Donaldson-Witten theory and indefinite theta functions

Abstract

We consider partition functions with insertions of surface operators of topologically twisted N=2, SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter $a$, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of a Q-exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate $\bar{a}$ using Zwegers' indefinite theta functions. In this way, we reproduce Göttsche's expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.

arXiv comment23 pages + appendices, comments welcome. v2: published version

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• Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD
• Gauge theories on compact toric manifolds
• Partition Functions and Fibering Operators on the Coulomb Branch of 5d SCFTs
• The $u$-plane integral, mock modularity and enumerative geometry
• Squashed Toric Manifolds and Higher Depth Mock Modular Forms
• Refinement and modularity of immortal dyons
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