AdS/CFT duality

The Anti–de Sitter/Conformal Field Theory (AdS/CFT) correspondence—often called gauge/gravity duality—posits an exact equivalence between a quantum theory of gravity in $(d{+}1)$-dimensional asymptotically AdS spacetime and a $d$-dimensional CFT living on its boundary. This maps bulk fields to boundary operators and relates bulk on‑shell actions to generating functionals of CFT correlators, enabling computations at strong coupling via weakly curved gravity. The canonical example identifies type IIB string theory on $\mathrm{AdS}_5 \times S^5$ with four‑dimensional $\mathcal{N}{=}4$ $SU(N)$ SYM in the large‑$N$, large ’t Hooft coupling limit. Though still a conjecture in full generality, AdS/CFT is supported by a wide array of nontrivial checks and has been extended to many less symmetric settings.

Beyond its conceptual unification of quantum gravity and quantum field theory, AdS/CFT is a practical tool. Black holes in AdS encode thermal states and hydrodynamics of the dual field theory, illuminating black hole entropy and information, transport, and far‑from‑equilibrium dynamics. The correspondence also yields geometric formulae for quantum information measures—most famously, the Ryu–Takayanagi prescription for entanglement entropy—linking spacetime geometry to entanglement structure. Applications range from strongly coupled QCD‑like plasmas to emergent phenomena in condensed matter systems. Current research pushes the duality beyond strict AdS or exact conformality, develops bulk reconstruction and quantum error–correction viewpoints, and leverages numerical and analytic methods to probe new corners of holography.

Learning resources

Books

Gauge/Gravity Duality: Foundations and Applications. By M. Ammon and J. Erdmenger. Cambridge University Press, 2015.

Introduction to the AdS/CFT Correspondence. By H. Năstase. Cambridge University Press, 2015.

AdS/CFT Duality User Guide. By M. Natsuume. Springer, 2015 [arXiv:1409.3575].

Holographic Duality in Condensed Matter Physics. By J. Zaanen, Y. Liu, Y. W. Sun, and K. Schalm. Cambridge University Press, 2015.
全息对偶中的凝聚态物理. 科学出版社.

Holographic Quantum Matter. By S. A. Hartnoll, A. Lucas and S. Sachdev. MIT Press, 2018 [arXiv:1612.07324].

Video lectures

String Theory and Holographic Duality. By Hong Liu.

Quantum Gravity and Black Holes. By Tom Hartman.

AdS/CFT入门中文讲义. By Wen-Cong Gan.

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ODEs

Learn more in Heun functions.