Donaldson-Witten theory and indefinite theta functions

TL;DR

This paper demonstrates how adding a specific surface operator to Donaldson-Witten theory allows the partition function to be expressed as a total derivative involving indefinite theta functions, linking to G"ottsche's formulas for rational surfaces.

Contribution

It introduces a method to rewrite the Donaldson-Witten partition function with surface operators as a total derivative using indefinite theta functions, generalizing previous results.

Findings

Partition functions can be expressed as total derivatives with indefinite theta functions.

Reproduces G"ottsche's formulas for Donaldson invariants of rational surfaces.

Connects wall-crossing behavior to 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.