Advances in Theoretical and Mathematical Physics

Volume 25 (2021)

Number 2

Rank $N$ Vafa–Witten invariants, modularity and blow-up

Pages: 275 – 308



Sergei Alexandrov (Laboratoire Charles Coulomb (L2C), Université de Montpellier, France)


We derive explicit expressions for the generating functions of refined Vafa–Witten invariants $\Omega(\gamma,y)$ of $\mathbb{P}^2$ of arbitrary rank $N$ and for their non-holomorphic modular completions. In the course of derivation we also provide: i) a generalization of the recently found generating functions of $\Omega(\gamma,y)$ and their completions for Hirzebruch and del Pezzo surfaces in the canonical chamber of the moduli space to a generic chamber; ii) a version of the blow-up formula expressed directly in terms of these generating functions and its reformulation in a manifestly modular form.

Published 17 February 2022