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

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.