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

TL;DR

This paper derives explicit formulas for the generating functions of refined Vafa-Witten invariants of projective plane for any rank, explores their modular properties, and generalizes blow-up formulas to a broader context.

Contribution

It provides explicit expressions for the generating functions of Vafa-Witten invariants for arbitrary rank and extends blow-up formulas and modularity results to general chambers.

Findings

Explicit formulas for generating functions of invariants.

Generalized blow-up formula in modular form.

Extended results to generic chambers of the moduli space.

Abstract

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.