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

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 Kähler surfaces with $b_2^{+}>1$, the moduli spaces become empty as soon as we perturb with a non-vanishing holomorphic 2-form.

17 pages