The Geometric Dual of a-maximisation for Toric Sasaki-Einstein Manifolds

TL;DR

The paper shows that for toric Calabi–Yau cones, the Reeb vector and Sasaki–Einstein volume can be determined without the explicit metric by minimizing a convex function Z over the dual toric cone, establishing a geometric dual to a-maximisation in AdS/CFT. It proves that volumes depend only on the Reeb vector and derives a Monge–Ampère equation capturing the Ricci-flat condition, with c1(X)=0 enforcing strong constraints. In complex dimension three, it provides explicit computations for examples like conifold, Y^{p,q}, SPP, and dP2, matching field-theoretic predictions from a-maximisation. The results decouple Reeb data from the metric problem, offering a practical, metric-free route to R-symmetry and volume data and hinting at possible three-dimensional analogues.

Abstract

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.

Figures (4)

• Figure 1: Toric diagram for the conifold.
• Figure 2: Toric diagram for $Y^{5,3}$.
• Figure 3: Toric diagram for the SPP.
• Figure 4: Toric diagram for the complex cone over $dP_2$.