10/23 at 17:30 in 507

Andrey Smirnov

Title: Frobenius structures for quantum differential equations and mirror symmetry.
Abstract:  

There exists a well-known connection between the Kloosterman sum in number theory, and the Bessel differential equation. This connection was explained by B. Dwork in 1974 by discovering Frobenius structures in the p-adic theory of the Bessel equation. In my talk I will speculate that this connection extends to the quantum differential equations in quantum cohomology of Nakajima varieties. As an example, I will give an explicit conjectural description of the Frobenius structures for the quantum connections of T*Gr(k,n) and also Gr(k,n).   The traces of Frobenius structures are natural finite field analogs of the integral solutions of quantum differential equations known in mirror symmetry. In particular, for Gr(k,n)  we arrive at the exact B-model description of quantum connection discovered by Marsh and Rietsch.