Gauged Gromov-Witten theory for small spheres

TL;DR

This paper establishes a relationship between gauged Gromov-Witten invariants for small areas and equivariant invariants, and explores their implications in symplectic geometry, including a gauged abelianization and vortex convergence.

Contribution

It introduces a new equivalence for gauged Gromov-Witten invariants at small areas and extends abelianization to this context, with applications in symplectic vortex theory.

Findings

Gauged Gromov-Witten invariants for small areas match invariant parts of equivariant invariants.

Proves a gauged abelianization principle for Gromov-Witten invariants.

Shows convergence of genus zero symplectic vortices with vanishing area to holomorphic maps.

Abstract

For smooth projective G-varieties, we equate the gauged Gromov-Witten invariants for sufficiently small area and genus zero with the invariant part of equivariant Gromov-Witten invariants. As an application we deduce a gauged version of abelianization for Gromov-Witten invariants. In the symplectic setting, we prove that any sequence of genus zero symplectic vortices with vanishing area has a subsequence that converges after gauge transformation to a holomorphic map with zero average moment map.