Numerical Sasaki--Einstein metrics and harmonic forms on del Pezzo links

Abstract: We numerically construct the Sasaki--Einstein metric on the link of the cone over the second del Pezzo surface $\mathrm{dP}_2$ and two primitive harmonic basic $(1,1)$-forms at its irregular volume-minimizing Reeb vector. Being toric, the metric in symplectic coordinates is encoded in a single convex function on a polygon. We approximate the correction to the canonical Guillemin potential in two ways: polynomial expansion and neural networks. The polynomial fit achieves a held-out mean-squared Monge--Ampère residual below $10<sup>{-13}$, in contrast to the $10<sup>{-2}$ plateau for the non-volume-minimizing regular Reeb vector. We validate our method against closed-form metrics of $Y<sup>{p,q}$ using curvature invariants, and also against the numerical result of Doran et al. (2007) for the Kähler--Einstein metric of $\mathrm{dP}_3$ using the Laplacian spectrum of low torus-invariant modes. Our data for the metric and harmonic forms can be used to study warped non-conformal holographic IIB backgrounds, the analogues of the Klebanov--Tseytlin solution on the conifold.