THE PHYSICAL MATHEMATICS OF VARIOUS QUANTUM FIELD THEORIES

Abstract

We explain how topologically-twisted N=2 gauge theory on an open four-manifold, will allow physical proofs of various mathematical conjectures and theorems. When the boundary is a Seifert manifold, one can also relate its instanton Floer-homology to an affine-algebra via a two-dimensional A-model with target based loop group. We further demonstrate an action of the affine-algebra on the quantum-cohomology of the moduli space of flat connections on a Riemann-surface, and derive the Verlinde-formula.

We also show that four-dimensional Chern-Simons theory is, with Nahm-pole-type boundary-conditions, dual to three-dimensional Toda theory with a novel three-dimensional W-algebra. Embedding four-dimensional Chern-Simons theory in a partial-twist of the five-dimensional maximally-supersymmetric-Yang-Mills theory on a manifold with corners reveals that three-dimensional Toda theory is dual to two-dimensional A-model on Bogomolny moduli space. This novel 3d-2d correspondence also reveals that representations of 3d W-algebra are acted on by the quantized-algebra of certain holomorphic functions on the Bogomolny moduli space.