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arXiv · 2109.00394

Higher Depth False Modular Forms

Kathrin Bringmann·Jonas Kaszian·Antun Milas·Caner Nazaroglu
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Abstract

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invariants of $3$-manifolds associated with gauge group $\mathrm{SU}(3)$.

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Kathrin Bringmann, Jonas Kaszian, Antun Milas, Caner Nazaroglu. 2022-06-28. Higher Depth False Modular Forms. https://arxiv.org/abs/2109.00394

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