Title: Generalised Seiberg-Witten equations and almost-Hermitian geometry,
Abstract: In this article, we study a generalisation of the Seiberg-Witten equations,
replacing the spinor representation with a hyperKahler manifold equipped with
certain symmetries. Central to this is the construction of a (non-linear) Dirac
operator acting on the sections of the non-linear fibre-bundle. For hyperKahler
manifolds admitting a hyperKahler potential, we derive a transformation formula
for the Dirac operator under the conformal change of metric on the base
manifold.
As an application, we show that when the hyperKahler manifold is of dimension
four, then away from a singular set, the equations can be expressed as a second
order PDE in terms of almost-complex structure on the base manifold and a
conformal factor. This extends a result of Donaldson to generalised
Seiberg-Witten equations.