[Paper Review] Supersymmetric Gauge Theories and the AdS/CFT Correspondence

Eric D’Hoker, Daniel Z. Freedman|ArXiv.org|Jan 30, 2002

Black Holes and Theoretical PhysicsPhysics and Astronomy210 references287 citations

TL;DR

This paper provides a comprehensive pedagogical introduction to the AdS/CFT correspondence, focusing on N=4 super Yang-Mills theory and its duality with Type IIB superstring theory on AdS₅×S⁵. It details how correlation functions in the conformal field theory are computed via holographic methods in anti-de Sitter space, and demonstrates the derivation of conformal anomalies and the holographic c-theorem using gravity duals, confirming field theory results through explicit calculations in the large-N and 't Hooft limit.

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.

Motivation & Objective

• To provide a self-contained, pedagogical introduction to the AdS/CFT correspondence for researchers in high-energy theory.
• To explain how correlation functions in N=4 super Yang-Mills theory are computed via holography in AdS₅×S⁵.
• To derive the conformal anomaly in four-dimensional CFTs using five-dimensional gravity, confirming field theory results.
• To establish the holographic c-theorem via domain wall solutions in AdS, showing monotonic decrease of the c-function along RG flows.
• To connect supersymmetric deformations of N=4 SYM to gravity duals and verify consistency with field theory expectations.

Proposed method

• Utilizes the large-N and 't Hooft limits to connect strongly coupled gauge theory to classical supergravity.
• Applies holographic renormalization to compute divergences in the on-shell action and extract conformal anomalies.
• Derives the holographic c-function from the warp factor A(r) in D=5, N=8 gauged supergravity, with C′(r) ≥ 0 implying monotonic decrease.
• Computes 2- and 3-point correlation functions in both CFT and AdS using Witten diagrams and AdS propagators.
• Uses the boundary-to-bulk propagator and conformal structure to match operator dimensions and OPE coefficients.
• Applies the method of diffeomorphism invariance and counterterm variations to isolate the conformal anomaly from the action's cutoff dependence.

Experimental results

Research questions

• RQ1 How do correlation functions in N=4 SYM theory map to Witten diagrams in AdS₅?
• RQ2 What is the holographic origin of the conformal anomaly in 4D CFTs, and how is it computed from 5D gravity?
• RQ3 Can the holographic c-theorem be derived from gravity duals, and does it reproduce the field theory c-theorem?
• RQ4 How do supersymmetric deformations of N=4 SYM correspond to domain wall solutions in AdS?
• RQ5 What is the precise relation between the central charge c and the AdS radius in the gravity dual?

Key findings

• The conformal anomaly in 4D N=4 SYM is reproduced holographically via the on-shell action with counterterms, yielding c ∝ L³/G, matching field theory results.
• The holographic c-function C(r) = π/(8G) A′⁻³ is monotonic decreasing along RG flows due to A′′ ≤ 0, confirming the c-theorem in gravity duals.
• Extremal and non-extremal 3-point functions in N=4 SYM are non-renormalized, consistent with superconformal symmetry and the AdS/CFT map.
• The ratio of IR to UV AdS radii for the mass-deformed N=4 theory gives L_IR/L_UV = (27/32)^{1/3}, leading to c_IR = a_IR, matching field theory.
• The logarithmic divergence in the on-shell action corresponds to the conformal anomaly, with explicit counterterms removing power and logarithmic divergences.
• The holographic c-theorem is proven via gravity duals, showing that C(r) decreases monotonically from UV to IR, with equality only at fixed points.