Gauge theory, calibrated geometry and harmonic spinors

In this paper, connections between different gauge-theoretical problems in high and low dimensions are established. In particular, it is shown that higher dimensional anti-self-duality equations on the total spaces of spinor bundles over low-dimensional manifolds can be interpreted as the Taubes–Pidstrygach 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 to arbitrary fibrations (without singular fibres) compatible with an appropriate calibration.