Sasaki-Einstein Manifolds and Volume Minimisation

TL;DR

This paper investigates a variational approach to determine Reeb vector fields on Sasaki-Einstein manifolds, linking volume minimization to topological data and extending previous toric results to more general cases.

Contribution

It generalizes volume minimization techniques for Sasaki-Einstein manifolds beyond toric cases, connecting geometric analysis with topological fixed point data and algebraic number properties.

Findings

Volume of Sasaki-Einstein manifolds is always an algebraic number.

Derived a formula for the volume function using localization and fixed point data.

Connected the variational problem to the vanishing of the Futaki invariant.

Abstract

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat-Heckman formula and also to a limit of a certain equivariant index on X that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a Sasaki-Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n=3 these results provide, via AdS/CFT, the geometric counterpart of a-maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki invariant of the transverse space, the latter being an obstruction to the existence of a Kahler-Einstein metric.