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 dP2 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 10−13, in contrast to the 10−2 plateau for the non-volume-minimizing regular Reeb vector. We validate our method against closed-form metrics of Yp,q using curvature invariants, and also against the numerical result of Doran et al. (2007) for the Kähler--Einstein metric of dP3 using the Laplacian spectrum of low torus-invariant modes. The integrated Riemann-squared curvature of the dP2 link agrees to two parts in 109 with the closed form conjectured from the dual field theory. 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.
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BibTeX
@misc{kim2026numerical,
title = {{Numerical Sasaki--Einstein metrics and harmonic forms on del Pezzo links}},
author = {Nakwoo Kim and Sejin Kim and Hoseob Shin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25857},
eprint = {2609.25857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25857},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Numerical Sasaki--Einstein metrics and harmonic forms on del Pezzo links
AU - Nakwoo Kim
AU - Sejin Kim
AU - Hoseob Shin
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.25857
UR - https://arxiv.org/abs/2609.25857
AB - We numerically construct the Sasaki--Einstein metric on the link of the cone over the second del Pezzo surface 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^−13, in contrast to the 10^−2 plateau for the non-volume-minimizing regular Reeb vector. We validate our method against closed-form metrics of Y^p,q using curvature invariants, and also against the numerical result of Doran et al. (2007) for the Kähler--Einstein metric of dP_3 using the Laplacian spectrum of low torus-invariant modes. The integrated Riemann-squared curvature of the dP_2 link agrees to two parts in 10^9 with the closed form conjectured from the dual field theory. 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.
ER -