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 covariance 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 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 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 includes 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.
Highlights
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
Effective actions satisfying these two symmetry requirements involve the addition of both extra, non-physical, bosonic and fermionic degrees of freedom. To preserve their non-physical nature, these supersymmetric brane effective actions must be invariant under additional gauge symmetries
Summary: We have constructed an effective action describing the propagation of Dp-branes in 10-dimensional Minkowski spacetime being invariant under p + 1 dimensional diffeomorphisms, 10-dimensional supersymmetry and kappa symmetry
Summary
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. The main concepts I want to stress in this part are 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,. 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 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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