# Spectral flow and localisation in $\text{AdS}_{3}$ string theory

Bob Knighton a Sean Seet b Vit Sriprachyakul<EMAIL_ADDRESS><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 string theory in three-dimensional Anti-de Sitter spacetime in the
path integral formalism. We derive expressions for generic spectrally-flowed
near-boundary vertex operators in the Wakimoto representation, and relate
their correlation functions to covering maps from the worldsheet to the target
space boundary. We show that the path integral structurally reproduces
correlation functions of the dual symmetric orbifold theory. By rephrasing
spectral flow as the introduction of a background gauge field, we provide a
path integral derivation of the localisation property of the near boundary
theory. We then focus on the case of IIB string theory on
$\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ with $k=1$ units of NS-
NS flux, where the relationship between correlation functions and covering
maps can be made sharp. We also comment on the relation of the $k=1$ theory
and twistor theory.

##  1 Introduction

The understanding of string theory on Anti-de Sitter (AdS) backgrounds is
crucial in a top-down approach to the AdS/CFT correspondence Maldacena:1997re
; Witten:1998qj . Of these backgrounds, the three-dimensional $\text{AdS}_{3}$
spacetime represents a particularly tractable option since string theory on
$\text{AdS}_{3}$ with pure NS-NS flux can be described by a WZW model on the
non-compact group $\text{SL}(2,\mathbb{R})$ Maldacena:2000hw ;
Maldacena:2000kv ; Maldacena:2001km . As such, string theory on
$\text{AdS}_{3}$ in the absence of RR flux is largely dictated by the affine
$\mathfrak{sl}(2,\mathbb{R})$ symmetry on the worldsheet.

Due to the non-compactness of the group $\text{SL}(2,\mathbb{R})$, this WZW
model is somewhat more subtle than those based on compact groups like
$\text{SU}(2)$. Specifically, the spectrum of the $\text{SL}(2,\mathbb{R})$
WZW model consists of usual highest-weight representations of
$\mathfrak{sl}(2,\mathbb{R})_{k}$, as well as novel spectrally-flowed
representations. These spectrally-flowed representations are highest-weight
with respect to the worldsheet Virasoro algebra, but are not highest-weight
with respect to the affine $\mathfrak{sl}(2,\mathbb{R})_{k}$ symmetry. As
such, correlation functions of spectrally-flowed states are notoriously
difficult to compute (although there has been great progress in recent years
Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr ; Bufalini:2022toj ; Iguri:2022pbp ;
Fiset:2022erp ).

Spectrally-flowed states are labelled by three quantum numbers: their
$\text{SL}(2,\mathbb{R})$ spin $j$, the eigenvalue $h$ of the cartan
$J^{3}_{0}$ and the amount of spectral flow $w\in\mathbb{Z}$. For holographic
applications, it is desirable to include yet another auxiliary label $x$,
which labels the location of emission of the string on the boundary of
$\text{AdS}_{3}$. A vertex operator labelled by $x$ is holographically dual to
a local operator in the boundary CFT placed at the position $x$. Thus, the
holographically interesting correlation functions in the
$\text{SL}(2,\mathbb{R})$ model take the form

$\Braket{V_{m_{1},j_{1}}^{w_{1}}(x_{1},z_{1})\cdots
V_{m_{n},j_{n}}^{w_{n}}(x_{n},z_{n})}_{\Sigma}\,,$ (1)

where $m=h-kw/2$. Correlators of this form turn out to be surprisingly
difficult to compute for two reasons. First, since the states have nonzero
spectral flow, they are not primary with respect to the
$\mathfrak{sl}(2,\mathbb{R})_{k}$ current algebra (specifically they have
higher-order poles in the OPE with $J^{+}$). Second, since the states are
located at nonzero $x$, they are no longer eigenstates with respect to the
$J^{3}_{0}$ Cartan of $\mathfrak{sl}(2,\mathbb{R})_{k}$. Both of these effects
mean that the local Ward identities associated to spectrally-flowed
correlation functions become rather complicated, and finding explicit
solutions is a very difficult problem. A third difficulty arises from the fact
that, while vertex operators in unflowed sectors admit explicit construction
in terms of fundamental fields (see for example Maldacena:2000hw ;
Maldacena:2001km ; deBoer:1998gyt ; Giveon:1998ns ; Naderi:2022bus ),
spectrally-flowed operators are usually specified implicitly by their OPEs
with the $\mathfrak{sl}(2,\mathbb{R})_{k}$ currents, making a path integral
approach somewhat difficult (although not impossible, see for example
Giribet:1999ft ; Giribet:2000fy ; Giribet:2001ft ; Iguri:2007af ;
Hikida:2007tq ; Hikida:2008pe ; Hikida:2020kil ; Iguri:2022eat ).

While, on the one hand, correlation functions in the $\text{SL}(2,\mathbb{R})$
model are hard to compute, there are hints of a subsector of the theory which
radically simplifies. In Maldacena:2001km , it was shown that correlation
functions of spectrally-flowed vertex operators diverge when certain linear
relations among the $\text{SL}(2,\mathbb{R})$ spins are satisfied. These
divergences were interpreted in terms of strings which can approach the
boundary of $\text{AdS}_{3}$ with finite energy cost, causing the path
integral to diverge. Such contributions were dubbed ‘worldsheet instantons’.
Geometrically, these instantons receive contributions from worldsheets which
holomorphically cover the boundary of $\text{AdS}_{3}$, i.e. for which there
exists a holomorphic map $\gamma:\Sigma\to\partial\text{AdS}_{3}$ such that

$\gamma(z)\sim x_{i}+\mathcal{O}((z-z_{i})^{w_{i}})\,.$ (2)

These covering maps represent, in turn, strings which wind the boundary of
$\text{AdS}_{3}$ a certain number of times with specific monodromies around
the points $x_{i}$.

This picture has recently been strengthened by the remarkable series of papers
Eberhardt:2019ywk ; Eberhardt:2020akk ; Dei:2021xgh ; Dei:2021yom ;
Dei:2022pkr . The authors were able to constrain the forms of
$\text{SL}(2,\mathbb{R})$ correlation functions almost entirely on the grounds
of affine symmetry and holomorphicity. They found that
$\text{SL}(2,\mathbb{R})$ correlators contain an infinite series of
divergences labeled by positive integers. One such class, and the one we will
be interested in, occurs when the $\text{SL}(2,\mathbb{R})$ spins satisfy the
relation

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

for some integer $m\geq 0$.111This result was established in Eberhardt:2019ywk
; Eberhardt:2020akk  for $m=0$ at arbitrary genus and number of insertion
points. For $m>0$, it has so far only been proven for genus zero correlation
functions with $n\leq 4$ Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr . Viewed as a
function of $\sum_{i}j_{i}$, the $\text{SL}(2,\mathbb{R})$ correlation
functions have poles at these values, and the residues of these poles can be
computed precisely. They can be shown to arises at points in the moduli space
$\mathcal{M}_{g,n}$ for which a holomorphic covering map of the form (2)
exists, and for which there are exactly $m$ extra simple branch points
Eberhardt:2021vsx ; Dei:2022pkr .

Recently an exact (perturbative) proposal for the CFT dual of bosonic string
theory on $\text{AdS}_{3}\times X$ at generic level $k$ was proposed
Dei:2022pkr ; Eberhardt:2021vsx  which schematically takes the form of a
deformed symmetric orbifold. Explicitly, the proposed dual CFT is222This
proposal improved on the earlier work of Eberhardt:2019qcl ; Dei:2019osr ,
which was based on the symmetric orbifold of Liouville theory rather than a
linear dilaton CFT. From the level of the one-loop long-string spectrum, these
proposals are indistinguishable.

$\text{Sym}^{K}(X\times\mathbb{R}_{\mathcal{Q}})+\mu\int\sigma_{2,\sigma}\,,$
(4)

where $\mathbb{R}_{\mathcal{Q}}$ is a linear dilaton with slope $\mathcal{Q}$,
and $\sigma_{2,\alpha}$ is a marginal operator in the twist-2 sector of the
symmetric orbifold which breaks the orbifold structure. From the point of view
of this duality, the worldsheet instantons of Maldacena:2001km  admit a
natural interpretation: they are the poles of correlation functions in the
dual CFT which contribute at order $\mu^{m}$ in conformal perturbation theory.
The role of the covering maps $\gamma$ then play the role of Feynman diagrams
in the theory Lunin:2000yv ; Lunin:2001ne ; Pakman:2009zz , with the $m$ extra
branch points coming from the $m$ insertions of the twist-2 field. Thus, an
understanding of the worldsheet instantons is fundamental to an understanding
of the perturbative structure of the dual CFT.

The goal of the present paper is to shed light on spectrally-flowed
correlators in the $\text{SL}(2,\mathbb{R})$ WZW model in the path integral
formalism. Our approach is to consider the worldsheet theory in the limit near
the boundary of $\text{AdS}_{3}$, since this is the regime which contributes
to the divergences in $\text{SL}(2,\mathbb{R})$ correlation functions. In this
limit, we write down explicit forms of spectrally-flowed vertex operators and
compute the schematic form of correlation functions of these operators. We
find that the holomorphic covering maps are naturally reproduced in the path
integral language, and that the $j$-constraint (3) follows from a
straightforward charge-conservation argument of worldsheet correlators (as was
argued in Eberhardt:2019ywk ; Hikida:2020kil ).

Let us now be more specific about the results of this paper. In Section 2, we
work in the Wakimoto representation, for which the $\text{SL}(2,\mathbb{R})$
sigma model takes the form

$S=\frac{1}{2\pi}\int_{\Sigma}\mathrm{d}^{2}z\left(\frac{1}{2}\partial\Phi\,\overline{\partial}\Phi+\beta\overline{\partial}\gamma+\overline{\beta}\partial\overline{\gamma}-\frac{1}{k}\beta\overline{\beta}\,e^{-Q\Phi}-\frac{Q}{4}R\Phi\right)\,,$
(5)

where $\Phi$ is a scalar representing the radial coordinate of
$\text{AdS}_{3}$, $\gamma,\bar{\gamma}$ are holomorphic coordinates of the
boundary, and $\beta,\bar{\beta}$ are formal Lagrange multipliers which allow
us to study the action in the near boundary $(\Phi\to\infty)$ limit. In this
limit, the above action reduces to that of a free theory, and thus the path
integral can be computed exactly. In terms of these free fields, we find
closed-form expressions for spectrally-flowed highest-weight states of
$\mathfrak{sl}(2,\mathbb{R})_{k}$, following a recent analysis of spectrally-
flowed states in the tensionless worldsheet theory Dei:2023ivl , which can be
written compactly in the $x$-basis:

$V_{h,j}^{w}(x,z)=e^{(w/Q-wj)\Phi}\left(\frac{\partial^{w}(\gamma-x)}{w!}\right)^{-m-j}\delta_{w}(\gamma-x)\,,$
(6)

where $\delta_{w}(\gamma-x)$ is a formal delta-function operator which imposes
that $\gamma-x$ has a zero of order $w$ at the insertion point (see Section 2
for a precise definition). Given their simple form, these states are can be
employed in the path integral formalism.

We also argue in Section 2 that, in order to study string theory correlators
in Euclidean $\text{AdS}_{3}$, we need to introduce a screening operator of
the form333Near the end stages of writing this paper, we became aware of the
work of Hikida and Schomerus Hikida:2023jyc , which introduces the same
screening operator to study correlation functions in the
$\text{SL}(2,\mathbb{R})$ WZW model.

$D=V_{\frac{k}{2},\frac{k-2}{2}}^{w=-1}=e^{-2\Phi/Q}\left(\oint\gamma\right)^{-(k-1)}\delta(\beta)\,,$
(7)

which effectively compactifies the target of the field $\gamma$ from
$\mathbb{C}$ to $\mathbb{CP}^{1}$.444See Frenkel:2005ku  for an analogous
construction in the large-radius limit of the topological A-model. This
operator can also be thought of as a screening operator for the field $\Phi$,
so that correlation functions are computed in a Coulomb gas formalism. This
operator $D$ is similar to, yet distinct from, other screening operators which
have previously been considered in the literature Giribet:2001ft ;
Iguri:2007af  and, as we will argue in Section 2, plays a natural geometric
role in the path integral computation of correlation functions. Using the
explicit forms of spectrally-flowed vertex operators and the screening
operator $D$, we find a straightforward explanation of why the worldsheet
theory is dominated by holomorphic covering maps, provided that the constraint
(3) is satisfied. We also show that these correlation functions schematically
reproduce the perturbative structure of the CFT dual (4).

In Section 3 we provide a novel interpretation of spectral flow in terms of a
nonlocal operator wrapping an unflowed state. This nonlocal operator is built
from a line integral of $\mathfrak{sl}(2,\mathbb{R})_{k}$ currents, and we
interpret it as the inclusion of a nontrivial background gauge field in the
$\text{SL}(2,\mathbb{R})$ WZW model which couples to the worldsheet currents.
The inclusion of this background modifies the classical equations of motion of
the WZW model, and solutions to these equations of motion are given precisely
by holomorphic maps $\gamma$ from the worldsheet to the $\text{AdS}_{3}$
boundary. In this sense, we can think of the holomorphic covering maps as
instantons of the worldsheet theory in a nontrivial background, thus making
concrete in what sense the worldsheet instantons of Maldacena:2001km  are
classical saddles of the string path integral.555A semiclassical justification
for these solutions was also given in Eberhardt:2019ywk .

### A simplification at $k=1$

The above discussion admits a drastic simplification in the case of the
superstring on $\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ with one
unit of pure NS-NS flux – known as the so-called ‘minimal tension’ or
‘tensionless’ string. This theory is desribed in the hybrid formalism of
Berkovits, Vafa and Witten Berkovits:1999im ; Gaberdiel:2022als  in terms of a
sigma model on the supergroup $\text{PSU}(1,1|2)$ at level $k=1$. An
interesting feature of this sigma model is that the correlation functions
analogous to (1) always localise to holomorphic covering maps Dei:2020zui ;
Knighton:2020kuh , and the $j$-constraint (3) for $m=0$ is always satisfied.
The path integral of this theory can be shown to localise completely onto
holomorphic covering maps whose only critical points are at $z=z_{i}$, and one
has the exact duality Gaberdiel:2018rqv ; Eberhardt:2018ouy ; Hikida:2020kil ;
Dei:2020zui ; Knighton:2020kuh ; Bertle:2020sgd ; Gaberdiel:2021njm ;
Gaberdiel:2021kkp ; Gaberdiel:2022oeu ; Fiset:2022erp ; Naderi:2022bus ;
Dei:2023ivl

$\begin{gathered}\text{IIB strings on
AdS}_{3}\times\text{S}_{3}\times\mathbb{T}^{4}\text{ with minimal (}k=1\text{)
units of NS-NS flux}\\\ \Longleftrightarrow\\\ \text{the symmetric orbifold
}(\mathbb{T}^{4})^{K}/S_{K}\text{ at large }K\end{gathered}$ (8)

with no perturbation away from the orbifold point.

In Section 4, we specialise the technology developed in Sections 2 and 3 to
this tesnionless $\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$
theory.666A similar analysis was also considered recently in a Wakimoto-like
representation Dei:2023ivl . The worldsheet theory of this background admits a
free field realisation which is exact. This free field realisation has been
used recently to show that the correlation functions of spectrally-flowed
vertex operators localise onto holomorphic covering maps Dei:2020zui ;
Knighton:2020kuh . In the case of the $k=1$ theory, we again find compact
expressions for spectrally-flowed vertex operators, and show that the
localisation property is a direct consequence of the forms of these operators
in the path integral formalism. We also show in this section how the idea of
inducing a background gauge field works in this particular model, and that the
localised solutions can be directly read off from the zero-modes of the
twisted kinetic operator $\overline{\partial}+A$ acting on a particular
holomorphic vector bundle.

As was noted in Dei:2020zui , the tensionless superstring on
$\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ is conceptually similar
to the Berkovits twistor string Berkovits_2004 . In Section 5, we sharpen this
analogy and interpret the vertex operators found in Section 4 as natural
objects in twistor space. We briefly introduce the relevant twistor space and
explain why the S2 twistor incidence relation appears in the model. In the
SL$(2)$ invariant notation which we introduce, we find that the vertex
operators are related to S2 twistor elementary states, a basis for meromorphic
twistor functions PENROSE1973241 . Finally, we collect various technical
results and discussions in Appendices A and B.

##  2 Bosonic strings on $\text{AdS}_{3}$

###  2.1 The worldsheet theory

String theory on $\text{AdS}_{3}$ can be described in terms of a Wess-Zumino-
Witten (WZW) model on the universal cover of $\text{SL}(2,\mathbb{R})$
Maldacena:2000hw ; Maldacena:2000kv ; Maldacena:2001km . The fundamental
fields are maps $g:\Sigma\to\text{SL}(2,\mathbb{R})$ governed by the classical
action

$S_{\text{WZW}}=\frac{k}{4\pi}\int_{\Sigma}\text{Tr}[g^{-1}\mathrm{d}g\wedge
g^{-1}\mathrm{d}g]+\frac{k}{2\pi i}\int_{B}\text{Tr}[g^{-1}\mathrm{d}g\wedge
g^{-1}\mathrm{d}g\wedge g^{-1}\mathrm{d}g]\,.$ (9)

Here, $B$ is any 3-manifold with boundary $\partial B=\Sigma$. We note that
since $\text{SL}(2,\mathbb{R})$ is non-compact, the level $k$ does not need to
be integer in order to produce a well-defined path integral.777Concretely,
since $\text{H}^{3}(\text{SL}(2,\mathbb{R}),\mathbb{Z})=0$, the Wess-Zumino
term is independent of the choice of the bulk manifold $B$ Witten:1983ar ;
Dijkgraaf:1989pz .

The above action admits holomorphic and anti-holomorphic conserved currents

$J=k\,g^{-1}\partial g\,,\quad\bar{J}=k\,\overline{\partial}g\,g^{-1}\,.$ (10)

Given generators $t^{a}$ of $\mathfrak{sl}(2,\mathbb{R})$, we can define

$J^{a}=\frac{1}{k}\text{Tr}[t^{a}J]\,,$ (11)

and similarly for $\bar{J}$. The currents $J^{a}$ satisfy the algebra
$\mathfrak{sl}(2,\mathbb{R})_{k}$. Specifically, in a convenient basis, we
have

$\begin{gathered}J^{3}(z)J^{3}(w)\sim-\frac{k/2}{(z-w)^{2}}\,,\quad
J^{3}(z)J^{\pm}(w)\sim\pm\frac{J^{\pm}(w)}{z-w}\,,\\\
J^{+}(z)J^{-}(w)\sim\frac{k}{(z-w)^{2}}-\frac{2J^{3}(w)}{z-w}\,.\end{gathered}$
(12)

#### The Wakimoto representation

To study the $\text{SL}(2,\mathbb{R})$ theory more explicitly, it is
convenient to choose an explicit set of coordinates on $\text{AdS}_{3}$. A
particularly convenient description is that of global $\text{AdS}_{3}$ in
Poincaré coordinates:

$\mathrm{d}s^{2}=\frac{\mathrm{d}r^{2}+\mathrm{d}\gamma\,\mathrm{d}\overline{\gamma}}{(r/L)^{2}}\,,$
(13)

where $r\in(0,\infty)$ is the radial coordinate, $(\gamma,\overline{\gamma})$
are complex coordinates of the boundary sphere, and $L$ is the AdS
radius.888In Euclidean signature, $\gamma,\overline{\gamma}$ are complex
conjugates of one another, while in Lorenzian signature they are independent
and real. The $\text{AdS}_{3}$ boundary is identified with the limit $r\to 0$.
In order for a string in this background to be anomaly-free, it must be
supported by a Kalb-Ramond $B$-field999Of course, we must also introduce
compact directions $X$ such that $c(\text{AdS}_{3})+c(X)=26$.

$B=-\left(\frac{L^{2}}{r^{2}}\right)\mathrm{d}\gamma\wedge\mathrm{d}\overline{\gamma}\,.$
(14)

The above metric and $B$-field give rise to the Polyakov action101010We use
the conventions of Polchinski Polchinski:1998rq  for complex coordinates,
namely $z=x+iy$, $\overline{z}=x-iy$,
$\partial=(\partial_{x}-i\partial_{y})/2$, and
$\overline{\partial}=(\partial_{x}+i\partial_{y})/2$.

$\begin{split}S&=\frac{1}{2\pi\alpha^{\prime}}\int_{\Sigma}\mathrm{d}^{2}z\left(G_{\mu\nu}(X)\partial
X^{\mu}\overline{\partial}X^{\nu}+B_{\mu\nu}(X)\partial
X^{\mu}\overline{\partial}X^{\nu}\right)\\\
&=\frac{L^{2}}{2\pi\alpha^{\prime}}\int_{\Sigma}\mathrm{d}^{2}z\left(\frac{1}{r^{2}}\partial
r\overline{\partial}r+\frac{1}{r^{2}}\partial\overline{\gamma}\,\overline{\partial}\gamma\right)\\\
&=\frac{k}{4\pi}\int_{\Sigma}\mathrm{d}^{2}z\left(2\partial\Phi\overline{\partial}\Phi+2e^{2\Phi}\partial\overline{\gamma}\,\overline{\partial}\gamma\right)\,,\end{split}$
(15)

where $k=L^{2}/\alpha^{\prime}$ measures the AdS radius in string units and we
have defined $r=e^{-\Phi}$, so that the boundary of $\text{AdS}_{3}$ lies at
$\Phi\to\infty$. The above action can equivalently be recovered from the
$\text{SL}(2,\mathbb{R})$ model by the identification

$g=\begin{pmatrix}e^{\Phi}&e^{\Phi}\overline{\gamma}\\\ e^{\Phi}\gamma&\qquad
e^{\Phi}\gamma\overline{\gamma}+e^{-\Phi}\end{pmatrix}\,.$ (16)

The constant $k$ reproduces the level of the WZW model, and the path integral
measure is taken to be Ishibashi:2000fn

$\mathcal{D}\Phi\,\mathcal{D}\left(e^{\Phi}\gamma\right)\mathcal{D}\left(e^{\Phi}\overline{\gamma}\right)\,,$
(17)

which reproduces the Haar measure on $\text{SL}(2,\mathbb{R})$.

In order to study strings near the boundary of $\text{AdS}_{3}$, we pass to a
first-order formulation. We introduce a worldsheet $(1,0)$-form $\beta$ and an
analogous $(0,1)$ form $\overline{\beta}$ and consider the action

$S=\frac{1}{4\pi}\int_{\Sigma}\mathrm{d}^{2}z\left(2(k-2)\partial\Phi\overline{\partial}\Phi+2\beta\overline{\partial}\gamma+2\overline{\beta}\partial\overline{\gamma}-\frac{2}{k}\beta\overline{\beta}e^{-2\Phi}-R\Phi\right)\,.$
(18)

The equations of motion for $\beta$ (resp. $\overline{\beta}$) fix
$\overline{\beta}=ke^{2\Phi}\overline{\partial}\gamma$ (resp.
$\beta=ke^{2\Phi}\partial\overline{\gamma}$). The shift in the coefficient of
the kinetic term for $\Phi$, as well as the linear dilaton term $-R\Phi$, come
from the change in the path integral measure upon integrating out $\beta$
deBoer:1998gyt ; Giveon:1998ns ; Kutasov:1999xu . We should emphasise that the
equivalence of the first- and second-order actions is only valid if the path
integral is dominated by large values of $\Phi$ deBoer:1998gyt . For later
convenience, we rescale the radial field $\Phi$ and define

$\Phi^{\prime}=\sqrt{2(k-2)}\Phi$ (19)

so that the action becomes

$S=\frac{1}{2\pi}\int_{\Sigma}\mathrm{d}^{2}z\left(\frac{1}{2}\partial\Phi^{\prime}\,\overline{\partial}\Phi^{\prime}+\beta\overline{\partial}\gamma+\overline{\beta}\partial\overline{\gamma}-\frac{1}{k}\beta\overline{\beta}\,e^{-Q\Phi^{\prime}}-\frac{Q}{4}R\Phi^{\prime}\right)\,,$
(20)

where we have defined the background charge $Q=\sqrt{2/(k-2)}$. From now on,
we will drop the ′ and simply refer to $\Phi^{\prime}$ as $\Phi$.

It should be noted that the Wakimoto representation of the $\text{AdS}_{3}$
worldsheet theory described above is only a good approximation to the physics
in the limit $\Phi\to\infty$, i.e. near the boundary of $\text{AdS}_{3}$
Giveon:1998ns ; Kutasov:1999xu . However, since in this work we are only
interested in the near-boundary behaviour of the theory, this point will not
be important.

#### Quantisation and OPEs

In the limit of large $\Phi$, we can treat the worldsheet theory as a free
field theory consisting of a chiral and anti-chiral $\beta\gamma$ system and a
linear dilaton $\Phi$. These fields satisfy the OPEs111111Throughout this
paper we will focus exclusively on the left-moving sector of the worldsheet
theory. All formulae will have completely analogous forms in the right-moving
sector.

$\beta(z)\gamma(w)\sim-\frac{1}{z-w}\,,\quad\Phi(z)\Phi(w)\sim-\log|z-w|^{2}\,.$
(21)

These free fields give rise to the so-called Wakimoto representation of
$\mathfrak{sl}(2,\mathbb{R})_{k}$ Wakimoto:1986gf :

$J^{+}=\beta\,,\quad J^{3}=-\frac{1}{Q}\partial\Phi+(\beta\gamma)\,,\quad
J^{-}=-\frac{2}{Q}(\partial\Phi\,\gamma)+(\beta(\gamma\gamma))-k\partial\gamma\,,$
(22)

where parentheses denote normal-ordering. Indeed, it is readily checked that
the above currents satisfy the current algebra
$\mathfrak{sl}(2,\mathbb{R})_{k}$.

Finally, we note that the stress tensor of the theory is given by

$T=-\frac{1}{2}(\partial\Phi)^{2}-\frac{Q}{2}\partial^{2}\Phi-(\beta\partial\gamma)\,,$
(23)

which can be shown to coincide with the Sugawara stress tensor of
$\mathfrak{sl}(2,\mathbb{R})_{k}$. Specifically, the central charge of the
theory is

$c=c(\beta,\gamma)+c(\Phi)=2+1+3Q^{2}=\frac{3k}{k-2}\,,$ (24)

which is the central charge of the $\text{SL}(2,\mathbb{R})$ WZW model at
level $k$.

###  2.2 Spectrally-flowed vertex operators

Highest-weight representations of $\mathfrak{sl}(2,\mathbb{R})_{k}$ fall into
several categories. In this paper, we will be focused on the so-called
continuous representations $\mathcal{C}_{j}^{\lambda}$ Maldacena:2000hw . The
highest-weight states of such representations consist of states of the form
$\ket{m,j}$ with $m\in\mathbb{Z}+\lambda$ and
$j=\tfrac{1}{2}+is,s\in\mathbb{R}$. The action of the zero-mode algebra on
these states is taken to be

$J^{3}_{0}\ket{m,j}=m\ket{m,j}\,,\quad J^{\pm}_{0}\ket{m,j}=(m\pm j)\ket{m\pm
1,j}\,.$ (25)

In terms of the Wakimoto fields, we equivalently have

$\begin{gathered}\beta_{0}\ket{m,j}=(m+j)\ket{m+1,j}\,,\quad\gamma_{0}\ket{m,j}=\ket{m-1,j}\,,\\\
(\partial\Phi)_{0}\ket{m,j}=Qj\ket{m,j}\,.\end{gathered}$ (26)

The worldsheet conformal dimension of the state $\ket{m,j}$ is read off from
the stress tensor (23) and is given by

$L_{0}\ket{m,j}=\frac{j(1-j)}{k-2}\,.$ (27)

The current algebra $\mathfrak{sl}(2,\mathbb{R})_{k}$ admits a nontrivial
automorphism known as spectral flow, which acts on the current algebra as

$\sigma^{w}(J^{\pm}_{n})=J^{\pm}_{n\mp
w}\,,\quad\sigma^{w}(J^{3}_{n})=J^{3}_{n}+\frac{kw}{2}\delta_{n,0}\,.$ (28)

The spectral flow operation $\sigma^{w}$ is defined for any integer $w$. As
shown in Maldacena:2000hw , the spectrum of $\text{AdS}_{3}$ string theory
consists of highest-weight representations of
$\mathfrak{sl}(2,\mathbb{R})_{k}$, as well as so-called spectrally-flowed
representations, obtained by composing highest-weight representations with
$\sigma^{w}$. In terms of the Wakimoto variables, spectral flow can be
realised by the automorphism

$\sigma^{w}(\beta_{n})=\beta_{n-w}\,,\quad\sigma^{w}(\gamma_{n})=\gamma_{n+w}\,,\quad\sigma^{w}((\partial\Phi)_{n})=(\partial\Phi)_{n}-\frac{w}{Q}\delta_{n,0}$
(29)

of the free field OPE algebra (21).

Let $\ket{m,j}^{(w)}$ denote the spectrally-flowed image of $\ket{m,j}$. This
state satisfies

$\begin{split}J^{3}_{0}\ket{m,j}^{(w)}&=(m+\tfrac{kw}{2})\ket{m,j}^{(w)}\,,\\\
J^{\pm}_{n}\ket{m,j}^{(w)}&=0\,,\quad n>\pm w\,.\end{split}$ (30)

Since, holographically, $J^{3}_{0}$ is identified with $L_{0}^{\text{CFT}}$ in
the spacetime CFT dual, the spacetime dimension of a state will be its
eigenvalue under $J^{3}_{0}$. Thus, the conformal dimension of
$\ket{m,j}^{(w)}$ is

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

The worldsheet conformal dimension of $\ket{m,j}^{(w)}$ reads

$L_{0}\ket{m,j}^{(w)}=\left(\frac{j(1-j)}{k-2}-hw+\frac{kw^{2}}{4}\right)\ket{m,j}^{(w)}\,,$
(32)

which again can either be read off from the Sugawara stress tensor of
$\mathfrak{sl}(2,\mathbb{R})_{k}$ or from the free field stress tensor (23).

#### Unflowed vertex operators

We want to write down expressions for the vertex operators of spectrally-
flowed states. In order to do this, let us first consider unflowed states.
These are highest-weight with respect to the free field algebra (21). We write
$V_{m,j}^{0}$ for the vertex operator associated to $\ket{m,j}$. The zero-mode
actions (26) then become the OPEs

$\begin{gathered}\beta(z)V_{m,j}^{0}(w)\sim\frac{m+j}{z-w}V_{m+1,j}^{0}(w)\,,\quad\gamma(z)V_{m,j}^{0}(0)\sim
V_{m-1,j}^{0}(w)\\\
\partial\Phi(z)V_{m,j}^{0}(w)\sim\frac{Qj}{z-w}V_{m,j}^{0}(w)\,.\end{gathered}$
(33)

A vertex operator satisfying these OPEs can easily be written down, and takes
the form

$V_{m,j}^{0}=e^{-Qj\Phi}\gamma^{-m-j}\,.$ (34)

Indeed, these vertex operators have $J^{3}_{0}$ eigenvalue $m$ and worldsheet
conformal dimension121212The conformal dimension of $e^{\alpha\Phi}$ is given
by $-\alpha(\alpha+Q)/2$

$-\frac{Q^{2}j(j-1)}{2}=\frac{j(1-j)}{k-2}$ (35)

in agreement with (27). These vertex operators are not new, and have appeared
in previous literature (see specifically Giveon:1998ns ; Kutasov:1999xu ).

#### Flowed vertex operators

Now that we have expressions for unflowed vertex operators in the Wakimoto
representation, let us derive expressions for spectrally-flowed operators. We
follow the strategy of Dei:2023ivl .

Spectral flow acts on the Wakimoto free fields as (29). In the case of the
$\beta\gamma$ system, this means that spectrally-flowed vertex operators
corresponding to $\ket{m,j}^{(w)}$ must satisfy the OPEs

$\begin{split}\beta(z)V_{m,j}^{w}(y)&\sim\frac{m+j}{(z-y)^{w+1}}V_{m+1,j}^{w}+\cdots\,,\\\
\gamma(z)V_{m,j}^{w}(y)&\sim(z-y)^{w}V_{m-1,j}^{w}(y)+\cdots\,.\end{split}$
(36)

In order to construct such states, we first ‘bosonise’ the $\beta\gamma$
system in terms of free chiral scalars $\phi,\kappa$:

$\beta=e^{\phi+i\kappa}\partial(i\kappa)\,,\quad\gamma=e^{-\phi-i\kappa}\,.$
(37)

The current $(\beta\gamma)$ is nothing more than $-\partial\phi$. The scalars
$\phi,\kappa$ are taken to satisfy the OPEs

$\phi(z)\phi(w)\sim-\log(z-w)\,,\quad\kappa(z)\kappa(w)\sim-\log(z-w)\,.$ (38)

Furthermore, $\phi$ and $\kappa$ have background charges
$Q_{\phi}=Q_{\kappa}=1$. In terms of the scalars $\phi,\kappa$, we can
construct the state

$e^{(w+m+j)\phi+i(m+j)\kappa}$ (39)

which satisfies the OPEs (36). Demanding in addition that the eigenvalue of
$(\partial\Phi)_{0}$ is $Qj-w/Q$ (as required by (26) and (29)), we arrive at
the expression Hikida:2000ry ; Iguri:2007af

$V_{m,j}^{w}(y)=e^{(w/Q-Qj)\Phi}e^{(w+m+j)\phi+i(m+j)\kappa}(y)\,.$ (40)

Indeed, this state has $J^{3}_{0}$ eigenvalue

$h=-\frac{1}{Q}\left(Qj-\frac{w}{Q}\right)+w+m+j=m+\frac{kw}{2}$ (41)

and worldsheet conformal dimension131313The worldsheet conformal dimension of
$e^{\alpha\phi+i\beta\kappa}$ is
$\Delta=-\frac{\alpha(\alpha-1)}{2}+\frac{\beta(\beta-1)}{2}\,.$

$\Delta=\frac{j(1-j)}{k-2}-hw+\frac{kw^{2}}{4}\,,$ (42)

in agreement with eq. (32).

#### The $x$-basis

In Lorenzian signature, the vertex operators $V_{m,j}^{w}$ correspond to the
emission of a string produced from the asymptotic past of $\text{AdS}_{3}$. In
Euclidean signature, the asymptotic past corresponds to the south pole of the
boundary sphere of Euclidean $\text{AdS}_{3}$, which we take to be the point
$x=0$.141414As is standard in the literature, we will use $z$ for worldsheet
coordinates and $x$ for spacetime coordinates whenever possible. In general,
we want to consider vertex operators emitted at arbitrary points on the
$\text{AdS}_{3}$ boundary in order to compare to correlation functions in the
dual CFT. This can be achieved by noting that, via the holographic dictionary,
$J^{+}_{0}=L_{-1}^{\text{CFT}}$ is the translation operator in the spacetime
CFT. Thus, we can use $J^{+}_{0}$ to bring vertex operators into the so-called
$x$-basis via the similarity transformation

$V_{m,j}^{w}(x,z)=e^{xJ^{+}_{0}}V_{m,j}^{w}(z)e^{-xJ^{+}_{0}}\,.$ (43)

In the bosonised form (40), explicitly constructing vertex operators in the
$x$-basis is rather subtle, since the action of $J^{+}_{0}$ on $\phi,\kappa$
is not local. However, there is an alternative representation of (40) which
lends itself extremely well to being written in the $x$-basis. Following the
analysis of Dei:2023ivl , we can write

$e^{(w+m+j)\phi+i(m+j)\kappa}=\left(\frac{\partial^{w}\gamma}{w!}\right)^{-m-j}\delta_{w}(\gamma)\,,$
(44)

where $\delta_{w}(\gamma)$ is a formal delta function operator defined as

$\delta_{w}(\gamma(z))=\prod_{i=0}^{w-1}\delta(\partial^{i}\gamma(z))\,.$ (45)

One should think of $\delta_{w}(\gamma(z))$ as an operator which, in the path
integral, demands that $\gamma$ is taken to have a zero of order $w$ at $z$.
The conformal weight of the delta-function operator can also be read off by
the inverse of the classical scaling dimensions of its arguments, i.e.

$\Delta(\delta_{w}(\gamma))=-\sum_{i=0}^{w-1}i=-\frac{w(w-1)}{2}\,.$ (46)

In terms of delta function operators, we can write the spectrally-flowed
vertex operators in a compact form as

$V_{m,j}^{w}=e^{(w/Q-Qj)\Phi}\left(\frac{\partial^{w}\gamma}{w!}\right)^{-m-j}\delta_{w}(\gamma)\,.$
(47)

Now, using the OPEs (21), one can show that the Wakimoto fields obey the
transformation rules

$e^{x\beta_{0}}\Phi e^{-x\beta_{0}}=\Phi\,,\quad e^{x\beta_{0}}\gamma
e^{-x\beta_{0}}=\gamma-x\,,$ (48)

and thus, we can immediately read off151515Since all factors in this
expression have regular OPEs among each other, we do not need to be careful
about normal ordering.

$V_{m,j}^{w}(x,z)=e^{-(Qj-w/Q)\Phi}(z)\left(\frac{\partial^{w}(\gamma(z)-x)}{w!}\right)^{-m-j}\delta_{w}(\gamma(z)-x)\,.$
(49)

The intuition of this vertex operator is that, in the path integral formalism,
the delta function restricts us to configurations of $\gamma$ such that

$\gamma(y)\sim x+\mathcal{O}((y-z)^{w})\,.$ (50)

This meshes well with the geometric understanding of a spectrally-flowed state
as describing a string which winds around $x$ with winding number $w$ near the
boundary of $\text{AdS}_{3}$.

###  2.3 The ‘secret’ representation

In the above, we have treated the Wakimoto field $\gamma$ as a free scalar,
i.e. a complex-valued field on the worldsheet. However, as was pointed out in
Dei:2023ivl  (see also McStay:2023thk  for related discussions), this
description of the worldsheet theory is not complete. Specifically, in the
Poincaré metric, the pair $(\gamma,\overline{\gamma})$ is meant to describe a
set of complex coordinates on the boundary. In Euclidean signature, the
asymptotic boundary of $\text{AdS}_{3}$ is the Riemann sphere
$\mathbb{CP}^{1}$, which does not admit a global set of coordinates.
Specifically, for any continuous map $\gamma:\Sigma\to\mathbb{CP}^{1}$, there
will generically be points on the worldsheet for which $\gamma$ diverges. In
the treatment of $\gamma$ as a free field, however, $\gamma$ cannot have a
pole unless it has a nontrivial OPE with a vertex operator at some point on
$\Sigma$.

In Dei:2023ivl  (following the treatment of Frenkel:2005ku ), it was suggested
that a natural way to implement the compactification of the target space of
$\gamma,\overline{\gamma}$ is to deform the worldsheet theory by adding a term

$p\int_{\Sigma}D\overline{D}$ (51)

to the action. Here, $D$ is an operator satisfying the properties

  * •

$D$ lives in the $w=-1$ spectrally-flowed sector.

  * •

$D$ has trivial OPEs with $J^{a}(z)$, up to a total derivative. Specifically,
$h(D)=0$.

  * •

$D$ has worldsheet conformal dimension $\Delta(D)=1$, so that it is
classically marginal.

A natural candidate for such a field is the combination161616The contour
integral $(\oint\gamma)^{-(k-1)}$ can be interpreted as the action of the mode
$(\gamma_{1})^{1-k}$ on the state associated to $\delta(\beta)$.

$\begin{split}D&=e^{-2\Phi/Q}e^{(k-2)\phi+i(k-1)\kappa}\\\
&=e^{-2\Phi/Q}\left(\oint\gamma\right)^{-(k-1)}\delta(\beta)\,.\end{split}$
(52)

Here, the contour integral is taken around the insertion point of $D$. This
state $D$ is a singlet with respect to $\mathfrak{sl}(2,\mathbb{R})_{k}$ up to
a total derivative and has a simple pole with respect to $\gamma$, so that
inserting several copies of $D$ into a correlation function allows $\gamma$ to
have poles. We should note that, while $D$ is not exactly an
$\mathfrak{sl}(2,\mathbb{R})_{k}$ singlet, it effectively is since

$J^{3}(z)D(w)\sim J^{+}(z)D(w)\sim\text{regular}\,,\quad
J^{-}(z)D(w)\sim\partial_{w}\left(\frac{[J^{-}_{1},D](w)}{(z-w)}\right)\,.$
(53)

In conformal perturbation theory, this deformation requires us to consider
correlators

$\Braket{\mathcal{O}_{1}\cdots\mathcal{O}_{n}}_{p}=\sum_{N=0}^{\infty}\frac{p^{N}}{N!}\prod_{a=1}^{N}\int\mathrm{d}^{2}\lambda_{a}\Braket{\mathcal{O}_{1}\cdots\mathcal{O}_{n}\prod_{a=1}^{N}(D\overline{D})(\lambda_{a})}\,.$
(54)

This effectively requires us to consider, in the path integral, field
configurations for which $\gamma$ is allowed to have an arbitrary number of
poles, labeled by the positions $\lambda_{a}$. In the language of
Frenkel:2005ku , the introduction of this deformation effectively acts to
compactify the target space from $\mathbb{C}$ to $\mathbb{CP}^{1}$. The
addition of $\int D\bar{D}$ is similar to the addition of screening fields in
the Coulomb gas formalism of $\text{AdS}_{3}$ string theory Giribet:2001ft .
Specifically, the operator $\mathcal{S}_{-}$ of Giribet:2001ft ; Iguri:2007af
shares many similarities to our deformation, but is not equivalent.

In Giribet:2019new ; Eberhardt:2019ywk ; Hikida:2020kil , a similar idea was
presented, for which a field (which they called the ‘secret representation’)
was considered. This field is meant to have worldsheet conformal weight
$\Delta=0$ and be a true singlet with respect to
$\mathfrak{sl}(2,\mathbb{R})_{k}$, but be non-trivial with respect to the
Wakimoto representation. Specifically, their state was identified with the
lift of the $\mathfrak{sl}(2,\mathbb{R})_{k}$ state

$\ket{\varphi}=\ket{\tfrac{k}{2},\tfrac{k}{2}}^{(-1)}$ (55)

to a state in the Wakimoto representation. The state $\ket{\varphi}$ is indeed
the vacuum with respect to $\mathfrak{sl}(2,\mathbb{R})_{k}$, but is
nontrivial with respect to the Wakimoto fields (specifically, $\gamma_{1}$
does not annihilate it). Our state, on the other hand, is

$\ket{D}=\ket{\tfrac{k}{2},\tfrac{k-2}{2}}^{(-1)}\,.$ (56)

While $\ket{D}$ shares many properties with $\ket{\varphi}$ (it has spacetime
conformal dimension $h=0$ and has a pole with $\gamma$), they are crucially
different in that $L_{0}\ket{D}=\ket{D}$ while $L_{0}\ket{\varphi}=0$. This
difference is due to the fact that $\ket{D}$ should be integrated over the
worldsheet, while $\ket{\varphi}$ was proposed in Eberhardt:2019ywk  to be
inserted by hand at specific points where $\gamma$ has poles. Despite the
difference between $\ket{D}$ and $\ket{\varphi}$, we will continue to refer to
$\ket{D}$ as the ‘secret representation’ field.171717As noted in the
introduction, this screening operator was independently discovered by Hikida
and Schomerus in their recent work Hikida:2023jyc .

###  2.4 Correlators and covering maps

We can now finally turn to the main problem of this section: the computation
of correlation functions of spectrally-flowed vertex operators. We consider
the correlator

$\Braket{\prod_{i=1}^{n}V_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}_{p}\,,$ (57)

where, as discussed above, we have included the deformation by the secret
representation field $D$ in the definition of the action. In conformal
perturbation theory, we have

$\sum_{N=0}^{\infty}\frac{p^{N}}{N!}\left(\prod_{a=1}^{N}\int\mathrm{d}^{2}\lambda_{a}\right)\Braket{\prod_{a=1}^{N}(D\overline{D})(\lambda_{a})\prod_{i=1}^{n}V_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\,.$
(58)

Note, however, since any correlation function which does not satisfy the
(anomalous) conservation of $-\partial\Phi$ must vanish, we do not need to
consider all values of $N$, only the one which satisfies the conservation law

$\frac{2N}{Q}+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)=Q(1-g)\,.$
(59)

We will find it useful in the following to rewrite this conservation law as

$N=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}-\frac{Q^{2}}{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)\,,$
(60)

where we recall that $Q^{2}=2/(k-2)$.

Let us now consider the path integral of this correlator. As mentioned above,
by the anomalous charge conservation of $\partial\Phi$, we can restrict $N$ to
be given by (60). Let us also for the moment ignore the factor of $p^{N}/N!$
and the integration over the points $\lambda_{a}$’s. Since all of the vertex
operators factor into operators depending on $\Phi$ and operators depending on
$\beta\gamma$, we can factorise the correlator into a $\Phi$ correlator and a
$\beta\gamma$ correlator. The $\Phi$ correlator is simply

$\Braket{\prod_{a=1}^{N}e^{-2\Phi/Q}(\lambda_{a})\prod_{i=1}^{n}e^{(w_{i}/Q-Qj_{i})\Phi}(z_{i})}\,,$
(61)

which can be computed by Wick contractions. This correlator will be
proportional to an overall momentum-conserving delta-function:

$\delta\left(\frac{2N}{Q}+\sum_{i=1}^{n}\left(Qj_{i}-\frac{w_{i}}{Q}\right)-Q(1-g)\right)\,,$
(62)

such that when the conservation law (60) is satisfied, the correlation
functions of the near-boundary theory will diverge. Fundamentally, this
divergence comes from the integral over the zero mode of $\Phi$, and explains
the divergences in the correlators $\text{SL}(2,\mathbb{R})$ WZW model for
certain values of the spins $j_{i}$.

In addition to the $\Phi$ correlator, the (chiral half of the) $\beta\gamma$
correlator is given by

$\int\mathcal{D}(\beta,\gamma)\,e^{-S[\beta,\gamma]}\prod_{a=1}^{N}\left(\oint_{\lambda_{a}}\gamma\right)^{-(k-1)}\delta(\beta(\lambda_{a}))\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(63)

We can rewrite the delta functions of $\beta$ using the formal identity

$\delta(\beta(\lambda_{a}))=\int\frac{\mathrm{d}\xi_{a}}{2\pi}e^{i\beta(\lambda_{a})\xi_{a}}\,.$
(64)

This amounts to modifying the action of the $\beta\gamma$ system to be

$S[\beta,\gamma]\to\frac{1}{2\pi}\int_{\Sigma}\beta\left(\overline{\partial}\gamma-\sum_{a}2\pi
i\xi_{a}\delta^{(2)}(z,\lambda_{a})\right)$ (65)

and integrating over the points $\xi_{a}$’s. Integrating out $\beta$ then
effectively amounts to imposing the constraint that $\gamma$ is holomorphic
with poles of residue $\xi_{a}$ at $\lambda_{a}$. Integrating over
$\lambda_{a}$ then amounts to restricting $\gamma$ to the space

$\mathcal{F}_{N}=\left\\{\gamma\text{ meromorphic with }N\text{
poles}\right\\}\,.$ (66)

This is a space of complex dimension $\text{dim}(\mathcal{F}_{N})=2N+1-g$.
Since the poles are taken to be unordered, we can remove the $1/N!$ in the
path integral, and we are left with

$\left(\frac{p}{2\pi}\right)^{N}\int_{\mathcal{F}_{N}}\mathrm{d}\gamma\,\left(\prod_{a=1}^{N}\mathop{\mathrm{Res}}_{z=\lambda_{a}}(\gamma)\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\delta_{w_{i}}(\gamma(z_{i})-x_{i})\,.$
(67)

Note that since $\mathcal{F}_{N}$ is a finite-dimensional space, we can
actually define a measure $\mathrm{d}\gamma$ in the path integral.181818There
is a natural measure on $\mathcal{F}_{N}$ induced from the measures on
$\Sigma$ and $\mathbb{CP}^{1}$ Baptista:2010rv ; Alqahtani_2013 . However, one
has to be careful about the Jacobian factors taken from integrating out
$\beta$ in the path integral to define the measure $\mathrm{d}\gamma$.

The delta functions in the integral over $\mathcal{F}_{N}$ restrict us further
to the subspace $\mathcal{G}_{N}$ of maps in $\mathcal{F}_{N}$ satisfying

$\gamma(z)\sim x_{i}+\mathcal{O}((z-z_{i})^{w_{i}})\,,\quad z\to z_{i}\,.$
(68)

This is a space of dimension

$\begin{split}\text{dim}(\mathcal{G}_{N})&=2N+1-g-\sum_{i=1}^{n}w_{i}\\\
&=-Q^{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)-(n+3g-3)\,,\end{split}$
(69)

where we have used the conservation law (60). Thus, the path integral reduces
to

$\int_{\mathcal{G}_{N}}\mathrm{d}\gamma\,\left(\prod_{a=1}^{N}\mathop{\mathrm{Res}}_{z=\lambda_{a}}(\gamma)\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\,.$
(70)

Note that we have dropped the factor of $(p/2\pi)^{N}$. Again, we have not
been careful about the measure of $\mathcal{G}_{N}$, which will contain
nontrivial Jacobian factors coming from the integration of the delta
functions.

In full string theory, we will want to multiply the above integral by the
$\Phi$ correlator and integrate over the moduli space $\mathcal{M}_{g,n}$ of
curves. For a fixed surface $\Sigma$, we can think of the space
$\mathcal{G}_{N}(\Sigma)$ as defining the fibre of a bundle
$\mathcal{H}_{N}\to\mathcal{M}_{g,n}$ whose total space is

$\mathcal{H}_{N}=\left\\{(\gamma,\Sigma)\,|\,\gamma:\Sigma\to\mathbb{CP}^{1}\text{
holomorphic, degree }N\text{, with }\gamma(z)\sim
x_{i}+\mathcal{O}((z-z_{i})^{w_{i}})\right\\}\,.$ (71)

The total space $\mathcal{H}_{N}$ has dimension

$\begin{split}\text{dim}(\mathcal{H}_{N})&=\text{dim}(\mathcal{G}_{N})+\text{dim}(\mathcal{M}_{g,n})\\\
&=-Q^{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)\,,\end{split}$
(72)

and the string amplitudes will schematically be given as an integral over
$\mathcal{H}_{N}$:

$\int_{\mathcal{H}_{N}}\mathrm{d}\mu\,f(\Sigma,\gamma)\,\left(\prod_{a=1}^{N}\mathop{\mathrm{Res}}_{z=\lambda_{a}}(\gamma)\right)^{-(k-1)}\prod_{i=1}^{n}\left(\frac{\partial^{w_{i}}(\gamma(z_{i})-x_{i})}{w_{i}!}\right)^{-m_{i}-j_{i}}\,.$
(73)

Here, $f(\Sigma,\gamma)$ is some function on $\mathcal{H}_{N}$ which will come
from the $\Phi$ correlator as well as any Jacobians we have ignored.

As a consequence, we see that if the $\text{SL}(2,\mathbb{R})$ spins are
chosen such that

$\sum_{i=1}^{n}j_{i}=\frac{k}{2}(n+2g-2)-(n+3g-3)\,,$ (74)

the dimension of $\mathcal{H}_{N}$ vanishes, and the integral (73) becomes a
discrete sum, that is, a zero dimensional integral. This discrete sum is
precisely over the set of branched covering maps $\Gamma\to\mathbb{CP}^{1}$
which are branched over $x_{i}$ with order $w_{i}$, and with no other
ramification points. Let us denote the local behaviour of such maps as

$\Gamma(z)\sim x_{i}+a_{i}^{\Gamma}(z-z_{i})^{w_{i}}+\cdots\,.$ (75)

Furthermore, let us write $\xi_{a}^{\Gamma}$ as the residues of $\Gamma$ at
$\lambda_{a}$. Then the worldsheet correlation function becomes

$\sum_{\Gamma}f(\Sigma,\Gamma)\prod_{a=1}^{N}(\xi_{a}^{\Gamma})^{-(k-1)}\prod_{i=1}^{n}(a_{i}^{\Gamma})^{-m_{i}-j_{i}}\,,$
(76)

where, again, $f(\Sigma,\Gamma)$ is some function depending on the covering
map data which we have not determined.

The above form of the string correlation functions was found in
Eberhardt:2019ywk  as a consistent solution to the Ward identities of the
$\text{SL}(2,\mathbb{R})$ WZW model. Specifically, they were able to show that
the worldsheet correlator can only localise to holomorphic covering maps when
the constraint (74) was satisfied. Furthermore, when the correlators do
localise, it was found in Eberhardt:2019ywk  that the correlators depend on
$m_{i}$ via the combination $\prod_{i=1}^{n}(a_{i}^{\Gamma})^{-m_{i}}$. The
above discussion then should be taken as an independent path integral
derivation of the results of Eberhardt:2019ywk  in the Wakimoto
representation.

Note that while we have not fixed the overall function $f(\Sigma,\Gamma)$, it
should in principle be possible to do so (see the discussion in Section
6).191919One such method of computing these Jacobians was found in
Hikida:2023jyc , based on the computational techniques of Hikida:2007tq ;
Hikida:2008pe ; Hikida:2020kil  We thank the Volker Schomerus and Yasuaki
Hikida for pointing this out to us.

###  2.5 Reading off the dual CFT

What happens when the constraint (74) is not satisfied?

The moduli space $\mathcal{H}_{N}$ is the moduli space of all holomorphic
covering maps with a fixed degree $N$ and which branch over $x_{i}$ with
degree $w_{i}$. If $m:=\text{dim}\mathcal{H}_{N}$ is positive, then
generically there will exist $m$ points $\xi_{a}$ on $\mathbb{CP}^{1}$ such
that $\gamma$ has a simple branch point over $\xi_{a}$, i.e. such that

$\gamma(z)\sim\xi_{i}+\mathcal{O}((z-\zeta_{i})^{2})$ (77)

near some point $\zeta_{i}$ on $\Sigma_{g}$. The degree of the covering map
will be

$N=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}+\frac{m}{2}\,,$ (78)

and for each fixed choice of $\xi_{i}$, the number of such covering maps is
finite. Thus, we can locally parametrise the space $\mathcal{H}_{N}$ by the
locations $\xi_{i}$ of the $m$ ‘extra’ branch points. This implies the
following schematic form for the correlation function of spectrally-flowed
states

$\begin{split}&\int\prod_{i=1}^{m}\mathrm{d}^{2}\xi_{i}\,\mathcal{O}(x_{1},\ldots,x_{n},\xi_{1},\ldots,\xi_{r})\,\\\
&\hskip
56.9055pt\times\delta\left(m+Q^{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)\right)\,,\end{split}$
(79)

where $\mathcal{O}$ is a function on $\mathcal{H}_{N}$ which depends on the
insertion points $x_{i},\xi_{i}$ on $\mathbb{CP}^{1}$, as well as some data of
the covering maps $\gamma$. Here, we have introduced the delta-function
imposing the $\partial\Phi$ conservation law, which we have previously
ignored.

In Eberhardt:2021vsx , a perturbative CFT dual to bosonic string theory on
$\text{AdS}_{3}\times X$ was proposed. This CFT is given by the symmetric
product orbifold, which is then perturbed by a certain twist-2 operator:

$\text{Sym}^{K}(X\times\mathbb{R}_{\mathcal{Q}})+\mu\int\sigma_{2,\alpha}\,.$
(80)

Here, $\mathbb{R}_{\mathcal{Q}}$ denotes a linear dilaton CFT with action

$S=\frac{1}{2\pi}\int\left(\frac{1}{2}\partial\varphi\,\overline{\partial}\varphi-\frac{\mathcal{Q}}{4}R\varphi\right)\,.$
(81)

The deformation operator $\sigma_{2,\alpha}$ is defined in the twist-2 twisted
sector of the symmetric orbifold,202020See Dei:2019iym  for a pedagogical
introduction to symmetric orbifolds and their twist fields. and has linear
dilaton momentum

$\alpha=\frac{1}{Q}=\sqrt{\frac{k-2}{2}}\,.$ (82)

In this CFT, we can compute correlation functions of the form

$\Braket{\prod_{i=1}^{n}\mathcal{O}_{w_{i},\alpha_{i}}(x_{i})}_{\mu}\,,$ (83)

where $\mathcal{O}_{w_{i},\alpha_{i}}$ lives in the $w_{i}$-twisted sector of
the symmetric orbifold theory and has $\varphi$-momentum $\alpha_{i}$. In
naive perturbation theory, one drops down $m$ copies of the perturbing field
$\sigma_{2,\alpha}$ and computes integrals of the form

$\int\prod_{i=1}^{m}\frac{(-\mu)^{m}}{m!}\Braket{\prod_{i=1}^{n}\mathcal{O}_{w_{i},\alpha_{i}}(x_{i})\prod_{i=1}^{m}\sigma_{2,\alpha_{i}}(\xi_{i})}_{\mu=0}$
(84)

computed in the undeformed symmetric orbifold. As is standard in symmetric
orbifold theories Lunin:2000yv ; Lunin:2001ne , in order to compute the above
correlation function, one sums over all branched covering maps satisfying

$\begin{split}\Gamma(z)\sim x_{i}+\mathcal{O}((z-z_{i})^{w_{i}})\,,&\quad z\to
z_{i}\,,\\\ \Gamma(z)\sim\xi_{i}+\mathcal{O}((z-\zeta_{i})^{2})\,,&\quad
z\to\zeta_{i}\,,\end{split}$ (85)

for points $z_{i},\zeta_{i}$ on some auxiliary covering surface $\Sigma$. The
number $m$ of perturbing fields is found by demanding that, upon lifting to
the covering surface, the linear dilaton momenta $\alpha_{i}$ satisfy the
charge conservation law

$\sum_{i=1}^{n}\alpha_{i}+m\alpha=\mathcal{Q}(g-1)\,,$ (86)

where $g$ is the genus of the covering surface. Amazingly, if one solves for
$m$, the result is given precisely by (72) if we identify the linear dilaton
momenta to the $\text{SL}(2,\mathbb{R})$ spins via

$\alpha_{i}=Qj_{i}-\frac{1}{Q}\,,$ (87)

and with $\mathcal{Q}=\sqrt{2}(k-3)/\sqrt{k-2}$. This is in fact precisely the
dictionary proposed by Eberhardt:2019qcl ; Eberhardt:2021vsx  for relating
linear dilaton momenta to $\text{SL}(2,\mathbb{R})$ spins, and so $m$ indeed
is the dimension of the worldsheet path integral, if one assumes this
dictionary. We shall note here that our linear dilaton conventions differ from
those of Eberhardt:2019qcl ; Eberhardt:2021vsx .212121Specifically, the two
conventions of linear dilaton charges and momenta are related by
$\mathcal{Q}_{\text{theirs}}=\mathcal{Q}_{\text{ours}}/\sqrt{2}$ and
$\alpha_{\text{theirs}}=(\alpha_{\text{ours}}+\mathcal{Q}_{\text{ours}})/\sqrt{2}$,
the latter difference being a combination of reflection symmetry $j\to 1-j$
and a minus sign.

We thus conclude, since the number of extra branch points $\xi_{i}$ in both
the worldsheet computation (79) and the dual CFT computation (84) are the
same, that the worldsheet path integral naturally reproduces the naive
perturbative structure of the dual CFT of Eberhardt.222222Strictly speaking,
the naive perturbative answer (84) is not correct, but rather reads off the
residues of poles in the configuration space of the momenta $\alpha_{i}$. This
is similar to the case of Liouville theory Dorn:1994xn ; Zamolodchikov:1995aa
. One would expect that a careful treatment of $\text{AdS}_{3}$ string theory
would find that, as one moves away from the boundary, the delta functions of
(79) would be smoothed out into poles. This is indeed the result of the Ward
identity analysis of Dei:2021xgh ; Dei:2021yom ; Dei:2022pkr . Of course, an
exact matching of both sides would require a careful study of the various
Jacobians arising in the worldsheet path integral. We will return to this
point in Section 6.

##  3 Spectral flow as a background gauge field

In this section, we will explore spectral flow from a different perspective to
that considered in the previous section. We will argue that spectral flow can
be thought of as the result of the insertion of a non-local operator wrapping
unflowed vertex operators on the worldsheet. As we will explain, this non-
local operator can be thought of as a constant background gauge field in the
$\text{SL}(2,\mathbb{R})$ WZW model whose curvature is concentrated at the
insertion points of the spectrally-flowed vertex operators. This background
gauge field couples to the $\mathfrak{sl}(2)$ currents and is a constant
nondynamical $(0,1)$-form partial connection on the worldsheet, whose form is
wholly determined by the spectrally-flowed vertex operator insertions. The
introduction of this gauge field modifies the equations of motion in the
worldsheet sigma model, and we show that the solutions to these equations of
motion naturally reproduce holomorphic covering maps from the worldsheet to
the boundary of $\text{AdS}_{3}$.

###  3.1 Spectral flow as a non-local operator

Consider an operator $\mathcal{O}$ which lives in the unflowed sector of the
$\text{SL}(2,\mathbb{R})$ model. Now, assume that this operator is placed at
the location $y$ on the worldsheet. We can consider a keyhole contour $P$
wrapping around $\mathcal{O}(y)$, whose shape is chosen to skirt around the
branch cut of the function $\log(z-y)$, see Figure 1. Let us define the
operator

$\mathcal{O}^{(w)}(y)=\exp\left(w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)\mathcal{O}(y)\,.$ (88)

We claim that $\mathcal{O}^{(w)}$ lives in the $w^{\text{th}}$ spectrally-
flowed sector.

$y$$P$$y$$P^{0}$$P^{1}$ Figure 1: The keyhole contour $P$ is contracted to a
circular contour $P^{0}$ and a contour $P^{1}$ along the branch cut.

This definition is justified as follows. Let us consider the OPE between, say,
$J^{+}$ and $\mathcal{O}^{(w)}$. In order to find the short-distance
behaviour, we need to ‘push’ the insertion point of $J^{+}$ through the
contour $P$. As shown in Figure 2, the result is that the OPE between
$J^{+}(\zeta)$ and $\mathcal{O}^{(w)}(y)$ is given by

$\exp\left(w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)\left(e^{-w\oint_{\zeta}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)}J^{+}(\zeta)\mathcal{O}(y)\right)\,,$ (89)

where the integral around $\zeta$ is taken to lie entirely inside the contour
$P$ and runs counter-clockwise. We can evaluate the given expression (recall
the $JJ$ OPEs given in (12)) and we find

$\exp\left(-w\oint_{\zeta}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)J^{+}(\zeta)=(\zeta-y)^{-w}J^{+}(\zeta)\,,$ (90)

so that

$J^{+}(\zeta)\mathcal{O}^{(w)}(y)\sim(\zeta-y)^{-w}\exp\left(w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)\left(J^{+}(\zeta)\mathcal{O}(y)\right)\,.$ (91)

If we write the OPE between $J^{+}$ and $\mathcal{O}$ as

$J^{+}(\zeta)\mathcal{O}(y)=\sum_{n}\frac{(J^{+}_{n}\mathcal{O})(y)}{(\zeta-y)^{n+1}}\,,$
(92)

then we can read off

$J^{+}(\zeta)\mathcal{O}^{(w)}(y)=\sum_{n}\frac{(J^{+}_{n}\mathcal{O})^{(w)}(y)}{(\zeta-y)^{n+w+1}}=\sum_{n}\frac{(J^{+}_{n-w}\mathcal{O})^{(w)}(y)}{(\zeta-y)^{n+1}}\,.$
(93)

Thus, we see that $J^{+}$ acts on $\mathcal{O}^{(w)}$ in the same way as it
would act on the spectrally-flowed image of $\mathcal{O}$. A completely
analogous argument shows

$J^{-}(\zeta)\mathcal{O}^{(w)}(y)=\sum_{n}\frac{(J^{-}_{n+w}\mathcal{O})^{(w)}(y)}{(\zeta-y)^{n+1}}\,,\quad
J^{3}(\zeta)\mathcal{O}^{(w)}(y)=\sum_{n}\frac{((J^{3}_{n}+\frac{kw}{2}\delta_{n,0})\mathcal{O})^{(w)}(y)}{(\zeta-y)^{n+1}}\,.$
(94)

Thus, we can identify $\mathcal{O}^{(w)}$ as the spectrally-flowed image of
$\mathcal{O}$. We emphasise that this contour prescription for spectral flow
is purely based on the existence of an $\mathfrak{sl}(2,\mathbb{R})_{k}$
affine symmetry, and thus does not rely on taking the near-boundary limit as
in Section 2.

$y$$\zeta$$y$$\zeta$$y$$\zeta$ Figure 2: Pushing an operator located at
$\zeta$ through the integration contour $P$. The result is that the operator
located at $\zeta$ is wrapped by a closed contour upon being pushed through.

Now, we can deform the contour $P$ by bringing its tails infinitesimally close
to the branch cut of the logarithm. Let $P_{+}$ and $P_{-}$ represent the
parts of the contour directly above and below the branch cut. Since the
logarithm jumps by a factor of $2\pi i$ above and below a branch cut, the
contributions of $P_{+},P_{-}$ to the contour integral limit onto the
following integral along the branch cut

$w\left(\int_{P_{+}}\frac{\mathrm{d}z}{2\pi
i}-\int_{P_{-}}\frac{\mathrm{d}z}{2\pi
i}\right)\log(z-y)J^{3}(z)=-w\int_{P^{1}}\mathrm{d}z\,J^{3}(z)\,,$ (95)

where $P^{1}$ is the contour shown on the right of of Figure 1. Thus, we have

$w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)=w\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)-w\int_{P^{1}}\mathrm{d}z\,J^{3}(z)\,,$ (96)

where $P^{0}$ is a closed contour which wraps around $y$, as shown in Figure
1. Now, we note that

$e^{-w\int_{P^{1}}\mathrm{d}z\,J^{3}(z)}=\exp\left(w\int_{P^{1}}\mathrm{d}z\,\left(\frac{\partial\Phi}{Q}-\beta\gamma\right)\right)=e^{w\Phi(y^{\prime})/Q}\delta_{w}(\gamma(y^{\prime}))\,,$
(97)

where $y^{\prime}$ is the position where $P^{0}$ and $P^{1}$ meet. Hence, we
can write the spectrally-flowed operator $\mathcal{O}^{(w)}$ in the form

$\mathcal{O}^{(w)}(y)=e^{w\Phi(y^{\prime})/Q}\delta_{w}(\gamma(y^{\prime}))\exp\left(w\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)\mathcal{O}(y)\,.$ (98)

As an example, if we take $\mathcal{O}=V_{m,j}^{0}=e^{-Qj\Phi}\gamma^{-m-j}$,
we have

$\mathcal{O}^{(w)}(y)=e^{w\Phi(y^{\prime})/Q}\delta_{w}(\gamma(y^{\prime}))\exp\left(w\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)\mathcal{O}(y)\,,$ (99)

The OPE between $J^{3}$ and $V^{0}_{m,j}$ is
$J^{3}(z)V_{m,j}^{0}(y)=mV_{m,j}^{0}(y)/(z-y)$. The contour integral around
$P^{0}$ then gives

$\displaystyle\exp\left(w\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)V_{m,j}^{0}(y)\sim$
$\displaystyle\exp\left(wm\oint_{y}\frac{dz}{2\pi
i}\frac{\log(z-y)}{z-y}\right)V_{m,j}^{0}(y),$ (100) $\displaystyle\sim$
$\displaystyle\exp(wm\log(\epsilon))V=\epsilon^{wm}V_{m,j}^{0}(y)\,,$

where $\epsilon=|y-y^{\prime}|$ is the radius of the contour $P^{0}$. This
approximation holds in the limit of small $\epsilon$. Collecting everything,
we can plug in the expression (34), and we get

$\mathcal{O}^{(w)}(y)\sim\epsilon^{wm}e^{w\Phi(y^{\prime})/Q}\delta_{w}(\gamma(y^{\prime}))e^{-Qj\Phi(y)}\gamma(y)^{-m-j}\,.$
(101)

In the end, we would like to shrink the contour around $y$. To do this, we
need to know the various short-distance behaviours of our fields in the limit
$y\to y^{\prime}$. This can be read off by the OPEs

$\begin{split}e^{w\Phi(y^{\prime})/Q}e^{-Qj\Phi(y)}&\sim\epsilon^{wj}e^{(w/Q-jQ)\Phi(y)}\\\
\delta_{w}(\gamma(y^{\prime}))\gamma(y)^{-m-j}&\sim\epsilon^{-wm-
wj}\left(\frac{\partial^{w}\gamma(y)}{w!}\right)^{-m-j}\delta_{w}(\gamma(y))\,.\end{split}$
(102)

Thus, upon taking $\epsilon\to 0$ (i.e. $y\to y^{\prime}$) the operator
$\mathcal{O}^{w}$ becomes

$\mathcal{O}^{(w)}(y)=e^{(w/Q-wj)\Phi(y)}\left(\frac{\partial^{w}\gamma(y)}{w!}\right)^{-m-j}\delta_{w}(\gamma(y))\,,$
(103)

which coincides with the result (47) for the spectrally-flowed vertex operator
in the Wakimoto representation.

The above discussion tells us that we can view spectral flow as the inclusion
of a non-local operator on the worldsheet. For holographic applications,
however, we would like to consider vertex operators in the $x$-basis. As
discussed in Section 2, such operators are defined by

$\mathcal{O}^{(w)}(x,y)=e^{xJ^{+}_{0}}\mathcal{O}^{(w)}(y)e^{-xJ^{+}_{0}}\,.$
(104)

Using the relation

$e^{xJ^{+}_{0}}J^{3}(z)e^{-xJ^{+}_{0}}=J^{3}(z)-xJ^{+}(z)\,,$ (105)

we have

$\begin{split}\mathcal{O}^{(w)}(x,y)&=e^{xJ^{+}_{0}}\exp\left(w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)e^{-xJ^{+}_{0}}e^{xJ^{+}_{0}}\mathcal{O}(y)e^{-xJ^{+}_{0}}\\\
&=\exp\left(w\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)(J^{3}(z)-xJ^{+}(z))\right)\mathcal{O}(y,x)\,.\end{split}$ (106)

That is, we can obtain spectrally-flowed vertex operators in the $x$-basis by
wrapping the unflowed operator $\mathcal{O}(y,x)$ with the non-local operator
constructed using $J^{3}-xJ^{+}$ as opposed to simply $J^{3}$.

For completeness, we also include the expression for the secret representation
operator $D$. As we mentioned in Section 2.3, the operator $D$ can be obtained
from the state $\ket{\frac{k}{2},\frac{k-2}{2}}^{(-1)}$, i.e. as a state in
the $w=-1$ spectrally-flowed sector. Thus, we can write it as

$\begin{split}D(y)&=\exp\left(-\int_{P}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)e^{-\Phi(y)/Q}\gamma(y)^{-k+1}\\\
&=\exp\left(\int_{P^{1}}\mathrm{d}z\,J^{3}(z)\right)\exp\left(-\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)e^{-\Phi(y)/Q}\gamma(y)^{-k+1}\,.\end{split}$ (107)

Analogously to the above calculations, we have

$\begin{split}\exp\left(-\oint_{P^{0}}\frac{\mathrm{d}z}{2\pi
i}\log(z-y)J^{3}(z)\right)e^{-\Phi(y)/Q}\gamma(y)^{-(k-1)}&\sim\epsilon^{-k/2}e^{-\Phi(y)/Q}\gamma(y)^{-(k-1)}\,,\\\
\exp\left(\int_{P^{1}}\mathrm{d}z\,J^{3}(z)\right)&=e^{-\Phi(y^{\prime})/Q}\delta(\beta(y^{\prime}))\,,\end{split}$
(108)

where again we have taken $P^{1}$ to end at $y^{\prime}$. Using the OPEs

$\begin{split}e^{-\Phi(y^{\prime})/Q}e^{-\Phi(y)/Q}&\sim\epsilon^{-(k-2)/2}e^{-2\Phi(y)/Q}\,,\\\
\delta(\beta(y^{\prime}))\gamma(y)^{-(k-1)}&\sim\epsilon^{k-1}\left(\oint_{y}\gamma\right)^{-(k-1)}\delta(\beta(y))\,,\end{split}$
(109)

we find

$D=e^{-2\Phi/Q}\left(\oint\gamma\right)^{-(k-1)}\delta(\beta)\,.$ (110)

This agrees precisely with the form (52), as expected.

### Locality

The non-local behaviour of the spectrally-flowed states constructed in this
fashion is somewhat surprising from the worldsheet perspective. Let us expand
on this a bit. Let us consider an operator $V$ in the
$\text{SL}(2,\mathbb{R})$ model. If we consider the OPE of $V$ with the
spectrally-flowed operator $\mathcal{O}^{(w)}$, we will find that this OPE has
a branch cut. Specifically,

$V(e^{2\pi i}z)\,\mathcal{O}^{(w)}(0)=e^{-2\pi
iwh}\,V(z)\,\mathcal{O}^{(w)}(0)\,,$ (111)

where $h$ is the $J^{3}_{0}$ charge of $V$. The phase comes from pushing $V$
through $P_{1}$.

In a full non-chiral analysis, the spectrally-flowed state $\mathcal{O}^{(w)}$
will be defined with respect to both $J^{3}_{0}$ and $\bar{J}^{3}_{0}$, and
the analogous statement will be

$V(e^{2\pi i}z)\,\mathcal{O}^{(w)}(0)=e^{-2\pi
iw(h-\bar{h})}\,V(z)\,\mathcal{O}^{(w)}(0)\,,$ (112)

where $\bar{h}$ is the $\bar{J}^{3}_{0}$ eigenvalue of $V$. This relation
suggests that states in the $\text{SL}(2,\mathbb{R})$ WZW model which have
local OPEs with spectrally-flowed states must obey the relation

$h-\bar{h}\in\mathbb{Z}/w\,.$ (113)

Indeed, this will be the case for all states we will consider. Specifically,
we will always take the left- and right-moving components of vertex operators
in the $\text{SL}(2,\mathbb{R})$ model to lie in the same representation of
$\mathfrak{sl}(2,\mathbb{R})_{k}$. Specifically, this means that $h$ and
$\bar{h}$ will have the same fractional part, and so in particular232323More
precisely, we consider the charge conjugation modular invariant spectrum, and
thus $j=\bar{j}$ for the discrete representations
$\mathcal{D}_{j}\otimes\bar{\mathcal{D}}_{\bar{j}}$ and $\alpha=\bar{\alpha}$
for the continuous representations
$\mathcal{C}^{\alpha}_{j}\otimes\bar{\mathcal{C}}^{\bar{\alpha}}_{j}$. In both
cases the $J^{3}_{0}$ and $\bar{J}^{3}_{0}$ eigenvalues therefore differ by an
integer.

$h-\bar{h}\in\mathbb{Z}\,.$ (114)

Thus, all states we will consider will have a local OPE with
$\mathcal{O}^{(w)}$, and all correlators we consider will be free of branch
cuts.

###  3.2 Interpretation of the non-local term

The above discussion suggests that we can consider the correlation function of
spectrally-flowed states in the $\text{SL}(2,\mathbb{R})$ WZW model in terms
of unflowed states in the presence of the extended operator

$\begin{split}&\prod_{a=1}^{N}\exp\left(-\int_{P_{a}}\frac{\mathrm{d}z}{2\pi
i}\log(z-\lambda_{a})J^{3}(z)\right)\\\ &\hskip
28.45274pt\times\prod_{i=1}^{n}\exp\left(w_{i}\int_{P_{i}}\frac{\mathrm{d}z}{2\pi
i}\log(z-z_{i})(J^{3}(z)-x_{i}J^{+}(z))\right)\,,\end{split}$ (115)

where $P_{a}$ and $P_{i}$ are keyhole contours like those in Figure 1 which
wrap around $\lambda_{a}$ and $z_{i}$, respectively, while avoiding the branch
cuts of the logarithms.

Now, we can deform the various contours in the above operator as follows.
First, we pick a point $p$ on the worldsheet where all of the branch cuts are
chosen to meet. We then deform the contours $P_{i}$ and $P_{a}$ as in the
right side of Figure 1, so that we are left with circular contours wrapping
$z_{i}$ and $\lambda_{a}$, as well as operators localised on the branch cuts
of the logarithms.

$\lambda_{1}$$z_{1}$$z_{2}$$z_{3}$$\lambda_{2}$$\infty$ Figure 3: The support
of the gauge field $A^{(0,1)}$ is localised to paths connecting the insertion
points $z_{i}$ and $\lambda_{a}$ to a base point $p$, taken to be the north
pole of the Riemann sphere.

The spectrally-flowed correlator can now be written in the form

$\Braket{\prod_{a=1}^{N}e^{-\oint_{\lambda_{a}}\frac{\mathrm{d}z}{2\pi
i}\log(z-\lambda_{a})J^{3}(z)}V_{\frac{k}{2},\frac{k-2}{2}}^{0}(\lambda_{a})\prod_{i=1}^{n}e^{w_{i}\oint_{z_{i}}\frac{\mathrm{d}z}{2\pi
i}\log(z-z_{i})(J^{3}(z)-x_{i}J^{+}(z))}V_{m,j}^{0}(z_{i},x_{i})}^{\prime}\,,$
(116)

where prime indicates that we evaluate the above correlator with the inclusion
of the operator

$\exp\left(\sum_{a=1}^{N}\int_{p}^{\lambda_{a}}J^{3}-\sum_{i=1}^{n}w_{i}\int_{p}^{z_{i}}(J^{3}-x_{i}J^{+})\right)\,,$
(117)

represented by the ‘tails’ of the contours. Here, $p$ is some basepoint where
we have chosen to end the contour integrals. For the sphere, it is convenient
to pick the branch cuts to end at the north pole, i.e. at $p=\infty$ as in
Figure 3. However, in the following, we will be agnostic about the basepoint
$p$, since we are mainly interested in the local behaviour near $z=z_{i}$ and
$z=\lambda_{a}$. We also recall that the number $N$ of secret representation
fields $D$ is dictated by the $\text{SL}(2,\mathbb{R})$ spins via
$\partial\Phi$ charge conservation as in equation (60).

We can understand this operator as a kind of modification of the action of the
worldsheet theory. Indeed, one can equivalently view the insertion of this
operator as a deformation of the action of the WZW model:

$S^{\prime}=S-\sum_{a=1}^{N}\int_{p}^{\lambda_{a}}J^{3}+\sum_{i=1}^{n}w_{i}\int_{p}^{z_{i}}(J^{3}-x_{i}J^{+})\,.$
(118)

We can now trade the integration over the various contours for an integration
over the full worldsheet by introducing a $(0,1)$-form valued distribution
$\Delta(x\to y)$ which has support on a curve connecting the points $x$ to
$y$. Its defining property is

$\int_{\Sigma}\omega\wedge\Delta(x\to y)=\int_{x}^{y}\omega$ (119)

for any $(1,0)$-form $\omega$. In terms of this distribution, we have

$S^{\prime}=S+\int_{\Sigma}\left(\sum_{i=1}^{n}w_{i}(J^{3}-x_{i}J^{+})\wedge\Delta(p\to
z_{i})-\sum_{a=1}^{N}J^{3}\wedge\Delta(p\to\lambda_{a})\right)\,.$ (120)

So far, we have simply defined a modified action with which spectrally-flowed
correlation functions are to be computed. We can now give this action a nice
physical interpretation. Let us write

$S^{\prime}=S+\frac{1}{2\pi}\int_{\Sigma}\text{Tr}[J\wedge A^{(0,1)}]\,.$
(121)

Here, $A^{(0,1)}$ is the $\mathfrak{sl}(2,\mathbb{R})$-valued $(0,1)$-form
distribution

$A^{(0,1)}=2\pi\sum_{i=1}^{n}w_{i}(t^{3}-x_{i}t^{+})\Delta(p\to
z_{i})-2\pi\sum_{a=1}^{N}t^{3}\Delta(p\to\lambda_{a})\,,$ (122)

where $t^{a}$ are the generators of $\mathfrak{sl}(2,\mathbb{R})$ and we have
defined $J$ so that

$J^{a}=\text{Tr}[J\,t^{a}]\,.$ (123)

Choosing the basis

$t^{3}=\begin{pmatrix}1/2&0\\\ 0&-1/2\end{pmatrix}\,,\quad
t^{+}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\,,\quad
t^{-}=\begin{pmatrix}0&0\\\ -1&0\end{pmatrix}\,,$ (124)

we can write $A^{(0,1)}$ in the form

$A^{(0,1)}=2\pi\sum_{i=1}^{n}\begin{pmatrix}w_{i}/2&-w_{i}x_{i}\\\
0&-w_{i}/2\end{pmatrix}\Delta(p\to
z_{i})-2\pi\sum_{a=1}^{N}\begin{pmatrix}1/2&0\\\
0&-1/2\end{pmatrix}\Delta(p\to\lambda_{a})\,.$ (125)

In this notation, the action (121) has the form of a WZW model with background
gauge field $A^{(0,1)}$. Thus, we propose that:

> The computation of a spectrally-flowed correlation function in the
> $\text{SL}(2,\mathbb{R})$ WZW model can be rewritten as the computation of
> an unflowed correlation function in a background gauge field $A^{(0,1)}$.

We should note that the notion of spectral flow as a background gauge field is
not a new one (see Nakatsu:1993np ; Hori:1994nc ). However, as far as we know,
this is the first time the idea has been used in the $\text{AdS}_{3}$
literature. As we will see below, this background gauge field modifies the
equations of motion of the Wakimoto fields such that the solutions to the
classical equations of motion impose $\gamma$ to be a holomorphic covering map
branched at $z_{i}$ with poles at $\lambda_{a}$.

###  3.3 Equations of motion and covering maps

Given a function $f$ on the worldsheet, we have

$\int_{\Sigma}\partial f\wedge\Delta(x\to y)=\int_{x}^{y}\partial
f=f(y)-f(x)\,.$ (126)

Integrating the first integral by parts, we see that the distribution
$\Delta(x\to y)$ must satisfy the differential equation

$\partial\Delta(x\to y)=-\delta^{(2)}(z,y)+\delta^{(2)}(z,x)\,.$ (127)

On the sphere, the solution to this equation is simply242424The complex
function $\overline{\partial}\log(z)$ does not vanish everywhere, but rather
has a delta-function singularity along the branch cut of the logarithm.

$\Delta(p\to
y)=-\frac{1}{2\pi}\,\overline{\partial}\left(\log(z-y)-\log(z-p)\right)\,.$
(128)

If we specify the basepoint to be $p=\infty$, then we simply have

$\Delta(\infty\to y)=-\frac{1}{2\pi}\overline{\partial}\log(z-y)\,.$ (129)

Thus, picking $p=\infty$, we can write the deformed action of the
$\mathfrak{sl}(2,\mathbb{R})_{k}$ WZW model as

$S^{\prime}=S-\frac{1}{2\pi}\int\mathrm{d}^{2}z\left(\sum_{i=1}^{n}w_{i}(J^{3}(z)-x_{i}J^{+}(z))\overline{\partial}\log(z-z_{i})-\sum_{a=1}^{N}J^{3}(z)\,\overline{\partial}\log(z-\lambda_{a})\right)\,.$
(130)

Now, let us consider the Wakimoto representation (20) in the near-boundary
limit $\Phi\to\infty$. The modified worldsheet action has the form

$\begin{split}S^{\prime}&=\frac{1}{2\pi}\int\left(\frac{1}{2}\partial\Phi\,\overline{\partial}\Phi+\sum_{i=1}^{n}\frac{w_{i}}{Q}\partial\Phi\,\overline{\partial}\log(z-z_{i})-\sum_{a=1}^{N}\frac{1}{Q}\partial\Phi\,\overline{\partial}\log(z-\lambda_{a})\right)\\\
&\hskip
14.22636pt+\frac{1}{2\pi}\int\left(\beta\overline{\partial}\gamma-\sum_{i=1}^{n}w_{i}\beta\,(\gamma-
x_{i})\,\overline{\partial}\log(z-z_{i})+\sum_{a=1}^{N}\beta\,\gamma\,\overline{\partial}\log(z-\lambda_{a})\right)\end{split}$
(131)

The top integral can be computed via integration by parts, and has the effect
of introducing the operator

$\prod_{i=1}^{n}e^{-w_{i}\Phi/Q}(z_{i})\prod_{a=1}^{N}e^{\Phi/Q}(\lambda_{a})\,.$
(132)

into the path integral. Meanwhile, the bottom line of (131) has the effect of
changing the equations of motion for $\gamma$. Specifically, integrating out
$\beta$ restricts the path integral to an integral over fields $\gamma$
satisfying

$\overline{\partial}\gamma(z)=\sum_{i=1}^{n}w_{i}(\gamma(z)-x_{i})\,\overline{\partial}\log(z-z_{i})-\sum_{a=1}^{N}\gamma(z)\,\overline{\partial}\log(z-\lambda_{a})\,.$
(133)

The solutions to this differential equation are precisely those for which
$\gamma$ has the local behavior

$\begin{split}\gamma(z)\sim x_{i}+\mathcal{O}((z-z_{i})^{w_{i}})\,,&\qquad
z\to z_{i}\,,\\\
\gamma(z)\sim\mathcal{O}\left(\frac{1}{z-\lambda_{a}}\right)\,,&\qquad
z\to\lambda_{a}\,.\end{split}$ (134)

As noted in Section 2, the number $N$ of poles is determined uniquely by the
$\text{SL}(2,\mathbb{R})$ spins $j_{i}$ by (60). If the value of $N$
determined by (60) is an integer, then the unique solutions to the equations
of motion for $\gamma$ are branched covering maps with branch points at
$z=z_{i}$, as well as $m$ extra branch points, where252525Strictly speaking,
our analysis in this section has only been for genus $g=0$. However, since the
conditions (134) are entirely local, it is natural to assume that this
analysis will extend to higher-genus surfaces as well.

$m=-Q^{2}\left(\sum_{i=1}^{n}j_{i}-\frac{k}{2}(n+2g-2)+(n+3g-3)\right)\,.$
(135)

Thus, we see from another perspective the conclusion of Section 2 – if the
above constraint on $j_{i}$ is chosen, then the correlation functions of the
near-boundary ($\Phi\to\infty$) sector of the
$\mathfrak{sl}(2,\mathbb{R})_{k}$ WZW model localise to holomorphic covering
maps.

##  4 The $k=1$ string on $\rm AdS_{3}\times S^{3}\times\mathbb{T}^{4}$

We begin this section with an overview of the $k=1$ string theory on $\rm
AdS_{3}\times S^{4}\times\mathbb{T}^{4}$, written in the hybrid formalism. The
string theory can be described using the level $k=1$ current algebra
$\mathfrak{psu}(1,1|2)_{1}$. At this level, the current algebra admits a free
field realisation and we use essentially the same free field realisation as in
Dei:2020zui ; Gaberdiel:2022als  but with a slightly different notation which
hopefully will make the connection to twistor variables more apparent as we
will discuss in the next section. We will again focus mainly on the $\rm
AdS_{3}$ part of the string theory as we did in the previous section. This
amounts to focusing on the $\mathfrak{sl}(2,\mathbb{R})_{1}$ subalgebra of the
$\mathfrak{psu}(1,1|2)_{1}$ superalgebra. A good set of references for the
interested reader includes, but is not restricted to, Dei:2020zui ;
Gaberdiel:2022als ; Maldacena:2000hw ; Berkovits:1999im ; Eberhardt:2018ouy ;
Gaberdiel:2018rqv . We then write down an explicit form of the spectrally-
flowed vertex operators. We will give multiple forms of the vertex operators,
each will have its own advantages that will be discussed in this section.

###  4.1 Review of the $k=1$ theory

Strings propagating in $\rm AdS_{3}\times S^{3}\times\mathbb{T}^{4}$
background with one unit of NS-NS flux and zero units of RR flux can be
conveniently described by the hybrid formalism. In this formalism, the
worldsheet theory is described by a WZW model of the supergroup
$\text{PSU}(1,1|2)$ and a topologically twisted $\mathbb{T}^{4}$
Berkovits:1999im . The level $k$, which also represents the amount of NS-NS
flux, in the WZW action is chosen to be 1. Additionally, as is usual in the
superstring, the worldsheet theory has ghosts which in this case are bosonised
and are denoted by $\rho,\sigma$.262626The $\sigma$ ghost is the bosonising
field of the $b,c$ ghost, however, the definition of $\rho$ is more
complicated, see Berkovits:1999im . Furthermore, the level one algebra
possesses a free field description in terms of two pairs of symplectic bosons
$(\mu,\pi),(\lambda,\omega)$,272727The symplectic bosons notation used in
Gaberdiel:2022als ; Fiset:2022erp ; Dei:2020zui  is related to our notation
here by the identification  $\displaystyle\lambda$ $\displaystyle\sim\xi^{+},$
$\displaystyle\omega$ $\displaystyle\sim\eta^{-}$ $\displaystyle\mu$
$\displaystyle\sim-\xi^{-},$ $\displaystyle\pi$ $\displaystyle\sim\eta^{+}.$
as well as two pairs of complex fermions $(\psi^{i},\eta_{i})$, with $i=1,2$
Dei:2020zui ; Gaberdiel:2022als . For reasons that will become clear in the
next section, we will package the free fields as follows

$Z^{\mathcal{A}}=\begin{pmatrix}Z^{A}\\\
Z^{I}\end{pmatrix}=\begin{pmatrix}\mu\\\ \lambda\\\ \psi^{1}\\\
\psi^{2}\end{pmatrix},\quad
Y_{\mathcal{A}}=(Y_{A},Y_{I})=(\pi,\omega,\eta_{1},\eta_{2}),$ (136)

where $\mathcal{A}=(A|I)$ and $A,I=0,1$. The worldsheet action is then a sum
of three independent parts

$S=S_{\text{matter}}[Z^{\mathcal{A}},Y_{\mathcal{B}}]+S_{\text{ghosts}}[\rho,\sigma]+S_{\mathbb{T}^{4}}\
.$ (137)

For the purposes of this paper, the $(\rho,\sigma)$ ghosts and the
$\mathbb{T}^{4}$ can be neglected since they decouple from all computations.
Indeed, the usual ansatz Gerigk_2012  for a full vertex operator reads

$\displaystyle
V_{\text{full}}=V(Z^{A},Y^{A})V(Z^{I},Y^{I})e^{2\rho+i\sigma}V_{\mathbb{T}^{4}}.$
(138)

Since the vertex operators and the action functional factorise, computations
of any correlators will also factorise into uncoupled pieces and one can focus
on each of the factors in equation (138).

The ‘matter’ fields $Y_{\mathcal{A}},Z^{\mathcal{A}}$ are taken to satisfy the
free field OPEs

$Y_{\mathcal{B}}(z)Z^{\mathcal{A}}(w)\sim-\frac{\delta\indices{{}_{\mathcal{B}}^{\mathcal{A}}}}{z-w}$
(139)

and have worldsheet conformal dimension
$\Delta(Y_{\mathcal{A}})=\Delta(Z^{\mathcal{A}})=\frac{1}{2}$. Note that the
order of the indices matters, i.e.

$\begin{split}\delta\indices{{}_{\mathcal{B}}^{\mathcal{A}}}&=\textrm{diag}(1,1,1,1)\,,\\\
\delta\indices{{}^{\mathcal{A}}_{\mathcal{B}}}&=\textrm{diag}(1,1,-1,-1)\end{split}$
(140)

because of the Grassmann statistics of the complex fermions. Bilinears in
these fields have conformal weight $\Delta=1$, and thus define worldsheet
currents. The algebra generated by these currents is $\mathfrak{gl}(2|2)_{1}$.
In order to restrict to $\mathfrak{psu}(1,1|2)$, we gauge the current

$\mathcal{Z}=\frac{1}{2}Y_{\mathcal{A}}Z^{\mathcal{A}}\,,$ (141)

which generates simultaneous scalings $Y\to\alpha Y$, $Z\to\alpha^{-1}Z$. See
Dei:2020zui  and Gaberdiel:2022als  for more details.

The matter action is a first-order action in the fields $Y_{\mathcal{A}}$ and
$Z^{\mathcal{A}}$, namely

$S_{\text{matter}}=\frac{1}{2\pi}\int_{\Sigma}Y_{\mathcal{A}}(\overline{\partial}+a)Z^{\mathcal{A}}\,.$
(142)

Here, we are only considering the chiral half of the theory, although
everything we discuss will extend to the full non-chiral theory. Here, $a$ is
$(0,1)$-form which acts as a $\mathfrak{u}(1)$ connection gauging the symmetry
$\mathcal{Z}$. In the path integral of the theory, we will be instructed to
integrate over all such connections, parametrised by the Jacobian
$\text{Jac}(\Sigma)$. For the moment, however, we will consider the theory
with the gauge fixing $a=0$ (such a choice is always possible on the sphere),
and return to the discussion of nonzero $a$ in Section 4.6.

From the free fields above, we can write down the
$\mathfrak{sl}(2,\mathbb{R})_{1}$ currents as bilinears of such fields.
Writing again $Z^{A},Y_{A}$ as the bosonic parts of
$Z^{\mathcal{A}},Y_{\mathcal{A}}$, we can write the
$\mathfrak{sl}(2,\mathbb{R})_{1}$ currents as

$J^{a}=\text{Tr}[Yt^{a}Z]\,,$ (143)

where $t^{a}$ are the $\mathfrak{sl}(2,\mathbb{R})$ generators given in eq.
(124). Explicitly, this reads

$\displaystyle J^{3}=$
$\displaystyle\frac{1}{2}\left(\pi\mu-\omega\lambda\right)$ (144)
$\displaystyle J^{+}=$ $\displaystyle\lambda\pi$ $\displaystyle J^{-}=$
$\displaystyle-\mu\omega\,.$

One can check that they satisfy the correct OPEs for the current algebra
$\mathfrak{sl}(2,\mathbb{R})_{1}$, namely,

$\displaystyle J^{3}(z)J^{3}(w)\sim$
$\displaystyle-\frac{1}{2}\frac{1}{(z-w)^{2}}$ (145) $\displaystyle
J^{3}(z)J^{\pm}(w)\sim$ $\displaystyle\pm\frac{J^{\pm}}{z-w}$ $\displaystyle
J^{+}(z)J^{-}(w)\sim$
$\displaystyle-\frac{2J^{3}(w)}{z-w}+\frac{1}{(z-w)^{2}}\,.$

Furthermore, the current $\mathcal{Z}$ can be written as $U+V$ with

$U=\frac{1}{2}(\omega\lambda+\pi\mu)\,,\quad
V=\frac{1}{2}(\eta_{1}\psi^{1}+\eta_{2}\psi^{2})\,.$ (146)

Highest-weight states in the $\mathfrak{sl}(2,\mathbb{R})_{1}$ model are
obtained from highest-weight Ramond-sector states in the free field algebra.
Specifically, we can introduce a basis of quantum numbers $m,j$ and define the
Ramond-sector highest-weight representation via the actions

$\displaystyle\mu_{0}\ket{m,j}=$
$\displaystyle\ket{m+\tfrac{1}{2},j-\tfrac{1}{2}}$ (147)
$\displaystyle\lambda_{0}\ket{m,j}=$
$\displaystyle\ket{m-\tfrac{1}{2},j-\tfrac{1}{2}}$
$\displaystyle\pi_{0}\ket{m,j}=$
$\displaystyle(j-m)\ket{m-\tfrac{1}{2},j+\tfrac{1}{2}}$
$\displaystyle\omega_{0}\ket{m,j}=$
$\displaystyle(j+m)\ket{m+\tfrac{1}{2},j+\tfrac{1}{2}}\,.$

Note that the above definition of ground states induces exactly the same
action of $\mathfrak{sl}(2,\mathbb{R})$ zero modes as in equation (25).
Indeed, using the definitions of $J^{a}$ in terms of $\mu,\lambda,\pi,\omega$,
we find

$J^{3}_{0}\ket{m,j}=m\ket{m,j}\,,\quad J^{\pm}_{0}\ket{m,j}=(m\pm j)\ket{m\pm
1,j}\,.$ (148)

Note furthermore that

$U_{0}\ket{m,j}=\left(j-\frac{1}{2}\right)\ket{m,j}\,,\quad
V_{0}\ket{m,j}=0\,,$ (149)

so that the $\mathcal{Z}_{0}=0$ constraint enforces $j=1/2$ for these states.

###  4.2 Spectral flow

Spectral flow in the $\mathfrak{sl}(2,\mathbb{R})_{1}$ theory can be similarly
defined as in Section 2, that is, as an automorphism of the mode algebra of
the worldsheet free fields. Specifically, it acts on the bosonic fields
$Z^{A},Y_{A}$ as Eberhardt:2018ouy ; Dei:2020zui

$\displaystyle\sigma^{w}(\lambda_{r})=\lambda_{r-\tfrac{w}{2}}\,,$
$\displaystyle\quad\sigma^{w}(\pi_{r})=\pi_{r-\tfrac{w}{2}}\,,$ (150)
$\displaystyle\sigma^{w}(\mu_{r})=\mu_{r+\tfrac{w}{2}}\,,$
$\displaystyle\quad\sigma^{w}(\omega_{r})=\omega_{r+\tfrac{w}{2}}\,.$

One can check easily that this definition reproduces equation (28) when $k=1$.
Furthermore, this action also preserves the commutation relations between the
symplectic bosons and hence, is indeed an automorphism at the level of
symplectic bosons. The action of spectral flow on the modes then induces the
action on the states. Let us again denote by $\ket{m,j}^{(w)}$ the image of a
state $\ket{m,j}$ under the action of spectral flow $\sigma^{w}$. These states
must satisfy the following property

$\displaystyle A_{m}\ket{m,j}^{(w)}=$
$\displaystyle(\sigma^{w}(A_{m})\ket{m,j})^{(w)}\,,$ (151)

where $A_{m}$ is the $m$th mode of an operator $A(z)$. Applying eq. (151) to
the modes of symplectic bosons, we deduce that

$\displaystyle A_{m}\ket{m,j}^{(w)}=$ $\displaystyle 0,\qquad m>\frac{w}{2}$
(152) $\displaystyle B_{m}\ket{m,j}^{(w)}=$ $\displaystyle 0,\qquad
m>-\frac{w}{2}\,,$

where $A\in\\{\lambda,\pi\\}$ and $B\in\\{\mu,\omega\\}$.

###  4.3 Bosonisation

Shortly, we will write down an explicit form of spectrally-flowed vertex
operators. It turns out that the crucial ingredient in doing so is the
bosonisation of the symplectic bosons. Note that a pair of symplectic bosons
has exactly the same OPE as a $\beta\gamma$ system. Since the $\beta\gamma$
system admits a bosonisation, so do the symplectic bosons. Let

$\displaystyle\phi_{i}(z)\phi_{j}(w)\sim-\delta_{ij}\ln(z-w)\sim\kappa_{i}(z)\kappa_{j}(w)$
(153)

where $i=1,2$. The bosonisation of the symplectic bosons reads

$\displaystyle\lambda$ $\displaystyle=e^{-\phi_{2}-i\kappa_{2}},$
$\displaystyle\omega$ $\displaystyle=e^{\phi_{2}+i\kappa_{2}}\partial
i\kappa_{2},$ (154) $\displaystyle\mu$
$\displaystyle=e^{-\phi_{1}-i\kappa_{1}},$ $\displaystyle\pi$
$\displaystyle=e^{\phi_{1}+i\kappa_{1}}\partial i\kappa_{1}\ .$

In this bosonisation, the $\mathfrak{sl}(2,\mathbb{R})_{1}$ currents become

$\displaystyle J^{3}=$ $\displaystyle-\frac{1}{2}\partial(\phi_{1}-\phi_{2})$
(155) $\displaystyle J^{+}=$ $\displaystyle
e^{\phi_{1}+i\kappa_{1}-\phi_{2}-i\kappa_{2}}\partial i\kappa_{1}$
$\displaystyle J^{-}=$
$\displaystyle-e^{\phi_{2}+i\kappa_{2}-\phi_{1}-i\kappa_{1}}\partial
i\kappa_{2}\ .$

To facilitate interchanging between the bosonised and the rebosonised forms,
we include here some identities of the exponentials of the bosonising fields
Witten:2012bh .

$\displaystyle e^{(l+w)\phi_{1}+il\kappa_{1}}=$
$\displaystyle\left(\frac{\partial^{w}\mu}{w!}\right)^{-l}\delta_{w}(\mu),$
$\displaystyle e^{(l+w)\phi_{2}+il\kappa_{2}}=$
$\displaystyle\left(\frac{\partial^{w}\lambda}{w!}\right)^{-l}\delta_{w}(\lambda)\
,$ (156) $\displaystyle e^{(l-w)\phi_{1}+il\kappa_{1}}=$ $\displaystyle
R\left[\left(\frac{\partial^{w}\mu^{-1}}{w!}\right)^{l}\delta_{w}(\pi)\right],$
$\displaystyle e^{(l-w)\phi_{2}+il\kappa_{2}}=$ $\displaystyle
R\left[\left(\frac{\partial^{w}\lambda^{-1}}{w!}\right)^{l}\delta_{w}(\omega)\right]\
,$

where $w$ is a positive integer and $l$ is a real number. The notation
$\delta_{w}(f)$ is the same as defined in Section 2. Finally, we put the
radial ordering symbol $R[\cdots]$ in the second line of equation (156) to
emphasise that the last line is really a radial ordering between the two
terms. However, in the following we will largely ignore this subtlety since we
are going to work with path integrals and radial ordering becomes a normal
product.

###  4.4 Spectrally-flowed vertex operators

Our goal in this section is to find closed-form expressions for spectrally-
flowed vertex operators, but now in the free field realisation of the
$\mathfrak{psu}(1,1|2)_{1}$ theory. We will make use of the idea of wrapping a
non-local operator to create spectrally-flowed vertex operators. Firstly,
observe that the operator

$\mu^{-j-m}(z)\lambda^{-j+m}(z)$ (157)

satisfies the OPE version of eq. (148). However, this vertex operator is in
the NS sector but the vertex operator we want is in the R sector. In order to
obtain the R sector vertex operator, we wrap it with a non-local operator
constructed by using the current $-\frac{1}{2}Y\cdot Z$.282828Note that this
current has a trivial OPEs with respect to the
$\mathfrak{sl}(2,\mathbb{R})_{1}$ currents and hence, does not change the
$\mathfrak{sl}(2,\mathbb{R})_{1}$ zero mode action in (25). The R sector
vertex operator is

$\displaystyle
V^{0}_{m,j}(y,x=0)=\exp\left(-\frac{1}{2}\int_{P}\frac{\mathrm{d}z}{2\pi
i}(Z\cdot Y)(z)\log(z-y)\right)\mu^{-j-m}(y)\lambda^{-j+m}(y).$ (158)

Using the bosonisation we introduced and repeating the manipulation in Section
3.1, this operator can be written as

$\displaystyle V^{0}_{m,j}(y,x=0)=\exp$
$\displaystyle\left(\left(j+m-\frac{1}{2}\right)\phi_{1}(y)+(j+m)i\kappa_{1}(y)\right.$
$\displaystyle\left.+\left(j-m-\frac{1}{2}\right)\phi_{2}(y)+(j-m)i\kappa_{2}(y)\right).$
(159)

To obtain a $w$ spectrally-flowed vertex operator, we then wrap by the non-
local operator (88) around the vertex operator in eq. (159). The final result
reads292929This is the same as equation (2.16) of Naderi:2022bus , with the
identification $\phi_{2,\rm\text{here}}=-\phi_{2,\rm there}$ and
$\kappa_{2,\rm here}=-\kappa_{2,\rm there}$ and with $\phi_{1},\kappa_{1}$
unchanged.

$\displaystyle V_{m,j}^{w}=$
$\displaystyle\exp\left(\left(j+m+\frac{w-1}{2}\right)\phi_{1}+(j+m)i\kappa_{1}\right.$
(160) $\displaystyle\hskip
28.45274pt+\left.\left(j-m-\frac{w+1}{2}\right)\phi_{2}+(j-m)i\kappa_{2}\right)\,.$

This expression holds for generic $w$. In the case in which $w$ is odd, using
eq. (156) we can rebosonise the vertex operator (160) and write it in terms of
the free fields as

$\displaystyle
V_{m,j}^{w}=\left(\frac{\partial^{\frac{w-1}{2}}\mu}{(\frac{w-1}{2})!}\right)^{-j-m}\delta_{\frac{w-1}{2}}(\mu)\times\left(\frac{\partial^{\frac{w+1}{2}}\lambda^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}\delta_{\frac{w+1}{2}}(\omega)\,.$
(161)

Note that this vertex operator bears a strong resemblance to the Wakimoto
vertex operators in Section 2.

There is another form of eqs.(160) and (161) that will be useful later when
considering the localisation property. Substituting the relations
$\partial\phi_{1}=-\mu\pi,\,\partial\phi_{2}=-\omega\lambda$ into eq. (161),
we have

$\displaystyle V_{m,j}^{w}(z)=$
$\displaystyle\left(\frac{\partial^{\frac{w-1}{2}}\mu(z)}{(\frac{w-1}{2})!}\right)^{-j-m}\exp\left(-\frac{w-1}{2}\int^{z}\mu\pi\right)$
(162)
$\displaystyle\times\left(\frac{\partial^{\frac{w+1}{2}}\lambda(z)^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}\exp\left(\frac{w+1}{2}\int^{z}\omega\lambda\right)\,.$

#### The $x$ basis

Just as in the case of general $k$, we are ultimately interested in computing
correlation functions of the form

$\Braket{\prod_{i=1}^{n}V_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\,,$ (163)

where $x_{i}$ are coordinates on the boundary sphere. As described in Section
2, we define the coordinate dependence of $x$ by taking a vertex operator $V$
and conjugating it with the boundary translation operator $e^{xJ^{+}_{0}}$.
Specifically, we define

$e^{xJ^{+}_{0}}V(z)e^{-xJ^{+}_{0}}=V(x,z)\,.$ (164)

Since we can write $J^{+}$ in terms of the symplectic bosons as
$J^{+}=\pi\lambda$, we can compute the effect of $x$-translation on the free
fields very easily. Specifically, we have

$\displaystyle e^{xJ^{+}_{0}}\lambda e^{-xJ^{+}_{0}}=\lambda,$
$\displaystyle\quad e^{xJ^{+}_{0}}\pi e^{-xJ^{+}_{0}}=\pi\,,$ (165)
$\displaystyle e^{xJ^{+}_{0}}\mu e^{-xJ^{+}_{0}}=\mu-x\lambda,$
$\displaystyle\quad e^{xJ^{+}_{0}}\omega e^{-xJ^{+}_{0}}=\omega+x\pi\,.$

From these relations, we can immediately read off the form of the vertex
operator $V_{m,j}^{w}$ in the $x$-basis. Specifically, equations (161), (162)
become

$\displaystyle V_{m,j}^{w}(x,z)=$
$\displaystyle\left(\frac{\partial^{\frac{w-1}{2}}(\mu-x\lambda))}{(\frac{w-1}{2})!}\right)^{-j-m}\delta_{\frac{w-1}{2}}(\mu-x\lambda)$
(166) $\displaystyle\hskip
42.67912pt\times\left(\frac{\partial^{\frac{w+1}{2}}\lambda^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}\delta_{\frac{w+1}{2}}(\omega+x\pi)\,,$
$\displaystyle=$
$\displaystyle\left(\frac{\partial^{\frac{w-1}{2}}(\mu-x\lambda)}{(\frac{w-1}{2})!}\right)^{-j-m}\exp\left(-\frac{w-1}{2}\int^{z}(\mu-x\lambda)\pi\right)$
$\displaystyle\hskip
42.67912pt\times\left(\frac{\partial^{\frac{w+1}{2}}\lambda^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}\exp\left(\frac{w+1}{2}\int^{z}(\omega+x\pi)\lambda\right)\,.$

Let us briefly comment on the form of these vertex operators. First, the first
line is only valid for $w$ odd. For $w$ even, the vertex operators live in the
Ramond sector and need to be treated somewhat more carefully. Second, the
above vertex operators, when inserted into the path integral at $z_{i}$,
restrict the integration to field configurations which satisfy

$\mu(z)-x\lambda(z)\sim\mathcal{O}\left((z-z_{i})^{\frac{w-1}{2}}\right)\,,\quad\omega(z)+x\pi(z)\sim\mathcal{O}\left((z-z_{i})^{\frac{w+1}{2}}\right)\,.$
(167)

This tells us that spectrally-flowed vertex operators in the $k=1$ theory are
to be interpreted as operators which apply constraints on the space of allowed
field configurations of fundamental fields, analogously to the condition that
spectral flow places on the Wakimoto fields for
$\mathfrak{sl}(2,\mathbb{R})_{k}$ discussed in Section 2.

#### The $W$ field

In order to calculate correlators of spectrally-flowed states, there is one
more subtlety that needs to be addressed in $k=1$ theory. In the hybrid
formalism, one must dress $n+2g-2$ of the fields in any correlation function
with operators $\mathcal{Q}_{-1}$ analogous to picture-changing operators in
the RNS formalism. However, as pointed out in Dei:2020zui , the effect of
introducing these operators is to lower the value of $j_{i}$ by one unit.
Thus, the overall sum of the charges $j_{i}$ is modified to

$\sum_{i=1}^{n}\left(j_{i}-\frac{1}{2}\right)\to\sum_{i=1}^{n}\left(j_{i}-\frac{1}{2}\right)-(n+2g-2)\,.$
(168)

However, the physical state conditions in the symplectic boson theory
restricts states to have $\mathfrak{sl}(2,\mathbb{R})$ spin $j=1/2$, see
Section 4.1. Thus, after acting with the $n+2g-2$ copies of
$\mathcal{Q}_{-1}$, the total sum on the right-hand side of (168) becomes
$-(n+2g-2)$.

However, such a correlator must vanish identically, since $j_{i}-1/2$ is the
eigenvalue of the current

$U=\frac{1}{2}(\omega\lambda+\pi\mu)\,,$ (169)

and therefore the sum of $j_{i}-1/2$ must vanish in any nonvanishing
correlator by charge conservation of $U_{0}$.

The resolution, as was explained in Dei:2020zui  is to insert a certain number
($n+2g-2$) of so-called ‘$W$ fields’. These fields live in the vacuum
representation with respect to $\mathfrak{sl}(2,\mathbb{R})_{1}$, but are non-
trivial with respect to the free fields. In order to account for the
nonvanishing $U_{0}$ charge, $W$ must satisfy $[U_{0},W]=1$. One can construct
$W$ as the unique state satisfying

$\displaystyle\pi(z)W(y)$
$\displaystyle\sim\mathcal{O}\left(\frac{1}{z-y}\right),$
$\displaystyle\mu(z)W(y)$ $\displaystyle\sim\mathcal{O}\left(z-y\right),$
(170) $\displaystyle\lambda(z)W(y)$
$\displaystyle\sim\mathcal{O}\left(z-y\right),$ $\displaystyle\omega(z)W(y)$
$\displaystyle\sim\mathcal{O}\left(\frac{1}{z-y}\right)\ ,$

with $[J^{3}_{0},W]=0$. This identifies $W$ with

$\displaystyle W=e^{\phi_{1}+\phi_{2}}=\delta(\mu)\delta(\lambda)\,.$ (171)

in the bosonised language. Equivalently, we can write

$W=\exp\left(\displaystyle-\int(\omega\lambda+\pi\mu)\right).$ (172)

Therefore, as was proposed in Dei:2020zui , the correct correlators to
calculate take the form

$\Braket{\prod_{\alpha=1}^{n+2g-2}W(u_{\alpha})\prod_{i=1}^{n}V_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\,,$
(173)

where the locations $u_{\alpha}$ of the $W$ fields are taken to be arbitrary,
and should drop out of any final calculation.

Readers may notice a parallel between the $W$ field of Dei:2020zui  discussed
in this section and the ‘secret’ representation $D$ of Eberhardt:2019ywk
discussed in Section 2. Both are singlets with respect to
$\mathfrak{sl}(2,\mathbb{R})_{1}$ but nontrivial with respect to the free
field algebras which generate $\mathfrak{sl}(2,\mathbb{R})_{1}$, and both are
required to define nonvanishing correlation functions. However, both operators
should be thought of as signalling a quirk in their respective free field
theories. In the Wakimoto construction of Section 2, the $D$ field signals the
need to compactify the target space of $\gamma$ from $\mathbb{C}$ to
$\mathbb{CP}^{1}$, while in the symplectic boson realisation of this section,
the $W$ field is an artifact of the gauging from $\mathfrak{gl}(2|2)_{1}$ to
$\mathfrak{psu}(1,1|2)_{1}$, for which $W$ signals a nontrivial gauge field
configuration around some point. As far as we know, there are no further
similarities between these two ‘auxilliary’ fields.

###  4.5 Localisation of correlation functions

Just as correlators of the $\mathfrak{sl}(2,\mathbb{R})_{k}$ WZW model
localise onto holomorphic covering maps provided that the $j$-constraint eq.
(74) is satisfied, the $\mathfrak{psu}(1,1|2)_{1}$ WZW model also admits such
a localisation property. In fact, _all_ correlation functions in the
$\mathfrak{psu}(1,1|2)_{1}$ localise as was shown in Dei:2020zui ;
Knighton:2020kuh . Here we will explain how this localisation property arises
in the path integral.

The localising correlators in the $\mathfrak{psu}(1,1|2)_{1}$ model are
described by holomorphic covering maps $\Gamma$ which satisfy the local
behaviour

$\Gamma(z)\sim x_{i}+a_{i}(z-z_{i})^{w_{i}}+\cdots\,,\quad z\to z_{i}\,,$
(174)

where $z_{i}$ is the insertion point of a spectrally-flowed operator on the
worldsheet, and $x_{i}$ is its location on the boundary sphere. In Dei:2020zui
; Knighton:2020kuh , this localisation property was shown via a Ward identity
analysis on the worldsheet. Let us briefly describe how this works in the path
integral prescription we have developed.

As mentioned above, spectrally-flowed vertex operators in the $k=1$ theory
take the form (166)

$V_{m,j}^{w}(x,z)=\mathcal{O}^{w}_{m,j}(x,z)\delta_{\frac{w-1}{2}}(\mu-x\lambda)\delta_{\frac{w+1}{2}}(\omega+x\pi)\,.$
(175)

where we have defined the operator

$\mathcal{O}^{w}_{m,j}(x,z)=\left(\frac{\partial^{\frac{w-1}{2}}(\mu-x\lambda)}{(\frac{w-1}{2})!}\right)^{-j-m}\left(\frac{\partial^{\frac{w+1}{2}}\lambda^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}$
(176)

for notational convenience. The correlation function we want to consider is
eq. (173), which in this notation takes the form

$\Braket{\prod_{\alpha=1}^{n+2g-2}\delta(\mu(u_{\alpha}))\delta(\lambda(u_{\alpha}))\prod_{i=1}^{n}\mathcal{O}^{w_{i}}_{m_{i},j_{i}}(x_{i},z_{i})\delta_{\frac{w_{i}-1}{2}}(\mu(z_{i})-x_{i}\lambda(z_{i}))\delta_{\frac{w_{i}+1}{2}}(\omega(z_{i})+x_{i}\pi(z_{i}))}\,.$
(177)

Now, considering this correlator in the path integral, we would like to
integrate out the fields $\pi,\omega$. This can be done by using the formal
delta-function identity

$\delta_{\frac{w_{i}+1}{2}}(\omega(z_{i})+x_{i}\pi(z_{i}))=\int\mathrm{d}\zeta^{i}_{0}\ldots\mathrm{d}\zeta^{i}_{\frac{w_{i}-1}{2}}\exp\Bigg{(}i\sum_{\ell_{i}=0}^{\frac{w_{i}-1}{2}}(-1)^{\ell}\zeta^{i}_{\ell_{i}}\partial^{\ell_{i}}(\omega(z_{i})+x_{i}\pi(z_{i}))\Bigg{)}\,.$
(178)

Inserting this identity into the above correlation function has the effect of
shifting the free action (142) to303030Relative to (142), we have set $a=0$
for simplicity. We will see below how this argument is modified if $a\neq 0$.

$\frac{1}{2\pi}\int_{\Sigma}\Bigg{(}\pi\Bigg{(}\overline{\partial}\mu-2\pi
i\sum_{i=1}^{n}\sum_{\ell_{i}=0}^{\frac{w_{i}-1}{2}}\zeta^{i}_{\ell_{i}}x_{i}\partial^{\ell_{i}}\delta(z,z_{i})\Bigg{)}+\omega\Bigg{(}\overline{\partial}\lambda-2\pi
i\sum_{i=1}^{n}\sum_{\ell_{i}=0}^{\frac{w_{i}-1}{2}}\zeta^{i}_{\ell_{i}}\partial^{\ell_{i}}\delta(z,z_{i})\Bigg{)}\Bigg{)}\,.$
(179)

Integrating out $\omega,\pi$ then imposes the equations of motion

$\begin{split}\overline{\partial}\mu&=2\pi
i\sum_{i=1}^{n}\sum_{\ell_{i}=0}^{\frac{w_{i}-1}{2}}\zeta^{i}_{\ell_{i}}x_{i}\partial^{\ell_{i}}\delta(z,z_{i})\,,\\\
\overline{\partial}\lambda&=2\pi
i\sum_{i=1}^{n}\sum_{\ell_{i}=0}^{\frac{w_{i}-1}{2}}\zeta^{i}_{\ell_{i}}\partial^{\ell_{i}}\delta(z,z_{i})\,.\end{split}$
(180)

That is to say, the path integral is taken to be over meromorphic
$\mu,\lambda$ which have poles of order $\frac{w_{i}+1}{2}$, and no other
poles. Furthermore, the coefficients of the poles are precisely chosen so that

$\mu(z)-x_{i}\lambda(z)\sim\mathcal{O}((z-z_{i})^{0})\,,$ (181)

i.e. so that $\mu(z)-x_{i}\lambda(z)$ has no pole near $z_{i}$. Let
$\mathcal{F}$ be the space of all such pairs, i.e.

$\mathcal{F}:=\left\\{(\mu,\lambda)\,\bigg{|}\,\mu,\lambda\sim\mathcal{O}\left((z-z_{i})^{-\frac{w_{i}+1}{2}}\right)\,,\quad\mu(z)-x_{i}\lambda(z)\sim\mathcal{O}((z-z_{i})^{0})\text{
near }z_{i}\right\\}\,.$ (182)

This space has complex dimension313131This can be computed from the Riemann-
Roch theorem as follows. Let $\mathscr{L}$ be the holomorphic line bundle
defined by spinors with poles of order $\frac{w_{i}+1}{2}$ at $z=z_{i}$. This
bundle has degree $\text{deg}(\mathscr{L})=g-1+\sum_{i}\frac{w_{i}+1}{2}$, and
so by Riemann-Roch (assuming no correction term), the space
$\text{H}^{0}(\mathscr{L},\Sigma)$ of holomorphic sections of $\mathscr{L}$
has dimension
$\text{dim}\,\text{H}^{0}(\mathscr{L},\Sigma)=1-g+\text{deg}(\mathscr{L})=\sum_{i}\frac{w_{i}+1}{2}\,.$
Now, $\mathcal{F}$ is the set of all pairs of sections of $\mathscr{L}$ such
that $\mu-x_{i}\lambda$ has no pole at $z=z_{i}$. This space has dimension
$2\,\text{dim}\,\text{H}^{0}(\mathscr{L},\Sigma)-\sum_{i}\frac{w_{i}+1}{2}=\sum_{i}\frac{w_{i}+1}{2}$,
as claimed. If there is a nonzero correction term in the Riemann-Roch theorem,
then nontrivial $\pi,\omega$ zero-modes must be included in the path integral
for a non-vanishing result.

$\text{dim}(\mathcal{F})=\sum_{i=1}^{n}\frac{w_{i}+1}{2}\,,$ (183)

and is parametrised by the Lagrange multipliers $\zeta^{i}_{\ell_{i}}$.
Ignoring Jacobian factors, integrating out $\pi,\omega$ effectively reduces
the path integral to

$\int_{\mathcal{F}}\mathcal{D}(\lambda,\mu)\,\prod_{\alpha=1}^{n+2g-2}\delta(\mu(u_{\alpha}))\delta(\lambda(u_{\alpha}))\prod_{i=1}^{n}\mathcal{O}^{w_{i}}_{m_{i},j_{i}}(x_{i},z_{i})\delta_{\frac{w_{i}-1}{2}}(\mu(z_{i})-x_{i}\lambda(z_{i}))\,.$
(184)

The delta functions in the path integral demand that the first
$\frac{w_{i}-1}{2}$ derivatives of $\mu-x_{i}\lambda$ vanish near $z=z_{i}$,
and furthermore that $\mu,\lambda$ both have simple zeroes near
$z=u_{\alpha}$. This reduces the above integral to a space of (virtual)
dimension

$\text{dim}(\mathcal{F})-2(n+2g-2)-\sum_{i=1}^{n}\frac{w_{i}-1}{2}=-(n+4g-4)\,.$
(185)

Thus, a generic correlator in the $k=1$ worldsheet theory will vanish, unless
$n+4g-4$ of the parameters in the theory are fine-tuned. Of these, $n+3g-3$
are taken care of by the moduli space $\mathcal{M}_{g,n}$ which we will
integrate over in the path integral. We will comment in the next section on
the role of the other $g-1$ moduli.

The localised values of the worldsheet fields will be pairs $(\mu,\lambda)$
satisfying all of the above constraints. Consider the ratio

$\Gamma(z)=\frac{\mu(z)}{\lambda(z)}\,,$ (186)

where we understand this ratio to be evaluated on the solution to the above
constraints. Since $\lambda,\mu$ are chiral fields of conformal dimension
$\Delta=1/2$, $\Gamma$ is a meromorphic function. Since $\lambda,\mu$ share
all of the same poles and $n+2g-2$ of the same zeroes, the poles of $\Gamma$
are given by the zeroes of $\lambda$ which are not located at $z=u_{\alpha}$.
The number of such zeroes is found to be

$\begin{split}Z(\lambda)-(n+2g-2)&=P(\lambda)+g-1-(n+2g-2)\\\
&=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}\,,\end{split}$ (187)

where $Z(\lambda),P(\lambda)$ are the total number of zeroes and poles of
$\lambda$, respectively, and we used the fact that spinors satisfy
$Z(\lambda)-P(\lambda)=g-1$. Thus, as a meromorphic function, $\Gamma$ has
degree

$\text{deg}(\Gamma)=1-g+\sum_{i=1}^{n}\frac{w_{i}-1}{2}\,.$ (188)

Next, near $z=z_{i}$, we have

$\Gamma(z)-x_{i}=\frac{\mu(z)-x_{i}\lambda(z)}{\lambda(z)}\sim\mathcal{O}((z-z_{i})^{w_{i}})\,.$
(189)

Hence, viewed as a topological map $\Gamma:\Sigma\to\mathbb{CP}^{1}$, $\Gamma$
has a branch point at $z=z_{i}$. Therefore, $\Gamma$ satisfies all of the
properties of a holomorphic covering map from the worldsheet to the boundary
of $\text{AdS}_{3}$. The fact that the degree of $\Gamma$ is given by the
Riemann-Hurwitz formula eq. (188) implies that $\Gamma$ has no other such
branch points. This reproduces the result of Dei:2020zui ; Knighton:2020kuh
that the correlator of $k=1$ theory localises onto holomorphic covering maps.

In the next section we will interpret this localisation property in terms of
the set of zero modes of a certain kinetic operator, and we will explicitly
calculate the result of the path integral, up to an overall Jacobian coming
from the delta functions.

###  4.6 Localisation as a zero-mode condition

To get another perspective on localisation in the $k=1$ worldsheet theory, we
can use the idea of Section 3 and re-express spectral flow as a background
gauge field in the path integral.

As discussed above, we can write a state in the spectrally-flowed sector of
the $k=1$ theory in the form (166)

$\begin{split}V^{w}_{m,j}(x,y)=&\left(\frac{\partial^{\frac{w-1}{2}}(\mu-x\lambda)}{(\frac{w-1}{2})!}\right)^{-j-m}\left(\frac{\partial^{\frac{w+1}{2}}\lambda^{-1}}{(\frac{w+1}{2})!}\right)^{j-m}\\\
&\times\exp\left(\int^{y}\left(\left(\frac{w+1}{2}\right)(\omega+x\pi)\lambda-\left(\frac{w-1}{2}\right)(\mu-x\lambda)\pi\right)\right)\,.\end{split}$
(190)

The factors in the top line are simply local functions of the symplectic
bosons, while the bottom line is a non-local factor which is responsible for
the localisation property. Note that we can write the bottom line as

$\exp\left(\int^{y}\left(-w(J^{3}-xJ^{+})+U\right)\right)\,.$ (191)

Up to the factor of the current $U$ (which appears as a consequence of gauging
$\mathfrak{gl}(2)$ to $\mathfrak{sl}(2,\mathbb{R})$), this is precisely the
nonlocal tail of vertex operators explored in Section 3.

Alternatively, we can write this nonlocal tail as

$\exp\left(-\int^{y}Y\begin{pmatrix}\frac{w-1}{2}&-wx\\\
0&-\frac{w+1}{2}\end{pmatrix}Z\right)=\exp\left(-\int^{y}\begin{pmatrix}\pi&\omega\end{pmatrix}\begin{pmatrix}\frac{w-1}{2}&-wx\\\
0&-\frac{w+1}{2}\end{pmatrix}\begin{pmatrix}\mu\\\
\lambda\end{pmatrix}\right)\,,$ (192)

where we are focusing only on the bosonic components of $Y$ and $Z$. Now, this
integral is formally defined by integrating along a path on the worldsheet
$\Sigma$ which ends at $y$. Up to a choice of basepoint, we can use the
arguments in Section 3 to replace this integral with an integral over the full
worldsheet weighted against a distribution which has support only on that
path. We thus write

$\int^{z}Y\begin{pmatrix}\frac{w-1}{2}&-wx\\\
0&-\frac{w+1}{2}\end{pmatrix}Z=\frac{1}{2\pi}\int_{\Sigma}YA_{w}(x,y)Z\,,$
(193)

where $A_{w}(x,z)$ is some appropriate distribution. On the sphere we can
explicitly write

$A_{w}(x,y)=-\begin{pmatrix}\frac{w-1}{2}&-wx\\\
0&-\frac{w+1}{2}\end{pmatrix}\overline{\partial}\log(z-y)\,,$ (194)

such that the support of $A_{w}(x,y)$ is precisely the branch cut of the
logarithm.323232Again, we are ignoring the contribution from the basepoint of
the integral.

In order to compute correlation functions, we will also need an expression for
the $W$ field. This can be written as

$W(y)=\exp\left(-\int^{y}Y\cdot
Z\right)=\exp\left(\frac{1}{2\pi}\int_{\Sigma}Y\cdot
Z\,\overline{\partial}\log(z-y)\right)\,.$ (195)

Now, let us now consider a full spectrally-flowed correlator

$\Braket{\prod_{\alpha=1}^{n+2g-2}W(u_{\alpha})\prod_{i=1}^{n}V_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})}\,.$
(196)

In the path integral, we can write this correlator as

$\int\mathcal{D}(Y,Z)\,e^{-S[Y,Z]}\prod_{i=1}^{n}\mathcal{O}^{w_{i}}_{m_{i},j_{i}}(x_{i},z_{i})\,,$
(197)

where the action is

$S[Y,Z]=\frac{1}{2\pi}\int_{\Sigma}Y(\overline{\partial}+a)Z+\frac{1}{2\pi}\int_{\Sigma}YAZ\,,$
(198)

and $\mathcal{O}^{w}_{m,j}(x,z)$ is defined in (176). Here, $a$ is a flat
$\mathfrak{u}(1)$ connection (see the discussion around equation (142)), and
$A$ is given by

$\begin{split}A&=-\sum_{i=1}^{n}A_{w_{i}}(x_{i},z_{i})-\sum_{\alpha=1}^{n+2g-2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\overline{\partial}\log(z-u_{\alpha})\\\
&=-\sum_{i=1}^{n}\begin{pmatrix}\frac{w_{i}-1}{2}&-w_{i}x_{i}\\\
0&-\frac{w_{i}+1}{2}\end{pmatrix}\overline{\partial}\log(z-z_{i})-\sum_{\alpha=1}^{n+2g-2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\overline{\partial}\log(z-u_{\alpha})\,.\end{split}$ (199)

Thus, we see again that the computation of spectrally-flowed correlation
functions can be recast into the problem of computing correlators of local
operators in the presence of a background gauge field $a+A$.

Since the worldsheet theory is Gaussian in the $k=1$ case, we can compute the
path integral using the saddle-point approximation. Specifically, we can
integrate out $Y$ in the path integral, and we are left with the delta
functional

$\int\mathcal{D}Z\,\delta\left((\overline{\partial}+a+A)Z\right)\,\prod_{i=1}^{n}\mathcal{O}^{w_{i}}_{m_{i},j_{i}}(x_{i},z_{i})\,,$
(200)

where $\mathcal{O}^{w}_{m,j}(x,z)$ was defined in eq. (176). We can compute
this path integral by 1) finding the set of solutions to the delta function,
2) inserting those solutions into the integrand, weighted by an appropriate
Jacobian and 3) sum/integrate over the space of all such solutions.

Algebraically, the solutions to the delta function are given by the set of
zero modes of the elliptic operator $\overline{\partial}+a+A$ acting on the
bundle $S\oplus S$, where $S$ is the spinor bundle of $\Sigma$. Assuming that
this operator has trivial cokernel, the dimension of its kernel can be worked
out by the Hirzebruch-Riemann-Roch theorem. First, we can think of
$\overline{\partial}+a+A$ acting on $S\oplus S$ as the operator
$\overline{\partial}$ acting on a rank-2 bundle $E$ with curvature

$\begin{split}F&=F_{S\oplus S}+\partial A+\partial a\\\ &=F_{S\oplus S}-2\pi
i\sum_{i=1}^{n}\begin{pmatrix}\frac{w_{i}-1}{2}&-w_{i}x_{i}\\\
0&-\frac{w_{i}+1}{2}\end{pmatrix}\delta^{(2)}(z-z_{i})-2\pi
i\sum_{\alpha=1}^{n+2g-2}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\delta^{(2)}(z-u_{\alpha})\,,\end{split}$ (201)

where $F_{S\oplus S}$ is the curvature of $S\oplus S$. The virtual dimension
of the space of holomorphic sections of this bundle is then given by the
index333333Here, the term ‘virtual dimension’ is meant in the usual sense of
the number of degrees of freedom minus the number of constraints, and can
therefore be negative. More precisely, it is given by the combination
$\text{dim}\,\text{H}^{0}(E,\Sigma)-\text{dim}\,\text{H}^{1}(E,\Sigma)$.
Alternatively, by Serre duality, one can think of the index as computing the
number of $Z$ zero modes minus the number of $Y$ zero modes.

$\text{vdim}\,\text{H}^{0}(E,\Sigma)=\int_{\Sigma}\text{ch}(E)\,\text{td}(\text{T}\Sigma)\,.$
(202)

The Chern character and Todd class are given by

$\text{ch}(E)=2+c_{1}(F)\,,\quad\text{td}(\text{T}\Sigma)=1+\frac{1}{2}c_{1}(\text{T}\Sigma)\,.$
(203)

The number of zero modes is therefore

$\text{vdim}\,\text{H}^{0}(E,\Sigma)=\int_{\Sigma}(c_{1}(\text{T}\Sigma)+c_{1}(F))=-(n+4g-4)\,,$
(204)

where we have used

$\int_{\Sigma}c_{1}(\text{T}\Sigma)=2-2g\,,\quad\int_{\Sigma}c_{1}(F_{S\oplus
S})=2g-2\,.$ (205)

Thus, the integral (200) localises to a space of zero modes with virtual
dimension $-(n+4g-4)$. However, in the full string theory calculation, we will
need to integrate over the moduli space $\mathcal{M}_{g,n}$ of complex
structures as well as the moduli space $\text{Jac}(\Sigma)$ of flat
$\mathfrak{u}(1)$ connections $a$ (the Jacobian of $\Sigma$). Thus, the total
dimension of the path integral increases to

$-(n+4g-4)+\text{dim}(\mathcal{M}_{g,n})+\text{Jac}(\Sigma)=1\,.$ (206)

In Appendix A, we explicitly compute this zero mode, and find that it takes
the form

$Z=\frac{\omega}{\sqrt{\partial\Gamma}}\begin{pmatrix}\Gamma\\\
1\end{pmatrix}\,,$ (207)

where $\Gamma$ is a holomorphic covering map and $\omega$ is a particular
$(1,0)$-form with simple poles at $z=z_{i}$ and simple zeroes at
$z=u_{\alpha}$. These are precisely the localising solutions found in
Eberhardt:2021jvj , and only exist for a discrete set of worldsheet moduli and
$\mathfrak{u}(1)$ connections $a$. Note that on these localising solutions we
have

$\begin{split}\frac{\partial^{\frac{w_{i}-1}{2}}(\mu(z_{i})-x_{i}\lambda(z_{i}))}{(\frac{w_{i}-1}{2})!}&=\left(\mathop{\mathrm{Res}}_{z=z_{i}}\omega\right)\left(\frac{a_{i}^{\Gamma}}{w_{i}}\right)^{1/2}\,,\\\
\frac{\partial^{\frac{w_{i}+1}{2}}\lambda(z_{i})^{-1}}{\left(\frac{w_{i}+1}{2}\right)!}&=\left(\mathop{\mathrm{Res}}_{z=z_{i}}\omega\right)^{-1}(w_{i}a_{i}^{\Gamma})^{1/2}\,.\end{split}$
(208)

Thus, we can compute the integrand of (200) and we find

$\prod_{i=1}^{n}\mathcal{O}_{m_{i},j_{i}}^{w_{i}}(x_{i},z_{i})=\prod_{i=1}^{n}\left(w_{i}^{-1/2}\mathop{\mathrm{Res}}_{z=z_{i}}\omega\right)^{-2j_{i}}(a_{i}^{\Gamma})^{-m_{i}}\,.$
(209)

The $m_{i}$ dependence reproduces exactly that of the dual symmetric orbifold
CFT Lunin:2001ne ; Dei:2019iym , and agrees with the Ward identity analyses of
Eberhardt:2019ywk ; Eberhardt:2020akk ; Dei:2020zui ; Knighton:2020kuh . While
the above analysis was only done for the bosonic half of the
$\mathfrak{psu}(1,1|2)_{1}$ WZW model, we expect this strategy to be easily
generalisable to include the worldsheet fermions as well.

##  5 Relationship to twistor theory

The relationship (186) between the worldsheet free fields and covering map
$\Gamma$ has a natural geometric interpretation. We can think of
$[\mu:\lambda]$ has homogeneous coordinates in $\mathbb{CP}^{1}$ and $\Gamma$
as a local coordinate on the same manifold. Equation (186) then tells us that
these two coordinates must coincide. This relation was first found in
Dei:2020zui , where the authors noticed that the equation

$\mu(z)-\Gamma(z)\lambda(z)=0$ (210)

holds inside of every correlation function. Due to the similarity of this
relation with the ‘incidence’ relation of twistor theory, it was suggested by
the authors of Dei:2020zui  that the $k=1$ worldsheet theory on
$\text{AdS}_{3}\times\text{S}^{3}\times\mathbb{T}^{4}$ may share a close
relation with a 2-dimensional version of the Berkovits-Witten twistor string
Witten:2003nn ; Berkovits_2004 .

In this section, we collect the features of the $k=1$ string model that have
an immediate twistorial interpretation and discuss their relation to twistor
theory. We introduce the salient features of the twistor space for
$\text{S}^{2}$. Then we discuss how the target space and the functional form
of the vertex operators relate to known twistor spaces and twistor objects,
and discuss the appearance of the incidence relation.

This twistorial interpretation was a motivation for the proposal
Gaberdiel:2021qbb ; Gaberdiel:2021jrv  in the setting of AdS5/CFT4, in which
they obtain a candidate for the string theory dual to free 4d $\mathcal{N}=4$
super Yang-Mills. The correspondence between these free field models and the
twistor string promises to be fruitful, and more work is needed to better
understand the precise connection.

###  5.1 Twistors for $\text{S}^{2}$

Consider the embedding of $\text{S}^{2}$ as the celestial sphere, the
projectivised null cone of the origin of $\mathbb{R}^{3,1}$:

$\begin{split}X_{\mu}&:=(t,x,X_{1},...,X_{2})\in\mathbb{R}^{1,1}\times\mathbb{R}^{2}\,,\\\
\mathrm{d}s^{2}&=\eta_{\mu\nu}\mathrm{d}X^{\mu}\mathrm{d}X^{\nu}=-\mathrm{d}t^{2}+\mathrm{d}x^{2}+(\mathrm{d}X_{i})^{2}\,,\\\
S^{2}&\cong\frac{\\{X^{2}=0\\}\setminus\\{\text{the origin}\\}}{X_{\mu}\sim
rX_{\mu},\quad r\in\mathbb{R}^{*}}\,.\end{split}$ (211)

In words, this means that each null generator through the origin corresponds
to a point on $S^{2}$. In order to identify a particular locus in
$\mathbb{R}^{3,1}$ as the $S^{2}$, we must gauge fix the scaling redundancy by
picking one point on each null generator. This is often done by choosing a
Cauchy slice and taking its intersection with the null cone. The resulting
space is conformal to $S^{2}$. Although the action of $\text{SO}(3,1)$ on
points on the $\mathbb{R}^{3,1}$ embedding space is linear, the realisation of
$\text{SO}(3,1)$ as the (double cover of the global) conformal group acting on
points in $\text{S}^{2}$ is nonlinear. In this context, this can be read off
from the fact that Lorentz transformations will in general move points away
from the slice that supplies our gauge fixing condition, and the scaling
transformation required to move them back onto the slice, resulting in the
nonlinearity.

In contrast, spinors for $\mathbb{R}^{3,1}$ will of course transform linearly
under the (global part of the) $2$ dimensional conformal group. The spinor for
the 2 dimension higher embedding space is called a twistor343434There is a
pedagogical introduction to the twistor space for S2 in the appendix of
bu2023celestial  for the interested reader. for $\text{S}^{2}$. The embedding
space Lorentz group is
$\text{SO}(1,3)\cong\text{SL}(2,\mathbb{C})/\mathbb{Z}_{2}$ whose Weyl spinors
are 2 component objects transforming in the fundamental of
$\text{SL}(2,\mathbb{C})$. These 2 component objects are the twistors for
$\text{S}^{2}$

$Z^{A}=(Z^{0},Z^{1})\in\mathbb{C}^{2}\,.$ (212)

A twistor is related to the 2d spacetime via the incidence relation

$Z^{0}=xZ^{1},\quad x\in\mathbb{C}\cong\mathbb{R}^{2}\subset\text{S}^{2}\,.$
(213)

Since the zero twistor has no physical information and the scale of the
twistor components is irrelevant in the relation to spacetime, we frequently
work with projective twistor space $\mathbb{PT}:=\mathbb{CP}^{1}$

$\mathbb{PT}:=\\{Z^{A}\in\mathbb{C}^{2}\setminus\\{0\\}\\}/\\{Z^{A}\sim
rZ^{A},\quad r\in\mathbb{C}^{*}\\}\,.$ (214)

There is a global description of the S2 spacetime as a $\mathbb{CP}^{1}$ via
Bloch sphere coordinates (see eq. (252)) that is more convenient to work with.
Define homogenous coordinates $x^{A}:=(x^{0},x^{1})\in\mathbb{CP}^{1}$.
Indices can be raised and lowered by the $\text{SL}(2)$ invariant two-index
Levi-Civita symbol, with the “upper left - lower right” index raising and
lowering convention

$1=\epsilon_{01}=\epsilon^{10},\quad x^{A}=\epsilon^{AB}x_{B}\,.$ (215)

We define the $\text{SL}(2)$ invariant contraction in the usual way as

$\langle
xy\rangle:=\epsilon^{AB}x_{B}y_{A}=x^{A}y_{A}=-x_{B}y^{B}=\epsilon_{BC}x^{C}y^{B}\,.$
(216)

The incidence relation can be written in the following SL(2) covariant way

$\langle xZ\rangle=0\,.$ (217)

We see that the previous given form of the incidence relation is equivalent on
the patch of $\mathbb{CP}^{1}$ on which $x^{1}\neq 0$ and under the
identification $x^{0}/x^{1}=:x\in\mathbb{C}\cong\mathbb{R}^{2}$, the local
coordinate on the patch. Explicitly, we have that

$\langle xZ\rangle=0\iff x^{1}Z^{0}-x^{0}Z^{1}=0\iff
Z^{0}=\frac{x^{0}}{x^{1}}Z^{1},\quad x_{1}\neq 0\,.$ (218)

The patch of $\mathbb{CP}^{1}$ in which $x^{1}\neq 0$ precisely misses the
point $(x^{0},0)\sim(1,0)\in\mathbb{CP}^{1}$, which is the point at $\infty$
in the $\mathbb{C}\cong\mathbb{R}^{2}$ patch of S2 coordinatised by $x$. On
the other canonical patch $x^{0}\neq 0$, the incidence relation reads $\langle
xZ\rangle=0\iff\frac{x^{1}}{x^{0}}Z^{0}=Z^{1}$, with local coordinate
$x^{\prime}:=x^{1}/x^{0}$. The transition function between the patches under
$x^{\prime}=1/x$, the inversion map, is holomorphic away from the points
$x^{1}=0$ and $x^{0}=0$. It is a straightforward exercise to check that under
this identification, the action of the global conformal group on the local
coordinate $x$ can be realised by the natural action of SL(2,$\mathbb{C}$)
acting on the homogenous coordinate $x^{A}$.

It is a striking feature of twistors in (conformally flat) 2d that the twistor
space and the real Euclidean spacetime are equivalent (see the appendix of
bu2023celestial ). The incidence relation defines a bijection between the
spaces. Given a twistor $Z^{A}$, the locus of the incidence relation is the
unique point in $x^{A}\in\mathbb{CP}^{1}$ given by $x^{A}\sim Z^{A}$.
Similarly, given a spacetime point $x^{A}$, the locus of the incidence
relation is the unique twistor $Z^{A}$ given by $Z^{A}\sim x^{A}$. Once we add
supersymmetry, it is no longer true that the incidence relation defines a
bijection. Rather, the locus of the incidence relation is a $0|N$-dimensional
space.

###  5.2 Target space interpretation

The matter action we are considering in eq. (142) bears a striking resemblance
to the Berkovits-Witten twistor string Berkovits_2004  (for $\mathcal{N}=4$ in
4d), and in fact admits an interpretation as a twistor string for
$\mathcal{N}=2$ in 2d Dei:2020zui .

Analogously to the bosonic case, the $\mathcal{N}=2$ supertwistors are
projectivised spinors for the complexified global $\mathcal{N}=2$
superconformal group (transform in the fundamental of
$\text{SL}_{\mathbb{C}}(2|2)$), which means that the $\mathcal{N}=2$ 2d
supertwistor space is $\mathbb{PT}:=\mathbb{CP}^{1|2}$. Our $Z^{\mathcal{A}}$
variables can therefore be interpreted as homogenous coordinates on
$\mathbb{PT}$, i.e whose components are sections of the $\mathbb{C}^{*}$ line
bundle $\mathcal{O}(1)$ over $\mathbb{CP}^{1|2}$. $Y_{\mathcal{A}}$ can
therefore be interpreted as a dual supertwistor, whose components are sections
of $\mathcal{O}(-1)$ over $\mathbb{CP}^{1|2}$. In each case, the rescaling
weight $1,-1$ can be read off from the charge under the gauged symmetry
current $Z\cdot Y$, which identifies $Z\sim rZ,Y\sim\frac{1}{r}Y$.

However $Z^{\mathcal{A}}(z)$, $Y_{\mathcal{B}}(z)$ are sections of
$K^{1/2}_{\Sigma}$ and therefore transform nontrivially under worldsheet
conformal transformations. The worldsheet Weyl rescaling (that rescales them
paralelly) and the $Z\cdot Y$ rescaling (that rescales them oppositely)
together amount to allowing $Z^{\mathcal{A}},Y_{\mathcal{B}}$ to scale
independently. The argument can therefore be made that
$Z^{\mathcal{A}},Y_{\mathcal{B}}$ should be thought of as homogenous
coordinates on $\mathbb{CP}^{1|2}\times\mathbb{CP}^{1|2}$. Together with
interpreting the gauge field $a$ as a Lagrange multiplier enforcing $Z\cdot
Y=0$, the target space can be interpreted as the quadric $Z\cdot Y=0$ inside
$\mathbb{CP}^{1|2}\times\mathbb{CP}^{1|2}$. This is the real Euclidean
ambitwistor space of the boundary $\text{S}^{2}$, which is also the (complex
codimension 1) boundary of the AdS3 minitwistor space Geyer_2023 , and
probably has holographic implications that await further investigation.

Note that this observed ambiguity in the target space interpretation is also
present in the Berkovits-Witten twistor string. In their analysis, they work
with gauge fixings that trivialise $a$ (gauge fixing $a$ to a flat connection)
and subscribe to the former interpretation, and we will follow suit.

###  5.3 Vertex operator interpretation

At zero units of spectral flow, the bosonic part of vertex operators have an
interpretation in terms of meromorphic twistor functions called elementary
states PENROSE1973241 . Consider homogenous coordinates
$x^{A}:=(x^{0},x^{1})\in\mathbb{CP}^{1}$, which is to be interpreted as the
real boundary 2-sphere (see eq. (252)). The antipodal map on $\mathbb{CP}^{1}$
is denoted by $x^{A}\rightarrow\hat{x}^{A}$

$\hat{x}^{A}=\begin{pmatrix}-\bar{x}_{1}\\\
\bar{x}_{0}\end{pmatrix},\quad\langle
x\hat{x}\rangle=|x_{0}|^{2}+|x_{1}|^{2}\geq 0\,.$ (219)

Here, the bar $\bar{.}$ denotes the usual complex conjugate. We may define the
origin to be the point $o^{A}:=(o^{0},o^{1})=(0,1)$ and the point at infinity
as $\iota^{A}:=-\hat{o}^{A}=(1,0)$, where the names are with respect to the
local coordinate $x:=x^{0}/x^{1}$ on the patch $x^{1}\neq 0$. We can therefore
rewrite the equation (157) as

$\begin{split}V^{0}_{(\text{NS})m,j}(z,0)&=(Z^{0}(z))^{-j-m}(Z^{1}(z))^{-j+m}\\\
&=\langle
oZ(z)\rangle^{-j-m}\langle\hat{o}Z(z)\rangle^{-j+m}(-1)^{-j+m}\,.\end{split}$
(220)

The map between $V^{0}_{(\text{NS})m,j}$ and the Ramond sector vertex
operators $V^{0}_{m,j}$ that we use to compute correlators is given in eq.
(158). The advantage of working in this manifestly $\text{SL}(2)$ covariant
formalism is that we can trade off $\text{SL}(2)$ rotations between $Z,x$:

$\begin{split}\langle
Mx,Z\rangle&=M\indices{{}^{A}_{B}}x^{B}\epsilon_{AC}Z^{C}=x^{B}\epsilon_{BD}M^{AD}\epsilon_{AC}Z^{C}\\\
&=x^{B}\epsilon_{BD}(M^{T})\indices{{}^{D}_{C}}Z^{C}=\langle
x,M^{T}Z\rangle\,,\end{split}$ (221)

for $M\in\text{SL}(2)$. For instance, consider a bilinear corresponding to the
action of some (traceless) $t\indices{{}^{A}_{B}}\in\mathfrak{sl}(2)$
generator

$-t\indices{{}^{A}_{B}}Z^{B}Y_{A}(z)Z^{C}(w)\sim\frac{t\indices{{}^{C}_{D}}Z^{D}(w)}{z-w}\,.$
(222)

The exponentiated relation on
$M(\alpha^{a})=e^{\alpha^{a}t^{a}}\in\text{SL}(2)$,
$\alpha^{a}\in\mathbb{C}^{3}$ (where $a$ is the adjoint index) reads

$\exp\left(-\alpha^{a}\oint_{w}(t^{a})\indices{{}^{A}_{B}}Z^{B}Y_{A}\right)Z^{C}(w)=M\indices{{}^{C}_{D}}(\alpha^{a})Z^{D}(w)\,.$
(223)

To go to the $x$-basis vertex operators located at points $x_{i}$ on the S2,
we will consider the action of the $\text{SL}(2,\mathbb{C})$ generator that
corresponds to the translation generator of the global part of the boundary 2D
conformal group. This is

$t\indices{{}^{A}_{B}}=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\rightarrow(t\indices{{}^{A}_{B}}Z^{B}Y_{A})=Y_{0}Z^{1}=J^{+}\,,$
(224)

which acts as

$(e^{x\oint J^{+}})\,Z^{A}=\begin{pmatrix}Z^{0}-xZ^{1}\\\
Z^{1}\end{pmatrix}=\begin{pmatrix}1&-x\\\
0&1\end{pmatrix}\begin{pmatrix}Z^{0}\\\
Z^{1}\end{pmatrix}=\left(\exp{\left(-x\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\right)}\right)Z^{A}\,.$ (225)

Acting on our vertex operators with all free indices tied up with $o,\hat{o}$,
it can be traded for the action of the transpose on
$o_{A},\hat{o}_{A}\propto\iota_{A}$. This leaves $\iota_{A}$ invariant, as it
should intuitively, because translations do not move the point at infinity. It
sends $o^{A}=(0,1)\rightarrow(x,1)=:x^{A}$. Therefore we have

$e^{x\oint J^{+}}\begin{pmatrix}\langle oZ\rangle\\\
\langle\hat{o}Z\rangle\end{pmatrix}=\begin{pmatrix}\langle xZ\rangle\\\
\langle\hat{o}Z\rangle\end{pmatrix}\,.$ (226)

The action of the translation generator on the vertex operator can be traded
for the action of the translation generator on the external data

$\begin{split}e^{x\oint J^{+}}\langle
oZ(z)\rangle^{-j-m}&\langle\hat{o}Z(z)\rangle^{-j+m}(-1)^{-j+m}\\\ &=\langle
xZ(z)\rangle^{-j-m}\langle\hat{o}Z(z)\rangle^{-j+m}(-1)^{-j+m}=:V^{0}_{(\text{NS})m,j}(z,x)\,.\end{split}$
(227)

In this way, the action of the abstract $\text{SL}(2,\mathbb{C})$ on the
twistor variables is realised as the action of the global conformal
transformations on the boundary 2-sphere coordinatised by $x^{A}$ and vice-
versa. These vertex operators are built of meromorphic twistor functions known
as elementary states

$\frac{1}{\langle xZ\rangle^{j+m}\langle\hat{o}Z\rangle^{j-m}}\,,$ (228)

which appeared in the context of 4d twistor theory PENROSE1973241  as Čech
cohomology representatives that Penrose transform to a useful basis for
massless on-shell wavefunctions. For us, they arise because they are the
natural building block for conformally covariant $n$-point functions. Consider
for example the following contour integral (in which the contour below
separates the two poles353535Note that the integration variable is understood
to be the local coordinate $Z^{0}/Z^{1}$ and not the homogeneous coordinates.)
that reproduces the 2 point function at $m=j=\frac{1}{2}$

$\oint\langle Z\mathrm{d}Z\rangle\left(\frac{1}{\langle
x_{1}Z\rangle}\frac{1}{\langle x_{2}Z\rangle}\right)=\frac{1}{\langle
x_{1}x_{2}\rangle}\,.$ (229)

Integrands made of elementary states give conformally covariant functions
because everything other than the elementary state insertions (which are
covariant) are $\text{SL}(2)$ invariant. As a suggestive aside, integral
formulae that reproduce conformally covariant functions often arise from
integrating over several twistors $Z_{i}^{A}$ (see e.g Berkovits_2004 ),

$\oint\bigwedge_{i=1}^{n+1}\langle
Z_{i}\mathrm{d}Z_{i}\rangle\left(\cdots\right)\,.$ (230)

These integral formulae naturally appear as the result of holomorphic
localisation in a twistor string (i.e $\bar{\partial}Z^{A}=0$ for
$Z^{A}\in\mathcal{O}(n)$) and having to do a finite dimensional set of
integrals over the zero modes of $Z^{A}$ (being the moduli space of
holomorphic maps of degree $n$), which can be specified by the values that
$Z^{A}_{i}:=Z^{A}(z_{i})$ that the map $Z^{A}$ takes at $n+1$ points on the
worldsheet. In our context, the localisation is to a zero dimensional set of
covering maps rather than a $n+1$ dimensional space of curves, but the role of
the elementary states as conformally covariant building blocks is the same.
The antichiral part of the correlator is supplied by the decoupled antichiral
sector, which has vertex operators that depend on $\hat{x}_{i}$.

In the special case in which
$p:=-j-m\in\mathbb{Z}^{+},q:=-j+m\in\mathbb{Z}^{+}$, the NS sector vertex
operator has an interpretation in terms of spin-weighted spherical harmonics
${}_{m}Y_{-j,l}$ on $\mathbb{CP}^{1}_{x}$, which are known to be a basis for
smooth data on S2 TorresdelCastillo2003 , see Appendix B for a short
introduction. Consider the twistor function $y$

$\begin{split}y_{p,q}(x,Z)&:=\langle
xZ\rangle^{p}\langle\hat{x}Z\rangle^{q}\\\ &\in
H^{0,0}(\mathbb{CP}^{1}_{Z},\mathcal{O}(p+q)\otimes\Gamma(\mathbb{CP}^{1}_{x},\mathcal{O}(p)\otimes\bar{\mathcal{O}}(q)))\,.\end{split}$
(231)

The $Z$ dependence can be removed by integrating over the
$\mathbb{CP}^{1}_{Z}$. There are $p+q+1$ independent choices of $\\{A_{i}\\}$,