Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-03-03
Commun.Math.Phys.280:611-673,2008
Physics
High Energy Physics
High Energy Physics - Theory
82 pages, 9 figures; homogeneity of the (n,0)-form now derived from the Einstein-Hilbert action of the link, example of an orb
Scientific paper
10.1007/s00220-008-0479-4
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.
Martelli Dario
Sparks James
Yau Shing-Tung
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