The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds

• DOI: 10.1007/s00220-006-0087-0
• Publication Date: 2006-08-23T10:17:22Z
• Authors: Dario Martelli; James Sparks; Shing-Tung Yau
• Keywords: High Energy Physics - Theory; Mathematics - Differential Geometry; High Energy Physics - Theory (hep-th); Differential Geometry (math.DG); 0103 physical sciences; FOS: Mathematics; FOS: Physical sciences; 01 natural sciences

35 pages, 4 figures; v2 minor changes; v3 typos corrected, eqn 2.60 removed, published version<br/>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 R^n 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 Y^{p,q} singularities and the complex cone over the second del Pezzo surface.<br/>