Machine Learned Calabi–Yau Metrics and Curvature
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Finding Ricci-flat (Calabi–Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi–Yau metric within a given Kähler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi–Yau threefolds. Using these Ricci-flat metric approximations for the Cefalú and Dwork family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points, but also elsewhere. For our neural network approximations, we observe a Bogomolov–Yau type inequality 3c_2 ≥ c_1^2 and observe an identity when our geometries have isolated A_1 type singularities. We sketch a proof that χ(X ∖ Sing X) + 2 |Sing X| = 24 also holds for our numerical approximations.
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