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It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition.
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S. P. Zwegers, Mock Theta Functions , Ph.D. thesis, Universiteit Utrecht (2002)
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D. Zagier, Ramanujan’s mock theta functions and their applications [d’après Zwegers and Bringmann–Ono] , Séminaire Bourbaki 986
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