# Generalised Seiberg-Witten equations and almost-Hermitian geometry

## TL;DR

This paper generalizes Seiberg-Witten equations using hyperKahler manifolds, deriving a non-linear Dirac operator and expressing the equations as second order PDEs related to almost-complex structures, extending Donaldson's results.

## Contribution

It introduces a new framework for generalized Seiberg-Witten equations with hyperKahler targets and derives transformation formulas for the Dirac operator under conformal changes.

## Key findings

- Derived a non-linear Dirac operator for hyperKahler targets.
- Expressed generalized equations as second order PDEs involving almost-complex structures.
- Extended Donaldson's results to a broader class of equations.

## 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.