Andrey Smirnov (Columbia University), Physically Inspired Mathematics

Title:  Quantum difference equations for quiver varieties

Abstract:  Let QH(X) be a quantum cohomology ring of some variety X. The operation of quantum multiplication defines a flat connection on H^2(X) also known as quantum differential equation. In this talk I will discuss the generalization of this picture to the quantum K-theory of X given by a quiver variety. The corresponding differential equation is now substituted by a difference equation, which can be considered as a “flat difference connection” on a lattice ( Picard group of X ). I will explain the relation of this difference connection with K-theoretic J-function of Givental, qKZ-equations, monodromy problem (for the quantum connection), and dynamical Weyl group.