Gauged Floer homology and spectral invariants

TL;DR

This paper introduces a new spectral invariant in vortex Floer theory for G-Hamiltonian manifolds, explores its implications for symplectic quasi-morphisms, and connects vortex Floer homology with quasimap Floer homology to address non-displaceability issues.

Contribution

It defines a novel spectral invariant in vortex Floer theory and establishes a link with quasimap Floer homology, expanding tools for symplectic topology in non-semi-positive settings.

Findings

Potential new symplectic quasi-morphisms and quasi-states are identified.

A closed-open string map relates vortex Floer homology to Woodward's quasimap Floer homology.

Applications to non-displaceability problems in symplectic geometry are demonstrated.

Abstract

We define a version of spectral invariant in the vortex Floer theory for a $G$-Hamiltonian manifold $M$. This defines potentially new (partial) symplectic quasi-morphism and quasi-states when $M//G$ is not semi-positive. We also establish a relation between vortex Hamiltonian Floer homology and Woodward's quasimap Floer homology by constructing a closed-open string map between them. This yields applications to study non-displaceability problems of subsets in $M//G$