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arXiv · 1911.06734

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

Gerard Freixas I Montplet·Dennis Eriksson·Christophe Mourougane
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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.

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Gerard Freixas I Montplet, Dennis Eriksson, Christophe Mourougane. 2019-11-15. On genus one mirror symmetry in higher dimensions and the BCOV conjectures. https://doi.org/10.1017/fmp.2022.13

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