Anti de Sitter space and holography. (English) Zbl 0914.53048

Witten, Edward

Adv. Theor. Math. Phys. 2, No. 2, 253-291 (1998).

Summary: Recently, it has been proposed by J. Maldacena [ibid., 231-252 (1998; Zbl 0914.53047), see the preceding review] that large \(N\) limits of certain conformal field theories in \(d\) dimensions can be described in terms of supergravity (and string theory) on the product of \(d+1\)-dimensional \(AdS\) space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on \(AdS_5\times {\mathbf S}^5\) match with the chiral operators of \({\mathcal N}=4\) super Yang-Mills theory in four dimensions.
With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the \({\mathcal N}=4\) theory has a large \(N\) phase transition related to the thermodynamics of \(AdS\) black holes.

MSC:

53Z05	Applications of differential geometry to physics
81T40	Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
83E50	Supergravity

Keywords:

conformal field theory observables; supergravity; correlation functions; asymptotic behavior at infinity

Citations:

Zbl 0914.53047