Gauged Floer theory of toric moment fibers

TL;DR

This paper introduces quasimap Floer cohomology, a new invariant for Lagrangians in symplectic quotients, extending non-displaceability results to toric orbifolds without virtual chains.

Contribution

It develops quasimap Floer cohomology as a novel approach to study Lagrangian displaceability in symplectic quotients, generalizing previous results to orbifolds.

Findings

Reproduces non-displaceability of toric moment fibers in Fano and non-Fano cases

Extends non-displaceability results to toric orbifolds

Proposes a conjectural link between quasimap Floer cohomology and quotient Floer cohomology

Abstract

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya-Oh-Ohta-Ono and Cho-Oh on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without virtual fundamental chains. Finally we describe a conjectural relationship to Floer cohomology in the quotient.