††institutetext: Physics Department, Ariel University, Ariel 40700, Israel.

# Wilson lines construction of $\mathfrak{osp}(1|2)$ conformal blocks

Vladimir Belavin, J. Ramos Cabezas<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study N=1 superconformal theory in the context of AdS/CFT correspondence in
the large central charge limit using Chern-Simons formulation of $3d$ gravity.
In this limit conformal dimensions of a subclass of so-called light primary
superfields remain finite and are governed by $\mathfrak{osp}(1|2)$ subalgebra
of N=1 super-Virasoro algebra. We describe the construction of
$\mathfrak{osp}(1|2)$ conformal blocks in terms of Wilson lines of the Chern-
Simons $3d$ gravity. We consider examples of two and three-point blocks on the
sphere and one-point torus blocks of light superprimary fields, which belong
to finite-dimensional representations of $\mathfrak{osp}(1|2)$. We study the
correlation function for lower and upper components of the primary
$\mathfrak{osp}(1|2)$ doublets and show that the associated conformal blocks
are obtained via Wilson line construction in Chern-Simons theory.

##  1 Introduction

In recent works [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], in the
context of the AdS/CFT correspondence, conformal blocks in the large $c$ limit
have been identified with geodesic networks in the $AdS$ spacetime. An
alternative description of conformal blocks via the Wilson lines of the Chern-
Simons (CS) $3d$ gravity was considered in [15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. In this work we use the latter
approach in order to study super-Virasoro conformal blocks.

In the Virasoro case the four-point correlation function on the sphere of four
primary fields $\varphi_{i}$ with conformal dimensions $h_{i},\bar{h}_{i}$ is

$\langle\varphi_{1}(z_{1},\bar{z}_{1})...\varphi_{4}(z_{4},\bar{z}_{4})\rangle=\sum_{h,\bar{h}}C_{12h}C_{h34}|w(h,h_{i},z_{i})|^{2},$
(1.1)

where $C_{12h}$, $C_{h34}$ are the structure constants, $(z_{i},\bar{z}_{i})$
are coordinates on the complex plane and $w(h_{s},h_{i},z_{i})$ is the four-
point holomorphic conformal block111For the holographic description of higher
point conformal blocks see [33, 34, 35, 36, 37, 38, 39]..

Similarly, the correlation functions are defined on the torus, for example,
the one-point correlation function of a primary field $\varphi_{1}$ on the
torus is 222Throughout this paper we omit factor $(q\bar{q})^{-\frac{c}{24}}$
which can be easily restored.

$\langle\varphi_{1}(z_{1},\bar{z}_{1})\rangle=\text{\Large{Tr}}_{h}\left(q^{\mathbf{L}_{0}}\bar{q}^{\bar{\mathbf{L}}_{0}}\varphi_{1}(z_{1},\bar{z}_{1})\right)=\sum_{h}C_{hh_{1}h}|\mathcal{F}(h,h_{1},q)|^{2},$
(1.2)

where $Tr_{h}$ is the trace taken over descendent states associated with the
intermediate primary field $\varphi_{h}$, $q$ is the elliptic parameter of the
torus $q=e^{2\pi i\tau}$ and $\mathbf{L}_{0}$ is the Virasoro generator,
$\mathbf{L}_{0}|h\rangle=h|h\rangle$. Here $\mathcal{F}(h,h_{1},q)$ is the
one-point holomorphic torus conformal block (for details see [40, 41, 42,
43])333For AdS/CFT correspondence in thermal AdS, relevant for the torus
topology, see [12, 44, 45, 46, 47]..

In [20, 26, 48, 49] the dual description of $\mathfrak{sl}(2)$ and
$\mathfrak{sl}(3)$ conformal blocks (which are obtained from the Virasoro and
$W_{3}$ conformal blocks respectively in the large central charge limit for
light primary fields444The conformal dimension $h$ of a light primary field
(resp. heavy primary field) $\varphi_{h}$ by definition behaves like
$h=\mathcal{O}(c^{0})$ (resp. $h=\mathcal{O}(c^{1})$) in the limit
$c\rightarrow\infty$. For the holographic description of conformal blocks in
the context of Virasoro heavy-light fields see [50, 51, 52, 53, 54, 55, 56,
57].) on the sphere in the context of CS theory was found. It was shown that
$\mathfrak{sl}(2)$ 2-point, 3-point and 4-point conformal blocks on the sphere
can be represented as follows

$\langle
s|\prod_{i=1}^{n}e^{\int_{z_{i}}^{z_{b}}J_{1}^{(i)}dz}|hw\rangle_{i},$ (1.3)

where $n=2,3,4$ give 2-point, 3-point and 4-point holomorphic conformal blocks
respectively. Here $J_{1}^{(i)}dz$ is the $\mathfrak{sl}(2)$ CS connection on
the sphere acting in spin-$j_{i}$ representation, $|hw\rangle_{i}$ is the
highest weight state of the representation with spin $j_{i}=-h_{i}$, and
$z_{b}$ is an arbitrary point in the bulk where the Wilson line operators
$Pe^{\int_{z_{i}}^{z_{b}}J_{1}^{(i)}dz}$ are joined together. The state
$|s\rangle$ is a singlet state which belongs to the tensor product of the
representations $j_{i}$ ($i=1,2,...,n$). The relation between conformal blocks
and the expression (1.3) involves two basic ingredients: the singlet state
$|s\rangle$ representing a specific channel of OPE and the boundary state
$|hw\rangle_{i}$ associated with the boundary primary field $\varphi_{i}$

$|hw\rangle_{i}\rightarrow\varphi_{i}(z_{i}).$ (1.4)

In [45] (see also [58]) it was proposed that $\mathfrak{sl}(2)$ one-point
torus conformal block $\mathcal{F}(-j,-j_{1},q)$, where $-j$ and $-j_{1}$ are
respectively the intermediate and external conformal dimensions555In the dual
description $j$ and $j_{1}$ are the spins of finite-dimensional
representations of $\mathfrak{sl}(2)$ CS gauge group., can be computed by a
prescription similar to (1.3) by introducing the following modifications:
Wilson line operators at the point $z_{b}$ are joined using the intertwining
operator and two boundary points are identified modulo $2\pi\tau$, and the
trace is taken over corresponding representation. Thus the proposal reads

$Tr_{j}\Bigg{(}W_{j}[z_{b},z_{b}+2\pi\tau]I_{j;j,j_{1}}\Bigg{)}\otimes
W_{j_{1}}[z_{b},z_{1}]|lw\rangle_{1}=\mathcal{F}(-j,-j_{1},q).$ (1.5)

Here $Tr_{j}$ is taken over the representation with spin $j$, $z_{1}$ is the
point on the boundary of the solid torus which is a geometric representation
of the thermal AdS3, $z_{b}$ is an arbitrary point in the bulk of AdS3,
$|lw\rangle_{1}$ is the lowest weight state of the representation with spin
$j_{1}$ and $I_{j;j,j_{1}}$ is the intertwining operator associated with the
representations $j$ and $j_{1}$. The factors $W_{a}[x,y]$ (for $a=j,j_{1}$;
$x=z_{b}$ and $y=z_{b}+2\pi\tau,z_{1}$) denote the Wilson line operators

$W_{a}[x,y]=\mathcal{P}\exp\bigg{(}-\int_{x}^{y}\Omega\bigg{)}=\exp\left((x-y)(J_{1}+\frac{1}{4}J_{-1})\right),$
(1.6)

where $\Omega$ is the flat connection

$\Omega=\bigg{(}J_{1}+\frac{1}{4}J_{-1}\bigg{)}dz,$ (1.7)

where $J_{1}$ and $J_{-1}$ are the lowering and raising operators of
$\mathfrak{sl}(2)$ in the representation $a$.

The purpose of this work is to investigate the generalization of this
construction on the $\mathfrak{osp}(1|2)$ algebra. As explained in A, this
algebra is relevant in the large $c$ limit of the $N=1$ super CFT. We are
interested in studying the correlation functions of superprimary fields666See
[59, 60, 61, 62, 63, 64, 65] for details of $N=1$ super CFT. For the AdS/CFT
in the context of superfields see also [66, 23]. in the Neveu-Schwarz (NS)
sector

$\Phi_{i}(z,\theta,\bar{z},\bar{\theta})=\varphi_{i}(z,\bar{z})+\theta\psi_{i}(z,\bar{z})+\bar{\theta}\bar{\psi}_{i}(z,\bar{z})+\bar{\theta}\theta\widetilde{\varphi}_{i}(z,\bar{z}),$
(1.8)

where $(z,\bar{z})$ are holomorphic and antiholomorphic coordinates, and
$(\theta,\bar{\theta})$ are Grassmann variables, $\varphi_{i}$ is a primary
field with conformal dimensions $(h_{i},\bar{h}_{i})$,
$\psi_{i}=[\mathbf{G}_{-\frac{1}{2}},\varphi_{i}]$,
$\bar{\psi}_{i}=[\mathbf{\bar{G}}_{-\frac{1}{2}},\varphi_{i}]$,
$\widetilde{\varphi}_{i}=\\{\mathbf{G}_{-\frac{1}{2}},\bar{\psi}_{i}\\}$. We
call $(\varphi_{i},\psi_{i},\bar{\psi}_{i},\widetilde{\varphi}_{i})$
components of the superfield $\Phi_{i}$. We are interested in the correlation
functions of light superfields in the spherical and toroidal topologies. The
conformal dimensions of degenerate superfields are given by the Kac formula
[60], in the large $c$ limit these conformal dimensions become

$h_{1,m}=-\frac{m-1}{4}+\frac{1-m^{2}}{8c}+\mathcal{O}(1/c^{2}),$ (1.9)

thus light degenerate NS superfields in the limit $c\rightarrow\infty$ have
conformal dimensions $-j=\frac{1-m}{4}$, where $m$ is odd, therefore $j$ can
take integer or half-integer values. Below, $j$ will be identified with the
superspin of a finite-dimensional representation of $\mathfrak{osp}(1|2)$.

On the sphere, we are interested in studying the Wilson lines formulation for
the correlation functions of light superfields. By using the expansion (1.8)
we can express the correlation function

$\langle\Phi_{1}(z_{1},\theta_{1},\bar{z}_{1},\bar{\theta}_{1})...\Phi_{n}(z_{n},\theta_{n},\bar{z}_{n},\bar{\theta}_{n})\rangle$
(1.10)

in terms of components, thus (1.10) contains $4^{n}$ terms, we call each of
these terms components of the correlation function. We will concentrate on the
components to which $\varphi_{i}$ and $\psi_{i}$ contribute, and we will
investigate only their holomorphic dependence (due to the factorization of the
symmetry into the holomorphic and antiholomorphic sectors, the consideration
of the antiholomorphic dependence and of the components $\bar{\psi}_{i}$,
$\widetilde{\varphi}_{i}$ is identical, see [41]). To this end it is
convenient to formally set $\bar{\theta}_{i}=0$, then we get

$\langle\Phi_{1}...\Phi_{n}\rangle=\langle\varphi_{1}...\varphi_{n}\rangle+...+\theta_{1}...\theta_{n}\langle\psi_{1}...\psi_{n}\rangle.$
(1.11)

The correlation functions on the torus can be also expressed in terms of the
components of the rhs of (1.8). We restrict ourselves to considering the
contributions of $\varphi_{i}$ and $\psi_{i}$ in the one-point correlation
function. By setting $\bar{\theta}_{1}=0$ we obtain

$\begin{split}\text{\Large{Tr}}_{h}\left[q^{\mathbf{L}_{0}}\bar{q}^{\mathbf{\bar{L}}_{0}}\Phi_{1}(z_{1},\theta_{1},\bar{z}_{1},\bar{\theta}_{1})\right]=\text{\Large{Tr}}_{h}\left[q^{\mathbf{L}_{0}}\bar{q}^{\mathbf{\bar{L}}_{0}}\varphi_{1}(z_{1},\bar{z}_{1})\right]+\theta_{1}\text{\Large{Tr}}_{h}\left[q^{\mathbf{L}_{0}}\bar{q}^{\mathbf{\bar{L}}_{0}}\psi_{1}(z_{1},\bar{z}_{1})\right],\end{split}$
(1.12)

where $\mathbf{L}_{0}$ is the generator of Virasoro subalgebra of Neveu-
Schwarz algebra. In the large $c$ limit, for light fields the trace $Tr_{h}$
in (1.12) is reduced to the $\mathfrak{osp}(1|2)$ subalgebra. We will see that
this trace splits into the even and odd parts

$\text{\Large{Tr}}_{h}[...]=\text{\Large{Tr}}_{h}^{e}[...]+\text{\Large{Tr}}_{h}^{o}[...],$
(1.13)

where $Tr_{h}^{e}$ is the trace over the states with weights $h+n$, $n$
belongs to the set of nonnegative integers, $Tr_{h}^{o}$ is the trace over the
states with weights $h+n+1/2$.

Below, we state the result of this work.

On the sphere, in the case of the two-point and three-point correlation
functions, the conformal block of each component can be computed by means of
equation (1.3) with the following prescription: $|s\rangle$ is a singlet which
belongs to the tensor product of superspin-$j_{i}$ representations of
$\mathfrak{osp}(1|2)$; $J_{1}^{(i)}$ has to be replaced by $L_{1}^{(i)}$-the
lowering generator of the CS gauge algebra $\mathfrak{osp}(1|2)$ acting in the
representation $j_{i}$; We will have two sorts of states, the highest weight
states $|hw\rangle_{i}$ with weights $j_{i}$ and states with weights
$j_{i}-1/2$ which we denote by $|hw-1/2\rangle_{i}$. For a component that
contains the fields $(\varphi_{i},\psi_{i})$, each field $\varphi_{i}$ in this
component will correspond to a state $|hw\rangle_{i}$ and each field
$\psi_{i}$ will correspond to a state $|hw-1/2\rangle_{i}$ in (1.3), thus
schematically we have

$\begin{split}&|hw\rangle_{i}\rightarrow\varphi_{i}(z_{i}),\\\
&|hw-1/2\rangle_{i}\rightarrow\psi_{i}(z_{i}).\end{split}$ (1.14)

In the case of the one-point correlation function on the torus, we show that
the conformal block of each component of the correlation function (1.12) can
be computed by means of the lhs of equation (1.5) using following
modifications: $Tr_{j}$ is the trace taken over the states of a
$\mathfrak{osp}(1|2)$ finite-dimensional representation with superspin $j=-h$;
$j_{1}$ is the superspin of a $\mathfrak{osp}(1|2)$ finite-dimensional
representation (the conformal dimension $h_{1}$ = $-j_{1}$); $I_{j;j,j_{1}}$
is the $\mathfrak{osp}(1|2)$ intertwining operator (this operator can be
expressed in terms of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients);
$W_{a}[z_{b},z]$ (for $a=j,j_{1}$; $z=z_{b}+2\pi\tau,z_{1}$) are the Wilson
line operators only this time ($J_{1},J_{-1}$) have to be replaced by
($L_{1},L_{-1}$)-the generators of the $\mathfrak{osp}(1|2)$ gauge algebra;
Similarly to the sphere, on the lhs of (1.5) we will have two sorts of states,
the lowest weight states $|lw\rangle_{1}$ with weight $-j_{i}$ and the states
with weight $-j_{1}+1/2$ which we denote by $|lw+1/2\rangle_{1}$. For a
component that contains the field $\varphi_{1}$, the field $\varphi_{1}$ in
this component will correspond to the state $|lw\rangle_{1}$ and for a
component that contains the field $\psi_{1}$, the field $\psi_{1}$ will
correspond to the state $|lw+1/2\rangle_{1}$ on the lhs of (1.5), thus
schematically we have

$\begin{split}&|lw\rangle_{i}\rightarrow\varphi_{i}(z_{i}),\\\
&|lw+1/2\rangle_{i}\rightarrow\psi_{i}(z_{i}).\end{split}$ (1.15)

The outline of the paper is the following: In section 2 we give a brief
framework of the $\mathfrak{osp}(1|2)$ CFT and $\mathfrak{osp}(1|2)$ finite-
dimensional representation theory. In section 3 we present the Wilson lines
formulation of $\mathfrak{osp}(1|2)$ conformal blocks in the spherical and
toroidal topology. In section 4 we give our conclusions and comment on further
developments of the Wilson lines formulation of $\mathfrak{osp}(1|2)$
conformal blocks. Appendixes A and B describe the notations of the Neveu-
Schwarz algebra and the explicit form of $\mathfrak{sl}(2)$ one-point torus
block respectively.

##  2 $\mathfrak{osp}(1|2)$ conformal field theory

The Neveu-Schwarz algebra (also termed $N=1$ NS super-Virasoro algebra)
contains the $\mathfrak{osp}(1|2)$ subalgebra. We are interested in studying
the limit $c\rightarrow\infty$ of $N=1$ NS super-Virasoro algebra, in this
limit, see appendix A, we need to keep only the generators of the
$\mathfrak{osp}(1|2)$ in order to have finite inner products of states
(similar to the Virasoro and $W_{3}$ algebras in the large $c$ limit, which
reduce to $sl(2)$ and $sl(3)$ algebras respectively). In this sense, in the
large $c$ limit, the $N=1$ NS super-Virasoro algebra reduces to the
$\mathfrak{osp}(1|2)$ algebra777For a review of $\mathfrak{osp}(1|2)$ CFT see
[63, 67, 68].. In the $N=1$ super CFT we have conformal superfields $\Phi_{i}$
(1.8) and in the sequel we restrict ourselves to studying the
$(\varphi_{i},\psi_{i})$ components and omit the antiholomorphic contribution,
thus for simplicity, we will write superfields as follows

$\Phi_{i}(z,\theta)=\varphi_{i}(z)+\theta\psi_{i}(z).$ (2.1)

The $\mathfrak{osp}(1|2)$ algebra has three even generators $\mathbf{L}_{\pm
1,0}$ and two odd generators $\mathbf{G}_{\pm\frac{1}{2}}$, we have the
following commutation and anticommutation relations

$[\mathbf{L}_{m},\mathbf{L}_{n}]=(m-n)\mathbf{L}_{m+n},\qquad[\mathbf{L}_{n},\mathbf{G}_{\pm\frac{1}{2}}]=\left(\frac{n}{2}\mp\frac{1}{2}\right)\mathbf{G}_{n\pm\frac{1}{2}},\qquad\\{\mathbf{G}_{r},\mathbf{G}_{s}\\}=2\mathbf{L}_{r+s},$
(2.2)

where $[,],\\{,\\}$ are the commutator and anticommutator respectively.
$m,n=0,\pm 1$. $r,s=\pm\frac{1}{2}$. It is assumed the conjugation rules
$\mathbf{L}_{-m}=(\mathbf{L}_{m})^{\dagger},\mathbf{G}_{-s}=(\mathbf{G}_{s})^{\dagger}$.
We see that generators $\mathbf{L}_{m}$ form $\mathfrak{sl}(2)$ subalgebra.

The states in the representation $\mathcal{V}_{h_{i}}$ are obtained by
applying the generators $(\mathbf{G}_{-\frac{1}{2}},\mathbf{L}_{-1})$ to the
hwv $|h_{i}\rangle$ associated with the field $\Phi_{i}$, thus any state can
be written as follows

$|M,h_{i}\rangle=\mathbf{L}_{-1}^{m}\mathbf{G}_{-\frac{1}{2}}^{k}|h_{i}\rangle,\qquad
M=\left(m,k\right),\quad m\in\mathbb{N}_{0},\quad k=0,1,$ (2.3)

where $\mathbb{N}_{0}$ is the set of nonnegative integers. Generators
$(\mathbf{L}_{1},\mathbf{G}_{+\frac{1}{2}})$ annihilate the hwv and

$\mathbf{L}_{0}|h_{i}\rangle=h_{i}|h_{i}\rangle.$ (2.4)

The supermodule $\mathcal{V}_{h_{i}}$ can be written as a direct sum of two
subspaces

$\mathcal{V}_{h_{i}}=V_{h_{i}}\oplus V_{h_{i}+\frac{1}{2}},$ (2.5)

where $(V_{h_{i}},V_{h_{i}+\frac{1}{2}})$ are two $\mathfrak{sl}(2)$ Verma
modules associated with the hwv $|h_{i}\rangle,|h_{i}+\frac{1}{2}\rangle$
respectively, where
$|h_{i}+\frac{1}{2}\rangle=\mathbf{G}_{-\frac{1}{2}}|h_{i}\rangle$.

###  2.1 Spherical and toroidal conformal blocks

In section 3.1 we will show how the conformal blocks (CB) of the components of
the two-point and three-point correlation functions are obtained in CS theory.
These two examples clarify how the idea proposed in [20, 26, 48, 49]
generalizes to the $\mathfrak{osp}(1|2)$ algebra and provide evidence that any
superconformal block could be obtained in CS theory with $\mathfrak{osp}(1|2)$
gauge symmetry. The CBs under consideration are presented.

The two-point correlation function of superfields $(\Phi_{1},\Phi_{2})$
($h_{1}=h_{2}=h$) is given by

$\langle\Phi_{1}(z_{1},\theta_{1})\Phi_{2}(z_{2},\theta_{2})\rangle=\frac{C_{12}}{(z_{1}-z_{2}-\theta_{1}\theta_{2})^{2h}}=C_{12}\left(\frac{1}{(z_{1}-z_{2})^{2h}}+\frac{2h\theta_{1}\theta_{2}}{(z_{1}-z_{2})^{2h+1}}\right).\\\
$ (2.6)

Using (2.1) we have

$\langle\Phi_{1}(z_{1},\theta_{1})\Phi_{2}(z_{2},\theta_{2})\rangle=\langle\varphi_{1}(z_{1})\varphi_{2}(z_{2})\rangle+\theta_{1}\theta_{2}\langle\psi_{1}(z_{1})\psi_{2}(z_{2})\rangle,\quad$
(2.7)

where

$\begin{split}&\langle\varphi_{1}(z_{1})\varphi_{2}(z_{2})\rangle\propto\frac{1}{(z_{1}-z_{2})^{2h}},\\\
&\langle\psi_{1}(z_{1})\psi_{2}(z_{2})\rangle\propto\frac{1}{(z_{1}-z_{2})^{2h+1}}.\end{split}$
(2.8)

The three-point correlation function of superfields
$(\Phi_{1},\Phi_{2},\Phi_{3})$ is given by

$\langle\Phi_{1}(z_{1},\theta_{1})\Phi_{2}(z_{2},\theta_{2})\Phi_{3}(z_{3},\theta_{3})\rangle=\frac{1}{Z_{12}^{\gamma_{123}}Z_{13}^{\gamma_{132}}Z_{23}^{\gamma_{231}}}\left(C_{123}+\eta\widetilde{C}_{123}\right),$
(2.9)

where

$\begin{split}Z_{ij}=(z_{i}-z_{j}-\theta_{i}\theta_{j}),\quad\gamma_{ijk}=h_{j}+h_{i}-h_{k},\quad\\\
\eta=(Z_{12}Z_{13}Z_{23})^{-1/2}(Z_{23}\theta_{1}+Z_{31}\theta_{2}+Z_{12}\theta_{3}+\theta_{1}\theta_{2}\theta_{3}).\end{split}$
(2.10)

Similarly, using (2.1) and expanding the rhs of (2.9), we have

$\Big{\langle}\left(\varphi_{1}(z_{1})+\theta_{1}\psi_{1}(z_{1})\right)\left(\varphi_{2}(z_{2})+\theta_{2}\psi_{2}(z_{2})\right)\left(\varphi_{3}(z_{3})+\theta_{3}\psi_{3}(z_{3})\right)\Big{\rangle}=\Gamma_{0}+\Gamma_{1},$
(2.11)

where

$\begin{split}&\Gamma_{0}=\frac{C_{123}}{(z_{1}-z_{2})^{\gamma_{123}}(z_{1}-z_{3})^{\gamma_{132}}(z_{2}-z_{3})^{\gamma_{231}}}\left(1+\gamma_{231}\frac{\theta_{2}\theta_{3}}{(z_{2}-z_{3})}+\gamma_{132}\frac{\theta_{1}\theta_{3}}{(z_{1}-z_{3})}+\gamma_{123}\frac{\theta_{1}\theta_{2}}{(z_{1}-z_{2})}\right),\\\
&\\\
&\Gamma_{1}=\widetilde{C}_{123}\frac{(z_{2}-z_{3})\theta_{1}+(z_{3}-z_{1})\theta_{2}+(z_{1}-z_{2})\theta_{3}+(-1/2+h_{1}+h_{2}+h_{3})\theta_{1}\theta_{2}\theta_{3}}{(z_{1}-z_{2})^{\gamma_{123}+1/2}(z_{1}-z_{3})^{\gamma_{132}+1/2}(z_{2}-z_{3})^{\gamma_{231}+1/2}}.\end{split}$
(2.12)

The one-point correlation function on the torus is given by equation (1.12).
Each term of the rhs of (1.12) has been found in [69] and look as follows

$\text{\Large{Tr}}_{h}\left[q^{\mathbf{L}_{0}}\varphi_{1}(z_{1})\right]=C_{hh_{1}h}B_{0}(h,h_{1},q),$
(2.13)

$\text{\Large{Tr}}_{h}\left[q^{\mathbf{L}_{0}}\psi_{1}(z_{1})\right]=C_{hh_{1}+\frac{1}{2}h}B_{1}(h,h_{1},q).$
(2.14)

For our purposes it is convenient to keep structure constants $C_{hh_{1}h}$
and $C_{hh_{1}+\frac{1}{2}h}$ in the definition of CBs. $B_{0}(h,h_{1},q)$ and
$B_{1}(h,h_{1},q)$ are the lower and upper torus superblocks, given by

$\begin{split}B_{0}(h,h_{1},q)=&\frac{q^{h}}{(1-q)^{h_{1}}}{}_{2}F_{1}(2h-h_{1},1-h_{1},2h|q)-\\\
&-\frac{2h-h_{1}}{2h}\frac{q^{h+1/2}}{(1-q)^{h_{1}}}{}_{2}F_{1}(2h-h_{1}+1,1-h_{1},2h+1|q),\end{split}$
(2.15)

$\begin{split}B_{1}(h,h_{1},q)=&\frac{q^{h}}{(1-q)^{h_{1}+\frac{1}{2}}}{}_{2}F_{1}(2h-h_{1}-\frac{1}{2},-h_{1}+\frac{1}{2},2h|q)-\\\
&-\frac{2h+h_{1}-\frac{1}{2}}{2h}\frac{q^{h+1/2}}{(1-q)^{h_{1}+\frac{1}{2}}}{}_{2}F_{1}(2h-h_{1}+\frac{1}{2},-h_{1}+\frac{1}{2},2h+1|q),\end{split}$
(2.16)

where ${}_{2}F_{1}(a,b,c|z)$ is the hypergeometric function. We notice that
(2.15, 2.16) have poles when $h$ takes negative integer or half-integer
values, at these values we need to truncate the series expansion of BHS in
(2.15), (2.16) according to the standard null-vector condition for degenerate
fields.

###  2.2 Finite-dimensional representations

In this section we recall some facts of $\mathfrak{osp}(1|2)$ finite-
dimensional representation theory which are relevant for our discussion in the
subsequent sections.

A finite-dimensional representation of $\mathfrak{osp}(1|2)$ is labelled by a
nonnegative integer or half-integer superspin $j$ and parity $\lambda$
($\lambda=0$ if $j$ is integer, or $\lambda=1$ if $j$ is half-integer), we
call these representations superspin-$j$ representations of
$\mathfrak{osp}(1|2)$, each representation decomposes into two normal
subspaces as follows

$\mathcal{V}_{j}=V_{j}\oplus V_{j-\frac{1}{2}},$ (2.17)

where $V_{j}$ stands for the state space of a spin-$j$ representation of
$\mathfrak{sl}(2)$ which is $2j+1$ dimensional, thus the direct sum (2.17) is
$4j+1$ dimensional. Any state of the representation (2.17) is characterized by
two variables $(l(j),m)$, one can denote any state of (2.17) as follows

$|l(j),m\rangle,$ (2.18)

where $l(j)$ denotes the subspace ($V_{j}$ or $V_{j-\frac{1}{2}}$) to which
the state (2.18) belongs, thus $l(j)$ takes two values

$l(j)=\begin{cases}j,\\\ j-1/2,\end{cases}$ (2.19)

$m$ is the $\mathfrak{sl}(2)$ spin projection of $l(j)$, hence

$\begin{split}&m\in[-j,-j+1,...,j-1,j]\qquad\text{if $l(j)=j$ or,}\\\
&m\in[-j+\frac{1}{2},-j+\frac{3}{2},...,j-\frac{3}{2},j-\frac{1}{2}]\qquad\text{if
$l(j)=j-\frac{1}{2}$}.\end{split}$ (2.20)

In the sequel we will use the notation

$l_{i}=l_{i}(j_{i}).$ (2.21)

The generators of the $\mathfrak{osp}(1|2)$ algebra in a superspin-$j$
representation will be denoted as follows

$L_{0},L_{\pm 1},G_{\pm\frac{1}{2}},$ (2.22)

with the commutation relations (2.2)888To distinguish conformal algebra and CS
gauge algebra generators we use bold and regular fonts respectively.. The
generators (2.22) act on the states999In the sequel, the superscript $i$ over
generators labels superspins-$j_{i}$ representation. (2.18) as follows [67,
70]

$\begin{split}&L_{0}|l(j),m\rangle=m|l(j),m\rangle,\\\ &L_{\mp
1}|l(j),m\rangle=\mp\sqrt{[j\mp m][j\pm m+1]}|l(j),m\pm 1\rangle,\end{split}$
(2.23) $G_{\mp\frac{1}{2}}|\widetilde{l}_{1}(j),m\rangle=\begin{cases}\pm
i\sqrt{j\mp m}|\widetilde{l}_{2}(j),m\pm\frac{1}{2}\rangle,&\text{if
$j-m\in\mathbb{Z}$ },\\\ i\sqrt{j\pm
m+\frac{1}{2}}|\widetilde{l}_{2}(j),m\pm\frac{1}{2}\rangle,&\text{If
$j-m\in\mathbb{Z}+\frac{1}{2}$},\end{cases}$ (2.24)

where $[x]$ represents the integer part of the number $x$ ($2x\in\mathbb{Z}$
101010$\mathbb{Z}$ is the set of integers.), and if $\widetilde{l}_{1}(j)=j$
then $\widetilde{l}_{2}(j)=j-\frac{1}{2}$ or vice-versa.

The second fact is related to the structure of the Clebsch-Gordan coefficients
(CGC) of $\mathfrak{osp}(1|2)$. The tensor product of two representations
$(\mathcal{R}_{j_{1}},\mathcal{R}_{j_{2}})$ decomposes in the direct sums

$\mathcal{R}_{j_{1}}\otimes\mathcal{R}_{j_{2}}=\bigoplus_{j_{3}=|j_{1}-j_{2}|\atop
2(j_{3}-j_{1}-j_{2})\in\mathbb{Z}}^{j_{1}+j_{2}}\mathcal{R}_{j_{3}},$ (2.25)

any state in the tensor product can be expressed as follows

$\big{|}l_{3},m_{3}\big{\rangle}=\sum_{l_{1},l_{2}}\sum_{m_{1},m_{2}}C(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3})\big{|}l_{1},m_{1}\big{\rangle}\otimes\big{|}l_{2},m_{2}\big{\rangle},$
(2.26)

where

$C(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3})$ (2.27)

are the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients, from (2.25) it
follows that $j_{3}$ in $l_{3}=l_{3}(j_{3})$ can take values

$j_{3}\in\Big{[}|j_{1}-j_{2}|,|j_{1}-j_{2}|+\frac{1}{2},...,j_{1}+j_{2}-\frac{1}{2},j_{1}+j_{2}\Big{]}.$
(2.28)

In [71, 70, 72] has been shown that the CGC (2.27) can be factorized as
follows

$C(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3})=\begin{bmatrix}j_{1}&j_{2}&j_{3}\\\
l_{1}&l_{2}&l_{3}\end{bmatrix}C_{\mathfrak{sl}(2)}\left(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3}\right),$
(2.29)

where $C_{\mathfrak{sl}(2)}\left(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3}\right)$
are the CGC of $\mathfrak{sl}(2)$ (3.27) and

$\begin{bmatrix}j_{1}&j_{2}&j_{3}\\\ l_{1}&l_{2}&l_{3}\end{bmatrix}$ (2.30)

is termed the symmetrical scalar factor. The scalar factor (2.30) does not
depend on the projections $(m_{1},m_{2},m_{3})$, it has been computed in [72].

The singlet state (the state with the total superspin and projection equal to
zero) in the tensor product of superspin representations will be denoted as
follows

$|0,0\rangle=|s\rangle.$ (2.31)

In the case of the tensor product of two representations (2.25) with the same
superspin $j_{1}=j_{2}=j$ the CGC of the singlet are given by [72]

$C(l_{1},m_{1};l_{2},m_{2}|0,0)=(-1)^{2(\lambda+1)(j-l_{1})+l_{1}-m_{1}}\delta_{l_{1}l_{2}}\delta_{m_{1},-m_{2}},$
(2.32)

where $\lambda$ is the parity of $j$.

Similarly to $\mathfrak{sl}(2)$, in $\mathfrak{osp}(1|2)$ it is defined the
super $3-j$ symbol given by the following formula [72]

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
l_{1}m_{1}&l_{2}m_{2}&l_{3}m_{3}\end{pmatrix}=\begin{bmatrix}j_{1}&j_{2}&j_{3}\\\
l_{1}&l_{2}&l_{3}\end{bmatrix}\begin{pmatrix}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix},$ (2.33)

where on the rhs the second factor is the Wigner $3-j$ symbol of
$\mathfrak{sl}(2)$ algebra. Using the super $3-j$ symbol we can express the
singlet state of the tensor product of three representations as follows

$|s\rangle=\sum_{l_{i},m_{i}}(-1)^{l_{1}-l_{2}+l_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
l_{1}m_{1}&l_{2}m_{2}&l_{3}m_{3}\end{pmatrix}|l_{1},m_{1}\rangle\otimes|l_{2},m_{2}\rangle\otimes|l_{3},m_{3}\rangle,$
(2.34)

where the sums are over all possible values of $l_{i},m_{i}$ according to
(2.19, 2.20, 2.21). Notice that for fixed values of $(l_{1},l_{2},l_{3})$ in
(2.34) the sums over $(m_{1},m_{2},m_{3})$ yield a singlet state in the tensor
product of three $\mathfrak{sl}(2)$ finite-dimensional representations.

##  3 Wilson line construction

In this section we will compute using the Wilson lines formulation the CBs of
the components of the two-point, three-point correlation functions on the
sphere and one-point correlation function on the torus.

###  3.1 Two- and three-point spherical conformal blocks

Here we show that the CBs of each component of the two-point correlation
function (2.8) and three-point correlation function (2.11) can be obtained by
an ansatz similar to (1.3). We introduce the following notation

$A_{i}^{k}(z)=\begin{cases}\varphi_{i}(z)\quad\text{if $k=0$,}\\\
\psi_{i}(z)\quad\text{if $k=1$.}\end{cases}$ (3.1)

As discussed in [20, 26, 48, 49, 58] the Wilson line description of CBs
implies relations between conformal dimensions $h_{i}$ and spins $j_{i}$ (in
our case superspins $j_{i}$) as well as relations between primary fields
$\varphi_{i}$ and states of $\mathcal{R}_{j_{i}}$, in the case of
$\mathfrak{osp}(1|2)$ we find the following relations111111For the other two
components of (1.8) the relation is
$\bar{\psi}_{i}(z)\rightarrow|j_{i},j_{i}\rangle$ and
$\widetilde{\varphi}_{i}(z)\rightarrow|j_{i}-\frac{1}{2},j_{i}-\frac{1}{2}\rangle$.
(the arrows stand for corresponds)

$\begin{split}&h_{i}=-j_{i},\\\
&\varphi_{i}(z)\rightarrow|j_{i},j_{i}\rangle,\\\
&\psi_{i}(z)\rightarrow|j_{i}-\frac{1}{2},j_{i}-\frac{1}{2}\rangle.\end{split}$
(3.2)

On the sphere, by taking into account the relations (3.2) and using the
notations (3.1) we will have that the components of the correlation functions
(2.7, 2.9) can be expressed by the following formula

$\bigg{\langle}\prod_{i=1}^{n}A_{i}^{k_{i}}(z_{i})\bigg{\rangle}=c\bigg{\langle}s\bigg{|}\prod_{i=1}^{n}e^{\int_{z_{i}}^{z_{b}}L_{1}^{(i)}dz}\bigg{|}j_{i}-\frac{k_{i}}{2},j_{i}-\frac{k_{i}}{2}\bigg{\rangle},$
(3.3)

where, $n=2$ (resp. $3$) in the case of two-point (resp. three-point)
correlation function, $c$ is a constant irrelevant to our discussions,
$\langle s|$ is a singlet state in the tensor product of superspin
representations $\mathcal{R}_{j_{i}}$ of $\mathfrak{osp}(1|2)$, $L_{1}^{(i)}$
is the lowering generator acting on the states of $\mathcal{R}_{j_{i}}$,
$|j_{i}-k_{i}/2,j_{i}-k_{i}/2\rangle$ (for $k_{i}=0,1$) is a state in
$\mathcal{R}_{j_{i}}$ according to the notation (2.18) and $z_{b}$ is an
arbitrary point (the CBs do not depend on it). To see that indeed (3.3) does
not depend on $z_{b}$ let us choose other point $z_{b^{\prime}}$ in (3.3) and
see that it gives the same result, thus we have

$\begin{split}&\bigg{\langle}s\bigg{|}\prod_{i=1}^{n}e^{\int_{z_{i}}^{z_{b^{\prime}}}L_{1}^{(i)}dz}\bigg{|}j_{i}-\frac{k_{i}}{2},j_{i}-\frac{k_{i}}{2}\bigg{\rangle}=\bigg{\langle}s\bigg{|}\left(\prod_{i=1}^{n}e^{\int_{z_{b}}^{z_{b^{\prime}}}L_{1}^{(i)}dz}\right)\prod_{i=1}^{n}e^{\int_{z_{i}}^{z_{b}}L_{1}^{(i)}dz}\bigg{|}j_{i}-\frac{k_{i}}{2},j_{i}-\frac{k_{i}}{2}\bigg{\rangle}=\\\
&=\bigg{\langle}s\bigg{|}\prod_{i=1}^{n}e^{\int_{z_{i}}^{z_{b}}L_{1}^{(i)}dz}\bigg{|}j_{i}-\frac{k_{i}}{2},j_{i}-\frac{k_{i}}{2}\bigg{\rangle},\end{split}$
(3.4)

where we used the property of the singlet $\langle s|$ that it is annihilated
by the action of the element $\widehat{L}_{1}=\sum_{i=1}^{n}L_{1}^{(i)}$
because the singlet has total superspin and projection equal to zero in the
tensor product, and hence the singlet $\langle s|$ is invariant under the
action of the group element $\langle
s|e^{(z_{b^{\prime}}-z_{b})\widehat{L}_{1}}=\langle s|$.

The two-point correlation function. By using the relations (3.2) and the
formula (3.3) we find that the CBs of the two components (2.8) can be computed
by

$\langle\varphi_{1}(z_{1})\varphi_{1}(z_{2})\rangle\propto\langle
s|e^{(z_{1}-z_{b})L_{1}^{(1)}}|j_{1},j_{1}\rangle\otimes
e^{(z_{2}-z_{b})L_{1}^{(1)}}|j_{1},j_{1}\rangle,$ (3.5)

$\langle\psi_{1}(z_{1})\psi_{1}(z_{2})\rangle\propto\langle
s|e^{(z_{b}-z_{1})L_{1}^{(1)}}|j_{1}-\frac{1}{2},j_{1}-\frac{1}{2}\rangle\otimes
e^{(z_{b}-z_{2})L_{1}^{(1)}}|j_{1}-\frac{1}{2},j_{1}-\frac{1}{2}\rangle,$
(3.6)

where $\langle s|$ is the singlet of the tensor product
$\mathcal{R}_{j_{1}}\otimes\mathcal{R}_{j_{1}}$. Choosing $z_{b}=z_{2}$ and
considering the coefficients of the singlet (2.32), the rhs of (3.5) reduces
(up to irrelevant constant) to

$\langle-j_{1},j_{1}|e^{-z_{12}L_{1}^{(1)}}|j_{1},j_{1}\rangle,$ (3.7)

where $z_{ij}=z_{i}-z_{j}$, by applying the generator $L_{1}^{(1)}$ to the
states in $\mathcal{R}_{j_{1}}$ according to (2.23) we obtain

$\langle\varphi_{1}(z_{1})\varphi_{1}(z_{2})\rangle\propto(z_{1}-z_{2})^{2j_{1}},$
(3.8)

which confirms (3.5). Similarly for (3.6) we obtain

$\langle\psi_{1}(z_{1})\psi_{1}(z_{2})\rangle\propto(z_{1}-z_{2})^{2j_{1}-1}.$
(3.9)

The three-point correlation function. According to (3.3) and the singlet
(2.34), we have in this case the following equation

$\begin{split}&\langle
A_{1}^{k_{1}}(z_{1})A_{2}^{k_{2}}(z_{2})A_{3}^{k_{3}}(z_{3})\rangle=C_{1}(j_{i},k_{i})\sum_{l_{i},m_{i}}\Bigg{[}(-1)^{l_{1}-l_{2}+l_{3}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
l_{1}m_{1}&l_{2}m_{2}&l_{3}m_{3}\end{pmatrix}\times\\\
&\times\Big{\langle}m_{1},l_{1}\big{|}e^{z_{b1}L_{1}^{(1)}}\big{|}j_{1}-\frac{k_{1}}{2},j_{1}-\frac{k_{1}}{2}\Big{\rangle}\times\Big{\langle}m_{2},l_{2}\big{|}e^{z_{b2}L_{1}^{(2)}}\big{|}j_{2}-\frac{k_{2}}{2},j_{2}-\frac{k_{2}}{2}\Big{\rangle}\times\\\
&\times\Big{\langle}m_{3},l_{3}\big{|}e^{z_{b3}L_{1}^{(3)}}\big{|}j_{3}-\frac{k_{3}}{2},j_{3}-\frac{k_{3}}{2}\Big{\rangle}\Bigg{]},\end{split}$
(3.10)

where in this sum $l_{i}$ takes values of $j_{i}$ and $j_{i}-1/2$ , and
$m_{i}$ takes values according to (2.20). By taking into account the
expression (2.33) of super $3-j$ symbol and since the action of the generator
$L_{1}^{(i)}$ on $|l_{i},m_{i}\rangle$ does not change the value of $l_{i}$,
the rhs of the above equation reduces to

$\begin{split}&C_{1}(j_{i},k_{i})(-1)^{j_{1}-j_{2}+j_{3}+\frac{k_{1}-k_{2}+k_{3}}{2}}\begin{bmatrix}j_{1}&j_{2}&j_{3}\\\
j_{1}-\frac{k_{1}}{2}&j_{2}-\frac{k_{2}}{2}&j_{3}-\frac{k_{3}}{2}\end{bmatrix}\times\\\
&\sum_{m_{i}}\Bigg{[}\begin{pmatrix}j_{1}-\frac{k_{1}}{2}&j_{2}-\frac{k_{2}}{2}&j_{3}-\frac{k_{3}}{2}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}\Big{\langle}m_{1},j_{1}-\frac{k_{1}}{2}\big{|}e^{z_{b1}L_{1}^{(1)}}\big{|}j_{1}-\frac{k_{1}}{2},j_{1}-\frac{k_{1}}{2}\Big{\rangle}\times\\\
&\times\Big{\langle}m_{2},j_{2}-\frac{k_{2}}{2}\big{|}e^{z_{b2}L_{1}^{(2)}}\big{|}j_{2}-\frac{k_{2}}{2},j_{2}-\frac{k_{2}}{2}\Big{\rangle}\times\Big{\langle}m_{3},j_{3}-\frac{k_{3}}{2}\big{|}e^{z_{b3}L_{1}^{(3)}}\big{|}j_{3}-\frac{k_{3}}{2},j_{3}-\frac{k_{3}}{2}\Big{\rangle}\Bigg{]},\end{split}$
(3.11)

this expression can be shown [48] to be proportional to
$z_{12}^{S_{1}+S_{2}-S_{3}}z_{13}^{S_{1}+S_{3}-S_{2}}z_{23}^{S_{2}+S_{3}-S_{1}}$,
where $S_{i}=j_{i}-k_{i}/2$, thus we obtain

$\langle
A_{1}^{k_{1}}(z_{1})A_{2}^{k_{2}}(z_{2})A_{3}^{k_{3}}(z_{3})\rangle=C(j_{i},k_{i},\lambda_{i})z_{12}^{S_{1}+S_{2}-S_{3}}z_{13}^{S_{1}+S_{3}-S_{2}}z_{23}^{S_{2}+S_{3}-S_{1}},$
(3.12)

where $C(j_{i},k_{i},\lambda_{i})$ is a constant. Notice that (3.11) does not
vanish only when $S_{1}+S_{2}+S_{3}$ is an integer that is the condition when
the structure constant $C_{123}$ (or $\widetilde{C}_{123}$) does not vanish.

###  3.2 One-point torus conformal block

In this section we will compute the lower and upper superblocks (2.15, 2.16)
for the case where the conformal dimensions $(h,h_{1})$ are nonpositive
integers or half-integers (these superblocks are associated with finite-
dimensional representations). We will denote them by
$B_{0}(h,h_{1})_{f},B_{1}(h,h_{1})_{f}$ where the subscript $f$ denotes
finite-dimensional representation. As mentioned above the superblocks (2.15,
2.16) have poles when $h$ is a nonpositive integer or nonpositive half-
integer, hence we can not obtain $B_{0}(h,h_{1})_{f}$, $B_{1}(h,h_{1})_{f}$ by
substituting $h$ with integer or half-integer nonpositive values. In order to
compute them we substitute $h$ with integer or half-integer nonpositive values
into (2.15, 2.16) and subtract the infinite part121212See [58] for details.,
the resulting expressions correspond to the superblocks associated with
finite-dimensional representations, thus these superblocks are computed as
follows

$B_{0}(h=-j,h_{1}=-j_{1},q)_{f}=B_{0}(-j,-j_{1},q)-B_{0}(-(j+\frac{1}{2}),-j_{1},q),$
(3.13)

$B_{1}(h=-j,h_{1}=-j_{1},q)_{f}=B_{1}(-j,-j_{1},q)-B_{1}(-(j+\frac{1}{2}),-j_{1},q).$
(3.14)

By computing these differences and taking into account only the first $4j+1$
terms131313Because there are only $4j+1$ states in finite-dimensional
representations. in the expansion in $q$ we obtain the following expressions

$B_{0}(-j,-j_{1},q)_{f}=\mathcal{F}(-j,-j_{1},q)-\frac{2j-j_{1}}{2j}\mathcal{F}(-(j-1/2),-j_{1},q),$
(3.15)

$B_{1}(-j,-j_{1},q)_{f}=\mathcal{F}(-j,-j_{1}+\frac{1}{2},q)-\frac{2j+j_{1}+\frac{1}{2}}{2j}\mathcal{F}(-(j-1/2),-j_{1}+\frac{1}{2},q),$
(3.16)

where $\mathcal{F}(-j,-j_{1},q)$ is the $\mathfrak{sl}(2)$ one-point torus
block given by (B.2).

Now, we want to prove that superblocks (3.15, 3.16) can be computed by the
following expression (up to an irrelevant constant $c(j,j_{1},k)$)

$\text{\Large{Tr}}_{j}\Bigg{(}W_{j}[z_{b},z_{b}+2\pi\tau]I_{j;j,j_{1}}\Bigg{)}\otimes
W_{j_{1}}[z_{b},z_{1}]|j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\rangle=c(j,j_{1},k)B_{k}(-j,-j_{1},q)_{f},$
(3.17)

where $k=0,1$. $|j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\rangle$ is a state of
the superspin representation $\mathcal{R}_{j_{1}}$ of $\mathfrak{osp}(1|2)$,
Trj denotes the trace over the supermodule $\mathcal{V}_{j}$ (2.17) of the
superspin representation $\mathcal{R}_{j}$, $z_{b}$ is an arbitrary point in
the bulk, $z_{1}$ is a point corresponding to the position of the fields
$(\varphi_{1},\psi_{1})$ at the boundary141414The boundary corresponds to
$\rho=\infty$ (for details, see [44, 46, 47]). (the one-point blocks do not
depend on $z_{1}$). $W_{a}[z_{1},z_{2}]$ are the Wilson line operators

$W_{a}[z_{1},z_{2}]=\exp\left((z_{1}-z_{2})(L_{1}+\frac{1}{4}L_{-1})\right),\,$
(3.18)

here $L_{1}$ and $L_{-1}$ are the generators of the $\mathfrak{osp}(1|2)$
gauge algebra which act on the states of the respective representation
$\mathcal{R}_{a}$. $I_{j;j,j_{1}}$ in (3.17) is the intertwining operator. In
general one can have the intertwining operator $I_{j_{3};j_{1},j_{2}}$
associated with representations
$(\mathcal{R}_{j_{1}},\mathcal{R}_{j_{2}},\mathcal{R}_{j_{3}})$, it acts as
follows

$I_{j_{3};j_{1},j_{2}}:\qquad\mathcal{R}_{j_{1}}\otimes\mathcal{R}_{j_{2}}\rightarrow\mathcal{R}_{j_{3}},$
(3.19)

and satisfies the following defining condition

$I_{j_{3};j_{1},j_{2}}U_{j_{1}}U_{j_{2}}=U_{j_{3}}I_{j_{3};j_{1},j_{2}},$
(3.20)

where $U_{j_{i}}$ (for $i=1,2,3$) are the elements of the $OSP(1|2)$ group in
representations $\mathcal{R}_{j_{i}}$. By taking into account these defining
properties of the intertwining operator we have the following matrix elements
in terms of the CGC

$\bigg{\langle}m_{3},l_{3}\bigg{|}I_{j_{3};j_{1},j_{2}}\bigg{|}l_{1},m_{1}\bigg{\rangle}\otimes\bigg{|}l_{2},m_{2}\bigg{\rangle}=C(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3}),$
(3.21)

where the CGC are given by (2.29). The proof of the relation (3.17) is based
on the structure (2.29) of the CGC and the Wilson lines formulation of the
$\mathfrak{sl}(2)$ torus block (see [58]).

We decompose the trace $Tr_{j}$ in (3.17) into two subtraces according to
(2.17), thus the lhs of (3.17) becomes

$\begin{split}&\sum_{m\in
M_{1}}\bigg{\langle}m,j\bigg{|}W_{j}[z_{b},z_{b}+2\pi\tau]I_{j;j,j_{1}}\bigg{|}j,m\bigg{\rangle}\otimes
W_{j_{1}}[z_{b},z_{1}]\big{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle}+\\\
&+\sum_{m\in
M_{2}}\bigg{\langle}m,j-\frac{1}{2}\bigg{|}W_{j}[z_{b},z_{b}+2\pi\tau]I_{j;j,j_{1}}\bigg{|}j-\frac{1}{2},m\bigg{\rangle}\otimes
W_{j_{1}}[z_{b},z_{1}]\big{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle},\end{split}$
(3.22)

where

$M_{1}=\Big{[}-j,-j+1,...,j-1,j\Big{]},$ (3.23)

$M_{2}=\Big{[}-j+1/2,-j+3/2,...,j-3/2,j-1/2\Big{]}.$ (3.24)

By taking into account (2.29, 3.21) and that the action of the operator
$L_{1}$ of the Wilson line operators $W_{j},W_{j_{1}}$ on the states
$|l_{i},m_{i}\rangle$ do not change the values of $l_{i}$ (see 2.23), we can
rewrite (3.22) as follows

$\begin{split}&\mathbf{F}_{1}\sum_{m\in
M_{1}}\bigg{\langle}m,j\bigg{|}W_{j}[z_{b},z_{b}+2\pi\tau]I^{\mathfrak{sl}_{2}}_{j;j,j_{1}}\bigg{|}j,m\bigg{\rangle}\otimes
W_{j_{1}}[z_{b},z_{1}]\big{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle}+\\\
&+\mathbf{F}_{2}\sum_{m\in
M_{2}}\bigg{\langle}m,j-\frac{1}{2}\bigg{|}W_{j}[z_{b},z_{b}+2\pi\tau]I^{\mathfrak{sl}_{2}}_{j;j,j_{1}}\bigg{|}j-\frac{1}{2},m\bigg{\rangle}\otimes
W_{j_{1}}[z_{b},z_{1}]\big{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle},\end{split}$
(3.25)

where $(\mathbf{F}_{1},\mathbf{F}_{2})$ are two factors (2.30)

$\begin{split}&\mathbf{F}_{1}=\begin{bmatrix}j&j_{1}&j\\\
j&j_{1}-\frac{k}{2}&j\end{bmatrix}=(2j+j_{1}+1-k/2)^{\frac{1}{2}},\\\
&\mathbf{F}_{2}=\begin{bmatrix}j&j_{1}&j\\\
j-\frac{1}{2}&j_{1}-\frac{k}{2}&j-\frac{1}{2}\end{bmatrix}=(-1)^{\lambda+1}(2j-j_{1}+k/2)^{\frac{1}{2}},\end{split}$
(3.26)

and $I^{\mathfrak{sl}_{2}}_{j_{3};j_{1},j_{2}}$ in (3.25) acts as follows (for
fixed values of $l_{i}$)

$\begin{split}\langle
l_{3},m_{3}|I^{\mathfrak{sl}_{2}}_{j_{3};j_{1},j_{2}}|l_{1},m_{1}\rangle\otimes|l_{2},m_{2}\rangle&=C_{\mathfrak{sl}(2)}\left(l_{1},m_{1};l_{2},m_{2}|l_{3},m_{3}\right)=\\\
&=(-1)^{-j_{1}+j_{2}-m_{3}}\sqrt{2j_{3}+1}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&-m_{3}\end{pmatrix}.\end{split}$ (3.27)

As the spherical CBs (3.25) does not depend on $z_{b}$ and due to the
translational invariance of the one-point torus block we can choose
$z_{1}=z_{b}=0$. In order to simplify the computation of (3.25) we diagonalize
$W_{j}[z_{b},z_{b}+2\pi\tau]=e^{2\pi\tau(L_{1}+\frac{1}{4}L_{-1})}$ expressing
it in the following way

$W_{j}[z_{b},z_{b}+2\pi\tau]=U_{j}e^{2\pi i\tau
L_{0}}U_{j}^{-1}=U_{j}q^{L_{0}}U_{j}^{-1},$ (3.28)

where $U_{j}=e^{\frac{i}{2}L_{-1}}e^{-iL_{1}}e^{-i\pi L_{0}}$ is a group
element in $\mathcal{R}_{j}$. Placing (3.28) in (3.25) and due to the property
(3.20) of the intertwining operator, after this replacement, (3.25) remains
unchanged except the boundary state
$|j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\rangle$ which transforms

$|j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\rangle\rightarrow
U_{j_{1}}^{-1}|j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\rangle=\sum_{s=-j_{1}+k/2}^{j_{1}-k/2}\beta_{s}|j_{1}-\frac{k}{2},s\rangle,$
(3.29)

where $\beta_{s}$ are some coefficients depending on ($j_{1},k,s$), in fact,
we will need only the state with projection $s=0$. By taking into account
(2.23, 3.27, 3.28, 3.29) we have that (3.25) becomes

$\begin{split}&{}\mathbf{F}_{1}\sum_{m\in
M_{1}}\bigg{\langle}m,j\bigg{|}q^{L_{0}}I^{\mathfrak{sl}_{2}}_{j;j,j_{1}}\bigg{|}j,m\bigg{\rangle}\otimes
U_{j_{1}}^{-1}{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle}+\\\
+&\mathbf{F}_{2}\sum_{m\in
M_{2}}\bigg{\langle}m,j-\frac{1}{2}\bigg{|}q^{L_{0}}I^{\mathfrak{sl}_{2}}_{j;j,j_{1}}\bigg{|}j-\frac{1}{2},m\bigg{\rangle}\otimes
U_{j_{1}}^{-1}\big{|}j_{1}-\frac{k}{2},-j_{1}+\frac{k}{2}\big{\rangle}=\\\
&=\mathbf{F}_{1}\sum_{m\in
M_{1},s}q^{m}(-1)^{-m-j+j_{1}-k/2}\sqrt{2j+1}\begin{pmatrix}j&j_{1}-\frac{k}{2}&j\\\
m&s&-m\end{pmatrix}\beta_{s}+\\\ &+\mathbf{F}_{2}\sum_{m\in
M_{2},s}q^{m}(-1)^{-m-j+\frac{1}{2}+j_{1}-k/2}\sqrt{2j}\begin{pmatrix}j-\frac{1}{2}&j_{1}-\frac{k}{2}&j-\frac{1}{2}\\\
m&s&-m\end{pmatrix}\beta_{s}.\end{split}$ (3.30)

Due to the property of Wigner $3-j$ symbol the nonzero terms correspond
to151515The zero projection $s=0$ only exists when $j_{1}-k/2$ is integer.
$s=0$, by taking into account this and collecting a common factor
$\widetilde{c}$ we have (3.30) is

$\widetilde{c}\bigg{[}\sum_{m\in
M_{1}}q^{m}(-1)^{m}\begin{pmatrix}j&j_{1}-\frac{k}{2}&j\\\
m&0&-m\end{pmatrix}+\frac{\mathbf{F}_{2}(-1)^{\frac{1}{2}}\sqrt{2j}}{\mathbf{F}_{1}\sqrt{2j+1}}\sum_{m\in
M_{2}}q^{m}(-1)^{m}\begin{pmatrix}j-\frac{1}{2}&j_{1}-\frac{k}{2}&j-\frac{1}{2}\\\
m&0&-m\end{pmatrix}\bigg{]}.$ (3.31)

Each sum in (3.31) can be shown [58] to be proportional to $\mathfrak{sl}(2)$
one-point torus block for finite-dimensional representations (B.2) denoted by
$\mathcal{F}$. By computing161616In (3.32) we set $j$ to be an integer. these
sums, (3.31) can be expressed as follows.

$c(j,j_{1},k)\left(\mathcal{F}(-j,-j_{1}+k/2,q)-\frac{2j-(-1)^{k}j_{1}+\frac{k}{2}}{2j}\mathcal{F}(-(j-1/2),-j_{1}+k/2,q)\right).$
(3.32)

which proves (3.17). We notice that when in (3.30) there is no such projection
$s=0$ then (3.30) vanishes, this is consistent with the fact that in this case
the structure constant $C_{j(j_{1}-\frac{k}{2})j}$ (2.13, 2.14) also vanishes.

##  4 Conclusions

In this work we studied the Wilson lines formulation of the
$\mathfrak{osp}(1|2)$ CBs. We considered CBs associated with finite-
dimensional representations and light $\Phi_{i}$ primary superfields. We used
the decomposition of superfields into ordinary primary fields (1.8) and we
concentrated only on the $(\varphi_{i},\psi_{i})$ sectors, thus we expressed
the correlation functions of superfields in terms of ordinary fields, (eqs.
(1.11, 1.12)). On the sphere, we showed that the CBs of the components of the
two and three-point correlation functions can be computed by the Wilson lines
formulation (3.3). On the torus, we computed the lower and upper superblocks
(3.15, 3.16) via the Wilson lines formulation (3.17). The ansatz (3.3, 3.17)
involve three main components: the singlet states belonging to the tensor
product of superspin$-j$ representations of $\mathfrak{osp}(1|2)$; Wilson line
operator built out of the elements of $\mathfrak{osp}(1|2)$ algebra; boundary
states of superspin$-j$ representations of $\mathfrak{osp}(1|2)$. The ansatz
(3.3, 3.17) is very similar to the ansatz of the ordinary Virasoro case, and
its proof essentially reduces to the $\mathfrak{sl}_{2}$ case. This reduction
is related to the facts that the state space (2.17) of a superspin
representation of $\mathfrak{osp}(1|2)$ splits into two normal subspaces of
two spin representations of $\mathfrak{sl}_{2}$ and that the CGC of
$\mathfrak{osp}(1|2)$ factorizes into the CGC of the $\mathfrak{sl}_{2}$ and
the scalar factor as in (2.29). We also found in the super-Virasoro case that
different components ($\varphi_{i},\psi_{i}$) of the superfield $\Phi_{i}$ are
related to different states of a superspin representation of
$\mathfrak{osp}(1|2)$ according to (1.15, 3.2), which make the dual
construction more general.

According to [73] there are seven classes of superalgebras that involve extra
internal degrees of freedom and realize the AdS3 gravity in the CS
formulation. In this work we have covered one of those cases, the
$\mathfrak{osp}(1|2)$. It might be interesting to extend the explicit
construction of CBs in the Wilson lines context to the remaining classes.
Another natural generalization consists in formulating the dual construction
in the full $N=1$ NS super-Virasoro CFT. In particular, this requires the
study of quantum corrections (for related consideration, see [24]). It is also
interesting to consider CBs of $\mathfrak{sl}_{3}$ algebra in higher genus
topologies. $\mathfrak{sl}_{3}$ algebra is relevant for $W_{3}$ CFT in the
large central limit. This consideration might be useful for a better
understanding of CBs which are not completely fixed by the symmetry algebra in
$W_{3}$ CFT (see, e.g. [74]).

Acknowledgments. J.R.C. is grateful to the organizers of the Workshop on
integrability held at the Leibniz University Hannover and the Israeli Physical
Society conference 2022 held at the Ben-Gurion University for giving the
opportunity to participate and report this work.

##  Appendix A $N=1$ superconformal field theory

The $N=1$ supersymmetric CFT2 is generated by the holomorphic and
antiholomorphic components of the stress-energy tensor $T$ and the spin-vector
current $J$. The operator product expansion has the following form:

$\begin{split}&T(z)T(w)=(3c/4)/(z-w)^{4}+\frac{2}{(z-w)^{2}}T(w)+\frac{1}{z-w}T^{\prime}(w)+...,\\\
&T(z)J(w)=\frac{3}{2}(z-w)^{-2}J(w)+(z-w)^{-1}J^{\prime}(w)+...,\\\
&J(z)J(w)=c(z-w)^{-3}+(z-w)^{-1}T(w)+...\quad.\end{split}$ (A.1)

Equations (A.1) imply that the Laurent components of $T$ and $J$ satisfy

$\begin{split}&[\mathbf{L}_{n},\mathbf{L}_{m}]=(n-m)\mathbf{L}_{m+n}+\frac{c}{8}(n^{3}-n)\delta_{n,-m},\\\
&\\{\mathbf{G}_{r},\mathbf{G}_{s}\\}=2\mathbf{L}_{r+s}+\frac{1}{2}c(r^{2}-\frac{1}{4})\delta_{r,-s},\\\
&[\mathbf{L}_{n},\mathbf{G}_{r}]=(\frac{n}{2}-r)\mathbf{G}_{n+r},\end{split}$
(A.2)

where $(n,m)$ take integer values and in the Neveu-Schwarz sector

$r,s\in\mathbb{Z}+1/2.$ (A.3)

The hwv $|h_{i}\rangle$ is generated by the action of the superfield (1.8) on
the vacuum $|0\rangle$

$|h_{i}\rangle=\Phi_{i}(0)|0\rangle.$ (A.4)

the action of $(\mathbf{L}_{n}$, $\mathbf{G}_{r})$ (for positive $n,r$) on the
hwv (A.4) vanishes, the other states of the supermodule are obtained by
applying the operators ($\mathbf{L}_{-n}$, $\mathbf{G}_{-r}$) on the hwv.

The inner product of a state of the supermodule can be computed using (A.2),
for example for positive integers $(n,r)$

$\begin{split}&\langle
h_{i}|\mathbf{L}_{n}\mathbf{L}_{-n}|h_{i}\rangle=(2nh_{i}+\frac{c}{8}(n^{3}-n))\langle
h_{i}|h_{i}\rangle,\\\ &\langle
h_{i}|\mathbf{G}_{r}\mathbf{G}_{-r}|h_{i}\rangle=(2h_{i}+\frac{c}{2}(r^{2}-\frac{1}{4}))\langle
h_{i}|h_{i}\rangle.\end{split}$ (A.5)

In order to have finite inner products (A.5) in the limit $c\rightarrow\infty$
we need to restrict the values of ($n,r$) to ($n=-1,0,1$; $r=-1/2,1/2$), this
restriction implies that (A.2) becomes the $\mathfrak{osp}(1|2)$ algebra
(2.2).

##  Appendix B $\mathfrak{sl}(2)$ one-point torus block

The $\mathfrak{sl}(2)$ one-point torus block, for general intermediate and
external conformal dimensions ($\widetilde{\Delta},\Delta$) respectively, can
be written in terms of the hypergeometric function as [43]

$\displaystyle\mathcal{F}(\widetilde{\Delta},\Delta,q)=\frac{\;\;q^{\widetilde{\Delta}}}{1-q}\,\;{}_{2}F_{1}(\Delta,1-\Delta,2\widetilde{\Delta}\,|\,\frac{q}{q-1})=$
(B.1)
$\displaystyle=1+\Big{[}1+\frac{(\Delta-1)\Delta}{2\widetilde{\Delta}}\Big{]}q+\Big{[}1+\frac{(\Delta-1)\Delta}{2\widetilde{\Delta}}+\frac{(\Delta-1)\Delta(\Delta^{2}-\Delta+4\widetilde{\Delta})}{4\widetilde{\Delta}(2\widetilde{\Delta}+1)}\Big{]}q^{2}+...\;.$

By using a reasoning similar to the used one in subsection 3.2, the
$\mathfrak{sl}(2)$ one-point torus block associated with nonpositive integer
external dimensions $-j_{1}$ and nonpositive integer intermediate dimensions
$-j$ can be extracted from (B.1), this CB is given by [58]

$\mathcal{F}(-j,-j_{1},q)=q^{-j}\left(f_{0}+qf_{1}+q^{2}f_{2}+...+q^{2j}f_{2j}\right),\qquad$
(B.2)

where $(j,j_{1})$ are nonnegative integers or half-integers, and coefficients
$f_{n}$ are given by

$f_{n}=\,_{3}F_{2}(-j_{1},j_{1}+1,-n;1,-2j;1)=\sum_{m=0}^{n}\,\frac{(-)^{m}n!}{(n-m)!(m!)^{2}}\,\frac{(j_{1}+m)_{2m}}{(2j)_{m}},$
(B.3)

where $(x)_{n}=x(x-1)...(x-n+1)$.

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