The AdS/CFT Correspondence (1501.00007v2)

Abstract: We give a brief review of the AdS/CFT correspondence, which posits the equivalence between a certain gravitational theory and a lower-dimensional non-gravitational one. This remarkable duality, formulated in 1997, has sparked a vigorous research program which has gained in breadth over the years, with applications to many aspects of theoretical (and even experimental) physics, not least to general relativity and quantum gravity. To put the AdS/CFT correspondence in historical context, we start by reviewing the relevant aspects of string theory (of which no prior knowledge is assumed). We then develop the statement of the correspondence, and explain how the two sides of the duality map into each other. Finally, we discuss the implications and applications of the correspondence, and indicate some of the current trends in this subject. The presentation attempts to convey the main concepts in a simple and self-contained manner, relegating supplementary remarks to footnotes.

• The paper presents a comprehensive review of the duality between AdS gravitational theories and boundary conformal field theories, highlighting its historical development and theoretical consistency.
• The paper demonstrates how matching observables, such as correlation functions and quasinormal modes, concretely link black hole dynamics in AdS space to thermal states in the dual CFT.
• The paper explores extensions and generalizations including deformations, dS/CFT proposals, and fluid/gravity correspondence, paving new pathways for future research in quantum gravity.

Review of "The AdS/CFT Correspondence" by Veronika E. Hubeny

The AdS/CFT correspondence, introduced by Juan Maldacena in 1997, posits a duality between a higher-dimensional gravitational theory described within Anti-de Sitter (AdS) space and a lower-dimensional Conformal Field Theory (CFT) that lacks gravitational interactions. This relation is a cornerstone of theoretical physics, impacting our understanding of quantum gravity, string theory, and gauge theories. Hubeny's paper provides a comprehensive review of the correspondence, exploring its historical roots, key features, and vast implications.

The paper begins by exploring the historical context, particularly emphasizing the role of black holes and string theory. The AdS/CFT correspondence emerged from efforts to understand black hole entropy through string theory. Specifically, this duality was born out of analyzing the low-energy limits of D-brane configurations in string theory, revealing deep insights into the connections between gravity and strongly coupled gauge theories.

Hubeny systematically unpacks the concept of AdS/CFT: the gravitational side is often represented in an AdS space, while the dual CFT typically resides on its boundary. This holographic principle indicates that a gravitational theory in $d+1$ dimensions could be entirely described by a $d$-dimensional non-gravitational theory, reflecting 't Hooft's holographic principle and Susskind's insights into quantum gravity. By comparing symmetries—such as $SO(4,2)$ symmetry of AdS space and the conformal symmetry of the dual CFT—Hubeny underscores the theoretical consistency and compelling nature of this correspondence.

The paper also explores the practical checks of this duality, such as matching correlation functions, and illustrates how various observables on the CFT side relate to properties and dynamics in the bulk gravity theory. An important element discussed is how phenomena such as black hole thermodynamics and quasinormal modes in AdS space have direct correspondents in thermal field states of the CFT.

In addressing black holes within AdS, Hubeny emphasizes the intriguing possibility that these correspond to thermal states in the dual CFT. This includes a discussion of the Hawking-Page transition, where AdS black holes undergo phase transitions paralleling the deconfinement transitions in gauge theory—a conceptual bridge between gravity, thermodynamics, and quantum field theory.

The review does not stop at static scenarios but extends to dynamical systems, describing how time-evolving black holes and their quasinormal modes provide insights into thermalization processes in quantum field theory.

Hubeny also considers various generalizations of the original correspondence, exploring different dimensional settings, deformations from conformal symmetry, and inclusions of additional fields, which broaden the application range of the AdS/CFT correspondence beyond its initial formulations.

The paper moves into speculative ventures such as dS/CFT aimed at extending holographic ideas into space-times with positive cosmological constants, probing the limits of holography and attempting to capture our universe's dynamics in a similar theoretical framework.

In terms of broader implications, the paper underscores how AdS/CFT has provided fruitful insights into complex systems, from heavy ion collisions in QCD to novel phases in condensed matter theory. This includes the connection between fluid dynamics and gravity, known as the fluid/gravity correspondence, revealing that gravitational equations in AdS can accurately map onto fluid dynamics on the AdS boundary.

On a theoretical level, the paper posits that AdS/CFT enriches our understanding of quantum gravity and offers potential pathways towards resolving long-standing puzzles, such as the black hole information paradox, by leveraging the unitarity of the dual CFT.

Ultimately, Hubeny's review encapsulates the profound influence of AdS/CFT correspondence on contemporary theoretical physics, illuminating its continued role in bridging disparate areas of paper and inspiring future research directions.