Higher depth quantum modular forms, multiple Eichler integrals, and sl3 false 
theta functions

Bringmann, Kathrin; Kaszian, Jonas; Milas, Antun

doi:10.1007/s40687-019-0182-4

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Higher depth quantum modular forms, multiple Eichler integrals, and sl₃ false theta functions

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Milas, Antun
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s40687-019-0182-4
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Citation

Bringmann, K., Kaszian, J., & Milas, A. (2019). Higher depth quantum modular forms, multiple Eichler integrals, and sl₃ false theta functions. Research in the Mathematical Sciences, 6(2): 20. doi:10.1007/s40687-019-0182-4.

Cite as: https://hdl.handle.net/21.11116/0000-0004-7573-7

Abstract

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose “companions” in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of “double error” functions as coefficients. In particular, we prove that the false theta functions of $\mathfrak {sl}_3$, appearing in the character of the vertex algebra $W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.