The Geometric Dual of a-maximisation for Toric Sasaki-Einstein Manifolds

TL;DR

This paper introduces a method to compute the Reeb vector and volume of toric Sasaki-Einstein manifolds by minimizing a function derived from toric data, providing a geometric dual to a-maximisation in AdS/CFT.

Contribution

It establishes a new geometric extremal principle for determining Reeb vectors and volumes without explicit metric construction, linking to a-maximisation in superconformal field theories.

Findings

Reeb vector and volume can be obtained by minimizing a toric data-dependent function

The method applies to complex dimension three, relating to R-symmetry and central charge

Examples include Y^{p,q} singularities and del Pezzo surfaces

Abstract

We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on R^n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki-Einstein manifold without finding the metric explicitly. For complex dimension n=3 the Reeb vector and the volume correspond to the R-symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a-maximisation. We illustrate our results with some examples, including the Y^{p,q} singularities and the complex cone over the second del Pezzo surface.