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

TL;DR

This work introduces cymyc, a high-performance Python framework for numerically exploring Calabi–Yau geometry and moduli spaces by constructing globally defined tensor-field approximations via a geometric ansatz and neural-network discretisation. It develops dd^c and global-section-based approaches to approximate Ricci-flat metrics, together with harmonic-form techniques anchored in Kodaira–Spencer theory, and validates them on complete-intersection CYs such as mirrors of P^5[3,3], P^7[2,2,2,2], and the Tian–Yau quotient. The library computes moduli-space metrics (Weil–Petersson) and canonically normalised Yukawa couplings, demonstrating close agreement with special-geometry results and revealing moduli-dependent curvature behaviour, including heavy-tailed distributions and singular-locus phenomena. By enabling efficient Monte Carlo evaluations of geometric and topological observables, cymyc provides a practical bridge from compactification geometry to low-energy phenomenology, and opens avenues for systematic exploration of the string landscape and mathematical conjectures in CY geometry.

Abstract

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.

Figures (10)

• Figure 1: Evolution of various geometric quantities over the optimisation process for the Ricci-flat metric on the $\mathbb{P}^5[3,3]$ mirror at the point $\psi=0.05$ in complex structure moduli space. Clockwise from top left: average Ricci scalar over $X$, $L_2$ norm of Ricci curvature, $L_2$ metric norm of Einstein tensor \ref{['eq:einstein_tensor']}, Euler characteristic, $\sigma$-measure \ref{['eq:sigma_measure']}, average $(d\omega, d\omega)$ over $X$. The solid lines indicate the average of five independent experiments, and the semi-transparent bands indicate the 95% confidence interval.
• Figure 2: Weil--Petersson metric (top), and normalised Yukawa coupling (bottom) for the mirror of $\mathbb{P}^5[3,3]$ along the $\mathfrak{I}(\psi)\,{=}\,0$ line in complex structure moduli space; the singular case of \ref{['eq:two_cubics']} at $\psi\,{=}\,0$ is at infinite distance in the moduli space. The two insets in the top figure exhibit the behaviour of the moduli space metric approaching the $\psi=0$ pole.
• Figure 3: Weil--Petersson metric for the $\mathbb{P}^5[3,3]$ mirror, close to the infinite-distance point in moduli space at $\psi=0$. Note the discrepancy of the non-globally defined ansatz from the true value worsens as one approaches the pole.
• Figure 4: Weil--Petersson metric (top), and normalised Yukawa coupling (bottom) for the mirror of $\mathbb{P}^7[2,2,2,2]$ along the $\mathfrak{I}(\psi)\,{=}\,0$ line in complex structure moduli space around the conifold point at $\psi = 1$.
• Figure 5: Spectrum of the Weil--Petersson metric (top), and normalised Yukawa coupling (bottom) for the Tian--Yau quotient along the $\mathfrak{I}(\psi)\,{=}\,0$ line in complex structure moduli space, removing degenerate eigenvalues by lexicographical priority. Note the behaviour of the eigenvalues around the singularities, indicated by the grey dashed lines.