Higher Depth False Modular Forms

TL;DR

This paper develops a theory of higher depth false modular forms by constructing modular completions for false theta functions, extending the understanding of their modular properties across various ranks and applications.

Contribution

It introduces a new framework for higher depth false modular forms, generalizing modular completions for false theta functions and applying them to algebraic and topological examples.

Findings

Established modular properties for false theta functions at all ranks.

Constructed explicit examples up to depth three from vertex algebra modules.

Connected the theory to invariants of 3-manifolds and vertex algebra characters.

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 $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invariants of $3$-manifolds associated with gauge group $\mathrm{SU}(3)$.