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

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.

23 pages; v2: 25 pages, the discussion of asymptotic expansion extended, to appear in Transactions of the AMS

References in corpus (2)

• Vafa-Witten theory and iterated integrals of modular forms
• Indefinite Theta Series on Cones