Recently, it has been proposed by Maldacena 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 $AdS_5\times {\bf S}^5$ match with the chiral operators of ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

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Anti De Sitter Space And Holography

Recently, it has been proposed by Maldacena 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 $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

Anti de sitter space and holography

Extended Theories of Gravity

Black holes in higher dimensional space-times

Unity of superstring dualities

Two branches of integrable open quantum-group invariant $D_{n+1}^{(2)}$ quantum spin chains are known. For one branch (epsilon=0), a complete Bethe ansatz solution has been proposed. However, the other branch (epsilon=1) has so far resisted solution. In an effort to address this problem, we consider here the simplest case n=1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

Towards the solution of an integrable $D_2^{(2)}$ spin chain

We express $$ {D}_2^{(2)} $$
 transfer matrices as products of $$ {A}_1^{(1)} $$
 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain with an even number of sites that depends on a continuous parameter, which we interpret as the rapidity of the boundary.

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Factorization identities and algebraic Bethe ansatz for D 2 2 $$ {D}_2^{(2)} $$ models

We consider the integrable open-chain transfer matrix corresponding to a Y=0 brane at one boundary, and a Y_theta=0 brane (rotated with the respect to the former by an angle theta) at the other boundary. We determine the exact eigenvalues of this transfer matrix in terms of solutions of a corresponding set of Bethe equations.

Bethe ansatz for an AdS/CFT open spin chain with non-diagonal boundaries

We compute by means of the Bethe ansatz the boundary S-matrix for the open anisotropic spin- chain with diagonal boundary magnetic fields in the noncritical regime . Our result, which is formulated in terms of q-gamma functions, agrees with the vertex-operator result of Jimbo et al.

Boundary S-matrix for the XXZ chain

We formulate $Q$-systems for the closed XXZ, open XXX and open quantum-group-invariant XXZ quantum spin chains. Polynomial solutions of these $Q$-systems can be found efficiently, which in turn lead directly to the admissible solutions of the corresponding Bethe ansatz equations.

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Rafael I. Nepomechie

Papers

Towards the solution of an integrable $D_2^{(2)}$ spin chain

Factorization identities and algebraic Bethe ansatz for D 2 2 $$ {D}_2^{(2)} $$ models

Bethe ansatz for an AdS/CFT open spin chain with non-diagonal boundaries

Boundary S-matrix for the XXZ chain

On Generalized $Q$-systems.