Vector-valued higher depth quantum modular forms and higher Mordell integrals

Abstract

We introduce vector-valued higher depth quantum modular forms and investigate examples coming from characters of representations of vertex algebras. These are expressed as rank two false theta series, generalizing unary false theta series studied from several standpoints. We also discuss certain double integrals, which may be viewed as the obstruction to modularity of the depth two quantum modular forms. We then find explicit formulas for the double error integrals in the form reminiscent of the classical Mordell integral.