---
abstract: 'We introduce a 3-manifold invariant $\hat{Z}_b^G(q)$ valued in integer-coefficient power series and an invariant of knot complements $F_K^G({\bf x},q)$ valued in multi-variable series that depend on the choice of gauge group $G$. This generalizes the earlier works of Gukov-Pei-Putrov-Vafa [@GPPV] and Gukov-Manolescu [@GM] which correspond to $G=SU(2)$ case. As in the $SU(2)$ case, there is a surgery formula relating $F_K^G({\bf x},q)$ to $\hat{Z}_b^G(q)$ of a Dehn surgery of the knot. We provide explicit calculations of $\hat{Z}_b^G(q)$ for negative definite plumbings and $F_K^G({\bf x},q)$ for torus knots. Furthermore, we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations which should satisfy a recurrence given by the quantum A-polynomial for symmetric representations.'
address: 'Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125'
author:
- Sunghyuk Park
bibliography:
- 'higher\_rank.bib'
date: 'September 19, 2019'
title: 'Higher rank $\hat{Z}$ and $F_K$'
---

Introduction
============

Categorification of the Chern-Simons theory is one of the most exciting open questions in quantum topology. While homology theories categorifying quantum link invariants are fairly well-understood by now, whether there is a homology theory categorifying the Witten-Reshetikhin-Tureav (WRT) invariants is still widely open. One approach to this problem comes from physics. According to recent works of Gukov-Putrov-Vafa [@GPV] and Gukov-Pei-Putrov-Vafa [@GPPV], we can decompose WRT invariants into categorifiable “homological blocks” often denoted by $\hat{Z}_b(q)$. These are integer coefficient $q$-series and are supposed to be the (graded) Euler characteristic of a conjectural homological invariant $\mathcal{H}^{i,j}_{b,{\rm BPS}}$ so that $$\hat{Z}_b(q) = \sum_{i,j}(-1)^i q^j \text{rank }\mathcal{H}^{i,j}_{b,{\rm BPS}}.$$ More recently Gukov-Manolescu [@GM] studied an analog of $\hat{Z}_b(q)$ for knot complements, denoted by $F_K(x,q)$, and demonstrated that $\hat{Z}_b(q)$ behaves well under cutting and gluing along a torus boundary.

More than a decade ago, Dunfield-Gukov-Rasmussen [@DGR] made a fascinating conjecture, also partly motivated from physics, that $\mathfrak{sl}(N)$ link homologies should be equipped with a family of differentials $d_N$ for $N\in \mathbb{Z}$. This conjecture revealed many hidden structures of the link homologies and could be used to compute Khovanov homology very efficiently. Now, a very natural (and ambitious) question is this : Is there an analogous story for 3-manifolds?

The purpose of this paper is to define categorifiable objects $\hat{Z}_b^G(q)$ and $F_K^G({\bf x},q)$ for arbitrary gauge group $G$. We believe that our work will serve as a stepping stone towards the analogous story of large $N$ on 3-manifold side.

Organization of the paper {#organization-of-the-paper .unnumbered}
-------------------------

In Section 2 we list a few notational conventions we will use throughout this paper.

In Section 3 we define higher rank $\hat{Z}_b(q)$ for (weakly) negative definite plumbed 3-manifolds. We provide many examples, hoping it will serve as a good reference.

In Section 4 we introduce higher rank $F_K({\bf x},q)$ and give an explicit formula for torus knots. Higher rank surgery formula (without proof) is also presented.

In Section 5 we specialize our higher rank $F_K$ to symmetric representations. We numerically check for some examples that it satisfies the quantum volume conjecture.

Appendix A is on a geometric meaning of the label $b$ of $\hat{Z}_b$, and Appendix B is a derivation of definition (\[integralZhat\]).

Acknowledgments {#acknowledgments .unnumbered}
---------------

We would like to thank Sergei Gukov for his invaluable guidance and Nikita Sopenko for his kind help with Mathematica, as well as Francesca Ferrari, Sarah Harrison and Ciprian Manolescu for helpful conversations.

The author was supported by Kwanjeong Educational Foundation.

Notations and conventions
=========================

We follow the convention used in [@GM] for knots, 3-manifolds, and colored Jones polynomials. Throughout this article, $G$ is a semisimple Lie group, $Q$ is the root lattice, $P$ is the weight lattice, and $W$ is the Weyl group. $\rho$ denotes the Weyl vector (half-sum of positive roots), and $\alpha$ and $\omega$ will be reserved for roots and fundamental weights. We use $\rm B$ for the linking matrix of a plumbed 3-manifold. For a multi-index monomial, we use the following notation $$x^\beta := \prod_{1\leq i\leq r}x_i^{(\beta,\omega_i)}$$ where $r = \text{rank }G$ and $\beta\in P$. When it comes to $q$-series, we mostly don’t bother to fix the overall $q$-power, and just use the notation $\cong$ for equivalence up to sign and overall $q$-power.

Higher rank $\hat{Z}_b$
=======================

Definition for plumbings
------------------------

We present here a formula for $\hat{Z}$ for (weakly) negative definite plumbed manifolds, with arbitrary gauge group $G$. We use the same good old Gauss sum reciprocity to deduce this definition. (See appendix \[Gauss\] for derivation.) :

\[Zhatsgm\] For a plumbed 3-manifold $Y$ with weakly negative definite linking matrix $\mathrm{B}$, define[^1] $$\boxed{
\hat{Z}_b^G(Y;q) := \pm \frac{q^{\frac{3\sigma-\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\int_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq r}\frac{dx_{vi}}{2\pi i x_{vi}}\left(\sum_{w\in W}(-1)^{l(w)} x_v^{w(\rho)}\right)^{2-\deg v}\, \Theta_b^{-\mathrm{B}}(x^{-1},q) \label{integralZhat}
}$$ where $$\Theta_b^{-\mathrm{B}}(x^{-1},q) := \sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}x_v^{-\ell_v}. \label{Theta}$$ In particular, in case $G = SU(N)$, this takes the following simple form : $$\label{Zhatb}
\hat{Z}_b^{SU(N)}(Y;q) := \pm \frac{q^{\frac{3\sigma -\mathrm{Tr}\,\mathrm{B}}{2}\frac{N^3-N}{12}}}{|W|^{b_1(\Gamma)}} {\rm v.p.}\oint_{|x_{vi}|=1}\prod_{v\in V}\prod_{1\leq i\leq N-1}\frac{dx_{vi}}{2\pi i x_{vi}}\, F_{3d}(x)\Theta_{2d}^{b}(x,q)$$ with $$\begin{aligned}
    F_{3d}(x) &:= \prod_{v\in V}\left(\sum_{w\in W}(-1)^{l(w)} \prod_{1\leq i\leq N-1}x_{vi}^{(\omega_i,w(\rho))}\right)^{2-\deg v}\nonumber\\
    &= \prod_{v\in V}\left( \prod_{1\leq i < j\leq N}(y_{vi}^{1/2}y_{vj}^{-1/2}-y_{vi}^{-1/2}y_{vj}^{1/2}) \right)^{2-\deg v}\\
    \Theta_{2d}^{b}(x,q)&:=\sum_{\ell\in \mathrm{B} Q^{V}+b}q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\prod_{v\in V}\prod_{1\leq i\leq N-1}x_{vi}^{-(\omega_i,\ell_{v})},   \end{aligned}$$ where $x_{i} = \frac{y_{i}}{y_{i+1}}$.

Here “v.p.” denotes the principal value integral. That is, taking average over $W$ number of deformed contours, each corresponding to a Weyl chamber. For instance, the deformed contour corresponding to a permutation $\sigma\in W \cong S_{N}$ is $$|y_{\sigma(1)}| < |y_{\sigma(2)}| < \cdots < |y_{\sigma(N)}|. \label{SU(N)contour}$$ These $\hat{Z}_b^G$ are topological invariants.

$\hat{Z}_b^G$ defined above is invariant under Neumann moves (a.k.a. 3d Kirby moves).

The proof is analogous to that of the $SU(2)$ version. See Proposition 4.6 in [@GM].

Examples and explicit calculations
----------------------------------

[**Comparison with Hee-Joong Chung’s paper.**]{} We find agreement with Chung’s result [@C] for many examples. (In the following, we present $\hat{Z}$’s up to $\cong$ equivalence.)

-   $Y = M(-1;\frac{1}{2},\frac{1}{3},\frac{1}{7})$. In this case $|H_1(Y)| = 1$ and there’s only one homological block.

         $G$                                            $\hat{Z}_0(Y)$
      --------- ----------------------------------------------------------------------------------------------
       $SU(2)$   $1 -q -q^5 +q^{10} -q^{11} +q^{18} +q^{30} -q^{41} +q^{43} -q^{56} -q^{76} +q^{93} -\cdots$
       $SU(3)$   $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$
       $SU(4)$        $1 -3q +q^2 +3q^3 -3q^5 -q^6 -q^7 -5q^8 +15q^9 +5q^{10} -11q^{11} -q^{12} +\cdots$

-   $Y = M(-1;\frac{1}{2},\frac{1}{5},\frac{2}{7})$. Again, $|H_1(Y)| = 1$ and there’s only one homological block.

         $G$                                                $\hat{Z}_0(Y)$
      --------- ------------------------------------------------------------------------------------------------------
       $SU(2)$   $1 -q^{3} -q^{5} +q^{12} -q^{23} +q^{36} +q^{42} -q^{59} +q^{81} -q^{104} -q^{114} +q^{141} -\cdots$
       $SU(3)$    $1 -2q^3 -2q^5 +2q^6 +2q^9 +q^{12} -2q^{14} +2q^{15} -2q^{18} -3q^{20} +6q^{21} -4q^{23} -\cdots$
       $SU(4)$        $1 -3q^3 -3q^5 +5q^6 -q^7 +2q^8 +3q^9 -q^{10} -q^{12} +2q^{13} -6q^{14} +2q^{15} -\cdots$

-   $Y = M(-1;\frac{1}{3},\frac{1}{5},\frac{3}{7})$. In this case $|H_1(Y)| = 4$.

         $G$                                                           $\hat{Z}_b(Y)$
      --------- ----------------------------------------------------------------------------------------------------------------------------
       $SU(2)$            $1 +q^4 +q^{16} -q^{68} +q^{144} -q^{260} -q^{320} -q^{356} +q^{484} +q^{528} +q^{612} -q^{832} +\cdots$
                 $-\frac{1}{2}q^{15/4}(1 +q^6 +q^{10} +q^{12} -q^{44} -q^{48} -q^{58} -q^{88} +q^{122} +q^{164} +q^{182} +q^{190} -\cdots)$
                   $q^{13/2}(1 -q^{32} -q^{56} -q^{72} +q^{136} +q^{160} +q^{208} -q^{344} +q^{496} -q^{696} -q^{792} -q^{848} +\cdots)$
       $SU(3)$               $1 +3q^4 +2q^{12} +3q^{16} +2q^{28} +2q^{48} +2q^{52} +q^{64} +4q^{68} +4q^{80} +4q^{92} - \cdots$
                          $-\frac{1}{6}q^{-7/4}(2 +2q +2q^3 -4q^5 +2q^6 -2q^7 +4q^9 +2q^{10} +4q^{12} +2q^{13} -2q^{14} +\cdots)$
                     $-\frac{1}{3}q^{-7}(1 +2q^6 -2q^8 +2q^{12} -2q^{14} +2q^{18} -2q^{20} +q^{24} -2q^{26} +2q^{28} +4q^{30} +\cdots)$
                    $\frac{1}{3}q^{-21/4}(1 +q -q^2 +q^4 -q^5 +q^7 +q^{10} +q^{13} -2q^{14} -q^{15} +2q^{16} +q^{19} +\cdots) \times 2$
       $SU(4)$                $1 +q^{12} +8q^{16} +3q^{20} +16q^{24} +11q^{28} +15q^{32} +4q^{36} +26q^{40} +5q^{44} +\cdots$
                                $-\frac{1}{12}q^{-15/4}(1 +2q -2q^2 +2q^4 -2q^5 +q^6 +4q^7 -2q^8 +3q^{10} +6q^{13} -\cdots)$
                           $\frac{1}{6}q^{-1}(2 -2q^2 -2q^4 -3q^6 -2q^8 -5q^{10} -14q^{12} -5q^{14} -4q^{16} -12q^{18} +\cdots)$
                               $\frac{1}{12}q^{-1/4}(2 +2q -2q^2 +2q^3 +q^4 -2q^5 +6q^6 +2q^7 +8q^8 +6q^9 -\cdots) \times 2$
                         $-q^{3/2}(1 +3q^4 +2q^8 +6q^{12} +6q^{16} +4q^{20} +9q^{24} +9q^{28} +11q^{32} +9q^{36} +\cdots) \times 2$
                             $q^{-2}(1 +2q^4 +3q^8 +6q^{16} +6q^{20} +11q^{24} +17q^{32} +10q^{36} +9q^{40} +14q^{48} +\cdots)$
                                    $-\frac{1}{12}q^{-15/4}(2 +q -2q^2 +q^3 +2q^4 -q^5 +4q^6 +4q^7 +2q^8 -2q^9 +\cdots)$
                        $\frac{1}{6}q^{-11/2}(1 +3q^6 -4q^8 +7q^{12} -2q^{14} -q^{16} +6q^{18} +3q^{20} +7q^{22} +4q^{24} +\cdots)$

Observe that [@C] agrees with our example computations except in the $SU(4)$ case of the last example. Our homological blocks are more refined in a sense that Chung’s $\hat{Z}_{1}$ is the sum of our 2nd and 7th blocks and Chung’s $\hat{Z}_{3}$ is the sum of our 1st and 6th blocks. This example illustrates that in general we can’t simply decompose $Z_a$ into $\hat{Z}_b$’s by just collecting terms whose $q$-powers differ by an integer.[^2]

[**Other examples and higher rank false theta functions.** ]{}

-   $Y=S_0^3(K_n)$. The 0-surgery on twist knots are probably the simplest examples. For instance,

         $G$                                            $\hat{Z}_0(S_0^3(\mathbf{5}_2))$
      --------- ----------------------------------------------------------------------------------------------------------------
       $SU(2)$   $\frac{1}{2!}(1 -q +q^3 -q^6 +q^{10} -q^{15} +q^{21} -q^{28} +q^{36} -q^{45} +q^{55} -q^{66} +q^{78} -\cdots)$
       $SU(3)$     $\frac{1}{3!}(1 -2q +2q^3 +q^4 -4q^6 +2q^9 +2q^{10} +q^{12} -2q^{13} -4q^{15} +2q^{18} +2q^{19} +\cdots)$
       $SU(4)$           $\frac{1}{4!}(1 -3q +q^2 +4q^3 -2q^4 +q^5 -5q^6 -2q^7 +3q^8 +2q^9 +9q^{10} -2q^{11} -\cdots)$
       $SU(5)$       $\frac{1}{5!}(1 -4q +3q^2 +6q^3 -7q^4 -2q^5 +2q^7 -2q^8 +6q^9 +15q^{10} -12q^{11} -23q^{12} +\cdots)$

    Indeed, for every positive twist knot $K_p$ the following is easy to deduce from our definition (\[integralZhat\]).

    \[twist knot prop\] $$\label{twist knot Zhat}
            \hat{Z}_0^G(S_0^3(K_p)) \cong \frac{1}{|W|}\sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\rho)||^2} =: \frac{1}{|W|}\chi_{p,\rho}$$ where $$N_\ell := \sum_{w\in W}(-1)^{l(w)}K(w(\ell))$$ and $K(\beta)$ denotes the Kostant partition function.[^3]

    Note that $\chi_{p,\rho}$ is exactly the higher rank false theta function (a character of the log-VOA $W^0(p)_Q$) given in equation (1.2) of [@BM]! Similarly for double twist knots $K_{m,n}$ with $m,n>0$,[^4] $$\label{double twist knot Zhat}
        \hat{Z}_0^G(S_0^3(K_{m,n})) \cong \frac{1}{|W|}\chi_{m,\rho}\chi_{n,\rho}.$$

    The 0-surgery on $K_p$ has a simple plumbing description as shown in Figure \[double twist knot figure\].

    (-2,0) node\[above\][$m$]{} circle(0.5ex)– (0,0) node\[above\][$0$]{} circle(0.5ex) (2,0) node\[above\][$0$]{} circle(0.5ex)– (4,0) node\[above\][$n$]{} circle(0.5ex); (0,0) edge\[bend left\] node\[midway,above\][$+$]{} (2,0); (0,0) edge\[bend right\] node\[midway,below\][$-$]{} (2,0);

    (7,0) node\[above\][$-1$]{} circle(0.5ex) (9,0) node\[above\][$0$]{} circle(0.5ex)– (11,0) node\[above\][$p$]{} circle(0.5ex); (7,0) edge\[bend left\] node\[midway,above\][$+$]{} (9,0); (7,0) edge\[bend right\] node\[midway,below\][$-$]{} (9,0);

    The linking matrix and its inverse are $$\mathrm{B} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & p\end{pmatrix}
            \quad,\quad
            \mathrm{B}^{-1} = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -p & 1 \\ 0 & 1 & 0\end{pmatrix}$$ There is a single trivalent vertex with $0$ framing. This contributes the following factor in $F_{3d}(x)$ : $$\qty(\sum_{w\in W}(-1)^{l(w)}x_0^{w(\rho)})^{-1} = \frac{1}{|W|}\sum_{\ell_0\in P_+ \cap (Q+\rho)}N_{\ell_0} \sum_{w\in W}(-1)^{l(w)}x_0^{w(\ell_0)}.$$ For $\ell = (0,\ell_0,\ell_p)^t$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2}||\sqrt{p}\ell_0 - \frac{1}{\sqrt{p}}\ell_p||^2 -\frac{1}{2p}||\ell_p||^2}.$$ Applying (\[integralZhat\]), it is straightforward to get (\[twist knot Zhat\]).

    Using a plumbing description of the 0-surgery on $K_{m,n}$ (Figure \[double twist knot figure\]), it is easy to derive (\[double twist knot Zhat\]) as well.

-   $Y=\Sigma(p,q,r)$. For convenience let’s define the following notation for higher rank false theta functions : $$\chi_{p,\beta}^{G} := \sum_{\ell\in P_+\cap (Q+\rho)} N_\ell \sum_{w\in W}(-1)^{l(w)}q^{\frac{1}{2}||\sqrt{p}\ell - \frac{1}{\sqrt{p}}w(\beta)||^2}$$ Note that for $SU(2)$, this notation is related to the usual notation of false theta functions as follows : $$\chi_{p,n\rho}^{SU(2)} = \Psi_{p,p-n},\text{ for }n = 1, \cdots, p-1.$$ For every Brieskorn sphere $Y = \Sigma(p,q,r)$ with $0 < p < q < r$ pairwise relatively prime, we have

    \[Brieskorn prop\] $$\hat{Z}_0^G(\Sigma(p,q,r)) \cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{pqr,qr\rho + prw_1(\rho) + pqw_2(\rho)}$$

    That is, it is a sum of $|W|^2$ number of higher rank false theta functions.[^5]

    The proof is analogous to that of Proposition 4.8 in [@GM].

    Note that we didn’t have to treat $\Sigma(2,3,5)$ separately. In this sense, using $\chi_{p,\beta}$ as false theta functions is more natural than using $\Psi_{p,n}$.

-   $Y = M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})$. Let $b_1,b_2,b_3>0$ and assume that $Y$ has negative orbifold number; i.e. $$e = a_0 + \sum_{i=1}^{3}\frac{a_i}{b_i} < 0.$$ Assume further that the central meridian is trivial in homology; i.e. $$e\, \mathrm{lcm}(b_1,b_2,b_3) = -1.$$ Then their $\hat{Z}_b$’s can be expressed as signed sum of higher rank false theta functions :

    \[some Seifert prop\] $$\label{some Seifert Zhat}
            \hat{Z}_b^G(M(a_0;\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3})) \cong \sum_{(w_1,w_2)\in W^2}\mathbf{1}_b(w_1,w_2) (-1)^{l(w_1w_2)}\chi_{\frac{b_1b_2b_3}{|H_1|},\frac{b_2b_3}{|H_1|}\rho + \frac{b_1b_3}{|H_1|}w_1(\rho) + \frac{b_1b_2}{|H_1|}w_2(\rho)}$$ where $$\mathbf{1}_b(w_1,w_2) := \begin{cases} 1 &\text{ if } \ell(\rho,\rho,w_1(\rho),w_2(\rho)) \in \mathrm{B}Q^V+b\\ 0 &\text{ otherwise}\end{cases}$$

    Observe that Proposition \[some Seifert prop\] is a slight generalization of Proposition \[Brieskorn prop\].

    $Y$ can be described as a star-shaped plumbing with 3 legs. The only vertices whose degree is not 2 are the central vertex and the terminal vertices. Denote by $\ell(\ell_0,\ell_1,\ell_2,\ell_3)$ an element $\ell\in \mathrm{B}Q^V + b$ such that $$\ell_v = \begin{cases} \ell_0 & v \text{ is the central vertex}\\ \ell_1, \ell_2, \ell_3 & v \text{ is the corresponding terminal vertex} \\ 0 & \text{Otherwise}\end{cases}$$ Then for any $\ell$ with $\ell_1,\ell_2,\ell_3 \in W(\rho)$, $$q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)} = q^{\frac{1}{2|H_1|}||\sqrt{b_1b_2b_3}\ell_0 - \frac{1}{\sqrt{b_1b_2b_3}}(b_2b_3\ell_1 + b_3b_1\ell_2 + b_1b_2\ell_3)||^2 + C}$$ for some constant $C$ independent of $\ell$. Applying the definition (\[integralZhat\]), it is straightforward to obtain (\[some Seifert Zhat\]). Note that the assumption $e\,\text{lcm}(b_1,b_2,b_3)=-1$ was introduced so that $$\ell(\rho,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b \Leftrightarrow \ell(\rho + Q,\rho,w_1(\rho),w_2(\rho))\in \mathrm{B}Q^V+b$$

Naturally, there are some open questions regarding modularity of these $q$-series :

What are the quantum modularity properties of the higher rank $\hat{Z}$?

More specifically we can ask

What are the “mock side” of the higher rank false theta functions? (i.e. what is the higher rank $\hat{Z}$ of the orientation reversed 3-manifold?)

Higher rank $F_K$
=================

Review of $SU(2)$ case
----------------------

In [@GM], a new knot invariant $F_K(x,q)$ was defined for weakly negative definite knots and conjectured for all knots. Roughly, $F_K$ is a series $$F_K(x,q) = \frac{1}{2!}\sum_{m=1}^{\infty} (x^{m/2}-x^{-m/2})f_m(q)$$ where $m$ is odd and each $f_m(q)$ is a Laurant series with $\mathbb{Z}$-coefficient. One of the most important property of $F_K(x,q)$ is that $$F_K(x,q) \xrightarrow{q\rightarrow 1} \frac{x^{1/2}-x^{-1/2}}{\Delta_K(x)}$$ where $\Delta_K(x) = \nabla_K(x^{1/2}-x^{-1/2})$ is the Alexander polynomial of $K$, and $\frac{1}{\Delta_K(x)}$ should be understood as an average over expansions near $x=0$ and $x=\infty$.

Higher rank $F_K$
-----------------

Let’s study higher rank $F_K(\mathbf{x},q)$. The higher rank analog of $F_K$ should be of the following form : $$F_{K}^G(\mathbf{x},q) = \frac{1}{|W|}\sum_{\beta\in P_+\cap (Q+\rho)}f_\beta^G(q)\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}.$$ This can be computed by ’t Hooft resumming the (higher rank) colored Jones polynomials and should be annihilated by the (higher rank) quantum A-polynomial.

[**Right-handed trefoil with $G = SU(3)$.**]{} The first few $f_\beta^{SU(3)}$ are (up to overall sign and $q$-power) $$\begin{aligned}
    f_{(1,1)} &= -q,\quad 
    f_{(4,1)} = -q^2,\quad
    f_{(5,2)} = -2q^3,\quad
    f_{(7,1)} = -q^4,\quad
    f_{(5,5)} = q^5,\quad \nonumber\\
    f_{(7,4)} &= 2q^6,\quad 
    f_{(10,1)} = -q^7,\quad
    f_{(8,5)} = q^8,\quad
    f_{(11,2)} = -2q^9,\quad
    f_{(7,7)} = -q^9,\quad \nonumber\\
    f_{(13,1)} &= -q^{11},\quad
    f_{(11,5)} = q^{12},\quad 
    \cdots\end{aligned}$$ where we have written $\beta$ in the fundamental weights basis. (Because $f_{(m,n)} = f_{(n,m)}$, we have only written those terms with $m \geq n$.) The $q$-power of this $f_\beta$ is, up to overall constant, $$\frac{(\beta,\beta)}{12}.\label{q-power}$$ In the $q\rightarrow 1$ limit we have $$F_{\mathbf{3}_1^r}^{SU(3)}(x_1,x_2,1) = \frac{x_1^{1/2}-x_1^{-1/2}}{x_1+x_1^{-1}-1}\frac{x_2^{1/2}-x_2^{-1/2}}{x_2+x_2^{-1}-1}\frac{x_1^{1/2}x_2^{1/2}-x_1^{-1/2}x_2^{-1/2}}{x_1x_2+x_1^{-1}x_2^{-1}-1}.\label{classicallimit}$$ Alternatively, we can start with $SU(3)$ colored Jones polynomials of the trefoil [@GMV] and ‘t Hooft resum : $$J_{\mathbf{3}_1^r,n_1,n_2}^{SU(3)}(q=e^\hbar) \xrightarrow{\text{`t Hooft resum}}\xrightarrow{\hbar \rightarrow 0} \frac{1}{x_1+x_1^{-1}-1}\frac{1}{x_2+x_2^{-1}-1}\frac{1}{x_1x_2+x_1^{-1}x_2^{-1}-1}.$$ Note that (\[q-power\]) and (\[classicallimit\]) completely characterize $F_{\mathbf{3}_1^r}(x_1,x_2,q)$.

In general the following version of Melvin-Morton conjecture was proven by D. Bar-Natan and S. Garoufalidis in [@BG] :

For arbitrary gauge group $G$ of rank $r$, the semi-classical limit of $F_K(x_1,\cdots,x_r,q)$ should be $$\boxed{
        F_K^G(x_1, \cdots, x_r,1) = \prod_{\alpha\in \Delta^+}\frac{x^{\alpha/2}-x^{-\alpha/2}}{\Delta_K(x^\alpha)}
        }$$

For torus knots, we found the following explicit expression :

For $K = T_{s,t}$, $f_\beta(q)$ is a monomial of degree $\frac{(\beta,\beta)}{2st}$, up to an overall $q$-power. More precisely, $$\label{torus knot F_K}
    F_{T_{s,t}}^G \cong \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\mathbf{1}(\beta,w_1,w_2)N_{\frac{1}{st}(\beta + tw_1(\rho)+sw_2(\rho))} q^{\frac{(\beta,\beta)}{2st}}$$ where $$\mathbf{1}(\beta,w_1,w_2):=\begin{cases} 1 &\text{ if }\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho)) \in P_+\cap (Q+\rho)\\ 0&\text{ otherwise}\end{cases}$$

This can be derived either directly from (\[integralZhat\]) by using plumbing description or by reverse-engineering using the higher rank surgery formula that we discuss below.

Here we present a direct derivation. Recall from [@GM] that the complement of $T_{s,t}$ has a plumbing description as in Figure \[torus knot figure\], where $0<t'<t$, $0<s'<s$ are chosen such that $st'\equiv -1 (\text{mod }t)$ and $ts' \equiv -1 (\text{mod }s)$.

(-2,0) node\[above\][$-\frac{t}{t'}$]{} circle(0.5ex)– (0,0) node\[above\][$-1$]{} circle(0.5ex) – (2,0) node\[above\][$-\frac{s}{s'}$]{} circle(0.5ex); (0,-2) node\[below\][$-st$]{} circle(0.5ex)–(0,0);

The linking matrix is $$\mathrm{B} = \begin{pmatrix} -st & 1 & 0 & 0 \\ 1 & -1 & 1 & 1 \\ 0 & 1 & (-\frac{t}{t'}) & 0 \\ 0 & 1 & 0 & (-\frac{s}{s'})\end{pmatrix}$$ where $(-\frac{t}{t'})$ and $(-\frac{s}{s'})$ should be understood as block matrices corresponding to the continued fractions. To compute the integral (\[integralZhat\]) with $x_{-st}$ left unintegrated, we just have to replace the theta function $\Theta^{-\mathrm{B}}(x^{-1},q)$ with $$\Theta^{-\mathrm{B}'}(x^{-1},q) \cong \sum_{\alpha\in Q^{V'}}q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha)-(\alpha,\delta)}\prod_{v\in V'}x_v^{-(\mathrm{B}'\alpha + \delta)}\cdot x_{-st}^{-\alpha_{-1}-\rho}$$ where $V' = V \setminus \{v_{-st}\}$ and $\mathrm{B}'$ is the corresponding sub-linking matrix. Set $\beta = -\alpha_{-1}-\rho$. We need to multiply $\Theta^{-\mathrm{B}'}(x^{-1},q)$ with $$\prod_{v\in V'}\prod_{1\leq i\leq r}\qty(\sum_{w\in W}(-1)^{l(w)}x_v^{w(\rho)})^{2-\deg v}$$ and take the constant term with respect to variables $x_v$, $v\in V'$. As $2-\deg v$ is non-zero for only 3 vertices (the central vertex $v_{-1}$ and the 2 terminal vertices) it is pretty easy to compute. The only contributions come from those $\alpha$’s such that $\mathrm{B}'\alpha + \delta$ takes values $w_1(\rho), w_2(\rho)$ on the terminal vertices for some $w_1,w_2\in W$, a value in $Q+\rho$ in the central vertex, and $0$ on all the other vertices. Using simple linear algebra, it is easy to check that for those $\alpha$’s, $$q^{-\frac{1}{2}(\alpha,\mathrm{B}'\alpha) - (\alpha,\delta)} = q^{\frac{(\beta,\beta)}{2st}+C}$$ for some constant $C$ independent of $\alpha$, and that $\frac{1}{st}(\beta + tw_1(\rho) + sw_2(\rho))$ is the value of $\mathrm{B}'\alpha + \delta$ on the central vertex. This proves (\[torus knot F\_K\]).

Just as in [@GM], we can use surgery formula for these higher rank $F_K$ to compute higher rank $\hat{Z}_b(S_{p/r}^3(K))$. For instance, surgery on $\mathbf{3}_1^r$ gives us the following $\hat{Z}$’s ($SU(3)$ analog of the table 1 in [@GM]) :

   $r$   $S_{-1/r}^3(\mathbf{3}_1^r)$                            $\hat{Z}_0^{SU(3)}(S_{-1/r}^3(\mathbf{3}_1^r))$
  ----- ------------------------------ ----------------------------------------------------------------------------------------------------
   $1$         $\Sigma(2,3,7)$             $1 -2q +2q^3 +q^4 -2q^5 -2q^8 + 4q^9 + 2q^{10} - 4q^{11} +2q^{13} -6q^{14} +2q^{15} -\cdots$
   $2$         $\Sigma(2,3,13)$         $1 -2q +2q^3 -q^4 +2q^{10} -2q^{11} -2q^{14} +2q^{16} +2q^{19} -2q^{20} +4q^{21} -4q^{23} -\cdots$
   $3$         $\Sigma(2,3,19)$         $1 -2q +2q^3 -q^4 +2q^{16} -2q^{17} -2q^{20} +2q^{22} +2q^{25} -2q^{26} +4q^{33} -4q^{35} -\cdots$
   $4$         $\Sigma(2,3,25)$         $1 -2q +2q^3 -q^4 +2q^{22} -2q^{23} -2q^{26} +2q^{28} +2q^{31} -2q^{32} +4q^{45} -4q^{47} -\cdots$
   $5$         $\Sigma(2,3,31)$         $1 -2q +2q^3 -q^4 +2q^{28} -2q^{29} -2q^{32} +2q^{34} +2q^{37} -2q^{38} +4q^{57} -4q^{59} -\cdots$
   $r$        $\Sigma(2,3,6r+1)$        $\sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{36r+6, 3(6r+1)w_1(\rho)+  2(6r+1)w_2(\rho)+ 6\rho}$

In fact it is easy to check that for $K=T_{s,t}$, $$\begin{aligned}
\mathcal{L}_{-1/r}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K(\mathbf{x},q) \right] &\cong \sum_{(w_1,w_2)\in W^2}(-1)^{l(w_1w_2)}\chi_{st(rst+1),t(rst+1)w_1(\rho) + s(rst+1)w_2(\rho) +st\rho}\\
 &\cong \hat{Z}_0^G(\Sigma(s,t,rst+1))\end{aligned}$$ We conjecture the following surgery formula (analogous to Conjecture 1.7 of [@GM]) :

Let $K\subset S^3$ be a knot. Then $$\boxed{
        \hat{Z}_b^G(S_{p/r}^3(K)) \cong \mathcal{L}_{p/r}^{(b)}\left[\prod_{\alpha\in \Delta^+}(x^{\frac{\alpha}{2r}}-x^{-\frac{\alpha}{2r}})F_K^G(\mathbf{x},q) \right]
        }$$ whenever the RHS makes sense.

Moreover we conjecture that our 0-surgery formula in [@CGPS] holds for higher rank as well :

Let $K\subset S^3$ be a knot. Then $$\boxed{
        \hat{Z}_0^G(S_{0}^3(K)) \cong \frac{1}{|W|}f_\rho^G(K)
        }$$

Symmetric representations and large $N$
=======================================

Specialization to symmetric representations
-------------------------------------------

In this section we present a specialization of $F_K^G({\bf x},q)$ to symmetric representations. We restrict our attention to $G=SU(N)$. Although the unreduced version seems to be better-behaving, to get $F_K$ with symmetric colorings (symmetric powers of the defining representation), we need to use the reduced version : $$F_K^{\rm red}(\mathbf{x},q) := \frac{1}{|W|}\sum_{\beta\in P_+ \cap (Q+\rho)}f_\beta(q)\frac{\sum_{w\in W}(-1)^{l(w)}x^{w(\beta)}}{\sum_{w\in W}(-1)^{l(w)}x^{w(\rho)}}$$ In particular, the symmetrically colored $F_K$ corresponds to the following specialization : $$F_K^{\rm sym}(x,q) := F_K^{\rm red}((x,q,\cdots,q),q)$$ That is, we set $x_2 = \cdots = x_r = q$. A version of quantum volume conjecture [@FGS] states that this should be annihilated by the symmetrically colored quantum A-polynomial[^6] : $$\label{QuantVolConj}
    \boxed{
    \hat{A}^{\rm sym}(\hat{x},\hat{y},q)F_K^{\rm sym}(x,q) = 0
    }$$

[**Right-handed trefoil.**]{} For the right-handed trefoil, $F_{\mathbf{3}_1^r}^{\rm sym}(x,q)$ for $SU(N)$ with the first few values of $N$ look like the following :

-   $SU(2)$ $$\begin{aligned}
            F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[ (-q + q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots) \right.\\
            &\quad + (x + x^{-1})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
            &\quad + (x^2 + x^{-2})(q^2 + q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
            &\quad + (x^3 + x^{-3})(q^3 - q^6 -q^8 + q^{13} + q^{16} - \cdots)\\
            &\quad + (x^4 + x^{-4})(-q^6 - q^8 + q^{13} + q^{16} - \cdots)\\
            &\quad\left. + \cdots \right]
        \end{aligned}$$

-   $SU(3)$ $$\begin{aligned}
            F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(-2q -2q^2 +2q^4 +4q^5 +4q^6 +4q^7 +2q^8 -2q^{10} -4q^{11} -6q^{12} +\cdots) \right.\\
            &\quad + (q^{1/2}x+q^{-1/2}x^{-1})q^{1/2}(-1 -2q -q^2 +q^3 +3q^4 +4q^5 +4q^6 +3q^7 +q^8 -q^9 +\cdots)\\
            &\quad + (qx^2 + q^{-1}x^{-2})(-q -q^2 +2q^4 +3q^5 +4q^6 +3q^7 + 2q^8 -2q^{10} +\cdots)\\
            &\quad + (q^{3/2}x^3 + q^{-3/2}x^{-3})q^{1/2}(q^3 +2q^4 +3q^5 +3q^6 +2q^7 +q^8 +\cdots)\\
            &\quad + (q^2x^4 + q^{-2}x^{-4})(q^3 +q^4 +2q^5 +2q^6 +2q^7 +q^8 +\cdots)\\
            &\quad\left. +\cdots \right]
        \end{aligned}$$

-   $SU(4)$ $$\begin{aligned}
            F_{\mathbf{3}_1^r}^{\rm sym}(x,q) &\cong \frac{1}{2}\left[(q^{-2} +q^{-1} -2 -4q -8q^2 -7q^3 -7q^4 +\cdots) \right.\\
            &\quad + (qx + q^{-1}x^{-1})(q^{-2} -1 -5q -6q^2 -8q^3 -5q^4 -2q^5 +\cdots)\\
            &\quad + (q^2x^2 + q^{-2}x^{-2})(-2 -3q -6q^2 -5q^3 -5q^4 +4q^6 +\cdots)\\
            &\quad + (q^3x^3 + q^{-3}x^{-3})(-q^{-1} -1 -3q -3q^2 -4q^3 -2q^4 +5q^6 +9q^7 +\cdots)\\
            &\quad + (q^4x^4 + q^{-4}x^{-4})(-1 -q -2q^2 -q^3 -q^4 +2q^5 +4q^6 +8q^7 +11q^8 +\cdots)\\
            &\quad\left. + \cdots \right]
        \end{aligned}$$

Note that the overall factor is $\frac{1}{2}$ instead of $\frac{1}{N!}$. This is due to reduction of the Weyl symmetry to $\mathbb{Z}_2$.

It is easy to verify (\[QuantVolConj\]) numerically in this case, using the super-A-polynomial for the right-handed trefoil $$\hat{A}^{\rm super}(\hat{x},\hat{y},a,q) = a_0 + a_1\hat{y} + a_2\hat{y}^2$$ where $$\begin{aligned}
    a_0 &= -\frac{(-1+\hat{x})(-1+aq\hat{x}^2)}{a\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\
    a_1 &= \frac{(-1+a\hat{x}^2)(-a^2\hat{x}^2 + aq^3\hat{x}^2 +aq\hat{x}(1+\hat{x}+a(-1+\hat{x})\hat{x})-q^2(1+a^2\hat{x}^4))}{a^2q\hat{x}^3(-1+a\hat{x})(-q + a\hat{x}^2)}\\
    a_2 &= 1\end{aligned}$$ with $a$ specialized to $q^N$.

Future direction : large $N$
----------------------------

We end with some intriguing open questions.

Is there a HOMFLYPT version of $F_K$?

Conjecturally, there is a HOMFLYPT version $F_K^{\rm super}(x,q,a)$ such that we can recover $F_K^{SU(N),{\rm sym}}(x,q)$ as a certain specialization (e.g. $a\rightarrow q^N$), and $$\boxed{\hat{A}^{\rm super}(\hat{x},\hat{y},a,q)F_K^{\rm super}(x,q,a) = 0.}$$ We expect to have an expansion of the form $$F_{K}^{\rm super}(x,q,a) \cong \frac{1}{2}\sum_{n\in \mathbb{Z}}(q^{-1} a^{1/2} x)^n f_n(q,a)$$ with $f_{-n}(q,a)=f_{n}(q,a)$, and that it has the following Weyl symmetry : $$F_{K}^{\rm super}(x^{-1},q,a) = F_{K}^{\rm super}(a^{-1}q^2x,q,a).$$

We can also ask another, much conjectural question regarding homology theories :

What’s $\mathcal{H}_{b,{\rm BPS}}^G$ categorifying $\hat{Z}_b^G(q)$? Is there a family of differentials analogous to that of [@DGR]?

We will pursue these questions in our future work.

$\mathrm{Spin}^{T^\vee}$-structures {#spint}
===================================

In this section we define a generalization of $\mathrm{Spin}^c$-structures which we will call $\mathrm{Spin}^{T^\vee}$-structures. These are what the labels $b$ of $\hat{Z}_b^G$ are geometrically.

Let $G$ be a Lie group and let $T < G$ be a maximal torus, and let $T^\vee \cong \mathfrak{h}^*/P$ be its dual.

First, the group $\mathrm{Spin}^{T^\vee}(n)$ is defined to be $$\mathrm{Spin}^{T^\vee}(n) := \mathrm{Spin}(n) \times_{\mathbb{Z}_2}T^\vee.$$ A $\mathrm{Spin}^{T^\vee}$-structure $\mathfrak{s}$ on an $n$-manifold $M^n$ is a principal $\mathrm{Spin}^{T^\vee}(n)$-bundle with an isomorphism between the natural $SO(n)$-bundle associated to $\mathfrak{s}$ and the frame bundle of $M^n$.

Note that the Weyl group naturally acts on $\mathrm{Spin}^{T^\vee}(n)$. It is important that we regard $T^\vee \cong \mathfrak{h}^*/P$ as a variety, not just a topological torus. There’s a natural first Chern class map $$\begin{aligned}
\mathrm{Spin}^{T^\vee}(M) &\rightarrow H^2(M;P) \\
\mathfrak{s}&\mapsto c_1(\mathfrak{s}).\end{aligned}$$ The following propositions are analogous to those of $\mathrm{Spin}^c$-structures.

$\mathrm{Spin}^{T^\vee}(M)$ is affinely isomorphic to $H^2(M;Q)$

On a 4-manifold $W$ with boundary $Y$, $\mathrm{Spin}^{T^\vee}(W)$ is canonically isomorphic to the set $\mathcal{C}_W\subseteq H^2(W;P)$ of characteristic covectors of $H_2(W,Y;Q)$; i.e. $\nu\in \mathcal{C}_W\subseteq H^2(W;P)$ iff $$(\nu,x) = \frac{1}{2}\langle x,x\rangle\,\mathrm{mod}\,2$$ for any $x\in H_2(W,Y;Q)$.

Good old Gauss sum reciprocity {#Gauss}
==============================

In this section we derive a formula which generalizes appendix A of [@GPPV] to any graph manifold with weakly negative definite linking matrix, with arbitrary gauge group $G$. Let $Y$ be a graph manifold obtained by a plumbing on a connected decorated graph $\Gamma$. (Each vertex $v$ of $\Gamma$ is decorated by an integer $a_v$ ‘framing’ and a nonnegative integer $g_v$ ‘genus’. We have set $g_v=0$ in Definition \[integralZhat\].) Then the WRT invariant $Z_{G_k}(Y)$ can be computed by [^7] $$Z_{G_k}(Y) \cong \sum_{\rm colorings}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \label{eq2.1}$$ where the vertex and the edge factors are[^8] $$\mathcal{V} = t_{\lambda\lambda}^{a_v} s_{\rho\lambda}^{2-2g_v-\deg v}$$ $$\mathcal{E} = s_{\mu\lambda}.$$ Here the $s, t$ matrices are as usual [^9] $$s_{\lambda \mu} = \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda),\mu)},$$ $$t_{\lambda \mu} = \delta_{\lambda \mu} q^{\frac{1}{2}(\lambda,\lambda)}q^{-\frac{1}{2}(\rho,\rho)}$$ with $q=e^{\frac{2\pi i}{m k}}$ and set of (shifted) colors being $$\lambda,\mu\in C = \{\lambda\in P_+ + \rho \,\vert\, (\lambda,\theta^\vee) < k\}.$$ Then these $s,t$ matrices are invariant (up to sign) under the action of the affine Weyl group $$W^a = W \ltimes kQ^\vee.$$ In fact, $C$ is simply the fundamental domain $P/W^a$. Using this fact, we can manipulate the form of $Z_{G_k}(Y)$ to write it in a Gauss sum reciprocity-friendly way : $$\begin{aligned}
(\ref{eq2.1}) &= \frac{1}{|W^V|} \sum_{\text{coloring}\in W(C)^{V}}\prod_{v\in V}\mathcal{V}_v\prod_{e\in E}\mathcal{E}_e \nonumber\\
&= \frac{1}{|W^V|} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \nonumber\\
&\quad\times \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}q^{\frac{a_v}{2}(\lambda_v,\lambda_v)} \prod_{(u_1,u_2)\in E}\sum_{w\in W}(-1)^{l(w)}q^{(w(\lambda_{u_1}),\lambda_{u_2})} \nonumber\\
&= \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \sum_{\lambda\in W(C)^{V}}\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} \nonumber\\
&\quad \times\sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}q^{\frac{1}{2}(\lambda, \mathrm{B}_s\lambda)} \label{midstep}\end{aligned}$$ Here, the self-adjoint matrix $\mathrm{B}_s$ is the symmetrization $\frac{1}{2}(\mathrm{B}_s^0 + {\mathrm{B}_s^0}^{\dagger})$ of a bilinear form $\mathrm{B}_s^0$ on $P^{V}$ characterized by the following properties :

-   For each $v$, $(\mu_v,\mathrm{B}_s^0\lambda_v) = a_v(\mu_v,\lambda_v)$

-   For each $u\neq v\in V$ with $(u,v)\not\in E$, $(\mu_u, \mathrm{B}_s^0 \lambda_v) = 0$

-   For each $u\neq v\in V$ with $(u,v)\in E$, $(\mu_u,\mathrm{B}_s^0\lambda_v) = (\mu_u,w_{uv}(\lambda_v))$ for some $w_{uv}\in W$

-   For each cycle $c=[v_1, \cdots, v_m]\in H_1(\Gamma)$, $\prod_{i=1}^{m}w_{v_i v_{i+1}} = s_c\in W$.

So far we have expressed the WRT invariant in the following form : $$Z_{G_k}(Y) = \sum_{s\in W^{b_1(\Gamma)}}(-1)^{l(s)}Z_{G_k}(Y,s).$$ Each $Z_{G_k}(Y,s)$ can be considered as the contribution from almost-Abelian flat connections twisted by $s$ [@CGPS]. In the following, let’s restrict our attention to the Abelian sector ($s=\text{id}$) for simplicity. In this case $\mathrm{B}_s = \mathrm{B}$ is the usual linking matrix.

To extend the range of summation $W(C)$ to $P/kQ^\vee$, i.e. to make sense of the summation even for colors upon which the action of $W^a$ is not free, we need to regularize the linear term $\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v}$. Let $\omega_1, \cdots, \omega_r\in P$ be a $\mathbb{Z}$-linear basis of $P$ (e.g. fundamental weights). We can then write $\lambda_v = \sum_{i = 1}^{r}n_{vi}\omega_i$ for some $n_{v1}, \cdots, n_{vr}\in \mathbb{Z}$. Thanks to Weyl denominator formula, we have $$\begin{aligned}
    \prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} 
    &= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(q^{\frac{(\lambda_v,\alpha)}{2}} - q^{-\frac{(\lambda_v,\alpha)}{2}}) \right)^{2-2g_v-\deg v} \nonumber\\
    &= \prod_{v\in V} \left( \prod_{\alpha\in \Delta_+}(\prod_{1\leq i\leq r}x_{vi}^{\frac{(\omega_i,\alpha)}{2}} - \prod_{1\leq i\leq r}x_{vi}^{-\frac{(\omega_i,\alpha)}{2}}) \right)^{2-2g_v-\deg v}\bigg|_{x_{vi} = q^{n_{vi}}}\label{regularize}\end{aligned}$$ In case $\deg v > 2$, this expression can be singular only when $$\left| \prod_{1\leq i\leq r}y_i^{\frac{(\omega_i,\alpha)}{2}}\right| = 1$$ for some $\alpha\in \Delta_+$. In terms of new variables $z_i := \log |x_i|$, this is simply $$\sum_{1\leq i\leq r} (\omega_i,\alpha) z_i = 0.$$ These are precisely the walls (hyperplanes) for Weyl reflections. Deforming the origin $z_1 = \cdots = z_r = 0$ to a complement of these walls is the same as a choice of a Weyl chamber. Moreover, for each choice of such a Weyl chamber, we can expand (\[regularize\]) as a geometric series. Therefore, we can regularize the linear term by taking an average of $|W|$ number of $q$-series, each determined by a choice of a Weyl chamber. To sum up, we can re-express the linear term in the following form : $$\prod_{v\in V} \left(\sum_{w\in W}(-1)^{l(w)}q^{(\lambda_v,w(\rho))}\right)^{2-2g_v-\deg v} 
    \overset{\text{regularize}}{=\joinrel=\joinrel=} \frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}} n_\ell \,q^{(\lambda,\ell)}$$ with $\delta_v = (2-2g_v-\deg v)\rho \mod Q$ and $n_\ell\in \mathbb{Z}$.

With this regularization in hand, we extend the range of summation and apply the Gauss sum reciprocity.[^10] Let $n\in \mathbb{Z}^+$ be such that $nP\subseteq Q^\vee\subseteq P$. Then we have[^11] $$\begin{aligned}
Z_{G_k}(Y,\text{id}) &\Rightarrow \frac{1}{|W|^{b_1(\Gamma)+1}} q^{-\frac{\sum a_i}{2}(\rho,\rho)}\left( \frac{i^{|\Delta_+|}}{|P/kQ^\vee|^{1/2}}\right)^{|V| + 1 - b_1(\Gamma)} \frac{1}{|(Q^\vee/nP)^V|} \nonumber\\
&\quad\times \sum_{\lambda \in P^{V}/nkP^{V}}q^{\frac{1}{2}(\lambda,\mathrm{B}\lambda)} \cdot \left(\frac{1}{|W^V|} \sum_{\ell\in \delta + Q^{V}}n_\ell q^{(\lambda,\ell)}\right) \nonumber\\
&= \frac{(-1)^{|\Delta_+||V|}}{|W|^{|V|+b_1(\Gamma)+1}}\left(\frac{i^{|\Delta_+|}}{|P/Q^\vee|^{1/2}} \right)^{-|V|-b_1(\Gamma)+1} q^{-\frac{\sum a_i}{2}(\rho,\rho)}k^{\frac{r}{2}(b_1(\Gamma) - 1)} \nonumber\\
&\quad\times \frac{e^{\frac{\pi i}{4}\sigma(\mathrm{B})}}{|\det \mathrm{B}|^{1/2}}\sum_{a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}}e^{-\pi i k(a,\mathrm{B}^{-1}a)}\sum_{b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}}e^{-2\pi i(a,\mathrm{B}^{-1}b)} \sum_{\ell\in \mathrm{B}Q^{V}+b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\label{rawresult}\end{aligned}$$ Note that $a$ and $b$ takes values in different sets : $a\in (P^\bullet)^{V}/\mathrm{B}(P^\bullet)^{V}$ while $b\in (Q^{V}+\delta)/\mathrm{B}Q^{V}$. The $a$ labels should be understood as ‘Abelian flat connections’ and the $b$ labels are ‘$\mathrm{Spin}^{T^\vee}$-structures’ of Section \[spint\].

Generalizing the definition of $\hat{Z}_b$ in appendix A of [@GPPV], we define $\hat{Z}_b^G$ for a weakly negative definite graph manifold $Y$ with gauge group $G$ as follows : $$\hat{Z}_b^G(Y;q) :\cong |W|^{-|V|-b_1(\Gamma)}q^{-\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho)}\sum_{\ell\in \mathrm{B}Q^{V} + b}n_\ell q^{-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)}\in |W|^{-|V|-b_1(\Gamma)} q^{\Delta_b}\mathbb{Z}[[q]] \label{Zhatdef}$$ where $$b\in (Q^{V}+\delta)/\mathrm{B}Q^{V},$$ $$\Delta_b = -\frac{\mathrm{Tr}\,\mathrm{B}}{2}(\rho,\rho) + \min_{\ell\in \mathrm{B}Q^{V}+b}-\frac{1}{2}(\ell,\mathrm{B}^{-1}\ell)\in \mathbb{Q}.$$

Here $\Delta_b$ should be thought of as a relative conformal weight among different $b$’s for the same 3-manifold. This is because we ignored the framing factor in (\[eq2.1\]), which shifts the overall $q$-power. It is easy to check that this definition is equivalent to Definition \[Zhatsgm\]. Finally, it is also clear from our derivation (\[rawresult\]) that the higher rank analog of the unfolded $S_{ab}$ matrix (appearing in Conjecture 2.6 of [@GPPV]) for the Abelian sector is simply $$S_{ab} = e^{-2\pi i (a,\mathrm{B}^{-1}b)}.$$ Of course for graph manifolds with $b_1(\Gamma)>0$ there are different sectors corresponding to each $s\in W^{b_1(\Gamma)}$, and we have to combine all of them to compare with the WRT invariant. See [@CGPS].

[^1]: By considering homology orientations, we should be able to fix the sign as well.

[^2]: Still, it is possible to derive our formula (\[integralZhat\]) from Mariño’s Chern-Simons matrix model or vice versa in case of Seifert manifolds. This is because Gaussian measure is the same as Laplace transform accompanied by $S_{ab}$ matrix.

[^3]: E.g. for $SU(2)$, $N_\ell^{A_1} = \text{sgn}((\ell,\alpha_1))$, and for $SU(3)$, $N^{A_2}_\ell = \mathrm{sgn}(\prod_{\alpha\in \Delta_+}(\ell,\alpha)) \min\{|(\ell,\alpha_1)|,|(\ell,\alpha_2)|\}$.

[^4]: In our notation, $K_{m,n}$ denotes the double twist knot with $m$ and $n$ full twists.

[^5]: That $\hat{Z}$’s for Brieskorn spheres should be expressed as sums of higher rank false theta functions was envisaged earlier in [@CCFGH].

[^6]: Of course in principle there should be quantum A-polynomials for all colors, with rank number of $x$ and $y$ variables, which annihilate $F_K^G({\bf x},q)$. However for simplicity we only consider symmetric representations here.

[^7]: Up to a framing factor, which is an integer power of $\zeta^3 = e^{2\pi i \frac{(k-h^\vee)\dim G/k}{24}} = e^{\frac{\dim G}{12} \pi i} q^{-\frac{m h^\vee\dim  G}{24}}$.

[^8]: Here, $2-2g_v-\deg v$ should probably be understood as the Euler characteristic of the Riemann surface $\Sigma_{g_v,\deg v}$ of genus $g_v$ with $\deg v$ punctures.

[^9]: We follow the convention and notations used in [@BK]. In particular, $P$ is the weight lattice, $Q$ is the root lattice, and $Q^\vee$ is the coroot lattice.

[^10]: In fact, this process of extending the range of colors and regularizing it is quite subtle, and in fact this may not be a strict equality for some values of $k$. We will deal with this subtlety in more detail elsewhere. Anyway the end result will be a topological invariant.

[^11]: See [@DT] for the version of Gauss sum reciprocity formula we use.
