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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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


Authors: Kathrin Bringmann and Caner Nazaroglu
Journal: Trans. Amer. Math. Soc. 372 (2019), 6135-6159
MSC (2010): Primary 11F37, 11P82, 14D21, 14F05, 14J60
DOI: https://doi.org/10.1090/tran/7714
Published electronically: August 1, 2019
MathSciNet review: 4024516
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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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Additional Information

Kathrin Bringmann
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: kbringma@math.uni-koeln.de

Caner Nazaroglu
Affiliation: Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
Email: cnazarog@math.uni-koeln.de

DOI: https://doi.org/10.1090/tran/7714
Received by editor(s): April 4, 2018
Received by editor(s) in revised form: September 3, 2018
Published electronically: August 1, 2019
Additional Notes: The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation, and the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement no. 335220 - AQSER
The research of the second author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement no. 335220 - AQSER
Article copyright: © Copyright 2019 by the authors