Brane Effective Actions, Kappa-Symmetry and Applications Joan Sim´on School of Mathematics and Maxwell Institute for Mathematical Sciences email: [email protected] Abstract This is a review on brane effective actions, their symmetries and some of its applications. Its first part uncovers the Green-Schwarz formulation of single M- and D-brane effective actions focusing on kinematical aspects : the identification of their degrees of freedom, the importance of world volume diffeomorphisms and kappa symmetry, to achieve manifest spacetime covari- ance and supersymmetry, and the explicit construction of such actions in arbitrary on-shell supergravity backgrounds. Its second part deals with applications. First, the use of kappa symmetry to determine supersymmetric world volume solitons. This includes their explicit construction in flat and curved backgrounds, their interpretation as 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 non-trivial dynamics of strongly coupled gauge theories using the AdS/CFT correspondence. This in- cludes expectation values of Wilson loop operators, spectrum information and the general use of D-brane probes to approximate the dynamics of systems with small number of degrees of freedom interacting with larger systems allowing a dual gravitational description. Its final part briefly discusses effective actions for N D-branes and M2-branes. This in- cludes both SYM theories, their higher order corrections and partial results in covariantising these couplings to curved backgrounds, and the more recent supersymmetric Chern-Simons matter theories describing M2-branes using field theory, brane constructions and 3-algebra considerations. arXiv:1110.2422v3 [hep-th] 22 Nov 2011 1 Contents 1 Introduction 3 2 The Green-Schwarz superstring: a brief motivation 7 3 Brane effective actions 11 3.1 Degrees of freedom & world volume supersymmetry . 12 3.1.1 Supergravity Goldstone modes . 15 3.2 Bosonic actions . 19 3.3 Consistency checks . 24 3.3.1 M2-branes and its classical reductions . 27 3.3.2 T-duality covariance . 29 3.3.3 M5-brane reduction . 32 3.4 Supersymmetric brane effective actions in Minkowski . 33 3.4.1 D-branes in flat superspace . 34 3.4.2 M2-brane in flat superspace . 39 3.5 Supersymmetric brane effective actions in curved backgrounds . 40 3.5.1 M2-branes . 41 3.5.2 D-branes . 41 3.5.3 M5-branes . 42 3.6 Symmetries : spacetime vs world volume . 43 3.6.1 Supersymmetry algebras . 44 3.6.2 World volume supersymmetry algebras . 47 3.7 Regime of validity . 50 4 World volume solitons : generalities 52 4.1 Supersymmetric bosonic configurations & kappa symmetry . 53 4.2 Hamiltonian formalism . 56 4.2.1 D-brane hamiltonian . 57 4.2.2 M2-brane hamiltonian . 58 4.2.3 M5-brane hamiltonian . 59 4.3 Calibrations . 60 5 World volume solitons : applications 62 5.1 Vacuum infinite branes . 63 5.2 Intersecting M2-branes . 64 5.3 Intersecting M2 and M5-branes . 66 5.4 BIons . 69 5.5 Dyons . 71 5.6 Branes within branes . 73 5.6.1 Dp-D(p+4) systems . 74 5.6.2 Dp-D(p+2) systems . 75 5.6.3 F-Dp systems . 76 5.7 Supertubes . 76 5.8 Baryon vertex . 79 5.9 Giant gravitons & superstars . 85 5.9.1 Giant gravitons as black hole constituents . 87 5.10 Deconstructing black holes . 89 2 6 Some AdS/CFT related applications 92 6.1 Wilson loops . 93 6.2 Quark energy loss in a thermal medium . 94 6.3 Semiclassical correspondence . 96 6.4 Probes as deformations & gapless excitations in complex systems . 98 7 Multiple branes 101 7.1 D-branes . 102 7.2 M2-branes . 111 8 Related topics 114 9 Acknowledgements 116 A Target superspace formulation & constraints 117 A.1 N = 2 type IIA/B superspace . 117 A.2 N = 1 d=11 supergravity conventions . 120 B Cone construction & supersymmetry 120 B.1 (M; g) riemannian . 122 B.2 (M; g) of signature (1; d − 1) . 122 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 [313, 496] 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 pertur- bative description in terms of open strings [424]. 3. The existence of decoupling limits in string theory providing non-perturbative formulations in different backgrounds. This gave rise to Matrix theory [49] and the AdS/CFT correspondence [367]. 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 hyper- surfaces where open strings can end. Their weakly coupled dynamics is controlled by the microscopic conformal field theory description of open strings satisfying Dirichlet boundary conditions. Their spectrum contains massless gauge fields. Thus, D-branes provide a window into non-perturbative string theory that, at low energies, is governed by supersymmetric gauge theories in different dimensions. 1 Non-perturbative in the sense that their mass goes like 1=gs, where gs is the string coupling constant. 3 3. On the other hand, any source of energy interacts with gravity. Thus, if the number of branes is large enough, one expects a closed string description of the same system. The crucial realisations in [49] and [367] are the existence of kinematical and dynamical regimes in which the full string theory is governed by either of these descriptions : the open or the closed string ones. The purpose of this review is to describe the kinematical properties characterising the super- symmetric gauge theories emerging as brane effective field theories in string and M-theory, and some of their important applications. In particular, I will focus on D-branes, M2-branes and M5-branes. For a schematical 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 a) the identification of their dynamical degrees of freedom, providing a geometrical interpreta- tion when available, b) the discussion of the world volume gauge symmetries required to achieve spacetime covariance and supersymmetry. These will include world volume diffeomorphisms and kappa symmetry, c) the description of the couplings governing the interactions in these effective actions, their global symmetries and their interpretation in spacetime, d) the connection between spacetime and world volume supersymmetry through gauge fixing, e) the description of the regime of validity of these effective actions. For multiple coincident branes, these theories are supersymmetric 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 supergrav- ity 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 a) stress the technical importance of kappa symmetry in determining these configurations, link- ing hamiltonian methods with supersymmetry algebra considerations, b) prove the existence of string theory BPS states carrying different (topological) charges, c) briefly mention microscopic constituent models for certain black holes. Regarding the dynamical applications, the intention is to provide some dynamical interpretation to specific probe calculations appealing to the AdS/CFT correspondence [13] in two main situations a) classical 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, 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. 4 string theory open strings closed strings U-duality interactions D-branes supergravity kinematics Applications: probe approx multiple Susy solitons susy algebras branes single brane Dynamics & AdS/CFT kappa symmetry spacetime covariance & supersymmetry world volume diffeos ?? gauge fixing world volume supersymmetry non-abelian susy geometrisation of field theories gauge theories Figure 1: Layout of the main relations uncovered 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.