Living Rev. Relativity, 15, (2012), 3 LIVINGREVIEWS http://www.livingreviews.org/lrr-2012-3 in relativity Brane Effective Actions, Kappa-Symmetry and Applications Joan Sim´on School of Mathematics, The University of Edinburgh, and Maxwell Institute for Mathematical Sciences Edinburgh EH9 3JZ, U.K. email: [email protected] Accepted on 9 January 2012 Published on 27 February 2012 Abstract This is a review on brane effective actions, their symmetries and some of their applications. Its first part covers 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 Bogomol'nyi{Prasad–Sommerfield (BPS) states carrying (topological) charges in the supersymmetry algebra and the connection between su- persymmetry 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 approx- imations to infer about the non-trivial dynamics of strongly-coupled 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 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 Super-Yang-Mills 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. This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0 Germany License. http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Imprint / Terms of Use Living Reviews in Relativity is a peer reviewed open access journal published by the Max Planck Institute for Gravitational Physics, Am M¨uhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351. 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Contents 1 Introduction 5 2 The Green{Schwarz Superstring: A Brief Motivation 9 3 Brane Effective Actions 14 3.1 Degrees of freedom and world volume supersymmetry . 14 3.1.1 Supergravity Goldstone modes ......................... 17 3.2 Bosonic actions ...................................... 21 3.3 Consistency checks .................................... 26 3.3.1 M2-branes and their classical reductions .................... 30 3.3.2 T-duality covariance ............................... 32 3.3.3 M5-brane reduction ............................... 35 3.4 Supersymmetric brane effective actions in Minkowski . 35 3.4.1 D-branes in flat superspace ........................... 37 3.4.2 M2-brane in flat superspace ........................... 41 3.5 Supersymmetric brane effective actions in curved backgrounds . 42 3.5.1 M2-branes ..................................... 43 3.5.2 D-branes ..................................... 43 3.5.3 M5-branes ..................................... 44 3.6 Symmetries: spacetime vs world volume ........................ 45 3.6.1 Supersymmetry algebras ............................. 46 3.6.2 World volume supersymmetry algebras ..................... 49 3.7 Regime of validity .................................... 52 4 World Volume Solitons: Generalities 55 4.1 Supersymmetric bosonic configurations and kappa symmetry . 55 4.2 Hamiltonian formalism .................................. 58 4.2.1 D-brane Hamiltonian ............................... 60 4.2.2 M2-brane Hamiltonian .............................. 61 4.2.3 M5-brane Hamiltonian .............................. 61 4.3 Calibrations ........................................ 62 5 World Volume Solitons: Applications 66 5.1 Vacuum infinite branes .................................. 66 5.2 Intersecting M2-branes .................................. 67 5.3 Intersecting M2 and M5-branes ............................. 69 5.4 BIons ........................................... 72 5.5 Dyons ........................................... 75 5.6 Branes within branes ................................... 77 5.6.1 Dp-D(p + 4) systems ............................... 77 5.6.2 Dp-D(p + 2) systems ............................... 78 5.6.3 F-Dp systems ................................... 79 5.7 Supertubes ........................................ 80 5.8 Baryon vertex ....................................... 83 5.9 Giant gravitons and superstars ............................. 88 5.9.1 Giant gravitons as black-hole constituents ................... 90 5.10 Deconstructing black holes ................................ 93 6 Some AdS/CFT Related Applications 97 6.1 Wilson loops ....................................... 98 6.2 Quark energy loss in a thermal medium ........................ 99 6.3 Semiclassical correspondence .............................. 100 6.4 Probes as deformations and gapless excitations in complex systems . 102 7 Multiple Branes 106 7.1 D-branes .......................................... 106 7.2 M2-branes ......................................... 115 8 Related Topics 120 9 Acknowledgements 122 A Target superspace formulation and constraints 123 A.1 N = 2 type IIA/B superspace .............................. 123 A.2 N = 1 d = 11 supergravity conventions ......................... 126 B Cone Construction and Supersymmetry 127 B.1 (M; g) Riemannian .................................... 128 B.2 (M; g) of signature (1; d − 1) .............................. 128 References 128 List of Tables 1 Scalar multiplets with X scalars in p + 1 worldvolume dimensions. 16 2 Vector multiplets with X scalar degrees of freedom in p + 1 worldvolume dimensions. 17 3 Tensor multiplets with X scalar degrees of freedom in p + 1 world volume dimensions. 17 4 Summary of supergravity Goldstone modes. ...................... 21 5 Set of kappa symmetry matrices Γ휅 evaluated in the bosonic subspace of configura- tions B. .......................................... 58 6 Half-BPS branes and the supersymmetries they preserve. 67 Brane Effective Actions, Kappa-Symmetry and Applications 5 1 Introduction Branes have played a fundamental role in the main string theory developments of the last twenty years: 1. The unification of the different perturbative string theories using duality symmetries [312, 495] relied strongly on the existence of 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 [423]. 3. 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 for- mulation of string theory in Minkowski spacetime and the latter in AdS Ö 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 win- dow 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