Virtual Coulomb branch and vertex functions

TL;DR

This paper introduces a new virtual Coulomb branch construction using $K$-theoretic methods, relating it to quasimaps, Whittaker functions, and quantum $q$-difference modules, with applications to wall-crossing phenomena.

Contribution

It develops a novel virtual intersection theory for Coulomb branches, defines Verma modules in this context, and connects them to Higgs branch invariants and quantum difference modules.

Findings

Defined virtual Coulomb branches using new intersection theory

Connected vertex functions to Whittaker functions via quasimaps

Proved wall-crossing for quantum $q$-difference modules

Abstract

We introduce a variant of the $K$-theoretic quantized Coulomb branch constructed by Braverman--Finkelberg--Nakajima, by application of a new virtual intersection theory. In the abelian case, we define Verma modules for such virtual Coulomb branches, and relate them to the moduli spaces of quasimaps into the corresponding Higgs branches. The descendent vertex functions, defined by $K$-theoretic quasimap invariants of the Higgs branch, can be realized as the associated Whittaker functions. The quantum $q$-difference modules and Bethe algebras (analogue of quantum $K$-theory rings) can then be described in terms of the virtual Coulomb branch. As an application, we prove the wall-crossing result for quantum $q$-difference modules under the variation of GIT. Nonabelian cases are also treated via abelianization.