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Higher depth quantum modular forms, multiple Eichler integrals, and \(\mathfrak {sl}_3\) false theta functions

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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.

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Correspondence to Kathrin Bringmann.

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Bringmann, K., Kaszian, J. & Milas, A. Higher depth quantum modular forms, multiple Eichler integrals, and \(\mathfrak {sl}_3\) false theta functions. Res Math Sci 6, 20 (2019). https://doi.org/10.1007/s40687-019-0182-4

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