Resonant Hypergeometric Systems and Mirror Symmetry

Abstract

The Gamma-series of Gel'fand-Kapranov-Zelevinsky are adapted so that they give solutions for certain resonant systems of GKZ hypergeometric differential equations. For this some complex parameters in the Gamma-series are replaced by nilpotent elements from a ring $R_{A,T}$. The adapted Gamma-series is a function $Ψ$ with values in the finite dimensional vector space $R_{A,T}\otimes C$. Applications of these results in the context of toric Mirror Symmetry are described. Building on work of Batyrev we show that the relative cohomology module of a certain hypersurface in a torus is a GKZ hypergeometric $D$-module which over an appropriate domain is isomorphic to the trivial $D$-module $R_{A,T}\otimes O_T$, where $O_T$ is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Gamma-series. As a result one finds the periods of a holomorphic differential form of degree $d$ on a $d$-dimensional Calabi-Yau manifold, needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov we interpret the ring $\cR_{\sA,\gT}$ as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.