# Unravelling AdS3/CFT2 near the boundary

Bob Knighton b Vit Sriprachyakul<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Applied Mathematics & Theoretical Physics, University of
Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United KingdomInstitut für Theoretische
Physik, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland

###### Abstract

We study correlation functions of spectrally-flowed vertex operators in
bosonic string theory on $\text{AdS}_{3}\times X$ in the path integral
formalism. By restricting the path integral to only include worldsheets which
live near the asymptotic boundary, we compute correlation functions of
spectrally-flowed vertex operators and find a precise agreement with the
perturbative correlators in the recently-proposed dual CFT at all orders in
conformal perturbation theory. We thus provide highly nontrivial evidence for
the bulk/boundary duality.

##  1 Introduction

The AdS/CFT correspondence relates string theories on Anti-de Sitter
spacetimes to conformal field theories on their boundary. Due to the
strong/weak nature of this duality, examples for which both sides are
analytically solvable are invaluable for a deeper understanding of the
holographic principle. A specifically nice set of such dualities are the
$\text{AdS}_{3}/\text{CFT}_{2}$ dualities, for which both the bulk worldsheet
theories and boundary CFTs are under much better control than in higher
dimensions. The simplification of the worldsheet theory lies in the accidental
isometry $\text{AdS}_{3}\cong\text{SL}(2,\mathbb{R})$, so that the worldsheet
CFT is described by a Wess-Zumino-Witten (WZW) model on
$\text{SL}(2,\mathbb{R})$, assuming that the background is supported purely by
NS-NS flux Balog:1988jb ; Petropoulos:1989fc ; Hwang:1990aq ;
Henningson:1991jc ; Gawedzki:1991yu ; Bars:1995mf ; Evans:1998qu ;
Giveon:1998ns ; Kutasov:1999xu ; Maldacena:2000hw ; Maldacena:2000kv ;
Maldacena:2001km .

Recently, a family of holographic CFTs have been proposed to be dual to pure
NS-NS $\text{AdS}_{3}\times X$ backgrounds Eberhardt:2019qcl ;
Balthazar:2021xeh ; Martinec:2021vpk ; Eberhardt:2021vsx . The proposed field
theories are modeled by a symmetric product orbifold of a non-compact CFT,
deformed by a marginal twist-2 field. The duality we will be most interested
in in this paper is that of Eberhardt:2021vsx , which relates bosonic string
theory on $\text{AdS}_{3}\times X$ to the symmetric orbifold of a linear
dilaton theory, perturbed by an exponential ‘wall’. Specifically, the proposal
is111The dual CFT is strictly speaking only defined in the large $K$ limit,
and so should be thought of as dual to perturbative string theory.

$\begin{gathered}\text{Bosonic string theory on AdS}_{3}\times X\\\
\Longleftrightarrow\\\ \text{Sym}^{K}(\mathbb{R}_{\varphi}\times
X)+\mu\int\sigma_{2,\alpha}\,,\end{gathered}$ (1)

where $\mathbb{R}_{\varphi}$ is a linear dilaton of background charge
$\mathcal{Q}=\sqrt{2}(3-k)/\sqrt{k-2}$ and $\sigma_{2,\alpha}$ is a marginal
field in the twist-2 sector whose profile is shown in Figure 1. In this way,
the dual CFT is morally analogous to the definition of Liouville theory as a
deformation of a linear dilaton, except that the exponential wall now lives in
the twisted sector of an orbifold group.

In Eberhardt:2021vsx , evidence for this duality was provided in the form of a
matching of spectrally-flowed 2- and 3-point functions in $\text{AdS}_{3}$
with twisted-sector correlators in the dual CFT up to fourth order in $\mu$.
Matching of the 4-point functions at zeroth order in $\mu$ was found in
Dei:2022pkr .222The $\text{AdS}_{3}$ correlation functions in question were
computed in Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr  by solving the
$\mathfrak{sl}(2,\mathbb{R})_{k}$ Ward identities and the
Knizhnik–Zamolodchikov equations explicitly for a large number of spectral
flows $w_{i}$ and proposing a general formula for all $w_{i}$. No proof of the
general formulae in Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr  is known, except
for in the case of 3-point functions Bufalini:2022toj . The matching of the
AdS3 and dual CFT correlation functions is highly nontrivial, and required the
conjectural relations between various analytic data of certain branched
holomorphic covering maps $\gamma:\text{S}^{2}\to\text{S}^{2}$. However,
matching $n$-point functions at all orders in $\mu$ is still an open problem.

The primary bottleneck in matching the two sides of (1) is the technical
difficulty of calculating correlation functions in $\text{AdS}_{3}$ string
theory. While correlation functions of the $\text{SL}(2,\mathbb{R})$ WZW model
have been discussed at great length in the literature,333See Teschner:1997ft ;
Teschner:1999ug ; Giribet:1999ft ; Giribet:2000fy ; Maldacena:2001km ;
Giribet:2001ft ; Ribault:2005ms ; Giribet:2005ix ; Hikida:2007tq ;
Iguri:2007af ; Baron:2008qf ; Iguri:2009cf ; Cagnacci:2013ufa ;
Giribet:2015oiy ; Hikida:2020kil ; Eberhardt:2019ywk ; Eberhardt:2020akk ;
Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr ; Bufalini:2022toj  for a non-
exhaustive list of references. a full solution is still an open problem.
Particularly difficult is the calculation of correlators of spectrally-flowed
vertex operators in the so-called $x$-basis, which are still not known in
closed form. The complication is two-fold. From the operator formalism, since
spectrally-flowed states in $\text{AdS}_{3}$ live in non-highest-weight
representations of $\mathfrak{sl}(2,\mathbb{R})_{k}$, their correlation
functions obey very complicated Ward identities, which have only been solved
analytically in very special cases Eberhardt:2019ywk ; Eberhardt:2020akk ;
Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr ; Bufalini:2022toj . From the path
integral, the difficulty is that spectrally-flowed vertex operators do not
admit obvious representations in terms of the fundamental fields of the
worldsheet CFT, and so it is difficult to construct an integrand in the
Polyakov path integral.

$V(\varphi)$$\varphi$NonperturbativeregimePerturbativeregimeDeep
interiorAsymptotic boundary Figure 1: The linear dilaton $\varphi$ of the dual
CFT is deformed by an exponential wall. In the regime $\varphi\to\infty$
(corresponding to the boundary of $\text{AdS}_{3}$) this wall can be regarded
as a perturbation, while in the opposite regime $\varphi\to-\infty$
(corresponding to the deep interior of $\text{AdS}_{3}$) it must be treated
non-perturbatively.

In Knighton:2023mhq ; Hikida:2023jyc , progress was made toward matching
correlation functions of the dual CFT from the worldsheet in full generality.
In Knighton:2023mhq , it was shown how the the structure of the correlators in
the CFT at order $\mu^{m}$ in perturbation theory can be reproduced by the
path integral of $\text{AdS}_{3}$ string theory. The key ingredient was an
explicit form of spectrally-flowed vertex operators in the worldsheet CFT. The
resulting path integral was shown to localize to an integral over exactly $m$
points on the boundary of $\text{AdS}_{3}$, reflecting the integrated
deformation $\int\sigma_{2,\alpha}$ in the dual CFT. However, this computation
was schematic, and the form of the integrand of the path integral was not
determined. Independently, the authors of Hikida:2023jyc  were able to use CFT
technology developed in the context of the $\mathbb{H}_{3}^{+}$/Liouville
duality Hikida:2007tq ; Hikida:2008pe ; Hikida:2020kil ; Hikida:2023jyc  to
show how the analytic forms of the dual CFT correlators can be derived from
the $\text{AdS}_{3}$ path integral, although did not provide an explanation
for the localization of the worldsheet path integral.

The goal of this paper is to provide a first principles derivation of the
perturbative CFT correlation functions from the $\text{AdS}_{3}$ path integral
at tree level in the worldsheet genus expansion. This is achieved by
considering an effective theory of $\text{AdS}_{3}$ string theory near the
asymptotic boundary. In this limit, the worldsheet theory simplifies, and it
is possible to compute correlation functions in a Coulomb-gas-like formalism
Knighton:2023mhq . By a careful analysis of the path integral measure, as well
as the integral over moduli space $\mathcal{M}_{0,n}$ on the worldsheet, we
precisely reproduce the symmetric orbifold answer perturbatively in $\mu$.

The fact that we can reproduce the full perturbative expansion in $\mu$ by
only considering a near-boundary approximation in $\text{AdS}_{3}$ reflects
the fact that the perturbative expansion in $\mu$ is not the full boundary CFT
answer. Just like in Liouville theory, the fact that the marginal operator in
the symmetric orbifold is not normalizable means that perturbation theory
breaks down and there are non-perturbative effects that must be taken into
account. These effects come from the region of finite $\varphi$ in Figure 1,
which holographically correspond to contributions from the deep interior of
$\text{AdS}_{3}$. Thus, the matching of the perturbative CFT correlators from
the near-boundary theory on $\text{AdS}_{3}$ should be thought of as a tree-
level proof of the slightly weaker duality:

$\begin{gathered}\text{Bosonic string theory on AdS}_{3}\times X\text{ near
the boundary}\\\ \Longleftrightarrow\\\
\text{Sym}^{K}(\mathbb{R}_{\varphi}\times X)+\mu\int\sigma_{2,\alpha}\text{ in
the perturbative regime}\,.\end{gathered}$ (2)

Although this subsector of the full $\text{AdS}_{3}/\text{CFT}_{2}$ duality
fails to capture nonperturbative (in $\mu$) effects, we view the work in this
paper as highly nontrivial evidence for the full duality (1).

This paper is organized as follows. In Sections 2 and 3, we review bosonic
string theory on $\text{AdS}_{3}$ and its proposed CFT dual, respectively.
While these sections contain no novel material, we include them for the sake
of completeness, and expert readers should feel free to skip over them. In
Section 4, we explicitly compute correlation functions of spectrally-flowed
states on $\text{AdS}_{3}$ in the path integral formalism, and demonstrate
that they exactly agree with the perturbative correlation functions of the
proposed dual CFT at all orders in conformal perturbation theory. We conclude
in Section 5 with a brief summary of the main text and a list of potential
future directions. We also include a technical appendix which computes the
Jacobian of a certain change of variables necessary for the evaluation of the
$\text{AdS}_{3}$ path integral.

##  2 Strings on AdS3

The main goal of this paper is to compute string correlation functions in
$\text{AdS}_{3}$ and compare them to dual CFT correlators. On the worldsheet
side, this amounts to computing correlators of the form

$\sum_{g=0}^{\infty}g_{s}^{2g-2}\int_{\mathcal{M}_{g,n}}\Braket{V_{m_{1},\bar{m}_{1},j_{1}}^{w_{1}}(x_{1},z_{1})\cdots
V_{m_{n},\bar{m}_{n},j_{n}}^{w_{n}}(x_{n},z_{n})}\,,$ (3)

where the vertex operators $V_{m,\bar{m},j}^{w}(x,z)$ are associated with
spectrally-flowed ground states in the worldsheet theory. In this section, we
will review the basic notions of $\text{AdS}_{3}$ string theory necessary to
compute the above correlators, and return to the actual computation later in
Section 4.

###  2.1 The sigma model

Euclidean $\text{AdS}_{3}$ in Poincaré coordinates has the metric tensor

$\mathrm{d}s^{2}=\frac{k}{r^{2}}(\mathrm{d}r^{2}+\mathrm{d}\gamma\,\mathrm{d}\bar{\gamma})\,,$
(4)

where $r\in\mathbb{R}_{+}$ is a radial coordinate such that $r\to 0$ is the
conformal boundary of $\text{AdS}_{3}$ and the complex coordinates
$(\gamma,\bar{\gamma})$ parametrize the boundary, whose induced metric is flat
everywhere except at the point $\gamma=\infty$. Here, $k$ is a length scale
related to the AdS radius $L$ by444Note that we set $\alpha^{\prime}=1$. If we
were to reinstate units, we would write $k=L^{2}/\alpha^{\prime}$.

$k=L^{2}\,.$ (5)

In order to describe a consistent string background, we must also introduce a
$B$-field

$B=-\frac{k}{r^{2}}\mathrm{d}\gamma\wedge\mathrm{d}\bar{\gamma}\,.$ (6)

The inclusion of the $B$-field (as well as an internal CFT $X$ with central
charge $c(X)=26-3k/(k-2)$) is required for the vanishing beta function of the
worldsheet.555Alternatively, it can be seen as the Wess-Zumino term of the
$\text{SL}(2,\mathbb{R})$ WZW model, which is guaranteed to be conformally
invariant at the quantum level.

To study string theory on this background, we find it convenient to introduce
a scalar field $\Phi$ such that

$r=r_{0}e^{-Q\Phi/2}\,,$ (7)

where $r_{0}$ is an arbitrary radius that sets a length scale. The constant
$Q$ is related to the level $k$ by

$Q=\sqrt{\frac{2}{k-2}}\,,$ (8)

and is chosen for later convenience. In these coordinates, it is possible to
immediately write down the Polyakov action for the worldsheet theory, which
will be a second-order action in the variables $(\Phi,\gamma,\bar{\gamma})$.
However, for the purposes of this paper, we will be interested in the first-
order formalism of $\text{AdS}_{3}$ strings, where one introduces Lagrange
multipliers $\beta,\bar{\beta}$ of weight $(1,0)$ and $(0,1)$ respectively.
The resulting action is deBoer:1998gyt ; Giveon:1998ns ; Kutasov:1999xu

$S=\frac{1}{2\pi}\int_{\Sigma}\left(\frac{1}{2}\partial\Phi\,\overline{\partial}\Phi+\beta\overline{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\nu^{-1}\beta\bar{\beta}e^{-Q\Phi}-\frac{Q}{4}R\Phi\right)\,.$
(9)

The linear dilaton term $-QR\Phi/4$ is present due to a redefinition of the
path integral measure that has been made upon introducing the Lagrange
multipliers $\beta,\bar{\beta}$. The constant $\nu$ is a dimensionful
parameter proportional to $r_{0}^{-2}$ (known as the ‘cosmological constant’
in the literature), and can be set to any value by performing a shift in
$\Phi$. While the shift $\Phi\to\Phi+\phi_{0}$ is not a symmetry of the above
action, the combined shift

$\Phi\to\Phi+\phi_{0}\,,\quad\nu\to e^{-Q\phi_{0}}\nu$ (10)

changes the action by a constant

$S\to S-Q\phi_{0}(1-g)\,,$ (11)

as can be seen by the Gauss-Bonnet theorem. This implies that correlators of
fields $\mathcal{O}_{i}$ carrying charge $q_{i}$ with respect to this symmetry
satisfy

$e^{\phi_{0}(\sum_{i}q_{i}+Q(1-g))}\Braket{\mathcal{O}_{1}\cdots\mathcal{O}_{n}}_{e^{Q\phi_{0}}\nu}=\Braket{\mathcal{O}_{1}\cdots\mathcal{O}_{n}}_{\nu}\,.$
(12)

Thus, the overall shift symmetry implies that correlators scale with $\nu$
like a power law:

$\Braket{\mathcal{O}_{1}\cdots\mathcal{O}_{n}}_{\nu}\propto\nu^{-\frac{1}{Q}\sum_{i}q_{i}-(1-g)}\,.$
(13)

#### Quantization

To quantize the $\text{AdS}_{3}$ sigma model, we can treat the action as a
perturbation of a free field theory. In canonical quantization, the worldsheet
fields can be treated to zeroth order as free fields satisfying the OPEs

$\begin{gathered}\Phi(z,\bar{z})\Phi(w,\bar{w})\sim-\log|z-w|^{2}\,,\\\
\beta(z)\gamma(w)\sim-\frac{1}{z-w}\,,\quad\bar{\beta}(\bar{z})\bar{\gamma}(\bar{w})\sim-\frac{1}{\bar{z}-\bar{w}}\,.\end{gathered}$
(14)

Here, we have explicitly included the (anti-)holomorphic behavior of the
various fields. Later, we will be less careful and simply denote points on the
worldsheet by a single coordinate $z$, and only re-introduce
$\bar{z}$-dependence when there is ambiguity.

The fields above can be used to construct a left-moving current algebra
generated by the currents

$J^{+}=\beta\,,\quad J^{3}=-\frac{1}{Q}\partial\Phi+(\beta\gamma)\,,\quad
J^{-}=-\frac{2}{Q}(\partial\Phi\,\gamma)+(\beta(\gamma\gamma))-k\partial\gamma\,,$
(15)

where $()$ denotes normal ordering. These currents satisfy the
$\mathfrak{sl}(2,\mathbb{R})_{k}$ algebra666In Euclidean signature, the
correct current algebra is technically $\mathfrak{sl}(2,\mathbb{C})_{k}$.
However, it is common in the literature to be agnostic about the choice of
real form and simply refer to the algebra as $\mathfrak{sl}(2,\mathbb{R})_{k}$
in both Euclidean and Lorenzian signature.

$\begin{gathered}J^{3}(z)J^{\pm}(w)\sim\pm\frac{J^{\pm}(w)}{z-w}\,,\quad
J^{+}(z)J^{-}(w)\sim\frac{k}{(z-w)^{2}}-\frac{2J^{3}(w)}{z-w}\,,\\\
J^{3}(z)J^{3}(w)\sim-\frac{k/2}{(z-w)^{2}}\,.\end{gathered}$ (16)

The zero modes of these currents generate the $\text{SL}(2,\mathbb{R})_{L}$
isometry of $\text{AdS}_{3}$ and their antiholomorphic counterparts generate
the $\text{SL}(2,\mathbb{R})_{R}$ isometry.

###  2.2 States and vertex operators

Vertex operators in the $\text{AdS}_{3}$ sigma model fall into representations
of the
$\mathfrak{sl}(2,\mathbb{R})_{k}\times\overline{\mathfrak{sl}(2,\mathbb{R})}_{k}$
current algebra generated by the left- and right-moving currents
$J^{a},\bar{J}^{a}$. Focusing only on the left-movers, highest-weight
representations of $\mathfrak{sl}(2,\mathbb{R})$ are spanned by states
$\ket{m,j}$ with

$J^{3}_{0}\ket{m,j}=m\ket{m,j}\,,\quad J^{\pm}_{0}\ket{m,j}=(m\pm j)\ket{m\pm
1,j}\,,$ (17)

and which are annihilated by the positive modes $J^{a}_{n}$, $n>0$. Such
representations are labeled by the $\mathfrak{sl}(2,\mathbb{R})$ spin $j$ and
the fractional part $\alpha$ of $m$. There are two cases of interest:

  * •

Discrete representations: These are representations $\mathcal{D}^{+}_{j}$ and
$\mathcal{D}^{-}_{j}$ which occur when there exists a highest/lowest weight
state with respect to the raising, lowering operators $J^{\pm}_{0}$.
Specifically,

$\begin{split}\mathcal{D}^{+}_{j}&=\\{\ket{m,j}\,,\,\,m=j,j+1,j+2,\ldots\\}\,,\\\
\mathcal{D}^{-}_{j}&=\\{\ket{m,j}\,,\,\,m=-j,-j-1,-j-2,\ldots\\}\,,\end{split}$
(18)

along with all of their $J^{a}_{-n}$ descendants. As we will explain below, it
is sufficient to only consider $\mathcal{D}^{+}_{j}$.

  * •

Continuous representations: These are representations
$\mathcal{C}_{\alpha}^{j}$ spanned by states of the form

$\mathcal{C}_{j}^{\alpha}=\\{\ket{m,j}\,,\,\,m\in\mathbb{Z}+\alpha\\}\,,$ (19)

along with all of their $J^{a}_{-n}$ descendants. These representations enjoy
a reflection symmetry, and the representations $\mathcal{C}^{\alpha}_{j}$ and
$\mathcal{C}^{\alpha}_{1-j}$ are equivalent.

The discrete representations $\mathcal{D}^{+}_{j}$ are unitary for
$\frac{1}{2}<j<\frac{k-1}{2}$ and the continuous representations
$\mathcal{C}_{\alpha}^{j}$ are unitary for $j\in\frac{1}{2}+i\mathbb{R}$.
Including the right-moving representations as well, the spectrum of highest-
weight representations takes the form777By the reflection symmetry $j\to 1-j$,
we can restrict $j\in\frac{1}{2}+i\mathbb{R}_{+}$ for the continuous
representations.

$\left(\int_{0}^{1}\mathrm{d}\alpha\int_{\frac{1}{2}+i\mathbb{R}_{+}}\mathrm{d}j\,\mathcal{C}^{j}_{\alpha}\otimes\overline{\mathcal{C}}^{j}_{\alpha}\right)\oplus\left(\int_{\frac{1}{2}}^{\frac{k-1}{2}}\mathrm{d}j\,\mathcal{D}^{+}_{j}\otimes\overline{\mathcal{D}}^{+}_{j}\right)\,.$
(20)

By the state-operator correspondence, we can introduce vertex operators
$V_{m,\bar{m},j}$ for the state $\ket{m,j}_{L}\otimes\ket{\bar{m},j}_{R}$. The
representation theory described above can be repackaged into the OPEs

$\begin{split}J^{3}(z)V_{m,\bar{m},j}(y)&\sim\frac{mV_{m,\bar{m},j}(y)}{z-y}\,,\\\
J^{\pm}(z)V_{m,\bar{m},j}(y)&\sim\frac{(m\pm j)V_{m\pm
1,\bar{m},j}(y)}{z-y}\,.\end{split}$ (21)

The action of the right-moving currents is completely analogous.

In this paper, we will primarily be interested in the continuous
representations $\mathcal{C}^{j}_{\alpha}$, as they play a more concrete role
in the dual CFT.888As explained in Eberhardt:2021vsx , discrete
representations of $\mathfrak{sl}(2,\mathbb{R})_{k}$ appear as bound states or
‘LSZ poles’ in the dual CFT, see Aharony:2004xn . Because of the equivalence
between the representations with spins $j$ and $1-j$, the vertex operators
should obey a reflection relation

$V_{m,\bar{m},1-j}=R(m,\bar{m},1-j)V_{m,\bar{m},j}\,,$ (22)

where $R(m,\bar{m},j)$ is a reflection coefficient satisfying

$R(m,\bar{m},1-j)R(m,\bar{m},j)=1\,.$ (23)

While we will mostly keep $R$ undetermined, it can be determined by requiring
compatibility with the OPEs (21). This is sufficient to determine $R$ up to a
function $R_{j}$ depending purely on the spins as

$R(m,\bar{m},j)=R_{j}\nu^{2j-1}(2j-1)\frac{\Gamma(m+j)\Gamma(-\bar{m}+j)\Gamma(1-2j)}{\Gamma(m+1-j)\Gamma(-\bar{m}+1-j)\Gamma(2j)}\,,$
(24)

where $R_{j}$ is constrained to satisfy the constraint $R_{j}R_{1-j}=1$. The
power of $\nu$ is conventional, but is chosen so that the full vertex operator
has definite transformation properties under the shift symmetry (10).

A convenient realization of the vertex operators $V_{m,\bar{m},j}$ in terms of
the worldsheet free fields takes the form

$\begin{split}V_{m,\bar{m},j}=\,&e^{-Qj\Phi}\gamma^{-m-j}\bar{\gamma}^{-\bar{m}-j}\\\
&\hskip
14.22636pt+R(m,\bar{m},j)\,e^{-Q(1-j)\Phi}\gamma^{-m-(1-j)}\bar{\gamma}^{-\bar{m}-(1-j)}\,.\end{split}$
(25)

This expression is strictly speaking only valid in the limit of large $\Phi$
and the ‘true’ vertex operators have a series of subleading corrections, as
explained in Giveon:1998ns ; Kutasov:1999xu ; Aharony:2004xn ; Iguri:2007af ;
Hikida:2000ry . These operators satisfy the reflection property under $j\to
1-j$ and have worldsheet conformal weight

$\Delta=\overline{\Delta}=\frac{j(1-j)}{k-2}=\frac{\frac{1}{4}+s^{2}}{k-2}\,,$
(26)

with $j=\frac{1}{2}+is$.

#### Spectrally-flowed states

In addition to highest-weight representations, the algebra
$\mathfrak{sl}(2,\mathbb{R})_{k}$ admits an infinite class of non-highest-
weight representations. These are obtained by composing the highest-weight
representations with the automorphism

$\sigma^{w}(J^{\pm}_{m})=J^{\pm}_{m\mp
w}\,,\quad\sigma^{w}(J^{3}_{m})=J^{3}_{m}+\frac{kw}{2}\delta_{m,0}\,,$ (27)

defined for any $w\in\mathbb{Z}$. The full string spectrum is generated by the
spectrally-flowed images $\sigma^{w}(\mathcal{C}^{j}_{\alpha})$ and
$\sigma^{w}(\mathcal{D}^{\pm}_{j})$ of the highest-weight representations
discussed above. Coincidentally, there is an equivalence

$\sigma^{w}(\mathcal{D}_{j}^{-})=\sigma^{w-1}(\mathcal{D}^{+}_{\frac{k}{2}-j})\,,$
(28)

and so we can safely ignore the discrete representations
$\sigma^{w}(\mathcal{D}^{-}_{j})$ in the spectrum. That is, the full string
spectrum takes the form

$\bigoplus_{w\in\mathbb{Z}}\left[\left(\int_{0}^{1}\mathrm{d}\alpha\int_{\frac{1}{2}+i\mathbb{R}_{+}}\mathrm{d}j\,\sigma^{w}(\mathcal{C}^{j}_{\alpha})\otimes\sigma^{w}(\overline{\mathcal{C}}^{j}_{\alpha})\right)\oplus\left(\int_{\frac{1}{2}}^{\frac{k-1}{2}}\mathrm{d}j\,\sigma^{w}(\mathcal{D}^{+}_{j})\otimes\sigma^{w}(\overline{\mathcal{D}}^{+}_{j})\right)\right]\,.$
(29)

Focusing on $w\geq 0$, vertex operators $V_{m,\bar{m},j}^{w}$ satisfy the OPEs

$\begin{split}J^{3}(z)V_{m,\bar{m},j}^{w}(y)&\sim\frac{(m+kw/2)V_{m,\bar{m},j}(y)}{z-y}\,,\\\
J^{\pm}(z)V_{m,\bar{m},j}^{w}(y)&\sim\frac{(m\pm j)V_{m\pm
1,\bar{m},j}(y)}{(z-y)^{1\pm w}}\,.\end{split}$ (30)

Since $J^{3}_{0}$ ($\bar{J}^{3}_{0}$) reads off the conformal weight $h$
($\bar{h}$) of the dual CFT operator, we have

$h=m+\frac{kw}{2}\,,\quad\bar{h}=\bar{m}+\frac{kw}{2}\,.$ (31)

Furthermore, the vertex operators $V_{m,\bar{m},j}^{w}$ have worldsheet
conformal weight

$\begin{split}\Delta&=\frac{j(1-j)}{k-2}-hw+\frac{kw^{2}}{4}\,,\\\
\overline{\Delta}&=\frac{j(1-j)}{k-2}-\bar{h}w+\frac{kw^{2}}{4}\,.\end{split}$
(32)

The spectrally-flowed vertex operators $V_{m,\bar{m},j}^{w}$ do not admit a
clean expression as an analytic function of the free fields. However, as was
pointed out in Knighton:2023mhq , one can formally express spectrally-flowed
vertex operators as formal distributions in the fundamental fields and their
derivatives, namely

$\begin{split}V_{m,\bar{m},j}^{w}=\,&e^{(w/Q-Qj)\Phi}\bigg{(}\frac{\partial^{w}\gamma}{w!}\bigg{)}^{-m-j}\bigg{(}\frac{\overline{\partial}^{w}\bar{\gamma}}{w!}\bigg{)}^{-\bar{m}-j}\delta_{w}^{(2)}(\gamma)\\\
&\hskip
28.45274pt+R(m,\bar{m},j)\,e^{(w/Q-Q(1-j))\Phi}\bigg{(}\frac{\partial^{w}\gamma}{w!}\bigg{)}^{-m-(1-j)}\bigg{(}\frac{\overline{\partial}^{w}\bar{\gamma}}{w!}\bigg{)}^{-\bar{m}-(1-j)}\delta_{w}^{(2)}(\gamma)\,.\end{split}$
(33)

The novelty of this expression lies in the delta function operators

$\delta_{w}^{(2)}(\gamma):=\delta(\gamma)\delta(\partial\gamma)\cdots\delta\left(\frac{\partial^{w-1}\gamma}{(w-1)!}\right)\delta(\bar{\gamma})\delta(\overline{\partial}\bar{\gamma})\cdots\delta\left(\frac{\overline{\partial}^{w-1}\bar{\gamma}}{(w-1)!}\right)\,.$
(34)

Intuitively, these operators constrain the path integral to fields $\gamma$
which have a zero of order $w$ at the insertion point of
$V_{m,\bar{m},j}^{w}$.999See Frenkel:2006fy ; Frenkel:2008vz  for a similar
construction in the context of topological sigma models. Just as in the case
of the unflowed operators (25), one should think of equation (33) as defining
nontrivial terms in an asymptotic expansion valid at large $\Phi$.

For ease of notation, we will define the combination

$\Phi_{m,\bar{m},j}^{w}:=e^{(w/Q-Qj)\Phi}\bigg{(}\frac{\partial^{w}\gamma}{w!}\bigg{)}^{-m-j}\bigg{(}\frac{\overline{\partial}^{w}\bar{\gamma}}{w!}\bigg{)}^{-\bar{m}-j}\delta_{w}^{(2)}(\gamma)\,,$
(35)

so that the full vertex operators take the form

$V_{m,\bar{m},j}^{w}=\Phi_{m,\bar{m},j}^{w}+R(m,\bar{m},j)\Phi_{m,\bar{m},1-j}^{w}\,.$
(36)

#### The $x$ basis

The vertex operators described above correspond to strings emitted from the
asymptotic past of $\text{AdS}_{3}$. In Euclidean language, this means that
the strings are emitted from the point $\gamma=\bar{\gamma}=0$ on the
conformal boundary. In order to describe strings emitted from an arbitrary
point $x$ on the boundary, we can use the translation operator $L^{\rm
CFT}_{-1}=J^{+}_{0}=\beta_{0}$. That is, we define

$V_{m,\bar{m},j}^{w}(x,z):=e^{xJ^{+}_{0}}V_{m,\bar{m},j}^{w}(z)e^{-xJ^{+}_{0}}\,.$
(37)

Using the identities

$e^{x\beta_{0}}\Phi e^{-x\beta_{0}}=\Phi\,,\quad e^{x\beta_{0}}\gamma
e^{-x\beta_{0}}=\gamma-x\,,$ (38)

we can readily write down the $x$ basis vertex operators as

$V_{m,\bar{m},j}^{w}(x,z)=\Phi_{m,\bar{m},j}^{w}(x,z)+R(m,\bar{m},j)\Phi_{m,\bar{m},1-j}^{w}(x,z)\,,$
(39)

with

$\Phi_{m,\bar{m},j}^{w}(x,z)=e^{(w/Q-Qj)\Phi}\bigg{(}\frac{\partial^{w}\gamma}{w!}\bigg{)}^{-m-j}\bigg{(}\frac{\overline{\partial}^{w}\bar{\gamma}}{w!}\bigg{)}^{-\bar{m}-j}\delta_{w}^{(2)}(\gamma-x)\,.$
(40)

###  2.3 Screening operators

Constant $\Phi$Constant $(\gamma,\bar{\gamma})$ Figure 2: Euclidean
$\text{AdS}_{3}$ in Poincaré coordinates. The shaded orange region denotes the
set $\Phi>\Phi_{0}$ for some large $\Phi_{0}$. Near the north pole
$\gamma=\infty$, the coordinates are not defined.

We are primarily interested in computing correlators in the asymptotic region
of $\text{AdS}_{3}$ near the boundary, which is specified by $\Phi>\Phi_{0}$
for some large value of $\Phi_{0}$. However, as shown in Figure 2, this subset
does not include the entire conformal boundary, but rather misses the point at
infinity. This is fundamentally due to working in Poincaré coordinates, which
do not fully cover the conformal boundary. This problem must be remidied if we
want to describe a CFT which lives on the full sphere $\text{S}^{2}$ instead
of the complex plane $\mathbb{C}$.

One way around this would be to cover $\text{AdS}_{3}$ in two coordinate
patches which together cover the full conformal boundary. For example, these
coordinate patches could cover the sets $|\gamma|\leq 1/\varepsilon$ and
$|\gamma|>1/\varepsilon$ for some $\varepsilon$. Taking $\varepsilon\to 0$
results in two patches: one which covers all finite values of $\gamma$ and one
which covers $\gamma=\infty$. From the point of view of the worldsheet sigma
model, including the second patch can be achieved by ‘allowing’ $\gamma$ to
diverge at certain points on the worldsheet. As was argued in Dei:2023ivl ;
Knighton:2023mhq ; Hikida:2023jyc , these divergences can be put in by hand by
inserting vertex operators which transform trivially under all worldsheet
symmetry currents $J^{a}$ but which have a simple pole with $\gamma$.

Such a vertex operator can indeed be constructed and takes the form
Giveon:2019gfk ; Halder:2022ykw ; Hikida:2023jyc ; Knighton:2023mhq 101010This
screening operator is related to the so-called ‘secret’ representations of
Eberhardt:2019ywk , as explained in Knighton:2023mhq . It also shares many
features with the so-caled $W$ field of the minimal tension worldsheet theory
on $\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ Dei:2020zui .

$D=V_{m=\frac{k}{2},\bar{m}=\frac{k}{2},j=\frac{k-2}{2}}^{w=-1}\,.$ (41)

This operator has spacetime conformal weight $h=\bar{h}=0$ and worldsheet
conformal weight $\Delta=\overline{\Delta}=1$. However, it does not exactly
live in the trivial representation of $\mathfrak{sl}(2,\mathbb{R})_{k}$, but
its integral does. Indeed, $D$ has trivial OPEs with $J^{3}$ and $J^{+}$,
while its OPE with $J^{-}$ is a total derivative

$J^{-}(z)D(w,\bar{w})\sim\partial_{w}(\cdots)\,,\quad\bar{J}^{-}(\bar{z})D(w,\bar{w})\sim\overline{\partial}_{w}(\cdots)\,.$
(42)

Thus, the nonlocal operator

$\mathcal{O}=\int_{\Sigma}D$ (43)

is completely invisible to the $\mathfrak{sl}(2,\mathbb{R})_{k}$ current
algebra. Furthermore, $D$ satisfies

$\gamma(z)D(w,\bar{w})\sim\mathcal{O}\left(\frac{1}{z-w}\right)\,,\quad\bar{\gamma}(\bar{z})D(w,\bar{w})\sim\left(\frac{1}{\bar{z}-\bar{w}}\right)\,.$
(44)

For use in the path integral, it is helpful to write $D$ out in an explicit
field-dependent form:

$D=e^{-2\Phi/Q}\left|\oint\frac{\mathrm{d}z}{2\pi
i}\,\gamma\right|^{-2(k-1)}\delta^{(2)}(\beta)\,.$ (45)

The contour integral should be taken around the insertion point of $D$, and
can equivalently be thought of as reading off the residue of $\gamma$ at the
pole enforced by the insertion of $D$. Again, we have defined the delta
function operator

$\delta^{(2)}(\beta):=\delta(\beta)\delta(\bar{\beta})\,.$ (46)

For the rest of the paper, we will refer to $\mathcal{O}$ as a screening
operator, keeping in tradition with the $\text{AdS}_{3}$ string theory
literature Giribet:2001ft ; Iguri:2007af . We should also note that the
operator $\mathcal{O}$ shares similarities with the interaction operator
$\nu^{-1}\beta\bar{\beta}e^{-Q\Phi}$ in that they both have conformal weights
$h=\bar{h}=0$, $\Delta=\overline{\Delta}=1$, have (almost) trivial OPEs with
the currents $J^{a}$, and have a simple pole in their OPEs with $\gamma$ and
$\bar{\gamma}$. However, they are crucially not equivalent operators (for one,
they have different charges under $\partial\Phi$).

With the above definitions in mind, we propose that the correct prescription
for computing correlators in the near-boundary region of Euclidean
$\text{AdS}_{3}$ is to compute correlators in the free field theory with an
arbitrary number of copies of $\mathcal{O}$ inserted. That is, we want to
compute

$\begin{split}\sum_{N=0}^{\infty}\frac{p^{N}}{N!}&\Braket{\mathcal{O}^{N}\prod_{i=1}^{n}V_{m_{i},\bar{m}_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\\\
&=\sum_{N=0}^{\infty}\frac{p^{N}}{N!}\int\mathrm{d}^{2}\lambda_{1}\cdots\mathrm{d}^{2}\lambda_{N}\Braket{D(\lambda_{1},\bar{\lambda_{1}})\cdots
D(\lambda_{N},\bar{\lambda}_{N})\prod_{i=1}^{n}V_{m_{i},\bar{m}_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\,.\end{split}$
(47)

The factor of $1/N!$ is included since the positions of the poles of $\gamma$
are indistinguishable. We have also included a fugacity $p$ which counts the
number of poles in $\gamma$. On dimensional grounds we have
$p\propto\nu^{2-k}$, but we will keep it generic for now. Performing the sum
over $N$, we see that including the screening operator $\mathcal{O}$ amounts
to changing the action to

$S=\frac{1}{2\pi}\int\left(\frac{1}{2}\partial\Phi\,\overline{\partial}\Phi+\beta\overline{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}+\nu\beta\bar{\beta}e^{-Q\Phi}-\frac{Q}{4}R\Phi-
pD\right)\,,$ (48)

so that (47) is reproduced by conformal perturbation theory in $p$. In Section
4 we will return to the problem of calculating the correlation functions in
equation (47) directly in the path integral formalism.

##  3 The CFT dual

Let us now turn our attention to the other side of the duality before actually
computing worldsheet correlation functions in Section 4. The CFT is based on a
non-normalizable deformation of a symmetric orbifold Eberhardt:2021vsx ;
Dei:2022pkr :

$\text{Sym}(\mathbb{R}_{\mathcal{Q}}\times X)+\mu\int\sigma_{2,\alpha}\,,$
(49)

where $\mathbb{R}_{\mathcal{Q}}$ is a linear dilaton with background charge
$\mathcal{Q}$ (which we specify below), and $\sigma_{2,\alpha}$ is a marginal
operator in the twist-2 sector of the symmetric orbifold which carries
momentum $\alpha$ with respect to the linear dilaton. We will review the
details of this CFT as well as the computation of residues of poles in its
correlators in the next few pages.

###  3.1 The symmetric orbifold

The dual CFT is based on a (non-normalizable) deformation of the symmetric
product orbifold of $\mathbb{R}_{\mathcal{Q}}\times X$. The linear dilaton
$\mathbb{R}_{\mathcal{Q}}$ is described by a scalar field $\varphi$ with
action

$S[\varphi]=\frac{1}{2\pi}\int_{B}\left(\frac{1}{2}\partial\varphi\,\overline{\partial}\varphi-\frac{\mathcal{Q}}{4}R\varphi\right)\,.$
(50)

Here, we have denoted by $B$ the conformal boundary
$B\cong\partial\text{AdS}_{3}$. For the purpose of this paper, we are
interested in the case $B\cong\text{S}^{2}$. The background charge
$\mathcal{Q}$ is related to the level $k$ of the $\text{AdS}_{3}$ string
theory by

$\mathcal{Q}=Q-\frac{2}{Q}=-\frac{\sqrt{2}(k-3)}{\sqrt{k-2}}\,.$ (51)

The central charge of the theory is

$c(\mathbb{R}_{\mathcal{Q}})=1+3\mathcal{Q}^{2}\,.$ (52)

The central charge of the internal CFT $X$ is determined by the consistency of
the string background $\text{AdS}_{3}\times X$ to be

$c(X)=26-\frac{3k}{k-2}\,,$ (53)

such that the total central charge is

$c(\mathbb{R}_{\mathcal{Q}}\times X)=6k\,,$ (54)

as expected from the asymptotic symmetry algebra Giveon:1998ns . Primary
states in the $\mathbb{R}_{\mathcal{Q}}\times X$ theory are given by tensor
products of exponentials $e^{-q\varphi}$ with a state $\mathcal{O}_{X}$ of the
internal CFT. We call $q$ the ‘momentum’ of the linear dilaton $\varphi$, and
the minus sign is chosen so that

$\partial\varphi(z)\,e^{-q\varphi}(w)\sim\frac{q}{z-w}e^{-q\varphi}(w)\,.$
(55)

The state $e^{-q\varphi}\otimes\mathcal{O}_{X}$ has conformal weight

$h=\frac{q(\mathcal{Q}-q)}{2}+h_{X}\,,\quad\bar{h}=\frac{q(\mathcal{Q}-q)}{2}+\bar{h}_{X}\,.$
(56)

The symmetric product orbifold $\text{Sym}^{K}(\mathbb{R}_{\mathcal{Q}}\times
X)$ is obtained by orbifolding $K$ copies of $\mathbb{R}_{\mathcal{Q}}\times
X$ by the $S_{K}$ permutation symmetry interchanging them. As in any orbifold,
the spectrum is organized into twisted and untwisted sectors labeled by
conjugacy classes of the orbifold group. The single-cycle sectors are
generated by permutations of the form $[(1\ldots w)]$ for $w\geq 1$. Given a
vertex operator in $\mathbb{R}_{\mathcal{Q}}\times X$ obeying $h-\bar{h}\equiv
0\,\,(\text{mod }w)$, there exists an operator in the $w$-twisted sector with
conformal weight

$h_{w}=\frac{h}{w}+\frac{k(w^{2}-1)}{4w}\,,\quad\bar{h}_{w}=\frac{\bar{h}}{w}+\frac{k(w^{2}-1)}{4w}\,.$
(57)

The restriction on the allowed values of $(h,\bar{h})$ is such that
$h_{w}-\bar{h}_{w}$ is integral, a necessity for locality of the theory. For
the case of primary fields in $\mathbb{R}_{\mathcal{Q}}\times X$, we have

$\begin{split}h_{w}&=\frac{q(\mathcal{Q}-q)}{2w}+\frac{h_{X}}{w}+\frac{k(w^{2}-1)}{4w}\,,\\\
\bar{h}_{w}&=\frac{q(\mathcal{Q}-q)}{2w}+\frac{\bar{h}_{X}}{w}+\frac{k(w^{2}-1)}{4w}\,.\end{split}$
(58)

The spectrum of single-cycle states can be recovered from the worldsheet
theory on $\text{AdS}_{3}\times X$ by considering a state of the form
$V_{m,\bar{m},j}^{w}\otimes\mathcal{O}_{X}$ and imposing the mass-shell
condition $\Delta=\overline{\Delta}=1$. Using equation (32), the mass-shell
equation is

$\begin{split}\frac{j(1-j)}{k-2}-h_{w}w+\frac{kw^{2}}{4}+h_{X}&=1\,,\\\
\frac{j(1-j)}{k-2}-\bar{h}_{w}w+\frac{kw^{2}}{4}+\bar{h}_{X}&=1\,.\end{split}$
(59)

Solving for $(h_{w},\bar{h}_{w})$ gives equation (58) if one makes the
identification

$q=Qj-\frac{1}{Q}\,.$ (60)

For strings living in the continuous representation
$\sigma^{w}(\mathcal{C}^{j}_{\alpha})\otimes\sigma^{w}(\overline{\mathcal{C}}^{j}_{\alpha})$,
we see that the worldsheet theory nearly recovers the single-cycle spectrum of
the symmetric product orbifold. However, there are two subtleties:

  * •

The worldsheet spin $j$ enjoys the reflection symmetry $j\to 1-j$. In terms of
the dual CFT, this would naively imply the reflection symmetry

$q\to\mathcal{Q}-q$ (61)

on the momenta of the linear dilaton $\varphi$.

  * •

The spins in the continuous representations are taken to live on the strip
$j\in\frac{1}{2}+i\mathbb{R}$. In terms of the momenta $q$, this condition
would translate to

$q\in\frac{\mathcal{Q}}{2}+i\mathbb{R}\,.$ (62)

As was pointed out by the authors of Eberhardt:2019qcl , both of these
conditions can be satisfied if instead of a linear dilaton
$\mathbb{R}_{\mathcal{Q}}$ one takes the symmetric orbifold of Liouville
theory with background charge $\mathcal{Q}$. However, as was pointed out in
Eberhardt:2021vsx , the proposal of Eberhardt:2019qcl  does not produce the
correct worldsheet $n$-point functions, and thus cannot be the correct
description of the dual CFT.

###  3.2 Perturbing the symmetric orbifold

The proposal of Eberhardt:2021vsx  was to instead consider the symmetric
orbifold $\text{Sym}(\mathbb{R}_{\mathcal{Q}}\times X)$ deformed by an
operator of the form

$S\to S+\mu\int\sigma_{2,\alpha}\,.$ (63)

Here, $\sigma_{2,\alpha}$ is a state in the $w=2$ twisted sector with linear
dilaton momentum $\alpha$. This operator is classically marginal if
$h(\sigma_{2,\alpha})=1$, which is solved by

$\alpha=\frac{1}{Q}=\sqrt{\frac{k-2}{2}}\,.$ (64)

Just as in the case of Liouville theory, the operator $\sigma_{2,\alpha}$
creates an exponential potential in $\varphi$ of which states can scatter off.
The result is that states with momenta $q$ and $\mathcal{Q}-q$ are no longer
linearly independent. This leads to the identification of vertex operators
with momentum $q$ and $\mathcal{Q}-q$, as predicted from the string theory.
Focusing purely on the linear dilaton factor, states $\sigma_{w,q}$ in the
single-cycle sector of the deformed theory are labeled by their twist $w$ and
momentum $q$, and satisfy the reflection property

$\sigma_{w,\mathcal{Q}-q}=R(h,\mathcal{Q}-q)\sigma_{w,\mathcal{Q}}\,,$ (65)

where $R(h,q)$ is a reflection coefficient which depends on the conformal
weight $h$ of the state and satisfies the reflection property

$R(h,q)R(h,\mathcal{Q}-q)=1\,.$ (66)

###  3.3 Correlators in the dual CFT

We consider correlation functions of the form

$\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})}_{\mu}\,,$ (67)

where the subscript reminds us that we are computing these correlation
functions in the deformed theory. Naively, this would be computed in conformal
perturbation theory as

$\sum_{m=0}^{\infty}\frac{(-\mu)^{m}}{m!}\int\mathrm{d}^{2}\xi_{1}\cdots\mathrm{d}^{2}\xi_{m}\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})\prod_{\ell=1}^{m}\sigma_{2,\alpha}(\xi_{\ell})}_{\mu=0}\,.$
(68)

However, like in Liouville theory, the non-normalizability of the operator
$\sigma_{2,\alpha}$ means that naive perturbation theory is not valid.

The correct statement is as follows: the correlator (67) scales with a power
law in $\mu$, namely Eberhardt:2021vsx

$\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})}_{\mu,g}\propto\mu^{-(\sum_{i=1}^{n}q_{i}-\mathcal{Q}(1-g))/\alpha}\,,$
(69)

where $g$ is the genus of the contribution, determined by the order in the
$1/K$ expansion. This correlator has poles when the exponent of $\mu$ is a
non-negative integer, i.e. when

$\sum_{i=1}^{n}q_{i}-\mathcal{Q}(1-g)=-\alpha m\,,\quad m\in\mathbb{Z}_{\geq
0}\,.$ (70)

Viewing (67) as an analytic function in the sum $\sum_{i=1}^{n}q_{i}$ of the
momenta, the residues of these poles will be given by Eberhardt:2021vsx

$\begin{split}&\mathop{\text{Res}}_{\sum_{i}q_{i}-\mathcal{Q}(1-g)=-\alpha
m}\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})}_{\mu}\\\ &\hskip
56.9055pt=\frac{(-\mu)^{m}}{m!}\int\mathrm{d}^{2}\xi_{1}\cdots\mathrm{d}^{2}\xi_{m}\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})\prod_{\ell=1}^{m}\sigma_{2,\alpha}(\xi_{\ell})}_{\mu=0}\,.\end{split}$
(71)

Thus, while the form of the full interacting correlator (67) is not known, by
naive perturbation theory we can extract the residues of the poles in momentum
space. Currently, this is all of the analytic data that can be explicitly
calculated in the CFT proposal of Eberhardt:2021vsx , and it is precisely the
data we will recover below from the near-boundary limit of AdS3 string theory.

#### Symmetric orbifold correlators

Let us work out the naive perturbative answer (68) explicitly. For this, we
need to know how to compute correlators of the form111111In general, we could
also consider nontrivial states in the seed CFT $X$. While we focus
exclusively on states which are in the vacuum of $X$ here, the following
discussion, as well as the calculations in Section 4, generalize in a
straightforward manner.

$\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})\prod_{\ell=1}^{m}\sigma_{2,\alpha}(\xi_{\ell})}_{\mu=0}\,.$
(72)

Such correlation functions can be computed using the covering space method
Hamidi:1986vh ; Lunin:2001ne . This method begins by noting that the path
integral computing the above correlator is an integral over fields which are
not single-valued. In order to compute such a correlator, it is simpler to
move to a covering space on which the fundamental fields lift to single-valued
fields. Each covering space is realized by a surface $\Sigma$ along with a
holomorphic map $\gamma:\Sigma\to\text{S}^{2}$ satisfying

$\begin{split}\gamma(z)\sim
x_{i}+a^{\gamma}_{i}(z-z_{i})^{w_{i}}+\cdots\,,&\quad z\to z_{i}\\\
\gamma(z)\sim\xi_{\ell}+b^{\gamma}_{\ell}(z-\zeta_{\ell})^{2}+\cdots\,,&\quad
z\to\zeta_{\ell}\,,\end{split}$ (73)

where $z_{i},\zeta_{\ell}$ are marked points on $\Sigma$ and
$a^{\alpha}_{i},b^{\gamma}_{\ell}$ constants determined by the points
$(z_{i},\zeta_{\ell})$ and $(x_{i},\xi_{\ell})$. For the moment, let us assume
that $\Sigma$ is a connected surface of genus $g$ – this turns out to capture
the contributions to (72) which come from a single string worldsheet in the
bulk. Requiring the map $\gamma$ to be holomorphic puts severe restrictions on
the allowed moduli of the surface $\Sigma$ and marked points
$z_{i},\zeta_{\ell}$. Indeed, it can be shown that the allowed set
$(\Sigma,z_{i},\zeta_{\ell})\in\mathcal{M}_{g,n+m}$ is finite for a given
fixed genus $g$. The genus $g$ of the covering space is determined by the
number of poles $N$ of $\gamma$ by the Riemann-Hurwitz relation

$N=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}+\frac{m}{2}\,.$ (74)

The locations $\lambda_{a}$ of the poles are determined uniquely once a
covering space is fixed, and we denote by $c^{\gamma}_{a}$ the residues of
$\gamma$ at $z=\lambda_{a}$, i.e.

$\gamma(z)\sim\frac{c^{\gamma}_{a}}{z-\lambda_{a}}\,,\quad z\to\lambda_{a}\,.$
(75)

We emphasize that the values of $c^{\gamma}_{a}$ are determined uniquely once
a covering surface is chosen. Finally, it is convenient to introduce a
constant $C^{\gamma}$ such that

$\partial\gamma(z)=C^{\gamma}\frac{\prod_{i=1}^{n}(z-z_{i})^{w_{i}-1}\prod_{\ell=1}^{m}(z-\zeta_{\ell})}{\prod_{a=1}^{N}(z-\lambda_{a})^{2}}\,.$
(76)

Again, $C^{\gamma}$ is determined uniquely by the covering surface data.

The correlation function (72) is then determined purely in terms of the
analytic data of the covering maps $\gamma:\Sigma\to\text{S}^{2}$ as well as
the data of the seed CFT $\mathbb{R}_{\mathcal{Q}}\times X$. Specifically
focusing on covering spaces of genus $g=0$ the correlator (72) can be written
in terms of the data
$(a_{i}^{\gamma},b_{\ell}^{\gamma},c_{a}^{\gamma},C^{\gamma})$ as Dei:2019iym
; Hikida:2020kil ; Eberhardt:2021vsx 121212The factor of $|C^{\gamma}|^{k}$ is
usually omitted in the literature, but can be derived from a careful path
integral analysis of the symmetric orbifold. Alternatively, it is necessary
for the correlator to obey the conformal Ward identities. See Dei:2023ivl  for
a discussion on this point.

$\begin{split}&\sum_{\begin{subarray}{c}\text{connected covering}\\\
\text{spaces
}\gamma\end{subarray}}K^{1-\frac{n+m}{2}}|C^{\gamma}|^{k}\prod_{a=1}^{N}|c^{\gamma}_{a}|^{-k}\prod_{i=1}^{n}\left(w_{i}^{\frac{1}{2}-\frac{k(w_{i}+1)}{2}}|a_{i}^{\gamma}|^{-2h_{i}+\frac{k(w_{i}-1)}{2}}\right)\\\
&\hskip
113.81102pt\times\prod_{\ell=1}^{m}\left(2^{\frac{1}{2}-\frac{3k}{2}}|b_{\ell}^{\gamma}|^{\frac{k-4}{2}}\right)\Braket{\prod_{i=1}^{n}e^{q_{i}\varphi}(z_{i})\prod_{\ell=1}^{m}e^{\alpha\varphi}(\zeta_{\ell})}\,.\end{split}$
(77)

The various factors of $2$ and $w_{i}$ arise from a careful analysis of the
normalization of twist fields and the combinatorial factors arising in the
large $K$ expansion.131313We note that, crucially, equation (77) should be
thought of as asymptotic in $K$. At finite $K$, the dependence on $K$ is much
more complicated. This is in contrast to, say, $\text{U}(N)$ gauge theories,
for which a single Feynman diagram admits a unique power of $N$, even at
finite $N$. Here, $h_{i}$ are the conformal weights of the states in the seed
CFT, i.e. those given by equation (58). The power of $K$ is $\frac{1}{2}\chi$,
where $\chi$ is the Euler character of a sphere with $n+m$ punctures.

Still focusing on $g=0$, the seed CFT correlators are readily computed by Wick
contractions:

$\begin{split}\Braket{\prod_{i=1}^{n}e^{-q_{i}\varphi}(z_{i})\prod_{\ell=1}^{m}e^{-\alpha\varphi}(\zeta_{\ell})}=&\prod_{i<j}|z_{i}-z_{j}|^{-q_{i}q_{j}}\prod_{i,\ell}|z_{i}-\zeta_{\ell}|^{-q_{i}\alpha}\prod_{\ell<k}|\zeta_{\ell}-\zeta_{k}|^{-\alpha^{2}}\\\
&\hskip
85.35826pt\times\delta\bigg{(}\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}(1-g)\bigg{)},\end{split}$
(78)

so that the naive perturbative answer (68) evaluates to

$\begin{split}K\sum_{m=0}^{\infty}&\frac{\left(-K^{-\frac{1}{2}}2^{\frac{1}{2}-\frac{3k}{2}}\mu\right)^{m}}{m!}\sum_{\begin{subarray}{c}\text{connected
covering}\\\ \text{spaces
}\gamma\end{subarray}}\prod_{i=1}^{n}\left(K^{-\frac{1}{2}}w_{i}^{\frac{1}{2}-\frac{k(w_{i}+1)}{2}}\right)\\\
&\times\int\mathrm{d}^{2}\xi_{1}\ldots\mathrm{d}^{2}\xi_{m}\,|C^{\gamma}|^{k}\prod_{a=1}^{N}|c^{\gamma}_{a}|^{-k}\prod_{i=1}^{n}|a_{i}^{\gamma}|^{-2h_{i}+\frac{k(w_{i}-1)}{2}}\prod_{\ell=1}^{m}|b_{\ell}^{\gamma}|^{\frac{k-4}{2}}\\\
&\times\prod_{i<j}|z_{i}-z_{j}|^{-q_{i}q_{j}}\prod_{i,\ell}|z_{i}-\zeta_{\ell}|^{-q_{i}\alpha}\prod_{\ell<k}|\zeta_{\ell}-\zeta_{k}|^{-\alpha^{2}}\delta\bigg{(}\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}(1-g)\bigg{)}.\end{split}$
(79)

Note that we have collected all of the numerical prefactors into the first
line. Since $m\geq 0$ is an integer, we see that this correlation function
diverges whenever the momenta $q_{i}$ satisfy

$\sum_{i=1}^{n}q_{i}=\mathcal{Q}(1-g)-m\alpha\,,\quad m\in\mathbb{Z}_{\geq
0}\,.$ (80)

As mentioned above, in the full interacting CFT, the correlators also diverge
at these values of the momenta. In fact, the correlators (67) are analytic
functions in the momenta $q_{i}$ which have poles at these points. The
residues of these poles are given by141414The fact that residues are related
to coefficients of delta functions can also be understood from the formula
$\frac{1}{x-i\epsilon}=\pi\delta(x)+\mathcal{P}(\frac{1}{x})$, where
$\mathcal{P}(\frac{1}{x})$ denotes the principal value. Note also that the
divergence on the right-hand-side comes from the delta function since
$\mathcal{P}(\frac{1}{0})=0$.

$\mathop{\mathrm{Res}}_{\sum_{i}q_{i}-\mathcal{Q}(1-g)=-m\alpha}\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})}_{\mu}=\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})\prod_{\ell=1}^{m}\sigma_{2,\alpha}}_{\mu=0}^{\prime}\,,$
(81)

where the prime means the correlator without the momentum-conserving delta
function inserted. Intuitively, the full interacting correlation function has
all of the divergences of (79), but with the delta functions ‘smoothed out’
into poles in momentum space.

In addition to the poles located at (80), the reflection symmetry
$q_{i}\to\mathcal{Q}-q_{i}$ means that there are poles located at all of the
reflected values of the momenta. The residues of these poles will be weighted
with appropriate factors of the reflection coefficients
$R(h_{i},\mathcal{Q}-q_{i})$.

###  3.4 Differences at $k\leq 3$ vs $k>3$

In the above discussion, we were simply interested in the poles of CFT
correlators as analytic functions of the momenta $q_{i}$. These poles occur
when the momenta $q_{i}$ satisfy the linear relation

$\sum_{i=1}^{n}q_{i}+m\alpha=\mathcal{Q}\,,\quad m\in\mathbb{Z}_{\geq 0}\,,$
(82)

as well as all images under the reflection symmetry
$q_{i}\to\mathcal{Q}-q_{i}$. However, the spectrum of the theory only includes
momenta on the critical line $\text{Re}(q_{i})=\mathcal{Q}/2$ in the complex
plane. This implies that

$\text{Re}\left(\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}\right)=\frac{\mathcal{Q}}{2}(n-2)+\frac{m}{Q}\,.$
(83)

Note that if $\mathcal{Q}>0$, then there is no way to satisfy the momentum
conservation (82) if $n>2$. From the path integral, this can be seen as
follows. After passing to the covering space, the integral over the zero mode
$\varphi_{\infty}$ of $\varphi$ takes the form

$\int^{\infty}\mathrm{d}\varphi_{\infty}\,\exp\left(-\left(\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}(1-g)\right)\varphi_{\infty}\right)\,,$
(84)

where we are agnostic about the lower-bound of integration. For

$\text{Re}\left(\sum_{i=1}^{n}q_{i}+m-\mathcal{Q}(1-g)\right)=\frac{\mathcal{Q}}{2}(n+2g-2)+\frac{m}{Q}>0\,,$
(85)

the worldsheet integral is well-defined and there are no poles in the
correlation functions. For generic correlators, this inequality is satisfied
only if $\mathcal{Q}\geq 0$ or, in $\text{AdS}_{3}$ terms, $k\leq 3$. If
$k>3$, then the above inequality is in general not satisfied and the path
integral is not well-defined. In Balthazar:2021xeh , it was argued the
divergence of the linear dilaton path integral is a sign that the boundary CFT
is only well-defined for $k<3$, corresponding to ‘sub-stringy’ values of the
$\text{AdS}_{3}$ radius.

However, as was noted in Eberhardt:2021vsx ; Dei:2022pkr , worldsheet
correlators in the dual CFT are analytic in the momenta $q_{i}$. Thus, while
the zero-mode integral formally diverges for the values of $q_{i}$ which live
in the CFT spectrum, one can analytically continue the answer of the integral
to be:

$\begin{split}&\int^{\infty}\mathrm{d}\varphi_{\infty}\,\exp\left(-\left(\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}(1-g)\right)\varphi_{\infty}\right)\\\
&\hskip
142.26378pt\sim\left(\sum_{i=1}^{n}q_{i}+m\alpha-\mathcal{Q}(1-g)\right)^{-1}\,.\end{split}$
(86)

The divergences now manifest as poles in the correlation functions. Thus,
while the theory naively seems to only be well-defined for $k\leq 3$, the
analyticity of the correlators seems to be enough to define correlators in the
regime $k>3$, at least perturbatively in $\mu$. Thus, since we are only
interested in the analytic behavior of the correlators in $q_{i}$, we remain
agnostic as to whether the CFT is defined in the regime $k>3$.

##  4 Computing the string correlators

In this section we will show that the worldsheet correlators in the near-
boundary limit of $\text{AdS}_{3}$ string theory computes the naive
perturbative correlation functions in the dual CFT described in the previous
section. We will focus entirely on the leading contribution in the string
coupling, which is dominated on the worldsheet side by genus-zero worldsheets,
and on the CFT side by covering maps of genus zero, see equation (77).

The statement of the duality (at tree-level) is that the integrated worldsheet
correlators should precisely reproduce the (connected) CFT correlators.
Schematically,

$\displaystyle
g_{s}^{-2}\int_{\mathcal{M}_{0,n}}\Braket{\prod_{i=1}^{n}V_{i}(x_{i},z_{i})}=\Braket{\prod_{i=1}^{n}\mathcal{O}_{i}^{\rm
CFT}(x_{i})}_{\text{tree-level}}\,,$ (87)

where $\mathcal{M}_{0,n}$ is the moduli space of $n$-punctured spheres and the
subscript on the CFT correlator reminds us that we are taking connected
components which contribute at leading order in $1/K$. Here, $V_{i}$ are
vertex operators on the worldsheet and $\mathcal{O}_{i}^{\rm CFT}$ are the
corresponding states in the dual CFT. Actually, the result we want to show is
slightly weaker, since we should allow for relative normalizations
$\mathcal{N}_{i}$ of the vertex operators on both sides, as well as an overall
relative normalization $C_{\text{S}^{2}}$ related to the definitions of the
path integrals on the string theory and CFT sides:

$\displaystyle
g_{s}^{-2}\int_{\mathcal{M}_{0,n}}\Braket{\prod_{i=1}^{n}V_{i}(x_{i},z_{i})}=\left(C_{\text{S}^{2}}\prod_{i=1}^{n}\mathcal{N}_{i}\right)\Braket{\prod_{i=1}^{n}\mathcal{O}_{i}^{\rm
CFT}(x_{i})}_{\text{tree-level}}\,.$ (88)

Ultimately, proving equation (88) is out of our reach, since neither the
worldsheet string theory nor the dual CFT have been completely solved.
However, as we have reviewed in Sections 2 and 3, both theories have
‘subsectors’ which are solvable. On the string theory side, this is the limit
$\Phi\to\infty$, in which strings are taken to lie near the asymptotic
boundary of $\text{AdS}_{3}$, and for which the worldsheet sigma model becomes
a free theory (with a screening operator). On the dual CFT side, the simple
subsector is the limit $\varphi\to\infty$, for which we can treat the
perturbative expansion (79) seriously.

The goal of this section is to show that the correlators computed from the
respective weak-coupling sectors precisely agree. The strategy is to take the
near-boundary limit $\Phi\to\infty$ seriously and compute correlation
functions in the resulting worldsheet theory in the path integral. Since the
worldsheet theory becomes free, and the $\beta\gamma$ integral contains a
large number of delta-functions, we find that this calculation can be done in
a straightforward manner. As we will show below, the integrated correlator in
the near-boundary theory precisely reproduces the naive perturbation series
(68) in the dual CFT.

###  4.1 Setting the stage

Let us walk through the steps of computing the worldsheet path integral. We
largely follow the exposition of Knighton:2023mhq . We will postpone the main
technical challenges to the next subsection.

Throughout this section, we focus on genus zero worldsheet correlators, but
will keep the genus generic in formulae which apply more generally. At genus
zero, the only worldsheet moduli are the $n-3$ cross ratios parametrizing
$\mathcal{M}_{0,n}$. One can use the $\rm SL(2,\mathbb{C})$ symmetry to fix
three positions of the vertex operators with the price of introducing three
conformal $c$ ghost insertions at those points. Thus, a worldsheet amplitude
takes the form

$\displaystyle\int\mathrm{d}^{2}z_{4}\cdots\mathrm{d}^{2}z_{n}\Braket{c\bar{c}V_{1}(x_{1},z_{1})c\bar{c}V_{2}(x_{2},z_{2})c\bar{c}V_{3}(x_{3},z_{3})\prod_{i=4}^{n}V_{i}(x_{i},z_{i})}\,,$
(89)

where we keep the coordinates $z_{1},z_{2},z_{3}$ fixed. For simplicity, we
will consider vertex operators of the form

$V(x,z)=\Phi_{m,\bar{m},j}^{w}(x,z)V^{X}(z)\,,$ (90)

where $\Phi_{m,\bar{m},j}^{w}(x,z)$ is defined in (40)151515We do not lose any
generality by assuming this since the full $\rm AdS_{3}$ vertex operator is
given in terms of $\Phi_{m,\bar{m},j}^{w}(x,z)$ as in (39). and $V^{X}(z)$ is
a vertex operator in the $X$ theory, chosen so that the full vertex operator
$V_{i}$ has worldsheet conformal weight $(1,1)$. This, and the fact that the
action is a sum of $\rm AdS_{3}$ and $X$ actions, means that we can compute
worldsheet correlators by separately computing each contribution. That is, the
worldsheet correlator factorizes as

$|z_{12}z_{23}z_{13}|^{2}\int\mathrm{d}^{2}z_{4}\cdots\mathrm{d}^{2}z_{n}\Braket{\prod_{i=1}^{n}\Phi_{m_{i},\bar{m}_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\Braket{\prod_{i=1}^{n}V^{X}_{i}(z_{i})}\,.$
(91)

The prefactor $|z_{12}z_{23}z_{13}|^{2}$ comes from the Wick contraction of
the $c$ ghosts. For the rest of this section, we will implicitly consider only
correlators which are trivial with respect to the compact CFT $X$. This is
done for notational convenience, and re-introducing the $X$-dependence of
correlators is straightforward.

Let us now consider the contribution from $\rm AdS_{3}$. Since the expression
(39) is the leading order in the large $\Phi$ expansion, we should therefore
keep only the leading term in the action (48) at large $\Phi$. This amounts to
dropping the interaction term $\nu^{-1}\beta\bar{\beta}e^{-Q\Phi}$ and working
with the resulting action

$S_{\rm
AdS_{3}}=\frac{1}{2\pi}\int\left(\frac{1}{2}\partial\Phi\,\overline{\partial}\Phi+\beta\overline{\partial}\gamma+\bar{\beta}\partial\bar{\gamma}-\frac{Q}{4}R\Phi-
pD\right)\,.$ (92)

With this action, let us now massage the integrand of (91) into a more
suggestive form. By expanding in powers of the screening operator $D$, the
$\rm AdS_{3}$ correlator in (91) becomes (at order $N$)

$\frac{p^{N}}{N!}\int\prod_{a=1}^{N}\mathrm{d}^{2}\lambda_{a}\Braket{\prod_{a=1}^{N}D(\lambda_{a})\prod_{i=1}^{n}\Phi_{m_{i},\bar{m}_{i},j_{i}}^{w_{i}}(x_{i};z_{i})}_{\text{free}}\,,$
(93)

where the subscript _free_ on the correlator reminds us that we are
calculating the correlator in the free theory, i.e. with the action (92) with
$p=0$.

This string correlator further factorizes into three components: namely, that
of the scalar $\Phi$, that of the $\beta\gamma$ system of the Wakimoto
representation and that of the compact CFT $X$. The compact CFT $X$ simply
comes along for the ride and plays no central role until we discuss the dual
CFT. The scalar correlator in the near-boundary (large $\Phi$) limit takes the
form

$\Braket{\prod_{a=1}^{N}e^{-2\Phi/Q}(\lambda_{a})\prod_{i=1}^{n}e^{-(Qj_{i}-w_{i}/Q)\Phi}(z_{i})}_{\text{free}}\,,$
(94)

while the $\beta\gamma$ correlator is better written in the path integral
language:161616From now on, we set $m_{i}=\bar{m}_{i}$. The generalization to
$m_{i}\neq\bar{m}_{i}$ is immediate but clutters notation significantly.

$\int\mathcal{D}(\beta,\gamma)\,e^{-S[\beta,\gamma]}\prod_{a=1}^{N}\left|\oint_{\lambda_{a}}\gamma\right|^{-2(k-1)}\delta^{(2)}(\beta(\lambda_{a}))\prod_{i=1}^{n}\left|\frac{\partial^{w_{i}}\gamma(z_{i})}{w_{i}!}\right|^{-2m_{i}-2j_{i}}\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(95)

Due to the symmetry $\Phi\to\Phi+\Phi_{0}$, the correlator (94) vanishes
unless the charge conservation condition

$\frac{2N}{Q}+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)=Q(1-g)\,,$
(96)

is satisfied. The right-hand-side comes from the background charge in the
action (92). This conservation law determines the allowed value of $N$ in
terms of the quantum numbers of the vertex operators:

$N=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}-\frac{Q^{2}}{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)\,.$
(97)

Since $N,g,w_{i}$ are integers, we immediately have

$\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)=-\frac{m}{Q^{2}}\,,$ (98)

where $m\in\mathbb{Z}$. As we will see below, the worldsheet path integral
vanishes unless $m\geq 0$.

Let us now turn to the $\beta\gamma$ path integral (95). Following
Knighton:2023mhq , we can formally consider the Fourier transform

$\delta^{(2)}(\beta(\lambda_{a}))=\int\mathrm{d}^{2}\xi_{a}\,e^{i\beta(\lambda_{a})\xi_{a}+i\bar{\beta}(\bar{\lambda}_{a})\bar{\xi}_{a}}$
(99)

and thus we can think of the insertion of the delta functions
$\delta^{(2)}(\beta(\lambda_{a}))$ in the path integral (95) as a modification
of the action of the $\beta\gamma$ system. The modified action reads

$S_{\text{new}}=\frac{1}{2\pi}\int_{\Sigma}\left[\beta\left(\bar{\partial}\gamma-2\pi
i\sum_{a}\xi_{a}\delta^{(2)}(z,\lambda_{a})\right)+\bar{\beta}\left(\partial\bar{\gamma}-2\pi
i\sum_{a}\bar{\xi}_{a}\delta^{(2)}(z,\lambda_{a})\right)\right].$ (100)

Integrating out $\beta,\bar{\beta}$ in (95) introduces a Dirac functional

$\int\mathcal{D}\beta\exp(-S_{\text{new}})=\delta^{(2)}\bigg{(}\bar{\partial}\gamma-2\pi
i\sum_{a}\xi_{a}\delta^{(2)}(z-\lambda_{a})\bigg{)}$ (101)

which imposes that $\gamma$ is meromorphic with $N$ simple poles at
$\lambda_{a}$ with residue $\xi_{a}$ as well as analogous conditions for
$\bar{\gamma}$. However, $\lambda_{a}$ and $\xi_{a}$ are not fixed but are
integrated over as can be seen from (99) and (93). Hence, the remaining path
integral over $\gamma$ is an integral over the space of meromorphic functions
with $N$ simple poles and we denote this space by $\mathcal{F}_{N}$. This
space has (complex) dimension

$\text{dim}(\mathcal{F}_{N})=2N+1-g\,,$ (102)

which can be derived from the Riemann-Roch theorem.171717See for example
footnote 20 of Dei:2023ivl . Schematically, we can thus think of the string
path integral as

$|z_{12}z_{23}z_{13}|^{2}\int_{\mathcal{F}_{N}}\braket{\beta\gamma}\braket{\Phi}\braket{X}\,,$
(103)

where $\braket{\beta\gamma}$ is the integrand coming from the Wakimoto field
$\gamma$, $\braket{\Phi}$ is the correlator (94), and $\braket{X}$ is the
compact CFT correlator. As mentioned above, we will set $\braket{X}=1$ for
simplicity, although it is trivial to generalize the discussion to nontrivial
correlation functions in the compact CFT.

Now, the delta functions $\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})$ in the
path integral (95) impose $\sum_{i}w_{i}$ additional constraints on the space
of functions $\gamma$. These constraints require that $\gamma$ obeys

$\gamma(z)=x_{i}+\mathcal{O}((z-z_{i})^{w_{i}}),\text{ as }z\to z_{i}$ (104)

for all $i=1,\ldots,n$. The space of functions that satisfy these conditions
now has dimension

$\text{dim}(\mathcal{F}_{N})-\sum_{i}w_{i}=-Q^{2}\left(\sum_{i}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)-(n+3g-3)\,,$
(105)

where we have used the charge conservation (97) to rewrite $N$ in terms of the
$\text{SL}(2,\mathbb{R})$ spins $j_{i}$. In the end, we also want to integrate
not only over the functions $\gamma$ but also over the worldsheet moduli, as
in equation (87). This means that, after computing the $\Phi$ correlator (94),
the worldsheet path integral reduces to an integral of dimension

$\text{dim}(\mathcal{F}_{N})-\sum_{i=1}^{n}w_{i}+\text{dim}(\mathcal{M}_{g,n})=2N+2g-2-\sum_{i=1}^{n}(w_{i}-1)$
(106)

In order for the path integral to make any sense, this dimension needs to be a
non-negative integer, which we call $m\in\mathbb{Z}_{\geq 0}$. Using the
charge conservation (97), we can write $m$ as a function of the
$\text{SL}(2,\mathbb{R})$ spins $j_{i}$:

$m=-Q^{2}\left(\sum_{i}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)$ (107)

The number $m$ has a simple interpretation from the point of view of
integration over maps $\gamma$. Since $\gamma$ has precisely

$N=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}+\frac{m}{2}\,,$ (108)

poles, the Riemann-Hurwitz formula tells us that $m$ counts the number of
‘extra’ branch points of $\gamma$, i.e. the number of points $\zeta_{\ell}$ on
the worldsheet where $\gamma$ takes the local form

$\gamma(\zeta_{\ell})\sim\xi_{\ell}+\mathcal{O}((z-\zeta_{\ell})^{2})\,,\quad
z\to\zeta_{\ell}\,.$ (109)

Hence, the worldsheet map $\gamma$ has precisely the behavior as the covering
maps which calculate the dual CFT correlators at $m^{\text{th}}$ order in
perturbation theory (see Section 3.3).

###  4.2 Evaluating the path integral

Now that we have set the stage, we can evaluate the path integral (93). This
calculation has three steps:

  1. 1.

For a fixed value of $N$, evaluate the $\Phi$ correlator (94).

  2. 2.

Evaluate the $\beta\gamma$ path integral (95).

  3. 3.

Massage the end result to take the form (79) of the dual CFT correlators.

As we outlined above, the result of Step 1 will be a set of Wick contractions,
along with a delta function enforcing the charge conservation (96), which
picks out the specific value (97) for the number of operators $D$ inserted
into the path integral. For Step 2, we then integrate out $\beta,\bar{\beta}$
and we are left with an integral over the set of meromorphic functions
$\gamma$ with poles at $\lambda_{a}$. The delta functions in the path integral
reduce this to a finite-dimensional integral of dimension $m$, defined in
(107). Upon calculating the appropriate Jacobians from the delta function,
multiplying by $|z_{12}z_{23}z_{13}|^{2}$, and integrating over the worldsheet
moduli $z_{4},\ldots,z_{n}$ then gives the full string correlator. Finally,
for Step 3, we will find that the Wick contractions of the $\Phi$ correlator
can be repackaged nicely into a simple function of the covering map data which
enters the symmetric orbifold calculation (79).

#### Step 1:

The scalar field $\Phi$ enters the string correlator through the contribution
(94). As $\Phi$ is a free scalar, this correlation function can be computed
via Wick contractions:

$\displaystyle\Braket{\prod_{a=1}^{N}e^{-2\Phi/Q}(\lambda_{a})\prod_{i=1}^{n}e^{(w_{i}/Q-Qj_{i})\Phi}(z_{i})}_{\text{free}}=\prod_{a<b}|\lambda_{a}-\lambda_{b}|^{-8/Q^{2}}\prod_{i,a}|z_{i}-\lambda_{a}|^{4(w_{i}/Q^{2}-j_{i})}$
(110) $\displaystyle\hskip
71.13188pt\times\prod_{i<j}|z_{i}-z_{j}|^{-2(w_{i}/Q-j_{i}Q)(w_{j}/Q-j_{j}Q)}\delta\bigg{(}\frac{2N}{Q}+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)-Q\bigg{)}.$

The delta-function comes from the integral over the zero mode of $\Phi$, and
ensures that the operators in (94) satisfy the anomalous charge conservation
(97).

#### Step 2:

Let us now treat the $\beta\gamma$ path integral. In order to make any sense
of the integral (103), we need to specify a measure on the space of
meromorphic functions. Here, we only work for genus $g=0$ worldsheets, but we
expect that it is possible to generalize it to higher genus.

As we have seen from the previous subsection, integrating out the $\beta$
field restricts $\gamma$ to lie in the space $\mathcal{F}_{N}$ of meromorphic
functions with $N$ poles. Thus, by picking a suitable basis for this class of
functions, we can deduce the path integral measure. After integrating out
$\beta$, the path integral looks like

$p^{N}\int_{\mathcal{A}_{N}}\mathcal{D}\gamma\prod_{a=1}^{N}\left|\oint_{\lambda_{a}}\gamma\right|^{-2(k-1)}\prod_{i=1}^{n}\left|\frac{\partial^{w_{i}}\gamma(z_{i})}{w_{i}!}\right|^{-2m_{i}-2j_{i}}\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})\,,$
(111)

where $\mathcal{A}_{N}$ denotes the space of meromorphic functions with $N$
simple poles at the fixed points $\lambda_{a}$, but with arbitrary residues.
These points are fixed since we have not yet integrated over $\lambda_{a}$.

Focusing on $g=0$, the most general form of such a function is

$\gamma(z)=b+\sum_{a=1}^{N}\frac{c_{a}^{\gamma}}{z-\lambda_{a}}\,.$ (112)

Note that there can be no polynomial in $z$ since it would imply that
$z=\infty$ has a pole singularity. Hence, the space $\mathcal{A}_{N}$ is
parametrized by $b,c_{a}^{\gamma}$ and thus it is natural to define the path
integral measure181818One may argue that there should be some nontrivial
Jacobian at this step. We refer to Appendix A.1 for a more detailed argument
why there should not be a nontrivial Jacobian associated to this change of
variable.

$\int_{\mathcal{A}_{N}}\mathcal{D}\gamma=\int\mathrm{d}^{2}b\prod_{a=1}^{N}\mathrm{d}^{2}c_{a}^{\gamma}\,.$
(113)

With this, we can now write eq. (111) as

$p^{N}\int\mathrm{d}^{2}b\,\mathrm{d}^{2N}c_{a}^{\gamma}\prod_{a=1}^{N}\left|c_{a}^{\gamma}\right|^{-2(k-1)}\prod_{i=1}^{n}\left|\frac{\partial^{w_{i}}\gamma(z_{i})}{w_{i}!}\right|^{-2m_{i}-2j_{i}}\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})\,,$
(114)

with $\gamma(z)$ given by (112). If we now include the integration over
$\lambda_{a}$ from (93), we obtain

$\frac{p^{N}}{N!}\int\mathrm{d}^{2}b\,\mathrm{d}^{2N}\lambda_{a}\,\mathrm{d}^{2N}c_{a}^{\gamma}\,\prod_{a=1}^{N}\left|c_{a}^{\gamma}\right|^{-2(k-1)}\prod_{i=1}^{n}\left|\frac{\partial^{w_{i}}\gamma(z_{i})}{w_{i}!}\right|^{-2m_{i}-2j_{i}}\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(115)

The worldsheet amplitude (91) now schematically takes the form

$\displaystyle\frac{p^{N}}{N!}\int\prod_{j=4}^{n}\mathrm{d}^{2}z_{j}\,\mathrm{d}^{2}b\prod_{a=1}^{N}\mathrm{d}^{2}\lambda_{a}\,\mathrm{d}^{2}c_{a}^{\gamma}\braket{\Phi}\braket{\beta\gamma}|z_{12}z_{23}z_{13}|^{2}\,,$
(116)

where $\braket{\Phi}$ is given by (110), $\braket{X}$ is the correlator in the
compact direction and $\braket{\beta\gamma}$ is the integrand of (115). The
integral is taken over the $2N+n-2$ parameters
$z_{j},b,\lambda_{a},c_{a}^{\gamma}$, whereas the integrand is a delta
function of the form

$\prod_{i=1}^{n}\prod_{\ell_{i}=0}^{w_{i}-1}\delta^{(2)}\left(\frac{\partial^{\ell_{i}}(\gamma(z_{i})-x_{i})}{\ell_{i}!}\right)\,.$
(117)

Thus, the delta functions eliminate $\sum_{i}w_{i}=2N+n-2-m$ of the $2N+n-2$
integrals, and we are left with an $m$-dimensional integral. As mentioned in
Section 4.1, $m$ is precisely the number of ‘extra’ branch points of the
covering map $\gamma$. Thus, a natural set of variables to integrate over are
the arguments of the delta functions (117) and the values
$\gamma(\zeta_{\ell})$ of $\gamma$ at the $m$ branch points $\zeta_{\ell}$.
That is, we consider the change of variables

$(z_{j},b,\lambda_{a},c_{a}^{\gamma})\to\left(\frac{\partial^{\ell_{i}}(\gamma(z_{i})-x_{i})}{\ell_{i}!},\gamma(\zeta_{\ell})\right)$
(118)

where $j=4,\ldots,n$, $a=1,\ldots,N$, $i=1,\ldots,n$,
$\ell_{i}=0,\ldots,w_{i}-1$ and $l=1,\ldots,m$. Under this change of
variables, the path integral (116) becomes191919The factor of $1/m!$ comes
from the fact that the coordinates on the RHS of (118) cover the moduli space
in question $m!$ times.

$\displaystyle
p^{N}\int\prod_{j=4}^{n}\mathrm{d}^{2}z_{j}\,\mathrm{d}^{2}b\prod_{a=1}^{N}\mathrm{d}^{2}\lambda_{a}\,\mathrm{d}^{2}c_{a}^{\gamma}\braket{\Phi}\braket{\beta\gamma}|z_{12}z_{23}z_{13}|^{2}$
(119) $\displaystyle\hskip
28.45274pt=\frac{p^{N}}{m!}\int\mathrm{d}^{2m}\gamma(\zeta_{\ell})\,\mathcal{J}^{-1}\prod_{a=1}^{N}|c_{a}|^{-2(k-1)}\prod_{i=1}^{n}|a^{\gamma}_{i}|^{-2m_{i}-2j_{i}}\braket{\Phi}|z_{12}z_{23}z_{13}|^{2}\,,$

where we have defined $a^{\gamma}_{i}:=\partial^{w_{i}}\gamma(z_{i})/w_{i}!$
and $\mathcal{J}$ is the Jacobian of the change of variables. This Jacobian
can be found by a straightforward, but tedious, calculation and we spell out
the details in Appendix A. The end result is that $\mathcal{J}$ is given by
(see (180))

$\mathcal{J}=|z_{12}z_{13}z_{23}|^{2}|C^{\gamma}|^{-4}\prod_{i=1}^{n}|A^{\gamma}_{i}|^{w_{i}+1}\prod_{\ell=1}^{m}|B^{\gamma}_{\ell}|\,,$
(120)

with $A^{\gamma}_{i}$ and $B^{\gamma}_{\ell}$ are defined below as in (124).
Defining $\xi_{\ell}:=\gamma(\zeta_{\ell})$, we can rewrite the worldsheet
amplitude as

$\begin{split}&\frac{p^{N}}{m!}\bigg{(}\prod_{i=1}^{n}\frac{1}{w_{i}^{w_{i}+1}2^{m}}\bigg{)}\int\mathrm{d}^{2m}\xi_{\ell}\,|C^{\gamma}|^{4}\prod_{a=1}^{N}|c_{a}^{\gamma}|^{-2(k-1)}\prod_{i=1}^{n}|a^{\gamma}_{i}|^{-2(m_{i}+j_{i})-(w_{i}+1)}\prod_{\ell=1}^{m}|b^{\gamma}_{\ell}|^{-1}\braket{\Phi}\,.\end{split}$
(121)

Here, $C^{\gamma}$ is a constant defined by

$\partial\gamma(z)=C^{\gamma}\frac{\prod_{i=1}^{n}(z-z_{i})^{w_{i}-1}\prod_{l=1}^{m}(z-\zeta_{l})}{\prod_{a=1}^{N}(z-\lambda_{a})^{2}}\,,$
(122)

and $b_{\ell}^{\gamma}$ is the first nontrivial Taylor coefficient of $\gamma$
around the point $\zeta_{\ell}$, i.e.

$\gamma(z)\sim\xi_{\ell}+b_{\ell}^{\gamma}(z-\zeta_{\ell})^{2}\,.$ (123)

Using (122), we also have

$\displaystyle A^{\gamma}_{i}:=w_{i}a^{\gamma}_{i}=$ $\displaystyle
C^{\gamma}\frac{\prod_{k\neq
i}^{n}(z_{i}-z_{k})^{w_{k}-1}\prod_{l=1}^{m}(z_{i}-\zeta_{l})}{\prod_{a=1}^{N}(z_{i}-\lambda_{a})^{2}}$
(124) $\displaystyle c_{a}^{\gamma}=$
$\displaystyle-C^{\gamma}\frac{\prod_{i=1}^{n}(\lambda_{a}-z_{i})^{w_{i}-1}\prod_{l=1}^{m}(\lambda_{a}-\zeta_{l})}{\prod_{b\neq
a}^{N}(\lambda_{a}-\lambda_{b})^{2}}$ $\displaystyle
B^{\gamma}_{l}:=2b^{\gamma}_{l}=$ $\displaystyle
C^{\gamma}\frac{\prod_{i=1}^{n}(\zeta_{l}-z_{i})^{w_{i}-1}\prod_{p\neq
l}^{m}(\zeta_{l}-\zeta_{p})}{\prod_{a=1}^{N}(\zeta_{l}-\lambda_{a})^{2}}\,,$

where we have defined $A^{\gamma}_{i},B^{\gamma}_{\ell}$ in terms of factors
of $2$ and $w_{i}$ and $a_{i}^{\gamma},b^{\gamma}_{\ell}$. Note that these are
precisely the covering map data used to define symmetric orbifold correlators
as in Section 3.

#### Step 3:

We have now computed the $\beta\gamma$ path integral in terms of the covering
map data $a_{i}^{\gamma},b_{\ell}^{\gamma},c_{a}^{\gamma},C^{\gamma}$. The
final step is to include the $\Phi$ correlator (110) to obtain the full string
correlator.

First, note that we can replace the various factors of $|z_{i}-z_{j}|$,
$|z_{i}-\lambda_{a}|$, and $|\lambda_{a}-\lambda_{b}|$ in (110) with the
covering map data. Indeed, using (124), we can rewrite the correlator (110) as

$\displaystyle\Braket{\Phi}=$
$\displaystyle|C^{\gamma}|^{\frac{2}{Q^{2}}-2}\prod_{a}|c_{a}^{\gamma}|^{\frac{2}{Q^{2}}}\prod_{\ell}|A_{\ell}^{\gamma}|^{\frac{1}{Q^{2}}}\prod_{i}|A^{\gamma}_{i}|^{-(\frac{w_{i}+1}{Q^{2}}-2j_{i})}$
(125)
$\displaystyle\times\prod_{\ell<p}|\zeta_{\ell}-\zeta_{p}|^{-\frac{2}{Q^{2}}}\prod_{i,\ell}|z_{i}-\zeta_{\ell}|^{-2j_{i}+\frac{2}{Q^{2}}}\prod_{i<k}|z_{i}-z_{k}|^{-2(Qj_{i}-\frac{1}{Q})(Qj_{k}-\frac{1}{Q})}$
$\displaystyle\hskip
156.49014pt\times\delta\bigg{(}\frac{2N}{Q}+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)-Q\bigg{)}.$

Inserting the $\Phi$ correlator into (121) (and removing the delta function),
we finally arrive at the expression

$\begin{gathered}\sum_{\begin{subarray}{c}\text{connected covering}\\\
\text{maps
}\gamma\end{subarray}}p^{N}\left(\prod_{i=1}^{n}w_{i}^{-\frac{k(w_{i}+1)}{2}+2j_{i}}2^{\frac{(k-4)m}{2}}\right)\int\frac{\mathrm{d}^{2m}\xi_{\ell}}{m!}\sum_{\gamma}|C^{\gamma}|^{k}\prod_{a=1}^{N}|c^{\gamma}_{a}|^{-k}\\\
\prod_{i=1}^{n}|a_{i}^{\gamma}|^{-2h_{i}+\frac{k(w_{i}-1)}{2}}\prod_{\ell=1}^{m}|a_{\ell}^{\gamma}|^{\frac{k-4}{2}}\prod_{i<j}|z_{i}-z_{j}|^{-2q_{i}q_{j}}\prod_{i,\ell}|z_{i}-\zeta_{\ell}|^{-2q_{i}\alpha}\prod_{\ell<k}|\zeta_{\ell}-\zeta_{k}|^{-2\alpha^{2}}.\end{gathered}$
(126)

Here, $h_{i}$ is defined in (31) and

$q_{i}:=Qj_{i}-\frac{1}{Q}\,,\quad\alpha:=\frac{1}{Q}\,.$ (127)

Note that in (126), we have also included the possibility that the space of
covering maps is not connected, and so we sum over non-continuously related
maps. Aside from the overall normalization (which we will discuss below), this
expression has precisely the form of (79) with $c=6k$.

###  4.3 Matching the normalization

To obtain a precise agreement between the CFT answer (79) and the worldsheet
answer (126), we need to introduce various normalization constants.
Specifically, we need to determine the constants $\mathcal{N}(j_{i},w_{i})$
and $C_{\text{S}^{2}}$ on the right-hand-side of equation (88), as well as a
dictionary relating the parameters $(\nu,p)$ of the worldsheet CFT to the
parameter $\mu$ of the boundary CFT. It is straightforward to verify that the
appropriate choice of constants is

$\begin{gathered}\mathcal{N}(j,w)=K^{1/2}\nu^{\frac{(2-k)(w-1)}{2}}w^{2j-\frac{1}{2}}\,,\quad\mu=-K^{1/2}2^{2k-\frac{5}{2}}\nu^{\frac{2-k}{2}}\,,\\\
p=\nu^{2-k}\,,\quad C_{\text{S}^{2}}=\nu^{2-k}\,,\quad
g_{s}=\frac{1}{\sqrt{K}}\,.\end{gathered}$ (128)

With these normalizations, we arrive at a precise match between the string
correlators (126) and the CFT correlators (79).202020The various
normalizations differ from those in Dei:2022pkr , see their equation (3.2). We
suspect that this difference is due to a difference in conventions for the
$\text{SL}(2,\mathbb{R})$ WZW model.

##  5 Discussion

In the previous section, we calculated worldsheet correlation functions of
spectrally-flowed vertex operators in $\text{AdS}_{3}\times X$ in the limit
that the strings are localized near the boundary. This region in the bulk is
dual to the regime of the dual CFT where correlators can be computed using
naive perturbation theory. The analysis of Section 4 showed that the string
theory correlators in the near-boundary limit indeed reproduce the
corresponding perturbative correlators of the dual CFT. We emphasize that this
result holds for 1) all spectral flows/twists $w_{i}$, 2) all numbers $n$ of
vertex operator insertions, and 3) at all orders in conformal perturbation
theory in $\mu$. This effectively proves the proposed near-boundary/weak-
coupling duality (2) and constitutes extremely strong evidence for the full
$\text{AdS}_{3}/\text{CFT}_{2}$ duality (1).

Let us recap how the worldsheet calculation went. First, we had to integrate
out the worldsheet Wakimoto scalar $\Phi$, which introduced various analytic
factors in the form of Wick contractions. We then re-wrote these Wick
contractions with the help of certain analytic data of the covering map
$\gamma$. Upon computing the path integral for the $\beta\gamma$ system, we
found that the worldsheet path integral localizes to covering maps with
certain branching structures (precisely those appearing in the symmetric
orbifold calculations). Upon computing the Jacobian in the $\beta\gamma$ path
integral and combining it with the correlator of the scalar $\Phi$, we
miraculously found a formula which matches the dual CFT correlator precisely.
While this computation certainly worked, it is conceivable that there is a
much more straightforward way to get the answer. We will now briefly speculate
how this might go.

The worldsheet correlation functions of the $\Phi$ scalar take the form

$\Braket{\prod_{a=1}^{N}e^{-2\Phi/Q}(\lambda_{a})\prod_{i=1}^{n}e^{(w_{i}/Q-Qj_{i})\Phi}(z_{i})}\,,$
(129)

whereas the correlators of the dual linear dilaton $\varphi$ take the form
(after passing to the covering space)

$\Braket{\prod_{i=1}^{n}e^{(1/Q-Qj_{i})\varphi}(z_{i})\prod_{\ell=1}^{m}e^{-\varphi/Q}(\zeta_{\ell})}\,.$
(130)

By analyzing the OPE structure of the currents $\partial\Phi,\partial\varphi$,
we see that within correlation functions we can make the replacements

$\begin{split}\partial\Phi(z)&=\sum_{a=1}^{N}\frac{2/Q}{z-\lambda_{a}}+\sum_{i=1}^{n}\frac{Qj_{i}-w_{i}/Q}{z-z_{i}}\,,\\\
\partial\varphi(z)&=\sum_{i=1}^{n}\frac{Qj_{i}-1/Q}{z-z_{i}}+\sum_{\ell=1}^{m}\frac{1/Q}{z-\zeta_{\ell}}\,.\end{split}$
(131)

Taking the difference yields

$\partial\Phi(z)-\partial\varphi(z)=\frac{1}{Q}\left(\sum_{a=1}^{N}\frac{2}{z-\lambda_{a}}-\sum_{i=1}^{n}\frac{(w_{i}-1)}{z-z_{i}}-\sum_{\ell=1}^{m}\frac{1}{z-\zeta_{\ell}}\right)\,.$
(132)

Defining $\gamma(z)$ through (122), we have

$\partial\Phi(z)=\partial\varphi(z)-\frac{1}{Q}\partial\log\partial\gamma\,.$
(133)

The same analysis for the right-moving sector gives the relation

$\Phi=\varphi-\frac{1}{Q}\Phi_{\gamma}\,,$ (134)

where we have defined

$\Phi_{\gamma}=\log|\partial\gamma|^{2}\,.$ (135)

We note that the field redefinition (134) is similar to the shift of the
Wakimoto scalar performed in Hikida:2007tq ; Hikida:2008pe ; Hikida:2020kil ;
Hikida:2023jyc . It is also similar to the anomalous transformation derived in
the Appendix C of Eberhardt:2019ywk .

The reason that the shift (134) is useful is that
$\text{Sym}(\mathbb{R}_{\varphi}\times X)$ correlators are determined entirely
by the linear dilaton field $\varphi$ and the scalar $\Phi_{\gamma}$ in a
straightforward way Lunin:2000yv ; Lunin:2001ne :212121We are assuming all
vertex operators are in the vacuum of $X$.

$\Braket{\prod_{i=1}^{n}\sigma_{w_{i},q_{i}}(x_{i})\prod_{\ell=1}^{m}\sigma_{2,\alpha}}=\sum_{\begin{subarray}{c}\text{connected
covering}\\\ \text{maps
}\gamma\end{subarray}}e^{-S_{L}[\Phi_{\gamma}]}\Braket{\prod_{i=1}^{n}e^{-q_{i}\varphi}(z_{i})\prod_{\ell=1}^{m}e^{-\alpha\varphi}(\zeta_{\ell})}\,,$
(136)

where the linear dilaton correlator on the right-hand-side is computed on the
covering space and $S_{L}$ is the Liouville action

$S_{L}[\Phi_{\gamma}]=\frac{c}{48\pi}\int\mathrm{d}^{2}z\sqrt{g}\left(2\partial\Phi_{\gamma}\overline{\partial}\Phi_{\gamma}+R\Phi_{\gamma}\right)\,.$
(137)

One might therefore hope that the substitution (134) would allow one to write
the string path integral manifestly in the form (136) without the need of
systematically computing Wick contractions and determinants of the
$\beta\gamma$ system. This would have the advantage of 1) laying bare the
precise mechanism of how the string path integral is repackaged into a
symmetric orbifold correlation function and 2) working for worldsheets of
generic genus, i.e. at all orders in the $g_{s}$ expansion. This would
effectively amount to a direct derivation of the
$\text{AdS}_{3}/\text{CFT}_{2}$ correspondence (2) in the perturbative regime.
As we still do not fully understand the details of how such a derivation would
go, we leave this to future work.

### Future directions

We will now discuss several future directions that this work could take, as
well as problems which we believe are now approachable given the progress made
in this paper.

##### Worldsheet perturbations and Coulomb gas integrals:

In this paper we matched the near-boundary worldsheet path integral with the
naive perturbative expansion of the dual CFT. On the worldsheet, considering
the near-boundary limit effectively amounts to ignoring the interaction term
$\nu^{-1}\beta\bar{\beta}e^{-Q\Phi}$ in the AdS Lagrangian. The perturbative
expansion of the dual CFT is then reproduced by the perturbative expansion in
the screening operator $D$ on the worldsheet. A natural extension of the
results of this paper would be to consider the effect of perturbatively
introducing the interaction term in the $\text{AdS}_{3}$ sigma model. From the
worldsheet perspective, perturbing in $\beta\bar{\beta}e^{-Q\Phi}$ would
introduce a 2-dimensional lattice of poles where

$\frac{2N}{Q}+MQ+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)=Q(1-g)\,,\quad
M,N\in\mathbb{Z}_{\geq 0}\,.$ (138)

Of this set of poles, only the set $M=0$ is understood from the dual CFT
perspective. Under the assumption that the full duality (1) holds, it would be
interesting to understand the roles that the poles $M>0$ play in the dual CFT.
The situation is likely to be similar to Liouville theory, where the duality
$b\to 1/b$ implies the existence of a two-dimensional lattice of poles which
are not visible in naive perturbation theory. It would thus be fruitful to
explore whether a technique analogous to the coulomb gas formalism for
Liouville theory Goulian:1990qr  exists for the CFT dual of $\text{AdS}_{3}$
string theory. If such a formalism does exist, then it would be in principle
possible to match the residues at all of the poles in (138), and not just
those with $M=0$.

##### Worldsheet perturbation and the tensionless limit

It was speculated in McStay:2023thk ; Dei:2023ivl  that the interaction term
$\beta\bar{\beta}e^{-Q\Phi}$ does not give any contribution in the tensionless
limit. In other words, the near-boundary limit becomes exact in the
tensionless limit. However, the precise mechanism that explains why this is
the case is still lacking in the literature. Indeed, by treating the
interaction term perturbatively and restricting ourselves to the tensionless
limit ($k=3$), it can be shown that the interaction gives no contribution in
generic string amplitude. This is currently under investigation
Sriprachyakul:2024abc .

##### Solving the $\text{SL}(2,\mathbb{R})$ WZW model:

String theory on $\text{AdS}_{3}$ is based on the $\text{SL}(2,\mathbb{R})$
WZW model (or $\text{SU}(2,\mathbb{C})/\text{SU}(2)$ in Euclidean signature).
As we have seen in this paper, a careful treatment of the free field
realization allows one to exactly calculate spectrally-flowed correlation
functions in the near-boundary limit. As mentioned above, a Coulomb gas-type
formalism including the interaction term $\nu^{-1}\beta\bar{\beta}e^{-Q\Phi}$
could allow us to identify all of the poles in the correlators of the
$\text{SL}(2,\mathbb{R})$ WZW model. Another, more ambitious, goal would be to
use the resulting Coulomb gas formalism to completely solve the $x$-basis
correlators of the $\text{SL}(2,\mathbb{R})$ WZW model, similar to the
solution of Liouville theory Dorn:1994xn ; Zamolodchikov:1995aa .

##### Superstrings on $\text{AdS}_{3}\times X$:

Another natural direction to persue is to supersymmetrize our results and
check the validity of the superstring proposal by Eberhardt Eberhardt:2021vsx
. In fact, one can already see from our result that a naive guess for the CFT
dual to superstring in ${\rm AdS_{3}}\times X$ is

${\rm Sym}^{N}(\mathbb{R}_{\mathcal{Q}}\times\text{3
fermions}\times\beta\gamma\text{ ghosts}\times
X)+\int\sigma_{2,\tilde{\alpha},\bar{\tilde{\alpha}}}\,,$ (139)

where $\tilde{\alpha},\bar{\tilde{\alpha}}$ denote collectively the linear
dilaton momentum and other labellings. Effectively, the $\beta\gamma$
ghosts222222These are the usual superconformal ghosts and should not be
confused with the $\beta\gamma$ in the Wakimoto representation. kill two of
the fermions and a tentative candidate for a dual CFT is thus

${\rm Sym}^{N}(\mathbb{R}_{\mathcal{Q}}\times\text{1 fermion}\times
X)+\int\sigma_{2,\tilde{\alpha},\bar{\tilde{\alpha}}}\,.$ (140)

Specializing to the case where $X={\rm S^{3}}\times\tilde{X}$ with
$\tilde{X}=K3$ or $\mathbb{T}^{4}$, we see that the dual CFT can be written as

${\rm Sym}^{N}(\mathbb{R}_{\mathcal{Q}}\times\text{4
fermions}\times\mathfrak{su}(2)_{k-2}\times\tilde{X})+\int\sigma_{2,\tilde{\alpha},\bar{\tilde{\alpha}}}\,,$
(141)

where we exmphasize that $k$ in the $\mathfrak{su}(2)$ level is the same as
the supersymmetric level of $\mathfrak{sl}(2,\mathbb{R})$ as is required by
string criticality. This is the proposal of Eberhardt:2021vsx . However, it
remains to be seen precisely how the $\beta\gamma$ ghost cancellation works in
any generic correlators. This is currently under investigation
Sriprachyakul:2024def .

##### String theory on $G^{\,\mathbb{C}}/G$ coset spaces:

The fundamental tool used throughout this paper was the realization of string
theory on Euclidean $\text{AdS}_{3}$ in terms of the Wakimoto free fields
$\Phi,\gamma,\bar{\gamma},\beta,\bar{\beta}$. A more general class of string
backgrounds admitting analogous free field realizations are the coset spaces
$G^{\,\mathbb{C}}/G$, where $G$ is some real Lie group and $G^{\,\mathbb{C}}$
is its complexification Gawedzki:1991yu . If $G=\text{SU}(2)$, then
$G^{\,\mathbb{C}}/G$ is simply Euclidean $\text{AdS}_{3}$. It would be
interesting to explore whether the techniques developed in this paper would
allow one to calculate correlation functions in such coset spaces for more
general groups $G$ and, if so, whether there is some ‘holographic’ relation to
a QFT on the asymptotic boundary.

##### Relation to Gromov-Witten theory:

In the calculation of the worldsheet path integral, the computation of the
$\beta\gamma$ system reduced to an integral over the moduli space of
holomorphic maps $\gamma:\Sigma\to B$ of degree $N$ with $B$ the boundary of
$\text{AdS}_{3}$. One can take the integration over such holomorphic maps and
the integral over the worldsheet moduli into an integral over the Kontsevich
moduli space $\mathcal{M}_{g,n}(B,N)$ of holomorphic curves in $B$ of degree
$N$ with $n$ marked points. This is a moduli space of (virtual) dimension

$\begin{split}\text{dim}(\mathcal{M}_{g,n}(B,N))&=n+2g-2+N\int_{B}c_{1}(B)\\\
&=n+2g-2+N(2-2G)\,,\end{split}$ (142)

where $G$ is the genus of the boundary $B$. From this point of view, we can
think of the delta function operators introduced in Section 2, namely

$\delta^{(2)}_{w_{i}}(\gamma(z_{i})-x_{i})$ (143)

as distributions on the moduli space $\mathcal{M}_{g,n}(B,N)$ which localize
to codimension $w_{i}$ subvarieties. The computation of spectrally-flowed
correlators can thus be rephrased in terms of intersection numbers of
subvarieties on the Kontsevich moduli space. Mathematically, the study of
these intersection numbers is the realm of Gromov-Witten theory. Given that
Gromov-Witten theory with a curve as a target is known to be equivalent to
Hurwitz theory Okounkov:2002cja , understanding the relation between
$\text{AdS}_{3}$ string theory and Gromov-Witten theory could help elucidate
the relationship between $\text{AdS}_{3}$ string theory and symmetric
orbifolds.232323We suspect the that work of Frenkel:2006fy ; Frenkel:2008vz ;
janda2023gromovwitten ; Lerche:2023wkj  will be instrumental to understanding
this relationship in more detail.

A topological version of this idea has recently been made concrete
Lerche:2023wkj , where the author argued that the correspondence between the
quantum cohomology of the Hilbert scheme $\text{Hilb}(\text{K}3)$ and Gromov-
Witten theory on $\mathbb{CP}^{1}\times\text{K}3$ corresponds to a topological
subsector of the $\text{AdS}_{3}/\text{CFT}_{2}$ correspondence. It would be
nice to see how this mathematical result fits into the language of
$\text{AdS}_{3}$ string theory.

###### Acknowledgements.

We thank Bruno Balthazar, Andrea Dei, Lorenz Eberhardt, Matthias Gaberdiel,
Yasuaki Hikida, Edward Mazenc, Volker Schomerus, Sean Seet and David Skinner
for useful discussions. We especially thank Andrea Dei, Matthias Gaberdiel,
and Nicolas Kovensky for helpful comments on an early version of the draft.
The work of BK is supported by STFC consolidated grants ST/T000694/1 and
ST/X000664/1. The work of VS is supported by the NCCR SwissMAP that is also
supported by the Swiss National Science Foundation.

##  Appendix A The $\beta\gamma$ path integral

In this appendix, we derive the expression (120) for the Jacobian of the
$\beta\gamma$ path integral under the change of variables (118), which is
necessary in obtaining a matching between the worldsheet and spacetime
correlation functions. The computation involves a complicated row reduction on
large matrices, which we show to end in a finite number of steps. The end
result is a simple epxression in terms of analytic data of the corresponding
branched covering maps. We work purely at genus $g=0$ and only focus on the
left-moving sector of the theory for simplicity.

###  A.1 Deriving the measure

In this subsection, we give another derivation of the $\gamma$ path integral
measure. We start by noticing that the path integral of $\beta,\gamma$ is over
the space of functions that depends only on $z$ (and not on $\bar{z}$) and
have pole singularities at isolated points. The most general form of such a
function is

$\displaystyle\gamma(z)=\sum_{n>0,w\in\mathbb{C}}\frac{c_{n}^{w}}{(z-w)^{n}}+\sum_{n\geq
0}b_{n}z^{n}\,,$ (144)

where the sum over $w$ runs through all the locations of poles of $\gamma(z)$.
Let us simplify our discussion further by assuming that $z=\infty$ is a
regular point.242424If $\gamma$ has a pole at $z=\infty$, we can simply
perform a Möbius transformation $\gamma\to(a\gamma+b)/(c\gamma+d)$ so that
$\gamma(\infty)$ becomes finite. This means that we restrict ourselves to the
expansion

$\displaystyle\gamma(z)=\sum_{n>0,w\in\mathbb{C}}\frac{c_{n}^{w}}{(z-w)^{n}}+b\,.$
(145)

Hence, instead of taking the path integral measure to be

$\displaystyle\mathcal{D}\gamma=\prod_{z\in\mathbb{C}}\mathrm{d}\gamma(z),$
(146)

we should instead change the variable so that the path integral measure is

$\displaystyle\mathcal{D}^{\prime}\gamma=\left(\prod_{n>0,w\in\mathbb{C}}\mathrm{d}c^{w}_{n}\right)\mathrm{d}b\,.$
(147)

Indeed, the Jacobian of this change of variable is the determinant of the
matrix with components252525Here, the rows are labelled by $z$ and the columns
are labelled by $w,n$.

$\displaystyle\begin{pmatrix}\displaystyle\frac{\mathrm{d}\gamma(z)}{\mathrm{d}c^{w}_{n}}&&\displaystyle\frac{\mathrm{d}\gamma(z)}{\mathrm{d}b}\end{pmatrix}=\begin{pmatrix}\displaystyle\frac{1}{(z-w)^{n}}&&\displaystyle
1\end{pmatrix}\,.$ (148)

And since this does not depend on $c^{w}_{n}$ or $b$, it is just an overall
constant which can be absorbed into the normalization of the path integral.
Thus the $\beta\gamma$ path integral with the insertions of
$V^{w_{i}}_{m_{i},j_{i}}$ and $D$ reads

$\displaystyle\int\mathcal{D}\beta\left(\prod_{n>0,w\in\mathbb{C}}\mathrm{d}c^{w}_{n}\right)\mathrm{d}b\,e^{-S[\beta,\gamma]}\prod_{a=1}^{N}\left(\oint_{\lambda_{a}}\gamma\right)^{-(k-1)}\delta(\beta(\lambda_{a}))$
(149) $\displaystyle\hskip
85.35826pt\times\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$

By writing $\delta(\beta)$ using the (formal) identity

$\delta(\beta(\lambda_{a}))=\int\frac{\mathrm{d}\xi_{a}}{2\pi}e^{i\beta(\lambda_{a})\xi_{a}}\,$
(150)

and performing the integration over $\mathcal{D}\beta(z)$, we obtain

$\mathcal{D}\beta\exp\left(-\frac{1}{2\pi}\int_{\Sigma}\beta\left(\overline{\partial}\gamma-\sum_{a}2\pi
i\xi_{a}\delta^{(2)}(z,\lambda_{a})\right)\right)=\delta\left(\bar{\partial}\gamma-\sum_{a}2\pi
i\xi_{a}\delta^{(2)}(z,\lambda_{a})\right)\,$ (151)

where the RHS is a Dirac functional. This Dirac functional restricts
$\gamma(z)$ to have $N$ simple poles at $\lambda_{a}$ with residue $i\xi_{a}$.
Hence, the path integral (149) becomes

$\displaystyle\int\left(\prod_{n>0,w\in\mathbb{C}}\mathrm{d}c^{w}_{n}\right)\mathrm{d}b\prod_{a=1}^{N}\mathrm{d}\xi_{a}\,\prod_{n>2,w\in\mathbb{C}}\delta(c^{w}_{n})\prod_{w\neq\lambda_{a}}\delta(c^{w}_{1})\prod_{w=\lambda_{a}}\delta(c^{w}_{1}-i\xi_{a})$
(152) $\displaystyle\hskip
56.9055pt\times\prod_{a=1}^{N}\left(\oint_{\lambda_{a}}\gamma\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$

Notice that in the last line, we have have rewritten the Dirac functional
(151) as the product of delta functions imposing the equivalent conditions on
the coefficients $c^{w}_{n}$. Performing the integrals for $c^{w}_{n}$, we
obtain

$\displaystyle\int\left(\prod_{a=1}^{N}\mathrm{d}\xi_{a}\right)\mathrm{d}b\,\prod_{a=1}^{N}\left(\oint_{\lambda_{a}}\gamma\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(153)

Redefining $c^{\gamma}_{a}:=\xi_{a}=\text{Res}_{z=\lambda_{a}}\gamma(z)$ , we
have

$\displaystyle\int\left(\prod_{a=1}^{N}\mathrm{d}c^{\gamma}_{a}\right)\mathrm{d}b\,\prod_{a=1}^{N}\left(\oint_{\lambda_{a}}\gamma\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(154)

We propose that this is the correct measure for the path integral over
$\gamma$, once we have restricted the integral over holomorphic functions.
Taking into account right-moving degrees of freedom, we obtain the form (115)
of the path integral used in the main text to match worldsheet correlators to
$\text{CFT}_{2}$ ones.

###  A.2 Computing the $\beta\gamma$ integral Jacobian

In this section we describe in details how we obtain various Jacobians in the
main text. We note here that we will generally ignore any minus signs and
phases since we eventually are interested in amplitudes which are modulus
squares of the quantity we are computing in this paper. As a warm up, we
consider the simplest case and build our way up to the most general case.

#### The simplest case

To start, let us consider the case of 3-point function whose
$\text{SL}(2,\mathbb{R})$ spins are chosen so that the number of extra branch
points is $m=0$, see (98). In this case, there are no worldsheet moduli and
the $\gamma$ field takes the form

$\displaystyle\gamma(z)=\sum_{a=1}^{N}\frac{c^{\gamma}_{a}}{z-\lambda_{a}}+b,$
(155)

where we have $2N+1=\sum_{i}w_{i}$. The Jacobian we want to compute is

$\displaystyle\mathcal{J}=$ $\displaystyle\left|\frac{\partial}{\partial
b}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)~{}~{}\frac{\partial}{\partial
c^{\gamma}_{a}}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)~{}~{}\frac{\partial}{\partial\lambda_{a}}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)\right|\,,$
(156)

where $i$ runs from 1 to 3. Plugging eq.(155) into the Jacobian, we can write
$\mathcal{J}$ as the determinant

$\displaystyle\mathcal{J}=$
$\displaystyle\begin{vmatrix}1&\frac{1}{z_{1}-\lambda_{1}}&...&\frac{1}{z_{1}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{w_{1}}}&...&\frac{1}{(z_{1}-\lambda_{N})^{w_{1}}}&\frac{w_{1}c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{w_{1}+1}}&...&\frac{w_{1}c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{w_{1}+1}}\\\
1&\frac{1}{z_{2}-\lambda_{1}}&...&\frac{1}{z_{2}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{2}-\lambda_{1})^{w_{2}}}&...&\frac{1}{(z_{2}-\lambda_{N})^{w_{2}}}&\frac{w_{2}c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{w_{2}+1}}&...&\frac{w_{2}c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{w_{2}+1}}\\\
1&\frac{1}{z_{3}-\lambda_{1}}&...&\frac{1}{z_{3}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{3}-\lambda_{1})^{w_{3}}}&...&\frac{1}{(z_{3}-\lambda_{N})^{w_{3}}}&\frac{w_{3}c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{w_{3}+1}}&...&\frac{w_{3}c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{w_{3}+1}}\\\
\end{vmatrix}\,.$ (157)

We will now compute this determinant and we divide the calculation into
various simple steps.

##### Step 1:

First, we factor out the common factors of $c^{\gamma}_{a}$ in the last $N$
columns. Then, we get rid of the 1’s in the first column by subtracting the
first row from the $(w_{1}+1)^{th}$ and $(w_{1}+w_{2}+1)^{th}$ rows. We obtain

$z_{12}z_{13}\prod_{a=1}^{N}c^{\gamma}_{a}\\\
\begin{vmatrix}1&...&\frac{1}{z_{1}-\lambda_{a}}&...&\frac{1}{(z_{1}-\lambda_{a})^{2}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{m}}&...&\frac{m}{(z_{1}-\lambda_{a})^{m+1}}&...\\\
\vdots&\vdots&...&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{w_{1}}}&...&\frac{w_{1}}{(z_{1}-\lambda_{a})^{w_{1}+1}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})}&...&\frac{1}{(z_{2}-\lambda_{a})^{2}(z_{1}-\lambda_{a})}+\frac{1}{(z_{2}-\lambda_{a})(z_{1}-\lambda_{a})^{2}}&...\\\
0&...&\frac{1}{(z_{2}-\lambda_{a})^{2}}&...&\frac{2}{(z_{2}-\lambda_{a})^{3}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{2}-\lambda_{a})^{w_{2}}}&...&\frac{w_{2}}{(z_{2}-\lambda_{a})^{w_{2}+1}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})}&...&\frac{1}{(z_{3}-\lambda_{a})^{2}(z_{1}-\lambda_{a})}+\frac{1}{(z_{3}-\lambda_{a})(z_{1}-\lambda_{a})^{2}}&...\\\
0&...&\frac{1}{(z_{3}-\lambda_{a})^{2}}&...&\frac{2}{(z_{3}-\lambda_{a})^{3}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{3}-\lambda_{a})^{w_{3}}}&...&\frac{w_{3}}{(z_{3}-\lambda_{a})^{w_{3}+1}}&...\\\
\end{vmatrix}.$ (158)

##### Step 2:

Next, we subtract the row that contains
$(\tfrac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})},\tfrac{1}{(z_{2}-\lambda_{a})^{2}(z_{1}-\lambda_{a})}+\tfrac{1}{(z_{2}-\lambda_{a})(z_{1}-\lambda_{a})^{2}})$
from the row that contains
$(\tfrac{1}{(z_{2}-\lambda_{a})^{2}},\tfrac{2}{(z_{2}-\lambda_{a})^{3}})$. We
then use the resulting expression to subtract from the row that contains
$(\tfrac{1}{(z_{2}-\lambda_{a})^{3}},\tfrac{3}{(z_{2}-\lambda_{a})^{4}})$ and
we repeat the process until we subtract from the row that contains
$\tfrac{1}{(z_{2}-\lambda_{a})^{w_{2}}}$. We then do the same for the rows
that depend on $z_{3}$. The result is

$z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}c^{\gamma}_{a}\\\
\begin{vmatrix}1&...&\frac{1}{z_{1}-\lambda_{a}}&...&\frac{1}{(z_{1}-\lambda_{a})^{2}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{2}}&...&\frac{2}{(z_{1}-\lambda_{a})^{3}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{w_{1}}}&...&\frac{w_{1}}{(z_{1}-\lambda_{a})^{w_{1}+1}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})}&...&\frac{1}{(z_{2}-\lambda_{a})^{2}(z_{1}-\lambda_{a})}+\frac{1}{(z_{2}-\lambda_{a})(z_{1}-\lambda_{a})^{2}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{m}}&...&\frac{m}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{m+1}}+\frac{1}{(z_{1}-\lambda_{a})^{2}(z_{2}-\lambda_{a})^{m}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{w_{2}}}&...&\frac{w_{2}}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{w_{2}+1}}+\frac{1}{(z_{1}-\lambda_{a})^{2}(z_{2}-\lambda_{a})^{w_{2}}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})}&...&\frac{1}{(z_{3}-\lambda_{a})^{2}(z_{1}-\lambda_{a})}+\frac{1}{(z_{3}-\lambda_{a})(z_{1}-\lambda_{a})^{2}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{m}}&...&\frac{m}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{m+1}}+\frac{1}{(z_{1}-\lambda_{a})^{2}(z_{3}-\lambda_{a})^{m}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{w_{3}}}&...&\frac{w_{3}}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{w_{3}+1}}+\frac{1}{(z_{1}-\lambda_{a})^{2}(z_{3}-\lambda_{a})^{w_{3}}}&...\\\
\end{vmatrix}.$ (159)

##### Step 3:

We then factor out the common factor $\tfrac{1}{(z_{1}-\lambda_{a})}$,
obtaining

$z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{c^{\gamma}_{a}}{z_{1}-\lambda_{a}}\\\
\begin{vmatrix}1&...&\frac{1}{z_{1}-\lambda_{a}}&...&\frac{1}{(z_{1}-\lambda_{a})}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{2}}&...&\frac{2}{(z_{1}-\lambda_{a})^{2}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})^{w_{1}}}&...&\frac{w_{1}}{(z_{1}-\lambda_{a})^{w_{1}}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})}&...&\frac{1}{(z_{2}-\lambda_{a})^{2}}+\frac{1}{(z_{2}-\lambda_{a})(z_{1}-\lambda_{a})}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{m}}&...&\frac{m}{(z_{2}-\lambda_{a})^{m+1}}+\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{m}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{w_{2}}}&...&\frac{w_{2}}{(z_{2}-\lambda_{a})^{w_{2}+1}}+\frac{1}{(z_{1}-\lambda_{a})(z_{2}-\lambda_{a})^{w_{2}}}&...\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})}&...&\frac{1}{(z_{3}-\lambda_{a})^{2}}+\frac{1}{(z_{3}-\lambda_{a})(z_{1}-\lambda_{a})}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{m}}&...&\frac{m}{(z_{3}-\lambda_{a})^{m+1}}+\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{m}}&...\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&...&\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{w_{3}}}&...&\frac{w_{3}}{(z_{3}-\lambda_{a})^{w_{3}+1}}+\frac{1}{(z_{1}-\lambda_{a})(z_{3}-\lambda_{a})^{w_{3}}}&...\\\
\end{vmatrix}\,.$ (160)

##### Step 4:

We now subtract the columns shown in (160) to obtain

$z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{c^{\gamma}_{a}}{z_{1}-\lambda_{a}}\begin{vmatrix}1&\frac{1}{z_{1}-\lambda_{1}}&...&\frac{1}{z_{1}-\lambda_{N}}&0&...&0\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{m}}&...&\frac{1}{(z_{1}-\lambda_{N})^{m}}&\frac{m-1}{(z_{1}-\lambda_{1})^{m}}&...&\frac{m-1}{(z_{1}-\lambda_{N})^{m}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{w_{1}}}&...&\frac{1}{(z_{1}-\lambda_{N})^{w_{1}}}&\frac{w_{1}-1}{(z_{1}-\lambda_{1})^{w_{1}}}&...&\frac{w_{1}-1}{(z_{1}-\lambda_{N})^{w_{1}}}\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{2}-\lambda_{1})}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{2}-\lambda_{N})}&\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{2}-\lambda_{1})^{m}}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{2}-\lambda_{N})^{m}}&\frac{m}{(z_{2}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{2}-\lambda_{N})^{m+1}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{2}-\lambda_{1})^{w_{2}}}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{2}-\lambda_{N})^{w_{2}}}&\frac{w_{2}}{(z_{2}-\lambda_{1})^{w_{2}+1}}&...&\frac{w_{2}}{(z_{2}-\lambda_{N})^{w_{2}+1}}\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{3}-\lambda_{1})}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{3}-\lambda_{N})}&\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{3}-\lambda_{1})^{m}}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{3}-\lambda_{N})^{m}}&\frac{m}{(z_{3}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{3}-\lambda_{N})^{m+1}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})(z_{3}-\lambda_{1})^{w_{3}}}&...&\frac{1}{(z_{1}-\lambda_{N})(z_{3}-\lambda_{N})^{w_{3}}}&\frac{w_{3}}{(z_{3}-\lambda_{1})^{w_{3}+1}}&...&\frac{w_{3}}{(z_{3}-\lambda_{N})^{w_{3}+1}}\\\
\end{vmatrix}\,.$ (161)

Factoring out $\tfrac{1}{z_{1}-\lambda_{a}}$ from the various rows, we obtain

$z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{c^{\gamma}_{a}}{(z_{1}-\lambda_{a})^{2}}\begin{vmatrix}\frac{1}{(z_{1}-\lambda_{1})}&...&\frac{1}{(z_{1}-\lambda_{N})}&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{1}-\lambda_{1})^{m-1}}&...&\frac{1}{(z_{1}-\lambda_{N})^{m-1}}&\frac{m-1}{(z_{1}-\lambda_{1})^{m}}&...&\frac{m-1}{(z_{1}-\lambda_{N})^{m}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{1}-\lambda_{1})^{w_{1}-1}}&...&\frac{1}{(z_{1}-\lambda_{N})^{w_{1}-1}}&\frac{w_{1}-1}{(z_{1}-\lambda_{1})^{w_{1}}}&...&\frac{w_{1}-1}{(z_{1}-\lambda_{N})^{w_{1}}}\\\
\frac{1}{(z_{2}-\lambda_{1})}&...&\frac{1}{(z_{2}-\lambda_{N})}&\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{2}-\lambda_{1})^{m}}&...&\frac{1}{(z_{2}-\lambda_{N})^{m}}&\frac{m}{(z_{2}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{2}-\lambda_{N})^{m+1}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{2}-\lambda_{1})^{w_{2}}}&...&\frac{1}{(z_{2}-\lambda_{N})^{w_{2}}}&\frac{w_{2}}{(z_{2}-\lambda_{1})^{w_{2}+1}}&...&\frac{w_{2}}{(z_{2}-\lambda_{N})^{w_{2}+1}}\\\
\frac{1}{(z_{3}-\lambda_{1})}&...&\frac{1}{(z_{3}-\lambda_{N})}&\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{3}-\lambda_{1})^{m}}&...&\frac{1}{(z_{3}-\lambda_{N})^{m}}&\frac{m}{(z_{3}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{3}-\lambda_{N})^{m+1}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{3}-\lambda_{1})^{w_{3}}}&...&\frac{1}{(z_{3}-\lambda_{N})^{w_{3}}}&\frac{w_{3}}{(z_{3}-\lambda_{1})^{w_{3}+1}}&...&\frac{w_{3}}{(z_{3}-\lambda_{N})^{w_{3}+1}}\\\
\end{vmatrix}\,.$ (162)

##### Step 5:

We now repeat the steps $1)-4)$ above,

$\displaystyle\mathcal{J}=$
$\displaystyle\prod_{a=1}^{N}c^{\gamma}_{a}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}$
(163) $\displaystyle\begin{vmatrix}1&...&1&0&...&0\\\
\frac{1}{(z_{1}-\lambda_{1})}&...&\frac{1}{(z_{1}-\lambda_{N})}&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{1}-\lambda_{1})^{m-1}}&...&\frac{1}{(z_{1}-\lambda_{N})^{m-1}}&\frac{m-1}{(z_{1}-\lambda_{1})^{m}}&...&\frac{m-1}{(z_{1}-\lambda_{N})^{m}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{1}-\lambda_{1})^{w_{1}-2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{w_{1}-2}}&\frac{w_{1}-2}{(z_{1}-\lambda_{1})^{w_{1}-1}}&...&\frac{w_{1}-2}{(z_{1}-\lambda_{N})^{w_{1}-1}}\\\
\frac{1}{(z_{2}-\lambda_{1})}&...&\frac{1}{(z_{2}-\lambda_{N})}&\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}\\\
\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}&\frac{2}{(z_{2}-\lambda_{1})^{3}}&...&\frac{2}{(z_{2}-\lambda_{N})^{3}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{2}-\lambda_{1})^{m}}&...&\frac{1}{(z_{2}-\lambda_{N})^{m}}&\frac{m}{(z_{2}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{2}-\lambda_{N})^{m+1}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{2}-\lambda_{1})^{w_{2}}}&...&\frac{1}{(z_{2}-\lambda_{N})^{w_{2}}}&\frac{w_{2}}{(z_{2}-\lambda_{1})^{w_{2}+1}}&...&\frac{w_{2}}{(z_{2}-\lambda_{N})^{w_{2}+1}}\\\
\frac{1}{(z_{3}-\lambda_{1})}&...&\frac{1}{(z_{3}-\lambda_{N})}&\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}\\\
\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}&\frac{2}{(z_{3}-\lambda_{1})^{3}}&...&\frac{2}{(z_{3}-\lambda_{N})^{3}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{3}-\lambda_{1})^{m}}&...&\frac{1}{(z_{3}-\lambda_{N})^{m}}&\frac{m}{(z_{3}-\lambda_{1})^{m+1}}&...&\frac{m}{(z_{3}-\lambda_{N})^{m+1}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{3}-\lambda_{1})^{w_{3}}}&...&\frac{1}{(z_{3}-\lambda_{N})^{w_{3}}}&\frac{w_{3}}{(z_{3}-\lambda_{1})^{w_{3}+1}}&...&\frac{w_{3}}{(z_{3}-\lambda_{N})^{w_{3}+1}}\\\
\end{vmatrix}\,.$

##### Step 6:

The idea now is to subtract so that only the first column has 1 in the first
row and other columns have 0 in the first row. We do so by subtracting the
first column from the other columns. A generic form of column $a$ ($a\neq 1$),
after subtractions, is

$\displaystyle\frac{1}{(z_{i}-\lambda_{a})^{m}}-\frac{1}{(z_{i}-\lambda_{1})^{m}}=$
$\displaystyle-\lambda_{1a}\left(\sum_{l=0}^{m-1}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{a})^{m-l}}\right).$
(164)

And the determinant becomes

$\displaystyle\prod_{a=1}^{N}c^{\gamma}_{a}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}\prod_{a=2}^{N}(-\lambda_{1a})$
(165) $\displaystyle\begin{vmatrix}\vdots&...&\vdots&\vdots&...&\vdots\\\
\sum_{l=0}^{m_{i}-2}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{2})^{m_{i}-l-1}}&...&\sum_{l=0}^{m_{i}-2}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{N})^{m_{i}-l-1}}&\frac{m_{i}-1}{(z_{i}-\lambda_{1})^{m_{i}}}&...&\frac{m_{i}-1}{(z_{i}-\lambda_{N})^{m_{i}}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\ \end{vmatrix}.$

Here, $m_{1}=2,\ldots,w_{1}-1$ and $m_{2,3}=2,\ldots,w_{2,3}+1$. We then
subtract the combination
$\sum_{l=0}^{m_{i}-2}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{a})^{m_{i}-l-1}}$
from $\tfrac{m_{i}-1}{(z_{i}-\lambda_{a})^{m_{i}}}$ to obtain

$\displaystyle\frac{m_{i}-1}{(z_{i}-\lambda_{a})^{m_{i}}}-\sum_{l=0}^{m_{i}-2}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{a})^{m_{i}-l-1}}$
(166)
$\displaystyle=-\lambda_{1a}\sum_{l=0}^{m_{i}-2}\sum_{k=0}^{l}\frac{1}{(z_{i}-\lambda_{1})^{k+1}(z_{i}-\lambda_{a})^{m_{i}-k}}=-\lambda_{1a}\sum_{k=0}^{m_{i}-2}\frac{m_{i}-1-k}{(z_{i}-\lambda_{1})^{k+1}(z_{i}-\lambda_{a})^{m_{i}-k}}.$

We then notice that the difference between the
$m_{i}^{th}\times(z_{i}-\lambda_{1})^{-1}$ and $(m_{i}+1)^{th}$ rows can be
written as

$\displaystyle\left(\sum_{l=0}^{m_{i}-2}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{a})^{m_{i}-l-1}}\right)-\frac{1}{(z_{i}-\lambda_{1})}\left(\sum_{l=0}^{m_{i}-3}\frac{1}{(z_{i}-\lambda_{1})^{l+1}(z_{i}-\lambda_{a})^{m_{i}-2-1}}\right)$
(167)
$\displaystyle=\frac{1}{(z_{i}-\lambda_{1})(z_{i}-\lambda_{a})^{m_{i}-1}}\ ,$
$\displaystyle\sum_{k=0}^{m_{i}-2}\frac{m_{i}-1-k}{(z_{i}-\lambda_{1})^{k+1}(z_{i}-\lambda_{a})^{m_{i}-k}}-\frac{1}{(z_{i}-\lambda_{1})}\left(\sum_{k=0}^{m_{i}-3}\frac{m_{i}-2-k}{(z_{i}-\lambda_{1})^{k+1}(z_{i}-\lambda_{a})^{m_{i}-k-1}}\right)$
$\displaystyle=\frac{m_{i}-1}{(z_{i}-\lambda_{1})(z_{i}-\lambda_{a})^{m_{i}}},$

and

$\displaystyle\frac{m_{i}-1}{(z_{i}-\lambda_{1})^{m_{i}}}-\frac{m_{i}-2}{(z_{i}-\lambda_{1})^{m_{i}}}=\frac{1}{(z_{i}-\lambda_{1})^{m_{i}}}.$
(168)

Factoring out the $\tfrac{1}{z_{i}-\lambda_{1}}$ then gives

$\displaystyle\prod_{a=1}^{N}c^{\gamma}_{a}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}z_{12}^{w_{2}}z_{13}^{w_{3}}\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2}}\prod_{a=2}^{N}(\lambda_{1a})^{2}$
(169)
$\displaystyle\times\frac{1}{(z_{1}-\lambda_{1})^{w_{1}-2}}\prod_{i=2}^{3}\frac{1}{(z_{i}-\lambda_{1})^{w_{i}}}\,\begin{vmatrix}\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{i}-\lambda_{1})^{m-1}}&...&\frac{1}{(z_{i}-\lambda_{N})^{m-1}}&\frac{m-1}{(z_{i}-\lambda_{2})^{m}}&...&\frac{m-1}{(z_{i}-\lambda_{N})^{m}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\ \end{vmatrix}\,.$

Note that maximum value of $m_{1}$ is $w_{1}-1$ and of $m_{2,3}$ is
$w_{2,3}+1$ while the minimum value of $m_{i}$ is 2 for all $i$. Note also the
fact that there is only one column depending on $\lambda_{1}$ (which is the
first column) and two each which depend on $\lambda_{i},i>1$. This agrees with
the counting since $w_{1}-2+w_{2}+w_{3}=2N+1-2=2N-1=N+(N-1)$.

##### Step 7:

We now perform steps $1)-6)$. The determinant is simplified to

$\displaystyle\prod_{a=1}^{N}c^{\gamma}_{a}z_{12}^{3w_{2}}z_{13}^{3w_{3}}\prod_{i=1}^{3}\frac{1}{(z_{i}-\lambda_{1})^{2w_{i}}}\prod_{a=2}^{N}\frac{1}{(z_{1}-\lambda_{a})^{6}}\prod_{a=2}^{N}\lambda_{1a}^{4}$
(170) $\displaystyle\hskip
28.45274pt\times\begin{vmatrix}\vdots&...&\vdots&\vdots&...&\vdots\\\
\frac{1}{(z_{i}-\lambda_{2})^{m-1}}&...&\frac{1}{(z_{i}-\lambda_{N})^{m-1}}&\frac{m-1}{(z_{i}-\lambda_{2})^{m}}&...&\frac{m-1}{(z_{i}-\lambda_{N})^{m}}\\\
\vdots&...&\vdots&\vdots&...&\vdots\\\ \end{vmatrix}\,.$

Here $m_{1}$ runs from $2$ to $w_{1}-2$ and $m_{2,3}$ from $2$ to $w_{2,3}+1$.
It is then not hard to do the computation inductively from this point on. The
result is

$\displaystyle\mathcal{J}=$
$\displaystyle\left(\prod_{a=1}^{N}c^{\gamma}_{a}\right)\left(\prod_{a=1}^{N}\frac{1}{(z_{1}-\lambda_{a})^{2w_{1}}(z_{2}-\lambda_{a})^{2w_{2}}(z_{3}-\lambda_{a})^{2w_{3}}}\right)$
(171)
$\displaystyle\times\left(\prod_{a<b}(\lambda_{a}-\lambda_{b})^{4}\right)z_{12}^{w_{2}w_{1}}z_{13}^{w_{3}w_{1}}z_{23}^{w_{2}w_{3}}.$

Written in terms of covering map data $(C^{\gamma},A^{\gamma}_{i})$, we obtain

$\displaystyle\mathcal{J}=$
$\displaystyle\prod_{i=1}^{3}(A^{\gamma}_{i})^{\frac{w_{i}+1}{2}}(C^{\gamma})^{-2}z_{12}z_{13}z_{23}\,.$
(172)

This recovers precisely the proposal we had in the main text for this special
case. We note one important observation that the steps taken in deriving
eq.(LABEL:finalans) need not assume that $i$ takes 3 values. As long as the
matrix is square, that is $\sum_{i=1}^{n}w_{i}=2N+1$, we can always apply the
method we described inductively. The corresponding modification is then to
allow $i$ to vary from $1,\ldots,n$. This observation will be important below.

#### The more general case

Let us now drop the assumption about the number of insertions and only assume
that the covering maps have no extra branch points. This requires only a minor
modification to the $n=3$ case. We have

$\displaystyle\mathcal{J}=$
$\displaystyle\left(\prod_{a=1}^{N}c^{\gamma}_{a}\right)\left(\prod_{a,i}\frac{1}{(z_{i}-\lambda_{a})^{2w_{i}}}\right)\left(\prod_{a<b}(\lambda_{a}-\lambda_{b})^{4}\right)\prod_{i<j}z_{ij}^{w_{i}w_{j}-1}z_{12}z_{13}z_{23}(C^{\gamma})^{n-3}\,.$
(173)

We start by writing the Jacobian is in a block lower triangular form where the
upper left $(2N+1)\times(2N+1)$ part looks like the one in the previous
section but with effective branching $w^{\prime}_{i}=w_{i}$ for $i=1,2,3$ and
$w^{\prime}_{i}=w_{i}-1$ for $i=4,\ldots,n$. The lower right
$(n-3)\times(n-3)$ is then just a diagonal matrix with elements
$\partial^{w_{i}}\gamma(z_{i})/(w_{i}-1)!$ for $i=4,\ldots,n$. Explicitly, we
have

$\mathcal{J}=\begin{vmatrix}1&\frac{1}{z_{1}-\lambda_{1}}&...&\frac{1}{z_{1}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{2}}&0&...&0\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{2}}&...&\frac{1}{(z_{1}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{1}-\lambda_{1})^{w_{1}}}&...&\frac{1}{(z_{1}-\lambda_{N})^{w_{1}}}&\frac{w_{1}c^{\gamma}_{1}}{(z_{1}-\lambda_{1})^{w_{1}+1}}&...&\frac{w_{1}c^{\gamma}_{N}}{(z_{1}-\lambda_{N})^{w_{1}+1}}\\\
1&\frac{1}{z_{2}-\lambda_{1}}&...&\frac{1}{z_{2}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{2}-\lambda_{1})^{2}}&...&\frac{1}{(z_{2}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{2}-\lambda_{1})^{w_{2}}}&...&\frac{1}{(z_{2}-\lambda_{N})^{w_{2}}}&\frac{w_{2}c^{\gamma}_{1}}{(z_{2}-\lambda_{1})^{w_{2}+1}}&...&\frac{w_{2}c^{\gamma}_{N}}{(z_{2}-\lambda_{N})^{w_{2}+1}}\\\
1&\frac{1}{z_{3}-\lambda_{1}}&...&\frac{1}{z_{3}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{3}-\lambda_{1})^{2}}&...&\frac{1}{(z_{3}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots&\vdots&\vdots&\vdots\\\
0&\frac{1}{(z_{3}-\lambda_{1})^{w_{3}}}&...&\frac{1}{(z_{3}-\lambda_{N})^{w_{3}}}&\frac{w_{3}c^{\gamma}_{1}}{(z_{3}-\lambda_{1})^{w_{3}+1}}&...&\frac{w_{3}c^{\gamma}_{N}}{(z_{3}-\lambda_{N})^{w_{3}+1}}\\\
1&\frac{1}{z_{4}-\lambda_{1}}&...&\frac{1}{z_{4}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{4}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{4}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{4}-\lambda_{1})^{2}}&...&\frac{1}{(z_{4}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{4}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{4}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{4}-\lambda_{1})^{w_{4}-1}}&...&\frac{1}{(z_{4}-\lambda_{N})^{w_{4}-1}}&\frac{(w_{4}-1)c^{\gamma}_{1}}{(z_{4}-\lambda_{1})^{w_{4}}}&...&\frac{(w_{4}-1)c^{\gamma}_{N}}{(z_{4}-\lambda_{N})^{w_{4}}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
1&\frac{1}{z_{n}-\lambda_{1}}&...&\frac{1}{z_{n}-\lambda_{N}}&\frac{c^{\gamma}_{1}}{(z_{n}-\lambda_{1})^{2}}&...&\frac{c^{\gamma}_{N}}{(z_{n}-\lambda_{N})^{2}}\\\
0&\frac{1}{(z_{n}-\lambda_{1})^{2}}&...&\frac{1}{(z_{n}-\lambda_{N})^{2}}&\frac{2c^{\gamma}_{1}}{(z_{n}-\lambda_{1})^{3}}&...&\frac{2c^{\gamma}_{N}}{(z_{n}-\lambda_{N})^{3}}\\\
\vdots&\vdots&...&\vdots&\vdots&...&\vdots\\\
0&\frac{1}{(z_{n}-\lambda_{1})^{w_{n}-1}}&...&\frac{1}{(z_{n}-\lambda_{N})^{w_{n}-1}}&\frac{(w_{n}-1)c^{\gamma}_{1}}{(z_{n}-\lambda_{1})^{w_{n}}}&...&\frac{(w_{n}-1)c^{\gamma}_{N}}{(z_{n}-\lambda_{N})^{w_{n}}}&0&...&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\frac{\partial^{w_{4}}\gamma(z_{4})}{(w_{4}-1)!}&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&...&\ddots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&...&...&\frac{\partial^{w_{n}}\gamma(z_{n})}{(w_{n}-1)!}\\\
\end{vmatrix}\,.$ (174)

Hence, the determinant is the product of the generalization of (171) and
$\prod_{i=4}^{n}(w_{i}a^{\gamma}_{i})$. Indeed, using eq.(171) with the
effective branchings gives

$\displaystyle\prod_{a=1}^{N}(c^{\gamma}_{a})\prod_{a<b}(\lambda_{a}-\lambda_{b})^{4}\prod_{a,i}\left(\frac{1}{(z_{i}-\lambda_{a})^{2w^{\prime}_{i}}}\right)\prod_{i<j}z_{ij}^{w^{\prime}_{i}w^{\prime}_{j}}\,.$
(175)

And the $\prod_{i=4}^{n}(w_{i}a^{\gamma}_{i})$ gives

$\displaystyle(C^{\gamma})^{n-3}\prod_{a,i=4}^{n}\left(\frac{1}{(z_{i}-\lambda_{a})^{2}}\right)\prod_{i=4}^{n}\prod_{j\neq
i}z_{ij}^{w_{j}-1}\,.$ (176)

By combining (175) and (176), this can be shown to reduce to (LABEL:genern).
Writing (LABEL:genern) in terms of covering map data therefore gives

$\displaystyle\mathcal{J}=$
$\displaystyle\prod_{i=1}^{n}(A^{\gamma}_{i})^{\frac{w_{i}+1}{2}}(C^{\gamma})^{-2}z_{12}z_{13}z_{23}.$
(177)

###  A.3 The most general case

We can now compute the Jacobian in the most general case of arbitrary
$n$-point functions with arbitrary spin, assuming that $m$ defined in equation
(98) is a non-negative integer. The Jacobian can be computed in essentially
the same way as the previous case. Again, the Jacobian is simpler to work out
for a 3-point function, this is given by

$\displaystyle\mathcal{J}=$
$\displaystyle\begin{vmatrix}\frac{\partial}{\partial
b}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)&\frac{\partial}{\partial
c^{\gamma}_{a}}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)&\frac{\partial}{\partial\lambda_{a}}\left(\frac{\partial^{l_{i}}\gamma(z_{i})}{l_{i}!}\right)\\\
\frac{\partial}{\partial b}\gamma(\zeta_{l})&\frac{\partial}{\partial
c^{\gamma}_{a}}\gamma(\zeta_{l})&\frac{\partial}{\partial\lambda_{a}}\gamma(\zeta_{l})\end{vmatrix}$
(178) $\displaystyle=$ $\displaystyle
z_{12}^{w_{1}w_{2}}z_{13}^{w_{1}w_{3}}z_{23}^{w_{2}w_{3}}\prod_{a=1}^{N}c^{\gamma}_{a}\prod_{a,i}\frac{1}{(z_{i}-\lambda_{a})^{2w_{i}}}\prod_{a,l}\frac{1}{(\zeta_{l}-\lambda_{a})^{2}}$
$\displaystyle\hskip
85.35826pt\times\prod_{a<b}(\lambda_{a}-\lambda_{b})^{4}\prod_{i,l}(z_{i}-\zeta_{l})^{w_{i}}\prod_{l<k}\zeta_{lk}\,\,.$

Indeed, the expression above can be obtained from (175) with the effective
branching $w_{l}=1$ for $l=1,\ldots,m$. Writing in terms of covering map data,
this gives

$\mathcal{J}=z_{12}z_{13}z_{23}(C^{\gamma})^{-2}\prod_{i=1}^{3}(A^{\gamma}_{i})^{\frac{w_{i}+1}{2}}\prod_{l=1}^{m}(A^{\gamma}_{l})^{\frac{1}{2}}\,.$
(179)

The generalization to $n>3$ is then straightforward by using effective
branching explained in the previous section and one can show that the Jacobian
takes the form

$\mathcal{J}=z_{12}z_{13}z_{23}(C^{\gamma})^{-2}\prod_{i=1}^{3}(A^{\gamma}_{i})^{\frac{w_{i}+1}{2}}\prod_{l=1}^{m}(A^{\gamma}_{l})^{\frac{1}{2}}\,.$
(180)

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