cymyc -- Calabi-Yau Metrics, Yukawas, and Curvature

We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

Machine Learned Calabi--Yau Metrics and Curvature

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 K\"ahler 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 Cefal\'u 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 $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations.