Mocking the u-plane integral

TL;DR

This paper investigates the u-plane integral in N=2 gauge theories on four-manifolds, demonstrating its computation via mock modular forms and Appell-Lerch sums, and exploring the asymptotic behavior of correlators.

Contribution

It introduces a method to compute u-plane correlators using mock modular forms and Appell-Lerch sums, and analyzes their asymptotic properties as functions of observables.

Findings

u-plane correlators are determined by mock modular forms and Appell-Lerch sums

the vev of exponentiated observables is an entire function of fugacities

the u-plane integral matches Donaldson invariants in certain cases

Abstract

The u-plane integral is the contribution of the Coulomb branch to correlation functions of N=2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU(2), for an arbitrary four-manifold with (b1,b2+)=(0,1). The u-plane contribution equals the full correlator in the absence of Seiberg-Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell-Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities.