Instability of Reducible Critical Points of the Seiberg-Witten Functional

TL;DR

This paper investigates the stability properties of reducible solutions to the Seiberg-Witten equations, showing that under certain geometric conditions, these solutions are unstable, which impacts the understanding of the solution space.

Contribution

It demonstrates the instability of reducible critical points of the Seiberg-Witten functional under specific geometric assumptions, such as the existence of a parallel spinor or negative invariants.

Findings

Reducible solutions can be unstable under certain conditions.

Existence of a parallel spinor implies instability.

Negative Perelman-Yamabe type invariants lead to instability.

Abstract

The Euler-Lagrange equations for the variational approach to the Seiberg-Witten equations always admit reducible solutions. In this context, the existence of unstable reducible solutions is achieved by assuming the existence of a parallel spinor or the negativeness of a Perelman-Yamabe type of invariant defined for a $\spinc$-structure.