On genus one mirror symmetry in higher dimensions and the BCOV conjectures
Résumé
The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed, in a seminal article from '94, an extension of genus zero mirror symmetry to higher genera. We offer a mathematical treatment of the BCOV conjecture at genus one, based on the usage of the arithmetic Riemann-Roch theorem. As an application of our previous results on the BCOV invariant, we establish this conjecture for Calabi-Yau hypersurfaces in projective space. This seems to be the first example of higher dimensional mirror symmetry, of BCOV type, at genus one. The case of quintic threefolds was studied by Fang-Lu-Yoshikawa. Our contribution takes place on the B-side, and the relation to the A-side is provided by Zinger.
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 Γ values for certain Calabi-Yau manifolds with complex multiplication. Finally, we put forward a variant of the BCOV program at genus one, as a conjectured functorial Grothendieck-Riemann-Roch relationship.
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