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

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

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

- Dualities guarantee that fundamental strings are no more fundamental than other dynamical extended objects in the theory, called branes.
- D-branes, a subset of the latter, are non-perturbative states
^{1}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. - On the other hand, any source of energy interacts with gravity. Thus, if the number of branes is large enough, one expects a closed string description of the same system. The crucial realisations in [48] and [366] are the existence of kinematical and dynamical regimes in which the full string theory is governed by either of these descriptions: the open or the closed string ones.

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

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

- the identification of their dynamical degrees of freedom, providing a geometrical interpretation when available,
- the discussion of the world volume gauge symmetries required to achieve spacetime covariance and supersymmetry. These will include world volume diffeomorphisms and kappa symmetry,
- the description of the couplings governing the interactions in these effective actions, their global symmetries and their interpretation in spacetime,
- the connection between spacetime and world volume supersymmetry through gauge fixing,
- the description of the regime of validity of these effective actions.

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

The covariant abelian brane actions provide a generalisation of the standard charged particle effective actions describing geodesic motion to branes propagating on arbitrary on-shell 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

- stress the technical importance of kappa symmetry in determining these configurations, linking Hamiltonian methods with supersymmetry algebra considerations,
- prove the existence of string theory Bogomol’nyi–Prasad–Sommerfield (BPS) states carrying different (topological) charges,
- briefly mention microscopic constituent models for certain black holes.

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

- classical on-shell probe action calculations providing a window to strongly coupled dynamics, spectrum and thermodynamics of non-abelian gauge theories by working with appropriate backgrounds with suitable boundary conditions,
- probes approximating the dynamics of small systems interacting among themselves and with larger systems, when the latter can be reliably replaced by supergravity backgrounds.

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 (214). 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 D-branes and M2-branes. Regarding D-branes, this includes an introduction to super-Yang–Mills theories in 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.

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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