An exact formula for 𝐔(3) Vafa-Witten invariants on ℙ²

Bringmann, Kathrin; Nazaroglu, Caner

doi:10.1090/tran/7714

An exact formula for $\mathbf {U (3)}$ Vafa-Witten invariants on $\mathbb {P}^2$
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by Kathrin Bringmann and Caner Nazaroglu PDF

Trans. Amer. Math. Soc. 372 (2019), 6135-6159

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.

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