Introduction to AdS-CFT

TL;DR

This collection of lectures presents AdS-CFT as a holographic duality between string theory in AdS_5×S^5 and 4D N=4 SU(N) SYM, illustrating how gravitational dynamics in a curved higher-dimensional spacetime encode strongly coupled gauge theory data. The notes develop the necessary QFT and gravity background, introduce D-branes and the AdS/CFT dictionary, and demonstrate concrete calculations such as bulk-to-boundary correlators, three-point functions, and Wilson-loop energies, revealing a nonperturbative window into gauge dynamics. They also discuss finite-temperature extensions and the Polchinski–Strassler program for QCD-like behavior, and conclude with the PP-wave (BMN) limit that renders string theory on a highly symmetric background solvable and connectable to spin chains. Overall, the work highlights how holography provides computational access to strongly coupled gauge theories, with implications for nonperturbative QCD phenomena and beyond.

Abstract

These lectures present an introduction to AdS-CFT, and are intended both for begining and more advanced graduate students, which are familiar with quantum field theory and have a working knowledge of their basic methods. Familiarity with supersymmetry, general relativity and string theory is helpful, but not necessary, as the course intends to be as self-contained as possible. I will introduce the needed elements of field and gauge theory, general relativity, supersymmetry, supergravity, strings and conformal field theory. Then I describe the basic AdS-CFT scenario, of ${\cal N}=4 $ Super Yang-Mills's relation to string theory in $AdS_5\times S_5$, and applications that can be derived from it: 3-point functions, quark-antiquark potential, finite temperature and scattering processes, the pp wave correspondence and spin chains. I also describe some general properties of gravity duals of gauge theories.

Figures (13)

• Figure 1: a) "Setting sun" diagram in x-space. b) "Setting sun" diagram in momentum space. c)anomalous diagram in 2 dimensions; d)anomalous diagram (triangle) in 4 dimensions; e)anomalous diagram (box) in 6 dimensions.
• Figure 2: a) Setting sun diagram in x space; b) Triangle diagram in x space; c) Star diagram in p space
• Figure 3: a) curved space. The functional form of the distance between 2 points depends on local coordinates. b) A triangle on a sphere, made from two meridian lines and a seqment of the equator has two angles of $90^0$ ($\pi/2$). c) The same triangle, drawn for a general curved space of positive curvature, emphasizing that the sum of the angles of the triangle exceeds $180^0$ ($\pi$). d) In a space of negative curvature, the sum of the angles of the triangle is below $180^0$ ($\pi$).
• Figure 4: Penrose diagrams. a) Penrose diagram of 2 dimensional Minkowski space. b) Penrose diagram of 3 dimensional Minkowski space. c) Penrose diagram of the é patch of Anti de Sitter space. d) Penrose diagram of global $AdS_2$ (2 dimensional Anti de Sitter), with the Poincaré patch emphasized; $x_0=0$ is part of the boundary, but $x_0=\infty$ is a fake boundary (horizon). e) Penrose diagram of global $AdS_d$ for $d\geq 2$. It is half the Penrose diagram of $AdS_2$ rotated around the $\theta=0$ axis.
• Figure 5: a) Kruskal diagram of the Schwarzschild black hole. b) Penrose diagram of the eternal Schwarzschild black hole (time independent solution). The dotted line gives the completion to the Penrose diagram of flat 2 dimensional (Minkowski) space. c) Penrose diagram of a physical black hole, obtained from a collapsing star (the curved line). The dotted line gives the completion to the Penrose diagram of flat $d>2$ dimensional (Minkowski) space.