A "Strange" Vector-Valued Quantum Modular Form

TL;DR

This paper constructs a new vector-valued quantum modular form inspired by Zagier's work, revealing complex 'strange' behaviors and connections to various topics in number theory and modular forms.

Contribution

It introduces a novel vector-valued quantum modular form with components exhibiting 'strange' properties, expanding the understanding of quantum modularity.

Findings

Constructed a natural vector-valued quantum modular form.

Demonstrated 'strange' behavior in the components of the form.

Connected the form to existing topics like mock theta functions and asymptotics.

Abstract

Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of $\Q$, and whose obstruction to modularity is some analytically "nice" function. Motivated by Zagier's example of the quantum modularity of Kontsevich's "strange" function $F(q)$, we revisit work of Andrews, Jimenez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly "strange".