An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on $\mathbb{P}^2$

TL;DR

This paper derives an exact formula for the U(3) Vafa-Witten invariants on the complex projective plane, revealing their mock modular form structure and employing the Circle Method for the first time on higher-depth mock modular forms.

Contribution

It provides the first Rademacher expansion for a higher-depth mock modular form arising from Vafa-Witten invariants on .

Findings

Derived an exact formula for  Vafa-Witten invariants with gauge group U(3)

Applied the Circle Method to a higher-depth mock modular form

Established the modular properties and holomorphic anomaly of the partition function

Abstract

Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth.