A Quantum Kirwan Map, I: Fredholm Theory

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]).

This article has been merged with arXiv:1106.1729. The new article is: A Quantum Kirwan Map: Bubbling and Fredholm Theory for Symplectic Vortices over the Plane, arXiv:1209.5866

References in corpus (1)

• Yang-Mills-Higgs theory for symplectic fibrations

Cited by in corpus (1)

• A Quantum Kirwan Map, II: Bubbling