Mocking the u-plane integral

Georgios Korpas,Jan Manschot,Iurii Nidaiev,Gregory W. Moore

doi:10.1007/s40687-021-00280-5

Georgios Korpas, Jan Manschot + Show 2 more

https://doi.org/10.1007/s40687-021-00280-5

Copy DOI

Abstract

The u-plane integral is the contribution of the Coulomb branch to correlation functions of {mathcal {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 mathrm{SU}(2), for an arbitrary four-manifold with (b_1,b_2^+)=(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.

Highlights

A powerful approach to understand the dynamics of supersymmetric field theories is to consider such theories on a compact four-manifold without boundary [1,2,3,4,5,6,7]
We consider in this paper the topologically twisted counterpart of N = 2 supersymmetric Yang–Mills theory with gauge group SU(2) and in the presence of arbitrary ’t Hooft flux [8]
The gauge group is broken to U(1) on the Coulomb branch B, which is parametrized by the vacuum expectation value u

Summary

Introduction

A powerful approach to understand the dynamics of supersymmetric field theories is to consider such theories on a compact four-manifold without boundary [1,2,3,4,5,6,7]. We consider in this paper the topologically twisted counterpart of N = 2 supersymmetric Yang–Mills theory with gauge group SU(2) and in the presence of arbitrary ’t Hooft flux [8]. The gauge group is broken to U(1) on the Coulomb branch B, which is parametrized by the vacuum expectation value u =. The Coulomb branch, known as the “u-plane,”. The vev O can be expressed as a sum of two contributions: the Seiberg–Witten contribution O SW from the strong coupling singularities u = ± 2, and the contribution from the u-plane [O],

Objectives

Results

Conclusion