Vafa-Witten theory and iterated integrals of modular forms

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.

44 pages, comments welcome; v2: minor changes

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• Scaling Black Holes and Modularity
• Donaldson-Witten theory, surface operators and mock modular forms
• Temperature-reflection I: field theory, ensembles, and interactions
• Three Avatars of Mock Modularity
• Mock modularity at work, or black holes in a forest
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on ${\mathbb{P}}^2$