What to do with a Ricci-flat Calabi--Yau metric?

Abstract

Numerical approximations to Ricci-flat Calabi--Yau metrics make it possible to move beyond the topological and holomorphic data that have traditionally dominated explicit string compactifications. This article explains what new physics and mathematics become accessible once the metric, and eventually the associated Hermitian Yang--Mills bundle data, can be computed. In heterotic compactifications, such data are needed to determine matter Kähler metrics, canonically normalized Yukawa couplings, Kaluza--Klein spectra, threshold effects, soft terms, and other non-holomorphic ingredients of the four-dimensional effective action. More broadly, numerical Calabi--Yau geometry provides quantitative input for moduli stabilization, $α'$-corrected backgrounds, de~Sitter model building, axion physics, swampland distance tests, and compactifications in which the internal geometry varies over spacetime. Geometric data permit a computational approach to long-standing mathematical questions involving special Lagrangian submanifolds, SYZ fibrations, mirror symmetry, calibrated geometry, metric degeneration, restrictions of Ricci-flat metrics to fibers, and the search for analytic or semi-analytic structures. We present these directions as a roadmap for future work.