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Vafa–Witten Theory and Iterated Integrals of Modular Forms

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Abstract

Vafa–Witten (VW) theory is a topologically twisted version of \({\mathcal{N}=4}\) 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 \({\mathbb{P}^2}\). 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.

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Acknowledgements

I would like to thank Sergey Alexandrov, Sibasish Banerjee, and Boris Pioline for collaborating on related subjects and in particular on Ref. [34]. I moreover wish to thank Sergey Alexandrov, Kathrin Bringmann, Greg Moore, Boris Pioline, Cumrun Vafa, and Sander Zwegers for discussions and correspondence.

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  1. School of Mathematics, Trinity College, Dublin 2, Ireland

    Jan Manschot

  2. Hamilton Mathematical Institute, Trinity College, Dublin 2, Ireland

    Jan Manschot

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Manschot, J. Vafa–Witten Theory and Iterated Integrals of Modular Forms. Commun. Math. Phys. 371, 787–831 (2019). https://doi.org/10.1007/s00220-019-03389-5

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