3-manifold topology and the Donaldson-Witten partition function

Abstract

We consider Donaldson-Witten theory on four-manifolds of the form $X=Y\times {\mathbf{S}}^1$ 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.

35 pages, harvmac b-mode, 3 figures, minor result added

References in corpus (3)

• Brane Dynamics and Gauge Theory
• Donaldson invariants for nonsimply connected manifolds
• On the Generalized Casson Invariant

Cited by in corpus (8)

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• Duality in Topological Quantum Field Theories
• From Exact Results to Gauge Dynamics on ${\mathbb{R}}^3\times S^1$
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• The partition function of a 3-dimensional topological scalar-vector model
• On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants