Kahler and symplectic structures on 4-manifolds and hyperKahler geometry

Abstract

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold. For a class of hyperKahler manifolds obtained via hyperKahler reduction, we also show that a harmonic spinor, under mild conditions, defines a symplectic structure. Finally, we show that if a covariantly constant, generalized spinor satisfies generalized Seiberg-Witten equations, the metric on the base manifold has a constant scalar curvature.

arXiv comment15 pages, added the case of hyperKahler manifolds obtained via hyperKahler reduction and rectified a couple of embarrassing mistakes in the previous version of the paper. Comments and suggestions are welcome!