An index relation for the quilted Atiyah-Floer conjecture

TL;DR

This paper proves that the chain level gradings of instanton Floer groups and quilted Lagrangian Floer groups coincide for certain 3-manifolds, advancing the understanding of the Atiyah-Floer conjecture.

Contribution

It establishes the equality of gradings between two Floer theories, providing a key step towards the quilted Atiyah-Floer conjecture.

Findings

Gradings of instanton and quilted Floer groups are shown to agree.

Supports the conjecture relating different Floer theories for 3-manifolds.

Enhances the mathematical framework connecting gauge theory and symplectic geometry.

Abstract

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. Each of these is equipped with a chain level grading. We show that the gradings agree.