A Quantum Kirwan Map, I: Fredholm Theory

TL;DR

This paper establishes the Fredholm property of the vortex equations' vertical differential in the context of Hamiltonian group actions on symplectic manifolds, laying groundwork for defining a quantum Kirwan map.

Contribution

It proves the Fredholm property of the vortex equations' vertical differential, enabling rigorous definition of the quantum Kirwan map in symplectic geometry.

Findings

Vertical differential of vortex equations is a Fredholm operator.

Main result provides the index of this Fredholm operator.

Lays foundation for computing quantum cohomology of symplectic quotients.

Abstract

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,\omega)$. Under some assumptions on $(M,\omega)$ and the action, D. A. Salamon conjectured that counting gauge equivalence classes of symplectic vortices on the plane $R^2$ gives rise to a quantum deformation $Q\kappa_G$ of the Kirwan map. This article is the first of three, whose goal is to define $Q\kappa_G$ rigorously. Its main result is that the vertical differential of the vortex equations over $R^2$ (at the level of gauge equivalence) is a Fredholm operator of a specified index. Potentially, the map $Q\kappa_G$ can be used to compute the quantum cohomology of many symplectic quotients. Conjecturally it also gives rise to quantum generalizations of non-abelian localization and abelianization (see [Woodward-Ziltener]).