Kahler and symplectic structures on 4-manifolds and hyperKahler geometry

TL;DR

This paper explores how generalized Dirac operators linked to hyperKahler manifolds induce Kahler and symplectic structures on 4-manifolds, revealing new geometric insights and conditions for constant scalar curvature.

Contribution

It introduces a non-linear Dirac operator framework using hyperKahler manifolds and demonstrates how covariantly constant spinors define Kahler and symplectic structures, and relate to scalar curvature.

Findings

Covariantly constant spinors define Kahler structures.

Harmonic spinors induce symplectic structures under certain conditions.

Generalized Seiberg-Witten equations imply constant scalar curvature.

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.