Quantum modularity of partial theta series with periodic coefficients

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 $F(q)$.


Introduction
In [30], Zagier introduced the notion of a quantum modular form of weight k ∈ 1  2 Z as a function g : Q → C for which the function r γ : Q \ {γ −1 (i∞)} → C given by extends to a real-analytic function on P 1 (R) \ S γ , where S γ is a finite set, for each γ = a b c d ∈ SL 2 (Z).Suitable modifications can be made to restrict the domain of r γ to appropriate subsets of Q and allow both multiplier systems and transformations on subgroups of SL 2 (Z).Since their inception, there has been substantial interest in studying these modular objects which emerge in diverse contexts: Maass forms [9], supersymmetric quantum field theory [12], topological invariants for plumbed 3-manifolds [5], [10], [11], combinatorics [13], [20], unified Witten-Reshetikhin-Turaev invariants [19] and L-functions [24], [26].For more examples, see Chapter 21 in [4].One of the most influential of the original five examples from [30] is the Kontsevich-Zagier "strange" series [29] F (q) := n≥0 (q) n (1.1) where (a 1 , a 2 , . . ., a j ) n = (a 1 , a 2 , . . ., a j ; q) n := is the standard q-hypergeometric notation, valid for n ∈ N 0 ∪ {∞}.F (q) is "strange" in the sense that it does not converge on any open subset of C, but is well-defined when q is a root of unity (where it is finite).Zagier proves that for α ∈ Q, φ(α) := e πiα 12 F (e 2πiα ) is a quantum modular form of weight 3/2 on Q with respect to SL 2 (Z).The key to proving this result is the "strange identity" F (q)" = " − 1 2 n≥1 n 12 n q n 2 −1 24 (1.2) where " = " means that the two sides agree to all orders at every root of unity (for further details, see Sections 2 and 5 in [29]) and 12 * is the quadratic character of conductor 12.The idea is to prove quantum modular properties for the right-hand side of (1.2) which are then inherited by F (q).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.Before stating our main result, we introduce some notation.
Let f : Z → C be an even or odd function with period M ≥ 2. For any fixed 1 ≤ k 0 < 2M , consider the set where q = e 2πiz , z ∈ H.
and let Γ M be defined as Γ 1 (2M ) if M is even and and let Here and throughout, We also employ the convention that f where τ ∈ C and χ is a multiplier.Finally, we write • • for the extended Jacobi symbol and let ε d = 1 or i according as d ≡ 1 or 3 (mod 4).Our main result is now as follows.
Theorem 1.1.Let f be a function with period M ≥ 2 and support S f (k 0 ).Let α ∈ Q.If f is even, then Θ f (α) is a quantum modular form of weight 3/2 on A M with respect to Γ M .If f is odd, then θ f (α) is a "strong" quantum modular form of weight 1/2 on Q with respect to Γ M and is a quantum modular form of weight 1/2 on B M with respect to Γ M .Remark 1.2.(i) The main novelty of Theorem 1.1 is that one does not require Θ f (z) or θ f (z) to be a cusp form.For example, consider θ ψ (z) where ψ is given in Section 4.2 (cf.[20]).Otherwise, one can invoke (2.4), (2.9) and Theorem 1.1 in [8]. (ii Here, r γ,f : R → C is a C ∞ function which is real-analytic in R \ {γ −1 (i∞)} and χ is a multiplier given by In Theorem 1.1, θ f (α) is a "strong" quantum modular form in the following sense (see [22] or [30]): (1) θ f and Θf "agree to infinite order" at all rational numbers (see Lemma 2.6), (2) for τ ∈ H − and γ ∈ Γ M , we have Here, r γ,f (τ ) is a holomorphic function in H − , extends as a C ∞ function to R and is real-analytic in R \ {γ −1 (i∞)}.Also, χ is the multiplier as in (1.6).A close inspection of the techniques in [22] reveals that one needs convergence of Θf (τ ) for τ ∈ H − (and not necessarily at rational points) to deduce the strong quantum modularity property for θ f (z).To ensure this condition, Θ f (z) does not have to be a cusp form.For a similar approach, see [3] and [17].
(iv) If f is a function with period M ≥ 2 and support S f (k 0 ), then θ f (z) is a sum of a modular form and a (strong) quantum modular form both of weight 1/2 and Θ f (z) is a sum of a modular form and a quantum modular form both of weight 3/2.To see this, write f as where Clearly, f e (n) (respectively, f o (n)) is an even (respectively, odd) function of period M with support contained in S f (k 0 ).Indeed, if S f,e (k 0 ) denotes (respectively, S f,o (k 0 )) the support of As pointed out by the referee, there is a "duality" in the proof of Theorem 1.1.If f is even, then the quantum modularity of Θ f (z) is driven by the modularity of θ f (z) and if f is odd, then the (strong) quantum modularity of θ f (z) is driven by the modularity of Θ f (z).See Lemmas 2.1 and 2.2.
The paper is organized as follows.In Section 2, we carefully study some important transformation and limiting properties of θ f (z) and Θ f (z).In Section 3, we prove Theorem 1.1.In Section 4, we give some examples, including the quantum modularity of the Kontsevich-Zagier series F t (q) associated to the family of torus knots T (3, 2 t ), t ≥ 2. This latter result generalizes the quantum modularity of F (q).

Preliminaries
We begin with transformation properties of the partial theta series θ f (z) and Θ f (z) in (1.3).Lemma 2.1.Let f be an even function with period M ≥ 2 and support S f (k 0 ).For all γ = a b c d ∈ Γ M , we have Proof.From (1.3), we have where where θ(z; k, M ) is the theta series .
By Proposition 2.1 in [28], we see that θ(z; k, M ) satisfies for all γ = a b c d ∈ Γ M .Also, since γ ∈ Γ M , we have for some integer j that where n has been replaced by n − 2jk in the second sum in (2.6).Noting that (2.1) now follows from (2.4) and (2.5)-(2.7).
Lemma 2.2.Let f be an odd function with period M ≥ 2 and support S f (k 0 ).For all γ = a b c d ∈ Γ M , we have Proof.If M is even, then f M 2 = 0 for odd f .So, we have where .
Lemma 2.3.Let f be an even function with period M ≥ 2 and support S f (k 0 ).Then where for Proof.We have where (2.13) follows from Jacobi's triple product identity with z → −q k and q → q M 2 in (2.14).As in the proof of Lemma 2.1, we have Lemma 2.4.Let f be an even function with period M ≥ 2 and support S f (k 0 ).Assume (2.16) Proof.For any 1 ≤ k < M , we obtain the following upon using [27, Chapter 5, pp.76] with u = (y − iα)M and x = k M : (2.17) As in the proof of Lemma 2.1, we have and so (2.16) follows from (2.17), (2.18) and Corollary 2.5.Let f be an even function with period M ≥ 2 and support S f (k 0 ).
Let C : Z → C be a periodic function with mean value zero and consider the L-series which has an analytic continuation to C [22, Proposition, page 98].
Lemma 2.6.Let f be an odd function with period M ≥ 2 and support S f (k 0 ).Then as t → 0 + , we have for (p, q) = 1 that where Thus, in the sense of Lawrence and Zagier [22, page 103], (2.20) and (2.21) imply that θ f and Θf "agree to infinite order" at all rational numbers.
Proof.For t > 0, we have ) is an odd function with period M q or 2M q according as 2 divides p or does not divide p.Also, C f (n, k) has mean value zero.This implies that C f,k 0 (n) is an odd function with period M q or 2M q according as 2 divides p or does not divide p and with mean value zero.Thus, by [22, Proposition, page 98] and (2.22), we obtain Next, we turn to Θf (τ ) where τ = x + iy with y < 0. First, we have for w ∈ H Thus, by the change of variable w → w + τ and contour integration, it follows that To evaluate the integral on the right-hand side of (2.23), we let w → iM w πn 2 .This yields where Γ(a, x) is the upper incomplete gamma function defined by Γ(a, x) := ∞ x w a−1 e −w dw.

Proof of Theorem 1.1
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.Let f be an even function with period M ≥ 2 and support S f (k 0 ).Define the Eichler integral of θ f (z) as follows: In view of Lemma 2.3 and Corollary 2.5, we see that θf (z) is well-defined on H ∪ A M .Thus, for z = x + iy ∈ H ∪ A M , it follows using contour integration that For α ∈ A M , we see from (3.1) and the fact that Γ(− It only remains to observe that r γ,f (z) is C ∞ and real-analytic in R \ {γ −1 (i∞)}.Let f be an odd function with period M ≥ 2 and support S f (k 0 ).For τ ∈ H − and γ ∈ Γ M , it follows from (1.7) and Lemma 2.2 that where Since by Lemma 2.6, θ f and Θf "agree to infinite order" at all rational numbers and Θf (τ ) satisfies the transformation property in (3.3) for all τ ∈ H − , it follows in the sense of Lawrence and Zagier [22, Page 103] that θ f (z) is a strong quantum modular form of weight 1/2 on Q with respect to Γ M .It is also clear that r γ,f (τ ) is a holomorphic function in H − , extends as a C ∞ function to R and is real-analytic in R \ {γ −1 (i∞)}.Here, χ is the multiplier given by (1.6).
4.2.Generating function for odd balanced unimodal sequences.Let v(n) denote the number of odd-balanced unimodal sequences of weight 2n + 2 and v(m, n) the number of such sequences having rank m.In [20], the authors study the bivariate generating function and prove that for α ∈ Q, q −7 V(−1, q −8 ) z→α is a quantum modular form of weight 3/2 on A = {α ∈ Q : α is Γ 0 (16)-equivalent to i∞} with respect to Γ 0 (16).A slight variant of this result is as follows.If we let q → q 2 in the identity (see [20, page 3693]) 1 For t = 1, one may define the sum over the j ℓ to be 1 in (4.2) and (4.3) to recover (4.1) and (1.1).
4.4.Rogers' false theta function.For M ∈ N and 1 ≤ j < M with j = M 2 , consider the false theta function of Rogers: where f (n) is the function defined by 1 or −1 according as n ≡ j or −j (mod M ) and 0 otherwise.Note that F M 2 ,M (z) = 0. Here, f is an odd function with period M .In this case, M(k 0 ) = {j} (respectively, M(k 0 ) = {M − j}) for 1 ≤ j < M 2 (respectively, M 2 < j < M ) with k 0 = j 2 (mod 2M ) (respectively, k 0 = (M − j) 2 (mod 2M )).So, S f (k 0 ) = {j, M − j}.Thus, for α ∈ Q, Theorem 1.1 implies that F j,M (α) is a strong quantum modular form of weight 1/2 on Q with respect to Γ M (given by (1.4)).This result (with z replaced by z M and M even) was discussed in [6, Theorem 4.1] (see [7] for a vector-valued version).More generally, for 1 ≤ k 0 < 2M , if where h(n) is an odd function with period M and support S h (k 0 ), then Theorem 1.1 shows that F M (z) is a strong quantum modular form of weight 1/2 on Q with respect to Γ M .Finally, F j,M (α) and, more generally, F M (α) are quantum modular forms of weight 1/2 on B M with respect to Γ M .