On genus one mirror symmetry in higher dimensions and the BCOV conjectures
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-Soulé 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2019
- DOI:
- 10.48550/arXiv.1911.06734
- arXiv:
- arXiv:1911.06734
- Bibcode:
- 2019arXiv191106734M
- Keywords:
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- Mathematics - Algebraic Geometry;
- Mathematics - Differential Geometry;
- Primary: 14J32;
- 14J33;
- 58J52. Secondary: 32G20
- E-Print:
- Final version, to appear in Forum of Mathematics, Pi