Quantum modularity of partial theta series with periodic coefficients

TL;DR

This paper proves the quantum modularity of certain partial theta series with periodic coefficients and demonstrates that specific series related to torus knots are quantum modular forms, extending previous results by Zagier.

Contribution

It explicitly establishes the quantum modularity of partial theta series with periodic coefficients and links these to knot invariants as quantum modular forms.

Findings

Partial theta series with periodic coefficients are quantum modular.

The Kontsevich-Zagier series for torus knots is a weight 3/2 quantum modular form.

Generalizes Zagier's quantum modularity result for the series F(q).

Abstract

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity) the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t \geq 2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series $F(q)$.