Sasaki-Einstein Manifolds and Volume Minimisation

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.

82 pages, 9 figures; homogeneity of the (n,0)-form now derived from the Einstein-Hilbert action of the link, example of an orbifold resolution added, various clarifications and references added; minor changes; published version

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• N=2 supersymmetric AdS_4 solutions of M-theory
• Geometries with Killing Spinors and Supersymmetric AdS Solutions
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• AdS_5 Solutions of Type IIB Supergravity and Generalized Complex Geometry
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• The central charge of supersymmetric AdS_5 solutions of type IIB supergravity
• Holographic formula for the determinant of the scattering operator in thermal AdS
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• 3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry
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• On Type IIA AdS${}_3$ solutions and massive GK geometries
• Fano 3-Folds, Reflexive Polytopes and Brane Brick Models
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• Holography, Matrix Factorizations and K-stability
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