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.

Final version, to appear in Forum of Mathematics, Pi

References in corpus (16)

• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Gromov-Witten invariants in algebraic geometry
• Enumerative geometry of Calabi-Yau 4-folds
• Twisted jets, motivic measure and orbifold cohomology
• Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties
• Mirror duality and string-theoretic Hodge numbers
• Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
• Standard vs. Reduced Genus-One Gromov-Witten Invariants
• On the existence of crepant resolutions of Gorenstein Abelian quotient singularities in dimensions \geq 4
• BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
• Periods of Hodge structures and special values of the gamma function
• Motivic integration and the birational invariance of BCOV invariants
• Analytic torsion for log-Enriques surfaces and Borcherds product
• Canonical isomorphisms of determinant line bundles
• Resultants and the Borcherds Phi-function

Cited by in corpus (5)

• Motivic integration and the birational invariance of BCOV invariants
• BCOV invariant and blow-up
• Elliptic Virtual Structure Constants and Generalizations of BCOV-Zinger Formula to Projective Fano Hypersurfaces
• An extension of BCOV invariant
• Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $t{t}^{\ast }$ Structures