Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors

TL;DR

This paper develops an equivariant Lagrangian Floer theory for Liouville sectors with Lie group symmetry and establishes a correspondence relating equivariant and quotient Floer cohomologies, confirming a conjecture by Lekili-Segal.

Contribution

It introduces an equivariant Floer theory for symmetric Liouville sectors and proves an isomorphism between equivariant and quotient Floer cohomologies in singular cases.

Findings

Established an equivariant Floer theory for Liouville sectors with symmetry.

Proved the isomorphism between equivariant and quotient Floer cohomologies.

Confirmed the Lekili-Segal conjecture in the presence of nodal singularities.

Abstract

We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.