Donaldson invariants for nonsimply connected manifolds

TL;DR

This paper derives explicit formulas for Donaldson invariants of certain 4-manifolds with positive first Betti number using Coulomb branch integrals and wall-crossing, potentially impacting quantum cohomology research.

Contribution

It provides new explicit expressions for Donaldson invariants on non-simply connected 4-manifolds with positive first Betti number, expanding the understanding of gauge theory invariants.

Findings

Derived formulas for Donaldson invariants using wall-crossing

Explicit calculations for manifolds like = P^1 imes F_g

Potential applications in quantum cohomology

Abstract

We study Coulomb branch (``u-plane'') integrals for $\mathcal{N}=2$ supersymmetric $SU(2),SO(3)$ Yang-Mills theory on 4-manifolds $X$ of $b_1(X)>0, b_2^+(X)=1$. Using wall-crossing arguments we derive expressions for the Donaldson invariants for manifolds with $b_1(X)>0, b_2^+(X)>0$. Explicit expressions for $X=\IC P^1 \times F_g$, where $F_g$ is a Riemann surface of genus $g$ are obtained using Kronecker's double series identity. The result might be useful in future studies of quantum cohomology.