both theories. This leaves the free theory spectrum invariant [337], but it has been shown to be an exact perturbative symmetry when including interactions [400, 241]. Despite its stringy nature, there exists a clean field theoretical realisation of this symmetry. The main point is that any field theory on a circle of radius R has a discrete momentum spectrum. Thus, in the limit R → 0, all non-vanishing momentum modes decouple, and one only keeps the original vanishing momentum sector. Notice this is effectively implementing a KK compactification on this circle. This is in contrast with the stringy realisation where in the same limit, the spectrum of winding modes opens up a dual circle of radius R ′.
Since Type IIA and Type IIB supergravities are field theories, the above field theoretical realisation applies. Thus, the R → 0 compactification limit should give rise to two separate \({\mathcal N} = 2 \, d = 9\) supergravity theories. But it is known [388] that there is just such a unique supergravity theory. In other words, given the type IIA/B field content {φa/b} and their KK reduction to d = 9 dimensions, i.e., φa = φa (φ9) and φβ = φβ (φ9), the uniqueness of \({\mathcal N} = 2 \, d = 9\) supergravity guarantees the existence of a non-trivial map between type IIA and type IIB fields in the subset of backgrounds allowing an S1 compactification
$${\varphi _A} = {\varphi _A}({\varphi _B})\,.$$
(57)
This process is illustrated in the diagram of Figure 4. These are the T-duality rules. When expressed in terms of explicit field components, they become [82, 388]
$$\begin{array}{*{20}c} {{g_{zz}} = {1 \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad} \\ {\phi = \phi {\prime} - {1 \over 2}\log \vert {{g{\prime}}_{z{\prime} z{\prime}}}\vert} \\ {{B_{nz}} = - {{{{g{\prime}}_{nz{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad} \\ \end{array}$$
$$\begin{array}{*{20}c} {{g_{nz}} = - {{{{B{\prime}}_{nz{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{g_{mn}} = {{g{\prime}}_{mn}} - {{{{g{\prime}}_{mz{\prime}}}{{g{\prime}}_{nz{\prime}}} - {{B{\prime}}_{mz{\prime}}}{{B{\prime}}_{nz{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad} \\ {{B_{mn}} = {{B{\prime}}_{mn}} - {{{{B{\prime}}_{mz{\prime}}}{{g{\prime}}_{nz{\prime}}} - {{B{\prime}}_{nz{\prime}}}{{g{\prime}}_{mz{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad} \\ {C_{{m_1} \ldots {m_p}z}^{(p + 1)} = C{\prime} _{{m_1} \ldots {m_p}}^{(p)} - p{{C{\prime} _{\left[ {{m_1} \ldots {m_{p - 1}}z{\prime}} \right.}^{(p)}{{g{\prime}}_{\left. {{m_p}} \right]z{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad} \\ {C_{{m_1} \ldots {m_p}}^{(p)} = C{\prime} _{{m_1} \ldots {m_p}z{\prime}}^{(p + 1)} - pC{\prime} _{\left[ {{m_1} \ldots {m_{p - 1}}} \right.}^{(p - 1)}{{B{\prime}}_{\left. {{m_p}} \right]z{\prime}}}\quad \quad \quad \quad \quad \quad} \\ {- p(p - 1){{C{\prime} _{\left[ {{m_1} \ldots {m_{p - 2}}z{\prime}} \right.}^{(p - 1)}{{B{\prime}}_{{m_{p - 1}}z{\prime}}}{{g{\prime}}_{\left. {{m_p}} \right]z{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\,.} \\ \end{array}$$
(58)
These correspond to the bosonic truncations of the superfields introduced in Appendix A.1. Prime and unprimed fields correspond to both T-dual theories. The same notation applies to the tensor components where {z, z′} describe both T-dual circles. Notice the dilaton and the gzz transformations do capture the worldsheet relations (56).
Figure 4

Schematic diagram describing the derivation of Buscher’s T-duality rules using type IIA/IIB supergravities.

Let me move to the brane transformation. A D(p + 1)-brane wrapping the original circle is mapped under T-duality to a Dp-brane where the dual circle is transverse to the brane [424]. It must be the case that one of the gauge field components in the original brane maps into a transverse scalar field describing the dual circle. At the level of the effective action, implementing the R → 0 limit must involve, first, a partial gauge fixing of the world volume diffeomorphisms, to explicitly make the physical choice that the brane wraps the original circle, and second, keeping the zero modes of all the remaining dynamical degrees of freedom. This is precisely the procedure described as a double dimensional reduction. The two differences in this D-brane discussion will be the presence of a gauge field and the fact that the KK reduced supergravity fields {φ9} will be rewritten in terms of the T-dual ten-dimensional fields using the T-duality rules (58).

In the following, it will be proven that the classical effective actions described in the previous section are interconnected in a way consistent with our T-duality and strongly-coupled considerations. Our logic is as follows. The M2-brane is linked to our starting worldsheet action through double-dimensional reduction. The former is then used to derive the D2-brane effective by direct dimensional reduction. T-duality covariance extends this result to any non-massive D-brane. Finally, to check the consistency of the PST covariant action for the M5-brane, its double dimensional reduction will be shown to match the D4-brane effective action. This will complete the set of classical checks on the bosonic brane actions discussed so far.

It is worth mentioning that the self-duality of the D3-brane effective action under S-duality could also have been included as a further test. For discussions on this point, see [483, 252].

3.3.1 M2-branes and their classical reductions

In the following, I discuss the double and direct dimensional reductions of the bosonic M2-brane effective action (40) to match the bosonic worldsheet string action (6) and the D2-brane effective action, i.e., the p = 2 version of Eq. (47). This analysis will also allow us to match/derive the tensions of the different branes.

Connection to the string worldsheet: Consider the propagation of an M2-brane in an 11-dimensional background of the form (54). Decompose the set of scalar fields as {XM} = {Xm, Y}, identify one of the world volume directions (ρ) with the KK circle, i.e., partially gauge fix the world volume diffeomorphisms by imposing Y = ρ, and keep the zero modes in the Fourier expansion of all remaining scalar fields {Xm} along the world volume circle, i.e., ρXm = 0. Under these conditions, which mathematically characterise a double dimensional reduction, the Wess-Zumino coupling becomes
$$\int\nolimits_{{\Sigma _2} \times {S_1}} {{{\mathcal A}_3}} = \int\nolimits_{{\Sigma _2} \times {S_1}} {{d^3}} \sigma \,{1 \over 2}{\epsilon ^{\hat \mu \hat \nu \rho}}{\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}{X^n}{A_{mny}} = \left({\int\nolimits_{{S_1}} d \rho} \right)\int\nolimits_{{\Sigma _2}} {{{\mathcal B}_2}} ,$$
(59)
where I already used the KK reduction ansatz (54). Here, \({{\mathcal B}_2}\) stands for the pull-back of the NS-NS two form into the surface Σ2 parameterised by \(\{{\sigma ^{\hat \mu}}\}\). The DBI action is reduced using the identity satisfied by the induced world-volume metric
$${{\mathcal G}_{\mu \nu}} = \left({\begin{array}{*{20}c} {{e^{- 2\phi /3}}({{\mathcal G}_{\hat \mu \hat \nu}} + {e^{2\phi}}{{\mathcal C}_{\hat \mu}}{{\mathcal C}_{\hat \nu}})} & {{e^{4\phi /3}}{{\mathcal C}_{\hat \mu}}} \\ {{e^{4\phi /3}}{{\mathcal C}_{\hat \nu}}\quad} & {{e^{4\phi /3}}} \\ \end{array}} \right)\quad \Rightarrow \quad \det {{\mathcal G}_{\mu \nu}} = \det {{\mathcal G}_{\hat \mu \hat \nu}}\,.$$
(60)
Since the integral over ρ equals the length of the M-theory circle,
$$\int\nolimits_{{S_1}} d \rho = 2\pi R = 2\pi {g_s}{l_s}\quad \Rightarrow \quad {T_f} = {T_{M2}}\int\nolimits_{{S_1}} d \rho = {1 \over {2\pi \alpha {\prime}}},$$
(61)
where I used Eq. (55), \({T_{M2}} = 1/{(2\pi)^2}l_p^3\) and absorbed the overall circle length, expressed in terms of type IIA data, in a new energy density scale, matching the fundamental string tension Tf defined in Section 2. The same argument applies to the charge density leading to Qf = Qm 2 2πR.

Altogether, the double reduced action reproduces the bosonic effective action (6) describing the string propagation in a type IIA background. Thus, our classical bosonic M2-brane action is consistent with the relation between half-BPS M2-brane and fundamental strings in the spectrum of M-theory and type IIA.

Connection to the D2-brane: The direct dimensional reduction of the bosonic M2 brane describes a three-dimensional diffeomorphism invariant theory propagating in 10 dimensions, with eleven scalars as its field content. The latter disagrees with the bosonic field content of a D2-brane, which includes a vector field. Fortunately, a scalar field is Hodge dual, in three dimensions, to a one form. Thus, one expects that by direct dimensional reduction of the bosonic M2-brane action and after world volume dualisation of the scalar field Y along the M-theory circle, one should reproduce the classical D2-brane action [439, 477, 93, 480].

To describe the direct dimensional reduction, consider the Lagrangian [480]
$$S = {{{T_{{\rm{M2}}}}} \over 2}\int {{d^3}} \sigma \left({{v^{- 1}}\det {\mathcal G}_{\mu \nu}^{(11)} - v + {1 \over 3}{\epsilon ^{\mu \nu \rho}}{{\mathcal A}_{\mu \nu \rho}}} \right).$$
(62)
This is classically equivalent to Eq. (40) after integrating out the auxiliary scalar density υ by solving its algebraic equation of motion. Notice I already focused on the relevant case for later supersymmetric considerations, i.e., QM2 = TM2. The induced world volume fields are
$${\mathcal G}_{\mu \nu}^{(11)} = {e^{- {2 \over 3}\phi}}{{\mathcal G}_{\mu \nu}} + {e^{{4 \over 3}\phi}}{Z_\mu}{Z_\nu}$$
(63)
$${{\mathcal A}_{\mu \nu \rho}} = {{\mathcal C}_{\mu \nu \rho}} + 3{{\mathcal B}_{\left[ {\mu \nu} \right.}}{Z_{\left. \rho \right]}} - 3{{\mathcal B}_{\left[ {\mu \nu} \right.}}{{\mathcal C}_{\left. \rho \right]}}\,,$$
(64)
where
$$Z \equiv dY + {{\mathcal C}_1}\;.$$
(65)
Using the properties of 3 × 3 matrices,
$$\det {\mathcal G}_{\mu \nu}^{(11)} = {e^{- 2\phi}}\det [{{\mathcal G}_{\mu \nu}} + {e^{2\phi}}{Z_\mu}{Z_\nu}] = (\det {{\mathcal G}_{\mu \nu}})[{e^{- 2\phi}} + \vert Z{\vert ^2}]\,,$$
(66)
where \(\vert Z{\vert ^2} = {Z_\mu}{Z_\nu}{{\mathcal G}^{\mu \nu}}\), the action (62) can be written as
$$\begin{array}{*{20}c} {S = {{{T_{{\rm{M2}}}}} \over 2}\int {d^3}\sigma \,\left({{v^{- 1}}{e^{- 2\phi}}\det {{\mathcal G}_{\mu \nu}} - v + {1 \over 3}{\epsilon ^{\mu \nu \rho}}[{{\mathcal C}_{\mu \nu \rho}} - 3{{\mathcal B}_{\mu \nu}}{{\mathcal C}_\rho}]} \right.} \\ {\left. {+ \;{v^{- 1}}(\det {{\mathcal G}_{\mu \nu}})\vert Z{\vert ^2} + {\epsilon ^{\mu \nu \rho}}{{\mathcal B}_{\mu \nu}}{Z_\rho}} \right).\quad \quad \quad} \\ \end{array}$$
(67)
The next step is to describe the world volume dualisation and the origin of the U(1) gauge symmetry on the D2 brane effective action [480]. By definition, the identity
$$d(Z - {{\mathcal C}_1}) \equiv 0$$
(68)
holds. Adding the latter to the action through an exact two-form F = dV Lagrange multiplier
$$- {1 \over {2\pi}}\int F \wedge (Z - {{\mathcal C}_1}),$$
(69)
allows one to treat Z as an independent field. For a more thorough discussion on this point and the nature of the U(1) gauge symmetry, see [480]. Adding Eq. (69) to Eq. (67), one obtains
$$\begin{array}{*{20}c} {S = {{{T_{{\rm{M2}}}}} \over 2}\int {d^3}\sigma \,\left({{v^{- 1}}{e^{- 2\phi}}\det {{\mathcal G}_{\mu \nu}} - v + {1 \over 3}{\epsilon ^{\mu \nu \rho}}[{{\mathcal C}_{\mu \nu \rho}} + 3{{\mathcal F}_{\mu \nu}}{{\mathcal C}_\rho}]} \right.\quad \quad} \\ {\left. {+ \;{v^{- 1}}(\det {{\mathcal G}_{\mu \nu}})\vert {Z^2}\vert - {\epsilon ^{\mu \nu \rho}}{{\mathcal F}_{\mu \nu}}{Z_\rho}} \right).} \\ \end{array}$$
(70)
Notice I already introduced the same gauge invariant quantity introduced in D-brane Lagrangians
$${{\mathcal F}_{\mu \nu}} = {F_{\mu \nu}} - {{\mathcal B}_{\mu \nu}}.$$
(71)
Since Y is now an independent field, it can be eliminated by solving its algebraic equation of motion
$${Z^\mu} = {v \over {2\det {\mathcal G}}}{\epsilon ^{\mu \nu \rho}}{{\mathcal F}_{\mu \nu}}.$$
(72)
Inserting this back into the action and integrating out the auxiliary field \(\tilde \upsilon = - \det ({{\mathcal G}_{\mu \nu}})/\upsilon\) by solving its equation of motion, yields
$$S = - {T_{{\rm{D2}}}}\int {{d^3}} \sigma \,{e^{- \phi}}\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}})} \; + \;{T_{{\rm{D2}}}}\int\nolimits_w ({{\mathcal C}_3} + {\mathcal F} \wedge {{\mathcal C}_1}){.}$$
(73)
This matches the proposed D2-brane effective action, since TM2 = TD2 as a consequence of Eq.s (55) and (48).

3.3.2 T-duality covariance

In this section, I extend the D2-brane’s functional form to any Dp-brane using T-duality covariance. My goal is to show that the bulk T-duality rules (58) guarantee the covariance of the D-brane effective action functional form [453] and to review the origin in the interchange between scalar fields and gauge fields on the brane19.

The second question can be addressed by an analysis of the D-brane action bosonic symmetries. These act infinitesimally as
$$s\,{X^M} = {\xi ^\nu}{\partial _\nu}{X^M} + \Delta {X^M},$$
(74)
$$s\,{V_\mu} = {\xi ^\nu}{\partial _\nu}{V_\mu} + {V_\nu}{\partial _\mu}{\xi ^\nu} + {\partial _\mu}c + \Delta {V_\mu}.$$
(75)
They involve world volume diffeomorphisms ξν, a U(1) gauge transformation c and global transformations Δϕi. Since the background will undergo a T-duality transformation, by assumption, this set of global transformations must include translations along the circle, i.e., ΔZ = ϵ, ΔXm = ΔVμ = 0, where the original XM scalar fields were split into {Xm, Z}.
I argued that the realisation of T-duality on the brane action requires one to study its double-dimensional reduction. The latter involves a partial gauge fixing Z = σp = ρ, identifying one world volume direction with the starting S1 bulk circle and a zero-mode Fourier truncation in the remaining degrees of freedom, ρXm = ρVμ = 0. Extending this functional truncation to the p-dimensional diffeomorphisms \({\xi ^{\hat \mu}}\) where I split the world volume indices according to \(\{\mu \} = \{\hat \mu, \rho \}\) and the space of global transformations, i.e.,z ΔxM = z Δ = 0, the consistency conditions requiring the infinitesimal transformations to preserve the subspace of field configurations defined by the truncation and the partial gauge fixing, i.e., zi|g.f.+trunc = 0, determines
$$c({\sigma ^{\hat \mu}},\rho) = \tilde c({\sigma ^{{\hat \mu}}}) + a + {{\epsilon {\prime}} \over {2\pi \alpha {\prime}}}\rho$$
(76)
where a, ϵ′ are constants, the latter having mass dimension minus one. The set of transformations in the double dimensional reduction are
$$\tilde s{X^m} = {\xi ^{\hat \nu}}{\partial _{\hat \nu}}{X^m} + \tilde \Delta {X^m}$$
(77)
$$\tilde s{V_{\hat \mu}} = {\xi ^{\hat \nu}}{\partial _{\hat \nu}}{V_{\hat \mu}} + {V_{\hat \nu}}{\partial _{\hat \mu}}{\xi ^{\hat \nu}} + {\partial _{\hat \mu}}\tilde c + \tilde \Delta {V_{\hat \mu}}$$
(78)
$$\tilde s{V_\rho} = {\xi ^{\hat \nu}}{\partial _{\hat \nu}}{V_\rho} + \tilde \Delta {V_\rho}$$
(79)
where \(\tilde \Delta {V_{\hat \mu}} = \Delta {V_{\hat \mu}} - {V_\rho}{\partial _{\hat \mu}}\Delta Z,\,\tilde \Delta {V_\rho} = \Delta {V_\rho} + {\epsilon {\prime}}/2\pi {\alpha {\prime}}\) and \(\tilde \Delta {x^m}\) satisfies \({\partial _z}\tilde \Delta {x^m} = 0\).
Let me comment on Eq. (79). Vρ was a gauge field component in the original action. But in its gauge-fixed functionally-truncated version, it transforms like a world volume scalar. Furthermore, the constant piece ϵ ′ in the original U(1) transformation (76), describes a global translation along the scalar direction. The interpretation of both observations is that under double-dimensional reduction
$$(2\pi \alpha {\prime})\,{V_\rho} \equiv Z\prime$$
(80)
Z ′ becomes the T-dual target space direction along the T-dual circle and ϵ′ describes the corresponding translation isometry. This discussion reproduces the well-known massless open string spectrum when exchanging a Dirichlet boundary condition with a Neumann boundary condition.
Having clarified the origin of symmetries in the T-dual picture, let me analyse the functional dependence of the effective action. First, consider the couplings to the NS sector in the DBI action. Rewrite the induced metric \({\mathcal G}\) and the gauge invariant \({\mathcal F}\) in terms of the T-dual background (g ′, B ′) and degrees of freedom ({XM} = {Xm′, Z}), which will be denoted by primed quantities. This can be achieved by adding and subtracting the relevant pullback quantities. The following identities hold
$${{\mathcal G}_{\hat \mu \rho}} = {\partial _{\hat \mu}}{X^m}{g_{mz}}$$
(81)
$${{\mathcal G}_{\rho \rho}} = {g_{zz}}$$
(82)
$$\begin{array}{*{20}c} {{{\mathcal G}_{\hat \mu \hat \nu}} = {{{\mathcal G}{\prime}}_{\hat \mu \hat \nu}} + {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}{X^n}({g_{mn}} - {{g{\prime}}_{mn}}) - {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}Z\prime {{g{\prime}}_{z{\prime} m}}}\\ {- {\partial _{\hat \mu}}Z\prime {\partial _{\hat \nu}}{X^{M{\prime}}}{{g{\prime}}_{z{\prime} M{\prime}}}\quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\ \end{array}$$
(83)
$${{\mathcal F}_{\hat \mu \rho}} = {\partial _{\hat \mu}}Z\prime - {\partial _{\hat \mu}}{X^m}{B_{mz}}$$
(84)
$$\begin{array}{*{20}c} {{{\mathcal F}_{\hat \mu \hat \nu}} = {{{\mathcal F}{\prime}}_{\hat \mu \hat \nu}} - {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}{X^n}({B_{mn}} - {{B{\prime}}_{mn}}) + {\partial _{\hat \mu}}Z\prime {\partial _{\hat \nu}}{X^n}{{B{\prime}}_{z{\prime} n}}} \\ {+ {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}Z\prime {{B{\prime}}_{mz{\prime}}}\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(85)
It is a consequence of our previous symmetry discussion that Xm = Xm and \({V_{\hat \mu}} = V_{\hat \mu}{\prime}\) i.e., there is no change in the description of the dynamical degrees of freedom not involved in the circle directions. The determinant appearing in the DBI action can now be computed to be
$$\begin{array}{*{20}c} {\det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}) = {g_{zz}}\,\det \left({{{{\mathcal G}{\prime}}_{\hat \mu \hat \nu}} + {{{\mathcal F}{\prime}}_{\hat \mu \hat \nu}}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.}\\ {+ {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}{X^n}\,[({g_{mn}} - {{g{\prime}}_{mn}}) - ({B_{mn}} - {{B{\prime}}_{mn}}) - ({g_{mz}} - {B_{mz}})({g_{nz}} + {B_{nz}})/{g_{zz}}]}\\ {- {\partial _{\hat \mu}}{X^m}{\partial _{\hat \nu}}Z\prime \,[({{g{\prime}}_{mz{\prime}}} - {{B{\prime}}_{mz{\prime}}}) - ({g_{mz}} - {B_{mz}})/{g_{zz}}]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {- {\partial _{\hat \mu}}Z\prime {\partial _{\hat \nu}}{X^n}\,[({{g{\prime}}_{z{\prime} n}} + {{B{\prime}}_{nz{\prime}}}) + ({g_{nz}} + {B_{nz}})/{g_{zz}}]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;}\\ {\left. {- {\partial _{\hat \mu}}Z\prime {\partial _{\hat \nu}}Z\prime \,\left({{{g{\prime}}_{z{\prime} z{\prime}}} - {1 \over {{g_{zz}}}}} \right)} \right)\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;}\\ \end{array}$$
(86)
Notice that whenever the bulk T-duality rules (58) are satisfied, the functional form of the effective action remains covariant, i.e., of the form
$$- {T{\prime} _{{\rm{D}}({\rm{p - 1}})}}\int {{d^p}} \sigma \,{e^{- \phi {\prime}}}\,\sqrt {- \det ({{{\mathcal G}{\prime}}_{\hat \mu \hat \nu}} + {{{\mathcal F}{\prime}}_{\hat \mu \hat \nu}})} \,.$$
(87)
This is because all terms in the determinant vanish except for those in the first line. Finally, \({e^{- \phi}}\sqrt {{g_{zz}}}\) equals the T-dual dilaton coupling eϕ ′ and the original Dp-brane tension \({T_{{{\rm{D}}_{\rm{p}}}}}\) becomes the D(p-1)-brane tension in the T-dual theory due to the worldsheet defining properties (56) after the integration over the world volume circle
$${T_{{\rm{Dp}}}}\int d \rho = {1 \over {{{(2\pi)}^p}\,{g_s}l_s^{p + 1}}}\,2\pi \,R = {1 \over {{{(2\pi)}^{p - 1}}\,g{\prime _s}l_s^p}} = {T{\prime} _{{\rm{D}}({\rm{p - 1}})}}\,.$$
(88)
Just as covariance of the DBI action is determined by the NS-NS sector, one expects the RR sector to do the same for the WZ action. Here I follow similar techniques to the ones developed in [255, 453]. First, decompose the WZ Lagrangian density as
$${{\mathcal L}_{WZ}} = {\mathcal L}_{WZ}^ + + {\mathcal L}_{WZ}^ - \equiv d\rho \wedge {i_{{\partial _\rho}}}{{\mathcal L}_{WZ}} + {i_{{\partial _\rho}}}(d\rho \wedge {{\mathcal L}_{WZ}})\,.$$
(89)
Due to the functional truncation assumed in the double dimensional reduction, the second term vanishes. The D-brane WZ action then becomes
$${T_p}\int\nolimits_{{\Sigma _{p + 1}}} {{{\mathcal L}_{WZ}}} = {T_p}\int\nolimits_{{\Sigma _{p + 1}}} d \rho \wedge \,{e^{{{\mathcal F}^ -}}} \wedge ({i_{{\partial _\rho}}}C + {i_{{\partial _\rho}}}{\mathcal F} \wedge \,{C^ -})$$
(90)
where \({{\mathcal F}^ -} \equiv {i_{{\vartheta _\rho}}}(d\rho \wedge{\mathcal F})\) and the following conventions are used
$$\begin{array}{*{20}c} {{i_{{\partial _\rho}}}{\Omega _{(n)}} = {1 \over {(n - 1)!}}{\Omega _{\rho {\mu _2} \ldots {\mu _n}}}d{\sigma ^{{\mu _2}}} \wedge \ldots d{\sigma ^{{\mu _n}}}} \\ {{i_{{\partial _\rho}}}({\Omega _{(m)}} \wedge {\Omega _{(n)}}) = {i_{{\partial _\rho}}}{\Omega _{(m)}} \wedge {\Omega _{(n)}} + {{(- 1)}^m}{\Omega _{(m)}} \wedge {i_{{\partial _\rho}}}{\Omega _{(n)}}\,.\quad \;} \\ \end{array}$$
(91)
Using the T-duality transformation properties of the gauge invariant quantity \({\mathcal F}\), derived from our DBI analysis,
$${{\mathcal F}^ -} \rightarrow {\mathcal F}{\prime} - \left({{i_{{\partial _{z{\prime}}}}}B{\prime} \wedge {i_{{\partial _{z{\prime}}}}}g{\prime}} \right)/{g{\prime} _{z{\prime} z{\prime}}}$$
(92)
$${i_{{\partial _\rho}}}{\mathcal F} \rightarrow - \,{i_{{\partial _{z{\prime}}}}}g{\prime} /{g{\prime} _{z{\prime} z{\prime}}}$$
(93)
it was shown in [453] that the functional form of the WZ term is preserved, i.e., \(T_{{\rm{D(p - 1)}}}{\prime}\int\nolimits_{\partial \Sigma} {{e^{{{\mathcal F}{\prime}}}}} \wedge{C{\prime}}\), whenever the condition
$${(- 1)^p}{C{\prime} _p} = {i_{{\partial _\rho}}}{C_{p + 1}} - {{{i_{{\partial _{z{\prime}}}}}B{\prime} \wedge {i_{{\partial _{z{\prime}}}}}g{\prime}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}} \wedge {i_{{\partial _\rho}}}{C_{p - 1}} - {{{i_{{\partial _{z{\prime}}}}}g{\prime}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}} \wedge C_{p - 1}^ -$$
(94)
holds (the factor (− 1)p is due to our conventions (91) and the choice of orientation \({\epsilon ^{{{\tilde \mu}_1} \ldots {{\tilde \mu}_p}}} \equiv {\epsilon ^{{\mu _1} \ldots {\mu _p}\rho}}\) and ϵ01…p = 1).
Due to our gauge-fixing condition, Z = ρ, the ± components of the pullbacked world volume forms appearing in Eq. (94) can be lifted to ± components of the spacetime forms. The condition (94) is then solved by
$${i_{{\partial _z}}}{C_{p + 1}} = {(- 1)^p}\left({{{C{\prime}}_{(p)}} - {{{i_{{\partial _{z{\prime}}}}}g{\prime}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}} \wedge {i_{{\partial _{z{\prime}}}}}{{C{\prime}}_p}} \right)$$
(95)
$$C_{p - 1}^ - = {(- 1)^{(p - 1)}}\left({{i_{{\partial _{z{\prime}}}}}{{C{\prime}}_p} - {i_{{\partial _{z{\prime}}}}}B{\prime} \wedge \left({{{C{\prime}}_{p - 2}} - {{{i_{{\partial _{z{\prime}}}}}g{\prime}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}} \wedge {i_{{\partial _{z{\prime}}}}}{{C{\prime}}_{p - 2}}} \right)} \right)\,.$$
(96)
These are entirely equivalent to the T-duality rules (58) but written in an intrinsic way.
The expert reader may have noticed that the RR T-duality rules do not coincide with the ones appearing in [208]. The reason behind this is the freedom to redefine the fields in our theory. In particular, there exist different choices for the RR potentials, depending on their transformation properties under S-duality. For example, the 4-form C4 appearing in our WZ couplings is not S self-dual, but transforms as
$${C_4} \rightarrow {C_4} - {C_2} \wedge {B_2}\,.$$
(97)
Using a superindex S to denote an S-dual self-dual 4-form, the latter must be
$$C_4^S = {C_4} - {1 \over 2}{C_2} \wedge {B_2}\,.$$
(98)
Similarly, C6 does not transform as a doublet under S-duality, whereas
$$C_6^S = {C_6} - {1 \over 4}{C_2} \wedge {B_2} \wedge {B_2}$$
(99)
does. It is straightforward to check that Eqs. (95) and (96) are equivalent to the ones appearing in [208] using the above redefinitions. Furthermore, one finds
$$\begin{array}{*{20}c} {C_{{m_1} \ldots {m_6}}^S = {{C{\prime}}_{{m_1} \ldots {m_6}z{\prime}}} - 6{{C{\prime}}_{\left[ {{m_1} \ldots {m_5}} \right.}}{{{{g{\prime}}_{\left. {{m_6}} \right]z{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad \quad}\\ {- 45\left({{{C{\prime}}_{\left[ {{m_1}} \right.}} - {{C{\prime}}_{z{\prime}}}{{{{g{\prime}}_{\left[ {{m_1}z{\prime}} \right.}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}} \right){{B{\prime}}_{{m_2}{m_3}}}{{B{\prime}}_{{m_4}{m_5}}}{{B{\prime}}_{\left. {{m_6}} \right]z{\prime}}}}\\ {- 45{{C{\prime}}_{\left[ {{m_1}{m_2}z{\prime}} \right.}}{{B{\prime}}_{{m_3}{m_4}}}\left({{{B{\prime}}_{{m_5}{m_6}}} - 4{{B{\prime}}_{{m_5}z{\prime}}}{{{{g{\prime}}_{\left. {{m_6}} \right]z{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}} \right)}\\ {- 30{{C{\prime}}_{\left[ {{m_1} \ldots {m_4}z{\prime}} \right.}}{{B{\prime}}_{{m_5}z{\prime}}}{{{{g{\prime}}_{\left. {{m_6}} \right]z{\prime}}}} \over {{{g{\prime}}_{z{\prime} z{\prime}}}}}\quad \quad \quad \quad \quad \quad}\\ \end{array}$$
(100)
, which was not computed in [208].

In Section 7.1, I will explore the consequences that can be extracted from the requirement of T-duality covariance for the covariant description of the effective dynamics of N overlapping parallel D-branes in curved backgrounds, following [395].

3.3.3 M5-brane reduction

The double dimensional reduction of the M5-brane effective action, both in its covariant [417, 8] and non-covariant formulations [420, 420, 78] was checked to agree with the D4-brane effective action. It is important to stress that the outcome of this reduction may not be in the standard D4-brane action form given in Eq. (47), but in the dual formulation. The two are related through the world volume dualisation procedure described in [483, 7].

3.4 Supersymmetric brane effective actions in Minkowski

In the study of global supersymmetric field theories, one learns the superfield formalism is the most manifest way of writing interacting manifestly-supersymmetric Lagrangians [491]. One extends the manifold ℝ1,3 to a supermanifold through the addition of Grassmann fermionic coordinates θ. Physical fields ϕ (x) become components of superfields Φ(x, θ), the natural objects in this mathematical structure defined as finite polynomials in a Taylor-like θ expansion
$$\Phi (x,\theta) = \phi (x) + {\theta ^\alpha}{\phi _\alpha}(x) + \ldots$$
(101)
that includes auxiliary (non-dynamical) components allowing one to close the supersymmetry algebra off-shell. Generic superfields do not transform irreducibly under the super-Poincaré group. Imposing constraints on them, i.e., fi (Φ) = 0, gives rise to the different irreducible supersymmetric representations. For a standard reference on these concepts, see [491].
These considerations also apply to the p + 1 dimensional supermultiplets describing the physical brane degrees of freedom propagating in ℝ1,9, since these correspond to supersymmetric field theories in ℝ1,p. The main difference in the GS formulation of brane effective actions is that it is spacetime itself that must be formulated in a manifestly supersymmetric way. By the same argument used in global supersymmetric theories, one would be required to work in a 10- or 11-dimensional superspace, with standard bosonic coordinates xm and the addition of fermionic ones θ, whose representations will depend on the dimension of the bosonic submanifold. There are two crucial points to appreciate for our purposes
  1. 1.

    the superspace coordinates {xm, θ} will become the brane dynamical degrees of freedom {Xm (σ), θ (σ)}, besides any additional gauge fields living on the brane;

     
  2. 2.

    the couplings of the latter to the fixed background where the brane propagates must also be described in a manifestly spacetime supersymmetric way. The formulation achieving precisely that is the superspace formulation of supergravity theories [491].

     
Both these points were already encountered in our review of the GS formulation for the superstring. The same features will hold for all brane effective actions discussed below. After all, both strings and branes are different objects in the same theory. Consequently, any manifestly spacetime supersymmetric and covariant formulation should refer to the same superspace. It is worth emphasising the world volume manifold Σp+1 with local coordinates σμ remains bosonic in this formulation. This is not what occurs in standard superspace formulations of supersymmetric field theories. There exists a classically equivalent formulation to the GS one, the superembedding formulation that extends both the spacetime and the world volume to supermanifolds. Though I will briefly mention this alternative and powerful formulation in Section 8, I refer readers to [460].
Figure 5

Kappa symmetry and world volume diffeomorphisms allow one to couple the brane degrees of freedom to the superfield components of supergravity in a manifestly covariant and supersymmetric way. Invariance under kappa symmetry forces the background to be on-shell. The gauge fixing of these symmetries connects the GS formulation with world volume supersymmetry, whose on-shell degrees of freedom match the Goldstone modes of the brane supergravity configurations.

As in global supersymmetric theories, supergravity superspace formulations involve an increase in the number of degrees of freedom describing the spacetime dynamics (to preserve supersymmetry covariance). Its equivalence with the more standard component formalism is achieved through the satisfaction of a set of non-trivial constraints imposed on the supergravity superfields. These guarantee the on-shell nature of the physical superfield components. I refer the reader to a brief but self-contained Appendix A where this superspace formulation is reviewed for \({\mathcal N} = 2\) type IIA/B d =10 and \({\mathcal N} = 1 \, d = 11\) supergravities, including the set of constraints that render them on-shell. These will play a very important role in the self-consistency of the supersymmetric effective actions I am about to construct.

Instead of discussing the supersymmetric coupling to an arbitrary curved background at once, my plan is to review the explicit construction of supersymmetric D-brane and M2-brane actions propagating in Minkowski spacetime, or its superspace extension, super-Poincaré.20 The logic will be analogous to that presented for the superstring. First, I will construct these supersymmetric and kappa invariant actions without using the superspace formulation, i.e., using a more explicit component approach. Afterwards, I will rewrite these actions in superspace variables, pointing in the right direction to achieve a covariant extension of these results to curved backgrounds in Section 3.5.

3.4.1 D-branes in flat superspace

In this section, I am aiming to describe the propagation of D-branes in a fixed Minkowski target space preserving all spacetime supersymmetry and being world volume kappa symmetry invariant. Just as for bosonic open strings, the gauge field dependence was proven to be of the DBI form by explicit open superstring calculations [482, 389, 87].21

Here I follow the strategy in [9]. First, I will construct a supersymmetric invariant DBI action, building on the superspace results reported in Section 2. Second, I will determine the WZ couplings by requiring both supersymmetry and kappa symmetry invariance. Finally, as in our brief review of the GS superstring formulation, I will reinterpret the final action in terms of superspace quantities and their pullback to p + 1 world volume hypersurfaces. This step will identify the correct structure to be generalised to arbitrary curved backgrounds.

Let me first set my conventions. The field content includes a set of p + 1 dimensional world volume scalar fields {ZM (σ)} = {Xm (σ), θα (σ)} describing the embedding of the brane into the bulk supermanifold. Fermions depend on the theory under consideration
  • \(\bullet {\mathcal N} = 2 \, d = 10\) type IIA superspace involves two fermions of different chiralities θ±, i.e., Γ#θ± = ±θ±, where Γ# = Γ0Γ1 … Γ9. I describe them jointly by a unique fermion θ, satisfying θ = θ+ + θ.

  • \(\bullet {\mathcal N} = 2 \, d = 10\) type IIB superspace contains two fermions of the same chirality (positive by assumption), θii = 1, 2. The index is an internal SU(2) index keeping track of the doublet structure on which Pauli matrices τa act.

In either case, one defines \(\bar \theta = {\theta ^t}C\), in terms of an antisymmetric charge conjugation matrix C satisfying
$$\Gamma _m^t = - C{\Gamma _m}{C^{- 1}}\,,\quad \quad {C^t} = - C\,,$$
(102)
with Γm satisfying the standard Clifford algebra {Γm Γn} = 2ηmn with mostly plus eigenvalues. I am not introducing a special notation above to refer to the tangent space, given the flat nature of the bulk. This is not accurate but will ease the notation below. I will address this point when reinterpreting our results in terms of a purely superspace formulation.
Let me start the discussion with the DBI piece of the action. This involves couplings to the NS-NS bulk sector, a sector that is also probed by the superstring. Thus, both the supervielbein (Em, Eα) and the NS-NS 2-form B2 were already identified to be
$${E^m} = {\Pi ^m} = d{X^m} + d\bar \theta {\Gamma ^m}d\theta \quad ,\quad {E^\alpha} = d{\theta ^\alpha}$$
(103)
$${B_2} = - \bar \theta {\Gamma _\sharp}{\Gamma _m}d\theta \,(d{X^m} + {1 \over 2}\bar \theta {\Gamma ^m}d\theta)\,,$$
(104)
in type IIA, whereas in type IIB one replaces Γ# by τ3. The DBI action
$${S_{{\rm{DBI}}}} = - {T_{{\rm{Dp}}}}\int {{d^{p + 1}}} \sigma \,\sqrt {- \det ({\mathcal G} + {\mathcal F})}$$
(105)
will therefore be invariant under the spacetime supersymmetry transformations
$${\delta _\epsilon}\theta = \epsilon \,,\quad {\delta _\epsilon}{X^m} = \bar \epsilon {\Gamma ^m}\theta$$
(106)
if both, the induced world volume metric \({\mathcal G}\) and the gauge invariant 2-form, \({\mathcal F}\), are. These are defined by
$${{\mathcal G}_{\mu \nu}} = \Pi _\mu ^m\Pi _\nu ^n{\eta _{mn}},\quad \Pi _\mu ^m = {\partial _\mu}{X^m} - \bar \theta {\Gamma ^m}{\partial _\mu}\theta$$
(107)
$${{\mathcal F}_{\mu \nu}} = 2\pi \alpha \prime {F_{\mu \nu}} - {{\mathcal B}_{\mu \nu}},$$
(108)
where \({\mathcal B}\) stands for the pullback of the superspace 2-form B2 into the worldvolume, i.e., \({{\mathcal B}_{\mu \nu}} = {\partial _\mu}{Z^M}{\partial _\nu}{Z^N}{B_{MN}}\). Since B2 is quasi-invariant under (106), one chooses
$${\delta _\epsilon}V = \bar \epsilon {\Gamma _\sharp}{\Gamma _m}\theta d{X^m} + {1 \over 6}(\bar \epsilon {\Gamma _\sharp}{\Gamma _m}\theta \bar \theta {\Gamma ^m}d\theta + \bar \epsilon {\Gamma _m}\theta \bar \theta {\Gamma _\sharp}{\Gamma ^m}d\theta),$$
(109)
so that \({\delta _\varepsilon}{\mathcal F} = 0\), guaranteeing the invariance of the action (105) since the set of 1-forms Πm are supersymmetric invariant.
Let me consider the WZ piece of the action
$${S_{{\rm{WZ}}}} = \int {{\Omega _{p + 1}}} \,.$$
(110)
Since invariance under supersymmetry allows total derivatives, the Lagrangian can be characterised in terms of p + 2)-form
$${I_{p + 2}} = d{\Omega _{p + 1}},$$
(111)
satisfying
$${\delta _\epsilon}{I_{p + 2}} = 0\quad \Rightarrow \quad {\delta _\epsilon}{\Omega _{p + 1}} = d{\Lambda _p}.$$
(112)
Thus, mathematically, I (p+2) must be constructed out of supersymmetry invariants \(\{{\Pi ^m}, \, d\theta, \, {\mathcal F}\}\).
The above defines a cohomological problem whose solution is not guaranteed to be kappa invariant. Since our goal is to construct an action invariant under both symmetries, let me first formulate the requirements due to the second invariance. The strategy followed in [9] has two steps:
  • First, parameterise the kappa transformation of the bosonic fields {Xm, V1} in terms of an arbitrary δκθ. Experience from supersymmetry and kappa invariance for the superparticle and superstring suggest
    $$\begin{array}{*{20}c} {{\delta _\kappa}{X^m} = - {\delta _\kappa}\bar \theta {\Gamma ^m}\theta \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\delta _\kappa}{V_1} = - {\delta _\kappa}\bar \theta {\Gamma _\sharp}{\Gamma _m}\theta {\Pi ^m} + {1 \over 2}{\delta _\kappa}\bar \theta {\Gamma _\sharp}{\Gamma _m}\theta \bar \theta {\Gamma ^m}d\theta - {1 \over 2}{\delta _\kappa}\bar \theta {\Gamma ^m}\theta \bar \theta {\Gamma _\sharp}{\Gamma _m}d\theta \,.} \\ \end{array}$$
    (113)
    Notice, δκV is chosen to remove the exact form coming from the kappa symmetry variation of B2, i.e., \({\delta _\kappa}{B_2} = - 2{\delta _\kappa}\bar \theta {\Gamma _\#}{\Gamma _m}d\theta {\Pi ^m} + d{\delta _k}{V_1}\).
  • Second, kappa symmetry must be able to remove half of the fermionic degrees of freedom. Thus, as in the superstring discussion, one expects δκθ to involve some non-trivial projector. This fact can be used to conveniently parameterise the kappa invariance of the total Lagrangian. The idea in [9] was to parameterise the DBI kappa transformation as
    $${\delta _\kappa}{{\mathcal L}_{{\rm{DBI}}}} = 2{\delta _\kappa}\bar \theta {\gamma ^{(p)}}T_{(p)}^\nu {\partial _\nu}\theta \,,\qquad {\rm{with}}\qquad {({\gamma ^{(p)}})^2} = 1\,,$$
    (114)
    requiring the WZ kappa transformation to be
    $${\delta _\kappa}{{\mathcal L}_{{\rm{WZ}}}} = 2{\delta _\kappa}\bar \theta T_{(p)}^\nu {\partial _\nu}\theta \,.$$
    (115)
    In this way, the kappa symmetry variation of the full Lagrangian equals
    $${\delta _\kappa}({{\mathcal L}_{{\rm{DBI}}}} + {{\mathcal L}_{{\rm{WZ}}}}) = 2{\delta _\kappa}\bar \theta (1 + {\gamma ^{(p)}})T_{(p)}^\nu {\partial _\nu}\theta \,.$$
    (116)
    This is guaranteed to vanish choosing \({\delta _\kappa}\bar \theta = \bar k(1 - {\gamma ^{(p)}})\), given the projector nature of \({1 \over 2}(1 \pm {\gamma ^{(p)}})\).
The question is whether \(T_{(p)}^\nu, {\gamma ^{(p)}}\) and I(p+ 2) exist satisfying all the above requirements. The explicit construction of these objects was given in [9]. Here, I simply summarise their results. The WZ action was found to be
$$d{{\mathcal L}_{{\rm{WZ}}}} = - {T_{{\rm{Dp}}}}{\mathcal R}{e^{\mathcal F}}\,,$$
(117)
where \({\mathcal R}\) is the pullback of the field strength of the RR gauge potential C, as defined in Eq. (523). Using Π = Πm Γm, this can be written as [293]
$${\mathcal R}={\bar E}{\mathcal C}_A({\rlap{/}\Pi})E,\qquad {\mathcal C}_A({\rlap{/}\Pi})\;\;=\;\;\sum\nolimits_{l=0}(\Gamma_\sharp)^{l+1}\frac{{\rlap{/}\Pi}^{2l}}{(2l)!}$$
(118)
in type IIA, whereas in type IIB [329]
$${\mathcal R} = - \bar E{{\mathcal S}_B}({\rlap{/}\Pi}){\tau _1}E,\quad \quad {{\mathcal S}_B}({\rlap{/}\Pi})\;\; = \;\;\sum\limits_{l = 0} {{{({\tau _3})}^l}} {{{\rlap{/}\Pi}^{2l + 1}} \over {(2l + 1)!}}\,.$$
(119)
Two observations are in order:
  1. 1.

    \(d{{\mathcal L}_{{\rm{WZ}}}}\) is indeed manifestly supersymmetric, since it only depends on supersymmetric invariant quantities, but \({{\mathcal L}_{{\rm{WZ}}}}\) is quasi-invariant. Thus, when computing the algebra closed by the set of conserved charges, one can expect the appearance of non-trivial charges in the right-hand side of the supersymmetry algebra. This is a universal feature of brane effective actions that will be conveniently interpreted in Section 3.6.

     
  2. 2.

    This analysis has determined the explicit form of all the RR potentials Cp as superfields in superspace. This was achieved by world volume symmetry considerations, but it is reassuring to check that the expressions found above do satisfy the superspace constraints reported in Appendix A.1. I will geometrically reinterpret the derived action as one describing a Dp-brane propagating in a fixed super-Poincaré target space shortly.

     
Let me summarise the global and gauge symmetry structure of the full action. The set of gauge symmetries involves world volume diffeomorphisms (ξμ), an abelian U(1) gauge symmetry (c) and kappa symmetry (κ). Their infinitesimal transformations are
$$s{X^m} = {{\mathcal L}_\xi}{X^m} + {\delta _\kappa}{X^m} = {\xi ^\mu}{\partial _\mu}{X^m} - {\delta _\kappa}\bar \theta {\Gamma ^m}\theta \,,$$
(120)
$$s{\theta ^\alpha} = {{\mathcal L}_\xi}{\theta ^\alpha} + {\delta _\kappa}{\theta ^\alpha} = {\xi ^\mu}{\partial _\mu}{\theta ^\alpha} + {\delta _\kappa}{\theta ^\alpha}\,,$$
(121)
$$s{V_\mu} = {{\mathcal L}_\xi}{V_\mu} + {\partial _\mu}c + {\delta _\kappa}{V_\mu} = {\xi ^\nu}{\partial _\nu}{V_\mu} + {V_\nu}{\partial _\mu}{\xi ^\nu} + {\partial _\mu}c + {\delta _\kappa}{V_\mu}\,,$$
(122)
where δκVμ is given in Eq. (113) and δκθ was determined in [9]
$${\delta _\kappa}\bar \theta = \bar \kappa (1 - {\gamma ^{(p)}}),\quad \quad {\gamma ^{(p)}} = {{{\rho ^{(p)}}} \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\,.$$
(123)
In type IIA, the matrix ρ(p) stands for the p + 1 world volume form coefficient of \({{\mathcal S}_A}(\not \Pi){e^{\mathcal F}}\), where
$${\rho ^{(p)}} = {[{{\mathcal S}_A}({\rlap{/}\Pi}){e^{\mathcal F}}]_{p + 1}},\quad \quad {{\mathcal S}_A}({\rlap{/}\Pi}) \;\;= \;\;\sum\limits_{l = 0} {{{({\Gamma _\sharp})}^{l + 1}}} {{{{\rlap{/}\Pi}^{2l + 1}}} \over {(2l + 1)!}}$$
(124)
, while in type IIB, it is given by
$${\rho ^{(p - 1)}} = - {[{{\mathcal C}_B}({\rlap{/}\Pi}){e^{\mathcal F}}{\tau _1}]_p},\quad \quad {{\mathcal C}_B}({\rlap{/}\Pi})\;\; = \;\;\sum\limits_{l = 0} {{{({\tau _3})}^{l + 1}}} {{{{\rlap{/}\Pi}^{2l}}} \over {(2l)!}}\,.$$
(125)
It was proven in [9] that \({\rho ^2} = - \det ({\mathcal G + \mathcal F})1\) This proves \(\gamma _{(p)}^2\) equals the identity, as required in our construction.
The set of global symmetries includes supersymmetry (), bosonic translations (am) and Lorentz transformations (ωmn). The field infinitesimal transformations are
$$\Delta {X^m} = {\delta _\epsilon}{X^m} + {\delta _{\rm{a}}}{X^m} + {\delta _\omega}{X^m} = \bar \epsilon {\Gamma ^m}\theta + {{\rm{a}}^m} + {\omega ^m}{\,_n}{X^n},$$
(126)
$$\Delta {\theta ^\alpha} = {\delta _\epsilon}{\theta ^\alpha} + {\delta _\omega}{\theta ^\alpha} = {\epsilon ^\alpha} + {1 \over 4}{\omega ^{mn}}{\left({{\Gamma _{mn}}\theta} \right)^\alpha},$$
(127)
$$\Delta {V_\mu} = {\delta _\epsilon}{V_\mu},$$
(128)
with δϵVμ given in Eq. (109) and \({\omega ^m}_n \equiv {\omega ^{mp}}{\eta _{pn}}\).
Geometrical reinterpretation of the effective action: the supersymmetric action was constructed out of the supersymmetric invariant forms \(\{{\Pi ^m}, d \theta, {\mathcal F}\}\). These can be reinterpreted as the pullback of 10-dimensional superspace tensors to the p + 1 brane world volume. To see this, it is convenient to introduce the explicit supervielbein components \(E_M^A(z)\), defined in Appendix A.1, where the index M stands for curved superspace indices, i.e., M = {m, α}, and the index A for tangent flat superspace indices, i.e., \(A = \{a,\underline {\alpha \}}\). In this language, the super-Poincaré supervielbein components equal
$$E_m^a = \delta _m^a\,,\quad \quad E_\alpha ^{\underline{\alpha}} = \delta _\alpha ^{\underline{\alpha}}\,,\quad \quad E_m^{\underline{\alpha}} = 0\,,\quad \quad E_\alpha ^a = {\left({\bar \theta {\Gamma ^a}} \right)_{\underline{\alpha}}}\delta _\alpha ^{\underline{\alpha}}\,.$$
(129)
manifest that all Clifford matrices Γa act in the tangent space, as they should. The components (129) allow us to rewrite all couplings in the effective action as pullbacks
$$\begin{array}{*{20}c} {{{\mathcal G}_{\mu \nu}}(Z) = {\partial _\mu}{Z^M}E_M^a(Z){\partial _\nu}{Z^N}E_N^b(Z){\eta _{ab}},\quad \quad \quad \quad \;\;} \\ {{{\mathcal B}_{\mu \nu}}(Z) = {\partial _\mu}{Z^M}E_M^A(Z){\partial _\nu}{Z^N}E_N^C(Z)\,{B_{AC}}(Z),\quad \quad \quad} \\ {{{\mathcal C}_{{\mu _1} \ldots {\mu _{p + 1}}}}(Z) = {\partial _{{\mu _1}}}{Z^{{M_1}}}E_{{M_1}}^{{A_1}}(Z) \ldots {\partial _{{\mu _{p + 1}}}}{Z^{{M_{p + 1}}}}E_{{M_{p + 1}}}^{{A_{p + 1}}}(Z){C_{{A_1} \ldots {A_{p + 1}}}}(Z),} \\ \end{array}$$
(130)
of the background superfields \(E_M^A\), Bac and \({C_{{A_1} \ldots {A_{p + 1}}}}\) to the brane world volume. Furthermore, the kappa symmetry transformations (113) and (123) also allow a natural superspace description as
$${\delta _\kappa}{Z^M}\,E_M^a = 0,\quad \quad {\delta _\kappa}{Z^M}\,E_M^{\underline{\alpha}} = (1 + {\Gamma _\kappa})\kappa$$
(131)
where the kappa symmetry matrix Γκ is nicely repackaged
$${({\Gamma _\kappa})_{(p + 1)}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\limits_{l = 0}^k {{\gamma _{(2l + 1)}}} \,\Gamma _\sharp ^{l + 1} \wedge {e^{\mathcal F}}\quad {\rm{type}}\,{\rm{IIA}}\;p = 2k$$
(132)
$${({\Gamma _\kappa})_{(p + 1)}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\limits_{l = 0}^{k + 1} {{\gamma _{(2l)}}} \tau _3^l \wedge {e^{\mathcal F}}\,i{\tau _2}\quad {\rm{type}}\,{\rm{IIB}}\;p = 2k + 1\,,$$
(133)
in terms of the induced Clifford algebra matrices γμ and the gauge invariant tensor \({\mathcal F}\)
$${\gamma _{(1)}} \equiv d{\sigma ^\mu}{\gamma _\mu} = d{\sigma ^\mu}{\partial _\mu}{Z^M}E_M^a(Z){\Gamma _a},$$
(134)
$${\mathcal F} = 2\pi \alpha \prime F - {{\mathcal B}_2},$$
(135)
whereas γ(l) stands for the wedge product of the 1-forms γ(1).

Summary: We have constructed an effective action describing the propagation of Dp-branes in 10-dimensional Minkowski spacetime being invariant under p + 1 dimensional diffeomorphisms, 10-dimensional supersymmetry and kappa symmetry. The final result resembles the bosonic action (47) in that it is written in terms of pullbacks of the components of the different superfields \(E_M^A(Z)\), Bac (Z) and \({C_{{A_1} \ldots {A_{p + 1}}}}\) encoding the non-trivial information about the non-dynamical background where the brane propagates in a manifestly supersymmetric way. These superfields are on-shell supergravity configurations, since they satisfy the set of constraints listed in Appendix A.1. It is this set of features that will allow us to generalise these couplings to arbitrary on-shell superspace backgrounds in Section 3.5, while preserving the same kinematic properties.

3.4.2 M2-brane in flat superspace

Let me consider an M2-brane as an example of an M-brane propagating in d = 11 super-Poincaré. Given the lessons from the superstring and D-brane discussions, my presentation here will be much more economical.

First, let me describe d = 11 super-Poincaré as a solution of eleven-dimensional supergravity using the superspace formulation introduced in Appendix A.2. In the following, all fermions will be 11-dimensional Majorana fermions θ as corresponds to \({\mathcal N} = 1 \, d = 11\) superspace. Denoting the full set of superspace coordinates as {ZM} = {Xm θα} with m = 0,…, 10 and α = 1,…, 32, the superspace description of \({\mathcal N} = 1 \, d = 11\) super-Poincaré is [165, 144]
$$\begin{array}{*{20}c} {{E^a} = d{X^a} + d\bar \theta {\Gamma ^a}\theta \,,\quad {E^{\underline{\alpha}}} = d{\theta ^{\underline{\alpha}}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{R_4} = {1 \over 2}{E^a} \wedge {E^b} \wedge d{\theta ^{\underline{\alpha}}} \wedge d{\theta ^{\underline{\beta}}}{{({\Gamma _{ab}})}_{\underline {\alpha \beta}}},\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{R_7} = {1 \over {5!}}{E^{{a_1}}} \wedge {E^{{a_2}}} \wedge {E^{{a_3}}} \wedge {E^{{a_4}}} \wedge {E^{{a_5}}} \wedge d{\theta ^{\underline{\alpha}}} \wedge d{\theta ^{\underline{\beta}}}{{({\Gamma _{{a_1}{a_2}{a_3}{a_4}{a_5}}})}_{\underline {\alpha \beta}}}.} \\ \end{array}$$
(136)
It includes the supervielbein EA = {Ea, Eα} and the gauge invariant field strengths R4 = dA3 and its Hodge dual \({R_7} = d{A_6} + {1 \over 2}{A_3}\wedge{R_4}\), defined as Eq. (552) in Appendix A.2.
The full effective action can be written as [91]
$$\begin{array}{*{20}c} {S = - {T_{{\rm{M2}}}}\int {{d^3}} \sigma \,\sqrt {- \det {{\mathcal G}_{\mu \nu}}} + {T_{{\rm{M2}}}}\int {{{\mathcal A}_3}} ,\quad \quad \quad \quad \quad \;} \\ {{{\mathcal G}_{\mu \nu}} = E_\mu ^a(X,\theta)E_\nu ^b(X,\theta){\eta _{ab}},\quad E_\mu ^A \equiv {\partial _\mu}{Z^M}\,E_M^A(X,\theta),} \\ {{{\mathcal A}_3} = {1 \over {3!}}{\varepsilon ^{\mu \nu \rho}}E_\mu ^BE_\nu ^CE_\rho ^D{A_{BCD}}(X,\theta){.}\quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(137)
Notice it depends on the supervielbeins \(E_M^A(X,\theta)\) and the three form potential Cabc (x, θ) super-fields only through their pullbacks to the world volume.
Its symmetry structure is analogous to the one described for D-branes. Indeed, the action (137) is gauge invariant under world volume diffeomorphisms (ξμ) and kappa symmetry (κ) with infinitesimal transformations given by
$$s{X^m} = {{\mathcal L}_\xi}{X^m} + {\delta _\kappa}{X^m} = {\xi ^\mu}{\partial _\mu}{X^m} + {\delta _\kappa}\bar \theta {\Gamma ^m}\theta \,,$$
(138)
$$s{\theta ^\alpha} = {{\mathcal L}_\xi}{\theta ^\alpha} + {\delta _\kappa}{\theta ^\alpha} = {\xi ^\mu}{\partial _\mu}{\theta ^\alpha} + (1 + {\Gamma _\kappa})\kappa \,,$$
(139)
$${\Gamma _\kappa} = {1 \over {3!\,\sqrt {- \det {\mathcal G}}}}{\varepsilon ^{\mu \nu \rho}}E_\mu ^aE_\nu ^bE_\rho ^c\,{\Gamma _{abc}}\,.$$
(140)
The kappa matrix (140) satisfies \(\Gamma _\kappa ^2 = 1\). Thus, δκθ is a projector that will allow one to gauge away half of the fermionic degrees of freedom.
The action (137) is also invariant under global super-Poincaré transformations
$$\delta \theta = \epsilon + {1 \over 4}{\omega _{mn}}{\Gamma ^{mn}}\theta ,$$
(141)
$$\delta {X^m} = \bar \epsilon {\Gamma ^m}\theta + {a^m} + {\omega ^m}{\,_n}{X^n}.$$
(142)
Supersymmetry quasi-invariance can be easily argued for since R4 is manifestly invariant. Thus, its gauge potential pullback variation will be a total derivative
$${\delta _\epsilon}{A_3} = d[\bar\epsilon {\Delta _2}]$$
(143)
for some spinor-valued two form Δ2.

It is worth mentioning that just as the bosonic membrane action reproduces the string world-sheet action under double dimensional reduction, the same statement is true for their supersymmetric and kappa invariant formulations [192, 476].

3.5 Supersymmetric brane effective actions in curved backgrounds

In this section, I extend the supersymmetric and kappa invariant D-brane and M2-brane actions in super-Poincaré to D-branes, M2-branes and M5-branes in arbitrary curved backgrounds. The main goal, besides introducing the formalism itself, is to highlight that the existence of kappa symmetry invariance forces the supergravity background to be on-shell.

In all effective actions under consideration, the set of degrees of freedom includes scalars ZM = {Xm, θα} and it may include some gauge field Vp, whose dependence is always through the gauge invariant combination \(d{V_p} - {{\mathcal B}_{p + 1}}\)22. The set of kappa symmetry transformations will universally be given by
$$\begin{array}{*{20}c} {{\delta _\kappa}{Z^M}\,E_M^a(X,\theta) = 0\,,\quad \quad \quad \quad \quad \quad \quad} \\ {{\delta _\kappa}{Z^M}\,E_M^{\underline{\alpha}}(X,\theta) = (1 + {\Gamma _\kappa})\kappa \,,\quad \quad \quad \quad} \\ {{\delta _\kappa}{V_p} = {Z^ \star}{i_\kappa}{B_{p + 1}}.} \\ \end{array}$$
(144)
The last transformation is a generalisation of the one encountered in super-Poincaré. Indeed, the kappa variation of the pullback of any n-form satisfies
$${\delta _\kappa}{{\mathcal T}_n} \equiv {\delta _\kappa}{Z^ \star}{T_n} = {Z^ \star}{{\mathcal L}_\kappa}{T_n} = {Z^ \star}\{d,\,{i_\kappa}\} {T_n},$$
(145)
where Z* stands for the pullback of to the world volume. The choice in Eq. (144) guarantees the kappa transformation of dVp removes the total derivative in \({\delta _\kappa}{{\mathcal B}_{p + 1}}\)
The structure of the transformations (144) is universal, but the details of the kappa symmetry matrix depend on the specific theory, as described below. A second universal feature, associated with the projection nature of kappa symmetry transformations, i.e.,\(\Gamma _\kappa ^2 = 1\), is the correlation between the brane charge density and the sign of Γκ in Eq. (144). More specifically, any brane effective action will have the structure
$${S_{{\rm{brane}}}} = - {T_{{\rm{brane}}}}\int {{d^{p + 1}}} \sigma ({{\mathcal L}_{DBI}} - {\epsilon_1}{{\mathcal L}_{WZ}}).$$
(146)
Notice this is equivalent to requiring Tbrane = |Qbrane|, a property that is just reflecting the half-BPS nature of these branes. It can be shown that
$${\delta _\kappa}{S_{{\rm{brane}}}} \propto (1 + {\epsilon _1}{\Gamma _\kappa}){\delta _\kappa}\theta \quad \Rightarrow \quad {\delta _\kappa}\theta = (1 - {\epsilon _1}{\Gamma _\kappa})\kappa .$$
(147)
The choice of ϵ1 is correlated to the distinction between a brane and an anti-brane. Both are supersymmetric, but preserve complementary supercharges. This ambiguity explains why some of the literature has apparently different conventions, besides the possibility of working with different Clifford algebra realisations23.

3.5.1 M2-branes

The effective action describing a single M2-brane in an arbitrary 11-dimensional background is formally the same as in Eq. (137)
$${S_{M2}} = - {T_{{\rm{M2}}}}\int {{d^3}} \sigma \,\sqrt {- \det {{\mathcal G}_{\mu \nu}}} + {T_{{\rm{M2}}}}\int {{{\mathcal A}_3}} ,$$
(148)
with the same definitions for the induced metric \({\mathcal G}\) and the pull back 3-form \({{\mathcal A}_3}\). The information regarding different 11-dimensional backgrounds is encoded in the different couplings described by the supervielbein \(E_M^A(X,\theta)\) and 3-form Aabd (X, θ) superfields.
The action (148) is manifestly 3d-diffeomorphism invariant. It was shown to be kappa invariant under the transformations (144), without any gauge field, whenever the background superfields satisfy the constraints reviewed in Appendix A.2, i.e., whenever they are on-shell, for a kappa symmetry matrix given by [90]
$${\Gamma _\kappa} = {1 \over {3!\sqrt {- \det {\mathcal G}}}}{\varepsilon ^{\mu \nu \rho}}E_\mu ^a(X,\theta)\,E_\nu ^b(X,\theta)\,E_\rho ^c(X,\theta)\,{\Gamma _{abc}},$$
(149)
where \(E_\mu ^a(X,\theta) = {\partial _m}u{X^m}E_m^a(X,\theta)\) is the pullback of the curved supervielbein to the world volume.

3.5.2 D-branes

Proceeding in an analogous way for Dp-branes, their effective action in an arbitrary type IIA/B background is
$${S_{{\rm{Dp}}}} = - {T_{{\rm{Dp}}}}\int {{d^{p + 1}}} \sigma \,{e^{- \phi}}\,\sqrt {- \det ({\mathcal G} + {\mathcal F})} + {T_{{\rm{Dp}}}}\int {\mathcal C} \wedge {e^{\mathcal F}},$$
(150)
$${{\mathcal F}_{\mu \nu}} = 2\pi \alpha \prime {F_{\mu \nu}} - E_\mu ^A\,E_\nu ^C\,{B_{AC}},\quad {{\mathcal C}_r} = {1 \over {r!}}{\epsilon^{{\mu _1} \ldots {\mu _r}}}E_{{\mu _1}}^{{A_1}} \ldots E_{{\mu _r}}^{{A_r}}{C_{{A_1} \ldots {A_r}}}.$$
(151)
It is understood that \({{\mathcal G}_{\mu \nu}}(X,\theta) = E_\mu ^aE_\nu ^b{\eta _{ab}}\) and \({\mathcal C}\) is defined using the same notation as in Eq. (523), i.e., as a formal sum of forms, so that the WZ term picks all contributions coming from the wedge product of this sum and the Taylor expansion of \({e^{\mathcal F}}\) that saturate the p + 1 world volume dimension. Notice all information on the background spacetime is encoded in the superfields \(E_M^A(X,\theta)\), ϕ (X, θ), Bac (X, θ) and the set of RR potentials \(\{{C_{{A_1} \ldots {A_r}}}(X,\theta)\}\).
The action (150) is p + 1 dimensional diffeomorphic invariant and it was shown to be kappa invariant under the transformations (144) for V1 in [141, 93] when the kappa symmetry matrix equals
$${({\Gamma _\kappa})_{(p + 1)}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\limits_{l = 0} {{\gamma _{(2l + 1)}}} \,\Gamma _{11}^{l + 1} \wedge {e^{\mathcal F}}\quad {\rm{type}}\,{\rm{IIA}}\;p = 2k,$$
(152)
$${({\Gamma _\kappa})_{(p + 1)}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\limits_{l = 0} {{\gamma _{(2l)}}} \tau _3^l \wedge {e^{\mathcal F}}\,i{\tau _2}\quad {\rm{type}}\,{\rm{IIB}}\;p = 2k + 1\,,$$
(153)
and the background is on-shell, i.e., satisfies the constraints reviewed in Appendix A.1. In the expressions above γ(1) stands for the pullback of the bulk tangent space Clifford matrices
$${\gamma _{(1)}} = d{\sigma ^\mu}{\gamma _\mu} = d{\sigma ^\mu}E_\mu ^a(X,\theta){\Gamma _a},$$
(154)
and γ(r) stands for the wedge product of r of these 1-forms. In [94], readers can find an extension of the results reviewed here when the background includes a mass parameter, i.e., it belongs to massive IIA [434].

3.5.3 M5-branes

The six-dimensional diffeomorphic and kappa symmetry invariant M5-brane [45] is a formal extension of the bosonic one
$$\begin{array}{*{20}c} {{S_{{\rm{M5}}}} = {T_{{\rm{M5}}}}\int {d^6}\xi \,({{\mathcal L}_0} + {{\mathcal L}_{{\rm{WZ}}}})\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;} \\ {{{\mathcal L}_0} = - \sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\tilde H}_{\mu \nu}})} + {{\sqrt {- \det {\mathcal G}}} \over {4(\partial a\cdot\partial a),}}({\partial _\mu}a){{({{\mathcal H}^{\ast}})}^{\mu \nu \rho}}{{\mathcal H}_{\nu \rho \iota}}({\partial ^\iota}a)} \\ \end{array}$$
(155)
$${{\mathcal L}_{{\rm{WZ}}}} = {{\mathcal A}_6} + {1 \over 2}{{\mathcal H}_3} \wedge {{\mathcal A}_3}\,,$$
(156)
where all pullbacks refer to superspace. This is kappa invariant under the transformations (144) for V2, including the extra transformation law
$${\delta _\kappa}a = 0,$$
(157)
for the auxiliary scalar field introduced in the PST formalism. These transformations are determined by the kappa symmetry matrix
$${\Gamma _\kappa} = {{{v_\mu}{\gamma ^\mu}} \over {\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\tilde H}_{\mu \nu}})}}}\left[ {{\gamma _\nu}{t^\nu} + {{\sqrt {- \det {\mathcal G}}} \over 2}\,{\gamma ^{\nu \rho}}{{\tilde H}_{\nu \rho}} - {1 \over {5!}}\,{\varepsilon ^{{\mu _1} \ldots {\mu _5}\nu}}{v_\nu}{\gamma _{{\mu _1} \ldots {\mu _5}}}} \right]\,,$$
(158)
where \({\gamma _\mu} = {E_\mu}^a{\Gamma _a}\) and the vector fields tμ and υμ are defined by
$${t^\mu} = {1 \over 8}\,{\varepsilon ^{\mu {\nu _1}{\nu _2}{\rho _1}{\rho _2}\iota}}{\tilde H_{{\nu _1}{\nu _2}}}{\tilde H_{{\rho _1}{\rho _2}}}{v_\iota}\quad {\rm{with}}\quad {v_\mu} \equiv {{{\partial _\mu}a} \over {\sqrt {- \partial a\cdot\partial a}}}.$$
(159)
Further comments on kappa symmetry:κ-symmetry is a fermionic local symmetry for which no gauge field is necessary. Besides its defining projective nature when acting on fermions, i.e., \({\delta _\kappa}\theta = {(1 + {\Gamma _\kappa})_\kappa}\) with \(\Gamma _\kappa ^2 = 1\), there are two other distinctive features it satisfies [449]:
  1. 1.

    the algebra of κ-transformations only closes on-shell,

     
  2. 2.

    κ-symmetry is an infinitely reducible symmetry.

     
The latter statement uses the terminology of Batalin and Vilkovisky [52] and it is a direct consequence of its projective nature, since the existence of the infinite chain of transformations
$$\kappa \rightarrow (1 - {\Gamma _\kappa}){\kappa _1}\,,{\kappa _1} \rightarrow (1 + {\Gamma _\kappa}){\kappa _2} \ldots$$
(160)
gives rise to an infinite tower of ghosts when attempting to follow the Batalin-Vilkovisky quantisation procedure, which is also suitable to handle the first remark above. Thus, covariant quantisation of kappa invariant actions is a subtle problem. For detailed discussions on problems arising from the regularisation of infinite sums and dealing with Stueckelberg type residual gauge symmetries, readers are referred to [326, 325, 254, 223, 84].

It was later realised, using the Hamiltonian formulation, that kappa symmetry does allow covariant quantisation provided the ground state of the theory is massive [327]. The latter is clearly consistent with the brane interpretation of these actions, by which these vacua capture the half-BPS nature of the (massive) branes themselves24.

For further interesting kinematical and geometrical aspects of kappa symmetry, see [449, 167, 166] and references therein.

3.6 Symmetries: spacetime vs world volume

The main purpose of this section is to discuss the global symmetries of brane effective actions, the algebra they close and to emphasise the interpretation of some of the conserved charges appearing in these algebras before and after gauge fixing of the world volume diffeomorphisms and kappa symmetry.
  • before gauge fixing, the p + 1 field theory will be invariant under the full superisometry of the background where the brane propagates. This is a natural extension of the super-Poincaré invariance when branes propagate in Minkowski. As such, the algebra closed by the brane conserved charges will be a subalgebra of the maximal spacetime superalgebra one can associate to the given background.

  • after gauge fixing, only the subset of symmetries preserved by the brane embedding will remain linearly realised. This subset determines the world volume (supersymmetry) algebra. In the particular case of brane propagation in Minkowski, this algebra corresponds to a subalgebra of the maximal super-Poincaré algebra in p + 1 dimensions.

To prove that background symmetries give rise to brane global symmetries, one must first properly define the notion of superisometry of a supergravity background. This involves a Killing superfield ξ (Z) satisfying the properties
$${{\mathcal L}_\xi}({E^a}{\otimes _s}{E^b}){\eta _{ab}} = 0\,,$$
(161)
$${{\mathcal L}_\xi}{R_4} = {{\mathcal L}_\xi}{R_7} = 0\,,\qquad {\rm{M - theory}}$$
(162)
$${{\mathcal L}_\xi}{H_3} = {{\mathcal L}_\xi}\phi = {{\mathcal L}_\xi}{R_k} = 0\,.\qquad {\rm{Type IIA/B}}$$
(163)
\({{\mathcal L}_\xi}\) denotes the Lie derivative with respect to ξ, η is either the d = 11 or d =10 Minkowski metric on the tangent space, depending on which superspace we are working on and {Rκ, H3} are the different M-theory or type IIA/B field strengths satisfying the generalised Bianchi identities defined in Appendix A. Notice these are the superfield versions of the standard bosonic Killing isometry equations. Invariance of the field strengths allows the corresponding gauge potentials to have non-trivial transformations
$${{\mathcal L}_\xi}{A_3} = d{\Delta _2}\,,\quad \quad {{\mathcal L}_\xi}{A_6} = d{\Delta _5} - {1 \over 2}{\Delta _2} \wedge {R_4}\,,\qquad {\rm{M - theory}}$$
(164)
$${{\mathcal L}_\xi}{B_2} = d{\lambda _1}\,,\qquad {{\mathcal L}_\xi}{C_{p + 1}} = d{\omega _p} - d{\omega _{p - 2}} \wedge {H_3}\,,\qquad {\rm{Type IIA/B}}$$
(165)
for some set of superfield forms {Δ2, Δ5, ωi}.
The invariance of brane effective actions under the global transformations
$${\delta _\xi}{Z^M} = {\xi ^M}(Z)\,,$$
(166)
was proven in [94]. The proof can be established by analysing the DBI and WZ terms of the action separately. If the brane has gauge field degrees of freedom, one can always choose its infinitesimal transformation
$$\delta {V_2} = {{\rm{Z}}^ \star}({\Delta _2})\,,\qquad {\rm{M - theory}}$$
(167)
$$\delta {V_1} = {{\rm{Z}}^ \star}({\lambda _1})\,,\qquad {\rm{Type IIA/B}}$$
(168)
where Z* stands for pullback to the world volume, i.e., Z*(λ1) = dZM (λ1)m. This guarantees the invariance of the gauge invariant forms, i.e., \({{\mathcal L}_\xi}{\mathcal F} = {{\mathcal L}_\xi}{{\mathcal H}_3} = 0\). Furthermore, the transformation of the induced metric
$${{\mathcal L}_\xi}{{\mathcal G}_{\mu \nu}} = {\partial _\mu}{Z^M}{\partial _\nu}{Z^N}\,{{\mathcal L}_\xi}(E_M^aE_N^b{\eta _{ab}})\,,$$
(169)
vanishes because of Eq. (161). This establishes the invariance of the DBI action. On the other hand, the WZ action is quasi-invariant by construction due to Eqs. (164) and (165). Indeed,
$$\begin{array}{*{20}c} {\delta {{\mathcal L}_{{\rm{WZ}}}} = {{\rm{Z}}^ \star}\left({d{\Delta _2}} \right)\,,\qquad {\rm{M2 - brane}}\quad \quad \quad \quad \quad \quad}\\ {\delta {{\mathcal L}_{{\rm{WZ}}}} = {{\rm{Z}}^ \star}\left({d({\Delta _5} + {1 \over 2}{{\mathcal H}_3} \wedge {\Delta _2})} \right)\,,\qquad {\rm{M5 - brane}}}\\ {\delta {{\mathcal L}_{{\rm{WZ}}}} = {{\mathcal L}_\xi}{\mathcal C} \wedge {e^{\mathcal F}} = {{\rm{Z}}^ \star}\left({d\omega} \right)\,.\qquad {\rm{D - branes}}\quad \quad}\\ \end{array}$$
(170)

Summary: Brane effective actions include the supergravity superisometries ξ (Z) as a subset of their global symmetries. It is important to stress that kappa symmetry invariance is necessary to define a supersymmetric field theory on the brane, but not sufficient. Indeed, any on-shell supergravity background having no Killing spinors, i.e., some superisometry in which fermions are shifted as δθ = ϵ, breaks supersymmetry, and consequently, will never support a supersymmetric action on the brane.

The derivation discussed above does not exclude the existence of further infinitesimal transformations leaving the effective action invariant. The question of determining the full set of continuous global symmetries of a given classical field theory is a well posed mathematical problem in terms of cohomological methods [50, 51]. Applying these to the bosonic D-string [111] gave rise to the discovery of the existence of an infinite number of global symmetries [113, 112]. These were also proven to exist for the kappa invariant D-string action [110].

3.6.1 Supersymmetry algebras

Since spacetime superisometries generate world-volume global symmetries, Noether’s theorem [406, 407] guarantees a field theory realisation of the spacetime (super)symmetry algebra using Poisson brackets. It is by now well known that such (super)algebras contain more bosonic charges than the ones geometrically realised as (super)isometries. There are several ways of reaching this conclusion:
  1. 1.
    Grouped theoretically, the anticommutator of two supercharges {Qα, Qβ} defines a symmetric matrix belonging to the adjoint representation of some symplectic algebra Sp(N, ℝ), whose order N depends on the spinor representation Qα. One can decompose this representation into irreducible representations of the bosonic spacetime isometry group. This can explicitly be done by using the completeness of the basis of antisymmetrised Clifford algebra gamma matrices as follows
    $$\{{Q_\alpha},\,{Q_\beta}\} = \sum\limits_p {{{({\Gamma ^{{m_1} \ldots {m_p}}}{C^{- 1}})}_{\alpha \beta}}} {Z_{{m_1} \ldots {m_p}}}\,,$$
    (171)
    where the allowed values of depend on symmetry considerations. The right-hand side defines a \(\{{Z_{{m_1} \ldots {m_p}}}\}\) that typically goes beyond the spacetime bosonic isometries.
     
  2. 2.

    Physically, BPS branes in a given spacetime background have masses equal to their charges by virtue of the amount of supersymmetry they preserve. This would not be consistent with the supersymmetry algebra if the latter would not include extra charges, the set \(\{{Z_{{m_1} \ldots {m_p}}}\}\) introduced above, besides the customary spacetime isometries among which the mass (time translations) always belongs to. Thus, some of the extra charges must correspond to such brane charges. The fact that these charges have non-trivial tensor structure means they are typically not invariant under the spacetime isometry group. This is consistent with the fact that the presence of branes breaks the spacetime isometry group, as I already explicitly discussed in super-Poincaré.

     
  3. 3.

    All brane effective actions reviewed above are quasi-invariant under spacetime superisometries, since the WZ term transformation equals a total derivative (170). Technically, it is a well-known theorem that such total derivatives can induce extra charges in the commutation of conserved charges through Poisson brackets. This is the actual field theory origin of the group theoretically allowed set of charges \(\{{Z_{{m_1} \ldots {m_p}}}\}\).

     
Let me review how these structures emerge in both supergravity and brane effective actions. Consider the most general superPoincaré algebra in 11 dimensions. This is spanned by a Majorana spinor supercharge Qα (α = 1,…, 32) satisfying the anti-commutation relations25 [487, 478, 481]
$$\{{Q_\alpha},{Q_\beta}\} = {({\Gamma ^m}{C^{- 1}})_{\alpha \beta}}{P_m} + {1 \over 2}{({\Gamma ^{mn}}{C^{- 1}})_{\alpha \beta}}{Z_{mn}} + {1 \over {5!}}{({\Gamma ^{{m_1} \ldots {m_5}}}{C^{- 1}})_{\alpha \beta}}{Y_{{m_1} \ldots {m_5}}}\,.$$
(172)
That this superalgebra is maximal can be argued using the fact that its left-hand side defines a symmetric tensor with 528 independent components. Equivalently, it can be interpreted as belonging to the adjoint representation of the Lie algebra of Sp(32, ℝ). The latter decomposes under its subgroup SO(1,10), the spacetime Lorentz isometry group, as
$${\bf{528}} \rightarrow {\bf{11}} \oplus {\bf{55}} \oplus {\bf{462}}\,.$$
(173)
The irreducible representations appearing in the direct sum do precisely correspond to the bosonic tensor charges appearing in the right-hand side: the 11-momentum Pm a 2-form charge Zmn, which is 55-dimensional, and a 5-form charge \({Y_{{m_1} \ldots {m_5}}}\), which is 462-dimensional.
The above is merely based on group theory considerations that may or may not be realised in a given physical theory. In 11-dimensional supergravity, the extra bosonic charges are realised in terms of electric Ze and magnetic Zm charges, the Page charges [410], that one can construct out of the 3-form potential A3 equation of motion, as reviewed in [467, 466]
$${Z_{\rm{e}}} = {1 \over {4{\Omega _7}}}\int\nolimits_{\partial {{\mathcal M}_8}} {(\star {R_4} + {1 \over 2}{A_3} \wedge {R_4})} \,,$$
(174)
$${Z_{\rm{m}}} = {1 \over {{\Omega _4}}}\int\nolimits_{\partial {{\mathcal M}_5}} {{R_4}} \,.$$
(175)
The first integral is over the boundary at infinity of an arbitrary infinite 8-dimensional spacelike manifold \({{\mathcal M}_8}\), with volume Ω7. Given the conserved nature of this charge, it does not depend on the time slice chosen to compute it. But there are still many ways of embedding in the corresponding ten-dimensional spacelike hypersurface \({{\mathcal M}_{10}}\). Thus, \({Z_{\rm{e}}}\) represents a set of charges parameterised by the volume element 2-form describing how \({{\mathcal M}_8}\) is embedded in \({{\mathcal M}_{10}}\). This precisely matches the 2-form Zmn in Eq. (172). There is an analogous discussion for Zm, which corresponds to the 5-form charge \({Y_{{m_1} \ldots {m_5}}}\). As an example, consider the M2 and M5-brane configurations in Eqs. (20) and (22). If one labels the M2-brane tangential directions as 1 and 2, there exists a non-trivial charge Z12 computed from Eq. (174) by plugging in Eq. (20) and evaluating the integral over the transverse 7-sphere at infinity. The reader is encouraged to read the lecture notes by Stelle [467] where these issues are discussed very explicitly in a rather general framework including all standard half-BPS branes. For a more geometric construction of these maximal superalgebras in AdS × S backgrounds, see [211] and references therein.

The above is a very brief reminder regarding spacetime superalgebras in supergravity. For a more thorough presentation of these issues, the reader is encouraged to read the lectures notes by Townsend [481], where similar considerations are discussed for both type II and heterotic supergravity theories. Given the importance given to the action of dualities on effective actions, the reader may wonder how these same dualities act on superalgebras. It was shown in [96] that these actions correspond to picking different complex structures of an underlying OSp(1|32) superalgebra.

Consider the perspective offered by the M5-brane effective action propagating in d = 11 super-Poincaré. The latter is invariant both under supersymmetry and bulk translations. Thus, through Noether’s theorem, there exist field theory realisations of these charges. Quasi-invariance of the WZ term will be responsible for the generation of extra terms in the calculation of the Poisson bracket of these charges [165]. This was confirmed for the case at hand in [464], where the M5-brane superalgebra was explicitly computed. The supercharges Qα are
$${Q_\alpha} = i\int {d^5}\sigma \,[{(\pi + \bar \theta {\Gamma ^m}{P_m})_\alpha} + i({{\mathcal P}^{{i_1}{i_2}}} + {1 \over 4}{{\mathcal H}^{{\ast}0{i_1}{i_2}}}){(\Delta _{{i_1}{i_2}}^2)_\alpha} - i{\varepsilon ^{{i_1} \ldots {i_5}}}{(\Delta _{{i_1} \ldots {i_5}}^5)_\alpha}]\,,$$
(176)
where π, Pm and \({{\mathcal P}^{ij}}\) are the variables canonically conjugate to θ, Xm and Vij. As in any Hamiltonian formalism, world volume indices were split according to σμ = {t, σi} i = 1, … 5. Notice that the pullbacks of the forms Δ2 and Δ5 appearing in \(\delta {{\mathcal L}_{{\rm{WZ}}}}\) for the M5-brane in Eq. (170) do make an explicit appearance in this calculation. The anti-commutator of the M5 brane world volume supercharges equals Eq. (172) with
$${P_m} = \int {{d^5}} \sigma \,{{\delta {\mathcal L}} \over {\delta ({\partial _t}{X^m})}}\,,$$
(177)
$${Z^{mn}} = - \int\nolimits_{{{\mathcal M}_5}} d {X^m} \wedge d{X^n} \wedge d{V_2}\,,$$
(178)
$${Y^{{m_1} \ldots {m_5}}} = \int\nolimits_{{{\mathcal M}_5}} d {X^{{m_1}}} \wedge \cdots \wedge d{X^{{m_5}}}\,,$$
(179)
where all integrals are computed on the 5-dimensional spacelike hypersurface \({{\mathcal M}_5}\) spanned by the M5-brane. Notice the algebra of supercharges depends on the brane dimensionality. Indeed, a single M2-brane has a two dimensional spacelike surface that cannot support the pullback of a spacetime 5-form as a single M5-brane can (see Eq. (179)). This conclusion could be modified if the degrees of freedom living on the brane would be non-abelian.

Even though my discussion above only applies to the M5-brane in the super-Poincaré background, my conclusions are general given the quasi-invariance of their brane WZ action, a point first emphasised in [165]. The reader is encouraged to read [165, 168] for similar analysis carried for super p-branes, [281] for D-branes in super-Poincaré and general mathematical theorems based on the structure of brane effective actions and [438, 437], for superalgebra calculations in some particular curved backgrounds.

3.6.2 World volume supersymmetry algebras

Once the physical location of the brane is given, the spacetime superisometry group G is typically broken into
$${\rm{G}} \rightarrow {{\rm{G}}_0} \times {{\rm{G}}_1}.$$
(180)
The first factor G0 corresponds to the world volume symmetry group in (p + 1)-dimensions, i.e., the analogue of the Lorentz group in a supersymmetric field theory in (p + 1)-dimensions, whereas the second factor G1 is interpreted as an internal symmetry group acting on the dynamical fields building (p + 1)-dimensional supermultiplets. The purpose of this subsection is to relate the superalgebras before and after this symmetry breaking process [328].26
The link between both superalgebras is achieved through the gauge fixing of world volume diffeomorphisms and kappa symmetry, the gauge symmetries responsible for the covariance of the original brane action in the GS formalism. Focusing on the scalar content in these theories {Xm,θ}, these transform as
$$s{X^m} = {k^m}(X) + {{\mathcal L}_\xi}{X^m} + {\delta _\kappa}{X^m} + {\delta _\epsilon}{X^m}\,,$$
(181)
$$s\theta = \epsilon+ {\delta _k}\theta + {{\mathcal L}_\xi}\theta + (1 + {\Gamma _\kappa})\kappa + {\delta _k}\theta \,.$$
(182)
The general Killing superfield was decomposed into a supersymmetry translation denoted by ϵ and a bosonic Killing vector fields kM (X). World volume diffeomorphisms were denoted as ξ. At this stage, the reader should already notice the inhomogeneity of the supersymmetry transformation acting on fermions (the same is true for bosons if the background spacetime has a constant translation as an isometry, as it happens in Minkowski).
Locally, one can always impose the static gauge: Xμ = σμ, where one decomposes the scalar fields Xm into world volume directions Xμ and transverse directions XI ≡ΦI. For infinite branes, this choice is valid globally and does describe a vacuum configuration. To diagnose which symmetries act, and how, on the physical degrees of freedom Φi, one must make sure to work in the subset of symmetry transformations preserving the gauge slice Xμ = σμ. This forces one to act with a compensating world volume diffeomorphism
$$s{X^\mu}{\vert _{{X^\mu} = {\sigma ^\mu}}} = 0\quad \Rightarrow \quad {\xi ^\mu} = - {k^\mu} - {\delta _\kappa}{X^\mu} - {\delta _\epsilon}{X^\mu}\,.$$
(183)
The latter acts on the physical fields giving rise to the following set of transformations preserving the gauge fixed action
$$s{\Phi ^I}{\vert _{{X^\mu} = {\sigma ^\mu}}} = {k^I} - {k^\mu}{\partial _\mu}{\Phi ^I} + \ldots \,,$$
(184)
$$s\theta {\vert _{{X^\mu} = {\sigma ^\mu}}} = - {k^\mu}{\partial _\mu}\theta + {{\mathcal L}_k}\theta + \ldots \,.$$
(185)
There are two important comments to be made at this point
  1. 1.

    The physical fields ΦI transform as proper world volume scalars [3]. Indeed, ΦI (σ) = (Φ′)I (σ ′) induces the infinitesimal transformation kμμ ΦI for any kμ (σ) preserving the p + 1 dimensional world volume. Below, the same property will be checked for fermions.

     
  2. 2.

    If the spacetime background allows for any constant kI isometry, it would correspond to an inhomogeneous symmetry transformation for the physical field ΦI. In field theory, the latter would be interpreted as a spontaneous broken symmetry and the corresponding ΦI would be its associated massless Goldstone field. This is precisely matching our previous discussions regarding the identification of the appropriate brane degrees of freedom.

     
There is a similar discussion regarding the gauge fixing of kappa symmetry and the emergence of a subset of linearly realised supersymmetries on the (p + 1)-dimensional world volume field theory. Given the projector nature of the kappa symmetry transformations, it is natural to assume \({\mathcal P}\theta = 0\) as a gauge fixing condition, where \({\mathcal P}\) stands for some projector. Preservation of this gauge slice, determines the kappa symmetry parameter κ as a function of the background Killing spinors ϵ
$$s\theta {\vert _{{\mathcal P}\theta = 0}} = 0\quad \Rightarrow \quad \kappa = \kappa (\epsilon)\,.$$
(186)
When analysing the supersymmetry transformations for the remaining dynamical fermions, only certain linear combinations of the original supersymmetries ϵ will be linearly realised. The difficulty in identifying the appropriate subset depends on the choice of \({\mathcal P}\).
Branes in super-Poincaré: The above discussion can be made explicit in this case. Consider a p + 1 dimensional brane propagating in d dimensional super-Poincaré. For completeness, let me remind the reader of the full set of transformations leaving the brane actions invariant
$$s{X^m} = {a^m} + {a^m}_n{X^n} + {{\mathcal L}_\xi}{X^m} + \bar\epsilon {\Gamma ^m}\theta + {\delta _\kappa}{X^m}\,,$$
(187)
$$s\theta = {1 \over 4}{a_{mn}}{\Gamma ^{mn}}\theta + {{\mathcal L}_\xi}\theta + \epsilon + {\delta _\kappa}\theta \,,$$
(188)
where I ignored possible world volume gauge fields. Decomposing the set of bosonic scalar fields Xm m = 0,1,…d − 1 into world volume directions \({X^\mu}\mu = 0,1, \ldots p\) and transverse directions XI ≡ ΦI I = p + 1,…d − 1, one can now explicitly solve for the preservation of the static gauge slice Xμ = σμ, which does globally describe the vacuum choice of a p-brane extending in the first p spacelike directions and time. This requires some compensating world volume diffeomorphism
$${\xi ^\mu} = - {a^\mu} - {a^\mu}_\nu {\sigma ^\nu} - {a^\mu}_I{\Phi ^I} - \bar\epsilon {\Gamma ^\mu}\theta - {\delta _\kappa}{X^\mu}\,,$$
(189)
inducing the following transformations for the remaining degrees of freedom
$$s{\Phi ^I} = - {a^\mu}{\partial _\mu}{\Phi ^I} - {a^\mu}_\nu {\sigma ^\nu}{\partial _\mu}{\Phi ^I} - {a^\mu}_J{\Phi ^J}{\partial _\mu}{\Phi ^I} + {a^J} + {a^I}_J{\Phi ^J} + {a^I}_\mu {\sigma ^\mu} + {\rm{fermions}}\,,$$
(190)
$$s\theta = - {a^\mu}_\nu {\sigma ^\nu}{\partial _\mu}\theta + {1 \over 4}{a_{\mu \nu}}{\Gamma ^{\mu \nu}}\theta + {1 \over 4}{a_{IJ}}{\Gamma ^{IJ}}\theta .$$
(191)
The subset of linearly realised symmetries is ISO(1,p) × SO (D − (p + 1)). The world volume “Poincaré” group is indeed ISO(1,p), under which are scalars, whereas θ are fermions, including the standard spin connection transformation giving them their spinorial nature. SO(D − (p + 1)), the transverse rotational group to the brane is reinterpreted as an internal symmetry, under which transforms as a vector. The parameters aμI describing the coset SO(1,D − 1)/(SO(1,p) × SO(Dp − 1)) are generically non-linearly realised, whereas the transverse translations aI act inhomogeneously on the dynamical fields ΦI, identifying the latter as Goldstone massless fields, as corresponds to the spontaneous symmetry breaking of these symmetries due to the presence of the brane in the chosen directions.
There is a similar discussion for the 32 spacetime supersymmetries (ϵ). Before gauge fixing all fermions θ transform inhomogeneously under supersymmetry. After gauge fixing \({\mathcal P}\theta = 0\), the compensating kappa symmetry transformation κ (ϵ) required to preserve the gauge slice in configuration space will induce an extra supersymmetry transformation for the dynamical fermions, i.e., \((1 - {\mathcal P})\theta\). On general grounds, there must exist sixteen linear combinations of supersymmetries being linearly realised, whereas the sixteen remaining will be spontaneously broken by the brane. There are many choices for \({\mathcal P}\theta = 0\). In [10], where they analysed this aspect for D-branes in super-Poincaré, they set one of the members of the \({\mathcal N} = 2\) fermion pair to zero, leading to fairly simple expressions for the gauge fixed Lagrangian. Another natural choice corresponds to picking the projector describing the preserved supersymmetries by the brane from the spacetime perspective. For instance, the supergravity solution describing M2-branes has 16 Killing spinors satisfying
$${\Gamma _{012}}\epsilon = \pm \epsilon \,,$$
(192)
where the ± is correlated with the R4 flux carried by the solution. If one fixes kappa symmetry according to
$${\mathcal P}\theta = (1 + {\Gamma _ \star})\theta = 0\,,\qquad {\rm{with}}\qquad {\Gamma _ \star} = {\Gamma _3} \ldots {\Gamma _9}{\Gamma _\sharp}\,,$$
(193)
where Γ# stands for the 11-dimensional Clifford algebra matrix, then the physical fermionic degrees of freedom are not only 3-dimensional spinors, but they are chiral spinors from the internal symmetry SO(8) perspective. They actually transform in the (2, 8s) [91]. Similar considerations would apply for any other brane considered in this review.
Having established the relation between spacetime and world volume symmetries, it is natural to close our discussion by revisiting the superalgebra closed by the linearly realised world volume (super)symmetries, once both diffeomorphisms and kappa symmetry have been fixed. Since space-time superalgebras included extra bosonic charges due to the quasi-invariance of the brane WZ action, the same will be true for their gauge fixed actions. Thus, these (p + 1)-dimensional world volume superalgebras will include as many extra bosonic charges as allowed by group theory and by the dimensionality of the brane world spaces [81]. Consider the M2-brane discussed above. Supercharges transform in the (2, 8s) representation of the SO(1, 2) × SO(8) bosonic isometry group. Thus, the most general supersymmetry algebra compatible with these generators, \({\mathcal N} = 8 \, d = 3\), is [81]
$$\{Q_\alpha ^I,\,Q_\beta ^J\} = {\delta ^{IJ}}{P_{(\alpha \beta)}} + Z_{(\alpha \beta)}^{(IJ)} + {\varepsilon _{\alpha \beta}}{Z^{[IJ]}}\qquad {\rm{with}}\qquad ({\delta _{IJ}}{Z^{(IJ)}} = 0)\,.$$
(194)
P(αβ) stands for a 3-dimensional one-form, the momentum on the brane; \(Z_{(\alpha \beta)}^{(IJ)}\) transforms in the 35+ under the R-symmetry group SO(8), or equivalently, as a self-dual 4-form in the transverse space to the brane; Z[IJ ] is a world volume scalar, which transforms in the 28 of SO(8), i.e., as a 2-form in the transverse space. The same superalgebra is realised on the non-abelian effective action describing N coincident M2-branes [415] to be reviewed in Section 7.2. Similar structures exist for other infinite branes. For example, the M5-brane gives rise to the d =6(2, 0) superalgebra [81]
$$\{Q_\alpha ^I,\,Q_\beta ^J\} = {\Omega ^{IJ}}{P_{[\alpha \beta ]}} + Z_{(\alpha \beta)}^{(IJ)} + Y_{[\alpha \beta ]}^{[IJ]}\qquad {\rm{with}}\qquad ({\Omega _{IJ}}{Y^{[IJ]}} = 0)\,.$$
(195)
Here α, β = 1,…, 4 is an index of SU* (4) ≃ Spin(1, 5), the natural Lorentz group for spinors in d = 6 dimensions, I, J = 1,…4 is an index of Sp(2) ≃ Spin(5), which is the double cover of the geometrical isometry group SO(5) acting on the transverse space to the M5-brane and ΩIJ is an Sp(2) invariant antisymmetric tensor. Thus, using appropriate isomorphisms, these superalgebras allow a geometrical reinterpretation in terms of brane world volumes and transverse isometry groups becoming R-symmetry groups. The last decomposition is again maximal since P[αβ] stands for 1-form in d =6 (momentum), \(Z_{(\alpha \beta)}^{(IJ)}\) transforms as a self-dual 3-form in d =6 and a 2-form in the transverse space and \(Z_{[\alpha \beta ]}^{(IJ)}\) as a 1-form both in d =6 and in the transverse space. For an example of a non-trivial world volume superalgebra in a curved background, see [152].

I would like to close this discussion with a remark that is usually not stressed in the literature. By construction, any diffeomorphism and kappa symmetry gauge fixed brane effective action describes an interacting supersymmetric field theory in p +1 dimensions.27 As such, if there are available superspace techniques in these dimensions involving the relevant brane supermultiplet, the gauge fixed action can always be rewritten in that language. The matching between both formulations generically involves non-trivial field redefinitions. To be more precise, consider the example of \({\mathcal N} = 1 \, d = 4\) supersymmetric abelian gauge theories coupled to matter fields. Their kinetic terms are fully characterised by a Kähler potential. If one considers a D3-brane in a background breaking the appropriate amount of supersymmetry, the expansion of the gauge fixed D3-brane action must match the standard textbook description. The reader can find an example of the kind of non-trivial bosonic field redefinitions that is required in [321]. The matching of fermionic components is expected to be harder.

3.7 Regime of validity

After thoroughly discussing the kinematic structure of the effective action describing the propagation of single branes in arbitrary on-shell backgrounds, I would like to reexamine the regime of validity under which the dynamics of the full string (M-) theory reduces to Sbrane.

As already stressed at the beginning of Section 3, working at low energies allows us to consider the action
$$S \approx {S_{{\rm{SUGRA}}}} + {S_{{\rm{brane}}}}.$$
(196)
In string theory, low energies means energies E satisfying \(E\sqrt {{\alpha {\prime}}} \ll 1\). This guarantees that no on-shell states will carry energies above that scale allowing one to write an effective action in terms of the fields describing massless excitations and their derivatives. The argument is valid for both the open and the closed string sectors. Furthermore, to ensure the validity of this perturbative description, one must ensure the weak coupling regime is satisfied, i.e., gs ≪ 1, to suppress higher loop world sheet contributions.
Dynamically, all brane effective actions reviewed previously, describe the propagation of a brane in a fixed on-shell spacetime background solving the classical supergravity equations of motion. Thus, to justify neglecting the dynamics of the gravitational sector, focusing on the brane dynamics, one must guarantee condition (18)
$$\vert T_{mn}^{{\rm{background}}}\vert \gg \vert T_{mn}^{{\rm{brane}}}\vert ,$$
(197)
but also to work in a regime where the effective Newton’s constant tends to zero. Given the low energy and weak coupling approximations, the standard lore condition for the absence of quantum gravity effects, i.e., \(E\ell _p^{(10)} \ll 1\), is naturally satisfied since \(E\ell _p^{(10)} \sim (E\sqrt {{\alpha {\prime}}})g_{{g_s}}^{1/4} \ll 1\). The analogous condition for 11-dimensional supergravity is Eℓp ≪ 1.
The purpose of this section is to spell out more precisely the conditions that make the above requirements not sufficient. As in any effective field theory action, one must check the validity of the assumptions made in their derivation. In our discussions, this includes
  1. 1.

    conditions on the derivatives of brane degrees of freedom, both geometrical Xm and world volume gauge fields, such as the value of the electric field;

     
  2. 2.

    the reliability of the supergravity background;

     
  3. 3.

    the absence of extra massless degrees of freedom emerging in string theory under certain circumstances.

     
I will break the discussion below into background and brane considerations.
Validity of the background description: Whenever the supergravity approximation is not reliable, the brane description will also break down. Assuming no extra massless degrees of freedom arise, any on-shell \({\mathcal N} = 2\) type IIA/IIB supergravity configuration satisfying the conditions described above, must also avoid
$${e^\phi}\sim 1,\qquad {\mathcal R}\,{(\ell _p^{(10)})^2} \simeq 1.$$
(198)
Since the string coupling constant gs is defined as the expectation value of , the first condition determines the regions of spacetime where string interactions become strongly coupled. The second condition, or any dimensionless scalar quantity constructed out of the Riemann tensor, determines the regions of spacetime where curvature effects cannot be neglected. Whenever there are points in our classical geometry where any of the two conditions are satisfied, the assumptions leading to the classical equations of motion being solved by the background under consideration are violated. Thus, our approximation is not self-consistent in these regions.
Similar considerations apply to 11-dimensional supergravity. In this case, the first natural condition comes from the absence of strong curvature effects, which would typically occur whenever
$${\mathcal R}\ell _p^2 \simeq 1,$$
(199)
where once more the scalar curvature can be replaced by other curvature invariants constructed out of the 11-dimensional Riemann tensor in appropriate units of the 11-dimensional Planck scale p.
Since the strong coupling limit of type IIA string theory is M-theory, which at low energies is approximated by \({\mathcal N} = 1 \, d = 11\) supergravity, it is clear that there should exist further conditions. This connection involves a compactification on a circle, and it is natural to examine whether our approximations hold as soon as its size R is comparable to p. Using the relations (55), one learns
$$R\sim{\ell _p}\quad \Leftrightarrow \quad {g_s}\sim 1.$$
(200)
Thus, as soon as the M-theory circle explores subPlanckian eleven-dimensional scales, which would not allow a reliable eleven-dimensional classical description, the type IIA string coupling becomes weakly coupled, opening a possible window of reliable classical geometrical description in terms of the KK reduced configuration (54).

The above discussion also applies to type IIA and IIB geometries. As soon as the scale of some compact submanifold, such as a circle, explores substringy scales, the original metric description stops being reliable. Instead, its T-dual description (58) does, using Eq. (56).

Finally, the strong coupling limit of type IIB may also allow a geometrical description given the SL(2, ℝ) invariance of its supergravity effective action, which includes the S-duality transformation
$${e^\phi} \rightarrow {e^{- \phi}}.$$
(201)
The latter maps a strongly coupled region to a weakly coupled one, but it also rescales the string metric. Thus, one must check whether the curvature requirements \({\mathcal R}{(\ell _p^{(10)})^2} \ll 1\) 1 hold or not.

It is important to close this discussion by reminding the reader that any classical supergravity description assumes the only relevant massless degrees of freedom are those included in the supergravity multiplet. The latter is not always true in string theory. For example, string winding modes become massless when the circle radius the string wraps goes to zero size. This is precisely the situation alluded to above, where the T-dual description, in which such modes become momentum modes, provides a T-dual reliable description in terms of supergravity multiplet fluctuations. The emergence of extra massless modes in certain classical singularities in string theory is far more general, and it can be responsible for the resolution of the singularity. The existence of extra massless modes is a quantum mechanical question that requires going beyond the supergravity approximation. What certainly remains universal is the geometrical breaking down associated with the divergence of scalar curvature invariants due to a singularity, independently of whether the latter is associated with extra massless modes or not.

Validity of the brane description: Besides the generic low energy and weak coupling requirements applying to D-brane effective actions (150), the microscopic derivation of the DBI action assumed the world volume field strength Fμν was constant. Thus, kappa symmetric invariant D-brane effective actions ignore corrections in derivatives of this field strength, i.e., terms like ρFμν or higher in number of derivatives. Interestingly, these corrections map to acceleration and higher-order derivative corrections in the scalar fields Xm under T-duality, see Eq. (80). Thus, there exists the further requirement that all dynamical fields in brane effective actions are slowly varying. In Minkowski, this would correspond to conditions like
$$\sqrt {\alpha {\prime}} {\partial ^2}X \ll \partial X,$$
(202)
or similar tensor objects constructed with the derivative operator in appropriate string units. In a general curved background, these conditions must be properly covariantised, although locally, the above always applies.

Notice these conditions are analogous to the ones we would encounter in the propagation of a point particle in a fixed background. Any corrections to geodesic motion would be parameterised by an expansion in derivatives of the scalar fields parameterising the particle position, this time in units of the mass particle.

Brane effective actions carrying electric fields E can manifestly become ill defined for values above a certain critical electric field Ecrit for which the DBI determinant vanishes. It was first noticed for the bosonic string in [120, 403] that such critical electric field is the value for which the rate of Schwinger charged-string pair production [442] diverges. This divergence captures a divergent density of string states in the presence of such critical electric field. These calculations were extended to the superstring in [25]. The conclusion is the same, though in this latter case the divergence applies to any pair of charge-conjugate states. Thus, there exists a correlation between the pathological behaviour of the DBI action and the existence of string instabilities.28 Heuristically, one interprets the regime with E > Ecrit as one where the string tension can no longer hold the string together.29

4 World Volume Solitons: Generalities

Brane effective actions capture the relevant dynamics of M-theory or string theory in some appropriate regimes of validity. Thus, they contain reliable information about its spectrum and its dynamics in those regimes. In this section, I will develop the tools to study the world volume realisation of supersymmetric states carrying the extra bosonic (topological) charges appearing in the maximal supersymmetry algebras introduced in Sections 3.6.1 and 3.6.2.

One such realisation is in terms of classical bosonic on-shell configurations. As it often occurs with supersymmetric configurations, instead of focusing on the integration of the equations of motion, I will focus on the conditions ensuring preservation of supersymmetry and on their physical interpretation. In particular,
  • I will argue the existence of a necessary condition that any bosonic supersymmetric configuration must satisfy involving the kappa symmetry matrix Γκ and the background Killing spinors ϵ.

  • I will review the Hamiltonian formulation for brane effective actions to compute the energy of these configurations. The latter will minimise the energy for a given set of charges carried by the state. The existence of energy bounds can be inferred from merely algebraic considerations and I will discuss their field theory realisations as BPS bounds30. Furthermore, the relation between their saturation and the solution to the necessary kappa symmetry condition will also be explained.

  • I will discuss the relation between these physical considerations and the mathematical notion of calibration, which is a purely geometric formulation of the problem of finding volume minimising surfaces. Since the latter corresponds to a subset of bosonic brane supersymmetric configurations, this connection will allow us to review the notion of generalised calibration, which, in physical terms, includes world volume gauge field excitations.

The framework and set of relations covered in this section are summarised in Figure 6.
Figure 6

Set of relations involving kappa symmetry, spacetime supersymmetry algebras, their bounds and their realisation as field theory BPS bounds in terms of brane solitons using the Hamiltonian formulation of brane effective actions.

4.1 Supersymmetric bosonic configurations and kappa symmetry

To know whether any given on-shell bosonic brane configuration is supersymmetric, and if so, how many supersymmetries are preserved, one must develop some tools analogous to the ones for bosonic supergravity configurations. I will review these first.

Consider any supergravity theory having bosonic (\(({\mathcal B})\)) and fermionic (\(({\mathcal F})\)) degrees of freedom. It is consistent with the equations of motion to set \({\mathcal F} = 0\). The question of whether the configuration \({\mathcal B}\) preserves supersymmetry reduces to the study of whether there exists any supersymmetry transformation ϵ preserving the bosonic nature of the on-shell configuration, i.e., \(\delta {\mathcal F}{\vert _{{\mathcal F} = 0}} = 0\), without transforming \({\mathcal B}\), i.e., \(\delta {\mathcal B}{\vert _{{\mathcal F} = 0}} = 0\) Since the structure of the local supersymmetry transformations in supergravity is
$$\delta {\mathcal B} \propto {\mathcal F}\,,\qquad \delta {\mathcal F} = {\mathcal P}({\mathcal B})\,\epsilon ,$$
(203)
these conditions reduce to \({\mathcal P}({\mathcal B})\epsilon = 0\). In general, the Clifford valued operator \({\mathcal P}({\mathcal B})\) is not higher than first order in derivatives, but it can also be purely algebraic. Solutions to this equation involve
  1. 1.

    Differential constraints on the subset of bosonic configurations \({\mathcal B}\). Given the first-order nature of the operator \({\mathcal P}({\mathcal B})\), these are simpler than the second-order equations of motion and help to reduce the complexity of the latter.

     
  2. 2.

    Differential and algebraic constraints on ϵ. These reduce the infinite dimensional character of the original arbitrary supersymmetry transformation parameter ϵ to a finite dimensional subset, i.e., \(\epsilon = {f_{\mathcal B}}({x^m}){\epsilon _\infty}\), where the function \({f_{\mathcal B}}({x^m})\) is uniquely specified by the bosonic background \({\mathcal B}\) and the constant spinor ϵ typically satisfies a set of conditions \({{\mathcal P}_i}{\varepsilon _\infty} = 0\), where \({\mathcal P}\) are projectors satisfying \({\mathcal P}_i^2 = {{\mathcal P}_i}\) and tr \({{\mathcal P}_i} = 0\). These ϵ are the Killing spinors of the bosonic background \({\mathcal B}\). They can depend on the spacetime point, but they are no longer arbitrary. Thus, they are understood as global parameters.

     
This argument is general and any condition derived from it is necessary. Thus, one is instructed to analyse the condition \({\mathcal P}({\mathcal B})\varepsilon = 0\) before solving the equations of motion. As a particular example, and to make contact with the discussions in Section 3.1.1, consider \({\mathcal N} = 1 \, d = 11\) supergravity. The only fermionic degrees of freedom are the gravitino components \({\Psi _a} = {E^M}_a{\Psi _M}\). Their supersymmetry transformation is [466]
$$\delta {\Psi _a} = \left({{\partial _a} + {1 \over 4}{\omega _a}^{bc}{\Gamma _{bc}}} \right)\epsilon - {1 \over {288}}\left({{\Gamma _a}^{bcde} - 8{\delta _a}^b{\Gamma ^{cde}}} \right){R_{bcde}}\epsilon \,.$$
(204)
Solving the supersymmetry preserving condition δ Ψa = 0 in the M2-brane and M5-brane back-grounds determines the Killing spinors of these solutions to be [466, 467]
$${\rm M}2 - {\rm brane}\quad \epsilon = {U^{- 1/6}}{\epsilon _\infty}\qquad {\rm{with}}\qquad {\Gamma _{012}}{\epsilon _\infty} = \pm {\epsilon _\infty}\,,$$
(205)
$${\rm{M}}5 - {\rm{brane}}\quad \epsilon = {U^{- 1/12}}{\epsilon _\infty}\qquad {\rm{with}}\qquad {\Gamma _{012345}}{\epsilon _\infty} = \pm {\epsilon _\infty}\,.$$
(206)
A similar answer is found for all D-branes in \({\mathcal N} = 2 \, d = 10\) type IIA/B supergravities.
The same question for brane effective actions is treated in a conceptually analogous way. The subspace of bosonic configurations \({\mathcal B}\) defined by θ = 0 is compatible with the brane equations of motion. Preservation of supersymmetry requires \(s\theta {\vert_{\mathcal B}} = 0\). The total transformation is given by
$$s\theta = {\delta _\kappa}\theta + \epsilon + \Delta \theta + {\xi ^\mu}{\partial _\mu}\theta \,,$$
(207)
where δκθ and ξμμθ stand for the kappa symmetry and world volume diffeomorphism infinitesimal transformations and ∇θ for any global symmetries different from supersymmetry, which is generated by the Killing spinors ϵ. When restricted to the subspace \({\mathcal B}\) of bosonic configurations,
$${\delta _\kappa}\theta {\vert _{\mathcal B}} = \left({1 + {\Gamma _\kappa}{\vert _{\mathcal B}}} \right)\kappa ,$$
(208)
$$\Delta \theta {\vert _{\mathcal B}} = 0\,,$$
(209)
one is left with
$$s\theta {\vert _{\mathcal B}} = \left({1 + {\Gamma _\kappa}{\vert _{\mathcal B}}} \right)\kappa + \epsilon \,.$$
(210)
This is because ∇θ describes linearly realised symmetries. Thus, kappa symmetry and supersym-metry transformations do generically not leave the subspace \({\mathcal B}\) invariant.
We are interested in deriving a general condition for any bosonic configuration to preserve supersymmetry. Since not all fermionic fields θ are physical, working on the subspace θ = 0 is not precise enough for our purposes. We must work in the subspace of field configurations being both physical and bosonic [85]. This forces us to work at the intersection of θ = 0 and some kappa symmetry gauge fixing condition. Because of this, I find it convenient to break the general argument into two steps.
  1. 1.
    Invariance under kappa symmetry. Consider the kappa-symmetry gauge-fixing condition \({\mathcal P}\theta = 0\), where \({\mathcal P}\) stands for any field independent projector. This allows us to decompose the original fermions according to
    $$\theta = {\mathcal P}\theta + (1 - {\mathcal P})\theta \,.$$
    (211)
    To preserve the kappa gauge slice in the subspace \({\mathcal B}\) requires
    $$s{\mathcal P}\theta {\vert _{\mathcal B}} = {\mathcal P}(1 + {\Gamma _\kappa}{\vert _{\mathcal B}})\kappa + {\mathcal P}\epsilon = 0\,.$$
    (212)
    This determines the necessary compensating kappa symmetry transformation κ (ϵ) as a function of the background Killing spinors.
     
  2. 2.
    Invariance under supersymmetry. Once the set of dynamical fermions \((1 - {\mathcal P})\theta\) is properly defined, we ask for the set of global supersymmetry transformations preserving them
    $$s(1 - {\mathcal P})\theta {\vert _{\mathcal B}} = 0\,.$$
    (213)
    This is equivalent to
    $$(1 + {\Gamma _\kappa}{\vert _{\mathcal B}})\kappa (\epsilon) + \epsilon = 0$$
    (214)
    once Eq. (212) is taken into account. Projecting this equation into the \((1 - {\Gamma _\kappa}{\vert_{\mathcal B}})\) subspace gives condition
    $${\Gamma _\kappa}{\vert _{\mathcal B}}\epsilon = \epsilon .$$
    (215)
    No further information can be gained by projecting to the orthogonal subspace \((1 - {\Gamma _\kappa}{\vert_{\mathcal B}})\).
     

I will refer to Eq. (215) as the kappa symmetry preserving condition. It was first derived in [85]. This is the universal necessary condition that any bosonic on-shell brane configuration {ϕi} must satisfy to preserve some supersymmetry.

In Table 5, I evaluate all kappa symmetry matrices Γκ in the subspace of bosonic configurations \({\mathcal B}\) for future reference. This matrix encodes information
  1. 1.

    on the background, both explicitly through the induced world volume Clifford valued matrices \({\gamma _\mu} = {E_\mu}^a{\Gamma _a} = {\partial _\mu}{X^m}{E_m}^a{\Gamma _a}\) and the pullback of spacetime fields, such as \({\mathcal G},{\mathcal F}\) or \({\tilde H}\), but also implicitly through the background Killing spinors ϵ solving the supergravity constraints \({\mathcal P}(\varepsilon) = 0\), which also depend on the remaining background gauge potentials,

     
  2. 2.

    on the brane configuration {ϕi}, including scalar fields Xm (σ) and gauge fields, either V1 or V2, depending on the brane under consideration.

     
Table 5

Set of kappa symmetry matrices Γκ evaluated in the bosonic subspace of configurations \({\mathcal B}\).

Brane

Bosonic kappa symmetry matrix

M2-brane

\({\Gamma _\kappa}{\vert_{\mathcal B}} = {1 \over {3!\sqrt {- \det {\mathcal G}}}}{\epsilon ^{\mu \nu \rho}}{\gamma _{\mu \nu \rho}}\)

M5-brane

\({\Gamma _\kappa}{|_{\mathcal B}} = {{{\upsilon _\mu}{\gamma ^\mu}} \over {\sqrt {- \det ({\mathcal G} + \tilde H)}}}\left[ {{{\sqrt {- \det {\mathcal G}}} \over 2}{\gamma ^{{\mu _1}{\mu _2}}}{{\tilde H}_{{\mu _1}{\mu _2}}} + {\gamma _v}{t^v} - {1 \over {5!}}{\varepsilon ^{{\mu _1} \ldots {\mu _5}v}}{\upsilon _v}{\gamma _{{\mu _1} \ldots {\mu _5}}}} \right]\)

IIA Dp-branes

\({\Gamma _\kappa}{|_{\mathcal B}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\nolimits_{l = 0} {{\gamma _{2l + 1}}\Gamma _\# ^{l + 1}} \wedge{e^{\mathcal F}}\)

IIB Dp-branes

\({\Gamma _\kappa}{|_{\mathcal B}} = {1 \over {\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}\sum\nolimits_{l = 0} {{\gamma _{2l}}\tau _3^li{\tau _2}} \wedge{e^{\mathcal F}}\)

Just as in supergravity, any solution to Eq. (215) involves two sets of conditions, one on the space of configurations {ϕi} and one on the amount of supersymmetries. More precisely,
  1. 1.

    a set of constraints among dynamical fields and their derivatives, fj (ϕi, ∂ϕi) = 0,

     
  2. 2.

    a set of supersymmetry projection conditions, \({\mathcal P}_i\prime{\varepsilon _\infty} = 0\), with \({\mathcal P}_i\prime\) being projectors, reducing the dimensionality of the vector space spanned by the original ϵ.

     
The first set will turn out to be BPS equations, whereas the second will determine the amount of supersymmetry preserved by the combined background and probe system.

4.2 Hamiltonian formalism

In this subsection, I review the Hamiltonian formalism for brane effective actions. This will allow us not only to compute the energy of a given supersymmetric on-shell configuration solving Eq. (215), but also to interpret the constraints fj (ϕi) = 0 as BPS bounds [107, 429]. This will lead us to interpret these configurations as brane-like excitations supported on the original brane world volume.

The existence of energy bounds in supersymmetric theories can already be derived from purely superalgebra considerations. For example, consider the M-algebra (172). Due to the positivity of its left-hand side, one derives the energy bound
$${P_0} \geq f({P_i},{Z_{ij}},{Y_{{i_1} \ldots {i_5}}};{Z_{0i}},{Y_{0{i_1} \ldots {i_4}}}),$$
(216)
where the charge conjugation matrix was chosen to be C = Γ0 and the spacetime indices were split as m = {0, i}. For simplicity, let us set the time components \({Y_{0{i_1} \ldots {i_4}}}\) and Z0i to zero. The superalgebra reduces to
$$\{Q,Q\} = {P^0}(1 + \bar \Gamma)\,,\qquad {\rm{with}}\qquad \bar \Gamma = {({P^0})^{- 1}}[{\Gamma ^{0i}}{P_i} + {1 \over 2}{\Gamma ^{0ij}}{Z_{ij}} + {1 \over {5!}}{\Gamma ^{0{i_1} \ldots {i_5}}}{Y_{{i_1} \ldots {i_5}}}]\,.$$
(217)
The bound (216) is now equivalent to the statement that no eigenvalue of \({{\bar \Gamma}^2}\) can exceed unity. Any bosonic charge (or distribution of them) for which the corresponding \({\bar \Gamma}\) satisfies
$${\bar \Gamma ^2} = 1,$$
(218)
defines a projector \({1 \over 2}(1 + \bar \Gamma)\). The eigenspace of \({\bar \Gamma}\) with eigenvalue 1 coincides with the one spanned by the Killing spinors ϕ determining the supersymmetries preserved by supergravity configurations corresponding to individual brane states. In other words, there is a one-to-one map between half BPS branes, the charges they carry and the precise supersymmetries they preserve. This allows one to interpret all the charges appearing in \({\bar \Gamma}\) in terms of brane excitations: the 10-momentum Pi describes d =11 massless superparticles [93], the 2-form charges Zij M2-branes [90, 91], whereas the 5-form charges \({Y_{{i_1} \ldots {i_5}}}\), M5-branes [464]. This correspondence extends to the time components \(\{{Y_0}_{{i_1} \ldots {i_4}},{Z_{0i}}\}\). These describe branes appearing in Kaluza-Klein vacua [311, 481]. Specifically, \({Y_0}_{{i_1} \ldots {i_4}}\) is carried by type IIA D6-branes (the M-theory KK monopole), while Z0i can be related to type IIA D8-branes.
That these algebraic energy bounds should allow a field theoretical realisation is a direct consequence of the brane effective action global symmetries and Noether’s theorem [406, 407]. If the system is invariant under time translations, energy will be preserved, and it can be computed using the Hamiltonian formalism, for example. Depending on the amount and nature of the charges turned on by the configuration, the general functional dependence of the bound (216) changes. This is because each charge appears in \({\bar \Gamma}\) multiplied by different antisymmetric products of Clifford matrices. Depending on whether these commute or anticommute, the bound satisfied by the energy P0 changes, see for example a discussion on this point in [394]. Thus, one expects to be able to decompose the Hamiltonian density for these configurations as sums of the other charges and positive definite extra terms such that when they vanish, the bound is saturated. More precisely,
  1. 1.
    For non-threshold bound states, or equivalently, when the associated Clifford matrices anti-commute, one expects the energy density to satisfy
    $${{\mathcal E}^2} = {\mathcal Z}_1^2 + {\mathcal Z}_2^2 + \sum\limits_i {{{\left({{t^i}{f_i}({\phi ^j})} \right)}^2}} \,.$$
    (219)
     
  2. 2.
    For bound states at threshold, or equivalently, when the associated Clifford matrices commute, one expects
    $${{\mathcal E}^2} = {({{\mathcal Z}_1} + {{\mathcal Z}_2})^2} + \sum\limits_i {{{\left({{t^i}{f_i}({\phi ^j})} \right)}^2}} \,.$$
    (220)
     
In both cases, the set {ti} involves non-trivial dependence on the dynamical fields and their derivatives. Due to the positivity of the terms in the right-hand side, one can derive lower bounds on the energy, or BPS bounds,
$${\mathcal E} \geq \sqrt {{\mathcal Z}_1^2 + {\mathcal Z}_2^2}$$
(221)
$${\mathcal E} \geq \vert {{\mathcal Z}_1}\vert + \vert {{\mathcal Z}_2}\vert$$
(222)
being saturated precisely when fi (ϕj) = 0 are satisfied, justifying their interpretation as BPS equations [107, 429]. Thus, saturation of the bound matches the energy \({\mathcal E}\) with some charges that may usually have some topological origin [165].

In the current presentation, I assumed the existence of two non-trivial charges, \({-\!\!\!\!\! Z_1}\) and \({-\!\!\!\!\! Z_2}\). The argument can be extended to any number of them. This will change the explicit saturating function in Eq. (216) (see [394]), but not the conceptual difference between the two cases outlined above. It is important to stress that, just as in supergravity, solving the gravitino/dilatino equations, i.e., \(\delta {\mathcal F} = 0\), does not guarantee the resulting configuration to be on-shell, the same is true in brane effective actions. In other words, not all configurations solving Eq. (215) and saturating a BPS bound are guaranteed to be on-shell. For example, in the presence of non-trivial gauge fields, one must still impose Gauss’ law independently.

After these general arguments, I review the relevant phase space reformulation of the effective brane Lagrangian dynamics discussed in Section 3.

4.2.1 D-brane Hamiltonian

As in any Hamiltonian formulation31, the first step consists in breaking covariance to allow a proper treatment of time evolution. Let me split the world volume coordinates as σμ = {t,σi} for i = 1,…,p and rewrite the bosonic D-brane Lagrangian by singling out all time derivatives using standard conjugate momenta variables
$${\mathcal L} = {\dot X^m}{P_m} + {\dot V_i}{E^i} + \dot \psi {T_{{\rm{Dp}}}} - H.$$
(223)
Here Pm and Ei are the conjugate momentum to Xm and Vi, respectively, while H is the Hamiltonian density. ψ is the Hodge dual of a p-form potential introduced in [94] to generate the tension \({T_{{{\rm{D}}_{\rm{p}}}}}\) dynamically [86, 356]. It is convenient to study the tensionless limit in these actions as a generalisation of the massless particle action limit. It was shown in [94] that H can be written as a sum of constraints
$$H = {\psi ^i}{{\mathcal T}_i} + {V_t}{\mathcal K} + {s^i}{{\mathcal H}_i} + \lambda {\mathcal H}\,,$$
(224)
where
$$\begin{array}{*{20}c} {{{\mathcal T}_i} = - {\partial _i}{T_{{\rm{Dp}}}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\mathcal K} = - {\partial _i}{{\tilde E}^i} + {{(- 1)}^{p + 1}}{T_{{\rm{Dp}}}}{\mathcal S}\qquad {\rm{with}}\qquad {\mathcal S} = \ast {{({\mathcal R}{e^{\mathcal F}})}_p},} \\ {{{\mathcal H}_i} = {{\tilde P}_a}E_i^a + {{\tilde E}^j}{{\mathcal F}_{ij}}\qquad {\rm{with}}\qquad E_i^a = E_m^a{\partial _i}{X^m},\quad \quad \quad} \\ {{\mathcal H} = {1 \over 2}\left[ {{{\tilde P}^2} + {{\tilde E}^i}{{\tilde E}^j}{{\mathcal G}_{ij}} + T_{{\rm{Dp}}}^2{e^{- 2\phi}}\det ({{\mathcal G}_{ij}} + {{\mathcal F}_{ij}})} \right]\,.\quad \quad \quad \quad} \\ \end{array}$$
(225)
The first constraint is responsible for the constant tension of the brane. It generates abelian gauge transformations for the p-form potential generating the tension dynamically. The second generates gauge field transformations and it implements the Gauss’ law constraint \({\mathcal K} = 0\). Notice its dependence on \({\mathcal R}\), the pullback of the RR field strengths R = dCCH3, coming from the WZ couplings and acting as sources in Gauss’ law. Finally, \({{\mathcal H}_a}\) and \({\mathcal H}\) generate world-space diffeomorphisms and time translations, respectively.
The modified conjugate momenta \({{\mathcal P}_a}\) and \({{\tilde E}^i}\) determining all these constraints are defined in terms of the original conjugate momenta as
$$\begin{array}{*{20}c} {{{\tilde P}_a} = {E_a}^m({P_m} + {E^i}{Z^ \star}{{\left({{i_m}B} \right)}_i} + {T_{{\rm{Dp}}}}{{\mathcal C}_m}),\qquad {\rm{with}}\qquad {{\mathcal C}_m} = *{{\left({{Z^ \star}({i_m}C) \wedge {e^{\mathcal F}}} \right)}_p},} \\ {{{\tilde E}^i} = {E^i} + T{{\mathcal C}^i}\,,\qquad {\rm{with}}\qquad {{\mathcal C}^i} = {{[*{{({\mathcal C}{e^{\mathcal F}})}_{p - 1}}]}^i}.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(226)
Z* (imB)i stands for the pullback to the world volume of the contraction of B2 along the vector field ∂/∂Xm. Equivalently, Z* (imB)i = iXnBmn. Z* (imC) is defined analogously. Notice * stands for the Hodge dual in the p-dimensional D-brane world space.

In practice, given the equivalence between the Lagrangian formulation and the one above, one solves the equations of motion on the subspace of configurations solving Eq. (215) in phase space variables and finally computes the energy density of the configuration P0 = \({P_0} = {\mathcal E}\) by solving the Hamiltonian constraint, i.e., \({\mathcal H} = 0\), which is a quadratic expression in the conjugate momenta, as expected for a relativistic dynamical system.

4.2.2 M2-brane Hamiltonian

The Hamiltonian formulation for the M2-brane can be viewed as a particular case of the analysis provided above, but in the absence of gauge fields. It was originally studied in [88]. One can check that the full bosonic M2-brane Lagrangian is equivalent to
$${\mathcal L} = {\dot X^m}{P_m} - {s^i}{\tilde P_a}E_i^a - {1 \over 2}\lambda \left[ {{{\tilde P}^2} + T_{{\rm{M2}}}^2\,\det {{\mathcal G}_{ij}}} \right],$$
(227)
where the modified conjugate momentum \({{\tilde P}_a}\) is related to the standard conjugate momentum Pm by
$${\tilde P_a} = E_a^m\left({{P_m} + {T_{{\rm{M2}}}}\,{{\mathcal C}_m}} \right)\qquad {\rm{with}}\qquad {{\mathcal C}_m} = *\left({{Z^ \star}({i_m}{C_3})} \right),$$
(228)
where * describes the Hodge dual computed in the 2-dimensional world space spanned by i, j = 1, 2. Notice no dynamically-generated tension was considered in the formulation above.

As before, one usually solves the equations of motion \(\delta {\mathcal L}/\delta {s^a} = \delta {\mathcal L}/\delta \upsilon = 0\) in the subspace of phase space configurations solving Eq. (215), and computes its energy by solving the Hamiltonian constraint, i.e., \(\delta {\mathcal L}/\delta \lambda = 0\).

4.2.3 M5-brane Hamiltonian

It turns out the Hamiltonian formulation for the M5-brane dynamics is more natural than its Lagrangian one since it is easier to deal with the self-duality condition in phase space [92]. One follows the same strategy and notation as above, splitting the world volume coordinates as σμ = {t, σi} with i = 1, … 5. Since the Hamiltonian formulation is expected to break SO(1, 5) into SO(5), one works in the gauge a = σ0 = t. It is convenient to work with the world space metric \({{\mathcal G}_{ij}}\) and its inverse \({\mathcal G}_5^{ij}\)32. Then, the following identities hold
$$\begin{array}{*{20}c} {{{\tilde H}^{ij}} = {1 \over {6\,\sqrt {\det {{\mathcal G}_5}}}}\,{\varepsilon ^{ij{k_1}{k_2}{k_3}}}{{\mathcal H}_{{k_1}{k_2}{k_3}}}\,,} \\ {\det ({{\mathcal G}_{\mu \nu}} + {{\tilde H}_{\mu \nu}}) = ({{\mathcal G}_{00}} - {{\mathcal G}_{0i}}{\mathcal G}_5^{ij}{{\mathcal G}_{0j}}){{\det}^5}({\mathcal G} + \tilde H)\,,\quad \quad} \\ \end{array}$$
(229)
where det \({{\mathcal G}_5}\) is the determinant of the world space components \({{\mathcal G}_{ij}}\), det \(^5({\mathcal G} + \tilde H) = \det ({{\mathcal G}_{ij}} + {{\tilde H}_{ij}})\) and \({{\tilde H}_{ij}} = {{\mathcal G}_{ik}}{{\mathcal G}_{jl}}{{\tilde H}^{kl}}\).
It was shown in [92] that the full bosonic M5-brane Lagrangian in phase space equals
$${\mathcal L} = {\dot X^m}{P_m} + {1 \over 2}{\Pi ^{ij}}{\dot V_{ij}} - \lambda {\mathcal H} - {s^i}{{\mathcal H}_i} + {\sigma _{ij}}{{\mathcal K}^{ij}}\,,$$
(230)
where Pm and Πij are the conjugate momenta to Xm and the 2-form Vij
$$\begin{array}{*{20}c} {{P_m} = {E_m}^a{{\tilde P}_a} + {T_{{\rm{M5}}}}{{\hat{\mathcal C}}_m},\quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\,} \\ {{{\hat{\mathcal C}}_m} = *[{Z^ \star}\left({{i_m}{C_6}} \right) - {1 \over 2}{Z^ \star}\left({{i_m}{C_3}} \right) \wedge ({{\mathcal C}_3} + 2{{\mathcal H}_3})],} \\ {{\Pi ^{ij}} = {1 \over 4}{T_{{\rm{M5}}}}{\varepsilon ^{ij{k_1}{k_2}{k_3}}}{\partial _{{k_1}}}{V_{{k_2}{k_3}}}\,.\quad \quad \quad \quad \quad \quad \quad \quad \,\,} \\ \end{array}$$
(231)
Notice the last equation is equivalent to \(\Pi = {1 \over 2}{T_{{\rm{M}}{5^{\ast}}}}(dV)\), from which we conclude d* Π = 0, using the Bianchi identify for dV2. The last three functionals appearing in Eq. (230)
$$\begin{array}{*{20}c} {{\mathcal H} = {1 \over 2}\left[ {{{\mathcal P}^2} + T_{{\rm{M5}}}^2{{\det}^5}({\mathcal G} + \tilde H)} \right],} \\ {{{\mathcal H}_i} = {\partial _i}{X^m}{P_m} + {T_{{\rm{M5}}}}({V_i} - {{\hat{\mathcal C}}_i}),\quad \,\,} \\ {{{\mathcal K}^{ij}} = {\Pi ^{ij}} - {1 \over 4}{T_{{\rm{M5}}}}{\varepsilon ^{ij{k_1}{k_2}{k_3}}}{\partial _{{k_1}}}{V_{{k_2}{k_3}}},\quad} \\ \end{array}$$
(232)
correspond to constraints generating time translations, world space diffeomorphisms and the self-duality condition. The following definitions were used in the expressions above
$$\begin{array}{*{20}c} {{V_i} = {1 \over {24}}{\varepsilon ^{{i_1}{i_2}{i_3}{i_4}{i_5}}}{H_{{i_3}{i_4}{i_5}}}{H_{{i_1}{i_2}i}}\,,\quad \quad \quad \quad \quad} \\ {{{\mathcal P}_a} = {E_a}^m{P_m} + {T_{{\rm{M5}}}}({V^i}{\partial _i}{X^m}{E_m}^b{\eta _{ba}} - {{\hat{\mathcal C}}_a})\,,} \\ {{{\hat{\mathcal C}}_a} = E_a^m{{\hat{\mathcal C}}_m}\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(233)
As for D-branes and M2-branes, in practice one solves the equations of motion in the subspace of phase space configurations solving Eq. (215) and eventually computes the energy of the system by solving the quadratic constraint coming from the Hamiltonian constraint \({\mathcal H} = 0\).

4.3 Calibrations

In the absence of WZ couplings and brane gauge field excitations, the energy of a brane configuration equals its volume. The problem of identifying minimal energy configurations is equivalent to that of minimising the volumes of p-dimensional submanifolds embedded in an n-dimensional ambient space. The latter is a purely geometrical question that can, in principle, be mathematically formulated independently of supersymmetry, kappa symmetry or brane theory. This is what the notion of calibration achieves. In this subsection, I review the close relation between this mathematical topic and a subset of supersymmetric brane configurations [235, 228, 2]. I start with static brane solitons in ℝn, for which the connection is more manifest, leaving their generalisations to the appropriate literature quoted below.

Consider the space of oriented p dimensional subspaces of ℝn, i.e., the Grassmannian G (p, ℝn). For any ξ ∈ G(p, ℝn), one can always find an orthonormal basis {e1,…,en} in ℝn such that {e1,…,ep} is a basis in ξ so that its co-volume is
$$\vec \xi = {e_1} \wedge \ldots \wedge {e_p}\,.$$
(234)
A p-form φ on an open subset U of ℝn is a calibration of degree p if
  1. (i)

    = 0

     
  2. (ii)
    for every point x ∈ U, the form φx satisfies \({\varphi _x}(\overrightarrow \xi) \leq 1\) for all ξ ∈ G (p, ℝn) and such that the contact set
    $${\rm{G}}(\varphi)=\{\xi \in {\rm{G}}(p,{{\mathbb R}^n}):\varphi (\vec \xi) = 1\}$$
    (235)
    is not empty.
     

One of the applications of calibrations is to provide a bound for the volume of p-dimensional submanifolds of ℝn. Indeed, the fundamental theorem of calibrations [289] states

Theorem: Let φ be a calibration of degree p on ℝn. The p-dimensional submanifold N, for which
$$\varphi (\vec N) = 1\,,$$
(236)
is volume minimising. One refers to such minimal submanifolds as calibrated submanifolds, or as calibrations for short, of degree p.
The proof of this statement is fairly elementary. Choose an open subset U of N with boundary ∂U and assume the existence of a second open subset V in another subspace W of ℝn with the same boundary, i.e., ∂U = ∂V. By Stokes’ theorem,
$${\rm{vol}}(U) = \int\nolimits_U \varphi = \int\nolimits_V \varphi = \int \varphi (\vec V){\mu _V} \leq \int\nolimits_V {{\mu _V}} = {\rm{vol}}(V)\,,$$
(237)
where μV = α1 ∧ …αp is the volume form constructed out of the dual basis {α1,…,αp} to {e1,…,en}.
Two remarks can motivate why these considerations should have a relation to brane solitons and supersymmetry:
  1. 1.
    For static brane configurations with no gauge field excitations and in the absence of WZ couplings, the energy of the brane soliton equals the volume of the brane submanifold embedded in ℝn. Thus, bounds on the volume correspond to brane energy bounds, which are related to supersymmetry saturation, as previously reviewed. Indeed, the dynamical field Xi (σ) does mathematically describe the map from the world volume p into ℝn. The above bound can then be re-expressed as
    $$\int {{d^p}} \sigma \sqrt {\det {{\mathcal G}_{\mu \nu}}} \geq \int {{X^{\ast}}} \varphi \,,$$
    (238)
    where X*φ stands for the pullback of the p-form φ.
     
  2. 2.

    There exists an explicit spinor construction of calibrations emphasising the connection between calibrated submanifolds, supersymmetry and kappa symmetry.

     
Let me review this spinor construction [159, 287]. For p = 1, 2 mod 4, the p-form calibration takes the form
$$\varphi = d{X^{{i_1}}} \wedge \ldots \wedge d{X^{{i_p}}}{\epsilon ^T}{\Gamma _{0{i_1} \ldots {i_p}}}\epsilon ,$$
(239)
where the set Xi (i = 1,…,n) stands for the transverse scalars to the brane parameterising φn, ϵ is a constant real spinor normalised so that \({\epsilon ^T}\epsilon = 1\) and \({\Gamma _{{i_1} \ldots {i_k}}}\) are antisymmetrised products of Clifford matrices in ℝn. Notice that, given a tangent p-plane ξ, one can write φ |ξ as
$$\varphi {\vert _\xi} = \sqrt {\det {\mathcal G}} \,{\epsilon ^T}{\Gamma _\xi}\epsilon ,$$
(240)
where the matrix
$${\Gamma _\xi} = {1 \over {p!\sqrt {\det {\mathcal G}}}}\,{\varepsilon ^{{\mu _1} \ldots {\mu _p}}}{\partial _{{\mu _1}}}{X^{{i_1}}} \cdots {\partial _{{\mu _p}}}{X^{{i_p}}}{\Gamma _{0{i_1} \ldots {i_p}}}$$
(241)
is evaluated at the point to which ξ is tangent. Given the restriction on the values of p,
$$\Gamma _\xi ^2 = 1\,.$$
(242)
It follows that φ |ξ ≤ volξ for all ξ. Since φ is also closed, one concludes it is a calibration. Its contact set is the set of p-planes for which this inequality is saturated. Using Eq. (240), the latter is equally characterised by the set of p-planes ξ for which
$${\Gamma _\xi}\epsilon = \epsilon \,.$$
(243)
Because of Eq. (242) and the fact that tr Γξ = 0, the solution space to this equation is always half the dimension of the spinor space spanned by φ for any given tangent p-plane ξ. However, this solution space generally varies as ξ varies over the contact set, so that the solution space of the set is generally smaller.

So far the discussion involved no explicit supersymmetry. Notice, however, that the matrix Γξ in Eq. (241) matches the kappa symmetry matrix Γκ for branes in the static gauge with no gauge field excitations propagating in Minkowski. This observation allows us to identify the saturation of the calibration bound with the supersymmetry preserving condition (215) derived from the gauge fixing analysis of kappa symmetry.

Let me close the logic followed in Section 4 by pointing out a very close relation between the supersymmetry algebra and kappa symmetry that all my previous considerations suggest. Consider a single infinite flat M5-brane propagating in d =11 Minkowski and fix the extra gauge symmetry of the PST formalism by a (σμ) = t (temporal gauge). The kappa symmetry matrix (158) reduces to
$${\Gamma _\kappa} = {1 \over {\sqrt {\det ({\delta _{ij}} + {{\tilde H}_{ij}})}}}[{\Gamma ^0}{\Gamma _i}{t^i} + {1 \over 2}{\Gamma ^0}{\Gamma ^{ij}}{\tilde H_{ij}} - {1 \over {5!}}{\Gamma ^0}{\Gamma _{{i_1}, \ldots ,{i_5}}}{\varepsilon ^{{i_1} \ldots {i_5}}}]\,,$$
(244)
where all {i,j} indices stand for world space M5 indices. Notice that the structure of this matrix is equivalent to the one appearing in Eq. (217) for \({\bar \Gamma}\) by identifying
$$\begin{array}{*{20}c} {{Y^{{i_1} \ldots {i_5}}} = - {\varepsilon ^{{i_1} \ldots {i_5}}},\,\,\,{{\tilde H}_{ij}} = {Z_{ij}},} \\ {{P^i} = {1 \over 8}{\varepsilon ^{i\,{j_1}{j_2}{j_3}{j_4}}}\,{Z_{{j_1}{j_2}}}{Z_{{j_3}{j_4}}},\,\,\,{P^0} = \sqrt {\det ({\delta _{ij}} + {Z_{ij}})} \,.} \\ \end{array}$$
(245)
Even though, this was only argued for the M5-brane and in a very particular background, it does provide some preliminary evidence for the existence of such connection. In fact, a stronger argument can be provided by developing a phase space formulation of the kappa symmetry transformations that allows one to write the supersymmetry anticommutator as [278]
$$\{Q,Q\} = {\Gamma ^0}\int {{d^p}} \sigma \,\left[ {{\Gamma ^a}{{\tilde p}_a} + \gamma} \right]\,,\qquad {\rm{with}}\qquad \gamma = {1 \over {p!}}\,{\varepsilon ^{{a_1} \ldots {a_p}}}{\partial _{{a_1}}}{X^{{i_1}}} \cdots {\partial _{{a_p}}}{X^{{i_p}}}{\Gamma _{{i_1} \ldots {i_p}}}\,.$$
(246)
This result has not been established in full generality but it agrees with the flat space case [165] and those non-flat cases that have been analysed [438, 437]. I refer the reader to [278] where they connect the functional form in the right-hand side of Eq. (246) with the kappa symmetry transformations for fermions in its Hamiltonian form.

The connection between calibrations, supersymmetry and kappa symmetry goes beyond the arguments given above. The original mathematical notion of calibration was extended in [277, 278] relaxing its first condition ≠ 0. Physically, this allowed one to include the presence of non-trivial potential energies due to background fluxes coming from the WZ couplings. Some of the applications derived from this notion include [231, 229, 230, 373, 139]. Later, the notion of generalised calibration was introduced in [344], where it was shown to agree with the notion of calibration defined in generalised Calabi-Yau manifolds [267] following the seminal work in [298]. This general notion allows one to include the effect of non-trivial magnetic field excitations on the calibrated submanifold, but still assumes the background and the calibration to be static. Some applications of these notions in the physics literature can be found in [344, 377, 413]. More recently, this formalism was generalised to include electric field excitations [376], establishing a precise correspondence between generic supersymmetric brane configurations and generalised geometry.

Summary: A necessary condition for a bosonic brane configuration to preserve supersymmetry is to solve the kappa symmetry preserving condition (215). In general, this is not sufficient for being an on-shell configuration, though it can be, if there are no gauge field excitations. Solutions to Eq. (215) typically impose a set of constraints on the field configuration, which can be interpreted as BPS equations by computing the Hamiltonian of the configuration, and a set of projection conditions on the constant parts ϵ of the background Killing spinors ϵ. The energy bounds saturated when the BPS equations hold are a field theory realisation of the algebraic bounds derived from the supersymmetry algebra. An attempt to summarise the essence of these relations is illustrated in Figure 6.

5 World Volume Solitons: Applications

There are two natural sets of applications involving brane effective actions: kinematical and dynamical. In this section, I will discuss the application of the general formalism developed in Section 4 to study the existence of certain string theory BPS states realised as world volume supersymmetric bosonic solitons, leaving more AdS/CFT dynamically-oriented applications to Section 6. The main goals in this section include:
  1. 1.

    In a Minkowski background, the identification of the vacuum of all the p + 1 dimensional supersymmetric field theories discussed before as half-BPS flat infinite branes, and the discussions of some of their excitations carrying topological charges, which are interpretable as brane intersections or branes within branes.

     
  2. 2.

    Supertubes, as examples of supersymmetric bound states realised as expanded branes without carrying charge under the gauge potential, which the world volume brane minimally couples to.

     
  3. 3.

    As examples of solitons in curved backgrounds, I will discuss the baryon vertex and giant gravitons in AdS5 × S5.

     
  4. 4.

    I will stress the relevance of supertubes and giant gravitons as constituents of small supersymmetric black holes, their connection to fuzzball ideas and the general use of probe techniques to identify black hole constituents in more general situations.

     

5.1 Vacuum infinite branes

There exist half-BPS branes in 10- and 11-dimensional Minkowski spacetime. Since their effective actions were discussed in Section 3, we can check their existence and the amount of supersymmetry they preserve, by solving the brane classical equations of motion and the kappa symmetry preserving condition (215).

First, one works with the bosonic truncation θ = 0. The background, in Cartesian coordinates, involves the metric
$$d{s^2} = {\eta _{mn}}d{x^m}d{x^n},\qquad \qquad m,n = 0,1, \ldots D - 1$$
(247)
and all remaining bosonic fields vanish, except for the dilaton, in type IIA/B, which is constant. This supergravity configuration is maximally supersymmetric, i.e., it has Killing spinors spanning a vector space, which is 32-dimensional. In Cartesian coordinates, these are constant spinors ϵ = ϵ ∞.
Half-BPS branes should correspond to vacuum configurations in these field theories describing infinite branes breaking the isometry group ISO(1, D − 1) to ISO(1,p) × SO(Dp − 1) and preserving half of the supersymmetries. Geometrically, these configurations are specified by the brane location. This is equivalent to first splitting the scalar fields Xm (σ) into longitudinal Xμ and transverse XI directions, setting the latter to constant values XI = cI (the transverse brane location). Second, one identifies the world volume directions with the longitudinal directions, Xμ = σμ. The latter can also be viewed as fixing the world volume diffeomorphisms to the static gauge. This information can be encoded as an array
$$p{\rm -brane}:\,\,1\,\,2{.}\,{.}p\_\,\_\,\_\,\_$$
(248)

It is easy to check that the above is an on-shell configuration given the structure of the Euler-Lagrange equations and the absence of non-trivial couplings except for the induced world volume metric \({{\mathcal G}_{\mu \nu}}\), which equals ημν in this case.

To analyse the supersymmetry preserved, one must solve Eq. (215). Notice that in the static gauge and in the absence of any further excitations, the induced gamma matrices equal
$${\gamma _\mu} = {\partial _\mu}{x^m}E_m^a{\Gamma _a} = {\Gamma _\mu} \Rightarrow {\gamma _{{\mu _0} \ldots {\mu _p}}} = {\Gamma _{{\mu _0} \ldots {\mu _p}}},$$
(249)
\(E_m^a = \delta _m^a\). Thus, Γκ reduces to a constant Clifford valued matrix standing for the volume of the brane, Γvol, up to the chirality of the background spinors, which is parameterised by the matrix τ
$${\Gamma _\kappa} = {\Gamma _{{\rm{vol}}}}\tau \,.$$
(250)
The specific matrices for the branes discussed in this review are summarised in Table 6. Since \({\Gamma _\kappa} = 0\) and Tr Γκ =0, only half of the vector space spanned by ϵ preserves these bosonic configurations, i.e., all infinite branes preserve half of the supersymmetries. These projectors match the ones derived from bosonic supergravity backgrounds carrying the same charges as these infinite branes.
Table 6

Half-BPS branes and the supersymmetries they preserve.

BPS state

Projector

M2-brane

Γ012ϵ = ϵ

M5-brane

Γ012345ϵ = ϵ

IIA D2n-brane

\({\Gamma _{0 \ldots 2n}}\Gamma _\sharp ^{n + 1}\epsilon = \epsilon\)

IIB D2n−1-brane

\({\Gamma _{0 \ldots 2n - 1}}\tau _3^ni{\tau _2}\epsilon = \epsilon\)

All these configurations have an energy density equaling the brane tension T since the Hamiltonian constraint is always solved by
$${{\mathcal E}^2} = {{\rm{T}}^2}\,\det {\mathcal G} = {{\rm{T}}^2}\,.$$
(251)
From the spacetime superalgebra perspective, these configurations saturate a bound between the energy and the p-form bosonic charge carried by the volume form defined by the brane
$${\mathcal E} = {{\mathcal Z}_{{\mu _1} \ldots {\mu _p}}} = {\rm{T}}{\epsilon _{{\mu _1} \ldots {\mu _p}}}.$$
(252)
The saturation corresponds to the fact that any excitation above the infinite brane configuration would increase the energy. From the world-volume perspective, the solution is a vacuum, and consequently, it is annihilated by all sixteen world-volume supercharges. These are precisely the ones solving the kappa symmetry preserving condition (215).

5.2 Intersecting M2-branes

As a first example of an excited configuration, consider the intersection of two M2-branes in a point corresponding to the array
$$\begin{array}{*{20}c} {M2:1\,{2_{- \, - \, - \, - \, - \, - \, - \, -}}} \\ {M2{:_{- \, -}}\,3\,4{\,_{- \, - \, - \, - \, - \, -}}.} \\ \end{array}$$
(253)
In the probe approximation, the M2-brane effective action describes the first M2-brane by fixing the static gauge and the second M2-brane as an excitation above this vacuum by turning on two scalar fields (X3, X4) according to the ansatz
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu}\quad ,\quad {X^i} = {c^i},\quad \quad \quad} \\ {{X^3}({\sigma ^a}) \equiv y({\sigma ^a})\quad ,\quad {X^4}({\sigma ^a}) \equiv z({\sigma ^a}),} \\ \end{array}$$
(254)
where a runs over the spatial world volume directions and i over the transverse directions not being excited.
Supersymmetry analysis: Given the ansatz (254), the induced metric components equal \({{\mathcal G}_{00}} = - 1,\,{{\mathcal G}_{0a}} = 0,\,{{\mathcal G}_{ab}} = {\delta _{ab}} + {\partial _a}{X^r}{\partial _b}{X^s}{\delta _{rs}}\) (with r,s = 3,4), whereas its determinant and the induced gamma matrices reduce to
$$- \det {\mathcal G} = 1 + \vert \vec \nabla y{\vert ^2} + \vert \vec \nabla z{\vert ^2} + {(\vec \nabla y \times \vec \nabla z)^2},$$
(255)
$${\gamma _0} = {\Gamma _0}\quad ,\quad {\gamma _a} = {\Gamma _a} + {\partial _a}{X^r}{\Gamma _r}\,.$$
(256)
Altogether, the kappa symmetry preserving condition (215) is
$$\sqrt {- \det {\mathcal G}} \,\epsilon = \left({{\Gamma _{012}} + {\varepsilon ^{ab}}{\partial _a}y{\partial _b}z{\Gamma _{034}} - {\varepsilon ^{ab}}{\partial _a}{x^r}{\Gamma _{0br}}} \right)\epsilon \,.$$
(257)
If the excitation given in Eq. (254) must describe the array in Eq. (253), the subspace of Killing spinors ϵ spanned by the solutions to Eq. (257) must be characterised by two projection conditions
$${\Gamma _{012}}\epsilon = {\Gamma _{034}}\epsilon = \epsilon ,$$
(258)
one for each M2-brane in the array (253). Plugging these projections into Eq. (257)
$$\left({\sqrt {- \det {\mathcal G}} - (1 + {\varepsilon ^{ab}}{\partial _a}y{\partial _b}z)} \right)\,\epsilon = {\varepsilon ^{ab}}{\partial _a}{X^r}{\Gamma _{0br}}\epsilon ,$$
(259)
one obtains an identity involving two different Clifford-valued contributions: the left-hand side is proportional to the identity matrix acting on the Killing spinor, while the right-hand side involves some subset of antisymmetric products of gamma matrices. Since these Clifford valued matrices are independent, each term must vanish independently. This is equivalent to two partial differential equations
$${\partial _2}y = - {\partial _1}z\,,\qquad {\partial _1}y = {\partial _2}z\,.$$
(260)
Notice this is equivalent to the holomorphicity of the complex function U (σ+) = y + iz in terms of the complex world space coordinates σ± = σ1 ± 2, since Eqs. (260) are equivalent to the Cauchy-Riemann equations for U (σ+).

When conditions (260) are used in the remaining left-hand side of Eq. (259), one recovers an identity. Thus, the solution to Eq. (215) in this particular case involves the two supersymmetry projections (258) and the BPS equations (260) satisfied by holomorphic functions U (σ+).

Hamiltonian analysis: Since this is the first non-trivial example of a supersymmetric soliton discussed in this review, it is pedagogically constructive to rederive Eqs. (260) from a purely Hamiltonian point of view [225]. This will also convince the reader that holomorphicity is the only requirement to be on-shell. To ease notation below, rewrite Eq. (260) as
$$\vec \nabla y = \star \vec \nabla z\,,$$
(261)
where standard vector calculus notation for ℝ2 is used, i.e., \(\overrightarrow \nabla = ({\partial _1},{\partial _2})\) and \(_\ast \overrightarrow \nabla = ({\partial _2},{\partial _1})\)
Consider the phase space description for the M2-brane Lagrangian given in Eq. (227) in a Minkowski background. The Lagrange multiplier fields sa impose the world space diffeomorphism constraints. In the static gauge, these reduce to
$${P_a} = {P_I}\cdot{\partial _a}{X^I},$$
(262)
where PI are the conjugate momenta to the eight world volume scalars XI describing transverse fluctuations. For static configurations carrying no momentum, i.e., PI = 0, the world space momenta will also vanish, i.e., Pa = 0.
Solving the Hamiltonian constraint imposed by the Lagrange multiplier λ for the energy density \({\mathcal E} = {P_0}\), one obtains [225]
$${({\mathcal E}/{T_{{\rm{M2}}}})^2} = 1 + \vert \vec \nabla y{\vert ^2} + \vert \vec \nabla z{\vert ^2} + {(\vec \nabla y \times \vec \nabla z)^2} = {(1 - \vec \nabla y \times \vec \nabla z)^2} + \vert \vec \nabla y - \star \vec \nabla z{\vert ^2}\,.$$
(263)
This already involves the computation of the induced world space metric determinant and its rewriting in a suggestive way to derive the bound
$${\mathcal E}/{T_{{\rm{M2}}}} \geq 1 + \vert \vec \nabla y \times \vec \nabla z\vert \,.$$
(264)
The latter is saturated if and only if Eq. (261) is satisfied. This proves the BPS character of the constraint derived from solving Eq. (215) in this particular case and justifies that any solution to Eq. (261) is on-shell, since it extremises the energy and there are no further gauge field excitations.
Integrating over the world space of the M2 brane allows us to derive a bound on the charges carried by this subset of configurations
$$E \geq {E_0} + \vert Z\vert \,.$$
(265)
E0 stands for the energy of the infinite M2-brane vacuum, whereas Z is the topological charge
$$Z = {T_{M2}}\int\nolimits_{M2} d y \wedge dz = {T_{M2}}{i \over 2}\int\nolimits_{M2} d U \wedge d\bar U\,,$$
(266)
accounting for the second M2-brane in the system.

The bound (265) matches the spacetime supersymmetry algebra bound: the mass (E) of the system is larger than the sum of the masses of the two M2-branes. Field theoretically, the first M2-brane charge corresponds to the vacuum energy (E0), while the second corresponds to the topological charge (Z) describing the excitation. When the system is supersymmetric, the energy saturates the bound E = E0 + |Z | and preserves 1/4 of the original supersymmetry. From the world volume superalgebra perspective, the energy is always measured with respect to the vacuum. Thus, the bound corresponds to the excitation energy EE0 equalling |Z |. This preserves 1/2 of the world volume supersymmetry preserved by the vacuum, matching the spacetime 1/4 fraction.

For more examples of M2-brane solitons see [95] and for a related classification of D2-brane supersymmetric soltions see [33].

5.3 Intersecting M2 and M5-branes

As a second example of BPS excitation, consider the 1/4 BPS configuration M5 ⊥ M2(1) corresponding to the brane array
$$\begin{array}{*{20}c} {{\rm{M}}5:\,\,1\;\;2\,\,3\,\,4\,\,5\,\_\,\_\,\_\,\_\,\_} \\ {{\rm{M}}2:\,\,\_\,\_\,\_\,\_5\,6\,\_\,\_\,\_\,\_.} \\ \end{array}$$
(267)
The idea is to describe an infinite M5-brane by the static gauge and to turn on a transverse scalar field X6 to account for the M2-brane excitation. However, X6 is not enough to support an M2-brane interpretation, since the latter is electrically charged under the 11-dimensional supergravity three form A3. Thus, the sought M5-brane soliton must source the A056 components. From the Wess-Zumino coupling
$$\int d {V_2} \wedge {{\mathcal A}_3}\,,$$
(268)
one learns that the magnetic \({(dV)_{\hat a\hat b\hat c}}\) components, where hatted indices stand for world space directions different from σ5, i.e., \(\hat a \neq 5\), must also be excited.
The full ansatz will assume delocalisation along the σ5 direction, so that the string-like excitation in the X6 direction can be viewed as a membrane:
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu}\quad ,\quad {X^i} = {c^i},} \\ {{X^6}({\sigma ^{\hat a}}) = y({\sigma ^{\hat a}}),} \\ {{{\mathcal H}_{5\hat a\hat b}} = 0\,.} \\ \end{array}$$
(269)
Supersymmetry analysis: The M5-brane kappa symmetry matrix (158) in the temporal gauge a = τ reduces to
$${\Gamma _\kappa} = {1 \over {\sqrt {- \det ({\mathcal G} + \tilde H)}}}\left[ {{1 \over {5!}}{\epsilon ^{{a_1} \ldots {a_5}}}{\Gamma _0}{\gamma _{{a_1} \ldots {a_5}}} - {1 \over 2}\sqrt {- \det {\mathcal G}} \,{\Gamma _0}{\gamma _{ab}}{{\tilde H}^{ab}} - {\Gamma _0}{\gamma _a}{t^a}} \right].$$
(270)
for the subset of configurations described by the ansatz (269), it follows
$$\begin{array}{*{20}c} {{t_a} = 0\quad ,\quad {{\tilde H}^{\hat a\hat b}} = 0} \\ {{{\tilde H}^{5\hat a}} = {{{\Pi ^{\hat a}}} \over {\sqrt {- \det {\mathcal G}}}}\quad ,\quad {\Pi ^{\hat a}} = {1 \over {3!}}{\epsilon ^{\hat a{{\hat a}_1}{{\hat a}_2}{{\hat a}_3}}}{{\mathcal H}_{{{\hat a}_1}{{\hat a}_2}{{\hat a}_3}}}\,.} \\ \end{array}$$
(271)
This reduces Eq. (270) to
$${\Gamma _\kappa} = {1 \over {\sqrt {- \det ({\mathcal G} + \tilde H)}}}\left[ {{\Gamma _{012345}} + {\partial _{\hat a}}y{\Gamma _{05y}}{\Gamma _{05}}{\Gamma ^{\hat a}}{\Gamma _{012345}} - {\Gamma _{05}}{\Gamma _{\hat a}}{\Pi ^{\hat a}} - {\partial _{\hat a}}y{\Pi ^{\hat a}}{\Gamma _{05y}}} \right].$$
(272)
To solve the kappa symmetry preserving condition (215), I impose two projection conditions
$$\begin{array}{*{20}c} {{\Gamma _{012345}}\epsilon = \epsilon ,} \\ {{\Gamma _{05y}}\epsilon = \epsilon} \\ \end{array}$$
(273)
on the constant Killing spinors ϵ. The eight supercharges satisfying them match the ones preserved by M5 ⊥ M2(1). Using Eq. (273) in Eq. (272), Γκ keeps a non-trivial dependence on \({\Gamma _{05}}{\Gamma _{\hat a}}\). Requiring its coefficient to vanish gives rise to the BPS condition
$${\Pi _{\hat a}} = - {\partial _{\hat a}}y.$$
(274)
Overall, the kappa symmetry preserving condition (215) reduces to the purely algebraic condition
$$\sqrt {- \det ({\mathcal G} + \tilde H)} \epsilon = \left({1 + {\delta ^{\hat a\hat b}}{\partial _{\hat a}}y{\partial _{\hat b}}y} \right)\epsilon .$$
(275)
To check this holds, notice the only non-vanishing components of \({{\tilde H}_{\mu \nu}}\) are \({{\tilde H}_{5\hat a}}\)
$${\tilde H_{5\hat a}} = {{\mathcal G}_{55}}{{\mathcal G}_{\hat a\hat b}}{{{\Pi ^{\hat b}}} \over {\sqrt {- \det {\mathcal G}}}}\,.$$
(276)
This allows us to compute the determinant
$$- \det ({\mathcal G} + \tilde H) = \det ({{\mathcal G}_{\hat a\hat b}} + {\tilde H_{5\hat a}}{\tilde H_{5\hat b}}) = \det ({{\mathcal G}_{\hat a\hat b}})\left({1 + {{\mathcal G}^{\hat a\hat b}}{{\tilde H}_{5\hat a}}{{\tilde H}_{5\hat b}}} \right),$$
(277)
which becomes a perfect square once the BPS equation (274) is used
$$- \det ({\mathcal G} + \tilde H) = {\left({1 + {\delta ^{\hat a\hat b}}{\partial _{\hat a}}y{\partial _{\hat b}}y} \right)^2}\,.$$
(278)
This shows that Eq. (275) holds automatically. Thus, the solution to the kappa symmetry preserving condition (215) for the ansatz (269) on an M5-brane action is solved by the supersymmetry projection conditions (273) and the BPS equation (274). Since the soliton involves a non-trivial world volume gauge field, the Bianchi identity \(d{{\mathcal H}_3} = 0\) must still be imposed. This determines the harmonic character for the excited transverse scalar in the four dimensional world space ω4
$${\partial ^{\hat a}}{\partial _{\hat a}}y = 0.$$
(279)
Hamiltonian analysis: The Hamiltonian analysis for this system was studied in [225] following the M5-brane phase space formulation given in Eq. (230). For static configurations, the Hamiltonian constraint can be solved by the energy density \({\mathcal E}\) as
$${{{{\mathcal E}^2}} \over {T_{{\rm{M5}}}^2}} = 1 + {(\partial y)^2} + {1 \over 2}\vert \tilde {\mathcal H}{\vert ^2} + \vert \tilde {\mathcal H}\cdot\partial y{\vert ^2} + \vert V{\vert ^2}$$
(280)
where
$$\begin{array}{*{20}c} {\vert \tilde{\mathcal H}{\vert ^2} = {{\tilde{\mathcal H}}^{ab}}{{\tilde{\mathcal H}}^{cd}}{\delta _{ac}}{\delta _{bd}}\,,\qquad \qquad {{\tilde{\mathcal H}}^{ab}} = {1 \over 6}{\varepsilon ^{abcde}}{{\mathcal H}_{cde}},} \\ {\vert \tilde{\mathcal H}\cdot\partial y{\vert ^2} = {{\tilde{\mathcal H}}^{ab}}{{\tilde{\mathcal H}}^{cd}}{\partial _b}y{\partial _d}y{\delta _{ac}},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\vert V{\vert ^2} = {V_a}{V_b}{\delta ^{ab}}\,,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,} \\ \end{array}$$
(281)
and world space indices were denoted by latin indices σa a = 1,…,5. It was noted in [225] that by introducing a unit length world space 5-vector ζ, i.e., ζaζbδab = 1, the energy density could be written in the suggestive form
$$\begin{array}{*{20}c} {{{{{\mathcal E}^2}} \over {T_{{\rm{M5}}}^2}} = \vert {\zeta ^a} \pm {{\tilde{\mathcal H}}^{ab}}{\partial _b}y{\vert ^2} + 2{{\left\vert {{\partial _{\left[ a \right.}}y\zeta {}_{\left. b \right]} \pm {1 \over 2}{\delta _{ac}}{\delta _{bd}}{{\tilde{\mathcal H}}^{cd}}} \right\vert}^2}} \\ {+ \,{{({\zeta ^a}{\partial _a}y)}^2} + \vert V{\vert ^2}\,.\quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(282)
The unit vector provides a covariant way of introducing a preferred direction in the 5-dimensional world space. Choosing ζ5 = 1 and \({\zeta ^5} = 0\), to match the delocalisation direction in our bosonic ansatz, one derives the inequality
$${{\mathcal E} \over {{T_{{\rm{M5}}}}}} \geq 1 \pm {1 \over 6}{\Pi ^{\hat a}}{\partial _{\hat a}}y.$$
(283)
The latter is saturated if and only if
$${\partial _5}y = 0\qquad {{\mathcal H}_{5\hat a\hat b}} = 0$$
(284)
and
$${{\mathcal H}_3} = \pm \star dy$$
(285)
where \({{\mathcal H}_3}\) is only defined on the 4-dimensional subspace ω4, orthogonal to ζ, and * is its Hodge dual. This confirms the BPS nature of Eq. (274). Since \({{\mathcal H}_3}\) is closed, y is harmonic in ω4.
To regulate the divergent energy, one imposes periodic boundary conditions in the 5-direction making the orbits of the vector field ζ have finite length L. Then, the total energy satisfies
$$E \geq {E_0} + L\cdot\vert Z\vert ,$$
(286)
where Z is the topological charge
$$Z = \int\nolimits_{{w_4}} {{{\mathcal H}_3}} \wedge dy\,.$$
(287)
The tension of the soliton, i.e., energy per unit of length, equals T = EE0/L. It is bounded by Z. It only equals the latter for configurations satisfying Eq. (285). Singularities in the harmonic function match the strings found in [301]. To check this interpretation, consider a solution with a single isolated point singularity at the origin. Its energy can be rewritten as the small radius limit of a surface integral over a 3-sphere surrounding the origin. Since y is constant on this integration surface, one derives the string tension [225]
$$T = \mu \, \underset {\delta \rightarrow 0} {\lim}\, y(\delta)\qquad {\rm{where}}\qquad \mu = \int\nolimits_{{S^3}} {{{\mathcal H}_3}}$$
(288)
is the string charge. Even though this tension diverges, it does so consistently, being the boundary of a semi-infinite membrane.

5.4 BIons

Perhaps one of the most pedagogical examples of brane solitons are BIons. These were first described in [128, 234] and correspond to on-shell supersymmetric D-brane configurations representing a fundamental string ending on the D-brane, i.e., the defining property of the D-brane itself. They correspond to the array of branes
$$\begin{array}{*{20}c} {{\rm{D}}p:1 \ldots p\quad \_\quad \_\,\_\,\_\,} \\ {{\rm{F}}:\_\,\_\,\_\,\_\,\_p + 1\_\,\_\,\_\,.} \\ \end{array}$$
(289)
Working in the static gauge describes the vacuum infinite Dp-brane. The static soliton excites a transverse scalar field (y = y (σa)) and the electric field (V0= V0 (σa)), while setting the magnetic components of the gauge field (Va) to zero
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu}\quad ,\quad {X^i} = {c^i},} \\ {{X^{p + 1}}({\sigma ^a}) = y({\sigma ^a})\quad ,\quad {V_0} = {V_0}({\sigma ^a})}{.} \\ \end{array}$$
(290)
The gauge invariant character of the scalar ensures its physical observability as a deformation of the flat world volume geometry described by the global static gauge, whereas the electric field can be understood as associated to the end of the open string, which is seen as a charged particle from the world volume perspective. A second way of arguing the necessity for such electric charge is to remember that fundamental strings are electrically charged under the NS-NS two form. The latter appears in the effective action through the gauge invariant form \({\mathcal F}\). Thus, turning on V0 is equivalent to turning such charge33.
Supersymmetry analysis: Let me analyse the amount of supersymmetry preserved by configurations (290) in type IIA and type IIB, separately. In both cases, the matrix \({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}\) equals
$${{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}} = \left({\begin{array}{*{20}c} {- 1} & {{F_{0b}}} \\ {- {F_{0a}}} & {{\delta _{ab}} + {\partial _a}y{\partial _b}y} \\ \end{array}} \right) \Rightarrow - \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}) = \det ({\delta _{ab}} + {\partial _a}y{\partial _b}y - {F_{0a}}{F_{0b}})$$
(291)
while the induced gamma matrices are decomposed as
$${\gamma _0} = {\Gamma _0},\qquad \qquad {\gamma _a} = {\Gamma _a} + {\partial _a}y{\Gamma _y},$$
(292)
where a stands for world space indices. Due to the electric ansatz for the gauge field, the kappa symmetry matrix Γκ has only two contributions. In particular, for type IIA (p = 2k)
$${\Gamma _\kappa} = {1 \over {\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}})}}}{1 \over {(p + 1)!}}{\varepsilon ^{{\mu _0} \ldots {\mu _p}}}\left({{\gamma _{{\mu _0} \ldots {\mu _p}}}\Gamma _\sharp ^{k + 1} + \left({\begin{array}{*{20}c} {p + 1} \\ 2 \\ \end{array}} \right){F_{{\mu _0}{\mu _1}}}{\gamma _{{\mu _3} \ldots {\mu _p}}}\Gamma _\sharp ^k} \right)\,.$$
(293)
Summing over world volume time, one obtains
$${\Gamma _\kappa} = {1 \over {\sqrt {- \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}})}}}{1 \over {p!}}{\varepsilon ^{{a_1} \ldots {a_p}}}\left({{\Gamma _0}{\gamma _{{a_1} \ldots {a_p}}}\Gamma _\sharp ^{k + 1} + {p \over 2}{F_{0{a_1}}}{\gamma _{{a_2} \ldots {a_p}}}\Gamma _\sharp ^k} \right)\,.$$
(294)
Using the duality relation
$${\varepsilon ^{{i_1} \ldots {i_k}{j_{k + 1}} \ldots {j_{p + 1}}}}{\gamma _{{j_{k + 1}} \ldots {j_{p + 1}}}} = {(- 1)^{k(k - 1)/2}}(p + 1 - k)!{\gamma ^{{i_1} \ldots {i_k}}}\,\sqrt {- \det {\mathcal G}} \,{\Gamma _{0 \ldots p}},$$
(295)
one can write the first term on the right-hand side of Eq. (294) as
$${\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1} - {\Gamma ^b}{\partial _b}y{\Gamma _y}{\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}.$$
(296)
Using the same duality relation and proceeding in an analogous way, the second term equals
$${F_{0a}}{\Gamma ^a}{\Gamma _{1 \ldots p}}\Gamma _\sharp ^k + {F_{0a}}{\partial _b}y{\Gamma _y}{\Gamma ^{ab}}{\Gamma _{1 \ldots p}}\Gamma _\sharp ^k.$$
(297)
Inserting Eqs. (296) and (297), the kappa symmetry preserving condition can be expressed as
$$\sqrt {- \det ({\mathcal G} + {\mathcal F})} \epsilon = \left[ {1 + {\Gamma ^a}{\Gamma _0}{\Gamma _\sharp}({F_{0a}} - {\partial _a}y{\Gamma _{0y}}{\Gamma _\sharp}) - {\Gamma ^{ab}}{F_{0a}}{\partial _b}y{\Gamma _{0y}}{\Gamma _\sharp}} \right]{\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}\epsilon .$$
(298)
Given the physical interpretation of the sought soliton, one imposes the following two supersymmetry projection conditions
$${{\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}\epsilon = \epsilon}$$
(299)
$${{\Gamma _{0y}}{\Gamma _\sharp}\epsilon = \epsilon}$$
(300)
corresponding to having a type IIA Dp-brane along directions 1,…, p and a fundamental string along the transverse direction y. Since both Clifford valued matrices commute, the dimensionality of the subspace of solutions is eight, as corresponds to preserving ν = 1/4 of the bulk supersymmetry. Plugging these projections into Eq. (298), the kappa symmetry preserving condition reduces to
$$\sqrt {- \det ({\mathcal G} + {\mathcal F})} \epsilon = \left({1 + {\Gamma ^a}{\Gamma _0}{\Gamma _\sharp}({F_{0a}} - {\partial _a}y) - {\Gamma ^{ab}}{F_{0a}}{\partial _b}y} \right)\epsilon \,.$$
(301)
It is clear that the BPS condition
$${F_{0a}} = {\partial _a}y\,,$$
(302)
derived from requiring the coefficient of Γa Γ0Γ# to vanish, solves Eq. (301). Indeed, the last term in Eq. (301) vanishes due to antisymmetry, whereas the square root of the determinant equals one, whenever Eq. (302) holds.
The analysis for type IIB Dp-branes (p = 2k +1) works analogously by appropriately dealing with the different bulk fermion chiralities, i.e., one should replace \(\Gamma _\# ^k\) by \(\tau _3^ki{\tau _2}\). Thus, the supersymmetry projection conditions (299) and (300) are replaced by
$${\Gamma _{0 \ldots p}}\tau _3^{k + 1}\,i{\tau _2}\epsilon , = \epsilon$$
(303)
$${\Gamma _{0y}}{\tau _3}\epsilon = \epsilon ,$$
(304)
corresponding to having a type IIB Dp-brane along the directions 1,…,p and a fundamental string along the transverse direction y.
Satisfying the BPS equation (302) does not guarantee the on-shell nature of the configuration. Given the non-triviality of the gauge field, Gauss’ law aEa = 0 must be imposed, where Ea is the conjugate momentum to the electric field, which reduces to
$${E^a} = {{\partial {\mathcal L}} \over {\partial {{\dot V}_a}}} = {\delta ^{ab}}{F_{0b}},$$
(305)
when Eq. (302) is satisfied. Thus, the transverse scalar y must be a harmonic function on the p-dimensional D-brane world space
$${\partial _a}{\partial ^a}y = 0\,.$$
(306)
Hamiltonian analysis: Using the phase space formulation of the D-brane Lagrangian in Eqs. (223) and (224), I will reproduce the BPS bound (302) and interpret the charges carried by BIons. Working in static gauge, the world space diffeomorphism constraints are trivially solved for static configurations, i.e., Pi = 0, and in the absence of magnetic gauge field excitations, i.e., Fab = 0. The Hamiltonian constraint can be solved for the energy density [225]
$${{{{\mathcal E}^2}} \over {T_{{\rm{Dp}}}^2}} = {E^a}{E^b}{{\mathcal G}_{ab}} + \det {{\mathcal G}_{ab}}\,.$$
(307)
Since det \({{\mathcal G}_{ab}} = 1 + {(\partial y)^2}\), Eq. (307) is equivalent to [225]
$${{{{\mathcal E}^2}} \over {T_{{\rm{Dp}}}^2}} = {(1 \pm {E^a}{\partial _a}y)^2} + {(E \mp \partial y)^2}.$$
(308)
There exists an energy bound
$${{\mathcal E} \over {{T_{{\rm{Dp}}}}}} \geq 1 + \left\vert {{E^a}{\partial _a}y} \right\vert ,$$
(309)
being saturated if and only if
$${E_a} = \pm {\partial _a}y.$$
(310)
This is precisely the relation (302) derived from the solution to the kappa symmetry preserving condition (215) (the sign is related to the sign of the fundamental string charge). Thus, the total energy integrated over the D-brane world space ω satisfies
$$E \geq {E_0} + \vert {Z_{el}}\vert ,$$
(311)
where Zel is the charge
$${Z_{el}} = \int\nolimits_\omega {{E^a}} {\partial _a}y.$$
(312)
To interpret this charge as the charge carried by a string, consider the most symmetric solution to Eq. (306), for Dp-branes with p ≥ 3, depending on the radial coordinate in world space r, i.e., r2 = σaσbδab,
$$y({\sigma ^a}) = {q \over {{\Omega _{p - 1}}{r^{p - 2}}}},$$
(313)
where Ωp stands for the volume of the unit p-sphere. This describes a charge q at the origin. Gauss’s law allows us to express the energy as an integral over a (hyper)sphere of radius δ surrounding the charge. Since y = y (δ) is constant over this (hyper)sphere, one has
$$\begin{array}{*{20}c} {E = \underset {\delta \rightarrow 0} {\lim} \left\vert {y(\delta)\int\nolimits_{r = \delta} {\overrightarrow {dS}} \cdot\vec E} \right\vert} \\ {= q \underset {\delta \rightarrow 0} {\lim} y(\delta){.}\quad \quad \,} \\ \end{array}$$
(314)
Thus, the energy is infinite since y as δ → 0, but this divergence has its physical origin on the infinite length of a string of finite and constant tension q [128, 234]. See [164] for a discussion of the D-string case, corresponding to string junctions.

5.5 Dyons

Dyons are on-shell supersymmetric D3-brane configurations describing a (p, q) string bound state ending on the brane. They are described by the array
$$\begin{array}{*{20}c} {{\rm{D3:}}\,1\,\,2\,\,3\,\_\,\_\,\_\,\_\,\_\,\_\,} \\ {\,{\rm{F}}:\_\,\_\,\_4\_\,\_\,\_\,\_\,\_} \\ {{\rm{D}}1:\_\,\_\,\_4\_\,\_\,\_\,\_\,\_.} \\ \end{array}$$
(315)
Since the discussion is analogous to the one for BIons, I shall be brief. The ansatz is as in Eq. (290) but including some magnetic components for the gauge field. This is both because a (p, q) string is seen as a dyonic particle on the brane and a D-string is electrically charged under the RR two form. The latter can be induced from the Wess-Zumino coupling
$$\int {{{\mathcal C}_2}} \wedge {\mathcal F}\,.$$
(316)
This shows that magnetic components in \({\mathcal F}\) couple to electric components in C2. Altogether, the dyonic ansatz is
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu}\quad ,\quad {X^i} = {c^i},\quad \quad \quad \quad \quad \quad \quad \,\,\,} \\ {{X^4}({\sigma ^a}) = y({\sigma ^a})\quad ,\quad {V_0} = {V_0}({\sigma ^a})\quad ,\quad {V_a} = {V_a}({\sigma ^b})\,.} \\ \end{array}$$
(317)
Supersymmetry analysis: In this case, the matrix elements \({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}\) are
$$\begin{array}{*{20}c} {{{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}} = \left({\begin{array}{*{20}c} {- 1} & {{F_{0b}}} \\ {- {F_{0a}}} & {{\delta _{ab}} + {\partial _a}y{\partial _b}y + {F_{ab}}} \\ \end{array}} \right),} \\ {- \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}) = \det ({\delta _{ab}} + {\partial _a}y{\partial _b}y - {F_{0a}}{F_{0b}} + {F_{ab}}),\quad\,} \\ \end{array}$$
(318)
while the induced gamma matrices are exactly those of Eq. (292). Due to the electric and magnetic components of the gauge field, the bosonic kappa matrix has a quadratic term in
$${\Gamma _\kappa} = {1 \over {4!\sqrt {- \det ({\mathcal G} + {\mathcal F})}}}{\varepsilon ^{{\mu _0} \ldots {\mu _3}}}\left({{\gamma _{{\mu _0} \ldots {\mu _3}}}\,i{\tau _2} + 6{F_{{\mu _0}{\mu _1}}}{\gamma _{{\mu _3}{\mu _4}}}\,{\tau _1} + 3{F_{{\mu _0}{\mu _1}}}{F_{{\mu _2}{\mu _3}}}\,i{\tau _2}} \right)\,.$$
(319)
To correctly capture the supersymmetries preserved by such a physical system, we impose the projection conditions
$${\Gamma _{0123}}\,i{\tau _2}\epsilon = \epsilon ,$$
(320)
$${\Gamma _{0y}}(\cos \alpha \,{\tau _3} + \sin \alpha \,{\tau _1})\epsilon = \epsilon ,$$
(321)
on the constant Killing spinor ϵ, describing a D3-brane and a (p, q)-string bound state, respectively. Defining \({B^a} = {1 \over 2}{\varepsilon ^{abc}}{F_{bc}}\) as the magnetic field and inserting Eqs. (320) and (321) into the resulting kappa symmetry preserving condition, one obtains
$$\begin{array}{*{20}c} {\sqrt {- \det ({\mathcal G} + {\mathcal F})} \epsilon = \left[ {1 + {\Gamma ^a}{\Gamma _0}{\partial _a}y(\cos \alpha {\tau _3} + \sin \alpha {\tau _1}) - {\Gamma ^a}{\Gamma _0}{\tau _3}{F_{0a}}} \right.} \\ {+ {\Gamma ^{ab}}{F_{0a}}{\partial _b}y(\cos \alpha {\tau _3} + \sin \alpha {\tau _1}) - {\Gamma ^a}{\Gamma _0}{B_a}{\tau _1}} \\ {\left. {+ {B^a}{\partial _a}y(\cos \alpha {\tau _3} + \sin \alpha {\tau _1}) + {B^a}{F_{0a}}i{\tau _2}} \right].\quad} \\ \end{array}$$
(322)
This equation is trivially satisfied when the following BPS conditions hold
$${F_{0a}} = \cos \alpha \,{\partial _a}y,\qquad \qquad {B^a} = \sin \alpha \,{\delta ^{ab}}{\partial _b}y.$$
(323)
Hamiltonian analysis: Following [225], the Hamiltonian constraint can be solved and rewritten as a sum of positive definite terms34
$$\begin{array}{*{20}c} {{{\mathcal E}^2} = 1 + \vert \vec \nabla y{\vert ^2} + \vert \vec E{\vert ^2} + \vert \vec B{\vert ^2} + {{(\vec E\cdot\vec \nabla y)}^2} + {{(\vec B\cdot\vec \nabla y)}^2} + \vert \vec E \times \vec B{\vert ^2}\quad \quad \quad \quad \quad \,\,\,} \\ {= {{(1 + \sin \alpha \,\vec E\cdot\vec \nabla y + \cos \alpha \,\vec B\cdot\vec \nabla y)}^2} + \vert \vec E - \sin \alpha \,\vec \nabla y{\vert ^2} + \vert \vec B - \cos \alpha \,\vec \nabla y{\vert ^2}} \\ {+ \,\vert \cos \alpha \,\vec E\cdot\vec \nabla y - \sin \alpha \,\vec B\cdot\vec \nabla y{\vert ^2} + \vert \vec E \times \vec B{\vert ^2}\quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(324)
where the last equality holds for any angle α. This allows one to derive the bound
$${{\mathcal E}^2} \geq {(1 + \sin \alpha \,\vec E\cdot\vec \nabla y + \cos \alpha \,\vec B\cdot\vec \nabla y)^2}.$$
(325)
Thus, the total energy satisfies
$$E \geq {E_0} + \sin \alpha {Z_{{\rm{el}}}} + \cos \alpha {Z_{{\rm{mag}}}}\,,$$
(326)
with
$${Z_{{\rm{el}}}} = \int\nolimits_{{\rm{D3}}} {\vec E} \cdot\vec \nabla y,\qquad {Z_{{\rm{mag}}}} = \int\nolimits_{{\rm{D3}}} {\vec B} \cdot\vec \nabla y.$$
(327)
The bound (325) is extremised when
$$\tan \alpha = {Z_{{\rm{el}}}}/{Z_{{\rm{mag}}}},$$
(328)
for which the final energy bound reduces to
$$E \geq {E_0} + \sqrt {Z_{{\rm{el}}}^2 + Z_{{\rm{mag}}}^2} .$$
(329)
Here E0 corresponds to the energy of the vacuum configuration (infinite D3-brane). The bound (329) is saturated when
$$\vec E = \sin \alpha \,\vec \nabla y,\qquad \vec B = \cos \alpha \vec \nabla y.$$
(330)
These are precisely the conditions (323) derived from supersymmetry considerations, confirming their BPS nature. Using the divergence free nature of both \(\overrightarrow E\) and \({\vec B}\), y must be harmonic, i.e.,
$${\nabla ^2}y = 0.$$
(331)
The interpretation of the isolated point singularities in this harmonic function as the endpoints of (p, q) string carrying electric and magnetic charge is analogous to the BIon discussion.
In fact, all previous results can be understood in terms of the SL(2, ℤ) symmetry of type IIB string theory. In particular, a (1,0) string, or fundamental string, is mapped into a (p,q) string by an SO(2) transformation rotating the electric and magnetic fields. The latter is a non-local transformation in terms of the gauge field V, but leaves the energy density (324) invariant
$$\left({\begin{array}{*{20}c} {{{E{\prime a}}}} \\ {{{B{\prime a}}}} \\ \end{array}} \right) = \left({\begin{array}{*{20}c} {\cos \alpha} & {- \sin \alpha} \\ {\sin \alpha} & {\cos \alpha} \\ \end{array}} \right)\left({\begin{array}{*{20}c} {{E_a}} \\ {{B^a}} \\ \end{array}} \right).$$
(332)
Applying this transformation to the BIon solution, one reproduces Eq. (330).

5.6 Branes within branes

The existence of Wess-Zumino couplings of the form
$$\int\nolimits_{{{\rm{D}}_{{\rm{p + 4}}}}} {{{\mathcal C}_{p + 1}}} \wedge {\mathcal F} \wedge {\mathcal F}\,,\qquad \qquad \int\nolimits_{{{\rm{D}}_{{\rm{p + 2}}}}} {{{\mathcal C}_{p + 1}}} \wedge {\mathcal F}\,,$$
(333)
suggests that on-shell non-trivial magnetic flux configurations can source the electric components of the corresponding RR potentials. Thus, one may speculate with the existence of D (p + 4)-Dp and D (p + 2)-Dp bound states realised as on-shell solutions in the higher dimensional D-brane effective action. In this section, I will review the conditions the magnetic fluxes must satisfy to describe such supersymmetric bound states.

The analysis below should be viewed as a further application of the techniques described previously, and not as a proper derivation for the existence of such bound states in string theory. The latter can be a rather subtle quantum mechanical question, which typically involves non-abelian phenomena [496, 185]. For general discussions on D-brane bound states, see [447, 424, 425], on marginal D0-D0 bound states [445], on D0-D4 bound states [446, 486] while for D0-D6, see [470]. D0-D6 bound states in the presence of B-fields, which can be supersymmetric [391], were considered in [501]. There exist more general analysis for the existence of supersymmetric D-branes with non-trivial gauge fields in backgrounds with non-trivial NS-NS 2-forms in [372].

5.6.1 Dp−D(p + 4) systems

These are bound states at threshold corresponding to the brane array
$$\begin{array}{*{20}c} {{\rm{D}}(p + 4)\,:\,1.p\,.\,\,.\,\,.\,\,p + 4\_\,\_\quad \,} \\ {\quad {\rm{D}}p:\quad \,1\,.\,p\,\_\,\_\,\_\quad \_\quad \_\,\_\,.\quad} \\ \end{array}$$
(334)
Motivated by the Wess-Zumino coupling \({\mathcal C}\wedge {\mathcal F}\wedge {\mathcal F}\), one considers the ansatz on the D (p + 4)-brane effective action
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu},\quad \mu = 0, \ldots ,p + 4,\quad {X^i} = {c^i},\quad i = p + 5, \ldots ,9,} \\ {{V_a} = {V_a}({\sigma ^b}),\quad a,b = p + 1, \ldots ,p + 4.} \\ \end{array}$$
(335)
Let me first discuss when such configurations preserve supersymmetry. Consider type IIA (p = 2k), even though there is an analogous analysis for type IIB. Γκ reduces to
$$\begin{array}{*{20}c} {{\Gamma _\kappa} = {1 \over {\sqrt {- \det ({\eta _{\mu \nu}} + {F_{\mu \nu}})}}}{1 \over {(2k + 5)!}}{\varepsilon ^{{\mu _1} \ldots {\mu _{2k + 5}}}}\left({{\Gamma _{{\mu _1} \ldots {\mu _{2k + 5}}}}\Gamma _\sharp ^{k + 1}} \right.} \\ {+ \left({\begin{array}{*{20}c} {{\rm{2k + 5}}} \\ 2 \\ \end{array}} \right)\,{F_{{\mu _1}{\mu _2}}}{\Gamma _{{\mu _3} \ldots {\mu _{2k + 5}}}}\Gamma _\sharp ^k\quad \quad \quad \quad \quad \quad \quad} \\ {+ {1 \over 2}\left({\begin{array}{*{20}c} {{\rm{2k + 5}}} \\ 4 \\ \end{array}} \right)\, \,\left({\begin{array}{*{20}c} 4 \\ 2 \\ \end{array}} \right)\left. {{F_{{\mu _1}{\mu _2}}}{F_{{\mu _3}{\mu _4}}}{\Gamma _{{\mu _5} \ldots {\mu _{2k + 5}}}}\Gamma _\sharp ^{k + 1}} \right),\quad \,\,} \\ \end{array}$$
(336)
where I already used the static gauge and the absence of excited transverse scalars, so that γμ = Γμ. For the same reason, det(ημν + Fμν) = det(δab + Fab), involving a 4 × 4 determinant.
Given our experience with previous systems, it is convenient to impose the supersymmetry projection conditions on the constant Killing spinors that are appropriate for the system at hand. These are
$${\Gamma _{0 \ldots p + 4}}\Gamma _\sharp ^{k + 1}\epsilon = \epsilon ,$$
(337)
$${\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}\epsilon = \epsilon .$$
(338)
Notice that commutativity of both projectors is guaranteed due to the dimensionality of both constituents, which is what selects the Dp −D(p + 4) nature of the bound state in the first place. Inserting these into the kappa symmetry preserving condition, the latter reduces to
$$\sqrt {\det ({\delta _{ab}} + {F_{ab}})} \epsilon = \left({1 + {1 \over 4}{{\tilde F}^{ab}}{F_{ab}} - {1 \over 2}{\Gamma ^{\underline {ab}}}{\Gamma _\sharp}{F_{ab}}} \right)\epsilon ,$$
(339)
where \({{\tilde F}^{ab}} = {1 \over 2}{\varepsilon ^{abcd}}{F_{cd}}\). Requiring the last term in Eq. (339) to vanish is equivalent to the self-duality condition
$${\tilde F^{ab}} = {F^{ab}}\,.$$
(340)
When the latter holds, Eq. (339) is trivially satisfied. Eq. (340) is the famous instanton equation in four dimensions35. The Hamiltonian analysis done in [225] again confirms its BPS nature.

5.6.2 Dp −D(p + 2) systems

These are non-threshold bound states corresponding to the brane array
$$\begin{array}{*{20}c} {\,{\rm{D}}(p + 2):1.p\quad p + 1\quad p + 2\_\,\_\,\_\,\_} \\ {\quad \,\,{\rm{D}}p\,\,\,:\quad 1.p\quad \quad \_\quad \quad \_\,\,\,\,\,\_\,\_\,\_\,\_\,\,} \\ \end{array}$$
(341)
Motivated by the Wess-Zumino coupling \({\mathcal S}\wedge{\mathcal F}\), one considers the ansatz on the D (p + 4)-brane effective action
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu},\quad \mu = 0, \ldots ,p + 2,\quad {X^i} = {c^i},\quad i = p + 3, \ldots ,9,} \\ {{V_a} = {V_a}({\sigma ^b}){.}} \\ \end{array}$$
(342)
Since there is a single non-trivial magnetic component, I will denote it by Fab = F to ease the notation. The DBI determinant reduces to
$$- \det ({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}) = 1 + {F^2}\,,$$
(343)
whereas the kappa symmetry preserving condition in type IIA is
$$\sqrt {1 + {F^2}} \epsilon = \left({{\Gamma _{0 \ldots p + 2}}\Gamma _\sharp ^k + {\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}F} \right)\epsilon$$
(344)
for p = 2k. This is solved by the supersymmetry projection
$$\left({\cos \alpha {\Gamma _{0 \ldots p + 2}}\Gamma _\sharp ^k + \sin \alpha {\Gamma _{0 \ldots p}}\Gamma _\sharp ^{k + 1}} \right)\epsilon = \epsilon ,$$
(345)
for any α, for the magnetic flux satisfying
$$F = \tan \alpha .$$
(346)
To interpret the solution physically, assume the world space of the D (p + 2)-brane is of the form δp × T2. This will quantise the magnetic flux threading the 2-torus according to
$$\int\nolimits_{{T^2}} F = 2\pi k\quad \Rightarrow \quad F = {{{{(2\pi)}^2}k\alpha{\prime}} \over {{L_1}{L_2}}}.$$
(347)
To derive this expression, I used the fact that the 2-torus has area L1L2 and I rescaled the magnetic field according to F2πα′ F, since it is in the latter units that it appears in brane effective actions. Since the energy density satisfies \({{\mathcal E}^2} = T_{{\rm{D(p + 2)}}}^2(1 + {F^2})\), flux quantisation allows us to write the latter as
$${{\mathcal E}^2} = T_{{\rm{D}}(p + 2)}^2 + T_{{\rm{D}}p}^2{\left({{k \over {{L_1}{L_2}}}} \right)^2},$$
(348)
matching the non-threshold nature of the bound state
$$E = \sqrt {E_{{\rm{D}}(p + 2)}^2 + E_{k{\rm{D}}p}^2} \,,$$
(349)
where the last term stands for the energy of k Dp-branes.

5.6.3 F-Dp systems

These are non-threshold bound states corresponding to the brane array
$$\begin{array}{*{20}c} {{\rm{D}}p:\quad 1\,\,.\,\,.\,\,.\, p\_\,\_\,\_\,\_} \\ {\quad {\rm{F}}:\quad \_\,\_\,\_\,\_\,p\,\_\,\_\,\_\,\_\,.} \\ \end{array}$$
(350)
Following previous considerations, one looks for bosonic configurations with the ansatz
$$\begin{array}{*{20}c} {{X^\mu} = {\sigma ^\mu}\quad ,\quad {X^i} = {c^i},} \\ {{F_{0\rho}} = {F_{0\rho}}({\sigma ^a}).} \\ \end{array}$$
(351)
Given the absence of transverse scalar excitations, γμ = Γμ and \(\sqrt {- \det ({\mathcal G} + {\mathcal F})} = \sqrt {1 - {F^2}}\), where F0ρ = F. The kappa symmetry preserving condition reduces to
$$\sqrt {1 - {F^2}}\epsilon = \left({{\Gamma _{0 \ldots p}}\Gamma _\sharp ^k - F{\Gamma _{0\rho}}{\Gamma _\sharp}{\Gamma _{0 \ldots p}}\Gamma _\sharp ^k} \right)\;\epsilon + \left({1 - F{\Gamma _{0\rho}}{\Gamma _\sharp}} \right){\Gamma _{0 \ldots p}}\Gamma _\sharp ^k \epsilon\,.$$
(352)
This is solved by the supersymmetry projection condition
$$\left({\cos \alpha {\Gamma _{0 \ldots p}}\Gamma _\sharp ^k + \sin \alpha {\Gamma _{0\rho}}{\Gamma _\sharp}} \right)\;\epsilon = \epsilon \,,$$
(353)
whenever
$$F = - \sin \alpha \,.$$
(354)
To physically interpret the solution, compute its energy density
$${{\mathcal E}^2} = {\mathcal E}_0^2 + {F^2}\,,$$
(355)
where I already used that F0ρ = F = Eρ. These configurations are T-dual to a system of D0-branes moving on a compact space. In this T-dual picture, it is clear that the momentum along the compact direction is quantised in units of 1/L. Thus, the electric flux along the T-dual circle must also be quantised, leading to the condition
$$F = {1 \over {2\pi \alpha \prime}}{n \over L}\,,$$
(356)
where the world volume of the Dp-brane is assumed to be ℝp × S1. In this way, one can rewrite the energy for the F-Dp system as
$$E = \sqrt {E_{Dp}^2 + T_f^2{{\left({{n \over L}} \right)}^2}} \,,$$
(357)
which corresponds to the energy of a non-threshold bound state made of a Dp-brane and n fundamental strings (Tf).

5.7 Supertubes

All reviewed solitonic configurations carry charge under the p + 1-dimensional gauge potential they minimally couple to. In this section, I want to consider an example where this is not the case. This phenomena may occur when a collection of lower-dimensional branes finds it energetically favourable to expand into higher-dimensional ones. The stability of these is due to either an external force, typically provided by non-trivial fluxes in the background, or presence of angular momentum preventing the brane from collapse. A IIA superstring blown-up to a tubular D2-brane [200], a collection of D0-branes turning into a fuzzy 2-sphere [395] or wrapping D-branes with quantised non-trivial world volume gauge fields in AdSm × Sn [419] are examples of the first kind, whereas giant gravitons [386], to be reviewed in Section 5.9, are examples of the second.

Supertubes are tubular D2-branes of arbitrary cross-section in a Minkowski vacuum spacetime supported against collapse by the angular momentum generated by a non-trivial Poynting vector on the D2-brane world volume due to non-trivial electric and magnetic Born-Infeld (BI) fields. They were discovered in [381] and its arbitrary cross-section reported in [380], generalising some particular non-circular cross-sections discussed in [30, 32]. Their stability is definitely not due to an external force, since these states exist in Minkowski spacetime. Furthermore, supertubes can be supersymmetric, preserving 1/4 of the vacuum supersymmetry. At first, the presence of non-trivial angular momentum may appear to be in conflict with supersymmetry, since the latter requires a time-independent energy density. This point, and its connection with the expansion of lower-dimensional branes, will become clearer once I have reviewed the construction of these configurations.

Let me briefly review the arbitrary cross-section supertube from [380]. Consider a D2-brane with world volume coordinates σμ = {t, z, σ} in the type IIA Minkowski vacuum
$$ds_{10}^2 = - d{T^2} + d{Z^2} + d\vec Y\;\cdot\;d\vec Y\,,$$
(358)
where \(\vec Y = \{{Y^i}\}\) are Cartesian coordinates on ℝ8. We are interested in describing a tubular D2-brane of arbitrary cross-section extending along the Z direction. To do so, consider the set of bosonic configurations
$$\begin{array}{*{20}c} {T = t,\quad Z = z,\quad \vec Y = \vec y(\sigma),\quad \,} \\ {F = E\,dt \wedge dz + B(\sigma)\,dz \wedge d\sigma .} \\ \end{array}$$
(359)
The static gauge guarantees the tubular nature of the configuration, whereas the arbitrary embedding functions \(\vec Y = \vec y(\sigma)\) describe its cross-section. Notice the Poynting vector will not vanish, due to the choice of electric and magnetic components, i.e., the world volume electromagnetic field will indeed carry angular momentum.
To study the preservation of supersymmetry, one solves Eq. (215). Given the ansatz (359) and the flat background (358), this condition reduces to [380]
$${y_i}\prime \,{\Gamma _i}\,{\Gamma _\sharp}\,\left({{\Gamma _{TZ}}{\Gamma _\sharp} + E} \right)\;\epsilon + \left({B\,{\Gamma _T}{\Gamma _\sharp} - \sqrt {(1 - {E^2})\vert \vec y\prime {\vert ^2} + {B^2}}} \right)\;\epsilon = 0,$$
(360)
where the prime denotes differentiation with respect to σ. For generic curves, that is, without imposing extra constraints on the embedding functions \(\vec Y = \vec y(\sigma)\), supersymmetry requires both to set |E | = 1 and to impose the projection conditions
$${\Gamma _{TZ}}{\Gamma _\sharp}\,\epsilon = - {\rm{sgn}}(E)\epsilon\;,\qquad {\Gamma _T}{\Gamma _\sharp}\,\epsilon = {\rm{sgn}}(B)\,\epsilon$$
(361)
on the constant background Killing spinors ϵ. These conditions have solutions, preserving 1/4 of the vacuum supersymmetry, if B (σ) is a constant-sign, but otherwise completely arbitrary, function of σ. Notice the two projections 361 correspond to string charge along the Z-direction and to D0-brane charge, respectively.
In order to improve our understanding on the arbitrariness of the cross-section, it is instructive to compute the charges carried by supertubes and its energy momentum tensor, to confirm the absence of any pull (tension) along the different spacelike directions where the tube is embedded in 10 dimensions. First, the conjugate momentum Pi and the conjugate variable to the electric field, Π, are
$${P_i} = {{\partial {{\mathcal L}_{D2}}} \over {\partial {{\dot Y}^i}}} = {{BE{y_i}\prime} \over {\sqrt {(1 - {E^2})\vert \vec y\prime {\vert ^2} + {B^2}}}} = {\rm{sgn}}(\Pi B)\,{y_i}\prime \,,$$
(362)
$$\Pi (\sigma) = {{\partial {{\mathcal L}_{D2}}} \over {\partial E}} = {{E\vert \vec y\prime {\vert ^2}} \over {\sqrt {(1 - {E^2})\vert \vec y\prime {\vert ^2} + {B^2}}}} = {\rm{sgn}}(E){{\vert \vec y\prime {\vert ^2}} \over {\vert B\vert}}\,,$$
(363)
where in the last step the supersymmetry condition |E | = 1 was imposed. Notice supertubes satisfy the identity
$$\vert \vec P{\vert ^2} = \vert \Pi B\vert \,.$$
(364)
Second, the fundamental string q F1 and D0-brane qD 0 charges are
$${q_{F1}} = \int d \sigma \,\Pi \,,\quad {q_{D0}} = \int d \sigma \,B\,.$$
(365)
Finally, the supertube energy-momentum tensor [380]
$${T^{mn}}(x) = {\left. {{2 \over {\sqrt {- \det g}}}\,{{\delta {S_{{\rm{D}}2}}} \over {\delta {g_{mn}}(x)}}} \right\vert _{{g_{m\,n}} = {\eta _{m\,n}}}} = - \sqrt {- \det ({\mathcal G} + F)} \;\,{\left[ {{{({\mathcal G} + F)}^{- 1}}} \right]^{(\mu \nu)}}{\partial _\mu}{X^m}{\partial _\nu}{X^n},$$
(366)
with Xm = {T, Z, Yi}, has only non-zero components
$${{\mathcal T}^{TT}} = \;\vert \Pi \vert + \vert B\vert \,,\quad {{\mathcal T}^{ZZ}} = - \vert \Pi \vert \,,\quad {{\mathcal T}^{Ti}} = {\rm{sgn}}(\Pi B)\,{y_i}\prime .$$
(367)
Some comments are in order:
  1. 1.

    As expected, the linear momentum density (362) carried by the tube is responsible for the off-diagonal components \({\tau ^{Ti}}\).

     
  2. 2.

    The absence of non-trivial components \({\tau ^{ij}}\) confirms the absence of tension along the cross-section, providing a more technical explanation of why an arbitrary shape is stable.

     
  3. 3.

    The tube tension \(- {\tau ^{zz}} = \vert \Pi \vert\) in the Z-direction is only due to the string density, since D0-branes behave like dust.

     
  4. 4.

    The expanded D2-brane does not contribute to the tension in any direction.

     
Integrating the energy momentum tensor along the cross-section, one obtains the net energy of the supertube per unit length in the Z-direction
$${\mathcal E} = \int d \sigma \,{{\mathcal T}^{TT}} = \;\vert {q_{F1}}\vert + \vert {q_{D0}}\vert \,,$$
(368)
matching the expected energy bound from supersymmetry considerations.

Let me make sure the notion of supersymmetry is properly tied with the expansion mechanism. Supertubes involve a uniform electric field along the tube and some magnetic flux. Using the language and intuition of previous Sections 5.6.25.6.3, the former can be interpreted as “dissolved” IIA superstrings and the latter as “dissolved” D0-branes, that have expanded into a tubular D2-brane. Their charges are the ones appearing in the supersymmetry algebra allowing the energy to be minimised. Notice the expanded D2-brane couples locally to the RR gauge potential C3 under which the string and D0-brane constituents are neutral. This is precisely the point made at the beginning of the section: supertubes do not carry D2-brane charge.36 When the number of constituents is large, one may expect an effective description in terms of the higher-dimensional D2-brane in which the original physical charges become fluxes of various types.

The energy bound (368) suggests supertubes are marginal bound states of D0s and fundamental strings (Fs). This was further confirmed by studying the spectrum of BPS excitations around the circular shape supertube by quantising the linearised perturbations of the DBI action [123, 29]. The quantisation of the space of configurations with fixed angular momentum J [123, 29] allowed one to compute the entropy associated with states carrying these charges
$$S = 2\pi \sqrt {2({q_{D0}}{q_{F1}} - J)} \,.$$
(369)
This entropy reproduces the microscopic conjecture made in [364] where the Bekenstein-Hawking entropy was computed using a stretched horizon. These considerations do support the idea that supertubes are typical D0-F bound states.

Supergravity description and fuzzball considerations: The fact that world volume quantisation reproduces the entropy of a macroscopic configuration and the presence of arbitrary profiles, at the classical level, suggests that supersymmetric supertubes may provide a window to understand the origin of gravitational entropy in a regime of parameters where gravity is reliable. This is precisely one of the goals of the fuzzball programme [363, 361].37

A first step towards this connection was provided by the supergravity realisation of supertubes given in [205]. These are smooth configurations described in terms of harmonic functions whose sources allow arbitrary profiles, thus matching the arbitrary cross-section feature in the world volume description [380].

The notion of supertube is more general than the one described above. Different encarnations of the same stabilising mechanism provide U-dual descriptions of the famous string theory D1-D5 system. To make this connection more apparent, consider supertubes with arbitrary cross-sections in ℝ4 and with an S1 tubular direction, allowing the remaining 4-spacelike directions to be a 4-torus. These supertubes are U-dual to D1-D5 bound states with angular momentum J [361], or to winding undulating strings [362] obtained from the original work [129, 158]. It was pointed out in [361] that in the D1-D5 frame, the actual supertubes correspond to KK monopoles wrapping the 4-torus, the circle also shared by D1 and D5-branes and the arbitrary profile in ℝ438. Smoothness of these solutions is then due to the KK monopole smoothness.

Since the U-dual D1-D5 description involves an AdS3 × S3 near horizon, supertubes were interpreted in the dual CFT: the maximal angular momentum configuration corresponding to the circular profile is global AdS3, whereas non-circular profile configurations are chiral excitations above this vacuum [361].

Interestingly, geometric quantisation of the classical moduli space of these D1−D5 smooth configurations was carried in [435], using the covariant methods originally developed in [156, 503]. The Hilbert space so obtained produced a degeneracy of states that was compatible with the entropy of the extremal black hole in the limit of large charges, i.e., \(S = 2\pi \sqrt {2({q_{D0}}{q_{F1}})}\). Further work on the quantisation of supergravity configurations in AdS3 × S3 and its relation to chiral bosons can be found in [183]. The conceptual framework described above corresponds to a particular case of the one illustrated in Figure 7.
Figure 7

Relation between the quantisation of the classical moduli space of certain supersymmetric probe configurations, their supergravity realisations and their possible interpretation as black hole constituents.

5.8 Baryon vertex

As a first example of a supersymmetric soliton in a non-trivial background, I will review the baryon vertex [500, 265]. Technically, this will provide an example of how to deal with non-constant Killing spinors. Conceptually, it is a nice use of the tools explained in this review having an interesting AdS/CFT interpretation.

Let me first try to conceptually motivate the entire set-up. Consider a closed D5-brane surrounding N D3-branes, i.e., such that the D3-branes thread the D5-brane. The Hanany-Witten (HW) effect [282] allows us to argue that each of these N D3-branes will be connected to the D5-brane by a fundamental type IIB string. Consequently, the lowest energy configuration should not allow the D5-brane to contract to a single point, but should describe these N D3-branes with N strings attached to them allowing one to connect the D3 and D5-branes. In the large N limit, one can replace the D3-branes by their supergravity backreaction description. The latter has an AdS5 × S5 near horizon. One can think of the D5-brane as wrapping the 5-sphere and the N strings emanating from it can be pictured as having their endpoints on the AdS5 boundary. This is the original configuration interpreted in [500, 265] as a baryon-vertex of the \({\mathcal N} = 4 \, d = 4\) super-Yang-Mills (SYM) theory.

At a technical level and based on our previous discussions regarding BIons, one can describe the baryon vertex as a single D5-brane carrying N units of world volume electric charge [315, 125] to account for the N type IIB strings. If one assumes all the electric charge is concentrated at one point, then one expects the minimum energy configuration to preserve the SO(5) rotational invariance around it. Such configuration will be characterised by the radial position of the D5-brane in AdS5 as a function r (θ) of the co-latitude angle θ on S5. This is the configuration studied in [315, 125, 152]. Since it is, a priori, not obvious whether the requirement of minimal energy forces the configuration to be SO(5) invariant, one can relax this condition and look for configurations where the charge is distributed through different points. One can study whether these configurations preserve supersymmetry and saturate some energy bound. This is the approach followed in [248], where the term baryonic branes was coined for all these kinds of configurations, and the one I will follow below.

Set-up: One is interested in solving the equations of motion of a single D5-brane in the background of N D3-branes carrying some units of electric charge to describe type IIB strings. The background is described by a constant dilaton, a non-trivial metric and self-dual 5-form field strength R5 [195]
$$ds_{{\rm{10}}}^2 = {U^{- 1/2}}\,d{s^2}({{\mathbb E}^{(1,3)}}) + {U^{1/2}}\left[ {d{r^2} + {r^2}d\Omega _5^2} \right]$$
(370)
$${R_5} = 4{R^4}\left[ {{\omega _5} + \star {\omega _5}} \right]$$
(371)
where \(d\Omega _5^2\) is the SO(6)-invariant metric on the unit 5-sphere, ω5 is its volume 5-form and *ω5 its Hodge dual. The function U is
$$U = a + {\left({{{{L_4}} \over r}} \right)^4}\qquad \left({L_4^4 = 4\pi {g_s}N{{(\alpha \prime)}^2}} \right)\,.$$
(372)
Notice a =1 corresponds to the full D3-brane background solution, whereas a = 0 to its near-horizon limit.
Consider a probe D5-brane of unit tension wrapping the 5-sphere. Let ξμ = (t, θi) be the world volume coordinates, so that θi (i = 1,…,5) are coordinates for the worldspace 5-sphere. This will be achieved by the static gauge
$${X^0} = t\,,\quad {\Theta ^i} = {\theta ^i}\,.$$
(373)
Since one is only interested in radial deformations of the world space carrying electric charge, one considers the ansatz
$${X^1} = {X^2} = {X^3} = 0\,,\qquad \qquad r = r({\theta ^i})\,,\qquad \quad F = {1 \over 2}{F_{0i}}({\theta ^i})\,dt \wedge d{\theta ^i}\,.$$
(374)
Even though the geometry will be curved, it can give some intuition to think of this system in terms of the array
$$\begin{array}{*{20}c} {{\rm{D}}3:\,1\;2\;3\,\_\,\_\,\_\,\_\,\_\,\_\quad {\rm{background}}} \\ {{\rm{D}}5:\,\_\,\_\,\_\,4\;5\;6\;7\;8\,\_\quad {\rm{probe}}\quad \quad \,} \\ {{\rm{F}}1:\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,9\quad {\rm{soliton}}\quad \quad} \\ \end{array}$$
(375)
viewing the 9-direction as the radial one.
Supersymmetry analysis: Given the electric nature of the world volume gauge field, the kappa symmetry matrix reduces to
$${\Gamma _\kappa} = {1 \over {6!}}{1 \over {\sqrt {- \det ({\mathcal G} + F)}}}{\varepsilon ^{{\mu _1} \ldots {\mu _6}}}\;\,\left[ {{\gamma _{{\mu _1} \ldots {\mu _6}}}{\tau _1} + 15\;{F_{{\mu _1}{\mu _2}}}{\gamma _{{\mu _3} \ldots {\mu _6}}}(i{\tau _2})} \right]\,.$$
(376)
Given the ansatz (374) and the background (370), the induced world volume metric equals
$${{\mathcal G}_{\mu \nu}} = \left({\begin{array}{*{20}c} {- \,{U^{- 1/2}}} & 0 \\ 0 & {{g_{ij}}} \\ \end{array}} \right)$$
(377)
where
$${g_{ij}} = {U^{1/2}}\left({{r^2}{{\bar g}_{ij}} + {\partial _i}r{\partial _j}r} \right)\;,$$
(378)
and \({{\bar g}_{ij}}\) stands for the SO(6)-invariant metric on the unit 5-sphere. Taking into account the nontrivial vielbeins, the induced gamma matrices equal
$${\gamma _0} = {U^{- \,1/4}}{\Gamma _0}\,,\qquad \qquad {\gamma _i} = {U^{1/4}}\,r{\hat \gamma _i} + {U^{1/4}}\,{\partial _i}r{\Gamma _r}\,,$$
(379)
where the matrices \({{\hat \gamma}_i}\) are defined as
$${\hat \gamma _i} = {e_i}^a{\Gamma _a}\,,$$
(380)
in terms of the fönfbein \({e_i}^a\) in the 5-sphere. Thus, \(\{{{\hat \gamma}_i},{{\hat \gamma}_j}\} = 2{{\bar g}_{ij}}\).
To solve the kappa symmetry preserving condition (215), one requires the background Killing spinors. These are of the form
$$\epsilon = {U^{- {1 \over 8}}}\chi ,$$
(381)
where \(\chi\) is a covariantly constant spinor on \({{\rm{{\mathbb E}}}^{(1,3)}} \times {{\rm{{\mathbb E}}}^6}\) subject to the projection condition
$${\Gamma _{0123}}\,i\,{\tau _2}\chi = \chi \,,$$
(382)
describing the D3-branes in the background. Importantly, χ is not constant when using polar coordinates in \({{\rm{{\mathbb E}}}^6}\). Indeed, covariantly constant spinors on Sn were constructed explicitly in [359] for a sphere parameterisation obtained by iteration of \(ds_n^2 = d\theta _n^2 + {\sin ^2}{\theta _n}ds_{n - 1}^2\). The result can be written in terms of the n angles \({\theta ^i} = (\theta, {\theta ^i})\) and the antisymmetrised products of pairs of the constant d =10 Clifford matrices \({\Gamma _a} = ({\Gamma _\theta},{\Gamma _{\hat i}})\). For n = 5, defining \({\Gamma _{\hat 5}} \equiv {\Gamma _\theta}\), these equal
$$\chi \, = {e^{{\theta \over 2}\,{\Gamma _{r\theta}}}}\,\,\prod\limits_{\hat \jmath = 1}^4 {{e^{- {{{}_\theta \hat \jmath} \over 2}\;{\Gamma _{\hat \jmath \hat \jmath + 1}}}}\,\,{\epsilon_0}} ,$$
(383)
where ϵ0 satisfies Eq. (382). Even though there are additional Killing spinors in the near-horizon limit, the associated extra supersymmetries will be broken by the baryonic D5-brane probe configuration I am about to construct, so these can be ignored.
Plugging the ansatz into the kappa matrix (376), the supersymmetry preserving condition (215) reduces, after some algebra, to
$$\begin{array}{*{20}c} {U\,\sqrt {\det \left[ {{r^2}{{\bar g}_{ij}} + {\partial _i}r{\partial _j}r - {F_{0i}}{F_{0j}}} \right]} \, \epsilon=} \\ {\left[ {U{r^5}\sqrt {\det \bar g} \,{\Gamma _0}{\gamma _{\ast}}{\tau _1} - U{r^3}\sqrt {\det \bar g} \,{F_{0j}}{\partial _i}r{{\hat \gamma}^{ij}}{\gamma _{\ast}}{\Gamma _r}(i{\tau _2})} \right.} \\ {\left. {+ \,U{r^4}\sqrt {\det \bar g} \,{{\hat \gamma}^i}{\gamma _{\ast}}\left({{F_{0i}}(i{\sigma _2}) + {\partial _i}r{\Gamma _{0r}}{\tau _1}} \right)} \right]\;\epsilon} \\ \end{array}$$
(384)
where \({{\hat \gamma}^i} = {{\bar g}^{ij}}{{\hat \gamma}_j}\) and γ * = Γ45678.
Given the physical interpretation of the sought solitons, one imposes two supersymmetry projections on the constant Killing spinors ϵ0:
$${\Gamma _0}{\gamma _{\ast}}{\tau _1}{\epsilon_0} = {\epsilon_0}\,,$$
(385)
$${\Gamma _{0r}}\,{\tau _3}{\epsilon_0} = {\epsilon_0}\,.$$
(386)
These are expected from the local preservation of 1/2 supersymmetry by the D5-brane and the IIB string in the radial direction, respectively. These projections imply
$$\begin{array}{*{20}c} {{\Gamma _{0r}}\,{\tau _3}\,\epsilon \, = \left[ {\cos \theta - \sin \theta \,{\Gamma _{r\theta}}} \right]\;\epsilon ,} \\ {{\Gamma _0}\,{\gamma _{\ast}}\,{\tau _1}\,\epsilon \, = \left[ {\cos \theta - \sin \theta \,{\Gamma _{r\theta}}} \right]\;\epsilon ,\;\;\,} \\ {{\Gamma _i}\,{\gamma _{\ast}}\,i{\tau _2}\,\epsilon = - {\Gamma _{ri}}\,\epsilon ,\quad \quad \quad \quad \quad \quad \,} \\ {{\Gamma _i}\,{\gamma _{\ast}}\,{\Gamma _{0r}}\,{\tau _1}\,\epsilon = {\Gamma _{ri}}\,{e^{- \theta {\Gamma _{r\theta}}}}\,\epsilon ,\quad \quad \quad \quad \quad \;\;\,} \\ {{\gamma _{\ast}}\,{\Gamma _r}\,\,i{\tau _2}\,\epsilon = - \epsilon \,.\quad \quad \quad \quad \quad \quad \quad \;\;} \\ \end{array}$$
(387)
Using these relations, one can rewrite the right-hand side of Eq. (384) as
$$\begin{array}{*{20}c} {{\Delta _5}\,\left[ {\left({r\sin \theta} \right)\prime + {\Gamma _{r\theta}}\left({(r\cos \theta)\prime - {F_{0\theta}}} \right) + {\Gamma _r}{{\hat \gamma}^{\hat \imath}}\left({{\partial _{\hat \imath}}r\cos \theta - {F_{0\hat \imath}}} \right)} \right.} \\ {\left. {+ {{\hat \gamma}^{\hat \imath \hat \jmath}}\,{1 \over r}\left({{\partial _{\hat \imath}}r{F_{0\hat \jmath}} - {\partial _{\hat \jmath}}r{F_{0\hat \imath}}} \right) + {{\hat \gamma}^{\hat \imath}}{\Gamma _\theta}\,{1 \over r}\left({{\partial _{\hat \imath}}r{F_{0\theta}} - r\prime {F_{0\hat \imath}} + r{\partial _{\hat \imath}}r\sin \theta} \right)} \right]\,,} \\ \end{array}$$
(388)
where \({\Delta _5} = U{r^4}\sqrt {\det \bar g}\). The coefficients of Γ and \({\Gamma _r}{{\hat \gamma}^i}\) in Eq. (388) vanish when
$${F_{0i}} = {\partial _i}\left({r\,\cos \theta} \right).$$
(389)
Furthermore, the ones of \({{\hat \gamma}^{\hat i\hat j}}\) and \({{\hat \gamma}^{\hat i}}{\Gamma _\theta}\) also do. I will eventually interpret Eq. (389) as the BPS equation for a world volume BIon. One concludes that Eq. (384) is satisfied as a consequence of Eq. (389) provided that
$$U\,\sqrt {\det \left[ {{r^2}{{\bar g}_{ij}} + {\partial _i}r{\partial _j}r - {F_{0i}}{F_{0j}}} \right]} = {\Delta _5}(r\sin \theta)\prime .$$
(390)
It can be checked that this is indeed the case whenever Eq. (389) holds.
Hamiltonian analysis: Solving the Hamiltonian constraint \({\mathcal H} = 0\) in Eq. (225) allows to write the Hamiltonian density for static configurations as [248]
$${{\mathcal H}^2} = {U^{- {1 \over 2}}}\left[ {{{\tilde E}^i}{{\tilde E}^j}{g_{ij}} + \det g} \right],$$
(391)
where \({{\tilde E}^i}\) is a covariantised electric field density related to F0i by
$$(\det g)\,{F_{0i}} = \sqrt {- \det ({\mathcal G} + F)} \,{\tilde E^j}{g_{ij}}.$$
(392)
for the ansatz (374), this reduces to
$${\tilde E^i}\, = \,{U^{1/4}}\,\,{{\sqrt {\det g}} \over {\sqrt {1\, - \,{U^{1/2}}\,{g^{mn}}\,{F_{0m}}\,{F_{0n}}}}}\,\,{g^{ij}}\,{F_{0j}}.$$
(393)
It was shown in [248] that one can rewrite the energy density (391) as
$${{\mathcal H}^2} = {\mathcal Z}_5^2 + {\left[ {{\Delta _5}\left({r\cos \theta} \right)\prime - {{\tilde E}^i}{\partial _i}\left({r\sin \theta} \right)} \right]^2} + \vert {\Delta _5}{\bar g^{\hat \imath \hat \jmath}}{\partial _{\hat \jmath}}r - r\,{\tilde E^{\hat \imath}}{\vert ^2},$$
(394)
where ||2 indicates contraction with \({g^{\hat i\hat j}}\) and
$${{\mathcal Z}_5} = {\Delta _5}\left({r\sin \theta} \right)\prime + {\tilde E^i}{\partial _i}\left({r\cos \theta} \right).$$
(395)
To achieve this, the 5-sphere metric was written as
$$d{s^2} = d{\theta ^2} + {\sin ^2}\theta \,d\Omega _4^2,$$
(396)
where \(d\Omega _4^2\) is the SO(5) invariant metric on the 4-sphere, which one takes to have coordinates \({\theta ^{\hat i}}\). In this way, all primes above refer to derivatives with respect to θ and \({{\bar g}^{\hat i\hat j}}\) are the \(\hat i\hat j\) components of the inverse S5 metric \({{\bar g}^{ij}}\).
Using the Gauss’ law constraint
$${\partial _i}{\tilde E^i} = - 4\,{R^4}\sqrt {\det \bar g} \,,$$
(397)
which has a non-trivial source term due to the RR 5-form flux background, one can show that \({-\!\!\!\!\! Z_5} = {\partial _i}-\!\!\!\!\! Z_5^i\) where \({{\vec -\!\!\!\!\! Z}_5}\) has components
$$\begin{array}{*{20}c} {{\mathcal Z}_5^\theta = {{\tilde E}^\theta}\,r\cos \theta + \sqrt {\det \bar g} \,\sin \theta \;\left({a\,{{{r^5}} \over 5} + r\,{R^4}} \right)\;,} \\ {{\mathcal Z}_5^{\hat \imath} = \,{{\tilde E}^{\hat \imath}}\,r\cos \theta \;.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ \end{array}$$
(398)
From Eq. (394), and the divergent nature of Z5, one deduces the bound
$${\mathcal H} \geq \;\vert {{\mathcal Z}_5}\vert .$$
(399)
The latter is saturated when
$${\tilde E^{\hat \imath}} = {\Delta _5}{{{{\bar g}^{\hat \imath \hat \jmath}}{\partial _{\hat \jmath}}r} \over r},$$
(400)
$${\tilde E^\theta} = {{{\Delta _5}} \over {\left({r\sin \theta} \right)\prime}}\left({\left({r\cos \theta} \right)\prime - {{{{\bar g}^{\hat \imath \hat \jmath}}{\partial _{\hat \imath}}r{\partial _{\hat \jmath}}\left({r\sin \theta} \right)} \over r}} \right)\,\;.$$
(401)
Combining Eqs. (400) and (401) with the Gauss law (397) yields the equation
$${\partial _{\hat \imath}}\left({{\Delta _5}{{\bar g}^{\hat i\hat j}}{{{\partial _{\hat \jmath}}r} \over r}} \right) + {\partial _\theta}\left[ {{{{\Delta _5}} \over {(r\sin \theta)\prime}}\;\left({(r\cos \theta)\prime - {{\bar g}^{\hat i\hat j}}{\partial _{\hat \imath}}r{\partial _{\hat \jmath}}r{{\sin \theta} \over r}} \right)} \right] = - 4{R^4}\sqrt {\det \bar g} .$$
(402)
Any solution to this equation gives rise to a 1/4 supersymmetric baryonic brane.
For a discussion of the first-order equations (400) and (401) for a = 1, see [126, 133]. Here, I will focus on the near horizon geometry corresponding to a=0. The Hamiltonian density bound (399) allows us to establish an analogous one for the total energy E
$$E \geq \int {{d^5}} \sigma \vert {{\mathcal Z}_5}\vert \; \geq \left\vert {\int {{d^5}} \sigma {{\mathcal Z}_5}\,} \right\vert .$$
(403)
while the first inequality is saturated under the same conditions as above, the second requires Z5 to not change sign within the integration region. For this configuration to describe a baryonic brane, one must identify this region with a 5-sphere having some number of singular points removed. Assuming the second inequality is saturated when the first one is, the total energy equals
$$E = \underset {\delta \rightarrow 0}{\lim} \sum\limits_k {\int\nolimits_{{B_k}} d} \vec S\;\cdot\;\overrightarrow {\mathcal Z} ,$$
(404)
where Bk. is a 4-ball of radius δ having the k’th singular point as its center. This expression suggests that one interpret the k’ th term in the sum as the energy of the IIB string(s) attached to the k’th singular point. No explicit solutions to Eq. (402) with these boundary conditions are known though.
Consider SO(5) invariant configurations (for a discussion of less symmetric configurations, see [248]). In this case \({{\tilde E}^i} = 0\),
$${\tilde E^\theta} = \sqrt {\det {g^{(4)}}} \,\tilde E(\theta)\,,$$
(405)
and r = r (θ). The BPS condition (401) reduces to [315, 125, 152]
$${{r\prime} \over r} = {{\Delta \sin \theta + \tilde E\,\cos \theta} \over {\Delta \cos \theta - \tilde E\,\sin \theta}}\,,$$
(406)
where Δ = R4 sin4 θ, while the Gauss’ law (397) equals
$$\tilde E\prime = - 4\,{R^4}\,{\sin ^4}\theta .$$
(407)
Its solution was first found in [125]
$$\tilde E = {1 \over 2}{R^4}\left[ {3\left({\nu \pi - \theta} \right) + 3\sin \theta \cos \theta + 2{{\sin}^3}\theta \cos \theta} \right],$$
(408)
where ν is an integration constant restricted to lie in the interval [0,1].
Given this explicit solution, let me analyse whether the second inequality in Eq. (403) is saturated when the first one is, as I assumed before. Notice
$${{\mathcal Z}_5} = \sqrt {\det {g^{(4)}}} \,{\mathcal Z}(\theta)\quad {\rm{with}}\quad {\mathcal Z}(\theta) = r\,{{{{\left({\Delta \cos \theta - \tilde E\sin \theta} \right)}^2} + {{\left({\Delta \sin \theta + \tilde E\cos \theta} \right)}^2}} \over {\left({\Delta \cos \theta - \tilde E\sin \theta} \right)}},$$
(409)
where I used Eq. (406). The sign of \(-\!\!\!\!\! Z\) is determined by the sign of the denominator. Thus, it will not change if it has no singularities within the region θ ∈ [0, π ] (except, possibly, at the endpoints θ = 0,π). Since
$$\Delta \cos \theta - \tilde E\sin \theta = {3 \over 2}{R^4}\sin \theta \,\eta (\theta)\quad {\rm{with}}\quad \eta (\theta) \equiv \theta - \nu \pi - \sin \theta \cos \theta ,$$
(410)
one concludes that the denominator for \(-\!\!\!\!\! Z\) vanishes at the endpoints θ = 0, π but is otherwise positive provided η (θ) is. This condition is only satisfied for ν = 0, in which case Eq. (408) becomes
$$\tilde E = {1 \over 2}{R^4}\left[ {3\left({\sin \theta \cos \theta - \theta} \right) + 2{{\sin}^3}\theta \cos \theta} \right]\;\,.$$
(411)
Integrating the differential equation (406) for r (θ) after substituting Eq. (411), one finds [125]
$$r = {r_0}{\left({{6 \over 5}} \right)^{\;{1 \over 3}}}({\rm{cosec}}\,\theta){(\theta - \sin \theta \cos \theta)^{{1 \over 3}}}\;,$$
(412)
where r0 is the value of r at θ = 0. It was shown in [125] that this configuration corresponds to N fundamental strings attached to the D5-brane at the point θ = π, where r (θ) diverges.

Solutions to Eq. (406) for ν ≠ 0 were also obtained in [125]. The range of the angular variable θ for which these solutions make physical sense is smaller than [0, π ] because the D5-brane does not completely wrap the 5-sphere. Consequently, the D5 probe captures only part of the five form flux. This suggests that one interpret these spike configurations as corresponding to a number of strings less than N. In fact, it was argued in [109, 314] that baryonic multiquark states with k < N quarks in \({\mathcal N} = 4\) d = 4 SYM correspond to k strings connecting the D5-brane to r = ∞ while the remaining N — k strings connect it to r = 0. Since the ν = 0 D5-brane solutions do reach r = 0, it is tempting to speculate on whether they correspond to these baryonic multi-quark states.

Related work: There exists similar work in the literature. Besides the study of non-SO(5) invariant baryonic branes in AdS5 × S5, [248] also carried the analysis for baryonic branes in M-theory. Similar BPS bounds were found for D4-branes in D4-brane backgrounds or more generically, for D-branes in a D-brane background [126, 133] and D3-branes in (p,q)5-branes [452, 357]. Baryon vertex configurations have also been studied in AdS5 × T1,1 [19], AdS5 × Yp,q [134] and were extended to include the presence of magnetic flux [319]. For a more general analysis of supersymmetric D-brane probes either in AdS or its pp-wave limit, see [458].

5.9 Giant gravitons and superstars

It was mentioned in Section 5.7 that angular momentum can stabilise an expanded brane carrying the same quantum numbers as a lower dimensional brane. I will now review an example of such phenomena, involving supersymmetric expanding branes in AdS, the so called giant gravitons [386]. In this case, a rotating pointlike graviton in AdS expands into a rotating brane due to the RR flux supporting the AdS supergravity solution [395]. Its angular momentum prevents the collapse of the expanding brane and it can actually make it supersymmetric [264, 290].

Consider type IIB string theory in AdS5 × S5. It is well known that this theory has BPS graviton excitations rotating on the sphere at the speed of light. In the dual \({\mathcal N} = 4\) d =4 SYM theory, these states correspond to single trace operators belonging to the chiral ring [18, 150, 68]. When their momentum becomes of order N, it is energetically favourable for these gravitons to expand into rotating spherical D3-branes, i.e., giant gravitons. The N scaling is easy to argue for: the conformal dimension must be proportional to the D3-brane tension times the volume of the wrapped cycle, which is controlled by the AdS radius of curvature L4, thus giving
$$\Delta \propto {T_{{\rm{D}}3}}L_4^4 = N\;.$$
(413)
Similar considerations apply in different AdSp+1 realisations of this phenomena [264, 368]. The field theory interpretation of these states was given in [35] in terms of subdeterminant operators.
Let us construct these configurations in AdS5 × S5. The bosonic background has a constant dilaton and non-trivial metric and RR 4-form potential given by
$$\begin{array}{*{20}c} {ds_{{\rm{10}}}^2 = - \left({1 + {{{r^2}} \over {L_4^2}}} \right)\;d{t^2} + {{d{r^2}} \over {1 + {{{r^2}} \over {L_4^2}}}} + {r^2}\,d\tilde \Omega _3^2 + L_4^2\,\left({d{\theta ^2} + {{\cos}^2}\theta \,d{\phi ^2} + {{\sin}^2}\theta \,d\Omega _3^2} \right),} \\ {{C_4} = L_4^4\,{{\sin}^4}\theta \,d\phi \wedge {\omega _3},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;} \\ \end{array}$$
(414)
where ω3 stands for the volume form of the 3-sphere in S5 and it is understood dC4 is made self-dual to satisfy the type IIB equations of motion39. Giant gravitons consist of D3-branes wrapping such 3-spheres and rotating in the ϕ direction to carry R-charge from the dual CFT perspective. Thus, one considers the bosonic ansatz
$$\begin{array}{*{20}c} {{\sigma ^0} = t,\quad {\sigma ^i} = {\omega ^i},\quad \quad \quad \quad \;\;} \\ {\theta = {\theta _0},\quad \phi = \phi (\tau),\quad r = 0.} \\ \end{array}$$
(415)
The D3-brane Lagrangian density evaluated on this ansatz and integrating over the 3-sphere world volume is [264]
$${\mathcal L} = {N \over {{L_4}}}\left[ {- {{\sin}^3}\theta \,\sqrt {1 - L_4^2{{\cos}^2}\theta \,{{\dot \phi}^2}} + {L_4}{{\sin}^4}\theta \,\dot \phi} \right]\;\,.$$
(416)
Since k = ϕ is a Killing vector, the conjugate momentum Pϕ is conserved
$${P_\phi} = N\;\left[ {{{{L_4}{{\sin}^3}\theta {{\cos}^2}\theta \,\dot \phi} \over {\sqrt {1 - L_4^2{{\cos}^2}\theta \,{{\dot \phi}^2}}}} + {{\sin}^4}\theta} \right] \equiv N\,p\,,$$
(417)
where the constant p was defined. Computing the Hamiltonian density,
$${\mathcal E} = {P_\phi}\dot \phi - {\mathcal L} = {N \over {{L_4}}}\,\sqrt {{p^2} + {{\tan}^2}\theta \,{{(p - {{\sin}^2}\theta)}^2}} \,,$$
(418)
allows us to identify the stable configurations by extremising Eq. (418). Focusing on finite size configurations, one finds
$$\sin {\theta _0} = \sqrt p \quad \Rightarrow \quad \dot \phi = {1 \over {{L_4}}}\quad \Rightarrow \quad {\mathcal E} = {{{P_\phi}} \over {{L_4}}}\,.$$
(419)
Notice the latter equality saturates the BPS bound, \(\Delta \equiv {\mathcal E}{L_4} = {P_\phi}\), as expected from supersymmetry considerations.
To check whether the above configuration indeed preserves some supersymmetry, one must check whether there exists a subset of target space Killing spinors solving the kappa symmetry preserving condition (215). The 32 Killing spinors for the maximally-supersymmetric AdS5 × S5 background were computed in [359, 264]. They are of the form ϵ = M ϵ where M is a non-trivial Clifford valued matrix depending on the bulk point and ϵ is an arbitrary constant spinor. It was shown in [264] that Eq. (215) reduces to
$$({\Gamma _{t\phi}} - 1){\epsilon_\infty} = 0.$$
(420)
Thus, giant gravitons preserve half of the spacetime supersymmetry. Furthermore, they preserve the same supercharg es as a pointlike graviton rotating in the ϕ direction.
General supersymmetric giant graviton construction: There exist more general giant gravitons charged under the full U(1)3 Cartan subalgebra of the full R-symmetry group SO(6). The general construction of such supersymmetric probes was done in [392]. The main idea is to embed the bulk 5-sphere into an auxiliary embedding C3 space with complex coordinates zi i = 1, 2, 3 and AdS5 into ℂ1,2. In the probe calculation, the Zi become dynamical scalar fields subject to the defining quadric constraint ∑i |Zi2 = 1. To prove these configurations are supersymmetric one can use the well known isomorphism between geometric Killing spinors on both the 5-sphere and AdS5 and parallel spinors in ℂ3 and ℂ1,2, respectively. This is briefly reviewed in Appendix B. The conclusion of such analysis is that any holomorphic function F (Z1,Z2,Z3) gives rise to a supersymmetric giant graviton configuration [392] defined
$$\begin{array}{*{20}c} {\vert {Z_1}{\vert ^2} + \vert {Z_2}{\vert ^2} + \vert {Z_3}{\vert ^2} = 1,} \\ {F({e^{- it/{L_4}}}{Z_1},{e^{- it/{L_4}}}{Z_2},{e^{- it/{L_4}}}{Z_3}) = 0,\quad \quad \quad} \\ \end{array}$$
(421)
as the intersection of the 5-sphere with a holomorphic hypersurface properly evolved in world volume time. The latter involves rotations in each of the ℂ planes in ℂ3 at the speed of light (in 1/L4 units), which is a consequence of supersymmetry and a generalisation of the condition explicitly found in Eq. (419).

Geometric quantisation and BPS counting: The above construction is classical and applies to backgrounds of the form \({\rm{Ad}}{{\rm{S}}_5} \times {{\mathcal M}_5}\). In [54], the classical moduli space of holomorphic functions mentioned above was originally quantised and some of its BPS spectrum matched against the spectrum of chiral operators in \({\mathcal N} = 4\)d = 4 SYM. Later, in [104, 369], the full partition function was derived and seen to agree with that of N noninteracting bosons in a 3d harmonic potential. Similar work and results were obtained for the moduli space of dual giant gravitons40 when \({{\mathcal M}_5}\) is an Einstein-Sasaki manifold [374]. The BPS partition functions derived from these geometric quantisation schemes agree with purely gauge theory considerations [69, 341] and with the more algebraic approach to counting chiral operators followed in the plethystics program [67, 210].

Related work: There exists an extensive amount of work constructing world volume configurations describing giant gravitons in different backgrounds to the ones mentioned above. This includes non-supersymmetric giant gravitons with NS-NS fields [131], M-theory giants with 3-form potential field [132], giants in deformed backgrounds [422] or electric/magnetic field deformed giants in Melvin geometries [310]. For discussions on supersymmetric D3, fractional D5 and D7-brane probes in AdS5 × Labc, see [135]. There is also interesting work on bound states of giant gravitons [430] and on the effective field theory description of many such giants (a non-abelian world volume description) with the inclusion of higher moment couplings responsible for their physical properties [317, 318].

5.9.1 Giant gravitons as black-hole constituents

Individual giant gravitons carry conformal dimension of order N and according to the discussion above, they exhaust the spectrum of chiral operators in the dual CFT, whereas R-charged AdS black holes carry mass of order N2. The idea that supersymmetric R-charged AdS black holes could be interpreted as distributions of giant gravitons was first discussed in [397], where these bulk configurations were coined as superstars. The main idea behind this identification comes from two observations:
  1. 1.

    The existence of naked singularities in these black holes located where giant gravitons sit in AdS suggests the singularity is due to the presence of an external source.

     
  2. 2.

    Giant gravitons do not carry D3-brane charge, but they do locally couple to the RR 5-form field strength giving rise to some D3-brane dipole charge. This means [397] that a small (five-dimensional) surface enclosing a portion of the giant graviton sphere will carry a net five-form flux proportional to the number of D3-branes enclosed. If this is correct, one should be able to determine the local density of giant gravitons at the singularity by analysing the net RR 5-form flux obtained by considering a surface that is the boundary of a six-dimensional ball, which only intersects the three-sphere of the giant graviton once, at a point very close to the singularity.

     
To check this interpretation, let us review these supersymmetric R-charged AdS5 black holes. These are solutions to \({\mathcal N} = 2\) d =5 gauged supergravity with U(1)3 gauge symmetry [56, 57] properly embedded into type IIB [157]. Their metric is
$$\begin{array}{*{20}c} {ds_{10}^2 = \sqrt \Delta \left[ {- {{({H_1}{H_2}{H_3})}^{- 1}}fd{t^2} + ({f^{- 1}}d{r^2} + {r^2}d\Omega _3^2)} \right]\quad \quad \quad \quad} \\ {+ {1 \over {\sqrt \Delta}}\sum\limits_{i = 1}^3 {{H_i}} \left({{L^2}d\mu _i^2 + \mu _i^2{{[{L_4}d{\phi _i} + (H_i^{- 1} - 1)dt]}^2}} \right),} \\ \end{array}$$
(422)
with the different scalar functions defined as
$$\begin{array}{*{20}c} {f = 1 + {{{r^2}} \over {L_4^2}}{H_1}{H_2}{H_3}\qquad {\rm{with}}\qquad {H_i} = 1 + {{{q_i}} \over {{r^2}}}\,,} \\ {\Delta = {H_1}{H_2}{H_3}\sum\limits_{i = 1}^3 {{{\mu _i^2} \over {{H_i}}}} ,\qquad {\rm{with}}\qquad \sum\limits_{i = 1}^3 {\mu _i^2} = 1\,.\;\;} \\ \end{array}$$
(423)
All these metrics have a naked singularity at the center of AdS that extends into the 5-sphere. Depending on the number of charges turned on, the rate at which curvature invariants diverge changes with the 5-sphere direction. Besides a constant dilaton, these BPS configurations also have a non-trivial RR self-dual 5-form field strength R5 = dC4 + *dC4 with
$${C_4} = - {{{r^4}} \over {{L_4}}}\Delta \,dt \wedge {\omega _3} - {L_4}\sum\limits_{i = 1}^3 {{q_i}} \mu _i^2({L_4}\,d{\phi _i} - dt) \wedge {\omega _3}\,,$$
(424)
with ω3 being volume 3-form of the unit 3-sphere.
To test the microscopic interpretation for the superstar solutions, consider the single R-charged configuration with q2 = q3 = 0. This should correspond to a collection of giant gravitons rotating along ϕ1 with a certain distribution of sizes (specified by μ1 = cos θ1). To measure the density of giant gravitons sitting near a certain θ1, one must integrate R5 over the appropriate surface. Describing the 3-sphere in AdS5 by
$$d\Omega _3^2 = d\alpha _1^2 + {\sin ^2}{\alpha _1}(d\alpha _2^2 + {\sin ^2}{\alpha _2}d\alpha _3^2)\,,$$
(425)
one can enclose a point on the brane at θ1 with a small five-sphere in the {r, θ1, ϕ1 αi} directions. The relevant five-form component is
$${({R_5})_{{\theta _1}\,{\phi _1}\,{\alpha _1}\,{\alpha _2}\,{\alpha _3}}} = 2{q_1}L_4^2\sin {\theta _1}\cos {\theta _1}{\sin ^2}{\alpha _1}\sin {\alpha _2}\,,$$
(426)
and by integrating the latter over the smeared direction ϕ1 and the 3-sphere, one infers the density of giants at a point θ1 [397]
$${{d{n_1}} \over {d{\theta _1}}} = {N \over {4{\pi ^3}L_4^4}}\int {{{({R_5})}_{{\theta _1}\,{\phi _1}\,{\alpha _1}\,{\alpha _2}\,{\alpha _3}}}} d{\phi _1}{d^3}\alpha = N{{{q_1}} \over {L_4^2}}\sin 2{\theta _1}\,.$$
(427)
If this is correct, the total number of giant gravitons carried by the superstar is
$${n_1} = \int\nolimits_0^{\pi /2} d {\theta _1}\,{{d{n_1}} \over {d{\theta _1}}} = N{{{q_1}} \over {L_4^2}}.$$
(428)
The matching is achieved by comparing the microscopic momentum carried by a single giant at the location θ1, Pmicro = N sin2 θ1, with the total mass of the superstar
$$M = {{{N^2}} \over 2}{{{q_1}} \over {L_4^3}}\,.$$
(429)
Indeed, by supersymmetry, the latter should equal the total momentum of the distribution
$$M = {{{P_{{\phi _1}}}} \over {{L_4}}} = \int\nolimits_0^{\pi /2} d {\theta _1}\,{{d{n_1}} \over {d{\theta _1}}}{{{P_{{\rm{micro}}}}} \over {{L_4}}} = {{{N^2}} \over 2}{{{q_1}} \over {L_4^3}}\,,$$
(430)
which establishes the physical correspondence. There exist extensions of these considerations when more than a single R-charge is turned on, i.e., when q2,q3 ≠ 0. See [397] for the specific details, though the conclusion remains the same.

1/2 BPS superstar and smooth configurations: Just as supertubes have smooth super-gravity descriptions [205] with U-dual interpretations in terms of chiral states in dual CFTs [361] when some of the dimensions are compact, one may wonder whether a similar picture is available for chiral operators in \({\mathcal N} = 4\) d = 4 SYM corresponding to collections of giant gravitons. For 1/2 BPS states, the supergravity analysis was done in [355]. The classical moduli space of smooth configurations was determined: it is characterised in terms of a single scalar function satisfying a Laplace equation. When the latter satisfies certain boundary conditions on its boundary, the entire supergravity solution is smooth. Interestingly, such boundary could be interpreted as the phase space of a single fermion in a 1d harmonic oscillator potential, whereas the boundary conditions correspond to exciting coherent states on it. This matches the gauge theory description in terms of the eigenvalues of the adjoint matrices describing the gauge invariant operators in this 1/2 BPS sector of the full theory [150, 68]. Moreover, geometric quantisation applied on the subspace of these 1/2 BPS supergravity configurations also agreed with the picture of N free fermions in a 1d harmonic oscillator potential [251, 371]. The singular superstar was interpreted as a coarsegrained description of the typical quantum state in that sector [37], providing a bridge between quantum mechanics and classical geometry through the coarse-graining of quantum mechanical information. In some philosophically vague sense, these supergravity considerations provide some heuristic realisation of Wheeler’s ideas [492, 493, 39]. Some partial progress was also achieved for similar M-theory configurations [355]. In this case, the quantum moduli space of BPS gauge theory configurations was identified in [450] and some steps to identify the dictionary between these and the supergravity geometries were described in [184]. Notice this set-up is also in agreement with the general framework illustrated in Figure 7.

Less supersymmetric superstars: Given the robustness of the results concerning the partition functions of 1/4 and 1/8 chiral BPS operators in \({\mathcal N} = 4\) SYM and their description in terms of BPS giant graviton excitations, it is natural to study whether there exist smooth supergravity configurations preserving this amount of supersymmetry and the appropriate bosonic isometries to be interpreted as these chiral states. The classical moduli space of these configurations was given in [142], extending previous work [182, 181]. The equations describing these moduli spaces are far more complicated than its 1/2 BPS sector cousin,
  • 1/4 BPS configurations depend on a 4d Kähler manifold with Kähler potential satisfying a non-linear Monge-Ampere equation [142],

  • 1/8 BPS configuration depend on a 6d manifold, whose scalar curvature satisfies a non-linear equation in the scalar curvature itself and the square of the Ricci tensor [338].

Some set of necessary conditions for the smoothness of these configurations was discussed in [142]. A more thorough analysis for the 1/4 BPS configurations was performed in [360], where it was argued that a set of extra consistency conditions were required, the latter constraining the location of the sources responsible for the solutions. Interestingly, these constraints were found to be in perfect agreement with the result of a probe analysis. This reemphasises the usefulness of probe techniques when analysing supergravity matters in certain BPS contexts.

5.10 Deconstructing black holes

Both supertubes and giant gravitons are examples of supersymmetric states realised as classical solitons in brane effective actions and interpreted as the microscopic constituents of small black holes. The bulk entropy is matched after geometric quantisation of their respective classical moduli spaces. This framework, which is summarised in Figure 7, suggests the idea of deconstructing the black hole into zero-entropy, minimally-charged bits, reinterpreting the initial black-hole entropy as the ground-state degeneracy of the quantum mechanics on the moduli space of such deconstructions (bits).

In this subsection, I briefly mention some work in this direction concerning large supersymmetric AdS5 × S5 black holes, deconstructions of supersymmetric asymptotically-flat black holes in terms of constituent excitations living at the horizon of these black holes and constituent models for extremal static non-BPS black holes.

Large supersymmetric AdS5 black holes: Large supersymmetric AdS5 × S5 black holes require the addition of angular momentum in AdS5, besides the presence of R-charges, to achieve a regular macroscopic horizon while preserving a generic 1/16 of the vacuum supersymmetries. The first examples were reported in [280]. Subsequent work involving more general (non-)BPS black holes can be found in [279, 143, 350].

Given the success in identifying the degrees of freedom for R-charged black holes, it is natural to analyse whether the inclusion of angular momentum in AdS5 can be accomplished by more general (dual) giant graviton configurations carrying the same charges as the black hole. This task was initiated in [339]. Even though their work was concerned with configurations preserving 1/8 of the supersymmetry, the importance of a non-trivial Poynting vector on the D3-brane world volume to generate angular momentum was already pointed out, extending the mechanism used already for supertubes. In [340], the first extension of these results to 1/16 world volume configurations was considered. The equations satisfied for the most general 1/16 dual giant D3-brane probe in AdS5 × S5 were described in [22], whereas explicit supersymmetric electromagnetic waves on (dual) giants were constructed in [23]. Similar interesting work describing giant gravitons in the pp-wave background with non-trivial electric fields was reported in [15].

All these configurations have interest on their own, given their supersymmetry and the conserved charges they carry, but further evidence is required to interpret them as bulk black hole constituents. This task was undertaken in [456]. Instead of working in the vacuum, these authors studied the spectrum of classical supersymmetric (dual) giant gravitons in the near horizon geometries of these black holes in [457], following similar reasonings for asymptotically-flat black holes [174]. The partial quantisation of this classical moduli space [456] is potentially consistent with the identification of dual giants as the constituents of these black holes, but this remains an open question. In the same spirit, [22] quantised the moduli space of the wobbling dual giants, 1/8 BPS configurations with two angular momentum in AdS5 and one in S5 and agreement was found with the gauge theory index calculations carried out in [341].

There have also been more purely field theoretical approaches to this problem. In [250], cohomological methods were used to count operators preserving 1/16 of the supersymmetries in \({\mathcal N} = 4\) d = 4 SYM, whereas in [97] explicit operators were written down, based on Fermi surface filling fermions models and working in the limit of large angular momentum in AdS5. These attempted to identify the pure states responsible for the entropy of the black hole and their counting agreed, up to order one coefficients, with the Hawking-Bekenstein classical entropy.

Large asymptotically-flat BPS black holes: There exists a large literature on the construction of supersymmetric configurations with the same asymptotics and charges as a given large BPS black hole, but having the latter carried by different constituent charges located at different “centers”41. The center locations are non-trivially determined by solving a set of constraint equations, called the bubble equations. The latter is believed to ensure the global smoothness and lack of horizon of the configuration. These constraints do reflect the intrinsic bound state nature of these configurations. The identification of a subset of 1/2 BPS centers as the fundamental constituents for large black holes was further developed in [38].

One of the new features in these deconstructions is that the charges carried by the different constituents do not have to match the charges carried by the black hole, i.e., a constituent can carry D6-brane charge even if the black hole does not, provided there exists a second centre with anti-D6-brane charge, cancelling the latter.

This idea of deconstructing a given black hole in terms of maximally entropic configurations of constituent objects42 was tested for the standard D0-D4 black hole in [174]. The black hole was deconstructed in terms of D 6 and \({{\bar D}^6}\) branes with world volume fluxes turned on, inducing further D4-D2-D0 charges, and a large set of D0-branes. Working in a regime of charges where the distance between centres scales to zero, i.e., the scaling solution, all D0-branes become equidistant to the D6-branes, forming some sort of accretion disk and the geometry deep inside this ring of D0-branes becomes that of global AdS3 × S2, when lifting the configuration to M-theory. Using the microscopic picture developed in [219], where it was argued that the entropy of this black hole came from the degeneracy of states due to non-abelian D0-branes that expand into D2-branes due to the Myers’ effect [395], the authors in [174] manage to extend the near horizon wrapping M2-branes found in [455] to M2-branes wrapping supersymmetric cycles of the full geometry. It was then argued that the same counting done [219], based on the degeneracy of the lowest Landau level quantum mechanics problem emerging from the effective magnetic field on the transverse Calabi-Yau due to the coupling of the D2-D0 bound states to the background RR 4-form field strength, would apply in this case.

The same kind of construction and logic was applied to black rings [206, 199] in [239]. Further work on stable brane configurations in the near horizon on brane backgrounds can be found in [130].

Extremal non-BPS deconstructions: These ideas are also applicable to non-supersymmetric systems, though one expects to have less control there. For the subset of static extremal non-BPS black holes in the STU model [155, 194, 58], these methods turned out to be successful. The most general static black-hole solution, including non-trivial moduli at infinity, was found in [237, 358]. It was pointed out in [237] that the mass of these black holes equals the sum of four mutually local 1/2 BPS constituents for any value of the background moduli fields and in any U-duality frame. Using probe calculations, it was shown that such constituents do not feel any force in the presence of these black holes [238]. This suggested that extra quanta could be added to the system and located anywhere. This is consistent with the multi-center extremal non-BPS solutions found in [218]: their centres are completely arbitrary but the charge vectors carried by each centre are constrained to be the ones of the constituents identified in [238] (or their linear combinations). This model identifies the same constituents as the ones used to account for the entropy of neutral black holes in [204] and extends it to the presence of fluxes. No further dynamical understanding of the open string degrees of freedom is available in terms of non-supersymmetric quiver gauge theories.

As soon as angular momentum is added to the system, while keeping extremality, the location of the deconstructed constituents gets constrained according to non-linear bubble equations that ensure the global smoothness of the full supergravity solution [61, 62]. These are fairly recent developments and one expects further progress to be achieved in this direction in the future. For example, very recently, an analysis of stable, metastable and non-stable supertubes in smooth geometries being candidates for the microstates of black holes and black rings was presented in [63]. This includes configurations that would also be valid for non-extremal black holes.

6 Some AdS/CFT Related Applications

This section is devoted to more dynamical applications of brane effective actions. More specifically, I will describe some well-established reinterpretations of certain brane probe calculations in the context of the AdS/CFT correspondence [366, 269, 498, 13]. I will mainly focus on two aspects:
  • The use of classical solitons solving the brane (string) equations of motion in particular backgrounds and with specific boundary conditions, to holographically compute either the expectation value of certain gauge invariant operators or the spectrum in sectors of certain strongly coupled gauge theories.

  • The use of D-brane effective actions to describe the dynamics of a small number of degrees of freedom responsible either for deforming the original dual CFT to theories with less or no supersymmetry, or for capturing the interaction of massless modes among themselves and with other sectors of the system conveniently replaced by a supergravity background.

Covariance of brane effective actions allows one to couple them to any on-shell supergravity background. In particular, one can probe either AdS5 × S5, or black holes with these asymptotics, with branes, and according to the AdS/CFT correspondence, one will be studying properties of the strongly coupled holographic theory in the vacuum or at finite temperature and chemical potentials, respectively. This set-up is illustrated in Figure 8. The same interpretation will hold for non-relativistic versions of these backgrounds. Alternatively, and depending on the boundary conditions imposed on these probes, they can deform the theory towards less symmetric and more realistic physical systems.
Figure 8

General framework in which probe calculations in appropriate backgrounds with suitable boundary conditions can be reinterpreted as strongly coupled observables and spectrum in non-abelian gauge theories using the AdS/CFT correspondence.

In the following, I will review the calculation of Wilson loop expectation values, the use of worldsheet string solitons to study the spectrum of states with large charges in \({\mathcal N} = 4\) SYM and the use of D-brane probes to either add flavour to the AdS/CFT correspondence or describe the dynamics of massless excitations in non-relativistic (thermal) set-ups, which could be of relevance for strongly-coupled condensed-matter physics.

6.1 Wilson loops

As a first example of the use of classical solutions to brane effective actions to compute the expectation values of gauge invariant operators at strong coupling, I will review the prescription put forward in [367, 433] for Wilson loop operators in \({\mathcal N} = 4\) SYM.

Wilson loop operators [494] in SU(N) Yang-Mills theories are non-local gauge invariant operators
$$W({\mathcal C}) = {1 \over N}{\rm{TrP}}{e^{i\;\oint\nolimits_{\mathcal C} A}}\,,$$
(431)
depending on a closed loop in spacetime \({\mathcal C}\) and where the trace is over the fundamental representation of the gauge group. This operator allows one to extract the energy E (L) of a quark-antiquark pair separated a distance L. Indeed, consider a rectangular closed loop in which the pair evolves in Euclidean time T. In the limit T → ∞, the expectation value of this rectangular Wilson loop equals
$$\langle W({\mathcal C})\rangle = A(L){e^{- TE(L)}}\,.$$
(432)
To understand the prescription in [367, 433], one must first introduce massive quarks in the theory. This is achieved by breaking the original gauge symmetry of the original \({\mathcal N} = 4\) SYM according to
$${\rm{U}}(N + 1) \rightarrow {\rm{U}}(N) \times {\rm{U}}(1).$$
(433)
The massive W-bosons generated by this process have a mass proportional to the norm of the Higgs field expectation value responsible for the symmetry breaking \((\vert \vec \Phi \vert)\) and transform in the fundamental representation of the U (N) gauge symmetry, as required. Furthermore at energy scales much lower than \(\vert \vec \Phi \vert\), the U (N) theory decouples from the U(1) theory.
In this set-up, the massive W-boson interacts with the U(N) gauge fields, including the scalar adjoint fields XI [367], leading to the insertion of the operator
$$W({\mathcal C}) = {1 \over N}{\rm{TrP}}{e^{i\;\oint\nolimits_{\mathcal C} d s[{A_\mu}(\sigma){{\dot \sigma}^\mu} + {\theta ^I}(s){X^I}(\sigma)\sqrt {{{\dot \sigma}^2}} ]}}\,.$$
(434)
The contour \({\mathcal C}\) is parameterised by σμ (s) whereas the vector \(\vec \theta (s)\) maps each point on the loop to a point on the five-sphere.
The proposal made in [367, 433] to compute the expectation value of Eq. (434) was
$$\langle W({\mathcal C})\rangle \sim {e^{- {S_{{\rm{string}}}}}}\,.$$
(435)
This holds in the large \({g_s}N\) limit and Sstring stands for the proper area of a fundamental string describing the loop \({\mathcal C}\) at the boundary of AdS5 and lying along θI (s) on S5. Notice that a quantum mechanical calculation at strong coupling reduces to determining a minimal worldsheet surface in AdS5, i.e., solving the worldsheet equations of motion with appropriate boundary conditions, and then solving for the worldsheet energy as a function of the separation L between the quarkantiquark. After subtracting the regularised mass of the W-boson one obtains the quark-antiquark potential energy
$$E(L) = - {{4{\pi ^2}{{(2g_{{\rm{YM}}}^2N)}^{1/2}}} \over {\Gamma {{({1 \over 4})}^4}L}}\,,$$
(436)
which differs from the linear perturbative dependence on \(g_{{\rm{YM}}}^2N\).

If one considers multiply-wrapped Wilson loops, the many coincident strings will suffer from self-interactions. This may suggest that a more appropriate description of the system is in terms of a D3-brane with non-trivial world volume electric flux accounting for the fundamental strings. This is the approach followed in [189], where it was shown that for linear and circular loops the D3-brane action agreed with the string worldsheet result at weak coupling, but captures all the higher-genus corrections at leading order in α ′.

6.2 Quark energy loss in a thermal medium

Having learnt how to describe a massive quark in \({\mathcal N} = 4\,{\rm{SYM}}\) in terms of a string, this opens up the possibility of describing its energy loss as it propagates through a thermal medium. One can think of this process
  1. 1.

    either from the bulk perspective, where the thermal medium gets replaced by a black hole and energy flows down the string towards its horizon,

     
  2. 2.

    or from the gauge-theory perspective, where energy and momentum emanate from the quark and eventually thermalise.

     
In this section, I will take the bulk point of view originally discussed in [297, 268], with a related fluctuation analysis in [138]. The goal is to highlight the power of the techniques developed in Sections 4 and 5 rather than being self-contained. For a more thorough discussion, the reader should check the review on this particular topic [272].
The thermal medium is holographically described in terms of the AdS5-Schwarzschild black hole,
$$d{s^2} = {g_{mn}}d{x^m}d{x^n} = {{{L^2}} \over {{z^2}}}\;\left({- h(z)d{t^2} + d{{\vec x}^2} + {{d{z^2}} \over {h(z)}}} \right)\;\,,$$
(437)
where \(h(z) = 1 - {{{z^4}} \over {z_H^4}}\) determines the horizon size zH and the black-hole temperature \(T = {1 \over {\pi {z_H}}}\). The latter coincides with the gauge-theory temperature [498]. Notice z = 0 is the location of the conformai boundary and L is the radius of AdS5.
If one is interested in describing the dragging effect suffered by the quark due to the interactions with the thermal medium, one considers a non-static quark, whose trajectory in the boundary satisfies Xl (t) = υt, assuming motion takes place only in the x1 direction. One can parameterise the bulk trajectory as
$${X^1}(t,z) = vt + \xi (z),$$
(438)
where ξ (z) satisfies ξ → 0 as z → 0. To determine ξ (z), one must solve the classical equations of motion of the bosonic worldsheet action (16) in the background (437). These reduce to a set of conserved equations of the form
$${\nabla _\mu}{\pi ^\mu}_m = 0\,,\qquad {\rm{where}}\qquad {\pi ^\mu}_m \equiv - {1 \over {2\pi \alpha \prime}}{{\mathcal G}^{\mu \nu}}{g_{mn}}{\partial _\nu}{X^n}$$
(439)
is the worldsheet momentum current conjugate to the position Xm. Plugging the ansatz (438) into Eq. (439), one finds
$${{d\xi} \over {dz}} = {{{\pi _\xi}} \over h}\sqrt {{{h - {v^2}} \over {{{{L^4}} \over {{z^4}}}h - \pi _\xi ^2}}} \,,$$
(440)
where πξ is an integration constant. A priori, there are several allowed possibilities compatible with the reality of the trailing function ξ (z). These were analysed in [272] where it was concluded that the relevant physical solution is given by
$${\pi _\xi} = - {{{L^2}} \over {z_{\ast}^2}}\sqrt {h({z_{\ast}})} = - {v \over {\sqrt {1 - {v^2}}}}{{{L^2}} \over {z_H^2}}\qquad \Rightarrow \qquad \xi = - {{{z_H}v} \over {4i}}\;\left({\log {{1 - iy} \over {1 + iy}} + i\log {{1 + y} \over {1 - y}}} \right)\;\,,$$
(441)
where y is a rescaled depth variable y = z/zh.
To compute the rate at which quark momentum is being transferred to the bath, one can simply integrate the conserved current \({p^\mu}_m\) over a line-segment and given the stready-state nature of the trailing string configuration, one infers [272]
$${{d{p_m}} \over {dt}} = - \sqrt {- g} \,{p^z}_m.$$
(442)
This allows us to define the drag force as
$${F_{{\rm{drag}}}} = {{d{p_1}} \over {dt}} = - {{{L^2}} \over {2\pi z_H^2\alpha \prime}}{v \over {\sqrt {1 - {v^2}}}} = - {{\pi \sqrt \lambda} \over 2}{T^2}{v \over {\sqrt {1 - {v^2}}}}\quad {\rm{with}}\quad \lambda = g_{{\rm{YM}}}^2N = {{{L^4}} \over {\alpha {\prime 2}}}.$$
(443)
for a much more detailed discussion on the physics of this system see [272, 137]. The latter also includes a discussion of the same physical effect for a finite, but large, quark mass, and the possible implications of these results and techniques for quantum chromodynamics (QCD).

More recently, it was argued in [212] that one can compute the energy loss by radiation of an infinitely-massive half-BPS charged particle to all orders in 1/N using a similar construction to the one mentioned at the end of Section 6.1. This involved the use of classical D5-brane and D3-brane world volume reaching the AdS5 boundary to describe particles transforming in the antisymmetric and symmetric representations of the gauge group, respectively.

6.3 Semiclassical correspondence

It is an extended idea in theoretical physics that states in quantum mechanics carrying large charges can be well approximated by a classical or semiclassical description. This idea gets realised in the AdS/CFT correspondence too. Consider the worldsheet sigma model description of a fundamental string in AdS5 × S5. One expects its perturbative oscillations to be properly described by supergravity, whereas solitons with large conformal dimension,
$$\Delta \sim {1 \over {\sqrt \lambda}}\,,\qquad \qquad \lambda = g_{{\rm{YM}}}^2N = {g_s}N$$
(444)
and the spectrum of their semiclassical excitations may approximate the spectrum of highly excited string states in \({\mathcal N} = 4\,{\rm{SYM}}\). This is the approach followed in [270], where it was originally applied to rotating folded strings carrying large bare spin charge.
To get an heuristic idea of the analytic power behind this technique, let me reproduce the spectrum of large R-charge operators obtained in [70] using a worldsheet quantisation in the pp-wave background by considering the bosonic part of the worldsheet action describing the AdS5 × S5 sigma model [270]
$$S = {1 \over {2\alpha}}\int {{d^2}} \sigma \,\sqrt g ({({\nabla _\alpha}n)^2} + {({\nabla _\alpha}K)^2}) + \ldots ,$$
(445)
where n is a unit vector describing S5, K is a hyperbolic unit vector describing AdS5, the sigm model coupling α is \(\alpha = {1 \over {\sqrt \lambda}}\) 43 and I have ignored all fermionic and RR couplings.
Consider a solution to the classical equations of motion describing a collapsed rotating closed string at the equator
$$\theta = 0\,,\qquad \psi = \omega \tau \,,$$
(446)
where θ and ψ are the polar and azimuthal angles on S2 in S5. Its classical worldsheet energy is
$$E = {1 \over {2\alpha}}{\omega ^2} = {\alpha \over 2}{J^2}\qquad {\rm{where}}\qquad J = {\omega \over \alpha}\,.$$
(447)
Next, consider the harmonic fluctuations around this classical soliton. Focusing on the quadratic θ oscillations,
$$\alpha L = {1 \over 2}\left[ {{{(\nabla \theta)}^2} + {\omega ^2}{{\cos}^2}\theta} \right] \simeq {1 \over 2}\left[ {{{\left({\nabla \theta} \right)}^2} - {\omega ^2}{\theta ^2} + {\omega ^2}} \right]\;\,,$$
(448)
one recognises the standard harmonic oscillator. Using its spectrum, one derives the corrections to the classical energy
$$\delta = {\alpha \over 2}{J^2} + \sum\limits_n {{N_n}} \sqrt {{n^2} + {\alpha ^2}{J^2}} \,,$$
(449)
where Nn is the excitation number of the n-th such oscillator. There is a similar contribution from the AdS part of the action, obtained by the change α to −α. Both contributions must satisfy the on-shell condition
$$\delta ({{\rm{S}}^5}) + \delta ({\rm{AdS}}_5) \approx 0.$$
(450)
This is how one reproduces the spectrum derived in [70]
$$\Delta = J + \sum\limits_{n = - \infty}^\infty {{N_n}} \sqrt {1 + {{\lambda {n^2}} \over {{J^2}}}} .$$
(451)
The method outlined above is far more general and it can be applied to study other operators. For example, one can study the relation between conformal dimension and AdS5 spin, as done in [270], by analysing the behaviour of solitonic closed strings rotating in AdS. Using global AdS5,
$$d{s_{\rm{5}}} = {L^2}\left[ {- {{\cosh}^2}\rho \,d{t^2} + d{\rho ^2} + {{\sinh}^2}\rho \left({d{\theta ^2} + {{\sin}^2}\theta d{\phi ^2} + {{\cos}^2}\theta d{\psi ^2}} \right)} \right],$$
(452)
as the background where the bosonic string propagates and working in the gauge τ = t allows one to identify the worldsheet energy with the conformal dimension in the dual CFT. Consider a closed string at the equator of the 3-sphere while rotating in the azimuthal angle
$$\phi = \omega t.$$
(453)
for configurations ρ = ρ (σ), the Nambu-Goto bosonic action reduces to
$${S_{{\rm{string}}}} = - 4{{{L^2}} \over {2\pi \alpha \prime}}\int d t\int\nolimits_0^{{\rho _0}} d \rho \,\sqrt {{{\cosh}^2}\rho - {{(\dot \phi)}^2}{{\sinh}^2}\rho} ,$$
(454)
where σ0 stands for the maximum radial coordinate and the factor of 4 arises because of the four string segments stretching from 0 to σ0 determined by the condition
$${\coth ^2}{\rho _0} = {\omega ^2}.$$
(455)
The energy E and spin S of the string are conserved charges given by
$$E = 4{{{L^2}} \over {2\pi \alpha \prime}}\int\nolimits_0^{{\rho _0}} d \rho \,{{{{\cosh}^2}\rho} \over {\sqrt {{{\cosh}^2}\rho - {\omega ^2}{{\sinh}^2}\rho}}},$$
(456)
$$S = 4{{{L^2}} \over {2\pi \alpha \prime}}\int\nolimits_0^{{\rho _0}} d \rho \,{{\omega {{\sinh}^2}\rho} \over {\sqrt {{{\cosh}^2}\rho - {\omega ^2}{{\sinh}^2}\rho}}}.$$
(457)
Notice the dependence of \(E/\sqrt \lambda\) on \(S/\sqrt \lambda\) is in parametric form since L4 = λα′2. One can obtain approximate expressions in the limits where the string is much shorter or longer than the radius of curvature L of AdS5.
Short strings: For large ω, the maximal string stretching is ρ0 ≈ 1/ω. Thus, strings are shorter than the radius of curvature L. Calculations reduce to strings in flat space for which the parametric dependence is [270]
$$E = {{{L^2}} \over {\alpha \prime \omega}}\,,\qquad S = {{{L^2}} \over {2\alpha \prime {\omega ^2}}}\;,\qquad \Rightarrow \qquad {E^2} = {L^2}{{2S} \over {\alpha \prime}}.$$
(458)
Using the AdS/CFT correspondence, the conformal dimension equals the energy, i.e., A = E. Furthermore, \(S \ll \sqrt \lambda\) for large ω. Thus,
$${\Delta ^2} \approx {m^2}{L^2},\qquad {\rm{where}}\qquad {m^2} = {{2(S - 2)} \over {\alpha \prime}}$$
(459)
for the leading closed string Regge trajectory, which reproduces the AdS/CFT result.
Long strings: The opposite regime takes place when ω is close to one (from above)
$$\omega = 1 + 2\eta ,\quad \eta \ll 1\qquad \Rightarrow \qquad {\rho _0} \to {1 \over 2}\log {1 \over \eta},\,\,S \gg \sqrt \lambda ,$$
(460)
so that the string is sensitive to the AdS boundary metric. The string energy and spin become
$$E = {{{L^2}} \over {2\pi \alpha \prime}}\left({{1 \over \eta} + \log {1 \over \eta} + \ldots} \right),$$
(461)
$$S = {{{L^2}} \over {2\pi \alpha \prime}}\left({{1 \over \eta} - \log {1 \over \eta} + \ldots} \right),$$
(462)
so that its difference approaches
$$E - S = {{\sqrt \lambda} \over \pi}\log {S \over {\sqrt \lambda}} + \ldots$$
(463)
This logarithmic asymptotics is qualitatively similar to the one appearing in perturbative gauge theories. For a more thorough discussion on this point, see [270].

Applying semiclassical quantisation methods to these classical solitons [216], it was realised that one can interpolate the results for E — S to the weakly-coupled regime. It should be stressed that these techniques allow one to explore the AdS/CFT correspondence in non-supersymmetric sectors [217], appealing to the correspondence principle associated to large charges. It is also worth mentioning that due to the seminal work on the integrability of planar \({\mathcal N} = 4\,{\rm{SYM}}\) at one loop [393, 60], much work has been devoted to using these semiclassical techniques in relation to integrability properties [21]. Interested readers are encouraged to check the review [59] on integrability and references therein.

6.4 Probes as deformations and gapless excitations in complex systems

The dynamical regime in which brane effective actions hold is particularly suitable to describe physical systems made of several interacting subsystems in which one of them has a much smaller number of degrees of freedom. Assume the larger subsystems allow an approximate description in terms of a supergravity background. Then, focusing on the dynamics of this smaller subsector, while keeping the dynamics of the larger subsystems frozen, corresponds to probing the supergravity background with the effective action describing the smaller subsystem. This conceptual framework is illustrated in Figure 9.
Figure 9

Conceptual framework in which the probe approximation captures the dynamics of small subsystems interacting with larger ones that have reliable gravity duals.

This set-up occurs when the brane degrees of freedom are responsible for either breaking the symmetries of the larger system or describing an interesting isolated set of massless degrees of freedom whose interactions among themselves and with the background one is interested in studying. In the following, I very briefly describe how the first approach was used to introduce flavour in the AdS/CFT correspondence, and how the second one can be used to study physics reminiscent of certain phenomena in condensed-matter systems.

Probing deformations of the AdS/CFT: Deforming the original AdS/CFT allows one to come up with set-ups with less or no supersymmetry. Whenever there is a small number of degrees of freedom responsible for the dynamics (typically D-branes), one may approximate the latter by the effective actions described in this review. This provides a reliable and analytical tool for describing the strongly-coupled behaviour of the deformed gauge theory.

As an example, consider the addition of flavour in the standard AdS/CFT. It was argued in [333] that this could be achieved by adding κ D7-branes to a background of N D3-branes. The D7-branes give rise to κ fundamental hypermultiplets arising from the lightest modes of the 3–7 and 7–3 strings, in the brane array
$$\begin{array}{*{20}c} {{\rm{D}}3:\,\,\,1\,2\,3\,\_\,\_\,\_\,\_\,\_\,\_\,} \\ {{\rm{D}}7:\,\,\,1\,2\,3\,4\,5\,6\,7\,\_\,\_\,.\,} \\ \end{array}$$
(464)
The mass of these dynamical quarks is given by mq = L/ 2πα′, where L is the distance between the D3- and the D7-branes in the 89-plane. If gsN ≫ 1 the D3-branes may be replaced (in the appropriate decoupling limit) by an AdS5 × S5 geometry, as in the standard AdS/CFT argument, whereas if, in addition, Nκ then the back-reaction of the D7-branes on this geometry may be neglected. Thus, one is left, in the gravity description, with κ D7-brane probes in AdS5 × S5. In the particular case of κ = 1, one can use the effective action described before. This specific set-up was used in [348] to study the linearised fluctuation equations for the different excitations on the D7-probe describing different scalar and vector excitations to get analytical expressions for the spectrum of mesons in \({\mathcal N} = 2\,{\rm{SYM}}\), at strong coupling.

This logic can be extended to non-supersymmetric scenarios44. For example, using the string theory realisation of four-dimensional QCD with Nc colours and NfNc flavours discussed in [499]. The latter involves Nf D6-brane probes in the supergravity background dual to Nc D4-branes compactified on a circle with supersymmetry-breaking boundary conditions and in the limit in which all the resulting Kaluza-Klein modes decouple. For Nf = 1 and for massless quarks, spontaneous chiral symmetry breaking by a quark condensate was exhibited in [349] by working on the D6-brane effective action in the near horizon geometry of the Nc D4-branes.

Similar considerations apply at finite temperature by using appropriate black-hole backgrounds [499] in the relevant probe action calculations. This allows one to study phase transitions associated with the thermodynamic properties of the probe degrees of freedom as a function of the probe location. This can be done in different theories, with flavour [379], and for different ensembles [343, 378].

The amount of literature in this topic is enormous. I refer the reader to the reviews on the use of gauge-gravity duality to understand hot QCD and heavy ion collisions [137] and meson spectroscopy [207], and references therein. These explain the tools developed to apply the AdS/CFT correspondence in these set-ups.

Condensed matter and strange metallic behaviour: There has been a lot of work in using the AdS/CFT framework in condensed matter applications. The reader is encouraged to read some of the excellent reviews on the subject [283, 296, 385, 284, 285], and references therein. My goal in these paragraphs is to emphasise the use of IR probe branes to extract dynamical information about certain observables in quantum field theories in a state of finite charge density at low temperatures.

Before describing the string theory set-ups, it is worth attempting to explain why any AdS/CFT application may be able to capture any relevant physics for condensed matter systems. Consider the standard Fermi liquid theory, describing, among others, the conduction of electrons in regular metals. This theory is an example of an IR free fixed point, independent of the UV electron interactions, describing the lowest energy fermionic excitations taking place at the Fermi surface κ = κf. Despite its success, there is experimental evidence for the existence of different “states of matter”, which are not described by this effective field theory. This could be explained by additional gapless bosonic excitations, perhaps arising as collective modes of the UV electrons. For them to be massless, the system must either be tuned to a quantum critical point or there must exist a kinematical constraint leading to a critical phase.

One interesting possibility involving this mechanism consists on the emergence of gauge fields (“photons”) at the onset of such critical phases. For example, 2 + 1 Maxwell theory in the presence of a Fermi surface (chemical potential μ)
$${\mathcal L} = - {1 \over 4}{F^2} + \bar \psi \Gamma \cdot \left({(i\partial + A) + {\Gamma ^0}\,\mu} \right)\psi ,$$
(465)
is supposed to describe at energies below κ, the interactions between gapless bosons (photons) with the fermionic excitations of the Fermi surfaces. The one-loop correction to the classical photon propagator at low energy ω and momenta κ is
$$D{(\omega ,\,k)^{- 1}} = \gamma {\omega \over {|k|}} + \vert k{\vert^2}.$$
(466)
Due to the presence of the chemical potential, this result manifestly breaks Lorentz invariance, but there exists a non-trivial IR scaling symmetry (Lifshitz scale invariance)
$$t \to {\lambda ^3}\,t,\qquad \qquad \vert x \vert \to \lambda \vert x \vert\,,$$
(467)
with dynamical exponent z = 3, replacing the UV scaling {t, ∣x ∣} → λ {t, ∣x ∣}. Since these systems are believed to be strongly interacting, it is an extremely challenging theoretical task to provide a proper explanation for them. It is this strongly-coupled character and the knowledge of the relevant symmetris that suggest one search for similar behaviour in “holographic dual” descriptions.
The general set-up, based on the discussions appearing, among others, in [334, 398, 286], is as follows. One considers a small set of charged degrees of freedom, provided by the probe “flavour” brane, interacting among themselves and with a larger set of neutral quantum critical degrees of freedom having Lifshitz scale invariance with dynamical critical exponent z. As in previous applications, the latter is replaced by a gravitational holographic dual with Lifshitz asymptotics [324]
$$ds_{{\rm{IR}}}^2 = {L^2}\left({- {{d{t^2}} \over {{v^{2z}}}} + {{d{v^2}} \over {{v^2}}} + {{d{x^2} + d{y^2}} \over {{v^2}}}} \right),$$
(468)
where υ will play the role of the holographic radial direction. Turning on non-trivial temperature corresponds to considering black holes having the above asymptotics [162, 370, 102, 34]
$$ds_{{\rm{IR}}}^2 = {L^2}\left({- {{f(v)d{t^2}} \over {{v^{2z}}}} + {{d{v^2}} \over {f(v){v^2}}} + {{d{x^2} + d{y^2}} \over {{v^2}}}} \right),$$
(469)
where the function f (υ) depends on the specific solution and characterises the thermal nature of the system.
In practice, one embeds the probe “flavour” brane in the spacetime holographic dual, which may include some non-trivial cycle wrapping in internal dimensions when embedded in string theory, and turns on some non-trivial electric (Φ(υ)) and magnetic fluxes (B) on the brane
$$V = \Phi (v)dt + Bxdy.$$
(470)
At low energies and in a quantum critical system, the only available scales are external, i.e., given by temperature T, electric and magnetic fields {E, B} and the density of charge carriers Jt. Solving the classical equations of motion for the world volume gauge field, allows one to integrate Φ(υ); whose constant behaviour at infinity, i.e., at υ → 0 in the above coordinate system, defines the chemical potential μ of the system. Working in an ensemble of fixed charge carrier density Jt, which is determined by computing the variation of the action with respect to \(\delta V_t^{(0)} = \delta \mu\), the free energy density f is given by
$$f \equiv {F \over {{\rm{vo}}{{\rm{l}}_2}}} = {{T{S_{{\rm{D}}p}}} \over {{\rm{vo}}{{\rm{l}}_2}}} + \mu {J^t},$$
(471)
where vol2 stands for the volume of the non-compact 2-space spanned by {x, y} and SDp is the on-shell Dp-brane action. As in any thermodynamic system, observables such as specific heat or magnetic susceptibility can be computed from Eq. (471) by taking appropriate partial derivatives. Additionally, transport observables, such as DC, AC or DC Hall conductivities can also be computed and studied as a function of the background, probe embedding and the different constants controlling the world volume gauge field (470).

More than the specific physics, which is nicely described in [334, 398, 286], what is important to stress, once more, is that using the appropriate backgrounds, exciting the relevant degrees of freedom and considering the adequate boundary conditions make the methods described in this review an extremely powerful tool to learn about physics in regimes of parameters that would otherwise be very difficult to handle, both analytically and conceptually.

7 Multiple Branes

The physics of multiple overlapping branes provides a connection between braue physics and non-abelian supersymmetric field theories. Thus, it has played a crucial role in the geometrisation of the latter and the interplay between string and field theory dualities.

An heuristic argument suggesting that the abelian description may break down comes from the analysis of BIons. All half-BPS probe branes described in this review feel no force when probing the background describing N − 1 parallel branes of the same nature [484]. This means they can sit at any distance . Consider a Dp-brane in the background of N − 1 parallel Dp-branes. As soon as the probe approaches the location of the Dp-branes sourcing the geometry, the properly regularised mass of the open string (BIon) stretching between the probe D-brane and the background D-branes will tend to zero [227]. This suggests the potential emergence of extra massless modes in the spectrum of these open strings. If so, this would signal a breakdown in the effective action, since these extra modes were not included in the former. U-duality guarantees that similar considerations apply to other brane set-ups not having a microscopic theory with which to test this phenomena.

In this section, I will briefly discuss the supersymmetric effective actions describing N coincident Dp-branes and M2-branes in a Minkowski background. These correspond to non-abelian super-Yang-Mills (SYM) theories in different dimensions and certain d = 3 superconformal field theories with non-dynamical gauge fields having Chern-Simons actions, respectively.

7.1 D-branes

The perturbative description of D-branes in terms of opens strings [423] allows one to answer the question regarding the enhancement of massless modes raised above in a firmer basis, at least at weak coupling. Consider the spectrum of open strings in the presence of two parallel Dp-branes separated by a physical distance . As the latter approaches zero, i.e., it becomes smaller than the string scale, there is indeed an enhancement in the number of massless modes. Its origin is in the sector of open strings stretching between D-branes, which is precisely the one captured by the BIon argument. This enhancement is consistent with an enhancement in the gauge symmetry from U(1) × U(1), corresponding to the two separated D-branes, to U(2), corresponding to the overlapping D-branes. The spectrum of massless excitations is then described by a non-abelian vector supermultiplet in the adjoint representation. To understand how this comes about, consider the set of massless scalar excitations. These are described by (Xi)rs, where i labels the transverse directions to the brane, as in the abelian discussion, and the subindices r, s label the D-branes where the open strings are attached. This is illustrated in Figure 10. Since the latter are oriented, there exist N2N such excitations, which arrange themselves into a matrix \({X^i} = {X^i}_a{T^a}\), with Ta being generators of U(2) in the adjoint representation. The conclusion is valid for any number N of D-branes of world volume dimension p + 1 [496].
Figure 10

Open strings stretched between multiple branes and their matrix representation

Super-Yang-Mills action: The previous discussion identifies the appropriate degrees of freedom to describe the low energy dynamics of multiple D-branes in Minkowski at weak coupling as non-abelian vector supermultiplets. Thus, multiple brane effective actions must correspond to supersymmetric non-abelian gauge field theories in p +1 dimensions. At lowest order in a derivative expansion, these are precisely super-Yang-Mills (SYM) theories. For simplicity of notation, let me focus on d =10 U(N) SYM with classical action
$$S = \int {{d^{10}}} \sigma \left({- {1 \over 4}{\rm{Tr}}\,{F_{\mu \nu}}{F^{\mu \nu}} + {i \over 2}{\rm{Tr}}\,\bar \psi {\Gamma ^\mu}{D_\mu}\psi} \right)$$
(472)
where the field strength
$${F_{\mu \nu}} = {\partial _\mu}{A_\nu} - {\partial _\nu}{A_\mu} - i{g_{Y\,M}}[{A_\mu},{A_\nu}]$$
(473)
is the curvature of a U(N) hermitian gauge field Aμ and ψ is a 16-component Majorana-Weyl spinor of SO(1, 9). Both fields, Aμ and ψ are in the adjoint representation of U(N). The covariant derivative of ψ is given by
$${D_\mu}\psi = {\partial _\mu}\psi - i{g_{{\rm{YM}}}}[{A_\mu},\psi ],$$
(474)
where gYM is the Yang-Mills coupling constant. This action is also usually written in terms of rescaled fields, by absorbing a factor of gYM in both Aμ and ψ to pull an overall coupling constant dependence in front of the full action
$$S = {1 \over {4g_{{\rm{YM}}}^2}}\int {{d^{10}}} \sigma \,\left({- {\rm{Tr}}\,{F_{\mu \nu}}{F^{\mu \nu}} + 2i{\rm{Tr}}\,\bar \psi {\Gamma ^\mu}{D_\mu}\psi} \right),$$
(475)
where Dμ ψ = μ ψ − i [Aμ, ψ
The action (472) is invariant under the supersymmetry transformation
$$\begin{array}{*{20}c} {\delta {A_\mu} = {i \over 2}\bar \epsilon {\Gamma _\mu}\psi ,\quad \quad} \\ {\delta \psi = - {1 \over 4}{F_{\mu \nu}}{\Gamma ^{\mu \nu}}\epsilon ,} \\ \end{array}$$
(476)
where ϵ is a constant Majorana-Weyl spinor in SO(1, 9), giving rise to 16 independent supercharges. Classically, this is a well-defined theory; quantum mechanically, it is anomalous. From the string theory perspective, as explained in Section 3.7, this is just an effective field theory, valid at low energies \(E\sqrt {{\alpha {\prime}}} \ll 1\) and weak coupling gs ≪ 1. Dimensional reduction: The low energy effective action for multiple parallel Dp-branes in Minkowski is SYM in p + 1 dimensions. This theory can be obtained by dimensional reduction of the ten-dimensional super Yang-Mills theory introduced above. Thus, one proceeds as described in Section 3.3: assume all fields are independent of coordinates σp+1, …, σ9. After dimensional reduction, the 10-dimensional gauge field Aμ decomposes into a (p + 1)-dimensional gauge field Aα and 9 − p adjoint scalar fields XI = 2π α′ Φ45, describing the transverse fluctuations of the D-branes. The reduced action takes the form
$$S = {1 \over {4g_{{\rm{YM}}}^2}}\int {{d^{p + 1}}} \sigma \,{\rm{Tr}}(- {F_{\alpha \beta}}{F^{\alpha \beta}} - 2{({D_\alpha}{\Phi ^I})^2} + {[{\Phi ^I},{\Phi ^J}]^2} + {\rm{fermions}}).$$
(477)
The p + 1 dimensional YM coupling \(g_{{\rm{YM}}}^2\) can be fixed by matching the expansion of the square root in the gauge fixed abelian D-brane action in a Minkowski background (105) and comparing it with Maxwell’s theory in the field normalisation used in Eq. (475)
$$g_{{\rm{YM}}}^2 = {1 \over {4{\pi ^2}{\alpha {\prime 2}}{T_{{\rm{Dp}}}}}} = {{{g_s}} \over {\sqrt {\alpha \prime}}}{(2\pi \sqrt {\alpha \prime})^{p - 2}}.$$
(478)
Notice also the appearance of a purely non-abelian interaction term in Eq. (477), the commutator [ΦI, ΦJ ]2 that acts as a potential term. Indeed, its contribution is negative definite since [ΦI, ΦJ ]† = [ΦJ, ΦI ] = −[ΦI, ΦJ ].
The classical vacuum corresponds to static configurations minimising the potential. This occurs when both the curvature Fαβ and the fermions vanish, and for a set of commuting matrices, at each point of the p + 1 world volume. In this situation, the fields ΦI can be simultaneously diagonalised, so that one has
$${\Phi ^I} = \left({\begin{array}{*{20}c} {x_1^I} & 0 & 0 & \ddots \\ 0 & {x_2^I} & \ddots & 0 \\ 0 & \ddots & \ddots & 0 \\ \ddots & 0 & 0 & {x_N^I} \\ \end{array}} \right).$$
(479)
The N diagonal elements of the matrix ΦI are interpreted as the positions of N distinct D-branes in the I-th transverse direction [496]. Consider a vacuum describing N − 1 overlapping Dp-branes and a single parallel D-brane separated in a transverse direction ΦI. This is equivalent to breaking the symmetry group to U(N − 1) × U(1) by choosing a diagonal matrix for Φ with x0 eigenvalue in the first N − 1 diagonal entries and xNx0 in the last diagonal entry. The off-diagonal components δΦ will acquire a mass, through the Higgs mechanism. This can be computed by expanding the classical action around the given vacuum. One obtains that this mass is proportional to the distance ∣x0xN ∣ between the two sets of branes
$${M^2} = {{{{({x_0} - {x_N})}^2}} \over {2\pi {\alpha \prime}}},$$
(480)
according to the geometrical interpretation given to the eigenvalues characterising the vacuum. In light of the open string interpretation, these off-diagonal components do precisely correspond to the open strings stretching between the different D-branes. The latter allow an alternative description in terms of the BIon configurations described earlier, by replacing the N − 1 Dp-branes by its supergravity approximation, though the latter is only suitable at large distances compared to the string scale.
It can then be argued that the moduli space of classical vacua for (p + 1)-dimensional SYM is
$${{{{({\mathbb R^{9 - p}})}^N}} \over {{S_N}}}.$$
(481)
Each factor of ℝ stands for the position of the N D-branes in the (9 − p)-dimensional transverse space, whereas the symmetry group SN is the residual Weyl symmetry of the gauge group. The latter exchanges D-branes, indicating they should be treated as indistinguishable objects.

A remarkable feature of this D-brane description is that a classical geometrical interpretation of D-brane configurations is only available when the matrices ΦI are simultaneously diagonalisable. This provides a rather natural venue for non-commutative geometry to appear in D-brane physics at short distances, as first pointed out in [496].

The exploration of further kinematical and dynamical properties of these actions is beyond the scope of this review. There are excellent reviews on the subject, such as [424, 472, 320], where the connection to Matrix Theory [48] is also covered. If the reader is interested in understanding how T-duality acts on non-abelian D-brane effective actions, see [471, 221]. It is also particularly illuminating, especially for readers not used to the AdS/CFT philosophy, to appreciate that by integrating out N − 1 overlapping D-branes at one loop, one is left with an abelian theory describing the remaining (single) D-brane. The effective dynamics so derived can be reinterpreted as describing a single D-brane in the background generated by the integrated N − 1 D-branes, which is AdS5 × S5 [365]46. This is illustrated in Figure 11.
Figure 11

Integrating out the degrees of freedom at one loop corresponding to N − 1 of the D-branes gives rise to an effective action interpretable as an abelian gauge theory in an AdS throat.

Given the kinematical perspective offered in this review and the relevance of the higher order α ′ corrections included in the abelian DBI action, I want to discuss two natural stringy extensions of the SYM description
  1. 1.

    Keeping the background fixed, i.e., Minkowski, it is natural to consider the inclusion of higher-order corrections in the effective action, matching the perturbative scattering amplitudes computed in the CFT description of open strings theory, and

     
  2. 2.

    Allowing to vary the background or equivalently, coupling the non-abelian degrees of freedom to curved background geometries. This is towards the direction of achieving a hypothetical covariant formulation of these actions, a natural question to ask given its relevance for the existence of the kappa invariant formulation of abelian D-branes.

     
In the following, I shall comment on the progress and the important technical and conceptual difficulties regarding the extensions of these non-abelian effective actions.

Higher-order corrections: In the abelian theory, it is well known that the DBI action captures all the higher-order corrections in α ′ to the open string effective action in the absence of field strength derivative terms47 [214]. It was further pointed that such derivative corrections were compatible with a DBI expansion by requiring conformal invariance for the bosonic string in [1] and for the superstring in [87].

In the non-abelian theory, such distinction is ambiguous due to the identity
$$[{D_\mu},\,{D_\nu}]{F_{\rho \sigma}} = [{F_{\mu \nu}},\,{F_{\rho \sigma}}],$$
(482)
relating commutators with covariant derivatives. It was proposed by Tseytlin [482] that the non-abelian extension of SYM including higher-order α′ corrections be given in terms of the symmetrised prescription. The latter consists of treating all Fμν matrices as commuting. Equivalently, the action is completely symmetric in all monomial factors of of the form tr(FF). This reproduces the F2 and α2F4 terms of the full non-abelian action, but extends it to higher orders
$${{\mathcal L}_{{\rm{DBI}}}} \propto {\rm{Str}}\sqrt {{\eta _{\mu \nu}} + 2\pi \alpha \prime {F_{\mu \nu}}} .$$
(483)
The notation Str defines this notion of symmetrised trace for each of the monomials appearing in the expansion of its arguments. For an excellent review describing the history of these calculations, motivating this prescription and summarising the most relevant properties of this action, see [485].
It is important to stress that, a priori, worldsheet calculations involving an arbitrary number of boundary disk insertions could determine this non-abelian effective action. Since this is technically hard, one can perform other consistency checks. For example, one can compare the D-brane BPS spectrum on tori in the presence of non-trivial magnetic fluxes. This is T-dual to intersecting D-branes, whose spectrum can be independently computed and compared with the fluctuation analysis of the proposed symmetrised non-abelian prescription. It was found in [291, 175, 448] that the proposed prescription was breaking down at order (α′)4F6. Further checks at order α3 and α4 were carried over in [103, 346, 345, 347]. The proposal in [346] was confirmed by a first principle five-gluon scattering amplitude at tree level in [387]. The conclusion is that the symmetrised prescription only works up to F4
$${\mathcal L} = {\rm{Str}}\left[ {{1 \over 4}F_{\mu \nu}^2 - {1 \over 8}{{(2\pi \alpha \prime)}^2}\left({{F^4} - {1 \over 4}{{(F_{\mu \nu}^2)}^2}} \right) + O(\alpha {\prime 4})} \right]$$
(484)
$$\begin{array}{*{20}c} {= {\rm{tr}}\left[ {{1 \over 4}F_{\mu \nu}^2 - {1 \over {12}}{{(2\pi \alpha \prime)}^2}\left({{F_{\mu \nu}}{F_{\rho \nu}}{F_{\mu \lambda}}{F_{\rho \lambda}} + {1 \over 2}{F_{\mu \nu}}{F_{\rho \nu}}{F_{\rho \lambda}}{F_{\mu \lambda}}} \right.} \right.} \\ {\left. {- {1 \over 4}{F_{\mu \nu}}{F_{\mu \nu}}{F_{\rho \lambda}}{F_{\rho \lambda}} - {1 \over 8}{F_{\mu \nu}}{F_{\rho \lambda}}{F_{\mu \nu}}{F_{\rho \lambda}}} \right)\left. {+ O(\alpha {\prime 4})} \right]\,.\quad \quad} \\ \end{array}$$
(485)
These couplings were first found in its Str form in [266] and in its tr form in [482]. For further checks on Tseytlin’s proposal using the existence of bound states and BPS equations, see the analysis in [115, 114].
Coupling to arbitrary curved backgrounds: The above corrections attempted to include higher-order corrections describing the physics of multiple D-branes in Minkowski. More generally, one is interested in coupling D-branes to arbitrary closed string backgrounds. In such situations, one would like to achieve a covariant formulation. This is non-trivial because as soon as the degrees of freedom become non-abelian, they lose their geometrical interpretation. In the abelian case, XI described the brane location. In the non-abelian case, at most, only their eigenvalues \(x_i^I\) may keep their interpretation as the location of the ith brane in the Ith direction. Given the importance and complexity of the problem, it is important to list a set of properties that one would like such a formulation to satisfy. These are the D-geometry axioms [186]. For the case of D0-branes, these follow.
  1. 1.

    It must contain a unique trace since this is an effective action derived from string theory disk diagrams involving many graviton insertions in their interior and scalar/vector vertex operators on their boundaries. Since the disk boundary is unique, the trace must be unique.

     
  2. 2.

    It must reduce to N-copies of the particle action when the matrices XI are diagonal.

     
  3. 3.

    It must yield masses proportional to the geodesic distance for off-diagonal fluctuations.

     
Having in mind that we required spacetime gauge symmetries to be symmetries of the abelian brane effective actions, it would be natural to include in the above list invariance under target space diffeomorphisms. This was analysed for the effective action kinetic terms in [172]. Instead of discussing this here, I will discuss two non-trivial checks that any such formulation must satisfy.
  1. a)

    to match the Matrix theory linear couplings to closed string backgrounds, and

     
  2. b)

    to be T-duality covariant, extending the notion I discussed in Section 3.3.2 for single D-branes.

     
The first was studied in [473, 474] and the second in [395]. Since the results derived from the latter turned out to be consistent with the former, I will focus on the implementation of T-duality covariance for non-abelian D-branes below.
As discussed in Section 3.3, T-duality is implemented by a dimensional reduction. This was already applied for SYM in Eq. (477). Using the same notation introduced there and denoting the world volume direction along which one reduces by ρ, one learns that FμρDμΦp where Φp is the T-dual adjoint matrix scalar. Furthermore, covariant derivatives of transverse scalar fields ΦI become
$${D_\rho}{\Phi ^I} = {\partial _\rho}{\Phi ^I} + i[{A_\rho},{\Phi ^I}] = i[{A_\rho},{\Phi ^I}].$$
(486)
Notice this contribution is purely non-abelian and it can typically contribute non-trivially to the potential terms in the effective action. To properly include these non-trivial effects, Myers [395] studied the consequences of requiring T-duality covariance taking as a starting point a properly covariantised version of the multiple D9-brane effective action, having assumed the symmetrised trace prescription described above. Studying T-duality along 9-p directions and imposing T-duality covariance of the resulting action, will generate all necessary T-duality compatible commutators, which would have been missed otherwise. This determines the DBI part of the effective action to be [395]
$${S_{{\rm{DBI}}}} = - {T_{{\rm{D}}p}}\int {{d^{p + 1}}} \sigma \,{\rm{STr}}\left({{e^{- \phi}}\sqrt {- \det \left({P\left[ {{E_{\mu \nu}} + {E_{\mu I}}{{({Q^{- 1}} - \delta)}^{IJ}}{E_{J\nu}}} \right] + \lambda \,{F_{\mu \nu}}} \right)\,\det ({Q^I}_J)}} \right)\,,$$
(487)
with
$${E_{\mu \nu}} = {g_{\mu \nu}} + {B_{\mu \nu}},\qquad \qquad {Q^I}_J \equiv {\delta ^I}_J + i\lambda \,[{\Phi ^I},{\Phi ^K}]\,{E_{KJ}},\qquad {\rm{and}}\qquad \lambda = 2\pi {\alpha \prime}.$$
(488)
Here μ, ν indices stand for world volume directions, and I, J indices for transverse directions. To deal with similar commutators arising from the WZ term, one considers [395]
$${S_{{\rm{WZ}}}} = {T_{{\rm{Dp}}}}\int {{\rm{STr}}} \left({P\left[ {{e^{i\lambda \,{{\rm{i}}_\Phi}{{\rm{i}}_\Phi}}}(\sum {{C^{(n)}}} \,{e^B})} \right]{e^{\lambda \,F}}} \right),$$
(489)
where the interior product iΦ is responsible for their appearance, for example, as in,
$${{\rm{i}}_\Phi}{{\rm{i}}_\Phi}{C_2} = {\Phi ^J}{\Phi ^I}\,C_{IJ}^2 = {1 \over 2}C_{IJ}^2\,[{\Phi ^J},{\Phi ^I}].$$
(490)
Notice one regards ΦI as a vector field in the transverse space. In both actions (487) and (489), stands for pullback and it only applies to transverse brane directions since all longitudinal ones are non-physical. Its presence is confirmed by scattering amplitudes calculations [342, 271, 222]. Some remarks are in order.
  1. 1.
    There exists some non-trivial dependence on the scalars ΦI through the arbitrary bosonic closed backgrounds appearing in the action. The latter is defined according to
    $$\begin{array}{*{20}c} {{g_{\mu \nu}} = \exp \left[ {\lambda {\Phi ^i}\,{\partial _{{x^i}}}} \right]g_{\mu \nu}^0({\sigma ^a},{x^i}){\vert_{{x^i} = 0}}\quad \quad \quad \quad \quad \quad \quad} \\ {= \sum\limits_{n = 0}^\infty {{{{\lambda ^n}} \over {n!}}} \,{\Phi ^{{i_1}}} \cdots {\Phi ^{{i_n}}}\,({\partial _{{x^{{i_1}}}}} \cdots {\partial _{{x^{{i_n}}}}})g_{\mu \nu}^0({\sigma ^a},{x^i}){\vert_{{x^i} = 0}}.} \\ \end{array}$$
    (491)
    Analogous definitions apply to other background fields.
     
  2. 2.

    There exists a unique trace, because this is an open string effective action that can be derived from worldsheet disk amplitudes. The latter has a unique boundary. Thus, there must be a unique gauge trace [186, 188]. Above, the symmetrised prescription was assumed, not only because one is following Tseytlin and this was his prescription, but also because there are steps in the derivation of T-duality covariance that assumed this property and the scalar field ΦI dependence on the background fields (491) is symmetric, by definition.

     
  3. 3.

    The WZ term (489) allows multiple Dp-branes to couple to RR potentials with a form degree greater than the dimension of the world-volume. This is a purely non-abelian effect whose consequences will be discussed below.

     
  4. 4.

    There are different sources for the scalar potential: det QIj, its inverse in the first determinant of the DBI and contributions coming from commutators coupling to background field components in the expansion (491).

     
It was shown in detail in [395], that the bosonic couplings described above were consistent with all the linear couplings of closed string background fields with Matrix Theory degrees of freedom, i.e., multiple D0-branes. These couplings were originally computed in [473] and then extended to Dp-branes in [474] using T-duality once more. We will not review this check here in detail, but as an illustration of the above formalism, present the WZ term for multiple D0-branes that is required to do such matching
$$\begin{array}{*{20}c} {S{S_{{\rm{WZ}}}} = {\mu _0}\int {{\rm{Tr}}} \,\left(P \right.\left[ {{C_1} + i\lambda \,{{\rm{i}}_\Phi}{{\rm{i}}_\Phi}\left({{C_3} + {C_1} \wedge B} \right)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \right.} \\ {- {{{\lambda ^2}} \over 2}{{({{\rm{i}}_\Phi}{{\rm{i}}_\Phi})}^2}\left({{C_5} + {C_3} \wedge B + {1 \over 2}{C_1} \wedge B \wedge B} \right)\quad \quad \quad \quad \quad \quad \quad \quad} \\ {- i{{{\lambda ^3}} \over 6}{{({{\rm{i}}_\Phi}{{\rm{i}}_\Phi})}^3}\left({{C_7} + {C_5} \wedge B + {1 \over 2}{C_3} \wedge B \wedge B + {1 \over 6}{C_1} \wedge B \wedge B \wedge B} \right)} \\ {\left. {\left. {\quad \quad \quad \quad + {{{\lambda ^4}} \over {24}}{{({{\rm{i}}_\Phi}{{\rm{i}}_\Phi})}^4}\left({{C_9} + \left({{C_7} + {1 \over 2}{C_5} \wedge B + {1 \over 6}{C_3} \wedge B \wedge B + {1 \over {24}}{C_1} \wedge B \wedge B \wedge B} \right) \wedge B} \right)} \right]} \right)} \\ {= {\mu _0}\int d t\,{\rm{TR}}\,\left({C_t^1 + \lambda \,C_I^1{D_t}{\Phi ^I} + i{\lambda \over 2}(C_{tJK}^3\,[{\Phi ^K},{\Phi ^J}] + \lambda \,C_{IJK}^3\,{D_t}{\Phi ^I}\,[{\Phi ^K},{\Phi ^J}]) + \ldots} \right).} \\ \end{array}$$
(492)
Two points are worth emphasising about this matching:
  1. 1.

    There is no ambiguity of trace in the linear Matrix theory calculations. Myers’ suggestion is to extend this prescription to non-linear couplings.

     
  2. 2.

    Some transverse M5-brane charge couplings are unknown in Matrix theory, but these are absent in the Lagrangian above. This is a prediction of this formulation.

     
One of the most interesting physical applications of the couplings derived above is the realisation of the dielectric effect in electromagnetism in string theory. As already mentioned above, the non-abelian nature of the degrees of freedom turns on new commutator couplings with closed string fields that can modify the scalar potential. If so, instead of the standard SYM vacua, one may find new potential minima with Tr ΦI = 0 but Tr (ΦI)2 ≠ 0. As a toy illustrative example of this phenomenon, consider N D0-branes propagating in Minkowski but in a constant background RR four-form field strength
$$R_{tIJK}^4 = \left\{{\begin{array}{*{20}c} {- 2f{\epsilon _{IJK}}} \\ 0 \\ \end{array}} \right.\quad \;\begin{array}{*{20}c} {{\rm{for}}\,I,J,K \in \{1,\,2,\,3\}} \\ {{\rm{otherwise}}} \\ \end{array} .$$
(493)
Due to gauge invariance, one expects a coupling of the form
$${i \over 3}{\lambda ^2}{\mu _0}\int d t\,{\rm{Tr}}\,\,\left({{\Phi ^I}{\Phi ^J}{\Phi ^K}} \right)R_{tIJK}^4(t)\,.$$
(494)
Up to total derivatives, this can indeed be derived from the cubic terms in the WZ action above. This coupling modifies the scalar potential to
$$V(\Phi) = - {{{\lambda ^2}{T_0}} \over 4}{\rm{Tr}}\,({[{\Phi ^I},{\Phi ^J}]^2}) - {i \over 3}{\lambda ^2}{\mu _0}{\rm{Tr}}\,\,\left({{\Phi ^I}{\Phi ^J}{\Phi ^K}} \right)R_{tIJK}^4(t),$$
(495)
whose extremisation condition becomes
$$0 = [[{\Phi ^I},{\Phi ^J}],{\Phi ^K}] + i\,f{\varepsilon _{IJK}}[{\Phi ^J},{\Phi ^K}].$$
(496)
The latter allows SU(2) solutions
$${\Phi ^I} = {f \over 2}\,{\alpha ^I}\qquad {\rm{with}}\qquad [{\alpha ^I},{\alpha ^J}] = 2i\,{\varepsilon _{IJK}}\,{\alpha ^K},$$
(497)
having lower energy than standard commuting matrices
$${V_N} = - {{{\pi ^2}\ell _s^3{f^4}} \over {6{g_s}}}N({N^2} - 1).$$
(498)

It is reassuring to compare the description above with the one available using the abelian formalism describing a single brane explained in Section 3. I shall refer to the latter as dual brane description. For the particular example discussed above, since the D0-branes blow up into spheres due to the electric RR coupling, one can look for on-shell configurations on the abelian D2-brane effective action in the same background corresponding to the expanded spherical D0-branes in the non-abelian description. These configurations exist, reproduce the energy VN up to 1/N2 corrections and carry no D2-brane charge [395]. Having reached this point, I am at a position to justify the expansion of pointlike gravitons into spherical D3-branes, giant gravitons, in the presence of the RR flux supporting AdS5 × S5 described in Section 5.9. The non-abelian description would involve non-trivial commutators in the WZ term giving rise to a fuzzy sphere extremal solution to the scalar potential. The abelian description reviewed in Section 5.9 corresponds to the dual D3-brane description in which, by keeping the same background, one searches for on-shell spherical rotating D3-branes carrying the same charges as a pointlike graviton but no D3-brane charge. For a more thorough discussion of the comparison between non-abelian solitons and their “dual” abelian descriptions, see [147, 149, 148, 396].

Kappa symmetry and superembeddings: The covariant results discussed above did not include fermions. Whenever these were included in the abelian case, a further gauge symmetry was required, kappa symmetry, to keep covariance, manifest supersymmetry and describe the appropriate on-shell degrees of freedom. One suspects something similar may occur in the non-abelian case to reduce the number of fermionic degrees of freedom in a manifestly supersymmetric non-abelian formulation. It is important to stress that at this point world volume diffeomorphisms and kappa symmetry will no longer appear together. In all the discussions in this section, world volume diffeomorphisms are assumed to be fixed, in the sense that the only scalar adjoint matrices already correspond to the transverse directions to the brane.

Given the projective nature of kappa symmetry transformations, it may be natural to assume that there should be as many kappa symmetries as fermions. In [79], a perturbative approach to determining such transformation
$${\delta _\kappa}{\bar \theta ^A} = {\bar \kappa ^B}(\sigma)\left({1{\delta ^{BA}} + {\Gamma ^{BA}}(\sigma)} \right)\,,\qquad \qquad A,\,B = 1,2, \ldots {N^2}$$
(499)
was analysed for multiple D-branes in super-Poincaré. The idea was to expand the WZ term in covariant derivatives of the fermions and the gauge field strength F, involving some a priori arbitrary tensors. One then computes its kappa symmetry variation and attempts to identify the DBI term in the action at the same order by satisfying the requirement that the total action variation equals
$${\delta _\kappa}{\mathcal L} = - {\delta _\kappa}\bar \theta (1 - \Gamma)\,{\mathcal T},$$
(500)
order by order. In a sense, one is following the same strategy as in [9], determining the different unknown tensors order by order. Unfortunately, it was later concluded in [76] that such an approach could not work.

There exists some body of work constructing classical supersymmetric and kappa invariant actions involving non-abelian gauge fields representing the degrees of freedom of multiple D-branes. This started with actions describing branes of lower co-dimension propagating in lower dimensional spacetimes [461, 462, 190]. It was later extended to multiple D0-branes in an arbitrary number of dimensions, including type IIA, in [411]. Here, both world volume diffeomorphisms and kappa symmetry were assumed to be abelian. It was checked that when the background is super-Poincaré, the proposed action agreed with Matrix Theory [48]. Using the superembedding formalism [460], actions were proposed reproducing the same features in [40, 44, 42, 41, 43], some of them involving a superparticle propagating in arbitrary 11-dimensional backgrounds. Finally, there exists a slightly different approach in which, besides using the superembedding formalism, the world sheet Chan-Paton factors describing multiple D-branes are replaced by boundary fermions. The actions constructed in this way in [303], based on earlier work [304], have similar structure to the ones described in the abelian case, their proof of kappa symmetry invariance is analogous and they reproduce Matrix Theory when the background is super-Poincaré and most of the features highlighted above for the bosonic couplings described by Myers.

Relation to non-commutative geometry: There are at least two reasons why one may expect non-commutative geometry to be related to the description of multiple D-brane actions:
  1. 1.

    D-brane transverse coordinates being replaced by matrices,

     
  2. 2.

    the existent non-commutative geometry description of D-branes in the presence of a B-field in space-time (or a magnetic field strength on the brane) [187, 146, 444].

     
The general idea behind non-commutative geometry is to replace the space of functions by a non-commutative algebra. In the D-brane context, a natural candidate to consider would be the algebra
$${\mathcal A} = {C^\infty}(M) \otimes {M_N}(C).$$
(501)
As customary in non-commutative geometry, the latter does not yet carry any metric information. Following Connes [145], the construction of a Riemannian structure requires a spectral triple (\(({\mathcal A},{\mathcal H}, D)\)), which, in addition to \({\mathcal A}\), also contains a Hilbert space \({\mathcal H}\) and a self-adjoint operator D obeying certain properties. It would be interesting to find triples (\({\mathcal A},{\mathcal H}, D\)) that describe, in a natural way, metrics relevant for multiple D-branes, incorporating the notion of covariance.

Regarding D-branes in the presence of a B-field, the main observation is that the structure of an abelian non-commutative gauge theory is similar to that of a non-abelian commutative gauge theory. In both cases, fields no longer commute, and the field strengths are non-linear. Moreover, non-commutative gauge theories can be constructed starting from a non-abelian commutative theory by expanding around suitable backgrounds and taking N → ∞ [443]. This connection suggests it may be possible to relate the gravity coupling of non-commutative gauge theories to the coupling of non-abelian D-brane actions to curved backgrounds (gravity). This was indeed the approach taken in [163] where the stress-tensor of non-commutative gauge theories was derived in this way. In [151], constraints on the kinematical properties of non-abelian D-brane actions due to this connection were studied.

7.2 M2-branes

In this section, I would like to briefly mention the main results involving the amount of progress recently achieved in the description of N parallel M2-branes, referring to the relevant literature when appropriate. This will be done taking the different available perspectives on the subject: a purely kinematic approach, based on supersymmetry and leading to 3-algebras, a purely field theory approach leading to three dimensional CFTs involving Chern-Simons terms, a brane construction approach, in which one infers the low energy effective description in terms of an intersection of branes and the connection between all these different approaches.

The main conclusion is that the effective theory describing N M2-branes is a d =3, U(N)×U(N) gauge theory with four complex scalar fields CI (I =1, 2, 3,4) in the (\({\bf{N}},{\bf{\bar N}}\)) representation, their complex conjugate fields in the (\({\bf{\bar N}},{\bf{N}}\)) representation and their fermionic partners [12]. The theory includes non dynamical gauge fields with a Chern-Simons action with levels κ and −κ for the two gauge groups. This gauge theory is weakly coupled in the large κ limit (κ ≫ N) and strongly coupled in the opposite regime (κ ≪ N), for which a weakly coupled gravitational description will be available if N ≫ 1.

Supersymmetry approach: Inspection of the = 3 SYM supersymmetry transformations and the geometrical intuition coming from M2-branes suggest that one look for a supersymmetric field theory with field content involving eight scalar fields \({X^I} = X_a^I{T^a}\)48 and their fermionic partners Ψ = ΨaTa, and being invariant under a set of supersymmetry transformations whose most general form is
$$\begin{array}{*{20}c} {\delta X_d^I = i\bar \epsilon {\Gamma ^I}{\Psi _d},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\delta {\Psi _d} = {\partial _\mu}X_d^I{\Gamma ^\mu}{\Gamma ^I}\epsilon - {1 \over 6}X_a^IX_b^JX_c^K{f^{abc}}_d{\Gamma ^{IJK}}\epsilon + {1 \over 2}X_a^JX_b^JX_c^I{g^{abc}}_d{\Gamma ^I}\epsilon .} \\ \end{array}$$
This was the original approach followed in [26], based on a real vector space with basis Ta, a = 1, … N, endowed with a triple product
$$[{T^a},{T^b},{T^c}] = {f^{abc}}_d\,{T^d},$$
(502)
where the set of \({f^{abc}}_d\) are real, fully antisymmetric in a, b, c and satisfy the fundamental identity
$${f^{\left[ {abc} \right.}}_e{f^{\left. d \right]ef}}_g = 0\,.$$
(503)
Closure of the supersymmetry algebra requires Eq. (503), but also shows the appearance of an extra gauge symmetry [26]. To deal properly with the latter, one must introduce an additional (non-dynamical) gauge field \({{\tilde A}_\mu}^cd\) requiring one to consider a more general set of supersymmetry transformations [27, 274]
$$\begin{array}{*{20}c} {\delta X_d^I = i\bar \epsilon {\Gamma ^I}{\Psi _d}\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {,\,\delta {\Psi _d} = {D_\mu}X_d^I{\Gamma ^\mu}{\Gamma ^I}\epsilon - {1 \over 6}X_a^IX_b^JX_c^K{f^{abc}}_d{\Gamma ^{IJK}}\epsilon + {1 \over 2}X_a^JX_b^JX_c^I{g^{abc}}_d{\Gamma ^I}\epsilon} \\ {,\,\delta \tilde A_{\mu \,\,d}^{\,\,\,c} = i\bar \epsilon {\Gamma _\mu}{\Gamma _I}X_a^I{\Psi _b}{h^{abc}}_d\,.\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(504)
Here Dμ is a covariant derivative, whereas \({g^{abc}}_d\) and \({h^{abc}}_d\) define triple products on the algebra.

Closure of the supersymmetry algebra determines a set of equations of motion that can be derived, which form a Lagrangian. It was soon realised that under the assumptions of a real vector space, essentially the only 3-algebra is the one defined by \({f^{abc}} = {f^{abc}}_e{h^{ed}}\), with hab = Tr (Ta, Tb) defining an inner product, and satisfying \({f^{abcd}} \propto {\varepsilon ^{abcd}}\) [399, 412, 226]. Interestingly, it was pointed out in [488] that such supersymmetric field theory could be rewritten as a Chern-Simons theory. The latter provided a link between a purely kinematic approach, based on supersymmetry considerations, and purely field theoric results that had independently been developed.

Field theory considerations: Conformal field theories have many applications. In the particular context of Chern-Simons matter theories in d = 3, they can describe interesting IR fixed points in condensed matter systems. Here I am interested in their supersymmetric versions to explore the AdS4/CFT3 conjecture.

Let me start this overview with \({\mathcal N} = 2\) theories. \({\mathcal N} = 2\) Chern-Simons theories coupled to matter49 include a vector multiplet A, the dimensional reduction of the four dimensional \({\mathcal N} = 1\) vector multiplet, in the adjoint representation of the gauge group, and chiral multiplets Φi in representations Ri of the latter. Integrating out the D-term equation and the gaugino, one is left with the action
$$\begin{array}{*{20}c} {{S^{{\mathcal N} = 2}} = \int {{k \over {4\pi}}{\rm{Tr}}\,(A \wedge dA + {2 \over 3}{A^3}) + {D_\mu}{{\bar \phi}_i}{D^\mu}{\phi _i} + i{{\bar \psi}_i}{\gamma ^\mu}{D_\mu}{\psi _i}\quad \quad \quad \quad \quad \quad \quad \quad}} \\ {- {{16{\pi ^2}} \over {{k^2}}}({{\bar \phi}_i}T_{{R_i}}^a{\phi _i})({{\bar \phi}_j}T_{{R_j}}^b{\phi _j})({{\bar \phi}_k}T_{{R_k}}^aT_{{R_k}}^b{\phi _k}) - {{4\pi} \over k}({{\bar \phi}_i}T_{{R_i}}^a{\phi _i})({{\bar \psi}_j}T_{{R_j}}^a{\psi _j})} \\ {- {{8\pi} \over k}({{\bar \psi}_i}T_{{R_i}}^a{\phi _i})({{\bar \phi}_j}T_{{R_j}}^a{\psi _j}),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ \end{array}$$
(505)
where ϕi and ψi are the bosonic and fermionic components of the chiral superfield ψi and the gauge field A is non-dynamical.
There are \({\mathcal N} = 3\) generalisations, but since their construction is more easily argued for starting with the field content of an \({\mathcal N} = 4\) theory, let me review the latter first. The field content of the \({\mathcal N} = 4\) theories adds an auxiliary (non-dynamical) chiral multiplet φ in the adjoint representation of G and pairs chiral multiplets Φi, \({{\tilde \Phi}_i}\) into a set of hypermultiplets by requiring them to transform in conjugate representations, as the notation suggests. The theory does not contain Chern-Simons terms, but a superpotential \(W = {{\tilde \Phi}_i}\varphi {\Phi _i}\) for each pair. \({\mathcal N} = 3\) theories are constructed by the addition of Chern-Simons terms, as in Eq. (505), and the extra superpotential \(W = - {k \over {8\pi}}{\rm{Tr(}}{\varphi ^2})\). Integrating out φ leads to a superpotential
$$W = {{4\pi} \over k}({\tilde \Phi _i}T_{{R_i}}^a{\Phi _i})({\tilde \Phi _j}T_{{R_j}}^a{\Phi _j}).$$
(506)
The resulting \({\mathcal N} = 3\) theory has the same action as Eq. (505) with the addition of the above superpotential.
In [12], an \({\mathcal N} = 6\) theory based on the gauge group U (N) × U (N) was constructed. Its field content includes two hypermultiplets in the bifundamental and the Chern-Simons levels of the two gauge groups were chosen to be equal but opposite in sign. Denoting the bifundamental chiral superfields by A1, A2 and their anti-bifundamental by B1, B2, the superpotential then equals
$$W = {k \over {8\pi}}{\rm{Tr}}\,(\varphi _{(2)}^2 - \varphi _{(1)}^2) + {\rm{Tr}}\,({B_i}{\varphi _{(1)}}{A_i}) + {\rm{Tr}}\,({A_i}{\varphi _{(2)}}{B_i})\,.$$
(507)
After integrating out the auxiliary fields φ(i)
$$W = {{2\pi} \over k}{\rm{Tr}}\,({A_i}{B_i}{A_j}{B_j} - {B_i}{A_i}{B_j}{A_j}) = {{4\pi} \over k}{\rm{Tr}}\,({A_1}{B_1}{A_2}{B_2} - {A_1}{B_2}{A_2}{B_1}).$$
(508)
As discussed in [12], the four bosonic fields \({C_I} \equiv ({A_1},{A_2}, B_1^\ast, B_2^\ast)\) transform in the 4 of SU(4), matching the generic \({\rm{SO(}}{\mathcal N})\) R-symmetry in d =3 super-CFTs. For a more thorough discussion of global symmetries and gauge invariant observables, see [12].

It was argued in [12] that the \({\mathcal N} = 6\) theory constructed above was dual to N M2-branes on \({{\rm{{\mathbb C}}}^4}/{{\rm{{\mathbb Z}}}_k}\) for k≥ 3. Below, I briefly review the brane construction in which their argument is based. This will provide a nice example of the notion of geometrisation (or engineering) of supersymmetric field theories provided by brane configurations.

Brane construction: Following the seminal work of [282], one can associate low energy effective field theories with the dynamics of brane configurations stretching between branes. Consider a set of N D3-branes wrapping the x6 direction and ending on different NS5-branes according to the array
$$\begin{array}{*{20}c} {{\rm{NS}}5\,\,:\,1\,2\,3\,4\,5\,\_\,\_\,\_\,\_\quad \quad} \\ {{\rm{NS}}5\,\,:\,1\,2\,3\,4\,\_\,\_\,\_\,\_\,\_\quad \quad} \\ {{\rm{D}}3\,:\,\,1\,2\,\_\,\_\,\_\,6\,\_\,\_\,\_\,.\quad} \\ \end{array}$$
(509)
This gives rise to an \({\mathcal N} = 4\,{\rm{U(}}N) \times {\rm{U(}}N)\) gauge theory in d = 1 + 2 dimensions, along the {x1, x2} directions, whose field content includes a vector multiplet in the adjoint representation and 2 complex bifundamental hypermultiplets, describing the transverse excitations to both D3-branes and NS5-branes [282].
Adding κ D5-branes, as illustrated in the array below,
$$\begin{array}{*{20}c} {{\rm{NS}}5\,\,:\,1\,2\,3\,4\,5\,\_\,\_\,\_\,\_\quad \quad} \\ {{\rm{NS}}5\,\,:\,1\,2\,3\,4\,5\,\_\,\_\,\_\,\_\quad \quad} \\ {{\rm{D}}5\,:\,\,1\,2\,3\,4\_\,\_\,\_\,\,\_\,9\quad} \\ {\,{\rm{D}}3\,:\,\,1\,2\,\_\,\_\,\_\,6\,\_\,\_\,\_\,,\quad} \\ \end{array}$$
(510)
breaks supersymmetry to \({\mathcal N} = 2\) and adds κ massless chiral multiplets in the N and \({{\rm{\bar N}}}\) representation of each of the U (N) gauge group factors. Field theoretically, this \({\mathcal N} = 2\) construction allows a set of mass deformations that can be mapped to different geometrical notions [282, 72, 12]:
  1. 1.

    Moving the D5-branes along the 78-directions generates a complex mass parameter.

     
  2. 2.

    Moving the D5-branes along the 5-directions generates a real mass, of positive sign for the fields in the fundamental representation and of negative sign for the ones in the antifundamental.

     
  3. 3.

    Breaking the κ D5-branes and NS5-branes along the 01234 directions and merging them into an intermediate (1) 5-brane bound state generates a real mass of the same sign for both N and \({{\rm{\bar N}}}\) representations. This mechanism is a web deformation [72]. The merging is characterised by the angle θ relative to the original NS5-brane subtended by the bound state in the 59-plane. The final brane configuration is made of a single NS5-brane in the 012345 directions and a (1, κ) 5-brane in the 01234[5, 9]θ, where [5, 9]g stands for the x5 cos θ+x4 sin θ direction. θ is fixed by supersymmetry [14].

     

After the web deformation and at low energies, one is left with an \({\mathcal N} = 2 \, U{(N)_k} \times U{(N)_{- k}}\). Yang-Mills-Chern-Simons theory with four massless bi-fundamental matter multiplets (and their complex conjugates), and two massless adjoint matter multiplets corresponding to the motion of the D3-branes in the directions 34 common to the two 5-branes.

The enhancement to an \({\mathcal N} = 3\) theory described in the purely field theoretical context is realised in the brane construction by rotating the (1, κ) 5-brane in the 37 and 48-planes by the same amount as in the original deformation. Thus, one ends with a single NS5-brane in the 012345 and a (1, κ) 5-brane along 012[3, 7]θ[4, 8]θ[5, 9]θ.This particular mass deformation ensures all massive adjoint fields acquire the same mass, enhancing the symmetry to \({\mathcal N} = 3\). Equivalently, there must exist an SO(3)R R-symmetry corresponding to the possibility of having the same SO(3) rotations in the 345 and 789 subspaces. Thus, the d = 3 supersymmetric field theory must be \({\mathcal N} = 3\).

The connection to \({\mathcal N} = 6\) is obtained by flowing the \({\mathcal N} = 3\) theory to the IR [12]. Indeed, by integrating out all the massive fields, we recover the field content and interactions described in the field theoretical \({\mathcal N} = 6\) construction. The enhancement to \({\mathcal N} = 8\) for κ = 1, 2 was properly discussed in [276].

It was realised in [12] that under T-duality in the x6 direction and uplifting the configuration to M-theory, the brane construction gets mapped to N M2-branes probing some configuration of KK-monopoles. These have a supergravity description in terms of hyper-Kähler geometries [224]. Flowing to the IR in the dual gravitational picture is equivalent to probing the near horizon of these geometries, which includes the expected AdS4 factor times a quotient of the 7-sphere.

The Chern-Simons theory has a 1/κ coupling constant. Thus, large κ has a weakly coupled description. At large N, it is natural to consider the ’t Hooft limit: λ = N/k fixed. The gauge theory is weakly coupled for κ ≫ N and strongly coupled for κ ≪ N. In the latter situation, the supergravity description becomes reliable and weakly coupled for N ≫ 1 [12].

Matching field theory, branes and 3-algebra constructions: The brane derivation of the supersymmetric field theory relevant to describe multiple M2-branes raised the natural question for what the connection was, if any, with the 3-algebra formulation that stimulated all these investigations. The answer was found in [28]. The main idea was to consider a 3-algebra based on a complex vector space endowed with a triple product
$$[{T^a},{T^b};{\bar T^{\bar c}}] = {f^{ab\bar c}}_d\,{T^d},$$
(511)
and an inner product
$${h^{\bar ab}} = {\rm{Tr}}\left({{{\bar T}^a}{T^b}} \right).$$
(512)
The change in the notation points out antisymmetry only occurs in the first two indices. Furthermore, the constants \({f^{ab\bar c}}_d\) satisfy the following fundamental identity,
$${f^{ef\bar g}}_b{f^{cb\bar a}}_d + {f^{fe\bar a}}_b{f^{cb\bar g}}_d + {f^{*\bar g\bar af}}_{\bar b}{f^{ce\bar b}}_d + {f^{*\bar a\bar ge}}_{\bar b}{f^{cf\bar b}}_d = 0.$$
(513)
It was proven in [28] that this set-up manages to close the algebra on the different fields giving rise to some set of equations of motion. In particular, the \({\mathcal N} = 6\) conformal field theories described in [12] could be rederived for the particular choices
$${f^{ab\bar c\bar d}} = - {f^{ba\bar c\bar d}},\qquad {\rm{and}}\qquad {f^{ab\bar c\bar d}} = {f^{*\bar c\bar dab}}.$$
(514)
Thus, the 3-algebra approach based on complex vector spaces is also suitable to describe these string theory models. Furthermore, it provides us with a mathematical formalism capable of describing more general set-ups.

8 Related Topics

There are several topics not included in previous sections that are also relevant to the subjects covered in this review. The purpose of this last section is to mention some of them, mentioning their main ideas and/or approaches, and more importantly, referring the reader to some of the relevant references where they are properly developed and explained.

Superembedding approach: The GS formulation consists in treating the bulk spacetime as a supermanifold while keeping the bosonic nature of the world volume. The superembedding formalism is a more symmetric formulation, in which both bulk and world volume are described as super-manifolds. As soon as the world volume formulation is extended into superspace, it incorporates extra degrees of freedom, which are non-physical. There exists a geometrically natural interpretation for the set of constraints, first discussed in [463], imposed to remove them. Given a target space supervielbein EM (Z) = (Ea, Eα) and world volume superconnection eA (σ,η) = (ea, eα), where η stands for the new world volume fermionic coordinates, then the pullback of the bosonic component can be expanded as
$${E^a}(Z(\sigma ,\,\eta)) = {e^b}E_b^a + {e^\alpha}E_\alpha ^a.$$
(515)
The constraint consists in demanding
$$E_\alpha ^a(Z(\sigma ,\eta)) = 0.$$
(516)
This means that at any world volume point, the tangent space in the Grassmann directions forms a subspace of the Grassmann tangent space in the bulk.

There are many results in this subject, nicely reviewed in [460]. It is worth mentioning that some equations of motion for supersymmetric objects in different numbers of dimensions were actually first derived in this formalism rather than in the GS one, including [220] for the d =10 superparticle, [47] for the superstring and supermembrane, [306] for superbranes and [305] for the M5-brane50. It is particularly relevant to stress the work done in formulating the M5-brane equations of motion covariantly [307, 308] and their use to identify supersymmetric world volume solitons [301, 302], and in pointing out the relation between superembeddings and non-linear realisations of supersymmetry [5].

MKK-monopoles and other exotic brane actions: This review was focused on the dynamics of D-branes and M-branes. It is well known that string and M theory have other extended objects, such as KK-monopoles or NS5-branes. There is a nice discussion regarding the identification of the degrees of freedom living on these branes in [311]. Subsequently, effective actions were written down to describe the dynamics of its bosonic sectors in [83, 80, 208, 209]. In particular, it was realised that gauged sigma models are able to encapsulate the right properties for KK monopoles. The results obtained in these references are consistent with the action of T-duality and S-duality. Of course, it would be very interesting to include fermions in these actions and achieve kappa symmetry invariance.

Blackfolds: The blackfold approach is suitable to describe the effective world volume dynamics of branes, still in the probe approximation, having a thermal population of excitations. In some sense, it describes the dynamics of these objects on length scales larger than the brane thickness. This formalism was originally developed in [201, 202] and extended and embedded in string theory in [203]. It was applied to the study of hot BIons in [261, 262], emphasising the physical features not captured by the standard Dirac-Born-Infeld action, and to blackfolds in AdS [20].

Non-relativistic kappa invariant actions: All the branes described in this review are relativistic. It is natural to study their non-relativistic limits, both for its own sake, but also as an attempt to identify new sectors of string theory that may be solvable. The latter is the direction originally pursued in [246, 161] by considering closed strings in Minkowski. This was extended to closed strings in AdS5 × S5 in [244]. At the level of brane effective actions in Minkoswki, non-relativistic diffeomorphism and kappa symmetry invariant versions of them were obtained in [245] for D0-branes, fundamental strings and M2-branes, and later extended to general Dp-branes in [247]. The consistency of these non-relativistic actions under the action of duality transformations was checked in [330]. This work was extended to non-relativistic effective D-brane actions in AdS5 × S5 in [119, 436].

Multiple M5-branes: It is a very interesting problem to find the non-abelian formulation of the (2,0) tensor multiplet describing the dynamics of N M5-branes. Following similar ideas to the ones used in the construction of the multiple M2-brane action using 3-algebras, some non-abelian representation of the (2,0) tensor supermultiplet was found in [351]. Their formulation includes a non-abelian analogue of the auxiliary scalar field appearing in the PST formulation of the abelian M5-brane. Closure of the superalgebra provides a set of equations of motion and constraints. Expanding the theory around a particular vacuum gives rise to d = 5 SYM along with an abelian (2,0) d = 6 supermultiplet. This connection to d = 5 SYM was further studied in [352]. Some further work along this direction can be found in [299]. Some of the BPS equations derived from this analysis were argued to be naturally reinterpreted in loop space [414]. There has been a different approach to the problem involving non-commutative versions of 3-algebras [275], but it seems fair to claim that this remains a very important open problem for the field.

Footnotes

  1. 1.

    Non-perturbative in the sense that their mass goes like 1/gs, where gs is the string-coupling constant.

  2. 2.

    The discovery of the RNS model of interacting bosons and fermions in d =10 critical dimensions is due to joining the results of the original papers [432, 404]. This was developed further in [405, 232].

  3. 3.

    The calculations of beta functions in general nonlinear sigma models were done in [17, 215]. For a general discussion of string theory in curved backgrounds see [124] or the discussions in books [260, 425].

  4. 4.

    Polyakov used the formulation of classical string theory in terms of an auxiliary world sheet metric [116, 176] to develop the modern approach to the path integral formulation of string theory in [427, 428].

  5. 5.

    Recently, it was pointed out in [390] that there may exist quantum-mechanically consistent superstrings in d =3. It remains to be seen whether this is the case.

  6. 6.

    The existence of kappa symmetry as a fermionic gauge symmetry was first pointed out in superparticle actions in [169, 170, 451, 171]. Though the term kappa symmetry was not used in these references, since it was later coined by Townsend, the importance of WZ terms for its existence is already stated in these original works.

  7. 7.

    For a proper definition of these superfields, see Appendix A.1.

  8. 8.

    See Appendix A.1 for a better discussion of what this means.

  9. 9.

    See [490, 421] for reviews and textbooks on what an effective field theory is and what the principles behind them are.

  10. 10.

    I will introduce this notion more thoroughly in Section 3.4 and Appendix A.

  11. 11.

    Here, \({\mathcal N}\) stands for the number of world volume supersymmetries.

  12. 12.

    The first examples of this phenomena were reported by Nambu [401] and Goldstone [242].

  13. 13.

    For earlier reviews on D-brane effective actions and on M-brane interactions, see [320] and [101], respectively.

  14. 14.

    Since I am not considering supersymmetric branes at this point, |Qp| = Tp is not a necessary condition.

  15. 15.

    This is the correct way to compute the energy momentum tensor due to the coupling of branes to gravity. The energy carried by such a brane must be localised on its p + 1 dimensional world volume.

  16. 16.

    The importance of these assumptions will be stressed when discussing the regime of validity of brane effective actions in Section 3.7.

  17. 17.

    There actually exist further gravitational interaction terms necessary for the cancellation of anomalies [253], but we will always omit them in our discussions concerning D-brane effective actions.

  18. 18.

    For a discussion on the interpretation of an M5-brane as a ‘D-brane’ for an open membrane, see [55].

  19. 19.

    Relevant work on the subject includes [24, 77, 16, 75].

  20. 20.

    For a discussion of the supersymmetric and kappa invariant M5-brane covariant action propagating in super-Poincaré, see [144].

  21. 21.

    Following the same philosophy as for their bosonic truncations, this functional dependence can be derived from the double dimensional reduction of the supersymmetric M2-brane action to be discussed in Section 3.4.2 [477, 439]. This also provides a derivation of the WZ couplings to be constructed in this subsection. Of course, this consideration would only apply to the D2-brane, but T-duality would allow one to extend this conclusion for any Dp-brane [292, 331]

  22. 22.

    For p =1, B2 is the NS-NS 2-form, whereas for p = 2, B3 = A3 is the d =11 3-form gauge potential.

  23. 23.

    The derivation of the property (147) is made more manifest in formalisms in which the tension is generated dynamically by the addition of an auxiliary volume density [86, 356, 94].

  24. 24.

    I will prove this explicitly in Section 5.1.

  25. 25.

    All our charge conjugation matrices are antisymmetric and unitary, i.e., CT = −C and \({C^\dagger}C = 1\). Furthermore, all Clifford matrices satisfy the symmetry relation \(\Gamma _m^T = - C{{\rm{T}}_m}{C^{- 1}}\).

  26. 26.

    For earlier work, see [4], which extended the original Volkov-Akulov approach in [489].

  27. 27.

    I have assumed both the background and the brane preserve some supersymmetry.

  28. 28.

    There exists some similar phenomena on the M5-brane dynamics with the self-dual 3-form field strength. See [74] for a discussion on the emergence of noncommutative gauge theories when the self-dual 3-form field strength is close to its critical value.

  29. 29.

    There are several claims in the literature advocating that extra massless degrees of freedom emerge in brane effective actions when the latter probe black holes very close to their horizons. See [322, 323, 313, 300] for interesting work in this direction.

  30. 30.

    BPS stands for Bogomolny, Prasad and Sommerfield and their work on stable solitonic configurations [107, 429].

  31. 31.

    For a complete and detailed discussion of the supersymmetric and kappa invariant D-brane Hamiltonian formalism, see [94], which extends the bosonic results in [356, 327] and the type IIB superMinkowski ones in [329]. Here I follow [94] even though the analysis is restricted to the bosonic sector.

  32. 32.

    This notation is introduced to emphasise that \({\mathcal G}_5^{ij}\) does not correspond to the world space components of \({{\mathcal G}^{\mu \nu}}\), but to the inverse matrix of the restriction of \({{\mathcal G}^{\mu \nu}}\) to the world space subspace.

  33. 33.

    There are many papers studying the dynamics of BIons, including [353, 31, 335] and [475], where the solution to the Born-Infeld action reviewed here is proven to solve the equations of motion derived from higher-order corrections to the effective action.

  34. 34.

    For simplicity I am setting the D3-brane tension to one.

  35. 35.

    This equation has a huge history in mathematical physics. For a self-contained presentation on all the mathematical developments regarding this equation, see [178]. For generalisations to higher dimensions, see [180, 179].

  36. 36.

    Strictly speaking, if the supertube cross-section is open, they can carry D2-brane charge. The arguments given above only apply to closed cross-sections. The reader is encouraged to read the precise original discussion in [380] concerning this point and the bounds on angular momentum derived from it.

  37. 37.

    For a list of reviews on this subject, see [382, 383, 65, 459, 36, 454].

  38. 38.

    By arbitrary, it is meant a general curve that is not self-intersecting and whose tangent vector never vanishes.

  39. 39.

    I do not write this term explicitly here because it will not couple to our D3-brane probes.

  40. 40.

    Dual giant gravitons are spherical rotating D3-branes in which the 3-sphere wrapped by the brane is in AdS5. See [264] for a proper construction of these configurations and some of its properties.

  41. 41.

    These configurations appeared in [64, 71], extending earlier seminal work [173, 53].

  42. 42.

    What is meant here by maximally entropic is that, given a large black hole, there may be more than one possible deconstruction of the total charge in terms of constituents with different charge composition. By maximally entropic I mean the choice of charge deconstruction whose moduli space of configurations carries the largest contribution to the entropy of the system.

  43. 43.

    This overall coupling constant is derived from the string tension 1/2πα ′ and the overall \({L^2} \sim {g_s}N{\alpha {\prime}}\) scale from the AdS5 × S5 background geometry.

  44. 44.

    For an analysis of supersymmetric D5-branes in a supergravity background dual to \({\mathcal N} = 1\,{\rm{SYM}}\), see [408].

  45. 45.

    ΦI is the natural adjoint scalar field after dimensional reduction. The rescaling by 2π α′ is to match the natural scalar fields appearing in the abelian description provided by the DBI action. A similar rescaling occurs for the fermions omitted below, Ψ = 2π α′ ψ.

  46. 46.

    There is a lot of work in this direction. For a review on the emergence of geometry and gravity in matrix models, in particular in the context of the IKKT matrix conjecture [316], see [465]. For more recent discussions, see [106].

  47. 47.

    Using T-duality arguments this would also include acceleration and higher-derivative corrections in the scalar sector Xm describing the excitations of the D-brane along the transverse dimensions.

  48. 48.

    Eight is the number of transverse dimensions to the world volume of the M2-branes.

  49. 49.

    For a complete list of references, see [12].

  50. 50.

    The equivalence of the equations of motion obtained in the PST-formalism and the ones developed in the superembedding formalism was proven in [46].

  51. 51.

    The reader should keep in mind that the RR field strengths Rn with n ≥ 5 are non-physical, in the sense that they are Hodge duality related to the physical propagating degrees of freedom contained in the RR field strengths Rl0−n [185, 196].

  52. 52.

    The chirality can be α1 or αi, depending on the reality of the volume form eigenspace.

  53. 53.

    Same comments as above.

Notes

Acknowledgements

Some of the material covered in this review is based on the PhD thesis World Volume Approach to String Theory defended by the author under the supervision of Prof. J. Gomis at the University of Barcelona in May 2000. JS would like to thank his PhD advisor J. Gomis for introducing him to this subject, to P. K. Townsend for sharing his extensive knowledge on many topics reviewed here, and to F. Brandt, K. Kamimura, O. Lunin, D. Mateos, A. Ramallo and J.M. Figueroa-O’Farrill for discussions and collaboration on part of the material reported in this work. The work of JS was partially supported by the Engineering and Physical Sciences Research Council [grant number EP/G007985/1].

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