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

# Wilson lines construction of $\mathfrak{sl}_{3}$ toroidal conformal blocks

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

We study $\mathcal{W}_{3}$ toroidal conformal blocks for degenerate primary
fields in AdS/CFT context. In the large central charge limit $\mathcal{W}_{3}$
algebra reduces to $\mathfrak{sl}_{3}$ algebra and $\mathfrak{sl}_{3}$ blocks
are defined as contributions to $\mathcal{W}_{3}$ blocks coming from the
generators of $\mathfrak{sl}_{3}$ subalgebra. We consider the construction of
$\mathfrak{sl}_{3}$ toroidal blocks in terms of Wilson lines operators of $3d$
Chern-Simons gravity in the thermal AdS3 space-time. According to the
correspondence, degenerate primary fields are associated with Wilson lines
operators acting in the corresponding finite-dimensional $\mathfrak{sl}_{3}$
representations. We verify this dual construction for one-point toroidal block
using $\mathfrak{sl}_{3}$ tensor technique in the bulk theory and an algorithm
based on AGT correspondence in the boundary CFT.

##  1 Introduction

Conformal blocks (CB’s) determine the holomorphic contributions to the
correlation functions that appear after fixing the OPE channels [1]. CB’s are
generally fixed by the symmetry algebra and depend on the topology of the CFT
Riemann surface. In this work, we are interested in $\mathcal{W}_{3}$ one-
point conformal block on the torus in the large central charge limit.

The one-point correlation function of a primary field $\Phi_{\alpha_{1}}$ with
conformal dimension $h_{1}$ on the torus is defined as111Throughout this paper
we omit factor $(q\bar{q})^{-\frac{c}{24}}$ which can be easily restored.

$\braket{\Phi_{\alpha_{1}}(z_{1},\bar{z}_{1})}=\operatorname{Tr}_{\alpha}\Big{(}q^{\mathbf{L}_{0}}\bar{q}^{\bar{\mathbf{L}}_{0}}\Phi_{\alpha_{1}}(z_{1},\bar{z}_{1})\Big{)}=\sum_{\alpha}C_{\alpha\alpha_{1}\alpha}|\mathcal{F}(\alpha,\alpha_{1},q)|^{2}\;,$
(1.1)

where $\operatorname{Tr}_{\alpha}$ is the trace taken over a module of the
symmetry algebra associated with the primary field $\Phi_{\alpha}$ in the
intermediate OPE channel, $q$ is the elliptic parameter of the torus
$q=e^{2\pi i\tau}$ and $\mathbf{L}_{0}$ is the generator of the algebra
satisfying $\mathbf{L}_{0}\ket{h_{1}}=h_{1}\ket{h_{1}}$. Here
$\mathcal{F}(\alpha,\alpha_{1},q)$ is the one-point holomorphic toroidal
conformal block (for more details, see, e.g., [2]).

In recent years, the AdS3/CFT2 provided a new formulation of conformal blocks
in terms of geodesic networks in $\operatorname{AdS}$ space-time, see e.g.,
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], or _Wilson network
operators_ of Chern-Simons gravity, which will be the subject of the present
consideration.222For recent development on CB’s in the holography context,
see, e.g., [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35].

The latter approach is based on the higher-spin version of AdS/CFT
correspondence [36], which, in general, identifies the minimal model cosets

$\frac{\operatorname{SU}(N)_{k}\oplus\operatorname{SU}(N)_{1}}{SU(N)_{k+1}}$
(1.2)

in _’t Hooft limit_ , as the holographic dual of the higher spin theory [37].
Here we study a different limit, the so-called _semiclassical limit_ , where
we have, from the bulk side, an
$\operatorname{SL}(N)\times\operatorname{SL}(N)$ Chern-Simons theory, while
from the CFT side, a non-unitary $\mathcal{W}_{N}$ CFT model considered in the
large central charge limit, with $N$ fixed. We recall that primary fields in
the $\mathcal{W}_{N}$ model are labelled by a pair of $\operatorname{SL}(N)$
highest-weights $(\Lambda_{+},\Lambda_{-})$, which are both highest-weights of
a finite-dimensional representation of $\operatorname{SL}(N)$ [38]. One can
identify two different kinds of primaries fields:

  * •

_heavy operators_ , identified by $(0,\Lambda_{-})$, which have scale
dimensions $\Delta\sim c$, and correspond to flat
$\operatorname{SL}(N)\times\operatorname{SL}(N)$ connections in the bulk;

  * •

_light operators_ , labelled by $(\Lambda_{+},0)$, whose scale dimensions go
as $\Delta\sim\mathit{o}(1)$, and are related to perturbative matter in the
bulk.

In this paper we focus on the light operators. This allows us to consider the
large $c$ limit by restricting the set of generators, which in general can be
written as $\\{L_{n},\,W^{3}_{n},\,\dots,\,W^{N}_{n}\\}$, to those of the
types

$L_{n}\;\text{ for }\;|n|<2\;,\qquad W^{s}_{n}\;\text{ for }\;|n|<s\;,$ (1.3)

with $s=3,\dots,N$. By looking at the commutation relations (for explicit form
in $N=3$ case, see sec. 2), it can be seen that these operators can be
identified as the generators of the $\mathfrak{sl}_{N}$ algebra. In the
spherical topology (for $N=2,3$) this program has been implemented in [39].
Regarding the toroidal topology, it was shown in [40] that $\mathfrak{sl}_{2}$
one-point toroidal block $\mathcal{F}(-j,-j_{1},q)_{\mathfrak{sl}_{2}}$, where
$-j$ and $-j_{1}$ are respectively the intermediate and external conformal
dimensions, is given by

$\mathcal{F}(-j,-j_{1},q)_{\mathfrak{sl}_{2}}=\operatorname{Tr}_{j}\Big{(}W_{j}[z_{b},z_{b}+2\pi\tau]I_{j;j,j_{1}}\Big{)}\otimes
W_{j_{1}}[z_{b},z_{1}]\ket{lw}_{1}\;.$ (1.4)

In the dual description $j$ and $j_{1}$ are the spins of $\mathfrak{sl}_{2}$
Chern-Simons gauge group. The trace $\operatorname{Tr}_{j}(\cdots)$ 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, $\ket{lw}_{1}$ 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 $W_{i}[x,y]$ (for $i=j,j_{1}$;
$x=z_{b}$ and $y=z_{b}+2\pi\tau,z_{1}$) denote the Wilson line operators

$W_{i}[x,y]=\mathcal{P}\exp\bigg{(}-\int_{x}^{y}\Omega\bigg{)}=\exp\left((x-y)(L_{1}+\frac{1}{4}L_{-1})\right)\;.$
(1.5)

Here $L_{1}$ and $L_{-1}$ are the lowering and raising operators of
$\mathfrak{sl}_{2}$ algebra in the representation with spin $i$.

In this work we consider the $\mathcal{W}_{3}$ algebra; we are interested in
fully degenerate primary fields $\Phi_{\alpha}(z,\bar{z})$, characterized by a
pair of quantum numbers $(h_{\alpha},q_{\alpha})$ (_conformal dimension_ and
$\mathcal{W}_{3}$ _charge_ , respectively).333 Explicit form of
$(h_{\alpha},q_{\alpha})$ in terms of $\alpha$ will be given below. In the
semiclassical limit $c\rightarrow\infty$, or $b\rightarrow 0$ in Toda-like
parameterization [41], $c=2+24(b+1/b)^{2}$, the parameter $\alpha$
corresponding to the intermediate primary field $\Phi_{\alpha}$ in (1.1) is
given by

$\alpha\rightarrow-bj\,,\quad j=m_{1}w_{1}+m_{2}w_{2}\;,$ (1.6)

where $w_{1}$, $w_{2}$ are $\mathfrak{sl}_{3}$ fundamental weights and
$m_{1}$, $m_{2}$ are non-negative integers. For the external primary
$\Phi_{\alpha_{1}}$ characterized by $(h_{\alpha_{1}},q_{\alpha_{1}})$

$\alpha_{1}\rightarrow-bj_{1}\,,\quad j_{1}=aw_{1}\;,$ (1.7)

where $a$ is a non-negative integer. The external field $\Phi_{\alpha_{1}}$ is
restricted to the so-called semi-degenerate form, to avoid the multiplicities
problem of conformal blocks in $\mathcal{W}_{3}$ CFT, see, e.g., [41].

In what follows, we focus on the one-point blocks with external fields
satisfying (1.7), which can be equivalently written as

$\mathbf{W}_{-1}\Phi_{\alpha_{1}}(0,0)\ket{0}=\frac{3q_{1}}{2h_{1}}\mathbf{L}_{-1}\Phi_{\alpha_{1}}(0,0)\ket{0}\;.$
(1.8)

The outline of the paper is the following. In section 2, we recall the
necessary facts about $\mathcal{W}_{3}$ CFT and $\operatorname{AdS}_{3}$
Chern-Simons gravity. In sections 3 and 4, we compute
$\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$ and check it by
comparing with the large central charge limit of $\mathcal{W}_{3}$ block of
light operators, using the algorithm based on AGT relation. To this end we
propose a relation between the light $\mathcal{W}_{3}$ and $\mathfrak{sl}_{3}$
one-point blocks, which is similar to the one existing between
$\mathfrak{sl}_{2}$ and Virasoro light blocks in the large central charge
limit [42]. In section 5, we describe the dual construction for
$\mathfrak{sl}_{3}$ one-point block in terms of the Wilson lines operators. We
find that the expression obtained in section 3 can be represented by the lhs
of (1.4). Our conclusions are collected in section 6. Appendices A, B, C, D
contain some technical details related to $\mathfrak{sl}_{3}$ matrix elements,
$\mathfrak{sl}_{3}$ Weyl character formula, AGT relation, and Wilson lines
description of conformal blocks respectively.

##  2 Preliminaries: $\operatorname{AdS}_{3}$ and $\mathcal{W}_{3}$ CFT

###  2.1 $\mathcal{W}_{3}$ Conformal Field Theory

The symmetry of the $\mathcal{W}_{3}$ CFT is generated by the energy-momentum
tensor $\mathbf{T}(z)$ (a spin-$2$ current) and the additional spin-3 current
$\mathbf{W}(z)$. Their expansions as Laurent series read

$\mathbf{T}(z)=\sum_{n=-\infty}^{\infty}\frac{\mathbf{L}_{n}}{z^{n+2}},\qquad\mathbf{W}(z)=\sum_{n=-\infty}^{\infty}\frac{\mathbf{W}_{n}}{z^{n+3}}.$
(2.1)

The modes $\mathbf{L}_{n}$ and $\mathbf{W}_{m}$ generate the $\mathcal{W}_{3}$
algebra, that is, they satisfy the commutation relations

$\begin{split}\left[\mathbf{L}_{n},\mathbf{L}_{m}\right]&=(n-m)\mathbf{L}_{n+m}+\displaystyle\frac{c}{12}(n^{3}-n)\delta_{n+m,0}\;,\\\
\left[\mathbf{L}_{n},\mathbf{W}_{m}\right]&=(2n-m)\mathbf{W}_{n+m}\;,\\\
\left[\mathbf{W}_{n},\mathbf{W}_{m}\right]&=\frac{c}{3\cdot
5!}(n^{2}-1)(n^{2}-4)n\delta_{n+m,0}+\frac{16}{22+5c}(n-m)\mathbf{\Lambda}_{n+m}+\\\
&\qquad+\frac{(n-m)}{30}\left(2m^{2}+2n^{2}-mn-8\right)\mathbf{L}_{n+m}\;,\end{split}$
(2.2)

where

$\mathbf{\Lambda}_{m}=\sum_{p\,{\leqslant}\,-2}\mathbf{L}_{p}\mathbf{L}_{m-p}+\sum_{p\,{\geqslant}\,-1}\mathbf{L}_{m-p}\mathbf{L}_{p}-\frac{3(m+2)(m+3)}{10}\mathbf{L}_{m}\;.$
(2.3)

In the limit $c\rightarrow\infty$ these commutation relations reduce to the
ones of the $\mathfrak{sl}_{3}$ algebra, generated by

$\\{\mathbf{L}_{-1},\mathbf{L}_{0},\mathbf{L}_{1},\mathbf{W}_{-1},\mathbf{W}_{1},\mathbf{W}_{0},\mathbf{W}_{-2},\mathbf{W}_{2}\\}\;,$
(2.4)

that satisfy

$\begin{split}[\mathbf{L}_{n},\mathbf{L}_{m}]&=(n-m)\mathbf{L}_{n+m}\;,\\\
[\mathbf{L}_{n},\mathbf{W}_{m}]&=(2n-m)\mathbf{W}_{n+m}\;,\\\
[\mathbf{W}_{n},\mathbf{W}_{m}]&=(n-m)\Big{(}\frac{1}{15}(n+m+2)(n+m+3)-\frac{1}{6}(n+2)(m+2)\Big{)}\mathbf{L}_{n+m}\;.\end{split}$
(2.5)

#####  $\mathcal{W}_{3}$ primary fields.

The conformal dimension $h_{\alpha}$ and charge $q_{\alpha}$ of a primary
field $\Phi_{\alpha}$ in a $\mathcal{W}_{3}$ CFT are expressed as

$h_{\alpha}=\frac{1}{2}(\alpha,\,2Q-\alpha)\,,\qquad
q_{\alpha}=i\sqrt{\frac{48}{22+5c}}\prod_{i=1}^{3}(e_{i},\,\alpha-Q)\;,$ (2.6)

Here $Q=(b+\frac{1}{b})(w_{1}+w_{2})$, $e_{i}$ are the weights of the
fundamental representation

$e_{1}=w_{1}\,,\quad e_{2}=w_{2}-w_{1}\,,\quad e_{3}=-w_{2}\;,$ (2.7)

and $\alpha$ is a vector on the root space given by

$\alpha_{r_{1}r_{2}s_{1}s_{2}}=b\big{(}(1-r_{1})w_{1}+(1-r_{2})w_{2}\big{)}+\frac{1}{b}\big{(}(1-s_{1})w_{1}+(1-s_{2})w_{2}\big{)}\;,$
(2.8)

with $r_{1}$, $r_{2}$, $s_{1}$ and $s_{2}$ positive integers. In the large
central charge limit, we have that (2.8) becomes

$\alpha\rightarrow-bj,\quad j=m_{1}w_{1}+m_{2}w_{2}\;.$ (2.9)

As explained before, we are mostly interested in semi-degenerate fields (1.7)
in the large central charge limit, for which the conformal dimension and the
charge assume the values

$h_{\alpha}=-m_{1}-m_{2},\qquad
q_{\alpha}=\frac{i}{3}\sqrt{\frac{2}{5}}(m_{2}-m_{1})\;.$ (2.10)

#####  $\mathcal{W}_{3}$ module.

A $\mathcal{W}_{3}$ _highest-weight vector_ $\ket{h_{\alpha},q_{\alpha}}$
given by

$\ket{h_{\alpha},q_{\alpha}}=\lim_{z\rightarrow 0}\Phi_{\alpha}(z)\ket{0}\;,$
(2.11)

satisfies the conditions

$\displaystyle\mathbf{L}_{0}\ket{h_{\alpha},q_{\alpha}}=h_{\alpha}\ket{h_{\alpha},q_{\alpha}}\;,\qquad\mathbf{W}_{0}\ket{h_{\alpha},q_{\alpha}}=q_{\alpha}\ket{h_{\alpha},q_{\alpha}}\;,$
(2.12)
$\displaystyle\mathbf{L}_{n}\ket{h_{\alpha},q_{\alpha}}=\mathbf{W}_{n}\ket{h_{\alpha},q_{\alpha}}=0\;,\qquad
n>0\;.$ (2.13)

The $\mathcal{W}_{3}$ _module_ associated with this highest-weight vector is
spanned by the basis of descendant states

$\mathcal{L}_{-I}\ket{h_{\alpha},q_{\alpha}}=\mathbf{L}_{-i_{1}}\dots\mathbf{L}_{-i_{m}}\mathbf{W}_{-j_{1}}\dots\mathbf{W}_{-j_{n}}\ket{h_{\alpha},q_{\alpha}}\,,\quad
I=\\{i_{1},\dots,i_{m};j_{1},\dots,j_{n}\\}\;,\\\ $ (2.14)

with

$1\leq i_{1}\leq\dots\leq i_{m}\,,\quad 1\leq j_{1}\leq\dots\leq j_{n}\;.$
(2.15)

The sum

$N=\sum_{i_{a},j_{b}\in I}i_{a}+j_{b}$ (2.16)

is called level of the state. Similarly, the $\mathfrak{sl}_{3}$ _module_ is
defined when in the set $I$, the indices are restricted to

$i_{1},\dots,i_{m}=1\;,\quad j_{1},\dots,j_{n}=1,2\;,$ (2.17)

that is, when descendants are generated only by $\mathbf{L}_{-1}$,
$\mathbf{W}_{-1}$ and $\mathbf{W}_{-2}$. For the basis states of
$\mathcal{W}_{3}$ and $\mathfrak{sl}_{3}$ modules, we introduce the following
convenient notation

$\mathcal{L}_{-I}\ket{h_{\alpha},q_{\alpha}}=\ket{h_{\alpha},q_{\alpha},N}\;.$
(2.18)

###  2.2 Brief review of 3d Chern-Simons gravity theory

It is known [43] that the $2+1$-dimensional Einstein-Hilbert action $S$ with a
negative cosmological constant can be written in terms of the Chern-Simons
action

$S=S_{\footnotesize\text{CS}}[A]-S_{\footnotesize\text{CS}}[\bar{A}]\;,$
(2.19)

where

$S_{\footnotesize\text{CS}}[A]=\frac{k}{4\pi}\int_{\mathcal{M}^{3}}\operatorname{Tr}\Big{(}A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\Big{)}\;.$ (2.20)

The constant $k$ is related to the $3$-dimensional Newton constant $G_{3}$ as

$k=\frac{r}{4G_{3}}\;,$ (2.21)

with $r$ the $\operatorname{AdS}_{3}$ radius, $\operatorname{Tr}$ stands for
the invariant Killing form, $\mathcal{M}^{3}$ is the $3$-dimensional space-
time and $A$ ($\bar{A}$) is the (anti-) chiral gauge connection, that is, a
one-form valued in the gauge group, given by a composition of vielbein and
spin-connection. In the case we are interested in, that is the
$\operatorname{SL}(N)\times\operatorname{SL}(N)$ Chern-Simons theory, $A$ is
an $\operatorname{SL}(N)$ connection. The equations of motion that follow from
(2.20) are given by (the same equations are valid for the anti-chiral
connection)

$dA+A\wedge A=0\;,$ (2.22)

which are the flatness conditions on the connection. It can be shown [44]
that, by choosing the proper boundary conditions, in the local coordinates
$x^{\mu}=(\rho,z,\bar{z})$ with $\rho\geq 0$ the radial one, the solution of
(2.22) can be written as the gauge-transformed $\Omega$:

$A=U^{-1}\Omega U+U^{-1}dU\;,$ (2.23)

with

$\Omega=\Big{(}L_{1}-2\pi\frac{6T(z)}{c}L_{-1}\Big{)}dz\;,\qquad U=e^{\rho
L_{0}}\;.$ (2.24)

Here $T(z)$ is the holomorphic boundary energy-momentum tensor, while $c$ is
the central charge, defined through the Brown-Henneaux relation
$c=\frac{3r}{2G_{3}}$ [45]. $L_{1}$, $L_{-1}$ are two of the generators of the
$\mathfrak{sl}_{2}$ subalgebra of the gauge algebra $\mathfrak{sl}_{N}$. We
are interested in the space-time with periodic time conditions, that is to
say, the thermal $\operatorname{AdS}_{3}$, in which case the stress-energy
tensor is

$T(z)=-\frac{c}{48\pi}\;,$ (2.25)

and the connection $\Omega$ becomes

$\Omega=\Big{(}L_{1}+\frac{1}{4}L_{-1}\Big{)}dz\;,$ (2.26)

together with the identifications $z\sim z+2\pi$ and $z\sim z+2\pi\tau$, with
$i\tau\in\mathbb{R}_{<0}$.

Finally, we define the Wilson line operators. Given a chiral connection $A$
defined in a representation $R$ of $\mathfrak{sl}_{N}$ and a path $l$ that
connects two points $x_{1},x_{2}\in\mathcal{M}^{3}$, we define the Wilson line
operator $W_{R}[l]$ as

$W_{R}[l]=\mathcal{P}\exp\Big{[}\int_{l}A\Big{]}\;,$ (2.27)

where $\mathcal{P}$ denotes the path-ordering operator.

##  3 One-point conformal block from CFT

In this section, we define the holomorphic $\mathcal{W}_{3}$ and
$\mathfrak{sl}_{3}$ one-point conformal blocks. The result of the calculation
of the $\mathcal{W}_{3}$ block will be given in the next section. The
holomorphic $\mathcal{W}_{3}$ one-point conformal block
$\mathcal{F}(\alpha,\alpha_{1},q)$ is given by

$\mathcal{F}(\alpha,\alpha_{1},q)=\frac{1}{\bra{h_{0},q_{0}}\Phi_{\alpha_{1}}(z)\ket{h_{0},q_{0}}}\sum_{\begin{subarray}{c}M,N=0\\\
M=N\end{subarray}}\bra{h_{0},q_{0},M}q^{L_{0}}\Phi_{\alpha_{1}}(z)\ket{h_{0},q_{0},N}G_{MN}^{-1}\;,$
(3.1)

where $h_{\alpha}:=h_{0}$, $q_{\alpha}:=q_{0}$, the states
$\ket{h_{0},q_{0},N}$ belong to the $\mathcal{W}_{3}$ module (2.14), and
$G_{MN}^{-1}$ is the inverse of the Shapavalov matrix

$G_{MN}=\braket{h_{0},q_{0},M}{h_{0},q_{0},N}\;.$ (3.2)

For general $\Phi_{\alpha_{1}}$, the matrix elements

$\bra{h_{0},q_{0},M}q^{L_{0}}\Phi_{\alpha_{1}}(z)\ket{h_{0},q_{0},N}\;$ (3.3)

are not defined uniquely. For this reason, we focus on the case where the
external field $\Phi_{\alpha_{1}}(z)$ satisfies (1.8), from which it follows
that

$-\frac{h_{1}^{2}}{5}=\frac{9q_{1}^{2}}{2}\;.$ (3.4)

This constraint is satisfied when the vector $j_{1}$ corresponding to the
field $\Phi_{\alpha_{1}}$ is given by $j_{1}=aw_{1}$ or $j_{1}=aw_{2}$.
Without loss of generality, we choose $j_{1}=aw_{1}$, which implies

$h_{1}=-a\;,\qquad q_{1}=-\frac{i}{3}\sqrt{\frac{2}{5}}a\;.$ (3.5)

Similarly, the holomorphic $\mathfrak{sl}_{3}$ one-point conformal block
$\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$ is defined by (3.1),
but now the states $\ket{h_{0},q_{0},N}$ belong to the $\mathfrak{sl}_{3}$
module, obtained by restricting the set $I$ (2.14) to the conditions (2.17).
For the fields $\Phi_{\alpha},\Phi_{\alpha_{1}}$ with $h$’s and $q$’s given by
(2.10) and (3.5) respectively, we compute the contribution to the
$\mathfrak{sl}_{3}$ one-point conformal block up to the second level. The
matrix elements involved in this computation are listed in appendix A. Here we
present the result

$\begin{split}\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}&=1+\bigg{(}\frac{18m_{1}m_{2}-a^{2}(m_{1}+m_{2})-3a(m_{1}+m_{2})}{9m_{1}m_{2}}\bigg{)}q\,+\\\
&\quad+\frac{1}{162}\bigg{(}\frac{(a+6)(a+3)a(a-3)}{m_{2}-1}\,+\\\
&\quad+\frac{\big{(}(a+3)a-18m_{1}\big{)}\big{(}(a+3)a-36(m_{1}-1)\big{)}}{m_{1}(m_{1}-1)}\,+\\\
&\quad-\frac{(a+3)a\big{(}(a+3)a(m_{1}-1)+36(m_{1}+1)\big{)}}{(m_{1}+1)m_{2}}\,+\\\
&\quad+\frac{2(a+3)a(a-3m_{1})(a+3m_{1}+3)}{(m_{1}+1)m_{1}(m_{1}+m_{2}+1)}\bigg{)}q^{2}+\dots\;.\end{split}$
(3.6)

As we see from the matrix elements in appendix A, the computation becomes
quite tedious, even at the second level. For this reason, it is clear that
another method is needed to compute a general expression of the one-point
conformal block. Nevertheless, these first terms of (3.6) will help us to
compare with the computation from the Wilson lines approach.

Zero-point Conformal Block. In the case when $\Phi_{\alpha_{1}}$ is given by
the identity operator and $\Phi_{\alpha}$ is fully degenerate, the
$\mathfrak{sl}_{3}$ one-point conformal block
$\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$ reduces to the
$\mathfrak{sl}_{3}$ character $\chi^{\mathfrak{sl}_{3}}_{j}$ of the finite-
dimensional representation with highest-weight vector
$j=m_{1}w_{1}+m_{2}w_{2}$. This character can be computed using the Weyl
character formula (the details of this computation are given in appendix B),

$\chi^{\mathfrak{sl}_{3}}_{j}=\frac{q^{-m_{1}-m_{2}}(-1+q^{1+m_{1}})(-1+q^{1+m_{2}})(-1+q^{2+m_{1}+m_{2}})}{(-1+q)^{3}(1+q)}\;.$
(3.7)

##  4 One-point conformal blocks from AGT

In this section, we move to compute the $\mathcal{W}_{3}$ one-point conformal
block (3.1) in the limit $c\rightarrow\infty$ up to the second level of the
module (2.14). To this end we use the algorithm based on the AGT
correspondence (notations and some details of the algorithm are given in
appendix C), which leads to the following expression

$\displaystyle\mathcal{F}(\alpha,\alpha_{1},q)$
$\displaystyle=1+\bigg{(}\frac{18m_{1}m_{2}-a^{2}(m_{1}+m_{2})-3a(m_{1}+m_{2})}{9m_{1}m_{2}}\bigg{)}q\,+$
$\displaystyle+\bigg{(}\frac{(a+3)^{2}(a-3m_{1}-3)(a-3m_{2})}{81(m_{1}+1)m_{2}}\,+$
$\displaystyle+\frac{(a+1)\big{(}a^{2}(m_{1}+m_{2})+3a(m_{1}+m_{2})-9(a+3)m_{1}m_{2}\big{)}}{9m_{1}m_{2}}\,+$
$\displaystyle-\frac{(a+3)^{2}(a-3m_{1})}{27m_{1}}-\frac{(a+3)^{2}(a-3m_{2}-3)}{27(m_{2}+1)}+\frac{1}{2}(a^{2}-a-2)\,+$
$\displaystyle+\frac{(a+6)(a+3)(a-3m_{1}+3)(a-3m_{1})}{162m_{1}(m_{1}-1)}-\frac{(a+3)(a+3m_{2}+3)(a-3m_{2})}{27m_{2}(m_{2}+1)}\,+$
$\displaystyle+\frac{(a+6)(a+3)(a-3m_{2}+3)(a-3m_{2})}{162m_{2}(m_{2}-1)}\,+$
$\displaystyle+\frac{(a+3)(a-3m_{1})(a+3m_{1}+3)(a-3m_{1}-3m_{2}-3)}{81m_{1}(m_{1}+1)(m_{1}+m_{2}+1)}\,+$
$\displaystyle\frac{1}{18}(a+6)(a+3)+\frac{a}{3}+1\bigg{)}q^{2}+\dots\;.$
(4.1)

Now, we propose the following formula that relates the $\mathcal{W}_{3}$ block
to the $\mathfrak{sl}_{3}$ one in the following way

$\frac{\mathcal{F}(\alpha,\alpha_{1},q)}{\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}}=\frac{\chi^{\mathcal{W}_{3}}}{\chi^{\mathfrak{sl}_{3}}}\;,$
(4.2)

where $\chi^{\mathcal{W}_{3}}$ and $\chi^{\mathfrak{sl}_{3}}$ are
$\mathcal{W}_{3}$ and $\mathfrak{sl}_{3}$ (for infinite-dimensional
representations) characters respectively, given by

$\begin{split}\chi^{\mathcal{W}_{3}}&=\frac{1}{\prod_{i=1}^{\infty}(1-q^{i})^{2}}\;,\\\
\chi^{\mathfrak{sl}_{3}}&=\frac{1}{(1-q)^{3}(1+q)}\;,\end{split}$ (4.3)

where the second line is obtained by fixing the proper normalization of (3.7)
consistent with the standard normalization of CB’s and then taking the limit
($m_{1},m_{2}\rightarrow\infty$). By substituting (4.1) in (4.2) and solving
for $\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$, we recover the
expression (3.6). This confirms, up to the second level, our proposal. The
motivation of this conjectured formula comes from the fact that a similar
relation [42] holds for the Virasoro block $\mathcal{F}_{\text{Vir}}$ in the
large central charge limit and $\mathfrak{sl}_{2}$ block, namely

$\frac{\mathcal{F}_{\text{Vir}}}{\mathcal{F}_{\mathfrak{sl}_{2}}}=\frac{\chi^{\text{Vir}}}{\chi^{\mathfrak{sl}_{2}}}=\frac{1-q}{\prod_{i=1}^{\infty}(1-q^{i})}\;.$
(4.4)

The relation (4.2) allows effectively to compute higher-level contributions to
the $\mathfrak{sl}_{3}$ block. Note that the characters and blocks involved in
(4.2) refer to infinite-dimensional representations. Nevertheless, we can
still use this formula for finite-dimensional representations, truncating the
series up to an appropriate level.

##  5 Conformal Blocks through Wilson line operators

In [40], it was proved that the expectation value of the Wilson network
operator computes the $\mathfrak{sl}_{2}$ toroidal one-point conformal block.
This section aims to show that a similar result is obtained in the case of
$\mathfrak{sl}_{3}$ algebra.

In the case of the spherical topology and the $\mathfrak{sl}_{3}$ algebra, in
[39], it was given the Wilson lines description of the $\mathfrak{sl}_{3}$
four-point block. Here we will combine the ideas of [40] and [39] to proceed
with our task. Before introducing the object we start with some definitions
and notations. In the subsequent, we will work with finite-dimensional
representations $\mathcal{R}$ of $\mathfrak{sl}_{3}$ that, in general, are
characterized by a highest-weight vector $j$ given by (2.9). We will label the
finite-dimensional representations $\mathcal{R}$ of $\mathfrak{sl}_{3}$ by the
vectors $\alpha,\alpha_{1}$, having in mind that they correspond to highest-
weight vectors $j,j_{1}$ according to (2.9) and (1.7). The Wilson line
operators in thermal $\operatorname{AdS}_{3}$ are given by

$W_{\alpha}[x,y]=\mathcal{P}\exp\bigg{(}-\int_{x}^{y}\Omega\bigg{)}=\exp\left((x-y)(L_{1}+\frac{1}{4}L_{-1})\right)\;,$
(5.1)

where $L_{1},L_{-1}$ are defined in the representation $\mathcal{R}_{\alpha}$
of the $\mathfrak{sl}_{3}$ algebra.

The _Wilson network operator_ allows to compute the conformal block and
contains a number of elements (see, e.g., [40]) that are listed below for
completeness.

  1. 1.

$\Phi_{(h_{\alpha},q_{\alpha})}(z_{\alpha},\bar{z}_{\alpha})$ on the boundary
is attached to a bulk-to-boundary Wilson line operator
$W_{\alpha}[z_{b},z_{\alpha}]$ acting in the representation
$\mathcal{R}_{\alpha}$, which connects the boundary point $z_{\alpha}$ to the
bulk point $z_{b}$.

  2. 2.

Bulk-to-bulk Wilson line operator $W_{\beta}[z_{1},z_{2}]$ in the
representation $\mathcal{R}_{\beta}$, which connects two bulk points and acts
trivially for a contractible loop. In the toroidal topology, there exists a
non-trivial bulk-to-bulk Wilson loop operator
$W_{\beta}[z_{b},z_{b}+2\pi\tau]$, associated with a non-contractible cycle.

  3. 3.

A vertex in the bulk, obtained when three Wilson line operators, associated
with the representations $\mathcal{R}_{\alpha}$, $\mathcal{R}_{\beta}$ and
$\mathcal{R}_{\gamma}$, meet each other in the same point. This vertex is
described by the trivalent intertwining operator

$I_{\alpha;\beta,\gamma}\,:\quad\mathcal{R}_{\beta}\otimes\mathcal{R}_{\gamma}\;\longrightarrow\mathcal{R}_{\alpha}\;,$
(5.2)

defined by the invariance property

$I_{\alpha;\beta,\gamma}U_{\beta}U_{\gamma}=U_{\alpha}I_{\alpha;\beta,\gamma}\;,$
(5.3)

where $U_{a}$, ($a=\alpha,\beta,\gamma$), are elements of the gauge group.

With a proper choice of the gauge, we can simplify the expression of the
Wilson loop operator [40]. Due to the gauge covariance of the Wilson line
operator, we can perform a gauge transformation of the connection

$\Omega=U\widetilde{\Omega}U^{-1}\;,$ (5.4)

where, if we choose as gauge element

$U=e^{-\frac{i}{2}L_{-1}}e^{iL_{1}}e^{-i\frac{\pi}{2}L_{0}}\;,$ (5.5)

we obtain

$\widetilde{\Omega}=-iL_{0}\,dz\;.$ (5.6)

With this particular choice, known as _diagonal gauge_ , we have that the
Wilson loop reads

$W_{i}[0,2\pi\tau]=e^{2\pi i\tau L_{0}}=q^{L_{0}}\qquad\text{ with
}\,q=e^{2\pi i\tau}\;.$ (5.7)

The only element that is affected by this choice of the gauge is the external
field, which has to transform accordingly. We consider as the boundary state
the lowest-weight state of the representation that has to be transformed as

$\ket{lw}_{\alpha}\;\longrightarrow\;\ket{\widetilde{lw}}_{\alpha}\equiv
U_{\alpha}^{-1}\ket{lw}_{\alpha}\;,$ (5.8)

where $U_{\alpha}$ is the gauge group element (5.5) in the representation
$\mathcal{R}_{\alpha}$.

###  5.1 Computation of the Wilson line operators

We are interested in performing the computation of the Wilson network operator
exploiting the tensor products of representations of the $\mathfrak{sl}_{3}$
algebra. This is based on the fact that every state in a given representation
of the algebra can be represented as a symmetric traceless tensor. Let us show
how this procedure works.

The fundamental representation of $\mathfrak{sl}_{3}$ is characterized by the
highest-weight vector $j=w_{1}=(1,0)$. This representation contains three
states given by (2.7), which for convenience, we denote as

$\ket{e_{1}}=w_{1},\quad\ket{e_{2}}=w_{2}-w_{1},\quad\ket{e_{3}}=-w_{2}\;.$
(5.9)

Similarly, we denote the states of the anti-fundamental representation $(0,1)$
(the conjugated of the fundamental)

$\ket{\bar{e}^{1}}=-w_{1},\quad\ket{\bar{e}^{2}}=w_{1}-w_{2},\quad\ket{\bar{e}^{3}}=w_{2}\;.$
(5.10)

For a generic $\mathfrak{sl}_{3}$ representation with a highest-weight vector

$j=m_{1}w_{1}+m_{2}w_{2}\equiv(m_{1},m_{2})\;,$ (5.11)

the whole vector space of the representation is obtained by applying the
lowering generators $L_{1}$, $W_{1}$, $W_{2}$ to the highest-weight vector.
Each of these states can be written as the tensor product of the states of the
fundamental representation (5.9) and its conjugate (5.10) in the following way

$\begin{split}T^{k_{1}....k_{m_{2}}}_{i_{1}...i_{m_{1}}}&=\frac{1}{m_{1}!m_{2}!}\ket{e_{(i_{1}}}\otimes\ket{e_{i_{2}}}\otimes\cdots\otimes\ket{e_{i_{m_{1}})}}\,\ket{\bar{e}^{(k_{1}}}\otimes\ket{\bar{e}^{k_{2}}}\otimes\cdots\otimes\ket{\bar{e}^{k_{m_{2}})}}=\\\
&=\frac{1}{m_{1}!m_{2}!}\ket{e_{(i_{1}}...e_{i_{m_{1}})}\bar{e}^{(k_{1}}...\bar{e}^{k_{m_{2}})}}\;,\end{split}$
(5.12)

where the parentheses of lower and upper indices denote the symmetrization,
and the tensor $T^{k_{1}....k_{m_{2}}}_{i_{1}...i_{m_{1}}}$ has to be
traceless. The highest-weight vector corresponds to the case when all indices
$i$ are equal to 1 for the fundamental representation, while for the anti-
fundamental one, we require all the indices $k$ to be equal to $3$. Below, we
will need to compute the tensor product of the kind
$\mathcal{R}_{\alpha}\otimes\mathcal{R}_{\alpha_{1}}$. In particular, here we
need to extract the representation $\mathcal{R}_{\alpha}$, i.e., we need the
projection

$P:\mathcal{R}_{\alpha}\otimes\mathcal{R}_{\alpha_{1}}\rightarrow\mathcal{R}_{\alpha}\;.$
(5.13)

The role of this projector is played by the intertwining operator
$I_{\alpha;\alpha,\alpha_{1}}$ (5.2). In the way this intertwining operator is
defined, its matrix elements are given by the Clebsch-Gordan coefficients,
i.e.

$\bra{a}I_{\alpha;\alpha,\alpha_{1}}\ket{b}\otimes\ket{c}=C_{abc}\;,$ (5.14)

where $a,b\in\mathcal{R}_{\alpha}$, $c\in\mathcal{R}_{\alpha_{1}}$, and
$C_{abc}$ is the Clebsch-Gordan coefficient. However, in what follows, we are
going to define this projector explicitly, since for $\mathfrak{sl}_{3}$, the
Clebsch-Gordan coefficients are not known for generic representations.

We want to show that the following expression holds, where
$\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$ is the
$\mathfrak{sl}_{3}$ one-point conformal block introduced in (3.6)

$\begin{split}V_{\alpha|\alpha_{1}}(\tau)\mathrel{\mathop{:}}=\sum_{\ket{\alpha}\in\mathcal{R}_{\alpha}}\bra{\alpha}W_{\alpha}[z_{b},z_{b}+2\pi\tau]&I_{\alpha;\alpha,\alpha_{1}}\ket{\alpha}\otimes
W_{{\alpha}_{1}}[z_{b},z_{1}]\ket{lw}_{\alpha_{1}}=\\\
&=\widetilde{C}(\alpha,\alpha_{1})\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}\;,\end{split}$
(5.15)

where $z_{1}$ is the point on the boundary of the solid torus that is a
geometric representation of the thermal $\operatorname{AdS}_{3}$, $z_{b}$ is
an arbitrary point in the bulk of $\operatorname{AdS}_{3}$,
$\ket{lw}_{\alpha_{1}}$ is the lowest-weight state of the representation
$\mathcal{R}_{\alpha_{1}}$ with $j_{1}=(a,0)$ and
$\widetilde{C}(\alpha,\alpha_{1})$ is an overall constant irrelevant to our
discussion. By using the diagonal gauge (5.7) it can be shown that
$V_{\alpha|\alpha_{1}}(\tau)$ does not depend on $z_{b}$ and $z_{1}$, and it
can be expressed as

$V_{\alpha|\alpha_{1}}(\tau)=Tr_{\alpha}\,\bra{\alpha}q^{L_{0}}I_{\alpha;\alpha,\alpha_{1}}\ket{\alpha}\otimes
U_{\alpha_{1}}^{-1}\ket{lw}_{\alpha_{1}}\;,$ (5.16)

where $Tr_{\alpha}=\sum_{\ket{\alpha}\in\mathcal{R}_{\alpha}}$ and
$U_{\alpha_{1}}^{-1}$ is given by (5.5) in the representation
$\mathcal{R}_{\alpha_{1}}$. Now, properly inserting resolutions of identities

$\mathbf{1}=\sum_{\mathcal{R}_{\beta}}\ket{\beta}\bra{\beta}\;,$ (5.17)

where $\sum_{\mathcal{R}_{\beta}}$ stands for the sum over all the states in
the representation $\mathcal{R}_{\beta}$, we can write, in a symbolic way,

$V_{\alpha|\alpha_{1}}(\tau)=\sum_{\mathcal{R}_{\alpha}}\sum_{\mathcal{R}_{\gamma}}\bra{\alpha}q^{L_{0}}\ket{\alpha}\Big{(}\bra{\alpha}I_{\alpha;\alpha,\gamma}\ket{\alpha}\otimes\ket{\gamma}\Big{)}\bra{\gamma}U^{-1}_{\alpha_{1}}\ket{lw}_{\alpha_{1}}\;.$
(5.18)

Let us see how to represent each of the factors in (5.18).

  1. 1.

The first term can be represented in terms of the so-called Wigner D-matrix.
We will be interested in particular in the fundamental and anti-fundamental
representations, where the $D$-matrices read respectively

$D_{ij}\mathrel{\mathop{:}}=\bra{e_{i}}q^{L_{0}}\ket{e_{j}}=\begin{pmatrix}q&0&0\\\
0&1&0\\\
0&0&q^{-1}\end{pmatrix}\;,\qquad\bar{D}^{ij}\mathrel{\mathop{:}}=\bra{\bar{e}^{i}}q^{L_{0}}\ket{\bar{e}^{j}}=\begin{pmatrix}q^{-1}&0&0\\\
0&1&0\\\ 0&0&q\end{pmatrix}\;.$ (5.19)

  2. 2.

The second factor in (5.18), the one involving the intertwining operator, acts
as a projector. Its explicit action will be explained later.

  3. 3.

The third term consists in the coordinates of the transformed lowest-weight
state of the external representation. Again, focusing on the fundamental
representation, we have that the transformed lowest-weight state is444In the
fundamental representation, we use the following forms of the generators of
$\mathfrak{sl}_{2}$ subalgebra of $\mathfrak{sl}_{3}$
$L_{1}=\sqrt{2}\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&1&0\end{pmatrix}\;,\quad
L_{0}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{pmatrix}\;,\quad
L_{-1}=\sqrt{2}\begin{pmatrix}0&-1&0\\\ 0&0&-1\\\ 0&0&0\end{pmatrix}\;.$
(5.20)

$U^{-1}_{\footnotesize\text{fund}}\ket{e_{3}}\propto\ket{e_{1}}+\sqrt{2}\ket{e_{2}}+\ket{e_{3}}\;.$
(5.21)

This means that, in general, defining

$p_{k}\mathrel{\mathop{:}}=\bra{e_{k}}U^{-1}_{\footnotesize\text{fund}}\ket{e_{3}}=\delta_{k,1}+\sqrt{2}\delta_{k,2}+\delta_{k,3}\qquad
k=1,2,3\;,$ (5.22)

we have that

$\bra{e_{k_{1}}e_{k_{2}}\cdots
e_{k_{a}}}U^{-1}_{\alpha_{1}}\ket{lw}_{\alpha_{1}}\sim
p_{k_{1}}p_{k_{2}}\cdots p_{k_{a}}\;.$ (5.23)

At this stage, we state the main result of this section. Even though we do not
have, as of this writing, a general proof of (5.15) for generic
representations $\mathcal{R}_{\alpha},\mathcal{R}_{\alpha_{1}}$, we have
tested it for some non-trivial representations given by the following values

$(m_{1},m_{2},a)=\\{(2,2,3),(2,2,6),(2,3,3),(2,3,6),(3,3,3),(3,3,6),(4,4,3),(4,4,6)\\}\;,$
(5.24)

obtaining that in all these cases, (5.15) holds precisely.

Let us anticipate here that, to obtain a non-vanishing one-point toroidal
conformal block, we have to require $a\equiv 0\mod 3$. The details of the case
($2,2,3$) are given in appendix D as an example. Furthermore, as a
complementary motivation of (5.15) for the case when $a=0$ (i.e., when
$\mathcal{R}_{\alpha_{1}}$ is the trivial representation), we show explicitly
that for representations $\mathcal{R}_{\alpha}$ with highest-weight vector
$j=(m_{1},0)$ or $j=(0,m_{2})$, $V_{\alpha|\alpha_{1}}$ in (5.15) is equal to
the character (3.7).

####  5.1.1 Zero-point conformal block

The simplest case of (5.15) is when $\mathcal{R}_{\alpha_{1}}$ is the trivial
representation (one-dimensional), in which case we expect to find the
character (3.7) (or zero-point conformal block) of the $\mathfrak{sl}_{3}$
representation. Let us check this statement.

We show this by choosing as highest-weight vector of $\mathcal{R}_{\alpha}$
the vector $j=(m_{1},0)$555The same procedure holds in the case
$j=(0,m_{2}).$. Since $\mathcal{R}_{\alpha_{1}}$ is the trivial
representation, the intertwining operator $I_{\alpha;\alpha,0}$ (or the
projector (5.13)) reduces to the identity operator, thus

$\begin{split}V_{\alpha|0}(\tau)\equiv V_{\alpha}
&=\sum_{i_{1},\dots,i_{m_{1}}=1,2,3}\bra{e_{i_{1}}\dots
e_{i_{m_{1}}}}q^{L_{0}}\ket{e_{(i_{1}}\dots e_{i_{m_{1})}}}=\\\
&=\sum_{i_{1},\dots,i_{m_{1}}=1,2,3}D_{i_{1}(i_{1}}\dots
D_{i_{m_{1}}i_{m_{1}})}\;.\end{split}$ (5.25)

Due to the symmetrization of the indices, the above expression can be written
as

$V_{\alpha}=\sum_{\begin{subarray}{c}m,n,k=0\\\
m+n+k=m_{1}\end{subarray}}^{m_{1}}q^{m}q^{-n}(1)^{k}\;,$ (5.26)

and after some algebra, we can show that (5.26) gives

$V_{\alpha}=\sum_{r=1}^{m_{1}}\left(q^{-r}+q^{r}\right)\left(-M(m_{1}+r)+\frac{m_{1}+r}{2}-r+1\right)-M(m_{1})+\frac{m_{1}}{2}+1\;,$
(5.27)

where $M(r)=\frac{1-(-1)^{r}}{4}$. By expanding
$\chi^{\mathfrak{sl}_{3}}_{j=(m_{1},0)}$ of (3.7) in $q$, we can see that this
character is equal to (5.27).

We would like to obtain the same result in the general case when $m_{2}\neq
0$, i.e., we want to show the following equation

$\sum_{\begin{subarray}{c}i_{1},\dots,i_{m_{1}}=1,2,3\\\
k_{1},\dots,k_{m_{2}}=1,2,3\end{subarray}}\bra{\bar{e}^{k_{1}}\dots\bar{e}^{k_{m_{2}}}\,e_{i_{1}}\dots
e_{i_{m_{1}}}}q^{L_{0}}T^{k_{1}\dots
k_{m_{2}}}_{i_{1}...i_{m_{1}}}=\chi^{\mathfrak{sl}_{3}}_{j}\;,$ (5.28)

where $\chi^{\mathfrak{sl}_{3}}_{j}$ is the character (3.7), and the tensor
$T^{k_{1}....k_{m_{2}}}_{i_{1}...i_{m_{1}}}$ is the symmetric (in lower and
upper indices) and traceless tensor from (5.12). We do not have a general
proof of (5.28). However, the numerical results we have obtained for many
values of $j$ confirm this equation.

####  5.1.2 One-point conformal block

In this section, we elaborate on the expression (5.16) (which we claimed that
reproduces the $\mathfrak{sl}_{3}$ one-point conformal block
$\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}})$ for general
representations $\mathcal{R}_{\alpha}$ and $\mathcal{R}_{\alpha_{1}}$ labelled
by the highest-weight vectors $j=(m_{1},m_{2})$ and $j_{1}=(a,0)$,
respectively. The purpose is to simplify this expression in order to compare
it with $\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}$. In what
follows, we will not give too much attention to overall constants since they
are generally ineffective. For this reason, some of the following equalities
have to be considered to hold up to a constant.

The idea is to represent (5.16) as a symmetric traceless tensor. The first
step will be constructing a tensor with the right structure to represent the
one-point block but without the requirements of symmetry and tracelessness,
which will be imposed at the end. So, let us start defining the following
tensor

${}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}T^{r_{1}\cdots
r_{m_{2}}}_{s_{1}\cdots s_{m_{1}};k_{1}\cdots k_{a}}\mathrel{\mathop{:}}=\\\
=\bra{\bar{e}^{r_{1}}\cdots\bar{e}^{r_{m_{2}}}\,e_{s_{1}}\cdots
e_{s_{m_{1}}}}q^{L_{0}}\ket{e_{a_{1}}\cdots
e_{a_{m_{1}}}\bar{e}^{b_{1}}\cdots\bar{e}^{b_{m_{2}}}}\,\bra{e_{k_{1}}\cdots
e_{k_{a}}}U^{-1}_{\alpha_{1}}\ket{lw}_{\alpha_{1}}\,,$ (5.29)

which, thanks to the definitions (5.19), (5.22) and (5.23), can be rewritten
as

${}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}T^{r_{1}\cdots
r_{m_{2}}}_{s_{1}\cdots s_{m_{1}};k_{1}\cdots k_{a}}=D_{a_{1}s_{1}}\cdots
D_{a_{m_{1}}s_{m_{1}}}\bar{D}^{b_{1}r_{1}}\cdots\bar{D}^{b_{m_{2}}r_{m_{2}}}p_{k_{1}}\cdots
p_{k_{a}}\;.$ (5.30)

The tensor defined in (5.29) resembles the quantity (5.16) we are interested
in. However, we need to insert the intertwining operator. In order to do that,
let us recall that it acts as a projector (5.13), i.e., it translates in the
insertion of a tensor $P$, which allows us to define

${}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}M^{c_{1}\cdots
c_{m_{2}}}_{d_{1}\cdots d_{m_{1}}}=\big{(}P^{c_{1}\cdots
c_{m_{2}}}_{d_{1}\cdots d_{m_{1}}}\big{)}^{s_{1}\cdots s_{m_{1}};k_{1}\cdots
k_{a}}_{r_{1}\cdots r_{m_{2}}}\;{}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots
a_{m_{1}}}T^{r_{1}\cdots r_{m_{2}}}_{s_{1}\cdots s_{m_{1}};k_{1}\cdots
k_{a}}\;.$ (5.31)

In order to construct the tensor $P$, we follow the method described in [39].
It has to be a product of $\mathfrak{sl}_{3}$ invariant tensors, which are

$\delta^{i}_{j}\;,\quad\epsilon^{ijk}\;,\quad\epsilon_{ijk}\;.$ (5.32)

However, since the final result has to be traceless and symmetric, the only
possible contractions we are interested in are with

$\delta^{k}_{r}\;,\qquad\epsilon^{csk}\;,$ (5.33)

where we have labelled the indices coherently to (5.31). If we suppose that
$P$ is made out of $x$ contractions with $\delta^{k}_{r}$ and $y$ contractions
with $\epsilon^{csk}$, since the remaining indices have to be the same in
number as the expected ones, i.e., $m_{1}$ lower and $m_{2}$ upper indices, a
simple counting shows us that $a$ has to satisfy the condition

$a=3l\;,\qquad l\in\mathbb{Z}_{\geq 0}\;.$ (5.34)

In this way, we obtain that the tensor that represents the intertwining
operator is

$\big{(}P^{c_{1}\cdots c_{m_{2}}}_{d_{1}\cdots d_{m_{1}}}\big{)}^{s_{1}\cdots
s_{m_{1}};k_{1}\cdots k_{a}}_{r_{1}\cdots
r_{m_{2}}}=\delta^{s_{1}}_{d_{1}}\cdots\delta^{s_{m_{1}-l}}_{d_{m_{1}-l}}\delta^{k_{1}}_{d_{m_{1}-l+1}}\cdots\delta^{k_{l}}_{d_{m_{1}}}\delta^{c_{1}}_{r_{1}}\cdots\delta^{c_{m_{2}-l}}_{r_{m_{2}-l}}\times\\\
\times\delta^{k_{l+1}}_{r_{m_{2}-l+1}}\cdots\delta^{k_{2l}}_{r_{m_{2}}}\epsilon^{c_{m_{2}-l+1}s_{m_{1}-l+1}k_{2l+1}}\cdots\epsilon^{c_{m_{2}}s_{m_{1}}k_{3l}}\;.$
(5.35)

Inserting this projector into (5.31), we end up with

${}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}M^{c_{1}\cdots
c_{m_{2}}}_{d_{1}\cdots d_{m_{2}}}=D_{a_{1}d_{1}}\cdots
D_{a_{m_{1}-l}d_{m_{1}-l}}\,\bar{D}^{b_{1}c_{1}}\cdots\bar{D}^{b_{m_{2}-l}c_{m_{2}-l}}p_{d_{m_{1}-l+1}}\cdots
p_{d_{m_{1}}}\times\\\
\times\bar{p}^{b_{m_{2}-l+1}}\cdots\bar{p}^{b_{m_{2}}}\,A^{c_{m_{2}-l+1}}_{a_{m_{1}-l+1}}\cdots
A^{c_{m_{2}}}_{a_{m_{1}}},$ (5.36)

where we have introduced the new quantities

$\bar{p}^{b}\mathrel{\mathop{:}}=\bar{D}^{br}p_{r}\;,\qquad
A^{c}_{a}\mathrel{\mathop{:}}=\epsilon^{csk}D_{as}p_{k}\;.$ (5.37)

We want to make this tensor symmetric, and we do this procedure just formally,
remembering in what follows that we have to consider all the possible cyclic
permutations of the indices. Hence we write

$^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}M^{c_{1}\cdots
c_{m_{2}}}_{d_{1}\cdots d_{m_{1}}}\equiv\;^{b_{1}\cdots
b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}M^{(c_{1}\cdots c_{m_{2}})}_{(d_{1}\cdots
d_{m_{1}})}\;.$ (5.38)

The last step consists in making this tensor traceless. We do this by
exploiting the procedure described in [39], where it is shown that a symmetric
tensor can be made traceless as

${}^{b_{1}\cdots b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}\widetilde{M}^{c_{1}\cdots
c_{m_{2}}}_{d_{1}\cdots
d_{m_{1}}}=\sum_{n=0}^{\min}\mathcal{C}_{n}\delta^{(c_{1}}_{(d_{1}}\cdots\delta^{c_{n}}_{d_{n}}\;{}^{b_{1}\cdots
b_{m_{2}}}_{a_{1}\cdots a_{m_{1}}}M^{c_{n+1}\cdots c_{m_{2}})f_{1}\cdots
f_{n}}_{d_{n+1}\cdots d_{m_{1}})f_{1}\cdots f_{n}}\;,$ (5.39)

where

$\mathcal{C}_{n}=(-1)^{n}\frac{(m_{1}+m_{2}-n+1)!}{(m_{1}+m_{2}+1)!(m_{1}-n)!(m_{2}-n)!n!}\;,\qquad\min=\min(m_{1},m_{2})\;,$
(5.40)

and in each summand, we take the sums over repeated indices $f_{i}$.

Finally, (5.15) can be written as

${}_{a_{1}\cdots a_{m_{1}}}^{b_{1}\cdots b_{m_{2}}}\widetilde{M}_{(d_{1}\cdots
d_{m_{1}})}^{(c_{1}\cdots
c_{m_{2}})}\delta_{b_{1}c_{1}}\cdots\delta_{b_{m_{2}}c_{m_{2}}}\delta^{a_{1}d_{1}}\cdots\delta^{a_{m_{1}}d_{m_{1}}}=\widetilde{C}(\alpha,\alpha_{1})\mathcal{F}(\alpha,\alpha_{1},q)_{\mathfrak{sl}_{3}}\;.$
(5.41)

As of the date of this work, we have not found a general proof of (5.41), but
many numerical computations (which are partially described in the appendix D)
confirm this equality.

##  6 Discussion

In this work we studied the semiclassical limit of $\mathcal{W}_{3}$ CFT on
the torus, with a focus on the $\mathfrak{sl}_{3}$ one-point conformal blocks
of fully degenerate primary fields for an external field satisfying (1.8). We
compute the $\mathcal{W}_{3}$ block in the light $c\rightarrow\infty$ limit
using AGT correspondence. In order to obtain the $\mathfrak{sl}_{3}$ block we
propose the relation (4.2). The utility of this relation is twofold since it
serves as a check of the dual representation for $\mathfrak{sl}_{3}$ conformal
block and provides a computational method to calculate its higher-level
contributions.

The dual description of $\mathfrak{sl}_{3}$ one-point toroidal conformal block
is given in terms of the Wilson lines in $3d$ Chern-Simons gravity. The
proposal summarizes in (5.41). The dual approach seems to give more manageable
computations compared to those based on the original definition, at least in
its construction. The simplest test of this proposal is the case when the one-
point block reduces to the zero-point block, which we confirmed analytically
and numerically in section 5.1.1. Further, we have performed more general
tests for specific cases of representations (5.24), obtaining that all these
examples confirm (5.15). The absence of a general expression of
$\mathfrak{sl}_{3}$ Clebsch-Gordan coefficients forces us to use the tensor
technique in section 5.1, the general proof of (5.41) remains an open problem.

From the Wilson line construction a natural generalization follows, that is
the study of the dual description in the cases of higher-point conformal
blocks, which show additional features, like the presence of different
intermediate channels (usually called $s$ and $t$), as shown for the
$\mathfrak{sl}_{2}$ case in [40]. This generalization is straightforward in
the context of the Wilson lines formulation of the conformal blocks, according
to the construction of section 5. Also, it would be worth to extend this
approach to the whole $\mathcal{W}_{3}$ CFT, which would require to focus also
on quantum corrections (for related consideration see [26]).

Another interesting question is how the Wilson line approach works if
multiplicities in the representations under consideration are present. From
the dual construction it follows that in such cases we can obtain multiple
nonequivalent expressions for the projector (5.13). For instance, if the
external field is in the adjoint representation, there is a multiplicity two
for the conformal blocks in the $\mathcal{W}_{3}$ CFT [46], which corresponds
to the fact that Clebsch-Gordan coefficients are not uniquely defined [47]. In
this case, the multiplicity is present also in the definition of the
projector. It would be interesting to establish a precise correspondence
between uncertainty related to the OPE multiplicities for conformal blocks and
multiplicities arising in the definition of the projector in the dual
construction.

##  Appendix A Matrix Elements

Here, we give the matrix elements that enter in the computation of the
$\mathfrak{sl}_{3}$ conformal block (3.6) at the first and second levels of
$\mathfrak{sl}_{3}$ module. We use the algorithm described in [48] to compute
these matrix elements.

First level. From the $\mathfrak{sl}_{3}$ commutation relations, one can find
the following products

$\begin{split}&\langle
h_{0},q_{0}|\mathbf{L}_{1}\mathbf{L}_{-1}|h_{0},q_{0}\rangle=2h_{0}\;,\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{-1}|h_{0},q_{0}\rangle=3q_{0}\;,\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=-h_{0}/5\;,\end{split}$
(A.1)

thus the Shapavalov matrix $G$ at the first level is

$G=\left(\begin{array}[]{cc}2h_{0}&3q_{0}\\\ 3q_{0}&-\frac{h_{0}}{5}\\\
\end{array}\right).$ (A.2)

The matrix elements at the first level are666All the matrix elements are
proportional to $\langle h_{0},q_{0}|\Phi_{\alpha_{1}}|h_{0},q_{0}\rangle$.
But, for the sake of brevity, we omit this factor.

$\begin{split}&\langle
h_{0},q_{0}|\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{L}_{-1}|h_{0},q_{0}\rangle=2h_{0}+\left(h_{1}-1\right)h_{1},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\Phi_{\alpha_{1}}\mathbf{L}_{-1}|h_{0},q_{0}\rangle=3q_{0}+\frac{1}{2}\left(h_{1}-1\right)q_{1},\\\
&\langle
h_{0},q_{0}|\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=3q_{0}-\frac{1}{2}\left(h_{1}-1\right)q_{1},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\Phi_{\alpha_{1}}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=\frac{1}{5}\left(h_{1}-h_{0}\right)-\frac{\left(h_{1}-3\right)q_{1}^{2}}{4h_{1}}.\end{split}$
(A.3)

Second level. At this level, we consider the basis

$\mathbf{L}_{-1}^{2}\;,\quad\mathbf{W}_{-1}^{2}\;,\quad\mathbf{L}_{-1}\mathbf{W}_{-1}\;,\quad\mathbf{W}_{-2}\;.$
(A.4)

We obtain the following products

$\begin{split}&\langle
h_{0},q_{0}|\mathbf{L}^{2}_{1}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=4h_{0}\left(2h_{0}+1\right),\\\
&\langle
h_{0},q_{0}|\mathbf{W}^{2}_{1}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=18q_{0}^{2}-\frac{6h_{0}}{5},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{1}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=6\left(2h_{0}+1\right)q_{0},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{2}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=12q_{0},\\\
&\langle
h_{0},q_{0}|\mathbf{W}^{2}_{1}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle=\frac{1}{25}h_{0}\left(2h_{0}+1\right),\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{1}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle=\frac{1}{5}(-3)\left(2h_{0}+3\right)q_{0},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{2}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle=\frac{6q_{0}}{5},\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{1}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=9q_{0}^{2}-\frac{2}{5}h_{0}\left(h_{0}+1\right),\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{2}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=-\frac{1}{5}\left(4h_{0}\right),\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{2}\mathbf{W}_{-2}|h_{0},q_{0}\rangle=\frac{8h_{0}}{5}.\end{split}$
(A.5)

Hence the Shapavalov matrix $G$ at the second level is

$G=\left(\begin{array}[]{cccc}4h_{0}\left(2h_{0}+1\right)&18q_{0}^{2}-\frac{6h_{0}}{5}&6\left(2h_{0}+1\right)q_{0}&12q_{0}\\\
18q_{0}^{2}-\frac{6h_{0}}{5}&\frac{1}{25}h_{0}\left(2h_{0}+1\right)&\frac{1}{5}(-3)\left(2h_{0}+3\right)q_{0}&\frac{6q_{0}}{5}\\\
6\left(2h_{0}+1\right)q_{0}&\frac{1}{5}(-3)\left(2h_{0}+3\right)q_{0}&9q_{0}^{2}-\frac{2}{5}h_{0}\left(h_{0}+1\right)&-\frac{1}{5}\left(4h_{0}\right)\\\
12q_{0}&\frac{6q_{0}}{5}&-\frac{1}{5}\left(4h_{0}\right)&\frac{8h_{0}}{5}\\\
\end{array}\right).$ (A.6)

The matrix elements at the second level are

$\begin{split}&\langle
h_{0},q_{0}|\mathbf{L}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=8h_{0}^{2}+\left(8\left(h_{1}-1\right)h_{1}+4\right)h_{0}+\left(h_{1}-1\right)h_{1}\left(\left(h_{1}-1\right)h_{1}+2\right),\\\
&\begin{split}\langle
h_{0},q_{0}|\mathbf{L}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle&=18q_{0}^{2}-6\left(h_{1}-1\right)q_{1}q_{0}+\frac{\left(h_{1}^{3}-7h_{1}+6\right)q_{1}^{2}}{4h_{1}}\\\
&-\frac{1}{5}\left(h_{1}-1\right)h_{1}\left(h_{1}+4\right)-\frac{6h_{0}}{5},\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|\mathbf{L}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle&=6\left(2h_{0}+\left(h_{1}-1\right)h_{1}+1\right)q_{0}\\\
&-\frac{1}{2}\left(h_{1}-1\right)\left(4h_{0}+\left(h_{1}-1\right)h_{1}+2\right)q_{1},\end{split}\\\
&\langle
h_{0},q_{0}|\mathbf{L}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{W}_{-2}|h_{0},q_{0}\rangle=12q_{0}-2\left(h_{1}-1\right)h_{1}q_{1}.\end{split}$
(A.7) $\begin{split}&\begin{split}\langle
h_{0},q_{0}|\mathbf{W}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle&=18q_{0}^{2}+6\left(h_{1}-1\right)q_{1}q_{0}+\frac{\left(h_{1}^{3}-7h_{1}-30\right)q_{1}^{2}}{4h_{1}}\\\
&-\frac{1}{5}h_{1}\left(h_{1}\left(h_{1}+3\right)-2\right)-\frac{6h_{0}}{5},\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|&\mathbf{W}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle=\frac{\left(h_{1}-6\right)\left(h_{1}-3\right)\left(h_{1}+3\right)q_{1}^{4}}{16h_{1}^{3}}\\\
&+\frac{\left(2h_{0}\left(h_{1}-3\right)-3\left(h_{1}-2\right)h_{1}+12\right)q_{1}^{2}}{10h_{1}}+\frac{1}{25}\left(2h_{0}^{2}+\left(1-4h_{1}\right)h_{0}+h_{1}\left(3h_{1}+2\right)\right),\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|&\mathbf{W}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=\frac{12}{40}\left(\frac{15q_{1}^{2}}{h_{1}}-5q_{1}^{2}-4h_{0}+4h_{1}-6\right)q_{0}+\\\
&+\frac{1}{40}\left(12h_{1}^{2}-8\left(h_{0}+1\right)h_{1}+8h_{0}-\frac{5\left(\left(h_{1}-3\right)\left(h_{1}-1\right)h_{1}-36\right)q_{1}^{2}}{h_{1}^{2}}+4\right)q_{1},\end{split}\\\
&\langle
h_{0},q_{0}|\mathbf{W}^{2}_{1}\Phi_{\alpha_{1}}\mathbf{W}_{-2}|h_{0},q_{0}\rangle=-\frac{\left(h_{1}-6\right)\left(h_{1}-3\right)q_{1}^{3}}{2h_{1}^{2}}+\frac{2}{5}\left(2h_{1}-3\right)q_{1}+\frac{6q_{0}}{5}.\end{split}$
(A.8) $\begin{split}&\begin{split}\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle&=6\left(2h_{0}+\left(h_{1}-1\right)h_{1}+1\right)q_{0}+\\\
&+\frac{1}{2}\left(h_{1}-1\right)\left(4h_{0}+\left(h_{1}-1\right)h_{1}+2\right)q_{1},\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|&\mathbf{W}_{1}\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle=\frac{12}{40}\left(\frac{15q_{1}^{2}}{h_{1}}-5q_{1}^{2}-4h_{0}+4h_{1}-6\right)q_{0}+\\\
&+\frac{1}{40}\frac{\left(h_{1}-1\right)q_{1}\left(5\left(h_{1}-6\right)\left(h_{1}+3\right)q_{1}^{2}-4h_{1}^{2}\left(-2h_{0}+3h_{1}+2\right)\right)}{h_{1}^{2}},\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|&\mathbf{W}_{1}\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle=\frac{1}{5}\left(-2h_{0}^{2}-\left(h_{1}-2\right)\left(h_{1}-1\right)h_{0}+45q_{0}^{2}+h_{1}\left(\left(h_{1}-1\right)h_{1}+2\right)\right)\\\
&-\frac{\left(h_{1}-3\right)\left(2h_{0}+\left(h_{1}-1\right)h_{1}+2\right)q_{1}^{2}}{4h_{1}},\end{split}\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{1}\mathbf{L}_{1}\Phi_{\alpha_{1}}\mathbf{W}_{-2}|h_{0},q_{0}\rangle=\frac{2}{5}\left(h_{1}^{2}+h_{1}-2h_{0}\right)-\frac{\left(\left(h_{1}-6\right)h_{1}+3\right)q_{1}^{2}}{h_{1}}.\end{split}$
(A.9) $\begin{split}&\langle
h_{0},q_{0}|\mathbf{W}_{2}\Phi_{\alpha_{1}}\mathbf{L}^{2}_{-1}|h_{0},q_{0}\rangle=2\left(6q_{0}+\left(h_{1}-1\right)h_{1}q_{1}\right),\\\
&\begin{split}\langle
h_{0},q_{0}|\mathbf{W}_{2}\Phi_{\alpha_{1}}\mathbf{W}^{2}_{-1}|h_{0},q_{0}\rangle&=\frac{\left(h_{1}^{2}-9\right)q_{1}^{3}}{2h_{1}^{2}}-\frac{3}{5}\left(h_{1}-1\right)q_{1}+\frac{6q_{0}}{5},\end{split}\\\
&\begin{split}\langle
h_{0},q_{0}|\mathbf{W}_{2}\Phi_{\alpha_{1}}\mathbf{L}_{-1}\mathbf{W}_{-1}|h_{0},q_{0}\rangle&=\frac{1}{10}\left(\frac{5\left(h_{1}\left(3-2h_{1}\right)+3\right)q_{1}^{2}}{h_{1}}-8h_{0}+2h_{1}\left(h_{1}+3\right)\right),\end{split}\\\
&\langle
h_{0},q_{0}|\mathbf{W}_{2}\Phi_{\alpha_{1}}\mathbf{W}_{-2}|h_{0},q_{0}\rangle=\frac{8h_{0}}{5}-\frac{4\left(h_{1}-3\right)q_{1}^{2}}{h_{1}}.\end{split}$
(A.10)

##  Appendix B Weyl character formula

Here, we want to compute the character $\chi^{\mathfrak{sl}_{3}}_{j}$ (3.7) of
a finite-dimensional representation of $\mathfrak{sl}_{3}$ with highest-weight
vector given by $j=m_{1}w_{1}+m_{2}w_{2}$, where $w_{1,2}$ are the fundamental
weights and $m_{1},m_{2}$ are positive integers. Let us define some notations.
The fundamental weights satisfy the following products

$\begin{split}&(w_{1},w_{1})=(w_{2},w_{2})=\frac{2}{3},\qquad(w_{1},w_{2})=\frac{1}{3}\;,\end{split}$
(B.1)

The Weyl vector is defined as

$\rho=w_{1}+w_{2}\;,$ (B.2)

and the Weyl group $W$ of $\mathfrak{sl}_{3}$ is formed by 6 elements, which
are denoted as

$W=[\mathbf{1},\;s_{1},\;s_{2},\;s_{1}s_{2},\;s_{2}s_{1},\;s_{1}s_{2}s_{1}]\;.$
(B.3)

and they act on the vector $j$ as follows

$\begin{split}\mathbf{1}(j)&=j\;,\\\
s_{1}(j)&=-m_{1}w_{1}+(m_{1}+m_{2})w_{2}\;,\\\
s_{2}(j)&=(m_{1}+m_{2})w_{1}-m_{2}w_{2}\;,\\\
s_{1}s_{2}(j)&=-(m_{1}+m_{2})w_{1}+m_{1}w_{2}\;,\\\
s_{2}s_{1}(j)&=m_{2}w_{1}-(m_{1}+m_{2})w_{2}\;,\\\
s_{1}s_{2}s_{1}(j)&=-m_{1}w_{1}-m_{2}w_{2}\;.\end{split}$ (B.4)

The signature $\epsilon(w)$ of a element $w\in W$ is defined by

$\epsilon(w)=(-1)^{l(w)}\;,$ (B.5)

where $l(w)$ is the length of $w$, that is the number of $s_{i}$ that $w$
contains. For example, the signatures of the elements (B.3) are respectively

$\epsilon(W)=[1,\;-1,\;-1,\;1,\;1,\;-1]\;.$ (B.6)

We compute the character $\chi^{\mathfrak{sl}_{3}}_{j}$ from the Weyl
character formula

$\chi^{\mathfrak{sl}_{3}}_{j}=\frac{\sum_{w\in W}\epsilon(w)e^{(w(j+\rho),2\pi
i\tau\rho)}}{\sum_{w\in W}\epsilon(w)e^{(w(\rho),2\pi i\tau\rho)}}\;,$ (B.7)

where we use $q=e^{2\pi i\tau}$, obtaining exactly (3.7).

##  Appendix C AGT relation

In this section, we comment on some important details of the AGT relation we
used to compute the expression (4.1). According to the AGT relation, the
$\mathcal{W}_{3}$ conformal block $\mathcal{F}(\alpha,\alpha_{1},q)$ (4.1) can
be computed by

$\mathcal{F}(\alpha,\alpha_{1},q)=Z_{SU(3)}\prod^{\infty}_{i=1}(1-q^{i})^{1-2h_{1}}\;,$
(C.1)

where the infinite product is the so-called _Heisenberg factor_ , and
$Z_{SU(3)}$ is the $SU(3)$ instanton partition function. For the algorithm
which we used to compute $Z_{SU(3)}$ we refer to [49], specifically the
section 3.2.1. Here, we want to clarify the relations between the parameters
of $\mathcal{F}(\alpha,\alpha_{1},q)$ and $Z_{SU(3)}$.

The partition function $Z_{SU(3)}$ is given by equation (28) of [49] and
depends on the parameters
$x_{i}=(Q-\alpha,e_{i}),\mu,\epsilon_{1},\epsilon_{2}$ which are related to
our parameters as follows

$\mu=a/3,\quad\epsilon_{1}=b,\quad\epsilon_{2}=1/b,$ (C.2)

while $\alpha$ is given by (2.9) and $Q,e_{i}$ are according to our notations
above. Finally, the expression (4.1) is given in the limit
$c\rightarrow\infty$ (or equivalently $b\rightarrow 0$) of C.1.

##  Appendix D Examples

The simplest test777Since the expression (3.6) contains contributions of two
states to the first level and four states to the second level, then
$m_{1},m_{2}\geq 2$. of (5.41) we can perform is the case
$j=2w_{1}+2w_{2}=(2,2),j_{1}=(3,0)$. In this case the tensor (5.38) becomes

${}^{b_{1}b_{2}}_{a_{1}a_{2}}M^{c_{1}c_{2}}_{d_{1}d_{2}}=\frac{1}{4}\bar{p}^{b_{2}}(D_{a_{1}d_{1}}p_{d2}+D_{a_{1}d_{2}}p_{d1})(\bar{D}^{b_{1}c_{1}}A_{a_{2}}^{c_{2}}+\bar{D}^{b_{1}c_{2}}A_{a_{2}}^{c_{1}})\;.$
(D.1)

The next step is to make this tensor traceless according to (5.39) and then
take the contraction according to (5.41). We can show that there will be 16
terms proportional to $\mathcal{C}_{1}$, and all the terms proportional to
$\mathcal{C}_{2}$ vanish. To write the result, we use the notation of the
matrix $B$

$B^{a}_{b}=\bar{p}^{a}p_{b}\;.$ (D.2)

By using this notation, and denoting the trace of a generic matrix $C$ by
$\operatorname{Tr}[C]=[C]$, one can show that

$\displaystyle{}_{a_{1}a_{2}}^{b_{1}b_{2}}\widetilde{M}_{(d_{1}d_{2})}^{(c_{1}c_{2})}\delta_{b_{1}c_{1}}\delta_{b_{2}c_{2}}\delta^{a_{1}d_{1}}\delta^{a_{2}d_{2}}=$
$\displaystyle=\frac{1}{4}\bigg{(}\mathcal{C}_{0}\Big{(}[D]^{2}[AB]+[D][\bar{D}BA]+[\bar{D}][DAB]+[DA\bar{D}B]\Big{)}+\frac{\mathcal{C}_{1}}{4}\Big{(}[D\bar{D}BA^{T}]+[DAB\bar{D}]$
$\displaystyle\quad+[D\bar{D}BA]+[\bar{D}D][AB]+[D][\bar{D}BA^{T}]+[D][B\bar{D}A]+[\bar{D}BDA^{T}]+[D\bar{D}BAB]$
$\displaystyle\quad+[\bar{D}][DAB^{T}]+[\bar{D}][DA^{T}B]+[DA\bar{D}B^{T}]+[D\bar{D}BA^{T}B^{T}]+[\bar{D}][D][A^{T}B]$
$\displaystyle\quad+[D][\bar{D}A^{T}B]+[\bar{D}][BDA^{T}]+[D\bar{D}A^{T}B]\Big{)}+(0)\mathcal{C}_{2}\bigg{)}=$
$\displaystyle=\widetilde{C}(2,2,3)(1-\frac{8}{5}q^{2}-\frac{4}{5}q^{3}+\frac{4}{5}q^{5}+\frac{8}{5}q^{6}-q^{8})\;.$
(D.3)

The firs three terms in $q$, (i.e. up to $q^{2}$) of the above expression are
exactly the terms obtained by (3.6) in this case
($m_{1},m_{2},a\rightarrow(2,2,3)$). This validates (5.41).

It is clear that the difficulty of this “simple” problem increases
substantially when $m_{1}$, $m_{2}$, $a$ increase. We were able to work out
the cases (5.24). Here we give the results and let us emphasize that all these
cases confirm (5.41).

##### Cases :

$(m_{1},m_{2},a)=(2,2,6)$

${}_{a_{1}a_{2}}^{b_{1}b_{2}}\widetilde{M}_{(a_{1}a_{2})}^{(b_{1}b_{2})}=\widetilde{C}(2,2,6)(1-4q+4q^{2}+\dots)\;,$
(D.4)

$(m_{1},m_{2},a)=(2,3,3)$

${}_{a_{1}a_{2}}^{b_{1}b_{2}b_{3}}\widetilde{M}_{(a_{1}a_{2})}^{(b_{1}b_{2}b_{3})}=\widetilde{C}(2,3,3)(1+\frac{q}{3}-\frac{7q^{2}}{9}+\dots)\;,$
(D.5)

$(m_{1},m_{2},a)=(2,3,6)$

${}_{a_{1}a_{2}}^{b_{1}b_{2}b_{3}}\widetilde{M}_{(a_{1}a_{2})}^{(b_{1}b_{2}b_{3})}=\widetilde{C}(2,3,6)(1-3q+q^{2}+\dots)\;,$
(D.6)

$(m_{1},m_{2},a)=(3,3,3)$

${}_{a_{1}a_{2}a_{3}}^{b_{1}b_{2}b_{3}}\widetilde{M}_{(a_{1}a_{2}a_{3})}^{(b_{1}b_{2}b_{3})}=\widetilde{C}(3,3,3)(1+\frac{2q}{3}+\frac{2q^{2}}{21}+\dots)\;,$
(D.7)

$(m_{1},m_{2},a)=(3,3,6)$

${}_{a_{1}a_{2}a_{3}}^{b_{1}b_{2}b_{3}}\widetilde{M}_{(a_{1}a_{2}a_{3})}^{(b_{1}b_{2}b_{3})}=\widetilde{C}(3,3,6)(1-2q-\frac{10q^{2}}{7}+\dots)\,,$
(D.8)

$(m_{1},m_{2},a)=(4,4,3)$

${}_{a_{1}a_{2}a_{3}a_{4}}^{b_{1}b_{2}b_{3}b_{4}}\widetilde{M}_{(a_{1}a_{2}a_{3}d_{4})}^{(b_{1}b_{2}b_{3}b_{4})}=\widetilde{C}(4,4,3)(1+q+q^{2}+\dots)\;,$
(D.9)

$(m_{1},m_{2},a)=(4,4,6)$

${}_{a_{1}a_{2}a_{3}a_{4}}^{b_{1}b_{2}b_{3}b_{4}}\widetilde{M}_{(a_{1}a_{2}a_{3}a_{4})}^{(b_{1}b_{2}b_{3}b_{4})}=\widetilde{C}(4,4,6)(1-q-\frac{5q^{2}}{3}+\dots)\;.$
(D.10)

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