∎

11institutetext: Sachin Chauhan 22institutetext: Department of Physics, Indian
Institute of Technology Bombay
Powai, Mumbai, 400076, India
22email<EMAIL_ADDRESS>33institutetext: Pichai Ramadevi
44institutetext: Department of Physics, Indian Institute of Technology Bombay
Powai, Mumbai, 400076, India
44email<EMAIL_ADDRESS>
#  $\hat{Z}$\- invariant for $SO(3)$ and $OSp(1|2)$ Groups

Sachin Chauhan  Pichai Ramadevi

(Received: date / Accepted: date)

###### Abstract

Three-manifold invariants $\hat{Z}$ (“$Z$-hat”), also known as homological
blocks, are $q$-series with integer coefficients. Explicit $q$-series form for
$\hat{Z}$ is known for $SU(2)$ group, supergroup $SU(2|1)$ and ortho-
symplectic supergroup $OSp(2|2)$. We focus on $\hat{Z}$ for $SO(3)$ group and
orthosymplectic supergroup $OSp(1|2)$ in this paper. Particularly, the change
of variable relating $SU(2)$ link invariants to the $SO(3)$ & $OSp(1|2)$ link
invariants plays a crucial role in explicitly writing the $q$-series.

###### Keywords:

Chern-Simons theory topological field theories topological strings M-theory
3-manifold knot quantum invariant q-series colored Jones polynomial

##  1 Introduction

Knot theory has attracted attention from both mathematicians and physicists
during the last 40 years. The seminal work of WittenWitten:1988hf giving a
three-dimensional definition for Jones polynomials of knots and links, using
$SU(2)$ Chern-Simons theory on $S^{3}$, triggered a tower of new colored link
invariants. Such new invariants are given by expectation value of Wilson loops
carrying higher dimensional representation $R\in\mathcal{G}$ in Chern-Simons
theory where $\mathcal{G}$ denotes gauge group. These link invariants are in
variable ${\mathbbm{q}}$ which depends on the rank of the gauge group
$\mathcal{G}$ and the Chern-Simons coupling constant $k\in\mathbb{Z}$ (For eg:
when $\mathcal{G}=SU(N)$ then ${\mathbbm{q}}=\text{exp}\left(\frac{2\pi
i}{k+N}\right)$). Witten’s approach also gives three-manifold invariant
$Z_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]$, called Chern-Simons partition function
for manifold $M$, obtained from surgery of framed links on $S^{3}$(Lickorish-
Wallace theorem10.2307/1970373 ; wallace_1960 ). Witten-Reshitikhin-Turaev
(WRT) invariants $\tau_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]$ known in the
mathematics literature are proportional to the Chern-Simons partition
function:

$Z_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]={\tau_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]\over\tau_{k}^{\mathcal{G}}[S^{2}\times
S^{1};{\mathbbm{q}}]}~{}.$ (1)

These WRT invariants can be written in terms of the colored invariants of
framed links10.2307/1970373 ; wallace_1960 ; Kaul:2000xe ; Ramadevi:1999nd .

It was puzzling observation that the colored knot polynomials appear as
Laurent series with integer coefficients. There must be an underlying
topological interpretation of such integer coefficients. This question was
answered both from mathematics and physics perspective. Initial work of
Khovanovkhovanov2000categorification titled ‘cateforification’ followed by
other papers on bi-graded homology theory including Khovanov-Rozansky homology
led to new homological invariants. Thus the integer coefficients of the
colored knot polynomials are interpreted as the dimensions of vector space of
homological theory. From topological strings and intersecting
branesOoguri:1999bv ; Gopakumar:1998ii ; Gopakumar:1998jq , the integers of
HOMFLY-PT polynomials are interpretable as counting of BPS states. Further the
connections to knot homologies within topological string context was initiated
in Gukov:2004hz resulting in concrete predictions of homological invariants
for some knots (see review Nawata:2015wya and references therein). Such a
physics approach involving brane set up in $M$-theoryGukov:2017kmk ;
Gukov:2016gkn ; Mikhaylov:2014aoa ; Ferrari:2020avq suggests the plausibility
of categorification of WRT invariants
$\tau_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]$ for three-manifolds. However, the
WRT invariants for simple three-manifolds are not a Laurent series with
integer coefficients.

The detailed discussion on $U(N)$ Chern-Simons partition function on Lens
space $L(p,1)\equiv S^{3}/\mathbb{Z}_{p}$ (see section 6 of Gukov:2016gkn )
shows a basis transformation
$Z_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]\underrightarrow{~{}~{}\mathcal{S}~{}~{}}{\hat{Z}}^{\mathcal{G}}[M;q]$
so that $\hat{Z}$ are $q$-series(where variable $q$ is an arbitrary complex
number inside a unit disk) with integer coefficients (GPPV
conjectureGukov:2017kmk ). These $\hat{Z}$ are called the homological blocks
of WRT invariants of three-manifolds $M$. Physically, the new three-manifold
invariants $\hat{Z}^{\mathcal{G}}[M;q]$ is the partition function
$Z_{T^{\mathcal{G}}[M]}[D^{2}\times S^{1}]$ for simple Lie groups. Here
$T^{\mathcal{G}}[M]$ denote the effective 3d $\mathcal{N}=2$ theory on
$D^{2}\times S^{1}$ obtained by reducing 6d $(2,0)$ theory (describing
dynamics of coincident $M5$ branes) on $M$.

For a class of negative definite plumbed three-manifolds as well as link
complements Gukov:2017kmk ; Gukov:2019mnk ; park2020higher ; Chung:2018rea ,
$\hat{Z}^{SU(N)}$ has been calculated. Further, $\hat{Z}$ invariants for super
unitary group $SU(n|m)$ supergroup with explicit $q$-series for $SU(2|1)$ is
presented in Ferrari:2020avq . Generalisation to orthosymplectic supergroup
$OSp(2|2n)$ with explicit $q$-series for $Osp(2|2)$chae2021towards motivates
us to look at $\hat{Z}$ for other gauge groups.

Our goal in this paper is to extract $\hat{Z}$ for the simplest orthogonal
group $SO(3)$ and the simplest odd orthosymplectic supergroup $OSp(1|2)$. We
take the route of relating $SU(2)$ colored link invariants to the link
invariants for these two groups to obtain $\hat{Z}$ invariants.

The plan of the paper is organised as follows. In section 2, we will review
the developments on the invariants of knots, links and three-manifolds. We
will first briefly present Chern-Simons theory and colored link invariants
with explicit results for $SU(2)$ gauge group and indicate how colored $SO(3)$
and $OSp(1|2)$ link invariants can be obtained from the colored $SU(2)$
polynomials. Then, we will summarise the developments of the homological
invariants. In section 3, we briefly review $\hat{Z}$-series invariant for
$SU(2)$ group for the negative definite plumbed three-manifolds. This will
serve as a warmup to extend to $SO(3)$ and $OSp(1|2)$ group which we will
present in section 4. We summarize the results and also indicate future
directions to pursue in the concluding section 5.

##  2 Knots, Links and Three-manifold Invariants

In this section, we will briefly summarise new invariants in knot theory from
the physics approach as well as from the mathematics approach.

###  2.1 Chern-Simons Field Theory Invariants

Chern-Simons theory based on gauge group $\mathcal{G}$ is a Schwarz type
topological field theory which provides a natural framework for study of
knots, links and three-manifolds $M$. Chern-Simons action
$S_{CS}^{\mathcal{G}}(A)$ is explicitly metric independent:

$S_{CS}^{\mathcal{G}}(A)=\frac{k}{4\pi}\int_{M}Tr\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\right)~{}.$ (2)

Here $A$ is the matrix valued gauge connection based on gauge group
$\mathcal{G}$ and $k\in\mathbb{Z}$ is the coupling constant. In fact, the
expectation value of Wilson loop operators associated with any $m$-component
link $\mathcal{L}_{m}$ are the the link invariants:

$V_{R_{1},R_{2},\ldots
R_{m}}^{\mathcal{G}}[\mathcal{L}_{m};{\mathbbm{q}}]=\langle
W_{R_{1},R_{2},\ldots
R_{m}}[\mathcal{L}_{m}]\rangle={\int{\mathcal{D}}A\exp(iS_{CS})\overbrace{P\left(\prod_{i}\Tr_{R_{i}}exp\oint_{\mathcal{K}_{i}}A\right)}^{W_{R_{1},R_{2},\ldots
R_{m}}[\mathcal{L}_{m}]}\over{\underbrace{\int{\mathcal{D}}A\exp(iS_{CS})}_{Z^{\mathcal{G}}_{k}[M;{\mathbbm{q}}]}}}~{},$
(3)

where $\mathcal{K}_{i}$’s denote the component knots of link $\mathcal{L}_{m}$
carrying representations $R_{i}$’s of gauge group $\mathcal{G}$ and
$Z^{\mathcal{G}}_{k}[M;{\mathbbm{q}}]$ defines the Chern-Simons partition
function encoding the topology of the three-manifold $M$.

Exploiting the connection between Chern-Simons theory, based on group
$\mathcal{G}$, and the corresponding Wess-Zumino-Witten (WZW) conformal field
theory with the affine Lie algebra symmetry $\mathfrak{g}_{k}$, the invariants
of these links embedded in a three-sphere $M=S^{3}$ can be explicitly written
in variable ${\mathbbm{q}}$ :

${\mathbbm{q}}=\exp({2\pi i\over k+C_{v}})~{},$ (4)

which depends on the coupling constant $k$ and the dual Coxeter number $C_{v}$
of the group $\mathcal{G}$. These link invariants include the well-known
polynomials in the knot theory literature.

$\mathcal{G}$ | $R$ | Invariant
---|---|---
$SU(2)$ | $\yng(1)$ | Jones
$SU(N)$ | $\yng(1)$ | HOMFLY-PT
$SO(N)$ | defining | Kauffman

####  2.1.1 Link Invariants

As indicated in the above table, Jones’ polynomial corresponds to the
fundamental representation $R=\tiny{\yng(1)}\equiv 1\in SU(2)$ placed on all
the component knots:

$V_{1,1,1,\ldots 1}^{SU(2)}[\mathcal{L}_{m};{\mathbbm{q}}]\equiv J_{2,2,\ldots
2}\left[\mathcal{L}_{m};{\mathbbm{q}}=\exp({2\pi i\over k+2})\right]~{},$ (5)

where the subscript ‘2’ in Jones polynomial
$J_{2,2,2,\ldots}[\mathcal{L}_{m};{\mathbbm{q}}]$ denotes the dimension of
$R={\tiny{\yng(1)}}$. Higher dimensional representations placed on the
component knots $R_{i}=\underbrace{\tiny{\yng(4)}}_{n_{i}-1}\equiv n_{i}-1\in
SU(2)$ are the colored Jones invariants:

$V_{n_{1}-1,n_{2}-1,n_{3}-1,\ldots
n_{m}-1}^{SU(2)}[\mathcal{L}_{m};{\mathbbm{q}}]\equiv J_{n_{1},n_{2},\ldots
n_{m}}\left[\mathcal{L}_{m};{\mathbbm{q}}=\exp({2\pi i\over k+2})\right]~{},$
(6)

and the invariants with these representations belonging to $SU(N)$ ($(SO(N)$)
are known as colored HOMFLY-PT (colored Kauffman) invariants. For clarity, we
will restrict to $SU(2)$ group to write the invariants explicitly in terms of
${\mathbbm{q}}$ variable.

We work with the following unknot ($\bigcirc$) normalisation:

$J_{n+1}[\bigcirc;{\mathbbm{q}}]={\rm
dim}_{\mathbbm{q}}\underbrace{\yng(4)}_{n}={{\mathbbm{q}}^{(n+1)\over
2}-{\mathbbm{q}}^{-{(n+1)\over 2}}\over{\mathbbm{q}}^{1\over
2}-{\mathbbm{q}}^{-{1\over 2}}}={\sin({\pi(n+1)\over k+2})\over\sin({\pi\over
k+2})}={S_{0n}\over S_{00}},$ (7)

where ${\rm dim}_{\mathbbm{q}}\underbrace{\yng(3)}_{n}$ denotes quantum
dimension of the representation $\underbrace{\yng(3)}_{n}$ and
$S_{n_{1}n_{2}}$ are the modular transformation matrix elements of the
$\mathfrak{su}(2)_{k}$ WZW conformal field theory whose action on the
characters is
$\chi_{n_{1}}(\tau)~{}~{}\underrightarrow{~{}~{}~{}S~{}~{}~{}}~{}~{}\chi_{n_{2}}\left(-{1\over\tau}\right)~{},$
where $\tau$ denotes the modular parameter. These knot and link polynomials
with the above unknot normalisation are referred as unnormalised colored Jones
invariant.

For framed unknots with framing number $f$, the invariant will be

$J_{n+1}[\bigcirc_{f};{\mathbbm{q}}]={\mathbbm{q}}^{f[{(n+1)^{2}-1\over
4}]}{{\mathbbm{q}}^{(n+1)\over 2}-{\mathbbm{q}}^{-{(n+1)\over
2}}\over{\mathbbm{q}}^{1\over 2}-{\mathbbm{q}}^{-{1\over
2}}}\propto(T_{nn})^{f}{S_{0n}\over S_{00}},$ (8)

where the action of the modular transformation matrix $T$ on characters is
$\chi_{n}(\tau)~{}~{}\underrightarrow{~{}~{}~{}T~{}~{}~{}~{}}\chi_{n}(\tau+1)~{}.$
The colored Jones invariant for the Hopf link can also be written in terms of
$S$ matrix:

$J_{n_{1}+1,n_{2}+1}[H;{\mathbbm{q}}]=\left({{\mathbbm{q}}^{\frac{(n_{1}+1)(n_{2}+1)}{2}}-{\mathbbm{q}}^{-\frac{(n_{1}+1)(n_{2}+1)}{2}}\over{\mathbbm{q}}^{\frac{1}{2}}-{\mathbbm{q}}^{-\frac{1}{2}}}\right)={S_{n_{1}n_{2}}\over
S_{00}}.$ (9)

The invariant for a framed Hopf link $H(f_{1},f_{2})$, with framing numbers
$f_{1}$ and $f_{2}$ on the two component knots, in terms of $T$ and $S$
matrices is

$J_{n_{1}+1,n_{2}+1}[H(f_{1},f_{2});{\mathbbm{q}}]\propto(T_{n_{1}n_{1}})^{f_{1}}(T_{n_{2}n_{2}})^{f_{2}}{S_{n_{1}n_{2}}\over
S_{00}}~{}~{}.$ (10)

We will look at a class of links obtained as a connected sum of framed Hopf
links. For instance, the invariant for the connected sum of two framed Hopf
links $H(f_{1},f_{2})\\#H(f_{2},f_{3})$ will be

$\displaystyle
J_{n_{1}+1,n_{2}+1,n_{3}+1}[H(f_{1},f_{2})\\#H(f_{2},f_{3});{\mathbbm{q}}]$
$\displaystyle\propto$
$\displaystyle{\prod_{i=1}^{3}T_{n_{i}n_{i}}^{f_{i}}}{S_{n_{1}n_{2}}\over
S_{00}}{S_{n_{2}n_{3}}\over S_{n_{2}0}}$ $\displaystyle=$
$\displaystyle{\prod_{i=1}^{2}J_{n_{i}+1,n_{i+1}+1}[H(f_{i},f_{i+1});{\mathbbm{q}}]\over
J_{n_{2}+1}[\bigcirc;{\mathbbm{q}}]}~{}.~{}$

Such a connected sum of two framed Hopf links, which is a 3-component link,
can be denoted as a linear graph

$f_{1}$$f_{2}$$f_{3}$

with three vertices labeled by the framing numbers and the edges connecting
the adjacent vertices. These are known as ‘plumbing graphs’. Another plumbing
graph $\Gamma$ with 8 vertices denoting the link $\mathcal{L}(\Gamma)$ (the
connected sum of many framed Hopf links) is illustrated in Figure 1. The
colored invariant for these links $\mathcal{L}(\Gamma)$ can be written in
terms of $S$ and $T$ matrices.

Figure 1: An example of a plumbing graph $\Gamma$ (left) and the corresponding
link ${\mathcal{L}}(\Gamma)$ of framed unknots in $S^{3}$ (right).

For a general $m$ vertex plumbing graph with vertices $v_{1},v_{2},\ldots
v_{m}\in V$ labelled by framing numbers $f_{1},f_{2},\ldots f_{m}$, there can
be one or more edges connecting a vertex $v$ with the other vertices. The
degree of any vertex $v$ ($\text{deg}(v)$) is equal to the total number of
edges intersecting $v$. For the graph in Figure 1,
$\text{deg}(2)=\text{deg}(4)=\text{deg}(6)=3$. The colored Jones’ invariant
for any plumbing graph $\Gamma$ is

$J_{n_{1}+1,n_{2}+1,\ldots n_{m}+1}[\mathcal{L};{\mathbbm{q}}]\propto{1\over
S_{00}}\prod_{i=1}^{m}\\{(T_{n_{i}n_{i}})^{f_{i}}(S_{0n_{i}})^{1-\text{deg}(v_{i})}\\}\prod_{(v_{1},v_{2})\;\in\;\text{Edges}}(S_{n_{v_{1}}n_{v_{2}}})~{}.$
(12)

Even though we have presented the colored Jones invariants (10, 2.1.1, 12),
the formal expression of these link invariants in terms of $S$ and $T$
matrices are applicable for any arbitrary gauge group $\mathcal{G}$.

$\bullet$ $SO(3)$ and $OSp(1|2)$ Link invariants

Using group theory arguments, it is possible to relate colored link invariants
between different groups. For instance, the representations of the $SO(3)$ can
be identified with a subset of $SU(2)$ representations. As a consequence, the
$SO(3)$ link invariants can be related to the colored Jones invariants as
follows:

$V_{n_{1},n_{2},n_{3},\ldots n_{m}}^{SO(3)}\left[\mathcal{L}_{m};Q=\exp({2\pi
i\over K+1})\right]=J_{2n_{1}+1,2n_{2}+1,\ldots
2n_{m}+1}[\mathcal{L}_{m};{\mathbbm{q}}]{\big{|}}_{{\mathbbm{q}}^{2}=Q}~{},$
(13)

where the level $K$ of the affine $\mathfrak{so}(3)_{K}$ Lie algebra must be
an even integer( $K\in 2\mathbb{Z}$).

Similarly, the representations of the orthosymplectic supergroup $OSp(1|2)$
can be related to the representations of the $SU(2)$ group from the study of
$\mathfrak{osp}(1|2)_{\hat{K}}$ WZW conformal field theory and the link
invariants
$V_{n_{1},n_{2},n_{3},\ldots
n_{m}}^{OSp(1|2)}\left[\mathcal{L}_{m};\hat{Q}=\exp({2\pi i\over
2{\hat{K}}+3})\right]$ Ennes:1997kx . Particularly, there is a precise
identification of the polynomial variable $\hat{Q}$ to $SU(2)$ variable
${\mathbbm{q}}$. Further, the fusion rules of the primary fields of
$\mathfrak{osp}(1|2)_{\hat{K}}$ WZW conformal field theory can be compared to
integer spin primary fields of the $\mathfrak{su}(2)_{k}$. Particularly, the
$\hat{S}$ and $\hat{T}$-matrices of $\mathfrak{osp}(1|2)_{\hat{K}}$ :

$\displaystyle\hat{S}_{n_{1}n_{2}}$ $\displaystyle=$
$\displaystyle\sqrt{4\over
2\hat{K}+3}(-1)^{n_{1}+n_{2}}\cos\left[{(2n_{1}+1)(2n_{2}+1)\over
2(2\hat{K}+3)}\pi\right]~{}~{},$ (14) $\displaystyle\hat{T}_{n_{1},n_{2}}$
$\displaystyle\propto$
$\displaystyle\delta_{n_{1},n_{2}}{\hat{Q}}^{[\frac{(2n_{1}+1)^{2}-1]}{4}}~{}~{}~{},$
(15)

are related to the $S$ and $T$ matrices of $\mathfrak{su}(2)_{k}$ in the
following way:

$\hat{S}_{n_{1},n_{2}}=S_{2n_{1},2n_{2}}{\big{|}}_{{\mathbbm{q}}=-\hat{Q}}~{};~{}\hat{T}_{n_{1},n_{1}}=T_{2n_{1},2n_{1}}{\big{|}}_{{\mathbbm{q}}=-\hat{Q}}$
(16)

Using these relations, we can show that the $OSp(1|2)$ colored invariant match
the colored Jones invariant for any arbitrary link $\mathcal{L}_{m}$ in the
following way:

$V_{n_{1},n_{2},n_{3},\ldots
n_{m}}^{OSp(1|2)}\left[\mathcal{L}_{m};\hat{Q}=\exp({2\pi i\over
2{\hat{K}}+3})\right]=\epsilon J_{2n_{1}+1,2n_{2}+1,\ldots
2n_{m}+1}[\mathcal{L}_{m};{\mathbbm{q}}]{\big{|}}_{{\mathbbm{q}}=-\hat{Q}}~{},$
(17)

where $\epsilon$ could be $\pm 1$ depending on the link $\mathcal{L}$ and the
representations $n_{i}$’s. For example, the colored $OSp(1|2)$ invariant for
framed Hopf link is

$\displaystyle V_{n_{1},n_{2}}^{OSp(1|2)}[H(f_{1},f_{2});\hat{Q}]$
$\displaystyle=$
$\displaystyle{\hat{Q}}^{\frac{f_{1}((2n_{1}+1)^{2}-1)}{4}}{\hat{Q}}^{\frac{f_{2}((2n_{1}+1)^{2}-1)}{4}}(-1)^{(n_{1}+n_{2})}\times$
$\displaystyle\left({{\hat{Q}}^{\frac{(2n_{1}+1)(2n_{2}+1)}{2}}+{\hat{Q}}^{-\frac{(2n_{1}+1)(2n_{2}+1)}{2}}\over{\hat{Q}}^{\frac{1}{2}}+{\hat{Q}}^{-\frac{1}{2}}}\right)$
$\displaystyle=$ $\displaystyle
J_{2n_{1}+1,2n_{2}+1}[H(f_{1},f_{2}),-\hat{Q}]~{}.$

In fact, for any link $\mathcal{L}(\Gamma)$ denoted by the graph $\Gamma$, the
invariants will be

$\displaystyle V_{n_{1},n_{2},\ldots
n_{m}}^{OSp(1|2)}[\mathcal{L}(\Gamma);\hat{Q}]$ $\displaystyle=$
$\displaystyle{1\over{\hat{Q}}^{\frac{1}{2}}+{\hat{Q}}^{-\frac{1}{2}}}\prod_{i=1}^{m}(-1)^{n_{i}}{\hat{Q}}^{\frac{f_{i}((2n_{i}+1)^{2}-1)}{4}}$
$\displaystyle\left({\hat{Q}}^{\frac{2n_{i}+1}{2}}+{\hat{Q}}^{-{\frac{2n_{i}+1}{2}}}\right)^{\text{deg}(v_{i})-1}$
$\displaystyle\prod_{(v_{1},v_{2})\;\in\;{\rm
Edges}}{\left({\hat{Q}}^{\frac{(2n_{v_{1}}+1)(2n_{v_{2}}+1)}{2}}+{\hat{Q}}^{-\frac{(2n_{v_{1}}+1)(2n_{v_{2}}+1)}{2}}\right)}~{}.$
(19)

As three-manifolds can be constructed by a surgery procedure on any framed
link, the Chern-Simons partition function/WRT invariant (1) can be written in
terms of link invariants10.2307/1970373 ; wallace_1960 ; Kaul:2000xe ;
Ramadevi:1999nd . We will now present the salient features of such WRT
invariants.

####  2.1.2 Three-Manifold Invariants

Let us confine to the three-manifold $M[\Gamma]$ obtained from surgery of
framed link associated with $L$-vertex graph (an example illustrated in Figure
1). These kind of manifolds are known in the literature as plumbed three-
manifolds. The linking matrix $B$ is defined as

$B_{v_{1},v_{2}}=\left\\{\begin{array}[]{ll}1,&v_{1},v_{2}\text{
connected},\\\ f_{v},&v_{1}=v_{2}=v,\\\
0,&\text{otherwise}.\end{array}\right.\qquad v_{i}\in\text{Vertices of
}\Gamma\;\cong\;\\{1,\ldots,L\\}.$ (20)

The algebraic expression for the WRT invariant
$\tau_{k}^{\mathcal{G}}[M(\Gamma);{\mathbbm{q}}]$ is

$\tau_{k}^{\mathcal{G}}[M(\Gamma);{\mathbbm{q}}]=\frac{F^{\mathcal{G}}[{\mathcal{L}}(\Gamma);{\mathbbm{q}}]}{F^{\mathcal{G}}[{\mathcal{L}}(+1\bullet);{\mathbbm{q}}]^{b_{+}}F^{\mathcal{G}}[{\mathcal{L}}(-1\bullet);{\mathbbm{q}}]^{b_{-}}}$
(21)

where $b_{\pm}$ are the number of positive and negative eigenvalues of a
linking matrix $B$ respectively and
$F^{\mathcal{G}}[\mathcal{L}(\Gamma);{\mathbbm{q}}]$ is defined as

$F^{\mathcal{G}}[{\mathcal{L}}(\Gamma);{\mathbbm{q}}]=\sum_{R_{1},R_{2},\ldots
R_{L}}\left(\prod_{i=1}^{L}V_{R_{i}}^{\mathcal{G}}[\bigcirc;{\mathbbm{q}}]\right)V_{R_{1},R_{2}\ldots
R_{L}}^{\mathcal{G}}[\mathcal{L}(\Gamma);{\mathbbm{q}}]~{},$ (22)

where the summation indicates all the allowed integrable representations of
affine $\mathfrak{g}_{k}$ Lie algebra. By construction, any two homeomorphic
manifolds must share the same three-manifold invariant. There is a prescribed
set of moves called Kirby moves on links which gives the same three-manifold.
For framed links depicted as plumbing graphs, these moves are known as Kirby-
Neumann moves as shown in Figure 2. Hence, the three-manifold invariant must
obey

$\tau_{k}^{\mathcal{G}}[M(\Gamma);{\mathbbm{q}}]=\tau_{k}^{\mathcal{G}}[M(\Gamma^{\prime});{\mathbbm{q}}]~{},$
(23)

where the plumbing graphs $\Gamma,\Gamma^{\prime}$ are related by the Kirby-
Neumann moves.

Figure 2: Kirby-Neumann moves that relate plumbing graphs which result in
homeomorphic 3-manifolds.

Towards the end of 20th century, attempts to give a topological interpretation
for the integer coefficients in the Laurent series expression for Jones
polynomial (HOMFLY-PT) as well as the corresponding colored invariants for any
knot $\mathcal{K}$

$J_{n}[\mathcal{K};{\mathbbm{q}}]=\sum_{s}a_{s}{\mathbbm{q}}^{s}~{},~{}\\{a_{s}\\}\in\mathbb{Z}$
(24)

has resulted in developments on homology theories as well as physics
explanation. We will discuss these ‘homological invariants’ and their
appearance in string/M-theory in the following section.

###  2.2 Knot, Link and Three-manifold Homologies

We will first review the developments on homological invariants of knots
accounting for these integers $a_{s}$ (24) as dimension of the vector space
$\mathcal{H}_{\mathcal{K}}$ of a homological theory. Then, we will present the
topological string/M-theory approach where these integers count number of BPS
states.

####  2.2.1 Homological Invariants of Knots

The pioneering work of Khovanovkhovanov2000categorification on bi-graded
homology theory led to categorification of the Jones polynomial. This was
extended to colored $\mathfrak{sl2}$ knot homology
$\mathcal{H}^{\mathfrak{sl}_{2};n}_{i,j}$ webster2017knot ;
cooper2012categorification ; frenkel2012categorifying leading to new
homological invariants $\mathcal{P}_{n}^{\mathfrak{sl}_{2}}[\mathcal{K},q,t]$
which categorifies the colored Jones polynomial:

$\mathcal{P}_{n}^{\mathfrak{sl}_{2}}[\mathcal{K},{\mathbbm{q}},t]=\sum_{i,j}t^{j}{\mathbbm{q}}^{i}{\rm
dim}\mathcal{H}^{\mathfrak{sl}_{2};n}_{i,j}~{}.$ (25)

The subscripts $i$ and $j$ on the colored $\mathfrak{sl}_{2}$ homology
$\mathcal{H}^{\mathfrak{sl}_{2};n}_{i,j}$ are called the polynomial grading
and the homological grading respectively. In fact, the ${\mathbbm{q}}$-graded
Euler characteristic of the colored $\mathfrak{sl}_{2}$ knot homology gives
the colored Jones invariant:

$J_{n}[\mathcal{K};{\mathbbm{q}}]=\sum_{i,j}(-1)^{j}{\mathbbm{q}}^{i}{\rm
dim}\mathcal{H}^{\mathfrak{sl}_{2};n}_{i,j}~{},$ (26)

explaining the reasons behind the integers $a_{s}$(24). Khovanov and Rozansky
khovanov2004matrix constructed $\mathfrak{sl}_{N}$ homology using matrix
factorizations. This led to the categorification of colored HOMFLY-PT
polynomials of knots. There has been interesting insight on these homological
invariants within topological strings context and $M$-theory. We will now
discuss the essential features from physics approach.

####  2.2.2 Topological Strings and M-theory

The parallel developments from topological strings and intersecting branes in
$M$-theory Ooguri:1999bv ; Gopakumar:1998jq interpreted the integers of
unnormalised HOMFLY-PT (24) as counting of BPS states. Invoking topological
string duality in the presence of any knot $\mathcal{K}$, Ooguri-Vafa
conjectured

$V_{\tiny\yng(1)}^{SU(N)}[\mathcal{K};{\mathbbm{q}},\lambda={\mathbbm{q}}^{N}]={1\over({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\sum_{Q,s}N_{\tiny\yng(1),Q,s}\lambda^{Q}{\mathbbm{q}}^{s}~{},$
(27)

where the integers $N_{\tiny\yng(1),Q,s}$ count $D4-D2$ bound states in string
theory. Further the relation between the BPS spectrum and $sl_{N}$/Khovanov-
Rozansky knot homology was conjectured within the topological string
contextGukov:2004hz :

$N_{\tiny\yng(1),Q,s}=\sum_{j}(-1)^{j}D_{Q,s,j}~{},$ (28)

where the integers $D_{Q,s,j}$ are referred to as refined BPS invariants. The
extra charge/ homological grading $j$ are explainable by the appearance of
extra $U(1)$ symmetry in $M$-theory compactified on Calabi-Yau three-folds
$CY_{3}$. The topological string duality and the dualities of physical string
theories compactified on $CY_{3}$ implies that the vector space of knot
homologies are the Hilbert space of BPS states(see review Nawata:2015wya and
references therein):

$\mathcal{H}_{\mathcal{K}}\equiv\mathcal{H}_{BPS}~{}.$

The impact of knot homology on the categorification of the WRT invariants has
been studied in the last six years. We now present a concise summary of the
recent developments in this direction.

####  2.2.3 Three-Manifold Homology

As WRT invariants (21) of three-manifolds involves invariants of links,
logically we would expect the homology of three-manifold
$\mathcal{H}^{\mathcal{G};M}$ such that

$\tau_{k}^{\mathcal{G}}[M;{\mathbbm{q}}]\stackrel{{\scriptstyle\text{?}}}{{=}}\sum_{i,j}(-1)^{j}{\mathbbm{q}}^{i}{\rm
dim}\mathcal{H}^{\mathcal{G};M}_{i,j}~{}.$ (29)

However, the WRT invariants known for many three-manifolds are not seen as
${\mathbbm{q}}$-series (29). We will now review the necessary steps
Gukov:2017kmk of obtaining a new three-manifold invariant $\hat{Z}$, as
${\mathbbm{q}}$-series, from $U(N)$ Chern-Simons partition function for Lens
space $M=L(p,1)\equiv S^{3}/\mathbb{Z}_{p}$. The space of flat connections
$\\{a\\}$ denoted by
$\pi_{1}[{S^{3}\over\mathbb{Z}_{p}}]\equiv\mathbb{Z}_{p}$. Hence
$Z_{k}^{U(N)}[L[p,1];{\mathbbm{q}}]$ can be decomposed as a sum of
perturbative Chern-Simons $Z_{a}^{U(N)}[L[p,1];{\mathbbm{q}}]$ around these
abelian flat connections $a$ Gukov:2017kmk :

$Z_{k}^{U(N)}[L[p,1];{\mathbbm{q}}]=\sum_{a}\exp[iS_{CS}^{(a)}]Z_{a}^{U(N)}[L[p,1];{\mathbbm{q}}]~{},$
(30)

where $S_{CS}^{(a)}$ is the corresponding classical Chern-Simons action. The
following change of basis by $\mathcal{S}$ matrix of $\mathfrak{u}(1)^{N}_{p}$
affine algebra:

$Z_{a}^{U(N)}[L(p,1);{\mathbbm{q}}]=\sum_{b}\mathcal{S}_{ab}\hat{Z}_{b}^{U(N)}[\mathcal{L}[p,1];q]\Big{|}_{q\rightarrow{\mathbbm{q}}}~{},$
(31)

is required so that

$\hat{Z}_{b}^{U(N)}[\mathcal{L}[p,1];q]\in
q^{\Delta_{b}}\mathbb{Z}[[q]]~{},~{}~{}\Delta_{b}\in\mathbb{Q}~{}.$ (32)

Physically, the $\hat{Z}_{b}[\mathcal{L}[p,1];q]$ is also the vortex partition
function $\hat{Z}_{T[L[p,1]]}[D^{2}\times_{q}S^{1}]$ obtained from reducing 6d
$(2,0)$ theory (describing dynamics of $N$-coincident $M5$ branes on
$L[p,1]\times D^{2}\times_{q}S^{1}$) on $L[p,1]$. The effective 3-d
$\mathcal{N}=2$ theory on $D^{2}\times_{q}S^{1}$ (cigar geometry) is denoted
as $T^{U(N)}[L[p,1]]$.

For other three-manifolds $M$, $\mathcal{S}$ matrix depends only on
$H_{1}(M,\mathbb{Z})$. Further the Hilbert space of BPS states
$\mathcal{H}^{i,j}_{BPS}$ on the M5 brane system, in the ambient space-time
$T^{*}M\times TN\times S^{1}$, where $i,j$ gradings will keep track of both
spins associated with the rotational symmetry $U(1)_{q}\times U(1)_{R}$ on
$D^{2}\subset TN$. The Hilbert space of states for the theory
$T^{\mathcal{G}}[M]$ with boundary condition at $\partial D^{2}=S^{1}$ labeled
by $a\in({\rm Tor}H_{1}(M,\mathbb{Z}))^{N}/S_{N}$ leads to bi-graded
homological invariants of $M$:

${\mathcal{H}}_{a}^{U(N)}[M]={\mathcal{H}}_{T_{L[p,1]}^{U(N)}}[D^{2};a]=\bigoplus_{\begin{subarray}{c}i\in\mathbb{Z}+\Delta_{a},\\\
j\in\mathbb{Z}\end{subarray}}{\mathcal{H}}_{a}^{i,j}~{}.$ (33)

Note that the grading $i$ counts the charge under $U(1)_{q}$ rotation of
$D^{2}$ and homological grading $j$ is the R-charge of the $U(1)_{R}$
R-symmetry. In the following section, we will review the necessary steps of
obtaining $\hat{Z}$ invariants for $SU(2)$ group. This will provide clarity of
notations to investigate $\hat{Z}$ for $SO(3)$ and $OSp(1|2)$ group.

##  3 Review of $SU(2)$ $\hat{Z}$ invariant

As discussed in subsection 2.2.3Gukov:2016gkn , the expression for Lens space
partition function using eqns.(30-32)

$\displaystyle Z_{k}^{U(N)}[L(p,1),{\mathbbm{q}}]$ $\displaystyle=$
$\displaystyle\sum_{a,b\in\mathbb{Z}_{p}}\mathcal{S}_{ab}\exp[iS_{CS}^{(a)}]\hat{Z}_{b}^{U(N)}[\mathcal{L}(p,1);q]\Big{|}_{q\rightarrow{\mathbbm{q}}}~{},$
(34) $\displaystyle{\rm where}~{}\hat{Z}_{b}[\mathcal{L}[p,1];q]$
$\displaystyle\in$ $\displaystyle
q^{\Delta_{a}}\mathbb{Z}[[q]]~{},~{}~{}\Delta_{a}\in\mathbb{Q}~{}.$ (35)

led to the following conjecture Gukov:2017kmk ; Gukov:2019mnk for any closed
oriented three manifold $M$ known as GPPV conjecture:

$\displaystyle Z^{SU(2)}_{k}[M;{\mathbbm{q}}]$ $\displaystyle=$
$\displaystyle(i\sqrt{2(k+2}))^{b_{1}(M)-1}\sum_{a,b\;\in\;\atop\text{Spin}^{c}(M)/{\mathbb{Z}}_{2}}\exp[2\pi
i(k+2){\ell k}(a,a)]\,\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}|\mathcal{W}_{b}|^{-1}\mathcal{S}_{ab}\hat{Z}^{SU(2)}_{b}[M;q]|_{q\rightarrow{\mathbbm{q}}=\exp({\frac{2\pi
i}{k+2}})}$ (36)

where

$\hat{Z}^{SU(2)}_{b}[M;q]\in\,2^{-c}q^{\Delta_{b}}{\mathbb{Z}}[[q]]\qquad\Delta_{b}\in{\mathbb{Q}},\qquad
c\in{\mathbb{Z}}_{+}$ (37)

is convergent for $|q|<1$ and

$\mathcal{S}_{ab}=\frac{e^{2\pi i{\ell k}(a,b)}+e^{-2\pi i{\ell
k}(a,b)}}{|{\mathcal{W}}_{a}|\sqrt{|H_{1}(M,{\mathbb{Z}})|}}.$ (38)

Here $\mathcal{W}_{a}$ is the stabilizer subgroup defined as

${\mathcal{W}}_{a}\;\equiv\;\text{Stab}_{{\mathbb{Z}}_{2}}(a)\;=\;\left\\{\begin{array}[]{cl}{\mathbb{Z}}_{2},&a=-a\,,\\\
1,&\text{otherwise\,.}\end{array}\right.$ (39)

and $\ell k$ denotes the linking pairing on $H_{1}(M,\mathbb{Z})$:

$\begin{array}[]{cccc}{\ell k}:&H_{1}(M,\mathbb{Z})\otimes
H_{1}(M,\mathbb{Z})&\longrightarrow&{\mathbb{Q}}/{\mathbb{Z}}\\\
&[a]\otimes[b]&\longmapsto&{\\#(a\cap\hat{b})}/{n}\\\ \end{array}$ (40)

where $\hat{b}$ is a two-chain complex such that $\partial\hat{b}=nb$ with
$n\in{\mathbb{Z}}$. Such a $\hat{b}$ and $n$ exists because $[b]\in
H_{1}(M,\mathbb{Z})$. The number $\\#(a\cap\hat{b})$ counts the intersection
points with signs determined by the orientation. The set of orbits is the set
of $\text{Spin}^{c}$ structures on $M$, with the action of ${\mathbb{Z}}_{2}$
by conjugation.

Although, the relation(36) is true for any closed oriented three-manifold $M$,
the explicit $q$ series expression for $\hat{Z}$ is waiting to be discovered
for a general three-manifold.

In the following subsection, we will review the $\hat{Z}^{SU(2)}$ for the
plumbed manifolds. We begin with the WRT invariant for a plumbing graph, of
the type shown in Figure. 1, discussed in section.(2.1.2). Then analytically
continue ${\mathbbm{q}}\rightarrow q$ to get the $\hat{Z}^{SU(2)}$-invariant .
We will see that the analytic continuation procedure is doable only for
negative definite plumbed manifolds(i.e., the signature of linking matrix $B$,
$\sigma=b_{+}-b_{-}=-L$)111In principle, this procedure is also doable when
$B$ is negative on a certain subspace of $\mathbb{Z}^{L}$.. Moreover, as
explained inGukov:2019mnk , the $\text{Spin}^{c}$-structure in case of plumbed
3-manifold with $b_{1}(M)=0$, is given by
$H_{1}(M,\mathbb{Z})\cong\text{Coker}B=\mathbb{Z}^{L}/B\mathbb{Z}^{L}$.

###  3.1 $\hat{Z}_{b}^{SU(2)(q)}$

The WRT invariant $\tau_{k}^{SU(2)}[M(\Gamma);{\mathbbm{q}}]$,222normalized
such that $\tau_{k}^{\mathcal{G}}[S^{3};{\mathbbm{q}}]=1$ and $k$ is the bare
level for $SU(2)$ Chern-Simons theory for plumbed three-manifold
$M(\Gamma)$(21), obtained from surgery of framed link ${\mathcal{L}}(\Gamma)$
in $S^{3}$, is

$\displaystyle\tau_{k}^{SU(2)}[M(\Gamma);{\mathbbm{q}}]$ $\displaystyle=$
$\displaystyle\frac{F^{SU(2)}[{\mathcal{L}}(\Gamma);{\mathbbm{q}}]}{F^{SU(2)}[{\mathcal{L}}(+1\bullet);{\mathbbm{q}}]^{b_{+}}F^{SU(2)}[{\mathcal{L}}(-1\bullet);{\mathbbm{q}}]^{b_{-}}}$
$\displaystyle{\rm where}~{}F^{SU(2)}[{\mathcal{L}}(\Gamma);{\mathbbm{q}}]$
$\displaystyle=$
$\displaystyle\sum_{{n}\in\\{1,\ldots,k+1\\}^{L}}J[{\mathcal{L}}(\Gamma)]_{n_{1},\ldots,n_{L}}\prod_{v=1}^{L}\frac{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}}.$
(41)

Note $b_{\pm}$ are the number of positive and negative eigenvalues of a
linking matrix $B$ respectively and the colored Jones polynomial of link
$\mathcal{L}(\Gamma)$ (12) in variable ${\mathbbm{q}}=\exp({2i\pi/(k+2)})$ is

$\displaystyle J[{\mathcal{L}}(\Gamma)]_{n_{1},\ldots,n_{L}}$ $\displaystyle=$
$\displaystyle\frac{2i}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}}\prod_{v\;\in\;\text{Vertices}\;\cong\;\\{1,\ldots,L\\}}{\mathbbm{q}}^{\frac{f_{v}(n_{v}^{2}-1)}{4}}\,\times$
(42)
$\displaystyle\left(\frac{2i}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-1}\prod_{(v_{1},v_{2})\;\in\;\text{Edges}}\frac{{\mathbbm{q}}^{n_{v_{1}}n_{v_{2}}/2}-{\mathbbm{q}}^{-n_{v_{1}}n_{v_{2}}/2}}{2i}.$

Using the following Gauss sum reciprocity formula

$\sum_{n\;\in\;{\mathbb{Z}}^{L}/2k{\mathbb{Z}}^{L}}\exp\left(\frac{\pi
i}{2k}(n,Bn)+\frac{\pi i}{k}(\ell,n)\right)=\\\ \frac{e^{\frac{\pi
i\sigma}{4}}\,(2k)^{L/2}}{|\det
B|^{1/2}}\sum_{a\;\in\;{\mathbb{Z}}^{L}/B{\mathbb{Z}}^{L}}\exp\left(-2\pi
ik\left(a+\frac{\ell}{2k},B^{-1}\left(a+\frac{\ell}{2k}\right)\right)\right)$
(43)

where $\ell\in{\mathbb{Z}}^{L}$, $(\cdot,\cdot)$ is the standard pairing on
${\mathbb{Z}}^{L}$ and $\sigma=b_{+}-b_{-}$ is the signature of the linking
matrix $B$, we can sum

$F^{SU(2)}[{\mathcal{L}}(\pm
1\bullet);{\mathbbm{q}}]=\sum_{n=1}^{k+1}{\mathbbm{q}}^{\pm\frac{n^{2}-1}{4}}\,\left(\frac{{\mathbbm{q}}^{n/2}-{\mathbbm{q}}^{-n/2}}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}}\right)^{2}=\frac{[2(k+2)]^{1/2}\,e^{\pm\frac{\pi
i}{4}}\,{\mathbbm{q}}^{\mp\frac{3}{4}}}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}},$
(44)

for the unknot with framing $\pm 1$. Incorporating the above equation and the
fact that $L-|\text{Edges}|=1$ for the framed link ${\mathcal{L}}(\Gamma)$,
the WRT invariant simplifies to

$\tau_{k}^{SU(2)}[M(\Gamma);{\mathbbm{q}}]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma}{4}}}{2\,(2(k+2))^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n}\in{\mathbb{Z}}^{L}/2(k+2){\mathbb{Z}}^{L}}}^{\prime}\prod_{v\;\in\;\text{Vertices}}{\mathbbm{q}}^{\frac{f_{v}(n_{v}^{2}-1)}{4}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}\times\\\
\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}\frac{{\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2}}{2}$
(45)

where we used invariance of the summand under $n_{v}\rightarrow-n_{v}$. The
prime ′ in the sum means that the singular values $n_{v}=0,\,k+2$ are omitted.
Let us focus on the following factor for general plumbed graph:

$\displaystyle\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}({\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2})$
$\displaystyle=$ $\displaystyle\sum_{{p}\in\\{\pm
1\\}^{\text{Edges}}}\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}p_{(v^{\prime},v^{\prime\prime})}$
$\displaystyle{\mathbbm{q}}^{p_{(v^{\prime},v^{\prime\prime})}n_{v^{\prime}}n_{v^{\prime\prime}}/2}.$
(46)

Note that, under $n_{v}\rightarrow-n_{v}$ on any vertex $v$ of degree ${\rm
deg}(v)$, the factor with a given configuration of signs associated to edges
(i.e., $p\in\\{\pm 1\\}^{\text{Edges}}$) will transform into a term with a
different configuration times $(-1)^{\text{deg}(v)}$. For the class of graphs
$\Gamma$ (like Figure. 1), the sequence of such transforms can be finally
brought to the configuration with all signs $+1$. Hence, the WRT invariant
(45) for these plumbed three-manifolds can be reduced to this form:

$\tau_{k}^{SU(2)}[M(\Gamma)]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma-\sum_{v}f_{v}}{4}}}{2\,(2(k+2))^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n}\in{\mathbb{Z}}^{L}/2(k+2){\mathbb{Z}}^{L}}}^{\prime}\;\;{\mathbbm{q}}^{\frac{(n,Bn)}{4}}\prod_{v\;\in\;\text{Vertices}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}.$
(47)

In the above expression, the points $0$ and $k+2$ are excluded in the
summation but in the reciprocity formula (43) no point is excluded. So, to
apply the reciprocity formula we have to first regularize the sum. This is
achieved by introducing the following regularising parameters:

$\displaystyle\Delta_{v}\in{\mathbb{Z}}_{+}:\;\Delta_{v}$ $\displaystyle=$
$\displaystyle\text{deg}(v)-1\mod 2,\qquad\forall v\;\in\;\text{Vertices},$
(48) $\displaystyle\omega\in{\mathbb{C}}:$ $\displaystyle 0<|\omega|<1.$

so that the sum in eqn.(47) is rewritable as $\omega\rightarrow 1$:

${\sum_{{n}\in{\mathbb{Z}}^{L}/2(k+2){\mathbb{Z}}^{L}}}^{\prime}\;\;{\mathbbm{q}}^{\frac{(n,Bn)}{4}}\prod_{v\;\in\;\text{Vertices}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}=\\\
\lim_{\omega\rightarrow
1}\frac{1}{2^{L}}\sum_{{n}\in{\mathbb{Z}}^{L}/2(k+2){\mathbb{Z}}^{L}}{\mathbbm{q}}^{\frac{(n,Bn)}{4}}F_{\omega}(x_{1},\ldots,x_{L})|_{x_{v}={\mathbbm{q}}^{n_{v}/2}},$
(49)

where

$\displaystyle F_{\omega}(x_{1},\ldots,x_{L})$ $\displaystyle=$
$\displaystyle\prod_{v\;\in\;\text{Vertices}}\left({x_{v}-1/x_{v}}\right)^{\Delta_{v}}\times\,$
(50)
$\displaystyle\left\\{\left(\frac{1}{x_{v}-\omega/x_{v}}\right)^{\text{deg}(v)-2+\Delta_{v}}+\left(\frac{1}{\omega
x_{v}-1/x_{v}}\right)^{\text{deg}(v)-2+\Delta_{v}}\right\\}$

Note that, we can perform a binomial expansion taking $(\omega/x_{v}^{2})$
small in the first term and $(\omega x_{v}^{2})$ small in the second term to
rewrite $F_{\omega}(x_{1},\ldots,x_{L})$ as a formal power series:

$F_{\omega}(x_{1},\ldots,x_{L})=\sum_{\ell\in
2{\mathbb{Z}}^{L}+\delta}F_{\omega}^{\ell}\prod_{v}x_{v}^{\ell_{v}}\qquad\in{\mathbb{Z}}[\omega][[x_{1}^{\pm
1},\ldots,x_{1}^{\pm L}]]~{},$ (51)

where
$\delta\in{\mathbb{Z}}^{L}/2{\mathbb{Z}}^{L},~{}\delta_{v}\equiv\text{deg}(v)\mod
2$ and

$F_{\omega}^{\ell}=\sum_{m:\,\ell\in{\mathcal{I}}_{m}}N_{m,\ell}\,\omega^{m}\;\in{\mathbb{Z}}[\omega]$
(52)

with ${\mathcal{I}}_{m}$ being a finite set of elements from
${\mathbb{Z}}^{L}$. By definition, ${\rm lim}_{\omega\rightarrow
1}F_{\omega}^{\ell}$ is not dependent on $\Delta\in{\mathbb{Z}}^{L}$ (48).
However this $\omega\rightarrow 1$ limit in eqn. (3.1) will restrict the
binomial expansion range of the first term to be $x\rightarrow\infty$ and that
of the second term to $x\rightarrow 0$:

$\displaystyle F_{\omega\rightarrow 1}(x_{1},\ldots,x_{L})=\sum_{\ell\in
2{\mathbb{Z}}^{L}+\delta}F_{\omega\rightarrow
1}^{\ell}\prod_{v}x_{v}^{\ell_{v}}$ $\displaystyle=$ (53)
$\displaystyle~{}~{}{\rm lim}_{\omega\rightarrow
1}\prod_{v\,\in\,\text{Vertices}}\left\\{{\scriptsize\begin{array}[]{c}\text{Expansion}\\\
\text{as}x\rightarrow\infty\end{array}}\frac{1}{(x_{v}-\omega/x_{v})^{\text{deg}\,v-2}}\right.$
$\displaystyle+$
$\displaystyle\left.{\scriptsize\begin{array}[]{c}\text{Expansion}\\\ \text{as
}x\rightarrow 0\end{array}}\frac{1}{(\omega
x_{v}-1/x_{v})^{\text{deg}\,v-2}}\right\\}.$ (58)

Now let us assume that the quadratic form
$B:{\mathbb{Z}}^{L}\times{\mathbb{Z}}^{L}\rightarrow{\mathbb{Z}}$ is negative
definite i.e.,  $\sigma=-L$. Then we can define the following series in $q$
which is convergent for $|q|<1$:

$\hat{Z}_{b}^{SU(2)}[M(\Gamma);q]\stackrel{{\scriptstyle\text{Def}}}{{=\joinrel=}}2^{-L}q^{-\frac{3L+\sum_{v}f_{v}}{4}}\sum_{\ell\in
2B{\mathbb{Z}}^{L}+b}F^{\ell}_{\omega\rightarrow
1}\,q^{-\frac{(\ell,B^{-1}\ell)}{4}}\;\in\;2^{-c}q^{\Delta_{b}}{\mathbb{Z}}[[q]]$
(59)

where $c\in{\mathbb{Z}}_{+},c\leq L$ and

$\displaystyle b$ $\displaystyle\in$
$\displaystyle(2{\mathbb{Z}}^{L}+\delta)/2B{\mathbb{Z}}^{L}\,/{\mathbb{Z}}_{2}\cong(2\text{Coker}\,B+\delta)\,/{\mathbb{Z}}_{2}\stackrel{{\scriptstyle\text{Set}}}{{\cong}}H_{1}(M_{3},{\mathbb{Z}})\,/{\mathbb{Z}}_{2},$
(60) $\displaystyle\Delta_{b}$ $\displaystyle=$
$\displaystyle-\frac{3L+\sum_{v}f_{v}}{4}-\max_{\ell\in
2M{\mathbb{Z}}^{L}+b}\frac{(\ell,B^{-1}\ell)}{4}\,\in{\mathbb{Q}}$ (61)

where ${\mathbb{Z}}_{2}$ action takes $b\rightarrow-b$ and is the symmetry of
(59).

Using relation (49) and applying Gauss reciprocity formula (43) we arrive at
the following expression for the WRT invariant:

$\tau_{k}^{SU(2)}[M(\Gamma);{\mathbbm{q}}]=\\\ \\\
~{}~{}~{}~{}~{}\frac{e^{-\frac{\pi
iL}{4}}\,{\mathbbm{q}}^{-\frac{3L+\sum_{v}f_{v}}{4}}}{2\,(2(k+2))^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\,\lim_{\omega\rightarrow
1}\sum_{{n}\in{\mathbb{Z}}^{L}/2(k+2){\mathbb{Z}}^{L}}{\mathbbm{q}}^{\frac{(n,Bn)}{4}}F_{\omega}(x_{1},\ldots,x_{L})|_{x_{v}={\mathbbm{q}}^{n_{v}/2}}=\\\
\\\
~{}~{}\frac{2^{-L}{\mathbbm{q}}^{-\frac{3L+\sum_{v}f_{v}}{4}}}{2\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\sum_{\scriptsize\begin{array}[]{c}a\in\mathrm{Coker}\,B\\\ b\in
2\mathrm{Coker}\,B+\delta\end{array}}e^{-2\pi i(a,B^{-1}b)}e^{-2\pi
i(k+2)(a,B^{-1}a)}\times\\\ \lim_{\omega\rightarrow 1}\sum_{\ell\in
2B{\mathbb{Z}}^{L}+b}F^{\ell}_{\omega}\,{\mathbbm{q}}^{-\frac{(\ell,B^{-1}\ell)}{4}}.$
(62)

Assuming that the limit
$\lim_{q\rightarrow{\mathbbm{q}}}\hat{Z}_{b}^{SU(2)}(q)$ exists, where $q$
approaches $(k+2)$-th primitive root of unity from inside of the unit disc
$|q|<1$, we expect

$\lim_{\omega\rightarrow 1}\sum_{\ell\in
2B{\mathbb{Z}}^{L}+b}F^{\ell}_{\omega}\,{\mathbbm{q}}^{-\frac{(\ell,B^{-1}\ell)}{4}}=\lim_{q\rightarrow{\mathbbm{q}}}\sum_{\ell\in
2B{\mathbb{Z}}^{L}+b}F^{\ell}_{\omega\rightarrow
1}\,q^{-\frac{(\ell,B^{-1}\ell)}{4}}.$ (63)

Thus we obtain GPPV conjecture form:

$\tau_{k}^{SU(2)}[M(\Gamma),{\mathbbm{q}}]=\frac{1}{2\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\,\times\\\ \sum_{a\in\mathrm{Coker}\,B}e^{-2\pi
i(k+2)(a,B^{-1}a)}\sum_{b\in 2\mathrm{Coker}\,B+\delta}e^{-2\pi
i(a,B^{-1}b)}\lim_{q\rightarrow{\mathbbm{q}}}\hat{Z}_{b}^{SU(2)}[M(\Gamma);q].$
(64)

There is also an equivalent contour integral form for the homological
blocks(59):

$\hat{Z}_{b}^{SU(2)}[M(\Gamma);q]=q^{-\frac{3L+\sum_{v}f_{v}}{4}}\cdot\text{v.p.}\int\limits_{|z_{v}|=1}\prod_{v\;\in\;\text{Vertices}}\frac{dz_{v}}{2\pi
iz_{v}}\,\left({z_{v}-1/z_{v}}\right)^{2-\text{deg}(v)}\cdot\Theta^{-B}_{b}(z),$
(65)

where $\Theta^{-B}_{b}(x)$ is the theta function of the lattice corresponding
to minus the linking form $B$:

$\Theta^{-B}_{b}(x)=\sum_{\ell\in
2B{\mathbb{Z}}^{L}+b}q^{-\frac{(\ell,B^{-1}\ell)}{4}}\prod_{i=1}^{L}x_{i}^{\ell_{i}},$
(66)

and “v.p.” refers to principle value integral (i.e. take half-sum of contours
$|z_{v}|=1\pm\epsilon$). This prescription corresponds to the regularization
by $\omega$ made in eqn.(3.1).

Thus we can obtain explicit $SU(2)$ $q$-series for any negative definite
plumbed three-manifolds. For completeness, we present the $q$-series for some
examples.

###  3.2 Examples

$\bullet$ Poincare homology sphere is a well-studied three-manifold
corresponding to the graph:

(67)

As $H_{1}(M,{\mathbb{Z}})=0$, we obtain only single homological block
$\hat{Z}_{b_{1}}$. Solving eqns.(58,59), we get

$\hat{Z}_{b_{1}}^{SU(2)}=q^{-3/2}(1-q-q^{3}-q^{7}+q^{8}+q^{14}+q^{20}+q^{29}-q^{31}-q^{42}+\cdots).$
(68)

$\bullet$ The next familiar example with $H_{1}(M,{\mathbb{Z}})=0$ is
Brieskorn homology sphere. A particular example of this class is
$\Sigma(2,3,7)$ with the following equivalent graphs:

${\,\raisebox{-64.01869pt}{\includegraphics[width=128.0374pt]{example-3ii}}\,}\stackrel{{\scriptstyle\text{Kirby}}}{{\sim}}{\,\raisebox{-59.75095pt}{\includegraphics[width=106.69783pt]{example-3iii}}\,}$
(69)

The homological block turns out to be

$\hat{Z}_{b_{1}}^{SU(2)}=q^{1/2}(1-q-q^{5}+q^{10}-q^{11}+q^{18}+q^{30}-q^{41}+q^{43}-q^{56}-q^{76}\cdots).$
(70)

$\bullet$ For a three-manifold with non-trivial
$H_{1}(M,{\mathbb{Z}})={\mathbb{Z}}_{3}$ as drawn below,

${\,\raisebox{-56.9055pt}{\includegraphics[width=227.62204pt]{example-2a}}\,}\stackrel{{\scriptstyle\text{Kirby}}}{{\sim}}{\,\raisebox{-56.9055pt}{\includegraphics[width=113.81102pt]{example-2b}}\,}$
(71)

the three homological blocks are

$\hat{Z}^{SU(2)}=\left(\begin{array}[]{c}1-q+q^{6}-q^{11}+q^{13}-q^{20}+q^{35}+O\left(q^{41}\right)\\\
q^{5/3}\left(-1+q^{3}-q^{21}+q^{30}+O\left(q^{41}\right)\right)\\\
q^{5/3}\left(-1+q^{3}-q^{21}+q^{30}+O\left(q^{41}\right)\right)\\\
\end{array}\right),$ (72)

where two of them are equal.

Our focus is to obtain explicit $q$-series for $SO(3)$ and $OSp(1|2)$ groups.
Using the relation between $SU(2)$ and $SO(3)$, $SU(2)$ and $OSp(1|2)$ link
invariants(2.1.1), we will investigate the necessary steps starting from the
WRT invariant for $SO(3)$ and $OSp(1|2)$ eventually leading to the
$\hat{Z}$-invariant. This will be the theme of the following section.

##  4 $\hat{Z}$ for $SO(3)$ and $OSp(1|2)$

Our aim is to derive the $\hat{Z}$-invariant for $SO(3)$ and $OSp(1|2)$
groups. We will first look at the WRT invariants
$\tau_{K}^{SO(3)}[M(\Gamma);Q]$ for plumbed three-manifolds written in terms
of colored Jones invariants of framed links ${\mathcal{L}}[\Gamma]$ in the
following subsection and then discuss $OSp(1|2)$ $\hat{Z}$ in the subsequent
section.

###  4.1 $SO(3)$ WRT invariant and $\hat{Z}^{SO(3)}$ invariant

Recall that the framed link invariants are written in variable ${\mathbbm{q}}$
which is dependent on Chern-Simons coupling and the rank of the gauge group
$\mathcal{G}$. For $SO(3)$ Chern-Simons with coupling $K\in 2\mathbb{Z}$, the
variable $Q=e^{\frac{2\pi i}{K+1}}$. Hence $F^{SO(3)}[\mathcal{L}(\Gamma);Q]$
in WRT $\tau_{K}^{SO(3)}[M(\Gamma);Q]$ is

$F^{SO(3)}[\mathcal{L}(\Gamma);Q]=\\\
\sum_{n_{1},n_{2},\dots,n_{L}\in\\{0,1,\dots,K\\}}V_{n_{1},n_{2},\dots,n_{L}}^{SO(3)}(\mathcal{L}(\Gamma);Q)\prod_{v=1}^{L}V_{n_{1},n_{2},\dots,n_{L}}^{SO(3)}(\bigcirc;Q)~{}=\\\
\\\
\sum_{n_{1},n_{2},\ldots,n_{L}\in\\{1,3,\dots,2K+1\\}}J_{n_{1},n_{2},\dots,n_{L}}^{SU(2)}\left(\mathcal{L}(\Gamma);{\mathbbm{q}}=e^{\frac{2\pi
i}{2K+2}}\right)\prod_{v=1}^{L}\frac{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}}\bigg{|}_{{\mathbbm{q}}^{2}\rightarrow
Q}~{},$ (73)

where we have used the relation (13) to write $SO(3)$ link invariants in terms
of the colored Jones invariants. Notice that the summation is over only odd
integers and hence WRT invariant for $SO(3)$ is different from the WRT for
$SU(2)$ group. Further, the highest integrable representation in the summation
indicates that the Chern-Simons coupling for $SU(2)$ group is $2K+2$. After
performing the summation, we can convert the ${\mathbbm{q}}=Q^{1/2}$(13) to
obtain $SO(3)$ WRT invariant. We need to modify the Gauss sum reciprocity
formula to incorporate the summation over odd integers in
$F^{SO(3)}[\mathcal{L}(\Gamma);Q]$.

Using the following Gauss sum reciprocity formula

$\sum_{n\;\in\;{\mathbb{Z}}^{L}/k{\mathbb{Z}}^{L}}\exp\left(\frac{2\pi
i}{k}(n,Bn)+\frac{2\pi i}{k}(\ell,n)\right)=\\\ \frac{e^{\frac{\pi
i\sigma}{4}}\,(k/2)^{L/2}}{|\det
B|^{1/2}}\sum_{a\;\in\;{\mathbb{Z}}^{L}/2B{\mathbb{Z}}^{L}}\exp\left(\frac{-\pi
ik}{2}\left(a+\frac{\ell}{k},B^{-1}\left(a+\frac{\ell}{k}\right)\right)\right)~{},$
(74)

for $k=2K+2$, we can obtain the summation over odd integers by replacing
$n\longrightarrow\frac{n-1}{2}$ :

$\sum_{n_{1},n_{2},\dots,n_{L}\;\in\;\\{1,3,\dots,4K+3\\}}{\mathbbm{q}}^{\frac{(n,Bn)}{4}+\frac{(n,d)}{2}}=\frac{e^{\frac{\pi
i\sigma}{4}}\,(K+1)^{L/2}}{|\det
B|^{1/2}}{\mathbbm{q}}^{-\frac{(d,B^{-1}d)}{4}}\times\\\ \\\
\sum_{a\;\in\;{\mathbb{Z}}^{L}/2B{\mathbb{Z}}^{L}}\exp\left[-\pi
i(K+1)(a,B^{-1}a)\right]\exp\left[-\pi i(a,B^{-1}(d+BI))\right],$ (75)

where $d=\ell-BI$ with $I$ denoting $L$ component vector with entry $1$ on all
the components. That is, the transpose of the vector $I$ is

$I^{T}=[1,1,\ldots,1]~{}.$ (76)

For unknot with framing $\pm 1$, the
$F^{SO(3)}[{\mathcal{L}}(-1\bullet);Q={\mathbbm{q}}^{2}]$ involving summation
over odd integers simplifies to

$F^{SO(3)}[{\mathcal{L}}(\pm\bullet);Q={\mathbbm{q}}^{2}]=\frac{\sqrt{K+1}\;e^{\pm\pi
i/4}\;{\mathbbm{q}}^{\mp
3/4}}{{\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2}}\underbrace{(1+e^{\pi
iK})}_{2}~{},$ (77)

as the coupling $K\in 2\mathbb{Z}$ for the $SO(3)$ Chern-Simons theory. Hence,
the WRT invariant takes the following form:

$\tau_{K}^{SO(3)}[M(\Gamma);Q={\mathbbm{q}}^{2}]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma}{4}}}{2^{L}(K+1)^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n}\in\\{1,3,\ldots,2K+1\\}^{L}}}\prod_{v\;\in\;\text{Vertices}}{\mathbbm{q}}^{\frac{f_{v}(n_{v}^{2}-1)}{4}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}\times\\\
\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}\Big{(}{\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2}\Big{)}.$
(78)

In above equation, the terms involving edges of the graph $\Gamma$

$\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}\Big{(}{\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2}\Big{)}=2^{L-1}\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}\frac{\Big{(}{\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2}\Big{)}}{2},$

can also be rewritten as

$\prod_{(v^{\prime},v^{\prime\prime})\in\text{Edges}}({\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}-{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2})=\sum_{p\in\\{\pm
1\\}^{\text{Edges}}}\prod_{(v^{\prime},v^{\prime\prime})\in\text{Edges}}p_{(v^{\prime},v^{\prime\prime})}{\mathbbm{q}}^{p_{(v^{\prime},v^{\prime\prime})}n_{v^{\prime}}n_{v^{\prime\prime}}/2}~{}.$

Here again, if we make a change of variable as $n_{v}\longrightarrow-n_{v}$ at
any vertex, a term in the sum with a given configuration of signs associated
to edges (that is $p\in\\{\pm 1\\}^{\text{Edges}}$) will transform into a term
with a different configuration times $(-1)^{\text{deg}(v)}$. However, for
these plumbing graphs $\Gamma$, the signs of such configuration can be brought
to the configuration with all signs +1. Incorporating this fact, the WRT
invariant(78) simplifies to

$\tau_{K}^{SO(3)}[M(\Gamma);Q={\mathbbm{q}}^{2}]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma-\sum_{v}f_{v}}{4}}}{2\,(K+1)^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n}\in\\{1,3,\ldots,2K+1\\}^{L}}}\;\;{\mathbbm{q}}^{\frac{(n,Bn)}{4}}\prod_{v\;\in\;\text{Vertices}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}.$
(79)

Further, we double the range of summation so as to use the reciprocity
formula(75)

$\tau_{K}^{SO(3)}[M(\Gamma);Q={\mathbbm{q}}^{2}]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma-\sum_{v}f_{v}}{4}}}{4\,(K+1)^{L/2}\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n}\in\\{1,3,\ldots,4K+3\\}^{L}}}\;\;{\mathbbm{q}}^{\frac{(n,Bn)}{4}}\prod_{v\;\in\;\text{Vertices}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}-{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}.$
(80)

The steps discussed in the $SU(2)$ context to extract $\hat{Z}$ can be
similarly followed for $SO(3)$. This procedure leads to

$\tau_{K}^{SO(3)}[M(\Gamma);Q={\mathbbm{q}}^{2}]=\frac{1}{2\,({\mathbbm{q}}^{1/2}-{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\,\sum_{a\in\mathrm{Coker}\,B}e^{-\pi i(K+1)(a,B^{-1}a)}\\\ \\\
\sum_{b\in 2\mathrm{Coker}\,B+\delta}e^{-\pi
i\big{(}a,B^{-1}(b{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}+BI})\big{)}}\lim_{q\rightarrow{\mathbbm{q}}}\hat{Z}^{SO(3)}_{b}[M(\Gamma);q]~{}.$
(81)

We observe that the $SO(3)$ WRT invariant is different from the $SU(2)$
invariant due to the factor highlighted in blue color in the summand whereas
the $\hat{Z}^{SO(3)}_{b}[M(\Gamma);q]$ is exactly same as the $SU(2)$
$q$-series. Even though $SO(3)\equiv SU(2)/\mathbb{Z}_{2}$, it is surprising
to see that the factor group $SO(3)$ shares the same $\hat{Z}$ as that of the
parent group $SU(2)$. The case of $\hat{Z}^{SO(3)}$ was also considered in
Costantino:2021yfd but they took a different route by considering the refined
WRT invariant which is consistent with our result.

In the following subsection, we will extract $\hat{Z}$ from the WRT invariant
$\tau_{\hat{K}}^{OSp(1|2)}[M(\Gamma);\hat{Q}]$ for $OSp(1|2)$ supergroup. We
will see that the $OSp(1|2)$ $q$-series are related to
$\hat{Z}^{SU(2)}[M(\Gamma);q]$.

###  4.2 $OSp(1|2)$ WRT and $\hat{Z}^{OSp(1|2)}$ invariant

Using the relation between $OSp(1|2)$ and $SU(2)$ link invariants (17), the
WRT invariant can be written for plumbed manifolds $M(\Gamma)$ as

$\tau_{\hat{K}}^{OSp(1|2)}[M(\Gamma);\hat{Q}={\mathbbm{q}}]=\frac{e^{-\frac{\pi
i\sigma}{4}}\,{\mathbbm{q}}^{\frac{3\sigma}{4}}}{(2\hat{K}+3)^{L/2}\,({\mathbbm{q}}^{1/2}+{\mathbbm{q}}^{-1/2})}\times\\\
{\sum_{{n_{1},n_{2},\ldots,n_{L}}\in\\{1,3,\ldots,2\hat{K}+1\\}}}\;\;\prod_{v\;\in\;\text{Vertices}}{\mathbbm{q}}^{\frac{f_{v}(n_{v}^{2}-1)}{4}}\,\left(\frac{1}{{\mathbbm{q}}^{n_{v}/2}+{\mathbbm{q}}^{-n_{v}/2}}\right)^{\text{deg}(v)-2}\times\\\
\prod_{(v^{\prime},v^{\prime\prime})\;\in\;\text{Edges}}\Big{(}{\mathbbm{q}}^{n_{v^{\prime}}n_{v^{\prime\prime}}/2}+{\mathbbm{q}}^{-n_{v^{\prime}}n_{v^{\prime\prime}}/2}\Big{)}\Big{|}_{{\mathbbm{q}}=\hat{Q}}~{}.$
(82)

Here again we use the Gauss reciprocity(75) as the summation is over odd
integers to work out the steps leading to $\hat{Z}^{OSp(1|2)}[M(\Gamma);q]$.
Note that, the highest integrable representation $2\hat{K}+1$ which fixes the
${\mathbbm{q}}$ as $(2\hat{K}+2)$-th root of unity. However to compare the
result with $OSp(1|2)$ WRT, we have to replace $\hat{K}+1\rightarrow
2\hat{K}+3$ which is equivalent to ${\mathbbm{q}}=\hat{Q}$.

Following similar steps performed for $SU(2)$, we find the following
expression for $OSp(1|2)$ WRT invariant:

$\frac{1}{2\,({\mathbbm{q}}^{1/2}+{\mathbbm{q}}^{-1/2})\,|\det
B|^{1/2}}\,\sum_{a\in\mathrm{Coker}\,B}e^{-\pi
i(2\hat{K}+3)(a,B^{-1}a)}\times\\\ \sum_{b\in
2\mathrm{Coker}\,B+\delta}e^{-\pi
i\big{(}a,B^{-1}(b{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}+BI})\big{)}}\lim_{q\rightarrow\hat{Q}}\hat{Z}_{b}^{OSp(1|2)}[M(\Gamma);q],$
(83)

where $I$ is again the column vector (76) and
$\hat{Z}_{b}^{OSp(1|2)}[M(\Gamma);q]$ is given by the following algebraic
expression:

$\hat{Z}_{b}^{OSp(1|2)}[M(\Gamma);q]\;\;=\;\;2^{-L}q^{-\frac{3L+\sum_{v}f_{v}}{4}}\sum_{d\;\in\;2B{\mathbb{Z}}^{L}+b}F^{d}_{1}\,q^{-\frac{(d,B^{-1}d)}{4}},$
(84)

with coefficient $F_{1}^{d}$ is obtained by following relation

$\sum_{d\;\in\;2{\mathbb{Z}}^{L}+\delta}F_{1}^{d}\prod_{v}x_{v}^{d_{v}}=\\\
\prod_{v\,\in\,\text{Vertices}}\left\\{{\scriptsize\begin{array}[]{c}\text{Expansion}\\\
\text{at }x\rightarrow
0\end{array}}\frac{1}{(x_{v}+1/x_{v})^{\text{deg}\,v-2}}+{\scriptsize\begin{array}[]{c}\text{Expansion}\\\
\text{at
}x\rightarrow\infty\end{array}}\frac{1}{(x_{v}+1/x_{v})^{\text{deg}\,v-2}}\right\\}.$
(85)

Equivalently, $\hat{Z}^{OSp(1|2)}[M(\Gamma);q]$(84) can also represented as
the following contour integral:

$\hat{Z}_{b}^{OSp(1|2)}[M(\Gamma);q]=q^{-\frac{3L+\sum_{v}f_{v}}{4}}\cdot\text{v.p.}\int\limits_{|z_{v}|=1}\prod_{v\;\in\;\text{Vertices}}\frac{dz_{v}}{2\pi
iz_{v}}\,\left({z_{v}+1/z_{v}}\right)^{2-\text{deg}(v)}\cdot\Theta^{-B}_{b}(z)~{}.$
(86)

Here $\Theta^{-B}_{b}(x)$ is the theta function of the lattice corresponding
to minus the linking form $B$:

$\Theta^{-B}_{b}(x)=\sum_{d\;\in\;2B{\mathbb{Z}}^{L}+b}q^{-\frac{(d,B^{-1}d)}{4}}\prod_{i=1}^{L}x_{i}^{d_{i}}$
(87)

and “v.p.” again means that we take principle value integral (i.e. take half-
sum of contours $|z_{v}|=1\pm\epsilon$). Comparing eqns.(84,85) with the
$SU(2)$ expressions(58,59), we can see that the $\hat{Z}$ for $OSp(1|2)$ are
different from $SU(2)$ q-series. We will now present explicit $q$-series for
some examples.

####  4.2.1 Examples

$\bullet$ For the Poincare homology sphere(67), we find the following
$OSp(1|2)$ $q$-series

$\hat{Z}_{b_{1}}^{OSp(1|2)}=q^{-3/2}(1+q+q^{3}+q^{7}+q^{8}+q^{14}+q^{20}-q^{29}+q^{31}-q^{42}-q^{52}+\cdots).$
(88)

$\bullet$ In the case of Brieskorn homology sphere(69), the $OSp(1|2)$
$q$-series is

$\hat{Z}_{b_{1}}^{OSp(1|2)}=q^{1/2}(1+q+q^{5}+q^{10}+q^{11}+q^{18}+q^{30}+q^{41}-q^{43}-q^{56}-q^{76}\cdots).$
(89)

$\bullet$ For the case of plumbing graph(71), the three homological blocks are

$\hat{Z}^{OSp(1|2)}=\left(\begin{array}[]{c}1+q+q^{6}+q^{11}-q^{13}-q^{20}-q^{35}+O\left(q^{41}\right)\\\
q^{5/3}\left(1+q^{3}-q^{21}-q^{30}+O\left(q^{41}\right)\right)\\\
q^{5/3}\left(1+q^{3}-q^{21}-q^{30}+O\left(q^{41}\right)\right)\end{array}\right).$
(90)

After comparing the $q$-series for $SU(2)$ and $OSp(1|2)$, we noticed that
these two $q$-series are related by a simple change of variable which is
$q\longrightarrow-q$. This change of variable applies only to the series not
to the overall coefficient outside the series.
$\bullet$ Lens space $L(p,q)$ is a well studied three-manifold. For
$L(-5,11)\sim L(-13,29)$ whose plumbing graph is shown below, we obtain the
five homological blocks

${\,\raisebox{0.0pt}{\includegraphics[width=142.26378pt]{example4b}}\,}\;\;\stackrel{{\scriptstyle\text{Kirby}}}{{\sim}}\;\;{\,\raisebox{0.0pt}{\includegraphics[width=142.26378pt]{example4a}}\,}$

$\hat{Z}^{OSp(1|2)}=\left(\begin{array}[]{c}q^{1/10}\\\ q^{-1/10}\\\ 0\\\
q^{-1/10}\\\
q^{1/10}\end{array}\right)~{}~{}~{}~{}\text{as}~{}~{}H_{1}(M,\mathbb{Z})={\mathbb{Z}}_{5}.$
(91)

$\bullet$ For the following plumbing graph,
$H_{1}(M,{\mathbb{Z}})=\mathbb{Z}_{13}$ ,

$\hat{Z}^{OSp(1|2)}=\frac{1}{4}\left(\tiny\begin{array}[]{c}q^{-1/2}(2+2q+2q^{2}-2q^{4}-4q^{5}+6q^{10}-8q^{11}+4q^{13}+2q^{14}-4q^{15}+O\left(q^{18}\right))\\\
q^{5/26}(3+2q-2q^{2}-4q^{3}-2q^{7}+q^{8}+2q^{9}+q^{10}-2q^{12}-4q^{13}-2q^{16}+O\left(q^{18}\right))\\\
q^{5/26}(3+2q-2q^{2}-4q^{3}-2q^{7}+q^{8}+2q^{9}+q^{10}-2q^{12}-4q^{13}-2q^{16}+O\left(q^{18}\right))\\\
q^{7/26}(4-q-2q^{3}-2q^{4}-2q^{6}+3q^{7}-2q^{8}-2q^{10}+q^{11}+2q^{13}-4q^{14}+2q^{15}-4q^{16}+O\left(q^{18}\right))\\\
q^{7/26}(4-q-2q^{3}-2q^{4}-2q^{6}+3q^{7}-2q^{8}-2q^{10}+q^{11}+2q^{13}-4q^{14}+2q^{15}-4q^{16}+O\left(q^{18}\right))\\\
q^{-7/26}(3+3q^{2}-2q^{4}-2q^{5}+4q^{7}-2q^{8}+2q^{9}-2q^{10}-4q^{12}+4q^{13}-4q^{14}+2q^{15}+O\left(q^{18}\right))\\\
q^{-7/26}(3+3q^{2}-2q^{4}-2q^{5}+4q^{7}-2q^{8}+2q^{9}-2q^{10}-4q^{12}+4q^{13}-4q^{14}+2q^{15}+O\left(q^{18}\right))\\\
q^{-11/26}(1+2q+2q^{2}+4q^{3}+3q^{6}-2q^{7}-4q^{8}-2q^{9}+2q^{11}+2q^{13}-q^{14}+2q^{16}-2q^{17}+O\left(q^{18}\right))\\\
q^{-11/26}(1+2q+2q^{2}+4q^{3}+3q^{6}-2q^{7}-4q^{8}-2q^{9}+2q^{11}+2q^{13}-q^{14}+2q^{16}-2q^{17}+O\left(q^{18}\right))\\\
q^{-5/26}(2+2q^{2}+q^{3}+3q^{5}-2q^{6}-2q^{7}-4q^{8}-2q^{10}+2q^{11}-2q^{12}+2q^{13}+5q^{15}-2q^{16}+2q^{17}+O\left(q^{18}\right))\\\
q^{-5/26}(2+2q^{2}+q^{3}+3q^{5}-2q^{6}-2q^{7}-4q^{8}-2q^{10}+2q^{11}-2q^{12}+2q^{13}+5q^{15}-2q^{16}+2q^{17}+O\left(q^{18}\right))\\\
q^{-15/26}(1-2q-2q^{2}+q^{4}-2q^{6}-2q^{7}-2q^{8}-4q^{10}-2q^{12}+2q^{13}+2q^{15}+4q^{17}+O\left(q^{18}\right))\\\
q^{-15/26}(1-2q-2q^{2}+q^{4}-2q^{6}-2q^{7}-2q^{8}-4q^{10}-2q^{12}+2q^{13}+2q^{15}+4q^{17}+O\left(q^{18}\right))\end{array}\right)$
(92)

We have checked for many examples that under $q\rightarrow-q$ in the
$OSp(1|2)$ $q$-series(not affecting the overall coefficient), we obtain the
$SU(2)$ $q$-series.

##  5 Conclusions and future directions

Our goal was to investigate $\hat{Z}$ for $SO(3)$ and $OSp(1|2)$ groups for
negative definite plumbed three-manifolds. The change of variable and color
indeed relates invariants of framed links ${\mathcal{L}}[\Gamma]$(13,17) of
$SO(3)$ and $OSp(1|2)$ to colored Jones. Such a relation allowed us to go
through the steps of GPPV conjecture to extract $\hat{Z}$ from WRT invariants.

Interestingly, we observe that the $\hat{Z}^{SO(3)}$ is same as
$\hat{Z}^{SU(2)}$ even though the WRT invariants are different. We know that
$SU(2)/\mathbb{Z}_{2}\equiv SO(3)$ and it is not at all obvious that the
homological blocks are same for both the groups. It is important to explore
other factor groups and the corresponding $\hat{Z}$ invariants.

For the odd orthosympletic supergroup $OSp(1|2)$, we observe from our
computations for many negative definite plumbing graph $\Gamma$:

$\hat{Z}^{OSp(1|2)}_{b}(\Gamma;q)=2^{-c}q^{\Delta_{b}}\left(\sum_{n}a_{n}q^{n}\right)$

whereas their $SU(2)$ q-series is

$\hat{Z}^{SU(2)}_{b}(\Gamma;q)=2^{-c}q^{\Delta_{b}}\left(\sum_{n}a_{n}(-q)^{n}\right)$

where $c\in\mathbb{Z}_{+}$, $\Delta_{b}\in\mathbb{Q}$.

The brane setup in string theory for $U(N)$ gauge group gives a natural
interpretation for these $q$-series as partition function of the theory
$T^{\mathcal{G}}[M]$. In principle, there should be a natural generalisation
to orthogonal $SO(N)$ and symplectic group $Sp(2n)$ involving orientifolds. It
will be worth investigating such a construction to obtain $\hat{Z}$ for
$SO(N)$ group and compare with our $SO(N=3)$ results. Extension of $\hat{Z}$
to the two variable series for link complementsGukov:2019mnk is another
direction to pursue. We hope to report on these aspects in future.

###### Acknowledgements.

PR is grateful to ICTP senior associateship funding for visit where this work
was initiated with Pavel Putrov and Francesca Ferrari during summer 2019.
Unfortunately due to covid, we could not pursue the collaboration through
online mode. We would like to thank Pavel Putrov and Francesca Ferrari for
clarifying the notations during the initial stages. SC would like to thank
Sunghyuk Park for useful comments and discussions during the String-Math 2022
conference held at Univ. of Warsaw, Poland. SC is grateful to Dmitry
Noshchenko for his comments on the manuscript. We would also like to thank
Vivek Kumar Singh for his clarification on mathematica program. SC would also
like to thank the organisers of String-Math 2022 where a part of this work was
presented. SC is thankful for the MHRD fellowship from IIT Bombay providing
financial support to visit. PR would like to thank SERB (MATRICS)
MTR/2019/000956 funding. We would like to thank the reviewers for their
valuable comments and the suggestions.

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