Polynomial Invariants for SU(2) Monopoles

TL;DR

This paper derives explicit polynomial invariants for SU(2) monopoles on spin four-manifolds, linking them to Seiberg-Witten invariants through topological quantum field theory analysis.

Contribution

It provides a new explicit formula for topological invariants of SU(2) monopoles using recent supersymmetric gauge theory results.

Findings

Invariants expressed in terms of Seiberg-Witten invariants

Utilizes recent exact results on moduli space of vacua

Connects topological invariants with supersymmetric gauge theories

Abstract

We present an explicit expression for the topological invariants associated to $SU(2)$ monopoles in the fundamental representation on spin four-manifolds. The computation of these invariants is based on the analysis of their corresponding topological quantum field theory, and it turns out that they can be expressed in terms of Seiberg-Witten invariants. In this analysis we use recent exact results on the moduli space of vacua of the untwisted $N=1$ and $N=2$ supersymmetric counterparts of the topological quantum field theory under consideration, as well as on electric-magnetic duality for $N=2$ supersymmetric gauge theories.