3-manifold topology and the Donaldson-Witten partition function

TL;DR

This paper explores the relationship between four-dimensional Donaldson invariants on manifolds of the form Y×S^1 and topological invariants of the three-manifold Y, revealing new connections and subtleties in gauge theory compactification.

Contribution

It demonstrates how Donaldson-Witten invariants relate to three-manifold invariants like Casson-Walker-Lescop and Reidemeister torsion, extending understanding of gauge theories on product manifolds.

Findings

Partition function reduces to Casson-Walker-Lescop invariant for b_1(Y)>1

Reinterpretation of Meng-Taubes relation between Seiberg-Witten invariants and Reidemeister torsion

Identification of subtleties in Kaluza-Klein reduction for b_1(Y)=1

Abstract

We consider Donaldson-Witten theory on four-manifolds of the form $X=Y \times {\bf S}^1$ where $Y$ is a compact three-manifold. We show that there are interesting relations between the four-dimensional Donaldson invariants of $X$ and certain topological invariants of $Y$. In particular, we reinterpret a result of Meng-Taubes relating the Seiberg-Witten invariants to Reidemeister-Milnor torsion. If $b_1(Y)>1$ we show that the partition function reduces to the Casson-Walker-Lescop invariant of $Y$, as expected on formal grounds. In the case $b_1(Y)=1$ there is a correction. Consequently, in the case $b_1(Y)=1$, we observe an interesting subtlety in the standard expectations of Kaluza-Klein theory when applied to supersymmetric gauge theory compactified on a circle of small radius.