Gauged Floer Theory Of Toric Moment Fibers

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder Fr1]. 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 FuOOO3,1] and Cho–Oh CO] on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without using virtual fundamental chains. Finally, we describe a conjectural relationship with Floer cohomology in the quotient.