Vafa-Witten theory and iterated integrals of modular forms

TL;DR

This paper explores the modular anomalies in Vafa-Witten theory for gauge groups SU(2) and SU(3), revealing that the SU(3) anomaly involves iterated integrals of modular forms and relates to mock modular forms of depth two.

Contribution

It demonstrates that the SU(3) modular anomaly involves iterated integrals of modular forms and connects to mock modular forms of depth two, extending previous SU(2) results.

Findings

SU(3) modular anomaly involves iterated integrals of modular forms

SU(3) anomaly can be expressed as a holomorphic anomaly

Partition function for SU(3) is a mock modular form of depth two

Abstract

Vafa-Witten (VW) theory is a topologically twisted version of N=4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P2. This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU(3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU(3) partition function is mathematically an example of a mock modular form of depth two.