The Volume of some Non-spherical Horizons and the AdS/CFT Correspondence

TL;DR

This paper calculates volumes of Sasaki-Einstein manifolds to test and extend the AdS/CFT correspondence, linking geometric properties to field theory operators and D-brane flux quantization.

Contribution

It provides new volume calculations for non-spherical Einstein manifolds, extending central charge computations and establishing links to dual field theory operators.

Findings

Volumes enable precise D-brane flux quantization

Extended central charge calculations to generalized conifolds

Linked manifold volumes to chiral primary operators

Abstract

We calculate the volumes of a large class of Einstein manifolds, namely Sasaki-Einstein manifolds which are the bases of Ricci-flat affine cones described by polynomial embedding relations in C^n. These volumes are important because they allow us to extend and test the AdS/CFT correspondence. We use these volumes to extend the central charge calculation of Gubser (1998) to the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999). These volumes also allow one to quantize precisely the D-brane flux of the AdS supergravity solution. We end by demonstrating a relationship between the volumes of these Einstein spaces and the number of holomorphic polynomials (which correspond to chiral primary operators in the field theory dual) on the corresponding affine cone.