Sachin Chauhan*

]Department of Physics, Indian Institute of Technology, Powai, Mumbai, 400076,
Maharashtra, India

# Gukov-Pei-Putrov-Vafa conjecture for $SU(N)/\mathbb{Z}_{m}$

<EMAIL_ADDRESS>Pichai Ramadevi<EMAIL_ADDRESS>[

###### Abstract

In our earlier work, we studied the $\hat{Z}$-invariant(or homological blocks)
for $SO(3)$ gauge group and we found it to be same as $\hat{Z}^{SU(2)}$. This
motivated us to study the $\hat{Z}$-invariant for quotient groups
$SU(N)/\mathbb{Z}_{m}$, where $m$ is some divisor of $N$. Interestingly, we
find that $\hat{Z}$-invariant is independent of $m$.

###### keywords:

3-manifold invariant, Quantum invariant, WRT invariant, Categorification of
WRT invariant, $\hat{Z}$-invariant, Homological blocks

##  1 Introduction

Over the past few decades, the notion of string dualities has emerged as a
unifying thread connecting diverse branches of mathematics. These dualities,
when expressed mathematically, have given rise to profound conjectures,
fostering a deeper understanding of the connections between various
mathematical domains. One notable duality in this context is the 3d-3d
correspondence as documented in works by Dimofte and others[1, 2, 3]. Such a
correspondence establishes a link between complex Chern-Simons theory on a
3-manifold $M$ based on gauge group $G_{\mathbb{C}}$ and 3d $\mathcal{N}=2$
$T[M;G]$ theory.

This correspondence was a result of compactification of 6d (2,0)
superconformal field theory(SCFT) of type ADE Lie algebra via topological
twisting along 3-manifold $M$ so that $\mathcal{N}=2$ supersymmetry remains
preserved in the remaining three flat directions:

$\text{6d}\;(2,0)\;\text{SCFT}\;\text{of type ADE Lie
algebra}\;\stackrel{{\scriptstyle\text{compactification on
M}}}{{\xrightarrow{\hskip 85.35826pt}}}\;\text{3d}\;\mathcal{N}=2\;\;T[M;G].$
(1)

The data of 3d $\mathcal{N}=2$ $T[M;G]$ theory is given by manifold $M$. In
other words, for every 3-manifold $M$ there would be a corresponding 3d
$\mathcal{N}=2$ $T[M;G]$ theory encoding the geometry and topology of $M$.
Many numerical and homological invariants of $M$ have been predicted by
studying $T[M;G]$ on various backgrounds[4, 5]. Further, the topology and
geometry of 3-manifolds is fairly well understood now111complete topological
classification of 3-manifolds is still an open problem, but still there is no
known way to explicitly identify 3d $\mathcal{N}=2$ $T[M;G]$ for a general
$M$.

In the seminal paper[6], Witten showed that Chern-Simons theory is a
topological gauge theory whose partition function
$Z^{G}_{k^{\prime}}[M;{\mathbbm{q}}]$ contains the toplogical information of
the underlying manifold $M$. Inspired by the work of Witten, Reshetikhin and
Turaev constructed invariants of 3-manifolds using surgery prescription on
links in $S^{3}$, commonly referred as Witten-Reshetikhin-Turaev(WRT)
invariants[7].

A concrete manifestation of the 3d-3d correspondence is the Gukov-Pei-Putrov-
Vafa (GPPV) conjecture. This conjecture connects the partition function of
Chern-Simons theory (or the Witten-Reshetikhin-Turaev (WRT) invariant) when
evaluated on a 3-manifold $M$ to a specific type of invariant expressed in
terms of $q$-series. In mathematical literature, these $q$-series valued
invariants are commonly referred to as $\hat{Z}$ or homological blocks, and
they are represented as vectors in the realm of $q$-series.

Initially, the exploration of $\hat{Z}$ was focused on $SU(2)$ gauge group
through the analytic continuation of the WRT invariant defined for plumbed
3-manifold $M(\Gamma)$. This investigation led to the definition of $\hat{Z}$
for negative semidefinite plumbed 3-manifolds. Subsequent research efforts, as
outlined in works[8, 9] extended the study of $\hat{Z}$ by examining the GPPV
conjecture in the contexts of $SU(N)$, $SO(3)$, and $OSp(1|2)$ gauge groups.

Furthermore, it is essential to recognize that the WRT invariant for gauge
group $G$, $\tau^{G}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]$, is defined at a
root of unity denoted as ${\mathbbm{q}}\left(=\exp\left(\frac{2\pi
i}{k^{\prime}}\right)\right)$, where $k^{\prime}$ signifies the renormalized
Chern-Simons level. In contrast, the variable $q$ within
$\hat{Z}^{G}_{b}[M(\Gamma);q]$ can be any complex number. Interestingly, as
$q$ approaches ${\mathbbm{q}}$, the relationship between these two invariants
expressed through a $S$-transformation matrix[5, 4]:

$\tau^{G}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]\cong\sum_{a,b}S_{ab}\hat{Z}^{G}_{b}[M(\Gamma);q]\Big{|}_{q\rightarrow{\mathbbm{q}}},$
(2)

where $\hat{Z}_{b}^{G}[M(\Gamma);q]$ admits the following physical
categorification:

$\hat{Z}_{b}^{G}[M(\Gamma);q]=\sum_{\begin{subarray}{c}i\in\mathbb{Z}+\Delta_{b}\\\
j\in\mathbb{Z}\end{subarray}}q^{i}(-1)^{j}\text{dim}\;\mathcal{H}_{b,G}^{i,j}.$
(3)

In this equation, $\mathcal{H}_{b,G}^{i,j}$ corresponds to the BPS sector of
the Hilbert space of $T[M;G]$ and $\Delta_{b}$ is a rational number specific
to a 3-manifold $M$. These insights provide a promising avenue for addressing
the long-standing categorification problem associated with the WRT invariant.

In our previous study[9], the $\hat{Z}$-invariant for the $SO(3)$ gauge group
was investigated by performing an analytical continuation of
$\tau_{k}^{SO(3)}[M(\Gamma);{\mathbbm{q}}]$ within a unit circle. Remarkably,
it was discovered that $\hat{Z}^{SO(3)}_{b}[M(\Gamma);q]$ is equivalent to
$\hat{Z}^{SU(2)}_{b}[M(\Gamma);q]$. This finding implies that the
$\hat{Z}$-invariant depends on the Lie algebra rather than the Lie group, as
$SU(2)$ and $SO(3)$ share the same Lie algebra. Furthermore, it is worth
noting that $SO(3)\cong SU(2)/\mathbb{Z}_{2}$ is the Langlands dual group of
$SU(2).$ Therefore, it raises the question of whether the equality
$\hat{Z}^{SU(2)}_{b}[M(\Gamma);q]=\hat{Z}^{SO(3)}_{b}[M(\Gamma);q]$ can be
attributed to this Langlands dual group correspondence. Additionally, for
higher rank gauge group $SU(N)$ with $N>2$ and $N$ not being a prime number
($N\notin\mathbb{P}$), there exists gauge groups between $SU(N)$ and
$SU(N)/\mathbb{Z}_{N}$. These groups are formed by taking a quotient of
$SU(N)$ with a subgroup $\mathbb{Z}_{m}$ of $\mathbb{Z}_{N}$. All these
quotient groups share the same $\mathfrak{su}(N)$ Lie algebra. Exploring the
$\hat{Z}$-invariant for $SU(N)/\mathbb{Z}_{m}$ quotient groups will eventually
answer whether $\hat{Z}$-invariant depends on $m$.

While the physics perspective suggests that $\hat{Z}$-invariant should be Lie
algebra dependent only as 3d $\mathcal{N}=2$ $T[M;G]$ obtained by
compactifying 6d $\mathcal{N}=(2,0)$ SCFT of type ADE Lie algebra on
3-manifold $M$. But in certain cases compactified theories does have Lie group
dependence instead of Lie algebra[10]. In this paper, our primary objective is
to address the question of whether the $\hat{Z}$-invariant exhibits dependence
on Lie group or Lie algebra. To achieve this, we explicitly study the Gukov-
Pei-Putrov-Vafa conjecture for gauge groups of the form
$SU(N)/\mathbb{Z}_{m}$. For this, we must first define the WRT invariant for
$SU(N)/\mathbb{Z}_{m}$ gauge group and then proceed with an analysis similar
to that conducted in Ref[8].

The organization of this paper is as follows: In section (2), we provide an
overview of the GPPV conjecture and $\hat{Z}$-invariant. Section (3)
introduces the appropriate formula for the WRT invariant for the quotient
group $SU(N)/\mathbb{Z}_{m}$. In section (4), we demonstrate how to decompose
the WRT invariant into $\hat{Z}$-invariant. Finally, we conclude in section
(5) by discussing open problems and offering concluding remarks.

#### Notations and conventions:

We use the following notation throughout this paper.

$M$: | 3-manifold
---|---
$G$: | Gauge group
$G_{\mathbb{C}}$: | Complex gauge group
$\mathfrak{g}$: | Lie algebra
$P$: | Weight lattice
$P_{+}$: | Cone of dominant integer weights
$\Lambda_{i}$: | Fundamental weight vector where $i=1,2,\ldots,r$
$Q$: | Root lattice
$P^{\prime}$: | Intermediate lattice between root and weight lattice
$(P^{\prime})^{\bullet}$: | Dual of lattice $P^{\prime}$
$(\lambda,\mu)$: | Denotes the inner product between any two weight vectors, $\lambda$ and $\mu$
$\Gamma$: | Plumbing graph of tree type
$L$: | Number of vertices in a plumbing graph $\Gamma$
$B$: | Linking matrix associated to plumbing graph $\Gamma$
$b_{\pm}$: | Number of positive and negative eigenvalues of $B$
$\sigma$: | Signature of linking matrix $B$ i.e. $\sigma=b_{+}-b_{-}$
$W$: | Weyl group
$|W|$: | Order of the Weyl group
$\omega_{i}$: |  $i^{\text{th}}$ element of the Weyl group
$\ell(\omega)$: | Length of Weyl group element $\omega$
$\rho$: | Weyl vector
$M(\Gamma)$: | Plumbed 3-manifold
$k^{\prime}$: | Renormalized Chern-Simons level
$k$: | Bare Chern-Simons level
${\mathbbm{q}}$: | Root of unity, $\exp(\frac{2\pi i}{k^{\prime}})$
$q$: | An arbitrary complex number inside the unit circle
$\text{deg }v$: | Denotes the degree of vertex $v$ in a plumbing graph $\Gamma$
$\tau^{G}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]$: | WRT invariant for gauge group $G$
$Z^{G}_{k^{\prime}}[M;{\mathbbm{q}}]$: | Chern-Simons partition function which is related to WRT invariant as $Z_{k^{\prime}}^{G}[M;{\mathbbm{q}}]={\tau_{k^{\prime}}^{G}[M;{\mathbbm{q}}]\over\tau_{k^{\prime}}^{G}[S^{2}\times S^{1};{\mathbbm{q}}]}$
$\hat{Z}^{G}_{b}[M(\Gamma);q]$: |  $\hat{Z}$-invariant labelled by index $b$ for gauge group $G$

##  2 Review of GPPV conjecture and $\hat{Z}$-invariant

In this section, we will give a brief survey of GPPV conjecture and
$\hat{Z}$-invariant. Moreover, in this paper, we would limit ourselves to the
case of rational homology sphere or plumbed 3-manifold coming from tree type
diagrams only. Therefore, we first introduce the plumbed 3-manifolds:

###  2.1 Plumbed 3-manifold

In general, any connected, closed, orientable 3-manifold can be obtained by
surgery on a framed link in $S^{3}$[11, 12]. In this context, we focus our
attention on the $L$-component link $\mathcal{L}$, composed of unknots with
framing $f_{i}$. The resulting manifold, which emerges through a surgical
operation on $\mathcal{L}$, is referred to as a plumbed 3-manifold. For
example, we represent a six component link $\mathcal{L}$ through a graph
$\Gamma$ with six vertices and call it a plumbing graph as shown in Figure 1.

Figure 1: An example of a plumbing graph $\Gamma$ and the corresponding link
$\mathcal{L}$

The degree of any vertex $v$ ($\text{deg}\;v$) is equal to the total number of
edges intersecting $v$. In Figure 1, we can see that vertices 2 and 5 have a
degree of 3, while vertices 1, 3, 4, and 6 each have a degree of 1. Further,
we associate a linking matrix $B$ with graph $\Gamma$. It is defined as
follows:

$B_{v_{1},v_{2}}=\begin{cases}1,~{}~{}~{}v_{1},v_{2}~{}\text{connected},\\\
f_{v},~{}~{}v_{1}=v_{2}=v,\\\ 0,~{}~{}~{}\text{otherwise}\end{cases}$ (4)

Moreover, we define the signature of the linking matrix $B$ as
$\sigma=b_{+}-b_{-}$ where $b_{\pm}$ denotes the number of positive and
negative eigenvalues respectively. So, depending on the sign of $\sigma$ we
referred the corresponding 3-manifold as positive or negative semidefinite
plumbed 3-manifold $M$. Three-manifolds obtained from a plumbing graph
$\Gamma$ or $\Gamma^{\prime}$ using surgery prescription are homeomorphic if
$\Gamma$ can be converted into $\Gamma^{\prime}$ using the following set of
transformations[13, 14]:

Figure 2: Kirby-Neumann moves which results in homeomorphic three-manifolds

###  2.2 GPPV conjecture

In Ref[4] Gukov et al. proposed a relation between WRT and $\hat{Z}$-invariant
for negative-definite plumbed 3-manifold $M$. This relation was obtained using
analytic continuation of $\tau_{k}^{SU(2)}[M(\Gamma),{\mathbbm{q}}]$ and
employing the Gauss sum reciprocity formula. The relation is as follows:

$\tau_{k}^{SU(2)}[M(\Gamma),{\mathbbm{q}}]=\frac{1}{2\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\,\times\\\ \sum_{a\in\mathrm{Coker}\,B}e^{-2\pi
i(k+2)(a,B^{-1}a)}\sum_{b\in 2\mathrm{Coker}\,B+\delta}e^{-2\pi
i(a,B^{-1}b)}\lim_{q\rightarrow{\mathbbm{q}}\left(=\exp(\frac{2\pi
i}{k+2})\right)}\hat{Z}_{b}^{SU(2)}[M(\Gamma);q],$ (5)

where $\delta=\\{\delta_{1},\delta_{2},\ldots,\delta_{L}\\}$ and
$\delta_{v}=\text{deg }v\;(\text{mod 2})$. In literature, this relation is
referred to as Gukov-Pei-Putrov-Vafa conjecture. Subsequently, this conjecture
was studied for $SU(N)$,$SO(3)$ and $OSp(1|2)$ groups in Ref[8, 9, 15]. For
the $SO(3)$ group, we deduced the conjecture in the following form:

$\tau_{k}^{SO(3)}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{2\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\,\sum_{a\in\mathrm{Coker}\,B}e^{{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\pi
i(2k+1)}(a,B^{-1}a)}\\\ \\\ \sum_{b\in 2\mathrm{Coker}\,B+\delta}e^{-\pi
i\big{(}a,B^{-1}(b{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+BI})\big{)}}\lim_{q\rightarrow{\mathbbm{q}}\left(=\exp(\frac{2\pi
i}{4k+2})\right)}\hat{Z}^{SU(2)}_{b}[M(\Gamma);q]~{}.$ (6)

From this expression, we see that $\hat{Z}$-invariant is same for both $SU(2)$
and $SO(3)$ groups. However, from equation (6), we see that the overall factor
undergoes modification, as highlighted in blue. Further, a proof of this
conjecture for $SU(2)$ group appeared in Ref[16]. We will now present a brief
overview of the physical and mathematical definitions of the
$\hat{Z}$-invariant.

###  2.3 $\hat{Z}$-invariant

$\hat{Z}$-invariants are $q$-series valued topological invariants of
3-manifold $M$. They admit the physical definition as a partition function of
$T[M;SU(2)]$ evaluated on a cigar geometry $D^{2}\times_{q}S^{1}$ i.e.

$\hat{Z}^{SU(2)}_{b}[M;q]=Z_{T[M;SU(2)]}(D^{2}\times_{q}S^{1};b)$ (7)

where $b$ is the index for these invariants and it belongs to
$\text{Spin}^{c}(M)$[17]. This relation can also be written in the following
way:

$\hat{Z}^{SU(2)}_{b}[M;q]=\sum_{\begin{subarray}{c}i\in\mathbb{Z}+\Delta_{b}\\\
j\in\mathbb{Z}\end{subarray}}q^{i}(-1)^{j}\text{dim}\;\mathcal{H}_{b,SU(2)}^{i,j}$
(8)

where $\Delta_{b}$ is a rational number222usually referred to as delta
invariant of $M$, see[18, 19] for more labeled by $\text{Spin}^{c}(M)$ and
$\mathcal{H}_{b,SU(2)}^{i,j}$ corresponds to the BPS sector of the Hilbert
space of $T[M;SU(2)]$. $\mathcal{H}_{b,SU(2)}^{i,j}$ are doubly-graded
homological invariants of $M$. Hence the $\hat{Z}^{SU(2)}_{b}[M;q]$ are often
referred to as homological blocks. Moreover one can take direct sum of all
these homological invariants labelled by $b$ as follows:

$\mathcal{H}_{D^{2},SU(2)}[M]=\bigoplus_{b\in\text{Spin}^{c}(M)/\mathbb{Z}_{2}}\;\;\bigoplus_{\begin{subarray}{c}i\in\mathbb{Z}+\Delta_{b},\\\
j\in\mathbb{Z}\end{subarray}}\mathcal{H}_{b,SU(2)}^{i,j}$ (9)

then this vector space $\mathcal{H}_{D^{2},SU(2)}[M]$ is widely seen as the
closed 3-manifold analog of Khovanov-Rozansky knot homology. From these
definitions, we can interpret coefficients of powers of $q$ arising in
$\hat{Z}$-invariant as counting the dimension of the Hilbert space of
$T[M;SU(2)]$.

For negative semidefinite 3-manifold $M(\Gamma)$, $\hat{Z}$-invariant admits
the following integral form for $\mathfrak{su}(2)$ Lie algebra333we are using
the Lie algebraic notation because we know $\hat{Z}$ is same for $SU(2)$ and
$SO(3)$ group:

$\hat{Z}_{b}^{\mathfrak{su}(2)}[M(\Gamma);q]=(-1)^{b_{+}}q^{\frac{3\sigma-\sum_{v}f_{v}}{4}}\cdot\text{v.p.}\int\limits_{|z_{v}|=1}\prod_{v\;\in\;\text{Vertices}}\frac{dz_{v}}{2\pi
iz_{v}}\,\left({z_{v}-1/z_{v}}\right)^{2-\text{deg}(v)}\cdot\left(\sum_{s\in
2B{\mathbb{Z}}^{L}+b}q^{-\frac{(s,B^{-1}s)}{4}}\prod_{i=1}^{L}z_{i}^{s_{i}}\right),$
(10)

and “v.p.” means that we take principle value integral. Similar expression of
$\hat{Z}$ for $SU(N)$, $OSp(1|2)$, $OSp(2|2)$ and $SU(2|1)$ for the plumbed
3-manifolds was found in Ref[8, 9, 20, 21]. Moreover, these invariants
exhibits the quantum modularity[22, 23]. For $SU(N)$ group,
$\hat{Z}$-invariant can be described as[8, 24]:

$\displaystyle\hat{Z}_{b}^{{\rm
SU}(N)}[M(\Gamma);q]=(-1)^{\frac{N(N-1)}{2}b_{+}}q^{\frac{3\sigma-\operatorname{Tr}{\rm
B}}{2}\frac{N^{3}-N}{12}}$ (11) $\displaystyle\hphantom{\hat{Z}_{b}^{{\rm
SU}(N)}(Y;q)=}{}\times{\rm v.p.}\oint_{|z_{vj}|=1}\prod_{v\in V}\prod_{1\leq
j\leq N-1}\frac{{\rm d}z_{vj}}{2\pi{\rm
i}z_{vj}}F_{3d}(z)\Theta_{2d}^{b}(z,q)$ (12)

with

$\displaystyle F_{3d}(z):=\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}\prod_{1\leq j\leq
N-1}z_{vj}^{(\Lambda_{j},\omega(\rho))}\right)^{2-\deg v}$
$\displaystyle\hphantom{F_{3d}(z)}{}=\prod_{v\in V}\left(\prod_{1\leq j<k\leq
N}\big{(}y_{vj}^{1/2}y_{vk}^{-1/2}-y_{vj}^{-1/2}y_{vk}^{1/2}\big{)}\right)^{2-\deg
v},$ $\displaystyle\Theta_{2d}^{b}(z,q):=\sum_{s\in{\rm
B}Q^{L}+b}q^{-\frac{1}{2}(s,{\rm B}^{-1}s)}\prod_{v\in V}\prod_{1\leq j\leq
N-1}z_{vj}^{-(\Lambda_{j},s_{v})},$

where $z_{j}=\frac{y_{j}}{y_{j+1}}$. Here, the principal value integral “v.p.”
implies the average over $|W|$ number of deformed contours, each associated
with a Weyl chamber. In the following section, we will focus on the WRT
invariant for quotient group $SU(N)/\mathbb{Z}_{m}$ which is necessary to
study the corresponding $\hat{Z}$-invariant.

##  3 WRT invariant for $SU(N)/\mathbb{Z}_{m}$

For a 3-manifold $M(\Gamma)$, we define the WRT invariant444we use the
normalization $\tau^{G}_{k^{\prime}}[S^{3},M]=1$ for quotient group
$SU(N)/\mathbb{Z}_{m}$ as follows[25, 26, 27]:

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\mathcal{\widetilde{S}}_{\rho\rho}^{L-1}\frac{\sum_{C^{L}}\prod_{v\in
V}\mathcal{V}_{v}\prod_{e\in
E}\mathcal{E}_{e}}{\left(\sum_{C}\mathcal{V}(+1\bullet)\right)^{b_{+}}\left(\sum_{C}\mathcal{V}(-1\bullet)\right)^{b_{-}}},$
(13)

where $\mathcal{V}$, $\mathcal{E}$ denotes the vertex and edge factors of
$L$-component plumbing graph $\Gamma$, $b_{\pm}$ represents the number of
positive and negative eigenvalues of the linking matrix $B$ and $\pm 1\bullet$
denotes the single vertex with $\pm 1$ framing. The summation is performed
over the set of allowed representations of the $SU(N)/\mathbb{Z}_{m}$ group,
which are:

$C=\\{\lambda\in(P_{+}\cap
P^{\prime})+\rho\;|\;(\lambda,\theta^{\vee})<k^{\prime}\\}.$ (14)

Here, $P_{+}$ represents the set of dominant weights, $\theta^{\vee}$ refers
to the maximal root, $\rho$ denotes the Weyl vector and $P^{\prime}$ is a
sublattice of $P$ such that there is an isomorphism between abelian group
$P/P^{\prime}$ and cyclic group $\mathbb{Z}_{m}(Q\subseteq P^{\prime}\subseteq
P)$. Furthermore, when $m=N$, then $P^{\prime}$ is simply the root lattice
$Q$, and when $m=1$, then $P^{\prime}=P$. The vertex and edge factor can be
expressed in terms of $\mathcal{\widetilde{S}}$ and $\mathcal{\widetilde{T}}$
matrices:

$\mathcal{V}_{v}=\mathcal{\widetilde{T}}_{\lambda\lambda}^{f_{v}}\mathcal{\widetilde{S}}_{\rho\lambda}^{2-\text{deg}v}\;,\;\;\mathcal{E}=\mathcal{\widetilde{S}}_{\lambda\mu}.$
(15)

The $\mathcal{\widetilde{S}}$ and $\mathcal{\widetilde{T}}$ matrices
are:555these matrices are exactly the usual modular transformation matrices
when $m=1$

$\mathcal{\widetilde{S}}_{\lambda\mu}=\frac{i^{|\Delta_{+}|}}{|P^{\prime}/k^{\prime}Q|^{1/2}}\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\omega(\lambda),\mu)},\;\;\;\;\;\mathcal{\widetilde{T}}_{\lambda\mu}=\delta_{\lambda\mu}{\mathbbm{q}}^{\frac{1}{2}(\lambda,\lambda)}{\mathbbm{q}}^{-\frac{1}{2}(\rho,\rho)}$
(16)

with

${\mathbbm{q}}=\exp\left(\frac{2\pi
i}{k^{\prime}}\right)~{}~{}~{}\text{and}~{}~{}~{}k^{\prime}\in\mathbb{Z}^{+},$
(17)

$W$, $Q$ and $k^{\prime}$ denotes the Weyl group, root lattice and
renormalized Chern-Simons level respectively.

#### Chern-Simons level for $SU(N)/\mathbb{Z}_{m}$ WRT invariant

In Ref[28], it was shown that the three-dimensional Chern-Simons gauge
theories with compact gauge group $G$ are classified by fourth cohomology
group of the classifying space of the gauge group: $H^{4}(BG;\mathbb{Z})$. The
classification parameter is the Chern-Simons level of the theory which is most
commonly denoted by $k$. The level $k^{\prime}$ is the renormalised Chern-
Simons level which is related to the bare Chern-Simons level $k$ for $SU(N)$
gauge group as follows:

$k^{\prime}=k+N.$ (18)

However for $SU(N)/\mathbb{Z}_{m}$ group which is non-simply connected, the
relation between $k$ and $k^{\prime}$ is as follows:

$k^{\prime}=\gamma k+N,$ (19)

where $\gamma$ is some integer which can be calculated by considering the
following short exact sequence:

$1\longrightarrow\mathbb{Z}_{m}\longrightarrow
SU(N)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}SU(N)/\mathbb{Z}_{m}\longrightarrow
1,$ (20)

where $\mathbb{Z}_{m}$ is the subgroup of $\mathbb{Z}_{N}$. Let $\alpha$ and
$\tilde{\alpha}$ be the generators of $H^{4}(B(SU(N));\mathbb{Z})$ and
$H^{4}(B(SU(N)/\mathbb{Z}_{m});\mathbb{Z})$ respectively. Then we have the
following relation:

$B\pi^{*}(\tilde{\alpha})=\gamma\alpha,$ (21)

where $\pi^{*}$ is the pullback map of $\pi$ in equation (20). The factor
$\gamma$ is simply determined by comparing the images of $\tilde{\alpha}$ and
$\alpha$ in the cohomology group $H^{*}(BT;\mathbb{Z})$ where $T$ is the
maximal torus of rank $N-1$. So, the factor $\gamma$ is found to be the
smallest integer for which the following equation is satisfied[28]:

$\frac{\gamma}{2}(\Lambda_{a},\Lambda_{a})\in\mathbb{Z},\;\;\forall\;a,$ (22)

where $\Lambda_{a}$’s are the fundamental weight vectors corresponding to the
subgroup $\mathbb{Z}_{m}$ of $\mathbb{Z}_{N}$. For $SU(N)/\mathbb{Z}_{N}$
group, $\gamma$ is determined to be $2N$ when $N$ is even and $N$ when $N$ is
odd:

$k^{\prime}=\begin{cases}Nk+N,\;\;\;\;\;\;\text{when $N$ is odd}\\\
2Nk+N,\;\;\;\;\text{when $N$ is even}\end{cases}.$ (23)

For clarity, we have included the computation of sublattice $P^{\prime}$ and
Chern-Simons level $k^{\prime}$ for certain non-simply connected groups in
Appendix (B). With this prescription of WRT for $SU(N)/\mathbb{Z}_{m}$, we
will now focus on the corresponding $\hat{Z}$ by studying the GPPV conjecture.

##  4 GPPV conjecture for $SU(N)/\mathbb{Z}_{m}$

As discussed in the previous section, the WRT invariant
$\tau_{k^{\prime}}^{SU(N)/\mathbb{Z}_{m}}[M(\Gamma);{\mathbbm{q}}]$ associated
with $M(\Gamma)$ is given by

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\mathcal{\widetilde{S}}_{\rho\rho}^{L-1}\frac{\sum_{C^{L}}\prod_{v\in
V}\mathcal{V}_{v}\prod_{e\in
E}\mathcal{E}_{e}}{\left(\sum_{C}\mathcal{V}(+1\bullet)\right)^{b_{+}}\left(\sum_{C}\mathcal{V}(-1\bullet)\right)^{b_{-}}}.$
(24)

For the sake of convenience, let’s express the above equation in the following
manner:

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\frac{\mathcal{F}[M(\Gamma);{\mathbbm{q}}]}{\left(\mathcal{F}[M(+1\bullet);{\mathbbm{q}}]\right)^{b_{+}}\left(\mathcal{F}[M(-1\bullet);{\mathbbm{q}}]\right)^{b_{+}}},$
(25)

where

$\mathcal{F}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}}\sum_{\begin{subarray}{c}C^{L}\end{subarray}}\prod_{v\in
V}\mathcal{V}_{v}\prod_{e\in E}\mathcal{E}_{e}.$ (26)

Similar to the $SU(2)$ group, we will have to perform Gauss decomposition of
eqn.(24) to extract the homological blocks from it. Hence we rewrite the above
equation (26) in a form so that we can use Gauss sum reciprocity formula[29].
We achieve this by extending the summation range $C^{L}$ over all Weyl
chambers $W(C)^{L}$. Note that the matrices are invariant under the action of
Weyl group elements.666upto a sign but that will not affect our final answer
for $\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]$ Hence
we can sum over all the Weyl chambers and divide by the number of Weyl
chambers to rewrite (26) as

$\mathcal{F}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}|W|^{L}}\sum_{\begin{subarray}{c}W(C)^{L}\end{subarray}}\prod_{v\in
V}\mathcal{V}_{v}\prod_{e\in
E}\mathcal{E}_{e},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(27)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}|W|^{L}}{\mathbbm{q}}^{-\frac{\sum_{j=1}^{L}f_{j}}{2}(\rho,\rho)}\left(\frac{i^{|\Delta_{+}|}}{|P^{\prime}/k^{\prime}Q|^{\frac{1}{2}}}\right)^{L+1}\sum_{\lambda\in
W(C)^{L}}\underbrace{\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\right)^{2-\text{deg}\;v}}_{\text{linear
in }\lambda_{v}}\times\\\ \\\
\underbrace{{\mathbbm{q}}^{\frac{f_{v}}{2}(\lambda_{v},\lambda_{v})}\prod_{(e_{1},e_{2})\in
E}\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\omega(\lambda_{e_{1}}),\lambda_{e_{2}})}}_{{\mathbbm{q}}^{\frac{1}{2}(\lambda,B\lambda)}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
(28)

where we have used the fact that sum and product can be interchanged. The set
over which the summation is being performed in equation (28) has now become
$W(C)$. We further extend it to the whole lattice $((P\cap
P^{\prime})+\rho)/k^{\prime}Q$ which is just $(P^{\prime}+\rho)/k^{\prime}Q$.
In doing so, we observe for some representations $\lambda$, the term linear in
$\lambda_{v}$ term (28) will be zero:

$\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\right)^{2-\text{deg}\;v}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(29)

Using Weyl denominator formula, the expression can be rewritten as

$\prod_{v\in
V}\left(\prod_{\alpha\in\Delta_{+}}\left({\mathbbm{q}}^{\frac{(\lambda_{v},\alpha)}{2}}-{\mathbbm{q}}^{-\frac{(\lambda_{v},\alpha)}{2}}\right)\right)^{2-\text{deg}\;v}=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Further, expressing $\lambda_{v}$ in terms of fundamental weight vectors
$\Lambda_{i}$ i.e.$\lambda_{v}=\sum_{j=1}^{r}n_{v_{j}}\Lambda_{j}$, the above
equation becomes

$~{}~{}~{}~{}~{}~{}~{}~{}\prod_{v\in
V}\left(\prod_{\alpha\in\Delta_{+}}\left(\prod_{1\leq j\leq
N-1}x_{v_{j}}^{\frac{(\Lambda_{j},\alpha)}{2}}-\prod_{1\leq j\leq
N-1}x_{v_{j}}^{-\frac{(\Lambda_{j},\alpha)}{2}}\right)\right)^{2-\text{deg}\;v}\Bigg{|}_{x_{v_{j}}={\mathbbm{q}}^{n_{v_{j}}}}=0.$
(30)

Hence, the points for which linear term in $\lambda_{v}$ becomes zero satisfy
the following equation

$\sum_{1\leq j\leq N-1}n_{v_{j}}(\Lambda_{j},\alpha)=0.$ (31)

These points causes the singularity when $\text{deg }v>2$. Hence we first need
to regularise the sum over these points. We introduce a parameter $\beta$ such
that

$\beta\in\mathbb{C}~{}~{}~{}\text{and}~{}~{}~{}0<|\beta|<1.$

Using this parameter, we can rewrite the linear term in $\lambda_{v}$ as:

$~{}~{}~{}~{}~{}~{}~{}\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\right)^{2-\text{deg}\;v}=\frac{1}{|W|^{L}}\prod_{v\in
V}\Bigg{[}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\beta^{f(\omega,\omega_{1})}\right)^{2-\text{deg}\;v}+\\\
\\\ \left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\beta^{f(\omega,\omega_{2})}\right)^{2-\text{deg}\;v}+\ldots+\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\beta^{f(\omega,\omega_{|W|})}\right)^{2-\text{deg}\;v}\Bigg{]}\Bigg{|}_{\beta\rightarrow
1},$ (32)

in which the function $f(\omega,\omega_{i})$ is defined as follows:

$f(\omega,\omega_{i}):=\begin{cases}1,~{}~{}~{}~{}~{}\text{when
}\omega=\omega_{i}\\\ 0,~{}~{}~{}~{}~{}\text{otherwise}.\end{cases}$ (33)

The RHS of equation (32) can be expanded as $|\beta|<1$:

$\frac{1}{|W|^{L}}\sum_{m\geq
0}\left(\sum_{s}\chi_{s}^{m}{\mathbbm{q}}^{(\lambda,s)}\right)\beta^{m},$ (34)

where $s=\\{s_{1},s_{2},\ldots,s_{L}\\}$ is some subset of $P^{L}$. Further,
we interchange the summation to rewrite equation (34) as

$\frac{1}{|W|^{L}}\sum_{s\in Q^{L}+\delta}\underbrace{\left(\sum_{m\geq
0}\chi_{s}^{m}\beta^{m}\right)}_{\xi^{\beta}_{s}}{\mathbbm{q}}^{(\lambda,s)}=\frac{1}{|W|^{L}}\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)},$ (35)

where $\chi_{s}^{m}\in\mathbb{Z}$. Hence, we can rewrite the linear term as
series in ${\mathbbm{q}}$, and its coefficients $\xi_{s}^{1}$ can be
determined by the following equation:

$\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\right)^{2-\text{deg}\;v}=\frac{1}{|W|^{L}}\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)}\Big{|}_{\beta\longrightarrow
1},$ (36)

where $\delta_{v}=(2-\text{deg}~{}v)\rho~{}\text{mod}~{}Q$ and
$\xi_{s}^{1}\in\mathbb{Z}$. In summary, we have rewritten the linear term as
some series in ${\mathbbm{q}}$. This series is obtained by taking an average
of individual geometric series in ${\mathbbm{q}}$, each determined by a
specific selection of Weyl chamber. This completes our regularization of
linear term. For clarity we have provided a detailed example in the appendix
(A). This led us to the following equation (37):

$~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{F}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}|W|}q^{-\frac{\sum
f_{j}}{2}(\rho,\rho)}\left(\frac{i^{|\Delta_{+}|}}{|P^{\prime}/k^{\prime}Q|^{\frac{1}{2}}}\right)^{L+1}\times\\\
\\\
\sum_{\lambda\in(P^{\prime}+\rho)^{L}/k^{\prime}Q^{L}}\left(\frac{1}{|W|^{L}}\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)}\right){\mathbbm{q}}^{\frac{1}{2}(\lambda,B\lambda)}\Big{|}_{\beta\longrightarrow
1}.~{}~{}~{}~{}~{}~{}~{}~{}$ (37)

Now, in order to use the reciprocity formula we replace $Q$ with $\eta
P^{\prime}$ for some positive integer $\eta$ as $\eta P^{\prime}\subseteq
Q\subseteq P^{\prime}\subseteq P$ and subsequently multiply it by the suitable
factor given by the order of quotient of these two lattices. Hence, the above
equation becomes:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{F}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}|W|}q^{-\frac{\sum
f_{j}}{2}(\rho,\rho)}\left(\frac{i^{|\Delta_{+}|}}{|P^{\prime}/k^{\prime}Q|^{\frac{1}{2}}}\right)^{L+1}\underbrace{\left(\frac{1}{|Q/\eta
P^{\prime}|^{L}}\right)}_{\text{factor which compensates for
replacing}~{}Q~{}\text{with}~{}\eta P^{\prime}}\times\\\ \\\
\sum_{\lambda\in(P^{\prime}+\rho)^{L}/\eta
k^{\prime}(P^{\prime})^{L}}\left(\frac{1}{|W|^{L}}\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)}\right){\mathbbm{q}}^{\frac{1}{2}(\lambda,B\lambda)}\Big{|}_{\beta\longrightarrow
1}.~{}~{}~{}~{}~{}~{}~{}~{}$ (38)

Since, we are interested in non-simply connected group $SU(N)/\mathbb{Z}_{m}$
for which $\rho\notin P^{\prime}$, we have to do the following shift in
$\lambda$: $\lambda\longrightarrow\lambda+\rho$. Subsequently, we get the
following:

$~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{F}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{(\mathcal{\widetilde{S}}_{\rho\rho})^{L+1}|W|^{L+1}}{\mathbbm{q}}^{-\frac{\sum
f_{j}}{2}(\rho,\rho)}\left(\frac{i^{|\Delta_{+}|}}{|P^{\prime}/k^{\prime}Q|^{\frac{1}{2}}}\right)^{L+1}\left(\frac{1}{|Q/\eta
P^{\prime}|^{L}}\right){\mathbbm{q}}^{\frac{1}{2}(\rho,B\rho)}\times\\\ \\\
\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\rho,s)}\sum_{\lambda\in(P^{\prime})^{L}/\eta
k^{\prime}(P^{\prime})^{L}}{\mathbbm{q}}^{\frac{1}{2}(\lambda,B\lambda)}{\mathbbm{q}}^{(\lambda,s+B\rho)}\Big{|}_{\beta\longrightarrow
1}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (39)

Now using Gauss sum reciprocity formula[29] and with the assumption that the
quadratic form,
$B:\mathbb{Z}^{L}\times\mathbb{Z}^{L}\longrightarrow\mathbb{Z}$ is negative
definite777that is $\sigma=-L$,888in following equation $l$ denotes the rank
of the lattice $(P^{\prime})^{L}$ and $\ell$ represents the length of the Weyl
group element, $\mathcal{F}[M(\Gamma);{\mathbbm{q}}]$ equals:

$=\left(\frac{1}{|W|\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\rho,\omega(\rho))}}\right)^{L+1}{\mathbbm{q}}^{-\frac{\sum
f_{j}}{2}(\rho,\rho)}\left(\frac{1}{|Q/\eta
P^{\prime}|^{L}}\right)\left(\frac{\exp(\frac{\pi i\sigma}{4})\;(\eta
k^{\prime})^{l/2}}{|\text{det}(\eta
B)|^{1/2}\;\;\text{Vol}[((P^{\prime})^{\bullet})^{L}]}\right)\times\\\ \\\
\sum_{a\in((P^{\prime})^{\bullet})^{L}/\eta
B((P^{\prime})^{\bullet})^{L}}\exp[-\pi
ik^{\prime}(a,B^{-1}a)]\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp[-2\pi
i(a,B^{-1}(b+B\rho))]\sum_{s\in
BQ^{L}+b}\xi_{s}^{\beta}{\mathbbm{q}}^{-\frac{(s,B^{-1}s)}{2}}\Big{|}_{\beta\longrightarrow
1},$ (40)

where $(P^{\prime})^{\bullet}$ denotes the dual lattice of $P^{\prime}$. The
WRT invariant
$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]$ including
the framing factor reduces to

$=\frac{1}{|W|^{L+1}}{\mathbbm{q}}^{-\frac{(\rho,\rho)}{2}(3L+\text{Tr}B)}\underbrace{\left(\sum_{a\in(P^{\prime})^{\bullet}/\eta(P^{\prime})^{\bullet}}\exp(\pi
ik^{\prime}(a,a)-2\pi
i(a,\rho))\right)^{-L}}_{|(P^{\prime})^{\bullet}/\eta(P^{\prime})^{\bullet}|^{-L}}\frac{1}{|\text{det}B|^{1/2}\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\rho,\omega(\rho))}}\times\\\ \\\
\underbrace{\sum_{a\in((P^{\prime})^{\bullet})^{L}/\eta
B((P^{\prime})^{\bullet})^{L}}}_{|(P^{\prime})^{\bullet}/\eta(P^{\prime})^{\bullet}|^{L}\sum_{a\in((P^{\prime})^{\bullet})^{L}/B((P^{\prime})^{\bullet})^{L}}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi
i(a,B^{-1}(b+B\rho)))\times\\\ \\\ \sum_{s\in
BQ^{L}+b}\xi_{s}^{\beta}{\mathbbm{q}}^{-\frac{(s,B^{-1}s)}{2}}\Big{|}_{\beta\longrightarrow
1},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(41)

which simplifies to the following:

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{|W|^{L+1}}\;\frac{q^{-\frac{(\rho,\rho)}{2}(3L+\text{Tr}B)}}{|\text{det}B|^{1/2}\sum_{\omega\in
W}(-1)^{\ell(\omega)}q^{(\rho,\omega(\rho))}}\sum_{a\in((P^{\prime})^{\bullet})^{L}/B((P^{\prime})^{\bullet})^{L}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\times\\\ \\\
\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi i(a,B^{-1}(b+B\rho)))\sum_{s\in
BQ^{L}+b}\xi_{s}^{\beta}{\mathbbm{q}}^{-\frac{(s,B^{-1}s)}{2}}\Big{|}_{\beta\longrightarrow
1}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (42)

Now, assuming that the following holds:

$\lim_{\beta\longrightarrow 1}\sum_{s\in
BQ^{L}+b}\xi_{s}^{\beta}{\mathbbm{q}}^{-\frac{(s,B^{-1}s)}{2}}=\lim_{q\rightarrow{\mathbbm{q}}}\sum_{s\in
BQ^{L}+b}\xi_{s}^{1}q^{-\frac{(s,B^{-1}s)}{2}},$ (43)

we finally obtain,

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{|W|^{L+1}}\;\frac{q^{-\frac{(\rho,\rho)}{2}(3L+\text{Tr}B)}}{|\text{det}B|^{1/2}\sum_{\omega\in
W}(-1)^{\ell(\omega)}q^{(\rho,\omega(\rho))}}\sum_{a\in((P^{\prime})^{\bullet})^{L}/B((P^{\prime})^{\bullet})^{L}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\times\\\ \\\
\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi i(a,B^{-1}(b+B\rho)))\sum_{s\in
BQ^{L}+b}\xi_{s}^{1}q^{-\frac{(s,B^{-1}s)}{2}}\Big{|}_{q\longrightarrow{\mathbbm{q}}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(44)

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\frac{1}{|W||\text{det}B|^{1/2}\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\rho,\omega(\rho))}}\sum_{a\in((P^{\prime})^{\bullet})^{L}/B((P^{\prime})^{\bullet})^{L}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\times\\\ \\\
\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi
i(a,B^{-1}(b+B\rho)))\lim_{q\rightarrow{\mathbbm{q}}}\underbrace{\hat{Z}^{\mathfrak{su}(N)}_{b}[M(\Gamma),q]}_{\text{independent
of }m}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ (45)

From equation (44) explicit expression of $\hat{Z}$-invariant can be read off
as:

$\hat{Z}^{\mathfrak{su}(N)}_{b}[M(\Gamma),q]=|W|^{-L}q^{-\frac{(3L+\text{Tr}\;B)}{2}(\rho,\rho)}\sum_{s\in
BQ^{L}+b}\xi_{s}^{1}q^{-\frac{(s,B^{-1}s)}{2}}\;\;\in\;|W|^{-L}q^{\Delta_{b}}\mathbb{Z}[[q]],$
(46)

Thus we have shown that $\hat{Z}$-invariant does not depend on $m$. The
overall factor which relates the $\hat{Z}$ with
$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]$ has the
$m$ dependence. This led us to the following conjecture:

######  Conjecture.

Let $M(\Gamma)$ be a negative definite plumbed 3-manifold. For non-simply
connected group $SU(N)/\mathbb{Z}_{m}$, WRT invariant can be decomposed in the
following form:

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{|W||\text{det}B|^{1/2}\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\rho,\omega(\rho))}}\sum_{a\in((P^{\prime})^{\bullet})^{L}/B((P^{\prime})^{\bullet})^{L}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\times~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\ \\\
\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi
i(a,B^{-1}(b{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+B\rho})))\lim_{q\rightarrow{\mathbbm{q}}}\hat{Z}^{\mathfrak{su}(N)}_{b}[M(\Gamma),q],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(47)

where $\bullet$ denotes the dual operation on the lattice $P^{\prime}$,
$k^{\prime}=\gamma k+N$ and ${\mathbbm{q}}=\exp\left(\frac{2\pi
i}{k^{\prime}}\right)$.

Note that for simply connected $SU(N)$ group, appendix(C), there is no shift
in $b$. The shift here, $b\longrightarrow b+B\rho$, is attributed to the non-
simply connected nature of $SU(N)/\mathbb{Z}_{m}$ group.

##  5 Conclusions and future directions

In this paper, we have worked out the explict form of GPPV conjecture for the
case of $SU(N)/\mathbb{Z}_{m}$ gauge group. We have found that the
$\hat{Z}$-invariant is independent of $\mathbb{Z}_{m}$ factor. In fact, it
turns out that the dependence of $\mathbb{Z}_{m}$ arises as an overall factor
to WRT invariant:

$\tau^{SU(N)/\mathbb{Z}_{m}}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\sum_{b}c_{b}(\mathbb{Z}_{m})\hat{Z}_{b}^{\mathfrak{su}(N)}[M(\Gamma);q]\Big{|}_{q\rightarrow
e^{\frac{2\pi i}{k^{\prime}}}}.$ (48)

We list some of the issues which we encountered while doing this exercise. We
hope to resolve these in our future works:

  * •

In the process of Gauss decomposition of WRT invariant, there exists
singularities correspondingly to the walls of the Weyl group. For certain
cases of quotient groups $SU(N)/\mathbb{Z}_{m}$, these singularities do not
arise by definition(For eg. $SU(2)/\mathbb{Z}_{2}$). We are interested in
observing the progression of the proof for the GPPV conjecture in these
particular instances. Although, recently a proof of this conjecture appeared
for simply laced Lie alegbras[30] but the proof is not available for non-
simply connected groups or quotient groups.

  * •

It would be nice to explore the extension of our research to encompass
positive semi-definite plumbed 3-manifolds and knot complements.

Acknowledgments

SC expresses gratitude to Pavel Putrov and Sunghyuk Park for numerous fruitful
discussions. SC is also appreciative of the MoU involving CAS-IITB and ICTP,
which provided the opportunity for a two-month visit to ICTP, where
significant progress related to this research was achieved. SC extends thanks
to the organizers of String-Math 2023, where a portion of this work was
presented. Additionally, SC would like to acknowledge the IoE cell at IIT
Bombay for providing financial support during the visit to String-Math 2023.
SC and PR would like to thank all the speakers as well as the organisers of
the Learning workshop on BPS states and 3-manifolds for discussions and
interactions. PR would like to acknowledge the ICTP’s Associate programme
where we made some progress during her visit as senior associate.

##  Appendix A Explicit calculation of $\hat{Z}^{\mathfrak{su}(3)}$-invariant
for a particular plumbing graph

In this appendix, we work out explicitly the $\hat{Z}$ for $\mathfrak{su}(3)$
Lie algebra for the following plumbed 3-manifold:

For $\mathfrak{su}(3)$ case,
$W=\\{1,s_{1},s_{2},s_{1}s_{2},s_{2}s_{1},s_{1}s_{2}s_{1}\\}$,
$\rho=\Lambda_{1}+\Lambda_{2}$ and $(\rho,\rho)=2$.

So, the $\hat{Z}$-invariant is:

$\hat{Z}^{\mathfrak{g}}_{b}[M(\Gamma),q]=|W|^{-L}q^{-\frac{(3L+\text{Tr}\;B)}{2}(\rho,\rho)}\sum_{s\in
BQ^{L}+b}\xi_{s}^{1}q^{-\frac{(s,B^{-1}s)}{2}}$ (49)

which in this case:

$\hat{Z}^{\mathfrak{su}(3)}_{b}[M(\Gamma),q]=\frac{q}{6^{4}}\sum_{s\in
BQ^{L}+b}\xi_{s}^{1}q^{-\frac{(s,B^{-1}s)}{2}}$ (50)

where $\xi_{s}^{1}$ is determined from the following equation:

$\prod_{v\in V}\left(\sum_{\omega\in
W}(-1)^{\ell(\omega)}{\mathbbm{q}}^{(\lambda_{v},\omega(\rho))}\right)^{2-\text{deg}\;v}=\frac{1}{6^{4}}\sum_{s\in
Q^{L}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)}\Big{|}_{\beta\longrightarrow
1}$ (51)

We write the LHS of above equation as follows:

$\displaystyle\prod_{v\in
V}\Big{(}{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(\Lambda_{1}+\Lambda_{2})}-{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(-\Lambda_{1}+2\Lambda_{2})}-{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(2\Lambda_{1}-\Lambda_{2})}$
$\displaystyle+{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(-2\Lambda_{1}+\Lambda_{2})}+{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(\Lambda_{1}-2\Lambda_{2})}-{\mathbbm{q}}^{(n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2})(-\Lambda_{1}-\Lambda_{2})}\Big{)}^{2-\text{deg
}v}$ (52)

where $\lambda_{v}=n^{v}_{1}\Lambda_{1}+n^{v}_{2}\Lambda_{2}$. The above
equation simplies to the following using
$\Lambda_{1}^{2}=\Lambda_{2}^{2}=\frac{2}{3}$ and
$\Lambda_{1}\Lambda_{2}=\frac{1}{3}$:

$\displaystyle\prod_{v\in
V}\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}$ (53)

We write this equation using regularising parameter $\beta$ as follows

$\displaystyle=\lim_{\beta\rightarrow 1}\frac{1}{6^{4}}\prod_{v\in
V}\Bigg{(}\Big{(}\beta{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\times$
$\displaystyle\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-\beta{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\times$
$\displaystyle\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-\beta{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\times$
$\displaystyle\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+\beta{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\times$
$\displaystyle\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+\beta{\mathbbm{q}}^{-n^{v}_{2}}-{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\times$
$\displaystyle\Big{(}{\mathbbm{q}}^{(n^{v}_{1}+n^{v}_{2})}-{\mathbbm{q}}^{n^{v}_{2}}-{\mathbbm{q}}^{n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{1}}+{\mathbbm{q}}^{-n^{v}_{2}}-\beta{\mathbbm{q}}^{-(n^{v}_{1}+n^{v}_{2})}\Big{)}^{2-\text{deg
}v}\Bigg{)},$ (54)

now we expand the term inside the parenthesis in the above equation as
$|\beta|<1$, to get the following:

$=\lim_{\beta\rightarrow 1}\left(\frac{1}{6^{4}}\sum_{s\in
Q^{4}+\delta}\xi_{s}^{\beta}{\mathbbm{q}}^{(\lambda,s)}\right)$ (55)

Above equation fixes the $\xi_{s}^{1}$. Moreover,
$\delta_{v}=0~{}\forall~{}v$, for $\mathfrak{su}(3)$ Lie algebra as
$\rho(=\Lambda_{1}+\Lambda_{2}=\alpha_{1}+\alpha_{2})\in Q$. Further for this
plumbing graph there is only one homological block which corresponds to $b=0$.
Using all this we obtain the $\hat{Z}$-invariant (49) as follows:

$\displaystyle\hat{Z}_{0}^{\mathfrak{su}(3)}[M(\Gamma);q]=q^{3/2}\Big{(}1-2q+2q^{3}+q^{4}-2q^{5}-2q^{8}+4q^{9}+2q^{10}-4q^{11}+2q^{13}-6q^{14}+2q^{15}-2q^{16}+4q^{18}-q^{20}$
$\displaystyle+4q^{21}-2q^{22}-4q^{23}+2q^{24}+2q^{25}-4q^{26}+6q^{30}-2q^{31}+6q^{33}-2q^{34}-2q^{35}-2q^{38}+q^{40}+\mathcal{O}(q^{41})\Big{)}.$
(56)

Note that the variable $q$ here is the analytically continued variable inside
the unit circle.

##  Appendix B Sublattice $P^{\prime}$ and Chern-Simons level $k^{\prime}$ for
$SU(4)/\mathbb{Z}_{2}$,$SU(6)/\mathbb{Z}_{2}$ and $SU(6)/\mathbb{Z}_{3}$

In this appendix, we will present the sublattice $P^{\prime}$ and Chern-Simons
level $k^{\prime}$ for some non-simply connected groups. The $i^{\text{th}}$
fundamental weight vector for $\mathfrak{su}(N)$ Lie algebra is given by:

$\Lambda_{i}=\frac{1}{N}(\underbrace{N-i,N-i,\ldots,N-i}_{i-\text{times}},\underbrace{-i,-i,\ldots,-i}_{(N-i)-\text{times}})~{}~{}~{}\text{where}~{}i\in\\{1,2,\ldots,N-1\\}.$
(57)

#### $SU(4)/\mathbb{Z}_{2}:$

The center of $SU(4)$ is $\mathbb{Z}_{4}$ which is isomorphic to
$\\{e,\Lambda_{1},\Lambda_{2},\Lambda_{3}\\}$. Further, the center of
$SU(4)/\mathbb{Z}_{2}$ is $\mathbb{Z}_{2}$ which is isomorphic to
$\\{e,\Lambda_{2}\\}$. The root lattice $Q$ of $\mathfrak{su}(4)$ Lie algebra,
which also corresponds to the equivalence class for identity group element, is
given by adding the following vectors:

$\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3})\Lambda_{2},(-n_{2}+2n_{3})\Lambda_{3}|n_{1},n_{2},n_{3}\in\mathbb{Z}\\},$
(58)

where $\Lambda_{1},\Lambda_{2}$ and $\Lambda_{3}$ are weight vectors. The
equivalence class for the group element $\Lambda_{2}$ corresponds to

$\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3}+1)\Lambda_{2},(-n_{2}+2n_{3})\Lambda_{3}|n_{1},n_{2},n_{3}\in\mathbb{Z}\\}.$
(59)

Hence the lattice $P^{\prime}$ is given by taking the union of two equivalence
classes (58) and (59).

Chern-Simons level $k^{\prime}=\gamma k+4$ and factor $\gamma$ is fixed by
smallest integer which satisfies the following equation:

$\frac{\gamma}{2}(\Lambda_{2},\Lambda_{2})\in\mathbb{Z}\implies\gamma=2,$ (60)

therefore, $k^{\prime}=2k+4$.

#### $SU(6)/\mathbb{Z}_{2}:$

The center of $SU(6)/\mathbb{Z}_{2}$ is
$\mathbb{Z}_{3}\cong\\{e,\Lambda_{2},\Lambda_{4}\\}$. Therefore the sublattice
$P^{\prime}$ corresponding to the allowed representations of
$SU(6)/\mathbb{Z}_{2}$ is given by:

$\displaystyle\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3})\Lambda_{2},(-n_{2}+2n_{3}-n_{4})\Lambda_{3},(-n_{3}+2n_{4}-n_{5})\Lambda_{4},(-n_{4}+2n_{5})\Lambda_{5}\\}\cup$
$\displaystyle\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3}+1)\Lambda_{2},(-n_{2}+2n_{3}-n_{4})\Lambda_{3},(-n_{3}+2n_{4}-n_{5})\Lambda_{4},(-n_{4}+2n_{5})\Lambda_{5}\\}\cup$
$\displaystyle\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3})\Lambda_{2},(-n_{2}+2n_{3}-n_{4})\Lambda_{3},(-n_{3}+2n_{4}-n_{5}+1)\Lambda_{4},(-n_{4}+2n_{5})\Lambda_{5}\\}.$
(61)

Chern-Simons level $k^{\prime}=\gamma k+6$ where $\gamma$ is fixed by
requiring:

$\frac{\gamma}{2}(\Lambda_{3},\Lambda_{3})\in\mathbb{Z}\implies 4,$ (62)

therefore, $k^{\prime}=4k+6$.

#### $SU(6)/\mathbb{Z}_{3}:$

For $\mathbb{Z}_{2}\cong\\{e,\Lambda_{3}\\}$, $P^{\prime}$ is:

$\displaystyle\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3})\Lambda_{2},(-n_{2}+2n_{3}-n_{4})\Lambda_{3},(-n_{3}+2n_{4}-n_{5})\Lambda_{4},(-n_{4}+2n_{5})\Lambda_{5}\\}\cup$
$\displaystyle\\{(2n_{1}-n_{2})\Lambda_{1},(-n_{1}+2n_{2}-n_{3})\Lambda_{2},(-n_{2}+2n_{3}-n_{4}+1)\Lambda_{3},(-n_{3}+2n_{4}-n_{5})\Lambda_{4},(-n_{4}+2n_{5})\Lambda_{5}\\}.$
(63)

and Chern-Simons level $k^{\prime}=\gamma k+6$ is fixed as follows:

$\frac{\gamma}{2}(\Lambda_{2},\Lambda_{2})\in\mathbb{Z}\;\;\text{and}\;\;\frac{\gamma}{2}(\Lambda_{4},\Lambda_{4})\in\mathbb{Z}\implies\gamma=3,$
(64)

hence, $k^{\prime}=3k+6$.

##  Appendix C GPPV conjecture for simply connected case: $SU(N)$

In order to compare the GPPV conjecture with the simply connected $SU(N)$
group, we present the GPPV conjecture for the $SU(N)$ group. For the $SU(N)$
group, with the weight lattice being $P$ and the root lattice being $Q$, the
WRT invariant can be decomposed as follows:

$\displaystyle\tau^{SU(N)}_{k^{\prime}}[M(\Gamma);{\mathbbm{q}}]=\frac{1}{|W||\text{det}B|^{1/2}\sum_{w\in
W}(-1)^{\ell(w)}{\mathbbm{q}}^{(\rho,w(\rho))}}\sum_{a\in(Q)^{L}/B(Q)^{L}}\exp(-\pi
ik^{\prime}(a,B^{-1}a))\times$
$\displaystyle\sum_{b\in(Q^{L}+\delta)/BQ^{L}}\exp(-2\pi
i(a,B^{-1}b))\lim_{q\rightarrow{\mathbbm{q}}}\hat{Z}^{\mathfrak{su}(N)}_{b}[M(\Gamma),q].$
(65)

where $B$ is the linking matrix for $\Gamma$.

## References

  *   * Dimofte et al. [2014] Dimofte, T., Gaiotto, D., Gukov, S.: Gauge Theories Labelled by Three-Manifolds. Commun. Math. Phys. 325, 367–419 (2014) https://doi.org/10.1007/s00220-013-1863-2 arXiv:1108.4389 [hep-th]
  * Dimofte et al. [2013] Dimofte, T., Gaiotto, D., Gukov, S.: 3-Manifolds and 3d Indices. Adv. Theor. Math. Phys. 17(5), 975–1076 (2013) https://doi.org/10.4310/ATMP.2013.v17.n5.a3 arXiv:1112.5179 [hep-th]
  * Dimofte et al. [2011] Dimofte, T., Gukov, S., Hollands, L.: Vortex Counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287 (2011) https://doi.org/10.1007/s11005-011-0531-8 arXiv:1006.0977 [hep-th]
  * Gukov et al. [2020] Gukov, S., Pei, D., Putrov, P., Vafa, C.: BPS spectra and 3-manifold invariants. J. Knot Theor. Ramifications 29(02), 2040003 (2020) https://doi.org/10.1142/S0218216520400039 arXiv:1701.06567 [hep-th]
  * Gukov et al. [2017] Gukov, S., Putrov, P., Vafa, C.: Fivebranes and 3-manifold homology. Journal of High Energy Physics 2017(7) (2017) https://doi.org/%****␣submission1.bbl␣Line␣125␣****10.1007/jhep07(2017)071
  * Witten [1989] Witten, E.: Quantum field theory and the jones polynomial. Communications in Mathematical Physics 121(3), 351–399 (1989)
  * Reshetikhin and Turaev [1991] Reshetikhin, N., Turaev, V.: Invariants of 3 - manifolds via link polynomials and quantum groups invent. Inventiones Mathematicae 103, 547–597 (1991) https://doi.org/10.1007/BF01239527
  * Park [2020] Park, S.: Higher rank hat z and fk. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 16, 044 (2020)
  * Chauhan and Ramadevi [2023] Chauhan, S., Ramadevi, P.: Z^-invariant for so (3) and osp (1| 2) groups. In: Annales Henri Poincaré, pp. 1–25 (2023). Springer
  * Freed and Teleman [2014] Freed, D.S., Teleman, C.: Relative quantum field theory. Communications in Mathematical Physics 326(2), 459–476 (2014)
  * Lickorish [1962] Lickorish, W.B.R.: A representation of orientable combinatorial 3-manifolds. Annals of Mathematics 76(3), 531–540 (1962). Accessed 2022-06-20
  * Wallace [1960] Wallace, A.H.: Modifications and cobounding manifolds. Canadian Journal of Mathematics 12, 503–528 (1960) https://doi.org/10.4153/CJM-1960-045-7
  * Kirby [1978] Kirby, R.: A calculus for framed links in s 3. Inventiones mathematicae 45(1), 35–56 (1978)
  * Neumann [1981] Neumann, W.D.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Transactions of the American Mathematical Society 268(2), 299–344 (1981)
  * Costantino et al. [2023] Costantino, F., Gukov, S., Putrov, P.: Non-Semisimple TQFT’s and BPS $q$-Series. SIGMA 19, 010 (2023) https://doi.org/10.3842/SIGMA.2023.010 arXiv:2107.14238 [math.GT]
  * Murakami [2023] Murakami, Y.: A proof of a conjecture of gukov-pei-putrov-vafa. arXiv preprint arXiv:2302.13526 (2023)
  * Gukov and Manolescu [2021] Gukov, S., Manolescu, C.: A two-variable series for knot complements. Quantum Topology 12(1) (2021)
  * Gukov et al. [2021] Gukov, S., Park, S., Putrov, P.: Cobordism invariants from bps q-series. In: Annales Henri Poincaré, vol. 22, pp. 4173–4203 (2021). Springer
  * Harichurn [2023] Harichurn, S.: On the $\Delta_{a}$ invariants in non-perturbative complex Chern-Simons theory (2023) arXiv:2306.11298 [math.GT]
  * Chae [2021] Chae, J.: Towards a q-series for osp (2| 2n). arXiv preprint arXiv:2106.09868 (2021)
  * Ferrari and Putrov [2020] Ferrari, F., Putrov, P.: Supergroups, q-series and 3-manifolds. arXiv preprint arXiv:2009.14196 (2020)
  * Cheng et al. [2023] Cheng, M.C.N., Coman, I., Passaro, D., Sgroi, G.: Quantum Modular $\hat{Z}^{G}$-Invariants (2023) arXiv:2304.03934 [hep-th]
  * Cheng et al. [2019a] Cheng, M.C.N., Ferrari, F., Sgroi, G.: Three-Manifold Quantum Invariants and Mock Theta Functions. Phil. Trans. Roy. Soc. Lond. 378(2163), 20180439 (2019) https://doi.org/10.1098/rsta.2018.0439 arXiv:1912.07997 [math.NT]
  * Cheng et al. [2019b] Cheng, M.C.N., Chun, S., Ferrari, F., Gukov, S., Harrison, S.M.: 3d Modularity. JHEP 10, 010 (2019) https://doi.org/10.1007/JHEP10(2019)010 arXiv:1809.10148 [hep-th]
  * Dedushenko et al. [2020] Dedushenko, M., Gukov, S., Nakajima, H., Pei, D., Ye, K.: 3d TQFTs from Argyres–Douglas theories. J. Phys. A 53(43), 43–01 (2020) https://doi.org/10.1088/1751-8121/abb481 arXiv:1809.04638 [hep-th]
  * Jeffrey [1992] Jeffrey, L.C.: Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation. Commun. Math. Phys. 147, 563–604 (1992) https://doi.org/10.1007/BF02097243
  * Kaul and Ramadevi [2001] Kaul, R.K., Ramadevi, P.: Three-manifold invariants from Chern-Simons field theory with arbitrary semi-simple gauge groups. Commun. Math. Phys. 217, 295–314 (2001) https://doi.org/10.1007/s002200000347 arXiv:hep-th/0005096
  * Dijkgraaf and Witten [1990] Dijkgraaf, R., Witten, E.: Topological Gauge Theories and Group Cohomology. Commun. Math. Phys. 129, 393 (1990) https://doi.org/10.1007/BF02096988
  * Deloup and Turaev [2007] Deloup, F., Turaev, V.: On reciprocity. Journal of Pure and Applied Algebra 208(1), 153–158 (2007) https://doi.org/10.1016/j.jpaa.2005.12.008
  * Murakami and Terashima [2023] Murakami, Y., Terashima, Y.: Homological blocks with simple lie algebras and witten–reshetikhin–turaev invariants. arXiv preprint arXiv:2308.04010 (2023)


# Machine Learning based tool for CMS RPC currents quality monitoring

E. Shumka<EMAIL_ADDRESS>A. Samalan  M. Tytgat  M. El Sawy  G.A. Alves
F. Marujo  E.A. Coelho  E.M. Da Costa  H. Nogima  A. Santoro  S. Fonseca De
Souza  D. De Jesus Damiao  M. Thiel  K. Mota Amarilo  M. Barroso Ferreira
Filho  A. Aleksandrov  R. Hadjiiska  P. Iaydjiev  M. Rodozov  M. Shopova  G.
Soultanov  A. Dimitrov  L. Litov  B. Pavlov  P. Petkov  A. Petrov  S.J. Qian
H. Kou  Z.-A. Liu  J. Zhao  J. Song  Q. Hou  W. Diao  P. Cao  C. Avila  D.
Barbosa  A. Cabrera  A. Florez  J. Fraga  J. Reyes  Y. Assran  M.A. Mahmoud
Y. Mohammed  ,I. Crotty  I. Laktineh  G. Grenier  M. Gouzevitch  L. Mirabito
K. Shchablo  I. Bagaturia  I. Lomidze  Z. Tsamalaidze  V. Amoozegar  B.
Boghrati  M. Ebraimi  M. Mohammadi Najafabadi  E. Zareian  M. Abbrescia  G.
Iaselli  G. Pugliese  F. Loddo  N. De Filippis  ,R. Aly  D. Ramos  W.
Elmetenawee  S. Leszki  I. Margjeka  D. Paesani  L. Benussi  S. Bianco  D.
Piccolo  S. Meola  S. Buontempo  F. Carnevali  L. Lista  P. Paolucci  F.
Fienga  A. Braghieri  P. Salvini  P. Montagna  C. Riccardi  P. Vitulo  E.
Asilar  J. Choi  T.J. Kim  S.Y. Choi  B. Hong  K.S. Lee  H.Y. Oh  J. Goh  I.
Yu  C. Uribe Estrada  I. Pedraza  H. Castilla-Valdez  A. Sanchez-Hernandez  R.
L. Fernandez  M. Ramirez-Garcia  E. Vazquez  M. A. Shah  N. Zaganidis  A. Radi
H. Hoorani  S. Muhammad  A. Ahmad  I. Asghar  W.A. Khan  J. Eysermans  F.
Torres Da Silva De Araujo
on behalf of the CMS Muon Group  Ghent University, Dept. of Physics and
Astronomy, Proeftuinstraat 86, B-9000 Ghent, Belgium.  Université Libre de
Bruxelles, Avenue Franklin Roosevelt 50-1050 Bruxelles, Belgium.  Centro
Brasileiro Pesquisas Fisicas, R. Dr. Xavier Sigaud, 150 - Urca, Rio de Janeiro
- RJ, 22290-180, Brazil.  Dep. de Fisica Nuclear e Altas Energias, Instituto
de Fisica, Universidade do Estado do Rio de Janeiro, Rua Sao Francisco Xavier,
524, BR - Rio de Janeiro 20559-900, RJ, Brazil.  Bulgarian Academy of
Sciences, Inst. for Nucl. Res. and Nucl. Energy, Tzarigradsko shaussee
Boulevard 72, BG-1784 Sofia, Bulgaria.  Faculty of Physics, University of
Sofia,5 James Bourchier Boulevard, BG-1164 Sofia, Bulgaria.  School of
Physics, Peking University, Beijing 100871, China.  State Key Laboratory of
Particle Detection and Electronics, Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China.  University of Chinese Academy of
Sciences, No.19(A) Yuquan Road, Shijingshan District, Beijing 100049, China.
Universidad de Los Andes, Carrera 1, no. 18A - 12, Bogotá, Colombia  Egyptian
Network for High Energy Physics, Academy of Scientific Research and
Technology, 101 Kasr El-Einy St. Cairo Egypt.  Suez University, Elsalam City,
Suez - Cairo Road, Suez 43522, Egypt.  Center for High Energy Physics(CHEP-
FU), Faculty of Science, Fayoum University, 63514 El-Fayoum, Egypt.
Department of Physics, Faculty of Science, Ain Shams University, Cairo, Egypt.
Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR
5822,F-69622, Villeurbanne, France.  Georgian Technical University, 77 Kostava
Str., Tbilisi 0175, Georgia.  School of Particles and Accelerators, Institute
for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran.
School of Engineering, Damghan University, Damghan, 3671641167, Iran.  INFN,
Sezione di Bari, Via Orabona 4, IT-70126 Bari, Italy.  INFN, Laboratori
Nazionali di Frascati (LNF), Via Enrico Fermi 40, IT-00044 Frascati, Italy.
INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, Via Cintia, IT-80126
Napoli, Italy.  Dipartimento di Ingegneria Elettrica e delle Tecnologie
dell’Informazione - Università Degli Studi di Napoli Federico II, IT-80126
Napoli, Italy.  INFN, Sezione di Pavia, Via Bassi 6, IT-Pavia, Italy.  INFN,
Sezione di Pavia and University of Pavia, Via Bassi 6, IT-Pavia, Italy.
Hanyang University, 222 Wangsimni-ro, Sageun-dong, Seongdong-gu, Seoul,
Republic of Korea.  Korea University, Department of Physics, 145 Anam-ro,
Seongbuk-gu, Seoul 02841, Republic of Korea.  Kyung Hee University, 26
Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea.  Sungkyunkwan
University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, Republic of
Korea.  Benemerita Universidad Autonoma de Puebla, Puebla, Mexico.  Cinvestav,
Av. Instituto Politécnico Nacional No. 2508, Colonia San Pedro Zacatenco, CP
07360, Ciudad de Mexico D.F., Mexico.  Universidad Iberoamericana, Mexico
City, Mexico.  Sultan Qaboos University, Al Khoudh, Muscat 123, Oman.
National Centre for Physics, Quaid-i-Azam University, Islamabad, Pakistan.
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, United States.  III. Physikalisches Institut (A), RWTH Aachen
University, Sommerfeldstrasse D-52056, Aachen, Germany.  Physics Department,
Faculty of Science, Helwan University, Ain Helwan 11795, Cairo, Egypt.

###### Abstract

The muon system of the CERN Compact Muon Solenoid (CMS) experiment includes
more than a thousand Resistive Plate Chambers (RPC). They are gaseous
detectors operated in the hostile environment of the CMS underground cavern on
the Large Hadron Collider where pp luminosities of up to $2\times 10^{34}$
$\text{cm}^{-2}\text{s}^{-1}$ are routinely achieved. The CMS RPC system
performance is constantly monitored and the detector is regularly maintained
to ensure stable operation. The main monitorable characteristics are dark
current, efficiency for muon detection, noise rate etc. Herein we describe an
automated tool for CMS RPC current monitoring which uses Machine Learning
techniques. We further elaborate on the dedicated generalized linear model
proposed already and add autoencoder models for self-consistent predictions as
well as hybrid models to allow for RPC current predictions in a distant
future.

††journal: NIMA

##  1 Introduction

The muon system of the CMS experiment [1] includes 1056 Resistive Plate
Chambers (RPC) operated at nominal high voltages (HV) of 9-10 kV. Monitoring
their dark current evolution, spotting deviations from normal performance and
anticipating an HV failure that would immediately propagate to higher detector
control levels is an unfeasible task for an online operator. Detector
parameters are abundant [2], thus HV problems manifest differently, making the
human intervention inefficient. Therefore, an automated process with built-in
notification logic and mechanism is highly sought after for development and
further implementation. Being able to spot increasing current tendencies, for
example, before they lead to an error is very important for controlling
detector operation. An automated tool that performs anomaly detection for the
RPC currents by using Machine Learning (ML) methods is presented here.
Two types of ML approaches are used: Generalized Linear Models (GLM) and
Autoencoders.
In the GLM case, a set of parameters such as environmental conditions, LHC
parameters and detector working points are used to characterize the behavior
of the current.
In the autoencoder case, the full set of the RPC HV system currents is used as
an input and the autoencoder network is trained to reproduce these inputs onto
the output neurons.
Both approaches show very good predictive capabilities that are the basis for
the monitoring tool. All the developed tools are integrated in a framework
that can be easily accessed and controlled by a specially developed Web User
Interface that allows the end-users to work with the monitoring tool in a
simple manner. It is being deployed for use during the CERN LHC Run-3 data-
taking period.

##  2 Generalized Linear Model

The GLM depicted in Fig. 1 is a generalization of a simple linear regression

Figure 1: The structure of the GLM

used to model the current as a function of the following sets of parameters:

  * 1.

Environmental conditions: temperature (T), relative humidity (RH) and pressure
(P)

  * 2.

LHC parameters: instantaneous luminosity (L) and integrated luminosity
($\sum$L)

  * 3.

Applied HV

  * 4.

Combined terms: $\text{L}\times\text{exp(HV/P)}$ and
$(\sum\text{HV}/\text{P})\Delta t$, where $\Delta t$ is the length of the time
period with no luminosity

The $\sum$L term replaces the $\Delta t$ term used in the initially proposed
model [3]. The improvement is inspired by [4]. The first combined term is to
account for the exponential increase of gas multiplication with the raising of
HV while the second one is to account for the chamber relaxation and the drop
of the current baseline during cosmic data taking, when there is no beam
luminosity and the chambers are at their working point. All the remaining
terms and the motivation for including them are discussed in [3].

##  3 Autoencoder

In contrast to the GLM approach, where we use detailed knowledge for the
physical processes taking place in a particular type of detector in order to
build the ML model, in this section we take a more general approach, namely
develop an ML model based on cross-correlation between different detector
modules, thus applicable for detector systems consisting of a large number of
RPC chambers. We develop an ML algorithm based on an autoencoder model.
Autoencoders are neural networks that are trained to encode the input into a
number of neurons that is lower than the number of inputs themselves and then
decode that same information onto the output layer (Fig. 2).

Figure 2: Topology of an autoencoder

During the learning stage, the autoencoder is supposed to learn the collective
behavior of all the RPC chambers. Such an autoencoder could be used later on
to spot an anomalous behavior of a single or a small subset of RPC chambers.

In this work, the set of RPC currents at a given moment in time is given as an
input to the autoencoder and the network is trained to reproduce them on its
output layer. The number of input and output neurons is 773, which corresponds
to the number of HV channels in the RPC system. The hidden layers count
respectively: 512, 128, 64, 128 and 512 neurons.

##  4 Hybrid network

As discussed above, GLM describes individual RPC chamber behavior while the
autoencoder describes collective correlations of the whole system. In order to
use their best qualities, we combine the two approaches into a model, referred
to as a hybrid network. In this model, a set of GLM equal in size to the
number of HV channels provide as output the currents for a given moment in
time. These currents are then used as inputs for an autoencoder, as shown in
Fig. 3.

Figure 3: The hybrid network (the left and right parts of this figure are the
same as Figs. 1 and 2, respectively)

The hybrid network is tested in a distant prediction scenario, where the end
of the training period is separated in time (e.g. 1 year) from the beginning
of the prediction period. Its performance in such a scenario (Section 7) shows
that it can be used as indication for current values that we could expect on a
system level for some specified conditions (e.g. the luminosity of the High-
Luminosity LHC).

##  5 Monitoring tool

The accurate predictions of the currents performed by both the GLM and
autoencoder can be used to detect anomalies in the RPC detector current
performance. The implemented tool follows the workflow presented in the
flowchart in Fig. 4.

Figure 4: Monitoring tool workflow

Raw data coming from the CMS non-physics event bus, referred to as online
condition data, are written in the $\text{cms}\\_\text{omds}\\_\text{adg}$
database copy. For each point in time for which data is available, the tool
performs comparisons between the measured and predicted RPC currents. If
differences higher than some predetermined threshold values are detected for a
given HV channel, a flag is raised and the case of that particular channel is
followed. There are two thresholds, the lower one inducing a warning and the
higher one inducing an error. After a specified number of points in time, the
running average of the differences is calculated and if this average exceeds
the thresholds, a warning or an error is sent to the end-users. This allows
for the detection of problematic HV channels before they result in an HV
channel trip.

##  6 Software implementation

The monitoring tool is programmed in Python. Tensorflow [5] is used for the
implementation of ML. The software is conceptualized and implemented with
modularity in mind (Fig. 5).

Figure 5: Software structure

All modules communicate back-and-forth with a database. The ”Trainer Module”
reads the training data from a table and after performing the training, writes
back the ML model parameters in another database table. The ”Estimator Module”
loads the models and performs predictions, which are also stored into the
database. Finally, the ”Notifications Manager” searches for anomalies in the
current values, as described in the previous section and provides
notifications.

(a) Mean = $0.21$ $\mu$A; $\sigma=0.59$ $\mu$A

(b) MAE = $0.72$ $\mu$A; MSE = $3.24\mu\text{A}^{2}$

Figure 6: GLM LT performance

(a) Mean = $0.14$ $\mu$A; $\sigma$ = $0.83$ $\mu$A

(b) MAE = $0.49$ $\mu$A; MSE = $1.39\mu\text{A}^{2}$

Figure 7: Autoencoder ST performance

(a) Mean = $0.60$ $\mu$A; $\sigma$ = $2.49$ $\mu$A

(b) MAE = $2.09$ $\mu$A; MSE = $23.19\mu\text{A}^{2}$

Figure 8: Hybrid network performance

##  7 Performance results

ML model performance validation is done for three different training
scenarios:

  * 1.

Short-term training (ST), with data from May to September 2018. Such models
are able to spot a rapid increase in the RPC currents.

  * 2.

Mid-term training (MT), with data from July 2017 to July 2018, appropriate for
describing the seasonal behavior of the currents.

  * 3.

Long-term training (LT), with data from May 2016 to July 2018, appropriate for
modelling the overall RPC currents evolution.

All models are tested against the RPC currents measured in the two-month
period between September and October of 2018. These tests show that GLM
performs best in LT scenario (Fig. 6), while the autoencoder performs best in
ST (Fig. 7). The Mean Absolute Error (MAE) and Mean Squared Error (MSE), which
are used as performance metrics, are defined as:

$\text{MAE}=\displaystyle\sum_{i=1}^{N}\frac{|\text{I}_{mon}^{i}-\text{I}_{pred}^{i}|}{\text{N}}$
(1)

Table 1: Performance results Model class |  Training period |  Prediction period |  1D histo mean [$\mu\text{A}$] |  1D histo sigma [$\mu\text{A}$] |  2D histo MAE [$\mu\text{A}$] |  2D histo MSE [$\mu\text{A}^{2}$]
---|---|---|---|---|---|---
GLMv2 |  18-05-01 to 18-09-01 |  18-09-01 to 18-10-30 |  -0.02 |  1.65 |  1.23 |  7.62
GLMv2 |  17-07-01 to 18-07-01 |  18-09-01 to 18-10-30 |  0.33 |  1.66 |  1.23 |  7.42
GLMv2 |  16-05-01 to 18-07-01 |  18-09-01 to 18-10-30 |  0.21 |  0.59 |  0.72 |  3.24
Autoencoder |  18-05-01 to 18-09-01 |  18-09-01 to 18-10-30 |  0.14 |  0.83 |  0.49 |  1.39
Autoencoder |  17-07-01 to 18-07-01 |  18-09-01 to 18-10-30 |  0.69 |  1.44 |  0.96 |  4.18
Autoencoder |  16-05-01 to 18-07-01 |  18-09-01 to 18-10-30 |  0.42 |  1.40 |  0.85 |  3.16
GLMv2 |  16-05-01 to 17-07-01 |  18-09-01 to 18-09-30 |  -0.24 |  2.59 |  1.92 |  18.69
Autoencoder |  16-05-01 to 17-07-01 |  18-09-01 to 18-09-30 |  0.06 |  2.51 |  2.14 |  22.57
Hybrid |  16-05-01 to 17-07-01 |  18-09-01 to 18-09-30 |  0.60 |  2.49 |  2.09 |  23.19
Figure 9: Screenshot of the Web User Interface

$\text{MSE}=\displaystyle\sum_{i=1}^{N}\frac{(\text{I}_{mon}^{i}-\text{I}_{pred}^{i})^{2}}{\text{N}}$
(2)

The sigma of the histogram in both cases is $<1\mu A$, which shows that both
models have excellent predictive capabilities consistent with the uncertainty
in the current measurement which is also of that same order. All performance
results are shown in Table 1.
Fig. 10 shows the case of an RPC chamber where the predicted current
increasingly diverges from the measured one. It was found that this
discrepancy could be explained with the appearance of a gas leak in this
chamber around the same time.

Figure 10: Monitored and predicted currents for an RPC chamber in W-1 of the
CMS barrel

##  8 Deployment on the CERN PaaS platform

The monitoring tool is accessible through a Web User Interface that is being
deployed on the CERN Platform-as-a-Service (PaaS) virtual environment (Fig.
9). It is based on OpenShift [6], a platform which allows for containerized
application deployment.

##  9 Conclusions

We use Machine Learning (ML) methods for anomaly detection in the current
behavior of CMS Resistive Plate Chambers. The excellent accuracy of the ML
model predictions allow us to implement a powerful monitoring tool which
notifies the end-users about potential high-voltage channel deviations from
normal behavior and increased risk of operational failures. The monitoring
tool has been developed and will be fully deployed for use during the Year-End
Technical Stop (YETS22/23).

## Acknowledgments

We congratulate our colleagues in the CERN accelerator departments for the
excellent performance of the LHC and thank the technical and administrative
staffs at CERN and at other CMS institutes for their contributions to the
success of the CMS effort. In addition, we gratefully acknowledge the
computing centres and personnel of the Worldwide LHC Computing Grid and other
centres for delivering so effectively the computing infrastructure essential
to our analyses. Finally, we acknowledge the enduring support for the
construction and operation of the LHC, the CMS detector, and the supporting
computing infrastructure provided by the following funding agencies: FWO
(Belgium); CNPq, CAPES and FAPERJ (Brazil); MES and BNSF (Bulgaria); CERN;
CAS, MoST, and NSFC (China); MINCIENCIAS (Colombia); CEA and CNRS/IN2P3
(France); SRNSFG (Georgia); DAE and DST (India); IPM (Iran); INFN (Italy);
MSIP and NRF (Republic of Korea); BUAP, CINVESTAV, CONACYT, LNS, SEP, and
UASLP-FAI (Mexico); PAEC (Pakistan); DOE and NSF (USA).

## References

  * [1] CMS Collaboration, ”The CMS experiment at the CERN LHC”, JINST 3 (2008) S08004.
URL https://doi.org/10.1088/1748-0221/3/08/S08004

  * [2] CMS Collaboration, ”Performance of the CMS muon detector and muon reconstruction with proton-proton collisions at $\sqrt{s}$ = 13 TeV”, JINST 13 (2018) P06015.
URL https://doi.org/10.48550/arXiv.1804.04528

  * [3] A. Samalan et al., ”A new approach for CMS RPC currents monitoring using Machine Learning techniques”, JINST 15 (2020) C10009.
URL https://dx.doi.org/10.1088/1748-0221/15/10/C10009

  * [4] A. Gelmi, R. Guida, B. Mandelli, on behalf of the CMS Muon Group, ”Gas mixture quality studies for the CMS RPC detectors during LHC Run 2”, JINST 16 (2021) C04004.
URL https://dx.doi.org/10.1088/1748-0221/16/04/C04004

  * [5] Tensorflow (2022).
URL https://www.tensorflow.org/

  * [6] Red Hat, OpenShift documentation (2022).
URL https://docs.openshift.com/


# Improving the Quality of Commit Messages
in Students’ Projects

Iris Ma  Department of Informatics
University of California, Irvine
Irvine, USA
<EMAIL_ADDRESS>Cristina V. Lopes  Department of Informatics
University of California, Irvine
Irvine, USA
<EMAIL_ADDRESS>
###### Abstract

Commit messages play a crucial role in collaborative software development.
They provide a clear and concise description of the changes made to the source
code. However, many commit messages among students’ projects lack useful
information. This is a concern, as low-quality commit messages can negatively
impact communication of software development and future maintenance. To
address this issue, this research aims to help students write high-quality
commit messages by “nudging” them in the right direction. We modified the
GitHub Desktop application by incorporating specific requirements for commit
messages, specifically “what” and “why” parts. To test whether this affects
the quality of commit messages, we divided students from an Information
Retrieval class into two groups, with one group using the modified application
and the other using other interfaces. The results show that the quality of
commit messages is improved in terms of informativeness, clearness, and
length.

###### Index Terms:

Commit message quality, GitHub, Education

##  I Introduction

Software engineering is a field that heavily relies on human resources. As a
social activity, software projects sometimes suffer from poor communication,
which can be a challenge during development and maintenance. Commit messages
are an essential tool to address communication in software development,
because they provide a historical record of changes made to the source code,
the reasons for those changes, and any other relevant information that
developers need to understand and review the code.

Unfortunately, the quality of commit messages is often poor. Studies have
found that a significant proportion of commit messages in open-source projects
lacks enough information. For instance, Dyer et al. found that about 14% of
commit messages in more than 23,000 Java projects hosted on SourceForge are
empty, and only 10% of commit messages contain at least one sentence-long
descriptive information [1]. More recently, Tian et al. found that more than
40% of commit messages on popular open-source projects need to be improved
[2]. Low-quality commit messages can lead to confusion and difficulty in
understanding the changes made to the code, making it more difficult for
developers work effectively in the process of developing, reverting, and
maintaining the projects.

In the context of education, the quality of commit messages in student
projects can vary greatly, but overall, it tends to be poor. According to [3],
without proper training, the percentage of good quality commits was reported
to range from 27% to 38%. Learning how to compose these messages properly
should start in school. Previous research has focused on improving overall
commit behavior, with Berg et al. [3] proposing a teaching process that
emphasizes committing periodically, focusing on a single task, and providing
informative commit messages. The results of their approach are promising in
promoting better version control and commit behavior among students. However,
such process may take at least one-week long and hours of tutorial sessions.
It can be resource-intensive and may not be feasible for certain classes.

For the past several years, we have been teaching a large project-based class
on Information Retrieval at UC Irvine. This class is focused on understanding
search engines, and there is very little time for teaching students about good
software engineering practices. The students work in groups to develop a
complete Web search engine. At the end of the course, the groups meet with one
of the Teaching Assistants, and the members of the group are asked questions
individually. Commit messages are important for three reasons: (1) they are a
means for the members of the group to communicate with each other during the
project development, (2) they are a means for students to review what they did
before meeting with the Teaching Assistant; and (3) they are used by the
Teaching Assistants to determine the contributions of individual students.

Because there is no time to properly teach software engineering practices in
this course, we set out to explore lightweight methods that could nudge
students into writing good commit messages without requiring them to read
guideline documents. Tian et al.’s work [2] makes the observation that ”good”
commit messages typically consist of two parts: what changed and why. Inspired
by this observation, we decided to test whether a slightly modified user
interface of a popular version control tool, GitHub
Desktop,111https://desktop.github.com/ would be enough to guide them into
better commit messages. The results we collected indicate that is the case.

This paper describes the changes we made to GitHub desktop, how we designed
the study, and presents the results we obtained.

##  II Study Design

Our study was integrated in the Web search engine project in the Information
Retrieval class offered at UC Irvine in the Fall of 2022. The course focuses
on providing students with a basic understanding of web information retrieval
and the functioning of modern search engines. As part of the class, students
complete three coding projects, either individually or in small groups of no
more than four people. The assignments include tasks related to text
processing, web crawling, and building an inverted index and a UI for search.
There were 120+ students enrolled in this class divided into 48 groups. All
groups are required to host their projects on GitHub, and they are encouraged
(but not required) to use the GitHub Desktop application to commit changes.

The standard GitHub Desktop application (Fig. 1) requires users to provide a
summary and, optionally, a description of their changes. This follows the
widely accepted form of commit messages in every source control system. For
the purpose of our study, however, we modified the application in two subtle
ways: first, we changed the language of those two parts to “what” and “why”
(Tian et al. [2]); second, we made the second part (i.e. “why”) required – see
Fig. 2.

Figure 1: Original GitHub Desktop Application

Figure 2: Modified GitHub Desktop Application

Odd-numbered student groups were asked to use version A (the modified version
of the GitHub Desktop application), while the even-numbered groups were asked
to use version B (the official version). The students were encouraged, but not
required, to use the appropriate version. Additionally, students were not
provided with information regarding the modifications made to version A, nor
were they informed the differences between versions.

Some students followed our recommendations, while other students did not; a
few students used the command line interface to git rather than the GUI tool.
We also did not give them any instructions about how to write commit messages,
or even how to use GitHub or the GitHub Desktop application. We asked them to
submit a screenshot of the tool they used along with their final submission
package, so we could check. Again, some students submitted the screenshot,
while others did not.

At the end of the quarter, and as part of the normal grading duties of the
Teaching Assistants, we collected the commit messages from each group’s GitHub
repository as a factor to their individual score. Although the quality of the
commit messages did not affect their score (only their number did), we
compared the quality of commit messages made using the the modified version of
the application against the quality of the commit messages made using other
interfaces.

##  III Results

In total, there were 1,593 commits collected. Before doing any comparisons, we
filtered out the auto-generated merge-related commit messages that start with
“Merge branch,” “Merge pull request” and “Merge remote-tracking branch;”
Additionally, we removed commits made before the start date of using the
applications, November 4, 2022. Also, some students did not provide their
GitHub repository links or screenshots for the application used, so we also
excluded commits whose creation interface could not be determined. After these
filters were applied, 801 commits remained for analysis. Table 1 shows the
distribution of commits among the various interfaces used by the students.

TABLE 1: Distribution of commit interface tools Tool | # of commits
---|---
Command Line | 105 (13.1%)
Official Desktop App | 571 (71.3%)
Modified Desktop App | 125 (15.6%)

Following the method used in Tian et al.’s work [2], we manually classified
these messages into four categories depending on whether their contents
included “what” and “why” parts, and whether these were meaningful (according
to that paper’s criteria). Table 2 shows a few concrete examples of commit
messages, and our classifications for them.

TABLE 2: Examples of commit messages and our classifications Message | What | Why
---|---|---
M1.py update | $\times$ | $\times$
added STUFF (justification: xx) | $\times$ | $\times$
print 5 results | ✓ | $\times$
fix some bug | $\times$ | ✓
fixed indexer | $\times$ | ✓
remove test.py (unnecessary file) | ✓ | ✓

The results are shown in Table 3. We found that over 80% of the commit
messages made using the official application and 67.6% of commits made using
the command line did not mention any motivation (the “why”) behind the
changes. Only 17.2% of the commit messages made using the official
application, and 27.6% using the command line, included both the “what” and
the “why” of the commits. In contrast, 76% of the commit messages made using
the modified application included both “what” and “why” information. Only 7.2%
of the commit messages made with the modified application were considered
meaningless, which was a significant drop compared to 26.8% made via official
application and 20% made via command line. This data indicates that “nudging”
the students towards a better form of commit messages is an effective method
for them to write useful information about code changes.

TABLE 3: distribution of commit message types

| What & Why | No What | No Why | Neither
---|---|---|---|---
Command | 29 (27.6%) | 26 (24.8%) | 85 (67.6%) | 21 (20.0%)
Official | 98 (17.2%) | 163 (28.5%) | 463 (81.1%) | 153 (26.8%)
Modified | 95 (76.0%) | 10 (8%) | 29 (23.2%) | 9 (7.2%)

Additionally, the majority of commit messages included the “what” information,
regardless of the tool used to create them, whether it be the official
application (71.5%), command line (75.2%), or the modified version (92%). Only
23.2% of the messages made via the modified application did not include any
justification, showing a significant decrease compared to commits made using
the original application (67.6%). Despite this, the results of the study
suggest that while students may have an understanding of what should be
included in an informative commit message, as evidenced by what the students
entered using the modified application, the lack of a required justification
in other interfaces makes them cut corners, and ignore the justification for
the change.

TABLE 4: distribution of commit message lengths

| Median/Mean Length | $<15$ (words) |  $15-20$ (words) |  $>20$ (words)
---|---|---|---|---
Command | 5/6 (words) | 99 (94.3%) | 5 (4.8%) | 1 (1.0%)
Official | 4/6 (words) | 543 (95.1%) | 14 (2.5%) | 14 (2.5%)
Modified | 16/20 (words) | 55 (44.0%) | 30 (24.0%) | 40 (32.0%)

Previously, Dyer et al. found that more than two thirds commit messages on
open source projects were less than a sentence long [1]. For the purpose of
this study, we also evaluated the length of the commit messages (Tab. 4). The
median length of the commit messages made using the original application and
command line was similar to each other, both being around four and five words
long. About 95% of the messages made both using the official application and
command line were less than one descriptive sentence (less than 15 words),
which is even worse than in Dyer et al.’s paper (90%) [1]. In contrast, the
median length of commit messages made using the modified application was 16
words long, and 56% of the messages were at least a normal sentence long
(greater than or equal to 15 words). The observations indicate that students
tend to write longer and more informative commit messages when they are
“nudged” to do so.

We performed a two-sample Welch’s t-test to compare the mean lengths of commit
messages generated via different tools. The results indicate that there are
statistically significant differences between the modified application and
both the official application (p-value $<<0.001$) and the command line tool
(p-value $<<0.001$). There was no statistical significant difference between
the mean lengths of commit messages made via the official application and the
command line tool (p-value = 0.7286). This suggests that the two tools produce
commit messages of similar lengths.

Previous studies have suggested that developers tend to write low-quality
commit messages due to a lack of motivation and time [1, 2, 4]; our results
lead us to speculate that a better UI might help reduce the number of low-
quality messages.

##  IV Threats to validity

In classifying the students’ commit messages according to the criteria stated
in [2], we encountered many situations that were very difficult to classify –
mostly because the messages were too short. We mitigated this problem by
having both authors perform the assessment together, referring to the few
examples given in Tian et al.’s paper in order to resolve the conflicts of
opinion. While sometimes very short messages are warranted and meaningful,
other times they are neither; while length can be a hint for quality, it is
not 100% correlated with quality. Nevertheless, it is possible that other
judges might interpret Tian et al.’s criteria differently, leading to
different results and conclusions.

##  V Conclusion

It is essential for developers to understand the importance of writing high-
quality commit messages in order to ensure the longevity of a software
development project. This work has shown that providing guidance on the
components of a good commit message within version control tools can improve
the quality of commit messages written by students. This approach requires
minimal resources for training and can easily be scaled to larger groups. In
the future, it would be valuable to conduct further studies in real-world
developer communities to validate these findings.

## References

  * [1] R. Dyer, H. A. Nguyen, H. Rajan, and T. N. Nguyen, “Boa: A language and infrastructure for analyzing ultra-large-scale software repositories,” in 2013 35th International Conference on Software Engineering (ICSE), pp. 422–431, 2013.
  * [2] Y. Tian, Y. Zhang, K.-J. Stol, L. Jiang, and H. Liu, “What makes a good commit message?,” in Proceedings of the 44th International Conference on Software Engineering, ICSE ’22, (New York, NY, USA), p. 2389–2401, Association for Computing Machinery, 2022.
  * [3] A. Berg, S. Osnes, and R. Glassey, “If in doubt, try three: Developing better version control commit behaviour with first year students,” in Proceedings of the 53rd ACM Technical Symposium on Computer Science Education \- Volume 1, SIGCSE 2022, (New York, NY, USA), p. 362–368, Association for Computing Machinery, 2022.
  * [4] Z. Liu, X. Xia, A. E. Hassan, D. Lo, Z. Xing, and X. Wang, “Neural-machine-translation-based commit message generation: how far are we?,” in Proceedings of the 33rd ACM/IEEE International Conference on Automated Software Engineering, pp. 373–384, 2018.