Higher depth quantum modular forms, multiple Eichler integrals, and $\frak{sl}_3$ false theta functions

TL;DR

This paper introduces higher depth quantum modular forms derived from rank two false theta functions, connecting them with double Eichler integrals and non-holomorphic theta series, and analyzing their relation to $rak{sl}_3$ false theta functions.

Contribution

It constructs new examples of higher depth quantum modular forms from rank two false theta functions and relates them to double Eichler integrals and non-holomorphic theta series.

Findings

False theta of $rak{sl}_3$ expressed as sum of two depth two quantum modular forms.

Constructed families of examples from rank two false theta functions.

Established connections with double Eichler integrals and non-holomorphic theta series.

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 of $\frak{sl}_3$, 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.