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 $\left[\mathrm{SU}\right(2)\mathrm{}\times \mathrm{SU}(2\left)\mathrm{}\right]/\mathrm{U}\left(1\right)\mathrm{}.$ 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 $D3\mathrm{}$-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 $\mathcal{N}=2\mathrm{}{\mathbf{Z}}_2$ orbifold theory from which it descends via a renormalization group flow.