The Morse Index of Redutible Solutions Of The Seiberg-Witten Equations

TL;DR

This paper derives a second variation formula for the Seiberg-Witten functional to estimate the Morse index of reducible solutions, showing it equals the dimension of a specific negative eigenspace, thus finite.

Contribution

It provides a new formula for the Morse index of reducible solutions of the Seiberg-Witten equations, linking it to spectral properties of a differential operator.

Findings

Morse index is finite for reducible solutions.

Morse index equals the dimension of the largest negative eigenspace of a specific operator.

Second variation formula for the Seiberg-Witten functional is established.

Abstract

The $2^{nd}$ variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions $(A,0)$. It is shown that their Morse index is given by the dimension of the largest negative eigenspace of the operator $\triangle_{A} +\frac{k_{g}}{4}$, hence it is finite.