aainstitutetext: I.E. Tamm Department of Theoretical Physics,
P.N. Lebedev Physical Institute,
Leninsky ave. 53, 119991 Moscow, Russia

# Example of the $4$-pt non-vacuum $\mathcal{W}_{3}$ classical block

Mikhail Pavlov<EMAIL_ADDRESS>
###### Abstract

In this note we study a special case of the $4$-pt non-vacuum classical block
associated with the $\mathcal{W}_{3}$ algebra. We formulate the monodromy
problem for the block and fix remaining functional arbitrariness using
parameters of the $4$-pt vacuum block. Under the given conditions, the block
function is computed.

##  1 Introduction

Two-dimensional conformal theories with extended symmetry given by the algebra
$W_{N},N\geq 3$ [1] are a natural generalization of theories with the Virasoro
algebra. The main object of study in these theories are the basis elements in
the space of correlation functions - conformal blocks. In this paper, we
consider the non-vacuum $4$-pt $\mathcal{W}_{3}$ classical block within the
monodromy method. The monodromy properties of the given block are governed by
the fusion rules for the fully degenerate representation of the
$\mathcal{W}_{3}$ algebra.

In contrast to the Virasoro blocks, generally $\mathcal{W}_{3}$ blocks contain
functional arbitrariness. In other words, it is necessary to fix this
arbitrariness, which imposes additional requirements on a particular
$\mathcal{W}_{3}$ CFT2. However, the arbitrariness is absent for the $4$-pt
$\mathcal{W}_{3}$ vacuum block [2]. Considering the non-vacuum block as a
certain deformation of the vacuum block, we discuss ways of fixing this
arbitrariness in Section 3.

The paper is organized as follows. In Section 2 we discuss $\mathcal{W}_{3}$
algebra in the classical limit and study its degenerate representations along
with associated fusion rules. Section 3 is devoted to the monodromy method for
classical $\mathcal{W}_{3}$ blocks. Here, we focus on calculations of the
special $4$-pt non-vacuum block using fusion rules derived in Section 2.
Section 4 summarizes our results. Appendix A contains details about the
monodromy method related with the degenerate representation $V_{(3,1)}$ of the
Virasoro algebra.

##  2 Three-point functions and fusion rules of the $\mathcal{W}_{3}$ algebra

In this section we consider fusion rules for the fully degenerate
representation of the $\mathcal{W}_{3}$ algebra.

#### The $\mathcal{W}_{3}$ algebra and highest-weight representations.

$\mathcal{W}_{3}$ symmetry is generated by two currents $T(z)$ and $W(z)$, the
first of which is the energy-momentum tensor, and the second corresponds to
the spin-3 current. These currents can be decomposed into modes $L_{m}$ and
$W_{n}$ as follows

$T(z)=\sum_{-\infty}^{\infty}\frac{L_{m}}{z^{m+2}}\;,\qquad
W(z)=\sum_{-\infty}^{\infty}\frac{W_{n}}{z^{n+3}}\;.$ (2.1)

The modes satisfy commutation relations

$\begin{array}[]{c}\left[L_{n},L_{m}\right]=(n-m)L_{n+m}+\displaystyle\frac{c}{12}(n^{3}-n)\delta_{n+m,0}\;,\\\
\\\ \left[L_{n},W_{m}\right]=(2n-m)W_{n+m}\;,\\\ \\\
\displaystyle\left[W_{n},W_{m}\right]=\frac{c}{3\cdot
5!}(n^{2}-1)(n^{2}-4)n\delta_{n+m,0}+\frac{16}{22+5c}(n-m)\Lambda_{n+m}+\\\
\\\
\displaystyle\frac{(n-m)}{30}\left(2m^{2}+2n^{2}-mn-8\right)L_{n+m}\;,\end{array}$
(2.2)

where

$\Lambda_{m}=\sum_{p\leq-2}L_{p}L_{m-p}+\sum_{p\geq-1}L_{m-p}L_{p}-\frac{3(m+2)(m+3)}{10}L_{m}\;,$
(2.3)

and for a central element $c$ it is convenient to use the parametrization

$c=2\left(1+12(b^{1}+b^{-1})^{2}\right).$ (2.4)

The associative algebra given by the relations (2.2) is called the
$\mathcal{W}_{3}$ algebra [3, 4]. Due to the presence of the nonlinear term
(2.3) the $\mathcal{W}_{3}$ algebra is not a Lie algebra, but its generators
satisfy the Jacobi identities [3].

In this paper we focus on the classical limit $c\rightarrow\infty$ (or
$b\rightarrow 0$). In this limit, one can neglect the non-linear term (2.3) in
the commutators (2.2). In what follows, mentioning the $\mathcal{W}_{3}$
algebra we mean the classical limit. Note that the resulting algebra contains
the global subalgebra $sl(3)$ spanned by $8$ generators $(L_{0},L_{\pm
1},W_{0},W_{\pm 1},W_{\pm 2})$. In the light of this, the $\mathcal{W}_{3}$
algebra may be considered as a central extension of $sl(3)$. Finally, the
classical $\mathcal{W}_{3}$ algebra is finitely generated, more precisely, the
set $(L_{0},L_{\pm 1},L_{\pm 2},W_{0})$ generates the entire algebra through
relations (2.2).

The representation theory of the $\mathcal{W}_{3}$ algebra can be constructed
in analogy with that of the Virasoro algebra. A $\mathcal{W}_{3}$ highest-
weight vector $|\Delta,Q\rangle$ satisfies the following conditions

$L_{n}|\Delta,Q\rangle=\delta_{n,0}~{}\Delta|\Delta,Q\rangle\;,\qquad
W_{n}|\Delta,Q\rangle=\delta_{n,0}~{}Q|\Delta,Q\rangle\;,\quad\text{where}~{}~{}~{}n\geq
0\;,$ (2.5)

where $\Delta$ stands for the conformal dimension, $Q$ denotes the spin-$3$
charge. The $\mathcal{W}_{3}$ module is spanned by the basis states111For more
detailed analysis, see [5], Appendix E.

$\begin{array}[]{c}\mathcal{L}_{-I}|\Delta,Q\rangle\equiv
L_{-i_{1}}...L_{-i_{k}}W_{-j_{1}}...W_{-j_{p}}|\Delta,Q\rangle,\qquad
I\equiv\\{i_{1},....,i_{k};j_{1},...,j_{p}\\},\\\ \\\ 1\leq i_{1}\leq...\leq
i_{k},\qquad 1\leq j_{1}\leq...\leq j_{p}\;.\end{array}$ (2.6)

For states (2.5), we can define the primary operator $\mathcal{O}_{\Delta,Q}$
using

$|\Delta,Q\rangle=\lim_{z\rightarrow 0}\mathcal{O}_{\Delta,Q}(z)|0\rangle.$
(2.7)

In contrast to the Virasoro algebra generators $L_{m}$, the modes $W_{n}$
cannot be expressed in terms of differential operators with respect to the
variable $z$. The differential realization of (2.2) includes two additional
operators $\hat{W}_{-1}\mathcal{O}_{{}_{\Delta,Q}}(z)$ and
$\hat{W}_{-2}\mathcal{O}_{{}_{\Delta,Q}}(z)$ [6]

$\begin{array}[]{c}\mathcal{L}_{m}\mathcal{O}_{{}_{\Delta,Q}}(z)=\left(z^{m+1}\partial_{z}+\Delta(m+1)z^{m}\right)\mathcal{O}_{{}_{\Delta,Q}}(z),\\\
\\\
\mathcal{W}_{n}\mathcal{O}_{{}_{\Delta,Q}}(z)=\left(\displaystyle\frac{Q}{2}(m+2)(m+1)z^{m}+(m+2)z^{m+1}\hat{W}_{-1}+z^{m+2}\hat{W}_{-2}\right)\mathcal{O}_{{}_{\Delta,Q}}(z).\end{array}$
(2.8)

In what follows, we use a simplified form of the subscript for the primary
operators $\mathcal{O}_{1}$, where the index $1$ denotes $(\Delta_{1},Q_{1})$.

####  $2,3$-pt correlators.

The $\mathcal{W}_{3}$ symmetry imposes certain restrictions on $n$-pt
correlation functions of primary operators governed by conformal Ward
identities. The ones with currents $T(z)$ and $W(z)$ have the form

$\displaystyle\langle
T(z)\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=\sum_{i=1}^{n}\left(\frac{\Delta_{i}}{(z-z_{i})^{2}}+\frac{\partial_{i}}{(z-z_{i})}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle,$
(2.9) $\displaystyle\langle
W(z)\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=\sum_{i=1}^{n}\left(\frac{Q_{i}}{(z-z_{i})^{3}}+\frac{\hat{W}_{-1}^{(i)}}{(z-z_{i})^{2}}+\frac{\hat{W}_{-2}^{(i)}}{(z-z_{i})}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle,$
(2.10)

where we use the notation

$\hat{W}_{-p}^{(i)}\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle\equiv\langle\mathcal{O}_{1}(z_{1})\dots\hat{W}_{-p}\mathcal{O}_{i}(z_{i})\dots\mathcal{O}_{n}(z_{n})\rangle,\qquad
p=1,2.$ (2.11)

The asymptotic behavior of the currents at $z\rightarrow\infty$ is given by

$T(z)\rightarrow z^{-4}\;,\qquad W(z)\rightarrow z^{-6}\;,\quad\text{at}\quad
z\rightarrow\infty.$ (2.12)

Applying (2.12) to (2.9) and (2.10) one obtains $8$ global Ward identities.
Three of them associated with $T(z)$ have the form

$\begin{array}[]{c}\displaystyle\sum^{n}_{i=1}\partial_{i}\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}(z_{i}\partial_{i}+\Delta_{i})\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}(z^{2}_{i}\partial_{i}+2z_{i}\Delta_{i})\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0.\end{array}$
(2.13)

Five remaining identities following from (2.10) are

$\begin{array}[]{c}\displaystyle\sum^{n}_{i=1}\hat{W}_{{}_{-2}}^{(i)}\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}\left(z_{i}\hat{W}_{{}_{-2}}^{(i)}+\hat{W}^{(i)}_{{}_{-1}}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}\left(z^{2}_{i}\hat{W}_{{}_{-2}}^{(i)}+2z_{i}\hat{W}^{(i)}_{{}_{-1}}+Q_{i}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}\left(z^{3}_{i}\hat{W}_{{}_{-2}}^{(i)}+3z^{2}_{i}\hat{W}^{(i)}_{{}_{-1}}+3z_{i}Q_{i}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0,\\\
\displaystyle\sum^{n}_{i=1}\left(z^{4}_{i}\hat{W}_{{}_{-2}}^{(i)}+4z^{3}_{i}\hat{W}^{(i)}_{{}_{-1}}+6z^{2}_{i}Q_{i}\right)\langle\mathcal{O}_{1}(z_{1})\dots\mathcal{O}_{n}(z_{n})\rangle=0.\end{array}$
(2.14)

As in the case of the Virasoro algebra, the 2-point correlation function is
fixed (up to normalization) by the Ward identities

$\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\rangle=\frac{\delta_{\Delta_{1},\Delta_{2}}\delta_{Q_{1},-Q_{2}}}{(z_{1}-z_{2})^{2\Delta_{1}}}\;,$
(2.15)

where in addition to the condition $\Delta_{1}=\Delta_{2}$, the
$\mathcal{W}_{3}$ symmetry imposes $Q_{1}+Q_{2}=0$. In contrast, conditions
(2.13) and (2.14) do not completely determine the $3$-point correlation
function. Indeed, the $3$-pt function depends on nine variables: coordinates
$z_{j}$ and the operators $\hat{W}^{(j)}_{-1},\hat{W}^{(j)}_{-2}$, $j=1,2,3$.
So 8 Ward identities are not enough to completely eliminate functional
arbitrariness.222Nevertheless, these functions are explicitly known in some
special cases, see [7, 8, 9].  However, the conditions (2.13) fix the
coordinate dependence of the $3$-pt function, which has the form

$V_{123}\equiv\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\mathcal{O}_{3}(z_{3})\rangle\sim(z_{1}-z_{2})^{(\Delta_{3}-\Delta_{3}-\Delta_{1})}(z_{1}-z_{3})^{(\Delta_{2}-\Delta_{3}-\Delta_{1})}(z_{2}-z_{3})^{(\Delta_{2}+\Delta_{3}-\Delta_{1})}.$
(2.16)

#### Degenerate representations of the $\mathcal{W}_{3}$ algebra.

There are highest-weight vectors, which descendants are highest-weight vectors
themselves. These so-called null-vectors are determined by the following
conditions

$L_{1}|\tilde{\psi}\rangle=L_{2}|\tilde{\psi}\rangle=W_{1}|\tilde{\psi}\
\rangle=0\;.$ (2.17)

For the further analysis in Section 3 we study the fully degenerate
representation $|\psi\rangle$ with null-vectors at levels one, two and three.
The conformal dimension and the spin-$3$ charge of $|\psi\rangle$ are given by

$\displaystyle\Delta_{\psi}=-1-\frac{4b^{2}}{3},\qquad
Q^{2}_{\psi}=-\frac{2\Delta^{2}_{\psi}}{27}\frac{5b+3b^{-1}}{3b+5b^{-1}}.$
(2.18)

The explicit form of the null-vectors is

$|\psi_{1}\rangle=\left(W_{-1}-\frac{3Q_{\psi}}{2\Delta_{\psi}}L_{-1}\right)|\psi\rangle=0,$
(2.19a)
$|\psi_{2}\rangle=\left(W_{-2}-\frac{6Q_{\psi}}{\Delta_{\psi}(5\Delta_{\psi}+1)}\left(L_{-1}^{2}-(\Delta_{\psi}+1)L_{-2}\right)\right)|\psi\rangle=0,$
(2.19b)
$|\psi_{3}\rangle=\left(W_{-3}+\frac{Q_{\psi}}{\Delta_{\psi}(5\Delta_{\psi}+1)}\left(-\frac{16}{\Delta_{\psi}+1}L_{-1}^{3}+12L_{-1}L_{-2}+\frac{3(\Delta_{\psi}-3)}{2}L_{-3}\right)\right)|\psi\rangle=0.$
(2.19c)

Let us consider the $3$-pt correlation function $V_{\alpha\beta\psi_{{}_{3}}}$
where the operator $\psi_{3}$ corresponds to the null-vector (2.19c). The
relations above impose certain conditions on $V_{\alpha\beta\psi_{{}_{3}}}$,
in other words, fusion rules. More precisely, putting
$V_{\alpha\beta\psi_{{}_{3}}}=0$ due to $|\psi_{3}\rangle=0$ one finds [10, 4,
7]

$\begin{array}[]{c}32Q_{\psi}(\Delta_{\alpha}-\Delta_{\beta})^{3}-12Q_{\psi}(\Delta_{\psi}+1)(\Delta^{2}_{\alpha}-\Delta^{2}_{\beta})-Q_{\psi}(15\Delta^{2}_{\psi}-12\Delta_{\psi}-1)(\Delta_{\alpha}-\Delta_{\beta})+\\\
\\\
(\Delta^{2}_{\psi}+\Delta_{\psi})(5\Delta_{\psi}+1)(Q_{\alpha}-Q_{\beta})=0\;.\end{array}$
(2.20)

The equation (2.20) defines the fusion rule connecting dimensions
$\Delta_{\alpha,\beta}$ and the charges $Q_{\alpha,\beta}$ of the operators
$\mathcal{O}_{\alpha}$ and $\mathcal{O}_{\beta}$. It is important to
concentrate on the classical limit, when $\Delta$ and $Q$ are linear in the
central charge at $c\rightarrow\infty$, which can also be rewritten using
(2.4)

$\Delta=\frac{4\epsilon}{b^{2}},\qquad Q=\frac{4q}{b^{2}},\qquad b\rightarrow
0,$ (2.21)

where $\epsilon,q$ are called classical dimensions/charges respectively. In
order to analyze monodromy properties of the correlation functions with an
insertion of the operator $\psi_{3}$ we solve the equation (2.20) in terms of
a variable $\gamma=\Delta_{\beta}-\Delta_{\alpha}-\Delta_{\psi}$ at
$b\rightarrow 0$. The solutions are

$\begin{array}[]{c}\displaystyle\gamma_{1}=1-\frac{R^{1/3}}{3^{2/3}Q_{0}}-\frac{Q_{0}(1-4\epsilon_{\alpha})}{3^{1/3}R^{1/3}}\;,\\\
\\\
\displaystyle\gamma_{2,3}=1+\left(-\frac{1}{3}\right)^{1/3}\frac{Q_{0}(1-4\epsilon_{\alpha})}{R^{1/3}}+\left(\frac{1}{3}\right)^{1/3}\frac{(1\pm
i\sqrt{3})R^{1/3}}{6Q_{0}}\;,\end{array}$ (2.22)

where

$R=\sqrt{3}\sqrt{3Q_{0}^{4}\left(q_{\alpha}-q_{\beta}\right)^{2}-Q_{0}^{6}(1-4\epsilon_{\alpha})^{3}}+3\tilde{Q}^{2}_{\psi}(q_{\alpha}-q_{\beta})^{2}\;.$
(2.23)

and $Q^{2}_{0}=-2/45$.

The simplest fusion rule corresponds to the case $q_{\alpha}=q_{\beta}$ and
takes the form

$\begin{array}[]{c}\gamma_{1}=1\;,\qquad\gamma_{2,3}=\displaystyle\left(1\pm\sqrt{1-4\epsilon_{\alpha}}\right)\;.\end{array}$
(2.24)

Note that it does not depend on classical charges. In fact, fusion rules
(2.24) exactly reproduce ones corresponding to a singular operator $V_{(3,1)}$
of the Virasoro algebra (see Appendix A for details). Hence, the monodromy
properties of the $3$-pt function are governed by the Virasoro algebra despite
the fact that $q_{\alpha}\neq 0$.

##  3 $4$-pt $\mathcal{W}_{3}$ blocks: monodromy method

###  3.1 Classical conformal blocks and the monodromy problem

Here we consider $4$-pt $\mathcal{W}_{3}$ blocks in the classical limit within
the monodromy method.

#### Classical conformal blocks.

In conformal field theory with the $\mathcal{W}_{3}$ symmetry the $4$-pt
correlation function of primary operators $\mathcal{O}_{i},i=\overline{1,4}$
can be decomposed as follows

$\begin{array}[]{c}\displaystyle\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\mathcal{O}_{3}(z_{3})\mathcal{O}_{4}(z_{4})\rangle=\displaystyle\sum_{p\in
S}C^{p}_{12}C^{p}_{34}\mathcal{F}_{W}\left(z_{i}|\Delta_{i},Q_{i},\tilde{\Delta}_{p},\tilde{Q}_{p},c\right)\;,\\\
\\\
\mathcal{F}_{W}\left(z_{i}|\Delta_{i},Q_{i},\tilde{\Delta}_{p},\tilde{Q}_{p},c\right)\equiv\displaystyle\sum_{I}\displaystyle\frac{\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\mathcal{L}_{-I}|\tilde{\Delta}_{p},\tilde{Q}_{p}\rangle\langle\tilde{\Delta}_{p},\tilde{Q}_{p}|\mathcal{L}_{I}\mathcal{O}_{3}(z_{3})\mathcal{O}_{4}(z_{4})\rangle}{\langle\tilde{\Delta}_{p},\tilde{Q}_{p}|\mathcal{L}_{I}\mathcal{L}_{-I}|\tilde{\Delta}_{p},\tilde{Q}_{p}\rangle\;C^{p}_{12}C^{p}_{34}}\;,\\\
\\\
C^{p}_{12}=\langle\Delta_{1},Q_{1}|\mathcal{O}_{2}(1)|\tilde{\Delta}_{p},\tilde{Q}_{p}\rangle\;,\qquad
C^{p}_{34}=\langle\tilde{\Delta}_{p},\tilde{Q}_{p}|\mathcal{O}_{3}(1)|\Delta_{4},Q_{4}\rangle\;,\end{array}$
(3.1)

where
$\mathcal{F}_{W}\left(z_{i}|\Delta_{i},Q_{i},\tilde{\Delta},\tilde{Q},c\right)$
denotes a $4$-pt $\mathcal{W}_{3}$ block, $\mathcal{L}_{I}$ is given by (2.6)
and $C^{p}_{12},C^{p}_{34}$ are the structure constants. The block represents
the contribution of the conformal family of one primary operator
$\tilde{\mathcal{O}}$ to the $4$-pt correlation function. Unlike the Virasoro
blocks, the $\mathcal{W}_{3}$ conformal blocks are not completely determined
by the symmetry algebra and conformal dimensions/charges of external and
internal operators. More precisely, it was shown [10, 11] that ratios

$C_{n}=\frac{\langle\Delta_{1},Q_{1}|\mathcal{O}_{2}(1)(W_{-1})^{n}|\tilde{\Delta},\tilde{Q}\rangle}{\langle\Delta_{1},Q_{1}|\mathcal{O}_{2}(1)|\tilde{\Delta},\tilde{Q}\rangle}\;,\qquad\tilde{C}_{n}=\frac{\langle\tilde{\Delta},\tilde{Q}|(W_{1})^{n}\mathcal{O}_{3}(1)|\Delta_{4},Q_{4}\rangle}{\langle\tilde{\Delta},\tilde{Q}|\mathcal{O}_{3}(1)|\Delta_{4},Q_{4}\rangle}\;,$
(3.2)

where $n=1,2,...$ act as additional parameters on which the conformal block
depends. In other words, the conformal block depends on two arbitrary
functions, which correspond to two infinite sets of parameters (3.2). These
two functions govern the functional arbitrariness of the conformal block. Note
that in special cases this arbitrariness is fixed,333 The functional
arbitrariness is also absent for the so-called semi-degenerate external
operators , cf.[11, 9]. for example, for a vacuum block
($\tilde{\Delta}=\tilde{Q}=0$), all parameters (3.2) are equal to 0.

In the classical limit $c\rightarrow\infty$, assuming all conformal dimensions
and spin-$3$ charges are proportional to $c$, it has been argued that the
block exponentiates

$\lim_{c\to\infty}\mathcal{F}_{W}\left(z_{i}|\Delta_{i},Q_{i},\tilde{\Delta},\tilde{Q},c\right)=\exp\left[\frac{c}{6}f_{W}(z_{i}|\epsilon_{i},\tilde{\epsilon},q_{i},\tilde{q})\right]\;,$
(3.3)

where $f_{W}(z_{i}|\epsilon,\tilde{\epsilon},q,\tilde{q})$ stands for the
classical block. Classical dimensions/spin-3 charges (2.21) are assumed to be
finite in the limit $c\rightarrow\infty$. In what follows, we will omit the
dependence on these parameters in the classical block function.

The monodromy method for calculating the $\mathcal{W}_{3}$ block is formulated
by analogy to that for Virasoro blocks [12, 13, 14]. Let $\Psi_{W}(y,z_{i})$
be an auxiliary $5$-pt conformal block with the fully degenerate operator
$\psi_{3}(y)$ which corresponds to the null-vector $|\psi_{3}\rangle$. Since
in the classical limit $\Delta_{\psi},Q_{\psi}\sim 1$ while other classical
dimension and charges are linear in $c$, this block is factorized into a
product of the following form

$\Psi_{W}(y,z_{i})\Big{|}_{c\to\infty}\rightarrow\psi(y,z_{i})\exp\left[\frac{c}{6}f_{W}(z_{i})\right]\;.$
(3.4)

The function $\psi(y,z_{i})$ corresponds to the large-$c$ contribution of the
operator $\psi_{3}(y)$ into the classical conformal block. Due to the
condition (2.19c) the conformal block $\Psi_{W}(y,z_{i})$ satisfies the BPZ
equation [15], which can be rewritten as a differential equation for the
function $\psi(y,z_{i})$

$\left[\frac{d^{3}}{dy^{3}}+4T(y,z_{i})\frac{d}{dy}+\frac{2dT(y,z_{i})}{dy}-4W(y,z_{i})\right]\psi(y,z_{i})=0\;.$
(3.5)

The functions $T(z)$ and $W(z)$ read

$\begin{array}[]{c}\displaystyle
T(y,z_{i})=\sum^{4}_{i=1}\left(\frac{\epsilon_{i}}{(y-z_{i})^{2}}+\frac{c_{i}}{(y-z_{i})}\right)\;,\\\
\\\ \displaystyle
W(y,z_{i})=\sum^{4}_{i=1}\left(\frac{q_{i}}{(y-z_{i})^{3}}+\frac{a_{i}}{(y-z_{i})^{2}}+\frac{b_{i}}{(y-z_{i})}\right)~{},\end{array}$
(3.6)

where we use notations

$a_{i}=\lim_{c\to\infty}\frac{6}{c}\frac{W^{(i)}_{-1}\Psi(y,z_{i})}{\Psi(y,z_{i})}\;,\qquad
b_{i}=\lim_{c\to\infty}\frac{6}{c}\frac{W^{(i)}_{-2}\Psi(y,z_{i})}{\Psi(y,z_{i})};,\qquad
c_{i}=\partial_{i}f(z_{i})\;.$ (3.7)

A few comments are in order. First, (3.5) is the Fuchsian-type equation with
$4$ singular regular points. Second, assuming $q_{i}=0$ and $a_{i}=b_{i}=0$ we
see that the BPZ equation (3.5) reduces to the equation associated with the
degenerate operator $V_{(3,1)}$ of the Virasoro algebra (see Appendix A for
details). Third, the conditions (2.12) allow fixing out 8 of 12 parameters
$(a_{i},b_{i},c_{i})$. Here we leave $(c_{{}_{2}},a_{1},a_{2},b_{2})$
independent. Simultaneously setting
$(z_{1},z_{2},z_{3},z_{4})\rightarrow(1,z,0,\infty)$ the expressions (3.6)
become

$T(y,z)=\frac{\epsilon_{1}}{(y-1)^{2}}+\frac{\epsilon_{2}}{(y-z)^{2}}+\frac{\epsilon_{3}}{y^{2}}+\frac{c_{2}}{y-z}+\frac{c_{2}\left(z+y-1\right)+\epsilon_{2}+\epsilon_{3}}{(1-y)y}\;,$
(3.8) $\begin{array}[]{c}\displaystyle
W(y,z)=\frac{q_{1}}{(y-1)^{3}}+\frac{q_{2}}{\left(y-z\right){}^{3}}+\frac{q_{3}}{y^{3}}+\frac{a_{1}}{(y-1)^{2}}+\frac{a_{2}}{\left(y-z\right){}^{2}}+\frac{b_{2}}{y-z}\\\
\\\
\displaystyle-\frac{a_{2}\left(2z+y-1\right)+a_{1}(y+1)+b_{2}\left(z\left(z+y-1\right)+(y-1)y\right)+q_{1}+q_{2}+q_{3}+q_{4}}{(y-1)y^{2}}\;.\end{array}$
(3.9)

#### The monodromy problem for classical $\mathcal{W}_{3}$ blocks.

Let a string $\;\psi_{a}(y,z),\;a=1,2,3$ denote three independent solutions of
(3.5). These solutions of the Fuchsian-type differential equation have non-
trivial monodromy along contours enclosing singular points. Here we consider
the case the contour $\Gamma$ bypasses the points $(1,z)$. The corresponding
monodromy matrix $M_{ab}(\Gamma|z)$ is defined by

$\psi_{a}(\Gamma\circ y,z)=\psi_{b}(y,z)M_{ab}(\Gamma|z)\;,$ (3.10)

where $\Gamma\circ y$ denotes a traversal along the contour $\Gamma$. The
matrix depends on the functions $(c_{{}_{2}},a_{1},a_{2},b_{2})$ as
parameters.

On the other hand, the main contribution to the monodromy of the auxiliary
$5$-pt block along the contour $\Gamma$ comes from the term
$(z-y)^{\Delta_{\beta}-\Delta_{\alpha}-\Delta_{\psi}}$, where
$\Delta_{\alpha}$ and $\Delta_{\beta}$ are intermediate dimensions of the
$5$-pt block. Hence the solutions must have the monodromy matrix

$\tilde{M}_{ab}(\Gamma)=\exp[2\pi i\gamma_{a}]\delta_{ab}\;,$ (3.11)

in a diagonal basis. The coefficients $\gamma_{a}$ are given by (2.22). In
what follows, we always use the fusion rules (2.24) for simplicity.

The monodromy problem is to find solutions of the equation (3.5) with the
monodromy matrix (3.11). In other words, the eigenvalues of the monodromy
matrices $M_{ab}$ in (3.10) and $\tilde{M}_{ab}$ in (3.11) must coincide. This
imposes algebraic restrictions on the parameters
$(c_{{}_{2}},a_{1},a_{2},b_{2})$, the so-called monodromy equations. Solving
these equations and integrating the accessory parameter
$c_{{}_{2}}=\partial_{z}f(z)$, we obtain the classical $\mathcal{W}_{3}$
block. It is important to mention that monodromy equations are generally not
enough to determine all 4 parameters, which is consistent with the functional
arbitrariness of the block.

###  3.2 Solution of the monodromy problem

#### HL approximation.

Since the solutions of the equation (3.5) with tensors (3.8) and (3.9) are not
generally known, we consider the heavy-light (HL) approximation. More
precisely, we assume that the classical dimensions and charges of the
operators $\mathcal{O}_{1}$ and $\mathcal{O}_{4}$ are much larger than the
remaining ones

$\begin{array}[]{c}\epsilon_{1,2},\tilde{\epsilon}\ll\epsilon_{H}\;,\qquad
q_{1,2}\ll q_{H}\;.\end{array}$ (3.12)

By virtue of (2.15), the classical dimensions and $3$-charges of heavy
operators are related by $\epsilon_{3}=\epsilon_{4}$ and $q_{3}=-q_{4}$, and
henceforth will be denoted by $\epsilon_{H},q_{H}$ respectively. For the sake
of simplicity, we set $\epsilon_{1}=\epsilon_{2}\equiv\epsilon$ and
$q_{1}=-q_{2}\equiv q$.

Now, we implement the heavy-light expansion

$\begin{array}[]{c}\psi(y,z)=\psi^{(0)}(y)+\psi^{(1)}(y,z)+...\;,\qquad
T(y,z)=T^{(0)}(y)+T^{(1)}(y,z)+...\;,\\\ \\\
W(y,z)=T^{(0)}(y)+W^{(1)}(y,z)+...\;,\qquad
f(z)=f^{(0)}(z)+f^{(1)}(z)+...\,,\end{array}$ (3.13)

where zeroth-order parameters $a^{(0)}_{1,2},b^{(0)}_{2},c^{(0)}_{2}$
associated with light operators and the classical block $f^{(0)}(z)$ are equal
to 0. Because of this, we omit the superscript of the parameters, always
referring to them in the first order of the HL approximation.

The monodromy matrix along contour $\Gamma$ is also expanded up to the first
order in the HL approximation

$M_{ab}(\Gamma|z)=M_{ab}^{(0)}(\Gamma)+M_{ab}^{(1)}(\Gamma|z)+...\;.$ (3.14)

#### Monodromy matrix in the zeroth order.

In zeroth order, the equation (3.5) has the form

$D^{(0)}\psi^{(0)}(y)=0\;,\qquad
D^{(0)}=\left[\frac{d^{3}}{dy^{3}}+4T^{(0)}\frac{d}{dy}+\frac{2dT^{(0)}}{dy}-4W^{(0)}\right]\;,$
(3.15)

where

$T^{(0)}=\frac{\epsilon_{H}}{y^{2}}\;,\qquad W^{(0)}=\frac{q_{H}}{y^{3}}\;.$
(3.16)

There are three branches of solutions

$\psi_{a}^{(0)}(y)=y^{1+p_{a}},\qquad a=1,2,3\;,$ (3.17)

where the exponents satisfy the characteristic equation of the third degree

$p^{3}_{a}-\alpha^{2}p_{a}-4q_{H}=0\;,\qquad\alpha=\sqrt{1-4\epsilon_{H}}\;.$
(3.18)

In order not to clutter up the calculations, we do not write explicit
expressions for the roots of $p_{a}$. However due to Vieta’s theorem for the
cubic equation the exponents $p_{i}$ are satisfied by several relations, which
are used below. For instance, $p_{1}+p_{2}+p_{3}=0$ and
$p_{1}p_{2}p_{3}=4q_{H}$.

The contour $\Gamma$ does not encircle the branch points of the solutions
(3.17) hence the monodromy matrix in the zeroth order is given by the identity
matrix, $M_{ab}^{(0)}(\Gamma)=\delta_{ab}$. The monodromy matrix (3.11) in the
zeroth order of the HL approximation coincides with $M_{ij}^{(0)}(\Gamma)$.

#### First-order monodromy matrix and monodromy equations.

In the first order of the HL approximation, we have

$D^{(0)}\psi^{(1)}(y)=-D^{(1)}\psi^{(0)}(y)\;,\qquad
D^{(1)}=\left[4T^{(1)}\frac{d}{dy}+\frac{2dT^{(1)}}{dy}-4W^{(1)}\right]\;,$
(3.19)

where

$T^{(1)}=\frac{c_{2}(z-1)+2\epsilon}{y-1}-\frac{c_{2}z+2\epsilon}{y}+\frac{c_{2}}{y-z}+\frac{\epsilon}{(z-y)^{2}}+\frac{\epsilon}{(y-1)^{2}}\;,$
(3.20) $\begin{array}[]{c}\displaystyle
W^{(1)}=\frac{q}{(y-1)^{3}}-\frac{q}{\left(y-z\right){}^{3}}+\frac{a_{1}}{(y-1)^{2}}+\frac{a_{2}}{\left(y-z\right){}^{2}}+\frac{b_{2}}{y-z}\\\
\\\
\displaystyle\frac{\left(2a_{2}z-a_{2}-a_{1}+b_{2}z(z-1)\right)(z-1)}{y^{2}}-\frac{2a_{2}\left(z-1\right)-2a_{1}+b_{2}\left(z-1\right){}^{2}}{y-1}\;.\end{array}$
(3.21)

Note that equation (3.19) is the inhomogeneous differential equation with the
same left hand side as (3.15). Therefore, the first-order solutions can be
expressed in terms of zeroth-order solutions (3.17) and take the form [2]

$\psi^{(1)}_{a}(y,z)=\psi_{b}^{(0)}(y)T_{ab}(y,z)\;,$ (3.22)

where

$T_{ab}=2\begin{pmatrix}\displaystyle\frac{z\left(2zW^{(1)}-2p_{1}T^{(1)}\right)}{p_{12}p_{13}}\;\;&\displaystyle\frac{{}^{1-p_{12}}\left(p_{3}T^{(1)}+2W^{(1)}z\right)}{p_{12}p_{13}}\;\;&\displaystyle\frac{2z^{1-p_{13}}\left(p_{2}T^{(1)}+2W^{(1)}z\right)}{p_{12}p_{23}}\\\
\\\
\displaystyle-\frac{2z^{1+p_{12}}\left(p_{3}T^{(1)}+2W^{(1)}z\right)}{p_{12}p_{23}}\;\;&\displaystyle-\frac{z\left(2zW^{(1)}-2p_{2}T^{(1)}\right)}{p_{12}p_{13}}\;\;&\displaystyle-\frac{2z^{1-p_{23}}\left(p_{1}T^{(1)}+2W^{(1)}z\right)}{p_{12}p_{23}}\\\
\\\
\displaystyle\frac{2z^{1+p_{13}}\left(p_{2}T^{(1)}+2W^{(1)}z\right)}{p_{13}p_{23}}\;\;&\displaystyle\frac{2z^{1+p_{23}}\left(p_{1}T^{(1)}+2W^{(1)}z\right)}{p_{13}p_{23}}\;\;&\displaystyle\frac{z\left(2zW^{(1)}-2p_{3}T^{(1)}\right)}{p_{13}p_{23}}\end{pmatrix}\,,$
(3.23)

where we introduce $p_{ij}\equiv p_{i}-p_{j}$. Hence the monodromy matrix
$M^{(1)}_{ab}$ has the form

$M^{(1)}_{ab}=\int_{\Gamma}dyT_{ab}\equiv I_{ab}\;.$ (3.24)

Calculating the contour integrals using residues, one can obtain the elements
of the monodromy matrix $M^{(1)}_{ab}$. In particular, the diagonal elements
turn out to be $M^{(1)}_{aa}=0,\;a=1,2,3$. Other elements are linear functions
of $a_{1,2},b_{2}$ and $c_{2}$ which we have the form

$\begin{array}[]{c}\displaystyle\frac{I_{12}}{2\pi
i}=\displaystyle\frac{2z^{-p_{12}}\left(z\left(2a_{2}\left(2-p_{12}\right)+c_{2}p_{3}\right)+2b_{2}z^{2}-\left(p_{12}-1\right)\left(p_{3}\epsilon+\left(p_{12}-2\right)q\right)\right)}{p_{12}p_{13}}\\\
\\\
\displaystyle-\frac{2\left(p_{12}\left(2a_{1}+p_{3}\epsilon+\left(3-p_{12}\right)q\right)+4a_{2}z+2b_{2}z^{2}+c_{2}p_{3}z+p_{3}\epsilon-2q\right)}{p_{12}p_{13}}\;,\quad
I_{13}=I_{12}\left(p_{3}\leftrightarrow p_{2}\right)\;,\\\ \\\
\displaystyle\frac{I_{21}}{2\pi i}=-\frac{2p_{12}\left(2a_{1}+\epsilon
p_{3}+\left(p_{12}+3\right)q\right)-4a_{2}z-2b_{2}z^{2}-c_{2}p_{3}x_{2}-\epsilon
p_{3}+2q}{p_{12}p_{23}}-\\\
\displaystyle\frac{2\left(z^{p_{12}}\left(z\left(2a_{2}\left(p_{12}+2\right)+c_{2}p_{3}\right)+2b_{2}z^{2}+\left(p_{12}+1\right)\left(\epsilon
p_{3}-\left(p_{12}+2\right)q\right)\right)\right)}{p_{12}p_{23}}\;,\quad
I_{31}=I_{21}(p_{2}\leftrightarrow p_{3})\;,\\\ \\\
\displaystyle\frac{I_{32}}{2\pi
i}=\frac{2\left(z\left(4a_{2}+2b_{2}z+c_{2}p_{1}\right)-p_{23}\left(2a_{1}+\left(p_{23}+3\right)q\right)+\epsilon
p_{1}\left(1-p_{23}\right)-2q\right)}{p_{23}p_{13}}\\\
-\displaystyle\frac{z\left(2a_{2}\left(p_{23}+2\right)+c_{2}p_{1}\right)+2b_{2}z^{2}+\left(p_{23}+1\right)\left(\epsilon
p_{1}-\left(p_{23}+2\right)q\right)}{p_{23}p_{13}}\;,\quad
I_{23}=I_{32}(p_{1}\leftrightarrow p_{3})\;.\end{array}$ (3.25)

Since the monodromy matrix of zero-order solutions is given by the identity
matrix, in the first order the eigenvalues of the monodromy matrix
$M^{(1)}_{ab}$ are given

$\tilde{M}^{(1)}_{ab}=\gamma^{(1)}_{a}\delta_{ab}\;,\qquad\gamma^{(1)}_{1,2}=\pm
4\pi i\tilde{\epsilon}\;,\qquad\gamma^{(1)}_{3}=0\;.$ (3.26)

Eigenvalues of the monodromy matrix $M^{(1)}_{ab}$ should be equal to (3.26).
It yields the monodromy equations of the following form

$I_{12}I_{23}I_{31}+I_{13}I_{21}I_{32}=0\;,\qquad
I_{12}I_{21}+I_{13}I_{31}+I_{23}I_{32}=-16\pi^{2}\tilde{\epsilon}^{2}\;.$
(3.27)

A few comments are in order. First, the system (3.27) consists of two
equations for 4 parameters. Thus, the solution is paremetrized by two
functions which is a manifestation of the functional arbitrariness discussed
in Section 3.1. Second, the monodromy matrix of the $4$-pt vacuum block
($\tilde{\epsilon}=0$) in the first order of the HL approximation is given by
the zero matrix so we have 6 equations for 4 parameters. However, as shown in
[2], not all of them are independent and in the case of the vacuum $4$-pt
block the corresponding monodromy system has one solution. Integrating the
accessory parameter $c_{2}$ one finds the block function [2, 16]

$\begin{array}[]{c}f^{(v)}_{W}(z)=\displaystyle\frac{\left(3q-\epsilon\right)}{2}\log\left(p_{3}\left(2z^{2p_{2}+p_{3}}-z^{p_{2}+2p_{3}}-1\right)+p_{2}\left(z^{2p_{2}+p_{3}}-2z^{p_{2}+2p_{3}}+1\right)\right)-\epsilon(1+p_{1})\log
z\\\ \\\
-\displaystyle\frac{\left(\epsilon+3q\right)}{2}\left(-\log\left(p_{3}\left(z^{p_{2}}-2z^{p_{3}}+z^{2\left(p_{2}+p_{3}\right)}\right)-p_{2}\left(-2z^{p_{2}}+z^{p_{3}}+z^{2\left(p_{2}+p_{3}\right)}\right)\right)\right)\;.\end{array}$
(3.28)

###  3.3 $4$-pt non-vacuum $\mathcal{W}_{3}$ blocks

There are several ways to exclude functional arbitrariness beyond known cases.
For example, one can fix the parameters $a_{2}=0,\;b_{2}=0$ and solve (3.27)
for $c_{2}$ and $a_{1}$.444More precisely, any pair of parameters can be fixed
equal to 0, trivial configurations correspond to $c^{(n)}_{2}=0$.  Since the
integrals $I_{ab}$ are linear in $a_{1}$ and $c_{2}$, the monodromy system
contains one third-order equation and one second-order equation. However, the
solution of these equations cannot be found in radicals.

In this section, we consider another way of capturing functional
arbitrariness. Since for the vacuum block parameters $c_{2},a_{1,2},b_{2}$ are
known, we consider the following ansatz for the $4$-pt non-vacuum block

$c_{2}=c^{(v)}_{2}+c^{(n)}_{2}\;,\qquad a_{1,2}=a^{(v)}_{1,2}\;,\qquad
b_{2}=b^{(v)}_{2}\;,$ (3.29)

where for the accessory parameter $c_{2}$ the superscript $v$ refers to the
vacuum block and $n$ to the non-vacuum block. Fixing the functional
arbitrariness in this way, we assume that the intermediate operator with the
classical dimension $\tilde{\epsilon}$ contributes only to the classical
conformal block.

Despite such a fixation of three parameters instead of two, one can see that
under (3.29) the first equation in (3.27) becomes trivial. The second equation
is a quadratic equation in $c^{(n)}_{2}$ and its solution with the correct
asymptotic takes the form

$c^{(n)}_{2}=\frac{\tilde{\epsilon}\displaystyle}{2z}\prod_{1\leq i\leq j\leq
3}p^{-1/2}_{ij}\left(\frac{p^{2}_{3}\left(z^{-p_{12}/2}-z^{p_{12}/2}\right)^{2}}{p_{12}}-\frac{p^{2}_{2}\left(z^{-p_{13}/2}-z^{p_{13}/2}\right)^{2}}{p_{13}}-\frac{p^{2}_{1}\left(z^{-p_{23}/2}-z^{p_{23}/2}\right)^{2}}{p_{23}}\right)^{-\frac{1}{2}},$
(3.30)

where $p_{i}$ are defined by (3.18). Although we have not calculated this
integral analytically, one can expand in $q_{H}/\alpha^{2}\ll 1$. The
exponents $p_{i}$ have the form

$p_{1}=\frac{4q_{H}}{\alpha^{2}}+...\;,\qquad
p_{2,3}=\pm\alpha+\frac{2q_{H}}{\alpha^{2}}+...\;,$ (3.31)

and $f^{(n)}_{W}$ reads

$f^{(n)}_{W}=f_{0}+\frac{q^{2}_{H}}{4\alpha^{4}}f_{2}+...\;,$ (3.32)

where

$\begin{array}[]{c}f_{0}=2\tilde{\epsilon}\text{Arcth}\left(z^{\alpha/2}\right)\;,\\\
\\\
f_{2}=z^{a/2}\left(3\alpha\log\left(z\right)\left(\displaystyle\frac{28(1-z^{\alpha})+3\alpha\log
z}{\left(1-z^{\alpha}\right){}^{2}}-36\,_{3}F_{2}\left(\frac{1}{2},\frac{1}{2},1;\frac{3}{2},\frac{3}{2};z^{\alpha}\right)\right)+27\Phi\left(z^{\alpha},3,\frac{1}{2}\right)+8\right)+\\\
\\\ \displaystyle
27\alpha^{2}\log^{2}z\;\text{Arcth}\left(z^{\alpha/2}\right)+8z^{-\frac{\alpha}{2}}+\frac{9\alpha^{2}z^{\frac{3\alpha}{2}}\log^{2}z}{\left(z^{\alpha}-1\right){}^{2}}\;.\end{array}$
(3.33)

Here $\Phi(x,s,a)$ denotes the Lerch transcendent function (see [17] for
review)

$\Phi(x,s,a)=\sum_{k=0}^{\infty}\frac{x^{k}}{(a+k)^{s}}\;.$ (3.34)

##  4 Conclusion

In this work, the monodromy method for the $4$-pt non-vacuum classical block
associated with the $\mathcal{W}_{3}$ algebra was considered. Using the fusion
rules for the auxiliary $5$-pt block, we have obtained monodromy equations
that partially define the $4$-pt classical block. Having fixed the functional
arbitrariness in the way mentioned above, we have found the contribution of
the intermediate operator to the block. We also studied the reduction of this
monodromy method to the Virasoro algebra and showed that the results obtained
within its framework go over to the well-known expressions for the $4$-pt non-
vacuum block.

It would be of interest to apply the monodromy method to conformal blocks,
where the functional arbitrariness is fixed by the choice of special primary
operators [7, 5] or within the framework of the isomonodromic approach [18].
In this work, we did not touch upon the holographic interpretation of the
results obtained. It would be interesting to study dual objects in terms of
Wilson lines and establish a correspondence similar to the examples given in
[19].

Acknowledgements. The work was supported by the Russian Science Foundation
grant 18-72-10123.

##  Appendix A $4$-pt Virasoro block via $V_{(3,1)}$

Here we elaborate the monodromy method associated with the degenerate
representation $V_{(3,1)}$ of the Virasoro algebra.

#### Fusion rules.

The null-vector of the representation $V_{(3,1)}$ is given by

$|\psi_{3v}\rangle=\left(L_{-3}-\frac{2}{\Delta+2}L_{-1}L_{-2}+\frac{1}{(\Delta+1)(\Delta+2)}L^{3}_{-1}\right)|\Delta\rangle=0\;,$
(A.1)

where $\Delta=-1-\displaystyle\frac{12}{c}$ at $c\rightarrow\infty$.
Considering the $3$-pt function $V_{\alpha\beta\psi_{{}_{3v}}}=0$ with the
operator $\psi_{{}_{3v}}$ corresponding to (A.1), we obtain the following
fusion rule

$(\Delta_{\alpha}-\Delta_{\beta})(\Delta^{2}_{\alpha}-(\Delta_{\beta}-\Delta)(1-\Delta)-\Delta_{\alpha}(1+\Delta_{\beta}+\Delta))=0\;.$
(A.2)

Solving this equation with respect to the variable
$\gamma=\Delta_{\beta}-\Delta_{\alpha}-\Delta_{\psi}$ in the classical limit,
we find

$\gamma_{1}=1\;,\qquad\gamma_{2,3}=\displaystyle\left(1\pm\sqrt{1-4\epsilon_{\alpha}}\right)\;.$
(A.3)

#### BPZ equation.

The BPZ equation associated with (A.1) has the following form

$\left[\frac{d^{3}}{dy^{3}}+4T(y,z_{i})\frac{d}{dy}+\frac{2dT(y,z_{i})}{dy}\right]\psi(y,z_{i})=0\;,$
(A.4)

where $T(y,z_{i})$ is defined in the first line of (3.6). Using the HL
approximation described in Section 3.2, we see that the zeroth-order equation
takes the form

$\left[\frac{d^{3}}{dy^{3}}+\frac{4\epsilon_{H}}{y^{2}}\frac{d}{dy}-\frac{4\epsilon_{H}}{y^{3}}\right]\psi^{(0)}(y)=0\;,$
(A.5)

and solutions are

$\psi_{1,2}^{(0)}(y)=y^{1\pm\alpha},\qquad\psi_{3}^{(0)}(y)=y\;.$ (A.6)

Note that this equation is closely related to the previously studied BPZ
equation, which is associated with the degenerate operator $V_{(2,1)}$ [13,
20, 21]. In the zeroth order, the equation and its solutions read

$\left[\frac{d^{2}}{dy^{2}}+\frac{\epsilon_{H}}{y^{2}}\right]\phi^{(0)}(y)=0\;,\qquad\phi^{(0)}(y)=y^{\frac{1\pm\alpha}{2}}\;.$
(A.7)

We see that the solutions (A.6) can be constructed as quadratic combinations
of solutions (A.7).

Due to the role of zero-order solutions, the monodromy properties of the
equation (A.5) are governed by ones of the equation (A.7). More precisely, the
monodromy matrix of the first-order solutions of (A.4) has the form

$I_{ab}=\frac{2\pi i}{\alpha}\begin{pmatrix}\displaystyle 0\;\;&\displaystyle
2I_{1}\;\;&\displaystyle-2I_{2}\\\ \displaystyle-I_{2}\;\;&\displaystyle
0\;\;&\displaystyle 1\\\ \displaystyle I_{1}\;\;&\displaystyle
0\;\;&0\end{pmatrix}\,,\qquad\alpha=\sqrt{1-4\epsilon_{H}}\;,$ (A.8)

where

$I_{1}=c_{2}z\left(z^{\alpha}-1\right)+\epsilon\left((\alpha+1)z^{\alpha}+\alpha-1\right)\;,\qquad
I_{2}=I_{1}(\alpha\rightarrow-\alpha)\;.$ (A.9)

Comparing the eigenvalues of this matrix and (A.3) with
$\epsilon_{\alpha}=\tilde{\epsilon}$ at $\tilde{\epsilon}\ll\epsilon_{H}$, we
obtain one non-trivial monodromy equation

$I_{1}I_{2}=-\alpha^{2}\tilde{\epsilon}^{2}\;,$ (A.10)

which can be solved with respect to $c_{2}$. After integrating the accessory
parameter, we get

$f(z)=2\tilde{\epsilon}\text{Arcth}\left(z^{\alpha/2}\right)-2\epsilon\log\left(1-z^{\alpha}\right)+\epsilon(\alpha-1)\log(z)\;,$
(A.11)

what constitutes the $4$-pt non-vacuum Virasoro block.

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