The Volume of some Non-spherical Horizons and the AdS/CFT Correspondence
Aaron Bergman, Christopher P. Herzog
TL;DR
Bergman and Herzog derive explicit volume formulas for Sasaki–Einstein bases of Calabi–Yau cones defined by weighted homogeneous polynomials, enabling AdS/CFT tests without requiring explicit metrics. They express Vol(V) and Vol(X) in terms of the weights, degree, and index, and use these volumes to extend the Gubser–N Nekrasov central-charge calculation to generalized conifolds, verifying the predicted c_IR/c_UV ratios. They also propose a quantitative link between the volume and the number of holomorphic monomials (CPOs) on the affine cone, via Hirzebruch–Riemann–Roch, suggesting a deep topological handle on operator spectra. The work broadens the class of geometries amenable to AdS/CFT checks and highlights a topological route to holographic data.
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.