Bayesian Integrals on Toric Varieties

Borinsky, Michael; Sattelberger, Anna-Laura; Sturmfels, Bernd; Telen, Simon

doi:10.48550/arxiv.2204.06414

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Numeric Integration1

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2022

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“…Numeric integration can be performed using Monte-Carlo methods (for a method developed recently based on rejection sampling from tropical densities see [49]). The basic goal is to be able to numerically evaluate integrals of the form…”

Section: Numeric Integrationmentioning

confidence: 99%

Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds

Larfors

Lukas

Ruehle

et al. 2022

Mach. Learn.: Sci. Technol.

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We introduce neural networks to compute numerical Ricci-ﬂat CY metrics for complete intersection and Kreuzer-Skarke Calabi-Yau manifolds at any point in Kähler and complex structure moduli space, and introduce the package cymetric which provides computation realizations of these techniques. In particular, we develop and computationally realize methods for point-sampling on these manifolds. The training for the neural networks is carried out subject to a custom loss function. The Kähler class is ﬁxed by adding to the loss a component which enforces the slopes of certain line bundles to match with topological computations. Our methods are applied to various manifolds, including the quintic manifold, the bi-cubic manifold and a Kreuzer-Skarke manifold with Picard number two. We show that volumes and line bundle slopes can be reliably computed from the resulting Ricci-ﬂat metrics. We also apply our results to compute an approximate Hermitian-Yang-Mills connection on a speciﬁc line bundle on the bi-cubic.

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Section: Numeric Integrationmentioning

confidence: 99%