Wall-crossing for vortex partition function and handsaw quiver varierty

TL;DR

This paper studies vortex partition functions from handsaw quiver varieties, deriving functional equations, explicit formulas, and geometric interpretations, thus confirming physicists' conjectured formulas and connecting to hypergeometric functions.

Contribution

It provides explicit formulas and geometric insights into vortex partition functions from handsaw quiver varieties, confirming physicists' conjectures and linking to hypergeometric functions.

Findings

Derived functional equations for vortex partition functions.

Provided explicit formulas for these partition functions.

Established geometric interpretations of hypergeometric function formulas.

Abstract

We investigate vortex partition functions defined from integrals over the handsaw quiver varieties of type $A_{1}$ via wall-crossing phenomena. We consider vortex partition functions defined by two types of cohomology classes, and get functional equations for each of them. We also give explicit formula for these partition functions. This gives proofs to formula suggested by physicsts. In particular, we obtain geometric interpretation of formulas for multiple hypergeometric functions including rational limit of the Kajihara transformation formula.