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 G\"ottsche's expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.