Monopoles and Four-Manifolds

TL;DR

This paper explores a dual perspective on Donaldson invariants of four-manifolds using solutions to a new abelian gauge equation inspired by supersymmetric Yang-Mills theory, offering fresh insights into four-manifold topology.

Contribution

It introduces a dual abelian gauge equation approach to define four-manifold invariants, providing a novel perspective aligned with recent supersymmetric Yang-Mills theory developments.

Findings

Establishes equivalence between non-abelian and abelian gauge counting methods

Suggests new computational techniques for Donaldson invariants

Provides theoretical insights linking supersymmetry and four-manifold topology

Abstract

Recent developments in the understanding of $N=2$ supersymmetric Yang-Mills theory in four dimensions suggest a new point of view about Donaldson theory of four manifolds: instead of defining four-manifold invariants by counting $SU(2)$ instantons, one can define equivalent four-manifold invariants by counting solutions of a non-linear equation with an abelian gauge group. This is a ``dual'' equation in which the gauge group is the dual of the maximal torus of $SU(2)$. The new viewpoint suggests many new results about the Donaldson invariants.