Einstein manifolds and conformal field theories

Abstract

In light of the anti--de Sitter space conformal field theory correspondence, it is natural to try to define a conformal field theory in a large N, strong coupling limit via a supergravity compactification on the product of an Einstein manifold and anti--de Sitter space. We consider the five-dimensional manifolds ${T}^{\mathrm{pq}}$ which are coset spaces $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/\mathrm{U}(1).$ The central charge and a part of the chiral spectrum are calculated, respectively, from the volume of ${T}^{\mathrm{pq}}$ and the spectrum of the scalar Laplacian. Of the manifolds considered, only ${T}^{11}$ admits any supersymmetry: it is this manifold which characterizes the supergravity solution corresponding to a large number of $D3$-branes at a conifold singularity, discussed recently by Klebanov and Witten. Through a field theory analysis of anomalous three point functions we are able to reproduce the central charge predicted for the ${T}^{11}$ theory by supergravity: it is $\frac{27}{32}$ of the central charge of the $\mathcal{N}=2{\mathbf{Z}}_{2}$ orbifold theory from which it descends via a renormalization group flow.