Communications in Number Theory and Physics

Volume 15 (2021)

Number 1

Vafa–Witten invariants from modular anomaly

Pages: 149 – 219



Sergei Alexandrov (Laboratoire Charles Coulomb (L2C), Université de Montpellier, France; and Department of High Energy and Elementary Particle Physics, Saint Petersburg State University, St. Petersburg, Russia)


Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N = 2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa–Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the holomorphic functions themselves, thereby providing the Betti numbers of instanton moduli spaces on such surfaces. As a result, we derive a closed formula for the generating functions and their completions for all $N$. Besides, our construction reveals in a simple way instances of fiber-base duality, which can be used to derive new non-trivial identities for generalized Appell functions. It also suggests the existence of new invariants, whose meaning however remains obscure.

2010 Mathematics Subject Classification

Primary 11F37, 14N35. Secondary 14D21.

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Received 17 June 2020

Accepted 18 September 2020

Published 4 January 2021