Boundary N=2 Theory, Floer Homologies, Affine Algebras, and the Verlinde Formula

TL;DR

This paper uses topologically-twisted N=2 gauge theory on four-manifolds with boundary to provide physical proofs of key conjectures in Floer homology, affine algebras, and the Verlinde formula, connecting geometry, physics, and algebra.

Contribution

It introduces a physical framework for proving the Atiyah-Floer conjecture, Munoz's theorem, and their generalizations, linking Floer homologies with affine algebra modules and the Verlinde formula.

Findings

Physical proofs of the Atiyah-Floer conjecture and Munoz's theorem.

Relation of Floer homology to affine algebra modules via 2d A-model.

Derivation of the Verlinde formula from gauge theory.

Abstract

Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii) their monopole counterparts, and (iv) their higher rank generalizations. In the case where the boundary is a Seifert manifold, one can also relate its instanton Floer homology to modules of an affine algebra via a 2d A-model with target the based loop group. As an offshoot, we will be able to demonstrate an action of the affine algebra on the quantum cohomology of the moduli space of flat connections on a Riemann surface, as well as derive the Verlinde formula.