2005/03/31 by Dario Martelli, James Sparks, Shing-Tung Yau · 1 citation
Physics and Astronomy · Mathematics · #hep-th #math.DG
paper · pdf · doi:10.1007/s00220-006-0087-0
published as Commun.Math.Phys.268:39-65,2006 · 35 pages, 4 figures; v2 minor changes; v3 typos corrected, eqn 2.60 removed, published version
arxiv created 2006/08/14 · arxiv updated 2011/05/05
We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z on Rn which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki-Einstein manifold without finding the metric explicitly. For complex dimension n=3 the Reeb vector and the volume correspond to the R-symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a-maximisation. We illustrate our results with some examples, including the Yp,q singularities and the complex cone over the second del Pezzo surface.