aainstitutetext: Mathematical Sciences and STAG Research Centre, University of
Southampton, Southampton SO17 1BJ, United Kingdom.bbinstitutetext: Instituto
de Astronomía y Física del Espacio (IAFE) - CONICET and Facultad de Ciencias
Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428
Buenos Aires, Argentina.ccinstitutetext: Mathematics with Computer Science
Program, Guangdong Technion - Israel Institute of Technology, 515063 Shantou,
Guangdong, People’s Republic of China.ddinstitutetext: Institut de Physique
Théorique, Université Paris Saclay, CEA, CNRS, Orme des Merisiers, 91191 Gif-
sur-Yvette CEDEX, France.

# A proof for string three-point functions in AdS3

Davide Bufalini b,c Sergio Iguri d and Nicolas Kovensky
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Correlation functions of the SL(2,$\mathbb{R}$)-WZW model involving spectrally
flowed vertex operators are notoriously difficult to compute. An explicit
integral expression for the corresponding three-point functions was recently
conjectured in Dei:2021xgh . In this paper, we provide a proof for this
conjecture. For this, we extend the methods of Iguri:2022eat  based on the so-
called SL(2,$\mathbb{R}$) series identifications, which relate vertex
operators belonging to different spectral flow sectors. We also highlight the
role of holomorphic covering maps in this context. Our results constitute an
important milestone for proving this instance of the AdS3/CFT2 holographic
duality at finite ’t Hooft coupling.

##  1 Introduction

In most instances of the AdS/CFT duality, the bulk theory is approximated by
supergravity, due to the notorious difficulty in performing stringy
computations. The case of string theory in AdS3 with pure NS fluxes is a
notable exception. The dynamics of closed strings propagating on this
background can be described at the worldsheet level in terms of the Wess-
Zumino-Witten (WZW) model built upon the universal cover of
SL$(2,\mathbb{R})$. This model is believed to be exactly solvable, hence
providing a concrete scenario in which the AdS/CFT duality might be proven at
finite ’t Hooft coupling.

The worldsheet CFT, being both Lorentzian and non-compact, enjoys a number of
unusual features. Perhaps the most important one is the non-trivial action of
the so-called spectral flow outer-automorphisms of the affine
sl(2,$\mathbb{R}$)k algebra. Spectral flow plays a central role in the
determination of the string spectrum and partition function Maldacena:2000hw ;
Maldacena:2000kv . In a semi-classical description of long string states, the
spectral flow charge $\omega$ can be thought of as a winding number around the
asymptotic boundary of AdS3. However, it is not a conserved quantity since the
associated circle becomes contractible in the AdS3 interior.

Spectral flow also introduces important complications, especially concerning
the computation of worldsheet correlation functions Maldacena:2001km . Indeed,
while being Virasoro primaries, spectrally flowed vertex operators are not
affine primaries. Their operator product expansions (OPEs) with the conserved
currents become increasingly complicated with growing $\omega$, and contain
many unknown terms. Hence, some conventional techniques of two-dimensional CFT
can not be applied directly.

The interest in correlation functions of the SL(2,$\mathbb{R}$)-WZW model is
explained by their important holographic applications, especially in the
context of superstring theory in
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$ (or K3) Giveon:1998ns ;
Kutasov:1999xu ; Giribet:2007wp ; Gaberdiel:2007vu , and for the dynamics of
light probes in black hole microstates Martinec:2017ztd ; Martinec:2018nco ;
Martinec:2019wzw ; Martinec:2020gkv ; Martinec:2022okx ; Bufalini:2021ndn ;
Bufalini:2022wyp ; Bufalini:2022wzu . Worldsheet vertex operators are
functions of the worldsheet coordinate $z$, and of an additional continuous
parameter $x$, which plays the role of the holomorphic coordinate on the
conformal boundary. Hence, $z$-integrated correlators are identified with
$n$-point functions of local operators in the holographic CFT. Although the
precise definition of the latter remains elusive, a concrete albeit
perturbative proposal was put forward recently in Eberhardt:2021vsx ;
Balthazar:2021xeh . The generators of the spacetime Virasoro algebra are in
one-to-one correspondence with those of the $\text{sl}(2,\mathbb{R})_{k}$
algebra, and the worldsheet quantum numbers determine the conformal weight $h$
of dual CFT operators Giveon:1998ns . Suppressing the anti-holomorphic
quantities, we denote the spectrally-flowed worldsheet vertex operators in the
so-called $x$-basis as $V_{jh}^{\omega}(x,z)$, where $j$ is the
SL$(2,\mathbb{R})$ spin, while $h$ is the spacetime weight.

In this paper, we consider genus-zero worldsheet correlators of spectrally
flowed vertex operators. Recently, progress in the computation of three- and
four-point functions has been made in Eberhardt:2019ywk ; Dei:2021xgh ;
Dei:2021yom , thanks to the systematic use of the constraints deriving from
global and local symmetries. In particular, the latter imply a set of
complicated linear recursion relations among correlators involving different
assignments of spectral flow charges. These are difficult to derive in
general. However, in Dei:2021xgh  it has been shown that they can be recast in
the form of partial differential equations thanks to the introduction of the
so-called ‘$y$-variable’. In this new basis, a vertex operator is now defined
as a coherent superposition of states with different spacetime weights,
denoted as $V_{j}^{\omega}(x,y,z)$. The authors of Dei:2021xgh  were able to
infer a closed-form expression for all three-point functions

$\braket{V_{j_{1}}^{\omega_{1}}(x_{1};y_{1};z_{1})V_{j_{2}}^{\omega_{2}}(x_{2};y_{2};z_{2})V_{j_{3}}^{\omega_{3}}(x_{3};y_{3};z_{3})}\,.$
(1)

This proposal can be understood intuitively from the existence, in some cases,
of a holomorphic mapping of the worldsheet to the AdS3 boundary, which has
branching points of order $\omega_{i}$ at each of the corresponding
insertions. The same authors provided a closed-form expression for four-point
functions in a follow-up work Dei:2021yom . These proposals satisfy a number
of highly non-trivial consistency checks. However, in both cases, these
expressions were derived on a case-by-case basis, and for a finite set of
sufficiently low spectral flow charges.

In this work, we provide a proof for the conjecture of Dei:2021xgh  concerning
the three-point functions (1). Our methods rely heavily on the so-called
‘series identifications’ of SL(2,$\mathbb{R}$), originally formulated in
Fateev ; Maldacena:2000hw , which constitute a set of isomorphisms among
affine modules of the $\text{sl}(2,\mathbb{R})_{k}$ algebra with adjacent
spectral flow charges. The corresponding identities at the level of $y$-basis
operators were obtained recently in Iguri:2022eat . This allows us to derive
the differential equations satisfied by (and the resulting $y$-dependence of)
all spectrally flowed three-point functions, including those that do not admit
a holomorphic cover.

The structure of the paper is as follows. In section 2, we establish our
conventions and review the derivation of the recursion relations satisfied by
the SL(2,$\mathbb{R}$) correlation functions of Eberhardt:2019ywk . We also
introduce the $y$-basis operators of Dei:2021xgh  together with their
conjecture for three-point functions. Our main results are presented in
section 3, where derive the partial differential equations satisfied by all
$y$-basis three-point functions, and show that all the corresponding solutions
are compatible with the proposal of Dei:2021xgh . In particular, generic odd
parity correlators are considered in section 3.1, while even parity
correlators are obtained in section 3.2. Edge cases and correlators with
unflowed insertions are treated in sections 3.3 and 3.4, respectively.
Finally, in section 3.5 we fix the $y$-independent normalisation factors,
following the arguments presented in Iguri:2022eat . We conclude with some
discussions and outlook for future work in section 4.

##  2 Conventions and brief review of the conjecture

Let us consider the bosonic SL(2,$\mathbb{R}$)-WZW model at level $k>3$. In
this section, we introduce the spectrum and the associated vertex operators.
We also briefly review the analysis of Eberhardt:2019ywk  leading to the
recursion relations satisfied by the correlation functions of the model, and
present the conjecture put forward in Dei:2021xgh  for spectrally flowed
three-point functions.

###  2.1 Vertex operators

We focus mainly on the holomorphic sector. The conserved currents satisfy

$J^{a}(z)J^{b}(w)\sim\frac{\eta^{ab}k/2}{(z-w)^{2}}+\frac{f^{ab}_{\phantom{ab}c}J^{c}(w)}{z-w}\,,$
(2)

with $a=+,-,3$, $\eta^{+-}=-2\eta^{33}=2$, $f^{+-}_{\phantom{+-}3}=-2$ and
$f^{3+}_{\phantom{3+}+}=-f^{3-}_{\phantom{3-}-}=1$. In the so-called
$x$-basis, vertex operators are denoted as $V^{\omega}_{jh}(x,z)$. They depend
both on the worldsheet coordinate $z$ and on $x$, the coordinate on the
boundary of AdS3 associated with the holographic CFT. Moreover, $h$ and $j$
denote the spacetime weight and the SL(2,$\mathbb{R}$) spin, respectively. The
worldsheet conformal weight is

$\Delta=-\frac{j(j-1)}{k-2}-h\omega+\frac{k}{4}\omega^{2}\,,$ (3)

while $\omega\geq 0$ is the spectral flow charge.

Two types of states must be taken into account Maldacena:2000hw . For
operators defined upon unflowed states in the continuous representations
${\cal{C}}_{j}^{\alpha}$, the relevant quantum numbers are

$j\,\in\,\frac{1}{2}+i\mathbb{R}\quad\text{and}\quad m=\alpha\pm
n\,,\quad\text{with}\quad\alpha\in[0,1)\quad\text{and}\quad
n\in\mathbb{N}_{0}\,,$ (4)

where $m$ stands for the spin projection of the corresponding unflowed state.
On the other hand, for the unflowed discrete highest/lowest-weight
representations ${\cal{D}}_{j}^{\pm}$ we have

$\frac{1}{2}<j<\frac{k-1}{2}\,\quad\text{and}\quad
m=\pm(j+n)\,,\quad\text{with}\quad j\in\mathbb{R}\quad\text{and}\quad
n\in\mathbb{N}_{0}.$ (5)

In both cases we have $h=m+\frac{k}{2}\omega$ when $\omega>0$, while $h=j$ for
$\omega=0$.

The vertex operators are defined by means of their OPEs with the currents:

$\displaystyle J^{+}(w)V^{\omega}_{jh}(x,z)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\omega+1}\frac{\left(J_{n-1}^{+}V_{jh}^{\omega}\right)(x,z)}{(w-z)^{n}}+\cdots\,,$
(6a) $\displaystyle J^{3}(w)V^{\omega}_{jh}(x,z)$ $\displaystyle=$
$\displaystyle
x\sum_{n=2}^{\omega+1}\frac{\left(J_{n-1}^{+}V_{jh}^{\omega}\right)(x,z)}{(w-z)^{n}}+\frac{\left(J_{0}^{3}V_{jh}^{\omega}\right)(x,z)}{(w-z)}+\cdots\,,$
(6b) $\displaystyle J^{-}(w)V^{\omega}_{jh}(x,z)$ $\displaystyle=$
$\displaystyle
x^{2}\sum_{n=2}^{\omega+1}\frac{\left(J_{n-1}^{+}V_{jh}^{\omega}\right)(x,z)}{(w-z)^{n}}+\frac{\left(J_{0}^{-}V_{jh}^{\omega}\right)(x,z)}{(w-z)}+\cdots\,,$
(6c)

where the ellipsis indicates higher order terms in $(w-z)$. Unflowed vertex
operators will be denoted by $V_{j}(x,z)$. The zero modes act as differential
operators in $x$,

$\displaystyle\left(J_{0}^{+}V_{jh}^{\omega}\right)(x,z)$ $\displaystyle=$
$\displaystyle\partial_{x}V_{jh}^{\omega}(x,z)\,,$ (7a)
$\displaystyle\left(J_{0}^{3}V_{jh}^{\omega}\right)(x,z)$ $\displaystyle=$
$\displaystyle(x\partial_{x}+h)V_{jh}^{\omega}(x,z)\,,$ (7b)
$\displaystyle\left(J_{0}^{-}V_{jh}^{\omega}\right)(x,z)$ $\displaystyle=$
$\displaystyle(x^{2}\partial_{x}+2hx)V_{jh}^{\omega}(x,z)\,,$ (7c)

while

$\left(J_{\pm\omega}^{\pm}V_{jh}^{\omega}\right)(x,z)=\left[h-\frac{k}{2}\omega\pm(1-j)\right]V_{j,h\pm
1}^{\omega}(x,z)\,.$ (8)

Importantly, in terms of the currents

$J^{+}(x,z)=J^{+}(z)\,,\quad J^{3}(x,z)=J^{3}(z)-xJ^{+}(z)\,,\quad
J^{-}(x,z)=J^{-}(z)-2xJ^{3}(z)+x^{2}J^{+}(z)$ (9)

we get

$\displaystyle J^{3}(x,w)V^{\omega}_{jh}(x,z)$ $\displaystyle=$
$\displaystyle\frac{h}{(w-z)}V_{jh}^{\omega}(x,z)+\cdots\,,$ (10a)
$\displaystyle J^{-}(x,w)V^{\omega}_{jh}(x,z)$ $\displaystyle=$
$\displaystyle(w-z)^{\omega-1}\left(J_{-w}^{-}V_{jh}^{\omega}\right)(x,z)+\cdots\,.$
(10b)

An alternative (equivalent) definition was given recently in Iguri:2022eat ,
based on Maldacena:2001km . This is to be understood as a point-splitting
procedure between the corresponding unflowed vertex and the so-called
generalized spectral flow operator
$V_{\frac{k}{2},\frac{k}{2}\omega}^{\omega-1}(x,z)$, and reads

$V_{jh}^{\omega}(x,z)=\lim_{\varepsilon,\bar{\varepsilon}\to
0}\varepsilon^{m\omega}\bar{\varepsilon}^{\bar{m}\omega}\int
d^{2}y\,y^{j-m-1}\bar{y}^{j-\bar{m}-1}V_{j}(x+y,z+\varepsilon)V_{\frac{k}{2},\frac{k}{2}\omega}^{\omega-1}(x,z)\,.$
(11)

We will come back to this shortly in section 2.3.

###  2.2 Recursion relations among correlators

We now discuss the correlation functions of the model. It was shown in
Eberhardt:2019ywk  that they must satisfy a set of recursion relations. Let us
briefly review how this works for three-point functions of the form

$F=\left\langle\prod_{j=1}^{3}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle.$
(12)

We define

$F_{n}^{i}=\left\langle\left(J_{n}^{+}V_{j_{i}h_{i}}^{\omega_{i}}\right)(x_{i},z_{i})\prod_{j\neq
i}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle,$ (13)

so that, in particular,

$F_{0}^{i}=\partial_{x_{i}}F\,,$ (14)

and

$F_{\omega_{i}}^{i}=\left(h_{i}-\frac{k}{2}\omega_{i}+1-j_{i}\right)\left\langle
V_{j_{i},h_{i}+1}^{\omega_{i}}(x_{i},z_{i})\prod_{j\neq
i}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle\,,$ (15)

as implied by Eqs. (7a) and (8) respectively. Note that all $F^{i}_{n}$ with
$n=1,\dots,\omega_{i}-1$ are, in principle, unknown.

By using the OPEs in Eq. (6), one finds that correlators involving a current
insertion can be expanded as

$\displaystyle\left\langle
J^{+}(z)\prod_{j=1}^{3}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\left[\frac{\partial_{x_{i}}F}{z-z_{i}}+\sum_{n=1}^{\omega_{i}}\frac{F_{n}^{i}}{(z-z_{i})^{n+1}}\right]+\cdots\,,$
(16a) $\displaystyle\left\langle
J^{3}(z)\prod_{j=1}^{3}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\left[\frac{(h_{i}+x_{i}\partial_{x_{i}})F}{z-z_{i}}+\sum_{n=1}^{\omega_{i}}\frac{x_{i}F_{n}^{i}}{(z-z_{i})^{n+1}}\right]+\cdots\,,$
(16b) $\displaystyle\left\langle
J^{-}(z)\prod_{j=1}^{3}V_{j_{j}h_{j}}^{\omega_{j}}(x_{j},z_{j})\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\left[\frac{(2h_{i}x_{i}+x_{i}^{2}\partial_{x_{i}})F}{z-z_{i}}+\sum_{n=1}^{\omega_{i}}\frac{x_{i}^{2}F_{n}^{i}}{(z-z_{i})^{n+1}}\right]+\cdots.$
(16c)

Combining these expressions, we get

$G_{j}(z)\equiv\left\langle
J^{-}(x_{j},z)\prod_{l=1}^{3}V_{j_{l}h_{l}}^{\omega_{l}}(x_{l},z_{l})\right\rangle=\sum_{i\neq
j}\left[\frac{(2h_{i}x_{ij}+x_{ij}^{2}\partial_{x_{i}})F}{z-z_{i}}+\sum_{n=1}^{\omega_{i}}\frac{x_{ij}^{2}F_{n}^{i}}{(z-z_{i})^{n+1}}\right]+\cdots$
(17)

where $x_{ij}=x_{i}-x_{j}$.

On the other hand, Eq. (10b) imposes stringent restrictions on the behavior of
$G_{j}(z)$ when $z$ is close to $z_{j}$. More precisely, in this regime, we
must have

$(z-z_{j})^{1-\omega_{j}}G_{j}(z)=\left(h_{j}-\frac{k}{2}\omega_{j}-1+j_{j}\right)\left\langle
V_{j_{j},h_{j}-1}^{\omega_{j}}(x_{j},z_{j})\prod_{i\neq
j}V_{j_{i}h_{i}}^{\omega_{i}}(x_{i},z_{i})\right\rangle+\cdots,$ (18)

where we have used (8). As it turns out, the regularity of
$(z-z_{j})^{1-\omega_{j}}G_{j}(z)$ at $z=z_{j}$ as implied by (18) for all
$j=1,2,3$ provides enough information to solve for all the unknown $F_{n}^{i}$
in terms of $F$, its $x_{i}$-derivatives, and $F_{\omega_{i}}^{i}$. Upon
inserting the resulting expressions into Eq. (18), one obtains a set of
complicated linear relations between correlators involving spectrally flowed
vertex operators with consecutive spacetime weights and their $x_{i}$
derivatives. Note that the latter are under control since the global Ward
identities uniquely fix the $x_{i}$-dependence of all such correlators.

###  2.3 The conjectured solution

The non-trivial OPEs in Eqs. (6) render the computation of correlation
functions involving spectrally flowed insertions quite complicated. It will be
useful to work with somewhat unusual linear combinations of the operators
$V_{jh}^{\omega}(x,z)$ with the same $j$ and $\omega$ but different values of
$h$. These are the $y$-basis operators $V_{j}^{\omega}(x,y,z)$ introduced in
Dei:2021xgh . In Iguri:2022eat , these were shown to be precisely the
integrands on the RHS of Eq. (11), namely

$V_{j}^{\omega}(x,y,z)\equiv\lim_{\varepsilon,\bar{\varepsilon}\to
0}|\varepsilon|^{2j\omega}V_{j}(x+y\varepsilon^{\omega},z+\varepsilon)V_{\frac{k}{2},\frac{k}{2}\omega}^{\omega-1}(x,z)\,.$
(19)

This can be understood directly from Eq. (11) above. Indeed, upon changing
variables $y\to y\varepsilon^{\omega}$ this can be re-written as

$V_{jh}^{\omega}(x,z)=\int
d^{2}y\,y^{j-m-1}\bar{y}^{j-\bar{m}-1}V_{j}^{\omega}(x,y,z)\,,$ (20)

which coincides with the so-called $y$-transform of Dei:2021xgh . More details
can be found in Dei:2021xgh ; Iguri:2022eat .

The OPEs of $y$-basis operators with the conserved currents are

$\displaystyle J^{+}(w)V^{\omega}_{j}(x,y,z)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\omega+1}\frac{\left(J_{n-1}^{+}V_{jh}^{\omega}\right)(x,y,z)}{(w-z)^{n}}+\cdots\,,$
(21a) $\displaystyle J^{3}(x,w)V^{\omega}_{j}(x,y,z)$ $\displaystyle=$
$\displaystyle\frac{y\partial_{y}+j+\frac{k}{2}\omega}{(w-z)}\,V_{j}^{\omega}(x,y,z)+\cdots\,,$
(21b) $\displaystyle J^{-}(x,w)V^{\omega}_{j}(x,y,z)$ $\displaystyle=$
$\displaystyle(w-z)^{\omega-1}\left(J_{-w}^{-}V_{j}^{\omega}\right)(x,y,z)+\cdots\,.$
(21c)

While the zero modes still act as in (7), the main motivation for using the
$y$-variable is that $J_{\pm\omega}^{\pm}$ act as differential operators in
$y$. More precisely, we have

$\displaystyle\left(J_{\omega}^{+}V_{j}^{\omega}\right)(x,y,z)$
$\displaystyle=$ $\displaystyle\partial_{y}V_{j}^{\omega}(x,y,z)\,,$ (22a)
$\displaystyle\left(J_{-\omega}^{-}V_{j}^{\omega}\right)(x,y,z)$
$\displaystyle=$
$\displaystyle(y^{2}\partial_{y}+2jy)V_{j}^{\omega}(x,y,z)\,.$ (22b)

This allows one to re-write the recursion relations derived formally in the
previous section as differential equations for correlators of the form

$F_{y}\equiv\langle
V_{j_{1}}^{\omega_{1}}(x_{1},y_{1},z_{1})V_{j_{2}}^{\omega_{2}}(x_{2},y_{2},z_{2})V_{j_{3}}^{\omega_{3}}(x_{3},y_{3},z_{3})\rangle\,.$
(23)

The $x$-basis correlators in Eq. (12) can be obtained from these by means of
(20)111The procedure is slightly different for flowed discrete and continuous
states. The former arise as residues from poles of the integrand in (20)
around $y=0$ or $y=\infty$, depending on whether the corresponding unflowed
vertex operator belongs to a lowest- or highest-weight representation. For the
flowed continuous states one must integrate over the full complex plane.. This
integration procedure can be complicated, and was only carried out explicitly
in Dei:2021xgh  for a subset of cases.

In order to discuss the structure of these differential equations and their
solutions, it will be useful to make use of the conformal invariance on the
worldsheet and boundary CFT to fix $x_{1}=z_{1}=0$, $x_{2}=z_{2}=1$ and
$x_{3}=z_{3}=\infty$, and consider

$\hat{F}_{y}=\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle\equiv\langle
V_{j_{1}}^{\omega_{1}}(0,y_{1},0)V_{j_{2}}^{\omega_{2}}(1,y_{2},1)V_{j_{3}}^{\omega_{3}}(\infty,y_{3},\infty)\rangle\,.$
(24)

The latter is related to the original correlator in Eq. (23) by

$\displaystyle\begin{aligned} &\langle
V_{j_{1}}^{\omega_{1}}(x_{1},y_{1},z_{1})V_{j_{2}}^{\omega_{2}}(x_{2},y_{2},z_{2})V_{j_{3}}^{\omega_{3}}(x_{3},y_{3},z_{3})\rangle=\frac{x_{21}^{h^{0}_{3}-h^{0}_{1}-h^{0}_{2}}x_{31}^{h^{0}_{2}-h^{0}_{1}-h^{0}_{3}}x_{32}^{h^{0}_{1}-h^{0}_{2}-h^{0}_{3}}}{z_{21}^{\Delta^{0}_{1}+\Delta^{0}_{2}-\Delta^{0}_{3}}z_{31}^{\Delta^{0}_{1}+\Delta^{0}_{3}-\Delta^{0}_{2}}z_{32}^{\Delta^{0}_{2}+\Delta^{0}_{3}-\Delta^{0}_{1}}}\times\,\\\\[4.30554pt]
&\left\langle
V_{j_{1}}^{\omega_{1}}\left(0,y_{1}\frac{x_{32}z_{21}^{\omega_{1}}z_{31}^{\omega_{1}}}{x_{21}x_{31}z_{32}^{\omega_{1}}},0\right)V_{j_{2}}^{\omega_{2}}\left(1,y_{2}\frac{x_{31}z_{21}^{\omega_{2}}z_{32}^{\omega_{2}}}{x_{21}x_{32}z_{31}^{\omega_{2}}},1\right)V_{j_{3}}^{\omega_{3}}\left(\infty,y_{3}\frac{x_{21}z_{31}^{\omega_{3}}z_{32}^{\omega_{3}}}{x_{31}x_{32}z_{21}^{\omega_{3}}},\infty\right)\right\rangle,\end{aligned}$
(25)

where

$h^{0}_{i}=j_{i}+\frac{k}{2}\omega_{i}\,,\qquad\Delta^{0}_{i}=-\frac{j_{i}(j_{i}-1)}{k-2}-j_{i}\omega_{i}-\frac{k}{4}\omega_{i}^{2}\,.$
(26)

Except for a certain subfamily of correlators which will be discussed below,
spectrally flowed $y$-basis three-point functions and their associated
differential equations were studied in Dei:2021xgh  on a case-by-case basis.
This was done for sufficiently low values of the spectral flow charges
$\omega_{i}$, thus leading the authors to conjecture a general solution for
the $y$-dependence of these correlators. The proposed expressions (not
including the right-movers, and up to an overall normalization constant to be
discussed below) read as follows. For odd parity correlators, namely when
$\omega_{1}+\omega_{2}+\omega_{3}\in 2\mathbb{Z}+1$, one has

$\displaystyle\left\langle
V^{\omega_{1}}_{j_{1}}(0,y_{1},0)\,V^{\omega_{2}}_{j_{2}}(1,y_{2},1)\,V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)\right\rangle=X_{123}^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\prod_{i=1}^{3}X_{i}^{-\frac{k}{2}+j_{1}+j_{2}+j_{3}-2j_{i}}\
,$ (27)

while for the even parity case, i.e. when $\omega_{1}+\omega_{2}+\omega_{3}\in
2\mathbb{Z}$,

$\displaystyle\left\langle
V^{\omega_{1}}_{j_{1}}(0,y_{1},0)\,V^{\omega_{2}}_{j_{2}}(1,y_{2},1)\,V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)\right\rangle=X_{\emptyset}^{j_{1}+j_{2}+j_{3}-k}\prod_{i<\ell}X_{i\ell}^{j_{1}+j_{2}+j_{3}-2j_{i}-2j_{\ell}}\
.$ (28)

Here, for any subset $I\subset\\{1,2,3\\}$,

$X_{I}(y_{1},y_{2},y_{3})\equiv\sum_{i\in I:\ \varepsilon_{i}=\pm
1}P_{\boldsymbol{\omega}+\sum_{i\in I}\varepsilon_{i}e_{i}}\prod_{i\in
I}y_{i}^{\frac{1-\varepsilon_{i}}{2}}\ .$ (29)

The spectral flow parameters are chosen as
$\boldsymbol{\omega}=(\omega_{1},\omega_{2},\omega_{3})$, while
$e_{1}=(1,0,0)$, $e_{2}=(0,1,0)$ and $e_{3}=(0,0,1)$. The numbers
$P_{\boldsymbol{\omega}}$ are defined as

$P_{\boldsymbol{\omega}}=0\qquad\text{for}\qquad\sum_{j}\omega_{j}<2\max_{i=1,2,3}\omega_{i}\quad\text{or}\quad\sum_{i}\omega_{i}\in
2\mathds{Z}+1$ (30)

and

$P_{\boldsymbol{\omega}}=S_{\boldsymbol{\omega}}\frac{G\left(\frac{-\omega_{1}+\omega_{2}+\omega_{3}}{2}+1\right)G\left(\frac{\omega_{1}-\omega_{2}+\omega_{3}}{2}+1\right)G\left(\frac{\omega_{1}+\omega_{2}-\omega_{3}}{2}+1\right)G\left(\frac{\omega_{1}+\omega_{2}+\omega_{3}}{2}+1\right)}{G(\omega_{1}+1)G(\omega_{2}+1)G(\omega_{3}+1)}\
,$ (31)

otherwise, where $G(n)$ is the Barnes G function

$G(n)=\prod_{i=1}^{n-1}\Gamma(i)$ (32)

for positive integer values, while $S_{\boldsymbol{\omega}}$ is a phase
depending on $\boldsymbol{\omega}\bmod 2$. For more details, see Dei:2021xgh .

Regarding the overall constants, their precise form was also conjectured in
Dei:2021xgh  and later proven in Iguri:2022eat . These structure constants are

$C_{\boldsymbol{\omega}}(j_{1},j_{2},j_{3})=\left\\{\begin{array}[]{cc}C(j_{1},j_{2},j_{3}),&\text{if}\quad\omega_{1}+\omega_{2}+\omega_{3}\in
2\mathbb{Z},\\\
{\cal{N}}(j_{1})C\left(\frac{k}{2}-j_{1},j_{2},j_{3}\right),&\qquad\text{if}\quad\omega_{1}+\omega_{2}+\omega_{3}\in
2\mathbb{Z}+1,\end{array}\right.$ (33)

where $C(j_{1},j_{2},j_{3})$ are the structure constants of the unflowed
three-point functions defined in terms of Barnes double Gamma functions in
Teschner:1999ug ; Maldacena:2001km . Finally, ${\cal{N}}(j_{1})$ is defined in
terms of the reflection coefficient appearing the unflowed two-point
functions, namely

${\cal{N}}(j)=\sqrt{\frac{B(j)}{B\left(\frac{k}{2}-j\right)}}\,,$ (34)

with222To be precise, $\nu$ is actually a free parameter of the model, which
essentially plays the role of the string coupling. Here we simply reproduce
the value originally advocated in Teschner:1999ug . For a related discussion,
see Dabholkar:2007ey ; Eberhardt:2021vsx . We thank A. Dei and L. Eberhardt
for pointing this out.

$B(j)=\frac{2j-1}{\pi}\frac{\Gamma[1-b^{2}(2j-1)]}{\Gamma[1+b^{2}(2j-1)]}\,\nu^{1-2j}\,,\quad\nu=\frac{\Gamma[1-b^{2}]}{\Gamma[1+b^{2}]}\,,\quad
b^{2}=(k-2)^{-1}\,.$ (35)

As shown in Dei:2021xgh , the proposal given in Eqs. (27)-(33) passes a number
of non-trivial consistency checks, including bosonic exchange symmetry and
reflection symmetry for continuous states. However, no general expression for
the $y$-basis differential equations is known. Hence, so far, this conjecture
remains to be proven.

In the remainder of the paper, we prove that this solution is indeed correct.
In doing so, we highlight the role of holomorphic covering maps. Although
these maps appear to be essential for the study of four-point functions
Dei:2021yom , we show that they also play a key role in the present context.
Furthermore, when treating cases where there is no well-defined covering map
available, we will make use of the so-called series identifications for
spectrally flowed vertex operators constructed upon states belonging to the
discrete representations of SL(2,$\mathbb{R}$) Fateev ; Iguri:2022eat .

##  3 The proof for $y$-basis three-point functions

In this section we prove the conjecture put forward in Dei:2021xgh . It was
shown in Maldacena:2001km  that all non-vanishing spectrally flowed three-
point functions in the SL(2,$\mathbb{R}$) model must satisfy the following
condition:

$\omega_{i}+\omega_{j}\geq\omega_{k}-1\qquad\forall\,\,i\neq j\neq k.$ (36)

We first consider the subfamily of odd parity correlators for which an
associated holomorphic covering map exists Eberhardt:2019ywk ; Dei:2021xgh .
We then show how to compute all remaining non-vanishing correlators with three
non-trivial spectral flow charges. This includes even parity correlators, and
also those we denote as edge correlators. The latter correspond to correlators
for which either the inequality in (36) for odd parity assignments or the
analogous inequality for even parity cases saturate. These need to be treated
with special care. Finally, we also discuss correlators with unflowed
insertions and the overall normalizations.

###  3.1 Holomorphic covering maps and odd parity correlators

For concreteness, and with no loss of generality, we take $\omega_{3}$ to be
the largest spectral flow charge, i.e. $\omega_{3}\geq\omega_{1,2}$. Let us
consider correlators satisfying

$\omega_{1}+\omega_{2}+\omega_{3}\in
2\mathbb{Z}+1\;,\quad\qquad\omega_{1}+\omega_{2}>\omega_{3}-1\;,\quad\qquad\omega_{i}\geq
1\,,\,\,\forall\,i.$ (37)

It was shown in Lunin:2000yv  that there exists a unique holomorphic covering
map $\Gamma[\omega_{1},\omega_{2},\omega_{3}](z)\equiv\Gamma(z)$ from the
worldsheet to the AdS3 boundary such that

$\Gamma(z)\sim x_{i}+a_{i}(z-z_{i})^{\omega_{i}}+\cdots\qquad\text{when}\quad
z\sim z_{i}\,,$ (38)

where the ellipsis indicates higher order terms in $(z-z_{i})$. This is a
rational function which approaches a constant $\Gamma_{\infty}$ as
$z\to\infty$. One can show that it develops $N$ single poles, counted by the
Riemann-Hurwitz formula

$N=\frac{1}{2}\left(\omega_{1}+\omega_{2}+\omega_{3}-1\right).$ (39)

The coefficients $a_{i}$ appearing in Eq. (38) take the form

$a_{i}=\left(\begin{array}[]{c}\frac{\omega_{1}+\omega_{2}+\omega_{3}-1}{2}\\\
\frac{-\omega_{1}+\omega_{2}+\omega_{3}-1}{2}\end{array}\right)\left(\begin{array}[]{c}\frac{-\omega_{1}+\omega_{2}-\omega_{3}-1}{2}\\\
\frac{\omega_{1}+\omega_{2}-\omega_{3}-1}{2}\end{array}\right)^{-1}\frac{x_{i,i+1}x_{i+2,i}}{x_{i+1,i+2}}\left(\frac{z_{i+1,i+2}}{z_{i,i+1}z_{i+2,i}}\right)^{\omega_{i}}\,,$
(40)

where the subscripts are understood mod 3. Note that the last two factors in
(40) simplify to 1 upon setting $x_{1}=z_{1}=0$, $x_{2}=z_{2}=1$ and
$x_{3}=z_{3}=\infty$. We will use the same notation $a_{i}$ for the resulting
purely numerical coefficients.

We now derive the differential equations satisfied by $y$-basis three-point
functions satisfying (37). Although the presentation here is slightly
different, this was already done in Eberhardt:2019ywk . Consider the operator
$J^{-}(\Gamma(z),z)$, where we use the notation of Eq. (10b), namely

$J^{-}(\Gamma(z),z)=J^{-}(z)-2\Gamma(z)J^{3}(z)+\Gamma^{2}(z)J^{+}(z)\,.$ (41)

In order to obtain the recursion relations for the correlators under
consideration, we compute following the contour integral:

$\oint_{z_{i}}\frac{dz}{(z-z_{i})^{\omega_{i}}}\left\langle
J^{-}(\Gamma(z),z)\,V_{j_{1}}^{\omega_{1}}(x_{1},y_{1},z_{1})V_{j_{2}}^{\omega_{2}}(x_{2},y_{2},z_{2})V_{j_{3}}^{\omega_{3}}(x_{3},y_{3},z_{3})\right\rangle\,.$
(42)

Similarly to what was done in Sec. 2.2 above, we do this in two different
ways. First, we note that, near $z_{i}$, we have

$J^{-}(\Gamma(z),z)=J^{-}(x_{i},z)-2a_{i}(z-z_{i})^{\omega_{i}}J^{3}(x_{i},z)+a_{i}^{2}(z-z_{i})^{2\omega_{i}}J^{+}(z)+\cdots.$
(43)

Hence, by using (21) we find that

$\eqref{ointJgammazi}=\left[\left(2j_{i}y_{i}+y_{i}^{2}\partial_{y_{i}}\right)-2a_{i}\left(j_{i}+\frac{k}{2}\omega_{i}+y_{i}\partial_{y_{i}}\right)+a_{i}^{2}\partial_{y_{i}}\right]F_{y}\,,$
(44)

where $F_{y}$ was defined in Eq. (23). On the other hand, using (41) together
with the OPEs in Eq. (6), one finds that

$\displaystyle\begin{aligned} &\left\langle
J^{-}(\Gamma(z),z)\,V_{j_{1}}^{\omega_{1}}(x_{1},y_{1},z_{1})V_{j_{2}}^{\omega_{2}}(x_{2},y_{2},z_{2})V_{j_{3}}^{\omega_{3}}(x_{3},y_{3},z_{3})\right\rangle=\\\
&\qquad\qquad=\sum_{j=1}^{3}\left\\{-\frac{2[\Gamma(z)-x_{j}](y_{j}\partial_{y_{j}}+j_{j}+\frac{k}{2}\omega_{j})}{z-z_{j}}F_{y}+\sum_{n=1}^{\omega_{j}}\frac{[\Gamma(z)-x_{j}]^{2}}{(z-z_{j})^{n+1}}F_{y,n}^{i}\right\\}\,,\end{aligned}$
(45)

where $F_{y,n}^{i}$ stands for the $y$-basis analogues of the $F_{n}^{i}$
defined in Eq. (13). As discussed in Eberhardt:2019ywk , the RHS of (45) is a
rational function of $z$ which, as implied by the constraint equations, has
zeros of order $\omega_{i}-1$ at all $z_{i}$. It also has double poles at the
$N$ simple poles of $\Gamma(z)$, and further goes to zero as $z^{-2}$ for
$z\to\infty$ due to the global Ward identities. This implies that it must be
proportional to the derivative of the covering map, namely

$\left\langle
J^{-}(\Gamma(z),z)\,V_{j_{1}}^{\omega_{1}}(x_{1},y_{1},z_{1})V_{j_{2}}^{\omega_{2}}(x_{2},y_{2},z_{2})V_{j_{3}}^{\omega_{3}}(x_{3},y_{3},z_{3})\right\rangle=\alpha\,\partial\Gamma(z)\,,$
(46)

where $\alpha$ must be independent of $z$. This coefficient was also computed
in Eberhardt:2019ywk . When working in the $y$-basis it takes the following
form:

$\alpha=-\frac{1}{N}\sum_{j=1}^{3}\left((y_{j}-a_{j})\partial_{y_{j}}+j_{j}+\frac{k}{2}\omega_{j}\right)F_{y}\,.$
(47)

This allows us to provide an alternative expression for the contour integral
(42). Indeed, the behavior of the covering map near the insertion points
showcased in (38) implies that

$\eqref{ointJgammazi}=-\frac{a_{i}\omega_{i}}{N}\sum_{j=1}^{3}\left((y_{j}-a_{j})\partial_{y_{j}}+j_{j}+\frac{k}{2}\omega_{j}\right)F_{y}\,.$
(48)

By combining the results in Eqs. (44) and (48), and further fixing the
insertion points as in (24), one obtains the following differential equations:

$\left\\{(y_{i}-a_{i})^{2}\partial_{y_{i}}+2j_{i}(y_{i}-a_{i})+\frac{a_{i}\omega_{i}}{N}\left[\sum_{j=1}^{3}\left((y_{j}-a_{j})\partial_{y_{j}}+j_{j}\right)-\frac{k}{2}\right]\right\\}\hat{F}_{y}=0,$
(49)

for $i=1,2,3$. This was derived originally in this form in Dei:2021xgh 333Note
that we have corrected a couple of typos in their presentation.. In this way,
the use of the covering map and its derivative allows one to avoid dealing
with the cumbersome unknowns discussed in Sec. 2.2.

The system of equations encoded in (49) uniquely fixes the dependence of the
corresponding correlators on $y_{1}$, $y_{2}$ and $y_{3}$. Up to some overall
normalization, which will be discussed in section 3.5 below, the solution of
(49) is

$\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
$\displaystyle=$
$\displaystyle(y_{1}-a_{1})^{j_{2}+j_{3}-j_{1}-\frac{k}{2}}(y_{2}-a_{2})^{j_{3}+j_{1}-j_{2}-\frac{k}{2}}(y_{3}-a_{3})^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}$
(50) $\displaystyle\quad\times\left[\sum_{\varepsilon_{1,2,3}=\pm
1}2\frac{N_{1}^{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}}}{N}\prod_{i=1}^{3}a_{i}^{\frac{\varepsilon_{i}+1}{2}}y_{i}^{\frac{\varepsilon_{i}-1}{2}}\right]^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}.$

An equivalent, perhaps simpler expression for the same solution is given by

$\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
$\displaystyle=(y_{1}-a_{1})^{-2j_{1}}(y_{2}-a_{2})^{-2j_{2}}(y_{3}-a_{3})^{-2j_{3}}$
(51)
$\displaystyle\quad\times\left(\omega_{1}\frac{y_{1}+a_{1}}{y_{1}-a_{1}}+\omega_{2}\frac{y_{2}+a_{2}}{y_{2}-a_{2}}+\omega_{3}\frac{y_{3}+a_{3}}{y_{3}-a_{3}}-1\right)^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}.$

Note that this formulation makes the bosonic exchange symmetry manifest.

At first sight, this expression might seem very different from that in Eq.
(27). The connection comes from the fact that while the
$P_{\boldsymbol{\omega}}$ defined in (31) are somewhat complicated, their
ratios are actually much simpler. For instance, consider the $X_{i}$ term
appearing in (27). From Eqs. (29) and (31), up to an overall sign, we find
that

$X_{i}=P_{\boldsymbol{\omega}-e_{i}}y_{i}+P_{\boldsymbol{\omega}+e_{i}}=P_{\boldsymbol{\omega}-e_{i}}\left(y_{i}+\frac{P_{\boldsymbol{\omega}+e_{i}}}{P_{\boldsymbol{\omega}-e_{i}}}\right)=P_{\boldsymbol{\omega}-e_{i}}\left(y_{i}-a_{i}\right)\,.$
(52)

A similar result holds for $X_{123}$, showing that the $y$-dependence of the
expression in (51) is consistent with that of Eq. (27).

As shown in Eq. (51), $y$-basis three-point functions diverge whenever a
variable $y_{i}$ approaches the corresponding coefficient $a_{i}$. Thus, the
$a_{i}$ are very special points in the $y$-plane, which signal the existence
of an appropriate holomorphic covering map. An even more extreme situation
takes place in the tensionless limit, which corresponds to $k=3$ in the
bosonic language Giribet:2018ada ; Gaberdiel:2018rqv ; Eberhardt:2018ouy ;
Eberhardt:2019ywk . There, spectrally flowed correlators are non-vanishing
only when $y_{i}=a_{i}$ for all $i$. The corresponding recursion relations
(49) are then satisfied provided444In the supersymmetric case, the RNS
formalism breaks down for the tensionless theory. It was shown in
Eberhardt:2018ouy , using the so-called hybrid formalism, that in this case
only vertex operators with $j=\frac{1}{2}$ are allowed.
$j_{1}+j_{2}+j_{3}=\frac{3}{2}$. It would be interesting to fully understand
the relation between $y$-variables and covering map coefficients in the
general case.

We end this section by noting that the same discussion can not be applied
directly to odd parity correlators saturating the inequality in Eq. (36), i.e.
those with $\omega_{3}=\omega_{1}+\omega_{2}+1$. These will be discussed in
detail in section 3.3 below. Correlators with unflowed insertions are further
considered in section 3.4.

###  3.2 Series identification and even parity correlators

We now prove that the conjecture of Dei:2021xgh  also holds for correlators
satisfying

$\omega_{1}+\omega_{2}+\omega_{3}\,\in
2\mathbb{Z}\;,\quad\qquad\omega_{1}+\omega_{2}>\omega_{3}\;,\quad\qquad\omega_{i}\geq
1\,,\,\,\forall\,i.$ (53)

These include all non-vanishing spectrally flowed three-point functions for
which the total spectral flow charge is even, except for the edge cases where
$\omega_{3}=\omega_{1}+\omega_{2}$ and/or some of the vertex operators are
unflowed, which will be treated separately.

When the $\omega_{i}$ are as in (53), it is not possible to construct a
covering map such as the one used in the previous section. Thus, one might
wonder if differential equations similar to those in (49) can be deduced in
this context. Indeed, the corresponding recursion relations have only been
obtained on a case-by-case basis and for sufficiently low values of the
spectral flow charges Dei:2021xgh .

Nevertheless, we observe that the procedure outlined in Sec. 2.2 guarantees
that, provided the system is compatible, and once the $F_{n}^{i}$ have been
solved for, the resulting $y$-basis recursions must take the form

$\left[y_{i}(y_{i}\partial_{y_{i}}+2j_{i})+\sum_{j=1}^{3}(A_{ij}y_{j}-B_{ij})\partial_{y_{j}}+C_{i}\right]\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=0\,,$
(54)

where $A_{ij}$, $B_{ij}$ and $C_{i}$ are some numerical constants to be
determined, which depend on the spins $j_{i}$ and the charges $\omega_{i}$.
The rationale behind the structure of Eq. (54), which is the $y$-basis version
of Eq. (18), goes as follows. First, note that, upon using Eq. (22b), the
operator $y_{i}(y_{i}\partial_{y_{i}}+2j_{i})$ is identified with the RHS of
(18). Second, recall that the recursion relations were derived by expressing
the OPEs of the vertex operators with the conserved currents in terms of the
unknowns coming from the action of the modes of $J^{+}(z)$, see Eq. (17). This
implies that the term involving the action of $J^{-}(x,z)$ does not mix with
the rest. By using the Möbius-fixed expression for the three-point function in
Eq. (25), one can see that the terms in the recursion relations involving
unknowns and $x$-derivatives of the correlator are mapped to operators of the
schematic form $y\partial_{y}$ and $\partial_{y}$, as well as $y$-independent
multiplicative factors. In Eq. (54) we have allowed for generic coefficients
$A_{ij}$, $B_{ij}$ and $C_{i}$ in front of the corresponding contributions.

Moreover, we note that the way in which these equations are derived only
depends on the values of the spectral flow charges involved in a given
correlator. In other words, for a given set of $\omega_{i}$, the recursion
relations are independent of whether the vertex operators involved belong to
spectrally flowed discrete or continuous representations. These two
observations will allow us to obtain all $y$-basis differential equations
associated with even parity correlators in closed form.

As it turns out, even and odd parity cases are not completely disconnected.
For the discrete representations, affine modules in spectral flow sectors with
one unit of difference in the spectral flow charge are identifiable. For
$y$-basis operators the corresponding series identifications read
Iguri:2022eat

$V_{j}^{\omega}(x,y=0,z)={\cal{N}}(j)\lim_{y\to\infty}y^{k-2j}V_{\frac{k}{2}-j}^{\omega+1}(x,y,z),$
(55)

and

$\lim_{y\to\infty}y^{2j}V_{j}^{\omega}(x,y,z)={\cal{N}}(j)V_{\frac{k}{2}-j}^{\omega-1}(x,y=0,z),$
(56)

where ${\cal{N}}(j)$ was defined in Eq. (34). It was shown recently in
Iguri:2022eat  that, assuming the $y$-dependence proposed in Dei:2021xgh  for
all three-point functions, these relations fix the $y$-basis structure
constants in terms of the unflowed ones, which can be found in
Maldacena:2001km . Here we show that Eqs. (55) and (56) are actually much more
powerful: they allow us to fix the $y$-dependence as well. More explicitly, we
use them to derive all unknown coefficients $A_{ij}$, $B_{ij}$, and $C_{i}$
appearing in (54).

By means of Eqs. (55) and (56), we find that all even parity correlators can
be related to (at least) three different situations where a covering map
satisfying (38) and (40) exists. Explicitly, given
($\omega_{1},\omega_{2},\omega_{3}$) satisfying (53), it follows that the
adjacent assignments $(\omega_{1}+1,\omega_{2},\omega_{3})$,
$(\omega_{1},\omega_{2}+1,\omega_{3})$ and
$(\omega_{1},\omega_{2},\omega_{3}-1)$ satisfy all conditions in (37). Let us
denote the corresponding covering maps as follows:

$\Gamma_{1}^{+}\equiv\Gamma[\omega_{1}+1,\omega_{2},\omega_{3}]\;,\quad\qquad\Gamma_{2}^{+}\equiv\Gamma[\omega_{1},\omega_{2}+1,\omega_{3}]\;,\quad\qquad\Gamma_{3}^{-}\equiv\Gamma[\omega_{1},\omega_{2},\omega_{3}-1]\,.$
(57)

Then, the relations (55) and (56) provide the following set of identities:

$\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(0)V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=\lim_{y_{1}\to\infty}y_{1}^{k-2j_{1}}{\cal{N}}(j_{1})\langle
V_{\frac{k}{2}-j_{1}}^{\omega_{1}+1}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle,$
(58a) $\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(0)V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=\lim_{y_{2}\to\infty}y_{2}^{k-2j_{2}}{\cal{N}}(j_{2})\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{\frac{k}{2}-j_{2}}^{\omega_{2}+1}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle,$
(58b) $\displaystyle\lim_{y_{3}\to\infty}y_{3}^{2j_{3}}\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle={\cal{N}}(j_{3})\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{\frac{k}{2}-j_{3}}^{\omega_{3}-1}(0)\rangle.$
(58c)

Having excluded the even edge cases, the same holds for the adjacent
assignments $(\omega_{1}-1,\omega_{2},\omega_{3})$,
$(\omega_{1},\omega_{2}-1,\omega_{3})$ and
$(\omega_{1},\omega_{2},\omega_{3}+1)$, the corresponding maps being

$\Gamma_{1}^{-}\equiv\Gamma[\omega_{1}-1,\omega_{2},\omega_{3}]\;,\quad\qquad\Gamma_{2}^{-}\equiv\Gamma[\omega_{1},\omega_{2}-1,\omega_{3}]\;,\quad\qquad\Gamma_{3}^{+}\equiv\Gamma[\omega_{1},\omega_{2},\omega_{3}+1]\,.$
(59)

These lead to

$\displaystyle\lim_{y_{1}\to\infty}y_{1}^{2j_{1}}\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle={\cal{N}}(j_{1})\langle
V_{\frac{k}{2}-j_{1}}^{\omega_{1}-1}(0)V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
(60a) $\displaystyle\lim_{y_{2}\to\infty}y_{2}^{2j_{2}}\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle={\cal{N}}(j_{2})\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{\frac{k}{2}-j_{2}}^{\omega_{2}-1}(0)V_{j_{3}}^{\omega_{3}}(y_{3})\rangle,$
(60b) $\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(0)\rangle=\lim_{y_{3}\to\infty}y_{3}^{k-2j_{3}}{\cal{N}}(j_{3})\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{\frac{k}{2}-j_{3}}^{\omega_{3}+1}(y_{3})\rangle.$
(60c)

All expressions on the RHS of Eqs. (58) and (60) are limits of correlators
discussed in the previous section. Hence, they must satisfy the appropriate
limits of the differential equations given in (49). For instance, $\langle
V_{\frac{k}{2}-j_{1}}^{\omega_{1}-1}(0)V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
is annihilated by the differential operators

$\displaystyle\begin{aligned}
&y_{2}(y_{2}\partial_{y_{2}}+2j_{2})+(\omega_{1}-\omega_{2}-\omega_{3})^{-1}\big{\\{}(\omega_{1}+\omega_{2}-\omega_{3})a_{2}[\Gamma_{1}^{-}]^{2}\partial_{y_{2}}\\\
&-2a_{2}[\Gamma_{1}^{-}]\left[(\omega_{1}-\omega_{3})(j_{2}+y_{2}\partial_{y_{2}})+\omega_{2}(j_{3}+y_{3}\partial_{y_{3}}-a_{3}[\Gamma_{1}^{-}]\partial_{y_{3}}+j_{1}+j_{3})\right]\big{\\}}\end{aligned}$
(61)

and

$\displaystyle\begin{aligned}
&y_{3}(y_{3}\partial_{y_{3}}+2j_{3})+(\omega_{1}-\omega_{2}-\omega_{3})^{-1}\big{\\{}(\omega_{1}-\omega_{2}+\omega_{3})a_{3}[\Gamma_{1}^{-}]^{2}\partial_{y_{3}}\\\
&-2a_{3}[\Gamma_{1}^{-}]\left[(\omega_{1}-\omega_{2})(j_{3}+y_{3}\partial_{y_{3}})+\omega_{3}(j_{2}+y_{2}\partial_{y_{2}}-a_{2}[\Gamma_{1}^{-}]\partial_{y_{2}}+j_{1}+j_{2})\right]\big{\\}},\end{aligned}$
(62)

where $a_{i}[\Gamma_{1}^{-}]$ denotes the coefficient $a_{i}$ associated with
the map $\Gamma_{1}^{-}$. One can obtain analogous equations from the other
odd parity correlators involved in (58). This leads to twelve differential
operators, which must coincide with the appropriate limits of those provided
in Eqs. (54). As an example, in the situation considered above we should match
(61) and (62) with

$\displaystyle\begin{aligned}
y_{2}(y_{2}\partial_{y_{2}}+2j_{2})+A_{22}y_{2}\partial_{y_{2}}+A_{23}y_{3}\partial_{y_{3}}-B_{22}\partial_{y_{2}}-B_{23}\partial_{y_{3}}-2A_{21}j_{1}+C_{2},\\\
y_{3}(y_{3}\partial_{y_{3}}+2j_{3})+A_{33}y_{3}\partial_{y_{3}}+A_{32}y_{2}\partial_{y_{2}}-B_{33}\partial_{y_{3}}-B_{32}\partial_{y_{2}}-2A_{31}j_{1}+C_{3}.\end{aligned}$
(63)

After identifying all linearly independent terms in these 12 equations we find
a total of 60 conditions. In the end, 21 of these 60 conditions can be used to
solve explicitly for all the coefficients $A_{ij}$, $B_{ij}$ and $C_{i}$ in
(54). Consistency of the system demands that the remaining 39 conditions must
hold. We emphasize that the fact that these are identically satisfied is a
highly non-trivial check of the logic behind our proof, and also a remarkable
consequence of the identities relating the $a_{i}$ coefficients of the
different covering maps involved.

There are many equivalent ways to write the resulting coefficients. We find
that the simplest one is as follows:

$A=\left(\begin{array}[]{ccc}\frac{2(\omega_{3}-\omega_{2})}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]&\frac{2\omega_{1}}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]&\frac{2\omega_{1}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{1}[\Gamma_{2}^{-}]\\\
\frac{2\omega_{2}}{\omega_{1}+\omega_{2}-\omega_{3}}a_{2}[\Gamma_{3}^{-}]&\frac{2(\omega_{1}-\omega_{3})}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{2}[\Gamma_{1}^{-}]&\frac{2\omega_{2}}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{2}[\Gamma_{1}^{-}]\\\
\frac{2\omega_{3}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{3}[\Gamma_{2}^{-}]&\frac{2\omega_{3}}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{3}[\Gamma_{1}^{-}]&\frac{2(\omega_{1}-\omega_{2})}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{3}[\Gamma_{1}^{-}]\end{array}\right),$
(64)
$B=\left(\begin{array}[]{ccc}\frac{(\omega_{1}-\omega_{2}+\omega_{2})}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]^{2}&\frac{2\omega_{1}}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]a_{2}[\Gamma_{3}^{-}]&\frac{2\omega_{1}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{1}[\Gamma_{2}^{-}]a_{3}[\Gamma_{2}^{-}]\\\
\frac{2\omega_{2}}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]a_{2}[\Gamma_{3}^{-}]&\frac{(\omega_{1}+\omega_{2}-w_{2})}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{2}[\Gamma_{1}^{-}]^{2}&\frac{2\omega_{2}}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{2}[\Gamma_{1}^{-}]a_{3}[\Gamma_{1}^{-}]\\\
\frac{2\omega_{3}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{1}[\Gamma_{2}^{-}]a_{3}[\Gamma_{2}^{-}]&\frac{2\omega_{3}}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{2}[\Gamma_{1}^{-}]a_{3}[\Gamma_{1}^{-}]&\frac{(\omega_{1}-\omega_{2}+w_{2})}{-\omega_{1}+\omega_{2}+\omega_{3}}a_{3}[\Gamma_{1}^{-}]^{2}\end{array}\right),$
(65)

and

$C=\left(\begin{array}[]{c}\frac{4\omega_{1}j_{3}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{1}[\Gamma_{2}^{-}]+\frac{2\omega_{1}(j_{2}+j_{3})+2j_{1}(\omega_{3}-\omega_{2})}{\omega_{1}+\omega_{2}-\omega_{3}}a_{1}[\Gamma_{3}^{-}]\\\
\frac{4\omega_{2}j_{1}}{\omega_{1}+\omega_{2}-\omega_{3}}a_{2}[\Gamma_{3}^{-}]-\frac{2\omega_{2}(j_{1}+j_{3})+2j_{2}(\omega_{1}-\omega_{3})}{\omega_{1}-\omega_{2}-\omega_{3}}a_{2}[\Gamma_{1}^{-}]\\\
\frac{4\omega_{3}j_{1}}{\omega_{1}-\omega_{2}+\omega_{3}}a_{3}[\Gamma_{2}^{-}]-\frac{2\omega_{3}(j_{1}+j_{2})+2j_{3}(\omega_{1}-\omega_{2})}{\omega_{1}+\omega_{2}-\omega_{3}}a_{3}[\Gamma_{1}^{-}]\end{array}\right).$
(66)

Consequently, we can write the differential equations for the even parity
correlators as

$\displaystyle\begin{aligned}
&\Big{\\{}\left(y_{1}-a_{1}[\Gamma_{3}^{-}]\right)^{2}\partial_{y_{1}}+2j_{1}\left(y_{1}-a_{1}[\Gamma_{3}^{-}]\right)+\frac{2a_{1}[\Gamma_{3}^{-}]\omega_{1}}{\omega_{1}+\omega_{2}-\omega_{3}}\Big{[}(y_{1}-a_{1}[\Gamma_{3}^{-}])\partial_{y_{1}}+j_{1}\\\
&\,+\,(y_{2}-a_{2}[\Gamma_{3}^{-}])\partial_{y_{2}}+j_{2}-(y_{3}-a_{3}[\Gamma_{2}^{-}])\partial_{y_{3}}-j_{3}\Big{]}\Big{\\}}\,\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=0\,,\end{aligned}$
(67) $\displaystyle\begin{aligned}
&\Big{\\{}\left(y_{2}-a_{2}[\Gamma_{3}^{-}]\right)^{2}\partial_{y_{2}}+2j_{2}\left(y_{2}-a_{2}[\Gamma_{3}^{-}]\right)+\frac{2a_{2}[\Gamma_{3}^{-}]\omega_{2}}{\omega_{1}+\omega_{2}-\omega_{3}}\Big{[}(y_{1}-a_{1}[\Gamma_{3}^{-}])\partial_{y_{1}}+j_{1}\\\
&\,+\,(y_{2}-a_{2}[\Gamma_{3}^{-}])\partial_{y_{2}}+j_{2}-(y_{3}-a_{3}[\Gamma_{1}^{-}])\partial_{y_{3}}-j_{3}\Big{]}\Big{\\}}\,\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=0\,,\end{aligned}$
(68)

and

$\displaystyle\begin{aligned}
&\Big{\\{}\left(y_{3}-a_{3}[\Gamma_{1}^{-}]\right)^{2}\partial_{y_{3}}+2j_{3}\left(y_{3}-a_{3}[\Gamma_{1}^{-}]\right)+\frac{2a_{3}[\Gamma_{1}^{-}]\omega_{3}}{\omega_{1}-\omega_{2}-\omega_{3}}\Big{[}(y_{1}-a_{1}[\Gamma_{2}^{-}])\partial_{y_{1}}+j_{1}\\\
&\,-\,(y_{2}-a_{2}[\Gamma_{1}^{-}])\partial_{y_{2}}-j_{2}-(y_{3}-a_{3}[\Gamma_{1}^{-}])\partial_{y_{3}}-j_{3}\Big{]}\Big{\\}}\,\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle=0\,.\end{aligned}$
(69)

The structural similarity of these equations with those of the odd case in Eq.
(49) is striking. This suggest that there must be a way to derive Eqs. (67),
(68) and (69) directly from the adjacent covering maps along the lines of Sec.
3.1. We will not attempt to do this here.

By solving Eqs. (67), (68) and (69), we find that, up to an overall constant,
even parity correlators satisfying (53) take the form

$\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
$\displaystyle=$
$\displaystyle\left(1-\frac{y_{2}}{a_{2}[\Gamma_{3}^{+}]}-\frac{y_{3}}{a_{3}[\Gamma_{2}^{+}]}+\frac{y_{2}y_{3}}{a_{2}[\Gamma_{3}^{-}]a_{3}[\Gamma_{2}^{+}]}\right)^{j_{1}-j_{2}-j_{3}}$
$\displaystyle\times$
$\displaystyle\left(1-\frac{y_{1}}{a_{1}[\Gamma_{3}^{+}]}-\frac{y_{3}}{a_{3}[\Gamma_{1}^{+}]}+\frac{y_{1}y_{3}}{a_{1}[\Gamma_{3}^{-}]a_{3}[\Gamma_{1}^{+}]}\right)^{j_{2}-j_{3}-j_{1}}$
$\displaystyle\times$
$\displaystyle\left(1-\frac{y_{1}}{a_{1}[\Gamma_{2}^{+}]}-\frac{y_{2}}{a_{2}[\Gamma_{1}^{+}]}+\frac{y_{1}y_{2}}{a_{1}[\Gamma_{2}^{+}]a_{2}[\Gamma_{1}^{-}]}\right)^{j_{3}-j_{1}-j_{2}}.$

As in the odd case, one can check that this is consistent with the conjectured
expressions in Eq. (28) by using the relations between covering map
coefficients and ratios of the numbers $P_{\boldsymbol{\omega}}$ defined in
(31). It is also straightforward to see that (3.2) matches all relevant limits
of the corresponding odd cases in Eq. (50), as implied by the different series
identifications in Eqs. (58) and (60). We also note that, upon using some
identities among the $a_{i}$ coefficients such as
$a_{1}[\Gamma_{2}^{+}]a_{2}[\Gamma_{1}^{-}]=a_{1}[\Gamma_{2}^{-}]a_{2}[\Gamma_{1}^{+}]$,
the solution in Eq. (3.2) manifestly enjoys bosonic exchange symmetry.

###  3.3 Edge cases

In this subsection, we consider spectral flow assignments lying at the “edge”
of the inequalities displayed in Eqs. (37) and (53). More explicitly, we
consider three-point functions with spectral flow charges satisfying either

$\displaystyle\omega_{3}=\omega_{1}+\omega_{2}\quad\text{or}\quad\omega_{3}=\omega_{1}+\omega_{2}+1\,,\qquad\omega_{i}\geq
1\,,\,\forall\,i\,.$ (71)

Following the nomenclature of the previous sections, we refer to these as the
even and odd edge cases, respectively. Note that, according to the fusion
rules in Eq. (36), these include all possibly non-vanishing correlators which
were not included in sections 3.1 and 3.2 above, except for correlators with
unflowed insertions, which will be discussed later on.

The treatment of the edge cases is slightly different from what we have
discussed so far. Indeed, the general method based on series identifications
employed in Sec. 3.2 breaks down when considering the even edge cases. We find
that many of the coefficients in Eqs. (64), (65) and (66) become either
divergent or indeterminate when $\omega_{1}+\omega_{2}=\omega_{3}$, related to
the fact that three of the six adjacent maps cease to exist, namely those in
(59). Although, in principle, the existence of the three maps in (57) could
give enough constraints, in practice one runs into similar problems with
divergent or indeterminate coefficients. As for the odd edge cases, it turns
out that the associated covering map does not exist. Moreover, the former are
only related to even edge cases by the SL(2,$\mathbb{R}$) series
identifications.

We now show that alternative techniques involving current insertions can be
used to derive the relevant differential equations satisfied by the $y$-basis
edge correlators. Some of these equations are easier to derive in the limit
where the first two vertex operators collide. As discussed in Dei:2021xgh ,
one can take the vertex operators to be inserted at
$(x_{1},x_{2},x_{3})=(0,x,\infty)$, and then consider the limit $x\to 0$. More
explicitly, we have Dei:2021xgh

$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(x,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}$
$\displaystyle=x^{-j_{1}-j_{2}+j_{3}+\frac{k}{2}(-\omega_{1}-\omega_{2}+\omega_{3})}\braket{V^{\omega_{1}}_{j_{1}}\left(0,\frac{y_{1}}{x},0\right)V^{\omega_{2}}_{j_{2}}\left(1,\frac{y_{2}}{x},1\right)V^{\omega_{3}}_{j_{3}}\left(\infty,y_{3}x,\infty\right)}\,.$
(72)

The three-point functions that remain well-defined in this limit are those for
which

$|\omega_{1}+\omega_{2}-\omega_{3}|\leq 1\,,$ (73)

including both edge cases in (71). In these instances, it is possible to
obtain the correlator with vertex operators inserted at generic points
$(x_{1},x_{2},x_{3})$ from the one in the colliding limit with $(x,x,x_{3})$
by means of the global Ward identities. We will derive some of the relevant
differential equations satisfied by the edge correlators in the above
collision limit, and only recover the full correlators at the end. Note that
taking $x\to 0$ and $x_{3}\to\infty$ precisely corresponds to the limit in
which these correlators can be interpreted as $m$-basis correlators of flowed
primaries as in Maldacena:2001km ; Cagnacci:2013ufa , where they were denoted
as spectral flow conserving and spectral flow violating three-point functions,
depending on the overall parity of the spectral flow charges.

To illustrate how this works, let us derive a constraint that will be
satisfied by both edge cases. We set $x_{1}=x_{2}=x$ and consider the integral

$\oint_{{\cal{C}}}\braket{J^{3}(x,z)V^{\omega_{1}}_{j_{1}}(x,y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x,y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}dz\,,$
(74)

where ${\cal{C}}$ denotes a contour encircling all three insertion points.
This trivially vanishes as there is no residue at infinity. On the other hand,
the action of the current on the vertex operators at $x$ can be read off
directly from Eq. (21), while for the one inserted at $x_{3}$ we use

$J^{3}(x,z)=J^{3}(x_{3},z)+(x_{3}-x)J^{+}(x_{3},z)\,,$ (75)

which follows from (9). We obtain the following differential equation:

$\displaystyle\left[\sum_{i=1}^{3}\left(y_{i}\partial_{y_{i}}+j_{i}+\frac{k}{2}\omega_{i}\right)+(x_{3}-x)\partial_{x_{3}}\right]\braket{V^{\omega_{1}}_{j_{1}}(x,y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x,y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}=0\,.$
(76)

We now Möbius-fix the worldsheet insertions to $(0,1,\infty)$, and further
consider the limit $(x,x_{3})\to(0,\infty)$. The $y_{i}$ coordinates then get
rescaled according to Eq. (25). This leads to

$\displaystyle\begin{aligned}
&\Big{[}y_{1}\partial_{y_{1}}+y_{2}\partial_{y_{2}}-y_{3}\partial_{y_{3}}+j_{1}+j_{2}-j_{3}\\\
&\qquad+\frac{k}{2}(\omega_{1}+\omega_{2}-\omega_{3})\Big{]}\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(0,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}=0\,.\end{aligned}$
(77)

We find that Eq. (LABEL:eq:_charge_conservation_general) is the $y$-basis
version of the usual charge-conservation equation for $m$-basis three-point
functions of spectrally flowed primaries. This holds for all correlators
satisfying (73), including both edge cases.

####  3.3.1 Even edge cases

In this subsection we focus on the even edge cases, where
$\omega_{3}=\omega_{1}+\omega_{2}$. We will derive the remaining two
differential equations by considering correlators with an extra insertion of
the $J^{-}(x,z)$ current multiplied by the appropriate ratio of worldsheet
coordinates. This is similar to what was used in McElgin:2015eho  when
discussing the proof of the $m$-basis spectral flow violation rules, and also
more recently in Iguri:2022pbp  in the context of the supersymmetric version
of this model.

The integral

$\oint_{{\cal{C}}}\braket{J^{-}(x_{3},z)V^{\omega_{1}}_{j_{1}}(x_{1},y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x_{2},y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}\frac{(z-z_{1})^{\omega_{1}}(z-z_{2})^{\omega_{2}}}{(z-z_{3})^{\omega_{3}}}dz$
(78)

vanishes since there is no pole at infinity. Using the OPEs of
$J^{-}(x_{3},z)$ with the vertex operators, this yields

$\displaystyle\Big{[}x_{31}^{2}\frac{z_{12}^{\omega_{2}}}{z_{13}^{\omega_{3}}}\partial_{y_{1}}$
$\displaystyle+x_{32}^{2}\frac{z_{21}^{\omega_{1}}}{z_{23}^{\omega_{3}}}\partial_{y_{2}}+z_{31}^{\omega_{1}}z_{32}^{\omega_{2}}(y_{3}^{2}\partial_{y_{3}}+2j_{3}y_{3})\Big{]}$
$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(x_{1},y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x_{2},y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}=0\,.$
(79)

Proceeding similarly with

$\oint_{{\cal{C}}}\braket{J^{-}(x,z)V^{\omega_{1}}_{j_{1}}(x,y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x,y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}\frac{(z-z_{3})^{\omega_{3}}}{(z-z_{1})^{\omega_{1}}(z-z_{2})^{\omega_{2}}}dz\,,$
(80)

where we have imposed the collision limit mentioned above, we find

$\displaystyle\Big{[}\frac{z_{13}^{\omega_{3}}}{z_{12}^{\omega_{2}}}(y_{1}^{2}\partial_{y_{1}}+2j_{1}y_{1})$
$\displaystyle+\frac{z_{23}^{\omega_{3}}}{z_{21}^{\omega_{2}}}(y_{2}^{2}\partial_{y_{2}}+2j_{2}y_{2})+\frac{(x-x_{3})^{2}}{z_{31}^{\omega_{1}}z_{32}^{\omega_{2}}}\partial_{y_{3}}\Big{]}$
$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(x,y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x,y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}=0\,.$
(81)

We now fix the worldsheet coordinates to $(0,1,\infty)$ while sending $x\to 0$
and $x_{3}\to\infty$, and use Eq. (25) for the corresponding rescaling of the
$y$ variables. Including the charge conservation condition
(LABEL:eq:_charge_conservation_general), the system of differential equations
satisfied by the even edge correlator in the collision limit is then

$\displaystyle\begin{aligned}
0&=\Big{[}j_{1}+j_{2}-j_{3}+y_{1}\partial_{y_{1}}+y_{2}\partial_{y_{2}}-y_{3}\partial_{y_{3}}\Big{]}\braket{\dots}\,,\\\
0&=\Big{[}(-1)^{\omega_{1}}\partial_{y_{1}}+(-1)^{\omega_{3}}\partial_{y_{2}}+(y_{3}^{2}\partial_{y_{3}}+2j_{3}y_{3})\Big{]}\braket{\dots}\,,\\\
0&=\Big{[}(-1)^{\omega_{1}}(y_{1}^{2}\partial_{y_{1}}+2j_{1}y_{1})+(-1)^{\omega_{3}}(y_{2}^{2}\partial_{y_{2}}+2j_{2}y_{2})+\partial_{y_{3}}\Big{]}\braket{\dots}\,.\end{aligned}$
(82)

where $\braket{\dots}$ stands for
$\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(0,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}$.
Note that only the second equation in (82) remains valid away from the
collision limit.

Up to an overall $y$-independent constant, the general solution of the system
(82) can be written as follows:

$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(0,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}$
(83)
$\displaystyle=((-1)^{\omega_{1}}y_{1}-(-1)^{\omega_{3}}y_{2})^{j_{3}-j_{1}-j_{2}}(1+(-1)^{\omega_{3}}y_{2}y_{3})^{j_{1}-j_{2}-j_{3}}(1+(-1)^{\omega_{1}}y_{1}y_{3})^{j_{2}-j_{1}-j_{3}}\,.$

This matches the result of Dei:2021xgh , see their Eq. (5.37b).

As mentioned above, for more general values of the insertion points the
corresponding three-point functions follow from the global Ward identities. As
it turns out, we can infer the result in a heuristic way by looking at the
general expression given in Eq. (3.2). Indeed, one can verify that, upon
setting $\omega_{3}=\omega_{1}+\omega_{2}$, the coefficients
$a_{1}(\Gamma_{3}^{+}),a_{2}(\Gamma_{3}^{+})$ and $a_{2}(\Gamma_{1}^{-})$
diverge, which is a manifestation of the fact that the associated covering
maps do not exist. Since all other coefficients remain finite, we obtain

$\displaystyle\langle
V_{j_{1}}^{\omega_{1}}(y_{1})V_{j_{2}}^{\omega_{2}}(y_{2})V_{j_{3}}^{\omega_{3}}(y_{3})\rangle$
$\displaystyle=\left(1+(-1)^{\omega_{1}}\frac{(\omega_{1}+\omega_{2}-1)!}{(\omega_{1}-1)!\omega_{2}!}y_{3}+(-1)^{\omega_{3}}y_{2}y_{3}\right)^{j_{1}-j_{2}-j_{3}}$
$\displaystyle\times$
$\displaystyle\left(1+(-1)^{\omega_{1}+1}\frac{(\omega_{1}+\omega_{2}-1)!}{\omega_{1}!(\omega_{2}-1)!}y_{3}+(-1)^{\omega_{1}}y_{1}y_{3}\right)^{j_{2}-j_{3}-j_{1}}$
$\displaystyle\times$
$\displaystyle\left(1+(-1)^{\omega_{1}+1}\frac{\omega_{1}!\omega_{2}!}{(\omega_{1}+\omega_{2})!}((-1)^{\omega_{1}}y_{1}-(-1)^{\omega_{3}}y_{2})\right)^{j_{3}-j_{1}-j_{2}}.$

Up to the normalisation, to be discussed below, this precisely matches the
conjecture (28).

####  3.3.2 Odd edge cases

We now turn to the odd edge cases, where $\omega_{3}=\omega_{1}+\omega_{2}+1$.
Since the procedure is analogous to what we just described we will skip some
of the intermediate steps.

In addition to (LABEL:eq:_charge_conservation_general), we find two
differential equations by considering contour integrals very similar to that
in Eq. (78). We first take

$\oint_{{\cal{C}}}\braket{J^{-}(x_{3},z)V^{\omega_{1}}_{j_{1}}(x_{1},y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x_{2},y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}\frac{(z-z_{1})^{\omega_{1}+1}(z-z_{2})^{\omega_{2}}}{(z-z_{3})^{\omega_{3}}}dz\,,$
(85)

which again vanishes due to the absence of a residue at infinity. The same
holds for

$\oint_{{\cal{C}}}\braket{J^{-}(x_{3},z)V^{\omega_{1}}_{j_{1}}(x_{1},y_{1},z_{1})V^{\omega_{2}}_{j_{2}}(x_{2},y_{2},z_{2})V^{\omega_{3}}_{j_{3}}(x_{3},y_{3},z_{3})}\frac{(z-z_{1})^{\omega_{1}}(z-z_{2})^{\omega_{2}+1}}{(z-z_{3})^{\omega_{3}}}dz\,.$
(86)

Hence, we find the following system of differential equations:

$\displaystyle\begin{aligned}
0&=\Big{[}j_{1}+j_{2}-j_{3}-\frac{k}{2}+y_{1}\partial_{y_{1}}+y_{2}\partial_{y_{2}}-y_{3}\partial_{y_{3}}\Big{]}\braket{\dots}\,,\\\
0&=\Big{[}(-1)^{\omega_{3}}\partial_{y_{2}}+(y_{3}^{2}\partial_{y_{3}}+2j_{3}y_{3})\Big{]}\braket{\dots}\,,\\\
0&=\Big{[}(-1)^{\omega_{1}}\partial_{y_{1}}+(y_{3}^{2}\partial_{y_{3}}+2j_{3}y_{3})\Big{]}\braket{\dots}\,.\end{aligned}$
(87)

where $\braket{\dots}$ again denotes
$\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(0,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}$.
In this case, only the first of these equations gets eventually modified away
from the collision limit.

We find that, up to an overall normalization, the odd edge three-point
functions read

$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(0,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}$
(88)
$\displaystyle=y_{3}^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}(1+(-1)^{\omega_{1}}y_{1}y_{3}+(-1)^{\omega_{3}}y_{2}y_{3})^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,,$

in the collision limit, thus matching the result in Dei:2021xgh , see their
Eq. (5.37c). Moreover, as in the even edge case, we can infer the solution for
generic insertion points from the general expression in Eq. (51). For this, we
set $\omega_{3}=\omega_{1}+\omega_{2}+1$ and carefully take the limit
$a_{3}\to 0,\,a_{1,2}\to\infty$ with the products $a_{1}a_{3}$ and
$a_{2}a_{3}$ fixed. This yields

$\displaystyle\braket{V^{\omega_{1}}_{j_{1}}(0,y_{1},0)V^{\omega_{2}}_{j_{2}}(1,y_{2},1)V^{\omega_{3}}_{j_{3}}(\infty,y_{3},\infty)}=y_{3}^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}$
(89)
$\displaystyle\times\left(1+(-1)^{\omega_{1}+1}\frac{(\omega_{1}+\omega_{2})!}{\omega_{1}!\omega_{2}!}y_{3}+(-1)^{\omega_{1}}y_{1}y_{3}+(-1)^{\omega_{3}}y_{2}y_{3}\right)^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,.$

One can check that this matches the $y$-dependence given in the conjecture of
Eq. (27) for correlators with appropriate spectral flow assignments. Moreover,
upon using Eq. (55) and (56) we also see that, as expected, the expressions in
Eq. (3.2) and Eq. (89) are related via series identifications.

###  3.4 Three-point functions with unflowed insertions

So far, we have considered three-point functions where all vertex operators
had non-zero spectral flow charges. However, it is natural to expect that the
above results include the special cases where some of the insertions are
unflowed. We now show how the latter are obtained. Note that we still assume
$\omega_{3}\geq\omega_{1,2}$, as in the previous sections.

Let us start by discussing the case of a single unflowed insertion, namely
$\omega_{1}=0$. The fusion rules in Eq. (36) can then be satisfied iff
$\omega_{3}=\omega_{2}$ or $\omega_{3}=\omega_{2}+1$. These are exactly the
two cases that were computed in Cagnacci:2013ufa  in full generality from
$m$-basis techniques. In this sense, the results presented in this section are
not new, but we include them for completeness. Hence, the relevant correlators
correspond to the spectral flow assignments $(0,\omega,\omega+1)$ and
$(0,\omega,\omega)$. By means of

$V_{j}(x,z)={\cal{N}}\left(j\right)V_{\frac{k}{2}-j,h=j}^{1}(x,z)={\cal{N}}(j)\lim_{y\to\infty}y^{k-2j}V_{\frac{k}{2}-j}^{1}(x,y,z)\,,$
(90)

which is a particular case of (55), these can be obtained from the three-point
functions with charges $(1,\omega,\omega+1)$ and $(1,\omega,\omega)$,
respectively. More precisely, we have

$\langle
V_{j_{1}}V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega+1}(y_{3})\rangle={\cal{N}}(j_{1})\lim_{y_{1}\to\infty}y_{1}^{k-2j_{1}}\langle
V_{\frac{k}{2}-j_{1}}^{1}(y_{1})V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega+1}(y_{3})\rangle\,,$
(91)

and

$\langle
V_{j_{1}}V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega}(y_{3})\rangle={\cal{N}}(j_{1})\lim_{y_{1}\to\infty}y_{1}^{k-2j_{1}}\langle
V_{\frac{k}{2}-j_{1}}^{1}(y_{1})V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega}(y_{3})\rangle\,,$
(92)

where we have abbreviated $V_{j_{1}}(0,0)\equiv V_{j_{1}}$.

Focusing on (91), the RHS involves an even edge correlator. We thus need to
consider the appropriate limit of Eq. (3.3.1), which gives

$\langle
V_{j_{1}}V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega+1}(y_{3})\rangle=y_{3}^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}\left(1-y_{3}-(-1)^{\omega}y_{2}y_{3}\right)^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,.$
(93)

up to an overall constant. On the other hand, the RHS of (92) is obtained as
the appropriate limit of the solution in Eq. (50), namely

$\displaystyle\langle
V_{j_{1}}^{1}(y_{1})V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega}(y_{3})\rangle=(y_{1}-\omega)^{j_{2}+j_{3}-j_{1}-\frac{k}{2}}(y_{2}+(-1)^{\omega})^{j_{1}+j_{3}-j_{2}-\frac{k}{2}}(y_{3}-1)^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}$
$\displaystyle\times\Big{(}(-1)^{\omega+1}(\omega+1)+(-1)^{\omega}y_{1}-y_{2}+(-1)^{\omega}y_{3}-(\omega-1)y_{2}y_{3}+y_{1}y_{2}y_{3}\Big{)}^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,.$
(94)

Hence, we get

$\langle
V_{j_{1}}V_{j_{2}}^{\omega}(y_{2})V_{j_{3}}^{\omega}(y_{3})\rangle=\left(y_{2}+(-1)^{\omega}\right)^{j_{3}-j_{1}-j_{2}}\left(y_{3}-1\right)^{j_{2}-j_{1}-j_{3}}\left((-1)^{\omega}+y_{2}y_{3}\right)^{j_{1}-j_{2}-j_{3}}\,.$
(95)

up to the overall constant. Finally, we consider correlators with exactly two
unflowed insertions, $\omega_{1}=\omega_{2}=0$. According to (36) this can
only be non-trivial for $\omega_{3}=1$. By using again Eq. (90) we get

$\displaystyle\langle V_{j_{1}}V_{j_{2}}V_{j_{3}}^{1}(y_{3})\rangle$
$\displaystyle={\cal{N}}(j_{2})\lim_{y_{2}\to\infty}y_{2}^{k-2j_{2}}\langle
V_{j_{1}}V_{\frac{k}{2}-j_{2}}^{1}(y_{2})V_{j_{3}}^{1}(y_{3})\rangle$
$\displaystyle=y_{3}^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}(y_{3}-1)^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,,$
(96)

where in the last line we have ignored an overall normalization factor. In
this way, we match all the corresponding results of Dei:2021xgh , where the
authors showed that this further reproduces the original computations of
Maldacena:2001km ; Cagnacci:2013ufa .

###  3.5 Normalization

So far we have focused on the dependence of the $y$-basis spectrally flowed
correlators on the variables $y_{1}$, $y_{2}$ and $y_{3}$, and shown that it
matches precisely the predictions of Dei:2021xgh . We now describe how the
overall normalizations in Eqs. (27)-(33) are obtained555This was already
discussed in Iguri:2022eat , assuming the $y$-dependence of the correlators
was as in Dei:2021xgh . We reproduce the argument here for completeness.

Once again, the argument relies on the SL(2,$\mathbb{R}$) series
identifications. Indeed, identities such as those in Eqs. (58) and (60) must
hold exactly, including the normalization factors. Having fixed the
$y$-dependence, we can thus determine the normalizations recursively, starting
from the unflowed three-point functions of Teschner:1999ug ; Maldacena:2001km
. For instance, we consider the following identity:

$\lim_{y_{3}\rightarrow\infty}y_{3}^{2j_{3}}\left\langle
V^{\omega_{1}}_{j_{1}}(y_{1})\,V^{\omega_{2}}_{j_{2}}(y_{2})\,V^{\omega_{3}}_{j_{3}}(y_{3})\right\rangle={\cal{N}}(j_{3})\left\langle
V^{\omega_{1}}_{j_{1}}(y_{1})\,V^{\omega_{2}}_{j_{2}}(y_{2})\,V^{\omega_{3}-1}_{\frac{k}{2}-j_{3}}(0)\right\rangle\,,$
(97)

which will give us a recursion relation for
$C_{\boldsymbol{\omega}}(j_{1},j_{2},j_{3})$. Since the latter is independent
of the $y_{i}$, we can set $y_{1}=y_{2}=0$. Using the $y$-dependence derived
above, written as in Eqs. (27) and (28), one finds that the product of $X_{I}$
factors on the left- an right-hand sides of (97), both reduce to either

$P_{\boldsymbol{\omega}}^{j_{1}+j_{2}+j_{3}-k}\,P_{\boldsymbol{\omega}+e_{1}+e_{2}}^{j_{3}-j_{1}-j_{2}}\,P_{\boldsymbol{\omega}+e_{2}-e_{3}}^{j_{1}-j_{2}-j_{3}}\,P_{\boldsymbol{\omega}+e_{1}-e_{3}}^{j_{2}-j_{3}-j_{1}},$
(98)

or

$P_{\boldsymbol{\omega+e_{1}+e_{2}-e_{3}}}^{\frac{k}{2}-j_{1}-j_{2}-j_{3}}\,P_{\boldsymbol{\omega}+e_{1}}^{-j_{1}+j_{2}+j_{3}-\frac{k}{2}}\,P_{\boldsymbol{\omega}+e_{2}}^{j_{1}-j_{2}+j_{3}-\frac{k}{2}}\,P_{\boldsymbol{\omega}-e_{3}}^{j_{1}+j_{2}-j_{3}-\frac{k}{2}}\,,$
(99)

depending on the overall parity of the spectral flow charges. Consequently, in
both cases we find that Eq. (97) holds iff

$C_{\boldsymbol{\omega}}(j_{1},j_{2},j_{3})={\cal{N}}(j_{3})C_{\boldsymbol{\omega}-e_{3}}\left(j_{1},j_{2},\frac{k}{2}-j_{3}\right).$
(100)

Analogous statements can be derived by shifting the spectral flow charges
$\omega_{1}$ and $\omega_{2}$ instead. Moreover, one has the identity

${\cal{N}}(j_{1})C\left(\frac{k}{2}-j_{1},j_{2},j_{3}\right)={\cal{N}}(j_{2})C\left(j_{1},\frac{k}{2}-j_{2},j_{3}\right)={\cal{N}}(j_{3})C\left(j_{1},j_{2},\frac{k}{2}-j_{3}\right)\,.$
(101)

Since ${\cal{N}}(j){\cal{N}}(\frac{k}{2}-j)=1$, it follows that, as stated in
(33), $C_{\boldsymbol{\omega}}(j_{1},j_{2},j_{3})$ can only be
$C(j_{1},j_{2},j_{3})$, i.e. the unflowed three-point function, or
${\cal{N}}(j_{1})C(\frac{k}{2}-j_{1},j_{2},j_{3})$, depending on the parity of
$\omega_{1}+\omega_{2}+\omega_{3}$. To be precise, this argument is valid for
discrete representations, although we expect that it holds also for the
continuous series by analytic continuation in j Maldacena:2001km ;
Zamolodchikov:1995aa . This concludes our computation of three-point functions
with arbitrary spectral flow charges.

##  4 Discussion

In this paper, we have computed the $y$-basis string three-point function in
AdS3 involving vertex operators with arbitrary spectral flow charges. This
provides a proof for the conjecture put forward recently in Dei:2021xgh , thus
establishing integral expressions for all (primary) three-point functions of
the SL(2,$\mathbb{R}$)-WZW model at level $k$, for all $k>3$.

The subfamily of (odd parity) correlators for which a holomorphic covering map
from the worldsheet to the AdS3 boundary exists had been obtained in
Eberhardt:2019ywk ; Dei:2021xgh . Here we have relied on the general structure
of local Ward identities (in their $y$-basis formulation) and made extensive
use of the SL(2,$\mathbb{R}$) series identifications, whose importance was
recently highlighted in Iguri:2022eat . This allowed us to extend the methods
based on covering maps to all other non-vanishing correlators, as defined by
the fusion rules computed in Maldacena:2001km .

Our strategy can be summarised as follows. We first argued that the
differential equations satisfied by all $y$-basis three-point functions must
take the form given in Eq. (54). Obtaining the general expressions for these
equations for all even parity correlators then reduces to computing all
unknown coefficients in (54). We have provided the relations among correlators
with adjacent spectral flow assignments that follow from SL(2,$\mathbb{R}$)
series identifications in Eqs. (58) and (60). These provide a considerable
number of identities between even and odd parity correlators in the limit
where one of the $y$ variables is taken to either zero or infinity. This
allowed us to evaluate all relevant coefficients $A_{ij}$, $B_{ij}$ and
$C_{i}$ in closed form, as given in Eqs. (64)-(66). The derivation of these 21
coefficients involves solving a total of 60 conditions, of which 39 can be
taken as consistency checks. The latter turn out to be satisfied in a highly
non-trivial manner, related to the existence of a set of identities relating
the behaviour of different covering maps in the vicinity of the insertion
points.

The resulting differential equations satisfied by even parity correlators are
provided in Eqs. (67)-(69). These show a striking similarity with the cases of
odd total spectral flow, a hallmark of the existence of a more direct
derivation by means of adjacent covering maps. We leave this for future work.
Here we have shown that the general solution to these equations, namely Eq.
(3.2), is compatible with the proposal of Dei:2021xgh .

We have also discussed the so-called edge cases, whose spectral flow
assignments saturate the fusion rules in Eqs. (37) and (53). Some subtleties
arise when trying to apply the general method described above in this context.
For these cases we have provided an alternative approach, based on an improved
version of the $m$-basis methods Maldacena:2001km ; Cagnacci:2013ufa . We have
then described how to obtain correlators involving unflowed insertions.
Finally, we fixed the overall normalization of all $y$-basis three-point
functions, following the arguments of Iguri:2022eat .

At this point, it is natural to ask if an analogous story holds for four-point
functions, which encode crucial dynamical information about the theory. A
closed formula for four-point functions in the $y$-basis with arbitrary
spectral flow assignments in terms of the corresponding unflowed correlator
was conjectured in Dei:2021yom . Here the situation is more subtle: on top of
the four $y_{i}$ variables, four-point functions also depend non-trivially on
the worldsheet and spacetime cross-ratios, and must satisfy the corresponding
Knizhnik-Zamolodchikov equations. It has been known for some time
Minces:2005nb ; Cagnacci:2015pka  that the latter intertwine non-trivially
with the recursion relations of the type described in section 2.2. If the
structure put forward in Dei:2021yom  is correct, its proof is likely to work
in two steps. First, one should use arguments similar to those we have
considered in this paper to show that solutions to the $y$-basis differential
equations associated with the four-point functions consist of various
prefactors given by powers of the generalized differences $X_{I}$, defined in
Eq. (29), multiplied by an arbitrary function of the so-called generalized
cross-ratio. Second, one should prove that this arbitrary function must
satisfy the same Knizhnik-Zamolodchikov equation as the corresponding unflowed
four-point function. The extension of the proof for the case of correlation
functions involving four spectrally-flowed vertex operators is a work in
progress.

###### Acknowledgements.

It is a pleasure to thank Andrea Dei, Gabriele Di Ubaldo, Lorenz Eberhardt,
Stefano Massai, Julian H. Toro, David Turton. The work of D.B. is supported by
the Royal Society Research Grant RGF\R1\181019. The work of N.K. is supported
by the ERC Consolidator Grant 772408-Stringlandscape.

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