What to expect from $U(n)$ Seiberg-Witten monopoles for $n > 1$

TL;DR

This paper investigates the generalization of Seiberg-Witten monopole equations to U(n) gauge groups for n > 1, concluding that such extensions likely do not produce new gauge-theoretical invariants due to the emptiness of their moduli spaces under generic conditions.

Contribution

It provides a detailed analysis of U(n) monopole equations, showing that for n > 1, the associated moduli spaces are typically empty, indicating no new invariants arise from these generalizations.

Findings

Moduli spaces are empty for n > 1 under generic deformations.

No non-trivial gauge-theoretical invariants are expected for U(n) monopoles when n > 1.

On Kähler surfaces with b2+ > 1, moduli spaces vanish with non-zero holomorphic 2-form perturbations.

Abstract

We study generalisations to the structure groups U(n) of the familiar (abelian) Seiberg-Witten monopole equations on a four-manifold $X$ and their moduli spaces. For $n=1$ one obtains the classical monopole equations. For $n > 1$ our results indicate that there should not be any non-trivial gauge-theoretical invariants which are obtained by the scheme `evaluation of cohomology classes on the fundamental cycle of the moduli space'. For, if $b_2^+$ is positive the moduli space should be `cobordant' to the empty space because we can deform the equations so as the moduli space of the deformed equations is generically empty. Furthermore, on K\"ahler surfaces with $b_2^+ > 1$, the moduli spaces become empty as soon as we perturb with a non-vanishing holomorphic 2-form.