Abstract
This is a review on brane effective actions, their symmetries and some of their applications. Its first part covers the GreenSchwarz formulation of single M and Dbrane 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 onshell 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’nyiPrasadSommerfield (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 nontrivial dynamics of stronglycoupled 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 Dbrane 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 Dbranes and M2branes. This includes both SuperYangMills theories, their higherorder corrections and partial results in covariantising these couplings to curved backgrounds, and the more recent supersymmetric ChernSimons matter theories describing M2branes using field theory, brane constructions and 3algebra considerations.
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1 Introduction
Branes have played a fundamental role in the main string theory developments of the last twenty years:

1.
The unification of the different perturbative string theories using duality symmetries [312, 495] relied strongly on the existence of nonperturbative supersymmetric states carrying RamondRamond (RR) charge for their first tests.

2.
The discovery of Dbranes as being such nonperturbative states, but still allowing a perturbative description in terms of open strings [423].

3.
The existence of decoupling limits in string theory providing nonperturbative 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 nonperturbative 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.
Dualities guarantee that fundamental strings are no more fundamental than other dynamical extended objects in the theory, called branes.

2.
Dbranes, a subset of the latter, are nonperturbative states^{Footnote 1} defined as dynamical hypersurfaces where open strings can end. Their weaklycoupled dynamics is controlled by the microscopic conformal field theory description of open strings satisfying Dirichlet boundary conditions. Their spectrum contains massless gauge fields. Thus, Dbranes provide a window into nonperturbative string theory that, at low energies, is governed by supersymmetric gauge theories in different dimensions.

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 supersymmetric gauge theories emerging as brane effective field theories in string and Mtheory, and some of their important applications. In particular, I will focus on Dbranes, M2branes and M5branes. For a schematic representation of the review’s content, see Figure 1.
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 GreenSchwarz worldsheet one for the superstring. The main concepts I want to stress in this part are

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

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,

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

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

e)
the description of the regime of validity of these effective actions.
For multiple coincident branes, these theories are supersymmetric nonabelian gauge field theories. The second main difference from the abelian setup 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 onshell supergravity backgrounds. Thus, they offer powerful tools to study the dynamics of string/Mtheory 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

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

b)
prove the existence of string theory Bogomol’nyiPrasadSommerfield (BPS) states carrying different (topological) charges,

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

a)
classical onshell probe action calculations providing a window to strongly coupled dynamics, spectrum and thermodynamics of nonabelian gauge theories by working with appropriate backgrounds with suitable boundary conditions,

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 GreenSchwarz 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 selfcontained. 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 setup 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 onshell degrees of freedom. As a warmup 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 nontrivial 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 onshell supergravity background. After introducing the effective actions, I discuss both their global bosonic and fermionic symmetries in Section 3.6, emphasising the difference between spacetime 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 onshell classical brane action calculations in specific curved backgrounds and with appropriate boundary conditions, as holographic duals of stronglycoupled 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 nonrelativistic limits. This is intended to be an illustration of the power of the probe approximation technique, rather than a selfcontained 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 nonabelian description of N Dbranes and M2branes. Regarding Dbranes, this includes an introduction to superYangMills theories in p + 1 dimensions, a summary of statements regarding higherorder corrections in these effective actions and the more relevant results and difficulties regarding the attempts to covariantise these couplings to arbitrary curved backgrounds. Regarding M2branes, I briefly review the more recent supersymmetric ChernSimons matter theories describing their low energy dynamics, using field theory, 3algebra 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 NS5branes and KKmonopoles, nonrelatistivistic kappa symmetry invariant brane actions, blackfolds or the prospects to achieve a formulation for multiple M5branes.
In appendices, I provide a brief but selfcontained introduction to the superspace formulation of the relevant supergravity theories discussed in this review, describing the explicit constraints required to match the onshell 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 higherdimensional flat spaces. I establish a onetoone map between the geometrical Killing spinors in AdS and spheres and the covariantlyconstant Killing spinors in their embedding flat spaces.
2 The GreenSchwarz Superstring: A Brief Motivation
The purpose of this section is to briefly review the GreenSchwarz (GS) formulation of the superstring. This is not done in a selfcontained 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.
The worldsheet supersymmetry formulation, called the RamondNeveuSchwarz (RNS) formulation^{Footnote 2}, where supersymmetry in 1 + 1 dimensions is manifest [432, 404].

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 g_{mn}
The latter is a onedimensional diffeomorphic invariant action equaling the physical length of the particle trajectory times its mass m. Its degrees of freedom X^{m} (τ) are the set of maps describing the embedding of the trajectory with affine parameter τ into spacetime, i.e., the local coordinates x^{m} of the spacetime manifold become dynamical fields X^{m} (τ) on the world line. Diffeomorphisms correspond to the physical freedom in reparameterising the trajectory.
The bosonic string action equals its tension T_{f} times its area
This is the NambuGoto (NG) action [402, 249]: a 1 + 1 dimensional field theory with coordinates σ^{μ} μ = 0,1 describing the propagation of a Lorentzian worldsheet, through the set of embeddings X^{m} (σ) m = 0,1 …d − 1, in a fixed ddimensional Lorentzian spacetime with metric g_{mn} (X). Notice, this is achieved by computing the determinant of the pullback of the spacetime metric into the worldsheet
Thus, it is a nonlinear interacting theory in 1 + 1 dimensions. Furthermore, it is spacetime covariant, invariant under twodimensional diffeomorphisms and its degrees of freedom {X^{m}} are scalars in two dimensions, but transform as a vector in ddimensions.
Just as point particles can be charged under gauge fields, strings can be charged under 2forms. The coupling to this extra field is minimal, as corresponds to an electricallycharged object, and is described by a WessZumino (WZ) term
where the charge density Q_{f} was introduced and \({\mathcal B}\) stands for the pullback of the ddimensional bulk 2form B_{(2)}, i.e.,
Thus, the total bosonic action is:
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 NeveuSchwarz NeveuSchwarz (NSNS) 2form B_{2}(X)). In that case,
where ℳ_{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 tensor^{Footnote 3}. This is illustrated in Figure 2.
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 supersymmetry. Thus, besides the spacetime vector {X^{m}}, a set of 1 + 1 scalars fields θ^{α} 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
where ϵ^{A} 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
This uses the Polyakov form of the action^{Footnote 4} involving an auxiliary twodimensional metric h_{μν}. Π_{μν} stands for the components of the supersymmetric invariant 1forms
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 onshell 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
Invariance under global supersymmetry requires, up to total derivatives, the identity
for (Ψ_{1}, Ψ_{2}, Ψ_{3}) = (θ, θ′ = ∂θ/∂σ^{1}, θ = dθ/∂σ^{0}). 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 MajoranaWeyl.
Let us focus on the last case, which is well known to match the superspace formulation of \({\mathcal N} = 2\) type IIA/B^{Footnote 5} 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 onshell bosonic and fermionic degrees of freedom remains unequal. Indeed, MajoranaWeyl fermions in d =10 have 16 real components, which get reduced to 8 onshell components by Dirac’s equation. The extra \({\mathcal N} = 2\) gives rise to a total of 16 onshell fermionic degrees of freedom, differing from the 8 bosonic ones coming from the 10dimensional vector representation after gaugefixing 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.^{Footnote 6} This feature fixes the fermionic nature of the local parameter κ(σ) and requires θ to transform by some projector operator
Here is a Cliffordvalued matrix depending nontrivially on {X^{m}, 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 superPoincaré [259]. The latter is an example of a supermanifold with local coordinates Z^{M} = {X^{m}, θ^{α}}. 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 NSNS 2form superfield B_{ac}, 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}^{Footnote 7}.
In the case of superPoincaré, the components \(E_M^A\) are explicitly given by
These objects allow us to reinterpret the action S_{1} + S_{2} in terms of the pullbacks of these bulk objects into the worldsheet extending the bosonic construction
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 onshell solution with metric g_{mn} = η_{mn}, constant dilaton and vanishing gauge potentials, dilatino and gravitino. If we work in superspace, superPoincaré is a solution to the superspace constraints having nontrivial fermionic components. The ones appearing in the NSNS 2form 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 T_{f} should be equal to the charge density Q_{f}, its supersymmetric and kappa invariant extension fixes the relation T_{f} = Q_{f}. 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]
The dependence on the background is encoded both in the superfields \(E_M^A\) and B_{ac}.
The counting of degrees of freedom is not different from the one done for superPoincaré. Thus, the GS superstring (16) still requires to be kappa symmetry invariant to have an onshell 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 onshell^{Footnote 8}. In other words, superstrings can only propagate in properly onshell backgrounds in the same theory.
It is important to stress that in the GS formulation, kappa symmetry invariance requires the background fields to be onshell, whereas in the RNS formulation, it is quantum Weyl invariance that ensures this selfconsistency 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 halfBPS branes in string theory. More precisely, to M2branes, M5branes and Dbranes.
3 Brane Effective Actions
This review is concerned with the dynamics of low energy string theory, or Mtheory, in the presence of brane degrees of freedom in a regime in which the full string (M) theory effective action^{Footnote 9} reduces to
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 S_{brane} 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 S_{brane}. This requires one to freeze the gravitational sector to its classical onshell description and to neglect its backreaction into spacetime. Thus, one requires
where T_{mn} stands for the energymomentum 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 S_{brane}. 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 3.1–3.6.
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 Dpbranes, which allow a perturbative quantum open string description, and continue with M2 and M5branes, applying the lessons learnt from strings and Dbranes.
Dpbranes: Dpbranes are p + 1 dimensional hypersurfaces Σ_{p} _{+1} where open strings can end. One of the greatest developments in string theory came from the realisation that these objects are dynamical, carry RamondRamond (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 10dimensional ℝ^{1,9} Minkowski spacetime gives rise to a perturbative spectrum containing a set of massless states that fit into an abelian vector supermultiplet of the superPoincaré 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, g_{s} ≪ 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 nonphysical and can be gauged away. Notice the full ISO(1, 9) of the vacuum is broken by the presence of the Dpbrane 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 X^{I} and a gauge field V_{1} in p+ 1 dimensions, describing p −1 physical polarisations. Thus, the total number of massless bosonic degrees of freedom is
Notice the number of world volume scalars X^{I} matches the number of transverse translations broken by the Dpbrane and transform as a vector under the transverse Lorentz subgroup SO(9−p), which becomes an internal symmetry group. Geometrically, these modes X^{I} (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 Dpbranes propagate in 10 dimensions, any covariant formalism must involve a set of 10 scalar fields X^{m} (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 Σ_{p} _{+1} in ℝ^{1,9}. Physically, the Dirichlet boundary conditions used in the open string description did fix these diffeomorphisms, since they encode the brane location in ℝ^{1,9}.
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 MajoranaWeyl spinors in 10 dimensions. Thus, matching the eight onshell 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.
Mbranes: Mbranes 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 Mbrane, extending the notion already discussed for the superstring and Dpbranes. 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 M2brane. 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 M2brane 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 M2brane. To achieve spacetime covariance, one must increase the number of scalar fields to eleven X^{m} (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 dimensions^{Footnote 10}. Such formulation involves a single copy of d = 11 Majorana fermions, which gives rise to a pair of d =10 MajoranaWeyl 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 onshell ones. Once again, kappa symmetry will be required to achieve this goal.
What about the M5brane? The fermionic discussion is equivalent to the M2brane one. The bosonic one must contain a new ingredient. Indeed, geometrically, there are only five scalars describing the transverse M5brane excitations. These do not match the eight onshell 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 twoform gauge potential V_{2} with selfdual field strength. The latter involves 62 choose 2 physical polarisations, with selfduality reducing these to three onshell 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 2form gauge potential. I will later discuss how to keep covariance while satisfying the selfduality 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 Dbranes, 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 M2brane in d =3 and X = 8 and of the superstring in d =2 and X = 8.
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 Dpbranes, starting from the D0brane (p = 0) and finishing with the D9 brane (p = 9) filling in all spacetime. Thus, the field content of all Dpbranes 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.
Finally, there is just one interesting tensor multiplet with X = 5 scalars in six dimensions, corresponding to the aforementioned M5 brane, among the sixdimensional tensor supermultiplets listed in Table 3.
Summary: All halfBPS Dpbranes, M2branes and M5branes are described at low energies by effective actions written in terms of supermultiplets in the corresponding worldvolume dimension. The number of onshell bosonic degrees of freedom is 8. Thus, the fermionic content in these multiplets satisfies
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 dynamics^{Footnote 11}. The addition of interactions must be consistent with such d dimensional supersymmetries.
By construction, an effective action written in terms of these onshell 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, nonphysical, bosonic and fermionic degrees of freedom. To preserve their nonphysical 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 worldvolume degrees of freedom and symmetries from the analysis of massless fluctuations of these solitons, applying collective coordinate techniques that are a wellknown notion for solitons in standard, nongravitational, 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] ^{Footnote 12}. These symmetries are global, whereas all brane solitons are onshell 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 setup of 5brane solitons in heterotic and type II strings. It was later extended to M2branes and M5branes in [332]. In this section, I follow the general discussion in [6] for the M2, M5 and D3branes. 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 onshell bosonic configurations to be analysed below. All of them are described by a nontrivial metric and a gauge field carrying the appropriate brane charge. The multiple M2brane solution, first found in [198], is
Here, and in the following examples, describe the longitudinal brane directions, i.e., μ = 0,1,2 for the M2brane, 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 ℝ^{8}
The structure for the M5brane, 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 M2brane and magnetic for the M5brane below
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 ℝ^{5}
The D3brane, first found in [195], similarly has a nontrivial metric and selfdual five form RR field strength
with isometry group ISO(1, 3) × SO(6). It is characterised by a single harmonic function U in ℝ^{6}
All these brane configurations are halfBPS 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 ξ_{i} (y^{P}) in our discussion are those that act nontrivially at infinity, matching a broken global transformation asymptotically ϵ_{i}, but differing otherwise in the bulk of the background geometry
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, δ_{ξi}φ_{i} and finally introduce some world volume dependence on the parameter ϵ_{i}, i.e., ϵ_{i} (x^{μ}). At that point, the transformation δ_{ξi}φ_{i} 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
If we allow \({{\bar \phi}^p}\) to arbitrarily depend on the world volume coordinates x^{μ}\({{\bar \phi}^p} \rightarrow {\phi ^p}({x^\mu})\), the perturbation h_{mn} will no longer be pure gauge. If one computes the firstorder correction to Einstein’s equations in supergravity, including the perturbative analysis of the energy momentum tensor, one discovers the lowestorder equation of motion satisfied by ϕ^{p} is
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 lowestorder contribution (in number of derivatives) to the gaugefixed worldvolume action of M2, M5 and D3branes in flat space is indeed described by the KleinGordon equation.
Fermionic modes: These must correspond to the breaking of supersymmetry. Consider the supersymmetry transformation of the 11dimensional gravitino ψ_{m}
where \({\tilde D}\) is some nontrivial 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 M5brane, as an example. The preserved supersymmetries are those satisfying δ ψ_{m} = 0. This forces \(s =  {7 \over {12}}\) and fixes the sixdimensional 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 +(x^{μ}), δ ψ_{m} becomes nonpure gauge. Plugging the latter into the original RaritaSchwinger equation, the linearised equation for the perturbation reduces to
The latter is indeed the massless Dirac equation for a chiral sixdimensional fermion. A similar analysis holds for the M2 and D3branes. The resulting perturbations are summarised in Table 4.
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 oneform nature of the gauge field. Since this must hold for any Dbrane, the natural candidate is the abelian gauge symmetry associated with the NSNS twoform
Proceeding as before, one considers a transformation with \({\Lambda _1} = {U^k}{{\bar V}_1}\) for some number k and constant oneform \({{\bar V}_1}\) When \({{\bar V}_1}\) is allowed to depend on the world volume coordinates, the perturbation
becomes physical. Plugging this into the NSNS twoform equation of motion, one derives dF = 0 where F = dV_{1} for both of the fourdimensional 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 M5brane and the requirement of world volume supersymmetry in six dimensions allowed us to identify the presence of a twoform potential with selfdual field strength. This must have its supergravity origin in the breaking of the abelian gauge transformation
where indeed the gauge parameter is a twoform. 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
becomes physical. Plugging this into the A_{3} equation of motion, we learn that each world volume duality component *xF_{3} = ±F_{3} with F_{3} = dV_{2} satisfies the bulk equation of motion if dF_{3} = 0 for a specific choice of k. More precisely, selfdual components require k =1, whereas antiselfdual 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 11dimensional spinor λ while for the M5 and D3 branes, it describes positive sixdimensional and fourdimensional 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 nonperturbative symmetry considerations in string theory to determine some of the higherorder 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 supersymmetric 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 M2brane effective action, which is purely geometric, continuing with Dbranes and their one form gauge potentials and finishing with M5branes including their selfdual three form field strength^{Footnote 13}.
Bosonic M2brane: In the absence of world volume gauge field excitations, all brane effective actions must satisfy two physical requirements

1.
Geometrically, branes are p + 1 hypersurfaces Σ_{p} _{+1} propagating in a fixed background with metric g_{mn}. Thus, their effective actions should account for their world volumes.

2.
Physically, all branes are electrically charged under some appropriate spacetime p + 1 gauge form C_{p+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 X^{m} brane degrees of freedom must be of the form
where T_{p} and Q_{p} stand for the brane tension and charge density^{Footnote 14}. The first term computes the brane world volume from the induced metric \({{\mathcal G}_{\mu \nu}}\)
whereas the second WZ term C_{p+1} describes the pullback of the target space p + 1 gauge field C_{p+1}(X) under which the brane is charged
At this stage, one assumes all branes propagate in a background with Lorentzian metric g_{mn} (X) coupled to other matter fields, such as C_{p+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 M5branes propagate in d =11 supergravity backgrounds, i.e., m,n = 0,1,… 10, and they are electrically charged under the gauge potential A_{3}(X) and its sixform dual potential A_{6}, respectively (see Appendix A for conventions). Dbranes propagate in d =10 type IIA/B backgrounds and the set {C_{p+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 = dC_{p+1}, the combined action can be written as
Here \({{\mathcal M}_D}\) stands for the Ddimensional spacetime, whereas \({\hat n}\) is a (D − p − 1)form whose components are those of an epsilon tensor normal to the brane having a δfunction support on the world volume^{Footnote 15}. Thus, the bulk equation of motion for the gauge potential C_{p+1} acquires a source term whenever a brane exists. Since the brane charge is computed as the integral of *R over any topological (D − p − 2)sphere surrounding it, one obtains
where the equation of motion was used in the last step. Thus, minimal WZ couplings do capture the brane physical charge.
Since M2branes 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
in analogy with the bosonic worldsheet string action. If Eq. (40) is viewed as the bosonic truncation of a supersymmetric M2brane, then Qm_{2} = T m_{2}. Besides its manifest spacetime covariance and its invariance under worldvolume diffeomorphisms infinitesimally generated by
this action is also quasiinvariant (invariant up to total derivatives) under the target space gauge transformation δ_{Λ}A_{3} = d Λ_{2} 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 Dbranes: Due to the perturbative description in terms of open strings [423], Dbrane 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 nonlinear generalisation of Maxwell’s electromagnetism originally proposed by Born and Infeld in [108]:
I will refer to this nonlinear action as the DiracBornInfeld (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_{vρ} 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 nonlinear gauge field dependence, one is in a position to include the dependence on the embedding scalar fields X^{m} (σ) and the coupling with nontrivial background closed string fields. Since in the absence of worldvolume gaugefield excitations, Dbrane actions should reduce to Eq. (35), it is natural to infer the right answer should involve
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} _{ν}X^{m} terms and/or higher derivative terms.^{Footnote 16} This proposal has nice properties under Tduality [24, 77, 16, 75], which I will explore in detail in Section 3.3.2 as a nontrivial check on Eq. (43). In particular, it will be checked that absence of acceleration terms is compatible with Tduality.
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 M2branes. Let me stress the two main issues separately:

1.
The functional dependence on the gauge field V_{1} in a general closed string background. Dbranes 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 δB_{2} = d Λ_{1}
$$\delta \int\nolimits_{{\Sigma _2}} b = \int\nolimits_{{\Sigma _2}} d \Lambda = \int\nolimits_{\partial {\Sigma _2}} \Lambda ,$$(44)due to the presence of boundaries. These are the Dbranes 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 δV_{1} = dX^{m} (σ)Λ_{m} under the bulk gauge transformation. Since Dbrane effective actions must be invariant under these target space gauge symmetries, this physical argument determines that all the dependence on the gauge field V_{1} should be through the gauge invariant combination \({\mathcal F} = 2\pi {\alpha {\prime}}d{V_1}  {\mathcal B}\).

2.
The coupling to the dilaton. The Dbrane effective action is an open string tree level action, i.e., the selfinteractions 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 e^{−ϕ}. 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.
The WZ couplings. Dpbranes are charged under the RR potential C_{p} _{+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 Dbrane 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 C_{r} 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 C_{p} _{+1}. An important physical consequence of this fact will be that turning on nontrivial gauge fluxes on the brane can induce nontrivial lowerdimensional Dbrane 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 Dbrane action is:^{Footnote 17}
If one views this action as the bosonic truncation of a supersymmetric Dbrane, the Dbrane 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]
Bosonic covariant M5brane: The bosonic M5brane degrees of freedom involve scalar fields and a world volume 2form with selfdual 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 selfduality 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 selfduality of the field strength of the RR 4form gauge potential. There are several solutions in the literature based on different formalisms:

1.
One natural option is to giveup Lorentz covariance and work with nonmanifestly Lorentz invariant actions. This was the approach followed in [420] for the M5brane, building on previous work [213, 295, 441].

2.
One can introduce an infinite number of auxiliary (nondynamical) fields to achieve a covariant formulation. This is the approach followed in [384, 502, 375, 177, 66, 98, 99, 100].

3.
One can follow the covariant approach due to Pasti, Sorokin and Tonin (PSTformalism) [416, 418], in which a single auxiliary field is introduced in the action with a nontrivial nonpolynomial 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.
Another option is to work with a Lagrangian that does not imply the selfduality 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 nonlinear 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 M5brane [417]
As in previous effective actions, all the dependence on the scalar fields X^{m} is through the bulk fields and their pullbacks to the sixdimensional world volume. As in Dbrane physics, all the dependence on the world volume gauge potential V_{2} is not just simply through its field strength 2, but through the gauge invariant 3form
The physics behind this is analogous. \({\mathcal F}\) describes the ability of open strings to end on Dbranes, whereas describes the possibility of M2branes to end on M5branes [469, 479]^{Footnote 18}. Its world volume Hodge dual and the tensor \({{\tilde H}_{\mu \nu}}\) are then defined as
The latter involves an auxiliary field a (σ) responsible for keeping covariance and implementing the selfduality 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 selfduality condition. The full action also includes a DBIlike 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 3form gauge potential and its Hodge dual in \({\mathcal N} = 1 \, d = 11\) supergravity [11].
Besides being manifestly invariant under sixdimensional world volume diffeomorphisms and ordinary abelian gauge transformations δV_{2} = d Λ_{1}, the action (49) is also invariant under the transformation
Given the nondynamical 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 M5brane propagating in Minkowski, the nonmanifest 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 ∂_{μ}a is a world volume vector, sixdimensional Lorentz transformations do not preserve this gauge slice. One must use a compensating gauge transformation (53), which also acts on V_{μν}. The overall gauge fixed action is invariant under the full sixdimensional Lorentz group but in a nonlinear nonmanifestly Lorentz covariant way as discussed in [420].
As a final remark, notice the charge density Q_{M5} of the bosonic M5brane 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 nontrivial 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 Mtheory and the action of Tduality on type II string theories and Dbranes. Figure 3 summarises the set of relations between the brane tensions discussed in this review under these symmetries.
Mtheory as the strong coupling limit of type IIA: From the spectrum of 1/2BPS states in string theory and Mtheory, an M2/M5brane in ℝ^{1,9} × S^{1} has a weaklycoupled description in type IIA

either as a long string or a D4brane, if the M2/M5brane wraps the Mtheory circle, respectively

or as a D2brane/NS5 brane, if the Mtheory circle is transverse to the M2/M5brane 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 ddimensional supergravity description by a d1 one using a KaluzaKlein (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]
where the lefthand side 11dimensional 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 S^{1}. In terms of the parameters of the theory, the relation between the Mtheory circle R and the 11dimensional Planck scale ℳ_{p} with the type IIA string coupling g_{s} and string length ℳ_{s} is
The same principle should hold for the brane degrees of freedom {Φ^{A}}. 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 σ^{p}, i.e., Y (σ) = σ^{p}, and keeps the zero mode in a Fourier expansion of all the remaining brane fields, i.e., Φ^{A} = Φ^{A} (σ ′) where {σ} = {σ′, σ^{p}}. 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.
Tduality on closed and open strings: From the quantisation of open strings satisfying Dirichlet boundary conditions, all Dbrane dynamics are described by a massless vector supermultiplet, whose number of scalar fields depends on the number of transverse dimensions to the Dbrane. Since Dbrane states are mapped among themselves under Tduality [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. Tduality is a perturbative string theory duality [241]. It says that type IIA string theory on a circle of radius R and string coupling g_{s} is equivalent to type IIB on a dual circle of radius R ′ and string coupling g ′_{s} related as [121, 122, 240]
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 nonvanishing 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 {φ_{a/b}} and their KK reduction to d = 9 dimensions, i.e., φ_{a} = φ_{a} (φ_{9}) and φ_{β} = φ_{β} (φ_{9}), the uniqueness of \({\mathcal N} = 2 \, d = 9\) supergravity guarantees the existence of a nontrivial map between type IIA and type IIB fields in the subset of backgrounds allowing an S^{1} compactification
This process is illustrated in the diagram of Figure 4. These are the Tduality rules. When expressed in terms of explicit field components, they become [82, 388]
These correspond to the bosonic truncations of the superfields introduced in Appendix A.1. Prime and unprimed fields correspond to both Tdual theories. The same notation applies to the tensor components where {z, z′} describe both Tdual circles. Notice the dilaton and the g_{zz} transformations do capture the worldsheet relations (56).
Let me move to the brane transformation. A D(p + 1)brane wrapping the original circle is mapped under Tduality to a Dpbrane 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 Dbrane discussion will be the presence of a gauge field and the fact that the KK reduced supergravity fields {φ_{9}} will be rewritten in terms of the Tdual tendimensional fields using the Tduality 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 Tduality and stronglycoupled considerations. Our logic is as follows. The M2brane is linked to our starting worldsheet action through doubledimensional reduction. The former is then used to derive the D2brane effective by direct dimensional reduction. Tduality covariance extends this result to any nonmassive Dbrane. Finally, to check the consistency of the PST covariant action for the M5brane, its double dimensional reduction will be shown to match the D4brane effective action. This will complete the set of classical checks on the bosonic brane actions discussed so far.
It is worth mentioning that the selfduality of the D3brane effective action under Sduality could also have been included as a further test. For discussions on this point, see [483, 252].
3.3.1 M2branes and their classical reductions
In the following, I discuss the double and direct dimensional reductions of the bosonic M2brane effective action (40) to match the bosonic worldsheet string action (6) and the D2brane 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 M2brane in an 11dimensional background of the form (54). Decompose the set of scalar fields as {X^{M}} = {X^{m}, 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 {X^{m}} along the world volume circle, i.e., ∂_{ρ}X^{m} = 0. Under these conditions, which mathematically characterise a double dimensional reduction, the WessZumino coupling becomes
where I already used the KK reduction ansatz (54). Here, \({{\mathcal B}_2}\) stands for the pullback of the NSNS two form into the surface Σ_{2} parameterised by \(\{{\sigma ^{\hat \mu}}\}\). The DBI action is reduced using the identity satisfied by the induced worldvolume metric
Since the integral over ρ equals the length of the Mtheory circle,
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 T_{f} defined in Section 2. The same argument applies to the charge density leading to Q_{f} = Q_{m} _{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 M2brane action is consistent with the relation between halfBPS M2brane and fundamental strings in the spectrum of Mtheory and type IIA.
Connection to the D2brane: The direct dimensional reduction of the bosonic M2 brane describes a threedimensional diffeomorphism invariant theory propagating in 10 dimensions, with eleven scalars as its field content. The latter disagrees with the bosonic field content of a D2brane, 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 M2brane action and after world volume dualisation of the scalar field Y along the Mtheory circle, one should reproduce the classical D2brane action [439, 477, 93, 480].
To describe the direct dimensional reduction, consider the Lagrangian [480]
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., Q_{M2} = T_{M2}. The induced world volume fields are
where
Using the properties of 3 × 3 matrices,
where \(\vert Z{\vert ^2} = {Z_\mu}{Z_\nu}{{\mathcal G}^{\mu \nu}}\), the action (62) can be written as
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
holds. Adding the latter to the action through an exact twoform F = dV Lagrange multiplier
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
Notice I already introduced the same gauge invariant quantity introduced in Dbrane Lagrangians
Since Y is now an independent field, it can be eliminated by solving its algebraic equation of motion
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
This matches the proposed D2brane effective action, since T_{M2} = T_{D2} as a consequence of Eq.s (55) and (48).
3.3.2 Tduality covariance
In this section, I extend the D2brane’s functional form to any Dpbrane using Tduality covariance. My goal is to show that the bulk Tduality rules (58) guarantee the covariance of the Dbrane effective action functional form [453] and to review the origin in the interchange between scalar fields and gauge fields on the brane^{Footnote 19}.
The second question can be addressed by an analysis of the Dbrane action bosonic symmetries. These act infinitesimally as
They involve world volume diffeomorphisms ξ^{ν}, a U(1) gauge transformation c and global transformations Δϕ^{i}. Since the background will undergo a Tduality transformation, by assumption, this set of global transformations must include translations along the circle, i.e., ΔZ = ϵ, ΔX^{m} = ΔV_{μ} = 0, where the original X^{M} scalar fields were split into {X^{m}, Z}.
I argued that the realisation of Tduality on the brane action requires one to study its doubledimensional reduction. The latter involves a partial gauge fixing Z = σ^{p} = ρ, identifying one world volume direction with the starting S^{1} bulk circle and a zeromode Fourier truncation in the remaining degrees of freedom, ∂_{ρ}X^{m} = ∂_{ρ}V_{μ} = 0. Extending this functional truncation to the pdimensional 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} Δx^{M} = ∂_{z} ΔVμ = 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., ∂_{z}sϕ^{i}_{g.f.+trunc} = 0, determines
where a, ϵ′ are constants, the latter having mass dimension minus one. The set of transformations in the double dimensional reduction are
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). V_{ρ} was a gauge field component in the original action. But in its gaugefixed functionallytruncated 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 doubledimensional reduction
Z ′ becomes the Tdual target space direction along the Tdual circle and ϵ′ describes the corresponding translation isometry. This discussion reproduces the wellknown massless open string spectrum when exchanging a Dirichlet boundary condition with a Neumann boundary condition.
Having clarified the origin of symmetries in the Tdual 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 Tdual background (g ′, B ′) and degrees of freedom ({X^{M′}} = {X^{m′, Z′}}), which will be denoted by primed quantities. This can be achieved by adding and subtracting the relevant pullback quantities. The following identities hold
It is a consequence of our previous symmetry discussion that X^{m} ^{′} = X^{m} 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
Notice that whenever the bulk Tduality rules (58) are satisfied, the functional form of the effective action remains covariant, i.e., of the form
This is because all terms in the determinant vanish except for those in the first line. Finally, \({e^{ \phi}}\sqrt {{g_{zz}}}\) equals the Tdual dilaton coupling e^{−ϕ} ′ and the original Dpbrane tension \({T_{{{\rm{D}}_{\rm{p}}}}}\) becomes the D(p1)brane tension in the Tdual theory due to the worldsheet defining properties (56) after the integration over the world volume circle
Just as covariance of the DBI action is determined by the NSNS 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
Due to the functional truncation assumed in the double dimensional reduction, the second term vanishes. The Dbrane WZ action then becomes
where \({{\mathcal F}^ } \equiv {i_{{\vartheta _\rho}}}(d\rho \wedge{\mathcal F})\) and the following conventions are used
Using the Tduality transformation properties of the gauge invariant quantity \({\mathcal F}\), derived from our DBI analysis,
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
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 ϵ^{01…p} = 1).
Due to our gaugefixing 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
These are entirely equivalent to the Tduality rules (58) but written in an intrinsic way.
The expert reader may have noticed that the RR Tduality 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 Sduality. For example, the 4form C_{4} appearing in our WZ couplings is not S selfdual, but transforms as
Using a superindex S to denote an Sdual selfdual 4form, the latter must be
Similarly, C_{6} does not transform as a doublet under Sduality, whereas
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
, which was not computed in [208].
In Section 7.1, I will explore the consequences that can be extracted from the requirement of Tduality covariance for the covariant description of the effective dynamics of N overlapping parallel Dbranes in curved backgrounds, following [395].
3.3.3 M5brane reduction
The double dimensional reduction of the M5brane effective action, both in its covariant [417, 8] and noncovariant formulations [420, 420, 78] was checked to agree with the D4brane effective action. It is important to stress that the outcome of this reduction may not be in the standard D4brane 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 manifestlysupersymmetric Lagrangians [491]. One extends the manifold ℝ^{1,3} 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 Taylorlike θ expansion
that includes auxiliary (nondynamical) components allowing one to close the supersymmetry algebra offshell. Generic superfields do not transform irreducibly under the superPoincaré group. Imposing constraints on them, i.e., f_{i} (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 ℝ^{1,9}, since these correspond to supersymmetric field theories in ℝ^{1,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 11dimensional superspace, with standard bosonic coordinates x^{m} 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.
the superspace coordinates {x^{m}, θ} will become the brane dynamical degrees of freedom {X^{m} (σ), θ (σ)}, besides any additional gauge fields living on the brane;

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 Σ_{p+1} with local coordinates σ^{μ} 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].
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 nontrivial constraints imposed on the supergravity superfields. These guarantee the onshell nature of the physical superfield components. I refer the reader to a brief but selfcontained 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 onshell. These will play a very important role in the selfconsistency 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 Dbrane and M2brane actions propagating in Minkowski spacetime, or its superspace extension, superPoincaré.^{Footnote 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 Dbranes in flat superspace
In this section, I am aiming to describe the propagation of Dbranes 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].^{Footnote 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 {Z^{M} (σ)} = {X^{m} (σ), θ^{α} (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_{#} = Γ_{0}Γ_{1} … Γ_{9}. 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), θ^{i}_{i} = 1, 2. The index is an internal SU(2) index keeping track of the doublet structure on which Pauli matrices τ_{a} act.
In either case, one defines \(\bar \theta = {\theta ^t}C\), in terms of an antisymmetric charge conjugation matrix C satisfying
with Γ_{m} satisfying the standard Clifford algebra {Γ_{m} Γ_{n}} = 2η_{mn} 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 NSNS bulk sector, a sector that is also probed by the superstring. Thus, both the supervielbein (E^{m}, E^{α}) and the NSNS 2form B_{2} were already identified to be
in type IIA, whereas in type IIB one replaces Γ_{#} by τ_{3}. The DBI action
will therefore be invariant under the spacetime supersymmetry transformations
if both, the induced world volume metric \({\mathcal G}\) and the gauge invariant 2form, \({\mathcal F}\), are. These are defined by
where \({\mathcal B}\) stands for the pullback of the superspace 2form B_{2} into the worldvolume, i.e., \({{\mathcal B}_{\mu \nu}} = {\partial _\mu}{Z^M}{\partial _\nu}{Z^N}{B_{MN}}\). Since B_{2} is quasiinvariant under (106), one chooses
so that \({\delta _\varepsilon}{\mathcal F} = 0\), guaranteeing the invariance of the action (105) since the set of 1forms Π^{m} are supersymmetric invariant.
Let me consider the WZ piece of the action
Since invariance under supersymmetry allows total derivatives, the Lagrangian can be characterised in terms of p + 2)form
satisfying
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 {X^{m}, V_{1}} in terms of an arbitrary δ_{κ}θ. 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, δ_{κ}V is chosen to remove the exact form coming from the kappa symmetry variation of B_{2}, 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 δ_{κ}θ to involve some nontrivial 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
where \({\mathcal R}\) is the pullback of the field strength of the RR gauge potential C, as defined in Eq. (523). Using Π = Π^{m} Γ_{m}, this can be written as [293]
in type IIA, whereas in type IIB [329]
Two observations are in order:

1.
\(d{{\mathcal L}_{{\rm{WZ}}}}\) is indeed manifestly supersymmetric, since it only depends on supersymmetric invariant quantities, but \({{\mathcal L}_{{\rm{WZ}}}}\) is quasiinvariant. Thus, when computing the algebra closed by the set of conserved charges, one can expect the appearance of nontrivial charges in the righthand side of the supersymmetry algebra. This is a universal feature of brane effective actions that will be conveniently interpreted in Section 3.6.

2.
This analysis has determined the explicit form of all the RR potentials C_{p} 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 Dpbrane propagating in a fixed superPoincaré 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 (ξ^{μ}), an abelian U(1) gauge symmetry (c) and kappa symmetry (κ). Their infinitesimal transformations are
where δ_{κ}V_{μ} is given in Eq. (113) and δ_{κ}θ was determined in [9]
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
, while in type IIB, it is given by
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 (a^{m}) and Lorentz transformations (ω^{mn}). The field infinitesimal transformations are
with δ_{ϵ}V_{μ} 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 10dimensional 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 superPoincaré supervielbein components equal
manifest that all Clifford matrices Γ^{a} act in the tangent space, as they should. The components (129) allow us to rewrite all couplings in the effective action as pullbacks
of the background superfields \(E_M^A\), B_{ac} 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
where the kappa symmetry matrix Γ_{κ} is nicely repackaged
in terms of the induced Clifford algebra matrices γ_{μ} and the gauge invariant tensor \({\mathcal F}\)
whereas γ_{(l)} stands for the wedge product of the 1forms γ_{(1)}.
Summary: We have constructed an effective action describing the propagation of Dpbranes in 10dimensional Minkowski spacetime being invariant under p + 1 dimensional diffeomorphisms, 10dimensional 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 \(E_M^A(Z)\), B_{ac} (Z) and \({C_{{A_1} \ldots {A_{p + 1}}}}\) encoding the nontrivial information about the nondynamical background where the brane propagates in a manifestly supersymmetric way. These superfields are onshell 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 onshell superspace backgrounds in Section 3.5, while preserving the same kinematic properties.
3.4.2 M2brane in flat superspace
Let me consider an M2brane as an example of an Mbrane propagating in d = 11 superPoincaré. Given the lessons from the superstring and Dbrane discussions, my presentation here will be much more economical.
First, let me describe d = 11 superPoincaré as a solution of elevendimensional supergravity using the superspace formulation introduced in Appendix A.2. In the following, all fermions will be 11dimensional Majorana fermions θ as corresponds to \({\mathcal N} = 1 \, d = 11\) superspace. Denoting the full set of superspace coordinates as {Z^{M}} = {X^{m} θ^{α}} with m = 0,…, 10 and α = 1,…, 32, the superspace description of \({\mathcal N} = 1 \, d = 11\) superPoincaré is [165, 144]
It includes the supervielbein E^{A} = {E^{a}, E^{α}} and the gauge invariant field strengths R_{4} = dA_{3} 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]
Notice it depends on the supervielbeins \(E_M^A(X,\theta)\) and the three form potential C_{abc} (x, θ) superfields only through their pullbacks to the world volume.
Its symmetry structure is analogous to the one described for Dbranes. Indeed, the action (137) is gauge invariant under world volume diffeomorphisms (ξ^{μ}) and kappa symmetry (κ) with infinitesimal transformations given by
The kappa matrix (140) satisfies \(\Gamma _\kappa ^2 = 1\). Thus, δ_{κ}θ 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 superPoincaré transformations
Supersymmetry quasiinvariance can be easily argued for since R_{4} is manifestly invariant. Thus, its gauge potential pullback variation will be a total derivative
for some spinorvalued two form Δ^{2}.
It is worth mentioning that just as the bosonic membrane action reproduces the string worldsheet 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 Dbrane and M2brane actions in superPoincaré to Dbranes, M2branes and M5branes 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 onshell.
In all effective actions under consideration, the set of degrees of freedom includes scalars Z^{M} = {X^{m}, θ^{α}} and it may include some gauge field V_{p}, whose dependence is always through the gauge invariant combination \(d{V_p}  {{\mathcal B}_{p + 1}}\)^{Footnote 22}. The set of kappa symmetry transformations will universally be given by
The last transformation is a generalisation of the one encountered in superPoincaré. Indeed, the kappa variation of the pullback of any nform satisfies
where Z* stands for the pullback of to the world volume. The choice in Eq. (144) guarantees the kappa transformation of dV_{p} 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 Γ_{κ} in Eq. (144). More specifically, any brane effective action will have the structure
Notice this is equivalent to requiring T_{brane} = Q_{brane}, a property that is just reflecting the halfBPS nature of these branes. It can be shown that
The choice of ϵ_{1} is correlated to the distinction between a brane and an antibrane. 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 realisations^{Footnote 23}.
3.5.1 M2branes
The effective action describing a single M2brane in an arbitrary 11dimensional background is formally the same as in Eq. (137)
with the same definitions for the induced metric \({\mathcal G}\) and the pull back 3form \({{\mathcal A}_3}\). The information regarding different 11dimensional backgrounds is encoded in the different couplings described by the supervielbein \(E_M^A(X,\theta)\) and 3form A_{abd} (X, θ) superfields.
The action (148) is manifestly 3ddiffeomorphism 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 onshell, for a kappa symmetry matrix given by [90]
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 Dbranes
Proceeding in an analogous way for Dpbranes, their effective action in an arbitrary type IIA/B background is
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), B_{ac} (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 V_{1} in [141, 93] when the kappa symmetry matrix equals
and the background is onshell, 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
and γ_{(r)} stands for the wedge product of r of these 1forms. 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 M5branes
The sixdimensional diffeomorphic and kappa symmetry invariant M5brane [45] is a formal extension of the bosonic one
where all pullbacks refer to superspace. This is kappa invariant under the transformations (144) for V_{2}, including the extra transformation law
for the auxiliary scalar field introduced in the PST formalism. These transformations are determined by the kappa symmetry matrix
where \({\gamma _\mu} = {E_\mu}^a{\Gamma _a}\) and the vector fields t_{μ} and υ_{μ} are defined by
Further comments on kappa symmetry:κsymmetry 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.
the algebra of κtransformations only closes onshell,

2.
κsymmetry 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
gives rise to an infinite tower of ghosts when attempting to follow the BatalinVilkovisky 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 halfBPS nature of the (massive) branes themselves^{Footnote 24}.
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 superPoincaré 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 superPoincaré 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
\({{\mathcal L}_\xi}\) 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 {R_{κ}, H_{3}} are the different Mtheory 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 nontrivial transformations
for some set of superfield forms {Δ_{2}, Δ_{5}, ω_{i}}.
The invariance of brane effective actions under the global transformations
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
where Z* stands for pullback to the world volume, i.e., Z*(λ_{1}) = dZ^{M} (λ_{1})_{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
vanishes because of Eq. (161). This establishes the invariance of the DBI action. On the other hand, the WZ action is quasiinvariant by construction due to Eqs. (164) and (165). Indeed,
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 onshell 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 Dstring [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 Dstring action [110].
3.6.1 Supersymmetry algebras
Since spacetime superisometries generate worldvolume 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.
Grouped theoretically, the anticommutator of two supercharges {Q_{α}, Q_{β}} defines a symmetric matrix belonging to the adjoint representation of some symplectic algebra Sp(N, ℝ), whose order N depends on the spinor representation Q_{α}. 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 righthand side defines a \(\{{Z_{{m_1} \ldots {m_p}}}\}\) that typically goes beyond the spacetime bosonic isometries.

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 nontrivial 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 superPoincaré.

3.
All brane effective actions reviewed above are quasiinvariant under spacetime superisometries, since the WZ term transformation equals a total derivative (170). Technically, it is a wellknown 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 Q_{α} (α = 1,…, 32) satisfying the anticommutation relations^{Footnote 25} [487, 478, 481]
That this superalgebra is maximal can be argued using the fact that its lefthand 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
The irreducible representations appearing in the direct sum do precisely correspond to the bosonic tensor charges appearing in the righthand side: the 11momentum Pm a 2form charge Z_{mn}, which is 55dimensional, and a 5form charge \({Y_{{m_1} \ldots {m_5}}}\), which is 462dimensional.
The above is merely based on group theory considerations that may or may not be realised in a given physical theory. In 11dimensional supergravity, the extra bosonic charges are realised in terms of electric Z_{e} and magnetic Z_{m} charges, the Page charges [410], that one can construct out of the 3form potential A_{3} equation of motion, as reviewed in [467, 466]
The first integral is over the boundary at infinity of an arbitrary infinite 8dimensional spacelike manifold \({{\mathcal M}_8}\), with volume Ω_{7}. 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 tendimensional spacelike hypersurface \({{\mathcal M}_{10}}\). Thus, \({Z_{\rm{e}}}\) represents a set of charges parameterised by the volume element 2form describing how \({{\mathcal M}_8}\) is embedded in \({{\mathcal M}_{10}}\). This precisely matches the 2form Z_{mn} in Eq. (172). There is an analogous discussion for Z_{m}, which corresponds to the 5form charge \({Y_{{m_1} \ldots {m_5}}}\). As an example, consider the M2 and M5brane configurations in Eqs. (20) and (22). If one labels the M2brane tangential directions as 1 and 2, there exists a nontrivial charge Z_{12} computed from Eq. (174) by plugging in Eq. (20) and evaluating the integral over the transverse 7sphere 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 halfBPS 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(132) superalgebra.
Consider the perspective offered by the M5brane effective action propagating in d = 11 superPoincaré. The latter is invariant both under supersymmetry and bulk translations. Thus, through Noether’s theorem, there exist field theory realisations of these charges. Quasiinvariance 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 M5brane superalgebra was explicitly computed. The supercharges Q_{α} are
where π, P_{m} and \({{\mathcal P}^{ij}}\) are the variables canonically conjugate to θ, X^{m} and V_{ij}. As in any Hamiltonian formalism, world volume indices were split according to σ^{μ} = {t, σ^{i}} i = 1, … 5. Notice that the pullbacks of the forms Δ_{2} and Δ_{5} appearing in \(\delta {{\mathcal L}_{{\rm{WZ}}}}\) for the M5brane in Eq. (170) do make an explicit appearance in this calculation. The anticommutator of the M5 brane world volume supercharges equals Eq. (172) with
where all integrals are computed on the 5dimensional spacelike hypersurface \({{\mathcal M}_5}\) spanned by the M5brane. Notice the algebra of supercharges depends on the brane dimensionality. Indeed, a single M2brane has a two dimensional spacelike surface that cannot support the pullback of a spacetime 5form as a single M5brane can (see Eq. (179)). This conclusion could be modified if the degrees of freedom living on the brane would be nonabelian.
Even though my discussion above only applies to the M5brane in the superPoincaré background, my conclusions are general given the quasiinvariance 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 pbranes, [281] for Dbranes in superPoincaré 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
The first factor G_{0} 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 G_{1} 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].^{Footnote 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 {X^{m},θ}, these transform as
The general Killing superfield was decomposed into a supersymmetry translation denoted by ϵ and a bosonic Killing vector fields k^{M} (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: X^{μ} = σ^{μ}, where one decomposes the scalar fields X^{m} into world volume directions X^{μ} and transverse directions X^{I} ≡Φ^{I}. 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 Φ^{i}, one must make sure to work in the subset of symmetry transformations preserving the gauge slice X^{μ} = σ^{μ}. This forces one to act with a compensating world volume diffeomorphism
The latter acts on the physical fields giving rise to the following set of transformations preserving the gauge fixed action
There are two important comments to be made at this point

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

2.
If the spacetime background allows for any constant k^{I} isometry, it would correspond to an inhomogeneous symmetry transformation for the physical field Φ^{I}. In field theory, the latter would be interpreted as a spontaneous broken symmetry and the corresponding Φ^{I} 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
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 superPoincaré: The above discussion can be made explicit in this case. Consider a p + 1 dimensional brane propagating in d dimensional superPoincaré. For completeness, let me remind the reader of the full set of transformations leaving the brane actions invariant
where I ignored possible world volume gauge fields. Decomposing the set of bosonic scalar fields X^{m} m = 0,1,…d − 1 into world volume directions \({X^\mu}\mu = 0,1, \ldots p\) and transverse directions X^{I} ≡ Φ^{I} I = p + 1,…d − 1, one can now explicitly solve for the preservation of the static gauge slice X^{μ} = σ^{μ}, which does globally describe the vacuum choice of a pbrane extending in the first p spacelike directions and time. This requires some compensating world volume diffeomorphism
inducing the following transformations for the remaining degrees of freedom
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 a^{μ}I describing the coset SO(1,D − 1)/(SO(1,p) × SO(D − p − 1)) are generically nonlinearly realised, whereas the transverse translations a^{I} act inhomogeneously on the dynamical fields Φ^{I}, 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 Dbranes in superPoincaré, 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 M2branes has 16 Killing spinors satisfying
where the ± is correlated with the R_{4} flux carried by the solution. If one fixes kappa symmetry according to
where Γ_{#} stands for the 11dimensional Clifford algebra matrix, then the physical fermionic degrees of freedom are not only 3dimensional spinors, but they are chiral spinors from the internal symmetry SO(8) perspective. They actually transform in the (2, 8^{s}) [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 spacetime superalgebras included extra bosonic charges due to the quasiinvariance 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 M2brane discussed above. Supercharges transform in the (2, 8^{s}) 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]
P_{(αβ)} stands for a 3dimensional oneform, the momentum on the brane; \(Z_{(\alpha \beta)}^{(IJ)}\) transforms in the 35^{+} under the Rsymmetry group SO(8), or equivalently, as a selfdual 4form 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 2form in the transverse space. The same superalgebra is realised on the nonabelian effective action describing N coincident M2branes [415] to be reviewed in Section 7.2. Similar structures exist for other infinite branes. For example, the M5brane gives rise to the d =6(2, 0) superalgebra [81]
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 M5brane and Ω^{IJ} 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 Rsymmetry groups. The last decomposition is again maximal since P_{[αβ]} stands for 1form in d =6 (momentum), \(Z_{(\alpha \beta)}^{(IJ)}\) transforms as a selfdual 3form in d =6 and a 2form in the transverse space and \(Z_{[\alpha \beta ]}^{(IJ)}\) as a 1form both in d =6 and in the transverse space. For an example of a nontrivial 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.^{Footnote 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 nontrivial 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 D3brane in a background breaking the appropriate amount of supersymmetry, the expansion of the gauge fixed D3brane action must match the standard textbook description. The reader can find an example of the kind of nontrivial 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 onshell backgrounds, I would like to reexamine the regime of validity under which the dynamics of the full string (M) theory reduces to S_{brane}.
As already stressed at the beginning of Section 3, working at low energies allows us to consider the action
In string theory, low energies means energies E satisfying \(E\sqrt {{\alpha {\prime}}} \ll 1\). This guarantees that no onshell 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., g_{s} ≪ 1, to suppress higher loop world sheet contributions.
Dynamically, all brane effective actions reviewed previously, describe the propagation of a brane in a fixed onshell 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)
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 11dimensional supergravity is Eℓ_{p} ≪ 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.
conditions on the derivatives of brane degrees of freedom, both geometrical X^{m} and world volume gauge fields, such as the value of the electric field;

2.
the reliability of the supergravity background;

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 onshell \({\mathcal N} = 2\) type IIA/IIB supergravity configuration satisfying the conditions described above, must also avoid
Since the string coupling constant g_{s} is defined as the expectation value of eϕ, 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 selfconsistent in these regions.
Similar considerations apply to 11dimensional supergravity. In this case, the first natural condition comes from the absence of strong curvature effects, which would typically occur whenever
where once more the scalar curvature can be replaced by other curvature invariants constructed out of the 11dimensional Riemann tensor in appropriate units of the 11dimensional Planck scale ℓ_{p}.
Since the strong coupling limit of type IIA string theory is Mtheory, 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 ℓ_{p}. Using the relations (55), one learns
Thus, as soon as the Mtheory circle explores subPlanckian elevendimensional scales, which would not allow a reliable elevendimensional 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 Tdual 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 Sduality transformation
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 Tdual description, in which such modes become momentum modes, provides a Tdual 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 Dbrane effective actions (150), the microscopic derivation of the DBI action assumed the world volume field strength F_{μν} was constant. Thus, kappa symmetric invariant Dbrane effective actions ignore corrections in derivatives of this field strength, i.e., terms like ∂_{ρ}F_{μν} or higher in number of derivatives. Interestingly, these corrections map to acceleration and higherorder derivative corrections in the scalar fields X^{m} under Tduality, 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
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 E_{crit} 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 chargedstring 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 chargeconjugate states. Thus, there exists a correlation between the pathological behaviour of the DBI action and the existence of string instabilities.^{Footnote 28} Heuristically, one interprets the regime with E > E_{crit} as one where the string tension can no longer hold the string together.^{Footnote 29}
4 World Volume Solitons: Generalities
Brane effective actions capture the relevant dynamics of Mtheory 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 onshell 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 Γ_{κ} 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 bounds^{Footnote 30}. 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.
4.1 Supersymmetric bosonic configurations and kappa symmetry
To know whether any given onshell 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 onshell 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
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.
Differential constraints on the subset of bosonic configurations \({\mathcal B}\). Given the firstorder nature of the operator \({\mathcal P}({\mathcal B})\), these are simpler than the secondorder equations of motion and help to reduce the complexity of the latter.

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]
Solving the supersymmetry preserving condition δ Ψa = 0 in the M2brane and M5brane backgrounds determines the Killing spinors of these solutions to be [466, 467]
A similar answer is found for all Dbranes 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 sθ is given by
where δ_{κ}θ and ξ^{μ}∂_{μ}θ 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,
one is left with
This is because ∇θ describes linearly realised symmetries. Thus, kappa symmetry and supersymmetry 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.
Invariance under kappa symmetry. Consider the kappasymmetry gaugefixing 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.
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 onshell brane configuration {ϕ^{i}} must satisfy to preserve some supersymmetry.
In Table 5, I evaluate all kappa symmetry matrices Γ_{κ} in the subspace of bosonic configurations \({\mathcal B}\) for future reference. This matrix encodes information

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.
on the brane configuration {ϕ^{i}}, including scalar fields X^{m} (σ) and gauge fields, either V_{1} or V_{2}, depending on the brane under consideration.
Just as in supergravity, any solution to Eq. (215) involves two sets of conditions, one on the space of configurations {ϕ^{i}} and one on the amount of supersymmetries. More precisely,

1.
a set of constraints among dynamical fields and their derivatives, f_{j} (ϕ^{i}, ∂ϕ^{i}) = 0,

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 onshell configuration solving Eq. (215), but also to interpret the constraints f_{j} (ϕ^{i}) = 0 as BPS bounds [107, 429]. This will lead us to interpret these configurations as branelike 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 Malgebra (172). Due to the positivity of its lefthand side, one derives the energy bound
where the charge conjugation matrix was chosen to be C = Γ^{0} 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 Z_{0i} to zero. The superalgebra reduces to
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
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 onetoone 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 10momentum P_{i} describes d =11 massless superparticles [93], the 2form charges Z_{ij} M2branes [90, 91], whereas the 5form charges \({Y_{{i_1} \ldots {i_5}}}\), M5branes [464]. This correspondence extends to the time components \(\{{Y_0}_{{i_1} \ldots {i_4}},{Z_{0i}}\}\). These describe branes appearing in KaluzaKlein vacua [311, 481]. Specifically, \({Y_0}_{{i_1} \ldots {i_4}}\) is carried by type IIA D6branes (the Mtheory KK monopole), while Z_{0i} can be related to type IIA D8branes.
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 P_{0} 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.
For nonthreshold bound states, or equivalently, when the associated Clifford matrices anticommute, 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.
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 {t^{i}} involves nontrivial dependence on the dynamical fields and their derivatives. Due to the positivity of the terms in the righthand side, one can derive lower bounds on the energy, or BPS bounds,
being saturated precisely when f_{i} (ϕ^{j}) = 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 nontrivial 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 onshell, 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 onshell. For example, in the presence of nontrivial 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 Dbrane Hamiltonian
As in any Hamiltonian formulation^{Footnote 31}, the first step consists in breaking covariance to allow a proper treatment of time evolution. Let me split the world volume coordinates as σ^{μ} = {t,σ^{i}} for i = 1,…,p and rewrite the bosonic Dbrane Lagrangian by singling out all time derivatives using standard conjugate momenta variables
Here P_{m} and E^{i} are the conjugate momentum to X^{m} and V_{i}, respectively, while H is the Hamiltonian density. ψ is the Hodge dual of a pform 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
where
The first constraint is responsible for the constant tension of the brane. It generates abelian gauge transformations for the pform 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 = dC − C ∧ H_{3}, coming from the WZ couplings and acting as sources in Gauss’ law. Finally, \({{\mathcal H}_a}\) and \({\mathcal H}\) generate worldspace 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
Z* (i_{m}B)_{i} stands for the pullback to the world volume of the contraction of B_{2} along the vector field ∂/∂X^{m}. Equivalently, Z* (i_{m}B)_{i} = ∂_{i}X^{n}B_{mn}. Z* (i_{m}C) is defined analogously. Notice * stands for the Hodge dual in the pdimensional Dbrane 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 P_{0} = \({P_0} = {\mathcal E}\) 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 M2brane Hamiltonian
The Hamiltonian formulation for the M2brane 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 M2brane Lagrangian is equivalent to
where the modified conjugate momentum \({{\tilde P}_a}\) is related to the standard conjugate momentum P_{m} by
where * describes the Hodge dual computed in the 2dimensional world space spanned by i, j = 1, 2. Notice no dynamicallygenerated 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 M5brane Hamiltonian
It turns out the Hamiltonian formulation for the M5brane dynamics is more natural than its Lagrangian one since it is easier to deal with the selfduality condition in phase space [92]. One follows the same strategy and notation as above, splitting the world volume coordinates as σμ = {t, σ^{i}} with i = 1, … 5. Since the Hamiltonian formulation is expected to break SO(1, 5) into SO(5), one works in the gauge a = σ^{0} = t. It is convenient to work with the world space metric \({{\mathcal G}_{ij}}\) and its inverse \({\mathcal G}_5^{ij}\)^{Footnote 32}. Then, the following identities hold
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 M5brane Lagrangian in phase space equals
where P_{m} and Π^{ij} are the conjugate momenta to X^{m} and the 2form V_{ij}
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 dV_{2}. The last three functionals appearing in Eq. (230)
correspond to constraints generating time translations, world space diffeomorphisms and the selfduality condition. The following definitions were used in the expressions above
As for Dbranes and M2branes, 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 pdimensional submanifolds embedded in an ndimensional 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 ℝ^{n}, 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 ℝ^{n}, i.e., the Grassmannian G (p, ℝ^{n}). For any ξ ∈ G(p, ℝ^{n}), one can always find an orthonormal basis {e_{1},…,e_{n}} in ℝ^{n} such that {e_{1},…,e_{p}} is a basis in ξ so that its covolume is
A pform φ on an open subset U of ℝ^{n} is a calibration of degree p if

(i)
dφ = 0

(ii)
for every point x ∈ U, the form φ_{x} satisfies \({\varphi _x}(\overrightarrow \xi) \leq 1\) for all ξ ∈ G (p, ℝ^{n}) 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 pdimensional submanifolds of ℝ^{n}. Indeed, the fundamental theorem of calibrations [289] states
Theorem: Let φ be a calibration of degree p on ℝ^{n}. The pdimensional submanifold N, for which
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 ℝ^{n} with the same boundary, i.e., ∂U = ∂V. By Stokes’ theorem,
where μ_{V} = α_{1} ∧ …α_{p} is the volume form constructed out of the dual basis {α_{1},…,α_{p}} to {e_{1},…,e_{n}}.
Two remarks can motivate why these considerations should have a relation to brane solitons and supersymmetry:

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 ℝ^{n}. Thus, bounds on the volume correspond to brane energy bounds, which are related to supersymmetry saturation, as previously reviewed. Indeed, the dynamical field X^{i} (σ) does mathematically describe the map from the world volume ℝ^{p} into ℝ^{n}. The above bound can then be reexpressed 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 pform φ.

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 pform calibration takes the form
where the set X^{i} (i = 1,…,n) stands for the transverse scalars to the brane parameterising φ^{n}, 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 ℝ^{n}. Notice that, given a tangent pplane ξ, one can write φ _{ξ} as
where the matrix
is evaluated at the point to which ξ is tangent. Given the restriction on the values of p,
It follows that φ _{ξ} ≤ vol_{ξ} for all ξ. Since φ is also closed, one concludes it is a calibration. Its contact set is the set of pplanes for which this inequality is saturated. Using Eq. (240), the latter is equally characterised by the set of pplanes ξ for which
Because of Eq. (242) and the fact that tr Γ_{ξ} = 0, the solution space to this equation is always half the dimension of the spinor space spanned by φ for any given tangent pplane 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 Γ_{ξ} in Eq. (241) matches the kappa symmetry matrix Γ_{κ} 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 M5brane propagating in d =11 Minkowski and fix the extra gauge symmetry of the PST formalism by a (σ^{μ}) = t (temporal gauge). The kappa symmetry matrix (158) reduces to
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
Even though, this was only argued for the M5brane 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]
This result has not been established in full generality but it agrees with the flat space case [165] and those nonflat cases that have been analysed [438, 437]. I refer the reader to [278] where they connect the functional form in the righthand 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 dφ ≠ 0. Physically, this allowed one to include the presence of nontrivial 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 CalabiYau manifolds [267] following the seminal work in [298]. This general notion allows one to include the effect of nontrivial 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 onshell 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 dynamicallyoriented applications to Section 6. The main goals in this section include:

1.
In a Minkowski background, the identification of the vacuum of all the p + 1 dimensional supersymmetric field theories discussed before as halfBPS 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.
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.
As examples of solitons in curved backgrounds, I will discuss the baryon vertex and giant gravitons in AdS_{5} × S^{5}.

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 halfBPS branes in 10 and 11dimensional 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
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 32dimensional. In Cartesian coordinates, these are constant spinors ϵ = ϵ ∞.
HalfBPS 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(D − p − 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 X^{m} (σ) into longitudinal X^{μ} and transverse X^{I} directions, setting the latter to constant values X^{I} = c^{I} (the transverse brane location). Second, one identifies the world volume directions with the longitudinal directions, X^{μ} = σ^{μ}. The latter can also be viewed as fixing the world volume diffeomorphisms to the static gauge. This information can be encoded as an array
It is easy to check that the above is an onshell configuration given the structure of the EulerLagrange equations and the absence of nontrivial couplings except for the induced world volume metric \({{\mathcal G}_{\mu \nu}}\), which equals η_{μν} 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
\(E_m^a = \delta _m^a\). Thus, Γ_{κ} reduces to a constant Clifford valued matrix standing for the volume of the brane, Γ_{vol}, up to the chirality of the background spinors, which is parameterised by the matrix τ
The specific matrices for the branes discussed in this review are summarised in Table 6. Since \({\Gamma _\kappa} = 0\) and Tr Γ_{κ} =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.
All these configurations have an energy density equaling the brane tension T since the Hamiltonian constraint is always solved by
From the spacetime superalgebra perspective, these configurations saturate a bound between the energy and the pform bosonic charge carried by the volume form defined by the brane
The saturation corresponds to the fact that any excitation above the infinite brane configuration would increase the energy. From the worldvolume perspective, the solution is a vacuum, and consequently, it is annihilated by all sixteen worldvolume supercharges. These are precisely the ones solving the kappa symmetry preserving condition (215).
5.2 Intersecting M2branes
As a first example of an excited configuration, consider the intersection of two M2branes in a point corresponding to the array
In the probe approximation, the M2brane effective action describes the first M2brane by fixing the static gauge and the second M2brane as an excitation above this vacuum by turning on two scalar fields (X^{3}, X^{4}) according to the ansatz
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
Altogether, the kappa symmetry preserving condition (215) is
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
one for each M2brane in the array (253). Plugging these projections into Eq. (257)
one obtains an identity involving two different Cliffordvalued contributions: the lefthand side is proportional to the identity matrix acting on the Killing spinor, while the righthand 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
Notice this is equivalent to the holomorphicity of the complex function U (σ^{+}) = y + iz in terms of the complex world space coordinates σ^{±} = σ^{1} ± iσ^{2}, since Eqs. (260) are equivalent to the CauchyRiemann equations for U (σ^{+}).
When conditions (260) are used in the remaining lefthand 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 nontrivial 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 onshell. To ease notation below, rewrite Eq. (260) as
where standard vector calculus notation for ℝ^{2} 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 M2brane Lagrangian given in Eq. (227) in a Minkowski background. The Lagrange multiplier fields s^{a} impose the world space diffeomorphism constraints. In the static gauge, these reduce to
where P_{I} are the conjugate momenta to the eight world volume scalars X^{I} describing transverse fluctuations. For static configurations carrying no momentum, i.e., P_{I} = 0, the world space momenta will also vanish, i.e., P_{a} = 0.
Solving the Hamiltonian constraint imposed by the Lagrange multiplier λ for the energy density \({\mathcal E} = {P_0}\), one obtains [225]
This already involves the computation of the induced world space metric determinant and its rewriting in a suggestive way to derive the bound
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 onshell, 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_{0} stands for the energy of the infinite M2brane vacuum, whereas Z is the topological charge
accounting for the second M2brane 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 M2branes. Field theoretically, the first M2brane charge corresponds to the vacuum energy (E_{0}), while the second corresponds to the topological charge (Z) describing the excitation. When the system is supersymmetric, the energy saturates the bound E = E_{0} + 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 E − E_{0} 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 M2brane solitons see [95] and for a related classification of D2brane supersymmetric soltions see [33].
5.3 Intersecting M2 and M5branes
As a second example of BPS excitation, consider the 1/4 BPS configuration M5 ⊥ M2(1) corresponding to the brane array
The idea is to describe an infinite M5brane by the static gauge and to turn on a transverse scalar field X^{6} to account for the M2brane excitation. However, X^{6} is not enough to support an M2brane interpretation, since the latter is electrically charged under the 11dimensional supergravity three form A_{3}. Thus, the sought M5brane soliton must source the A_{056} components. From the WessZumino coupling
one learns that the magnetic \({(dV)_{\hat a\hat b\hat c}}\) components, where hatted indices stand for world space directions different from σ^{5}, i.e., \(\hat a \neq 5\), must also be excited.
The full ansatz will assume delocalisation along the σ^{5} direction, so that the stringlike excitation in the X^{6} direction can be viewed as a membrane:
Supersymmetry analysis: The M5brane kappa symmetry matrix (158) in the temporal gauge a = τ reduces to
for the subset of configurations described by the ansatz (269), it follows
This reduces Eq. (270) to
To solve the kappa symmetry preserving condition (215), I impose two projection conditions
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), Γ_{κ} keeps a nontrivial dependence on \({\Gamma _{05}}{\Gamma _{\hat a}}\). Requiring its coefficient to vanish gives rise to the BPS condition
Overall, the kappa symmetry preserving condition (215) reduces to the purely algebraic condition
To check this holds, notice the only nonvanishing components of \({{\tilde H}_{\mu \nu}}\) are \({{\tilde H}_{5\hat a}}\)
This allows us to compute the determinant
which becomes a perfect square once the BPS equation (274) is used
This shows that Eq. (275) holds automatically. Thus, the solution to the kappa symmetry preserving condition (215) for the ansatz (269) on an M5brane action is solved by the supersymmetry projection conditions (273) and the BPS equation (274). Since the soliton involves a nontrivial 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 ω_{4}
Hamiltonian analysis: The Hamiltonian analysis for this system was studied in [225] following the M5brane phase space formulation given in Eq. (230). For static configurations, the Hamiltonian constraint can be solved by the energy density \({\mathcal E}\) as
where
and world space indices were denoted by latin indices σ^{a} a = 1,…,5. It was noted in [225] that by introducing a unit length world space 5vector ζ, i.e., ζ^{a}ζ^{b}δ_{ab} = 1, the energy density could be written in the suggestive form
The unit vector provides a covariant way of introducing a preferred direction in the 5dimensional world space. Choosing ζ^{5} = 1 and \({\zeta ^5} = 0\), to match the delocalisation direction in our bosonic ansatz, one derives the inequality
The latter is saturated if and only if
and
where \({{\mathcal H}_3}\) is only defined on the 4dimensional subspace ω_{4}, 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 ω_{4}.
To regulate the divergent energy, one imposes periodic boundary conditions in the 5direction making the orbits of the vector field ζ have finite length L. Then, the total energy satisfies
where Z is the topological charge
The tension of the soliton, i.e., energy per unit of length, equals T = E − E_{0}/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 3sphere surrounding the origin. Since y is constant on this integration surface, one derives the string tension [225]
is the string charge. Even though this tension diverges, it does so consistently, being the boundary of a semiinfinite 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 onshell supersymmetric Dbrane configurations representing a fundamental string ending on the Dbrane, i.e., the defining property of the Dbrane itself. They correspond to the array of branes
Working in the static gauge describes the vacuum infinite Dpbrane. The static soliton excites a transverse scalar field (y = y (σ^{a})) and the electric field (V_{0}= V_{0} (σ^{a})), while setting the magnetic components of the gauge field (V_{a}) to zero
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 NSNS two form. The latter appears in the effective action through the gauge invariant form \({\mathcal F}\). Thus, turning on V_{0} is equivalent to turning such charge^{Footnote 33}.
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
while the induced gamma matrices are decomposed as
where a stands for world space indices. Due to the electric ansatz for the gauge field, the kappa symmetry matrix Γ_{κ} has only two contributions. In particular, for type IIA (p = 2k)
Summing over world volume time, one obtains
Using the duality relation
one can write the first term on the righthand side of Eq. (294) as
Using the same duality relation and proceeding in an analogous way, the second term equals
Inserting Eqs. (296) and (297), the kappa symmetry preserving condition can be expressed as
Given the physical interpretation of the sought soliton, one imposes the following two supersymmetry projection conditions
corresponding to having a type IIA Dpbrane 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
It is clear that the BPS condition
derived from requiring the coefficient of Γ^{a} Γ_{0}Γ_{#} 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 Dpbranes (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
corresponding to having a type IIB Dpbrane along the directions 1,…,p and a fundamental string along the transverse direction y.
Satisfying the BPS equation (302) does not guarantee the onshell nature of the configuration. Given the nontriviality of the gauge field, Gauss’ law ∂_{a}E^{a} = 0 must be imposed, where E^{a} is the conjugate momentum to the electric field, which reduces to
when Eq. (302) is satisfied. Thus, the transverse scalar y must be a harmonic function on the pdimensional Dbrane world space
Hamiltonian analysis: Using the phase space formulation of the Dbrane 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., P_{i} = 0, and in the absence of magnetic gauge field excitations, i.e., F_{ab} = 0. The Hamiltonian constraint can be solved for the energy density [225]
Since det \({{\mathcal G}_{ab}} = 1 + {(\partial y)^2}\), Eq. (307) is equivalent to [225]
There exists an energy bound
being saturated if and only if
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 Dbrane world space ω satisfies
where Z_{el} is the charge
To interpret this charge as the charge carried by a string, consider the most symmetric solution to Eq. (306), for Dpbranes with p ≥ 3, depending on the radial coordinate in world space r, i.e., r^{2} = σ^{a}σ^{b}δ_{ab},
where Ω_{p} stands for the volume of the unit psphere. 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
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 Dstring case, corresponding to string junctions.
5.5 Dyons
Dyons are onshell supersymmetric D3brane configurations describing a (p, q) string bound state ending on the brane. They are described by the array
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 Dstring is electrically charged under the RR two form. The latter can be induced from the WessZumino coupling
This shows that magnetic components in \({\mathcal F}\) couple to electric components in C_{2}. Altogether, the dyonic ansatz is
Supersymmetry analysis: In this case, the matrix elements \({{\mathcal G}_{\mu \nu}} + {{\mathcal F}_{\mu \nu}}\) are
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
To correctly capture the supersymmetries preserved by such a physical system, we impose the projection conditions
on the constant Killing spinor ϵ, describing a D3brane 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
This equation is trivially satisfied when the following BPS conditions hold
Hamiltonian analysis: Following [225], the Hamiltonian constraint can be solved and rewritten as a sum of positive definite terms^{Footnote 34}
where the last equality holds for any angle α. This allows one to derive the bound
Thus, the total energy satisfies
with
The bound (325) is extremised when
for which the final energy bound reduces to
Here E_{0} corresponds to the energy of the vacuum configuration (infinite D3brane). The bound (329) is saturated when
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.,
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 nonlocal transformation in terms of the gauge field V, but leaves the energy density (324) invariant
Applying this transformation to the BIon solution, one reproduces Eq. (330).
5.6 Branes within branes
The existence of WessZumino couplings of the form
suggests that onshell nontrivial 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 onshell solutions in the higher dimensional Dbrane 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 nonabelian phenomena [496, 185]. For general discussions on Dbrane bound states, see [447, 424, 425], on marginal D0D0 bound states [445], on D0D4 bound states [446, 486] while for D0D6, see [470]. D0D6 bound states in the presence of Bfields, which can be supersymmetric [391], were considered in [501]. There exist more general analysis for the existence of supersymmetric Dbranes with nontrivial gauge fields in backgrounds with nontrivial NSNS 2forms in [372].
5.6.1 Dp−D(p + 4) systems
These are bound states at threshold corresponding to the brane array
Motivated by the WessZumino coupling \({\mathcal C}\wedge {\mathcal F}\wedge {\mathcal F}\), one considers the ansatz on the D (p + 4)brane effective action
Let me first discuss when such configurations preserve supersymmetry. Consider type IIA (p = 2k), even though there is an analogous analysis for type IIB. Γ_{κ} reduces to
where I already used the static gauge and the absence of excited transverse scalars, so that γ_{μ} = Γ_{μ}. For the same reason, det(η_{μν} + F_{μν}) = det(δ_{ab} + F_{ab}), 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
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
where \({{\tilde F}^{ab}} = {1 \over 2}{\varepsilon ^{abcd}}{F_{cd}}\). Requiring the last term in Eq. (339) to vanish is equivalent to the selfduality condition
When the latter holds, Eq. (339) is trivially satisfied. Eq. (340) is the famous instanton equation in four dimensions^{Footnote 35}. The Hamiltonian analysis done in [225] again confirms its BPS nature.
5.6.2 Dp −D(p + 2) systems
These are nonthreshold bound states corresponding to the brane array
Motivated by the WessZumino coupling \({\mathcal S}\wedge{\mathcal F}\), one considers the ansatz on the D (p + 4)brane effective action
Since there is a single nontrivial magnetic component, I will denote it by F_{ab} = F to ease the notation. The DBI determinant reduces to
whereas the kappa symmetry preserving condition in type IIA is
for p = 2k. This is solved by the supersymmetry projection
for any α, for the magnetic flux satisfying
To interpret the solution physically, assume the world space of the D (p + 2)brane is of the form δ^{p} × T^{2}. This will quantise the magnetic flux threading the 2torus according to
To derive this expression, I used the fact that the 2torus has area L_{1}L_{2} and I rescaled the magnetic field according to F → 2πα′ 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
matching the nonthreshold nature of the bound state
where the last term stands for the energy of k Dpbranes.
5.6.3 FDp systems
These are nonthreshold bound states corresponding to the brane array
Following previous considerations, one looks for bosonic configurations with the ansatz
Given the absence of transverse scalar excitations, γ_{μ} = Γ_{μ} and \(\sqrt { \det ({\mathcal G} + {\mathcal F})} = \sqrt {1  {F^2}}\), where F_{0ρ} = F. The kappa symmetry preserving condition reduces to
This is solved by the supersymmetry projection condition
whenever
To physically interpret the solution, compute its energy density
where I already used that F_{0ρ} = F = E^{ρ}. These configurations are Tdual to a system of D0branes moving on a compact space. In this Tdual picture, it is clear that the momentum along the compact direction is quantised in units of 1/L. Thus, the electric flux along the Tdual circle must also be quantised, leading to the condition
where the world volume of the Dpbrane is assumed to be ℝ^{p} × S_{1}. In this way, one can rewrite the energy for the FDp system as
which corresponds to the energy of a nonthreshold bound state made of a Dpbrane and n fundamental strings (T_{f}).
5.7 Supertubes
All reviewed solitonic configurations carry charge under the p + 1dimensional 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 lowerdimensional branes finds it energetically favourable to expand into higherdimensional ones. The stability of these is due to either an external force, typically provided by nontrivial fluxes in the background, or presence of angular momentum preventing the brane from collapse. A IIA superstring blownup to a tubular D2brane [200], a collection of D0branes turning into a fuzzy 2sphere [395] or wrapping Dbranes with quantised nontrivial world volume gauge fields in AdS_{m} × S^{n} [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 D2branes of arbitrary crosssection in a Minkowski vacuum spacetime supported against collapse by the angular momentum generated by a nontrivial Poynting vector on the D2brane world volume due to nontrivial electric and magnetic BornInfeld (BI) fields. They were discovered in [381] and its arbitrary crosssection reported in [380], generalising some particular noncircular crosssections 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 nontrivial angular momentum may appear to be in conflict with supersymmetry, since the latter requires a timeindependent energy density. This point, and its connection with the expansion of lowerdimensional branes, will become clearer once I have reviewed the construction of these configurations.
Let me briefly review the arbitrary crosssection supertube from [380]. Consider a D2brane with world volume coordinates σ_{μ} = {t, z, σ} in the type IIA Minkowski vacuum
where \(\vec Y = \{{Y^i}\}\) are Cartesian coordinates on ℝ^{8}. We are interested in describing a tubular D2brane of arbitrary crosssection extending along the Z direction. To do so, consider the set of bosonic configurations
The static gauge guarantees the tubular nature of the configuration, whereas the arbitrary embedding functions \(\vec Y = \vec y(\sigma)\) describe its crosssection. 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]
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
on the constant background Killing spinors ϵ. These conditions have solutions, preserving 1/4 of the vacuum supersymmetry, if B (σ) is a constantsign, but otherwise completely arbitrary, function of σ. Notice the two projections 361 correspond to string charge along the Zdirection and to D0brane charge, respectively.
In order to improve our understanding on the arbitrariness of the crosssection, 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 P_{i} and the conjugate variable to the electric field, Π, are
where in the last step the supersymmetry condition E  = 1 was imposed. Notice supertubes satisfy the identity
Second, the fundamental string _{q} _{F1} and D0brane _{qD} _{0} charges are
Finally, the supertube energymomentum tensor [380]
with X^{m} = {T, Z, Y^{i}}, has only nonzero components
Some comments are in order:

1.
As expected, the linear momentum density (362) carried by the tube is responsible for the offdiagonal components \({\tau ^{Ti}}\).

2.
The absence of nontrivial components \({\tau ^{ij}}\) confirms the absence of tension along the crosssection, providing a more technical explanation of why an arbitrary shape is stable.

3.
The tube tension \( {\tau ^{zz}} = \vert \Pi \vert\) in the Zdirection is only due to the string density, since D0branes behave like dust.

4.
The expanded D2brane does not contribute to the tension in any direction.
Integrating the energy momentum tensor along the crosssection, one obtains the net energy of the supertube per unit length in the Zdirection
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 5.6.2 — 5.6.3, the former can be interpreted as “dissolved” IIA superstrings and the latter as “dissolved” D0branes, that have expanded into a tubular D2brane. Their charges are the ones appearing in the supersymmetry algebra allowing the energy to be minimised. Notice the expanded D2brane couples locally to the RR gauge potential C_{3} under which the string and D0brane constituents are neutral. This is precisely the point made at the beginning of the section: supertubes do not carry D2brane charge.^{Footnote 36} When the number of constituents is large, one may expect an effective description in terms of the higherdimensional D2brane 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
This entropy reproduces the microscopic conjecture made in [364] where the BekensteinHawking entropy was computed using a stretched horizon. These considerations do support the idea that supertubes are typical D0F 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].^{Footnote 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 crosssection 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 Udual descriptions of the famous string theory D1D5 system. To make this connection more apparent, consider supertubes with arbitrary crosssections in ℝ^{4} and with an S^{1} tubular direction, allowing the remaining 4spacelike directions to be a 4torus. These supertubes are Udual to D1D5 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 D1D5 frame, the actual supertubes correspond to KK monopoles wrapping the 4torus, the circle also shared by D1 and D5branes and the arbitrary profile in ℝ^{4}^{Footnote 38}. Smoothness of these solutions is then due to the KK monopole smoothness.
Since the Udual D1D5 description involves an AdS3 × S^{3} near horizon, supertubes were interpreted in the dual CFT: the maximal angular momentum configuration corresponding to the circular profile is global AdS_{3}, whereas noncircular 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 AdS_{3} × S^{3} 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.
5.8 Baryon vertex
As a first example of a supersymmetric soliton in a nontrivial background, I will review the baryon vertex [500, 265]. Technically, this will provide an example of how to deal with nonconstant 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 setup. Consider a closed D5brane surrounding N D3branes, i.e., such that the D3branes thread the D5brane. The HananyWitten (HW) effect [282] allows us to argue that each of these N D3branes will be connected to the D5brane by a fundamental type IIB string. Consequently, the lowest energy configuration should not allow the D5brane to contract to a single point, but should describe these N D3branes with N strings attached to them allowing one to connect the D3 and D5branes. In the large N limit, one can replace the D3branes by their supergravity backreaction description. The latter has an AdS_{5} × S^{5} near horizon. One can think of the D5brane as wrapping the 5sphere and the N strings emanating from it can be pictured as having their endpoints on the AdS^{5} boundary. This is the original configuration interpreted in [500, 265] as a baryonvertex of the \({\mathcal N} = 4 \, d = 4\) superYangMills (SYM) theory.
At a technical level and based on our previous discussions regarding BIons, one can describe the baryon vertex as a single D5brane 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 D5brane in AdS_{5} as a function r (θ) of the colatitude angle θ on S^{5}. 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.
Setup: One is interested in solving the equations of motion of a single D5brane in the background of N D3branes carrying some units of electric charge to describe type IIB strings. The background is described by a constant dilaton, a nontrivial metric and selfdual 5form field strength R_{5} [195]
where \(d\Omega _5^2\) is the SO(6)invariant metric on the unit 5sphere, ω_{5} is its volume 5form and *ω_{5} its Hodge dual. The function U is
Notice a =1 corresponds to the full D3brane background solution, whereas a = 0 to its nearhorizon limit.
Consider a probe D5brane of unit tension wrapping the 5sphere. Let ξ^{μ} = (t, θ^{i}) be the world volume coordinates, so that θ^{i} (i = 1,…,5) are coordinates for the worldspace 5sphere. This will be achieved by the static gauge
Since one is only interested in radial deformations of the world space carrying electric charge, one considers the ansatz
Even though the geometry will be curved, it can give some intuition to think of this system in terms of the array
viewing the 9direction as the radial one.
Supersymmetry analysis: Given the electric nature of the world volume gauge field, the kappa symmetry matrix reduces to
Given the ansatz (374) and the background (370), the induced world volume metric equals
where
and \({{\bar g}_{ij}}\) stands for the SO(6)invariant metric on the unit 5sphere. Taking into account the nontrivial vielbeins, the induced gamma matrices equal
where the matrices \({{\hat \gamma}_i}\) are defined as
in terms of the fönfbein \({e_i}^a\) in the 5sphere. 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
where \(\chi\) is a covariantly constant spinor on \({{\rm{{\mathbb E}}}^{(1,3)}} \times {{\rm{{\mathbb E}}}^6}\) subject to the projection condition
describing the D3branes in the background. Importantly, χ is not constant when using polar coordinates in \({{\rm{{\mathbb E}}}^6}\). Indeed, covariantly constant spinors on S^{n} 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
where ϵ_{0} satisfies Eq. (382). Even though there are additional Killing spinors in the nearhorizon limit, the associated extra supersymmetries will be broken by the baryonic D5brane 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
where \({{\hat \gamma}^i} = {{\bar g}^{ij}}{{\hat \gamma}_j}\) and γ * = Γ_{45678}.
Given the physical interpretation of the sought solitons, one imposes two supersymmetry projections on the constant Killing spinors ϵ_{0}:
These are expected from the local preservation of 1/2 supersymmetry by the D5brane and the IIB string in the radial direction, respectively. These projections imply
Using these relations, one can rewrite the righthand side of Eq. (384) as
where \({\Delta _5} = U{r^4}\sqrt {\det \bar g}\). The coefficients of Γ_{rθ} and \({\Gamma _r}{{\hat \gamma}^i}\) in Eq. (388) vanish when
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
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]
where \({{\tilde E}^i}\) is a covariantised electric field density related to F_{0i} by
for the ansatz (374), this reduces to
It was shown in [248] that one can rewrite the energy density (391) as
where ^{2} indicates contraction with \({g^{\hat i\hat j}}\) and
To achieve this, the 5sphere metric was written as
where \(d\Omega _4^2\) is the SO(5) invariant metric on the 4sphere, 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 S^{5} metric \({{\bar g}^{ij}}\).
Using the Gauss’ law constraint
which has a nontrivial source term due to the RR 5form flux background, one can show that \({\!\!\!\!\! Z_5} = {\partial _i}\!\!\!\!\! Z_5^i\) where \({{\vec \!\!\!\!\! Z}_5}\) has components
From Eq. (394), and the divergent nature of Z_{5}, one deduces the bound
The latter is saturated when
Combining Eqs. (400) and (401) with the Gauss law (397) yields the equation
Any solution to this equation gives rise to a 1/4 supersymmetric baryonic brane.
For a discussion of the firstorder 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
while the first inequality is saturated under the same conditions as above, the second requires Z_{5} to not change sign within the integration region. For this configuration to describe a baryonic brane, one must identify this region with a 5sphere having some number of singular points removed. Assuming the second inequality is saturated when the first one is, the total energy equals
where B_{k}. is a 4ball 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 \({{\tilde E}^i} = 0\),
and r = r (θ). The BPS condition (401) reduces to [315, 125, 152]
where Δ = R^{4} sin^{4} θ, while the Gauss’ law (397) equals
Its solution was first found in [125]
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
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
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
Integrating the differential equation (406) for r (θ) after substituting Eq. (411), one finds [125]
where r_{0} is the value of r at θ = 0. It was shown in [125] that this configuration corresponds to N fundamental strings attached to the D5brane 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 D5brane does not completely wrap the 5sphere. 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 D5brane to r = ∞ while the remaining N — k strings connect it to r = 0. Since the ν = 0 D5brane solutions do reach r = 0, it is tempting to speculate on whether they correspond to these baryonic multiquark states.
Related work: There exists similar work in the literature. Besides the study of nonSO(5) invariant baryonic branes in AdS_{5} × S^{5}, [248] also carried the analysis for baryonic branes in Mtheory. Similar BPS bounds were found for D4branes in D4brane backgrounds or more generically, for Dbranes in a Dbrane background [126, 133] and D3branes in (p,q)5branes [452, 357]. Baryon vertex configurations have also been studied in AdS_{5} × T^{1,1} [19], AdS_{5} × Y^{p,q} [134] and were extended to include the presence of magnetic flux [319]. For a more general analysis of supersymmetric Dbrane probes either in AdS or its ppwave 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 AdS_{5} × S^{5}. 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 D3branes, i.e., giant gravitons. The N scaling is easy to argue for: the conformal dimension must be proportional to the D3brane tension times the volume of the wrapped cycle, which is controlled by the AdS radius of curvature L_{4}, thus giving
Similar considerations apply in different AdS_{p+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 AdS_{5} × S^{5}. The bosonic background has a constant dilaton and nontrivial metric and RR 4form potential given by
where ω_{3} stands for the volume form of the 3sphere in S^{5} and it is understood dC_{4} is made selfdual to satisfy the type IIB equations of motion^{Footnote 39}. Giant gravitons consist of D3branes wrapping such 3spheres and rotating in the ϕ direction to carry Rcharge from the dual CFT perspective. Thus, one considers the bosonic ansatz
The D3brane Lagrangian density evaluated on this ansatz and integrating over the 3sphere world volume is [264]
Since k = ∂_{ϕ} is a Killing vector, the conjugate momentum P_{ϕ} is conserved
where the constant p was defined. Computing the Hamiltonian density,
allows us to identify the stable configurations by extremising Eq. (418). Focusing on finite size configurations, one finds
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 maximallysupersymmetric AdS_{5} × S^{5} background were computed in [359, 264]. They are of the form ϵ = M ϵ_{∞} where M is a nontrivial 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
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 Rsymmetry group SO(6). The general construction of such supersymmetric probes was done in [392]. The main idea is to embed the bulk 5sphere into an auxiliary embedding C^{3} space with complex coordinates z_{i} i = 1, 2, 3 and AdS_{5} into ℂ^{1,2}. In the probe calculation, the Z_{i} become dynamical scalar fields subject to the defining quadric constraint ∑_{i} Z_{i} ∑^{2} = 1. To prove these configurations are supersymmetric one can use the well known isomorphism between geometric Killing spinors on both the 5sphere and AdS_{5} and parallel spinors in ℂ^{3} and ℂ^{1,2}, respectively. This is briefly reviewed in Appendix B. The conclusion of such analysis is that any holomorphic function F (Z_{1},Z_{2},Z_{3}) gives rise to a supersymmetric giant graviton configuration [392] defined
as the intersection of the 5sphere with a holomorphic hypersurface properly evolved in world volume time. The latter involves rotations in each of the ℂ planes in ℂ^{3} at the speed of light (in 1/L_{4} 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 gravitons^{Footnote 40} when \({{\mathcal M}_5}\) is an EinsteinSasaki 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 nonsupersymmetric giant gravitons with NSNS fields [131], Mtheory giants with 3form 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 D7brane probes in AdS_{5} × L^{abc}, 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 nonabelian world volume description) with the inclusion of higher moment couplings responsible for their physical properties [317, 318].
5.9.1 Giant gravitons as blackhole 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 Rcharged AdS black holes carry mass of order N^{2}. The idea that supersymmetric Rcharged 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.
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.
Giant gravitons do not carry D3brane charge, but they do locally couple to the RR 5form field strength giving rise to some D3brane dipole charge. This means [397] that a small (fivedimensional) surface enclosing a portion of the giant graviton sphere will carry a net fiveform flux proportional to the number of D3branes 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 5form flux obtained by considering a surface that is the boundary of a sixdimensional ball, which only intersects the threesphere of the giant graviton once, at a point very close to the singularity.
To check this interpretation, let us review these supersymmetric Rcharged AdS_{5} 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
with the different scalar functions defined as
All these metrics have a naked singularity at the center of AdS that extends into the 5sphere. Depending on the number of charges turned on, the rate at which curvature invariants diverge changes with the 5sphere direction. Besides a constant dilaton, these BPS configurations also have a nontrivial RR selfdual 5form field strength R_{5} = dC_{4} + _{*}dC_{4} with
with ω_{3} being volume 3form of the unit 3sphere.
To test the microscopic interpretation for the superstar solutions, consider the single Rcharged configuration with q_{2} = q_{3} = 0. This should correspond to a collection of giant gravitons rotating along ϕ_{1} with a certain distribution of sizes (specified by μ_{1} = cos θ_{1}). To measure the density of giant gravitons sitting near a certain θ_{1}, one must integrate R_{5} over the appropriate surface. Describing the 3sphere in AdS_{5} by
one can enclose a point on the brane at θ_{1} with a small fivesphere in the {r, θ_{1}, ϕ_{1} α_{i}} directions. The relevant fiveform component is
and by integrating the latter over the smeared direction ϕ_{1} and the 3sphere, one infers the density of giants at a point θ_{1} [397]
If this is correct, the total number of giant gravitons carried by the superstar is
The matching is achieved by comparing the microscopic momentum carried by a single giant at the location θ_{1}, P_{micro} = N sin^{2} θ_{1}, with the total mass of the superstar
Indeed, by supersymmetry, the latter should equal the total momentum of the distribution
which establishes the physical correspondence. There exist extensions of these considerations when more than a single Rcharge is turned on, i.e., when q_{2},q_{3} ≠ 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 supergravity descriptions [205] with Udual 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 coarsegraining 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 Mtheory 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 setup 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 nonlinear MongeAmpere equation [142],

1/8 BPS configuration depend on a 6d manifold, whose scalar curvature satisfies a nonlinear 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 zeroentropy, minimallycharged bits, reinterpreting the initial blackhole entropy as the groundstate 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 AdS_{5} × S^{5} black holes, deconstructions of supersymmetric asymptoticallyflat black holes in terms of constituent excitations living at the horizon of these black holes and constituent models for extremal static nonBPS black holes.
Large supersymmetric AdS_{5} black holes: Large supersymmetric AdS_{5} × S^{5} black holes require the addition of angular momentum in AdS_{5}, besides the presence of Rcharges, 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 Rcharged black holes, it is natural to analyse whether the inclusion of angular momentum in AdS_{5} 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 nontrivial Poynting vector on the D3brane 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 D3brane probe in AdS_{5} × S^{5} were described in [22], whereas explicit supersymmetric electromagnetic waves on (dual) giants were constructed in [23]. Similar interesting work describing giant gravitons in the ppwave background with nontrivial 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 asymptoticallyflat 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 AdS_{5} and one in S^{5} 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 AdS_{5}. 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 HawkingBekenstein classical entropy.
Large asymptoticallyflat 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”^{Footnote 41}. The center locations are nontrivially 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 D6brane charge even if the black hole does not, provided there exists a second centre with antiD6brane charge, cancelling the latter.
This idea of deconstructing a given black hole in terms of maximally entropic configurations of constituent objects^{Footnote 42} was tested for the standard D0D4 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 D4D2D0 charges, and a large set of D0branes. Working in a regime of charges where the distance between centres scales to zero, i.e., the scaling solution, all D0branes become equidistant to the D6branes, forming some sort of accretion disk and the geometry deep inside this ring of D0branes becomes that of global AdS_{3} × S^{2}, when lifting the configuration to Mtheory. 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 nonabelian D0branes that expand into D2branes due to the Myers’ effect [395], the authors in [174] manage to extend the near horizon wrapping M2branes found in [455] to M2branes 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 CalabiYau due to the coupling of the D2D0 bound states to the background RR 4form 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 nonBPS deconstructions: These ideas are also applicable to nonsupersymmetric systems, though one expects to have less control there. For the subset of static extremal nonBPS black holes in the STU model [155, 194, 58], these methods turned out to be successful. The most general static blackhole solution, including nontrivial 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 Uduality 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 multicenter extremal nonBPS 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 nonsupersymmetric 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 nonlinear 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 nonstable 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 nonextremal 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 wellestablished 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 Dbrane 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 onshell supergravity background. In particular, one can probe either AdS_{5} × S^{5}, 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 setup is illustrated in Figure 8. The same interpretation will hold for nonrelativistic 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.
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 Dbrane probes to either add flavour to the AdS/CFT correspondence or describe the dynamics of massless excitations in nonrelativistic (thermal) setups, which could be of relevance for stronglycoupled condensedmatter 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) YangMills theories are nonlocal gauge invariant operators
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 quarkantiquark 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
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
The massive Wbosons 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 setup, the massive Wboson interacts with the U(N) gauge fields, including the scalar adjoint fields X^{I} [367], leading to the insertion of the operator
The contour \({\mathcal C}\) is parameterised by σ^{μ} (s) whereas the vector \(\vec \theta (s)\) maps each point on the loop to a point on the fivesphere.
The proposal made in [367, 433] to compute the expectation value of Eq. (434) was
This holds in the large \({g_s}N\) limit and S_{string} stands for the proper area of a fundamental string describing the loop \({\mathcal C}\) at the boundary of AdS_{5} and lying along θ^{I} (s) on S^{5}. Notice that a quantum mechanical calculation at strong coupling reduces to determining a minimal worldsheet surface in AdS_{5}, 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 Wboson one obtains the quarkantiquark potential energy
which differs from the linear perturbative dependence on \(g_{{\rm{YM}}}^2N\).
If one considers multiplywrapped Wilson loops, the many coincident strings will suffer from selfinteractions. This may suggest that a more appropriate description of the system is in terms of a D3brane with nontrivial 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 D3brane action agreed with the string worldsheet result at weak coupling, but captures all the highergenus 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.
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.
or from the gaugetheory 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 selfcontained. 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 AdS_{5}Schwarzschild black hole,
where \(h(z) = 1  {{{z^4}} \over {z_H^4}}\) determines the horizon size z_{H} and the blackhole temperature \(T = {1 \over {\pi {z_H}}}\). The latter coincides with the gaugetheory temperature [498]. Notice z = 0 is the location of the conformai boundary and L is the radius of AdS_{5}.
If one is interested in describing the dragging effect suffered by the quark due to the interactions with the thermal medium, one considers a nonstatic quark, whose trajectory in the boundary satisfies X^{l} (t) = υt, assuming motion takes place only in the x^{1} direction. One can parameterise the bulk trajectory as
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
is the worldsheet momentum current conjugate to the position X^{m}. Plugging the ansatz (438) into Eq. (439), one finds
where π_{ξ} 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
where y is a rescaled depth variable y = z/z_{h}.
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 linesegment and given the streadystate nature of the trailing string configuration, one infers [272]
This allows us to define the drag force as
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 infinitelymassive halfBPS 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 D5brane and D3brane world volume reaching the AdS_{5} 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 AdS_{5} × S^{5}. One expects its perturbative oscillations to be properly described by supergravity, whereas solitons with large conformal dimension,
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 Rcharge operators obtained in [70] using a worldsheet quantisation in the ppwave background by considering the bosonic part of the worldsheet action describing the AdS_{5} × S_{5} sigma model [270]
where n is a unit vector describing S^{5}, K is a hyperbolic unit vector describing AdS_{5}, the sigm model coupling α is \(\alpha = {1 \over {\sqrt \lambda}}\) ^{Footnote 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
where θ and ψ are the polar and azimuthal angles on S^{2} in S^{5}. Its classical worldsheet energy is
Next, consider the harmonic fluctuations around this classical soliton. Focusing on the quadratic θ oscillations,
one recognises the standard harmonic oscillator. Using its spectrum, one derives the corrections to the classical energy
where N_{n} is the excitation number of the nth 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 onshell condition
This is how one reproduces the spectrum derived in [70]
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 AdS_{5} spin, as done in [270], by analysing the behaviour of solitonic closed strings rotating in AdS. Using global AdS_{5},
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 3sphere while rotating in the azimuthal angle
for configurations ρ = ρ (σ), the NambuGoto bosonic action reduces to
where σ_{0} stands for the maximum radial coordinate and the factor of 4 arises because of the four string segments stretching from 0 to σ_{0} determined by the condition
The energy E and spin S of the string are conserved charges given by
Notice the dependence of \(E/\sqrt \lambda\) on \(S/\sqrt \lambda\) is in parametric form since L^{4} = 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 AdS_{5}.
Short strings: For large ω, the maximal string stretching is ρ_{0} ≈ 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]