Supersymmetric Gauge Theories and the AdS/CFT Correspondence

TL;DR

The notes survey the AdS/CFT correspondence, detailing how ${ m N}=4$ SYM in four dimensions is holographically dual to type IIB string theory on AdS$_5$×S$^5$. They develop the operator/state dictionary, show how to compute CFT correlators via AdS supergravity (and Witten diagrams), and establish non-renormalization results for protected quantities, including extremal and near-extremal correlators. A major emphasis is placed on holographic RG flows, where domain-wall geometries encode RG trajectories of deformed CFTs and the dynamics of scalar fields in five-dimensional gauged supergravity. The work also covers KK reduction, brane dynamics, and the intricate mapping of symmetries between the bulk and boundary, highlighting the practical procedures to extract CFT data from gravitational computations and the limitations of the various limits (large N, large λ, and their intersections). Overall, the notes provide a comprehensive, methodical framework for applying holography to the study of strongly coupled gauge theories and their RG flows.

Abstract

In these lecture notes we first assemble the basic ingredients of supersymmetric gauge theories (particularly N=4 super-Yang-Mills theory), supergravity, and superstring theory. Brane solutions are surveyed. The geometry and symmetries of anti-de Sitter space are discussed. The AdS/CFT correspondence of Maldacena and its application to correlation functions in the the conformal phase of N=4 SYM are explained in considerable detail. A pedagogical treatment of holographic RG flows is given including a review of the conformal anomaly in four-dimensional quantum field theory and its calculation from five-dimensional gravity. Problem sets and exercises await the reader.