A Quantum Kirwan Map, II: Bubbling

TL;DR

This paper studies the convergence of vortex sequences in symplectic geometry and aims to rigorously define a quantum deformation of the Kirwan map, which could impact the computation of quantum cohomology of symplectic quotients.

Contribution

It proves subsequential convergence of vortex sequences to stable maps, advancing the rigorous definition of the quantum Kirwan map in symplectic geometry.

Findings

Sequences of vortices with bounded energy converge to stable maps.

The quantum Kirwan map may compute quantum cohomology of symplectic quotients.

Potential for quantum generalizations of localization and abelianization.

Abstract

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,\omega)$. Under suitable assumptions, counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$ conjecturally gives rise to a quantum deformation $Qk_G$ of the Kirwan map. This is the second of a series of articles, whose goal is to define $Qk_G$ rigorously. The main result is that every sequence of vortices with uniformly bounded energies has a subsequence that converges to a genus 0 stable map of vortices on $R^2$ and holomorphic spheres in the symplectic quotient. Potentially, the map $Qk_G$ 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.