Abstract
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 ℱ t ( q ) {\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(3,2^{t})} , t ≥ 2 {t\geq 2} , is a weight 3 2 {\frac{3}{2}} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ( q ) {F(q)} .
Highlights
In [30], Zagier introduced the notion of a quantum modular form of weight k ∈ 1 Z as a function g : Q → C for which the function rγ : Q \ {γ−1(i∞)} → C given by g(α) −−k g( acαα ++ db ) =: rγ(α) extends to a real-analytic function on P1(R) \ Sγ, where Sγ is a finite set, for each γ = ∈ SL2(Z)
We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients
We show that the Kontsevich–Zagier series Ft(q) which matches the colored Jones polynomial for the family of torus knots T(3, 2t), t ≥ 2, is a weight 3 quantum modular form
Summary
The purpose of this paper is to place the right-hand side of (1.2) and other examples in the literature into the general context of quantum modularity of partial theta series with even or odd periodic coefficients. In Theorem 1.1, θf (α) is a “strong” quantum modular form in the following sense (see [22] or [30]): (1) θf and Θf “agree to infinite order” at all rational numbers (see Lemma 2.6);. This latter result generalizes the quantum modularity of F(q)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
