Brane Effective Actions, KappaSymmetry and Applications
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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.
1 Introduction
 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.
 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^{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.
 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.
 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.
 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 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.
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.
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.

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^{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 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}^{7}.
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.
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^{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
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.
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^{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].
Scalar multiplets with X scalars in p + 1 worldvolume dimensions.
p + 1  X  X  X  X 

1  1  2  4  8 
2  1  2  4  8 
3  1  2  4  8 
4  2  4  
5  4  
6  4 
Vector multiplets with X scalar degrees of freedom in p + 1 worldvolume dimensions.
p + 1  X  X  X  X 

1  2  3  5  9 
2  1  2  4  8 
3  0  1  3  7 
4  0  2  6  
5  1  5  
6  0  4  
7  3  
8  2  
9  1  
10  0 
Tensor multiplets with X scalar degrees of freedom in p + 1 world volume dimensions.
p + 1  X  X 

6  1  5 
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^{11}. The addition of interactions must be consistent with such d dimensional supersymmetries.

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] ^{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.
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.
Summary of supergravity Goldstone modes.
Symmetry  M2  M5  D3  

Reparametrisations:  ε^{m} =  \({U^{ 1}}{{\bar \phi}^m}\)  \({U^{ 1}}{{\bar \phi}^m}\)  \({U^{ 1}}{{\bar \phi}^m}\) 
Local supersymmetry:  ζ =  \({U^{ 2/3}}{{\bar \lambda}_ }\)  \({U^{ 7/12}}{{\bar \lambda}_ +}\)  \({U^{ 5/8}}{{\bar \lambda}_ +}\) 
Tensor gauge symmetry:  Λ =  \(_{(\ast\bar H = \bar H)}^{{U^{ 1}}{{\bar V}_{(2)}}}\)  \(_{(i\ast\bar F = \bar F)}^{{U^{ 1}}{{\bar V}_{(1)}}}\) 
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^{13}.
 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.
 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}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 \({\mathcal F} = 2\pi {\alpha {\prime}}d{V_1}  {\mathcal B}\).$$\delta \int\nolimits_{{\Sigma _2}} b = \int\nolimits_{{\Sigma _2}} d \Lambda = \int\nolimits_{\partial {\Sigma _2}} \Lambda ,$$(44)
 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 actionwhere \({T_{{{\rm{D}}_{\rm{p}}}}}\) tension.$${S_{{\rm{DBI}}}} =  {T_{{{\rm{D}}_{\rm{p}}}}}\int {{d^{p + 1}}} \sigma \,{e^{ \phi}}\sqrt { \det ({\mathcal G} + {\mathcal F})} ,$$(45)
 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 formwhere \({\mathcal C}\) 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].$$\int\nolimits_{{\Sigma _{p + 1}}} {\mathcal C} \wedge {e^{\mathcal F}},$$(46)
 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.
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.

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.

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 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.
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].
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^{19}.
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
 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].
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.^{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].^{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.

\(\bullet {\mathcal N} = 2 \, d = 10\) 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.
 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 suggestNotice, δ_{κ}V is chosen to remove the exact form coming from the kappa symmetry variation of B_{2}, i.e., \({\delta _\kappa}{B_2} =  2{\delta _\kappa}\bar \theta {\Gamma _\#}{\Gamma _m}d\theta {\Pi ^m} + d{\delta _k}{V_1}\).$$\begin{array}{*{20}c} {{\delta _\kappa}{X^m} =  {\delta _\kappa}\bar \theta {\Gamma ^m}\theta \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {{\delta _\kappa}{V_1} =  {\delta _\kappa}\bar \theta {\Gamma _\sharp}{\Gamma _m}\theta {\Pi ^m} + {1 \over 2}{\delta _\kappa}\bar \theta {\Gamma _\sharp}{\Gamma _m}\theta \bar \theta {\Gamma ^m}d\theta  {1 \over 2}{\delta _\kappa}\bar \theta {\Gamma ^m}\theta \bar \theta {\Gamma _\sharp}{\Gamma _m}d\theta \,.} \\ \end{array}$$(113)
 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 asrequiring the WZ kappa transformation to be$${\delta _\kappa}{{\mathcal L}_{{\rm{DBI}}}} = 2{\delta _\kappa}\bar \theta {\gamma ^{(p)}}T_{(p)}^\nu {\partial _\nu}\theta \,,\qquad {\rm{with}}\qquad {({\gamma ^{(p)}})^2} = 1\,,$$(114)In this way, the kappa symmetry variation of the full Lagrangian equals$${\delta _\kappa}{{\mathcal L}_{{\rm{WZ}}}} = 2{\delta _\kappa}\bar \theta T_{(p)}^\nu {\partial _\nu}\theta \,.$$(115)This is guaranteed to vanish choosing \({\delta _\kappa}\bar \theta = \bar k(1  {\gamma ^{(p)}})\), given the projector nature of \({1 \over 2}(1 \pm {\gamma ^{(p)}})\).$${\delta _\kappa}({{\mathcal L}_{{\rm{DBI}}}} + {{\mathcal L}_{{\rm{WZ}}}}) = 2{\delta _\kappa}\bar \theta (1 + {\gamma ^{(p)}})T_{(p)}^\nu {\partial _\nu}\theta \,.$$(116)
 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.
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.
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.
3.5.1 M2branes
3.5.2 Dbranes
3.5.3 M5branes
 1.
the algebra of κtransformations only closes onshell,
 2.
κsymmetry is an infinitely reducible symmetry.
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^{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

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.
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
 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 followswhere 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.$$\{{Q_\alpha},\,{Q_\beta}\} = \sum\limits_p {{{({\Gamma ^{{m_1} \ldots {m_p}}}{C^{ 1}})}_{\alpha \beta}}} {Z_{{m_1} \ldots {m_p}}}\,,$$(171)
 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 \(\{{Z_{{m_1} \ldots {m_p}}}\}\).
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.
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
 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.
I would like to close this discussion with a remark that is usually not stressed in the literature. By construction, any diffeomorphism and kappa symmetry gauge fixed brane effective action describes an interacting supersymmetric field theory in p +1 dimensions.^{27} As such, if there are available superspace techniques in these dimensions involving the relevant brane supermultiplet, the gauge fixed action can always be rewritten in that language. The matching between both formulations generically involves 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}.
 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.
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).
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.
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.^{28} Heuristically, one interprets the regime with E > E_{crit} as one where the string tension can no longer hold the string together.^{29}
4 World Volume Solitons: Generalities
Brane effective actions capture the relevant dynamics of 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.

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^{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.
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.
 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.
 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 toTo preserve the kappa gauge slice in the subspace \({\mathcal B}\) requires$$\theta = {\mathcal P}\theta + (1  {\mathcal P})\theta \,.$$(211)This determines the necessary compensating kappa symmetry transformation κ (ϵ) as a function of the background Killing spinors.$$s{\mathcal P}\theta {\vert _{\mathcal B}} = {\mathcal P}(1 + {\Gamma _\kappa}{\vert _{\mathcal B}})\kappa + {\mathcal P}\epsilon = 0\,.$$(212)
 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 themThis is equivalent to$$s(1  {\mathcal P})\theta {\vert _{\mathcal B}} = 0\,.$$(213)once Eq. (212) is taken into account. Projecting this equation into the \((1  {\Gamma _\kappa}{\vert_{\mathcal B}})\) subspace gives condition$$(1 + {\Gamma _\kappa}{\vert _{\mathcal B}})\kappa (\epsilon) + \epsilon = 0$$(214)No further information can be gained by projecting to the orthogonal subspace \((1  {\Gamma _\kappa}{\vert_{\mathcal B}})\).$${\Gamma _\kappa}{\vert _{\mathcal B}}\epsilon = \epsilon .$$(215)
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.
 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.
Set of kappa symmetry matrices Γ_{κ} evaluated in the bosonic subspace of configurations \({\mathcal B}\).
Brane  Bosonic kappa symmetry matrix 

M2brane  \({\Gamma _\kappa}{\vert_{\mathcal B}} = {1 \over {3!\sqrt { \det {\mathcal G}}}}{\epsilon ^{\mu \nu \rho}}{\gamma _{\mu \nu \rho}}\) 
M5brane  \({\Gamma _\kappa}{_{\mathcal B}} = {{{\upsilon _\mu}{\gamma ^\mu}} \over {\sqrt { \det ({\mathcal G} + \tilde H)}}}\left[ {{{\sqrt { \det {\mathcal G}}} \over 2}{\gamma ^{{\mu _1}{\mu _2}}}{{\tilde H}_{{\mu _1}{\mu _2}}} + {\gamma _v}{t^v}  {1 \over {5!}}{\varepsilon ^{{\mu _1} \ldots {\mu _5}v}}{\upsilon _v}{\gamma _{{\mu _1} \ldots {\mu _5}}}} \right]\) 
IIA Dpbranes  \({\Gamma _\kappa}{_{\mathcal B}} = {1 \over {\sqrt { \det ({\mathcal G} + {\mathcal F})}}}\sum\nolimits_{l = 0} {{\gamma _{2l + 1}}\Gamma _\# ^{l + 1}} \wedge{e^{\mathcal F}}\) 
IIB Dpbranes  \({\Gamma _\kappa}{_{\mathcal B}} = {1 \over {\sqrt { \det ({\mathcal G} + {\mathcal F})}}}\sum\nolimits_{l = 0} {{\gamma _{2l}}\tau _3^li{\tau _2}} \wedge{e^{\mathcal F}}\) 
 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_{∞}.
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.
 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 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
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
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
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.
 (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 setis not empty.$${\rm{G}}(\varphi)=\{\xi \in {\rm{G}}(p,{{\mathbb R}^n}):\varphi (\vec \xi) = 1\}$$(235)
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
 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 aswhere X*φ stands for the pullback of the pform φ.$$\int {{d^p}} \sigma \sqrt {\det {{\mathcal G}_{\mu \nu}}} \geq \int {{X^{\ast}}} \varphi \,,$$(238)
 2.
There exists an explicit spinor construction of calibrations emphasising the connection between calibrated submanifolds, supersymmetry and kappa symmetry.
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.
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
 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).
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.
HalfBPS branes and the supersymmetries they preserve.
BPS state  Projector 

M2brane  Γ_{012}ϵ = ϵ 
M5brane  Γ_{012345}ϵ = ϵ 
IIA D_{2n}brane  \({\Gamma _{0 \ldots 2n}}\Gamma _\sharp ^{n + 1}\epsilon = \epsilon\) 
IIB D_{2n−1}brane  \({\Gamma _{0 \ldots 2n  1}}\tau _3^ni{\tau _2}\epsilon = \epsilon\) 
5.2 Intersecting M2branes
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^{+}).
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
5.4 BIons
5.5 Dyons
5.6 Branes within branes
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
5.6.2 Dp −D(p + 2) systems
5.6.3 FDp systems
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.
 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.
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.^{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.
Supergravity description and fuzzball considerations: The fact that world volume quantisation reproduces the entropy of a macroscopic configuration and the presence of arbitrary profiles, at the classical level, suggests that supersymmetric supertubes may provide a window to understand the origin of gravitational entropy in a regime of parameters where gravity is reliable. This is precisely one of the goals of the fuzzball programme [363, 361].^{37}
A first step towards this connection was provided by the supergravity realisation of supertubes given in [205]. These are smooth configurations described in terms of harmonic functions whose sources allow arbitrary profiles, thus matching the arbitrary 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}^{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].
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.
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].
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^{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
 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.
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.

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].
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”^{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^{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

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.
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.
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
 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.
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
Applying semiclassical quantisation methods to these classical solitons [216], it was realised that one can interpolate the results for E — S to the weaklycoupled regime. It should be stressed that these techniques allow one to explore the AdS/CFT correspondence in nonsupersymmetric sectors [217], appealing to the correspondence principle associated to large charges. It is also worth mentioning that due to the seminal work on the integrability of planar \({\mathcal N} = 4\,{\rm{SYM}}\) at one loop [393, 60], much work has been devoted to using these semiclassical techniques in relation to integrability properties [21]. Interested readers are encouraged to check the review [59] on integrability and references therein.
6.4 Probes as deformations and gapless excitations in complex systems
This setup occurs when the brane degrees of freedom are responsible for either breaking the symmetries of the larger system or describing an interesting isolated set of massless degrees of freedom whose interactions among themselves and with the background one is interested in studying. In the following, I very briefly describe how the first approach was used to introduce flavour in the AdS/CFT correspondence, and how the second one can be used to study physics reminiscent of certain phenomena in condensedmatter systems.
Probing deformations of the AdS/CFT: Deforming the original AdS/CFT allows one to come up with setups with less or no supersymmetry. Whenever there is a small number of degrees of freedom responsible for the dynamics (typically Dbranes), one may approximate the latter by the effective actions described in this review. This provides a reliable and analytical tool for describing the stronglycoupled behaviour of the deformed gauge theory.
This logic can be extended to nonsupersymmetric scenarios^{44}. For example, using the string theory realisation of fourdimensional QCD with N_{c} colours and N_{f} ≪ N_{c} flavours discussed in [499]. The latter involves N_{f} D6brane probes in the supergravity background dual to N_{c} D4branes compactified on a circle with supersymmetrybreaking boundary conditions and in the limit in which all the resulting KaluzaKlein modes decouple. For N_{f} = 1 and for massless quarks, spontaneous chiral symmetry breaking by a quark condensate was exhibited in [349] by working on the D6brane effective action in the near horizon geometry of the N_{c} D4branes.
Similar considerations apply at finite temperature by using appropriate blackhole backgrounds [499] in the relevant probe action calculations. This allows one to study phase transitions associated with the thermodynamic properties of the probe degrees of freedom as a function of the probe location. This can be done in different theories, with flavour [379], and for different ensembles [343, 378].
The amount of literature in this topic is enormous. I refer the reader to the reviews on the use of gaugegravity duality to understand hot QCD and heavy ion collisions [137] and meson spectroscopy [207], and references therein. These explain the tools developed to apply the AdS/CFT correspondence in these setups.
Condensed matter and strange metallic behaviour: There has been a lot of work in using the AdS/CFT framework in condensed matter applications. The reader is encouraged to read some of the excellent reviews on the subject [283, 296, 385, 284, 285], and references therein. My goal in these paragraphs is to emphasise the use of IR probe branes to extract dynamical information about certain observables in quantum field theories in a state of finite charge density at low temperatures.
Before describing the string theory setups, it is worth attempting to explain why any AdS/CFT application may be able to capture any relevant physics for condensed matter systems. Consider the standard Fermi liquid theory, describing, among others, the conduction of electrons in regular metals. This theory is an example of an IR free fixed point, independent of the UV electron interactions, describing the lowest energy fermionic excitations taking place at the Fermi surface κ = κf. Despite its success, there is experimental evidence for the existence of different “states of matter”, which are not described by this effective field theory. This could be explained by additional gapless bosonic excitations, perhaps arising as collective modes of the UV electrons. For them to be massless, the system must either be tuned to a quantum critical point or there must exist a kinematical constraint leading to a critical phase.
More than the specific physics, which is nicely described in [334, 398, 286], what is important to stress, once more, is that using the appropriate backgrounds, exciting the relevant degrees of freedom and considering the adequate boundary conditions make the methods described in this review an extremely powerful tool to learn about physics in regimes of parameters that would otherwise be very difficult to handle, both analytically and conceptually.
7 Multiple Branes
The physics of multiple overlapping branes provides a connection between braue physics and nonabelian supersymmetric field theories. Thus, it has played a crucial role in the geometrisation of the latter and the interplay between string and field theory dualities.
An heuristic argument suggesting that the abelian description may break down comes from the analysis of BIons. All halfBPS probe branes described in this review feel no force when probing the background describing N − 1 parallel branes of the same nature [484]. This means they can sit at any distance ℓ. Consider a Dpbrane in the background of N − 1 parallel Dpbranes. As soon as the probe approaches the location of the Dpbranes sourcing the geometry, the properly regularised mass of the open string (BIon) stretching between the probe Dbrane and the background Dbranes will tend to zero [227]. This suggests the potential emergence of extra massless modes in the spectrum of these open strings. If so, this would signal a breakdown in the effective action, since these extra modes were not included in the former. Uduality guarantees that similar considerations apply to other brane setups not having a microscopic theory with which to test this phenomena.
In this section, I will briefly discuss the supersymmetric effective actions describing N coincident Dpbranes and M2branes in a Minkowski background. These correspond to nonabelian superYangMills (SYM) theories in different dimensions and certain d = 3 superconformal field theories with nondynamical gauge fields having ChernSimons actions, respectively.
7.1 Dbranes
A remarkable feature of this Dbrane description is that a classical geometrical interpretation of Dbrane configurations is only available when the matrices Φ^{I} are simultaneously diagonalisable. This provides a rather natural venue for noncommutative geometry to appear in Dbrane physics at short distances, as first pointed out in [496].
 1.
Keeping the background fixed, i.e., Minkowski, it is natural to consider the inclusion of higherorder corrections in the effective action, matching the perturbative scattering amplitudes computed in the CFT description of open strings theory, and
 2.
Allowing to vary the background or equivalently, coupling the nonabelian degrees of freedom to curved background geometries. This is towards the direction of achieving a hypothetical covariant formulation of these actions, a natural question to ask given its relevance for the existence of the kappa invariant formulation of abelian Dbranes.
Higherorder corrections: In the abelian theory, it is well known that the DBI action captures all the higherorder corrections in α ′ to the open string effective action in the absence of field strength derivative terms^{47} [214]. It was further pointed that such derivative corrections were compatible with a DBI expansion by requiring conformal invariance for the bosonic string in [1] and for the superstring in [87].
 1.
It must contain a unique trace since this is an effective action derived from string theory disk diagrams involving many graviton insertions in their interior and scalar/vector vertex operators on their boundaries. Since the disk boundary is unique, the trace must be unique.
 2.
It must reduce to Ncopies of the particle action when the matrices X^{I} are diagonal.
 3.
It must yield masses proportional to the geodesic distance for offdiagonal fluctuations.
 a)
to match the Matrix theory linear couplings to closed string backgrounds, and
 b)
to be Tduality covariant, extending the notion I discussed in Section 3.3.2 for single Dbranes.
 1.There exists some nontrivial dependence on the scalars Φ^{I} through the arbitrary bosonic closed backgrounds appearing in the action. The latter is defined according toAnalogous definitions apply to other background fields.$$\begin{array}{*{20}c} {{g_{\mu \nu}} = \exp \left[ {\lambda {\Phi ^i}\,{\partial _{{x^i}}}} \right]g_{\mu \nu}^0({\sigma ^a},{x^i}){\vert_{{x^i} = 0}}\quad \quad \quad \quad \quad \quad \quad} \\ {= \sum\limits_{n = 0}^\infty {{{{\lambda ^n}} \over {n!}}} \,{\Phi ^{{i_1}}} \cdots {\Phi ^{{i_n}}}\,({\partial _{{x^{{i_1}}}}} \cdots {\partial _{{x^{{i_n}}}}})g_{\mu \nu}^0({\sigma ^a},{x^i}){\vert_{{x^i} = 0}}.} \\ \end{array}$$(491)
 2.
There exists a unique trace, because this is an open string effective action that can be derived from worldsheet disk amplitudes. The latter has a unique boundary. Thus, there must be a unique gauge trace [186, 188]. Above, the symmetrised prescription was assumed, not only because one is following Tseytlin and this was his prescription, but also because there are steps in the derivation of Tduality covariance that assumed this property and the scalar field Φ^{I} dependence on the background fields (491) is symmetric, by definition.
 3.
The WZ term (489) allows multiple Dpbranes to couple to RR potentials with a form degree greater than the dimension of the worldvolume. This is a purely nonabelian effect whose consequences will be discussed below.
 4.
There are different sources for the scalar potential: det Q^{I}_{j}, its inverse in the first determinant of the DBI and contributions coming from commutators coupling to background field components in the expansion (491).
 1.
There is no ambiguity of trace in the linear Matrix theory calculations. Myers’ suggestion is to extend this prescription to nonlinear couplings.
 2.
Some transverse M5brane charge couplings are unknown in Matrix theory, but these are absent in the Lagrangian above. This is a prediction of this formulation.
It is reassuring to compare the description above with the one available using the abelian formalism describing a single brane explained in Section 3. I shall refer to the latter as dual brane description. For the particular example discussed above, since the D0branes blow up into spheres due to the electric RR coupling, one can look for onshell configurations on the abelian D2brane effective action in the same background corresponding to the expanded spherical D0branes in the nonabelian description. These configurations exist, reproduce the energy V_{N} up to 1/N^{2} corrections and carry no D2brane charge [395]. Having reached this point, I am at a position to justify the expansion of pointlike gravitons into spherical D3branes, giant gravitons, in the presence of the RR flux supporting AdS_{5} × S^{5} described in Section 5.9. The nonabelian description would involve nontrivial commutators in the WZ term giving rise to a fuzzy sphere extremal solution to the scalar potential. The abelian description reviewed in Section 5.9 corresponds to the dual D3brane description in which, by keeping the same background, one searches for onshell spherical rotating D3branes carrying the same charges as a pointlike graviton but no D3brane charge. For a more thorough discussion of the comparison between nonabelian solitons and their “dual” abelian descriptions, see [147, 149, 148, 396].
Kappa symmetry and superembeddings: The covariant results discussed above did not include fermions. Whenever these were included in the abelian case, a further gauge symmetry was required, kappa symmetry, to keep covariance, manifest supersymmetry and describe the appropriate onshell degrees of freedom. One suspects something similar may occur in the nonabelian case to reduce the number of fermionic degrees of freedom in a manifestly supersymmetric nonabelian formulation. It is important to stress that at this point world volume diffeomorphisms and kappa symmetry will no longer appear together. In all the discussions in this section, world volume diffeomorphisms are assumed to be fixed, in the sense that the only scalar adjoint matrices already correspond to the transverse directions to the brane.
There exists some body of work constructing classical supersymmetric and kappa invariant actions involving nonabelian gauge fields representing the degrees of freedom of multiple Dbranes. This started with actions describing branes of lower codimension propagating in lower dimensional spacetimes [461, 462, 190]. It was later extended to multiple D0branes in an arbitrary number of dimensions, including type IIA, in [411]. Here, both world volume diffeomorphisms and kappa symmetry were assumed to be abelian. It was checked that when the background is superPoincaré, the proposed action agreed with Matrix Theory [48]. Using the superembedding formalism [460], actions were proposed reproducing the same features in [40, 44, 42, 41, 43], some of them involving a superparticle propagating in arbitrary 11dimensional backgrounds. Finally, there exists a slightly different approach in which, besides using the superembedding formalism, the world sheet ChanPaton factors describing multiple Dbranes are replaced by boundary fermions. The actions constructed in this way in [303], based on earlier work [304], have similar structure to the ones described in the abelian case, their proof of kappa symmetry invariance is analogous and they reproduce Matrix Theory when the background is superPoincaré and most of the features highlighted above for the bosonic couplings described by Myers.
Regarding Dbranes in the presence of a Bfield, the main observation is that the structure of an abelian noncommutative gauge theory is similar to that of a nonabelian commutative gauge theory. In both cases, fields no longer commute, and the field strengths are nonlinear. Moreover, noncommutative gauge theories can be constructed starting from a nonabelian commutative theory by expanding around suitable backgrounds and taking N → ∞ [443]. This connection suggests it may be possible to relate the gravity coupling of noncommutative gauge theories to the coupling of nonabelian Dbrane actions to curved backgrounds (gravity). This was indeed the approach taken in [163] where the stresstensor of noncommutative gauge theories was derived in this way. In [151], constraints on the kinematical properties of nonabelian Dbrane actions due to this connection were studied.
7.2 M2branes
In this section, I would like to briefly mention the main results involving the amount of progress recently achieved in the description of N parallel M2branes, referring to the relevant literature when appropriate. This will be done taking the different available perspectives on the subject: a purely kinematic approach, based on supersymmetry and leading to 3algebras, a purely field theory approach leading to three dimensional CFTs involving ChernSimons terms, a brane construction approach, in which one infers the low energy effective description in terms of an intersection of branes and the connection between all these different approaches.
The main conclusion is that the effective theory describing N M2branes is a d =3, U(N)×U(N) gauge theory with four complex scalar fields C_{I} (I =1, 2, 3,4) in the (\({\bf{N}},{\bf{\bar N}}\)) representation, their complex conjugate fields in the (\({\bf{\bar N}},{\bf{N}}\)) representation and their fermionic partners [12]. The theory includes non dynamical gauge fields with a ChernSimons action with levels κ and −κ for the two gauge groups. This gauge theory is weakly coupled in the large κ limit (κ ≫ N) and strongly coupled in the opposite regime (κ ≪ N), for which a weakly coupled gravitational description will be available if N ≫ 1.
Closure of the supersymmetry algebra determines a set of equations of motion that can be derived, which form a Lagrangian. It was soon realised that under the assumptions of a real vector space, essentially the only 3algebra is the one defined by \({f^{abc}} = {f^{abc}}_e{h^{ed}}\), with h^{ab} = Tr (T^{a}, T^{b}) defining an inner product, and satisfying \({f^{abcd}} \propto {\varepsilon ^{abcd}}\) [399, 412, 226]. Interestingly, it was pointed out in [488] that such supersymmetric field theory could be rewritten as a ChernSimons theory. The latter provided a link between a purely kinematic approach, based on supersymmetry considerations, and purely field theoric results that had independently been developed.
Field theory considerations: Conformal field theories have many applications. In the particular context of ChernSimons matter theories in d = 3, they can describe interesting IR fixed points in condensed matter systems. Here I am interested in their supersymmetric versions to explore the AdS_{4}/CFT_{3} conjecture.
It was argued in [12] that the \({\mathcal N} = 6\) theory constructed above was dual to N M2branes on \({{\rm{{\mathbb C}}}^4}/{{\rm{{\mathbb Z}}}_k}\) for k≥ 3. Below, I briefly review the brane construction in which their argument is based. This will provide a nice example of the notion of geometrisation (or engineering) of supersymmetric field theories provided by brane configurations.
 1.
Moving the D5branes along the 78directions generates a complex mass parameter.
 2.
Moving the D5branes along the 5directions generates a real mass, of positive sign for the fields in the fundamental representation and of negative sign for the ones in the antifundamental.
 3.
Breaking the κ D5branes and NS5branes along the 01234 directions and merging them into an intermediate (1,κ) 5brane bound state generates a real mass of the same sign for both N and \({{\rm{\bar N}}}\) representations. This mechanism is a web deformation [72]. The merging is characterised by the angle θ relative to the original NS5brane subtended by the bound state in the 59plane. The final brane configuration is made of a single NS5brane in the 012345 directions and a (1, κ) 5brane in the 01234[5, 9]_{θ}, where [5, 9]g stands for the x^{5} cos θ+x^{4} sin θ direction. θ is fixed by supersymmetry [14].
After the web deformation and at low energies, one is left with an \({\mathcal N} = 2 \, U{(N)_k} \times U{(N)_{ k}}\). YangMillsChernSimons theory with four massless bifundamental matter multiplets (and their complex conjugates), and two massless adjoint matter multiplets corresponding to the motion of the D3branes in the directions 34 common to the two 5branes.
The enhancement to an \({\mathcal N} = 3\) theory described in the purely field theoretical context is realised in the brane construction by rotating the (1, κ) 5brane in the 37 and 48planes by the same amount as in the original deformation. Thus, one ends with a single NS5brane in the 012345 and a (1, κ) 5brane along 012[3, 7]_{θ}[4, 8]_{θ}[5, 9]_{θ}.This particular mass deformation ensures all massive adjoint fields acquire the same mass, enhancing the symmetry to \({\mathcal N} = 3\). Equivalently, there must exist an SO(3)_{R} Rsymmetry corresponding to the possibility of having the same SO(3) rotations in the 345 and 789 subspaces. Thus, the d = 3 supersymmetric field theory must be \({\mathcal N} = 3\).
The connection to \({\mathcal N} = 6\) is obtained by flowing the \({\mathcal N} = 3\) theory to the IR [12]. Indeed, by integrating out all the massive fields, we recover the field content and interactions described in the field theoretical \({\mathcal N} = 6\) construction. The enhancement to \({\mathcal N} = 8\) for κ = 1, 2 was properly discussed in [276].
It was realised in [12] that under Tduality in the x^{6} direction and uplifting the configuration to Mtheory, the brane construction gets mapped to N M2branes probing some configuration of KKmonopoles. These have a supergravity description in terms of hyperKähler geometries [224]. Flowing to the IR in the dual gravitational picture is equivalent to probing the near horizon of these geometries, which includes the expected AdS_{4} factor times a quotient of the 7sphere.
The ChernSimons theory has a 1/κ coupling constant. Thus, large κ has a weakly coupled description. At large N, it is natural to consider the ’t Hooft limit: λ = N/k fixed. The gauge theory is weakly coupled for κ ≫ N and strongly coupled for κ ≪ N. In the latter situation, the supergravity description becomes reliable and weakly coupled for N ≫ 1 [12].
8 Related Topics
There are several topics not included in previous sections that are also relevant to the subjects covered in this review. The purpose of this last section is to mention some of them, mentioning their main ideas and/or approaches, and more importantly, referring the reader to some of the relevant references where they are properly developed and explained.
There are many results in this subject, nicely reviewed in [460]. It is worth mentioning that some equations of motion for supersymmetric objects in different numbers of dimensions were actually first derived in this formalism rather than in the GS one, including [220] for the d =10 superparticle, [47] for the superstring and supermembrane, [306] for superbranes and [305] for the M5brane^{50}. It is particularly relevant to stress the work done in formulating the M5brane equations of motion covariantly [307, 308] and their use to identify supersymmetric world volume solitons [301, 302], and in pointing out the relation between superembeddings and nonlinear realisations of supersymmetry [5].
MKKmonopoles and other exotic brane actions: This review was focused on the dynamics of Dbranes and Mbranes. It is well known that string and M theory have other extended objects, such as KKmonopoles or NS5branes. There is a nice discussion regarding the identification of the degrees of freedom living on these branes in [311]. Subsequently, effective actions were written down to describe the dynamics of its bosonic sectors in [83, 80, 208, 209]. In particular, it was realised that gauged sigma models are able to encapsulate the right properties for KK monopoles. The results obtained in these references are consistent with the action of Tduality and Sduality. Of course, it would be very interesting to include fermions in these actions and achieve kappa symmetry invariance.
Blackfolds: The blackfold approach is suitable to describe the effective world volume dynamics of branes, still in the probe approximation, having a thermal population of excitations. In some sense, it describes the dynamics of these objects on length scales larger than the brane thickness. This formalism was originally developed in [201, 202] and extended and embedded in string theory in [203]. It was applied to the study of hot BIons in [261, 262], emphasising the physical features not captured by the standard DiracBornInfeld action, and to blackfolds in AdS [20].
Nonrelativistic kappa invariant actions: All the branes described in this review are relativistic. It is natural to study their nonrelativistic limits, both for its own sake, but also as an attempt to identify new sectors of string theory that may be solvable. The latter is the direction originally pursued in [246, 161] by considering closed strings in Minkowski. This was extended to closed strings in AdS_{5} × S^{5} in [244]. At the level of brane effective actions in Minkoswki, nonrelativistic diffeomorphism and kappa symmetry invariant versions of them were obtained in [245] for D0branes, fundamental strings and M2branes, and later extended to general Dpbranes in [247]. The consistency of these nonrelativistic actions under the action of duality transformations was checked in [330]. This work was extended to nonrelativistic effective Dbrane actions in AdS_{5} × S^{5} in [119, 436].
Multiple M5branes: It is a very interesting problem to find the nonabelian formulation of the (2,0) tensor multiplet describing the dynamics of N M5branes. Following similar ideas to the ones used in the construction of the multiple M2brane action using 3algebras, some nonabelian representation of the (2,0) tensor supermultiplet was found in [351]. Their formulation includes a nonabelian analogue of the auxiliary scalar field appearing in the PST formulation of the abelian M5brane. Closure of the superalgebra provides a set of equations of motion and constraints. Expanding the theory around a particular vacuum gives rise to d = 5 SYM along with an abelian (2,0) d = 6 supermultiplet. This connection to d = 5 SYM was further studied in [352]. Some further work along this direction can be found in [299]. Some of the BPS equations derived from this analysis were argued to be naturally reinterpreted in loop space [414]. There has been a different approach to the problem involving noncommutative versions of 3algebras [275], but it seems fair to claim that this remains a very important open problem for the field.
Footnotes
 1.
Nonperturbative in the sense that their mass goes like 1/g_{s}, where g_{s} is the stringcoupling constant.
 2.
 3.
 4.
 5.
Recently, it was pointed out in [390] that there may exist quantummechanically consistent superstrings in d =3. It remains to be seen whether this is the case.
 6.
The existence of kappa symmetry as a fermionic gauge symmetry was first pointed out in superparticle actions in [169, 170, 451, 171]. Though the term kappa symmetry was not used in these references, since it was later coined by Townsend, the importance of WZ terms for its existence is already stated in these original works.
 7.
For a proper definition of these superfields, see Appendix A.1.
 8.
See Appendix A.1 for a better discussion of what this means.
 9.
 10.
 11.
Here, \({\mathcal N}\) stands for the number of world volume supersymmetries.
 12.
 13.
 14.
Since I am not considering supersymmetric branes at this point, Q_{p} = T_{p} is not a necessary condition.
 15.
This is the correct way to compute the energy momentum tensor due to the coupling of branes to gravity. The energy carried by such a brane must be localised on its p + 1 dimensional world volume.
 16.
The importance of these assumptions will be stressed when discussing the regime of validity of brane effective actions in Section 3.7.
 17.
There actually exist further gravitational interaction terms necessary for the cancellation of anomalies [253], but we will always omit them in our discussions concerning Dbrane effective actions.
 18.
For a discussion on the interpretation of an M5brane as a ‘Dbrane’ for an open membrane, see [55].
 19.
 20.
For a discussion of the supersymmetric and kappa invariant M5brane covariant action propagating in superPoincaré, see [144].
 21.
Following the same philosophy as for their bosonic truncations, this functional dependence can be derived from the double dimensional reduction of the supersymmetric M2brane action to be discussed in Section 3.4.2 [477, 439]. This also provides a derivation of the WZ couplings to be constructed in this subsection. Of course, this consideration would only apply to the D2brane, but Tduality would allow one to extend this conclusion for any Dpbrane [292, 331]
 22.
For p =1, B_{2} is the NSNS 2form, whereas for p = 2, B_{3} = A_{3} is the d =11 3form gauge potential.
 23.
 24.
I will prove this explicitly in Section 5.1.
 25.
All our charge conjugation matrices are antisymmetric and unitary, i.e., C^{T} = −C and \({C^\dagger}C = 1\). Furthermore, all Clifford matrices satisfy the symmetry relation \(\Gamma _m^T =  C{{\rm{T}}_m}{C^{ 1}}\).
 26.
 27.
I have assumed both the background and the brane preserve some supersymmetry.
 28.
There exists some similar phenomena on the M5brane dynamics with the selfdual 3form field strength. See [74] for a discussion on the emergence of noncommutative gauge theories when the selfdual 3form field strength is close to its critical value.
 29.
 30.
 31.
 32.
This notation is introduced to emphasise that \({\mathcal G}_5^{ij}\) does not correspond to the world space components of \({{\mathcal G}^{\mu \nu}}\), but to the inverse matrix of the restriction of \({{\mathcal G}^{\mu \nu}}\) to the world space subspace.
 33.
 34.
For simplicity I am setting the D3brane tension to one.
 35.
 36.
Strictly speaking, if the supertube crosssection is open, they can carry D2brane charge. The arguments given above only apply to closed crosssections. The reader is encouraged to read the precise original discussion in [380] concerning this point and the bounds on angular momentum derived from it.
 37.
 38.
By arbitrary, it is meant a general curve that is not selfintersecting and whose tangent vector never vanishes.
 39.
I do not write this term explicitly here because it will not couple to our D3brane probes.
 40.
Dual giant gravitons are spherical rotating D3branes in which the 3sphere wrapped by the brane is in AdS_{5}. See [264] for a proper construction of these configurations and some of its properties.
 41.
 42.
What is meant here by maximally entropic is that, given a large black hole, there may be more than one possible deconstruction of the total charge in terms of constituents with different charge composition. By maximally entropic I mean the choice of charge deconstruction whose moduli space of configurations carries the largest contribution to the entropy of the system.
 43.
This overall coupling constant is derived from the string tension 1/2πα ′ and the overall \({L^2} \sim {g_s}N{\alpha {\prime}}\) scale from the AdS_{5} × S_{5} background geometry.
 44.
For an analysis of supersymmetric D5branes in a supergravity background dual to \({\mathcal N} = 1\,{\rm{SYM}}\), see [408].
 45.
Φ^{I} is the natural adjoint scalar field after dimensional reduction. The rescaling by 2π α′ is to match the natural scalar fields appearing in the abelian description provided by the DBI action. A similar rescaling occurs for the fermions omitted below, Ψ = 2π α′ ψ.
 46.
 47.
Using Tduality arguments this would also include acceleration and higherderivative corrections in the scalar sector X^{m} describing the excitations of the Dbrane along the transverse dimensions.
 48.
Eight is the number of transverse dimensions to the world volume of the M2branes.
 49.
For a complete list of references, see [12].
 50.
The equivalence of the equations of motion obtained in the PSTformalism and the ones developed in the superembedding formalism was proven in [46].
 51.
 52.
The chirality can be α1 or αi, depending on the reality of the volume form eigenspace.
 53.
Same comments as above.
Notes
Acknowledgements
Some of the material covered in this review is based on the PhD thesis World Volume Approach to String Theory defended by the author under the supervision of Prof. J. Gomis at the University of Barcelona in May 2000. JS would like to thank his PhD advisor J. Gomis for introducing him to this subject, to P. K. Townsend for sharing his extensive knowledge on many topics reviewed here, and to F. Brandt, K. Kamimura, O. Lunin, D. Mateos, A. Ramallo and J.M. FigueroaO’Farrill for discussions and collaboration on part of the material reported in this work. The work of JS was partially supported by the Engineering and Physical Sciences Research Council [grant number EP/G007985/1].
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