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

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 sl₃, appearing in the character of the vertex algebra W⁰ (p)_A2, can be written as the sum of two depth two quantum modular forms of positive integral weight.