Higher depth false modular forms

Abstract

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra [Formula: see text], [Formula: see text], and from [Formula: see text]-invariants of three-manifolds associated with gauge group [Formula: see text].