Higher depth quantum modular forms and plumbed $3$-manifolds

TL;DR

This paper explores new quantum invariants of plumbed 3-manifolds, showing that certain series are depth two quantum modular forms, advancing understanding of their structure and relation to WRT invariants.

Contribution

It proves that the series (q) for specific 3-manifolds are depth two quantum modular forms, providing new insights into their mathematical properties.

Findings

(q) is a depth two quantum modular form on  for positive definite unimodular plumbing matrices.

The study confirms conjectural links between -series and WRT invariants of plumbed 3-manifolds.

Results extend the understanding of quantum modularity in the context of 3-manifold invariants.

Abstract

In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$ attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable $q$-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate the series $\widehat{Z}_{0}(q)$ for unimodular plumbing ${\tt H}$-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, $\widehat{Z}_{0}(q)$ is a depth two quantum modular form on $\mathbb{Q}$.

Equations

$$f(\tau )-(c\tau +d{)}^{-k}f(\gamma \tau )\in {\mathcal{Q}}^1\mathcal{O}(R),\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z}),$$

$$\sum _{\mathbf{n}\in {\mathbb{N}}_0^2+\mathbf{\beta }}{q}^{a{n}_1^2+b{n}_2^2+c{n}_1{n}_2},$$

$${\hat{Z}}_{\mathbf{a}}(q):=\frac{q^{\frac{-3N+\mathrm{tr}(M)}{4}}}{(2\pi i{)}^N}\text{PV}{\int }_{|w_j|=1}\prod _{j=1}^{N}g(w_j)\prod _{(k,\ell )\in E}f(w_k,w_{\ell }){\Theta }_{-M,\mathbf{a}}(q;\mathbf{w})\frac{d{w}_j}{w_j},$$

$${\Theta }_{-M,\mathbf{a}}(q;\mathbf{w}):=\sum _{\mathbf{\ell }\in 2M{\mathbb{Z}}^N+\mathbf{a}}{q}^{\frac{1}{4}{\mathbf{\ell }}^T{M}^{-1}\mathbf{\ell }}{\mathbf{w}}^{\mathbf{\ell }},\mathbf{a}\in 2\mathrm{coker}(M)+\mathbf{\delta },$$

$$\sum _{n\in \mathbb{Z}}\mathrm{sgn}(n)q^{a{n}^2+bn},$$

$$M=\left(\begin{array}{cccccc}{b}_1& 0& -1& 0& 0& 0\\ 0& b_2& -1& 0& 0& 0\\ -1& -1& b_3& -1& 0& 0\\ 0& 0& -1& b_4& -1& -1\\ 0& 0& 0& -1& b_5& 0\\ 0& 0& 0& -1& 0& b_6\end{array}\right).$$

$$\sum _{\mathbf{n}\in {\mathbb{Z}}^2}{\mathrm{sgn}}^{\ast }(n_1){\mathrm{sgn}}^{\ast }(n_2)q^{a(n_1+{\beta }_1{)}^2+b(n_2+{\beta }_2{)}^2+c(n_1+{\beta }_1)(n_2+{\beta }_2)},$$

$$F_{\mathcal{S},Q,\epsilon }(\tau ):=\sum _{\mathbf{\alpha }\in \mathcal{S}}\epsilon (\mathbf{\alpha })\sum _{\mathbf{n}\in {\mathbb{N}}_0^2}{q}^{KQ(\mathbf{n}+\mathbf{\alpha })},$$

$$E_2(\kappa ;\mathbf{x}):={\int }_{{\mathbb{R}}^2}\mathrm{sgn}\left(w_1\right)\mathrm{sgn}\left(w_2+\kappa w_1\right)e^{-\pi \left({\left(w_1-x_1\right)}^2+{\left(w_2-x_2\right)}^2\right)}d{w}_{1}d{w}_2.$$

$$M_2(\kappa ;\mathbf{x}):=-\frac{1}{{\pi }^2}{\int }_{{\mathbb{R}}^2-i\mathbf{x}}\frac{e^{-\pi w_1^2-\pi w_2^2-2\pi i(x_1{w}_1+x_2{w}_2)}}{w_2(w_1-\kappa w_2)}d{w}_{1}d{w}_2.$$

$$\begin{gathered} M_2(\kappa ;x_1+\kappa x_2,x_2)=E_2(\kappa ;x_1+\kappa x_2,x_2)+\mathrm{sgn}(x_1)\mathrm{sgn}(x_2) \\ -\mathrm{sgn}(x_2)E(x_1+\kappa x_2)-\mathrm{sgn}(x_1)E\left(\frac{\kappa }{\sqrt{1+{\kappa }^2}}{x}_1+\sqrt{1+{\kappa }^2}{x}_2\right), \end{gathered}$$

$$M_2(\kappa ;x_1,x_2)=-\frac{x_1}{2\sqrt{v}}\frac{x_2}{\sqrt{v}}{q}^{\frac{x_1^2}{4v}+\frac{x_2^2}{4v}}{\int }_{-\overline{\tau }}^{i\infty }\frac{e^{\frac{\pi i{x}_1^2{w}_1}{2v}}}{\sqrt{-i(w_1+\tau )}}{\int }_{w_1}^{i\infty }\frac{e^{\frac{\pi i{x}_2^2{w}_2}{2v}}}{\sqrt{-i(w_2+\tau )}}d{w}_{2}d{w}_1$$

$$-\frac{x_2+\kappa x_1}{2\sqrt{(1+{\kappa }^2)v}}\frac{x_1-\kappa x_2}{\sqrt{(1+{\kappa }^2)v}}{q}^{\frac{(x_2+\kappa x_1{)}^2}{4\left(1+{\kappa }^2\right)v}+\frac{(x_1-\kappa x_2{)}^2}{4\left(1+{\kappa }^2\right)v}}{\int }_{-\overline{\tau }}^{i\infty }\frac{e^{\frac{\pi i(x_2+\kappa x_1{)}^2{w}_1}{2\left(1+{\kappa }^2\right)v}}}{\sqrt{-i(w_1+\tau )}}{\int }_{w_1}^{i\infty }\frac{e^{\frac{\pi i(x_1-\kappa x_2{)}^2{w}_2}{2\left(1+{\kappa }^2\right)v}}}{\sqrt{-i(w_2+\tau )}}d{w}_{2}d{w}_1.$$

$$B_m(1-x)=(-1{)}^m{B}_m(x).$$

$$\sum _{\mathbf{n}\in {\mathbb{N}}_0^2}F((\mathbf{n}+\mathbf{\alpha })t)$$

$$\sim \frac{{\mathcal{I}}_F}{t^2}-\sum _{n_2\ge 0}\frac{B_{n_2+1}({\alpha }_2)}{(n_2+1)!}{\int }_0^{\infty }{F}^{(0,n_2)}(x_1,0)d{x}_1{t}^{n_2-1}-\sum _{n_1\ge 0}\frac{B_{n_1+1}({\alpha }_1)}{(n_1+1)!}{\int }_0^{\infty }{F}^{(n_1,0)}(0,x_2)d{x}_2{t}^{n_1-1}$$

$$+\sum _{n_1,n_2\ge 0}\frac{B_{n_1+1}({\alpha }_1)}{(n_1+1)!}\frac{B_{n_2+1}({\alpha }_2)}{(n_2+1)!}{F}^{(n_1,n_2)}(0,0)t^{n_1+n_2},$$

$$G_c(a,b):=\sum _{n\left(\mathrm{mod}c\right)}{e}^{\frac{2\pi i}{c}\left(a{n}^2+bn\right)};$$

$${\vartheta }_{\nu }(A,h,N;\tau ):=\sum _{\begin{array}{c}m\in \mathbb{Z}\\ m\equiv h\left(\mathrm{mod}N\right)\end{array}}{m}^{\nu }{q}^{\frac{A{m}^2}{2N^2}}.$$

$$f\big{|}_{k}\gamma(\tau):=\left(\tfrac{c}{d}\right)^{2k}\varepsilon_{d}^{2k}(c\tau+d)^{-k}f(\gamma\tau),\qquad\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in{\rm SL}_{2}(\mathbb{Z}).$$

$${\vartheta }_{\nu }(A,h,N;\tau ){\mid }_{\frac{3}{2}}\gamma =e\left(\frac{abA{h}^2}{2N^2}\right)\left(\frac{-2A}{d}\right){\vartheta }_{\nu }(A,ah,N;\tau ).$$

$${\int }_{|w|=1}\left(w-w^{-1}\right)w^m\frac{dw}{w}={\int }_{|w|=1}{w}^{m}dw-{\int }_{|w|=1}{w}^{m-2}dw=2\pi i\left({\delta }_{m,-1}-{\delta }_{m,1}\right),$$

$$\frac{1}{2\pi i}\text{PV}{\int }_{|w|=1}\frac{w^m}{w-w^{-1}}\frac{dw}{w}=\frac{1}{2}{\mathrm{sgn}}_o(m),$$

$$f(\tau)-f\big{|}_{k}\gamma(\tau)$$

$$f-f\big{|}_{k}\gamma\in\bigoplus_{j}\mathcal{Q}_{\kappa_{j}}^{N_{j}}(\Gamma)\mathcal{O}(R),$$

$$I_f(\tau )$$

$$I_{f_1,f_2}(\tau )$$

$$r_{f,\varrho }(\tau )$$

$$r_{f_1,f_2,\varrho }(\tau )$$

$$I_{f_1,f_2}(\tau )-I_{f_1,f_2}{\mid }_{k_1+k_2-4}\gamma (\tau )=r_{f_1,f_2,\tau ,\frac{d}{c}}(\tau )+I_{f_1}(\tau )r_{f_2,\frac{d}{c}}(\tau ).$$

$$H_1(M_3(G),\mathbb{Z})\cong \mathrm{coker}(M)={\mathbb{Z}}^N/M{\mathbb{Z}}^N.$$

$$f(w_{j},w_{k}):=\frac{1}{\big{(}w_{j}-w_{j}^{-1}\big{)}\big{(}w_{k}-w_{k}^{-1}\big{)}}$$

$$g(w_{j}):=\big{(}w_{j}-w_{j}^{-1}\big{)}^{2}.$$

Full text

The research of the first author is supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results receives funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. The third author was supported by NSF-DMS grant 1601070 and a stipend from the Max Planck Institute for Mathematics, Bonn.

1. Introduction and statement of results

A quantum modular form is a complex-valued function defined on $\mathbb{Q}$ or subset thereof, called the quantum set, that exhibits modular-like transformation properties up to an obstruction term with “nice” analytic properties (for instance, it can be extended to a real-analytic function on some open subset of $\mathbb{R}$). Quantum modular forms were introduced by Zagier in [21], where he described several non-trivial examples. They have appeared in several areas including quantum invariants of knots and 3-manifolds [14, 15, 16, 17], mock modular forms [22], meromorphic Jacobi forms [7], mathematical physics [12], partial and false theta functions [8], and representation theory [8, 11].

Motivated on the one hand by the concept of higher depth mock modular forms and on the other hand by the appearance of higher rank false theta functions in representation theory, Kaszian and two of the authors [4] defined so-called higher depth quantum modular forms, and gave an infinite family of examples coming from characters of representations of vertex algebras. If the depth is two, these functions satisfy

[TABLE]

where $\mathcal{Q}^{1}$ is the space of quantum modular forms and $\mathcal{O}(R)$ is the space of real-analytic functions on $R$. All known examples of depth two quantum modular come from rank two partial theta functions ($q:=e^{2\pi i\tau},\tau\in\mathbb{H}$)

[TABLE]

where ${\bm{\beta}}\in\mathbb{Q}^{2}$ (throughout we write vectors in bold letters and their components with subscripts) and $a,4ab-c^{2}>0$. Further examples of this kind were studied in [3, 18]. Depth two quantum modular forms also appear as the coefficients of meromorphic Jacobi forms of negative matrix index [5].

In [13], as a part of the construction of homological invariants for closed 3-manifolds, Gukov, Pei, Putrov, and Vafa proposed a new approach to WRT invariants for a large class of $3$-manifolds. For any plumbed $3$-manifold, homeomorphically represented by a plumbing graph and positive definite linking matrix $M$ 111In [13], $M$ is negative definite, which we account for by replacing it with $-M$ when referring to their work., they [19] defined a certain family of $q$-series (called homological blocks)

[TABLE]

where PV denotes the Cauchy principle value, where throughout integrals are oriented counterclockwise and $\int_{|w_{j}|=1}$ indicates the integration $\int_{|w_{1}|=1}\ldots\int_{|w_{N}|=1}$. Moreover $g(w_{j})$ and $f(w_{k},w_{\ell})$ are certain simple rational functions defined in (2.7) and (2.8), respectively and

[TABLE]

where ${\bm{\delta}}:=(\delta_{j})$ such that $\delta_{j}\equiv{\rm deg}(v_{j})\ \,\left(\mathrm{mod}\,2\right)$ with $\delta_{j}$ denoting the degree (or valency) of $j$-th node. Conjecturally, a suitable (explicit) linear combination of $\widehat{Z}_{{\bm{a}}}(q)$, denoted by $\widehat{Z}(q)$ in [13], is the universal WRT invariant, that is, as $q\to e^{\frac{2\pi i}{k}}$ its limit coincides with the SU$(2)$ WRT invariant of $M$ at level $k$. This, in particular, leads to another conjecture (attributed in [6] to Gukov) that $\widehat{Z}_{{\bm{a}}}(q)$ and $\widehat{Z}(q)$ are quantum modular forms. This conjecture can be verified for specific $3$-manifolds obtained from unimodular 3-star plumbing graphs (e.g. the $E_{8}$ graph) [6, 9] due to the fact that $\widehat{Z}_{{\bm{a}}}(q)$ can be expressed via one-dimensional unary false theta functions

[TABLE]

whose quantum modularity properties are well-understood [8, 15, 16, 17, 22].

In this paper we investigate $\widehat{Z}_{{\bm{a}}}(q)$ for a family of non-Seifert plumbed $3$-manifolds. We consider the simplest plumbing graph of this kind obtained by splicing two $3$-star graphs. This way we obtain the so-called ${\tt H}$-graph with six vertices (Figure 1), with the linking matrix

[TABLE]

We only consider positive definite unimodular matrices whose $3$-manifolds are integral homology spheres (i.e., $H_{1}(M_{3},\mathbb{Z})=0$ as explained further in Section 2.7 below). Due to the invariance of $\widehat{Z}_{{{\bm{\delta}}}}(q)$ under a Kirby move [13], we may assume that $b_{j}\geq 2$, $j\in\{1,2,5,6\}$ (graphs with $b_{j}=1$, $j\in\{1,2,5,6\}$ reduce to $3$-star graphs whose quantum modularity is well-understood [6, 9]). With these assumptions $\widehat{Z}_{{\bm{\delta}}}(q)$ (also denoted by $\widehat{Z}_{0}(q)$ in [13]) is the only homological block and therefore it conjecturally gives WRT invariants at roots of unity. An important feature of this family of graphs is that $\widehat{Z}_{{\bm{\delta}}}(q)$ can be expressed via rank two false theta functions (${\bm{\beta}}\in\mathbb{Q}^{2}$, $a,b,c\in\mathbb{N}$)

[TABLE]

where ${\rm sgn}^{*}(x):={\rm sgn}(x)$ for $x\in\mathbb{R}\setminus\{0\}$ and ${\rm sgn}^{*}(0):=1$. Our first result is on quantum modularity of certain partial theta functions needed to study $\widehat{Z}_{{\bm{\delta}}}(q)$.

More generally, we prove quantum modularity of an infinite family of false theta functions which we now introduce. Define

[TABLE]

where $\mathcal{S}\subset\mathbb{Q}^{2}\cap(0,1)^{2}$ is a finite set with the property that $(1,1)-{\bm{\alpha}}$, $(1-\alpha_{1},\alpha_{2})$, $(\alpha_{1},1-\alpha_{2})\in\mathcal{S}$ for ${\bm{\alpha}}\in\mathcal{S}$, $\varepsilon:\mathcal{S}\to\mathbb{C}$ satisfies $\varepsilon({\bm{\alpha}})=\varepsilon((1,1)-{\bm{\alpha}})=\varepsilon((1-\alpha_{1},\alpha_{2}))$, and $K\in\mathbb{N}$ is minimal such that $K\mathcal{S}\subset\mathbb{N}^{2}$. For convenience, we extend the domain of $\varepsilon$ to $\mathcal{S}+\mathbb{Z}^{2}$ by letting $\varepsilon({\bm{\alpha}})=\varepsilon({\bm{\alpha}}+{\bm{n}})$, ${\bm{n}}\in\mathbb{Z}^{2}$.

Theorem 1.1**.**

The function $F_{\mathcal{S},Q,\varepsilon}$ is a quantum modular form of depth two, weight one, and quantum set $\mathcal{Q}_{\mathcal{S},Q,\varepsilon}$, defined in (3.1).

Theorem 1.1 is of independent interest and can be used to investigate other examples of quantum modular forms.

Next we move on to studying unimodular matrices arising from ${\tt H}$-graphs. Since the graph has six vertices it is not surprising that there are only finitely many positive definite unimodular matrices. We prove the following result.

Theorem 1.2**.**

There are, up to graph isomorphism, precisely $39$ equivalence classes of unimodular positive definite plumbing matrices (1.2) with $b_{j}\geq 2$, $j\in\{1,2,5,6\}$.

Then our main result is the following.

Theorem 1.3**.**

For any positive definite unimodular plumbing matrix as in Theorem 1.2 , $q^{c_{M}}\widehat{Z}_{0}(q)$, for some $c_{M}\in\mathbb{Q}$, is a quantum modular form of depth two, weight one, and quantum set $\mathbb{Q}$.

Based on our results here and in [6], we can slightly reformulate Gukov’s conjecture mentioned in [6] on the quantum modularity of $\widehat{Z}_{{\bm{a}}}(q)$ and $\widehat{Z}(q)$.

Conjecture 1.4**.**

Let $T$ be a plumbing graph (tree) with $r$ nodes of degree at least three. Then $\widehat{Z}_{\mathbb{a}}(q)$ is a depth $r$ quantum modular form whose quantum set is a subset of $\mathbb{Q}$. Moreover, for any unimodular plumbing matrix, $\widehat{Z}(q)$ is quantum of depth $r$ with quantum set $\mathbb{Q}$.

Combined with the conjecture on $\widehat{Z}(q)$ mentioned above, Conjecture 1.4 would imply that (unified) WRT invariants of plumbed $3$-manifolds are higher depth quantum modular forms. We expect that the higher depth property also holds true for higher rank ${\rm SU}(N)$ invariants (see [10]).

The paper is organized as follows. In Section 2, we discuss special functions, the Euler-Maclaurin summation formula, higher depth quantum modular forms, and double Eichler integrals. In Section 3 we show quantum modularity of $F_{\mathcal{S},Q_{1},\varepsilon}$ (see Theorem 3.1). In Section 4, we prove our main result on quantum modularity of ${Z}(q)$, defined in (2.9), for unimodular plumbing graphs (see Theorem 4.1). The proof of the classification of positive definite unimodular matrices (1.2) is given in Section 5. Finally, in the appendix we list data for all $39$ equivalence classes of positive unimodular matrices needed to compute ${Z}(q)$.

Acknowledgements: The authors thank S. Chun, S. Gukov, and C. Manolescu for helpful discussion on some aspects of [13] .

2. Preliminaries

2.1. Special functions

Following [1] (with slightly different notation), for each $\kappa\in\mathbb{R}$ we define a function $E_{2}:\mathbb{R}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ by

[TABLE]

For $x_{2},x_{1}-\kappa x_{2}\neq 0$, we set

[TABLE]

The following formula relates $M_{2}$ and $E_{2}$

[TABLE]

where for $x\in\mathbb{R}$, we set $E(x):=2\int_{0}^{x}e^{-\pi w^{2}}dw$.

The proof of the next result follows from the proof of [4, Lemma 6.1]. Here $\tau=u+iv$.

Proposition 2.1**.**

For $\kappa,x_{1},x_{2}\in\mathbb{R}$ we have

[TABLE]

2.2. Euler-Maclaurin summation formula

Let $B_{m}(x)$ be the $m$-th Bernoulli polynomial defined by $\frac{we^{xw}}{e^{w}-1}=:\sum_{m\geq 0}B_{m}(x)\frac{w^{m}}{m!}$. We require

[TABLE]

The Euler-Maclaurin summation formula implies the following lemma.

Lemma 2.2**.**

For $\bm{\alpha}\in\mathbb{R}^{2}$, $F:\mathbb{R}^{2}\rightarrow\mathbb{R}$ a $C^{\infty}$-function which has rapid decay, we have

[TABLE]

where $\mathcal{I}_{F}:=\int_{0}^{\infty}\int_{0}^{\infty}\allowbreak F(\bm{x})dx_{1}dx_{2}$. Here by $\sim$ we mean that the difference between the left- and the right-hand side is $O(t^{N})$ for any $N\in\mathbb{N}$.

2.3. Gauss sums

We define for $a,b,c\in\mathbb{Z}$ with $c>0$ the quadratic Gauss sums

[TABLE]

see [2, Section 1.5] for some basic properties. We use the following elementary result on the vanishing of $G_{c}(a,b)$.

Proposition 2.3**.**

If $\gcd(a,c)\nmid b$, then $G_{c}(a,b)=0$.

2.4. Shimura theta function

We require certain theta functions studied, for example, by Shimura [20]. For $\nu\in\{0,1\}$, $h\in\mathbb{Z}$, $N,A\in\mathbb{N}$, with $A|N$, $N|hA$, define

[TABLE]

Define the slash operator of weight $k\in\frac{1}{2}\mathbb{Z}$ ($(\frac{\,\cdot\,}{\,\cdot\,})$ the Jacobi symbol)

[TABLE]

Note that if $k\in\mathbb{Z}+\frac{1}{2}$, we require that $\gamma\in\Gamma_{0}(4)$. Recall that Shimura’s modular transformation formula [20, Proposition 2.1] states that for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(2N)$, with $2|b$, we have

[TABLE]

Here $e(x):=e^{2\pi ix}$, for odd $d$, $\varepsilon_{d}=1$ or $i$, depending on whether $d\equiv 1\ \,\left(\mathrm{mod}\,4\right)$ or $d\equiv 3\ \,\left(\mathrm{mod}\,4\right)$.

2.5. Integral evaluations

We require, for $m\in\mathbb{Z}$,

[TABLE]

where $\delta_{m,a}=0$ unless $m=a$ in which case it equals $1$ and

[TABLE]

where ${\rm sgn}_{o}(m):=\frac{1}{2}{\rm sgn}(m)(1-(-1)^{m})$.

2.6. Higher depth quantum modular forms

We now give the formal definition of quantum modular forms, following [21].

Definition 2.4**.**

A function $f:\mathcal{Q}\to\mathbb{C}$ ($\mathcal{Q}\subseteq\mathbb{Q}$) is called a quantum modular form of weight $k\in\frac{1}{2}\mathbb{Z}$ for a subgroup $\Gamma$ of ${\rm SL}_{2}(\mathbb{Z})$ (of $\Gamma_{0}(4)$ if $k\in\mathbb{Z}+\frac{1}{2}$) and quantum set $\mathcal{Q}$ if for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma$, the function

[TABLE]

can be extended to an open subset of $\mathbb{R}$ and is real-analytic there. We denote the vector space of such forms by $\mathcal{Q}_{k}(\Gamma)$.

We next turn to the definition of higher-depth quantum modular forms.

Definition 2.5**.**

A function $f:\mathcal{Q}\to\mathbb{C}$ ($\mathcal{Q}\subset\mathbb{Q}$) is called a quantum modular form of depth $N\in\mathbb{N}$, weight $k\in\frac{1}{2}\mathbb{Z}$, and quantum set $\mathcal{Q}$ for $\Gamma$ if for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma$

[TABLE]

where $j$ runs through a finite set, $\kappa_{j}\in\frac{1}{2}\mathbb{Z}$, $N_{j}\in\mathbb{N}$ with $\max_{j}(N_{j})=N-1$, $\mathcal{Q}_{k}^{1}(\Gamma):=\mathcal{Q}_{k}(\Gamma)$, $\mathcal{Q}_{k}^{0}(\Gamma):=1$, and $\mathcal{Q}_{k}^{N}(\Gamma)$ is the space of quantum modular forms of weight $k$ and depth $N$ for $\Gamma$.

For $f_{j}\in S_{k_{j}}(\Gamma)$, the space of cusp forms of weight $k_{j}$ for $\Gamma$ with $k_{j}>\frac{1}{2}$ define the (non-holomorphic) Eichler integrals

[TABLE]

and the errors of modularity, for $\varrho\in\mathbb{Q}$

[TABLE]

The next result is [4, Theorem 5.1].

Theorem 2.6**.**

We have, for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma^{\ast}:=\left(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\right)\Gamma\left(\begin{smallmatrix}-1&0\\ 0&1\end{smallmatrix}\right)$,

[TABLE]

Moreover $r_{f_{1},f_{2},\frac{d}{c}}\in\mathcal{O}(\mathbb{R})$.

2.7. Definitions and notation

In this section we recall the construction of $Z(q)$ following [6], which is another invariant that is closely related to $\widehat{Z}_{{\bm{a}}}(q)$ from (1.1). Consider a tree $G$ with $N$ vertices labeled by integers $m_{jj}$, $1\leq j\leq N$, which is called a plumbing graph. To this data we associate an $N\times N$ matrix $M=(m_{jk})_{1\leq j,k\leq N}$, called its linking (or plumbing) matrix, such that $m_{jk}=-1$ if vertex $j$ is connected to vertex $k$ and zero otherwise. We say that two plumbing matrices $M$ and $M^{\prime}$ are equivalent if their underlying graphs are isomorphic, and there is a graph isomorphism that maps $M$ to $M^{\prime}$. The first homology group of $M_{3}(G)$ (the plumbed $3$-manifold constructed from $G$ and $M$) is

[TABLE]

If $M$ is invertible, then this group is finite and if $M\in\text{SL}_{N}(\mathbb{Z})$, then $H_{1}(M_{3},\mathbb{Z})=0$; this is the case for the main results of this paper, as $M$ is positive definite and unimodular.

To each edge $j-k$ in $G$ we associate a rational function

[TABLE]

and to each vertex $w_{j}$ a Laurent polynomial

[TABLE]

For a fixed tree $G$ and positive definite $M$, set

[TABLE]

where we let $a_{j}:=m_{jj}$ for the vertex labels, $w_{j}:=e^{2\pi iz_{j}}$

[TABLE]

Note that we may write

[TABLE]

The following result is given in Proposition 3.4 of [4].

Proposition 2.7**.**

If $M$ is unimodular, then $Z(q)=\widehat{Z}_{{\bm{\delta}}}(q^{2})$, where $\widehat{Z}_{{\bm{\delta}}}(q)$ is defined in (1.1).

3. Some general construction

In this section we construct an infinite family of quantum modular forms of depth two closely following the arguments in [2]. Define

[TABLE]

We write $Q({\bm{n}})=:\sigma_{1}n_{1}^{2}+2\sigma_{2}n_{1}n_{2}+\sigma_{3}n_{2}^{2}$, and denote its discriminant by $D:=\sigma_{1}\sigma_{3}-\sigma_{2}^{2}.$ We also regularly use the relationship between the quadratic form and the associated bilinear form, namely

[TABLE]

Theorem 3.1**.**

The functions $F_{\mathcal{S},Q,\varepsilon}$ are quantum modular forms of depth two, weight one, on some congruence subgroup containing $\Gamma(8\cdot{\rm lcm}(\sigma_{1},\sigma_{3})KD)$, and quantum set $\mathcal{Q}_{\mathcal{S},Q,\varepsilon}$.

Before proving Theorem 3.1, we require some auxiliary lemmas. Set

[TABLE]

where

[TABLE]

with

[TABLE]

We begin by determining the asymptotic expansions of these functions.

Lemma 3.2**.**

If $\frac{h}{k}\in\mathcal{Q}_{\mathcal{S},\varepsilon}$, then we have the asymptotic expansions (as $t\to 0^{+}$)

[TABLE]

Proof.

For the proof we abbreviate

[TABLE]

We first determine the asymptotic expansion of $F$ using the Euler-Maclaurin summation formula. We let ${\bm{n}}\mapsto{\bm{\ell}}+k{\bm{n}}$ with $0\leq{\bm{\ell}}\leq k-1$ (i.e., $0\leq\ell_{j}\leq k-1$, $j\in\{1,2\}$), ${\bm{n}}\in\mathbb{N}_{0}^{2}$. The assumption that $K\mathcal{S}\subset\mathbb{N}^{2}$ implies that $\frac{h}{k}KQ({\bm{\ell}}+{\bm{\alpha}}+k{\bm{n}})\equiv\frac{h}{k}KQ({\bm{\ell}}+{\bm{\alpha}})\ \,\left(\mathrm{mod}\,1\right)$, thus

[TABLE]

where $g({\bm{x}}):=e^{-KQ({\bm{x}})}$. The main term in Lemma 2.2 is

[TABLE]

Using that $K\mathcal{S}\subset\mathbb{N}^{2}$ we may let ${\bm{\ell}}$ run$\ \,\left(\mathrm{mod}\,k\right)$. Since $\frac{h}{k}\in\mathcal{Q}$ the sum vanishes.

The second term in Lemma 2.2 yields

[TABLE]

Making the change of variables ${\bm{\ell}}\mapsto(k-1)(1,1)-{\bm{\ell}}$ and using that $(1,1)-{\bm{\alpha}}\in\mathcal{S}$ if ${\bm{\alpha}}\in\mathcal{S}$, (2.3) yields that only the odd values of $n_{2}$ survive, and (3.4) becomes

[TABLE]

In exactly the same way we obtain that the third term in Lemma 2.2 equals

[TABLE]

For the final term in Lemma 2.2 we obtain, pairing in exactly the same way

[TABLE]

In particular we obtain that the asymptotic expansion of $F$ has the shape as claimed in (3.3).

We now turn to the asymptotic behavior of $\mathbb{E}$. We use (2.1) and let $M_{2}^{*}$ denote the function such that the ${\rm sgn}$ in (2.1) is replaced by ${\rm sgn}^{*}$, where ${\rm sgn}^{*}(x):={\rm sgn}(x)$ if $x\in\mathbb{R}\setminus\{0\}$ and ${\rm sgn}^{*}(0):=1$. We obtain

[TABLE]

Proceeding as above

[TABLE]

where

[TABLE]

[TABLE]

We again use the Euler-Maclaurin summation formula. The main term in Lemma 2.2 is

[TABLE]

by conjugating the condition in $\mathcal{Q}$.

The second term in Lemma 2.2 is, pairing terms as before,

[TABLE]

It is now straightforward to verify, as in [4], that

[TABLE]

Via symmetry the third term in Lemma 2.2 is treated in exactly the same way.

The fourth term in Lemma 2.2 is, pairing as before,

[TABLE]

It can now be shown that

[TABLE]

Comparing terms gives the claim. ∎

Write $\mathcal{A}:=K\mathcal{S}$, and define

[TABLE]

The following lemma rewrites $\mathbb{E}$ as a two-dimensional theta integral, which is essential in order to calculate modular transformations.

Lemma 3.3**.**

We have

[TABLE]

where

[TABLE]

Proof.

Using (2.2) we obtain

[TABLE]

This yields

[TABLE]

where

[TABLE]

We now rewrite the $\theta_{j}$ in terms of the Shimura theta functions. Letting ${\bm{n}}\mapsto\frac{{\bm{n}}}{K}$, we obtain

[TABLE]

Set $\nu_{1}:=\sigma_{1}n_{1}+\sigma_{2}n_{2}$ and $\nu_{2}:=n_{2}$, so that $n_{1}=\frac{\nu_{1}-\sigma_{2}\nu_{2}}{\sigma_{1}}$. Plugging in the restrictions on ${\bm{n}}$ yields

[TABLE]

This shows that ${\bm{\nu}}\in\mathcal{B}.$ Furthermore, if ${\bm{\alpha}}\in\mathcal{A}$, there exists a corresponding ${\bm{B}}\in\mathcal{B}$ such that

[TABLE]

Overall, we therefore have

[TABLE]

In the same way, by setting $\nu_{1}:=\sigma_{2}n_{1}+\sigma_{3}n_{2}$ and $\nu_{2}:=n_{1}$, we can show that

[TABLE]

∎

We are now ready to prove Theorem 3.1.

Proof of Theorem 3.1.

Suppose that $f$ is one of the theta functions from Lemma 3.3 and $\gamma\in\Gamma(8\cdot{\rm lcm}(\sigma_{1},\sigma_{3})KD)$. Then the transformation (2.4) implies (after a short calculation) that $f\!\!\mid_{\frac{3}{2}}\!\!\gamma=\!f$. The theorem statement now follows from Lemmas 3.2 and 3.3, and Theorem 2.6. ∎

4. A family with quantum set $\mathbb{Q}$ and unimodular matrices

In this section we construct a family of depth two quantum modular forms with quantum set $\mathbb{Q}$. Let $N_{1},N_{2}\in 2\mathbb{N}$ and write $L:=\gcd(N_{1},N_{2})$, $N_{1}:=LR_{1},N_{2}:=LR_{2},$ so that $\gcd(R_{1},R_{2})=1$. Set $Q({\bm{n}})=\sigma_{1}n_{1}^{2}+2\sigma_{2}n_{1}n_{2}+\sigma_{3}n_{2}^{2}$. We assume the factorizations $\sigma_{1}=R_{1}\mu_{1},$ with $\gcd(R_{1},\mu_{1})=1$, and $\sigma_{3}=R_{2}\mu_{3}$, with $\gcd(\mu_{3},R_{2})=1$. Moreover we assume that $2\sigma_{2}=LR_{1}R_{2}={\rm lcm}(N_{1},N_{2})$ and that $\gcd(\mu_{1},\mu_{3})$ consists of at most one odd prime factor, and always satisfies $\gcd(L,\gcd(\mu_{1},\mu_{3}))=1$. If $4\nmid L$, then we also require that exactly one of $R_{1},R_{2},\mu_{3}$ is even. Set, with $r_{1},r_{2},s_{1},s_{2}\in\mathbb{N}$ satisfying $\gcd(r_{j},N_{j})=\gcd(s_{j},N_{j})=1$, $r_{j}^{2}\equiv s_{j}^{2}\ \,\left(\mathrm{mod}\,2N_{j}\right)$,

[TABLE]

We define

[TABLE]

where $\mathcal{S}:=\mathcal{S}_{1}\cup\mathcal{S}_{2}$ and $\varepsilon({\bm{\alpha}}):=(-1)^{j+1}$ if ${\bm{\alpha}}\in\mathcal{S}_{j}$. We see in the proof of Theorem 4.1 that the assumptions imply that the asymptotic expansion of $\mathcal{Z}_{Q,{\bm{r}},{\bm{s}}}(q)$ consists of several leading terms with identical Gauss sums that always cancel, and thus the series converges for all $\mathbb{Q}$.

Theorem 4.1**.**

Under the assumption above, the function $\mathcal{Z}_{Q,{\bm{r}},{\bm{s}}}(q)$ is a quantum modular form of depth two, weight one, group $\Gamma\left(8\cdot\text{\rm lcm}(\sigma_{1},\sigma_{2})LR_{1}R_{2}\right)$, and quantum set $\mathbb{Q}$.

Proof.

Note that the conditions of Theorem 3.1 are satisfied. We are left to show that we have quantum set $\mathbb{Q}$, which follows if we show that

[TABLE]

Write $L=2^{\Lambda}L_{1},k=gk_{1}$, where $L_{1},k_{1}$ are odd and where $g:=\gcd(k,L)$. We claim that the sum on ${\bm{\ell}}$ vanishes unless $\gcd(LR_{1}R_{2},k_{1})=1$ and $g\in\{1,2\}$. For this we first consider the (one-dimensional) Gauss sum in $\ell_{1}$, which is $(a_{j}:=N_{j}\alpha_{j})$

[TABLE]

The linear term reduces to $2\mu_{1}a_{1}\ \,\left(\mathrm{mod}\,R_{1}\right)$, and $\mu_{1}a_{1}$ is coprime to $R_{1}$ by assumption. Thus by Proposition 2.3 the expression in (4.3) is zero if $\gcd(R_{1},k_{1})>1$. Similarly, the linear term reduces to $2\mu_{1}a_{1}\ \,\left(\mathrm{mod}\,L\right)$. The Gauss sum (4.3) vanishes if $g>1$ and $g\nmid 2\mu_{1}$. Now write an alternative Gauss sum by grouping the $\ell_{2}$ terms in (4.2), obtaining an analogous version of (4.3). As before, this immediately shows that (4.2) is zero if $\gcd(R_{2},k_{1})>1$, and also vanishes if $g>1$ and $g\nmid 2\mu_{3}$. If $g>1$, then the only way the sum fails to vanish is if $g\mid\gcd(2\mu_{1},2\mu_{3}),$ which implies that $g=2$ by assumption. This shows that (4.2) vanishes if $4\mid L$.

Next, assuming $g=2$, $4\nmid L$, and $4\mid k$, we also show that (4.2) vanishes in this case. Recalling the corresponding assumptions on the $R_{j}$ and $\mu_{j}$, one possibility is that $2|R_{1}$ and $2\nmid R_{2}\mu_{1}\mu_{2}$ (or the analogous condition with $\ell_{1}$ and $\ell_{2}$ swapped if necessary). Then $4$ divides the factor in front of $\ell_{1}^{2}$ in (4.3), and the linear term is congruent to $2$ modulo $4$ since $a_{1}$ is odd. The sum therefore vanishes by Proposition 2.3. Otherwise the condition on $R_{j}$ and $\mu_{j}$ is that $2\nmid R_{1}R_{2}\mu_{1},2\mid\mu_{3}$, and we again consider the analog of (4.3) for the sum in $\ell_{2}$. Now $4$ divides the coefficient in front of $\ell_{2}^{2}$, and the linear term is congruent to $2\ \,\left(\mathrm{mod}\,4\right)$ so Proposition 2.3 again applies.

We next assume that $\gcd(LR_{1}R_{2},k_{1})=1$, and $g\in\{1,2\}$ and prove that the sum on ${\bm{\ell}}$ in (4.2) is the same for all choices of ${\bm{\alpha}}$. We note that the multiplicative inverses $\overline{N_{j}}\ \,\left(\mathrm{mod}\,k_{1}\right)$ exist. Using (3.2), we write

[TABLE]

Since $B({\bm{\ell}},{\bm{\alpha}})-B({\bm{\ell}},(\overline{N_{1}}a_{1},\overline{N_{2}}a_{2}))\equiv 0\ \,\left(\mathrm{mod}\,k_{1}\right)$ by construction, (4.4) implies that

[TABLE]

We now calculate

[TABLE]

where $X:=R_{2}\mu_{1}a_{1}^{2}+LR_{1}R_{2}a_{1}a_{2}+R_{1}\mu_{3}a_{2}^{2}-N_{1}N_{2}Q(\overline{N_{1}}a_{1},\overline{N_{2}}a_{2})$.

If $p$ is an odd prime such that $p^{\lambda}$ exactly divides $LR_{1}R_{2}$, then the assumptions on the parameters easily imply that

[TABLE]

is independent from ${\bm{\alpha}}$.

Finally, suppose that $2^{\lambda}$ exactly divides $LR_{1}R_{2}$. Then the final congruence is

[TABLE]

which is independent from ${\bm{\alpha}}$ due to the assumption that $r_{j}^{2}\equiv s_{j}^{2}\ \,\left(\mathrm{mod}\,2^{\lambda+1}\right).$

Therefore the sum on ${\bm{\ell}}$ in (4.2) equals

[TABLE]

by shifting ${\bm{\ell}}$; this overall expression is now clearly independent from choice of ${\bm{\alpha}}$.∎

5. Classification of positive unimodular ${\tt H}$-matrices and the proofs of Theorem 1.2 and Theorem 1.3

5.1. Proof of Theorem 1.2

Let

[TABLE]

In this section, we classify all positive, unimodular (PU) matrices $M$ with the additional property that $b_{j}\geq 2$ ($j\in\{1,2,5,6\}$). The determinant of $M$ can be written as follows:

[TABLE]

The goal of this section is to show the following.

Proposition 5.1**.**

If $M(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6})$ is a PU matrix with $b_{j}\geq 2$ ($j\in\{1,2,5,6\}$), then (up to equivalence)

[TABLE]

In particular, there are finitely many PU matrices $M$.

This then enables us to prove Theorem 1.2.

Proof of Theorem 1.2.

Proposition 5.1 together with a computer search quickly shows there are $312$ PU matrices. Since the group of automorphisms of an ${\tt H}$-graph is $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, we have $39$ equivalence classes of such matrices; these are listed in the appendix. This gives the claim. ∎

We now prove the main statement of this section, namely Proposition 5.1.

Proof of Proposition 5.1.

It is clear that $\gcd(b_{1},b_{2})\mid D$, thus $b_{1}$ and $b_{2}$ must be coprime and without loss of generality we may assume $b_{1}<b_{2}$ and $b_{5}<b_{6}$. This further implies that $b_{1}b_{2},b_{5}b_{6}\geq 6$, and $\frac{1}{b_{1}}+\frac{1}{b_{2}}\leq\frac{5}{6}$. Since $b_{3},b_{4}\geq 2$ we therefore have

[TABLE]

thus $M$ is not unimodular. Furthermore, the fact that $1-\frac{1}{b_{1}}-\frac{1}{b_{2}}<1$ immediately shows that if $b_{3}=b_{4}=1$, then $\det(M)<0$. Thus without loss of generality we assume that $b_{4}=1$ and $b_{3}\neq 1$. If $b_{3}\geq 8$, then

[TABLE]

Thus we must have $b_{3}\leq 7$.

Now suppose that $b_{5}\geq 14$. Then, since $b_{6}>b_{5}$,

[TABLE]

Thus we must have $b_{5}\leq 13$.

The remaining bounds require a case by case analysis based on the values of $b_{5}$. If $b_{5}=2$, then for $b_{2}=2$, $D\leq 0$, thus we must have $b_{3}\geq 3$. If $b_{6}\geq 28$, then

[TABLE]

We therefore conclude that $b_{6}\leq 27$. However, in order to have $D$ positive we also need

[TABLE]

which implies that $b_{6}\geq 7$.

We next determine the possible values of $b_{1}$. In order to have $D=1$, it must be true that $D>0$, thus

[TABLE]

Now suppose that $3\leq b_{3}\leq 7$ and $14\leq b_{6}\leq 27$ are fixed. Now suppose that $b_{1}\geq 11$. Then

[TABLE]

so we must have $b_{1}\leq 10.$

In this case a Maple calculation shows that the right-side is at most $5$ (which occurs for $b_{3}=3$ and $b_{6}=7$), and thus all $b_{1}>10$ are not possible; in other words, we must have $b_{1}\leq 10$. To complete this case, we now consider fixed $2\leq b_{1}\leq 10,3\leq b_{3}\leq 7,$ and $3\leq b_{6}\leq 27$. If there is a solution, then following (5.2), it must be for the minimal value of $b_{2}$ such that

[TABLE]

A Maple search shows that the maximum value of the right-side is 30 (which occurs with $b_{1}=6,b_{3}=3$ and $b_{6}=7$), $b_{2}\leq 31$.

Next, let $b_{5}=3$. If $b_{3}\geq 4$, then

[TABLE]

Thus $b_{3}\leq 3$, and we begin with $b_{3}=3$. Very similar calculations show, in turn, that $b_{6}\leq 5$, and $b_{1}\leq 3$. As in (5.3), checking

[TABLE]

in these ranges now gives a maximum right-side value of $10$ (with $b_{1}=2,b_{3}=3$, and $b_{6}=4$), then $b_{2}\leq 11$.

For the case $b_{5}=3$ and $b_{3}=2$, if $b_{6}\leq 6$, then

[TABLE]

and thus we must have $b_{6}\geq 7$. However, in order for $D>0$, it also must be true that

[TABLE]

The largest values of $b_{6}$ occurs when the left side is as close to $\frac{3}{2}$ as possible (while being larger, so $b_{1}\geq 3$), which occurs for $b_{1}=3$ and $b_{2}=7$ (and then $2-\frac{1}{3}-\frac{1}{7}=\frac{32}{21}$). Plugging in to (5.4), this implies that the first inequality holds for $b_{6}>96$, and again by monotonicity, this gives the bound $b_{6}\leq 97$.

Furthermore, if $b_{1}\geq 24$, then

[TABLE]

thus we must have $b_{1}\leq 23$. Finally, checking

[TABLE]

over the ranges $3\leq b_{1}\leq 23$, and $7\leq b_{6}\leq 97$ shows that the right-side is at most 132 (which occurs at $b_{1}=12$ and $b_{6}=7$), so $b_{2}\leq 133$.

For the remaining values $4\leq b_{5}\leq 13$, we proceed similarly. First, if $b_{3}\geq 3$, then

[TABLE]

thus we must have $b_{3}=2$. Furthermore, if $b_{1}\geq 11$, then

[TABLE]

thus $b_{1}\leq 10$.

Now we bound $b_{6}$ as in the previous case. For example, if $b_{5}=4$, then $D>0$ requires that

[TABLE]

This is only possible if $\frac{1}{b_{1}}+\frac{1}{b_{2}}<\frac{2}{3}$, and the largest value of $b_{6}$ occurs when the sum is as close as possible to $\frac{2}{3}$. This occurs with $b_{1}=2,b_{2}=7$, which implies that $b_{6}\leq 77$. Repeating the argument for $b_{5}\geq 5$ never gives a larger range for $b_{6}$ (and $b_{5}\geq 8$ can be treated as a single case, since then the maximal case is always $\frac{1}{2}+\frac{1}{3}<\frac{b_{5}-2}{b_{5}-1}$). Finally, plugging in $b_{1}\leq 10,4\leq b_{5}\leq 13$, and $b_{6}\leq 77$ to

[TABLE]

gives the bound $b_{2}\leq 71$. ∎

5.2. Calculation of $Z(q)$ and the proof of Theorem 1.3.

Let $M$ be as in (5.1), with inverse matrix $M^{-1}=(\ell_{jk})_{1\leq j,k\leq 6}$. We need the central $2\times 2$ sub-matrix of $M^{-1}$, which we write as

[TABLE]

In order to write $Z(q)$ as a double series of the type found in Section 4, we use a linear algebra identity, which can be verified by a Maple computation.

Lemma 5.2**.**

If ${\bm{r}}=(\varepsilon_{1},\varepsilon_{2},2n_{1}+1,2n_{2}+1,\varepsilon_{5},\varepsilon_{6})^{T}$ with $n_{1},n_{2}\in\mathbb{Z}$ and $\varepsilon_{j}\in\{\pm 1\}$, then

[TABLE]

where

[TABLE]

Remark**.**

Importantly, note that $c$ is independent of the $\varepsilon_{j}$’s.

We can now evaluate $Z(q)$ for any positive unimodular $M$.

Proposition 5.3**.**

With $\mathcal{S}:=\left\{{\bm{\alpha}}(\varepsilon)\right\}$, we have

[TABLE]

where $Q_{1}({\bm{n}}):=\frac{1}{2}{{\bm{m}}}^{T}M^{-1}{{\bm{m}}}$, with ${{\bm{m}}}:=(0,0,2n_{1},2n_{2},0,0)^{T}$.

Proof.

An application of formula (2.9) for the ${\tt H}$-graph gives

[TABLE]

where by (2.10) (because $M$ is unimodular) we have

[TABLE]

Applying (2.5) and (2.6) we find that

[TABLE]

Applying Lemma 5.2 completes the proof. ∎

We are now ready to prove Theorem 1.3.

Proof.

of Theorem 1.3. By splitting the summation over $\mathbb{Z}^{2}$ in (5.5) into summations over $\mathbb{N}_{0}\times\mathbb{N}_{0}$, $(-\mathbb{N})\times(-\mathbb{N})$, $\mathbb{N}_{0}\times(-\mathbb{N})$, and $(-\mathbb{N})\times\mathbb{N}_{0}$, a case-by-case computation for each unimodular matrix (5.1) gives

[TABLE]

where

[TABLE]

[TABLE]

and $Q^{*}({\bm{n}}):=Q(-n_{1},n_{2})$.

The quadratic form $Q$ and constants $N_{1},N_{2},r_{1},r_{2},s_{1},s_{2}$ (recall, $L={\rm gcd}(N_{1},N_{2})$) are given in the appendix. In Section 4, Theorem 4.1 establishes that $\mathcal{Z}_{1}(q)$ is a quantum modular form of weight one and depth two on $\mathbb{Q}$. The same result also applies to $\mathcal{Z}_{2}(q)$. Finally, we let $c_{M}:=9-\frac{1}{2}{\rm tr}(M)-c$, where $c$ is also listed in the appendix. ∎

Appendix: Data for positive unimodular matrices

Here we list all positive unimodular matrices of the form (5.1), and the corresponding parameters that appear in $Z(q)$ (see (4.1) and Proposition 5.3). In each case one can directly check that the assumptions in Section 4 are satisfied.

The value of $c$ and the quadratic form $Q$ are given below, and the data for $\mathcal{S}_{j}$ are presented in condensed form.

1. $M(2,3,7,1,2,3)$

$Q({\bm{n}})=n_{1}^{2}+12n_{1}n_{2}+37n_{2}^{2}$, $c=\frac{5}{6}$, $N_{1}=N_{2}=12$, $r_{1}=r_{2}=1$, $s_{1}=s_{2}=5$.

2. $M(2,7,4,1,5,2)$

$Q({\bm{n}})=21n_{1}^{2}+140n_{1}n_{2}+235n_{2}^{2}$, $c=\frac{47}{70}$, $N_{1}=28$, $N_{2}=20$, $r_{1}=5$, $s_{1}=9$, $r_{2}=3$, $s_{2}=7$.

3. $M(6,31,3,1,2,7)$

$Q({\bm{n}})=465n_{1}^{2}+2604n_{1}n_{2}+3647n_{2}^{2}$, $c=\frac{274}{651}$, $N_{1}=372$, $N_{2}=28$, $r_{1}=149$, $s_{1}=161$, $r_{2}=5$, $s_{2}=23$.

4. $M(7,18,3,1,2,7)$

$Q({\bm{n}})=45n_{1}^{2}+252n_{1}n_{2}+353n_{2}^{2}$, $c=\tfrac{53}{126}$, $N_{1}=252$, $N_{2}=28$, $r_{1}=101$, $s_{1}=115$, $r_{2}=5$, $s_{2}=9$.

5. $M(3,11,3,1,2,9)$

$Q({\bm{n}})=77n_{1}^{2}+396n_{1}n_{2}+510n_{2}^{2}$, $c=\tfrac{205}{396}$, $N_{1}=66$, $N_{2}=36$, $r_{1}=19$, $s_{1}=25$, $r_{2}=7$, $s_{2}=11$.

6. $M(2,19,3,1,2,11)$

$Q({\bm{n}})=171n_{1}^{2}+836n_{1}n_{2}+1023n_{2}^{2}$, $c=\frac{239}{418}$, $N_{1}=76$, $N_{2}=44$, $r_{1}=17$, $s_{1}=21$, $r_{2}=9$, $s_{2}=13$.

7. $M(2,3,3,1,2,27)$

$Q({\bm{n}})=25n_{1}^{2}+108n_{1}n_{2}+117n_{2}^{2}$, $c=\frac{37}{54}$, $N_{1}=12$, $N_{2}=108$, $r_{1}=1$, $s_{1}=5$, $r_{2}=25$, $s_{2}=29$.

8. $M(2,3,3,1,3,5)$

$Q({\bm{n}})=14n_{1}^{2}+60n_{1}n_{2}+65n_{2}^{2}$, $c=\frac{41}{60}$, $N_{1}=12$, $N_{2}=30$, $r_{1}=1$, $s_{1}=5$, $r_{2}=7$, $s_{2}=13$.

9. $M(2,11,3,1,3,4)$

$Q({\bm{n}})=55n_{1}^{2}+264n_{1}n_{2}+318n_{2}^{2}$, $c=\frac{155}{264}$, $N_{1}=44$, $N_{2}=24$, $r_{1}=9$, $s_{1}=13$, $r_{2}=5$, $s_{2}=11$.

10. $M(3,4,3,1,3,4)$

$Q({\bm{n}})=5n_{1}^{2}+24n_{1}n_{2}+29n_{2}^{2}$, $c=\tfrac{155}{264}$, $N_{1}=N_{2}=24$, $r_{1}=5$, $s_{1}=11$, $r_{2}=5$, $s_{2}=11$.

11. $M(3,7,2,1,3,97)$

$Q({\bm{n}})=1337n_{1}^{2}+4074n_{1}n_{2}+3104n_{2}^{2}$, $c=\frac{835}{2037}$, $N_{1}=42$, $N_{2}=582$, $r_{1}=11$, $s_{1}=17$, $r_{2}=191$, $s_{2}=197$.

12. $M(3,8,2,1,3,56)$

$Q({\bm{n}})=109n_{1}^{2}+336n_{1}n_{2}+259n_{2}^{2}$, $c=\tfrac{17}{42}$, $N_{1}=48$, $N_{2}=336$, $r_{1}=13$, $s_{1}=19$, $r_{2}=109$, $s_{2}=115$.

13. $M(3,47,2,1,3,17)$

$Q({\bm{n}})=1457n_{1}^{2}+4794n_{1}n_{2}+3944n_{2}^{2}$, $c=\frac{895}{2397}$, $N_{1}=282$, $N_{2}=102$, $r_{1}=91$, $s_{1}=97$, $r_{2}=31$, $s_{2}=37$.

14. $M(3,88,2,1,3,16)$

$Q({\bm{n}})=319n_{1}^{2}+1056n_{1}n_{2}+874n_{2}^{2}$, $c=\frac{391}{1056}$, $N_{1}=528$, $N_{2}=96$, $r_{1}=173$, $s_{1}=179$, $r_{2}=29$, $s_{2}=35$.

15.$M(4,5,2,1,3,47)$

$Q({\bm{n}})=1820n_{1}^{2}+5640n_{1}n_{2}+4371n_{2}^{2}$, $c=\frac{2263}{5640}$, $N_{1}=40$, $N_{2}=282$, $r_{1}=11$, $s_{1}=19$, $r_{2}=91$, $s_{2}=97$.

16. $M(4,77,2,1,3,11)$

$Q({\bm{n}})=532n_{1}^{2}+1848n_{1}n_{2}+1605n_{2}^{2},c=\frac{635}{1848}$, $N_{1}=616$, $N_{2}=66$, $r_{1}=227$, $s_{1}=235$, $r_{2}=19$, $s_{2}=25$.

17. $M(5,16,2,1,3,11)$

$Q({\bm{n}})=1520n_{1}^{2}+5280n_{1}n_{2}+4587n_{2}^{2},c=\frac{1813}{5280}$, $N_{1}=160$, $N_{2}=66$, $r_{1}=59$, $s_{1}=69$, $r_{2}=19$, $s_{2}=25$.

18. $M(7,92,2,1,3,8)$

$Q({\bm{n}})=2093n_{1}^{2}+7728n_{1}n_{2}+7134n_{2}^{2},c=\frac{2365}{7728}$, $N_{1}=1288$, $N_{2}=48$, $r_{1}=545$, $s_{1}=559$, $r_{2}=13$, $s_{2}=19$.

19. $M(8,35,2,1,3,8)$

$Q({\bm{n}})=455n_{1}^{2}+1680n_{1}n_{2}+1551n_{2}^{2}),c=\frac{257}{840}$, $N_{1}=560$, $N_{2}=48$, $r_{1}=237$, $s_{1}=253$, $r_{2}=13$, $s_{2}=19$.

20. $M(11,16,2,1,3,8)$

$Q({\bm{n}})=286n_{1}^{2}+1056n_{1}n_{2}+975n_{2}^{2},c=\frac{323}{1056}$, $N_{1}=352$, $N_{2}=48$, $r_{1}=149$, $s_{1}=171$, $r_{2}=13$, $s_{2}=19$.

21. $M(12,133,2,1,3,7)$

$Q({\bm{n}})=836n_{1}^{2}+3192n_{1}n_{2}+3047n_{2}^{2},c=\frac{905}{3192}$, $N_{1}=3192$, $N_{2}=42$, $r_{1}=1451$, $s_{1}=1475$, $r_{2}=11$, $s_{2}=17$.

22. $M(13,72,2,1,3,7)$

$Q({\bm{n}})=3432n_{1}^{2}+13104n_{1}n_{2}+12509n_{2}^{2},c=\frac{3715}{13104},N_{1}=1872,N_{2}=42,r_{1}=851,s_{1}=877,r_{2}=11,s_{2}=17$.

23. $M(3,4,2,1,4,23)$

$Q({\bm{n}})=195n_{1}^{2}+552n_{1}n_{2}+391n_{2}^{2}$, $c=\frac{121}{276}$, $N_{1}=24$, $N_{2}=184$, $r_{1}=5$, $s_{1}=11$, $r_{2}=65$, $s_{2}=73$.

24. $M(3,10,2,1,4,9)$

$Q({\bm{n}})=115n_{1}^{2}+360n_{1}n_{2}+282n_{2}^{2},c=\frac{143}{360}$, $N_{1}=60$, $N_{2}=72$, $r_{1}=17$, $s_{1}=23$, $r_{2}=23$, $s_{2}=31$.

25. $M(3,52,2,1,4,7)$

$Q({\bm{n}})=663n_{1}^{2}+2184n_{1}n_{2}+1799n_{2}^{2},c=\frac{407}{1092}$, $N_{1}=312$, $N_{2}=56$, $r_{1}=101$, $s_{1}=107$, $r_{2}=17$, $s_{2}=25$.

26. $M(6,67,2,1,4,5)$

$Q_{1}({\bm{n}})=2211n_{1}^{2}+8040n_{1}n_{2}+7310n_{2}^{2}$, $c=\tfrac{2539}{8040}$, $N_{1}=804$, $N_{2}=40$, $r_{1}=329$, $s_{1}=341$, $r_{2}=11$, $s_{2}=19$.

27. $M(2,7,2,1,4,77)$

$Q({\bm{n}})=227n_{1}^{2}+616n_{1}n_{2}+418n_{2}^{2}),c=\frac{279}{616}$, $N_{1}=28$, $N_{2}=616$, $r_{1}=5$, $s_{1}=9$, $r_{2}=227$, $s_{2}=235$.

28. $M(7,26,2,1,4,5)$

$Q({\bm{n}})=1001n_{1}^{2}+3640n_{1}n_{2}+3310n_{2}^{2},c=\frac{1149}{3640}$, $N_{1}=364$, $N_{2}=40$, $r_{1}=149$, $s_{1}=163$, $r_{2}=11$, $s_{2}=19$.

29. $M(2,11,2,1,4,25)$

$Q({\bm{n}})=781n_{1}^{2}+2200n_{1}n_{2}+1550n_{2}^{2},c=\frac{969}{2200}$, $N_{1}=44$, $N_{2}=200$, $r_{1}=9$, $s_{1}=13$, $r_{2}=71$, $s_{2}=79$.

30. $M(2,19,2,1,4,17)$

$Q({\bm{n}})=893n_{1}^{2}+2584n_{1}n_{2}+1870n_{2}^{2},c=\frac{1113}{2584}$, $N_{1}=76$, $N_{2}=136$, $r_{1}=17$, $s_{1}=21$, $r_{2}=47$, $s_{2}=55$.

31. $M(2,71,2,1,4,13)$

$Q({\bm{n}})=2485n_{1}^{2}+7384n_{1}n_{2}+5486n_{2}^{2},c=\frac{3105}{7384}$, $N_{1}=284$, $N_{2}=104$, $r_{1}=69$, $s_{1}=73$, $r_{2}=35$, $s_{2}=43$.

32. $M(3,7,2,1,5,7)$

$Q({\bm{n}})=69n_{1}^{2}+210n_{1}n_{2}+160n_{2}^{2},c=\frac{43}{105}$, $N_{1}=42$, $N_{2}=70$, $r_{1}=11$, $s_{1}=17$, $r_{2}=23$, $s_{2}=33$.

33. $M(2,5,2,1,5,33)$

$Q({\bm{n}})=254n_{1}^{2}+660n_{1}n_{2}+429n_{2}^{2},c=\frac{307}{660}$, $N_{1}=20$, $N_{2}=330$, $r_{1}=3$, $s_{1}=7$, $r_{2}=127$, $s_{2}=137$.

34. $M(2,7,2,1,5,16)$

$Q({\bm{n}})=413n_{1}^{2}+1120n_{1}n_{2}+760n_{2}^{2},c=\frac{507}{1120}$, $N_{1}=28$, $N_{2}=160$, $r_{1}=5$, $s_{1}=9$, $r_{2}=59$, $s_{2}=69$.

35.$M(2,21,2,1,5,9)$

$Q({\bm{n}})=434n_{1}^{2}+1260n_{1}n_{2}+915n_{2}^{2},c=\frac{541}{1260}$, $N_{1}=84$, $N_{2}=90$, $r_{1}=19$, $s_{1}=23$, $r_{2}=31$, $s_{2}=41$.

36. $M(2,55,2,1,5,8)$

$Q({\bm{n}})=297n_{1}^{2}+880n_{1}n_{2}+652n_{2}^{2},c=\frac{371}{880}$, $N_{1}=220$, $N_{2}=80$, $r_{1}=53$, $s_{1}=57$, $r_{2}=27$, $s_{2}=37$.

37. $M(2,3,2,1,8,57)$

$Q({\bm{n}})=391n_{1}^{2}+912n_{1}n_{2}+532n_{2}^{2},c=\frac{445}{912}$, $N_{1}=12$, $N_{2}=912$, $r_{1}=1$, $s_{1}=5$, $r_{2}=391$, $s_{2}=407$.

38. $M(2,3,2,1,9,32)$

$Q({\bm{n}})=247n_{1}^{2}+576n_{1}n_{2}+336n_{2}^{2},c=\frac{281}{576}$, $N_{1}=12$, $N_{2}=576$, $r_{1}=1$, $s_{1}=5$, $r_{2}=247$, $s_{2}=265$.

39. $M(2,3,2,1,12,17)$

$Q({\bm{n}})=175n_{1}^{2}+408n_{1}n_{2}+238n_{2}^{2},c=\frac{199}{408}$, $N_{1}=12$, $N_{2}=408$, $r_{1}=1$, $s_{1}=5$, $r_{2}=175$, $s_{2}=199$.