SasakiEinstein Manifolds and Volume Minimisation
Abstract
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 L in a CalabiYau 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 DuistermaatHeckman 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 SasakiEinstein 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 amaximisation 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 KählerEinstein metric.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2008
 DOI:
 10.1007/s0022000804794
 arXiv:
 arXiv:hepth/0603021
 Bibcode:
 2008CMaPh.280..611M
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Differential Geometry
 EPrint:
 82 pages, 9 figures