Modular linear differential equations for four-point sphere conformal blocks

Ratul Mahanta a and Tanmoy Sengupta b,c

a INFN, Sezione di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy
b The Institute of Mathematical Sciences, IV Cross Road, C.I.T. Campus,
Taramani, Chennai 600113, India
c Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India

<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

> We construct modular linear differential equations (MLDEs) w.r.t. subgroups
> of the modular group whose solutions are Virasoro conformal blocks appearing
> in the expansion of a crossing symmetric 4-point correlator on the sphere.
> This uses a connection between crossing transformations and modular
> transformations. We focus specifically on second order MLDEs with the cases
> of all identical and pairwise identical operators in the correlator. The
> central charge, the dimensions of the above operators and those of the
> intermediate ones are expressed in terms of parameters that occur in such
> MLDEs. In doing so, the $q$-expansions of the solutions to the MLDEs are
> compared with those of Virasoro blocks; hence, Zamolodchikov’s elliptic
> recursion formula provides an important input. Using the actions of
> respective subgroups, bootstrap equations involving the associated 3-point
> coefficients have been set up and solved as well in terms of the MLDE
> parameters. We present explicit examples of MLDEs corresponding to BPZ and
> novel non-BPZ equations, as well as unitary and non-unitary CFTs.

###### Contents

  1. 1 Introduction and summary
  2. 2 Review
    1. 2.1 Stripped correlators
    2. 2.2 Modular properties of stripped correlators
  3. 3 MLDE for sphere blocks and bootstrap
  4. 4 Case studies: second order MLDEs
    1. 4.1 All identical operators
    2. 4.2 Pairwise identical operators
  5. 5 Discussions
  6. A Zamolodchikov’s $H$-function
  7. B $q$-expansion of certain functions
  8. C Closed form solutions

##  1 Introduction and summary

There are infinite-dimensional local algebras in 2 dimensions, having Virasoro
algebra as a subalgebra. The conformal blocks on the sphere are determined by
these algebras at generic central charge $c$ and operator dimensions
$\\{h_{a},h^{\rm int}_{p}\\}$, where $h_{a}$ correspond to external operators
$O_{a}$ and $h^{\rm int}_{p}$ to the intermediate operator $O^{\rm int}_{p}$.
On the other hand, conformal field theories (CFTs) are specified by consistent
CFT data namely the central charge, the operator dimensions in the spectrum
and the 3-point structure constants. The latter occurs in the conformal block
expansion coefficients of the correlators. As one of the consistency criteria
we require crossing symmetry of the correlators on the sphere, and to set up
such bootstrap equations we need to know the matrices connecting the conformal
blocks at two singularities in the cross-ratio $x$. In plenty of cases, blocks
have been obtained in closed forms as solutions to differential equations
satisfied by the correlators, and then the said connection matrices have been
obtained, e.g. [1, 2, 3, 4]. They have employed null states w.r.t. the local
algebra to derive such differential equations; however, see the appendix of
[5] for a discussion on general construction of such differential equations on
the sphere.

This paper discusses the construction of differential equation for the
conformal blocks that appear in the expansion of a crossing symmetric 4-point
sphere correlator. Crossing symmetry translates as certain modular properties
of the correlator under a well-known map [6] between the cross-ratio space and
the complex upper half plane $\mathbb{H}_{+}$ [7, 8, 9]. This allows us to
write modular linear differential equations (MLDEs) in $\tau\in\mathbb{H}_{+}$
w.r.t. some subgroup $\tilde{\Gamma}$ of the modular group for the conformal
blocks appearing in the correlator. We determine $\tilde{\Gamma}$ based on
whether some operators in the correlator are identical or not. By construction
the order of the differential equation is equal to the number of blocks
appearing in the expansion of the correlator. Since we do not use null states,
our construction is general and includes null state equations as subcases.
This construction is in the spirit of [10, 11], which wrote MLDEs for the
characters in rational CFTs in keeping with the modular invariance of the
torus partition function formed by the characters.111See [12, 13, 14, 15, 16,
17, 18, 19, 20, 21] for recent works along this line. However, the group does
not have to be the full modular group in our case. And, we do not need to
assume a finite number of operators in the spectrum; we only need to assume a
finite number of conformal blocks in the expansion of a correlator $\langle
O_{1}O_{2}O_{3}O_{4}\rangle$ in consideration. The most general scenario for
the latter to happen is as follows.222An integral version can also be written
for this scenario resulting in finitely many blocks in the expansion of the
correlator $\langle O_{1}O_{2}O_{3}O_{4}\rangle$.

$O_{1}\times O_{2}=\sum_{p=1}^{N}O^{\rm int}_{p}+\sum_{q}\Xi^{\rm
int}_{q},\qquad O_{3}\times O_{4}=\sum_{p=1}^{N}O^{\rm
int}_{p}+\sum_{r}\tilde{\Xi}^{\rm int}_{r}\,,$ (1)

where in the fusion rules $O_{a}$ are the 4 external operators and “int”
denotes the intermediate operators. And, the operators $\Xi^{\rm int}_{q}$ are
all distinct from $\tilde{\Xi}^{\rm int}_{r}$.

Although $\tilde{\Gamma}$ differs depending on whether some or all of the
external operators are identical or not, they all contain $\Gamma(2)$ as a
subgroup. Thus, we begin with an ansatz for the MLDE w.r.t. $\Gamma(2)$ that
introduces a set of parameters $\vec{\alpha}$. Modular forms w.r.t.
$\Gamma(2)$ can be expanded in terms of specific powers of the Jacobi theta
functions, and $\vec{\alpha}$ are the expansion coefficients. The locus on the
parameter space is then found so that it forms an MLDE w.r.t.
$\tilde{\Gamma}$. The rationale behind such a construction is as follows. We
strip out a specific factor from the correlator that encodes the statistics in
2 dimensions, following [7]. The crossing symmetry of the stripped correlator
translates as the invariance under $\tilde{\Gamma}$-actions on $\tau$. The
holomorphic blocks that contribute to the expansion of the stripped correlator
are therefore generally allowed to transform as a vector valued modular
function under $\tilde{\Gamma}$, and should so satisfy such an MLDE. In
particular, the $\Gamma(2)$-actions on $\tau$ generate closed loops circling
$x=0,1$. Thus, the $\Gamma(2)$-invariance is interpreted as the single-
valuedness of the stripped correlator. Since the statistics are already
encoded in the stripped out factor, this should be the case.

In this paper, we further tune $\vec{\alpha}$ such that the solutions to above
MLDE are Virasoro conformal blocks. We organise our program as the following
series of steps. Solving the MLDE, we obtain the $q$-expansion coefficients of
trial solutions at generic $\vec{\alpha}$. The $q$-expansions of the Virasoro
blocks can be computed using Zamolodchikov’s elliptic recursion relation [6].
We identify $c,h_{a},h^{\rm int}_{p}$ in terms of $\vec{\alpha}$ by comparing
the leading behaviour as $q\to 0$ and the first few coefficients in the trial
solutions with those of Virasoro blocks. The parameter space is then
constrained such that the higher coefficients also match. In specific
examples, we check it numerically to a sufficiently high order in the
$q$-expansions. In latter checks, we numerically produce the $q$-expansion
coefficients in the solutions using known $q$-expansions of the modular forms
and solving the MLDE recursively. On the other hand, we use a Mathematica code
published in [22] for the numerical computations of $q$-expansion coefficients
in the Virasoro blocks. On the surviving parameter space $\vec{\alpha}$, the
solutions to MLDE are 4-point Virasoro blocks. We discuss the (hyper-)surfaces
in this parameter space on which the MLDE corresponds to BPZ equations and the
possibilities of new equations. Using the $\tilde{\Gamma}$-actions, bootstrap
equations involving the associated 3-point coefficients have been set up and
solved as well in terms of the MLDE parameters $\vec{\alpha}$. For above
implementations, we particularly focus on second order MLDEs with the cases of
all identical operators and pairwise identical operators in the correlator.
Let us summarize our main results in the following.

  * •

In the case of all identical primaries in the correlator and 2 intermediate
ones, one of which is identity and the other with dimension greater than zero,
we produce a MLDE w.r.t. the full modular group with a single parameter whose
solutions are the 4-point Virasoro blocks. In this case, the MLDE always
corresponds to a second order BPZ equation. For scalar operators, the square
of the associated non-trivial 3-point coefficient is expressed in terms of the
MLDE parameter. Correlator of 4 spinning operators vanishes in this case.

  * •

We also present explicit examples of MLDEs in both unitary and non-unitary
theories in the above case. By testing various unitarity bounds, we constrain
the parameter space. In the present case, the non-negativity of the
$q$-expansion coefficients of the pillow blocks (which are related to the
Virasoro blocks) applies as unitarity bound [23, 24]. We provide evidence that
this is not a sufficient condition for unitarity.

  * •

In the case of pairwise identical primaries in the correlator and 2
intermediate ones, one of which is identity and the other with dimension
greater than zero, starting from some specific examples of MLDEs (obtained by
a computer-assisted search) we use linear perturbations around them to
generate (infinite-)family of MLDEs w.r.t. a subgroup of the modular group
whose solutions are the 4-point Virasoro blocks. The product of the associated
non-trivial 3-point coefficients is expressed in terms of the MLDE parameters,
confining us to scalar operators. We present some MLDEs that do not correspond
to BPZ equations. These are new second order differential equations whose
solutions are Virasoro blocks. These specific cases are non-unitary.

  * •

We also discuss the importance to study MLDEs with meromorphic forms as
coefficients, since by tuning the parameters solutions to such MLDEs may as
well give 4-point Virasoro blocks. We present one such example where although
the differential equation has additional singularities than the expected ones
the Virasoro blocks are free from them.

As it stands now, our program focuses on constructing differential equations
for conformal blocks on a basis that is more general than the occurrence of
null states, rather than classifying CFTs.333However, a classification program
in the context of MLDEs for blocks may also be initiated by considering
correlators of the intermediate operators (which we obtain using our second
order MLDEs) with those of the external ones, and writing down MLDEs for the
blocks of these new correlators to see the dimensions of possible new(if any)
intermediate operators. In turn, we advance toward the full spectrum of the
CFT. But, the complexity lies in allowing higher order MLDEs for the blocks in
the new correlators. We do not bear any expectations regarding integrality of
the $q$-expansion coefficients in the conformal blocks, unlike characters,
which makes our program challenging. In fact, that is another place where our
program differs from that of [10, 11]. In the latter demanding integrality of
the coefficients greatly helps to obtain a finite number of values of the MLDE
parameter, whereas we get connected region(s) in the parameter space
$\vec{\alpha}$ where our required MLDEs exist. Nonetheless, we have reported
whether the coefficients are non-negative integers or not in each of our
examples presented, keeping in mind that the cases with integer coefficients
may be suitable for a sphere-torus correspondence as recently reported in [9].

This paper is organized as follows. In section 2, we review the relationship
between crossing symmetry and modular transformations. In section 3, we
formulate our program of the construction of MLDEs for the Virasoro blocks
appearing in the expansion of a crossing symmetric 4-point correlator on the
sphere. Section 4 deals with the studies of second order MLDEs with section
4.1 focusing on the case of all identical external operators and section 4.2
on the pairwise identical external operators. We end by discussing potential
interesting generalisations in section 5.

##  2 Review

Under a suitable map between the cross-ratio space and the complex upper half
plane, crossing symmetry of a 4-point correlator on the sphere translates as
certain modular transformation properties. In this section, we review the
necessary elements of this relationship. For more details, see [6, 7, 8, 9].

###  2.1 Stripped correlators

For 4 (quasi-)primary operators, [7] introduced stripped correlators whose
transformations under crossing are simple in the sense explained below.

Let $O_{1}(z_{1},\bar{z}_{1}),\dots,O_{4}(z_{4},\bar{z}_{4})$ be 4
(quasi-)primary operators located on the sphere. A 4-point correlator of these
operators is denoted as

$\langle
O_{a}(z_{a},\bar{z}_{a})O_{b}(z_{b},\bar{z}_{b})O_{c}(z_{c},\bar{z}_{c})O_{d}(z_{d},\bar{z}_{d})\rangle$

where $(abcd)$ is a permutation of $(1234)$. It is a certain function of the
operator locations and conformal dimensions, which remains invariant up to a
phase if we reorder the operators inside the angle brackets
$\langle\cdots\rangle$. The latter property is referred to as crossing
symmetry and the said phase encodes the statistics of the operators in 2
dimensions.

Let us label the 4 places from left to right inside the angle brackets
$\langle\cdots\rangle$ by $i=1,\dots,4$. With each ordering of the 4 operators
inside the angle brackets, we associate a cross-ratio of their locations, see
(2). Consider any given ordering of the 4 operators, w.r.t. which let
$(z(i),\bar{z}(i))$ and $(h(i),\bar{h}(i))$ respectively denote the location
and the dimension of the operator that sits at the $i$-th place. The
corresponding correlator can be factored into 2 components, one of which is a
function of the associated cross-ratio and the other satisfies the Ward
identities for Global conformal symmetry. I.e.,

$\displaystyle\big{\langle}\underbrace{}_{\color[rgb]{1,0,0}1}\underbrace{}_{\color[rgb]{1,0,0}2}\underbrace{}_{\color[rgb]{1,0,0}3}\underbrace{}_{\color[rgb]{1,0,0}4}\big{\rangle}=\text{function
of cross-
ratio}\times\underbrace{\prod_{i<j=2}^{4}(z_{ij}^{\mu_{ij}}\cdot\bar{z}_{ij}^{\mu_{ij}})}_{\equiv
G_{0}}\,,$ $\displaystyle
z_{ij}=z(i)-z(j),\quad\mu_{ij}=\frac{1}{3}\sum_{k=1}^{4}h(k)-h(i)-h(j),\quad\bar{\mu}_{ij}=\frac{1}{3}\sum_{k=1}^{4}\bar{h}(k)-\bar{h}(i)-\bar{h}(j)\,,$
$\displaystyle\text{cross-
ratio}\equiv\frac{\big{(}z(1)-z(2)\big{)}\big{(}z(3)-z(4)\big{)}}{\big{(}z(1)-z(3)\big{)}\big{(}z(2)-z(4)\big{)}}\
.$ (2)

A specific solution to the global Ward identities is used above which we
denote by $G_{0}$. Under permutations of operators inside the angle brackets,
$G_{0}$ picks the phase suited to the statistics of the operators. E.g., it
remains invariant for scalar operators.

W.r.t. above rule for cross-ratio, the stripped correlators $G_{abcd}$ as
functions of cross-ratios are defined as

$\displaystyle\big{\langle}\underbrace{O_{a}(z_{a},\bar{z}_{a})}_{\color[rgb]{1,0,0}1}\underbrace{O_{b}(z_{b},\bar{z}_{b})}_{\color[rgb]{1,0,0}2}\underbrace{O_{c}(z_{c},\bar{z}_{c})}_{\color[rgb]{1,0,0}3}\underbrace{O_{d}(z_{d},\bar{z}_{d})}_{\color[rgb]{1,0,0}4}\big{\rangle}=\
G_{abcd}\big{(}x_{abcd},\bar{x}_{abcd}\big{)}\times G_{0}\,,$ $\displaystyle
x_{abcd}=\frac{(z_{a}-z_{b})(z_{c}-z_{d})}{(z_{a}-z_{c})(z_{b}-z_{d})},\quad(abcd)=\text{permutation
of}\ (1234)\,.$ (3)

Now, denoting $x=x_{1234}$ crossing symmetry relates them as follows444Here,
$x$ lives on a 3-punctured sphere which we call as the cross-ratio space.

$\displaystyle{\text{\it non-trival}:}\quad G_{1234}(x,\bar{x})$
$\displaystyle=G_{1243}(\frac{x}{x-1},\frac{\bar{x}}{\bar{x}-1})=G_{4231}(\frac{1}{x},\frac{1}{\bar{x}})=G_{3214}(1-x,1-\bar{x})$
$\displaystyle=G_{3241}(\frac{x-1}{x},\frac{\bar{x}-1}{\bar{x}})=G_{4213}(\frac{1}{1-x},\frac{1}{1-\bar{x}})\,,$
$\displaystyle{\text{\it trival}:}\quad G_{1234}(x,\bar{x})$
$\displaystyle=G_{2143}(x,\bar{x})=G_{3412}(x,\bar{x})=G_{4321}(x,\bar{x})\,.$
(4)

Furthermore, each $G_{abcd}(x,\bar{x})$ is single-valued in $x$.555Here, we
mean on the slice $\bar{x}=x^{*}$ with $x^{*}$ denoting the complex conjugate
of $x$. Only 6 among all the $G_{abcd}$ are independent. From them all the
others can be obtained using the trivial relations in (4).

We emphasize that the specific selection of $G_{0}$ results in the aforesaid
simple crossing transformations of the stripped correlators. There are no
$x$-dependent prefactors in any of the equalities in (4), in contrast to [1,
4]. They reveal the role of the full modular group, as discussed in the
following subsection. However, the role of modular $S$-transformation is
observable with other choices of $G_{0}$ and has long been studied; for recent
considerations, see [25, 26, 27, 28].

We also note that

$\displaystyle\langle O_{1}(\infty)O_{2}(1)O_{3}(x)O_{4}(0)\rangle$
$\displaystyle\equiv\lim_{z_{1},\bar{z}_{1}\to\infty}z_{1}^{2h_{1}}\bar{z}_{1}^{2\bar{h}_{1}}\langle
O_{1}(z_{1},\bar{z}_{1})O_{2}(1,1)O_{3}(x,\bar{x})O_{4}(0,0)\rangle$
$\displaystyle=\
x^{\mu_{34}}(1-x)^{\mu_{23}}\bar{x}^{\bar{\mu}_{34}}(1-\bar{x})^{\bar{\mu}_{23}}G_{1234}(x,\bar{x})\,.$
(5)

The holomorphic conformal blocks that contribute to the stripped correlator
$G_{1234}(x,\bar{x})$ exhibit the following leading behaviour as $x\to
0$.666Since the holomorphic conformal blocks that contribute to $\langle
O_{1}(\infty)O_{2}(1)O_{3}(x)O_{4}(0)\rangle$ go as $x^{h^{\rm
int}_{p}-h_{3}-h_{4}}$ in the limit $x\to 0$ [1, 4].

$F_{p}(x)=x^{h^{\rm
int}_{p}-\frac{\mathfrak{H}}{3}}\left(1+\mathcal{O}(x)\right),\quad\mathfrak{H}=h_{1}+\cdots+h_{4}\,,$
(6)

where $h^{\rm int}_{p}$ denotes the dimension of the intermediate operator
$O^{\rm int}_{p}$ that appears in the fusion channel: $O_{3}\times
O_{4}=\sum_{p}O^{\rm int}_{p}$. As opposed to the intermediate ones, we call
the operators which sit inside the angle brackets as external operators.

###  2.2 Modular properties of stripped correlators

Consider the map from the complex upper half plane $\mathbb{H}_{+}$ to the
cross-ratio space777We take the cross-ratio space as
$\mathbb{C}\setminus\\{0,1\\}$ setting the 3 punctures at $0,1,\infty$ using
global conformal transformations. given by the elliptic lambda function as
follows.888$\theta_{2}(\tau),\theta_{3}(\tau),\theta_{4}(\tau)$ are the Jacobi
theta functions related by Jacobi Identity:
$\theta_{4}(\tau)^{4}=\theta_{3}(\tau)^{4}-\theta_{2}(\tau)^{4}$. See [29] for
further details about $\lambda(\tau),\theta_{r}(\tau)$.

$\displaystyle
x=\lambda(\tau)=\left(\frac{\theta_{2}(\tau)}{\theta_{3}(\tau)}\right)^{4},\quad\tau\in\mathbb{H}_{+}\,,$
$\displaystyle\theta_{2}(\tau)=\sum_{n\in\mathbb{Z}}e^{\pi
i\tau(n+\frac{1}{2})^{2}},\quad\theta_{3}(\tau)=\sum_{n\in\mathbb{Z}}e^{\pi
i\tau n^{2}}\,.$ (7)

The action of the modular group $PSL(2,\mathbb{Z})$ on $\tau$ generates all
the crossing transformations on $x$. In particular, under the actions of the 2
generators of the modular group which are $T:\tau\to\tau+1,\
S:\tau\to-\frac{1}{\tau}$, the images on the cross-ratio space respectively
have the transformations: $T\cdot x=\frac{x}{x-1},\ S\cdot x=1-x$.

$\lambda(\tau)$ remains invariant under the index 6 normal subgroup
$\Gamma(2)$ of the modular group generated by $T^{2}$ and $ST^{2}S$. In
particular, $T^{2},ST^{2}S$ generate simple loops around $x=0,1$ respectively.
Consequently, $\Gamma(2)$ plays a role in single-valuedness of each
$G_{abcd}(x,\bar{x})$ in $x$ as discussed below.

Arrange any 6 independent $G_{abcd}$ in a column and consider it to be a
vector-valued modular function (vvmf)999More generally, w.r.t. a subgroup
$\Gamma\subset PSL(2,\mathbb{Z})$ a vector-valued modular form $\vec{V}$ of
weight $k$ is defined as [30]
$\vec{V}(\frac{a\tau+b}{c\tau+d})=\sigma(\begin{pmatrix}a&b\\\
c&d\end{pmatrix})(c\tau+d)^{k}\vec{V}(\tau),\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\,,$ with $\sigma$ as linear a representation of
$\Gamma$. Hereafter, the shorthand vvmf refers to a vector-valued modular form
of weight $0$ w.r.t. some subgroup $\Gamma$. The subgroup should be clear from
the context unless specified. w.r.t. $PSL(2,\mathbb{Z})$ on $\mathbb{H}_{+}$
as follows.

$\displaystyle\vec{G}(\tau,\bar{\tau})=\begin{pmatrix}G_{1234}(\tau,\bar{\tau})\\\
G_{2134}(\tau,\bar{\tau})\\\ G_{4132}(\tau,\bar{\tau})\\\
G_{1432}(\tau,\bar{\tau})\\\ G_{2431}(\tau,\bar{\tau})\\\
G_{4231}(\tau,\bar{\tau})\end{pmatrix}:\quad\vec{G}(\gamma\tau,\gamma\bar{\tau})=\sigma(\gamma)\cdot\vec{G}(\tau,\bar{\tau})\text{,
}\quad\gamma\in PSL(2,\mathbb{Z})\,,$
$\displaystyle\sigma(S)=\begin{pmatrix}0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\end{pmatrix},\quad\sigma(T)=\begin{pmatrix}0&1&0&0&0&0\\\
1&0&0&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&0&0&1\\\
0&0&0&0&1&0\end{pmatrix}.$ (8)

More precisely, matrices $\sigma(\gamma)$ form a linear representation of
$PSL(2,\mathbb{Z})/\Gamma(2)$. The chosen matrices $\sigma(S),\sigma(T)$
ensure the non-trivial crossing transformations in (4). Also, we have
$\sigma(T^{2})=\sigma(ST^{2}S)=\mathds{1}$. As a result, each component of
$\vec{G}(\tau,\bar{\tau})$ has $\Gamma(2)$-invariance, which ensures its
single-valuedness in $x=\lambda(\tau)$.

A few comments are in order. In case all the 4 external operators are
distinct, all the 6 components of $\vec{G}(\tau,\bar{\tau})$ are independent
and each component is invariant under only $\Gamma(2)$. But, $\sigma(S),\
\sigma(T)$ are reducible when some or all of the 4 external operators are
identical. In the latter case, each component $G_{a}(\tau,\bar{\tau})$ is
invariant under a bigger subgroup $\tilde{\Gamma}_{a}$ that contains
$\Gamma(2)$. E.g., when there are 4 identical operators,
$\vec{G}(\tau,\bar{\tau})$ has only 1 independent component which is modular
invariant. When there are 2 pairs of identical operators,
$\vec{G}(\tau,\bar{\tau})$ has 3 independent components; the stabilizer
subgroup of each of them contains $\Gamma(2)$. In particular, the following
representation will be useful for the present work.

$\displaystyle O_{1}=O_{2}\equiv O_{L},\ O_{3}=O_{4}\equiv O_{R}\,,$
$\displaystyle\vec{G}(\tau,\bar{\tau})=\begin{pmatrix}G_{LLRR}(\tau,\bar{\tau})\\\
G_{LRLR}(\tau,\bar{\tau})\\\
G_{LRRL}(\tau,\bar{\tau})\end{pmatrix},\quad\sigma(S)=\begin{pmatrix}0&0&1\\\
0&1&0\\\ 1&0&0\end{pmatrix},\quad\sigma(T)=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}\,.$ (9)

In this case, the stabilizer of the vector $(1\ 0\ 0)^{t}$ is the subgroup of
$PSL(2,\mathbb{Z})$ generated by $T,ST^{2}S$.

Here, we tabulate the possible cases along with respective stabilizer
subgroups as per our cross-ratio convention and (8).

| Independent stripped correlators | Generators of stabilizer subgroup
---|---|---
All identical operators | $G_{OOOO}$ | $S,\ T$
Pairwise identical operators | $G_{LLRR}$ | $T,\ ST^{2}S$
$G_{RLRL}$ | $S,\ T^{2}$
$G_{LRRL}$ | $T^{2},\ STS$
3 identical operators | $G_{LRRR}$ | $S,\ T^{2}$
$G_{RLRR}$ | $T^{2},\ STS$
$G_{RRRL}$ | $T,\ ST^{2}S$
All distinct operators | $\begin{aligned} &\ \ \ \ \ G_{abcd}\ ,\\\\[-7.0pt] (abcd)=&\text{permutations of}\ (1234)\end{aligned}$ | $T^{2},\ ST^{2}S$

Table 1: List of possible stabilzer subgroups

##  3 MLDE for sphere blocks and bootstrap

Here, we consider the sphere correlator: $\langle
O_{1}(z_{1},\bar{z}_{1})O_{2}(z_{2},\bar{z}_{2})O_{3}(z_{3},\bar{z}_{3})O_{4}(z_{4},\bar{z}_{4})\rangle$.
From (3) and (7), the associated stripped correlator is
$G_{1234}(\tau,\bar{\tau})$, which is the first component of the vvmf
$\vec{G}(\tau,\bar{\tau})$ in (8). It has invariance under a subgroup
$\tilde{\Gamma}_{1}\supseteq\Gamma(2)$ of the modular group, as discussed in
section 2.2. We confine ourselves to the case where a finite number of
intermediate operators occur in the s-channel OPE $O_{3}\times
O_{4}$.101010More generally, the assumption (1) suffices for our analysis. We
denote the number by $N$.

Let the $N$ intermediate operators that contribute to the s-channel conformal
block expansion of the stripped correlator $G_{1234}(\tau,\bar{\tau})$ be:
$O_{p(k)},\ k=1,\dots,N$ with respective dimensions
$(h_{p(k)},\bar{h}_{p(k)})$. Since the real $(c,h)$ plane is the primary focus
of this article, we take $h_{p(k)}$ to be ordered as

$h_{p(1)}\leq h_{p(2)}\leq\cdots\leq h_{p(N)}\,,$ (10)

without loss of generality. Now, we have

$G_{1234}(\tau,\bar{\tau})=\sum_{k=1}^{N}C_{12p(k)}C_{34p(k)}F_{p(k)}(\tau,\bar{\tau})\,,$
(11)

where $C_{ijp(k)}$ are the 3-point coefficients and
$F_{p(k)}(\tau,\bar{\tau})$ are the non-holomorphic conformal blocks
respective to operators $O_{p(k)}$. Each $F_{p(k)}(\tau,\bar{\tau})$ further
factorizes as

$F_{p(k)}(\tau,\bar{\tau})=F_{p(k)}(\tau){\bar{F}}_{p(k)}(\bar{\tau})\,,$ (12)

where ${\bar{F}}_{p(k)}(\bar{\tau})$ is obtained with replacing
$\\{h_{1},\dots,h_{4},h_{p(k)},\tau\\}$ by
$\\{\bar{h}_{1},\dots,\bar{h}_{4},\bar{h}_{p(k)},\bar{\tau}\\}$ in the
holomorphic block $F_{p(k)}(\tau)$.

The holomorphic blocks $F_{p(k)}(\tau)$ mix among themselves under the action
of $\tilde{\Gamma}_{1}$ on the complex upper half plane $\mathbb{H}_{+}$,
while $G_{1234}(\tau,\bar{\tau})$ is an invariant.111111In particular, mixing
under $\Gamma(2)$ gives monodromies of the holomorphic blocks around $x=0,1$.
In contrast to the non-trivial monodromies around $x=1$, those around $x=0$
are only phases. I.e., the blocks $F_{p(k)}(\tau)$ can be arranged in a row
which transform as a vvmf w.r.t. $\tilde{\Gamma}_{1}$ as follows.121212For
definition, see footnote 9.

$\displaystyle
F_{p(k)}(\gamma\tau)=\sum_{k^{\prime}}F_{p(k^{\prime})}(\tau)M(\gamma)_{p(k^{\prime})p(k)},\quad\gamma\in\tilde{\Gamma}_{1}\supseteq\Gamma(2)\,,$
$\displaystyle M(\gamma_{2}\cdot\gamma_{1})=M(\gamma_{1})\cdot
M(\gamma_{2})\,,$ (13)

where matrices $M(\gamma)$ form a linear representation of
$\tilde{\Gamma}_{1}$. Now, we regard the $N$ holomorphic blocks
$F_{p(k)}(\tau)$ to be the independent solutions of a linear differential
equation in $\tau$ of order $N$. And, (13) requires the differential equation
to be invariant under $\tilde{\Gamma}_{1}$ i.e. an MLDE w.r.t.
$\tilde{\Gamma}_{1}$. Since $\tilde{\Gamma}_{1}\supseteq\Gamma(2)$131313Where
equality holds only when all the external operators are distinct., an MLDE
w.r.t. $\tilde{\Gamma}_{1}$ is always an MLDE w.r.t. $\Gamma(2)$, but the
converse is not true. Below, we take the strategy to start with the general
form of an MLDE w.r.t. $\Gamma(2)$ which introduces a set of parameters. Then,
we find the locus on the parameter space such that the differential equation
becomes an MLDE w.r.t. $\tilde{\Gamma}_{1}$.

To begin, the most generic ansatz of aforesaid differential equation that can
be written for the blocks is given by141414In general, the differential
operator in l.h.s. of (14) may act on a vector-valued modular form of weight
$k$ w.r.t. $\Gamma(2)$. But, for all our purposes, it operates on a vvmf.

$D^{N}\vec{F}+\sum_{s=1}^{N-1}\phi_{s}(\tau)D^{s}\vec{F}+\phi_{0}(\tau)\vec{F}=0,\quad\vec{F}=\left(F_{p(1)}(\tau)\
\cdots\ F_{p(N)}(\tau)\right),$ (14)

where $\phi_{r}(\tau)$ are some (meromorphic-)modular forms of weight $2(N-r)$
w.r.t. $\Gamma(2)$ and $D^{r}$ stands for the $r$ successive operations of the
Serre derivative $D$. At each operation, $D$ depends on the weight $k$ of the
form on which it operates as follows.

$D=\frac{d}{d\tau}-\frac{i\pi k}{6}E_{2}(\tau)\,,$ (15)

where $E_{2}(\tau)$ is the second Eisenstein series.151515Note the modular
transformation property of the second Eisenstein series [29]
$E_{2}(\frac{a\tau+b}{c\tau+d})=(c\tau+d)^{2}E_{2}(\tau)+\frac{6}{i\pi}c(c\tau+d),\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in PSL(2,\mathbb{Z})\,.$ E.g.,
$D\vec{F}=\frac{d}{d\tau}\vec{F}$ as $k=0$ in this case, while
$D^{2}\vec{F}=\left(\frac{d}{d\tau}-\frac{i\pi}{3}E_{2}(\tau)\right)\frac{d}{d\tau}\vec{F}$,
etc.

A few comments are in order. Firstly, the objective of using the Serre
derivate is to produce a $k+2$ form operating on a $k$ form w.r.t.
$\Gamma(2)$. Unlike the case of $PSL(2,\mathbb{Z})$, as there exist forms of
weight 2 w.r.t. $\Gamma(2)$ they may be added to modify the Serre derivative
as follows.

$\tilde{D}=D+\tilde{\alpha}_{1}\theta_{2}(\tau)^{4}+\tilde{\alpha}_{2}\theta_{3}(\tau)^{4},\quad\tilde{\alpha}_{1},\tilde{\alpha}_{2}\in\mathbb{C}\,.$
(16)

However, an MLDE written using $\tilde{D}$ can always be brought to the form
(14). Thus, we continue to use simply $D$. Secondly, the requirement of (14)
to be an MLDE w.r.t. $\tilde{\Gamma}_{1}$ ($\neq\Gamma(2)$ when some/all of
the operators are identical) restricts the locus of $\phi_{r}(\tau)$ on the
space of $2(N-r)$-forms w.r.t. $\Gamma(2)$.

In the following, we organise the present program in a series of
steps.161616We detail the construction of MLDE for the holomorphic blocks, but
similar steps can be followed with dimensions
$\\{\bar{h}_{1},\dots,\bar{h}_{4},\bar{h}_{p(k)}\\}$ producing the anti-
holomorphic blocks at $\tau$ i.e. ${\bar{F}}_{p(k)}(\tau)$. At the end, we
evaluate them at $\bar{\tau}(=\tau^{*})$. Hereafter, blocks refer to
holomorphic blocks; when we need to glue them with anti-holomorphic ones we
use specific words. We apply them to definite cases in the subsequent section.

Step 1: We fix $N\geq 2$ in (14). We expand each $\phi_{r}(\tau)$ on a given
basis of the space of (meromorphic-)modular forms of weight $2(N-r)$ w.r.t.
$\Gamma(2)$. The expansion coefficients, collectively called as
$\vec{\alpha}$, are complex-valued parameters. However, as the real $(c,h)$
plane is the primary focus of this article, scanning over real $\vec{\alpha}$
will often suffice.

Step 2: Based on which external operators are identical, we determine the
subgroup $\tilde{\Gamma}_{1}$. We find the locus on the parameter space
$\vec{\alpha}$ such that each $\phi_{r}(\tau)$ become a form of weight
$2(N-r)$ w.r.t. $\tilde{\Gamma}_{1}$. For the subsequent steps, we stay on
this locus.

Step 3: We take trial solutions to (14) to be of the form

$q^{s}\sum_{n=0}^{\infty}a_{n}q^{n}=a_{0}q^{s}\big{(}1+\sum_{n=1}^{\infty}\frac{a_{n}}{a_{0}}q^{n}\big{)},\quad
a_{0}\neq 0,\quad q=e^{i\pi\tau}\,.$ (17)

The indicial equation is a polynomial equation of degree $N$ in $s$, which we
solve to get $N$ exponents in terms of the parameters
$\vec{\alpha}$.171717This can always be done analytically for $N<5$. We narrow
the parameter space further such that each of these exponents leads to a
series solution consistent with $a_{0}\neq 0$.181818This entails removing
points $\vec{\alpha}$ where a pair of exponents are separated by an even
integer. On the surviving parameter space, for each of these exponents
$s(\vec{\alpha})$, (14) being a linear differential equation does not fix
$a_{0}$. However, it gives recursion relation(s) for the higher coefficients
$a_{n}$ in terms of the lower coefficients. The values $s(\vec{\alpha})$
encode information about the dimensions of the intermediate operators, while
the coefficients $a_{n},\ n\geq 2$ encode information about the central charge
as well.

The preceding 3 steps work for any 4 external (quasi-)primaries $O_{a}$, since
(8) functions. In particular for quasi-primaries, (14) is supposed to be an
MLDE for the global conformal blocks which have no dependencies on central
charge. For the subsequent steps, we confine ourselves to Virasoro primaries
$O_{a}$. In addition, we consider the expansion of the stripped correlator in
terms of Virasoro blocks.

Step 4: We consider the solutions of (14) to be 4-point Virasoro blocks. For
generic central charge $c$, external operator dimensions $h_{a}$ and internal
dimensions $h_{p(k)}$, the $q$-expansions of respective Virasoro blocks can be
computed [6]. We compare them with (17) to relate the quantities
$\\{c,h_{a},h_{p(k)}\\}$ with parameters $\vec{\alpha}$ as follows. In terms
of Zamolodchikov’s $H$-function, the Virasoro blocks that contribute to the
stripped correlator are given by191919Note that the $\lambda$-dependent
factors multiplying $H$ are in accordance with (5), and hence differ from [6].
Also, recall that $\mathfrak{H}=h_{1}+\cdots+h_{4}$.

$\displaystyle
F_{p(k)}(\tau)=(16q)^{h_{p(k)}-\frac{c-1}{24}}\lambda(\tau)^{\frac{c-1}{24}-\frac{\mathfrak{H}}{3}}(1-\lambda(\tau))^{\frac{c-1}{24}-\frac{\mathfrak{H}}{3}}\theta_{3}(\tau)^{\frac{c-1}{2}-4\mathfrak{H}}H(c,h_{a},h_{p(k)},q)\,,$
$\displaystyle
H(c,h_{a},h_{p(k)},q)=1+\sum_{n=1}^{\infty}b_{n}(c,h_{a},h_{p(k)})q^{n}\,,$
(18)

where the coefficients $b_{n}$ can be obtained using Zamolodchikov’s elliptic
recursion relation, see appendix A. $b_{1}$ is independent of $c$. Employing
the respective $q$-expansions of $\lambda(\tau),\theta_{3}(\tau)$ as given in
appendix B, we get

$\displaystyle F_{p(k)}(\tau)$
$\displaystyle=(16q)^{h_{p(k)}-\frac{\mathfrak{H}}{3}}\left(1+b_{1}q+\big{(}b_{2}+\frac{1-c+8\mathfrak{H}}{2}\big{)}q^{2}+\big{(}\frac{1-c+8\mathfrak{H}}{2}b_{1}+b_{3}\big{)}q^{3}+\cdots\right)$
$\displaystyle\equiv(16q)^{h_{p(k)}-\frac{\mathfrak{H}}{3}}\bigg{(}1+\sum_{n=1}^{\infty}f_{n}(c,h_{a},h_{p(k)})q^{n}\bigg{)}.$
(19)

The $q$-expansion coefficients of $F_{p(k)}$ are linearly related to those of
$H$. In fact, the following general patterns arise in (19) with some
coefficients $\beta_{nr}(c,\mathfrak{H}),\gamma_{nr}(c,\mathfrak{H})$.

$f_{2n-1}=\sum_{r=1}^{n}\beta_{nr}b_{2r-1},\quad
f_{2n}=\gamma_{n0}+\sum_{r=1}^{n}\gamma_{nr}b_{2r},\quad n\geq 1\,,$ (20)

which are obtained using the $q$-expansion of $q^{-1}\eta(\tau)$ with known
even coefficients $\sigma_{2n}$ while odd ones vanish as follows.202020Below,
we use (18), $\lambda=\theta_{2}^{4}/\theta_{3}^{4},\
\theta_{4}^{4}=\theta_{3}^{4}-\theta_{2}^{4}$ and
$\theta_{2}\theta_{3}\theta_{4}=2\eta^{3}$, where
$\eta(\tau)=q^{\frac{1}{12}}\prod_{r=1}^{\infty}(1-q^{2r})$ is the Dedekind
eta function. $\sigma_{2n}$ can be obtained by using (B.1) up to any desired
$n$. Thereby, the coefficients $\beta_{nr}(c,\mathfrak{H})$,
$\gamma_{nr}(c,\mathfrak{H})$ are known as well for any desired $n,r$.

$\displaystyle(16q)^{-h_{p(k)}+\frac{\mathfrak{H}}{3}}F_{p(k)}=(16q)^{-\delta}\theta_{2}^{4\delta}\theta_{3}^{4\delta}\theta_{4}^{4\delta}H,\quad\delta=\frac{c-1}{24}-\frac{\mathfrak{H}}{3}\,,$
$\displaystyle(16q)^{-1}\theta_{2}^{4}\theta_{3}^{4}\theta_{4}^{4}=q^{-1}\eta^{12}=1+\sum_{n=1}^{\infty}\sigma_{2n}q^{2n}\,.$
(21)

Note that if the odd coefficients vanish in $H$, the odd coefficients vanish
in $F_{p(k)}$. We need to equate quantities in (19) to respective quantities
in (17) which are obtained by solving (14):

$\displaystyle{\text{\it for arbitrary $q$ near $0$}:}$
$\displaystyle(16q)^{h_{p(k)}-\frac{\mathfrak{H}}{3}}=a_{0}q^{s(\vec{\alpha})}\,,$
(22) $\displaystyle{\text{\it for $n\geq 1$}:}$ $\displaystyle
f_{n}(c,h_{a},h_{p(k)})=\frac{a_{n}(\vec{\alpha})}{a_{0}}\,.$

At this stage, the number of independent CFT quantities
$\\{c,h_{a},h_{p(k)}\\}$ is $N+5$ or lower when some of the operators have
same dimensions. In order to generate this many independent equations, we use
(22) with the first few $n$. These in principle can be inverted to express the
aforesaid CFT quantities in terms of the parameters $\vec{\alpha}$. Then,
plugging these $c(\vec{\alpha}),h_{a}(\vec{\alpha}),h_{p(k)}(\vec{\alpha})$ we
check the required validity of (22) for higher $n$, which may further restrict
the parameter space $\vec{\alpha}$. In cases where the said inversion and/or
the checks for higher $n$ are cumbersome to perform analytically, we resort to
numerical techniques.

On the surviving parameter space, the solutions to MLDE (14) are 4-point
Virasoro blocks. We discuss the (hyper-)surfaces in this parameter space on
which the MLDE corresponds to BPZ equations and the possibilities of new
equations.

Step 5: In terms of the solutions to (14) (which are identified as Virasoro
blocks), the stripped correlator takes the form:

$G_{1234}(\tau,\bar{\tau})=\left(F_{p(1)}(\tau)\ \cdots\
F_{p(N)}(\tau)\right)\cdot\mathcal{C}\cdot\left(\bar{F}_{p(1)}(\bar{\tau})\
\cdots\ \bar{F}_{p(N)}(\bar{\tau})\right)^{t},$ (23)

where $\mathcal{C}$ is an $N\times N$ matrix containing the associated 3-point
coefficients. Now, the $\tilde{\Gamma}_{1}$-invariance of $G_{1234}$ sets up
the bootstrap equations:

$M(\gamma)\cdot\mathcal{C}\cdot\bar{M}(\gamma)^{\dagger}=\mathcal{C},\quad\gamma\in\tilde{\Gamma}_{1}\,,$
(24)

where $\bar{M}$ is obtained from $M$ using the replacement rule stated below
(12). Confining $\gamma$ over the generators of $\tilde{\Gamma}_{1}$ suffices
due to the multiplication rule in (13). Note that knowing the $q$-expansion of
the blocks around $q=0$ is not enough to derive the transformation matrices
$M(\gamma)$. Writing down the MLDE (14) for the blocks is useful in many
instances where we can solve it to get the blocks in closed forms. In latter
cases, we obtain $M(\gamma)$ in terms of $\vec{\alpha}$ for the holomorphic
blocks. For spinning operators, the anti-holomorphic conformal dimension
differs from the holomorphic one. Thus,
$\\{\bar{h}_{1},\dots,\bar{h}_{4},\bar{h}_{p(k)}\\}$ correspond to different
values of the parameters denoted by $\vec{\bar{\alpha}}$ such that
$c(\vec{\alpha})=c(\vec{\bar{\alpha}})$, and the solutions to (14) with
parameters $\vec{\bar{\alpha}}$ are the anti-holomorphic blocks at
$\tau$.212121See footnote 16. The complex conjugation acts in (24) as we
evaluate the anti-holomorphic blocks at $\bar{\tau}=\tau^{*}$. We solve the
above matrix equations to obtain $\mathcal{C}$ in terms of
$\vec{\alpha},\vec{\bar{\alpha}}$. Note that when we deal with only scalar
operators, we have $\vec{\alpha}=\vec{\bar{\alpha}}$ and $M=\bar{M}$.

Step 6: We attempt to differentiate unitary CFTs from non-unitary ones on the
surviving parameter space $\vec{\alpha}$ and present examples of each. For
unitary theories, we should necessarily set $c,h_{a},h_{p(k)}\geq 0$. In our
context, unitarity bounds on the $q$-expansion coefficients of the blocks will
be of importance. The only known such bounds are for the case
$h_{1}=h_{4},h_{2}=h_{3}$, as stated in the following in terms of the
quantities used in the present work. Consider the following $q$-expansion
coefficients $\tilde{f_{n}}$ in linear relation to $f_{r}$.

$\displaystyle{\text{\it pillow blocks}:}\ \ \ $
$\displaystyle\tilde{F}_{p(k)}(\tau)\equiv
16^{\frac{\mathfrak{H}}{3}-\frac{c}{24}}\eta(\tau)^{4\mathfrak{H}-\frac{c}{2}}F_{p(k)}(\tau)=(16q)^{h_{p(k)}-\frac{c}{24}}\bigg{(}1+\sum_{n=1}^{\infty}\tilde{f}_{n}q^{n}\bigg{)}\,,$
$\displaystyle{\text{\it first few coefficients}:}\ \ \ $
$\displaystyle\tilde{f}_{1}=f_{1},\quad\tilde{f}_{2}=f_{2}+\frac{1}{2}(c-8\mathfrak{H}),\quad\tilde{f}_{3}=f_{3}+\frac{1}{2}(c-8\mathfrak{H})f_{1}\,,$
$\displaystyle\tilde{f}_{4}=f_{4}+\frac{1}{2}(c-8\mathfrak{H})f_{2}+\frac{1}{8}(c-8\mathfrak{H})(c-8\mathfrak{H}+6),\quad\dots\,,$
$\displaystyle{\text{\it general patterns}:}\ \ \ $
$\displaystyle\tilde{f}_{2n-1}=\sum_{r=1}^{n}\tilde{\beta}_{nr}f_{2r-1},\quad\tilde{f}_{2n}=\tilde{\gamma}_{n0}+\sum_{r=1}^{n}\tilde{\gamma}_{nr}f_{2r},\quad
n\geq 1\,.$ (25)

In the above, $\tilde{F}_{p(k)}$ are known as the pillow blocks. The
coefficients $\tilde{f}_{n}$ can be obtained to any desired order $q^{n}$,
using (19) and (B.1), thereby the coefficients
$\tilde{\beta}_{nr}(c,\mathfrak{H})$, $\tilde{\gamma}_{nr}(c,\mathfrak{H})$
are known as well. For the case $h_{1}=h_{4},h_{2}=h_{3}$, all the
$\tilde{f}_{n}\geq 0$ [23, 24], as they can be interpreted as the norms on
some Hilbert space.222222The work [23] provides an alternative viewpoint on
$q$-expansions of Virasoro blocks, under the map from the Riemann sphere with
4 marked points $\infty,1,x,0$ to the “pillow” geometry $T^{2}/\mathbb{Z}_{2}$
with 4 corners. This enables to uncover the above non-negativity conditions.
In our fourth step, we have obtained $f_{r},c,\mathfrak{H}$ in terms of
$\vec{\alpha}$. Therefore, $\tilde{f}_{r}$ can be expressed in terms of
$\vec{\alpha}$ using (25). Hence, the non-negativity of
$\tilde{f}_{n}(\vec{\alpha})$ restricts the parameter space $\vec{\alpha}$
when we have $h_{1}=h_{4},h_{2}=h_{3}$ for unitary theories.

In case the Virasoro representations corresponding to
$\bar{h}_{a},\bar{h}_{p(k)}$ are unitary, similar constraints apply on
$\vec{\bar{\alpha}}$. In addition, the reality of the 3-point coefficients
associated with the stripped correlator (obtained in terms of
$\vec{\alpha},\vec{\bar{\alpha}}$ in our fifth step) may narrow both parameter
spaces $\vec{\alpha},\vec{\bar{\alpha}}$ further, in unitary CFTs.

##  4 Case studies: second order MLDEs

In the simplest scenario, in (14) $\phi_{r}(\tau),\ r=0,\dots,N-1$ are
holomorphic forms of respective weights $2(N-r)$ w.r.t. $\Gamma(2)$ on
$\mathbb{H}_{+}$. Since $\theta_{2}(\tau)^{4a}\theta_{3}(\tau)^{4b}$ with non-
negative integers $a,b$ form the basis of the space of holomorphic
$2(a+b)$-forms w.r.t. $\Gamma(2)$ on $\mathbb{H}_{+}$, we can write

$\phi_{r}=\sum_{\begin{subarray}{c}a,b=0,1,2,\dots\\\
a+b=N-r\end{subarray}}\alpha_{ab}\theta_{2}^{4a}\theta_{3}^{4b},\quad
r=0,\dots,N-1\,,$ (26)

for some coefficients $\alpha_{ab}$.

The present program has been organised as a series of steps in section 3. In
this section, we implement this for the case: $N=2$ with holomorphic forms
$\phi_{0},\phi_{1}$ of respective weights $4,2$ w.r.t. $\Gamma(2)$. In this
case, we write down the following ansatz introducing parameters
$\vec{\alpha}=(\alpha_{1},\dots,\alpha_{5})$, completing our first
step.232323The expansion coefficients $\vec{\alpha}$ on the basis (26) are
chosen in a specific way for later convenience.

$\displaystyle\frac{d^{2}\vec{F}}{d\tau^{2}}-\frac{i\pi}{3}E_{2}(\tau)\frac{d\vec{F}}{d\tau}+i\pi\left((\frac{1}{3}-2\alpha_{1})\theta_{2}(\tau)^{4}+(\frac{1}{3}-2\alpha_{2})\theta_{3}(\tau)^{4}\right)\frac{d\vec{F}}{d\tau}$
$\displaystyle\qquad\qquad+\pi^{2}\left(\alpha_{3}\theta_{2}(\tau)^{8}+\alpha_{4}\theta_{2}(\tau)^{4}\theta_{3}(\tau)^{4}+\alpha_{5}\theta_{3}(\tau)^{8}\right)\vec{F}=0\,,$
$\displaystyle\vec{F}=\left(F_{p(1)}(\tau)\ F_{p(2)}(\tau)\right).$ (27)

As shown in appendix C, (27) leads to the most general second order linear
differential equation with 3 regular singular points in variable
$x=\lambda(\tau)$.242424Hence, clearly meromorphic $\phi_{0},\phi_{1}$ should
result in more and/or new type singularities. See section 5, for a discussion.
In the following two subsections, we implement the remaining steps separately
to the cases $O_{1}=O_{2}=O_{3}=O_{4}$ and $O_{1}=O_{2},\ O_{3}=O_{4}$.

###  4.1 All identical operators

Let us take $O_{1}=O_{2}=O_{3}=O_{4}\equiv O$. Now, we implement the second
step. In the present case, $\tilde{\Gamma}_{1}$ is the full modular group. In
order for (27) to be an MLDE w.r.t. full modular group, we must have

$\alpha_{1}=\alpha_{2}=\frac{1}{6},\quad\alpha_{3}=\alpha_{5}=-\alpha_{4}\,.$
(28)

The rationale behind this is as follows. The space of holomorphic
$(4a+6b)$-forms w.r.t. the modular group is spanned by
$E_{4}(\tau)^{a}E_{6}(\tau)^{b}$ with non-negative integers $a,b$. The
Eisenstein series $E_{4},E_{6}$ being forms w.r.t. the modular group are also
forms w.r.t. any subgroup without altering respective weights. Thus, they can
be expressed in terms of $\theta_{2}^{4},\theta_{3}^{4}$. Clearly, there is no
holomorphic modular form of weight $2$ and only 1 holomorphic modular form of
weight $4$ which is $E_{4}$. And, the latter can be expressed as [29]

$E_{4}=\theta_{2}^{8}-\theta_{2}^{4}\theta_{3}^{4}+\theta_{3}^{8}\,.$ (29)

Therefore, in the present case, the MLDE (27) becomes

$\frac{d^{2}\vec{F}}{d\tau^{2}}-\frac{i\pi}{3}E_{2}(\tau)\frac{d\vec{F}}{d\tau}+\alpha_{3}\pi^{2}E_{4}(\tau)\vec{F}=0\,.$
(30)

In the third step, in (30) we substitute the trial solution (17) and the
respective $q$-expansions of $E_{2}(\tau),E_{4}(\tau)$ as given in appendix B.
Solving the indicial equation, we get252525It is worth noting that for real
$\alpha_{3}>-1/36$, we have $s_{1}<s_{2}$.

$\displaystyle s=s_{1},s_{2}\,,\quad
s_{1}=\frac{1}{6}\left(1-\sqrt{36\alpha_{3}+1}\right),\quad
s_{2}=\frac{1}{6}\left(1+\sqrt{36\alpha_{3}+1}\right)\,.$ (31)

Let us first take $s=s_{1}$, and consider the following points on the
parameter space.262626The statements that come next are based on studying (30)
at points (32) with $m=1,2,\dots,50$.

$\alpha_{3}=\frac{1}{36}\left(9m^{2}-1\right),\quad m=1,2,3,\dots\,.$ (32)

At any of the above points with even $m$, the coefficients in the
$q$-expansion of the l.h.s. of the MLDE (30) cannot be set to zero
consistently with $a_{0}\neq 0$. Therefore, by requiring the existence of a
solution with $s=s_{1}$, we constrain the parameter space as follows.

$\alpha_{3}\neq\frac{1}{36}\left(9m^{2}-1\right),\quad m=2,4,6,\dots\,.$ (33)

At any of the points (32) with odd $m$, (30) does not fix $a_{0},a_{m}$. In
this case, all the $a_{r}$ with odd $r<m$ are equal to zero, all the $a_{r}$
with odd $r>m$ are proportional to $a_{m}$.272727Thus, if we require $a_{m}$
to be zero, all the odd coefficients vanish. At a generic point $\alpha_{3}$
different from (32), only $a_{0}$ is unfixed. In this case, all the $a_{r}$
with odd $r$ vanish. In both cases, all the $a_{r}$ with even $r$ are
proportional to $a_{0}$. The first few are presented below.

$\displaystyle
a_{2}=-\frac{2a_{0}\left(\sqrt{36\alpha_{3}+1}+180\alpha_{3}-1\right)}{\sqrt{36\alpha_{3}+1}-6}\,,$
(34) $\displaystyle
a_{4}=\frac{a_{0}\left(7\left(\sqrt{36\alpha_{3}+1}-1\right)+36\left(25\sqrt{36\alpha_{3}+1}-1800\alpha_{3}-129\right)\alpha_{3}\right)}{18\sqrt{36\alpha_{3}+1}-36\alpha_{3}-73}\,,$
$\displaystyle
a_{6}=\frac{4a_{0}\left(91\left(\sqrt{36\alpha_{3}+1}-1\right)-6\alpha_{3}\left(180\alpha_{3}\left(1800\alpha_{3}-105\sqrt{36\alpha_{3}+1}+958\right)+117\sqrt{36\alpha_{3}+1}+1198\right)\right)}{397\sqrt{36\alpha_{3}+1}+36\left(\sqrt{36\alpha_{3}+1}-36\right)\alpha_{3}-1332}\,.$

Analytic expressions of higher even coefficients are more lengthy in terms of
$a_{0}$. However, since we will be using $a_{8}$ for analytic computations,
let us now provide the equation to be solved.

$\displaystyle\left(\sqrt{36\alpha_{3}+1}-1\right)\left(7a_{0}+4a_{2}+3a_{4}+a_{6}\right)+2\left(\sqrt{36\alpha_{3}+1}-24\right)a_{8}$
$\displaystyle\quad+180\left(73a_{0}+28a_{2}+9a_{4}+a_{6}\right)\alpha_{3}-12\left(4a_{2}+6a_{4}+3a_{6}\right)=0\,.$
(35)

Let us now take $s=s_{2}$. In this case, (30) does not fix $a_{0}$. All the
$a_{r}$ with odd $r$ vanish, and all the $a_{r}$ with even $r$ are
proportional to $a_{0}$. The first few even coefficients are presented below.

$\displaystyle
a_{2}=-\frac{2a_{0}\left(\sqrt{36\alpha_{3}+1}-180\alpha_{3}+1\right)}{\sqrt{36\alpha_{3}+1}+6}\,,$
(36) $\displaystyle
a_{4}=\frac{a_{0}\left(7\left(\sqrt{36\alpha_{3}+1}+1\right)+36\left(25\sqrt{36\alpha_{3}+1}+1800\alpha_{3}+129\right)\alpha_{3}\right)}{18\sqrt{36\alpha_{3}+1}+36\alpha_{3}+73}\,,$
$\displaystyle
a_{6}=\frac{4a_{0}\left(91\left(\sqrt{36\alpha_{3}+1}+1\right)+6\alpha_{3}\left(180\alpha_{3}\left(105\sqrt{36\alpha_{3}+1}+1800\alpha_{3}+958\right)-117\sqrt{36\alpha_{3}+1}+1198\right)\right)}{\left(\sqrt{36\alpha_{3}+1}+6\right)\left(\sqrt{36\alpha_{3}+1}+12\right)\left(\sqrt{36\alpha_{3}+1}+18\right)}\,.$

And, $a_{8}$ can be obtained by solving the following equation.

$\displaystyle\left(\sqrt{36\alpha_{3}+1}+1\right)\left(7a_{0}+4a_{2}+3a_{4}+a_{6}\right)+2\left(\sqrt{36\alpha_{3}+1}+24\right)a_{8}$
$\displaystyle\quad-180\left(73a_{0}+28a_{2}+9a_{4}+a_{6}\right)\alpha_{3}+12\left(4a_{2}+6a_{4}+3a_{6}\right)=0\,.$
(37)

In the fourth step, we tune $\alpha_{3}$ and the above unfixed coefficients so
that the 2 solutions found in the previous step are 4-point Virasoro blocks.
The relevant CFT quantities are $\\{c,h,h_{p(1)},h_{p(2)}\\}$.282828Here,
$h\equiv h_{1}=h_{2}=h_{3}=h_{4},\ \mathfrak{H}=4h$. We assume identity to be
an intermediate operator as compatible with the identical external
operators292929The necessary requirements for the occurance of the vacuum
block are: $h_{1}=h_{2},\ h_{3}=h_{4}$., and the dimension of the other
intermediate operator to be greater than zero. Thus, in keeping with our
convention (10), we take303030The exclusion of the equality
$h_{p(2)}=h_{p(1)}$ is due to the uniqueness of identity operator.

$h_{p(2)}>h_{p(1)}=0\,.$ (38)

By comparing the $q\to 0$ limits of the 2 solutions found in the third step
with those of (19), we see that the 2 values of $s$ in (31) are the candidates
for $h_{p(k)}-4h/3,\ k=1,2$. Since $s_{1}\ngtr s_{2}$, we equate the first
solution i.e. the one with $s=s_{1}$, with the vacuum block. Hence,
$h,h_{p(2)}$ can be expressed in terms of $\alpha_{3}$ as

$h=\frac{1}{8}\left(\sqrt{36\alpha_{3}+1}-1\right),\quad
h_{p(2)}=\frac{1}{3}\sqrt{36\alpha_{3}+1},\quad\alpha_{3}>-\frac{1}{36}\,.$
(39)

This also fixes $a_{0}$ of the respective solutions to be $16^{s}$. For the
Virasoro blocks in the present case, all the odd coefficients $f_{2r-1},\
r\geq 1$ in (19) vanish.313131When $h_{1}=h_{2},\ h_{3}=h_{4}$, all the odd
coefficients $b_{2r-1},\ r\geq 1$ in $H$ identically vanish [22]. See appendix
A for a discussion.  Both the series solutions are automatically tailored to
this, except for the first solution we need to fix $a_{m}=0$ when
$\alpha_{3}=\left(9m^{2}-1\right)/36,\ m=1,3,5,\dots$. Now, to express the
central charge $c$ in terms of $\alpha_{3}$, one may equate the ratio
$a_{2}/a_{0}$ for the first solution to the coefficient $f_{2}$ for the
Virasoro vacuum block:323232Here, we use (A.5) with $h_{L}=h_{R}=h$.

$\displaystyle
b_{2}(c,h,0)+\frac{1-c+32h}{2}=-\frac{2\left(\sqrt{36\alpha_{3}+1}+180\alpha_{3}-1\right)}{\sqrt{36\alpha_{3}+1}-6}\,.$
$\displaystyle\implies
c\doteq\frac{7-7\sqrt{36\alpha_{3}+1}-9\alpha_{3}\left(\sqrt{36\alpha_{3}+1}-10\right)}{9\alpha_{3}-2}\,,$
(40)

where the dotted equality indicates that the limiting value to be taken at
$\alpha_{3}=2/9$. It is straightforward to check that
$(c(\alpha_{3}),h(\alpha_{3}))$ lies on the locus of the level-2 Virasoro null
states:

$16h^{2}+h(2c-10)+c=0\,.$ (41)

However, only a portion of it gets traced when we vary $\alpha_{3}$, since
$c(\alpha_{3})\leq 1,\ h(\alpha_{3})>-1/8$, see Fig. 1.

Figure 1: Red dashed line represents the locus of
$(c(\alpha_{3}),h(\alpha_{3}))$. Blue dashed line represents the locus of the
level-2 Virasoro null states.

Therefore, for the case of 4 identical external primaries and 2 intermediate
ones, one of which is identity and the other with dimension greater than zero,
the MLDE (30) always corresponds to a second order BPZ equation.333333In the
opposite order, second order BPZ equations for identical operators containing
a level-2 null state have been brought to the form (30) in appendix A of [9].

Note that in the present case, the non-vanishing coefficients in the
$q$-expansions of the Virasoro blocks are the even ones $f_{2n}$, and they all
have $c$-dependencies. The MLDE, on the other hand, computes $a_{2n}/a_{0}$ in
terms of $\alpha_{3}$ as candidates for them. So far, we have only equated
$f_{2}$ of the vacuum block with $a_{2}/a_{0}$ of our first solution, yielding
$c(\alpha_{3})$ in (40). We have checked that plugging this $c(\alpha_{3})$
into $f_{4},f_{6},f_{8}$ of the vacuum block respectively gives
$a_{4}/a_{0},a_{6}/a_{0},a_{8}/a_{0}$ of the first solution (as in (34),(35)).
We have performed a similar check, relating the coefficients
$f_{2},f_{4},f_{6},f_{8}$ of the excited block (with dimension
$h_{p(2)}(\alpha_{3})$ as obtained in (39)) to the coefficients
$a_{2}/a_{0},a_{4}/a_{0},a_{6}/a_{0},a_{8}/a_{0}$ of our second solution (as
in (36),(37)). For these checks, we have used a Mathematica code published in
[22] to get the first few analytical coefficients $b_{2},b_{4},b_{6},b_{8}$ of
the $q$-expansion of $H$, and then used its relation to Virasoro blocks (18)
to generate the required coefficients $f_{2},f_{4},f_{6},f_{8}$, see appendix
A. We expect this to hold true for higher coefficients as well, in support of
which we present two numerical examples at the end of this subsection.

In the fifth step, we obtain the modular transformation matrices that act on
the above solutions to (30) and then set up equation for the 3-point
coefficient $c_{OOp(2)}$ associated with the stripped correlator $G_{OOOO}$.
Plugging (28) in (C.3), and then using (C.4) we
get343434$\Gamma(1–z)\Gamma(z)=\pi\csc\pi z$ has also been used to simplify
the expressions.

$\displaystyle
M(T)=\begin{pmatrix}e^{\frac{i\pi}{6}\left(1-\sqrt{36\alpha_{3}+1}\right)}&0\\\
0&e^{\frac{i\pi}{6}\left(1+\sqrt{36\alpha_{3}+1}\right)}\end{pmatrix},$ (42)
$\displaystyle
M(S)=\begin{pmatrix}\frac{1}{2}\csc\left(\frac{\pi}{6}\sqrt{36\alpha_{3}+1}\right)&\frac{\sqrt{\pi}2^{\frac{1}{3}\sqrt{36\alpha_{3}+1}}\csc\left(\frac{\pi}{3}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(1+\frac{1}{6}\sqrt{36\alpha_{3}+1}\right)}{\Gamma\left(1-\frac{1}{3}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(\frac{1}{2}+\frac{1}{2}\sqrt{36\alpha_{3}+1}\right)}\\\
-\frac{\sqrt{\pi}2^{-\frac{1}{3}\sqrt{36\alpha_{3}+1}}\csc\left(\frac{\pi}{3}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(1-\frac{1}{6}\sqrt{36\alpha_{3}+1}\right)}{\Gamma\left(\frac{1}{2}-\frac{1}{2}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(1+\frac{1}{3}\sqrt{36\alpha_{3}+1}\right)}&-\frac{1}{2}\csc\left(\frac{\pi}{6}\sqrt{36\alpha_{3}+1}\right)\end{pmatrix}.$

At this stage, we want to glue the holomorphic blocks with the anti-
holomorphic ones. When $-1/36<\alpha_{3}<20/9,\ \alpha_{3}\neq 2/9$, for each
$\alpha_{3}$ we have an $\bar{\alpha}_{3}\neq\alpha_{3}$ with
$c(\alpha_{3})=c(\bar{\alpha}_{3})$, thus, we have possibilities of spinning
operators. We take $\bar{\alpha}_{3}$ to correspond to the anti-holomorphic
dimensions, and from aforementioned formulae of dimensions it is clear that
the identical external operators and also the excited intermediate operator
are all spinning. However, in such cases, the bootstrap condition
corresponding to $T$ in (24) alone sets $\mathcal{C}=0$, thereby, the
correlator of 4 identical spinning operators vanish. Now, the non-trivial case
will be to deal with only scalar operators. In the latter case, with
$\alpha_{3}>-1/36$ the bootstrap condition corresponding to $T$ in (24) allows
only vanishing non-diagonal elements in $\mathcal{C}$:

$\mathcal{C}=\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix},\quad\beta=c^{2}_{OOp(2)}\,.$ (43)

Now, the $S$-invariance solves for $\beta$ as

$\beta=\frac{2^{\frac{1}{3}(-2)\sqrt{36\alpha_{3}+1}}\Gamma\left(-\frac{1}{3}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(1-\frac{1}{6}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(\frac{1}{2}\sqrt{36\alpha_{3}+1}+\frac{1}{2}\right)}{\Gamma\left(\frac{1}{3}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(\frac{1}{2}-\frac{1}{2}\sqrt{36\alpha_{3}+1}\right)\Gamma\left(\frac{1}{6}\sqrt{36\alpha_{3}+1}+1\right)}\,.$
(44)

In the sixth step, we constrain the parameter space $\alpha_{3}$ to meet
various unitarity bounds. Note that $\alpha_{3}>-1/36$ already maintains
$h_{p(2)}>0$, see (39). $h>0$ now further restricts $\alpha_{3}>0$.
Accordingly, $c$ takes the value $0$ only when $\alpha_{3}=35/36$, which (33)
rules out. As a result, we can only have $c>0$, which limits
$0<\alpha_{3}<35/36$. In fact, in this region of the parameter space, we have
$0<c\leq 1$, where the equality holds at $\alpha_{3}=2/9$.

We have 4 identical external operators in the present case, thus the non-
negativity of the $q$-expansion coefficients of the pillow blocks, as
discussed in section 3, applies. Since the odd coefficients $f_{2r+1}$ of the
Virasoro blocks vanish, so do the odd coefficients $\tilde{f}_{2n+1}$ of
respective pillow blocks in (25). Now, the non-negativity of the even
coefficients $\tilde{f}_{2n}$ of the pillow blocks requires

$\tilde{\gamma}_{n0}+\sum_{r=1}^{n}\tilde{\gamma}_{nr}\frac{a_{2r}}{a_{0}}\geq
0,\quad n=1,2,3,\dots\,,$ (45)

as per the identification of the Virasoro blocks with the 2 solutions of our
MLDE. Corresponding to each solution, by plugging respective $a_{2r}/a_{0}$ in
terms of $\alpha_{3}$ in (45) along with the substitutions (39) and (40), we
get a countable infinite set of non-negative functions:
$\tilde{f}_{2n}(\alpha_{3})\geq 0,\ n\geq 1$. For both the solutions, we
analyze the cases with $n=1,\dots,10$. For the first solution i.e. the one
identified with the vacuum block,
$\tilde{f}_{2}(\alpha_{3}),\dots,\tilde{f}_{20}(\alpha_{3})$ are non-negative
in the region $0<\alpha_{3}<35/36$, except
$\tilde{f}_{6}(\alpha_{3}),\tilde{f}_{14}(\alpha_{3})$. Now, the requirement
of the non-negativity of the last 2 uplifts the lower bound on $\alpha_{3}$
from $0$ to $\alpha_{3}^{*}\approx 0.00719982$.353535$\alpha_{3}^{*}$ is the
smallest positive real root of the polynomial  $\displaystyle
132860250000000000x^{7}-90541248876000000x^{6}+67714737664118400x^{5}-11791778905265760x^{4}$
$\displaystyle+581776809364116x^{3}+1455851807721x^{2}-375258931508x+2439586688\,.$
In fact, $\alpha_{3}^{*}$ comes from the condition
$\tilde{f}_{6}(\alpha_{3})\geq 0$ as the uplift of the lower bound from the
condition $\tilde{f}_{14}(\alpha_{3})\geq 0$ is smaller, see Fig. 2a. Now, the
consideration of the second solution i.e. the one identified with the excited
block provides a stronger lower bound on $\alpha_{3}$. In this case,
$\tilde{f}_{2}(\alpha_{3}),\dots,\tilde{f}_{20}(\alpha_{3})$ are non-negative
in the region $0<\alpha_{3}<35/36$, with the exceptions for
$\tilde{f}_{2}(\alpha_{3}),\tilde{f}_{8}(\alpha_{3}),\tilde{f}_{10}(\alpha_{3}),\tilde{f}_{12}(\alpha_{3})$.
Now, the requirement of the non-negativity of the last 4 uplifts the lower
bound on $\alpha_{3}$:

$\alpha_{3}^{**}\leq\alpha_{3}<\frac{35}{36}\,,$ (46)

where $\alpha_{3}^{**}\approx 0.0242458$ is the smallest positive real root of
the polynomial $2624400x^{3}-333720x^{2}+3249x+80$. In fact, $\alpha_{3}^{**}$
comes from the condition $\tilde{f}_{2}(\alpha_{3})\geq 0$ as the uplifts of
the lower bound from the conditions
$\tilde{f}_{8}(\alpha_{3}),\tilde{f}_{10}(\alpha_{3}),\tilde{f}_{12}(\alpha_{3})\geq
0$ are smaller, see Fig. 2b.

(a) For vaccum pillow block

(b) For excited pillow block

Figure 2: First few non-vanishing $q$-expansion coefficients in the pillow
blocks are plotted against $\alpha_{3}$. $y$-axes are taken to be logarithmic.

Fig. 2 shows the semi-logarithmic plots versus $\alpha_{3}$ of the first few
non-vanishing coefficients of the pillow blocks corresponding to the 2
solutions of our MLDE. As can be observed, each of these coefficients is non-
negative and continuous on some region of the type $[\hat{\alpha}_{3},35/36)$
with $\hat{\alpha}_{3}<2/9$. If this feature continues to occur in higher
coefficients, we may expect further improvement on the lower bound in (46),
but we do not obtain countable points in $\alpha_{3}$. In contrast to this, we
know that for $0<c<1$ the following discrete values are admissible for unitary
CFTs [31, 32].

$c=1-\frac{6}{l(l+1)},\quad l=3,4,5,\dots\,,$ (47)

which can be attained respectively at363636The lowest such $\alpha_{3}\ (l=3)$
equals $5/144\approx 0.0347222$, which is well above $\alpha_{3}^{**}$.

$\alpha_{3}=\frac{2}{9}-\frac{l}{(l+1)^{2}},\
\frac{2}{9}+\frac{l+1}{l^{2}},\quad l=3,4,5,\dots\,.$ (48)

Hence, the above discussion indicates that the non-negativity of the
$q$-expansion coefficients of pillow blocks is a necessary but not sufficient
condition for unitary CFTs. In support of this, we also present one numerical
non-unitary example at the end of this subsection in which the $q$-expansion
coefficients of both pillow blocks are tested to be non-negative to order
$q^{125}$.

Now in unitary CFTs, the 3-point coefficient $c_{OOp(2)}$ associated with the
stripped correlator $G_{OOOO}$ must be real. Thus, the non-negativity of
$c^{2}_{OOp(2)}$ obtained in (44) improves the upper bound on $\alpha_{3}$ as

$\alpha_{3}^{**}\leq\alpha_{3}\leq\frac{2}{3}\,,$ (49)

see Fig. 3. Note that $2/3$ is also the highest value of $\alpha_{3}$ allowed
in the discrete set (48).

Figure 3: $c^{2}_{OOp(2)}$ is plotted against $\alpha_{3}$.

#### Explicit examples

Non-unitary: Consider the MLDE (30) with $\alpha_{3}=1/2$. The associated CFT
quantities can be computed as

$\displaystyle c=\frac{1}{5}\left(104-23\sqrt{19}\right)\approx 0.749065,\quad
h=\frac{1}{8}\left(\sqrt{19}-1\right)\approx 0.419862,\quad
h_{p(2)}=\frac{\sqrt{19}}{3}\approx 1.45297\,,$ (50) $\displaystyle
c^{2}_{OOp(2)}=\frac{2^{-\frac{2\sqrt{19}}{3}}\Gamma\left(-\frac{\sqrt{19}}{3}\right)\Gamma\left(1-\frac{\sqrt{19}}{6}\right)\Gamma\left(\frac{1}{2}+\frac{\sqrt{19}}{2}\right)}{\Gamma\left(\frac{\sqrt{19}}{3}\right)\Gamma\left(\frac{1}{2}-\frac{\sqrt{19}}{2}\right)\Gamma\left(1+\frac{\sqrt{19}}{6}\right)}\approx
0.833669\,.$

The 2 solutions to the MLDE respectively go as
$(16q)^{\frac{1}{6}-\frac{\sqrt{19}}{6}}$ and
$(16q)^{\frac{1}{6}+\frac{\sqrt{19}}{6}}$ as $q\to 0$. We have solved the MLDE
recursion relations numerically and found the $q$-expansion coefficients
$a_{n}$ up to order $q^{125}$ for both solutions, with odd ones vanishing. On
the other hand, first using the Mathematica code in [22], the $q$-expansion
coefficients $b_{n}$ of the $H$-functions corresponding to $h_{p(1)}=0$ and
$h_{p(2)}=1.45297$ are numerically obtained up to order $q^{125}$, and then
the numerical coefficients $f_{n}$ of the 2 respective Virasoro blocks are
calculated coding the relation (21). Indeed, they respectively match the 2
solutions of our MLDE. In the present example, we have found the $q$-expansion
coefficients of the blocks to be non-integers.

Knowing above coefficients $a_{n}$, we have calculated the $q$-expansion
coefficients of respective pillow blocks up to order $q^{125}$, with odd ones
vanishing. We have found all the even ones (45) to be positive and non-
linearly growing with $n$ for both the pillow blocks, although the central
charge does not lie in (47).

Ising CFT: We consider the correlator $\langle\sigma\sigma\sigma\sigma\rangle$
where $\mathds{1}$ and $\varepsilon$ contribute in the intermediate channel.
The CFT parameters as well as the Virasoro blocks that contribute to the
associated stripped correlator are given by

$\displaystyle c=\frac{1}{2},\quad h_{\sigma}=\frac{1}{16},\quad
h_{\varepsilon}=\frac{1}{2},\quad
c^{2}_{\sigma\sigma\varepsilon}=\frac{1}{4}\,,$ (51) $\displaystyle
F_{\mathds{1}}(\tau)=\frac{\sqrt{\sqrt{1-\lambda(\tau)}+1}}{\sqrt{2}\sqrt[12]{(1-\lambda(\tau))\lambda(\tau)}},\quad
F_{\varepsilon}(\tau)=\frac{\sqrt{2}\sqrt{1-\sqrt{1-\lambda(\tau)}}}{\sqrt[12]{(1-\lambda(\tau))\lambda(\tau)}}\,.$

Here we record the first few $q$-expansion coefficients:

$\displaystyle
F_{\mathds{1}}(\tau)=(16q)^{-\frac{1}{12}}\left(1+3q^{2}+4q^{4}+7q^{6}+13q^{8}+19q^{10}+\cdots\right),$
$\displaystyle
F_{\varepsilon}(\tau)=(16q)^{\frac{5}{12}}\left(1+q^{2}+3q^{4}+4q^{6}+7q^{8}+10q^{10}+\cdots\right).$
(52)

The above $q$-expansions can be obtained to any desired order $q^{n}$
employing the $q$-expansion of $\lambda(\tau)$ as given in appendix B. The odd
coefficients are zero while all the even ones happen to be positive integers,
which we have tested up to order $q^{125}$.

Now considering the MLDE (30) with $\alpha_{3}=5/144$, the 2 series solutions
generate the aforesaid $q$-expansions, which we have tested up to order
$q^{125}$. Also the CFT quantities associated with the MLDE agree with those
mentioned above.

###  4.2 Pairwise identical operators

Let us take $O_{1}=O_{2}\equiv O_{L},\ O_{3}=O_{4}\equiv O_{R}$. In this case,
$\tilde{\Gamma}_{1}$ is the subgroup of the modular group generated by
$T,ST^{2}S$. In order for (27) to be an MLDE w.r.t. $\tilde{\Gamma}_{1}$, we
must have

$\alpha_{2}=\frac{1}{2}-2\alpha_{1},\quad\alpha_{5}=-\alpha_{4}\,.$ (53)

This is due to the following transformations and Jacobi identity [29]:

$\theta_{2}(\tau+1)=e^{\frac{i\pi}{4}}\theta_{2}(\tau),\quad\theta_{3}(\tau+1)=\theta_{4}(\tau),\quad\theta_{4}(\tau)^{4}=\theta_{3}(\tau)^{4}-\theta_{2}(\tau)^{4}\,.$
(54)

Therefore, in the present case, the MLDE (27) becomes

$\displaystyle\frac{d^{2}\vec{F}}{d\tau^{2}}-\frac{i\pi}{3}E_{2}(\tau)\frac{d\vec{F}}{d\tau}+i\pi(\frac{1}{3}-2\alpha_{1})\left(\theta_{2}(\tau)^{4}-2\theta_{3}(\tau)^{4}\right)\frac{d\vec{F}}{d\tau}$
$\displaystyle\qquad\qquad\qquad\qquad+\pi^{2}\left(\alpha_{3}\theta_{2}(\tau)^{8}-\alpha_{4}\theta_{3}(\tau)^{4}\theta_{4}(\tau)^{4}\right)\vec{F}=0\,.$
(55)

The residual parameter space $\vec{\alpha}=(\alpha_{1},\alpha_{3},\alpha_{4})$
is 3 dimensional.

The series solutions (17) satisfying (55) have the following
exponents.373737It is worth noting that for real $\alpha_{1}$ and
$\alpha_{4}<(\frac{1}{2}-2\alpha_{1})^{2}$, we have $s_{1}<s_{2}$.

$s=s_{1},s_{2},\quad
s_{1}=\frac{1}{2}-2\alpha_{1}-\sqrt{\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}},\quad
s_{2}=\frac{1}{2}-2\alpha_{1}+\sqrt{\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}}\,.$
(56)

For $s=s_{1}$, consider the following surfaces.

$\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}=\frac{m^{2}}{4},\quad
m=1,2,3,\dots\,.$ (57)

On these surfaces with even $m$, the $q$-expansion coefficients of the l.h.s.
of MLDE (55) cannot be set to zero consistently with $a_{0}\neq 0$. Therefore,
by requiring the existence of a solution with $s=s_{1}$, we constrain the
parameter space as follows.

$\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}\neq\frac{m^{2}}{4},\quad
m=2,4,6,\dots\,.$ (58)

On the surfaces (57) with odd $m$, (55) does not fix $a_{0},a_{m}$. In this
case, all the $a_{r}$ with odd $r<m$ are equal to zero, all the $a_{r}$ with
odd $r>m$ are proportional to $a_{m}$.383838Thus, if we require $a_{m}$ to be
zero, all the odd coefficients vanish. At a generic point $\vec{\alpha}$ away
from (57), only $a_{0}$ is unfixed. In this case, all the $a_{r}$ with odd $r$
vanish. In both cases, all the $a_{r}$ with even $r$ are proportional to
$a_{0}$. For $s=s_{2}$, (55) does not fix $a_{0}$. All the $a_{r}$ with odd
$r$ vanish, and all the $a_{r}$ with even $r$ are proportional to $a_{0}$.

We will only use $a_{2},a_{4},a_{6},a_{8}$ for analytic computations. Their
analytic expressions in terms of $a_{0}$ being lengthy, we record here the
equations to be solved in the cases $s=s_{1},s_{2}$.

$\displaystyle
a_{2}\left(-\alpha_{4}-\left((s+2)\left(4\alpha_{1}+s+1\right)\right)\right)+8a_{0}\left(32\alpha_{3}+2\alpha_{4}-12\alpha_{1}s+s\right)=0\,,$
(59) $\displaystyle
a_{4}\left(-\alpha_{4}-\left((s+4)\left(4\alpha_{1}+s+3\right)\right)\right)+8a_{2}\left(32\alpha_{3}+2\alpha_{4}-12\alpha_{1}(s+2)+s+2\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-8a_{0}\left(-256\alpha_{3}+14\alpha_{4}+12\alpha_{1}s+s\right)=0\,,$
$\displaystyle
a_{6}\left(-\alpha_{4}-s^{2}-4\alpha_{1}(s+6)-11s-30\right)+8a_{4}\left(32\alpha_{3}+2\alpha_{4}-12\alpha_{1}(s+4)+s+4\right)$
$\displaystyle-8a_{2}\left(-256\alpha_{3}+14\alpha_{4}+12\alpha_{1}(s+2)+s+2\right)+32a_{0}\left(224\alpha_{3}+14\alpha_{4}-12\alpha_{1}s+s\right)=0\,,$
$\displaystyle
a_{8}\left(-\alpha_{4}-s^{2}-4\alpha_{1}(s+8)-15s-56\right)+8a_{6}\left(32\alpha_{3}+2\alpha_{4}-12\alpha_{1}(s+6)+s+6\right)$
$\displaystyle-8a_{4}\left(-256\alpha_{3}+14\alpha_{4}+12\alpha_{1}(s+4)+s+4\right)+32a_{2}\left(224\alpha_{3}+14\alpha_{4}-12\alpha_{1}(s+2)+s+2\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-8a_{0}\left(-2048\alpha_{3}+142\alpha_{4}+12\alpha_{1}s+5s\right)=0\,.$

The recursion relations for even coefficients involve the lower even
coefficients only, which is owing to the fact that
$E_{2},\theta_{2}^{4}-2\theta_{3}^{4},\theta_{2}^{8},\theta_{3}^{4}\theta_{4}^{4}$
have $q$-expansions with even coefficients.

Now, we tune $\vec{\alpha}$ and the above unfixed coefficients so that the
above 2 series solutions are 4-point Virasoro blocks. The associated CFT
quantities are $\\{c,h_{L},h_{R},h_{p(1)},h_{p(2)}\\}$.393939Here,
$h_{L}\equiv h_{1}=h_{2},\ h_{R}\equiv h_{3}=h_{4},\
\mathfrak{H}=2(h_{L}+h_{R})$. We assume identity to be an intermediate
operator as compatible with the pairwise identical external operators, and the
dimension of the other intermediate operator to be greater than zero. In
keeping with our convention (10), we take

$h_{p(2)}>h_{p(1)}=0\,.$ (60)

Since $s_{1}\ngtr s_{2}$, the first series solution i.e. the one with
$s=s_{1}$ should be equated with the vacuum block, while the series solution
with $s=s_{2}$ should be equated with the excited block. Comparing the
exponents (56) with those in (19), we get

$\displaystyle
2(h_{L}+h_{R})=6\alpha_{1}-\frac{3}{2}+3\sqrt{\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}}\,,$
(61) $\displaystyle
h_{p(2)}(\vec{\alpha})=\sqrt{\left(1-4\alpha_{1}\right){}^{2}-4\alpha_{4}},\quad\alpha_{4}<\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2},\quad\alpha_{1}\in\mathbb{R}\,.$
(62)

This also fixes $a_{0}$ of the respective solutions to be $16^{s}$. For the
Virasoro blocks in case of pairwise identical external operators, all the odd
coefficients $f_{2r-1},\ r\geq 1$ in (19) vanish.404040See footnote 31. Both
the series solutions are automatically tailored to this, except for the first
solution we need to fix $a_{m}=0$ when
$\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}=m^{2}/4,\
m=1,3,5,\dots$. In order to obtain the 3 quantities $c,h_{L},h_{R}$ in terms
of $\vec{\alpha}$, one may solve the 3 equations namely (61),
$f_{2}=a_{2}/a_{0}$ and $f_{4}=a_{4}/a_{0}$, where in each of the last 2,
l.h.s. corresponds to the vacuum block (A.5) while r.h.s. corresponds to the
first series solution. However, due to the symmetry of these 3 equations under
$h_{L}\leftrightarrow h_{R}$,
$c(\vec{\alpha}),h_{L}(\vec{\alpha}),h_{R}(\vec{\alpha})$ cannot be determined
uniquely. In addition, unlike in the case of all identical external operators,
the demands to match the higher coefficients in the first solution with
respective ones in the vacuum block, as well as the coefficients in the second
solution with respective ones in the excited block turn out to be quite
restrictive for the parameter space $\vec{\alpha}$ in the present case. Thus,
we adopt a different strategy detailed below. For a computer-assisted search
for points $\vec{\alpha}$ where both solutions to the MLDE (55) are Virasoro
blocks, we first simultaneously satisfy the 6 equations in
$\\{\alpha_{1},\alpha_{3},\alpha_{4},c,h_{L},h_{R}\\}$ given below.

$\displaystyle{\text{\it for vacuum block}:}$ $\displaystyle
2(h_{L}+h_{R})-6\alpha_{1}-\frac{3}{2}+3\sqrt{\big{(}\frac{1}{2}-2\alpha_{1}\big{)}^{2}-\alpha_{4}}=0\,,$
(63) $\displaystyle
f_{2}\left(c,h_{L},h_{R},0\right)-\frac{a_{2}(\vec{\alpha})}{a_{0}}=0,\quad
f_{4}\left(c,h_{L},h_{R},0\right)-\frac{a_{4}(\vec{\alpha})}{a_{0}}=0\,,$
$\displaystyle
f_{6}\left(c,h_{L},h_{R},0\right)-\frac{a_{6}(\vec{\alpha})}{a_{0}}=0,\quad
f_{8}\left(c,h_{L},h_{R},0\right)-\frac{a_{8}(\vec{\alpha})}{a_{0}}=0\,,$
$\displaystyle{\text{\it for excited block}:}$ $\displaystyle
f_{2}\left(c,h_{L},h_{R},h_{p(2)}(\vec{\alpha})\right)-\frac{a_{2}(\vec{\alpha})}{a_{0}}=0\,.$

These equations can be written explicitly using (59), (62), (A.5)-(A.8). They
are symmetric under $h_{L}\leftrightarrow h_{R}$, involve radicals in
$\vec{\alpha}$ and ratios of polynomials with at most the powers
$c^{6},h_{L}^{4},h_{R}^{4}$. We provide a set of explicit points
$\\{\alpha_{1},\alpha_{3},\alpha_{4},c,h_{L},h_{R}\\}$ satisfying above. With
these values we numerically check the required validity of (22) for higher
coefficients in both solutions. In the next step, we use linear perturbations
around these points to generate new ones, eventually proving the set to be
infinite. The general procedure is as follows. The set of equations (63)
abstractly can be written as

$g^{I}(t_{J})=0,\quad I,J=1,\dots,6\,.$ (64)

Suppose a solution $\hat{t}_{J}$ is a part of a continuous family, then we
have

$g^{I}(\hat{t}_{J}+\delta
t_{J})=0\implies\sum_{J}\partial_{J}g^{I}\Big{|}_{\hat{t}_{J}}\delta
t_{J}=0\quad I=1,\dots,6\,,$ (65)

with ${\rm rank}(\partial_{J}g^{I}\big{|}_{\hat{t}_{J}})\leq 5$. Solving above
equations allows us to choose some of the $\delta t_{J}$, the remaining ones
are linearly related to them. In the case ${\rm
rank}(\partial_{J}g^{I}\big{|}_{\hat{t}_{J}})=6$, we have trivial solution:
$\delta t_{J}=0\ \forall J$, and the solution $\hat{t}_{J}$ is likely to be an
isolated point, unless non-linear perturbations come into play.

Let us present here 2 points satisfying (63). Subsequently we discuss linear
perturbations around them in detail.

$\displaystyle\alpha_{1}=-\frac{1}{3},\quad\alpha_{3}=-\frac{13}{198},\quad\alpha_{4}=-3\sqrt{\frac{2}{11}}-\frac{52}{99},\quad
c=82\sqrt{\frac{2}{11}}-34\approx 0.964917\,,$ $\displaystyle
h_{L}=\frac{5}{\sqrt{22}}-1\approx 0.0660036,\quad
h_{R}=2\sqrt{\frac{2}{11}}\approx 0.852803\,,$ (66) $\displaystyle
h_{p(2)}=6\sqrt{\frac{2}{11}}+1\approx 3.55841\,;$
$\displaystyle\alpha_{1}=\frac{1}{2},\quad\alpha_{3}=1,\quad\alpha_{4}=0,\quad
c=1,\quad h_{L}=\frac{1}{4},\quad h_{R}=\frac{5}{4},\quad h_{p(2)}=1\,.$ (67)

Non-BPZ cases: For the example (66), with the values of
$\alpha_{1},\alpha_{3},\alpha_{4}$ the 2 solutions to the MLDE (55)
respectively go as $(16q)^{\frac{2}{3}-3\sqrt{\frac{2}{11}}}$ and
$(16q)^{3\sqrt{\frac{2}{11}}+\frac{5}{3}}$ as $q\to 0$. We have solved the
MLDE recursion relations numerically and found the $q$-expansion coefficients
$a_{n}$ up to order $q^{125}$ for both solutions, with odd ones vanishing. On
the other hand with the values of $c,h_{L},h_{R}$, first using the Mathematica
code in [22], the $q$-expansion coefficients $b_{n}$ of the $H$-functions
corresponding to $h_{p(1)}=0$ and $h_{p(2)}=3.55841$ are numerically obtained
up to order $q^{125}$, and then the numerical coefficients $f_{n}$ of the 2
respective Virasoro blocks are calculated coding the relation (21). Indeed,
they respectively match to the 2 solutions of our MLDE. In the present
example, we have found the $q$-expansion coefficients of the blocks to be non-
integers.

Note that none of the $(c,h_{L}),(c,h_{R})$ lies on the locus of the level-2
Virasoro null states:

$16h_{L}^{2}+h_{L}(2c-10)+c\neq 0,\quad 16h_{R}^{2}+h_{R}(2c-10)+c\neq 0\,.$
(68)

Hence, corresponding to $\vec{\alpha}$ in (66), we have a second order MLDE
(55) whose solutions are 4-point Virasoro blocks and the MLDE does not
correspond to a BPZ equation.

At (66), ${\rm rank}(\partial_{J}g^{I})=5$ and we have

$\displaystyle\delta\alpha_{3}=\frac{1}{726}(385-18\sqrt{22})\delta\alpha_{1},\quad\delta\alpha_{4}=\frac{1}{726}(945\sqrt{22}+1364)\delta\alpha_{1},$
(69) $\displaystyle\delta
c=\frac{3}{11}(188-41\sqrt{22})\delta\alpha_{1},\quad\delta
h_{L}=-\frac{3}{44}(5\sqrt{22}-24)\delta\alpha_{1},$ $\displaystyle\delta
h_{R}=-\frac{3}{11}(\sqrt{22}+4)\delta\alpha_{1}\,.$

Knowing the above first derivatives w.r.t. $\alpha_{1}$ at one point, one
cannot trace the full curve away from (66), however, guided by the above
formulae nearby points satisfying (63) can be scanned. Here we present one new
point near (66) well approximated by (69) with $\delta\alpha_{1}=10^{-4}$, and
indeed the errors are of the order $(\delta\alpha_{1})^{2}$.

$\displaystyle\alpha_{1}=-\frac{9997}{30000},\quad\alpha_{3}=\frac{-54\sqrt{549835009}-7144225927}{108900000000}\,,$
(70)
$\displaystyle\alpha_{4}=\frac{-1484514\sqrt{549835009}-14294885369}{27225000000},\quad
c=\frac{41\sqrt{549835009}-934859}{27500}\,,$ $\displaystyle
h_{L}=\frac{5\sqrt{549835009}-109982}{110000},\quad
h_{R}=\frac{\sqrt{549835009}-3}{27500},\quad
h_{p(2)}=\frac{3\sqrt{549835009}+27491}{27500}\,.$

Moreover, with (70) we have numerically tested the required validity of (22)
for higher coefficients in both solutions up to order $q^{125}$, in a similar
way to the previous case. We have found the $q$-expansion coefficients of the
solutions to be non-integers. In this case as well, the MLDE is a non-BPZ one
since (68) holds.

BPZ cases: For the example (67), with the values of
$\alpha_{1},\alpha_{3},\alpha_{4}$ the 2 solutions to the MLDE (55)
respectively go as $(16q)^{-1}$ and $(16q)^{0}$ as $q\to 0$. We have solved
the MLDE recursion relations and found the $q$-expansion coefficients $a_{n}$
up to order $q^{21}$ for both solutions, with odd ones vanishing.414141For the
first solution, we set $a_{1}=0$ by hand as $\vec{\alpha}$ lies on the surface
(57) with $m=1$. On the other hand with the values of $c,h_{L},h_{R}$, first
using the Mathematica code in [22], the $q$-expansion coefficients $b_{n}$ of
the $H$-functions corresponding to $h_{p(1)}=0$ and $h_{p(2)}=1$ are obtained
up to order $q^{21}$, and then the coefficients $f_{n}$ of the 2 respective
Virasoro blocks are calculated coding the relation (21). Indeed, they
respectively match to the 2 solutions of our MLDE. In the present example, we
have found the $q$-expansion coefficients of the blocks to be non-integers.
The full computation has been done analytically.424242For $c=1$ i.e. $b=i$
[see, definition (A.1)], the numerical code in [22] breaks. Hence, taking the
external dimensions $h_{L}=1/4,h_{R}=5/4$, we use the code for the analytical
computations of the first 21 $q$-expansion coefficients in the $H$-functions
corresponding to 2 Virasoro blocks with above intermediate dimensions and
arbitrary $c$. Then we take $b\to i$ limit.

Clearly $(c,h_{L})$ lies on the locus of the level-2 Virasoro null states:

$16h_{L}^{2}+h_{L}(2c-10)+c=0,\quad 16h_{R}^{2}+h_{R}(2c-10)+c\neq 0\,.$ (71)

Hence, the MLDE corresponds to a BPZ equation in the present case.

At (67), ${\rm rank}(\partial_{J}g^{I})=4$ and we have

$\displaystyle\delta\alpha_{4}=3\delta\alpha_{1}-\frac{\delta\alpha_{3}}{2},\quad\delta
h_{L}=\frac{3}{8}(\delta\alpha_{3}-2\delta\alpha_{1}),\quad\delta
h_{R}=\frac{3}{8}(6\delta\alpha_{1}+\delta\alpha_{3}),\quad\delta c=0\,.$ (72)

This is a 2 dimensional space. The points on it are not guaranteed to
correspond BPZ cases. In addition, there could be higher order perturbations
that may alter $c$ from 1. However, the curve satisfying (63) and passing
trough (67) such that always $c=1$ and the MLDE corresponds to a BPZ equation
i.e. $h_{L}=1/4$ can be easily obtained as

$\alpha_{3}(\alpha_{1})=-\alpha_{1}\left(\alpha_{1}-3\right)-\frac{1}{4},\quad\alpha_{4}(\alpha_{1})=2\alpha_{1}\left(2\alpha_{1}-1\right),\quad
h_{R}(\alpha_{1})=3\alpha_{1}-\frac{1}{4}\,.$ (73)

Along the curve, $h_{p(2)}=1$. And, the 2 solutions to the MLDE (55)
respectively go as $(16q)^{-2\alpha_{1}}$ and $(16q)^{1-2\alpha_{1}}$ as $q\to
0$. As the above $\vec{\alpha}$ lies on the surface (57) with $m=1$, we set
$a_{1}=0$ by hand for the first solution. With this, the odd coefficients in
both solutions vanish as required for the 2 Virasoro blocks corresponding to
$h_{p(1)}=0$ and $h_{p(2)}=1$. We have analytically tested the required
validity of (22) in both solutions up to order $q^{21}$.

Now, we present one new point near (67) for which $c\neq 1$, nonetheless well
approximated by (72) with $\delta\alpha_{1}=10^{-4},\delta\alpha_{3}=10^{-4}$
up to errors of order
$(\delta\alpha_{1})^{2},(\delta\alpha_{3})^{2},\delta\alpha_{1}\delta\alpha_{3}$.

$\displaystyle\alpha_{1}=\frac{5001}{10000},\quad\alpha_{3}=\frac{10001}{10000},\quad\alpha_{4}=\frac{10003\sqrt{400080013}-200030003}{200000000}\,,$
(74) $\displaystyle
c=\frac{13002600310009-600060003\sqrt{400080013}}{1000200010000},\quad
h_{L}=\frac{3\sqrt{400080013}-40009}{80000}\,,$ $\displaystyle
h_{R}=\frac{3\sqrt{400080013}+40015}{80000},\quad
h_{p(2)}=\frac{\sqrt{400080013}-10003}{10000}\,.$

Moreover, with (74) we have numerically tested the required validity of (22)
for higher coefficients in both solutions up to order $q^{125}$. We have found
the $q$-expansion coefficients of the solutions to be non-integers. In this
case as well, the MLDE is a BPZ one since (71) holds.

Now, the modular transformation matrices that act on the above solutions to
(55) are434343Computed using (C.5).

$\displaystyle
M(T)=\begin{pmatrix}e^{\frac{i\pi}{2}\left(1-4\alpha_{1}-\sqrt{\kappa_{1}}\right)}&0\\\
0&e^{\frac{i\pi}{2}\left(1-4\alpha_{1}+\sqrt{\kappa_{1}}\right)}\end{pmatrix},$
(75) $\displaystyle
M(ST^{2}S)_{(1,1)}=-ie^{\frac{1}{2}i\pi\left(4\alpha_{1}+\sqrt{\kappa_{1}}\right)}\cos\left(2\pi\sqrt{\kappa_{2}}\right)\csc\big{(}\frac{\pi\sqrt{\kappa_{1}}}{2}\big{)}\,,$
$\displaystyle
M(ST^{2}S)_{(2,1)}=-\frac{2^{-2\sqrt{\kappa_{1}}-1}\left((-1)^{4\sqrt{\kappa_{2}}}-1\right)(-1)^{2\alpha_{1}-2\sqrt{\kappa_{2}}}\csc\left(2\pi\sqrt{\kappa_{2}}\right)\Gamma\big{(}1-\frac{\sqrt{\kappa_{1}}}{2}\big{)}\Gamma\big{(}-\frac{\sqrt{\kappa_{1}}}{2}\big{)}}{\Gamma\big{(}-\frac{\sqrt{\kappa_{1}}}{2}-2\sqrt{\kappa_{2}}+\frac{1}{2}\big{)}\Gamma\big{(}-\frac{\sqrt{\kappa_{1}}}{2}+2\sqrt{\kappa_{2}}+\frac{1}{2}\big{)}}\,,$
$\displaystyle M(ST^{2}S)_{(1,2)}=-\frac{2i\pi
e^{2i\pi\alpha_{1}}\Gamma\big{(}\sqrt{\kappa_{1}}+1\big{)}\Gamma\big{(}\sqrt{\kappa_{1}}\big{)}}{{\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}+1)\big{)}}^{2}\
\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}-4\sqrt{\kappa_{2}}+1)\big{)}\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}+4\sqrt{\kappa_{2}}+1)\big{)}}\,,$
$\displaystyle
M(ST^{2}S)_{(2,2)}=ie^{-\frac{1}{2}i\pi\left(\sqrt{\kappa_{1}}-4\alpha_{1}\right)}\cos\left(2\pi\sqrt{\kappa_{2}}\right)\csc\big{(}\frac{\pi\sqrt{\kappa_{1}}}{2}\big{)}\,,$
$\displaystyle\kappa_{1}=\left(1-4\alpha_{1}\right){}^{2}-4\alpha_{4},\quad\kappa_{2}=\alpha_{1}^{2}+\alpha_{3}\,.$
(76)

Now confining us to scalar operators, the bootstrap condition corresponding to
$T$ in (24) allows only vanishing non-diagonal elements in $\mathcal{C}$:

$\mathcal{C}=\begin{pmatrix}1&0\\\
0&\beta\end{pmatrix},\quad\beta=c_{LLp(2)}\cdot c_{RRp(2)}\,,$ (77)

where $c_{LLp(2)},c_{RRp(2)}$ are the 3-point coefficients associated with the
stripped correlator $G_{LLRR}$. Now, the $S$-invariance solves for $\beta$
as444444Here, ∗ denotes the complex conjugate. We know that $\kappa_{1}>0$ due
to (62).

$\displaystyle\beta=$
$\displaystyle-\frac{{\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}+1)\big{)}}^{4}\left(\cos\big{(}2\pi\sqrt{\kappa_{2}}\big{)}\csc^{2}\big{(}\frac{\pi\sqrt{\kappa_{1}}}{2}\big{)}\cos\big{(}2\pi{\sqrt{\kappa_{2}}}^{*}\big{)}-1\right)}{4\pi^{2}{\Gamma\big{(}\sqrt{\kappa_{1}}+1\big{)}}^{2}\
{\Gamma\big{(}\sqrt{\kappa_{1}}\big{)}}^{2}}$ (78)
$\displaystyle\times\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}-4\sqrt{\kappa_{2}}+1)\big{)}\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}+4\sqrt{\kappa_{2}}+1)\big{)}\Gamma\big{(}\frac{1}{2}(\sqrt{\kappa_{1}}-4{\sqrt{\kappa_{2}}}^{*}+1)\big{)}\Gamma\big{(}\frac{1}{2}(4{\sqrt{\kappa_{2}}}^{*}+\sqrt{\kappa_{1}}+1)\big{)}\,.$

Thus, in the aforesaid explicit examples we have

$\displaystyle{\text{\it for \eqref{eq:nonBPZex1LR}}},\quad\beta$
$\displaystyle=-\frac{16^{-6\sqrt{\frac{2}{11}}-1}{\Gamma\big{(}-3\sqrt{\frac{2}{11}}-\frac{1}{2}\big{)}}^{2}{\Gamma\big{(}\frac{1}{2}-3\sqrt{\frac{2}{11}}\big{)}}^{2}}{\bigg{(}\frac{1}{\left(1-2\cos\left(2\sqrt{\frac{2}{11}}\pi\right)\right)^{2}}-1\bigg{)}{\Gamma\big{(}-4\sqrt{\frac{2}{11}}\big{)}}^{2}{\Gamma\big{(}-2\sqrt{\frac{2}{11}}\big{)}}^{2}}$
(79) $\displaystyle\approx 0.0000419138\,;$ $\displaystyle{\text{\it for
\eqref{eq:BPZex1LR}}},\quad\beta$ $\displaystyle=\frac{5}{4}\,;$
$\displaystyle{\text{\it for \eqref{eq:nonBPZex2LR}}},\quad\beta$
$\displaystyle\approx 0.0000419882\,;$ $\displaystyle{\text{\it for
\eqref{eq:BPZex2LR}}},\quad\beta$ $\displaystyle\approx 1.24968\,.$

And, along the curve (73)

$\beta=\frac{1}{4}\left|1-12\alpha_{1}\right|\cot\big{(}\pi\sqrt{12\alpha_{1}-1}\big{)}\tan\big{(}\pi{\sqrt{12\alpha_{1}-1}}^{*}\big{)}\,.$
(80)

Note that with the pairing $LLRR$ of external operators the non-negativity of
the $q$-expansion coefficients of pillow blocks does not apply as unitarity
bound, see section 3. In all the examples above, $c,h_{L},h_{R},h_{p(2)}$ are
greater than zero, except in the case (73) we need to restrict
$\alpha_{1}>1/12$. However, the non-BPZ examples (66), (70) and the BPZ
example (74) are non-unitary since the respective central charges are less
than 1 but do not lie in (47). On the other hand, in the BPZ example (67) with
$c=1$, the representations with respective $h_{L},h_{R},h_{p(2)}$ are all
unitary.454545The Kac determinants for a representation with $(c=1,h>0)$ are
non-negative, and respective Gram matrices are positive semi-definite [4].

##  5 Discussions

In the above, we have constructed MLDEs w.r.t. subgroups of modular group
whose solutions are Virasoro conformal blocks that appear in a crossing
symmetric 4-point correlator on the sphere. We have particularly focused on
second order MLDEs with holomorphic coefficients in the cases of all identical
and pairwise identical ($LLRR$) external operators, namely (30) and (55). In
the first case, the relevant subgroup is the full modular group, and in the
second case, it is generated by $T,ST^{2}S$. The central charge, operator
dimensions of both external as well intermediate ones are functions of
parameters that appeared in the MLDEs. We are able to provide these functional
dependencies explicitly in the first case and in a subcase of the second case,
elsewhere they are implicit. This has been done by comparing the
$q$-expansions of the solutions to the MLDEs with those of Virasoro blocks.
Using actions of respective subgroups, bootstrap equations involving the
associated 3-point coefficients have been set up and solved as well in terms
of the MLDE parameters. For the case of all identical external operators, our
MLDE with one single parameter corresponds to BPZ equation. We have
illustrated one unitary example and one non-unitary example at the end of
section 4.1. The non-negativity of the $q$-expansion coefficients in the
pillow blocks applies in this case as unitarity bound, and we have shown this
to be an insufficient condition for unitarity by calculating the coefficients
in pillow blocks to be non-negative for our non-unitary example as well. For
the case of pairwise identical operators, starting from a few specific
examples of MLDEs we have used linear perturbations around them to generate
(infinite-)family of MLDEs in section 4.2. In this case, some of the MLDEs
presented do not correspond to BPZ equations. These are novel second order
differential equations whose solutions are Virasoro blocks. This itself is
quite exciting result in the context of building differential equations for
the blocks. Also, out of those blocks crossing symmetric correlators can be
written as our bootstrap equations possess nontrivial solutions in those
cases. In these specific examples, from the corresponding central charges it
is clear that the respective representations are non-unitary. With the
classification of non-unitary CFTs with $c<1$, one will be able to identify
the respective CFTs where our $(c,h)$-values lie.

Below, we discuss potential generalisations of the present analysis and
related future avenues, including the consideration of meromorphic forms as
the coefficients in our MLDEs in some detail.

1\. Meromorphic forms: In the above, as the coefficients of our MLDEs we have
considered holomorphic forms, see (27). The ansatz (27) leads to the most
general second order linear differential equation with 3 regular singular
points in variable $x=\lambda(\tau)$ namely at $x=0,1,\infty$. Thus,
considerations of meromorphic $\phi_{0},\phi_{1}$ should result in more and/or
new type singular points of the differential equation in $x$. However, here we
point out that such considerations will be important for writing down MLDEs
for 4-point Virasoro blocks. Consider the following MLDE w.r.t. the full
modular group.

$\displaystyle\frac{d^{2}\vec{F}}{d\tau^{2}}-\frac{i\pi}{3}E_{2}(\tau)\frac{d\vec{F}}{d\tau}+2i\pi\alpha_{1}\frac{E_{6}(\tau)}{E_{4}(\tau)}\frac{d\vec{F}}{d\tau}+\pi^{2}\left(\alpha_{3}\frac{E_{6}(\tau)^{2}}{E_{4}(\tau)^{2}}+\alpha_{2}E_{4}(\tau)\right)\vec{F}=0\,,$
(81)
$\displaystyle\alpha_{1}=\frac{2}{3},\quad\alpha_{2}=\frac{164}{225},\quad\alpha_{3}=-\frac{8}{9}\,.$

$E_{4}$ has zeros at $\tau=e^{\frac{i\pi}{3}},e^{\frac{2i\pi}{3}}$ which
correspond to $x=(-1)^{1/3},-(-1)^{2/3}$ respectively. $E_{6}$ is non-
vanishing at these points. The 2 solutions to the MLDE (81) respectively go as
$(16q)^{-\frac{4}{5}}$ and $(16q)^{-\frac{1}{5}}$ as $q\to 0$. Now considering
4 identical external operators of dimension $h=3/5$ in a CFT with central
charge $c=7/10$, the above solutions respectively coincide with the Virasoro
blocks corresponding to the intermediate dimensions $h_{p(1)}=0$ and
$h_{p(2)}=3/5$. We have analytically tested the required validity of (22) for
$q$-expansion coefficients in both solutions up to order $q^{15}$. In doing
so, the $q$-expansion coefficients in the solutions are obtained solving the
above MLDE recursively. For $c=7/10$ i.e. $b=i\sqrt{5}/2$ [see, definition
(A.1)], the numerical code in [22] breaks. Thus, taking external dimensions
$h=3/5$, we use the code for the analytical computations of the first 15
$q$-expansion coefficients in the $H$-functions corresponding to 2 Virasoro
blocks with above intermediate dimensions and arbitrary $c$. Then we take
$b\to i\sqrt{5}/2$ limit. These coefficients are then used to calculate the
coefficients in the Virasoro blocks coding the relation (21). In the present
example, we have found the $q$-expansion coefficients of the solutions to be
non-integers. The MLDE (81) is a non-BPZ one as $16h^{2}+h(2c-10)+c\neq 0$.

Once the above solutions are identified with the Virasoro blocks, one can
solve the MLDE recursively to obtain their $q$-expansions to any desired order
$q^{n}$. We have done this numerically and found the coefficients up to order
$q^{100}$ within an accuracy $10^{-50}$. Within this accuracy, the two
solutions vanishes at $q=ie^{-\sqrt{3}\pi/2},-ie^{-\sqrt{3}\pi/2}$ i.e.
respectively at $\tau=e^{\frac{i\pi}{3}},e^{\frac{2i\pi}{3}}$. Thus, the
solutions are analytic although the MLDE has singularities at these points. In
the variable $x=\lambda(\tau)$, (81) becomes

$\displaystyle\frac{d^{2}\vec{F}}{dx^{2}}+P(x)\frac{d\vec{F}}{dx}+Q(x)\vec{F}=0,\quad
P(x)=\frac{2(2x-1)}{x(x-1)\left(x^{2}-x+1\right)}\,,$ (82) $\displaystyle
Q(x)=\frac{2\left(2x^{6}-6x^{5}-63x^{4}+136x^{3}-63x^{2}-6x+2\right)}{25x^{2}(x-1)^{2}\left(x^{2}-x+1\right)^{2}}\,.$

Clearly $x=(-1)^{1/3},-(-1)^{2/3}$ are regular singular points of the above
ODE. Nonetheless, at these points the 2 general solutions given below are
analytic with vanishing constant term in respective Taylor series.

$\frac{((x-1)x+1)P_{-\frac{1}{5}}^{\frac{3}{5}}(2x-1)}{\sqrt{-((x-1)x)}},\quad\frac{((x-1)x+1)Q_{-\frac{1}{5}}^{\frac{3}{5}}(2x-1)}{\sqrt{-((x-1)x)}}\,,$
(83)

where $P^{m}_{n}(x)$ denotes the associated Legendre polynomial and
$Q^{m}_{n}(x)$ the associated Legendre function of the second kind [33]. As
the 2 aforesaid Virasoro blocks can be written as linear combinations of (83),
they are analytic at $x=(-1)^{1/3},-(-1)^{2/3}$ as well, and vanish at these
points.

The above discussion shows the importance to scan over MLDEs with meromorphic
coefficients in order to construct differential equations for 4-point Virasoro
blocks. Such an MLDE may lead to new singularities as well in the Virasoro
blocks in addition to the expected ones, unlike the above example.464646For
$W_{3}$ conformal blocks, such a situation has been reported earlier in [34].
Such a case will be interesting as this can not be observed from the
$q$-expansions of the Virasoro blocks due to Zamolodchikov. We leave the
systematic study for the future.

2\. MLDEs with $c>1,\ h_{a},h_{p(k)}>0$: The examples constructed in this
paper are with $c\leq 1$. It will be important to scan the parameter space
$\vec{\alpha}$ for points such that the MLDE (14) gives Virasoro blocks with
$c>1,\ h_{a},h_{p(k)}>0$ when $O_{p(k)}\neq\mathds{1}$. If it exists, the MLDE
will be a non-BPZ one, additionally the corresponding CFT will be unitary as
all representations with $(c>1,h>0)$ are unitary. Such a CFT contains infinite
number of Virasoro primaries in its spectrum [35]. Thus, above MLDE having a
finite order $N$ would reveal non-trivial information about the fusion rules,
such as (1) or a truncation.

To initiate, let us take the case of pairwise identical operators studied in
this paper. In this case, existence of an MLDE (55) that corresponds to $c>1,\
h_{L},h_{R},h_{p(2)}>0\ (h_{p(1)}=0)$ is not ruled out yet, unlike the case of
all identical operators. We have performed a preliminary search as follows.
Here, we deal with an inverse problem of the one discussed in section 4.2. We
use the first 3 equations in (63) to express
$\alpha_{1},\alpha_{3},\alpha_{4}$ in terms of $c,h_{L},h_{R}$, then try to
satisfy the remaining 3 equations in the region $c>1,\ h_{L},h_{R}>0$ and
single out the cases with $h_{p(2)}>0$. For this, we numerically vary $c$ from
$1.025$ to $3$ with step-size $0.025$, and each $h_{L},h_{R}$ from $0.025$ to
$2$ with step-size $0.025$. This generates $259200$ points in the above
region. For each of these, we numerically solve the first 3 equations in (63)
to get $\alpha_{1},\alpha_{3},\alpha_{4}$ and with these values we check the
validity of the remaining 3 equations within a tolerance $0.1$. We have taken
a large tolerance, keeping in mind that the desired point satisfying (63) can
possibly be an isolated one and may lie only near any of the generated points.
For none of the $259200$ points generated, (63) can be satisfied within above
tolerance. Increasing the upper limit of $c,h_{L},h_{R}$ and/or decreasing the
step-sizes make the numerics more involved, which we leave for the future.

3\. Sphere-torus correspondence: Recently [9] reported a new sphere-torus
correspondence that relates the characters of diagonal theories to specific
4-point conformal blocks on the sphere. Their proposed correspondence
considers on the torus-side a CFT denoted by $\mathcal{T}_{t}$ and on the
sphere-side a CFT given by
$\mathcal{T}_{t}\times\mathcal{T}_{t}/\mathbb{Z}_{2}$. In the present context
of MLDEs, a potential future direction will be to classify all MLDEs (second
or higher order, with holomorphic and meromorphic coefficients as well) in the
case of all identical operators (as here the relevant subgroup is the full
modular group as evident from Tab. 1) whose solutions are Virasoro blocks with
non-negative integer $q$-expansion coefficients, additionally the
$S$-transformation matrix $M(S)$ that acts on the blocks should satisfy some
integrality conditions coming from Verlinde’s formula [36]. Then we get
candidates for MLDEs for characters and will be able to study sphere-torus
correspondences more generally(if they exist) than the proposed one in certain
types of theories.

4\. Infinite number of Virasoro blocks: In unitary CFTs with $c>1$ whose
spectra necessarily contain infinite number of Virasoro primaries, the
conformal block expansions of some (crossing symmetric)correlators are likely
to have an infinite number of Virasoro blocks as well. Construction of MLDEs
for such blocks are out of the scope of the present program. However, such
blocks may be expressed in terms of eigen functions of Laplace-Beltrami
operator on the upper half plane with some choice of metric, generalising the
ideas in [37, 38, 39, 40].

It will be also interesting to study MLDEs in the other possible cases given
in Tab. 1. For this, one need to generalise the code in [22] for analytical
and/or numerical computations of the $q$-expansion coefficients in the
$H$-functions adapted in those cases. When all the operators are distinct, we
take a preliminary attempt as discussed at the end of appendix A. It will be
also interesting to consider higher order MLDEs and to generalize our
construction for other type of conformal blocks. Finally, we should mention
that the discrete values in (48) require an explanation in the alternative
viewpoint on $q$-expansions of Virasoro blocks provided in [23] on top of the
non-negativity of the $q$-expansion coefficients of pillow blocks.

## Acknowledgements

We would like to thank Sujay K. Ashok and Dileep P. Jatkar for useful comments
on the draft.

##  Appendix A Zamolodchikov’s $H$-function

The Zamolodchikov’s $H$-functions are related to the Virasoro blocks, see (18)
. The $q$-expansion coefficients in the $H$-functions can be obtained using
Zamolodchikov’s elliptic recursion formula [6]. We discuss relevant properties
of these coefficients in this appendix. We also obtain the $q$-expansion
coefficients of the Virasoro blocks.

To set up the recursion relation, we suitably adapt the parameterizations from
[22] for our correlator $\langle O_{1}(\infty)O_{2}(1)O_{3}(x)O_{4}(0)\rangle$
as follows.474747In contrast to us, [22] considers $\langle
O_{1}(0)O_{2}(x)O_{3}(1)O_{4}(\infty)\rangle$. And, as discussed in appendix
B, $x$ is the inverse elliptic nome.

$\displaystyle c=1+6\big{(}b+\frac{1}{b}\big{)}^{2},\qquad
h_{a}=\frac{1}{4}\big{(}b+\frac{1}{b}\big{)}^{2}-\lambda_{a}^{2}\,,$ (A.1)
$\displaystyle
h_{mn}=\frac{1}{4}\big{(}b+\frac{1}{b}\big{)}^{2}-\lambda_{mn}^{2},\qquad\lambda_{mn}=\frac{1}{2}\big{(}\frac{m}{b}+nb\big{)}\,,$
$\displaystyle
R_{mn}=2\prod_{p,q}(\lambda_{3}+\lambda_{4}-\lambda_{pq})(\lambda_{4}-\lambda_{3}-\lambda_{pq})(\lambda_{1}+\lambda_{2}-\lambda_{pq})(\lambda_{2}-\lambda_{1}-\lambda_{pq}){\prod_{r,s}}^{\prime}\lambda_{rs}^{-1}\,,$

where $p,q,r,s$ run over

$\displaystyle p=-m+1,-m+3,\dots,m-3,m-1,\quad q=-n+1,-n+3,\dots,n-3,n-1\,,$
(A.2) $\displaystyle r=-m+1,-m+2,\dots,m,\quad s=-m+1,-m+2,\dots,m\,,$

and the primed product means that $(r,s)=(0,0)$ and $(m,n)$ are excluded. The
recursion relation for $H$ is given by

$H(c,h_{a},h_{p},q)=1+\sum_{m,n=1,2,\dots}\frac{q^{mn}R_{mn}}{h_{p}-h_{mn}}H(c,h_{a},h_{mn}+mn,q)\,.$
(A.3)

Note that $R_{mn}$ is symmetric when we interchange both
$\lambda_{1}\leftrightarrow\lambda_{4}$ and
$\lambda_{2}\leftrightarrow\lambda_{3}$, as the following holds.

$\prod_{p,q}(\lambda_{4}-\lambda_{3}-\lambda_{pq})(\lambda_{2}-\lambda_{1}-\lambda_{pq})=\prod_{p,q}(\lambda_{1}-\lambda_{2}-\lambda_{pq})(\lambda_{3}-\lambda_{4}-\lambda_{pq})\,.$
(A.4)

Hence, (A.3) asserts that $H$ is invariant under the interchange:
$h_{1}\leftrightarrow h_{4},h_{2}\leftrightarrow h_{3}$.

One can obtain the higher coefficients $b_{n}$ from the knowledge of the lower
ones by evaluating (A.3) iteratively. All the odd coefficients $b_{n}$ contain
a factor of $R_{11}$. Therefore, they vanish when $h_{1}=h_{2}$ or
$h_{3}=h_{4}$ [22, 41].

In the case $h_{1}=h_{2}\equiv h_{L},\ h_{3}=h_{4}\equiv h_{R}$, [22]
implements Zamolochikov’s recursion relation for $H$ in a companion
Mathematica notebook. The code can be used for both numerical and analytical
computations of the $q$-expansion coefficients $b_{n}$ in $H$ yielding the
analytic even coefficients $b_{2n}$ in terms of $b,h_{L},h_{R},h_{p}$ with odd
ones vanishing. In the present paper, we need the first few analytic even
coefficients $f_{2n}$ in (19). $f_{2n}$ are linearly related to $b_{2n}$ (20).
Thus, we use the above code to get the first few analytic coefficients
$b_{2n}$, and replace the parameter $b$ in terms of $c$. Then, coding the
relation (21), we get the required even coefficients of the Virasoro blocks.
Below we present the first few. The higher analytic coefficients are very
lengthy. For the vacuum block we have

$\displaystyle f_{2}$
$\displaystyle=b_{2}+\frac{1}{2}\left(1-c+16h_{L}+16h_{R}\right)=\frac{512h_{L}h_{R}}{c}-8\left(h_{L}+h_{R}\right)\,,$
$\displaystyle f_{4}$
$\displaystyle=b_{4}+\frac{1}{8}\left(1-c+16h_{L}+16h_{R}\right)\left(4b_{2}+7-c+16h_{L}+16h_{R}\right)$
$\displaystyle=\frac{4}{c(5c+22)}\Big{[}8h_{L}^{2}\left(640h_{R}\left(2-c+32h_{R}\right)+c(5c+22)\right)+c(5c+22)h_{R}\left(8h_{R}-1\right)$
$\displaystyle\qquad\qquad+h_{L}\left(16h_{R}\left(128-320(c-2)h_{R}+c(5c-42)\right)-c(5c+22)\right)\Big{]}\,.$
(A.5) $\displaystyle f_{6}=$
$\displaystyle\frac{1}{3c(-1+2c)(22+5c)(68+7c)}\times$
$\displaystyle{\left(32\left(-c(-1+2c)(22+5c)(68+7c)h_{L}\left(1+h_{L}\left(-3+8h_{L}\right)\right)+\right.\right.}$
$\displaystyle{\left(-c(-1+2c)(22+5c)(68+7c)+2(-23296+c(155320+3c(12322+c(5113+70c))))h_{L}-\right.}$
$\displaystyle{24(68928+c(35640+c(46346+c(-279+70c))))h_{L}^{2}+}$
$\displaystyle{\left.512(376+c(15494+3c(57+70c)))h_{L}^{3}\right)h_{R}-}$
$\displaystyle{3\left(-c(-1+2c)(22+5c)(68+7c)+8(68928+c(35640+c(46346+c(-279+70c))))h_{L}-\right.}$
$\displaystyle{\left.1024(9256+c(9538+c(-839+70c)))h_{L}^{2}+32768(876+c(-251+70c))h_{L}^{3}\right)h_{R}^{2}-}$
$\displaystyle{8\left(c(-1+2c)(22+5c)(68+7c)-64(376+c(15494+3c(57+70c)))h_{L}+\right.}$
$\displaystyle{\left.\left.\left.12288(876+c(-251+70c))h_{L}^{2}-262144(29+70c)h_{L}^{3}\right)h_{R}^{3}\right)\right)}\,.$
(A.6) $\displaystyle f_{8}=$
$\displaystyle\frac{1}{3c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)}\times$
$\displaystyle{\left(2\left(c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)h_{L}\left(-3+4h_{L}\left(35+16h_{L}\left(-3+4h_{L}\right)\right)\right)+\right.\right.}$
$\displaystyle{(-3c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)+}$
$\displaystyle{8(-15630336+c(93118128+c(2402116+c(-65081200+c(3581843+5c(-163753+7350c))))))h_{L}-}$
$\displaystyle{64(82286592+c(-121751536+c(-129927860+c(56360240+c(-4282919+15c(72119+630c))))))}$
$\displaystyle{h_{L}^{2}+1024(-7839360+c(-97857360+c(-2557588+c(-10723064+5c(427349+c(3181+210c))))))}$
$\displaystyle{\left.h_{L}^{3}-65536(-42864+c(-2709332+c(-1710448+c(259441+5c(2173+210c)))))h_{L}^{4}\right)h_{R}+}$
$\displaystyle{4(35c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)-}$
$\displaystyle{16(82286592+c(-121751536+c(-129927860+c(56360240+c(-4282919+15c(72119+630c))))))}$
$\displaystyle{h_{L}+}$
$\displaystyle{128(-213555712+c(66109744+3c(76581868+c(-7652528+5c(1019645+c(2173+210c))))))h_{L}^{2}-}$
$\displaystyle{16384(-13406640+c(-5758900+c(-1753868+15c(195457+c(-2363+210c)))))h_{L}^{3}+}$
$\displaystyle{\left.524288(-831704+c(-410542+3c(180603+25c(-19+42c))))h_{L}^{4}\right)h_{R}^{2}+}$
$\displaystyle{64(-3c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)+}$
$\displaystyle{16(-7839360+c(-97857360+c(-2557588+c(-10723064+5c(427349+c(3181+210c))))))h_{L}-}$
$\displaystyle{1024(-13406640+c(-5758900+c(-1753868+15c(195457+c(-2363+210c)))))h_{L}^{2}+}$
$\displaystyle{196608(-386760+c(368134+5c(50751+c(-2783+210c))))h_{L}^{3}-}$
$\displaystyle{\left.4194304(-10686+c(24559+5c(-599+210c)))h_{L}^{4}\right)h_{R}^{3}+}$
$\displaystyle{256(c(-1+2c)(46+3c)(3+5c)(22+5c)(68+7c)-}$
$\displaystyle{256(-42864+c(-2709332+c(-1710448+c(259441+5c(2173+210c)))))h_{L}+}$
$\displaystyle{8192(-831704+c(-410542+3c(180603+25c(-19+42c))))h_{L}^{2}-}$
$\displaystyle{\left.\left.\left.1048576(-10686+c(24559+5c(-599+210c)))h_{L}^{3}+16777216(-251+5c(661+210c))h_{L}^{4}\right)h_{R}^{4}\right)\right)}\,.$
(A.7)

And, for the excited block corresponding to dimension $h_{p}$ we have

$\small
f_{2}=\frac{4h_{p}\left(3c-2(c-1)h_{p}\right)-8h_{L}\left(2h_{p}\left(c-8h_{p}-64h_{R}+11\right)+c-64h_{R}\right)-8h_{R}\left(2h_{p}\left(c-8h_{p}+11\right)+c\right)}{2h_{p}\left(c+8h_{p}-5\right)+c}\,.$
(A.8) $\displaystyle f_{4}=$
$\displaystyle\frac{2}{\left(c+8h_{p}-1\right)\left(5c\left(2h_{p}+3\right)+2\left(h_{p}-1\right)\left(8h_{p}-33\right)\right)\left(2ch_{p}+c+2h_{p}\left(8h_{p}-5\right)\right)}\times$
$\displaystyle{\left(4\left(-1+h_{p}\right)h_{p}\left(7h_{p}\left(-13+4h_{p}\right)\left(-3+8h_{p}\right)-1206h_{R}+64h_{p}\left(273-386h_{p}+64h_{p}^{2}\right)h_{R}+\right.\right.}$
$\displaystyle{\left.16\left(-645+8h_{p}\left(487+8h_{p}\left(-43+8h_{p}\right)\right)\right)h_{R}^{2}\right)+}$
$\displaystyle{64h_{L}^{2}\left(\left(-1+h_{p}\right)h_{p}\left(-645+8h_{p}\left(487+8h_{p}\left(-43+8h_{p}\right)\right)\right)+\right.}$
$\displaystyle{\left.64\left(-15+h_{p}\left(161+2h_{p}\left(191+64\left(-7+h_{p}\right)h_{p}\right)\right)\right)h_{R}+1024\left(-15+4h_{p}\left(26+h_{p}\left(-19+8h_{p}\right)\right)\right)h_{R}^{2}\right)+}$
$\displaystyle{8h_{L}\left(\left(-1+h_{p}\right)h_{p}\left(-603+32h_{p}\left(273-386h_{p}+64h_{p}^{2}\right)\right)+\right.}$
$\displaystyle{16\left(-96+h_{p}\left(1013+h_{p}\left(-6829+8h_{p}\left(1307+8h_{p}\left(-39+8h_{p}\right)\right)\right)\right)\right)h_{R}+}$
$\displaystyle{\left.512\left(-15+h_{p}\left(161+2h_{p}\left(191+64\left(-7+h_{p}\right)h_{p}\right)\right)\right)h_{R}^{2}\right)+}$
$\displaystyle{5c^{3}\left(16h_{L}^{2}\left(3+4h_{p}\left(2+h_{p}\right)\right)+h_{p}\left(1+2h_{p}\right)\left(9+2h_{p}\left(-7+4h_{p}\right)\right)-6h_{R}+8h_{p}\left(-20+h_{p}\left(-1+8h_{p}\right)\right)h_{R}+\right.}$
$\displaystyle{\left.16\left(1+2h_{p}\right)\left(3+2h_{p}\right)h_{R}^{2}+2h_{L}\left(3+2h_{p}\right)\left(-1+16h_{R}+2h_{p}\left(-13+8h_{p}+16h_{R}\right)\right)\right)-}$
$\displaystyle{c^{2}\left(h_{p}\left(-153-4h_{p}\left(-484+h_{p}\left(919+4h_{p}\left(-101+8h_{p}\right)\right)\right)\right)+\right.}$
$\displaystyle{2\left(51+4h_{p}\left(-83+h_{p}\left(33+16h_{p}\left(25+16h_{p}\right)\right)\right)\right)h_{R}+16\left(-51+4h_{p}\left(-235+h_{p}\left(7+72h_{p}\right)\right)\right)h_{R}^{2}+}$
$\displaystyle{2h_{L}\left(51+4h_{p}\left(-83+h_{p}\left(33+16h_{p}\left(25+16h_{p}\right)\right)\right)+2256h_{R}+64h_{p}\left(-467+h_{p}\left(71+232h_{p}\right)\right)h_{R}+\right.}$
$\displaystyle{\left.\left.5120\left(3+4h_{p}\left(2+h_{p}\right)\right)h_{R}^{2}\right)+16h_{L}^{2}\left(-51+1920h_{R}+4h_{p}\left(-235+7h_{p}+72h_{p}^{2}+640\left(2+h_{p}\right)h_{R}\right)\right)\right)+}$
$\displaystyle{2c\left(h_{p}\left(-99+2h_{p}\left(1238+h_{p}\left(-2771+4\left(499-64h_{p}\right)h_{p}\right)\right)\right)+66h_{R}-\right.}$
$\displaystyle{4h_{p}\left(748+h_{p}\left(-6929+8h_{p}\left(1415+8h_{p}\left(-73+8h_{p}\right)\right)\right)\right)h_{R}+}$
$\displaystyle{16\left(-33+2h_{p}\left(-728+h_{p}\left(-353+16h_{p}\left(73+12h_{p}\right)\right)\right)\right)h_{R}^{2}+}$
$\displaystyle{2h_{L}\left(33-2h_{p}\left(748+h_{p}\left(-6929+8h_{p}\left(1415+8h_{p}\left(-73+8h_{p}\right)\right)\right)\right)+4080h_{R}-\right.}$
$\displaystyle{\left.32h_{p}\left(176+h_{p}\left(369+16h_{p}\left(-189+4h_{p}\right)\right)\right)h_{R}+512\left(45+2h_{p}\left(-177+68h_{p}+64h_{p}^{2}\right)\right)h_{R}^{2}\right)+}$
$\displaystyle{16h_{L}^{2}\left(-33+2880h_{R}+2\left(h_{p}\left(-728+h_{p}\left(-353+16h_{p}\left(73+12h_{p}\right)\right)\right)+\right.\right.}$
$\displaystyle{\left.\left.64h_{p}\left(-177+68h_{p}+64h_{p}^{2}\right)h_{R}+5120\left(3+4h_{p}\left(2+h_{p}\right)\right)h_{R}^{2}\right)\right)}\,.$
(A.9)

It is straightforward to check the symmetry of the above coefficients under
$h_{L}\leftrightarrow h_{R}$, as previously argued.

The aforesaid code in [22] cannot be used for the calculations of the
coefficients $b_{n}$ when all the external operators are distinct, and needs
to be generalised. In a preliminary analysis, coding (A.3) for the general
case, we have calculated the first few coefficients $b_{n}$, which upon
setting $h_{1}=h_{2},\ h_{3}=h_{4}$ reproduce above results. Here, we present
the first few analytic coefficients $b_{n}$ in the general case. The higher
analytic coefficients are very lengthy.

$b_{1}=\frac{8(h_{1}-h_{2})(h_{3}-h_{4})}{h_{p}}\,.$ (A.10) $\displaystyle
b_{2}=$
$\displaystyle\frac{1}{2h_{p}\left(c+2h_{p}\left(-5+c+8h_{p}\right)\right)}\times$
$\displaystyle{\left(128c\left(h_{1}-h_{2}\right){}^{2}\left(h_{3}-h_{4}\right){}^{2}+\right.}$
$\displaystyle{((-1+c)c+}$
$\displaystyle{16\left(-h_{2}\left(c+48h_{3}^{2}+16h_{4}\left(-1+3h_{4}\right)-16h_{3}\left(1+6h_{4}\right)\right)+c\left(4h_{3}^{2}+h_{4}\left(-1+4h_{4}\right)-h_{3}\left(1+8h_{4}\right)\right)+\right.}$
$\displaystyle{4h_{1}^{2}\left(c+16h_{3}^{2}+4h_{4}\left(-3+4h_{4}\right)-4h_{3}\left(3+8h_{4}\right)\right)+4h_{2}^{2}\left(c+16h_{3}^{2}+4h_{4}\left(-3+4h_{4}\right)-4h_{3}\left(3+8h_{4}\right)\right)+}$
$\displaystyle{\left.\left.h_{1}\left(-c+16h_{3}-48\left(h_{3}-h_{4}\right){}^{2}+16h_{4}-8h_{2}\left(c+16h_{3}^{2}+4h_{4}\left(-3+4h_{4}\right)-4h_{3}\left(3+8h_{4}\right)\right)\right)\right)\right)h_{p}+}$
$\displaystyle{2\left(5+6c+c^{2}+64h_{1}^{2}+64h_{2}^{2}-48h_{3}-16ch_{3}+64h_{3}^{2}-16h_{2}\left(3+c-16h_{3}-16h_{4}\right)-\right.}$
$\displaystyle{\left.\left.16h_{1}\left(3+c+8h_{2}-16h_{3}-16h_{4}\right)-16\left(3+c+8h_{3}\right)h_{4}+64h_{4}^{2}\right)h_{p}^{2}\right)}\,.$
(A.11) $\displaystyle b_{3}=$
$\displaystyle\frac{4\left(h_{1}-h_{2}\right)\left(h_{3}-h_{4}\right)}{3h_{p}\left(2+c+h_{p}\left(-7+c+3h_{p}\right)\right)\left(c+2h_{p}\left(-5+c+8h_{p}\right)\right)}\times$
$\displaystyle{\left(c(2+c)\left(5+3c+64h_{3}^{2}+16h_{4}\left(-3+4h_{4}\right)-16h_{3}\left(3+8h_{4}\right)\right)+\right.}$
$\displaystyle{\left(-100+c\left(19+48c+9c^{2}\right)+64(-20+c(-1+3c))h_{3}^{2}+16h_{4}\left(60-9c(1+c)+4(-20+c(-1+3c))h_{4}\right)-\right.}$
$\displaystyle{\left.16h_{3}\left(-60+9c(1+c)+8(-20+c(-1+3c))h_{4}\right)\right)h_{p}+}$
$\displaystyle{3\left(-310+c(-5+c(49+2c))+64(6+5c)h_{3}^{2}+16h_{4}\left(42-c(27+2c)+4(6+5c)h_{4}\right)-\right.}$
$\displaystyle{\left.16h_{3}\left(-42+c(27+2c)+8(6+5c)h_{4}\right)\right)h_{p}^{2}+}$
$\displaystyle{6\left(167+c(154+3c)+192h_{3}^{2}-48\left(19+c-4h_{4}\right)h_{4}-48h_{3}\left(19+c+8h_{4}\right)\right)h_{p}^{3}+1152h_{p}^{4}+}$
$\displaystyle{48h_{2}\left(-c(2+c)-16h_{3}^{2}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)-16h_{4}^{2}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)+\right.}$
$\displaystyle{h_{p}\left(20-3c(1+c)-h_{p}\left(-42+c(27+2c)+6(19+c)h_{p}\right)\right)+}$
$\displaystyle{16h_{4}\left(c+h_{p}\left(3(-3+c)+h_{p}\left(9+2c+6h_{p}\right)\right)\right)+}$
$\displaystyle{\left.16h_{3}\left(c-9h_{p}+h_{p}\left(3+2h_{p}\right)\left(c+3h_{p}\right)+2h_{4}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)\right)\right)+}$
$\displaystyle{64h_{1}^{2}\left(c(2+c)+h_{p}\left(-20+c(-1+3c)+3h_{p}\left(6+5c+6h_{p}\right)\right)-12h_{4}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)+\right.}$
$\displaystyle{2h_{3}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)+2h_{4}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)+}$
$\displaystyle{\left.4h_{3}\left(-3\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)-h_{4}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)\right)\right)+}$
$\displaystyle{64h_{2}^{2}\left(c(2+c)+h_{p}\left(-20+c(-1+3c)+3h_{p}\left(6+5c+6h_{p}\right)\right)-12h_{4}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)+\right.}$
$\displaystyle{2h_{3}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)+2h_{4}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)+}$
$\displaystyle{\left.4h_{3}\left(-3\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)-h_{4}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)\right)\right)+}$
$\displaystyle{16h_{1}\left(-3c\left(2+c+16\left(-1+h_{4}\right)h_{4}\right)-3\left(-20+3c(1+c)+16h_{4}\left(9-3c+(-7+3c)h_{4}\right)\right)h_{p}+\right.}$
$\displaystyle{3\left(42-c(27+2c)+16\left(9+2c-9h_{4}\right)h_{4}\right)h_{p}^{2}-18\left(19+c-16h_{4}\right)h_{p}^{3}-48h_{3}^{2}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)+}$
$\displaystyle{48h_{3}\left(c-9h_{p}+h_{p}\left(3+2h_{p}\right)\left(c+3h_{p}\right)+2h_{4}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)\right)+}$
$\displaystyle{8h_{2}\left(-c(2+c)+h_{p}\left(20+c-3c^{2}-3h_{p}\left(6+5c+6h_{p}\right)\right)+12h_{4}\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)-\right.}$
$\displaystyle{2h_{3}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)-2h_{4}^{2}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)+}$
$\displaystyle{\left.\left.\left.4h_{3}\left(3\left(c+h_{p}\left(-7+3c+9h_{p}\right)\right)+h_{4}\left(c(8+c)+h_{p}\left(-26+11c+24h_{p}\right)\right)\right)\right)\right)\right)}\,.$
(A.12)

##  Appendix B $q$-expansion of certain functions

The $q$-expansions of the functions
$\lambda(\tau),\theta_{2}(\tau),\theta_{3}(\tau),\theta_{4}(\tau),\eta(\tau),E_{2}(\tau),E_{4}(\tau),E_{6}(\tau)$
are used in this paper. Exact formulae for the $q$-expansions of $\theta_{r}$
and $\eta$ are given by

$\displaystyle\theta_{2}(\tau)=2\sum_{n=1,3,5,\dots}q^{n^{2}/4}\,,$
$\displaystyle\theta_{3}(\tau)=1+2\sum_{n=1}^{\infty}q^{n^{2}}\,,$
$\displaystyle\theta_{4}(\tau)=1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}\,,$
$\displaystyle\eta(\tau)=\sum_{n=0,\mp 1,\mp
2,\dots}(-1)^{n}q^{(6n+1)^{2}/12}\,.$ (B.1)

For the functions $\lambda(\tau),E_{2}(\tau),E_{4}(\tau),E_{6}(\tau)$, below
we present formulae in terms of some Mathematica library functions. Then, it
is straightforward to write Mathematica routines to obtain their $q$-expansion
coefficients to any desired order $q^{n}$.

First note that the local inverse of (7) is given by[6, 29]

$\displaystyle\tau(\lambda)=i\frac{K(1-\lambda)}{K(\lambda)}\,,$
$\displaystyle
K(\lambda)\equiv\int_{0}^{1}\frac{dt}{\sqrt{\left(1-t^{2}\right)\left(1-\lambda
t^{2}\right)}}=\frac{1}{2}\pi\,_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1;\lambda\right)\,,$
(B.2)

where $K$ denotes the complete elliptic integral of the first kind. The
elliptic nome is given by

$q(\lambda)=e^{i\pi\tau(\lambda)}\,.$ (B.3)

The elliptic nome and its inverse are Mathematica library functions,
EllipticNomeQ[$\lambda$] and InverseEllipticNomeQ[$q$], respectively, with the
latter providing the elliptic lambda function $\lambda$ as a function of
$q$.484848On the other hand, the Mathematica library function
ModularLambda[$\tau$] provides $\lambda(\tau)$.

The library function EllipticTheta[$r,q$] provides the Jacobi theta functions
$\theta_{r},\ r=2,3,4$ as a function of $q$. As presented below, the
Eisenstein series $E_{2}$ can be expressed in terms of the complete elliptic
integral of the second kind $E$, the elliptic lambda function $\lambda$ and
the Jacobi theta functions $\theta_{r}$, and $E_{4},E_{6}$ in terms of
$\theta_{r}$ [29]. Using Mathematica library function EllipticE[$\lambda$] for
$E(\lambda)$, EllipticE[InverseEllipticNomeQ[$q$]] provides its
$q$-dependency.

$\displaystyle
E_{2}=\frac{6}{\pi}E(\lambda)\theta_{3}^{2}-\theta_{3}^{4}-\theta_{4}^{4},\quad
E(\lambda)\equiv\int_{0}^{1}\sqrt{\frac{1-\lambda t^{2}}{1-t^{2}}}dt\,,$
$\displaystyle
E_{4}=\frac{1}{2}(\theta_{2}^{8}+\theta_{3}^{8}+\theta_{4}^{8})\,,$
$\displaystyle
E_{6}=\frac{1}{2}\left(\theta_{3}^{12}+\theta_{4}^{12}-3\theta_{2}^{8}(\theta_{3}^{4}+\theta_{4}^{4}\right))\,.$
(B.4)

Now using Mathematica, the coefficients of the $q$-expansions of
$\lambda(\tau),E_{2}(\tau),E_{4}(\tau),E_{6}(\tau)$ can be obtained to any
desired order $q^{n}$. Here, we present the first few terms:

$\displaystyle\lambda(\tau)=16q-128q^{2}+704q^{3}-3072q^{4}+11488q^{5}-38400q^{6}+117632q^{7}-335872q^{8}+\cdots\,,$
$\displaystyle E_{2}(\tau)=1-24q^{2}-72q^{4}-96q^{6}-168q^{8}+\cdots\,,$
$\displaystyle
E_{4}(\tau)=1+240q^{2}+2160q^{4}+6720q^{6}+17520q^{8}+\cdots\,,$
$\displaystyle
E_{6}(\tau)=1-504q^{2}-16632q^{4}-122976q^{6}-532728q^{8}+\cdots\,.$ (B.5)

##  Appendix C Closed form solutions

In this appendix, we consider the second order MLDE w.r.t. $\Gamma(2)$ in (27)
and provide 2 solutions in closed form. The solutions are so chosen that they
shall be directly identified – whenever possible on the parameter space
$\vec{\alpha}$ – with conformal blocks from their $q\to 0$ behaviour.

Under the change of variable $x=\lambda(\tau)$, (27) leads to494949In order to
obtain closed form solutions to MLDEs of orders 2 and 3 for the characters,
[11] made a variable change from the torus parameter $\tau$ to an auxiliary
space $x=\lambda(\tau)$. Here, we follow the same path, except in our case
$x$-space should be interpreted as the cross-ratio space.

$x^{2}(1-x)^{2}\frac{d^{2}\vec{F}}{dx^{2}}+x(1-x)\big{(}1-2\alpha_{2}-(2\alpha_{1}+1)x\big{)}\frac{d\vec{F}}{dx}-(\alpha_{3}x^{2}+\alpha_{4}x+\alpha_{5})\vec{F}=0\,,$
(C.1)

where we have used the following relations.

$\frac{d\lambda}{d\tau}=i\pi\lambda(1-\lambda)\theta_{3}^{4},\quad
E_{2}=-(2\lambda-1)\theta_{3}^{4}+\frac{3}{i\pi\theta_{3}^{4}}\frac{d}{d\tau}\theta_{3}^{4}\,.$
(C.2)

(C.1) is the most general second order linear differential equation with 3
regular singular points viz. at $x=0,1,\infty$.505050In order for $\infty$ to
be a regular singular point of a differential equation given below,
$\lim_{y\to 0}yP(\frac{1}{y})$ and $\lim_{y\to 0}y^{2}Q(\frac{1}{y})$ should
be finite. Thus, $P$ should be a linear while $Q$ a quadratic polynomial in
$x$. (C.1) has the most general such $P,Q$.
$\frac{d^{2}f}{dx^{2}}+\frac{P(x)}{x(x-1)}\frac{df}{dx}+\frac{Q(x)}{x^{2}(x-1)^{2}}f=0\
.$ Therefore, the solutions can be obtained in terms of Gauss hypergeometric
function as

$\displaystyle
F_{p(1)}(x)=x^{\mu_{-}}(1-x)^{\nu_{-}}\,_{2}F_{1}(a_{-},b_{-};c_{-};x),\quad
F_{p(2)}(x)=x^{\mu_{+}}(1-x)^{\nu_{+}}\,_{2}F_{1}(a_{+},b_{+};c_{+};x)\,,$
$\displaystyle\mu_{\mp}=\alpha_{2}\mp\sqrt{\alpha_{2}^{2}+\alpha_{5}}\,,$
$\displaystyle\nu_{\mp}=\frac{1}{2}-\alpha_{1}-\alpha_{2}\mp\sqrt{\big{(}\alpha_{1}+\alpha_{2}-\frac{1}{2}\big{)}^{2}+(\alpha_{3}+\alpha_{4}+\alpha_{5})}\,,$
$\displaystyle
a_{\mp}=\frac{1}{2}\pm\sqrt{\alpha_{1}^{2}+\alpha_{3}}\mp\sqrt{\alpha_{2}^{2}+\alpha_{5}}\mp\sqrt{\big{(}\alpha_{1}+\alpha_{2}-\frac{1}{2}\big{)}^{2}+(\alpha_{3}+\alpha_{4}+\alpha_{5})}\,,$
$\displaystyle
b_{\mp}=\frac{1}{2}\mp\sqrt{\alpha_{1}^{2}+\alpha_{3}}\mp\sqrt{\alpha_{2}^{2}+\alpha_{5}}\mp\sqrt{\big{(}\alpha_{1}+\alpha_{2}-\frac{1}{2}\big{)}^{2}+(\alpha_{3}+\alpha_{4}+\alpha_{5})}\,,$
$\displaystyle c_{\mp}=1\mp 2\sqrt{\alpha_{2}^{2}+\alpha_{5}}\,.$ (C.3)

It is straightforward to obtain the actions of $T,S$ on the above solutions
using the following identities [29].

${}_{2}F_{1}(a,b;c;x)=(1-x)^{c-a-b}\,_{2}F_{1}(c-a,c-b;c;x)\,,$
${}_{2}F_{1}(a,b;c;T\cdot
x)=(1-x)^{a}\,_{2}F_{1}(a,c-b;c;x)=(1-x)^{b}\,_{2}F_{1}(c-a,b;c;x)\,,$
${}_{2}F_{1}(a,b;c;S\cdot
x)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\,_{2}F_{1}(a,b;a+b-c+1;x)$
$\displaystyle\qquad\qquad\quad+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}x^{c-a-b}\,_{2}F_{1}(c-a,c-b;c-a-b+1;x)\,,$
${}_{2}F_{1}(a,b;c;S\cdot
x)=\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}x^{c-a-b}(1-x)^{1-c}\,_{2}F_{1}(1-b,1-a;1+c-a-b;x)$
$\displaystyle\qquad\qquad\quad+\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}(1-x)^{1-c}\,_{2}F_{1}(1+b-c,1+a-c;1+a+b-c;x)\,.$
(C.4)

In particular, we compute that515151We also use the relation
$\Gamma(1–z)\Gamma(z)=\pi\csc\pi z$.

$\displaystyle F(ST^{2}S\cdot
x)=\frac{\left(e^{2i\pi(a+c)}+e^{2i\pi(b+c)}-e^{2i\pi(a+b)}-e^{2i\pi
c}\right)e^{-2i\pi(a+b-\nu)}}{e^{2i\pi c}-1}F(x)$
$\displaystyle-\frac{2i\pi\Gamma(c-1)\Gamma(c)e^{-i\pi(a+b-c-2\nu)}}{\Gamma(a)\Gamma(b)\Gamma(c-a)\Gamma(c-b)}x^{1-c+\mu}(1-x)^{c-a-b+\nu}\,_{2}F_{1}(1-a,1-b;2-c;x)\,,$
$\displaystyle F(x)=x^{\mu}(1-x)^{\nu}\,_{2}F_{1}(a,b;c;x)\,.$ (C.5)

And, note that the values of $\\{1-c+\mu,\ c-a-b+\nu,\ 1-a,\ 1-b,\ 2-c\\}$ for
any of the 2 solutions in (C.3) give the respective values of
$\\{\mu,\nu,a,b,c\\}$ for the other solution.

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