Degeneracy loci of families of Dirac operators

TL;DR

This paper computes the rational cohomology class of the degeneracy locus of Dirac operators over moduli spaces of anti-self-dual connections, aiming to connect Donaldson and spin invariants.

Contribution

It generalizes previous results to compute the Poincaré dual of degeneracy loci in the context of Dirac operators on moduli spaces, advancing the understanding of invariants in gauge theory.

Findings

Computed the Poincaré dual of the degeneracy locus in rational cohomology.

Established a foundation for relating Donaldson and spin invariants.

Extended previous work to a broader class of Dirac operator families.

Abstract

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\SO(3)$ connections. This is the first step in a program to derive a relation between the Donaldson and spin invariants.