Donaldson invariants of product ruled surfaces and two-dimensional gauge theories

TL;DR

This paper derives a unified formula for Donaldson invariants of product ruled surfaces using the u-plane integral, extending previous results to higher genus surfaces and applying it to moduli space intersection pairings and Fukaya-Floer cohomology.

Contribution

It generalizes Morgan and Szabo's theorem for genus g to all g and connects gauge theory invariants with moduli space and Floer cohomology computations.

Findings

Derived a simple expression for Donaldson invariants of lgebraic surfaces.

Extended Thaddeus' formulas to higher genus surfaces.

Computed the eigenvalue spectrum of Fukaya-Floer cohomology.

Abstract

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of $\Sigma_g \times S^2$, where $\Sigma_g$ is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabo for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya-Floer cohomology of $\Sigma_g \times S^1$.