A Necessary Condition for the Existence of SW-Monopoles

TL;DR

This paper establishes a necessary condition on spin^c classes for the existence of solutions called SW-monopoles in Seiberg-Witten theory, clarifying when these solutions can be minima of the SW-functional.

Contribution

It introduces a new necessary condition that must be satisfied for a spin^c class to admit SW-monopole solutions as minima, advancing understanding of solution existence criteria.

Findings

Identifies a necessary condition for SW-monopole existence.

Shows that only finitely many classes satisfy the condition.

Clarifies the relationship between solutions and minima of the SW-functional.

Abstract

Originally, the SW-equations discovered by Seiberg-Witten are 1st-order PDE, which solutions (A,\phi), with \phi\ne 0, are known as SW-monopoles. It is known that the solutions of these 1st-order eq correspond to the minimum of SW-functional. However, it is not true, that for all spin^{c} class \alpha, the minimum is always attained by this sort of solution. In fact, there are only a finite number of \alpha such that the minimum is a SW-monopole. We show a necessary condition to be satisfied by the class \alpha in order to the minimum be a SW-monopole.