Gauge theory, calibrated geometry and harmonic spinors

TL;DR

This paper explores the connections between gauge theory, calibrated geometry, and harmonic spinors, revealing how higher-dimensional anti-self-dual equations relate to generalized Seiberg-Witten equations and harmonic spinors through geometric fibrations.

Contribution

It establishes a novel interpretation of higher-dimensional anti-self-dual equations as generalized Seiberg-Witten equations and relates solutions to harmonic spinors via fiber collapsing techniques.

Findings

Higher-dimensional ASD equations can be viewed as generalized Seiberg-Witten equations.

Solutions of these equations relate to harmonic spinors through fiber collapsing.

The approach extends to arbitrary fibrations compatible with calibration.

Abstract

In this paper connections between different gauge-theoretical problems in high and low dimensions are established. In particular it is shown that higher dimensional asd equations on total spaces of spinor bundles over low dimensional manifolds can be interpreted as Taubes-Pidstrygach's generalization of the Seiberg-Witten equations. By collapsing each fibre of the spinor bundle to a point, solutions of the Taubes-Pidstrygach equations are related to generalized harmonic spinors. This approach is also generalized for arbitrary fibrations (without singular fibres) compatible with an appropriate calibration.