Abstract
We consider partition functions with insertions of surface operators of topologically twisted mathcal{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 overline{mathcal{Q}} -exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate ā 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.
Highlights
In to a continuous integral Zu over the Coulomb branch of the theory, where the gauge group is spontaneously broken to U(1) by a non-vanishing expectation value of the order parameter u
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
A famous aspect of Donaldson-Witten theory is that the correlation function (1.2) is a generating function of Donaldson polynomials defined in mathematics [9, 10], where they play an important role in the classification of smooth four-manifolds
Summary
We begin our discussion with a brief reminder of the solution of Seiberg and Witten of the SU(2) N = 2 super Yang-Mills gauge theory on R4 [34, 35] (see [36,37,38,39] for a review, [40, 41] for a more modern perspective, and [42, 44] for a more mathematically inclined discussion)
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