# zbMATH — the first resource for mathematics

TASI lectures: Introduction to the AdS/CFT correspondence. (English) Zbl 1131.81315
Harvey, Jeffrey (ed.) et al., Strings, branes and gravity. TASI 99. Papers from the Theoretical Advanced Study Institute (TASI) in elementary particle physics, Univ. of Colorado, Boulder, CO, USA, May 31–June 25, 1999. Singapore: World Scientific (ISBN 981-02-4774-5). 615-650 (2001).
The introductory review of the AdS/CFT correspondence and of the ideas that led to its formulation. It is shown how comparison of stacks of D3-branes with corresponding supergravity solutions leads to dualities between conformal large $$N$$ gauge theories in 4 dimensions and string backgrounds of the form $$\mathrm{AdS}_5\times X_5$$ where $$X_5$$ is an Einstein manifold. The gauge invariant chiral operators of the field theory are in one-to-one correspondence with the supergravity modes, and their correlation functions at strong ‘t Hooft coupling are determined by the dependence of the supergravity action on AdS boundary conditions. The simplest case is when $$X_5$$ is a 5-sphere and the dual gauge theory is the $${\mathcal N}=4$$ supersymmetric $$\mathrm{SU}(N)$$ Yang-Mills theory. The D3-branes on the conifold corresponding to $$X_5$$ being a coset space $$T^{1,1}=(\mathrm{SU}(2)\times \mathrm{SU}(2))/\mathrm{U}(1)$$ are discussed. This background is dual to a certain $${\mathcal N}=1$$ superconformal field theory with gauge group $$\mathrm{SU}(N)\times \mathrm{SU}(N)$$.
For the entire collection see [Zbl 0987.00054].

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 83E30 String and superstring theories in gravitational theory