Numerical Kaehler-Einstein metric on the third del Pezzo

TL;DR

This paper numerically computes the unique Kaehler-Einstein metric on the third del Pezzo surface using three algorithms, providing insights into its geometric properties and potential applications in supergravity solutions.

Contribution

It introduces three novel numerical algorithms for solving the Kaehler-Einstein equation on toric manifolds, applicable beyond the specific case studied.

Findings

Successfully computed the Kaehler-Einstein metric on the third del Pezzo surface.

Calculated geometric quantities like Laplacian eigenvalues and harmonic forms.

Demonstrated the applicability of algorithms to general toric manifolds.

Abstract

The third del Pezzo surface admits a unique Kaehler-Einstein metric, which is not known in closed form. The manifold's toric structure reduces the Einstein equation to a single Monge-Ampere equation in two real dimensions. We numerically solve this nonlinear PDE using three different algorithms, and describe the resulting metric. The first two algorithms involve simulation of Ricci flow, in complex and symplectic coordinates respectively. The third algorithm involves turning the PDE into an optimization problem on a certain space of metrics, which are symplectic analogues of the "algebraic" metrics used in numerical work on Calabi-Yau manifolds. Our algorithms should be applicable to general toric manifolds. Using our metric, we compute various geometric quantities of interest, including Laplacian eigenvalues and a harmonic (1,1)-form. The metric and (1,1)-form can be used to construct a Klebanov-Tseytlin-like supergravity solution.