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Quantum modularity of partial theta series with periodic coefficients

Ankush Goswami, Robert Osburn

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 $F(q)$.

Quantum modularity of partial theta series with periodic coefficients

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 which matches (at a root of unity) the colored Jones polynomial for the family of torus knots , , is a weight quantum modular form. This generalizes Zagier's result on the quantum modularity for the "strange" series .

Paper Structure

This paper contains 8 sections, 8 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['main']}
  4. Examples
  5. Kontsevich-Zagier series $\mathscr{F}_t(q)$ for torus knots $T(3,2^t)$
  6. Generating function for odd balanced unimodal sequences
  7. Kontsevich-Zagier series for torus knots $T(2,2m+1)$
  8. Rogers' false theta function

Key Result

Theorem 1.1

Let $f$ be a function with period $M \geq 2$ and support $S_f(k_0)$. Let $\alpha\in\mathbb{Q}$. If $f$ is even, then $\Theta_f(\alpha)$ is a quantum modular form of weight $3/2$ on $A_M$ with respect to $\Gamma_M$. If $f$ is odd, then $\theta_f(\alpha)$ is a "strong" quantum modular form of weight $

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 7 more