The Monopole Equations in Topological Yang-Mills

TL;DR

This paper explores the realization of Seiberg-Witten monopole equations within topological Yang-Mills theory, establishing connections through Floer cohomology and curvature conditions on specific bundles.

Contribution

It demonstrates how monopole equations are embedded in topological Yang-Mills theory and constructs a unitary transformation linking different Floer cohomologies.

Findings

Monopole equations are realized in topological Yang-Mills theory.

A Floer derivative and Morse functional are constructed.

Connections between Floer cohomologies are established.

Abstract

We twist the monopole equations of Seiberg and Witten and show how these equations are realized in topological Yang-Mills theory. A Floer derivative and a Morse functional are found and are used to construct a unitary transformation between the usual Floer cohomologies and those of the monopole equations. Furthermore, these equations are seen to reside in the vanishing self-dual curvature condition of an $OSp(1|2)$-bundle. Alternatively, they may be seen arising directly from a vanishing self-dual curvature condition on an $SU(2)$-bundle in which the fermions are realized as spanning the tangent space for a specific background.