On genus one mirror symmetry in higher dimensions and the BCOV conjectures

TL;DR

This paper mathematically formulates the genus one BCOV mirror symmetry conjecture, proves it for Calabi-Yau hypersurfaces, and explores arithmetic properties of the BCOV invariant in higher-dimensional mirror symmetry.

Contribution

It provides a mathematical proof of the genus one BCOV conjecture for Calabi-Yau hypersurfaces and advances the understanding of mirror symmetry in higher dimensions.

Findings

Proved the BCOV conjecture for Calabi-Yau hypersurfaces in projective spaces.

Established the first complete examples of higher-dimensional mirror symmetry.

Connected the BCOV invariant to special values of Gamma functions in arithmetic geometry.

Abstract

The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed, in a seminal article from '94, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck-Riemann-Roch theorem, we offer a mathematical description of the BCOV conjecture at genus one. As an application of the arithmetic Riemann-Roch theorem of Gillet-Soul\'e and of our previous results on the BCOV invariant, we establish this conjecture for Calabi-Yau hypersurfaces in projective spaces. Our contribution takes place on the $B$-side, and together with the work of Zinger on the $A$-side, it provides the first complete examples of the mirror symmetry program in higher dimensions. The case of quintic threefolds was studied by Fang-Lu-Yoshikawa. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla-Selberg type theorem expressing it in terms of special $\Gamma$ values for certain Calabi-Yau manifolds with complex multiplication.