aainstitutetext: Jefferson Physical Laboratory, Harvard University,
Cambridge, MA 02138 USAbbinstitutetext: School of Natural Sciences, Institute
for Advanced Study,
Princeton, NJ 08540, USA

#  String correlators on $\boldsymbol{\text{AdS}_{3}}$: Four-point functions

Andrea Dei b Lorenz Eberhardt<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose a closed-form formula for genus 0 four-point functions in
$\text{AdS}_{3}$ string theory with pure NS-NS flux including arbitrary
amounts of spectral flow. Our formula passes many non-trivial consistency
checks and has intriguing connections to Hurwitz theory. This paper is the
second in a series with several instalments.

##  1 Introduction

The duality between strings propagating on three-dimensional Anti-de Sitter
space ($\text{AdS}_{3}$) and two-dimensional conformal field theories
($\text{CFT}_{2}$) is one of the best understood incarnations of the AdS/CFT
correspondence Maldacena:1997re . Due to the presence of R-R fluxes, the study
of higher-dimensional holographic setups is much more difficult from a
worldsheet CFT perspective. Strings propagating on $\text{AdS}_{3}$ with pure
NS-NS flux can be studied via the $\text{SL}(2,\mathds{R})$ Wess-Zumino-Witten
(WZW) model. This has a long history Gawedzki:1991yu ; Teschner:1997ft ;
Giveon:1998ns ; deBoer:1998gyt ; Kutasov:1999xu ; Teschner:1999ug ;
Maldacena:2000hw ; Maldacena:2000kv ; Maldacena:2001km  and repeated advances
throughout the last three decades have raised the hope to completely solve the
worldsheet model and compute string theory observables exactly at tree level.

A complete understanding of the spectrum of the $\text{SL}(2,\mathds{R})$ WZW
model has not been reached until the need to introduce spectral flow was
realised Balog:1988jb ; Petropoulos:1989fc ; Hwang:1990aq ; Henningson:1991jc
; Gawedzki:1991yu ; Bars:1995mf ; Teschner:1997ft ; Evans:1998qu ;
Giveon:1998ns ; deBoer:1998gyt ; Teschner:1999ug ; Kutasov:1999xu ;
Giribet:1999ft ; Giribet:2000fy ; Maldacena:2000hw ; Maldacena:2000kv ;
Maldacena:2001km ; Giribet:2001ft . The presence of spectrally flowed
representations — corresponding to non highest-weight modules of the
worldsheet algebra — provides a rich and intriguing dynamics for
$\text{AdS}_{3}$ strings. At the same time, while the spectrum of the
worldsheet theory is under a firm control, the presence of spectral flow has
hampered a full understanding of $\text{AdS}_{3}$ string correlators. See
Giribet:2000fy ; Stoyanovsky:2000pg ; Fateev ; Maldacena:2001km ;
Giribet:2001ft ; Giribet:2005ix ; Ribault:2005ms ; Minces:2005nb ;
Ribault:2005wp ; Iguri:2007af ; Hikida:2007tq ; Baron:2008qf ; Iguri:2009cf ;
Giribet:2011xf ; Cagnacci:2013ufa ; Cagnacci:2015pka ; Giribet:2019new ;
Eberhardt:2019ywk ; Hikida:2020kil  for some important results.

Recently, a further step towards a full solution of the
$\text{SL}(2,\mathds{R})$ WZW model has been taken in Dei:2021xgh , where we
proposed a closed-form formula for three-point functions with an arbitrary
amount of spectral flow. As usual in conformal field theory, our formula for
the structure constants is just a collection of numbers since the coordinate
dependence of three-point functions is fixed by the global conformal symmetry
on the worldsheet and in spacetime. Four- and higher-point functions contain
much more interesting dynamical information. While in principle they can be
accessed by a conformal block expansion, it is presently not known how to
formulate such a conformal block expansion that fully incorporates spectral
flow in the $x$-basis. For this reason, it is very worth-while to study four-
point functions directly without making use of a spectrally flowed conformal
block expansion.

We exclusively study bosonic strings at genus 0 in this paper. On the
worldsheet, this means that we focus on the CFT describing bosonic strings on
Euclidean $\text{AdS}_{3}$ defined by analytic continuation of the
$\text{SL}(2,\mathds{R})_{k}$ WZW model — see Maldacena:2001km  for the
precise definition of the worldsheet theory. We follow the conventions of our
previous paper Dei:2021xgh . This is however not a restriction since the
description of superstrings in the RNS-formalism involves the same model at
level $k+2$, together with free fermions and at least at genus 0 the two
sectors are completely decoupled. Thus we think that most of our results carry
over to the superstring case with the replacement $k\to k+2$ in most of the
formulae.111When computing genus zero superstring $n$-point functions via the
RNS-formalism, one also needs to insert $n-2$ picture changing operators. This
is in general a non-trivial step, even though the necessary technology is
well-known in the literature. See for example Green-Schwarz-Witten ;
Gaberdiel:2007vu ; Dabholkar:2007ey .

We are then motivated to continue our study of spectrally flowed correlators
on $\text{AdS}_{3}$. Let us explain the most important features with the help
of Figure 1 — more precise definitions will be spelled out in the following
sections. Euclidean $\text{AdS}_{3}$ is topologically a three-dimensional ball
with the Riemann sphere $\text{S}^{2}$ as asymptotic boundary. The worldsheet
of the four-point string correlator whose vertex operators carry two units of
spectral flow is depicted in Figure 1. As one can see, there are two sheets of
the worldsheet touching each insertion point corresponding to the two units of
spectral flow. In total, there are three sheets, ensuring that the topology of
the worldsheet is a Riemann sphere.

Figure 1: A sketch of the string interaction in $\text{AdS}_{3}$.

Vertex operators describe the insertion of an asymptotic string state. From
the cartoon of Figure 1, one directly sees that there are two coordinates
associated with the vertex operator: the worldsheet coordinate $z$ and the
boundary coordinate $x$, where the worldsheet touches the boundary. Moreover,
one can see that there is another integer associated to each vertex operator —
the so-called spectral flow $w\in\mathds{Z}_{\geq 0}$. It corresponds to the
number of times the worldsheet is winding around the insertion point in the
boundary.222This is not a topological invariant and from the picture alone it
is not clear whether this number remains well-defined in the quantum theory,
since it seems that there is geometrically a continuous transition between the
different winding numbers. However, this integer will correspond to a certain
well-defined number that labels the representation of the symmetry algebra
$\mathfrak{sl}(2,\mathds{R})_{k}\times\mathfrak{sl}(2,\mathds{R})_{k}$. Beyond
these geometric quantities, the vertex operators have also two different
$\mathfrak{sl}(2,\mathds{R})$ spins associated to them. In spacetime, they are
primary operators of spacetime conformal weight $(h,\bar{h})$ and on the
worldsheet they are built on $\mathfrak{sl}(2,\mathds{R})$ representations of
spin $j$. While as usual in a unitary CFT $h-\bar{h}\in\mathds{Z}$ and
$h,\bar{h}\geq 0$, the $\mathfrak{sl}(2,\mathds{R})$ spin on the worldsheet
can fall into either the discrete series with $j\in\mathds{R}$ or the
principal continuous series with $j\in\tfrac{1}{2}+i\,\mathds{R}$.333The case
$w=0$ is exceptional, since $j$ and $h$ coincide.

To summarise, the worldsheet vertex operators take in general the following
form

$V_{j,h,\bar{h}}^{w}(x;z)\ .$ (1.1)

Thus, the objects that we want to study are the correlators

$\left\langle V_{j_{1},h_{1},\bar{h}_{1}}^{w_{1}}(x_{1};z_{1})\cdots
V_{j_{n},h_{n},\bar{h}_{n}}^{w_{n}}(x_{n};z_{n})\right\rangle\ .$ (1.2)

In this paper we take a step towards the full understanding of such
correlators. For genus 0 four-point functions, we propose a map that takes an
unflowed correlation function and transforms it into a flowed correlation
function with the desired spectral flow indices. Notice that this is not just
a rewriting of the problem, since in the unflowed sector (i.e. the case where
all vertex operators satisfy $w_{i}=0$) these correlators are very well-known
and have been thoroughly studied in the literature, see e.g. Teschner:1997ft ;
Teschner:1999ug ; Giribet:2000fy ; Giribet:2001ft ; Teschner:2001gi ;
Maldacena:2001km ; Ribault:2005wp ; Hikida:2007tq .

Our approach is almost entirely based on symmetry. We use all available
symmetries on the worldsheet to constrain correlators as much as possible.
These constraints come in the form of global and local Ward identities
Eberhardt:2019ywk , Knizhnik-Zamolodchikov Knizhnik:1984nr  equations and null
vector equations. Most of these are much more involved than for the unflowed
sector and not known in closed-form for generic choice of the spectral flow
parameters. Nevertheless, by explicitly solving these constraints we end up
with a relatively simple closed-form formula. For the four point function, it
is given in the form of an integral transform relating the unflowed sector to
the flowed sector,

$\displaystyle\left\langle
V_{j_{1},h_{1},\bar{h}_{1}}^{w_{1}}(0;0)V_{j_{2},h_{2},\bar{h}_{2}}^{w_{2}}(1;1)V_{j_{3},h_{3},\bar{h}_{3}}^{w_{3}}(\infty;\infty)V_{j_{4},h_{4},\bar{h}_{4}}^{w_{4}}(x;z)\right\rangle$
$\displaystyle=\int\prod_{i=1}^{4}\mathrm{d}^{2}y_{i}\
y_{i}^{\frac{kw_{i}}{2}+j_{i}-h_{i}-1}\bar{y_{i}}^{\frac{kw_{i}}{2}+j_{i}-\bar{h_{i}}-1}|X_{\emptyset}|^{2(j_{1}+j_{2}+j_{3}+j_{4}-k)}$
$\displaystyle\quad\quad\times|X_{12}|^{2(-j_{1}-j_{2}+j_{3}-j_{4})}\,|X_{13}|^{2(-j_{1}+j_{2}-j_{3}+j_{4})}|X_{23}|^{2(j_{1}-j_{2}-j_{3}+j_{4})}|X_{34}|^{-4j_{4}}$
$\displaystyle\quad\quad\times\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(\frac{X_{23}X_{14}}{X_{12}X_{34}};z\right)\right\rangle\,$
(1.3a)

for $\sum_{i}w_{i}\in 2\mathds{Z}$ and

$\displaystyle\left\langle
V_{j_{1},h_{1},\bar{h}_{1}}^{w_{1}}(0;0)V_{j_{2},h_{2},\bar{h}_{2}}^{w_{2}}(1;1)V_{j_{3},h_{3},\bar{h}_{3}}^{w_{3}}(\infty;\infty)V_{j_{4},h_{4},\bar{h}_{4}}^{w_{4}}(x;z)\right\rangle$
$\displaystyle=\mathcal{N}(j_{3})\int\prod_{i=1}^{4}\mathrm{d}^{2}y_{i}\
y_{i}^{\frac{kw_{i}}{2}+j_{i}-h_{i}-1}\bar{y_{i}}^{\frac{kw_{i}}{2}+j_{i}-\bar{h_{i}}-1}|X_{123}|^{2(\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4})}$
$\displaystyle\quad\quad\quad\times|X_{1}|^{2(-j_{1}+j_{2}+j_{3}+j_{4}-\frac{k}{2})}\,|X_{2}|^{2(j_{1}-j_{2}+j_{3}+j_{4}-\frac{k}{2})}|X_{3}|^{2(j_{1}+j_{2}-j_{3}+j_{4}-\frac{k}{2})}|X_{4}|^{-4j_{4}}$
$\displaystyle\quad\quad\quad\times\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{\frac{k}{2}-j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(\frac{X_{2}X_{134}}{X_{123}X_{4}};z\right)\right\rangle\,\
,$ (1.3b)

for $\sum_{i}w_{i}\in 2\mathds{Z}+1$.

Let us point the reader to the relevant definitions that enter these formulae.
First of all, the presence of the $y$-integral transform was explained in
detail in Dei:2021xgh  and we recall the main definition in (2.2). The
quantities $X_{I}$ are polynomials of degree 1 in each $y_{i}$ for $i\in
I\subset\\{1,2,3,4\\}$. They are defined in eq. (3.8). In that definition the
quantities $P_{\boldsymbol{w}}(x;z)$ enter; they are the (suitably normalised)
polynomials that carve out the Hurwitz space of ramified covers, see eqs.
(2.12) and (2.17). Finally, the prefactor $\mathcal{N}(j)$ in (1.3b) is
constructed out of the normalisation of the two-point function, see eq.
(2.20). The positions of the first three vertex operators can be chosen to be
arbitrary when making use of global Ward identities, see Section 2.4.

The correlator vanishes identically for $\sum_{i}w_{i}\leq 2\max_{i}(w_{i})-3$
Maldacena:2001km ; Eberhardt:2019ywk , which is reflected in the vanishing of
the quantities $X_{I}$. For the edge case $\sum_{i}w_{i}=2\max_{i}(w_{i})-2$,
the expression is still valid, provided it is interpreted correctly. In our
expression, both $X_{\emptyset}$ and the unflowed correlator that enters the
right-hand side are singular. Their product is still well-behaved and can be
defined by a limiting procedure as explained in Section 3.5. We should also
mention that this formula as written is valid for continuous representations
$j_{i}\in\frac{1}{2}+i\,\mathds{R}$. For discrete representations there is a
small modification in the integration contours as discussed in Section 2.1.

Eq. (1) essentially encodes the geometry of the problem. One can already guess
from Figure 1 that the relevant geometric notion that enters the stage is a
holomorphic branched covering map. In fact, it was argued in Maldacena:2001km
that configurations in the worldsheet field space for which the worldsheet
covers the boundary holomorphically are somewhat analogous to worldsheet
instanton sectors, although strictly speaking these are not different
topological sectors.

We will show that our formula (1) satisfies all the constraints that follow
from symmetry, namely

  * •

Global Ward identities (both in $x$\- and $z$-space),

  * •

Local Ward identities,

  * •

Knizhnik-Zamolodchikov equation,

  * •

Null vector decoupling (when applicable).

These constraints fix the result essentially uniquely up to a constant (i.e.
independent of the coordinates $x_{i}$, $y_{i}$ and $z_{i}$) prefactor. We
perform several consistency checks that strongly motivate the prefactor. These
are

  * •

Internal consistency,

  * •

Contact with previous results in the literature for special cases,

  * •

Reduction to our previous conjecture about the three-point function
Dei:2021xgh ,

  * •

Reflection symmetry for continuous representations,

  * •

Bosonic exchange statistics.

Some of these seem obvious (especially the last points), but it turns out that
they impose lots of non-trivial constraints on our proposal.

This paper is computationally quite heavy. Because of the way the various
constraints are formulated on the correlators we can only evaluate them for
fixed values of spectral flow. So our strategy is to systematically solve them
for a large number of choices of spectral flows (we have analyzed 721 cases)
and try to guess the general formula from these examples. While we have
managed to find a simple formula that works for all cases it might be hard for
the reader to understand where certain formulae come from. For this reason we
have included a Mathematica notebook that performs many of the computations
that are described in this paper. We have tried to keep the code general and
flexible in the hope that it will be useful to some of the readers.

The solution (1) depends very much on the existence of polynomials
$P_{\boldsymbol{w}}(x;z)$ whose zero-locus is the so-called _Hurwitz-space_ of
branched covers from the sphere to itself with ramification indices
$\boldsymbol{w}=(w_{1},w_{2},w_{3},w_{4})$ at four points. The precise
definition is given in eq. (2.17). These polynomials control the complete
flowed structure of the correlator in terms of the unflowed correlator.
Consistency of our proposal (1) implies various identities among these
polynomials, which we collect in Appendix C. We have tested them in multiple
examples in the ancillary Mathematica notebook and we have a certain degree of
confidence in their correctness. To the best of our knowledge these identities
are not known in the mathematics literature and it would be interesting to
deduce them from a more rigorous point of view in the context of Hurwitz
theory.

The paper is organised as follows. After briefly recalling the results of
Dei:2021xgh , in Section 2 we describe the various symmetry constraints that
spectrally flowed correlators must obey. As far as we are aware, the KZ-
equations and the null vector equations in the spectrally flowed sector have
not appeared before in the literature.444See however Ribault:2005ms  for an
analysis of the spectrally flowed KZ-equation in the $m$-basis. This
corresponds to a degenerate case of our KZ-equation where several vertex
operators collide in spacetime.  In Section 3 we then move on to the object of
ultimate interest — the four-point function. It depends non-trivially on _two_
crossratios (one on the worldsheet and one in spacetime). We fix the
dependence on the coordinates fully in terms of the corresponding unflowed
correlator. We analyse further properties and carry out various consistency
checks on our proposal in Section 4. In Section 5 we comment on various points
deserving a better understanding and suggest future directions of research.
Various appendices complement the discussion of some technical points we
encounter throughout the text.

##  2 Constraints

###  2.1 A short review of the three-point function

We will pick the discussion up more or less where we ended it in Dei:2021xgh .
To make the paper self-contained, we briefly recall the result that we
conjectured there for the three-point functions.

##### Definition of the vertex operators.

We want to compute correlators of the spectrally flowed affine primary vertex
operators

$V_{j,h,\bar{h}}^{w}(x,\bar{x};z,\bar{z})\ .$ (2.1)

Our definition coincides with that of most of the literature, for example of
Maldacena:2001km . For the details of how these operators can be constructed
we refer to Dei:2021xgh . Here, $j$ is the $\mathfrak{sl}(2,\mathds{R})_{k}$
spin of the vertex operator on the worldsheet. It can take either values in
$\frac{1}{2}<j<\frac{k-1}{2}$, corresponding to discrete representations, or
values in $\frac{1}{2}+i\mathds{R}$, corresponding to continuous
representations. The non-negative integer $w$ parametrises the amount of
spectral flow. In particular, $w=0$ labels the unflowed sector. $(h,\bar{h})$
are the conformal weights of the operator in _spacetime_ (they are denoted by
$(J,\bar{J})$ in Maldacena:2001km ). Finally, $x$ and $z$ are the insertion
points of the operator in spacetime and on the worldsheet respectively. In the
unflowed sector for $w=0$ the labels $j$ and $h=\bar{h}$ coincide and we will
hence omit the labels $h$ and $\bar{h}$. We will also frequently omit the
right-moving labels $\bar{h}$, $\bar{x}$ and $\bar{z}$ in the flowed sector.

#####  $\boldsymbol{y}$-basis.

In Dei:2021xgh  we introduced an integral transform that trades the conformal
weights $(h,\bar{h})$ for a complex variable $y$ as follows:

$V_{j,h,\bar{h}}^{w}(x;z)=\int_{\mathfrak{C}}\mathrm{d}^{2}y\
y^{\frac{kw}{2}+j-h-1}\bar{y}^{\frac{kw}{2}+j-\bar{h}-1}V_{j}^{w}(x;y;z)\ .$
(2.2)

The contour of integration $\mathfrak{C}$ depends on the type of
representation. For a continuous representation one integrates over the whole
complex plane, whereas for a discrete representation one takes a contour
integral around $0$ (in the case of a lowest weight representation
$\mathcal{D}^{+}$) or $\infty$ (in the case of a highest weight representation
$\mathcal{D}^{-}$) for both $y$ and $\bar{y}$. Most of our results look simple
when expressed in the $y$-basis, but computing the integral over $y$-space —
which is necessary in order to transform the expression back to the
$h$-variables — is usually complicated.

##### Reflection symmetry.

Let us mention one further aspect of the $y$-basis. The continuous
representations with spins $j$ and $1-j$ are equivalent. Similarly to what
happens for unflowed vertex operators in the $x$-basis Teschner:1997ft ,

$V^{0}_{1-j}(x;z)=R_{1-j}\frac{(1-2j)}{\pi}\int\mathrm{d}^{2}x^{\prime}\
|x-x^{\prime}|^{4j-4}\,V^{0}_{j}(x^{\prime};z)\ .$ (2.3)

this equivalence is expressed for flowed vertex operators in the $y$-basis via
a simple integral transform

$V^{w}_{1-j}(x;y;z)=R_{1-j}\frac{(1-2j)}{\pi}\int\mathrm{d}^{2}y^{\prime}\
|y-y^{\prime}|^{4j-4}\,V^{w}_{j}(x;y^{\prime};z)\ ,$ (2.4)

where $R_{j}$ is the reflection coefficient. It is related to the
normalisation $B(j)$ of the two-point function via

$R_{j}=\frac{\pi B(j)}{2j-1}$ (2.5)

and satisfies $R_{j}R_{1-j}=1$.

##### Three-point functions.

Our main conjecture in Dei:2021xgh  was a closed-form formula for the three-
point functions of these operators. To simplify, we used global Ward
identities in both $x$\- and $z$-space to put the operators at
$x_{1}=z_{1}=0$, $x_{2}=z_{2}=1$ and $x_{3}=z_{3}=\infty$. Our result was then

$\displaystyle\left\langle
V^{w_{1}}_{j_{1}}(0;y_{1};0)\,V^{w_{2}}_{j_{2}}(1;y_{2};1)\,V^{w_{3}}_{j_{3}}(\infty;y_{3};\infty)\right\rangle$
$\displaystyle\quad=\begin{cases}D(j_{1},j_{2},j_{3})|X_{\emptyset}|^{2\sum_{l}j_{l}-2k}\displaystyle\prod_{i<\ell}^{3}|X_{i\ell}|^{2\sum_{l}j_{l}-4j_{i}-4j_{\ell}}\
,&\sum_{i}w_{i}\in 2\mathds{Z}\ ,\\\
\mathcal{N}(j_{1})D(\tfrac{k}{2}-j_{1},j_{2},j_{3})|X_{123}|^{k-2\sum_{l}j_{l}}\displaystyle\prod_{i=1}^{3}|X_{i}|^{2\sum_{l}j_{l}-4j_{i}-k}\
,&\sum_{i}w_{i}\in 2\mathds{Z}+1\ .\end{cases}$ (2.6)

Here, $D(j_{1},j_{2},j_{3})$ are the structure constants of the unflowed
three-point functions. They are obtained from those of the $H_{3}^{+}$ model
by analytic continuation and were determined by Teschner in Teschner:1997ft ,
see Maldacena:2001km  for our conventions. The quantity
$\mathcal{N}(j)=\sqrt{B(j)\big{/}B\left(\frac{k}{2}-j\right)}$ is determined
through the normalisation of the unflowed two-point function. The quantities
$X_{I}$ for $I\subset\\{1,2,3\\}$ encode the dependence on the spectral flow.
They are linear functions in $y_{i}$ for $i\in I$ and we gave a very explicit
formula for the coefficients in Dei:2021xgh . For example, for
$\boldsymbol{w}=(6,7,9)$,

$X_{12}=116424-2520y_{1}-5292y_{2}-126y_{1}y_{2}\ ,$ (2.7)

so everything is very explicit. It follows from the explicit formula for
$X_{I}$ that the three-point functions are non-vanishing whenever the spectral
flow indices satisfy

$\sum_{i=1}^{3}w_{i}\geq 2\max_{i}(w_{i})-1\ .$ (2.8)

This is a manifestation of the fusion rules of the model and agrees with the
analysis of Maldacena:2001km  and Eberhardt:2019ywk . Our main objective in
this paper will be to demonstrate that the formula (2.6) admits a natural
generalisation to the case of four-point functions.

###  2.2 Covering maps

One of the main players in this paper will be branched covering maps, since it
will turn out that the generalisation of the formula (2.6) to four-point
functions heavily relies on them. We will collect here all the relevant
properties of such covering maps. For the purpose of this paper, we will only
consider branched coverings from the sphere to itself,
$\gamma:\text{S}^{2}\longrightarrow\text{S}^{2}$. When we want to emphasise
the complex structure on $\text{S}^{2}$ in the following, we write
$\text{S}^{2}=\mathbb{CP}^{1}$. $\gamma$ is a branched covering map, which
means that it is locally biholomorphic, except at finitely many points
$z_{1},\dots,z_{n}$, where $\gamma$ has ramification indices
$w_{i}\in\mathds{Z}_{\geq 0}$. In other words, the only critical points of
$\gamma$ are $z_{i}$, where it has the Taylor expansion

$\gamma(\zeta)=x_{i}+a_{i}(\zeta-z_{i})^{w_{i}}+\mathcal{O}\big{(}(\zeta-
z_{i})^{w_{i}+1}\big{)}\ .$ (2.9)

In the mathematics literature one usually assumes that $w_{i}\geq 2$ so that
the map is really ramified there. But to ensure uniformity of our discussion
it is also useful to allow for $w_{i}=0$ and $w_{i}=1$. Of course when
$w_{i}=1$, the point is unramified and the only requirement is that $z_{i}$ is
mapped to $x_{i}$. For $w_{i}=0$ there is no condition on the map $\gamma$ at
the point $z_{i}$ (and hence we could delete $x_{i}$ and $z_{i}$ from the list
of points).

In this paper we will only consider the case of four ramification points. In
this case, we can set $x_{1}=0$, $x_{2}=1$, $x_{3}=\infty$, $x_{4}=x$ and the
same for the $z_{i}$’s.555Around $\infty$, the relevant Laurent expansion
takes the form
$\gamma(\zeta)=\frac{(-1)^{w_{3}+1}}{a_{3}}\zeta^{w_{3}}+\mathcal{O}(\zeta^{w_{3}-1})\
.$ (2.10) Given $(w_{1},w_{2},w_{3},w_{4})$ and $x$, there are in general only
finitely many values of $z$ such that a covering map exists.

The simplest case is given by $w_{1}=w_{2}=w_{3}=w_{4}=1$, where the covering
map has no critical point and is hence a Möbius transformation. Since it has
to map $0$ to $0$, $1$ to $1$ and $\infty$ to $\infty$, the only possible
choice is $\gamma(\zeta)=\zeta$. This is only possible provided that $z=x$ and
hence only in this case a covering map exists. In general, the existence of a
covering map imposes a relation between $x$ and $z$. Since all the conditions
we imposed are algebraic, this relation can be written as

$\tilde{P}_{\boldsymbol{w}}(x;z)=0\ ,$ (2.11)

where $\tilde{P}_{\boldsymbol{w}}(x;z)$ is a polynomial in $x$ and $z$. Here
and in the following, we lighten the notation by writing
$\boldsymbol{w}=(w_{1},w_{2},w_{3},w_{4})$. This polynomial defines a
subvariety of $\mathbb{CP}^{1}\times\mathbb{CP}^{1}$, which in the mathematics
literature is known as the Hurwitz space.666Strictly speaking the Hurwitz
space should be defined as the corresponding subspace of
$\mathbb{CP}^{1}\setminus\\{0,1,\infty\\}\times\mathbb{CP}^{1}\setminus\\{0,1,\infty\\}$
and should be subsequently compactified. This issue will not play a role for
the four-point function. It is a known result that the Hurwitz space is
connected and irreducible and hence $\tilde{P}_{\boldsymbol{w}}(x;z)$ is an
irreducible polynomial Liu . In general, we can write

$\tilde{P}_{\boldsymbol{w}}(x;z)\equiv\prod_{\gamma^{-1}}\left(z-\gamma^{-1}(x)\right)\
,$ (2.12)

where the product runs over all possible preimages of all possible covering
maps. The number of terms in the product is known as the _Hurwitz number_ and
takes in this case the form Liu

$H_{\boldsymbol{w}}=\frac{1}{2}\min_{i=1,2,3,4}\Biggl{(}w_{i}\Biggl{(}\sum_{j}w_{j}-2w_{i}\Biggr{)}\Biggr{)}\
.$ (2.13)

This is the order of the polynomial $\tilde{P}_{\boldsymbol{w}}$ in $z$. The
order of the polynomial in $x$ is given by777This can be seen from the
symmetric product orbifold CFT, where this number corresponds to the number of
channels in the four-point function of four twist-operators. This was
discussed in Dei:2019iym .

$\frac{1}{2}\left(\min(w_{1}+w_{2},w_{3}+w_{4})-\max(|w_{1}-w_{2}|,|w_{3}-w_{4}|)\right)\
.$ (2.14)

We should note that integrality and non-negativity of $H_{\boldsymbol{w}}$
imposes constraints on $\boldsymbol{w}$. In general a covering map only exists
if

$H_{\boldsymbol{w}}\in\mathds{Z}_{\geq
1}\qquad\Longleftrightarrow\qquad\sum_{i=1}^{4}w_{i}\in 2\mathds{Z}_{\geq 1}\
\ \text{and}\ \ \sum_{i=1}^{4}w_{i}>2\max_{i}w_{i}\ .$ (2.15)

In case $H_{\boldsymbol{w}}=0$, we define $\tilde{P}_{\boldsymbol{w}}(x;z)=1$,
since the product in its definition (2.12) is empty. In case
$H_{\boldsymbol{w}}<0$ (or if $w_{i}<0$ for some $i$), we define
$\tilde{P}_{\boldsymbol{w}}(x;z)=0$.

For illustration, we list in the following a few examples of such polynomials.
The definition (2.12) is actually very impracticable to compute these
polynomial. In practice, we computed them using an algorithm inspired by
Pakman:2009zz , which is described in Appendix A and implemented in the
ancillary Mathematica notebook. We have e.g.

$\displaystyle\tilde{P}_{(1,1,1,1)}(x;z)$ $\displaystyle=z-x\,$ (2.16a)
$\displaystyle\tilde{P}_{(2,1,2,1)}(x;z)$ $\displaystyle=z^{2}-x\,$ (2.16b)
$\displaystyle\tilde{P}_{(2,2,2,2)}(x;z)$
$\displaystyle=z^{4}-4xz^{3}+6xz^{2}-4xz+x^{2}\ ,$ (2.16c)
$\displaystyle\tilde{P}_{(2,3,4,5)}(x;z)$
$\displaystyle=z^{10}-30xz^{8}+120xz^{7}-210xz^{6}+168xz^{5}-50xz^{4}$
$\displaystyle\hskip
30.0pt+50x^{2}z^{4}-120x^{2}z^{3}+105x^{2}z^{2}-40x^{2}z+6x^{2}\ .$ (2.16d)

For later convenience, it will be useful to include prefactors in the
definition of the polynomial. These prefactors do not change the
interpretation of $P_{\boldsymbol{w}}(x;z)$ as the polynomial defining the
Hurwitz space. We define

$P_{\boldsymbol{w}}(x;z)\equiv
f(\boldsymbol{w})\,(1-x)^{\frac{1}{2}s\left(-w_{1}+w_{2}-w_{3}+w_{4})\right)}(1-z)^{\frac{1}{4}s\left(\left(w_{1}+w_{2}-w_{3}-w_{4}\right)\left(w_{1}-w_{2}-w_{3}+w_{4}\right)\right)-\frac{1}{2}w_{2}w_{4}}\\\
\times
x^{\frac{1}{2}s\left(w_{1}-w_{2}-w_{3}+w_{4}\right)}z^{\frac{1}{4}s\left(\left(w_{1}+w_{2}-w_{3}-w_{4}\right)\left(-w_{1}+w_{2}-w_{3}+w_{4}\right)\right)-\frac{1}{2}w_{1}w_{4}}\tilde{P}_{\boldsymbol{w}}(x;z)\
,$ (2.17)

where $f(\boldsymbol{w})$ is a function of $w_{1},\dots,w_{4}$. The reader can
find its explicit form in Appendix C. Here, we used the function

$s(\alpha)=\begin{cases}\alpha\ ,\qquad\alpha>0\ ,\\\ 0\ ,\qquad\alpha\leq 0\
.\end{cases}$ (2.18)

The zero locus of these polynomials coincides with the one of
$\tilde{P}_{\boldsymbol{w}}(x;z)$, since we don’t allow collisions of $x$ or
$z$ with $0$, $1$ or $\infty$. The polynomials $P_{\boldsymbol{w}}(x;z)$
satisfy many surprising identities that are relevant for our purposes. We
collect them in Appendix C. We checked many examples in Mathematica and have a
degree of confidence in their correctness. We do not understand these
identities from first principles and a proof would probably enhance our
understanding of the story.

Let us remark at this point that upon putting $w_{4}=0$, one can check that
$P_{\boldsymbol{w}}(x;z)$ becomes independent of $x$ and $z$ and hence is a
simple number. This is the generalisation of the quantity $P_{\boldsymbol{w}}$
that appeared also in the solution for the three-point functions in
Dei:2021xgh .

###  2.3 The unflowed four-point function

Correlators of unflowed vertex operators have been studied thoroughly in the
literature. Let us briefly mention the main results and set up our
conventions. Two-point functions read Teschner:1997ft ; Fateev ;
Teschner:1999ug

$\left\langle
V^{0}_{j_{1}}(x_{1};z_{1})V^{0}_{j_{2}}(x_{2};z_{2})\right\rangle\\\
=\frac{1}{|z_{1}-z_{2}|^{4\Delta(j_{1})}}\left(\delta^{2}(x_{1}-x_{2})\delta(j_{1}+j_{2}-1)+\frac{B(j_{1})}{|x_{1}-x_{2}|^{4j_{1}}}\delta(j_{1}-j_{2})\right)\
,$ (2.19)

see e.g. Dei:2021xgh  for the explicit definition of $B(j)$ and
$\Delta(j)=-\frac{j(j-1)}{k-2}$. Let us also define

$\mathcal{N}(j)\equiv\sqrt{\frac{B(j)}{B\left(\frac{k}{2}-j\right)}}=\frac{\nu^{\frac{k}{2}-2j}}{\text{\textgamma}\left(\frac{2j-1}{k-2}\right)}\
,$ (2.20)

which in the following will enter various formulae. Three-point functions are
explicitly known Teschner:1997ft ; Fateev ; Teschner:1999ug  and we will
denote them by

$D(j_{1},j_{2},j_{3})=\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)\right\rangle\
.$ (2.21)

Since the generator of translations on the worldsheet can be written as a
bilinear in the $\mathfrak{sl}(2,\mathds{R})$ currents Sugawara:1967rw ,

$\displaystyle\partial_{z}V_{j}^{0}(0;z)$
$\displaystyle=L_{-1}V_{j}^{0}(0;z)=$
$\displaystyle=\frac{1}{k-2}\Bigl{(}-(J^{3}J^{3})_{-1}+\tfrac{1}{2}(J^{+}J^{-})_{-1}+\tfrac{1}{2}(J^{-}J^{+})_{-1}\Bigr{)}\,V_{j}^{0}(0;z)\
,$ (2.22)

a partial differential equation for correlators, known as Knizhnik-
Zamolodchikov (KZ) equation, can be derived Knizhnik:1984nr ; Frenkel:1991gx ;
Teschner:1999ug . For four-point functions involving only unflowed vertex
operators, the KZ equation reads Teschner:1999ug 888With respect to the
conventions adopted in Teschner:1999ug , we have $j_{1}^{\text{T}}=-j_{1}$,
$j_{2}^{\text{T}}=-j_{4}$, $j_{3}^{\text{T}}=-j_{2}$ and
$j_{4}^{\text{T}}=-j_{3}$.

$\partial_{z}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\\\
=\frac{1}{k-2}\Bigl{(}\frac{\mathcal{P}}{z}+\frac{\mathcal{Q}}{z-1}\Bigr{)}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\
,$ (2.23)

where $\mathcal{P}$ and $\mathcal{Q}$ are differential operators in $x$,

$\displaystyle\mathcal{P}$
$\displaystyle=x^{2}(x-1)\partial_{x}^{2}-\left((\kappa-1)x^{2}+2j_{1}x-2j_{4}x(x-1)\right)\partial_{x}-2\kappa
j_{4}x-2j_{1}j_{4}\ ,$ (2.24a) $\displaystyle\mathcal{Q}$
$\displaystyle=-(1-x)^{2}x\partial_{x}^{2}+(1-x)\left((\kappa-1)(1-x)+2j_{2}+2j_{4}x\right)\partial_{x}-2\kappa
j_{4}(1-x)-2j_{2}j_{4}\ ,$ (2.24b)

and

$\kappa\equiv j_{3}-j_{1}-j_{2}-j_{4}\ .$ (2.25)

Beyond the singularities one usually encounters when two insertion points
collide on the worldsheet or in spacetime (i.e. when $z=0$, $z=1$, $z=\infty$,
$x=0$, $x=1$ or $x=\infty$), the unflowed correlator features an additional
singularity when $z=x$ Maldacena:2001km . This can be seen by substituting the
ansatz

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\sim|z-x|^{2\delta}$
(2.26)

into (2.23). The first order in $(z-x)$ implies Maldacena:2001km

$\delta=0\quad\text{or}\quad\delta=k-j_{1}-j_{2}-j_{3}-j_{4}\ .$ (2.27)

Both solutions appear in the conformal block expansion and are in general
necessary to yield a single-valued four-point function.

Fusion rules and conformal blocks of unflowed correlators are also known
Teschner:1997ft ; Teschner:1999ug  and crossing symmetry has been proved in
Teschner:2001gi . In the following we will make use of the identity

$\mathcal{N}(j_{1})\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\\\
=\mathcal{N}(j_{3})|x|^{-4j_{4}}|z|^{2j_{4}}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}\Bigl{(}\frac{z}{x};z\Bigr{)}\right\rangle\
.$ (2.28)

Eq. (2.28) has been proved in Parnachev:2001gw . Since various ingredients
will be useful in the following, we give an alternative derivation in Appendix
E.

###  2.4 Global Ward identities

Let us now explain the various constraints four-point functions satisfy. We
start with the global Ward identities. They allow us to put $x_{1}=z_{1}=0$,
$x_{2}=z_{2}=1$ and $x_{3}=z_{3}=\infty$ in our formulae. We discussed them in
the $y$-space in Dei:2021xgh . One can always reinstate the dependence on all
four positions in $x$\- and $z$-space by using the solution to the global Ward
identities for four-point functions which read

$\displaystyle\langle
V_{j_{1}}^{w_{1}}(x_{1};y_{1};z_{1})V_{j_{2}}^{w_{2}}(x_{2};y_{2};z_{2})V_{j_{3}}^{w_{3}}(x_{3};y_{3};z_{3})V_{j_{4}}^{w_{4}}(x_{4};y_{4};z_{4})\rangle$
$\displaystyle=|x_{21}|^{2(-h_{1}^{0}-h_{2}^{0}+h_{3}^{0}-h_{4}^{0})}\,|x_{31}|^{2(-h_{1}^{0}+h_{2}^{0}-h_{3}^{0}+h_{4}^{0})}\,|x_{32}|^{2(h_{1}^{0}-h_{2}^{0}-h_{3}^{0}+h_{4}^{0})}\,|x_{34}|^{-4h_{4}^{0}}$
$\displaystyle\qquad\\!\times|z_{21}|^{2(-\Delta_{1}^{0}-\Delta_{2}^{0}+\Delta_{3}^{0}-\Delta_{4}^{0})}\,|z_{31}|^{2(-\Delta_{1}^{0}+\Delta_{2}^{0}-\Delta_{3}^{0}+\Delta_{4}^{0})}\,|z_{32}|^{2(\Delta_{1}^{0}-\Delta_{2}^{0}-\Delta_{3}^{0}+\Delta_{4}^{0})}\,|z_{34}|^{-4\Delta_{4}^{0}}$
$\displaystyle\qquad\\!\times\Bigg{\langle}V_{j_{1}}^{w_{1}}\left(0;\frac{y_{1}\,x_{32}\,z_{21}^{w_{1}}\,z_{31}^{w_{1}}}{x_{21}\,x_{31}\,z_{32}^{w_{1}}};0\right)V_{j_{2}}^{w_{2}}\left(1;\frac{y_{2}\,x_{31}\,z_{21}^{w_{2}}\,z_{32}^{w_{2}}}{x_{21}\,x_{32}\,z_{31}^{w_{2}}};1\right)V_{j_{3}}^{w_{3}}\left(\infty;\frac{y_{3}\,x_{21}\,z_{31}^{w_{3}}\,z_{32}^{w_{3}}}{x_{31}\,x_{32}\,z_{21}^{w_{3}}};\infty\right)$
$\displaystyle\qquad\qquad\qquad\qquad
V_{j_{4}}^{w_{4}}\left(\frac{x_{32}\,x_{14}}{x_{12}\,x_{34}};\frac{y_{4}\,x_{31}\,x_{32}\,z_{21}^{w_{4}}\,z_{34}^{2w_{4}}}{x_{21}\,x_{34}^{2}\,z_{31}^{w_{4}}\,z_{32}^{w_{4}}};\frac{z_{32}\,z_{14}}{z_{12}\,z_{34}}\right)\Bigg{\rangle}\
,$ (2.29)

where

$h^{0}\equiv j+\frac{kw}{2}\
,\qquad\Delta^{0}\equiv-\frac{j(j-1)}{k-2}-wj-\frac{kw^{2}}{4}\ .$ (2.30)

The prefactors are analogous to those we are used to in CFT. However, since
vertex operators in the $y$-basis don’t have a definite conformal weight on
the worldsheet or in spacetime also the $y$-coordinate transforms non-
trivially under conformal transformations.

###  2.5 Local Ward identities

Local Ward identities impose further constraints on correlators. They are a
consequence of the _affine_ symmetry algebra (whereas global Ward identities
only make use of the global subalgebra). In the unflowed sector this fact does
not lead to further constraints for affine primary fields. However, as
realised in Eberhardt:2019ywk , the situation is different in the spectrally
flowed sector. As we have mentioned in Section 2.1, unflowed vertex operator
depend on less quantum numbers because $j=h=\bar{h}$. This is compensated in
the flowed sector by the existence of local Ward identities that allow for
non-vanishing solutions whenever Eberhardt:2019ywk 999The same bound was
derived in Maldacena:2001km  by different techniques.

$\sum_{i=1}^{4}w_{i}\geq 2\max_{i=1,\dots 4}(w_{i})-2\ .$ (2.31)

When the bound (2.31) is obeyed, one can derive recursion relations in the
spacetime conformal weights $(h_{i},\bar{h}_{i})$ for the correlators

$\left\langle\prod_{i=1}^{4}V^{w_{i}}_{j_{i},h_{i}}(x_{i};z_{i})\right\rangle\
.$ (2.32)

After transforming to the $y$-basis, they become partial differential
equations in the variables $x_{i}$, $y_{i}$ and $z_{i}$. We will not review
the derivation of these constraints, it should suffice if the reader keeps in
mind that there is such a local Ward identity for every spectrally flowed
vertex operator in the correlator of interest. We will discuss one example in
detail in Section 3.1.

###  2.6 The Knizhnik-Zamolodchikov equation

We now derive the analogue of (2.23) for correlators containing flowed vertex
operators. See Giribet:2004qe ; Ribault:2005ms ; Giribet:2005mc ;
Minces:2005nb  for previous results on the KZ equation in the flowed sector.
Evaluating $(k-2)L_{-1}$ on the spectrally flowed affine primary state with
$w>0$ we find

$\displaystyle(k-2)\partial_{z}V_{j,h}^{w}(0;z_{i})$
$\displaystyle=-2hJ^{3}_{-1}V_{j,h}^{w}(0;z)+\sum_{n=0}^{w}J_{-1-n}^{-}J_{n}^{+}V_{j,h}^{w}(0;z)$
(2.33)
$\displaystyle=-2hJ^{3}_{-1}V_{j,h}^{w}(0;z)+\sum_{n=0}^{w-1}[J_{-1-n}^{-},J_{n}^{+}]V_{j,h}^{w}(0;z)$
$\displaystyle\qquad+J^{-}_{-w-1}J^{+}_{w}V_{j,h}^{w}(0;z)+J^{+}_{w-1}J^{-}_{-w}V_{j,h}^{w}(0;z)$
(2.34) $\displaystyle=-2(h-w)J^{3}_{-1}V_{j,h}^{w}(0;z)$
$\displaystyle\quad+\left(h-\tfrac{k}{2}w+j-1\right)J^{+}_{w-1}V_{j,h-1}^{w}(0;z)$
$\displaystyle\quad+\left(h-\tfrac{k}{2}w-j+1\right)J^{-}_{-w-1}V_{j,h+1}^{w}(0;z)\
.$ (2.35)

Translating this formula to the position $x$ leads to additional correction
factors and we obtain

$(k-2)\partial_{z}V_{j,h}^{w}(x;z)=-2(h-w)(J^{3}_{-1}-xJ^{+}_{-1})V_{j,h}^{w}(x;z)\\\
+\left(h-\tfrac{k}{2}w+j-1\right)J^{+}_{w-1}V_{j,h-1}^{w}(x;z)\\\
+\left(h-\tfrac{k}{2}w-j+1\right)(J^{-}_{-w-1}-2xJ^{3}_{-w-1}+x^{2}J^{+}_{-w-1})V_{j,h+1}^{w}(x;z)\
.$ (2.36)

Inserting eq. (2.36) into a correlation function leads to the KZ equation. For
simplicity, let us consider four-point functions and set $x=x_{4}$ and
$z=z_{4}$ in (2.36).101010Of course, one could also set $x=x_{i}$ and
$z=z_{i}$ for $i=1,2,3$. However, this is not important: all choices are
equivalent by Möbius symmetry. Applying contour deformation techniques and
making use of the OPEs of the $\mathfrak{sl}(2,\mathds{R})_{k}$ currents with
$V^{w}_{j,h}(x;z)$, correlators containing $J^{3}$ and $J^{-}$ modes can be
rewritten in terms of correlators containing only $J^{+}$ modes,

$\displaystyle\left\langle\prod_{i=1}^{3}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})J^{3}_{-m}V_{j_{4},h_{4}}^{w_{4}}(x_{4};z_{4})\right\rangle=-\sum_{i=1}^{3}\Biggl{[}\frac{h_{i}+x_{i}\partial_{x_{i}}}{(z_{i}-z_{4})^{m}}\left\langle\prod_{p=1}^{4}V_{j_{p},h_{p}}^{w_{p}}(x_{p};z_{p})\right\rangle$
$\displaystyle\hskip
40.0pt+\sum_{\ell=1}^{w_{i}}\binom{m+\ell-2}{\ell-1}\frac{(-1)^{\ell-1}\,x_{i}}{(z_{i}-z_{4})^{m+\ell-1}}\left\langle
J^{+}_{\ell}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})\prod_{p\neq
i}V_{j_{p},h_{p}}^{w_{p}}(x_{p};z_{p})\right\rangle\Biggr{]}\ ,$ (2.37a)
$\displaystyle\left\langle\prod_{i=1}^{3}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})J^{-}_{-m}V_{j_{4},h_{4}}^{w_{4}}(x_{4};z_{4})\right\rangle=-\sum_{i=1}^{3}\Biggl{[}\frac{2h_{i}\,x_{i}+x_{i}^{2}\partial_{x_{i}}}{(z_{i}-z_{4})^{m}}\left\langle\prod_{p=1}^{4}V_{j_{p},h_{p}}^{w_{p}}(x_{p};z_{p})\right\rangle$
$\displaystyle\hskip
40.0pt+\sum_{\ell=1}^{w_{i}}\binom{m+\ell-2}{\ell-1}\frac{(-1)^{\ell-1}\,x_{i}^{2}}{(z_{i}-z_{4})^{m+\ell-1}}\left\langle
J^{+}_{\ell}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})\prod_{p\neq
i}V_{j_{p},h_{p}}^{w_{p}}(x_{p};z_{p})\right\rangle\Biggr{]}\ .$ (2.37b)

Finally, making use of the local Ward identities of Eberhardt:2019ywk , one
can solve for correlators of the form

$\left\langle J^{+}_{\ell}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})\prod_{p\neq
i}V_{j_{p},h_{p}}^{w_{p}}(x_{p};z_{p})\right\rangle\ ,\qquad 0<\ell<w_{i}\ ,$
(2.38)

in terms of correlators (2.32), their $h_{i}$ shifted values and their
derivatives with respect to the $x_{i}$. Taking the $y$-transform we obtain a
partial differential equation which is first order in
$z_{4},y_{1},\dots,y_{4}$ and second order in $x_{4}$. Explicit examples can
be found in the ancillary Mathematica file and in Section 3.

###  2.7 Null vector constraints

An unflowed spin $j$ representation of $\mathfrak{sl}(2,\mathds{R})_{k}$ is
reducible whenever Kac:1979fz ; McElgin:2015eho

$j=j_{r,s}^{+}=\frac{1+r+s(k-2)}{2}\ ,\qquad r,s\in\mathds{Z}_{>0}$ (2.39)

or

$\hskip 35.0ptj=j_{r,s}^{-}=\frac{1-r-s(k-2)}{2}\ ,\qquad r\in\mathds{Z}_{>0}\
,\quad s\in\mathds{Z}_{\geq 0}\ .$ (2.40)

The corresponding null vector $\ket{\mathcal{N}}$ appears at level $rs$ in the
Verma module Malikov1984  (though as we shall explain momentarily, this does
not translate into the order of the resulting differential equation). Some
simple examples are

$\displaystyle j$ $\displaystyle=j_{1,0}^{-}=0:$
$\displaystyle\ket{\mathcal{N}}$ $\displaystyle=m\ket{j,m}\ ,$ (2.41a)
$\displaystyle j$ $\displaystyle=j_{2,0}^{-}=-\frac{1}{2}:$
$\displaystyle\ket{\mathcal{N}}$ $\displaystyle=(2m-1)(2m+1)\ket{j,m}\ ,$
(2.41b) $\displaystyle j$ $\displaystyle=j_{1,1}^{-}=1-\frac{k}{2}:$
$\displaystyle\ket{\mathcal{N}}$
$\displaystyle=J^{-}_{-1}\ket{j,m+1}-2J^{3}_{-1}\ket{j,m}+J^{+}_{-1}\ket{j,m-1}\
,$ (2.41c) $\displaystyle j$ $\displaystyle=j_{1,1}^{+}=\frac{k}{2}:$
$\displaystyle\ket{\mathcal{N}}$
$\displaystyle=(k+2m-2)(k+2m)J_{-1}^{-}\ket{j,m+1}$
$\displaystyle\qquad+2(k-2m-2)(k+2m-2)J^{3}_{-1}\ket{j,m}$
$\displaystyle\qquad+(k-2m-2)(k-2m)J^{+}_{-1}\ket{j,m-1}\ .$ (2.41d)

By a null vector $m\ket{0,m}$ we mean that the vector $\ket{0,m}$ is null
whenever $m\neq 0$. In other words, it is the vacuum representation. It will
be convenient in the following to think about these null vectors in this way.
Null vectors imply constraints in the form of differential equations on
correlation functions. Their derivation is somewhat different in the unflowed
and flowed sector.

##### Unflowed sector.

Let us start with the more conventional unflowed sector. Here, we just pick
the lowest weight state of the family of null vectors for the
$\mathcal{D}^{+}$ representation. Similar formulae can be derived for
$\mathcal{D}^{-}$ representations. In the above examples:

$\displaystyle j$ $\displaystyle=j_{1,0}^{-}=0:$
$\displaystyle\ket{\mathcal{N}}$ $\displaystyle=J_{0}^{+}\ket{j,j}\ ,$ (2.42a)
$\displaystyle j$ $\displaystyle=j_{2,0}^{-}=-\frac{1}{2}:$
$\displaystyle\ket{\mathcal{N}}$ $\displaystyle=J_{0}^{+}J_{0}^{+}\ket{j,j}\
,$ (2.42b) $\displaystyle j$ $\displaystyle=j_{1,1}^{-}=1-\frac{k}{2}:$
$\displaystyle\ket{\mathcal{N}}$
$\displaystyle=\left(J^{-}_{-1}J_{0}^{+}J_{0}^{+}+2(k-3)J^{3}_{-1}J_{0}^{+}+(k-3)(k-2)J^{+}_{-1}\right)\ket{j,j}\
,$ (2.42c) $\displaystyle j$ $\displaystyle=j_{1,1}^{+}=\frac{k}{2}:$
$\displaystyle\ket{\mathcal{N}}$ $\displaystyle=J^{-}_{-1}\ket{j,j}\ .$
(2.42d)

Following Malikov1984 , null vectors can be written in closed-form for each
choice of the spins in eqs. (2.39) and (2.40) at the price of allowing non-
integer exponents for the modes. We review this construction in Appendix D.
Null vectors can in turn be easily translated into differential equations. The
number of currents corresponds to the order of the differential equation.

The simplest example is given by $j_{4}=j_{1,0}^{-}=0$, where the null vector
(2.42a) gives immediately

$0=\partial_{x_{4}}\left\langle\prod_{i=1}^{3}V^{0}_{j_{i}}(x_{i};z_{i})\,V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle\
,$ (2.43)

with $j_{4}=0$.

Slightly less trivial is the case $j_{4}=j_{1,1}^{+}=\frac{k}{2}$. Standard 2D
CFT techniques imply the differential equation

$0=\left\langle\prod_{i=1}^{3}V^{0}_{j_{i}}(x_{i};z_{i})\,[J^{-}_{-1}V^{0}_{j_{4}}](x_{4};z_{4})\right\rangle\\\
=\sum_{i=1}^{3}\frac{[(x_{i}-x_{4})^{2}\partial_{x_{i}}+2(x_{i}-x_{4})j_{i}]}{z_{4}-z_{i}}\left\langle\prod_{i=1}^{4}V^{0}_{j_{i}}(x_{i};z_{i})\right\rangle$
(2.44)

Making use of global Ward identities and setting $x_{1}=0$, $x_{2}=1$,
$x_{3}=\infty$ and $x_{4}=x$ and similarly for $z_{i}$, eq. (2.44) simplifies
to

$\partial_{x_{4}}\left\langle\prod_{i=1}^{4}V^{0}_{j_{i}}(x_{i};z_{i})\right\rangle=\frac{1}{2x(x-1)(x-z)}\Bigl{[}-2j_{1}\left(x^{2}-2x+z\right)-2j_{2}\left(x^{2}-z\right)\\\
+2j_{3}\left(-2x\,z+x^{2}+z\right)-k\left(-2x\,z+x^{2}+\right)\Bigr{]}\left\langle\prod_{i=1}^{4}V^{0}_{j_{i}}(x_{i};z_{i})\right\rangle\
.$ (2.45)

In Appendix D we present the differential equations associated to the null
vectors (2.42b) and (2.42c) Teschner:1997ft . In general — see Appendix D —
the order of the null vector equation in the unflowed sector is Malikov1984

$\text{ord}(j^{\pm}_{r,s})=r(2s\mp 1)\ .$ (2.46)

##### Flowed sector.

In addition to unflowed representations, one also considers spectrally flowed
representations. Also spectrally flowed representations contain null
vectors.111111The spectrally flowed image of a null vector is itself null. In
fact, the image of a null vector under the spectral flow automorphism gives
rise to a descendant that is at the same time the spectrally flowed image of a
primary state (in the spectrally flowed sense). For example, for
$j=j^{-}_{1,1}=1-\frac{k}{2}$, the vector

$(J^{-}_{-w-1}V^{w}_{j,\,h+1})(x;z)-2(J^{3}_{-1}V^{w}_{j,\,h})(x;z)+(J^{+}_{w-1}V^{w}_{j,\,h-1})(x;z)$
(2.47)

is null, in analogy to the unflowed sector (2.41c). However, the logic how
these null vectors are translated into differential equations is slightly
different. The primary reason for this is that the Ward identities relate
correlators with different values of $h$ and thus we cannot simply restrict
ourselves to one state of the null representation. Instead, we proceed as in
the derivation of the KZ equation and rewrite $J^{3}$ and $J^{-}$ modes in
terms of $J^{+}$ modes only, see eqs. (2.37). After solving the local Ward
identities, this reduces to a differential equation for the correlator itself.
For example, in the case of $j_{4}=j^{+}_{1,1}=\frac{k}{2}$, the resulting
differential equation is first order (in both $x_{4}$ and $y_{i}$) because it
only involves one current insertion. Thus, the order of the differential
equation is the same as in the unflowed sector. This is no coincidence, since
we will later relate a spectrally flowed correlator to its unflowed
counterpart. We will discuss the spectrally flowed null vector equations
further in Section 3.3.

###  2.8 Summary of constraints

To keep the reader oriented, let us summarise the various constraints. All of
the constraints we discussed are partial differential equations in the
variables $x_{1},\dots,x_{4}$, $y_{1},\dots,y_{4}$ and $z_{1},\dots,z_{4}$.
There are 6 global Ward identities which allow us to put $x_{1}=z_{1}=0$,
$x_{2}=z_{2}=1$ and $x_{3}=z_{3}=\infty$. One can recover the general
dependence using (2.29). There are as many local Ward identities as there are
spectrally flowed vertex operators in the correlator. Thus assuming that
$w_{i}>0$ for all $i$, there are four local Ward identities which we will
solve below and one ends up with a function depending on two variables only.
This function satisfies a further constraint due to the KZ-equation. For
degenerate representations, null vectors can impose additional constraints
which determine the correlator fully, up to an overall constant. In the
following section, we will solve all these constraints and propose a closed-
form expression for the correlator.

##  3 A closed-form expression for the spectrally flowed four-point function

We will consider four-point functions of the form

$\left\langle\prod_{i=1}^{4}V_{j_{i},h_{i}}^{w_{i}}(x_{i};z_{i})\right\rangle\
.$ (3.1)

We will show that the constraints of the recursion relations can be completely
solved after transforming to $y$-space and that, as anticipated in the
Introduction, knowledge of the unflowed correlator leads to a complete
determination of the flowed correlator. Based on different consistency
conditions, we are able to give a conjecture for the prefactor.

###  3.1 The flowed correlator with $\boldsymbol{w_{1}=w_{2}=w_{3}=w_{4}=1}$

Before presenting our general result for the four-point functions, let us
treat the simplest case where all four vertex operators are flowed. We find it
convenient to perform the computation of the correlator in the $y$-space
introduced in Section 2.1, since in these variables all constraints can be
written as partial differential equations. We consider the following
correlator:

$\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}=1}(x_{i};y_{i};z_{i})\right\rangle\
.$ (3.2)

Even though this correlator depends on four new variables $y_{1}$, $y_{2}$,
$y_{3}$ and $y_{4}$ that did not enter the unflowed correlation function, we
also have four more constraints coming from the local Ward identities. We
again set $x_{1}=z_{1}=0$, $x_{2}=z_{2}=1$, $x_{3}=z_{3}=\infty$, $x_{4}=x$
and $z_{4}=z$. The four local Ward identities read in this case121212The
reader can reproduce these equations easily using the ancillary Mathematica
file.

$\displaystyle 0$
$\displaystyle=\Big{(}\left(1-y_{1}\right)y_{1}z^{2}\partial_{y_{1}}+\left(1-y_{2}\right)z^{2}\partial_{y_{2}}+\left(1-y_{3}\right)z^{2}\partial_{y_{3}}+\left(x^{2}-2\,z\,y_{4}\,x+z^{2}y_{4}\right)\partial_{y_{4}}$
$\displaystyle\quad+\left(-kx+zj_{1}(1-2y_{1})-zj_{2}-zj_{3}+j_{4}(z-2x)\right)z-x(x-z)z\partial_{x}\Big{)}\langle\cdots\rangle\
,$ (3.3a) $\displaystyle 0$
$\displaystyle=\Big{(}\left(1-y_{1}\right)(1-z)^{2}\partial_{y_{1}}+\left(1-y_{2}\right)y_{2}(1-z)^{2}\partial_{y_{2}}+\left(1-y_{3}\right)(1-z)^{2}\partial_{y_{3}}$
$\displaystyle\quad+\left(x^{2}-2\,z\,y_{4}\,x+2\,y_{4}\,x-2x+z^{2}y_{4}-y_{4}+1\right)\partial_{y_{4}}$
$\displaystyle\quad+\big{(}k(1-x)+j_{1}(1-z)+j_{2}(z-1)(1-2y_{2})+j_{3}(1-z)+j_{4}(1+z-2x)\big{)}(z-1)$
$\displaystyle\quad-(x-1)(x-z)(z-1)\partial_{x}\Big{)}\langle\cdots\rangle\ ,$
(3.3b) $\displaystyle 0$
$\displaystyle=\Big{(}\left(1-y_{1}\right)\partial_{y_{1}}+\left(1-y_{2}\right)\partial_{y_{2}}+\left(1-y_{3}\right)y_{3}\partial_{y_{3}}+\left(1-y_{4}\right)\partial_{y_{4}}$
$\displaystyle\quad+\left(-k-j_{1}-j_{2}+j_{3}(1-2y_{3})-j_{4}\right)+(z-x)\partial_{x}\Big{)}\langle\cdots\rangle\
,$ (3.3c) $\displaystyle 0$
$\displaystyle=\Big{(}(1-z)\left(x^{2}(1-z)+z(1-x)^{2}\,y_{1}\right)\partial_{y_{1}}+z\left(z(1-x)^{2}+(x^{2}-z)(1-z)\,y_{2}\right)\partial_{y_{2}}$
$\displaystyle\quad-(1-z)z\left(z(z-1)+(x^{2}-2zx+z)\,y_{3}\right)\partial_{y_{3}}$
$\displaystyle\quad+(1-z)z\,y_{4}\left(x^{2}-2\,z\,x+z+z(z-1)\,y_{4}\right)\partial_{y_{4}}$
$\displaystyle\quad-(1-z)z\big{(}kx(1-x)-(x^{2}-2x+z)\,j_{1}-(x^{2}-z)\,j_{2}+(x^{2}-2zx+z)\,(j_{3}+j_{4})$
$\displaystyle\quad\qquad+2\,z(1-z)\,j_{4}\,y_{4}\big{)}+(1-x)x(1-z)z(z-x)\partial_{x}\Big{)}\langle\cdots\rangle\
,$ (3.3d)

where $\langle\cdots\rangle$ is a shorthand notation for the correlator (3.2).
These equations look very complicated, but we should note that they are by
construction first order in all variables. They are all mutually compatible
and their common solution contains an undetermined function of two
variables.131313We started with the six variables $y_{1}$, $y_{2}$, $y_{3}$,
$y_{4}$, $x$ and $z$ and solving the recursion relations fixes the dependence
on four of those. We can take one variable to be $z$, since the equations do
not contain $z$-derivatives. The other variable is taken to be

$c=\frac{x-2z+z^{2}+z(1-z)y_{2}+z(1-x)y_{3}-x(1-z)y_{2}\,y_{3}}{x-2x\,z+z^{2}-(1-x)z\,y_{1}-x(1-z)y_{2}+z(1-z)y_{1}\,y_{2}}\\\
\times\frac{x(x-2z+z^{2})+(1-x)z^{2}\,y_{1}+z^{2}(1-z)y_{4}-z^{2}(1-z)y_{1}\,y_{4}}{x-2x\,z+z^{2}-x(1-x)y_{3}-z(1-z)y_{4}+z(1-z)y_{3}\,y_{4}}\
.$ (3.4)

By explicitly solving these equations one finds the solution

$\displaystyle\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}=1}(x_{i};y_{i};z_{i})\right\rangle=(1-z)^{j_{2}+j_{4}+\frac{k}{2}}z^{j_{1}+j_{4}+\frac{k}{2}}(x-z)^{j_{1}+j_{2}+j_{3}+j_{4}-k}$
$\displaystyle\qquad\times\left(x-2\,x\,z+z^{2}-(1-x)z\,y_{1}-x(1-z)y_{2}+z(1-z)y_{1}\,y_{2}\right){}^{-j_{1}-j_{2}+j_{3}-j_{4}}$
$\displaystyle\qquad\times\left(x-z^{2}-z(1-z)y_{1}\,z^{2}-x(1-z)y_{3}+(1-x)z\,y_{1}\,y_{3}\right){}^{-j_{1}+j_{2}-j_{3}+j_{4}}$
$\displaystyle\qquad\times\left(x-2\,z+z^{2}+z(1-z)y_{2}+z(1-x)y_{3}-x(1-z)y_{2}\,y_{3}\right){}^{j_{1}-j_{2}-j_{3}+j_{4}}$
$\displaystyle\qquad\times\left(x-2\,x\,z+z^{2}-x(1-x)y_{3}-z(1-z)y_{4}+z(1-z)y_{3}\,y_{4}\right){}^{-2j_{4}}F(c,z)\
.$ (3.5)

The recursion relations of the right-movers are analogous and their solution
would lead to the same right-moving structure. The undetermined function
$F(c,z)$ actually depends on $c$, $\bar{c}$, $z$ and $\bar{z}$. The solution
(3.5) is relatively simple, compared to the complexity of the differential
equations (3.3). The prefactors $z$ and $1-z$ are not fixed by the recursion
relations (since the recursion relations do not contain $z$-derivatives), but
are inserted for later convenience. The function $F(c,z)$ is further
restricted by the KZ-equation and possible null vector decoupling equations.
Since these are conceptually not more complicated for higher values of
spectral flow, we postpone their discussion to Section 3.3.

###  3.2 The general solution of the recursion relations

The solution we obtained in Section 3.1 for $w_{1}=w_{2}=w_{3}=w_{4}=1$ is
suggestive. Let us compare it to the solution of the global
$\mathfrak{sl}(2,\mathds{R})$ Ward identities in the unflowed sector, which is
given by (compare to eq. (2.29))

$\left\langle\prod_{i=1}^{4}V_{j_{i}}^{0}(x_{i};z_{i})\right\rangle=x_{12}^{-j_{1}-j_{2}+j_{3}-j_{4}}x_{13}^{-j_{1}+j_{2}-j_{3}+j_{4}}x_{23}^{j_{1}-j_{2}-j_{3}+j_{4}}x_{34}^{-2j_{4}}F\left(\frac{x_{23}\,x_{14}}{x_{12}\,x_{34}},z\right)\
,$ (3.6)

where $x_{ij}=x_{i}-x_{j}$ and we suppressed the other variables. For
$x_{1}=0$, $x_{2}=1$ and $x_{3}=\infty$, the prefactor trivialises. The
solution to the recursion relations has the same structure, except that we
have to replace the differences $x_{ij}$ by ‘generalised differences’
$X_{ij}(x,z,y_{i},y_{j})$ and include a couple of simple prefactors. The
general solution of the recursion relations is given by

$\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}}(x_{i};y_{i};z_{i})\right\rangle=X_{\emptyset}^{j_{1}+j_{2}+j_{3}+j_{4}-k}X_{12}^{-j_{1}-j_{2}+j_{3}-j_{4}}X_{13}^{-j_{1}+j_{2}-j_{3}+j_{4}}\\\
\times
X_{23}^{j_{1}-j_{2}-j_{3}+j_{4}}X_{34}^{-2j_{4}}F\left(\frac{X_{23}X_{14}}{X_{12}X_{34}},z\right)\
,$ (3.7)

where we omitted the dependence on anti-holomorphic variables and we define

$X_{I}(x,z,y_{1},y_{2},y_{3},y_{4})=z^{\frac{1}{2}\delta_{\\{1,4\\}\subset
I}}\left((1-z)^{\frac{1}{2}}(-1)^{w_{1}w_{2}+w_{1}w_{3}+w_{2}w_{4}+w_{3}w_{4}}\right)^{\delta_{\\{2,4\\}\subset
I}}\\\ \times\sum_{i\in I:\ \varepsilon_{i}=\pm 1}P_{\boldsymbol{w}+\sum_{i\in
I}\varepsilon_{i}e_{i}}(x;z)\prod_{i\in I}y_{i}^{\frac{1-\varepsilon_{i}}{2}}\
.$ (3.8)

In this definition, $I$ can be any subset of $\\{1,2,3,4\\}$. We made use of
the polynomial $P_{\boldsymbol{w}}(x;z)$ that was defined in Section 2.2. The
prefactor is again inserted to simplify various expressions later. It is
actually not needed presently, since we have not determined the $z$-dependence
anyway.

The definition of $X_{I}$ plays a central role in our story, so let us
elucidate it a bit further with an example. We have for example for
$\boldsymbol{w}=(1,1,1,1)$

$\displaystyle X_{12}$
$\displaystyle=P_{(2,2,1,1)}(x;z)+P_{(0,2,1,1)}(x;z)y_{1}+P_{(2,0,1,1)}(x;z)y_{2}+P_{(0,0,1,1)}(x;z)y_{1}y_{2}$
(3.9)
$\displaystyle=\frac{1}{z(1-z)}\left(x-2\,x\,z+z^{2}-(1-x)z\,y_{1}-x(1-z)y_{2}+z(1-z)y_{1}\,y_{2}\right)\
,$ (3.10)

which appears as the first of the factors in the solution (3.5). Let us recall
that the polynomials $P_{\boldsymbol{w}}(x;z)$ that appear in this expression
encode the condition on $x$ and $z$ for the existence of a branched covering
map from the worldsheet to the boundary of $\mathrm{AdS}_{3}$. In this case,
only $P_{(2,2,1,1)}(x;z)$ is interesting and if it vanishes the corresponding
covering map is

$\gamma(\zeta)=\frac{\zeta^{2}}{2\zeta-1}\ .$ (3.11)

One should also notice that by construction $X_{\emptyset}$ coincides with
$P_{\boldsymbol{w}}(x;z)$. It should now be clear why we included the
normalisation factors in (2.17). They are needed to generate the correct
solution of the local Ward identities.

Notice that at this stage the function $F(c,z)$ may depend on the choice of
spectral flow parameters. In fact, both the KZ equation and the null vector
equations depend on $w_{1},\dots,w_{4}$. However, we will see in a moment that
this is not the case and that $F(c,z)$ does not actually depend on the choice
of $w_{1},\dots,w_{4}$.

The solution (3.7) is only valid when $\sum_{i}w_{i}\in 2\mathds{Z}$ and the
bound (2.31) is not saturated. Correlators saturating this bound require extra
care and will be treated separately in Section 3.5. As we already saw for the
three-point function, the structure of the opposite parity correlators is
different and we discuss it in Section 3.4.

If in eq. (3.7) $w_{i}=0$ for some $i$, then the dependence on $y_{i}$ drops
out. This is as it should be, since in the unflowed sector $y_{i}$ coincides
with $x_{i}$ and we decided to keep the latter.

For $w_{4}=0$, the definition (3.8) reduces to the one we have given for the
three-point function in Dei:2021xgh . We shall later see that if one also
specialises $j_{4}=0$, eq. (3.7) reduces to the even parity solution in (2.6).
In fact, one can show that the remaining function $F(c,z)$ is $c$ and
$z$-independent. This is in agreement with $V^{w=0}_{j=0}(x;z)$ being the
identity operator on the worldsheet.

We should note that the ‘generalised differences’ $X_{ij}$ satisfy some
analogous relations that we would expect from ordinary differences
$x_{ij}=x_{i}-x_{j}$. In fact, while $X_{ij}\neq X_{i\ell}+X_{\ell j}$, we
have

$1-\frac{X_{23}X_{14}}{X_{12}X_{34}}=\frac{X_{13}X_{24}}{X_{12}X_{34}}\ ,$
(3.12)

which is the analogous relation that the $x_{ij}$’s would satisfy. Thus,
$\frac{X_{23}X_{14}}{X_{12}X_{34}}$ really behaves as a cross ratio. This
implies an identity on the polynomials $P_{\boldsymbol{w}}(x;z)$ that is
further discussed in Appendix C.

Hence, somewhat surprisingly, the structure of the flowed correlator resembles
very much the one of the unflowed one. While we do not know of an analytic
derivation of this result, we have checked it in Mathematica for all choices
of $(w_{1},w_{2},w_{3},w_{4})$ with $\sum w_{i}\leq 10$ (721 possibilities).
We view this as ample evidence that our conjectured solution is indeed
correct.

###  3.3 More constraints and the full parity even correlator

It turns out that the analogy between the unflowed correlator and the flowed
version extends beyond what was already discussed in the previous section. One
can inject the solution of the recursion relations into the KZ equation, which
leads to a differential equation for the function $F(c,z)$ entering
(3.7).141414The fact that the KZ equation is compatible with the recursion
relations is another cross-check of our analysis. We have done this in
Mathematica for all choices of $w_{i}$ with $\sum_{i}w_{i}\leq 10$. In all
those cases, it turns out that $F(c,z)$ obeys the _unflowed_ KZ equation
(2.23) with spins $j_{i}$.

Similarly, in cases where $j_{i}$ is degenerate, one can compute the null
vector equation that the correlator obeys. As explained in Section 2.7 this
again translates into a differential equation for the function $F(c,z)$. We
have checked in the cases (2.42a) – (2.42d) that the resulting differential
equation coincides with the unflowed differential equation with spins $j_{i}$,
see Appendix D.

This observation can be checked with the ancillary Mathematica notebook. In
the simplest case, it is easy to understand analytically. Let us assume
$w_{4}>0$ and consider the degenerate case of $j_{4}=0$. In this case, the
null field takes the following form (see eq. (2.41a))

$(h-\tfrac{kw}{2})V_{0,h}^{w}(x;z)\longrightarrow
y_{4}\partial_{y_{4}}V_{0}^{w}(x;y;z)\ .$ (3.13)

The first formula is the null vector in $h$-space, which translates to the
given formula in $y$-space. Hence, we obtain the differential equation

$\partial_{y_{4}}\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}}(x_{i};y_{i};z_{i})\right\rangle=0\
.$ (3.14)

For $j_{4}=0$, we see however that the prefactor of $F(c,z)$ in the solution
(3.7) does not depend on $y_{4}$ and thus the differential equation
immediately reduces to

$\partial_{c}F(c,z)=0\ ,$ (3.15)

which coincides with the corresponding unflowed null vector equation (2.43).

Since the function $F(c,z)$ seems to obey the same constraints as the unflowed
correlator with spins $j_{i}$, we are thus motivated to _identify_ $F(c,z)$
with the unflowed correlator, up to a normalisation constant. We can now state
our general conjecture that yields the flowed correlator once the unflowed
correlator is known

$\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1};0)V_{j_{2}}^{w_{2}}(1;y_{2};1)V_{j_{3}}^{w_{3}}(\infty;y_{3};\infty)V_{j_{4}}^{w_{4}}(x;y_{4};z)\right\rangle\\\
=C(w_{i};j_{i};k)|X_{\emptyset}|^{2(j_{1}+j_{2}+j_{3}+j_{4}-k)}\,|X_{12}|^{2(-j_{1}-j_{2}+j_{3}-j_{4})}|X_{13}|^{2(-j_{1}+j_{2}-j_{3}+j_{4})}\\\
\times|X_{23}|^{2(j_{1}-j_{2}-j_{3}+j_{4})}|X_{34}|^{-4j_{4}}\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}=0}(x_{i}^{\prime};z_{i}^{\prime})\right\rangle\
,$ (3.16)

where the locations of the unflowed correlator are
$x_{1}^{\prime}=z_{1}^{\prime}=0$, $x_{2}^{\prime}=z_{2}^{\prime}=1$,
$x_{3}^{\prime}=z_{3}^{\prime}=\infty$ and

$x_{4}^{\prime}=\frac{X_{23}X_{14}}{X_{12}X_{34}}\,\qquad z_{4}^{\prime}=z\ .$
(3.17)

Here, we also reinstated the dependence on the right-movers. The normalisation
constant will be discussed further below.

###  3.4 Parity odd case

One can perform a similar analysis for the parity odd case. Here the solution
is naturally written in terms of $X_{I}$ where $I$ has an odd number of
elements. The solution to the local Ward identities takes the form

$\left\langle\prod_{i=1}^{4}V_{j_{i}}^{w_{i}}(x_{i};y_{i};z_{i})\right\rangle=X_{123}^{\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4}}X_{1}^{-j_{1}+j_{2}+j_{3}+j_{4}-\frac{k}{2}}\\\
\times
X_{2}^{j_{1}-j_{2}+j_{3}+j_{4}-\frac{k}{2}}X_{3}^{j_{1}+j_{2}-j_{3}+j_{4}-\frac{k}{2}}X_{4}^{-2j_{4}}F\left(\frac{X_{2}X_{134}}{X_{123}X_{4}},z\right)\
.$ (3.18)

We have checked for $\sum_{i}w_{i}\leq 10$ that (3.18) indeed provides a
solution of the Ward identities.

One can additionally impose the KZ equation and possibly null vector equations
on this solution to further constrain the unknown function $F(c,z)$. The same
arguments as in the previous subsection show also that the remaining unknown
function $F(c,z)$ satisfies the unflowed KZ-equation with spins
$(j_{1},j_{2},\frac{k}{2}-j_{3},j_{4})$. If one of the spins is degenerate,
the resulting null vector equation also coincides with the unflowed null
vector equation for spins $(j_{1},j_{2},\frac{k}{2}-j_{3},j_{4})$.151515Note
that the null vector differential equations for spin $j_{r,s}^{\pm}$ and
$\frac{k}{2}-j_{r,s}^{\pm}=j_{r,s\mp 1}^{\mp}$ both have order $r(2s\mp 1)$,
which is a good consistency check on this assertion. Thus, we are again
motivated to identify the function $F(c,z)$ with the unflowed correlator, but
with spins $(j_{1},j_{2},\frac{k}{2}-j_{3},j_{4})$. We thus extend our
proposal for the correlation functions to the parity odd sector as follows

$\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1};0)V_{j_{2}}^{w_{2}}(1;y_{2};1)V_{j_{3}}^{w_{3}}(\infty;y_{3};\infty)V_{j_{4}}^{w_{4}}(x;y_{4};z)\right\rangle\\\
=C(w_{i};j_{i};k)|X_{123}|^{2(\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4})}|X_{1}|^{2(-j_{1}+j_{2}+j_{3}+j_{4}-\frac{k}{2})}\\\
\times|X_{2}|^{2(j_{1}-j_{2}+j_{3}+j_{4}-\frac{k}{2})}|X_{3}|^{2(j_{1}+j_{2}-j_{3}+j_{4}-\frac{k}{2})}|X_{4}|^{-4j_{4}}\\\
\times\left\langle
V_{\frac{k}{2}-j_{3}}^{w_{3}=0}(x_{3}^{\prime};z_{3}^{\prime})\prod_{i=1,2,4}V_{j_{i}}^{w_{i}=0}(x_{i}^{\prime};z_{i}^{\prime})\right\rangle\
.$ (3.19)

The insertion points are given by $x_{1}^{\prime}=z_{1}^{\prime}=0$,
$x_{2}^{\prime}=z_{2}^{\prime}=1$, $x_{3}^{\prime}=z_{3}^{\prime}=\infty$ and

$x_{4}^{\prime}=\frac{X_{2}X_{134}}{X_{123}X_{4}}\ ,\qquad z_{4}^{\prime}=z\
.$ (3.20)

Our formula looks asymmetric in the four insertion points, but as we shall
discuss below, this is actually not the case.

###  3.5 Four-point functions saturating the bound on spectral flow

As already anticipated in Section 3.2, four-point functions saturating the
bound (2.31) require particular care. It is the aim of this section to discuss
them in detail. We will refer to this case in the following as the edge case.

##### Local Ward identities.

There are several interesting things happening in this case. At a technical
level, it turns out that there is one more local Ward identity compared to the
generic case. Hence the correlator is much more constrained than generically.
Let us explain how this happens at the simple example
$\boldsymbol{w}=(0,0,0,2)$. One derives the recursion relations in this case
as discussed in Eberhardt:2019ywk . One considers an insertion of a current
$J^{a}(z)$ inside the correlator. These insertions are determined up to a
finite number of terms by holomorphicity. In this case, there is only one
unknown term – the action of the mode $J_{1}^{+}$ on the fourth field. By
imposing the various OPEs one can then derive two equations on the correlator.
One is used to eliminate the unknown and the remaining one is the local Ward
identity that determines the $y_{4}$ behaviour. In this special case, the
unknown action of $J_{1}^{+}$ on the fourth field drops actually completely
out when deriving the local Ward identities. Thus there is no need to
eliminate it and one ends up with two linearly independent constraints on the
correlator. This can be seen directly using the provided Mathematica code.

Counting variables, we therefore expect that the solution of the local Ward
identities only contains a function of a single cross-ratio, say $z=z_{4}$.
One finds that the general solution of the local Ward identities takes the
form

$\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1},0)V_{j_{2}}^{w_{2}}(1;y_{2},1)V_{j_{3}}^{w_{3}}(\infty;y_{3},\infty)V_{j_{4}}^{w_{4}}(x,y_{4};z)\right\rangle\\\
=X_{13}^{-j_{1}+j_{2}-j_{3}+j_{4}}X_{23}^{j_{1}-j_{2}-j_{3}+j_{4}}X_{12}^{2j_{3}-k}X_{34}^{j_{1}+j_{2}+j_{3}-j_{4}-k}X_{1234}^{-j_{1}-j_{2}-j_{3}-j_{4}+k}f(z)\
,$ (3.21)

for an arbitrary function $f(z)$ (that as usual can also depend on $j_{i}$,
$w_{i}$ and $k$). This form can already almost be guessed from the form of the
generic even-parity four-point function (3.16). Eq. (3.16) is no longer well-
defined in the edge case because $X_{\emptyset}=P_{\boldsymbol{w}}(x;z)=0$,
since $H_{\boldsymbol{w}}<0$, see eq. (2.13). Furthermore, the ‘generalised
crossratio’ degenerates as

$c=\frac{X_{23}X_{14}}{X_{12}X_{34}}\longrightarrow z_{4}\ .$ (3.22)

These two features can be combined to obtain still a well-defined result.
Using the general relation (that makes sense away from the edge case and is
discussed in Appendix C)

$z_{4}-c=z_{4}-\frac{X_{23}X_{14}}{X_{12}X_{34}}=\frac{X_{\emptyset}X_{1234}}{X_{12}X_{34}}\
,$ (3.23)

we see that we can set

$F(c,z)=f(z)|c-z|^{-2j_{1}-2j_{2}-2j_{3}-2j_{4}+2k}$ (3.24)

in (3.7) to cancel the $X$-dependence and obtain a result that continues to
make sense in the edge case. This then reproduces (3.21).

This then also suggests what the function $f(z)$ should be. $F(c,z)$ was
identified with the corresponding unflowed correlator and thus

$f(z)=\lim_{x\to z}|x-z|^{2j_{1}+2j_{2}+2j_{3}+2j_{4}-2k}\left\langle
V_{j_{1}}(0;0)V_{j_{2}}(1;1)V_{j_{3}}(\infty;\infty)V_{j_{4}}(x;z)\right\rangle\
,$ (3.25)

up to a possible overall constant. It is a consequence of the unflowed KZ-
equation that the unflowed correlator has precisely this type of singularity
so that the limit is well-defined, see Teschner:1997ft ; Maldacena:2001km  and
our discussion in Section 2.3.

##### Further constraints.

When computing the spectrally flowed KZ-equation, it turns out that the
unknown action of the mode $J_{1}^{+}$ on the fourth field (in the above
example $\boldsymbol{w}=(0,0,0,2)$) now enters the computation. Thus the KZ-
equation does not lead to a constraint on $f(z)$. This should not come as a
surprise in view of the identification, since the $x$-value of the unflowed
correlator is pinned to $z$. Hence also the unflowed KZ-equation does not give
any information on $f(z)$. To get a non-trivial equation on $f(z)$ we have to
assume that one of the fields is degenerate in which case there is also a null
vector equation. The two equations both involve the unknown mode $J_{1}^{+}$
on the fourth field that we were discussing. Hence this mode can be eliminated
and we obtain a new constraint on $f(z)$. This constraint coincides with the
constraint that we would find in the unflowed sector when combing the null
vector equation and the KZ-equation such that the $x$-derivatives are
eliminated. This gives further evidence for the identification (3.25).

Recall that the KZ-equation generically predicts one out of two possible
behaviours if $x$ approaches $z$. Either the correlator behaves singularly as
discussed above or it behaves regularly, see eqs. (2.26) and (2.27). We pick
out the singular behaviour in the edge case. There are however some degenerate
fields for which only one of the two channels is present. Among them is in
particular the identity field. Putting $j_{4}=0$ leads to the null vector
equation $\partial_{c}F(c,z)=0$ on the unflowed correlator, which together
with the KZ-equation leads to $F(x,z)=\text{const}$. This is obviously not
singular as $x\to z$ and hence the edge case correlator vanishes when one of
the fields is the identity. This explains why there is no non-vanishing three-
point function with e.g. $\boldsymbol{w}=(0,0,2)$ (that could be obtained from
the four point function with $\boldsymbol{w}=(0,0,2,0)$ by specifying
$j_{4}=0$).

Contrary to the generic case of eq. (3.16), the $y$-dependence of the edge
case is fully contained in its prefactor. Hence it is in principle possible to
compute the integrals over $y$-space (2.2) to obtain the $h$-dependence of the
correlators. We have not attempted to do this calculation.

##  4 Further properties and consistency checks

We have shown in the previous section that our conjecture (1.3a) and (1.3b)
follows by imposing all the symmetry constraints, up to an overall constant.
As we mentioned in the Introduction, we conjecture that the missing constant
is 1 in the parity even-case and $\mathcal{N}(j_{3})$ in the parity odd case.
In the following we will explain all further consistency checks that we
performed on our proposal that give us high confidence for its correctness.

###  4.1 Reduction to the three-point function

Let us explain the reduction to our formula for three-point functions that we
recalled in eq. (2.6). For this, we set $j_{4}=w_{4}=0$, which sets the fourth
field equal to the identity field. In this case, the unflowed four-point
function that enters the formula is just a three-point function and hence
equal to $D(j_{1},j_{2},j_{3})$ or $D(j_{1},j_{2},\frac{k}{2}-j_{3})$ for the
parity even and odd case, respectively. Here, $D(j_{1},j_{2},j_{3})$ are the
structure constants, see eq. (2.21). Next, we should explain how our
definition of $P_{\boldsymbol{w}}(x;z)$ reduces to the one for three-point
functions. For $w_{4}=0$, the Hurwitz number vanishes and so by definition
$\tilde{P}_{\boldsymbol{w}}(x;z)=1$. Hence the definition of
$P_{\boldsymbol{w}}(x;z)$ reduces to the prefactor in eq. (2.17). It is also
easy to see that the $x$\- and $z$-dependent part of the prefactor in (2.17)
are trivial and we simply have $P_{\boldsymbol{w}}(x;z)=f(\boldsymbol{w})$,
where the function $f(\boldsymbol{w})$ is given in Appendix B. For $w_{4}=0$,
this function further simplifies and can be written as in Dei:2021xgh . This
shows that the present definition of $X_{I}$ for $I\subset\\{1,2,3\\}$
coincides with the one we gave there. Finally, every $X_{I}$ for $4\in I$
drops out of the formula because $j_{4}=0$ and thus one obtains (2.6) after
using in the parity odd case the symmetry property

$\mathcal{N}(j_{1})D(\tfrac{k}{2}-j_{1},j_{2},j_{3})=\mathcal{N}(j_{2})D(j_{1},\tfrac{k}{2}-j_{2},j_{3})=\mathcal{N}(j_{3})D(j_{1},j_{2},\tfrac{k}{2}-j_{3})\
,$ (4.1)

whose generalisation to the four-point function we will discuss in Section
4.2.

###  4.2 Exchange symmetry

Let us discuss another non-trivial consistency check. Obviously, the
correlator in the $h$-basis has bosonic statistics and satisfies e.g.

$\left\langle
V_{j_{1},h_{1},\bar{h}_{1}}^{w_{1}}(x_{1};z_{1})V_{j_{2},h_{2},\bar{h}_{2}}^{w_{2}}(x_{2};z_{2})V_{j_{3},h_{3},\bar{h}_{3}}^{w_{3}}(x_{3};z_{3})V_{j_{4},h_{4},\bar{h}_{4}}^{w_{4}}(x_{4};z_{4})\right\rangle\\\
=\left\langle
V^{w_{3}}_{j_{3},h_{3},\bar{h}_{3}}(x_{3};z_{3})V^{w_{2}}_{j_{2},h_{2},\bar{h}_{2}}(x_{2};z_{2})V^{w_{1}}_{j_{1},h_{1},\bar{h}_{1}}(x_{1};z_{1})V^{w_{4}}_{j_{4},h_{4},\bar{h}_{4}}(x_{4};z_{4})\right\rangle\
,$ (4.2)

and similarly for the other fields. In our conjecture (1.3a) and (1.3b), it is
highly non-obvious that this property is satisfied, because the fields enter
in an asymmetric way into the formula. This is in particular true for the
parity odd case, where the third field enters in the unflowed correlator with
spin $\frac{k}{2}-j_{3}$, whereas the others enter with spin $j_{i}$.

Let us explain the exchange symmetry for the first and third field. Using
global Ward identities, it corresponds to the following statement in the
$y$-basis

$\displaystyle\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1};0)V_{j_{2}}^{w_{2}}(1;y_{2};1)V_{j_{3}}^{w_{3}}(\infty;y_{3};\infty)V_{j_{4}}^{w_{4}}(x;y_{4};z)\right\rangle$
$\displaystyle\qquad=|x|^{-4h_{4}^{0}}|z|^{-4\Delta_{4}^{0}}\Big{\langle}V_{j_{3}}^{w_{3}}(0;(-1)^{w_{3}-1}y_{3};0)V_{j_{2}}^{w_{2}}(1;(-1)^{w_{2}-1}y_{2};1)$
$\displaystyle\qquad\qquad\qquad\qquad
V_{j_{1}}^{w_{1}}(\infty;(-1)^{w_{1}-1}y_{1};\infty)V_{j_{4}}^{w_{4}}\left(x^{-1};(-1)^{w_{4}-1}y_{4}x_{4}^{-2}z^{2w_{4}};z^{-1}\right)\Big{\rangle}\
,$ (4.3)

where $h_{4}^{0}$ and $\Delta_{4}^{0}$ were introduced in (2.30). We checked
in Mathematica that in order for our conjecture to be consistent with this
exchange symmetry, the unflowed correlator that enters the formula has to
satisfy the following identity

$\displaystyle\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}(x;z)\right\rangle$
$\displaystyle\qquad=|x|^{-4j_{4}}|z|^{\frac{4j_{4}(j_{4}-1)}{k-2}}\left\langle
V_{j_{3}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{1}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(x^{-1};z^{-1}\right)\right\rangle\
,$ (4.4) $\displaystyle\mathcal{N}(j_{3})\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{\frac{k}{2}-j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}(x;z)\right\rangle$
$\displaystyle\qquad=\mathcal{N}(j_{1})|z|^{\frac{2j_{4}(2j_{4}-k)}{k-2}}\left\langle
V_{j_{3}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{\frac{k}{2}-j_{1}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(xz^{-1};z^{-1}\right)\right\rangle\
,$ (4.5)

in the parity even and odd case, respectively. The first identity is a direct
consequence of the global Ward identities in the unflowed sector. The second
identity is much more interesting. After combining it with the global Ward
identities, it reduces to

$\displaystyle\mathcal{N}(j_{3})\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{\frac{k}{2}-j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}(x;z)\right\rangle$
$\displaystyle\qquad=\mathcal{N}(j_{1})|x|^{-4j_{4}}|z|^{2j_{4}}\left\langle
V_{j_{3}}^{0}(\infty;\infty)V_{j_{2}}^{0}(1;1)V_{\frac{k}{2}-j_{1}}^{0}(0;0)V_{j_{4}}^{0}\left(x^{-1}z;z\right)\right\rangle$
(4.6)

This is identity (2.28), which is derived in Appendix E. This identity arises
in the unflowed sector thanks to the existence of the degenerate
representation with $j=j_{1,1}^{+}=\frac{k}{2}$ and doesn’t follow immediately
from Ward identities. We also remark that the prefactor $\mathcal{N}(j_{3})$
is necessary to obtain a symmetric answer in the spins. Exchanges of other
spins are similar to the parity even case that we discussed. They all follow
from global Ward identities in the unflowed sector, which we check directly in
the ancillary Mathematica file.

Finally, the reader might wonder why one of the spins is swapped
($j\to\frac{k}{2}-j$) in the parity odd sector and none in the parity even
sector. It turns out that one can write also alternative formulae for our
correlators where for the even-parity sector any even number of spins has been
replaced by $j\to\frac{k}{2}-j$ and for the odd-parity sector any odd number
of spins has been swapped. For each swapped spin $j_{i}\to\frac{k}{2}-j_{i}$,
one has to include a prefactor $\mathcal{N}(j_{i})$ in the formula. For
example, the identity (2.28) can be rewritten as

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\\\
=\mathcal{N}(j_{1})\mathcal{N}(j_{3})|x|^{-4j_{4}}|z|^{2j_{4}}\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}\Bigl{(}\frac{z}{x};z\Bigr{)}\right\rangle\
.$ (4.7)

Since the left-hand side enters our formula (1.3a) in the even-parity case, we
could alternatively also use the right hand side to express it in terms of the
correlator where both $j_{1}$ and $j_{3}$ have been swapped. Similarly we
could swap two other spins or all four spins. Similar reasoning applies in the
parity odd sector.

###  4.3 Four-point functions and reflection symmetry

We will now show that the reflection symmetry of the spectrally flowed
correlator for continuous representations is a consequence of properties of
the unflowed correlator that enters (1.3a) and (1.3b).

##### Parity even case.

Let us discuss the reflection symmetry first in the parity even case (i.e.
$\sum_{i}w_{i}\in 2\mathds{Z}$). We also restrict to the typical case where
$\sum_{i}w_{i}\geq 2\max_{i}w_{i}$. The edge case follows once we interpret it
correctly as a limiting case of the typical case. In this case, the four-point
function is given by eq. (3.16). To demonstrate reflection symmetry, we work
directly in the $y$-basis, where reflection symmetry acts according to (2.4).
Let us reflect $j_{4}$. This calculation implies that a similar result is true
also for $j_{1}$, $j_{2}$ and $j_{3}$ because by the exchange symmetry that we
discuss in Section 4.2, we can let any field play the role of the fourth
field.

So we want to compute the integral

$\int\mathrm{d}^{2}y_{4}\
|y-y_{4}|^{4j_{4}-4}|X_{34}|^{-4j_{4}}F\left(\frac{X_{23}X_{14}}{X_{12}X_{34}},z\right)\
,$ (4.8)

where we omitted all factors in (3.16) that do not depend on $y_{4}$ and
$F(x,z)$ denotes the unflowed four-point function. One can explicitly compute
this integral by changing variables to

$c=\frac{X_{23}X_{14}}{X_{12}X_{34}}\ .$ (4.9)

One gets

$|AD-BC|^{2-4j_{4}}|X_{34}(y)|^{4j_{4}-4}\int\mathrm{d}^{2}c\
\left|c-\frac{A+By}{C+Dy}\right|^{4j_{4}-4}F(c,z)\ ,$ (4.10)

where $X_{34}(y)$ is $X_{34}$ with $y_{4}$ replaced by $y$ and

$A+By_{4}=\frac{X_{23}X_{14}}{X_{12}}\ ,\quad C+Dy_{4}=X_{34}\ ,$ (4.11)

so that $A$, $B$, $C$ and $D$ are $y_{4}$-independent. The quadratic identity
(B.2i) yields

$\displaystyle AD-BC=\pm\frac{XX_{13}X_{23}}{X_{12}}\ .$ (4.12)

Moreover,

$\frac{A+By}{C+Dy}=\frac{X_{23}X_{14}(y)}{X_{12}X_{34}(y)}$ (4.13)

is the ‘new’ cross ratio that depends now on $y$ instead of $y_{4}$. At this
point the integral over $c$ is exactly the integral that one would compute for
the unflowed correlator to obtain the reflected correlator (2.4), see eq.
(2.3). Since the unflowed correlator is by assumption reflection symmetric,
the integral just evaluates to the reflected correlator (with reflection
coefficient). The extra factors of $X$ and $X_{ij}$ that we obtained through
the change of variables is exactly what is needed to also turn around the
$j_{4}$ in the prefactors of eq. (3.16). Thus we conclude that reflection
symmetry of the flowed correlators follows directly from the reflection
symmetry of the unflowed correlators.

##### Parity odd case.

The parity odd case is very similar. Essentially the same calculation reduces
the check of reflection symmetry for the flowed correlator to the reflection
symmetry of the unflowed correlator. In this case, one changes variables to

$c=\frac{X_{2}X_{134}}{X_{123}X_{4}}\ .$ (4.14)

and uses the quadratic identity (B.2k).

##### Edge case.

We already mentioned that the edge case (i.e. when the bound (2.31) is
saturated) can be understood as a limiting case of the parity even case. Let
us be more precise here. In the edge case, we found that the solution was
expressed in terms of the limit of the unflowed correlator (3.25)

$f_{j_{4}}(z)=\lim_{x\to z}|x-z|^{2j_{1}+2j_{2}+2j_{3}+2j_{4}-2k}\left\langle
V_{j_{1}}(0;0)V_{j_{2}}(1;1)V_{j_{3}}(\infty;\infty)V_{j_{4}}(x;z)\right\rangle\
,$ (4.15)

where we momentarily emphasised the dependence on $j_{4}$ (even though $f(z)$
also depends of course on the other spins). Using Mathematica one can check
that reflection symmetry in the flowed sector reduces to the following
identity for this limit of the unflowed correlator:

$\displaystyle
f_{1-j_{4}}(z)=R_{1-j_{4}}(1-2j_{4})f_{j_{4}}(z)\frac{\text{\textgamma}(2j_{4}-1)\text{\textgamma}(k-j_{1}-j_{2}-j_{3}-j_{4}+1)}{\text{\textgamma}(k-j_{1}-j_{2}-j_{3}+j_{4})}\
,$ (4.16)

where $R_{j}$ is the reflection coefficient (2.5) and
$\text{\textgamma}(x)=\Gamma(x)/\Gamma(1-x)$. This identity follows almost
directly from reflection symmetry of the unflowed correlator. It can be
demonstrated as follows.

$\displaystyle f_{1-j_{4}}(z)$
$\displaystyle=\frac{(1-2j_{4})R_{1-j_{4}}}{\pi}\lim_{x\to
z}|x-z|^{2j_{1}+2j_{2}+2j_{3}+2(1-j_{4})-2k}\int\mathrm{d}^{2}x^{\prime}\
|x-x^{\prime}|^{4j_{4}-4}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\left\langle
V_{j_{1}}(0;0)V_{j_{2}}(1;1)V_{j_{3}}(\infty;\infty)V_{j_{4}}(x^{\prime},z)\right\rangle$
(4.17) $\displaystyle=\frac{(1-2j_{4})R_{1-j_{4}}}{\pi}\int\mathrm{d}^{2}u\
|u|^{4j_{4}-4}\lim_{x\to z}|x-z|^{2j_{1}+2j_{2}+2j_{3}+2j_{4}-2k}$
$\displaystyle\times\left\langle
V_{j_{1}}(0;0)V_{j_{2}}(1;1)V_{j_{3}}(\infty;\infty)V_{j_{4}}(x+u(x-z),z)\right\rangle$
(4.18)

where we changed variables $x^{\prime}=x+u(x-z)$ in the integral and we
assumed that it is allowed to interchange the integral with the limit. The
limit yields now by definition
$|u+1|^{-2j_{1}-2j_{2}-2j_{3}-2j_{4}+2k}f_{j_{4}}(z)$. The remaining integral
can be evaluated and leads to the combination of -functions that appear in
(4.16). Thus, we have also checked reflection symmetry in the edge case.

###  4.4 Spectrally flowed correlators à la Fateev-Zamolodchikov-Zamolodchikov

Correlators with insertions of spectrally flowed vertex operators were first
computed in Fateev , see also Maldacena:2001km . In this section we will
compute the correlator

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{1}_{j_{4},h_{4}}(x;z)\right\rangle$
(4.19)

$\grave{\text{a}}$ la Fateev-Zamolodchikov-Zamolodchikov and show that the
result agrees with what follows from our proposal (1.3b). According to Fateev
; Maldacena:2001km , the vertex operator with $w=1$ can be defined as161616The
normalisation is usually neglected in the literature. It can be reinstated by
taking into account the various $k$-dependent contributions in
Maldacena:2001km .

$V^{1}_{j,h,\bar{h}}(x;z)\equiv\sqrt{\frac{B\left(0\right)}{B\left(\frac{k}{2}\right)}}\\\
\times\,\lim_{\epsilon\to
0}\epsilon^{h-\frac{k}{2}}\epsilon^{\bar{h}-\frac{k}{2}}\int\text{d}^{2}y\,y^{j-h+\frac{k}{2}-1}y^{j-\bar{h}+\frac{k}{2}-1}\,V^{0}_{j}(x+y,z+\epsilon)V^{0}_{\frac{k}{2}}(x;z)\
.$ (4.20)

Eq. (4.20) is understood to hold inside any correlation function. In order to
compute the correlator (4.19) one should then consider the five-point function

$\displaystyle\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle$
$\displaystyle\qquad=\frac{B\left(\frac{k}{2}\right)}{B\left(\frac{k}{2}-j_{3}\right)}|1-x_{5}|^{2(-\frac{k}{2}+j_{1}-j_{2}+j_{3}+j_{4})}\,|x_{5}|^{2(-\frac{k}{2}-j_{1}+j_{2}+j_{3}+j_{4})}\,|x_{4}-x_{5}|^{-4j_{4}}$
$\displaystyle\qquad\qquad\times\,|x_{5}-z_{5}|^{2(\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4})}|1-z_{5}|^{2j_{2}}|z_{5}|^{2j_{1}}|z_{4}-z_{5}|^{2j_{4}}$
$\displaystyle\qquad\qquad\times\,\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}\left(\frac{(x_{5}-1)(x_{5}z_{4}-x_{4}z_{5})}{(x_{5}-x_{4})(x_{5}-z_{5})};z_{4}\right)\right\rangle$
(4.21)

and set

$x_{4}=x+y\ ,\qquad z_{4}=z+\epsilon\ ,\qquad x_{5}=x\ ,\qquad z_{5}=z\ .$
(4.22)

Eq. (4.21) simply follows from injecting (B.2x) into (B.2h). After performing
the change of variable $y\to y\epsilon$ in (4.20) and taking the $\epsilon\to
0$ limit, we obtain

$\displaystyle\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{1}_{j_{4},h_{4}}(x;z)\right\rangle$
$\displaystyle\quad=\mathcal{N}(j_{3})\,|1-x|^{2(-\frac{k}{2}+j_{1}-j_{2}+j_{3}+j_{4})}\,|x|^{2(-\frac{k}{2}-j_{1}+j_{2}+j_{3}+j_{4})}$
$\displaystyle\
\quad\times\,|x-z|^{2(\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4})}\,|1-z|^{2j_{2}}\,|z|^{2j_{1}}$
$\displaystyle\
\quad\times\int\text{d}^{2}y\,y^{\frac{k}{2}-h_{4}-j_{4}-1}\bar{y}^{\frac{k}{2}-\bar{h}_{4}-j_{4}-1}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}\Bigl{(}\tfrac{(1-x)(x-yz)}{y(x-z)};z_{4}\Bigr{)}\right\rangle\
.$ (4.23)

Since

$\displaystyle X_{1}$ $\displaystyle=\frac{x}{\sqrt{z}}\ ,$ $\displaystyle
X_{2}$ $\displaystyle=\frac{1-x}{\sqrt{1-z}}\ ,$ $\displaystyle X_{3}$
$\displaystyle=1\ ,$ (4.24) $\displaystyle X_{4}$ $\displaystyle=y_{4}\ ,$
$\displaystyle X_{123}$ $\displaystyle=\frac{z-x}{\sqrt{z(1-z)}}\ ,$
$\displaystyle X_{134}$ $\displaystyle=\frac{yz-x}{\sqrt{z}}\ ,$ (4.25)

eq. (4.23) exactly reproduces what one would expect from our proposal (1.3b).

###  4.5 Coincidence limit and spectral flow violation

In this section, we make contact with previous results in the literature
through another limiting case. In the literature mostly the limit where all
$x_{i}$’s are either 0 or $\infty$ was considered. Here we analyse this limit
on our conjectured four-point functions and show that they reduce to known
correlators in the literature. As a first step, we use the solution to the
global Ward identity (2.29) to put $x_{2}$ at a generic position. We then
consider the limit

$\lim_{\genfrac{}{}{0.0pt}{}{x_{2}\to 0}{x_{4}\to 0}}\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1};0)V_{j_{2}}^{w_{2}}(x_{2};y_{2};1)V_{j_{3}}^{w_{3}}(\infty;y_{3};\infty)V_{j_{4}}^{w_{4}}(x_{4};y_{4};z_{4})\right\rangle\
.$ (4.26)

This limit is generically singular. However there are some exceptions where
one can take the limit. They occur for

$|w_{1}+w_{2}+w_{4}-w_{3}|\leq 2\ .$ (4.27)

In the cases $|w_{1}+w_{2}+w_{4}-w_{3}|\leq 1$ this is quite straightforward,
whereas the two edge cases are more subtle.

##### Spectral flow conserving correlator.

Let us start by discussing the spectral flow conserving correlator, i.e. the
correlator satisfying $w_{3}=w_{1}+w_{2}+w_{4}$. After a rescaling of the
$y_{i}$ and the change of variables $y_{3}\to-y_{3}^{-1}$, we find that the
correlator takes the following simple form in the $h$-basis (omitting the
right-moving $y$-dependence):

$\displaystyle\left\langle
V_{j_{1},h_{1}}(0;0)V_{j_{2},h_{2}}(0;1)V_{j_{3},h_{3}}(\infty;\infty)V_{j_{3},h_{3}}(0;z)\right\rangle=(-1)^{w_{1}h_{1}+w_{3}h_{2}+h_{3}+(w_{1}+w_{4})h_{4}}$
$\displaystyle\qquad\times
z^{\frac{kw_{1}w_{4}}{2}-w_{1}h_{4}-w_{4}h_{1}}(1-z)^{\frac{kw_{2}w_{4}}{2}-w_{2}h_{4}-w_{4}h_{2}}$
$\displaystyle\qquad\times\int\prod_{i=1}^{4}\mathrm{d}y_{i}\prod_{i=1,2,4}y_{i}^{\frac{kw_{i}}{2}-h_{i}+j_{i}-1}y_{3}^{\frac{kw_{3}}{2}+h_{3}+j_{3}-1}y_{12}^{-j_{1}-j_{2}+j_{3}-j_{4}}y_{13}^{-j_{1}+j_{2}-j_{3}+j_{4}}y_{23}^{j_{1}-j_{2}-j_{3}+j_{4}}$
$\displaystyle\qquad\qquad\times y_{34}^{-2j_{4}}\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(\frac{y_{32}\,y_{41}}{y_{21}\,y_{34}};z\right)\right\rangle\
.$ (4.28)

From our perspective, it is somewhat surprising that this expression depends
in a very simple way on the spectral flow — the dependence is fully contained
in the prefactor. We also notice that the $y$-dependence of the integral just
comes from the usual global Ward identities and we could write the integrand
as $\left\langle
V_{j_{1}}^{0}(y_{1};0)V_{j_{2}}^{0}(y_{2};1)V_{j_{3}}^{0}(y_{3};\infty)V_{j_{4}}^{0}(y_{4};z)\right\rangle$.
Hence it might be more appropriate to rename $y_{i}\to x_{i}$ in this context.
The integral then simply is the transform from the $x$-basis to a basis where
$J_{0}^{3}$ is diagonalised. This is exactly the same change of basis as going
from the $y$-basis to the $h$-basis. Finally we notice that we can write the
prefactor as a ratio of two free boson correlators with momenta

$m_{i}=\left(h_{i}-\frac{kw_{i}}{2}\right)\ .$ (4.29)

Then the prefactor takes the form

$\frac{z^{\frac{2m_{1}m_{4}}{k}}(1-z)^{\frac{2m_{2}m_{4}}{k}}}{z^{\frac{2h_{1}h_{4}}{k}}(1-z)^{\frac{2h_{2}h_{4}}{k}}}\
.$ (4.30)

This form can be derived by decomposing the algebra according to
$\mathfrak{sl}(2,\mathds{R})_{k}\supset\frac{\mathfrak{sl}(2,\mathds{R})_{k}}{\mathfrak{u}(1)}\times\mathfrak{u}(1)$
and using the fact that spectral flow only acts on the $\mathfrak{u}(1)$ part
Ribault:2005ms .

There is one further qualitative difference of these limiting correlators that
doesn’t occur in general. We notice that the integrand is homogeneous under
rescaling $y_{i}\to\lambda y_{i}$ with homogeneity degree

$h_{1}+h_{2}+h_{4}-h_{3}=m_{1}+m_{2}+m_{4}-m_{3}\ .$ (4.31)

Thus, the integral has to vanish except if this condition is true, which leads
to momentum conservation in the boson prefactors. This recovers
(Ribault:2005ms, , eq. (2.23)) and serves as a consistency check of our
analysis.

##### Spectral flow violation by two units.

Next we consider the maximally spectral flow violating case. Let us consider
the case $w_{1}+w_{2}+w_{4}=w_{3}-2$, which corresponds to the edge case of
the correlators discussed in Section 3.5. The case with
$w_{1}+w_{2}+w_{4}=w_{3}+2$ is again identical after the replacement
$w_{3}\to-w_{3}$ has been performed. As explained there only a limit of the
unflowed correlator enters in these correlators. As a consequence, the
integral over $y_{i}$ becomes completely explicit and beyond the prefactors
already present in (4.28) we obtain (omitting right-moving dependence)

$\displaystyle(-1)^{h_{1}+h_{2}+h_{3}}z^{h_{2}-j_{2}-\frac{kw_{2}}{2}-h_{3}-j_{3}+\frac{kw_{3}}{2}}(1-z)^{h_{1}-j_{1}-\frac{kw_{1}}{2}-h_{3}-j_{3}+\frac{kw_{3}}{2}}$
$\displaystyle\qquad\times\int\mathrm{d}y_{i}\
\prod_{i=1,2,4}y_{i}^{\frac{kw_{i}}{2}-h_{i}+j_{i}-1}y_{3}^{-\frac{kw_{3}}{2}+h_{3}+j_{i}-1}(y_{1}+y_{2}+y_{3}+y_{4})^{k-j_{1}-j_{2}-j_{3}-j_{4}}$
$\displaystyle\qquad\times\lim_{x\to
z}(x-z)^{j_{1}+j_{2}+j_{3}+j_{4}-k}\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}(x;z)\rangle\
.$ (4.32)

Of course for this limit to make sense we should rather use the non-chiral
analogue of this formula. One can in principle evaluate this formula further
because the $y$-integral is simple to compute, but we did not find it very
insightful to do so. A closed-form formula for correlators with maximal
spectral flow violation (2 in our case) in terms of Liouville correlators has
been proposed in Ribault:2005ms  and confirmed in Giribet:2011xf . One can
show the equivalence with the formula presented here as follows. We have
Ponsot:2002cp

$\lim_{x\to z}|x-z|^{2(j_{1}+j_{2}+j_{3}+j_{4}-k)}\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}(x;z)\rangle\\\
=C\,\text{\textgamma}(j_{1}+j_{2}+j_{3}+j_{4}-k)|z|^{2j_{2}+2j_{3}-k}|1-z|^{2j_{1}+2j_{3}-k}\langle
V_{\alpha_{1}}(0)V_{\alpha_{2}}(0)V_{\alpha_{3}}(0)V_{\alpha_{4}}(z)\rangle_{\text{L}}\
,$ (4.33)

where

$\displaystyle C$
$\displaystyle=-\frac{\nu^{-2-b^{-2}}}{2\pi^{2}b^{7}\text{\textgamma}(b^{2})^{2}}\
,$ (4.34) $\displaystyle\alpha_{i}$ $\displaystyle=b+\frac{1}{2b}-bj_{i}$
(4.35)

and $b^{-2}=k-2$. The correlator on the right hand side is a correlator in
Liouville theory. This can be derived as a special case from the
$H_{3}^{+}$-Liouville correspondence Ribault:2005wp . We follow the standard
conventions for Liouville theory Zamolodchikov:1995aa . The Liouville
cosmological constant $\mu_{\text{L}}$ is related to the parameter $\nu$ as
follows,

$\mu_{\text{L}}=\frac{1}{\pi\nu\,\text{\textgamma}(b^{2})}\ .$ (4.36)

Evaluating the integral above over the $y$-coordinates and assuming continuous
representations leads to

$\displaystyle\lim_{\genfrac{}{}{0.0pt}{}{x_{2}\to 0}{x_{4}\to 0}}\left\langle
V_{j_{1}}^{w_{1}}(0;y_{1};0)V_{j_{2}}^{w_{2}}(x_{2};y_{2};1)V_{j_{3}}^{w_{3}}(\infty;y_{3};\infty)V_{j_{4}}^{w_{4}}(x_{4};y_{4};z_{4})\right\rangle$
$\displaystyle\qquad=\pm
i(2\pi)^{2}C\delta^{(2)}(m_{3}-m_{1}-m_{2}-m_{4}+k)\,z^{\beta_{14}}\bar{z}^{\bar{\beta}_{14}}\,(1-z)^{\beta_{24}}(1-\bar{z})^{\bar{\beta}_{24}}$
$\displaystyle\qquad\qquad\times\prod_{i=1,2,4}\text{\textgamma}(j_{i}-m_{i})\,\text{\textgamma}(m_{3}-j_{3})\
\langle
V_{\alpha_{1}}(0)V_{\alpha_{2}}(0)V_{\alpha_{3}}(0)V_{\alpha_{4}}(z)\rangle_{\text{L}}\
,$ (4.37)

where $m_{i}=h_{i}-\frac{kw_{i}}{2}$ and

$\beta_{ij}=\frac{k}{2}(1-w_{i}w_{j})-m_{i}(w_{j}+1)-m_{j}(w_{i}+1)\ .$ (4.38)

Here, we suppressed the signs that arose as prefactors since they are
convention dependent anyway. This formula matches precisely with
Ribault:2005ms  and Giribet:2011xf .171717To match their convention one has to
replace $w_{3}$ by $-w_{3}$ and $j_{3}$ by $-j_{3}$. This is because we
inserted the third field at $x_{3}=\infty$ and used positive spectral flow,
whereas in Ribault:2005ms  and Giribet:2011xf  all fields are inserted at
$x_{i}=0$ and instead a negative amount of spectral flow is used. The two
pictures are completely equivalent. Our result also gives a value for the
unknown prefactor in their formula.

##### Spectral flow violation by one unit.

Finally, we look at the correlators that violate spectral flow by one unit. In
this case, we find the same almost trivial dependence of the correlator on
spectral flow as in the spectral flow conserving case. We get the same
prefactor in front of the integral, whereas the integral itself takes the form
(after a change of variables)

$\displaystyle\int\prod_{i=1}^{4}\mathrm{d}^{2}y_{i}\
\prod_{i=1,2,4}y_{i}^{-\frac{kw_{1}}{2}+h_{i}+j_{i}-1}y_{3}^{\frac{kw_{3}}{2}-h_{3}+j_{3}-1}(y_{1}+y_{2}+y_{3})^{\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4}}$
$\displaystyle\qquad\times\left\langle
V_{j_{1}}^{0}(0;0)V_{j_{2}}^{0}(1;1)V_{j_{3}}^{0}(\infty;\infty)V_{j_{4}}^{0}\left(\frac{y_{1}+zy_{3}-y_{4}}{y_{1}+y_{2}+y_{3}};z\right)\right\rangle\
.$ (4.39)

for $w_{1}+w_{2}+w_{4}=w_{3}+1$. The integrand for $w_{1}+w_{2}+w_{4}=w_{3}-1$
is identical, except that the replacement $w_{i}\to-w_{i}$ has to be
performed. We again note that the integrand is homogeneous in a joint
rescaling of the $y_{i}$’s. This implies again the conservation of the
$J_{0}^{3}$-eigenvalues

$h_{1}+h_{2}+h_{4}=h_{3}\ .$ (4.40)

The correlator violating spectral flow conservation by one unit has been
computed in Ribault:2005ms  in terms of correlators of Liouville theory.
Contrary to the maximally spectral flow violating case, the Liouville
correlator involves a five-point function and the relation of our formula to
the formula in terms of Liouville correlators becomes less straightforward. We
expect that the two results will agree, even though we have not tried to show
this explicitly.

##  5 Conclusions, discussion and open questions

In this paper we have studied correlators of the analytically continued
$\text{SL}(2,\mathds{R})$ WZW model and proposed a closed-form formula for
four-point functions with an arbitrary amount of spectral flow, see eq. (1).
We have shown that our proposal is consistent with all the symmetries of the
model, namely global and local Ward identities, Knizhnik-Zamolodchikov
equation, null vector equations (when applicable), reflection symmetry and
exchange symmetry.

Solving a 2D CFT requires, beyond knowledge of spectrum and structure
constants, a good control over the conformal block expansion of correlators
and a proof of crossing symmetry. Usually, formulae for four-point functions
in 2D CFTs are expressed in terms of conformal block expansions. However, in
eq. (1) our logic is different — we would first determine the unflowed
correlator through a conformal block expansion that is well-understood thanks
to the work of Teschner Teschner:1997ft . After the unflowed correlator has
been found the flowed one follows through the integral transform that we have
given. Nonetheless, it would be interesting to understand whether one can
obtain a conformal block expansion directly in the flowed sector. In fact, we
feel that the OPE of spectrally flowed vertex operators deserves a better
understanding. A related issue is crossing symmetry of the
$\text{SL}(2,\mathds{R})$ WZW model. This consistency requirement has been
shown only in the unflowed sector Teschner:2001gi  for the $H_{3}^{+}$ model.
Since eq. (1) gives the flowed four-point function in terms of the unflowed
correlator, one might be able to prove crossing symmetry in the flowed sector
by deducing it from the unflowed sector.

We saw that consistency of our construction implies a number of highly non-
obvious identities for the irreducible polynomials $P_{\boldsymbol{w}}(x;z)$
that encode the geometry of branched covers that we have collected in Appendix
C. As far as we are aware these identities are unknown in the mathematical
literature and for a further understanding a direct proof of them would be
beneficial. We also suspect that similar identities hold for the polynomials
that encode higher-point correlation functions.181818In general, there are
$n-3$ polynomials that encode the relation of the $n-3$ crossratios in $x$ and
$z$ space. However, we guessed the prefactors of the polynomials directly
which does not give us much intuition how this should generalise.

Knowing the higher-point generalisation of the polynomials
$P_{\boldsymbol{w}}(x;z)$ that we mentioned in the previous paragraph would be
a prerequisite to a generalisation of our formula to arbitrary $n$-point
functions. We suspect that a similar generalisation even exists for higher
genus correlators, where the polynomials $P_{\boldsymbol{w}}(x;z)$ should be
appropriately generalised to higher genera. Thus, we think that our formula
scratches only the tip of the iceberg and the relation between unflowed and
flowed correlators hides an intriguing mathematical structure.

As we already mentioned at various places, our study of $\text{AdS}_{3}$
correlators is motivated by holography. Despite various efforts Seiberg:1999xz
; Argurio:2000tb ; Eberhardt:2019qcl ; Dei:2019osr , the exact incarnation of
the $\text{CFT}_{2}$ dual to strings propagating on AdS3 with pure NS-NS flux
is still unclear. We believe that the findings of this paper, together with
Dei:2021xgh , furnish new important indications on the nature of the dual CFT
candidate. In particular, the correlators (1) feature an intriguing
singularity structure that we plan to explore in a future publication paper3 .
Beyond the usual singularities, due to the collision of vertex operators in
spacetime or on the worldsheet, we find a rich collection of singularities
located at the middle of the string moduli space. These are a manifestation of
the existence of worldsheet instantons as discussed in Maldacena:2001km  for
the unflowed sector. The behaviour of worldsheet correlators near these
singularities gives direct access to the string correlator and hence provides
an important peephole to learn about the nature of the dual $\text{CFT}_{2}$.

We should mention that in the regime where the string tension is minimal
($k=1$) the status of the $\text{AdS}_{3}/\text{CFT}_{2}$ duality is much
better understood than what we mentioned in the previous paragraph. In recent
years a series of publications collected strong evidence that tensionless
string theory on $\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ is
exactly dual to the symmetric product orbifold of $\mathbb{T}^{4}$
Giribet:2018ada ; Gaberdiel:2018rqv ; Eberhardt:2018ouy ; Eberhardt:2019ywk ;
Dei:2020zui ; Eberhardt:2020bgq ; Eberhardt:2020akk ; Knighton:2020kuh ;
Hikida:2020kil ; Gaberdiel:2020ycd ; Gaberdiel:2021njm . Beyond a complete
matching of the unprotected perturbative spectrum, also the structure of the
correlators has been matched. It is then natural to ask what our formula (1)
implies for the tensionless string. We plan to discuss this further in a
future publication paper3 , but we can already anticipate that by performing
the string integral and building on previous results Dei:2020zui  we exactly
reproduce symmetric orbifold correlators from the worldsheet.

###### Acknowledgements.

We would like to thank Andrea Cappelli, Matthias Gaberdiel, Gaston Giribet,
Sergio Iguri, Bob Knighton, Nicolas Kovensky, Juan Maldacena, Sylvain Ribault,
Cumrun Vafa, Edward Witten and Xi Yin for useful discussions and
correspondence. We are grateful to Sylvain Ribault for his comments on a
preliminary version of this paper. The work of A.D. is funded by the Swiss
National Science Foundation via the Early Postdoc.Mobility fellowship. LE is
supported by the IBM Einstein Fellowship at the Institute for Advanced Study.

##  Appendix A An algorithm for the polynomial
$\boldsymbol{\tilde{P}_{\boldsymbol{w}}(x;z)}$

The polynomials $\tilde{P}_{\boldsymbol{w}}(x;z)$ feature prominently in our
solution of the four-point function. In this appendix, we describe the
algorithm we used to compute them. We assume here that $w_{i}>0$ for every $i$
and $H_{\boldsymbol{w}}>0$. If this is not the case, we fix
$\tilde{P}_{\boldsymbol{w}}(x;z)$ as described in Section 2.2. We follow
closely Pakman:2009zz . The covering map can be written as

$\gamma(\zeta)=\frac{f_{1}(\zeta)}{f_{2}(\zeta)}$ (A.1)

for two polynomials $f_{1}$ and $f_{2}$ of degrees191919Recall that we put
$x_{3}=z_{3}=\infty$, so the degree of the denominator polynomial is lower.

$d_{1}=\frac{w_{1}+w_{2}+w_{3}+w_{4}}{2}-1\ ,\qquad
d_{2}=\frac{w_{1}+w_{2}-w_{3}+w_{4}}{2}-1\ .$ (A.2)

Both $f_{1}$ and $f_{2}$ satisfy Heun’s differential equation

$f^{\prime\prime}-\left(\frac{w_{1}-1}{\zeta}+\frac{w_{2}-1}{\zeta-1}+\frac{w_{4}-1}{\zeta-z}\right)f^{\prime}+\frac{d_{1}d_{2}\zeta+q}{\zeta(\zeta-1)(\zeta-z)}f=0\
.$ (A.3)

Here, $q$ is the so-called accessory parameter of Heun’s equation that is
unfixed at this point. To find the covering map, we need to determine
polynomial solutions of this equation as described in Pakman:2009zz . This
gives a recursion relation for the coefficients of the polynomial. Since $0$
is a ramification point of order $w_{1}$ of the map $\gamma$, we can take
$f_{1}(\zeta)=\zeta^{w_{1}}+\mathcal{O}(\zeta^{w_{1}+1})$. Requiring that
$f_{1}$ has indeed degree $d_{1}$ in the solution determines $q$ in terms of
$z$. The second independent solution is determined by requiring that it has a
non-vanishing constant term and is automatically a polynomial of order
$d_{2}$. Thus, this algorithm determines the covering map as a function of
$(z,q)$ and leads also to a polynomial relation $g(z,q)=0$ between $z$ and
$q$.202020This sometimes yields polynomials $g$ that are not irreducible. One
finds however that the true relation between $z$ and $q$ is always irreducible
and it can be found by checking the factors of $g(z,q)$ individually. In our
Mathematica notebook, the function zqRelPre[w_,z_,q_] computes the relation
between $z$ and $q$ using this algorithm and zqRel[w_,z_,q_] picks the correct
irreducible factor. We will in the following assume that $g(z,q)$ is the
irreducible factor.

To find $\tilde{P}_{w}(x;z)$, one only has to combine this information. $x$ is
determined to be $\gamma(z)$, which is a rational expression in $z$ and $q$.
We are thus given two polynomial conditions

$g(z,q)=0\ ,\qquad h(x,z,q)=0\ ,$ (A.4)

where the second condition comes from setting $x=\gamma(z)$ (and is linear in
$x$). All that remains to be done is to eliminate $q$ from these two
polynomial equations. Since both are non-linear in $q$, it is impracticable to
solve one equation for $q$ and insert it into the other. Instead we use some
basic commutative algebra to achieve this.

In the language of commutative algebra, $g$ and $h$ define an ideal in the
ring $\mathds{C}[x,z,q]$. We want to find a basis of this ideal where one of
the generators contains only $x$ and $z$. Such a basis is precisely given by
the Groebner basis. Thus, we simply have to find the Groebner basis and
extract the first basis element. Extra care is needed, since it can happen
that the resulting polynomial is not irreducible. Since we know that
$\tilde{P}_{\boldsymbol{w}}(x;z)$ is an irreducible polynomial, we are
guaranteed that it appears as one of the factors in this basis element. By
experimentation, we found that there is always only one irreducible factor
that contains both $x$ and $z$, which we can hence identify with
$\tilde{P}_{\boldsymbol{w}}(x;z)$. Finally, we normalise the polynomial such
that the highest power of $z$ has unit coefficient, which follows from our
definition (2.12).

This algorithm is implemented in the ancillary Mathematica file and is very
efficient at producing these polynomials. The function Ptilde[w_,x_,z_]
produces the polynomial and Ptildenorm[w_,x_,z_] normalises it correctly.
Finally, P[w_,x_,z_] inserts all the prefactors that were discussed in Section
2.2.

##  Appendix B The function $\boldsymbol{f(w)}$

The function $f(\boldsymbol{w})$, introduced in eq. (2.17) reads

$f(\boldsymbol{w})=\mathcal{S}(\boldsymbol{w})\frac{\displaystyle\prod_{p=1}^{(|w_{1}-w_{2}|-|w_{3}-w_{4}|)/2}\binom{\max(w_{3},w_{4})+p-1}{\frac{|w_{1}-w_{2}|+|w_{3}-w_{4}|}{2}}}{\displaystyle\prod_{\ell=1}^{\left||w_{1}-w_{2}|-|w_{3}-w_{4}|\right|/2-1}\binom{\frac{|w_{1}-w_{2}|+|w_{3}-w_{4}|}{2}+\ell}{\ell}}\\\
\times\,\prod_{q=1}^{(|w_{3}-w_{4}|-|w_{1}-w_{2}|)/2}\binom{\max(w_{1},w_{2})+q-1}{\frac{|w_{1}-w_{2}|+|w_{3}-w_{4}|}{2}}\
,$ (B.1)

where $\mathcal{S}(\boldsymbol{w})$ is a sign, depending on $\boldsymbol{w}$
mod $2$,

$\displaystyle\boldsymbol{w}$ $\displaystyle\sim(0,0,0,0):$
$\displaystyle(-1)^{\scaleto{\frac{s((-w_{1}+w_{2}+w_{3}-w_{4})(-w_{1}-w_{2}+w_{3}+w_{4}))}{4}+\frac{s(-w_{1}+w_{2}-w_{3}+w_{4})}{2}+\frac{s((-w_{1}-w_{2}+w_{3}+w_{4})^{2})}{8}+\frac{w_{1}+w_{2}+w_{3}+w_{4}}{4}}{10pt}}\
$ $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(0,0,1,1):$
$\displaystyle(-1)^{\scaleto{\frac{s((-w_{1}+w_{2}-w_{3}+w_{4})^{2})}{8}+\frac{s((-w_{1}+w_{2}-w_{3}+w_{4}))}{2}+\frac{w_{1}-w_{2}+w_{3}-w_{4}}{4}}{10pt}}\
,$ (B.2a) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(0,1,0,1):$
$\displaystyle(-1)^{\scaleto{-\frac{s((-w_{1}+w_{2}+w_{3}-w_{4})(-w_{1}-w_{2}+w_{3}+w_{4}))}{4}+\frac{s((w_{1}+w_{2}-w_{3}-w_{4})^{2})}{8}+\frac{-w_{1}-w_{2}-w_{3}+w_{4}}{4}}{10pt}}\
,$ (B.2b) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(0,1,1,0):$
$\displaystyle(-1)^{\scaleto{-\frac{s((-w_{1}-w_{2}+w_{3}+w_{4})^{2})}{8}+\frac{-w_{1}+w_{2}-w_{3}-w_{4}}{4}}{10pt}}\
,$ (B.2c) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(1,0,0,1):$
$\displaystyle(-1)^{\scaleto{-\frac{s((-w_{1}-w_{2}+w_{3}+w_{4})^{2})}{8}+\frac{w_{1}-w_{2}-w_{3}-w_{4}}{4}}{10pt}}\
,$ (B.2d) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(1,0,1,0):$
$\displaystyle(-1)^{\scaleto{-\frac{s((-w_{1}+w_{2}+w_{3}-w_{4})(-w_{1}-w_{2}+w_{3}+w_{4}))}{4}+\frac{s((w_{1}+w_{2}-w_{3}-w_{4})^{2})}{8}+\frac{w_{1}+w_{2}-w_{3}+w_{4}}{4}}{10pt}}\
,$ (B.2e) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(1,1,0,0:$
$\displaystyle(-1)^{\scaleto{\frac{s((-w_{1}+w_{2}-w_{3}+w_{4})^{2})}{8}+\frac{s((-w_{1}+w_{2}-w_{3}+w_{4}))}{2}+\frac{-w_{1}+w_{2}+w_{3}-w_{4}}{4}}{10pt}}\
,$ (B.2f) $\displaystyle\boldsymbol{w}$ $\displaystyle\sim(1,1,1,1):$
$\displaystyle(-1)^{\scaleto{\frac{s((-w_{1}+w_{2}+w_{3}-w_{4})(-w_{1}-w_{2}+w_{3}+w_{4}))}{4}+\frac{s(-w_{1}+w_{2}-w_{3}+w_{4})}{2}+\frac{s((-w_{1}-w_{2}+w_{3}+w_{4})^{2})}{8}-\frac{w_{1}+w_{2}-w_{3}-w_{4}}{4}}{10pt}}\
.$ (B.2g)

##  Appendix C Identities for the polynomials
$\boldsymbol{P_{\boldsymbol{w}}(x;z)}$

There is a quadratic relation between the polynomials
$P_{\boldsymbol{w}}(x;z)$ that ensures that combinations of $X_{ij}$’s behave
like cross-ratios. It reads

$\sqrt{z}P_{\boldsymbol{w}+\sum_{i=2,3}\varepsilon_{i}e_{i}}(x;z)P_{\boldsymbol{w}+\sum_{i=1,4}\varepsilon_{i}e_{i}}(x;z)-P_{\boldsymbol{w}+\sum_{i=1,2}\varepsilon_{i}e_{i}}(x;z)P_{\boldsymbol{w}+\sum_{i=3,4}\varepsilon_{i}e_{i}}(x;z)\\\
+(-1)^{w_{3}w_{4}+w_{2}w_{4}+w_{1}w_{3}+w_{1}w_{2}}\sqrt{1-z}P_{\boldsymbol{w}+\sum_{i=1,3}\varepsilon_{i}e_{i}}(x;z)P_{\boldsymbol{w}+\sum_{i=2,4}\varepsilon_{i}e_{i}}(x;z)=0\
,$ (B.2c)

where $\varepsilon_{i}\in\\{\pm 1\\}$ and $e_{i}$ is the unit vector in the
$i$-th direction. This is equivalent to the identity

$X_{32}X_{41}-X_{21}X_{34}+X_{24}X_{31}=0$ (B.2d)

that is satisfied by the ‘generalised differences’ $X_{ij}$, see (3.8) for
their definition. Another equivalent form is

$P_{\boldsymbol{w}+\sum_{i=2,3}\varepsilon_{i}e_{i}}(x;z)P_{\boldsymbol{w}+\sum_{i=1,4}\varepsilon_{i}e_{i}}(x;z)-\sqrt{z}P_{\boldsymbol{w}+\sum_{i=1,2}\varepsilon_{i}e_{i}}(x;z)P_{\boldsymbol{w}+\sum_{i=3,4}\varepsilon_{i}e_{i}}(x;z)\\\
+(-1)^{w_{3}w_{4}+w_{2}w_{4}+w_{1}w_{3}+w_{1}w_{2}}\sqrt{1-z}P_{\boldsymbol{w}}(x;z)P_{\boldsymbol{w}+\sum_{i=1,2,3,4}\varepsilon_{i}e_{i}}(x;z)=0\
,$ (B.2g)

that makes the equality

$X_{32}X_{41}-zX_{21}X_{34}+XX_{1234}=0$ (B.2h)

manifest, see (3.8) for the definition of these polynomials.

There is a second type of quadratic identity that involves no shifts in one of
the four spectral flows. We can write it as

$X_{1}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}+e_{4}}X_{3}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}-e_{4}}-X_{1}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}-e_{4}}X_{3}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}+e_{4}}=\pm
XX_{13}\ .$ (B.2i)

We haven’t tried to determine the sign in this identity since we won’t need
it. This holds provided that

$\sum_{i=1}^{4}w_{i}\in 2\mathds{Z}\quad\text{and}\quad\sum_{i=1}^{3}w_{i}\geq
1+2\max_{i=1,2,3}w_{i}\ .$ (B.2j)

This identity is required to prove reflection symmetry of the spectrally
flowed four-point function. There exist similar identities in the cases where
the roles of the indices 1, 2 and 3 in the above expression have been
permuted. There is also a corresponding identity for the other parity, which
takes the form

$X\big{|}_{\boldsymbol{w}\to\boldsymbol{w}-e_{4}}X_{13}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}+e_{4}}-X\big{|}_{\boldsymbol{w}\to\boldsymbol{w}+e_{4}}X_{13}\big{|}_{\boldsymbol{w}\to\boldsymbol{w}-e_{4}}=\pm
X_{1}X_{3}\ ,$ (B.2k)

which holds provided that

$\sum_{i=1}^{4}w_{i}\in
2\mathds{Z}+1\quad\text{and}\quad\sum_{i=1}^{3}w_{i}\geq
1+2\max_{i=1,2,3}w_{i}\ .$ (B.2l)

##  Appendix D Null vectors of $\boldsymbol{\mathfrak{sl}(2,\mathds{R})_{k}}$

In this appendix we review the construction of $\mathfrak{sl}(2,\mathds{R})$
null vectors and present examples that complement the discussion of Sections
2.7 and 3.3.

###  D.1 Null vectors in the unflowed sector

As already mentioned in the main text, $\mathfrak{sl}(2,\mathds{R})_{k}$ null
vectors appear in the unflowed sector for Kac:1979fz

$j=j_{r,s}^{+}=\frac{1+r+s(k-2)}{2}\ ,\qquad r,s\in\mathds{Z}_{>0}$ (B.2a)

or

$j=j_{r,s}^{-}=\frac{1-r-s(k-2)}{2}\ ,\qquad r\in\mathds{Z}_{>0}\ ,\quad
s\in\mathds{Z}_{\geq 0}\ .$ (B.2b)

The null vector with spin $j_{r,s}^{-}$ can be written as Malikov1984

$(J_{0}^{+})^{r+st}(J_{-1}^{-})^{r+(s-1)t}(J_{0}^{+})^{r+(s-2)t}(J_{-1}^{-})^{r+(s-3)t}\dots(J_{0}^{+})^{r-st}\ket{j,j}\
,$ (B.2c)

where $t=-k+2$. Similarly, for $j_{r,s}^{+}$ it reads

$(J_{-1}^{-})^{r+(s-1)t}(J_{0}^{+})^{r+(s-2)t}(J_{-1}^{-})^{r+(s-3)t}(J_{0}^{+})^{r+(s-4)t}\dots(J_{-1}^{-})^{r-(s-1)t}\ket{j,j}\
.$ (B.2d)

Note that for $s=0$, the general form (B.2c) reproduces the two low-lying null
vectors (2.42a) and (2.42b). Similarly, for $r=s=1$ eq. (B.2d) reproduces eq.
(2.42d). Moreover, it is straightforward to check that also
$\mathcal{N}=(J^{-}_{-1})^{2}\ket{j,j}$ with $j=j_{2,1}^{+}$ is null.

Notice that the exponents in (B.2c) and (B.2d) need not be integer. Following
Kac:1979fz ; Bauer:1993jj , let us review how by analytic continuation in the
exponents eqs. (B.2c) and (B.2d) give rise to null vectors. Let us consider
the example

$j=j^{-}_{1,1}\
,\qquad\ket{\mathcal{N}}=(J_{0}^{+})^{1+t}J_{-1}^{-}(J_{0}^{+})^{1-t}\ket{j,j}$
(B.2e)

where we have made use of (B.2c). The expression in (B.2e) should be
understood as follows. Expanding

$e^{xJ^{+}_{0}}J^{-}_{-1}e^{-xJ_{0}^{+}}=J_{-1}^{-}-2xJ^{3}_{-1}+x^{2}J^{+}_{-1}$
(B.2f)

in powers of $x$ one obtains

$(J_{0}^{+})^{p}J_{-1}^{-}=J_{-1}^{-}(J_{0}^{+})^{p}-2pJ_{-1}^{3}(J_{0}^{+})^{p-1}+p(p-1)J^{+}_{-1}(J_{0}^{+})^{p-2}\
.$ (B.2g)

Setting $p=t+1$ and analytically continuing in $t$, we find

$\displaystyle(J_{0}^{+})^{1+t}J_{-1}^{-}(J_{0}^{+})^{1-t}$
$\displaystyle\quad=\Bigl{(}J_{-1}^{-}(J_{0}^{+})^{t+1}-2(t+1)J_{-1}^{3}(J_{0}^{+})^{t}+t(t+1)J^{+}_{-1}(J_{0}^{+})^{t-1}\Bigr{)}(J_{0}^{+})^{1-t}$
(B.2h)
$\displaystyle\quad=J^{-}_{-1}(J_{0}^{+})^{2}+2(k-3)J^{3}_{-1}J^{+}_{0}+(k-2)(k-3)J^{+}_{-1}\
,$ (B.2i)

which, as expected, agrees with (2.42c). Let us consider an additional
example,

$j=j_{2,1}^{-}\
,\qquad\ket{\mathcal{N}}=(J_{0}^{+})^{2+t}(J^{-}_{-1})^{2}(J_{0}^{+})^{2-t}\ket{j,j}$
(B.2j)

With the help of Mathematica and making use of the Virasoro package developed
by Matthew Headrick Virasoro.package  one finds

$\displaystyle(J^{+}_{0})^{p}(J_{-1}^{-})^{2}=$ $\displaystyle\
(J_{-1}^{-})^{2}(J^{+}_{0})^{p}+2pJ^{-}_{-2}(J_{0}^{+})^{p-1}-4pJ^{-}_{-1}J^{3}_{-1}(J_{0}^{+})^{p-1}$
$\displaystyle-2p(p-1)J^{3}_{-2}(J_{0}^{+})^{p-2}+2p(p-1)J^{-}_{-1}J^{+}_{-1}(J_{0}^{+})^{p-2}$
$\displaystyle+4p(p-1)J^{3}_{-1}J^{3}_{-1}(J_{0}^{+})^{p-2}+2p(p-1)(p-2)J^{+}_{-2}(J_{0}^{+})^{p-3}$
$\displaystyle-4p(p-1)(p-2)J^{3}_{-1}J^{+}_{-1}(J_{0}^{+})^{p-3}$
$\displaystyle+p(p-1)(p-2)(p-3)J^{+}_{-1}J^{+}_{-1}(J_{0}^{+})^{p-4}\ .$
(B.2k)

Following the same strategy of the previous example, for $p=t+2$ we obtain

$\displaystyle\ket{\mathcal{N}}=$ $\displaystyle\
(J_{0}^{+})^{2+t}(J^{-}_{-1})^{2}(J_{0}^{+})^{2-t}\ket{j,j}$ (B.2l)
$\displaystyle=$
$\displaystyle\Biggl{(}(J_{-1}^{-})^{2}(J^{+}_{0})^{4}+2(t+2)J^{-}_{-2}(J_{0}^{+})^{3}$
$\displaystyle\quad-4(t-2)J^{-}_{-1}J^{3}_{-1}(J_{0}^{+})^{3}-2(t+1)(t+2)J^{3}_{-2}(J_{0}^{+})^{2}$
$\displaystyle\quad\quad+2(t+1)(t+2)J^{-}_{-1}J^{+}_{-1}(J_{0}^{+})^{2}+4(t+1)(t+2)J^{3}_{-1}J^{3}_{-1}(J_{0}^{+})^{2}$
$\displaystyle\quad\quad\quad+2t(t+1)(t+2)J^{+}_{-2}J_{0}^{+}-4t(t+1)(t+2)J^{3}_{-1}J^{+}_{-1}J_{0}^{+}$
$\displaystyle\hskip
175.0pt+(t-1)t(t+1)(t+2)J^{+}_{-1}J^{+}_{-1}\Biggr{)}\ket{j,j}\ .$ (B.2m)

Once more making use of Mathematica and of the Virasoro package
Virasoro.package , one can check that indeed the vector in eq. (B.2m) is null
and annihilated by $J^{-}_{0}$ for $j=j_{2,1}^{-}$ and $t=-k+2$.

From eqs. (B.2c) and (B.2d) it follows that null vectors with spin
$j=j^{\pm}_{r,s}$ appear at level $rs$. The degree of the associated
differential equation coincides with the number of currents. It is then given
by

$\displaystyle\text{ord}(j^{-}_{r,s})$
$\displaystyle=r\sum_{n=-s}^{s}1=r(2s+1)\ ,$ (B.2n)
$\displaystyle\text{ord}(j^{+}_{r,s})$
$\displaystyle=r\sum_{n=-(s-1)}^{s-1}1=r(2s-1)\ .$ (B.2o)

###  D.2 Null vector differential equations

Null vectors give rise to differential equations for correlators with one
degenerate field. Let us present a few examples that complement the discussion
of Sections 2.7 and 3.3. We discuss null vector equations both in the unflowed
Teschner:1997ft  and flowed sector and investigate their relation.

####  D.2.1 The unflowed sector

For $j_{4}=j_{2,0}^{-}=-\frac{1}{2}$, the differential equation associated to
the null vector (2.42b) is simply

$\partial_{x_{4}}^{2}\left\langle\prod_{i=1}^{3}V^{0}_{j_{i}}(x_{i};z_{i})\,V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle=0\
.$ (B.2p)

Slightly more involved is the derivation of the null equation for (2.42c).
Contour deformation arguments imply for $j_{4}=j_{1,1}^{-}=1-\frac{k}{2}$

$\sum_{i=1}^{3}\frac{1}{z_{4}-z_{i}}\Biggl{[}(x_{i}-x_{4})^{2}\,\partial_{x_{4}}^{2}\,\partial_{x_{i}}+2(x_{i}-x_{4})\,j_{i}\,\partial_{x_{4}}^{2}+2(k-3)(x_{i}-x_{4})\,\partial_{x_{4}}\,\partial_{x_{i}}\\\
+2(k-3)\,j_{i}\,\partial_{x_{4}}+(k-3)(k-2)\,\partial_{x_{i}}\Biggr{]}\left\langle\prod_{\ell=1}^{4}V^{0}_{j_{\ell}}(x_{\ell};z_{\ell})\right\rangle=0\
.$ (B.2s)

Making use of global Ward identities and choosing the insertion points as

$z_{1}=x_{1}=0\ ,\qquad z_{2}=x_{2}=1\ ,\qquad z_{3}=x_{3}=\infty\ ,$ (B.2t)

eq. (B.2s) can be rewritten as

$\displaystyle\Biggl{[}-2\left(x_{4}-1\right)x_{4}\left(x_{4}-z_{4}\right)\partial_{x_{4}}^{3}+\Bigl{(}x_{4}^{2}(-2j_{1}-2j_{2}+2j_{3}+5k-18)+$
$\displaystyle-2x_{4}(2j_{3}z_{4}+kz_{4}-6z_{4}-2j_{1}+2k-6)+z_{4}(-2j_{1}+2j_{2}+2j_{3}+k-6)\Bigr{)}\partial^{2}_{x_{4}}$
$\displaystyle-2(k-3)\Bigl{(}2x_{4}(-j_{1}-j_{2}+j_{3}+k-3)+2j_{1}-2j_{3}z_{4}-k+2z_{4}+2\Bigr{)}\partial_{x_{4}}$
$\displaystyle\hskip
65.0pt+(k^{2}-5k+6)(-2j_{1}-2j_{2}+2j_{3}+k-2)\Biggr{]}\left\langle\prod_{\ell=1}^{4}V^{0}_{j_{\ell}}(x_{\ell};z_{\ell})\right\rangle=0\
.$ (B.2u)

####  D.2.2 The flowed sector

Let us derive the null vector equation for $j_{4}=-\frac{1}{2}$ and
$\sum_{i}w_{i}\in 2\mathds{Z}$ , $w_{4}>0$. The null field is

$(h-\tfrac{kw}{2}-\tfrac{1}{2})(h-\tfrac{kw}{2}+\tfrac{1}{2})V_{-\frac{1}{2},h}^{w}(x;z)\longrightarrow
y_{4}^{2}\partial_{y_{4}}^{2}V_{-\frac{1}{2}}^{w}(x;y;z)$ (B.2v)

and in terms of the $y$-transform the differential equation reads, see eq.
(1.3a),

$\partial_{y_{4}}^{2}\Bigl{(}X_{34}\,F(c,z)\Bigr{)}=0\
,\qquad\text{with}\qquad c\equiv\frac{X_{23}X_{14}}{X_{12}X_{34}}\ .$ (B.2w)

Distributing derivatives and using the chain rule it follows

$\Bigl{(}X_{34}\,\partial^{2}_{y_{4}}c+2(\partial_{y_{4}}X_{34})\,\partial_{y_{4}}c\Bigr{)}\,\partial_{c}F(c,z)+X_{34}\,(\partial_{y_{4}}c)^{2}\,\partial_{c}^{2}F(c,z)=0\
.$ (B.2x)

Using that $c=\frac{X_{23}X_{14}}{X_{12}X_{34}}$, the first term in the
parenthesis vanishes. The differential equation obeyed by $F(c,z)$ then simply
becomes

$\partial_{c}^{2}F(c,z)=0\ ,$ (B.2y)

which coincides with the unflowed null vector equation (B.2p).

We now consider a more involved example, $j_{4}=1-\tfrac{k}{2}$ with
$\sum_{i}w_{i}\in 2\mathds{Z}$ and $w_{4}>0$. The null field is, see eq.
(2.41d)

$(k-2h+kw-2)(k-2h+kw)[J^{+}_{-2}V_{j,h-1}^{w}](x;z)\\\
+2(k-2h+kw-2)(k+2h-kw-2)[J^{3}_{-1}V_{j,h}^{w}](x;z)\\\
+(k+2h-kw-2)(k+2h-kw)[J_{0}^{-}V_{j,h+1}^{w}](x;z)\ .$ (B.2ac)

Given a polynomial $p(h)$, from (2.2) it follows

$p(h)V_{j,h,\bar{h}}^{w}(x;z)=\int\mathrm{d}^{2}y\,y^{\frac{kw}{2}+j-h-1}\,\bar{y}^{\frac{kw}{2}+j-\bar{h}-1}\,\mathcal{D}_{y}^{p}V_{j}^{w}(x;y;z)\
,$ (B.2ad)

where $\mathcal{D}_{y}^{p}$ is a differential operator in $y$. In particular,
we have

$\displaystyle V_{j,h-1,\bar{h}}^{w}(x;z)\longrightarrow y\ ,\qquad
V_{j,h,\bar{h}}^{w}(x;z)\longrightarrow 1\ ,\qquad
V_{j,h+1,\bar{h}}^{w}(x;z)\longrightarrow y^{-1}\ ,$ (B.2ae) $\displaystyle
h\,V_{j,h-1,\bar{h}}^{w}(x;z)\longrightarrow(j+\tfrac{kw}{2}+1)y+y^{2}\partial_{y}\
,$ (B.2af) $\displaystyle h\,V_{j,h,\bar{h}}^{w}(x;z)\longrightarrow
j+\tfrac{kw}{2}+y\partial_{y}\ ,$ (B.2ag) $\displaystyle
h\,V_{j,h+1,\bar{h}}^{w}(x;z)\longrightarrow(j-1+\tfrac{kw}{2})y^{-1}+\partial_{y}\
,$ (B.2ah) $\displaystyle
h^{2}V_{j,h-1,\bar{h}}^{w}(x;z)\longrightarrow(j^{2}+2j+jkw+\tfrac{k^{2}w^{2}}{4}+kw+1)y$
$\displaystyle\hskip 100.0pt+(3+2j+kw)y^{2}\partial_{y}+y^{3}\partial_{y}^{2}\
,$ (B.2ai) $\displaystyle
h^{2}V_{j,h,\bar{h}}^{w}(x;z)\longrightarrow(j^{2}+jkw+\tfrac{k^{2}w^{2}}{4})+(2j+kw+1)\partial_{y}+y^{2}\partial_{y}^{2}\
,$ (B.2aj) $\displaystyle
h^{2}V_{j,h+1,\bar{h}}^{w}(x;z)\longrightarrow\tfrac{1}{4}(2j+kw-2)^{2}y^{-1}+(2j+kw-1)\partial_{y}+y\partial_{y}^{2}\
,$ (B.2ak)

where the arrows are a shorthand for identities of the form (B.2ad). Making
use of equations (B.2ac) – (B.2ak) we obtain the differential equation

$4\,y_{4}\Bigl{(}y_{4}^{2}\partial_{y_{4}}^{2}-2(k-3)y_{4}\partial_{y_{4}}+k^{2}-5k+6\Bigr{)}\left\langle\prod_{\ell=1}^{3}V^{w_{\ell}}_{J_{\ell}}(x_{\ell};y_{\ell};z_{\ell})\Bigl{[}J^{+}_{-2}V^{w_{4}}_{1-\frac{k}{2}}\Bigr{]}(x_{4};y_{4};z_{4})\right\rangle\\\
+8\,y_{4}\Bigl{(}(k-3)\partial_{y_{4}}-y_{4}\partial_{y_{4}}^{2}\Bigr{)}\left\langle\prod_{\ell=1}^{3}V^{w_{\ell}}_{J_{\ell}}(x_{\ell};y_{\ell};z_{\ell})\Bigl{[}J^{3}_{-1}V^{w_{4}}_{1-\frac{k}{2}}\Bigr{]}(x_{4};y_{4};z_{4})\right\rangle\\\
+4\,y_{4}\,\partial_{y_{4}}^{2}\left\langle\prod_{\ell=1}^{3}V^{w_{\ell}}_{J_{\ell}}(x_{\ell};y_{\ell};z_{\ell})\Bigl{[}J^{-}_{0}V^{w_{4}}_{1-\frac{k}{2}}\Bigr{]}(x_{4};y_{4};z_{4})\right\rangle=0\
.$ (B.2ao)

We have checked extensively that upon making use of the recursion relations of
Eberhardt:2019ywk , (B.2ao) reduces to a differential equation for $F(c,z)$,
exactly reproducing (B.2u) with $F(c,z)$ in place of the unflowed correlator.
The details can be found in the ancillary Mathematica notebook.

##  Appendix E An identity for the unflowed correlator

In this appendix we derive the identity

$\mathcal{N}(j_{1})\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x;z)\right\rangle\\\
=\mathcal{N}(j_{3})|x|^{-4j_{4}}|z|^{2j_{4}}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}\Bigl{(}\frac{z}{x};z\Bigr{)}\right\rangle\
.$ (B.2c)

Eq. (B.2c) has been shown in Parnachev:2001gw . The derivation we present here
is alternative and some of the steps involved will be useful throughout the
text. Let us start by considering the five-point function

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle\
.$ (B.2d)

Making use of eq. (2.22), two independent KZ differential equations can be
derived along the lines of Teschner:1999ug . Moreover, since the spin
$j_{5}=j^{+}_{1,1}=\frac{k}{2}$ is degenerate, the correlator (B.2d) also
obeys one additional null vector equation. By solving one of the two KZ
equations and the null vector equation — see the ancillary Mathematica
notebook for the explicit computation — one finds

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle\\\
=|1-x_{5}|^{2(-\frac{k}{2}+j_{1}-j_{2}+j_{3}+j_{4})}|x_{5}|^{2(-\frac{k}{2}-j_{1}+j_{2}+j_{3}+j_{4})}|x_{4}-x_{5}|^{-4j_{4}}|x_{5}-z_{5}|^{2(\frac{k}{2}-j_{1}-j_{2}-j_{3}-j_{4})}\\\
\times|1-z_{5}|^{2j_{2}}|z_{5}|^{2j_{1}}|z_{4}-z_{5}|^{2j_{4}}f\left(\frac{(1-x_{5})(x_{5}z_{4}-x_{4}z_{5})}{(x_{4}-x_{5})(x_{5}-z_{5})},z_{4}\right)\
,$ (B.2h)

where $f(x,z)$ is an unknown function of two variables.

Let us now consider the $z_{5}\to 0$ limit of (B.2d) and fuse
$V^{0}_{\frac{k}{2}}(x_{5};z_{5})$ with $V^{0}_{j_{1}}(0;0)$. We find

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle\\\
\sim|z_{5}|^{2(\Delta(\frac{k}{2}-j_{1})-\Delta(j_{1})-\Delta(\frac{k}{2}))}\int\text{d}j\int\text{d}^{2}x\,|x_{5}|^{2(1-j_{1}-\frac{k}{2}-j)}|x|^{2(\frac{k}{2}-1-j_{1}+j)}|x_{5}-x|^{2(j_{1}-\frac{k}{2}-1+j)}\\\
\times\,D(1-j,j_{1},\tfrac{k}{2})\left\langle
V^{0}_{j}(x;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle\
,$ (B.2l)

where we used the explicit form of the OPEs given in Teschner:1997ft  and
$\Delta(j)$ is the conformal dimension on the worldsheet,

$\Delta(j)\equiv-\frac{j(j-1)}{k-2}\ .$ (B.2m)

It follows from (Dei:2021xgh, , eqs. (B.7) and (B.8c)) that

$D(1-j,j_{1},\tfrac{k}{2}+\epsilon)=-\frac{\epsilon\,B(j)}{\pi
B(0)}\,\delta(j-\tfrac{k}{2}+j_{1})\ ,$ (B.2n)

where following Teschner:1999ug  we regularised the singularity by the
replacement $j_{5}=\frac{k}{2}\to\frac{k}{2}+\epsilon$. Adopting the same
regularisation in (B.2l) and making use of (B.2n) we obtain for $z_{5}\to 0$

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle\\\
\sim-\frac{\epsilon\,B(j)}{\pi
B(0)}|z_{5}|^{2j_{1}}|x_{5}|^{2(-k+1)}\int\text{d}^{2}x|x|^{2(-2j_{1}+k-1)}|x_{5}-x|^{2(-1-\epsilon)}\\\
\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(x;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle\
.$ (B.2r)

We regulate the divergence in the integral as in Teschner:1999ug , and
formally write

$|x_{5}-x|^{2(-1-\epsilon)}=-\frac{\pi\delta^{2}(x_{5}-x)}{\epsilon}\ .$
(B.2s)

Hence, for $z_{5}\to 0$

$\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})V^{0}_{\frac{k}{2}}(x_{5};z_{5})\right\rangle\\\
\sim\frac{B(j_{1})}{B(0)}|z_{5}|^{2j_{1}}|x_{5}|^{-4j_{1}}\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(x_{5};0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle\
.$ (B.2v)

Finally, comparing with (B.2h) and setting $x_{5}=0$,

$\left\langle
V^{0}_{\frac{k}{2}-j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})$
(B.2w) =B(j1)B(0)|x4|-4j4|z4|2j4f(z4x4,z4).

By similar techniques one can consider the fusion of
$V^{0}_{\frac{k}{2}}(x_{5};z_{5})$ with $V_{j_{3}}^{0}(\infty;\infty)$ and
obtain

$f(x_{4},z_{4})=\frac{B(j_{3})}{B(0)}\left\langle
V^{0}_{j_{1}}(0;0)V^{0}_{j_{2}}(1;1)V^{0}_{\frac{k}{2}-j_{3}}(\infty;\infty)V^{0}_{j_{4}}(x_{4};z_{4})\right\rangle$
(B.2x)

Injecting (B.2x) into (E) and noticing that

$\frac{B\left(j_{1}\right)}{B\left(j_{3}\right)}=\frac{\mathcal{N}(j_{1})}{\mathcal{N}(j_{3})}$
(B.2y)

one recovers eq. (B.2c).

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