Xiaohan Yan: Quantum K-theory of flag varieties via non-abelian localization


Abstract: Quantum K-theory studies the K-theoretic analogue of Gromov-Witten invariants defined as holomorphic Euler characteristics of sheaves on the moduli space of stable maps. Generating functions of such invariants, which are called the (K-theoretic) big J-functions, play a crucial role in the theory. In this talk, we provide a reconstruction theorem of the permutation-invariant big J-function of partial flag varieties (regarded as GIT quotients of vector spaces) using a family of finite-difference operators, from the quantum K-theory of their associated abelian quotients which is well-understood. The idea is to use a recursive characterization of the big J-functions based on the geometry of isolated fixed points and connecting 1-dimensional orbits on the flag varieties in toric-equivariant settings, but along the way we will need to address the issue of possibly non-isolated fixed points on the abelian quotient. A portion of this talk is based on a joint work with Alexander Givental.