TASI lectures on AdS/CFT

TL;DR

The notes provide a thorough, pedagogical tour of AdS/CFT, starting from foundational CFT concepts and embedding-space techniques, then introducing AdS geometry and the holographic dictionary. They explain how bulk gravity and string theory in AdS encode the data of a boundary CFT, including finite-temperature physics and entanglement, and introduce Mellin amplitudes as a powerful framework to organize CFT correlators and their holographic duals. Key contributions include connecting large-N factorization to bulk gravity, illustrating explicit AdS/CFT realizations via D-branes, and detailing Witten diagrams and the flat-space limit within Mellin space. The exposition emphasizes both conceptual foundations and concrete calculational tools for exploring holography across high-energy and condensed-matter contexts.

Abstract

We introduce the AdS/CFT correspondence as a natural extension of QFT in a fixed AdS background. We start by reviewing some general concepts of CFT, including the embedding space formalism. We then consider QFT in a fixed AdS background and show that one can define boundary operators that enjoy very similar properties as in a CFT, except for the lack of a stress tensor. Including a dynamical metric in AdS generates a boundary stress tensor and completes the CFT axioms. We also discuss some applications of the bulk geometric intuition to strongly coupled QFT. Finally, we end with a review of the main properties of Mellin amplitudes for CFT correlation functions and their uses in the context of AdS/CFT.

Figures (7)

• Figure 1: Vacuum diagrams in the double line notation. Interaction vertices are marked with a small blue dot. The left diagram is planar while the diagram on the right has the topology of a torus (genus 1 surface).
• Figure 2: Curves of constant $\tau$ (in blue) and constant $\rho$ (in red) for AdS$_{2}$ stereographically projected to the unit disk (Poincaré disk). This shows how surfaces of constant $\tau$ converge to a boundary bound when $\tau\to-\infty$. The cartesian coordinates in the plane of the figure are given by $\vec{w}=\frac{(\cosh\rho\sinh\tau,\sinh\rho)}{1+\cosh\rho\cosh\tau}$ which puts the AdS$_{2}$ metric in the form $ds^{2}=\frac{4d\vec{w}^{2}}{1-\vec{w}^{2}}$.
• Figure 3: (a) Closed string scattering off branes in flat space. (b) Closed string propagating in a curved background.
• Figure 4: Integration contour for the Mellin variable $\gamma_{12}$. The crosses represent (double) poles of the $\Gamma$-functions given by \ref{['eq:gamma12poles1']} and \ref{['eq:gamma12poles2']}. In general, the Mellin amplitude has several semi-infinite sequence of poles. Each sequence should stay entirely on one side of the contour.
• Figure 5: Witten diagram for a $n$-point contact interaction in AdS. The interior of the disk represents the bulk of AdS and the circumference represents its conformal boundary. The lines connecting the boundary points $P_i$ to the bulk point $X$ represent bulk to boundary propagators.