We explain how the string spectrum in flat space and pp-waves arises from the large N limit, at fixed g 2 Y M , of U(N) N = 4 super Yang Mills. We reproduce the spectrum by summing a subset of the planar Feynman diagrams. We give a heuristic argument for why we can neglect other diagrams.We also discuss some other aspects of pp-waves and we present a matrix model associated to the DLCQ description of the maximally supersymmetric eleven dimensional pp-waves.

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We discuss the full nonlinear Kaluza-Klein (KK) reduction of the original formulation of d=11 supergravity on AdS 7 × S 4 to gauged maximal (N=4) supergravity in 7 dimensions. We derive the full nonlinear embedding of the d = 7 fields in the d = 11 fields ("the ansatz") and check the consistency of the ansatz by deriving the d=7 supersymmetry laws from the d=11 transformation laws in the various sectors. The ansatz itself is nonpolynomial but the final d = 7 results are polynomial. The correct d = 7 scalar potential is obtained. For most of our results the explicit form of the matrix U connecting the d = 7 gravitino to the Killing spinor is not needed, but we derive the equation which U has to satisfy and present the general solution. Requiring that the expression δF = dδA in d = 11 can be written as δd(f ields in d = 7), we find the ansatz for the 4-form F . It satisfies the Bianchi identities. The corresponding ansatz for the 3-form A modifies the geometrical proposal by Freed et al. by including d = 7 scalar fields. A first order formulation for A in d = 11 is needed to obtain the d=7 supersymmetry laws and the action for the nonabelian selfdual antisymmetric tensor field S αβγ,A . Therefore selfduality in odd dimensions originates from a first order formalism in higher dimensions.

We show that there exists a consistent truncation of 11 dimensional supergravity to the 'massless' fields of maximal (N=4) 7 dimensional gauged supergravity. We find the complete expressions for the nonlinear embedding of the 7 dimensional fields into the 11 dimensional fields, and check them by reproducing the d=7 susy transformation laws from the d=11 laws in various sectors. In particular we determine explicitly the matrix U which connects the Killing spinors to the gravitinos in the KK ansatz, and the dependence of the 4-index field strength on the scalars. This is the first time a complete nonlinear KK reduction of the original d=11 supergravity on a nontrivial compact space has been explicitly given. We need a first order formulation for the 3 index tensor field A ΛΠΣ in d=11 to reproduce the 7 dimensional result. The concept of 'self-duality in odd dimensions' is thus shown to originate from first order formalism in higher dimensions. For the AdS-CFT correspondence, our results imply that one can use 7d gauged supergravity (without further massive modes) to compute certain correlators in the d=6 (0,2) CFT at leading order in N. This eliminates an ambiguity in the formulation of the correspondence.The question whether in general a consistent Kaluza-Klein (KK) truncation exists at the nonlinear level is an old problem. For tori, the consistency is easy to prove, but for more complicated compact spaces little is known. In supergravity (sugra), the truncation of d=11 sugra on AdS 4 × S 7 to maximal d = 4 gauged sugra was intensively studied 15 years ago [1], culminating in a series of papers by de Wit and Nicolai [2,3]. The interest in those days was to find realistic 4 dimensional models from spontaneous compactification of maximal 11 dimensional sugra. Recent developments in the AdS-CFT correspondence [4,5,6,7,8] have renewed interest in AdS compactifications

We study the large N limit of a little string theory that reduces in the IR to U (N ) N = 1 supersymmetric Yang-Mills with Chern Simons coupling k. Witten has shown that this field theory preserves supersymmetry if k ≥ N/2 and he conjectured that it breaks supersymmetry if k < N/2. We find a non-singular solution that describes the k = N/2 case, which is confining. We argue that increasing k corresponds to adding branes to this solution, in a way that preserves supersymmetry, while decreasing k corresponds to adding anti-branes, and therefore breaking supersymmetry.

We fix the long-standing ambiguity in the 1-loop contribution to the mass of a 1+1-dimensional supersymmetric soliton by adopting a set of boundary conditions which follow from the symmetries of the action and which depend only on the topology of the sector considered, and by invoking a physical principle that ought to hold generally in quantum field theories with a topological sector: for vanishing mass and other dimensionful constants, the vacuum energies in the trivial and topological sectors have to become equal. In the two-dimensional N = 1 supersymmetric case we find a result which for the supersymmetric sine-Gordon model agrees with the known exact solution of the S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation between the quantum soliton mass and the quantum BPS bound and find a resolution. For N = 2 supersymmetric theories, there are no one-loop corrections to the soliton mass and to the central charge (and also no ambiguities) so that the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in any theory, which we explicitly check by a 2-loop calculation in the sine-Gordon model.

Providing a pedagogical introduction to the rapidly developing field of AdS/CFT correspondence, this is one of the first texts to provide an accessible introduction to all the necessary concepts needed to engage with the methods, tools and applications of AdS/CFT. Without assuming anything beyond an introductory course in quantum field theory, it begins by guiding the reader through the basic concepts of field theory and gauge theory, general relativity, supersymmetry, supergravity, string theory and conformal field theory, before moving on to give a clear and rigorous account of AdS/CFT correspondence. The final section discusses the more specialised applications, including QCD, quark-gluon plasma and condensed matter. This book is self-contained and learner-focused, featuring numerous exercises and examples. It is essential reading for both students and researchers across the fields of particle, nuclear and condensed matter physics.

Abstract:We investigate the origins and implications of the duality between topological insulators and topological superconductors in three and four spacetime dimensions. In the latter, the duality transformation can be made at the level of the path integral in the standard way, while in three dimensions, it takes the form of "self-duality in odd dimensions". In this sense, it is closely related to the particle-vortex duality of planar systems. In particular, we use this to elaborate on Son's conjecture that a three dimensional Dirac fermion that can be thought of as the surface mode of a four dimensional topological insulator is dual to a composite fermion.