1 Introduction

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

  1. The unification of the different perturbative string theories using duality symmetries [312Jump To The Next Citation Point, 495Jump To The Next Citation Point] relied strongly on the existence of non-perturbative supersymmetric states carrying Ramond–Ramond (RR) charge for their first tests.
  2. The discovery of D-branes as being such non-perturbative states, but still allowing a perturbative description in terms of open strings [423Jump To The Next Citation Point].
  3. The existence of decoupling limits in string theory providing non-perturbative formulations in different backgrounds. This gave rise to Matrix theory [48Jump To The Next Citation Point] and the anti de Sitter/conformal field theory (AdS/CFT) correspondence [366Jump To The Next Citation Point]. The former provides a non-perturbative formulation of string theory in Minkowski spacetime and the latter in AdS × M spacetimes.

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

  1. Dualities guarantee that fundamental strings are no more fundamental than other dynamical extended objects in the theory, called branes.
  2. D-branes, a subset of the latter, are non-perturbative states1 defined as dynamical hypersurfaces where open strings can end. Their weakly-coupled dynamics is controlled by the microscopic conformal field theory description of open strings satisfying Dirichlet boundary conditions. Their spectrum contains massless gauge fields. Thus, D-branes provide a window into non-perturbative string theory that, at low energies, is governed by supersymmetric gauge theories in different dimensions.
  3. 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 [48Jump To The Next Citation Point] and [366Jump To The Next Citation Point] are the existence of kinematical and dynamical regimes in which the full string theory is governed by either of these descriptions: the open or the closed string ones.

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

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

For multiple coincident branes, these theories are supersymmetric non-abelian gauge field theories. The second main difference from the abelian set-up is the current absence of a spacetime covariant and supersymmetric formulation, i.e., there is no known world volume diffeomorphic and kappa invariant formulation for them. As a consequence, we do not know how to couple these degrees of freedom to arbitrary (supersymmetric) curved backgrounds, as in the abelian case, and we must study these on an individual background case.

The covariant abelian brane actions provide a generalisation of the standard charged particle effective actions describing geodesic motion to branes propagating on arbitrary on-shell supergravity backgrounds. Thus, they offer powerful tools to study the dynamics of string/M-theory in regimes that will be precisely described. In the second part of this review, I describe some of their important applications. These will be split into two categories: supersymmetric world volume solitons and dynamical aspects of the brane probe approximation. Solitons will allow me to

Regarding the dynamical applications, the intention is to provide some dynamical interpretation to specific probe calculations appealing to the AdS/CFT correspondence [13Jump To The Next Citation Point] in two main situations

View Image

Figure 1: Layout of the main relations covered in this review.

Content of the review:
I start with a very brief review of the Green–Schwarz formulation of the superstring in Section 2. This is an attempt at presenting the main features of this formulation since they are universal in brane effective actions. This is supposed to be a reminder for those readers having a standard textbook knowledge of string theory, or simply as a brief motivation for newcomers, but it is not intended to be self-contained. It also helps to set up the notation for the rest of this review. Section 3 is fully devoted to the kinematic construction of brane effective actions. After describing the general string theory set-up where these considerations apply, it continues in Section 3.1 with the identification of the relevant dynamical degrees of freedom. This is done using open string considerations, constraints from world volume supersymmetry in p + 1 dimensions and the analysis of Goldstone mode in supergravity. A second goal in Section 3.1 is to convey the idea that spacetime covariance and manifest supersymmetry will require these effective actions to be both diffeomorphic and kappa symmetry invariant, where at this stage the latter symmetry is just conjectured, based on our previous world sheet considerations and counting of on-shell degrees of freedom. As a warm-up exercise, in Section 3.2, the bosonic truncations of these effective actions are constructed, focusing on diffeomorphism invariance, spacetime covariance, physical considerations and a set of non-trivial string theory duality checks that are carried in Section 3.3. Then, I proceed to discuss the explicit construction of supersymmetric brane effective actions propagating in a fixed Minkowski spacetime in Section 3.4. This has the virtue of being explicit and provides a bridge towards the more technical and abstract, but also more geometrical, superspace formalism, which provides the appropriate venue to covariantise the results in this particular background to couple the brane degrees of freedom to arbitrary curved backgrounds in Section 3.5. The main result of the latter is that kappa symmetry invariance is achieved whenever the background is an on-shell supergravity background. After introducing the effective actions, I discuss both their global bosonic and fermionic symmetries in Section 3.6, emphasising the difference between 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 (214View Equation). Reviewing the Hamiltonian formulation of these brane effective actions in 4.2, allows me to establish a link between supersymmetry, kappa symmetry, supersymmetry algebra bounds and their field theory realisations in terms of Hamiltonian BPS bounds in the space of bosonic configurations of these theories. The section finishes connecting these physical concepts to the mathematical notion of calibrations, and their generalisation, in Section 4.3.

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

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

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

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

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

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