SasakiEinstein manifolds and volume minimisation

Dario Martelli James Sparks Shing-Tung Yau

Mathematical Physics mathscidoc:1912.43474

Communications in Mathematical Physics, 280, (3), 611-673, 2008.6
We study a variational problem whose critical point determines the Reeb vector field for a SasakiEinstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the EinsteinHilbert action, restricted to a space of Sasakian metrics on a link <i>L</i> in a CalabiYau cone <i>X</i>, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the DuistermaatHeckman formula and also to a limit of a certain equivariant index on <i>X</i> 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 SasakiEinstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension <i>n</i>=3 these results provide, via AdS
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  title={SasakiEinstein manifolds and volume minimisation},
  author={Dario Martelli, James Sparks, and Shing-Tung Yau},
  booktitle={Communications in Mathematical Physics},
Dario Martelli, James Sparks, and Shing-Tung Yau. SasakiEinstein manifolds and volume minimisation. 2008. Vol. 280. In Communications in Mathematical Physics. pp.611-673.
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