Andrey Smirnov (UNC) “Frobenius structures for quantum differential equations and mirror symmetry”

Seminar Organizer

Event Details

Abstract: There exists a well-known similarity between the Kloosterman sum in number theory and the Bessel differential equation. This connection was explained by B. Dwork in 70s by discovering the Frobenius structures in the p-adic theory of the Bessel differential equation. In my talk I will speculate that this connection extends to the equivariant quantum differential equations for a wide class of varieties, which includes the Nakajima quiver varieties. As the main result, I will give an explicit conjectural description of the Frobenius structure for the equivariant quantum connection. The traces of the Frobenius structures are natural finite field analogs of the mirror integral representations of the J-function in quantum cohomology. As an example, I will discuss how, in the case of Grassmannians Gr(k,n), these considerations bring us to the B-model side of Gr(k,n) discovered earlier by Marsh and Rietsch.