Vafa-Witten theory and iterated integrals of modular forms

Understand

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.