Large N Field Theories, String Theory and Gravity

TL;DR

The paper provides a comprehensive review of the AdS/CFT correspondence, articulating how large N gauge theories map to string/M theory in AdS backgrounds, with a central focus on N=4 super-Yang–Mills and its dual IIB string theory on AdS_5×S^5. It lays out the dictionary between bulk fields and boundary operators, demonstrates highly nontrivial checks via spectra, correlators, and anomalies, and extends the framework to finite temperature, Wilson loops, and deformations. The survey further explores AdS backgrounds beyond AdS_5×S^5, including orbifolds, orientifolds, and conifold theories, and expands the holographic program to AdS_3/CFT_2 with D1–D5 systems, illustrating how black hole entropy and elliptic genera are captured by boundary CFT data. Beyond conformal cases, the authors discuss deformations, c-theorems from gravity, and potential routes toward connecting holography with QCD-like theories, highlighting both the successes and outstanding challenges in understanding quantum gravity via holography. Overall, the work codifies the holographic paradigm as a concrete, calculable, and highly predictive framework linking gauge theory dynamics to higher-dimensional gravity and string theory.

Abstract

We review the holographic correspondence between field theories and string/M theory, focusing on the relation between compactifications of string/M theory on Anti-de Sitter spaces and conformal field theories. We review the background for this correspondence and discuss its motivations and the evidence for its correctness. We describe the main results that have been derived from the correspondence in the regime that the field theory is approximated by classical or semiclassical gravity. We focus on the case of the N=4 supersymmetric gauge theory in four dimensions, but we discuss also field theories in other dimensions, conformal and non-conformal, with or without supersymmetry, and in particular the relation to QCD. We also discuss some implications for black hole physics.

Figures (25)

• Figure 1: Some diagrams in a field theory with adjoint fields in the standard representation (on the left) and in the double line representation (on the right). The dashed lines are propagators for the adjoint fields, the small circles represent interaction vertices, and solid lines carry indices in the fundamental representation.
• Figure 2: (a) The D-brane is where open strings can end. (b) The D-brane is a source of closed strings.
• Figure 3: An application of the optical theorem.
• Figure 4: Low energy dynamics of extremal or near-extremal black branes. $r_5$ denotes the typical gravitational size of the system, namely the position where the metric significantly deviates from Minkowski space. The Compton wavelength of the particles we scatter is much larger than the gravitational size, $\lambda \gg r_5$. In this situation we replace the whole black hole geometry (a) by a point-like system in the transverse coordinates with localized excitations (b). These excitations are the ones described by the field theory living on the brane.
• Figure 5: Two-dimensional Minkowski space is conformally mapped into the interior of the rectangle.