aainstitutetext: I.E. Tamm Department of Theoretical Physics,
P.N. Lebedev Physical Institute, 119991 Moscow, Russiabbinstitutetext:
Institute for Theoretical and Mathematical Physics,
Lomonosov Moscow State University, 119991 Moscow, Russia

# Torus conformal blocks and Casimir equations
in the necklace channel

Konstantin Alkalaev a Semyon Mandrygin a Mikhail Pavlov<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the conformal block decomposition in arbitrary exchange channels
of a two-dimensional conformal field theory on a torus. The channels are
described by diagrams built of a closed loop with external legs (a necklace
sub-diagram) and trivalent vertices forming trivalent trees attached to the
necklace. Then, the $n$-point torus conformal block in any channel can be
obtained by acting with a number of OPE operators on the $k$-point torus block
in the necklace channel at $k=1,...,n$. Focusing on the necklace channel, we
go to the large-$c$ regime, where the Virasoro algebra truncates to the
$sl(2,\mathbb{R})$ subalgebra, and obtain the system of the Casimir equations
for the respective $k$-point global conformal block. In the plane limit, when
the torus modular parameter $q\to 0$, we explicitly find the Casimir equations
on a plane which define the $(k+2)$-point global conformal block in the comb
channel.

##  1 Introduction

Correlation functions of conformal field theory (CFT) associated with a
particular algebra of local/global conformal symmetries can be decomposed into
conformal blocks which are completely fixed by the conformal symmetry. The
conformal blocks are instrumental in the conformal/modular bootstrap programs
[1, 2] (for recent developments see e.g. [3, 4]). The deep role of conformal
blocks was revealed in the context of AdS/CFT correspondence, where they are
dual to lengths of geodesic diagrams stretched in the bulk [5, 6, 7, 8, 9, 10,
11].

Two-dimensional conformal field theories (CFT2) on Riemannian surfaces are
interesting from different perspectives. The original motivation comes from
the study of critical phenomena in statistical physics and string theory
amplitudes, see e.g. [12, 13, 14]. The first cases of Riemannian surfaces –
sphere $\mathbb{S}^{2}$ and torus $\mathbb{T}^{2}$ – are both ubiquitous and
rather simple. On the other hand, while the sphere (plane) CFT2 is quite
well–studied, this is not so for the torus CFT2, especially, when it comes to
multipoint correlation functions and corresponding conformal blocks.111For a
general formulation of CFT2 on Riemannian surfaces see e.g. [13, 14].
Literature on multipoint ($n\geq 2$) torus blocks is rather scarce: $n$-point
torus blocks in the necklace channel were studied within the recursive
representation [15]; various large-$c$ torus blocks were studied from
AdS3/CFT2 perspective in [16, 17, 18, 19]. The large-$c$ regime can
significantly simplify the form of various conformal functions which are often
not even known for arbitrary values of the conformal dimensions and the
central charge. On the other hand, by the Brown-Henneaux formula $c\sim
1/G_{N}$, a large-$c$ CFT2 is holographically dual to AdS3 quantum gravity in
the semiclassical approximation [20].

In the torus CFT2 already $1$-point conformal block is not known in a closed
from. The global 1-point torus block (which is associated to the
$sl(2,\mathbb{R})\subset Vir$ subalgebra) is found to be a hypergeometric
function [21]. As in any CFT with a finite-dimensional symmetry group, the
global blocks can be equivalently described as eigenfunctions of the Casimir
operators associated with irreducible conformal families in the exchange
channels [22, 23]. In the case of the torus CFT2 the study of the Casimir
approach was initiated in [16], where the Casimir equation for the 1-point
torus block was found to be a 2nd order differential (hypergeometric) equation
in the modular parameter.222This result naturally extends to $n$-point blocks
in the so-called OPE channel [16]. The same technique was used to compute
thermal 1-point conformal blocks in CFTd [24] and torus 1-point
$\mathcal{N}=1$ superblocks [25].

In this paper, we analyze multipoint torus blocks in general channels with a
focus on the so-called necklace channel. This channel is represented by a
closed loop with a number of external legs and plays the role of a building
block of any other torus diagram. Going to the large-$c$ limit, we formulate
the Casimir equations for global torus blocks in the necklace channel.

Note that from the plane CFT2 perspective, the $n$-point torus block can be
understood by gluing the endpoints of the plane $(n+2)$-point block. In the
plane limit, when the torus modular parameter $q\rightarrow 0$, the leading
asymptotics of the torus block in the necklace channel defines the plane block
in the comb channel (see, e.g. [15]).333Multipoint blocks in the comb channel
were explicitly computed in [26] (see also [27]). We show that in this case
the limiting Casimir torus equations coincide with the Casimir plane equations
and, therefore, find explicit realization of the Casimir operators of the
plane CFT2.

The paper is organized as follows. In the beginning of Section 2 we fix our
notation and conventions related to the torus CFT2. In Section 2.1 we review
multipoint torus correlation functions. In Section 2.2 we consider the general
structure of the conformal block decomposition and introduce a classification
of torus diagrams by a number of external legs of the necklace sub-diagram.
Recognizing the necklace sub-diagram as the basic constituent, any multipoint
torus block can be built by acting with a number of OPE operators on the torus
block in the necklace channel with less points. Section 3 considers the global
torus blocks. In Section 3.1 we formulate the global $n$-point torus blocks
and further in Section 3.2 we describe in more detail how to obtain torus
blocks in general channels by acting with $sl(2,\mathbb{R})$ OPE operators on
the necklace global block. In Section 3.3 we present the system of the Casimir
equations for $n$-point global torus blocks in the necklace channel and
discuss their general properties. In particular, by change of coordinates, the
torus Casimir equations are conveniently expressed in terms of the counting
operators. In Section 4 we obtain the Casimir equations in the plane limit
$q\rightarrow 0$. Section 5 summarizes our results and discusses future
perspectives. In Appendix A we explicitly derive the 2-point Casimir equations
and consider the plane limit as well as the reduction to the 1-point case.

##  2 Torus conformal blocks

A two-dimensional torus $\mathbb{T}^{2}$ can be defined as
$\mathbb{T}^{2}\simeq\mathbb{R}^{2}/{\mathbb{L}}$, where $\mathbb{R}^{2}$ is a
two-dimensional Euclidean plane and $\mathbb{L}=\mathbb{Z}+\tau\mathbb{Z}$ is
a lattice generated by two fundamental periods $1,\tau\in\mathbb{R}^{2}$. The
modular invariant characterization of $\mathbb{T}^{2}$ is achieved in terms of
the modulus $\tau$ taking values in the fundamental domain
$\subset\mathbb{H}$. In local coordinates $(w,\bar{w})\in\mathbb{R}^{2}$ the
torus is therefore realized through the identifications $w\sim w+1$ and $w\sim
w+\tau$. Equivalently, introducing coordinates $z=e^{2\pi iw}$, the
identification reads $z\sim qz$, where $q=e^{2\pi i\tau}$, $q\bar{q}\leq 1$,
is the modular parameter. In the latter case, $\mathbb{T}^{2}$ can be viewed
as an annulus with identified and rotated boundaries. The above quotient
parameterization is extremely useful when considering the torus CFT2. It
follows that the basic techniques of the plane CFT2 can be directly used
provided that the double periodicity conditions are imposed on local operators
in $w$-coordinates or the scaling conditions in $z$-coordinates.

Local conformal symmetries in the torus CFT2 are governed by Virasoro algebra
$Vir\oplus\bar{V}ir$ spanned by $L_{m},\bar{L}_{n},m,n\in\mathbb{Z}$, the
central charge is $c$:
$[L_{m},L_{n}]=(m-n)L_{m+n}+(c/12)m(m^{2}-1)\delta_{m+n,0}$ plus the same
anti-chiral commutation relations. The only global symmetry is translational,
$u(1)\oplus u(1)\subset Vir\oplus\bar{V}ir$, which is generated by $L_{0}$ and
$\bar{L}_{0}$. The Verma module $\mathcal{V}_{h,c}$ is spanned by basis
descendants $|M,h\rangle=L_{-k_{1}}^{i_{1}}\cdots
L_{-k_{n}}^{i_{n}}|h\rangle$, where $|h\rangle$ is a highest-weight vector of
conformal dimension $h$ and $|M|=i_{1}k_{1}+\cdots+i_{n}k_{n}$ with $1\leq
k_{1}\leq\cdots\leq k_{n}$ denotes a level. The Gram matrix in
$\mathcal{V}_{h,c}$ is defined by the inner product of elements on each level
by $B_{M|N}=\langle h,M|N,h\rangle$.

###  2.1 Correlation functions

The Hilbert space of states is
$\mathcal{H}=\bigoplus_{D}\mathcal{V}_{h,c}\otimes\bar{\mathcal{V}}_{\bar{h},c}\,$,
where conformal dimensions $h,\bar{h}\in D$ and the domain $D$ depends on a
particular theory. The partition function on $\mathbb{T}^{2}$ is given by the
following trace over $\mathcal{H}$:

$Z=\operatorname{Tr}_{\mathcal{H}}\big{(}q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{c}{24}}\big{)}\,.$
(2.1)

Then, torus CFT2 (normalized) correlation functions of $n$ primary local
operators $\tilde{\mathcal{O}}_{h_{i},\bar{h}_{i}}(w_{i},\bar{w}_{i})\\\
\equiv\tilde{\mathcal{O}}_{i}(w_{i},\bar{w}_{i})$ with conformal dimensions
$(h_{i},\bar{h}_{i})$ are defined as

$\big{\langle}\tilde{\mathcal{O}}_{1}(w_{1},\bar{w}_{1})\cdots\tilde{\mathcal{O}}_{n}(w_{n},\bar{w}_{n})\big{\rangle}=\frac{(q\bar{q})^{-\frac{c}{24}}}{Z}\operatorname{Tr}_{\mathcal{H}}\Big{(}q^{L_{0}}\bar{q}^{\bar{L}_{0}}\tilde{\mathcal{O}}_{1}\left(w_{1},\bar{w}_{1}\right)\ldots\tilde{\mathcal{O}}_{n}\left(w_{n},\bar{w}_{n}\right)\Big{)}\,.$
(2.2)

For further analysis of the conformal block decomposition the prefactor in
front of the trace is inessential and can be dropped out. By the same reason,
we will suppress anti-chiral dependence. Furthermore, it is more convenient to
use $z$-coordinates so the correlation functions (2.2) take the form

$\big{\langle}\mathcal{O}_{1}(z_{1})\cdots\mathcal{O}_{n}(z_{n})\big{\rangle}=(2\pi
i)^{-\sum_{i=1}^{n}h_{i}}\,\Big{[}\prod_{i=1}^{n}z_{i}^{-h_{i}}\Big{]}\,\big{\langle}\tilde{\mathcal{O}}_{1}(w_{1}(z_{1}))\cdots\tilde{\mathcal{O}}_{n}(w_{n}(z_{n}))\big{\rangle}\,,$
(2.3)

where $\mathcal{O}_{i}(z_{i})$ are related to $\tilde{\mathcal{O}}_{i}(w_{i})$
by the conformal map, so that the difference between two parameterizations is
given by the power law prefactor.

The Virasoro symmetry acts on primary operators $\mathcal{O}_{h}(z)$ in the
standard fashion as444Note that $\mathcal{L}_{n}$ form the algebra with
opposite sign structure constants, i.e.
$[\mathcal{L}_{m},\mathcal{L}_{n}]=-(m-n)\mathcal{L}_{m+n}$.

$[L_{m},\mathcal{O}_{h}(z)]=\mathcal{L}_{m}\mathcal{O}_{h}(z)\,,\qquad\mathcal{L}_{m}=z^{m}\left[z\partial_{z}+h(m+1)\right]\,.$
(2.4)

It constrains the torus correlation functions (2.2) to satisfy the Ward
identities which generalize those on the plane and reproduce the standard OPE
with the stress tensor [14]. The global Ward identity which is associated with
the (chiral) translational $u(1)$ symmetry reads

$\sum_{i=1}^{n}\mathcal{L}_{0}^{(i)}\,\big{\langle}\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{i}(z_{i})\ldots\mathcal{O}_{n}(z_{n})\big{\rangle}=0\,.$
(2.5)

In particular, 1-point functions are obviously coordinate independent though
still have non-trivial modular covariance property [28] that stems from the
modular invariance of the partition function (2.1). In the $n$-point case, the
translational invariance fixes the form of correlation functions up to a leg
factor $\mathbb{L}$ thereby making it an arbitrary function $\mathbb{V}$ of
$u(1)$–invariant coordinate combinations $x_{i}=x_{i}(z_{i}/z_{i-1})$,
$i=2,...,n$. Namely,

$\big{\langle}\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{n}(z_{n})\big{\rangle}=\mathbb{L}({\bf
z})\mathbb{V}(q,{\bf x})\,,\qquad\mathbb{L}({\bf
z})=\prod_{i=1}^{n}z_{i}^{-h_{i}}\,,$ (2.6)

where ${\bf z}=\\{z_{1},\ldots,z_{n}\\}$ and ${\bf x}=\\{x_{2},\ldots
x_{n}\\}$. The above leg factor is somewhat canonical but can be redefined by
multiplying by a function of ${\bf x}$. Note that in $w$–coordinates this leg
factor cancels the same prefactor in (2.3) and the correlation function (2.2)
becomes a function of $u(1)$–invariant combinations $w_{i+1}-w_{i}$.

###  2.2 Conformal blocks and diagrams

The conformal block decomposition of torus correlation functions in a given
channel $\mathfrak{Ch}$ reads

$\big{\langle}\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{n}(z_{n})\big{\rangle}=\sum_{\tilde{h}\in
D}\mathfrak{Ch}_{h,\tilde{h}}(C)\;\mathcal{F}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}(q,{\bf
z})\,,$ (2.7)

where $\mathcal{F}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}(q,{\bf z})$ is the
(holomorphic) conformal block, coefficients $\mathfrak{Ch}_{h,\tilde{h}}(C)$
are $n$-th order monomials of structure constants555The structure constants
are defined through the matrix elements as
$C_{ijk}=z_{j}^{h_{k}+h_{j}-h_{i}}\langle
h_{i}|\mathcal{O}_{j}(z_{j})|h_{k}\rangle$. $C_{ijk}$ which form is completely
fixed by the choice of $\mathfrak{Ch}$, and $\\{{h}_{i},\tilde{h}_{j}\in
D|\,i,j=1,...,n\\}$ are external and intermediate dimensions.

The global Ward identity (2.5) fixes a given $n$-point torus block to be a
product of the leg factor and a bare conformal block,

$\mathcal{F}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}(q,{\bf z})=\mathbb{L}({\bf
z})\;\mathcal{V}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}(q,{\bf x})\,.$ (2.8)

cf. (2.6). The leg factor $\mathbb{L}({\bf z})$ here is entirely responsible
for the behaviour of the conformal block
$\mathcal{F}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}$ with respect to the
global conformal transformations. Thus, the bare conformal block
$\mathcal{V}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}$ is $u(1)$–invariant.
Moreover, by choosing appropriate variables $(q,{\bf
x})\mapsto{\bm{t}}\equiv(t_{1},...,t_{n})$ along with some new leg factor the
bare conformal block can be defined to satisfy the boundary condition
$\mathcal{V}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h},c}(\bm{t})=1$ at $\bm{t}=0$.

Evaluating the trace in (2.2) yields

$\big{\langle}\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{n}(z_{n})\big{\rangle}=\sum_{\tilde{h}_{1}\in
D}\;\;\;\sum_{m=0}^{\infty}\sum_{m=|M|=|N|}q^{\tilde{h}_{1}+m}\,B_{1}^{M|N}\big{\langle}\tilde{h}_{1},M|\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{n}(z_{n})|N,\tilde{h}_{1}\big{\rangle}\,,$
(2.9)

where $\tilde{h}_{1}$ stands for the first intermediate conformal dimension,
$B_{1}^{M|N}$ are elements of the inverse Gram matrix in the Verma module
$\mathcal{V}_{\tilde{h}_{1},c}$. Thus, $n$-point torus correlation functions
are expressed in terms of $(n+2)$-point matrix elements.666Of course, a
straightforward calculation of the torus matrix elements rapidly gets
complicated. One can use different approximation schemes when the conformal
dimensions and/or the central charge are large. For instance, in the large-$c$
regime one can use the monodromy method, see e.g. [29, 19, 30], which is also
relevant in the context of AdS3/CFT2 . This formula makes manifest the plane
limit $\mathop{\mathrm{Im}}\tau\to\infty$ or $q\to 0$, where the $n$-point
torus correlation function gives the plane $(n+2)$-correlation function
because the contribution of descendants is suppressed and the leading order
defines the product of $n$ primary operators sandwiched between two primary
states of weights $\tilde{h}_{1}$. Geometrically, the plane limit can be seen
as going to the infinitely thin and long torus which can effectively be
understood as an infinite cylinder conformally mapped onto the plane. In
Section 4 we will discuss the plane limit of conformal blocks in the necklace
channel.

The relation (2.9) is the first step in defining a channel $\mathfrak{Ch}$ of
the conformal block decomposition (2.7) as it singles out matrix elements with
some $\tilde{h}_{1}\in D$. The channel $\mathfrak{Ch}$ can be further
specified by doing OPE and/or inserting projectors in the matrix elements
(2.9):

$\sum_{m=0}^{\infty}\sum_{m=|M|=|N|}q^{\tilde{h}_{1}+m}\,B_{1}^{M|N}\big{\langle}\tilde{h}_{1},M|\mathcal{O}_{1}(z_{1})\ldots\mathcal{O}_{n}(z_{n})|N,\tilde{h}_{1}\big{\rangle}\,.$
(2.10)

Namely, any two primary operators $\mathcal{O}_{h_{i}}$ and
$\mathcal{O}_{h_{j}}$ in (2.10) can be fused by means of the OPE
$\mathcal{O}_{h_{i}}\mathcal{O}_{h_{j}}\subset[\mathcal{O}_{\tilde{h}}]$ to
produce a secondary operator belonging to the conformal family of dimension
$\tilde{h}\in D$ either one can insert between them a projector on the Verma
module $\mathcal{V}_{\tilde{h},c}$. A particular combination of these
operations defines a channel $\mathfrak{Ch}$ and, hence, the corresponding
conformal block. Each OPE introduces one intermediate dimension $\tilde{h}$
and reduces the number of primary operators by one so that after $(n-k)$ OPEs
there are $k$ primary operators left. Then, between the remaining operators
one inserts $(k-1)$ projectors with other intermediate dimensions. Taking into
account that one intermediate dimension $\tilde{h}_{1}$ comes from the trace
(2.9), there are $(n-k)+(k-1)+1=n$ intermediate dimensions characterising a
particular channel $\mathfrak{Ch}$, cf. (2.7).

Figure 1: The $n$-point torus block in the necklace channel (a). It can be
thought of as the $(n+2)$-point block in the comb channel of the plane CFT2
with identified endpoints (b). Note that $\tilde{h}_{1}$ on the necklace
diagram is always between $h_{1}$ and $h_{n}$.

#### Necklace channel.

Depending on whether only OPEs or projectors were used there are two
distinguished channels: the OPE channel and the projection channel. Let us
consider the projection channel which we will refer to as the necklace channel
following the form of its diagram shown in Fig. 1.777The OPE channel is when
only one operator along with its conformal family is connected to the necklace
which therefore has only one leg [16], see Fig. 2 (a). For blocks of this type
we omit the superscript $\mathfrak{Ch}$, for the sake of simplicity. The
corresponding block function is defined by inserting the projectors between
each pair of primary operators in (2.2) as

$\mathcal{F}_{h,\tilde{h},c}(q,z_{1},\ldots,z_{n})=\Big{(}\prod_{i=1}^{n}C_{\tilde{h}_{i},h_{i},\tilde{h}_{i+1}}\Big{)}^{-1}\operatorname{Tr}_{\tilde{h}_{1}}\left[q^{L_{0}}\mathcal{O}_{1}(z_{1})\mathbb{P}_{2}\mathcal{O}_{2}(z_{2})\ldots\mathbb{P}_{n}\mathcal{O}_{n}(z_{n})\right]\,,$
(2.11)

where, in the prefactor, we identify $\tilde{h}_{n+1}\equiv\tilde{h}_{1}$, the
trace $\operatorname{Tr}_{\tilde{h}_{1}}$ is taken over
$\mathcal{V}_{\tilde{h}_{1},c}$, and the projectors are given by

$\mathbb{P}_{i}=\sum_{m_{i}=0}^{\infty}\;\sum_{m_{i}=|M|=|N|}B_{i}^{M|N}|N,\tilde{h}_{i}\rangle\langle\tilde{h}_{i},M|,\qquad\mathbb{P}_{i}^{2}=\mathbb{P}_{i}\,,\qquad\sum_{\tilde{h}_{i}\in
D}\mathbb{P}_{i}=1,$ (2.12)

with $B_{i}^{M|N}$ being inverse Gram matrices in
$\mathcal{V}_{\tilde{h}_{i},c}$. The structure constants
$\mathfrak{Ch}_{h,\tilde{h}}(C)$ in the conformal block decomposition (2.7) in
this case read

$\mathfrak{Ch}_{h,\tilde{h}}(C)=\prod_{i=1}^{n}C_{\tilde{h}_{i},h_{i},\tilde{h}_{i+1}}\,.$
(2.13)

In fact, this function is generated by matrix elements
$\langle\tilde{h}_{i}|\mathcal{O}_{i}(z_{i})|\tilde{h}_{i+1}\rangle$ (see the
footnote 5) which arise as prefactors when calculating level-$M$ matrix
elements. Then, the bare torus block in the necklace channel is given by

$\mathcal{V}(q,{\bf
x})=\Big{(}\prod_{i=1}^{n}C_{\tilde{h}_{i},h_{i},\tilde{h}_{i+1}}\Big{)}^{-1}\Big{(}\prod_{i=1}^{n}z_{i}^{h_{i}}\Big{)}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}(z_{1})\mathbb{P}_{2}\mathcal{O}_{2}(z_{2})\ldots\mathbb{P}_{n}\mathcal{O}_{n}(z_{n})]\,.$
(2.14)

Figure 2:  (a) A particular OPE channel for 6-point torus block. (b) Arbitrary
torus channel $\mathfrak{Ch}$.

#### General channel.

Using the quotient description of $\mathbb{T}^{2}$, the necklace channel can
be viewed as obtained from the comb channel on the plane (Fig. 1) by gluing
together the endpoints $z=\infty$ and $z=0$ (first and last operators) and
summing up over all descendants that produces the trace
$\operatorname{Tr}_{\tilde{h}}$. Such a gluing can be done for any conformal
block diagram on $\mathbb{R}^{2}$ yielding a conformal block diagram on
$\mathbb{T}^{2}$ which is a necklace with $k\leq n$ legs to which $k$ graphs
of arbitrary topology888Here, by a topology we mean a plane graph obtained by
gluing single node trivalent trees in a particular order without making loops.
See [31, 32, 33] for discussion of the plane CFT2 conformal blocks described
by diagrams of arbitrary topology. are attached, see Fig. 2. Specific plane
graphs correspond to a particular ordering of OPEs in the torus correlation
function.

In what follows, we denote the channel topology described above as
$\mathfrak{Ch}$ = $(k-1,\mathfrak{ch})$, where the number of inserted
projectors comes first and the second symbol $\mathfrak{ch}$ stands for a
certain topology of trivalent trees obtained by doing $(n-k)$ ordered OPEs. It
is clear that the number of projectors producing a loop in respective torus
block diagrams is a natural characterization of all possible diagrams by a
number of intermediate lines forming a loop, see Fig. 2. On the other hand,
$\mathfrak{ch}$ embraces all possible OPEs with intermediate legs attached to
the loop.

A torus conformal block of an arbitrary channel topology can be generated from
the torus conformal block with less points in the necklace channel.999The comb
channel in the plane CFT2 plays the analogous role. One can consider a
$n$-point block, which is obtained from the $n$-point correlation function
(2.9) by inserting $(k-1)$ projectors where $k=1,2,\ldots$ and doing $(n-k)$
OPEs. On the other hand, each OPE contains Virasoro generators which can be
represented as differential operators acting on the correlation function [1].
Then, the resulting OPE coefficients can be transformed into some differential
operator $\hat{B}^{\mathfrak{ch}}_{h,\tilde{h},c}$ acting on a $k$-point torus
block $\mathcal{F}_{h,\tilde{h},c}(q,z^{\prime}_{1},\ldots,z^{\prime}_{k})$ in
the necklace channel:

$\mathcal{F}^{\,(k-1,\mathfrak{ch})}_{h,\tilde{h},c}(q,z_{1},\ldots,z_{n})=\hat{B}^{\mathfrak{ch}}_{h,\tilde{h},c}(z_{1},\ldots,z_{n},\partial_{z^{\prime}})\mathcal{F}_{h,\tilde{h},c}(q,z^{\prime}_{1},\ldots,z^{\prime}_{k})\,,$
(2.15)

where the set $(z^{\prime}_{1},...,z^{\prime}_{k})$ denotes coordinates left
after doing all OPEs. The explicit form of
$\hat{B}^{\mathfrak{ch}}_{h,\tilde{h},c}$ is directly determined by the
topology of a channel $\mathfrak{ch}$ which for the general Virasoro conformal
blocks can be quite complicated. However, in the $sl(2,\mathbb{R})$ case,
these operators take a simpler form, see Section 3.2.

To conclude, it is instructive to compare the mixed use of the projector/OPE
technique in torus and plane correlation functions. For instance, for the
4-point correlation function
$\langle\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})\mathcal{O}_{3}(z_{3})\mathcal{O}_{4}(z_{4})\rangle$
on the plane this formally yields several options. There are three standard
OPE channels ($s,t,u$) obtained by doing OPEs between pairs of operators,
depending on how the points $(z_{1}=0,z_{2}=1,z_{3}=z,z_{4}=\infty)$ are
ordered. However, one can see that the only non-trivial way to obtain the
projection channel is to insert the projector between pairs
$\mathcal{O}_{1}(z_{1})\mathcal{O}_{2}(z_{2})$ and
$\mathcal{O}_{3}(z_{3})\mathcal{O}_{4}(z_{4})$. On the other hand, it is
straightforward to show that the projection channel coincides with the
$s$-channel (see e.g. [34]). Thus, all channels on the plane are the OPE
channels. As mentioned above, $n$-point torus correlation functions can be
thought of as identifying points $z_{1}=0\sim z_{n+2}=\infty$ and taking the
trace over descendants in $(n+2)$-point plane correlation functions. It makes
one to distinguish between projectors and OPEs as operations leading to
(topologically) different channels. For example, the plane $t$-channel
directly yields the torus OPE channel, while the plane $s,u$-channels can
yield only one torus channel which is the projection (necklace) channel. In
the latter case the equivalence between the projector and the OPE on the plane
is broken in favour of the projector on the torus. This explains why $n$-point
torus correlators can be expanded in more (topologically inequivalent)
channels than $(n+2)$-point and, hence, $n$-point plane correlators.

##  3 Casimir equations for $n$-point global torus blocks

Let us consider the large-$c$ limit. It can be shown that the Virasoro algebra
and its representations can enjoy at least two consistent Inonu-Wigner
contractions in $1/c$, one of which yields global conformal blocks and another
one yields light conformal blocks which in the torus CFT2 are different but
related functions [35, 15]. In this section we focus on global conformal
blocks $\mathcal{F}^{\,{}^{\mathfrak{Ch}}}_{h,\tilde{h}}(q,z)$ which are
associated with the finite-dimensional subalgebra $sl(2,\mathbb{R})\subset
Vir$.

###  3.1 Global torus blocks in the necklace channel

The $sl(2,\mathbb{R})$ Verma modules $V_{h}$ are spanned by basis vectors
obtained from the highest weight vector $|h\rangle$ as
$\\{|m,h\rangle=L_{-1}^{m}|h\rangle,\;L_{0}|h\rangle=h|h\rangle\\}$, where
$L_{0,\pm 1}$ are $sl(2,\mathbb{R})$ generators. Thus, each level contains
just one basis element. The conformal dimensions are assumed to be arbitrary
real numbers, $h\in\mathbb{R}$. The inverse Gram matrix is diagonal,
$B^{M|N}:=B^{m}=(m!(2h)_{m})^{-1}$, where $(a)_{m}=a(a+1)\ldots(a+m-1)$ is the
Pochhammer symbol. Then, the projector (2.12) onto intermediate Verma modules
$V_{\tilde{h}_{i}}$ takes the form

$\mathbb{P}_{i}=\sum_{m=0}^{\infty}\frac{|m,\tilde{h}_{i}\rangle\langle\tilde{h}_{i},m|}{m!(2\tilde{h}_{i})_{m}}\,.$
(3.1)

All the constructions discussed in the previous section for the Virasoro
conformal blocks directly apply to the $sl(2,\mathbb{R})$ global blocks. In
particular, substituting the projectors (3.1) into the definition (2.11)
yields the following $n$-point torus block function in the necklace channel
[17]

$\mathcal{V}_{h,\tilde{h}}(x)=\Big{(}\prod_{j=1}^{n}x_{j}^{\tilde{h}_{j}}\Big{)}\sum_{s_{1},...,s_{n}=0}^{\infty}\;\;\prod_{m=1}^{n}\frac{\tau_{s_{m},s_{m+1}}(\tilde{h}_{m},h_{m},\tilde{h}_{m+1})}{s_{m}!\,(2\tilde{h}_{m})_{s_{m}}}\,x_{m}^{s_{m}}\,,$
(3.2)

where $x_{j}=z_{j}/z_{j-1}$, $j=2,\ldots,n$, $x_{1}=q/(x_{2}\ldots x_{n})$
with identifications $\tilde{h}_{n+1}=\tilde{h}_{1}$, $s_{n+1}=s_{1}$ and
$\tau$-coefficients are given by [36]

$\tau_{m,n}(a,b,c)=m!n!\sum_{p=0}^{min[m,n]}\frac{(2c+n-1)^{(p)}(c+b-a)_{n-p}(a+b-c+p-n)_{m-p}}{p!(m-p)!(n-p)!}\,.$
(3.3)

Note that it would be useful to find the block function (3.2) in the form
similar to that one in [26], where $n$-point global block of the plane CFT2
was represented as a special function generalizing the Appel series (see
(4.12) below). Hopefully, the Casimir approach developed in the next section
will provide more insight into the formulation of the global torus block
functions (see discussion in the end of Section 4).

(a)(b)(c)(d)(e)(f) Figure 3: 1-point block (a). 2-point blocks: in the
necklace channel (b), in the OPE channel (c). 3-point blocks: in the necklace
channel (d), in the mixed channel (e), in the OPE channel (f).

###  3.2 Global torus blocks in arbitrary channels

Let us consider the general prescription (2.15) in the case of the global
blocks. For the $sl(2,\mathbb{R})$ conformal symmetry, the OPE of two primary
operators is given by

$\mathcal{O}_{i}(z_{i})\mathcal{O}_{j}(z_{j})\sim\hat{\phi}_{h,\tilde{h}}(z_{i},z_{j},\partial_{z_{j}})\mathcal{O}_{\tilde{h}}(z_{j})\,,$
(3.4)

where (see e.g. [1, 37, 38])

$\hat{\phi}_{h,\tilde{h}}(z_{i},z_{j},\partial_{z_{j}})=(z_{i}-z_{j})^{\tilde{h}-h_{i}-h_{j}}\sum_{m=0}^{\infty}\frac{(\tilde{h}+h_{i}-h_{j})_{m}}{m!(2\tilde{h})_{m}}\,(z_{i}-z_{j})^{m}\,\partial_{j}^{m}\,.$
(3.5)

Then, the operators $\hat{B}^{\mathfrak{ch}}_{h,\tilde{h}}$ (2.15) are built
as sequences of the OPE operators $\hat{\phi}_{h,\tilde{h}}$ ordered according
to the particular OPE ordering of a given channel $\mathfrak{ch}$. This can be
illustrated by considering a few examples of low-point blocks shown in Fig. 3.

The 1-point block exists in the only channel (a). The first non-trivial
example is given by the $2$-point torus block which can be considered in the
necklace channel (b) and in the OPE channel (c). In the latter case, the
formula (2.15) takes the form

$\mathcal{F}^{\,(0,\mathfrak{ch}_{1})}_{h_{1}h_{2},\tilde{h}_{1}\tilde{h}_{2}}(q,z_{1},z_{2})=\hat{\phi}_{h_{1}h_{2},\tilde{h}_{2}}(z_{1},z_{2},\partial_{z_{1}})\mathcal{F}_{\tilde{h}_{2},\tilde{h}_{1}}(q,z_{1})\,.$
(3.6)

Thus, the $2$-point block is expressed in terms of the $1$-point block
$\mathcal{F}_{\tilde{h}_{2},\tilde{h}_{1}}(q,z_{1})$ (a). The next example is
the $3$-point block in three different channels (d), (e), (f). In the OPE
channel (f), which directly generalizes the OPE channel (c) by adding one more
OPE operator, one finds

$\mathcal{F}^{\,(0,\mathfrak{ch}_{2})}_{h_{1}h_{2}h_{3},\tilde{h}_{1}\tilde{h}_{2}\tilde{h}_{3}}(q,z_{1},z_{2},z_{3})=\hat{\phi}_{h_{2}h_{3},\tilde{h}_{3}}(z_{2},z_{3},\partial_{z_{2}})\hat{\phi}_{h_{1}\tilde{h}_{3},\tilde{h}_{2}}(z_{1},z_{2},\partial_{z_{1}})\mathcal{F}_{\tilde{h}_{2},\tilde{h}_{1}}(q,z_{1})\,.$
(3.7)

In the mixed channel (e) obtained by inserting two projectors and doing one
OPE one finds

$\mathcal{F}^{\,(1,\mathfrak{ch}_{1})}_{h_{1}h_{2}h_{3},\tilde{h}_{1}\tilde{h}_{2}\tilde{h}_{3}}(q,z_{1},z_{2},z_{3})=\hat{\phi}_{h_{2}h_{3},\tilde{h}_{3}}(z_{2},z_{3},\partial_{z_{2}})\mathcal{F}_{h_{1}\tilde{h}_{3},\tilde{h}_{1}\tilde{h}_{2}}(q,z_{1},z_{2})\,,$
(3.8)

where
$\mathcal{F}_{h_{1}\tilde{h}_{3},\tilde{h}_{1}\tilde{h}_{2}}(q,z_{1},z_{2})$
is a $2$-point block in the necklace channel (b). The 3-point necklace channel
(d) can be used to build torus block with 4 points and higher.

###  3.3 Casimir equations in the necklace channel

Since the conformal families defining exchange channels form
$sl(2,\mathbb{R})$ irreducible representations the global blocks can be viewed
as eigenfunctions of the Casimir operators [22] which in the present case is
given by

$\operatorname{C_{2}}=L_{1}L_{-1}-L_{0}-L_{0}^{2}\;.$ (3.9)

Its eigenvalue on irreducible modules $V_{h}$ equals $-h(h-1)$.

The associated Casimir equations can be obtained by inserting the Casimir
operator (3.9) between operators under the trace in the torus block function
(2.11). Using the obvious properties
$\text{C}_{2}\mathbb{P}_{i}=\mathbb{P}_{i}\text{C}_{2}=-\tilde{h}_{i}(\tilde{h}_{i}-1)\mathbb{P}_{i}$
and ${\rm Tr}_{h}\text{C}_{2}\hat{A}=-h(h-1){\rm Tr}_{h}\hat{A}$ one finds $n$
distinct relations which realize irreducibility of intermediate conformal
families

$\begin{split}&1)\quad\operatorname{Tr}_{\tilde{h}_{1}}[C_{2}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{2}\mathcal{O}_{2}\cdots\mathbb{P}_{n}\mathcal{O}_{n}]=-\tilde{h}_{1}(\tilde{h}_{1}-1)\mathcal{F}_{h,\tilde{h}}\,,\\\
&\quad\quad\ldots\\\
&j)\quad\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{2}\mathcal{O}_{2}\cdots\mathbb{P}_{j}C_{2}\mathcal{O}_{j}\cdots\mathbb{P}_{n}\mathcal{O}_{n}]=-\tilde{h}_{j}(\tilde{h}_{j}-1)\mathcal{F}_{h,\tilde{h}}\,,\\\
&\quad\quad\ldots\\\
&n)\quad\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{2}\mathcal{O}_{2}\cdots\mathbb{P}_{n}C_{2}\mathcal{O}_{n}]=-\tilde{h}_{n}(\tilde{h}_{n}-1)\mathcal{F}_{h,\tilde{h}}\,,\end{split}$
(3.10)

where $\mathcal{F}_{h,\tilde{h}}=\mathcal{F}_{h,\tilde{h}}(q,{\bf z})$ is the
$n$-point torus block in the necklace channel. Note that in this form the
torus Casimir equations are not immediately seen as the eigenvalue problem for
the conformal block functions as in the plane CFT2 case [22, 36, 26].

In Appendix A we explicitly demonstrate how to transform the above system to
the eigenvalue equations in the 2-point case. Generalizing that procedure to
the $n$-point case we find that the global torus block in the necklace channel
is subjected to the following Casimir equation system ($j=2,\ldots,n$):

$\begin{array}[]{l}\displaystyle\bigg{[}-q^{2}\partial_{q}^{2}+\frac{2q}{1-q}q\partial_{q}-\frac{q}{(1-q)^{2}}\sum_{i=1}^{n}\mathcal{L}_{-1}^{(i)}\sum_{k=1}^{n}\mathcal{L}_{1}^{(k)}\bigg{]}\mathcal{F}_{h,\tilde{h}}=-\tilde{h}_{1}(\tilde{h}_{1}-1)\mathcal{F}_{h,\tilde{h}}\,,\vspace{4mm}\\\
\displaystyle\bigg{[}-q^{2}\partial_{q}^{2}+\frac{2q}{1-q}q\partial_{q}-\frac{1}{(1-q)^{2}}\bigg{(}\sum_{k=1}^{j-1}\mathcal{L}_{-1}^{(k)}+q\sum_{k=j}^{n}\mathcal{L}_{-1}^{(k)}\bigg{)}\bigg{(}q\sum_{k=1}^{j-1}\mathcal{L}_{1}^{(k)}+\sum_{k=j}^{n}\mathcal{L}_{1}^{(k)}\bigg{)}\vspace{2mm}\\\
\displaystyle\hskip
8.53581pt+\frac{1+q}{1-q}\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}-\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}\sum_{l=j}^{n}\mathcal{L}_{0}^{(l)}-2\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}q\partial_{q}\bigg{]}\mathcal{F}_{h,\tilde{h}}=-\tilde{h}_{j}(\tilde{h}_{j}-1)\mathcal{F}_{h,\tilde{h}}\,,\end{array}$
(3.11)

where $\mathcal{L}_{m}^{(k)}$ are differential realizations of
$sl(2,\mathbb{R})$ generators $L_{m}$, $m=0,\pm 1$ (2.4) at points $z_{k}$. A
few comments are in order.

First, we note that the first equation in (3.11) differs from the other
equations due to the way the intermediate dimensions were introduced: the
first dimension $\tilde{h}_{1}$ arises from the common trace in the definition
of the torus function (2.9), while the other dimensions $\tilde{h}_{2,...,n}$
come from the projectors. The equations can be unified by applying the Ward
identity (2.5):

$\sum_{i=1}^{n}\mathcal{L}_{0}^{(i)}\mathcal{F}_{h,\tilde{h}}=0\,.$ (3.12)

Then, the first equation in (3.11) can be written in the form of the second
equation in (3.11) by setting $j=1$ and adding zero terms proportional (3.12).

Second, supplementing the Casimir equation system by the Ward identity one
gets $(n+1)$ 2nd order PDEs in the modular parameter $q$ and $n$ insertion
points ${\bf z}$ on a single block function $\mathcal{F}_{h,\tilde{h}}(q,{\bf
z})$. The Ward identity removes dependence on one of ${\bf z}$-coordinates
and, hence, the general solution is parameterized by $2n$ constants which
define an asymptotic behaviour of the block function. In general, one expects
that different branches would correspond to the original block in the necklace
channel along with the shadow blocks.101010For the shadow blocks in the plane
CFTd see e.g. [39], the shadow formalism was also used to study CFTd thermal
blocks in [24].

Third, the system (3.11) allows reducing a number of points from $n$ to
$(n-k)$. In order to do this, one has to take into account that $(n-k)$-point
blocks satisfy the Ward identity (3.12) with $n$ replaced by $(n-k)$ and
impose on the $n$-point conformal block $k$ conditions $h_{i}=0$,
$i=n,\ldots,(n-k+1)$, from which it follows that

$\partial_{z_{k}}\mathcal{F}_{h,\tilde{h}}=0\,,\qquad
k=n,\,(n-1),\,\ldots\,,(n-k+1)\,.$ (3.13)

In addition, one has to put $\tilde{h}_{j}=\tilde{h}_{1}$,
$j=n,\ldots,(n-k+1)$.

The Casimir equations (3.11) can be considerably simplified by switching to
the bare conformal block $\mathcal{V}(q,\bm{x})$ (2.8) and changing variables
from $z_{i}$, $i=1,\ldots,n$ to $x_{i}=z_{i}/z_{i-1}$, $i=2,\ldots,n$. We find
that the Casimir equation system can be cast into the form

$\mathbb{C}_{2}[n,j]\,\mathcal{V}_{h,\tilde{h}}(q,\bm{x})=-\tilde{h}_{j}(\tilde{h}_{j}-1)\,\mathcal{V}_{h,\tilde{h}}(q,\bm{x})\,,\qquad
j=1,...\,,n\,,$ (3.14)

where the 2nd order differential operators on the left-hand side are given by

$\mathbb{C}_{2}[n,j]=\frac{1+q}{1-q}\,(N+N_{j})-(N+N_{j})^{2}-\frac{1}{1-q}\bigg{(}\frac{q}{1-q}\sum_{i,k=1}^{n}\mathcal{D}_{i,k}+\sum_{i=1}^{j-1}\sum_{k=j}^{n}(\mathcal{D}_{i,k}-q\mathcal{D}_{k,i})\bigg{)}.$
(3.15)

Here, $N_{k}\equiv x_{k}\partial_{x_{k}}$ and $N\equiv q\partial_{q}$ are the
counting operators ($k=1,...,n+1$, with the convention $N_{1}\equiv 0\equiv
N_{n+1}$), and their combinations are

$\mathcal{D}_{i,k}\equiv\Big{(}\lambda_{i,k}(N_{i}-N_{i+1}-h_{i})+\delta_{i,k}\Big{)}\Big{(}N_{k}-N_{k+1}+h_{k}\Big{)}\,,$
(3.16)

where

$\lambda_{i,k}\equiv\frac{z_{k}}{z_{i}}=\left\\{\begin{split}&\prod_{l=i+1}^{k}x_{l},\quad\mbox{if
}k>i,\\\ &1,\quad\mbox{if }k=i,\\\ &\prod_{l=k+1}^{i}x_{l}^{-1},\quad\mbox{if
}k<i.\end{split}\right.$ (3.17)

Choosing different coordinates one can find other equivalent forms of the
Casimir equations. However, the system (3.14)–(3.17) is conveniently expressed
in terms of the counting operators that could be crucial from the perspective
of finding and classifying solutions in terms of special functions.

##  4 Plane limit

In order to consider the plane limit $q\rightarrow 0$ for $n$-point torus
blocks in the necklace channel, we expand the torus block with respect to the
modular parameter as

$\mathcal{F}_{h,\tilde{h}}(q,{\bf
z})=q^{\tilde{h}_{1}}\displaystyle\sum^{\infty}_{p=0}\mathcal{F}_{p}({\bf
z})q^{p}\,,$ (4.1)

where the leading asymptotics is always $q^{\tilde{h}_{1}}$ due to the
definition (2.2). In what follows we argue that the expansion coefficient in
the leading order $\mathcal{F}_{0}({\bf z})$ can be identified with a
particular conformal block of the plane CFT2 . The exact relation is given by

$\mathcal{F}_{0}({\bf z})=\lim\limits_{\begin{subarray}{c}\\\
z_{0}\to\infty\\\ z_{n+1}\to
0\end{subarray}}z^{2\tilde{h}_{1}}_{0}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\,.$ (4.2)

Here, $\tilde{\mathcal{F}}(z_{0},{\bf z},z_{n+1})$ is the $(n+2)$-point
conformal block in the comb channel with external dimensions
$(\tilde{h}_{1},h_{1},h_{2},\ldots,h_{n},\tilde{h}_{1})$ and intermediate
dimensions $(\tilde{h}_{2},\tilde{h}_{3},\ldots,\tilde{h}_{n})$. In other
words, the comb channel diagram is obtained from the necklace channel diagram
by cutting the intermediate line of dimension $\tilde{h}_{1}$ that produces
two external lines of the same equal dimensions, see Fig. 1.

Let us demonstrate that $\tilde{\mathcal{F}}(z_{0},{\bf z},z_{n+1})$ satisfies
the limiting Casimir equations (3.11) which are exactly the Casimir equations
in the comb channel of the plane CFT2. Indeed, substituting the expansion
(4.1) into the Casimir equation system (3.11) and keeping the leading order in
$q$ we find that the first equation in (3.11) becomes trivial, while the
remaining $(n-1)$ equations in (3.11) take the form

$\begin{array}[]{c}\displaystyle\bigg{[}\tilde{h}_{j}(\tilde{h}_{j}-1)-\tilde{h}_{1}(\tilde{h}_{1}-1)-\bigg{(}\sum_{k=1}^{j-1}\mathcal{L}_{-1}^{(k)}\sum_{k=j}^{n}\mathcal{L}_{1}^{(k)}\bigg{)}+\bigg{(}1-2\tilde{h}_{1}-\sum_{l=j}^{n}\mathcal{L}_{0}^{(l)}\bigg{)}\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}\bigg{]}\mathcal{F}_{0}({\bf
z})=0\,,\end{array}$ (4.3)

where $j=2,...,n$.111111The $1$-point torus block is determined only by the
first equation in (3.11) which is automatically satisfied at $q\rightarrow 0$.
Thus, $n\geq 2$. The case $n=2$ is worked out in Appendix A. On the other
hand, there are $(n-1)$ Casimir equations for $(n+2)$-point conformal block in
the comb channel that can be written as

$\tilde{\mathbb{C}}_{2}[n+2,j]\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})=-\tilde{h}_{j}(\tilde{h}_{j}-1)\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\,,\qquad j=2,...,n\,,$ (4.4)

where the 2nd order differential operators on the left-hand side are directly
read off from the $sl(2,\mathbb{R})$ Casimir operator (3.9) as [23, 36, 26]:

$\tilde{\mathbb{C}}_{2}[n+2,j]=\mathbb{L}_{1}\mathbb{L}_{-1}+\mathbb{L}_{0}-(\mathbb{L}_{0})^{2}\,,\qquad\quad\mathbb{L}_{m}=\displaystyle\sum_{k=j}^{n+1}\mathcal{L}^{(k)}_{m}\,,\quad
m=0,\pm 1\,,$ (4.5)

where the $sl(2,\mathbb{R})$ generators $\mathcal{L}^{(k)}_{m}$ are given by
(2.4). The above equations on the block function are supplemented by the Ward
identities of the global $sl(2,\mathbb{R})$ (chiral) symmetry of the plane
CFT2:

$\sum_{k=0}^{n+1}\mathcal{L}^{(k)}_{m}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})=0\,,\qquad m=0,\pm 1\,.$ (4.6)

Now, using the corollary of the Ward identity at $m=-1$ (4.6),

$\displaystyle\sum_{k=j}^{n+1}\mathcal{L}^{(k)}_{-1}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})=-\sum_{k=0}^{j-1}\mathcal{L}^{(k)}_{-1}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\,,$ (4.7)

we find that the plane Casimir equations (4.4) can be cast into the form

$\begin{array}[]{c}\displaystyle\bigg{[}\Big{(}-\sum_{k=1}^{j-1}\mathcal{L}_{-1}^{(k)}\sum_{k=j}^{n}\mathcal{L}_{1}^{(k)}+\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}-\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}\sum_{l=j}^{n}\mathcal{L}_{0}^{(l)}-2\sum_{k=j}^{n}\mathcal{L}_{0}^{(k)}\mathcal{L}^{(n+1)}_{0}\Big{)}-\mathcal{L}^{(0)}_{-1}\mathcal{L}^{(n+1)}_{1}+\vspace{2mm}\\\
\displaystyle+\mathcal{L}^{(n+1)}_{0}-\left(\mathcal{L}^{(n+1)}_{0}\right)^{2}-\sum_{k=1}^{j-1}\mathcal{L}_{-1}^{(k)}\mathcal{L}^{(n+1)}_{1}-\sum_{k=j}^{n}\mathcal{L}_{1}^{(k)}\mathcal{L}^{(0)}_{-1}+\tilde{h}_{j}(\tilde{h}_{j}-1)\bigg{]}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})=0\,,\end{array}$ (4.8)

where we separated groups of terms inside and outside the round brackets. In
view of the relation (4.2) we recall that the comb channel block function
$\tilde{\mathcal{F}}(z_{0},{\bf z},z_{n+1})$ is regular at $z_{n+1}\rightarrow
0$ and grows as $z^{-2\tilde{h}_{1}}_{0}$ at $z_{0}\rightarrow\infty$. In
particular, focusing on the terms outside the round brackets one can shown the
following asymptotic behaviour:

$\mathcal{L}^{(n+1)}_{0}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\Big{|}_{z_{n+1}\rightarrow
0}=\tilde{h}_{1}\tilde{\mathcal{F}}(z_{0},{\bf z},0)\,,$

$\Big{(}\mathcal{L}^{(n+1)}_{0}-\big{(}\mathcal{L}^{(n+1)}_{0}\big{)}^{2}\Big{)}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\Big{|}_{z_{n+1}\rightarrow
0}=\tilde{h}_{1}(1-\tilde{h}_{1})\tilde{\mathcal{F}}(z_{0},{\bf z},0)\,,$
(4.9)

$\mathcal{L}^{(n+1)}_{1}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\Big{|}_{z_{n+1}\rightarrow
0}=0\,,\quad\displaystyle\mathcal{L}^{(0)}_{-1}\tilde{\mathcal{F}}(z_{0},{\bf
z},z_{n+1})\Big{|}_{z_{0}\rightarrow\infty}\sim z^{-2\tilde{h}_{1}-1}_{0}\,.$

Taking the limit $z_{0}\to\infty$ and $z_{n+1}\to 0$ in the plane Casimir
equation (4.8) and using the definition (4.2) along with the relations (4.9)
one finally arrives at the limiting torus Casimir equations (4.3). Thus, this
proves the relation (4.2) between the leading $q\to 0$ asymptotics of the
$n$-point torus block $\mathcal{F}_{0}({\bf z})$ and the $(n+2)$-point plane
block $\tilde{\mathcal{F}}(z_{0},{\bf z},z_{n+1})$.

Note that in the limit $q\to 0$ the Casimir equations in the
$\bm{x}$-parameterization (3.14) take the form

$\mathbb{C}^{{}^{q\to
0}}_{2}[n,j]\,\mathcal{V}_{0}(\bm{x})=-\tilde{h}_{j}(\tilde{h}_{j}-1)\,\mathcal{V}_{0}(\bm{x})\,,\qquad
j=2,\ldots,n\,,$ (4.10)

where the limiting 2nd order differential operators on the left-hand side are
given by

$\mathbb{C}^{{}^{q\to
0}}_{2}[n,j]=(1-2\tilde{h}_{1})N_{j}-N^{2}_{j}+\tilde{h}_{1}(1-\tilde{h}_{1})-\sum_{k=1}^{j-1}\sum_{k=j}^{n}\mathcal{D}_{i,k}\,,$
(4.11)

and $\mathcal{V}_{0}(\bm{x})$ is the leading asymptotics of the bare conformal
block, which, therefore, defines the comb channel conformal block through the
relations (2.8) and (4.2). Equivalently, one can start with the limiting
equations (4.3) and change variables ${\bf z}\mapsto{\bm{x}}$.

Remarkably, the equations (4.10) provide the explicit realization of the
Casimir equations for multipoint conformal blocks of the plane CFT2 in the
comb channel. On the other hand, there is a nice parameterization in the plane
CFT2, in which $(n+2)$-point conformal blocks in the comb channel take the
form [26]

$\mathcal{V}_{0}(\bm{x})=\tilde{\mathbb{L}}(\bm{x})\left(\prod_{i=1}^{n-1}\eta_{i}\right)F_{K}\left[\begin{array}[]{l
l}\tilde{h}_{1}+\tilde{h}_{2}-h_{1},\tilde{h}_{2}+\tilde{h}_{3}-h_{2},\ldots,\tilde{h}_{n}+\tilde{h}_{1}-h_{n}\\\
\qquad\qquad\qquad
2\tilde{h}_{2},\ldots,2\tilde{h}_{n}\end{array}\bigg{|}\eta_{1},\ldots,\eta_{n-1}\right]\,,$
(4.12)

where $F_{K}$ is the so-called Comb Function of the cross-ratios
$\bm{\eta}=\\{\eta_{a}({\bm{x}}),a=1,...,n-1\\}$ given by

$\eta_{1}=x_{2}\frac{1-x_{3}}{1-x_{2}x_{3}}\,,\quad\eta_{i}=x_{i+1}\frac{(1-x_{i})(1-x_{i+2})}{(1-x_{i}x_{i+1})(1-x_{i+1}x_{i+2})}\,,\quad\eta_{n-1}=x_{n}\frac{1-x_{n-1}}{1-x_{n}x_{n-1}}\,,$
(4.13)

while the leg factor reads

$\tilde{\mathbb{L}}(\bm{x})=(1-x_{2})^{\tilde{h}_{1}-h_{1}}(1-x_{n})^{\tilde{h}_{1}-h_{n}}\prod_{j=2}^{n}\left(x_{j}^{-\tilde{h}_{1}}\right)\prod_{i=2}^{n-1}\left(\frac{1-x_{i}x_{i+1}}{(1-x_{i})(1-x_{i+1})}\right)^{h_{i}}\,.$
(4.14)

The Comb Function satisfies simple recursive equations in $\bm{\eta}$
variables [26]. Thus, we conclude that changing variables
$\bm{x}\mapsto\bm{\eta}$ the Casimir equations (4.10) must take the form of
the equations that determine the Comb function.121212It can be directly
checked for small $n$, although it is not quite clear how to prove this
statement for any $n$. However, using $\bm{\eta}$-parameterization instead of
$\bm{x}$-parameterization significantly complicates the torus Casimir
equations (3.14)–(3.15) beyond the $q\to 0$ limit. (In this respect, it would
be crucial to connect the eigenvalue torus Casimir equations and their
solutions to Hamiltonians of integrable models and the theory of special
functions [40], that, hopefully, will help to clarify the structure of torus
conformal blocks.)

##  5 Conclusion and outlooks

In this paper we elaborated on the conformal block decomposition in the torus
CFT2 and recognized the essential role of the necklace channel in building
torus conformal blocks in general channels. We explicitly found the Casimir
equations for the multipoint global torus block in the necklace channel
(3.14). The resulting PDE system is given by 2nd order differential operators
both in the modular parameter and local coordinates. The conformal blocks
diagonalize these operators with eigenvalues $-\tilde{h}(\tilde{h}-1)$ defined
by dimensions of exchange channels.

In particular, the limiting $q\rightarrow 0$ torus blocks with $n$ points in
the necklace channel are recognized as plane blocks with $n+2$ points in the
comb channel. Such an identification can be justified by simple counting of
independent variables. The $n$-point torus block depends on $(n-1)$ variables
because of global $u(1)$ symmetry. At the same time, the $(n+2)$-point plane
block depends on the same $(n-1)$ variables because originally it depends on
$(n+2)$ variables but 3 points can be fixed by the global $sl(2,\mathbb{R})$
symmetry. There is also a simple geometric argument underlying this relation:
cutting a necklace diagram yields a comb diagram. As a by-product, we obtained
the plane Casimir equations as explicit 2nd order differential operators in
given coordinates and conformal blocks are their eigenfunctions (4.10).

Our study of global torus conformal blocks was partly motivated by Wilson
network realization of conformal blocks in the context of AdS3/CFT2
correspondence [41, 42, 43, 16, 44, 45, 46, 47, 48]. On the other hand,
semiclassical heavy-light torus blocks dual to geodesic networks in the
thermal AdS3 can be obtained from the large-$h$ (“heavy-light”) expansion of
global torus blocks [49, 36, 35, 50]. The Casimir approach could be useful
here, see e.g. [16].

An important issue completely left beyond the scope of this paper is the
Casimir approach for torus blocks in general channels described by diagrams
containing the necklace as a sub-diagram of lowest order, see Fig. 2 (b). The
only non-necklace channel for which the Casimir approach has been explicitly
formulated is the OPE channel [16] (see Fig. 2 (a)), which is opposite to the
necklace channel in the sense of maximum/minimum number of used
OPE/projectors, see our discussion in Section 2.2.

Also, from the AdS3/CFT2 perspective it would be useful to study
$\mathcal{N}=1$ supersymmetric extensions of the Casimir approach for
$n$-point superblocks similar to the case of 1-point torus superblocks [25].

#### Acknowledgements.

Our work was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

##  Appendix A Derivation of the torus Casimir equations for 2-point blocks

In what follows we explicitly derive the Casimir equations in the case $n=2$
and analyze their limits $n=2\to n=1$ at any $q$ and $q\to 0$ at $n=2$.

#### Derivation of 2-point Casimir equations.

In the 2-point case, the Casimir irreducibility conditions (3.10) are given by
two relations

$\begin{array}[]{c}\displaystyle
1)\;\;\operatorname{Tr}_{\tilde{h}_{1}}[\text{C}_{2}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]=-\tilde{h}_{1}(\tilde{h}_{1}-1)\mathcal{F}_{h,\tilde{h}}\,,\vspace{2mm}\\\
\displaystyle
2)\;\;\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\text{C}_{2}\,\mathcal{O}_{2}]=-\tilde{h}_{2}(\tilde{h}_{2}-1)\mathcal{F}_{h,\tilde{h}}\,.\end{array}$
(A.1)

In order to represent (A.1) as the eigenvalue equations for the 2-point block
function $\mathcal{F}_{h,\tilde{h}}$ in the necklace channel (2.11) one
substitutes the Casimir operator
$\operatorname{C_{2}}=L_{1}L_{-1}-L_{0}-L_{0}^{2}$ (3.9) into the both
relations. Each of the three terms in $\operatorname{C_{2}}$ is to be treated
separately.

Let us begin with the first relation in (A.1) and consider the term
$L_{1}L_{-1}$. The trick is to commute $L_{-1}$ to the right to obtain the
following relation

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}L_{-1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]&=\frac{1}{q}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}L_{-1}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]=\\\
&=\frac{1}{q}\mathcal{L}_{-1}^{(1)}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]+\frac{1}{q}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{-1}\mathcal{O}_{2}]\,,\end{split}$
(A.2)

where we used $L_{n}q^{L_{0}}=q^{L_{0}+n}L_{n}$,
$L_{n}\mathbb{P}=\mathbb{P}L_{n}$, and
$[L_{n},\mathcal{O}_{i}(z_{i})]=\mathcal{L}_{n}^{(i)}\mathcal{O}_{i}(z_{i})$
(2.4). Commuting $L_{-1}$ with $\mathcal{O}_{2}$ in the last trace in (A.2),
using the cyclic property of the trace and then commuting operators $L_{\pm
1}$, we obtain

$\begin{array}[]{l}\displaystyle\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}L_{-1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]=\frac{1}{q}(\mathcal{L}_{-1}^{(1)}+\mathcal{L}_{-1}^{(2)})\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]\vspace{2mm}\\\
\displaystyle\hskip
122.34685pt-\frac{2}{q}\operatorname{Tr}_{\tilde{h}_{1}}[L_{0}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]+\frac{1}{q}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}L_{-1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]\,.\end{array}$
(A.3)

The last step is to calculate the trace
$\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]$
in the first line of (A.3). Using the same procedure one finds

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]&=q\left(\mathcal{L}_{1}^{(1)}+\mathcal{L}_{1}^{(2)}\right)\mathcal{F}_{h,\tilde{h}}+q\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}L_{1}]\,.\end{split}$
(A.4)

By the cyclic property of the trace the last term is similar to the left-hand
side term. Substituting it back into (A.3) one obtains

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[L_{1}L_{-1}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]&=\frac{2}{1-q}q\partial_{q}\mathcal{F}_{h,\tilde{h}}-\frac{q}{(1-q)^{2}}\left(\mathcal{L}_{-1}^{(1)}+\mathcal{L}_{-1}^{(2)}\right)\left(\mathcal{L}_{1}^{(1)}+\mathcal{L}_{1}^{(2)}\right)\mathcal{F}_{h,\tilde{h}}\,.\end{split}$
(A.5)

Now, $L_{0}$ and $L_{0}^{2}$ terms in the Casimir operator are calculated by
using the following obvious relations

$\operatorname{Tr}_{\tilde{h}_{1}}[L_{0}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]=q\partial_{q}\mathcal{F}_{h,\tilde{h}}\,,\qquad\operatorname{Tr}_{\tilde{h}_{1}}[L^{2}_{0}q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}\mathcal{O}_{2}]=(q\partial_{q})^{2}\mathcal{F}_{h,\tilde{h}}\,.$
(A.6)

Finally, the first Casimir equation is cast into the form

$\begin{split}\left[-q^{2}\partial_{q}^{2}+\frac{2q}{1-q}q\partial_{q}-\frac{q}{(1-q)^{2}}\left(\mathcal{L}_{-1}^{(1)}+\mathcal{L}_{-1}^{(2)}\right)\left(\mathcal{L}_{1}^{(1)}+\mathcal{L}_{1}^{(2)}\right)\right]\mathcal{F}_{h,\tilde{h}}=-\tilde{h}_{1}(\tilde{h}_{1}-1)\mathcal{F}_{h,\tilde{h}}\,,\end{split}$
(A.7)

which is now the eigenvalue equation for $\mathcal{F}_{h,\tilde{h}}$. This is
the first equation in (3.11) at $n=2$.

The same procedure applies to the second relation in (A.1). Isolating the term
$L_{1}L_{-1}$ in the Casimir operator (3.9) one obtains

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{1}L_{-1}\mathcal{O}_{2}]&=\frac{1}{q}(\mathcal{L}_{-1}^{(1)}+q\mathcal{L}_{-1}^{(2)})\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{1}\mathcal{O}_{2}]-\\\
&-\frac{2}{q}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{0}\mathcal{O}_{2}]+\frac{1}{q}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{1}L_{-1}\mathcal{O}_{2}]\,,\end{split}$
(A.8)

where first and second traces on the right-hand side can be cast into the form

$\begin{array}[]{l}\displaystyle\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{1}\mathcal{O}_{2}]=\frac{1}{1-q}\left(\mathcal{L}_{1}^{(2)}+q\mathcal{L}_{1}^{(1)}\right)\mathcal{F}_{h,\tilde{h}}\,,\vspace{2mm}\\\
\displaystyle\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{0}\mathcal{O}_{2}]=\mathcal{L}_{0}^{(2)}\mathcal{F}_{h,\tilde{h}}+q\partial_{q}\mathcal{F}_{h,\tilde{h}}\,.\end{array}$
(A.9)

Substituting these expressions into (A.8) yields

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{1}L_{-1}\mathcal{O}_{2}]&=\frac{2}{(1-q)}q\partial_{q}\mathcal{F}_{h,\tilde{h}}+\mathcal{L}_{0}^{(2)}\mathcal{F}_{h,\tilde{h}}-\\\
&-\frac{1}{(1-q)^{2}}\left(\mathcal{L}_{-1}^{(1)}+q\mathcal{L}_{-1}^{(2)}\right)\left(\mathcal{L}_{1}^{(2)}+q\mathcal{L}_{1}^{(1)}\right)\mathcal{F}_{h,\tilde{h}}\,.\end{split}$
(A.10)

Using (A.9) one derives the last term in Casimir operator (3.9) as

$\begin{split}\operatorname{Tr}_{\tilde{h}_{1}}[q^{L_{0}}\mathcal{O}_{1}\mathbb{P}_{\tilde{h}_{2}}L_{0}^{2}\mathcal{O}_{2}]=\left(\mathcal{L}_{0}^{(2)}\right)^{2}\mathcal{F}_{h,\tilde{h}}+2\mathcal{L}_{0}^{(2)}q\partial_{q}\mathcal{F}_{h,\tilde{h}}+q\partial_{q}(q\partial_{q}\mathcal{F}_{h,\tilde{h}})\,.\end{split}$
(A.11)

Substituting (A.10), (A.9), (A.11) into (A.1) one finally obtains

$\begin{array}[]{l}\displaystyle\left[-q^{2}\partial_{q}^{2}+\frac{2q}{1-q}q\partial_{q}-\frac{1}{(1-q)^{2}}\left(\mathcal{L}_{-1}^{(1)}+q\mathcal{L}_{-1}^{(2)}\right)\left(\mathcal{L}_{1}^{(2)}+q\mathcal{L}_{1}^{(1)}\right)\right.\vspace{2mm}\\\
\displaystyle\hskip
113.81102pt\left.+\frac{1+q}{1-q}\mathcal{L}_{0}^{(2)}-\left(\mathcal{L}_{0}^{(2)}\right)^{2}-2\mathcal{L}_{0}^{(2)}q\partial_{q}\right]\mathcal{F}_{h,\tilde{h}}=-\tilde{h}_{2}(\tilde{h}_{2}-1)\mathcal{F}_{h,\tilde{h}}\,,\end{array}$
(A.12)

which is the second equation in (3.11) at $n=2$. It is straightforward to
check that the 2-point torus block function (3.2) does solve the Casimir
equation system (A.7) and (A.12). The same procedure in the $n$-point case
yields the Casimir system (3.11).

#### Reduction to the 1-point Casimir equation.

Let us show that the 2-point Casimir equations (A.7) and (A.12) are reduced to
the 1-point Casimir equation [16]. To this end, we choose the second primary
operator to be the identity operator, $\mathcal{O}_{2}(z_{2})=\mathbb{I}$,
i.e. $h_{2}=0$. It follows that the intermediate dimensions must be equated,
$\tilde{h}_{1}=\tilde{h}_{2}$.

The 2-point conformal block
$\mathcal{F}_{h,\tilde{h}}\equiv\mathcal{F}_{h_{1}h_{2},\tilde{h}_{1}\tilde{h}_{2}}$
satisfies the Ward identity (3.12):
$(\mathcal{L}_{0}^{(1)}+\mathcal{L}_{0}^{(2)})\mathcal{F}_{h_{1}h_{2},\tilde{h}_{1}\tilde{h}_{2}}=0$,
which by using (2.4) takes the form

$\left(h_{1}+h_{2}+z_{1}\partial_{1}+z_{2}\partial_{2}\right)\mathcal{F}_{h_{1}h_{2},\tilde{h}_{1}\tilde{h}_{2}}=0\,.$
(A.13)

On the other hand, the $1$-point block
$\mathcal{F}_{h_{1},\tilde{h}_{1}}\equiv\mathcal{F}_{h_{1}h_{2},\tilde{h}_{1}\tilde{h}_{2}}\big{|}_{h_{2}=0,\tilde{h}_{2}=\tilde{h}_{1}}$
satisfies the Ward identity
$\mathcal{L}_{0}^{(1)}\mathcal{F}_{h_{1},\tilde{h}_{1}}=0$. Then, together
with the condition $h_{2}=0$, equation (A.13) gives

$\partial_{2}\mathcal{F}_{h_{1},\tilde{h}_{1}}=0\,.$ (A.14)

Thus, the first Casimir equation (A.7) takes the form [16]

$\left[-q^{2}\partial_{q}^{2}+\frac{2q}{1-q}q\partial_{q}+\frac{q}{(1-q)^{2}}h_{1}(h_{1}-1)\right]\mathcal{F}_{h_{1},\tilde{h}_{1}}=-\tilde{h}_{1}(\tilde{h}_{1}-1)\mathcal{F}_{h_{1},\tilde{h}_{1}}\,.$
(A.15)

Similarly, using (A.14) and $\tilde{h}_{1}=\tilde{h}_{2}$ one shows that the
second Casimir equation (A.12) also reduces to (A.15).

#### Reduction from the $2$-point torus block to the $4$-point plane block.

Let us illustrate the relation (4.2) for the $2$-point torus block with
external dimensions $(h_{1},h_{2})$ and intermediate dimensions
$(\tilde{h}_{1},\tilde{h}_{2})$ in the plane limit. Setting $n=j=2$ in (4.3)
we find

$\bigg{[}\tilde{h}_{1}(1-\tilde{h}_{1})-\mathcal{L}_{-1}^{(1)}\mathcal{L}_{1}^{(2)}+(1-2\tilde{h}_{1})\mathcal{L}_{0}^{(2)}-\left(\mathcal{L}_{0}^{(2)}\right)^{2}+\tilde{h}_{2}(\tilde{h}_{2}-1)\bigg{]}\mathcal{F}_{0}(z_{1},z_{2})=0\,.$
(A.16)

Using (2.4) one directly shows that (A.16) is reduced to the hypergeometric
equation and its solution has the following form

$\mathcal{F}_{0}(z_{1},z_{2})=z_{1}^{-h_{1}}z_{2}^{-h_{2}}\;x^{\tilde{h}_{2}-\tilde{h}_{1}}\;{}_{2}F_{1}\left(\tilde{h}_{2}-\tilde{h}_{1}+h_{1},\tilde{h}_{2}-\tilde{h}_{1}+h_{2},2\tilde{h}_{2}\,\big{|}\,x\right),$
(A.17)

where $x=z_{2}/z_{1}$. On the other hand, the 4-point plane block is known to
be given by the hypergeometric function of external dimensions
$(h_{0},h_{1},h_{2},h_{3})$ and intermediate dimension $\tilde{h}$:

$\begin{array}[]{l}\displaystyle\tilde{\mathcal{F}}(z_{0},z_{1},z_{2},z_{3}|h,\tilde{h})=z_{01}^{-h_{0}-h_{1}}z_{23}^{-h_{2}-h_{3}}z_{02}^{-h_{0}+h_{1}}z_{12}^{h_{0}-h_{1}-h_{2}+h_{3}}z_{13}^{h_{2}-h_{3}}\vspace{2mm}\\\
\displaystyle\hskip
85.35826pt\times\,\eta^{\tilde{h}}(1-\eta)^{h_{2}-h_{3}-h_{0}+h_{1}}\;{}_{2}F_{1}\left(\tilde{h}-h_{0}+h_{1},\tilde{h}+h_{2}-h_{3},2\tilde{h}\,\big{|}\,\eta\right),\end{array}$
(A.18)

where $z_{ij}\equiv z_{i}-z_{j}$ and the cross-ratio
$\eta=(z_{01}z_{23})/(z_{02}z_{13})$. Then, (A.17) is indeed related to (A.18)
by (4.2) provided that the external/intermediate dimensions in (A.18) are
chosen as $(\tilde{h}_{1},h_{1},h_{2},\tilde{h}_{1})$ and $\tilde{h}_{2}$.

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