aainstitutetext: Mathematical Sciences and STAG Research Centre, University of
Southampton, Southampton SO17 1BJ, United Kingdom.bbinstitutetext: Instituto
de Astronomía y Física del Espacio (IAFE), CONICET - Universidad de Buenos
Aires, C. C. 67, Suc. 28, 1428 Buenos Aires, Argentina.ccinstitutetext:
Mathematics with Computer Science Program, Guangdong Technion - Israel
Institute of Technology, 515063 Shantou, Guangdong, People’s Republic of
China. ddinstitutetext: Institut de Physique Théorique, Université Paris
Saclay, CEA, CNRS, Orme des Merisiers, 91191 Gif-sur-Yvette CEDEX, France.

# Worldsheet computation of heavy-light correlators

Davide Bufalini b,c Sergio Iguri d Nicolas Kovensky a David Turton
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute a large collection of string worldsheet correlators describing
light probes interacting with heavy black hole microstates. The heavy states
consist of NS5 branes carrying momentum and/or fundamental string charge. In
the fivebrane decoupling limit, worldsheet string theory on a family of such
backgrounds is given by exactly solvable null-gauged WZW models. We construct
physical vertex operators in these cosets, including all massless
fluctuations. We first compute a large class of novel heavy-light-light-heavy
correlators in the AdS3 limit, where the light operators include those dual to
chiral primaries of the holographically dual CFT. We compare a subset of these
correlators to the holographic CFT at the symmetric product orbifold point,
and find precise agreement in all cases, including for light operators in
twisted sectors of the orbifold CFT. The agreement is highly non-trivial, and
includes amplitudes that describe the analogue of Hawking radiation for these
microstates. We further derive a formula for worldsheet correlators consisting
of $n$ light insertions on these backgrounds, and discuss which subset of
these correlators are likely to be protected. As a test, we compute a heavy-
light five-point function, obtaining precisely the same result both from the
worldsheet and the symmetric orbifold CFT. This paper is a companion to and
extension of Bufalini:2022wyp .

##  1 Introduction

String Theory provides a microscopic description of black holes as being bound
states of strings and branes with an exponentially large number of internal
microstates Strominger:1996sh . Amongst these microstates, there are coherent
pure states, large families of which have been shown to be well-described by
smooth and horizonless supergravity solutions, see e.g. Lunin:2001jy ;
Lunin:2002iz ; Bena:2016ypk ; Bena:2017xbt ; Ceplak:2018pws ; Heidmann:2019zws
; Ganchev:2022exf . Upon taking an appropriate AdS decoupling limit, these
solutions are proposed to correspond to specific families of pure states in
the holographically dual CFT (HCFT); precision holography has provided sharp
evidence supporting this correspondence Kanitscheider:2006zf ;
Kanitscheider:2007wq ; Giusto:2015dfa ; Giusto:2019qig ; Rawash:2021pik .

While supergravity constructions provide valuable insight into the structure
of black hole microstates, it is natural to expect that string-theoretic
physics beyond supergravity will be necessary to obtain a complete description
of black hole microstructure. A fruitful arena in which to investigate such
stringy physics is provided by bound states of NS5 branes carrying fundamental
string (F1) and/or momentum charge (P). More specifically, we work in Type IIB
compactified on $\mathbb{S}^{1}\times\mathbb{T}^{4}$, with $n_{5}$ NS5 branes
wrapped on $\mathbb{S}^{1}\times\mathbb{T}^{4}$, $n_{1}$ units of F1 winding
on $\mathbb{S}^{1}$, and $n_{P}$ units of momentum charge along
$\mathbb{S}^{1}$.

Upon taking the fivebrane decoupling limit, one obtains asymptotically linear
dilaton configurations, which are holographically dual to (doubly scaled)
Little String Theory Giveon:1999px ; Giveon:2001up . In an appropriate region
of the parameter space, there is an AdS3 regime in the IR, and one can take a
further AdS3 decoupling limit Maldacena:1997re . Upon doing so, one obtains
the well-studied NS5-F1 instance of AdS3/CFT2 holography Giveon:1998ns ;
Kutasov:1999xu .

The NSNS vacuum of the holographic CFT corresponds to the global
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$ background, whose
worldsheet theory involves an SL(2,$\mathbb{R}$)$\times$SU(2) Wess-Zumino-
Witten (WZW) model Zamolodchikov:1986bd ; Teschner:1999ug ; Maldacena:2000hw ;
Maldacena:2000kv ; Maldacena:2001km . In recent work, a family of gauged WZW
models has been constructed and studied involving the same Lie groups,
providing an exact worldsheet description of a set of NS5-F1-P black hole
microstates Martinec:2017ztd ; Martinec:2018nco ; Martinec:2019wzw ;
Martinec:2020gkv ; Bufalini:2021ndn .

Processes in which light probes interact with a heavy background such as a
black hole or a black hole microstate give rise to interesting and computable
dynamical observables. Mixed heavy-light (HL) correlators have been previously
studied in holography, see e.g. Galliani:2016cai ; Fitzpatrick:2016mjq ;
Balasubramanian:2017fan ; Galliani:2017jlg ; Bombini:2019vnc . In the NS5-F1
system, there is a locus in moduli space at which the holographic CFT is
conjectured to be the ${\cal{N}}=(4,4)$ symmetric product orbifold CFT with
target space $\left(\mathbb{T}^{4}\right)^{N}/S_{N}$, where $N=n_{1}n_{5}$.
There is now a substantial body of evidence for this conjecture, see e.g.
Kanitscheider:2006zf ; Kanitscheider:2007wq ; Giusto:2015dfa ; Giusto:2019qig
; Rawash:2021pik ; Argurio:2000tb ; Gaberdiel:2007vu ; Dabholkar:2007ey ;
Giribet:2007wp . For recent discussions of holography in related systems, see
Eberhardt:2021vsx ; Eberhardt:2019qcl ; Balthazar:2021xeh .

For instance, heavy-light-light-heavy (HLLH) four-point functions have been
computed in the supergravity approximation and/or in the symmetric product
orbifold CFT, for particular sets of heavy and light operators
Galliani:2016cai ; Galliani:2017jlg ; Bombini:2019vnc . Having solvable
worldsheet models associated to black hole microstates means we can go much
further by taking into account $\alpha^{\prime}$ corrections Martinec:2017ztd
. Given a worldsheet model describing string dynamics on a heavy background,
the relevant quantities correspond to (a particular limit of) integrated
correlators of light operators in the worldsheet vacuum.

Worldsheet correlators in global AdS3 were first studied in Maldacena:2001km ,
building in part on Fateev ; Teschner:1999ug , and the role of the vertex
operators associated with spectrally flowed representations was highlighted.
Further studies include Giribet:2001ft ; Cardona:2010qf ; Cagnacci:2013ufa ;
Dei:2021xgh ; Dei:2021yom . The spectrum of chiral primaries and their three-
point functions in global AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$
were computed in Argurio:2000tb ; Gaberdiel:2007vu ; Dabholkar:2007ey ;
Giribet:2007wp , and shown to match those of the symmetric product orbifold
CFT, as studied in Lunin:2000yv ; Lunin:2001pw .

The supergravity backgrounds we consider are known as NS5-F1 circular
supertubes and spectral flows thereof Lunin:2004uu ; Giusto:2004id ;
Giusto:2004ip ; Jejjala:2005yu ; Giusto:2012yz ; Chakrabarty:2015foa . This
includes non-BPS spectrally flowed supertubes, known as the JMaRT solutions,
after the authors of Jejjala:2005yu . The associated worldsheet models are
null-gauged WZW models, where before gauging one considers a
(10+2)-dimensional target space
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{R}_{t}\times\mathbb{S}^{1}_{y}\times\mathbb{T}^{4}$.
Roughly speaking, in the IR AdS3 regime, the gauging is concentrated mostly
along the $t$ and $y$ directions, while in the linear dilaton regime the
gauging is concentrated mostly in the time and angular directions of
SL(2,$\mathbb{R}$).

These coset models can also be thought of as marginal current-current
deformations of the worldsheet theory for strings in AdS3. These are instances
of a larger class of deformations that undo the decoupling limit with respect
to the F1 harmonic function, i.e. they “add back the 1+” in that function,
leading to linear dilaton asymptotics; see e.g. Martinec:2020gkv . At the
level of the dual field theory, a closely related procedure has been argued to
correspond to the so-called single-trace $T\bar{T}$ irrelevant deformation of
the original holographic CFT Giveon:2017myj ; Giveon:2017nie , flowing towards
a non-local Little String Theory.

In this paper we study string correlators in these highly excited backgrounds.
To do so, we first compute a large set of physical vertex operators, in both
NSNS and RR sectors, building on Martinec:2018nco ; Martinec:2020gkv . These
describe linearized perturbations of the background configurations. We focus
primarily on coset states in discrete series representations, including
worldsheet spectral flow, that are dual to chiral primary operator excitations
in the HCFT. When the background is BPS, a subset of these are BPS
fluctuations.

The currents being gauged in these cosets are linear combinations of the
Cartan generators of the symmetry algebra. Therefore the “$m$-basis” for
vertex operators, in which the actions of these currents are diagonalized, is
the natural framework to use. In the IR AdS3 limit, we describe how these
operators are related to their global AdS${}_{3}\times\mathbb{S}^{3}$
counterparts.

We then compute a large set of correlators in the AdS3 limit. It is well known
that in worldsheet models of global AdS3, one can define an “$x$” variable
that corresponds to the local coordinate of the holographic CFT Kutasov:1999xu
. One of the main novelties of our approach is the identification of the
analogous $x$ variable in the coset models we study. This identification
requires some care due to the gauging. Indeed, the construction of
Kutasov:1999xu  breaks down, because the SL(2,$\mathbb{R}$) raising and
lowering operators do not commute with the BRST charge. A considerable amount
of interesting physics follows from this step. It leads, for instance, to the
combination of seemingly simple $m$-basis two-point functions into spacetime-
local $x$-basis correlators with highly non-trivial $x$-dependence.

Our first main result is of a family of HLLH correlators, for which we obtain
fully explicit expressions. In doing so, we show that these correlators assume
a remarkably simple structure when written in terms of a covering space
related only to the heavy states. From this observation, we obtain our second
main result: a closed-form expression for a set of HL worldsheet correlators
with an arbitrary number $n$ of massless insertions, in terms of a correlator
consisting of $n$ light insertions in global AdS${}_{3}\times\mathbb{S}^{3}$.
For $n=3$ this result can be made completely explicit, and we present a
particular example in full detail. This constitutes the first correlator in
the literature involving three light worldsheet vertices on a black hole
microstate background, dual to a heavy-light five-point function of the
holographic CFT.

A priori, our worldsheet correlators give predictions for correlators of the
dual holographic CFT at strong coupling. Generically, four-point correlators
are not protected across moduli space, however, a specific set of HLLH
correlators have been shown to precisely agree between supergravity and the
symmetric product orbifold CFT Galliani:2016cai . Similarly, the emission
spectrum and rate for the unitary analog of Hawking radiation from the JMaRT
solutions agrees between supergravity and symmetric product orbifold CFT
Chowdhury:2007jx ; Avery:2009tu ; Avery:2009xr ; Chakrabarty:2015foa . Thus it
is natural to investigate more generally which HL correlators are protected
(at large $N$) between worldsheet and symmetric product orbifold CFT, and
which are not.

We carry out this comparison for three sub-families of our worldsheet
correlators. Firstly, we compare various sets of HLLH correlators to the
symmetric product orbifold CFT, finding exact agreement in all cases for which
the orbifold CFT correlator is available in the literature. Importantly, this
matching holds at leading order in large $N$, but exactly in
$\alpha^{\prime}$. This comparison includes a substantial generalization of
the supergravity and holographic CFT correlators computed in Galliani:2016cai
. Our comparisons notably include an example in which the light operators in
the symmetric orbifold CFT are twist-two. In this case, and as shown recently
in Lima:2021wrz ; AlvesLima:2022elo , the Lunin-Mathur covering map used in
the symmetric orbifold computation is different to the one appearing in the
worldsheet computation, making the comparison highly non-trivial. Remarkably,
both results agree exactly in the large $N$ limit.

Secondly, we compute the five-point HLLLH symmetric orbifold CFT correlator
corresponding to the three-point worldsheet correlator mentioned above, and
also find exact agreement. While most of our main results were announced in
the short paper Bufalini:2022wyp , this five-point correlator is completely
new.

Finally, we compute the analogue of the Hawking radiation rate for the JMaRT
solutions. Once again, we find perfect agreement with the dual symmetric
product orbifold CFT, extending the supergravity and holographic CFT results
of Chowdhury:2007jx ; Avery:2009tu ; Avery:2009xr ; Chakrabarty:2015foa .

A likely explanation for this remarkable agreement is that the heavy states we
consider are quite special. Specifically, the heavy backgrounds are related to
the global AdS${}_{3}\times\mathbb{S}^{3}$ vacuum via orbifolding and
fractional spectral flow Giusto:2012yz ; Chakrabarty:2015foa . This fact also
underlies our general formula for the HL correlators with $n$ light
insertions. When $n>3$, we do not expect these HL correlators to be
generically protected across moduli space; we shall discuss this in detail in
due course.

The structure of the paper is as follows. In Section 2 we review the null-
gauged WZW models we study. We present the supergravity fields in the
fivebrane decoupling limit, take the AdS3 limit, and describe the dual heavy
states of the holographic CFT. In Section 3 we present the light operators we
are interested in. We review the chiral primaries of the symmetric product
orbifold CFT. We then describe how the corresponding operators are constructed
in the worldsheet theory for strings in AdS${}_{3}\times\mathbb{S}^{3}$ in the
RNS formalism, including spectrally flowed states. In Section 4 we construct a
large set of vertex operators of the worldsheet cosets we study, both in the
NS and in the R sectors. We then examine their AdS3 limit and relate these
vertices to those constructed in Sec. 3.

In Sections 5 and 6, we present our main results. We identify the “$x$”
variable dual to the local coordinate of the holographic CFT, and obtain an
extensive set of novel HLLH correlators, including massless insertions with
arbitrary spacetime weights and charges. The final results are presented in
Eqs. (208) and (220). We then compare a subset of these results to the
symmetric product orbifold CFT, finding exact agreement for all correlators
available in the literature. We present a closed formula for a large class of
worldsheet correlators with an arbitrary number of massless insertions, Eq.
(6.3). We compute a five-point correlator in the symmetric orbifold CFT and
find agreement with our general worldsheet formula. Finally, we compute the
amplitude describing the unitary analogue of Hawking radiation for the JMaRT
microstates. We discuss our results in Section 7.

##  2 The heavy background states

In this section we introduce the heavy backgrounds known as NS5-F1 circular
supertubes and spectral flows thereof Lunin:2004uu ; Giusto:2004id ;
Giusto:2004ip ; Jejjala:2005yu ; Giusto:2012yz ; Chakrabarty:2015foa ,
including the general NS5-F1-P JMaRT solutions and their BPS limits. We work
in the fivebrane decoupling limit, and review the worldsheet description of
tree-level string theory on these backgrounds in terms of null-gauged WZW
models Martinec:2017ztd ; Martinec:2018nco ; Bufalini:2021ndn . In the IR, the
backgrounds become asymptotically AdS${}_{3}\times\mathbb{S}^{3}$, and we
review the corresponding heavy states of the holographically dual CFT at the
symmetric orbifold point Chakrabarty:2015foa .

###  2.1 JMaRT backgrounds from the worldsheet

We begin by reviewing the family of coset CFT models that describe strings
probing the JMaRT backgrounds (and their BPS limits), introduced in
Martinec:2017ztd  and analyzed in Martinec:2018nco ; Bufalini:2021ndn . We
will mostly use the notation and conventions from Bufalini:2021ndn , to which
we refer the reader who is interested in more details. We work in units in
which $\alpha^{\prime}=1$.

The null-gauged WZW model relevant for the present work has the following
coset as a target space:

$\mathcal{G}/\mathcal{H}\;\times\mathbb{T}^{4}=\frac{\text{SL}(2,\mathbb{R})\times\text{SU}(2)\times\mathbb{R}_{t}\times\text{U}(1)_{y}}{\text{U}(1)_{L}\times\text{U}(1)_{R}}\;\times\mathbb{T}^{4}\,.$
(1)

To be precise, globally we work with the universal cover of
SL(2,$\mathbb{R}$), and we gauge $\mathbb{R}\times U(1)$. The line element and
NSNS three-form flux of the 10+2-dimensional “upstairs” model before gauging
are given in local coordinates by

$\displaystyle ds^{2}$
$\displaystyle=n_{5}\big{(}-\cosh^{2}\\!\rho\>\\!\,d\tau^{2}+d\rho^{2}+\sinh^{2}\\!\rho\>\\!\,d\sigma^{2}+d\theta^{2}+\cos^{2}\theta
d\psi^{2}+\sin^{2}\theta d\phi^{2}\big{)}-dt^{2}+dy^{2},$ $\displaystyle H$
$\displaystyle=n_{5}\big{(}\sinh 2\rho\;d\rho\wedge d\tau\wedge d\sigma+\sin
2\theta\;d\theta\wedge d\psi\wedge d\phi\big{)}.$ (2)

The Killing vectors associated to the group action being gauged are111Note
that some conventions differ from our letter Bufalini:2022wyp . In the latter,
the following changes must be performed to make contact with our current
notation: $s_{+}\mapsto l_{2},s_{-}\mapsto-r_{2},\mu\mapsto l_{3},k_{+}\mapsto
l_{4},k_{-}\mapsto r_{4}$.

$\displaystyle\begin{aligned}
\xi_{\scriptscriptstyle{\text{L}}}&=(\partial_{\tau}-\partial_{\sigma})-l_{2}(\partial_{\psi}-\partial_{\phi})+l_{3}\partial_{t}-l_{4}\partial_{y}\,,\cr\xi_{\scriptscriptstyle{\text{R}}}&=(\partial_{\tau}+\partial_{\sigma})+r_{2}(\partial_{\psi}+\partial_{\phi})+r_{3}\partial_{t}-r_{4}\partial_{y}\,.\end{aligned}$
(3)

We could be slightly more general and include similar parameters $l_{1}$,
$r_{1}$ in (3), however we have assumed these to be non-vanishing and have set
$l_{1}=r_{1}=1$ by a choice of normalization. The corresponding currents are

$\mathcal{J}=\mathsf{j}^{3}_{\text{sl}}+l_{2}\,\mathsf{j}^{3}_{\text{su}}+l_{3}\mathsf{P}^{t}_{\scriptscriptstyle{\text{L}}}+l_{4}\mathsf{P}^{y}_{\scriptscriptstyle{\text{L}}}\;,\quad\qquad\bar{\mathcal{J}}=\bar{\,\mathsf{j}}^{3}_{\text{sl}}+r_{2}\bar{\,\mathsf{j}}^{3}_{\text{su}}+r_{3}\mathsf{P}^{t}_{\scriptscriptstyle{\text{R}}}+r_{4}\mathsf{P}^{y}_{\scriptscriptstyle{\text{R}}}\,,$
(4)

where222Compared to Bufalini:2021ndn  we have implemented the change
$\theta\mapsto\frac{\pi}{2}-\theta,\phi\leftrightarrow-\psi$. This effectively
exchanges the sign of $r_{2}$.

$\displaystyle\begin{aligned}
&\,\mathsf{j}^{3}_{\text{sl}}=n_{5}\big{(}\cosh^{2}\\!\rho\>\\!\,\partial\tau+\sinh^{2}\\!\rho\>\\!\,\partial\sigma\big{)}\;,\quad\qquad\bar{\,\mathsf{j}}^{3}_{\text{sl}}=n_{5}\big{(}\cosh^{2}\\!\rho\>\\!\,\bar{\partial}\tau-\sinh^{2}\\!\rho\>\\!\,\bar{\partial}\sigma\big{)}\,,\\\
&\mathsf{j}^{3}_{\text{su}}=n_{5}\big{(}\cos^{2}\theta\,\partial\psi-\sin^{2}\theta\,\partial\phi\big{)}\;,\quad\qquad\quad\bar{\,\mathsf{j}}^{3}_{\text{su}}=-n_{5}\big{(}\cos^{2}\theta\,\bar{\partial}\psi+\sin^{2}\theta\,\bar{\partial}\phi\big{)}\,,\end{aligned}$
(5)

and

$\displaystyle\mathsf{P}^{t}_{L}=\partial t\
,\qquad\mathsf{P}^{t}_{R}=\bar{\partial}t\ ,\qquad\mathsf{P}^{y}_{L}=\partial
y\ ,\qquad\mathsf{P}^{y}_{R}=\bar{\partial}y\,.$ (6)

For the currents in Eq. (4) to be null, we impose the constraints

$n_{5}(1-l_{2}^{2})+l_{3}^{2}-l_{4}^{2}=0\;,\quad\qquad
n_{5}(1-r_{2}^{2})+r_{3}^{2}-r_{4}^{2}=0\,.$ (7)

Upon integrating out the gauge fields, the gauging procedure effectively adds
a term quadratic in the currents, resulting in an action of the schematic form

$\displaystyle
S_{\text{\tiny{WZW}}}+\frac{2}{\pi}\int\frac{\mathcal{J}\>\\!\bar{\mathcal{J}}}{\Sigma}\,d^{2}z\,,$
(8)

with

$\Sigma\;\equiv\;-\frac{1}{2}\xi_{1}^{i}G_{ij}\xi_{2}^{j}\;,$ (9)

where $G_{ij}$ is the metric in Eq. (2.1). One can then read off the resulting
line element and $B$-field of the gauged model. The change in the measure also
generates a non-trivial dilaton, which can be obtained by solving for the
vanishing of the appropriate worldsheet one-loop beta function, see
Bufalini:2021ndn .

The geometry obtained from the WZW model is free of horizons and closed
timelike curves (CTCs) if and only if Bufalini:2021ndn

$l_{3}\;=\;r_{3}\,.$ (10)

To obtain smooth geometries up to orbifold singularities or NS5 sources, we
further impose333There are also consistent models with $l_{2}=r_{2}=0$, which
we do not consider in this work.

$l_{2}={\mathsf{m}}+{\mathsf{n}}\in 2\mathbb{Z}+1\ ,\qquad
r_{2}={\mathsf{m}}-{\mathsf{n}}\in
2\mathbb{Z}+1\,,\qquad{\mathsf{m}},{\mathsf{n}}\in\mathbb{Z}\,,$ (11)

and

$l_{4}\,=\,-\Big{(}{\mathsf{k}}R_{y}-\frac{{\mathsf{p}}}{R_{y}}\Big{)}\,,\qquad
r_{4}\,=\,{\mathsf{k}}R_{y}+\frac{{\mathsf{p}}}{R_{y}}\;,\quad\qquad{\mathsf{k}},{\mathsf{p}}\in\mathbb{Z}\,.$
(12)

Combining these expression with the null constraints Eq. (7) leads to

$l_{3}\;=\;r_{3}\;=\;-\sqrt{{\mathsf{k}}^{2}R_{y}^{2}+\frac{{\mathsf{p}}^{2}}{R_{y}^{2}}+n_{5}\left({\mathsf{m}}^{2}+{\mathsf{n}}^{2}-1\right)}\,,$
(13)

and

${\mathsf{k}}\,{\mathsf{p}}=n_{5}\,{\mathsf{m}}\,{\mathsf{n}}\,,$ (14)

hence that only three of the four integers
${\mathsf{k}},{\mathsf{m}},{\mathsf{n}},{\mathsf{p}}$ are independent. The
very same conditions are necessary and sufficient for the consistency of the
spectrum of the worldsheet CFT Bufalini:2021ndn . For the AdS3 limit of these
backgrounds to be dual to pure states of the holographic CFT, there is an
additional requirement from momentum quantization in the
${\mathsf{k}}$-twisted sectors that we shall review in Section 2.3 (see e.g.
Chakrabarty:2015foa ),

$\frac{{\mathsf{m}}\>\\!{\mathsf{n}}}{{\mathsf{k}}}\,\in\,\mathbb{Z}\,.$ (15)

Without loss of generality, we work in the range of parameters
${\mathsf{k}}\geq 0,\,{\mathsf{m}}>{\mathsf{n}}\geq 0$.

###  2.2 Supergravity configurations and AdS3 limit

The family of coset CFTs we have just defined corresponds precisely to the
NS5-decoupled JMaRT configurations and their limits Bufalini:2021ndn . Using
the integer parametrization introduced above, the supergravity fields are
given by

$\displaystyle ds^{2}\,=$
$\displaystyle\,\;n_{5}(d\theta^{2}+d\rho^{2})+\frac{1}{\Sigma_{0}}\Bigg{[}-\left(\sinh^{2}\\!\rho\>\\!+({\mathsf{m}}^{2}-{\mathsf{n}}^{2})\cos^{2}\theta+1-{\mathsf{m}}^{2}-\frac{{\mathsf{p}}^{2}}{n_{5}R_{y}^{2}}\right)dt^{2}$
$\displaystyle+\left(\sinh^{2}\\!\rho\>\\!+({\mathsf{m}}^{2}-{\mathsf{n}}^{2})\cos^{2}\theta+{\mathsf{n}}^{2}+\frac{{\mathsf{p}}^{2}}{n_{5}R_{y}^{2}}\right)dy^{2}-2\>\\!\frac{{\mathsf{p}}}{n_{5}R_{y}}\;\\!\Delta\;\\!dtdy$
$\displaystyle+\left(n_{5}\sinh^{2}\\!\rho\>\\!+n_{5}{\mathsf{m}}^{2}+{\mathsf{k}}^{2}R_{y}^{2}\right)\>\\!\sin^{2}\theta\;\\!d\phi^{2}+\left(n_{5}\sinh^{2}\\!\rho\>\\!+n_{5}{\mathsf{n}}^{2}+{\mathsf{k}}^{2}R_{y}^{2}\right)\;\\!\cos^{2}\theta\>\\!d\psi^{2}$
$\displaystyle+2\left({\mathsf{m}}\>\\!\Delta\>\\!dt-\Big{(}{\mathsf{m}}\frac{{\mathsf{p}}}{R_{y}}+{\mathsf{n}}\>\\!{\mathsf{k}}R_{y}\Big{)}dy\right)\sin^{2}\theta\>\\!d\phi-2\left({\mathsf{n}}\>\\!\Delta\>\\!dt-\Big{(}{\mathsf{n}}\frac{{\mathsf{p}}}{R_{y}}+{\mathsf{m}}\>\\!{\mathsf{k}}R_{y}\Big{)}dy\right)\cos^{2}\theta\;\\!d\psi\Bigg{]},$
$\displaystyle B\>=$
$\displaystyle\,\;\frac{1}{\Sigma_{0}}\Bigg{[}-\frac{{\mathsf{k}}R_{y}}{n_{5}}\;\\!\Delta\>\\!dt\wedge
dy+\left(n_{5}\sinh^{2}\\!\rho\>\\!+n_{5}\,{\mathsf{m}}^{2}+{\mathsf{k}}^{2}R_{y}^{2}\right)\,\cos^{2}\\!\theta\,d\phi\wedge
d\psi$
$\displaystyle+\left({\mathsf{m}}\>\\!\Delta\>\\!dt-\Big{(}{\mathsf{m}}\frac{{\mathsf{p}}}{R_{y}}+{\mathsf{n}}\>\\!{\mathsf{k}}R_{y}\Big{)}dy\right)\wedge\cos^{2}\theta\>\\!d\psi-\left({\mathsf{n}}\>\\!\Delta\>\\!dt-\Big{(}{\mathsf{n}}\frac{{\mathsf{p}}}{R_{y}}+{\mathsf{m}}\>\\!{\mathsf{k}}R_{y}\Big{)}dy\right)\wedge\sin^{2}\theta\>\\!d\phi\Bigg{]}\>\\!,$
(16)

where $R_{y}$ is the asymptotic proper radius of the $\mathbb{S}^{1}_{y}$
circle, and where

$\displaystyle\Sigma_{0}$
$\displaystyle=\sinh^{2}\rho+({\mathsf{m}}^{2}-{\mathsf{n}}^{2})\cos^{2}\theta+{\mathsf{n}}^{2}+\frac{{\mathsf{k}}^{2}R_{y}^{2}}{n_{5}}\,,$
(17) $\displaystyle\Delta$
$\displaystyle=\sqrt{n_{5}({\mathsf{m}}^{2}+{\mathsf{n}}^{2}-1)+{\mathsf{k}}^{2}R_{y}^{2}+\frac{{\mathsf{p}}^{2}}{R_{y}^{2}}}\,.$
(18)

We also note the relations between the supergravity charges and integer charge
quanta,

$Q_{1}=n_{1}\frac{g_{s}^{2}}{V_{4}}\;,\quad\qquad
Q_{p}=\frac{n_{p}}{R_{y}^{2}}\frac{g_{s}^{2}}{V_{4}}\;,$ (19)

where $(2\pi)^{4}V_{4}$ is the coordinate volume of the $\mathbb{T}^{4}$. Note
that the three-charge NS5-decoupled JMaRT solutions are specified by the
integers $n_{5},{\mathsf{k}},{\mathsf{m}},{\mathsf{n}}$, the modulus $R_{y}$,
and the charge $Q_{1}$ appearing in the dilaton.

One can take a further IR AdS3 decoupling limit by taking $R_{y}$ to be large,
keeping fixed the charge $Q_{1}$ and the rescaled energy $ER_{y}$ and momentum
$P_{y}R_{y}$. We define rescaled coordinates

$\tilde{t}=\frac{t}{R_{y}}\;,\quad\qquad\tilde{y}=\frac{y}{R_{y}}\,.$ (20)

and perform the large-$R_{y}$ expansion at the level of the coefficients in
Eqs. (11)–(14), such that the leading terms in $l_{3},r_{3},l_{4}$ and $r_{4}$
become independent of ${\mathsf{p}}$. However, the product
${\mathsf{k}}{\mathsf{p}}$ is kept fixed and the relation Eq. (14) still
holds, defining the momentum per strand for the holographic CFT
Chakrabarty:2015foa . Order-by-order in $1/R_{y}$, the coefficients still
satisfy the null conditions Eq. (7). The six-dimensional fields in (2.2) then
become

$\displaystyle ds^{2}$
$\displaystyle=n_{5}\bigg{[}-\frac{1}{{\mathsf{k}}^{2}}\cosh^{2}\rho\,d\tilde{t}\;\\!^{2}+\frac{1}{{\mathsf{k}}^{2}}\sinh^{2}\rho\,d\tilde{y}^{2}+d\rho^{2}$
(21)
$\displaystyle\qquad~{}+d\theta^{2}+\sin^{2}\theta\left(d\phi-\frac{{\mathsf{n}}}{{\mathsf{k}}}d\tilde{t}+\frac{{\mathsf{m}}}{{\mathsf{k}}}d\tilde{y}\right)^{2}+\cos^{2}\theta\left(d\psi+\frac{{\mathsf{m}}}{{\mathsf{k}}}d\tilde{t}-\frac{{\mathsf{n}}}{{\mathsf{k}}}d\tilde{y}\right)^{2}\bigg{]},$
$\displaystyle B$
$\displaystyle=n_{5}\Bigg{[}\frac{\sinh^{2}\\!\rho\>\\!+({\mathsf{m}}^{2}-{\mathsf{n}}^{2})\cos^{2}\theta}{{\mathsf{k}}^{2}}\,d\tilde{t}\wedge
d\tilde{y}+\,\cos^{2}\theta\,d\phi\wedge d\psi$ (22) $\displaystyle\hskip
42.67912pt+\sin^{2}\theta\left(-\frac{{\mathsf{n}}}{{\mathsf{k}}}\,d\tilde{t}+\frac{{\mathsf{m}}}{{\mathsf{k}}}\,d\tilde{y}\right)\wedge
d\phi+\cos^{2}\theta\left(\frac{{\mathsf{m}}}{{\mathsf{k}}}\,d\tilde{t}-\frac{{\mathsf{n}}}{{\mathsf{k}}}\,d\tilde{y}\right)\wedge
d\psi\Bigg{]}\,,$ $\displaystyle e^{2\Phi}$
$\displaystyle=\frac{n_{5}}{Q_{1}}=\frac{Q_{5}}{Q_{1}}\,,$ (23)

where a trivial gauge transformation has been performed on the $B$-field.

These solutions are related by a large coordinate transformation to
$\mathbb{Z}_{\mathsf{k}}$ orbifolds of global AdS${}_{3}\times\mathbb{S}^{3}$,
which are the decoupling limits of the supertube solutions of
Balasubramanian:2000rt ; Maldacena:2000dr . This large coordinate
transformation is known as spacetime spectral flow, and takes the form

$\tilde{\psi}\;=\;\psi+\frac{{\mathsf{m}}}{{\mathsf{k}}}\,\tilde{t}-\frac{{\mathsf{n}}}{{\mathsf{k}}}\,\tilde{y}\;,\quad\qquad\tilde{\phi}\;=\;\phi-\frac{{\mathsf{n}}}{{\mathsf{k}}}\,\tilde{t}+\frac{{\mathsf{m}}}{{\mathsf{k}}}\,\tilde{y}\,.$
(24)

For the special case ${\mathsf{m}}={\mathsf{n}}=0$, spectral flow is not
relevant and the solutions are already $\mathbb{Z}_{\mathsf{k}}$ orbifolds of
global AdS${}_{3}\times\mathbb{S}^{3}$. When ${\mathsf{m}},{\mathsf{n}}$ are
not both zero, one typically works in the range ${\mathsf{m}}>{\mathsf{n}}\geq
0$ without loss of generality. For ${\mathsf{m}}=1,{\mathsf{n}}=0$, the
solutions (21)–(23), are the AdS decoupling limits of the two-charge circular
supertube solutions of Balasubramanian:2000rt ; Maldacena:2000dr . For
${\mathsf{m}}={\mathsf{n}}+1$ with ${\mathsf{n}}>0$, the solutions the AdS
limit of the supersymmetric spectral flowed solutions of Lunin:2004uu ;
Giusto:2004id ; Giusto:2004ip ; Giusto:2012yz , and for other values of
${\mathsf{m}},{\mathsf{n}}$ one obtains the AdS limit of the non-
supersymmetric JMaRT solutions Jejjala:2005yu . For ${\mathsf{k}}=1$ the
solutions are smooth; for ${\mathsf{k}}>1$ the solutions have orbifold
singularities near $\tilde{y}=0$, the details of which depend on the common
divisors of ${\mathsf{m}},{\mathsf{n}},{\mathsf{k}}$ Jejjala:2005yu ;
Chakrabarty:2015foa ; Martinec:2018nco .

###  2.3 Holographic description and boundary spectral flow

As mentioned in the Introduction, the holographic CFT that corresponds to the
AdS3 limit of the system in which we work is an $\mathcal{N}=(4,4)$ symmetric
product orbifold CFT with target space
$\left(\mathbb{T}^{4}\right)^{N}/S_{N}$, where $N=n_{1}n_{5}$. To make the
presentation self-contained, we now review some aspects of this theory.

Recall that we work in Type IIB compactified on
$\mathbb{S}^{1}\times\mathbb{T}^{4}$, with $n_{5}$ NS5 branes wrapped on
$\mathbb{S}^{1}\times\mathbb{T}^{4}$, $n_{1}$ units of F1 winding on
$\mathbb{S}^{1}$, and $n_{P}$ units of momentum charge along $\mathbb{S}^{1}$.
The moduli space is $20$-dimensional and the symmetric product orbifold CFT
lies at a particular locus of this moduli space Larsen:1999uk , see also
Bena:2017xbt . The configuration breaks the $\text{SO}(1,9)$ Lorentz group to
$\text{SO}(1,1)\times\text{SO}(4)_{E}\times\text{U}(1)^{4}$, where the
external R-symmetry
$\text{SO}(4)_{E}\simeq\text{SU}(2)_{L}\times\text{SU}(2)_{R}$ corresponds to
rotations in the spatial $\mathbb{R}^{4}$ transverse to the branes (in the IR
limit, rotations of the $\mathbb{S}^{3}$). It is customary to introduce an
approximate internal
$\text{SO}(4)_{I}\simeq\text{SU}(2)_{1}\times\text{SU}(2)_{2}$, which is
broken to $\text{U}(1)^{4}$ by the compactification, but which is useful for
classifying states and organizing fields David:2002wn ; Avery:2010qw .

In the symmetric product orbifold theory, for each copy of $\mathbb{T}^{4}$
there are four free bosons, together with their left and right-moving
fermionic superpartners. Indices $\alpha,\dot{\alpha},A,\dot{A}$ correspond
respectively to
$\text{SU}(2)_{L}\>\\!,\text{SU}(2)_{R}\>\\!,\text{SU}(2)_{1}\>\\!,\text{SU}(2)_{2}$.
The free fields are denoted as (we use the conventions of Avery:2009tu )

$X_{A\dot{A}\,(r)}(z,\bar{z})\;,\quad\qquad\psi^{\alpha\dot{A}}_{(r)}(z)\;,\quad\qquad\bar{\psi}^{\dot{\alpha}\dot{A}}_{(r)}(\bar{z})\,,$
(25)

where the subscript $(r)$ denotes the $r$-th copy of the seed $\mathbb{T}^{4}$
theory. Omitting this copy subscript and focusing on the holomorphic sector,
the energy-momentum tensor $\mathbb{T}(z)$, the supercurrents
$\mathbb{G}^{\alpha A}(z)$ and the SU$(2)_{L}$ currents $\mathbb{J}^{a}$
generate the small (4,4) supersymmetric algebra. We denote holographic CFT
conformal weights by $h$ and R-symmetry quantum numbers by
$(\mathbbl{j},m^{\prime})$ and $(\bar{\mathbbl{j}},\bar{m}^{\prime})$,
respectively.

The heavy states we are interested in are obtained by fractional spectral flow
Chakrabarty:2015foa , see also Martinec:2001cf ; Giusto:2012yz . We start from
the NSNS vacuum in the ${\mathsf{k}}$-twisted sector,
$\ket{0_{{\mathsf{k}}}}_{\text{NS}}$. In order to have a gauge invariant state
we consider $n_{1}n_{5}/{\mathsf{k}}$ identical strands of length
${\mathsf{k}}$. Its dimension is

$h=\bar{h}=\frac{c}{24}\left[1-\frac{1}{{\mathsf{k}}^{2}}\right],$ (26)

where the central charge $c=6N$. The R-charges of this state are zero. Because
all strands are of length ${\mathsf{k}}$ there is an enhancement of the usual
spectral flow, such that one can perform spectral flow with fractional
parameters,

$\alpha=\frac{{\mathsf{m}}+{\mathsf{n}}}{{\mathsf{k}}}=\frac{2s+1}{{\mathsf{k}}}\;,\quad\qquad\bar{\alpha}=\frac{{\mathsf{m}}-{\mathsf{n}}}{{\mathsf{k}}}=\frac{2\bar{s}+1}{{\mathsf{k}}}\,,$
(27)

where $s,\bar{s}\in\mathbb{Z}$ and the range ${\mathsf{m}}>{\mathsf{n}}\geq 0$
is the range $s\geq\bar{s}\geq 0$. This generates a new state with quantum
numbers

$\displaystyle h$
$\displaystyle=\frac{c}{24}\left[1-\frac{1}{{\mathsf{k}}^{2}}+\alpha^{2}\right]\;,\quad\qquad
m^{\prime}=\frac{\alpha c}{12}\,,$ $\displaystyle\bar{h}$
$\displaystyle=\frac{c}{24}\left[1-\frac{1}{{\mathsf{k}}^{2}}+\bar{\alpha}^{2}\right]\;,\quad\qquad\bar{m}^{\prime}=\frac{\bar{\alpha}c}{12}\,.$
(28)

These states are “heavy” in the sense that their conformal dimensions and
charges scale linearly with the large central charge $c=6N$. In the dual
theory they correspond to the classical configurations presented in Eqs.
(21)–(23). By constrast, the “light” perturbative string states probing these
backgrounds will correspond to holographic CFT states with conformal
dimensions that are independent of $c$.

We conclude this section by summarizing how the bulk, the worldsheet, and the
dual CFT encode in different ways the same information about the heavy state.

  * •

In the worldsheet model, the heavy state defines the theory itself by means of
the gauging parameters $l_{i},r_{i}$ appearing in Eq. (4) and the radius
$R_{y}$.

  * •

In supergravity, the information about the heavy state is contained in the
integers ${\mathsf{m}},{\mathsf{n}},{\mathsf{k}}$ parameterizing the fields in
Eq. (2.2), together with $R_{y}$, which gets scaled out in the AdS limit.

  * •

In the symmetric orbifold CFT, the information about the heavy states is
contained in the spectral flow parameters $\alpha$ and $\bar{\alpha}$, and the
twist index ${\mathsf{k}}$.

The map between the three descriptions, in the AdS limit, is then

$-R_{y}\,\frac{l_{2}}{l_{4}}~{}\xrightarrow{R_{y}\to\infty}~{}\frac{{\mathsf{m}}+{\mathsf{n}}}{{\mathsf{k}}}\,=\,\alpha\;,\quad\qquad
R_{y}\,\frac{r_{2}}{r_{4}}~{}\xrightarrow{R_{y}\to\infty}~{}\frac{{\mathsf{m}}-{\mathsf{n}}}{{\mathsf{k}}}\,=\,\bar{\alpha}\;.$
(29)

##  3 The light probe states

We now introduce the light states that we will study. These correspond to
chiral primary operators of the boundary theory. We focus first on
fluctuations around the global AdS${}_{3}\times\mathbb{S}^{3}$ vacuum. We
review the dictionary between holographic CFT operators and their counterparts
in the worldsheet theory, following Dabholkar:2007ey ; Giribet:2007wp .

###  3.1 Chiral primaries in the D1D5 CFT

We first briefly review the construction of chiral primary operators in the
symmetric orbifold CFT Lunin:2001pw . We focus primarily on the holomorphic
sector in the following; the antiholomorphic sector is entirely analogous. In
the untwisted sector, on each copy of the seed $\mathbb{T}^{4}$ theory, the
chiral primary operators correspond to the states (suppressing the copy $(r)$
label)

$|0_{\mathrm{NS}}\rangle\;,\quad\qquad\psi_{-\frac{1}{2}}^{+\dot{A}}|0_{\mathrm{NS}}\rangle\;,\quad\qquad\mathbb{J}_{-1}^{+}|0_{\mathrm{NS}}\rangle\,=\,\psi_{-\frac{1}{2}}^{+\dot{1}}\psi_{-\frac{1}{2}}^{+\dot{2}}|0_{\mathrm{NS}}\rangle\,,$
(30)

where $|0_{\mathrm{NS}}\rangle$ is the NS vacuum. The corresponding weights
and R-charges are $h=m^{\prime}=0,\frac{1}{2},1$, respectively. Physical
configurations in the orbifold theory are obtained by symmetrizing the states
in (30) over the different copies of the seed theory.

By including the antiholomorphic sector we can obtain, for instance, the
dimension $(\frac{1}{2},\frac{1}{2})$ operator (see e.g. Moscato:2017usq )

$O^{++}=\sum_{r=1}^{N}O^{++}_{(r)}=\frac{-i}{\sqrt{2}}\sum_{r=1}^{N}\psi^{+\dot{A}}_{(r)}\varepsilon_{\dot{A}\dot{B}}\bar{\psi}^{+\dot{B}}_{(r)}\;,\quad\qquad(O^{++})^{\dagger}=O^{--}\,.$
(31)

We will use this operator in an explicit example later in the paper.

In order to construct more general chiral primaries one needs to consider the
twisted sectors of the theory. Consider the ‘bare’ twist operators
$\sigma_{n}$, defined on the cylinder, that impose the following boundary
conditions corresponding to a single-cycle permutation,

$\displaystyle\begin{aligned} X_{(1)}\to X_{(2)}\to\cdots\to X_{(n)}\to
X_{(1)}\,,\\\
\psi_{(1)}\to\psi_{(2)}\to\cdots\to\psi_{(n)}\to-\psi_{(1)}\,,\end{aligned}$
(32)

and likewise for the antiholomorphic fermions. The bare twist operators are
defined to be the lowest-dimension twist operators that impose the above
boundary conditions; they have dimension
$h=\bar{h}=\frac{1}{4}\left(n-\frac{1}{n}\right)$ and zero R-charge. Chiral
operators are obtained by exciting the bare twist operators operators to add
R-charge. The lowest-dimension chiral operators have
$h=m^{\prime}=\frac{n-1}{2}$. For $n$ odd, these operators are obtained by
acting with modes of the SU(2) currents, which are bilinears in the free
fermions. Due to the twist operator, the SU(2) currents are fractional-moded
in units of $1/n$. The relation between these modes and those of free fermions
on the $n$ copies of the seed theory can be found in Lunin:2001pw . To
construct the chiral operators, one acts with the currents
$\mathbb{J}^{+}_{-l/n}$ for which $l$ is odd and $l<n$,

$n~{}\mathrm{odd}:\qquad\quad\sigma_{n}^{-}\,=\,\prod_{p=1}^{(n-1)/2}\mathbb{J}^{+}_{-\frac{2p-1}{n}}{\sigma}_{n}~{}=~{}\mathbb{J}^{+}_{-\frac{n-2}{n}}\cdots\mathbb{J}^{+}_{-\frac{3}{n}}\mathbb{J}^{+}_{-\frac{1}{n}}{\sigma}_{n}\,.$
(33)

For $n$ even, one first acts with a spin field $S^{+}_{n}$, which has weight
$\frac{1}{4n}$ and charge $\frac{1}{2}$, putting the fermions into the Ramond
sector (i.e. their boundary conditions are similar to Eq. (32) but with the
final sign being $+\psi_{(1)}$). One then acts with the currents
$\mathbb{J}^{+}_{-l/n}$ for which $l$ is even and $l<n$,

$n~{}\mathrm{even}:\qquad\quad\sigma_{n}^{-}\,=\,\prod_{p=1}^{(n-2)/2}\mathbb{J}^{+}_{-\frac{2p}{n}}S^{+}_{n}{\sigma}_{n}~{}=~{}\mathbb{J}^{+}_{-\frac{n-2}{n}}\cdots\mathbb{J}^{+}_{-\frac{4}{n}}\mathbb{J}^{+}_{-\frac{2}{n}}S^{+}_{n}{\sigma}_{n}\,.$
(34)

As in the untwisted case, for both odd and even $n$ we can act with
$\psi_{-\frac{1}{2}}^{+\dot{A}}\equiv\sum\limits_{r=1}^{n}\psi_{-\frac{1}{2}(r)}^{+\dot{A}}$
to obtain a chiral operator $\psi_{-\frac{1}{2}}^{+\dot{A}}\sigma_{n}^{-}$
which has $h=m^{\prime}=\frac{n}{2}$. Similarly we can act with
$\mathbb{J}_{-1}^{+}$ to obtain a chiral operator
$\mathbb{J}^{+}_{-1}\sigma_{n}^{-}$ which has $h=m^{\prime}=\frac{n+1}{2}$.
Together with the analogous antiholomorphic operators, this exhausts the
single-cycle chiral operators. Indeed, by making use of anti-commutators of
the supercurrent modes $\mathbb{G}_{-m/n}^{\pm A}$ in the corresponding
twisted sectors, one can show that chiral weights are bounded by Avery:2010qw

$\frac{n-1}{2}\,\leq\,h\,\leq\,\frac{n+1}{2}\,.$ (35)

In the twisted sectors, it is often convenient to work in a basis that
diagonalizes the twisted boundary conditions. We shall make use of this basis
in Section 6.4. One defines

$\tilde{\psi}_{\rho}^{\alpha\dot{A}}\,=\,\frac{1}{\sqrt{n}}\sum_{r=1}^{n}e^{\alpha\frac{2\pi
ir\rho}{n}}\psi_{(r)}^{\alpha\dot{A}}\;,\quad\qquad\rho=0,\dots,n-1\,,$ (36)

where $\alpha=\pm$ should not be confused with the spectral flow parameter in
Eq. (27). These are mutually orthogonal, and diagonalize the twisted boundary
conditions as

$\tilde{\psi}_{\rho}^{\alpha\dot{A}}(e^{2\pi i}z)\,=\,e^{-\alpha\frac{2\pi
i\rho}{n}}\tilde{\psi}_{\rho}^{\alpha\dot{A}}(z)\,.$ (37)

These fermions can be bosonized to construct an explicit expression for the
spin fields mentioned above. Note that the fields
$\tilde{\psi}_{\rho=0}^{\alpha\dot{A}}$ are invariant under the twisting. For
further discussion, see Dabholkar:2007ey .

We now combine the above holomorphic construction with its antiholomorphic
counterpart and define the complete list of scalar left-right chiral primaries
we will be interested in:

$O_{n}^{--}=\sigma_{n}^{--}\;,\quad\qquad
O_{n}^{\dot{A}\dot{B}}=\tilde{\psi}_{\\!\\!\;\rho=0}^{+\dot{A}}\;\\!\bar{\tilde{\psi}}_{\\!\\!\;\rho=0}^{+\dot{B}}\;\\!\sigma_{n}^{--}\;,\quad\qquad
O_{n}^{++}=\tilde{\psi}_{\\!\\!\;\rho=0}^{+\dot{1}}\;\\!\tilde{\psi}_{\rho=0}^{+\dot{2}}\;\\!\bar{\tilde{\psi}}_{\\!\\!\;\rho=0}^{+\dot{1}}\;\\!\bar{\tilde{\psi}}_{\\!\\!\;\rho=0}^{+\dot{2}}\;\\!\sigma_{n}^{--}\,,$
(38)

where $\sigma_{n}^{--}$ is defined similarly to Eqs. (33), (34) but now also
with the same construction in the antiholomorphic sector. The operators in
(38) are normalized such that they have unit two-point functions.

For later reference, we note that in each case the respective weights and
twist numbers can be written in terms of $j=\frac{n+1}{2}$ as

$h\left[O_{n}^{--}\right]=j-1\;,\quad\qquad
h\left[O_{n}^{\dot{A}\dot{B}}\right]=j-\frac{1}{2}\;,\quad\qquad
h\left[O_{n}^{++}\right]=j\,.$ (39)

An analogous list of anti-chiral primaries (which have $h=-m^{\prime}$) is
obtained by acting on the bare twist fields with current and fermion modes
with opposite charge, i.e. $\mathbb{J}_{-l/n}^{-}$ and $\psi^{-\dot{A}}$. As
we will shortly review, and up to a shift related to spectral flow, this $j$
will be identified with the principal quantum number of the bosonic (global)
SL(2,$\mathbb{R}$) algebra of the worldsheet theory, to which we now turn.

###  3.2 Superstring theory on
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$

We now review the basics of superstring theory on
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$ using the RNS formalism
with BRST quantization. We first discuss the bosonic SL($2,\mathbb{R}$) and
SU$(2)$ WZW models and then present their supersymmetric counterparts. We
present the current algebra and review the spectrum, including states arising
from worldsheet spectral flow.

####  3.2.1 Bosonic WZW model for SL(2,$\mathbb{R}$)

The SL(2,$\mathbb{R}$) WZW model was studied in detail in Maldacena:2000hw ;
Maldacena:2000kv ; Maldacena:2001km . In what follows we will mostly follow
the notation of Dabholkar:2007ey ; Giribet:2007wp ; Dei:2021xgh , and normal
ordering will be implicitly assumed. The holomorphic SL(2,$\mathbb{R}$)
currents will be denoted $j^{a}(z)$. They satisfy the OPEs

$j^{a}(z)j^{b}(w)\,\sim\,\frac{k}{2}\frac{\eta^{ab}}{(z-w)^{2}}+\frac{f^{ab}_{\phantom{ab}c}\;\\!j^{c}(w)}{z-w}\,,$
(40)

where $k$ is the level of the affine algebra, and where

$-2\eta^{33}=\eta^{+-}=2\;,\quad\qquad f^{+-}_{\phantom{+-}3}=-2\,,\quad
f^{3+}_{\phantom{3+}+}=-f^{3-}_{\phantom{3-}-}=1\,.$ (41)

The holomorphic stress tensor and the central charge follow from the Sugawara
construction, and are given by (likewise for the antiholomorphic sector)

$T_{\mathrm{sl}}(z)=\frac{1}{k-2}\left[-j^{3}(z)j^{3}(z)+\frac{1}{2}\,j^{+}(z)j^{-}(z)+\frac{1}{2}\,j^{-}(z)j^{+}(z)\right]\,,\qquad
c_{\mathrm{sl}}=\frac{3k}{k-2}\,.$ (42)

We denote bosonic SL$(2,\mathbb{R})$ primary vertex operators by
$V_{j,m,\bar{m}}(z,\bar{z})$. Their zero-mode wavefunctions do not factorize
between holomorphic and antiholomorphic sectors, however as is often done we
shall work primarily with the holomorphic sector, and suppress the $\bar{m}$
and $\bar{z}$ dependence. The relevant representations of the holomorphic
zero-mode algebra are as follows. The principal series discrete
representations of lowest (highest) weight are spanned by

${\cal{D}}_{j}^{\pm}=\\{|j,m\rangle\ ,\ m=\pm j,\pm j\pm 1,\pm j\pm
2,\cdots\\}\,,$ (43)

respectively, where $j_{0}^{3}|j,m\rangle=m|j,m\rangle$. These are unitary
representations for any positive real $j$, and one is the charge conjugate of
the other (we will restrict the range of $j$ momentarily). There are also the
principal continuous series representations, spanned by

${\cal{C}}_{j}^{\hat{\alpha}}=\\{|j,\hat{\alpha},m\rangle\ ,\
0\leq\hat{\alpha}<1\ ,\ j=\frac{1}{2}+is\ ,\ s\in\mathbb{R}\ ,\
m=\hat{\alpha},\hat{\alpha}\pm 1,\hat{\alpha}\pm 2,\cdots\\}\,.$ (44)

The particular case $\hat{\alpha}=1/2=j$ is actually reducible. It was shown
in Maldacena:2000hw  that the spectrum of the model is built out of continuous
and lowest weight representations with

$\frac{1}{2}<j<\frac{k-1}{2}\,,$ (45)

together with their spectrally flowed images, to be introduced below. The
allowed range (45) follows from $L^{2}$ normalization conditions, no-ghost
theorems and spectral flow considerations.

Before considering worldsheet spectral flow (we refer to this as the
“unflowed” sector), the action of the currents on the primary states is given
by

$\displaystyle j^{3}_{0}|j,m\rangle$ $\displaystyle=$ $\displaystyle
m|j,m\rangle\;,$ (46a) $\displaystyle j^{\pm}_{0}|j,m\rangle$ $\displaystyle=$
$\displaystyle\begin{cases}\,(m\mp(j-1))|j,m\pm
1\rangle&\text{if}~{}~{}m\neq\mp j\\\ ~{}0&\text{if}~{}~{}m=\mp
j\;,\end{cases}$ (46b) $\displaystyle j^{a}_{n}|j,m\rangle$ $\displaystyle=$
$\displaystyle 0\ \qquad\forall n>0\,.$ (46c)

These vertex operators can be obtained from those of the Euclidean counterpart
of the model, namely the $H_{3}^{+}$ WZW model Teschner:1997ft ;
Teschner:1997fv ; Teschner:1999ug  (see also McElgin:2015eho ), as follows.
One introduces a set of operators depending on a complex label $x$, written as
$V_{j}(x|z)$, and having conformal weight

$\Delta\,=\,-\frac{j(j-1)}{k-2}\,.$ (47)

The action of the currents on $V_{j}(x,z)$ is given by

$j^{a}(z)V_{j}(x,w)\,\sim\,\frac{D_{j}^{a}V_{j}(x,w)}{(z-w)}\,,$ (48)

where

$D_{j}^{+}=\partial_{x}\;,\quad\qquad D_{j}^{3}=x\partial_{x}+j\;,\quad\qquad
D_{j}^{-}=x^{2}\partial_{x}+2jx\,.$ (49)

The two-point function is given by Teschner:1999ug

$\langle
V_{j_{1}}(x_{1},z_{1})V_{j_{2}}(x_{2},z_{2})\rangle\,=\,\frac{1}{|z_{12}|^{4\Delta_{1}}}\left[\delta^{2}(x_{1}-x_{2})\delta(j_{1}+j_{2}-1)+\frac{B(j_{1})}{|x_{12}|^{4j_{1}}}\delta(j_{1}-j_{2})\right],$
(50)

with

$B(j)=\frac{2j-1}{\pi}\frac{\Gamma[1-b^{2}(2j-1)]}{\Gamma[1+b^{2}(2j-1)]}\,\nu^{1-2j}\;,\quad\qquad\nu=\frac{\Gamma[1-b^{2}]}{\Gamma[1+b^{2}]}\
,\ b^{2}=(k-2)^{-1}\,.$ (51)

The operators $V_{j,m}(z)$ are related to $V_{j}(x|z)$ by means of the
following Mellin-like transform:

$V_{j,m}(z)=\int_{\mathbb{C}}d^{2}x\,x^{j-m-1}\bar{x}^{j-\bar{m}-1}V_{j}(x,z)\,.$
(52)

In the Euclidean $H_{3}^{+}$ model, $j$ takes values $j=1/2+is$. To obtain the
unflowed $V_{j,m}$ for Lorentzian AdS3, one assumes a well-defined
analytically continuation to real values of $j$. This procedure was discussed
in Maldacena:2001km , which identified the physical origin of the different
divergences arising in correlation functions. For related work, see
Harlow:2011ny . The two-point functions in the $m$-basis then follow from
(50), (52). Using the shorthand $V_{i}\equiv V_{j_{i},m_{i}}$, one finds

$\langle
V_{1}V_{2}\rangle\,=\,\frac{\delta^{2}(m_{1}+m_{2})}{|z_{12}|^{4\Delta_{1}}}\left[\delta(j_{1}+j_{2}-1)+\delta(j_{1}-j_{2})\frac{\pi
B(j_{1})}{\gamma(2j_{1})}\frac{\gamma(j_{1}+m_{1})}{\gamma(1-j_{1}+m_{1})}\right],$
(53)

where $\gamma(x)=\Gamma(x)/\Gamma(1-\bar{x})$, and where $\delta^{2}(m)$ is a
Dirac delta in $m+\bar{m}$ times a Kroenecker delta in $m-\bar{m}$.

At first sight, the complex variable $x$ may appear simply as an
SL(2,$\mathbb{R}$) version of the isospin variables defined for SU(2) in
Zamolodchikov:1986bd . However, given that the integrated zero modes of the
currents realize the spacetime Virasoro modes $L_{0}$ and $L_{\pm 1}$, and by
examining the expressions of the associated differential operators (49), one
is led to interpret $x$ as the local coordinate on the boundary theory
Giveon:1998ns . According to (50), in the bosonic theory a $z$-integrated
vertex operator $V_{j}(x)$ is identified with a local operator on the boundary
theory with weight $j$. Conversely, the corresponding boundary modes are given
by the $m$-basis operators. Indeed, for states in the discrete sector, the
transform in Eq. (52) can be inverted, giving

$V_{j}(x,z)\;=\sum_{m=j+n,\,n\in\mathbb{N}_{0}}x^{m-j}\>\\!\bar{x}^{\bar{m}-j}\,V_{j,m\bar{m}}(z)\,.$
(54)

The vertex $V_{j}(x,z)$ is realized via Eq. (54) as $V_{j,j}(z)$ translated
from the origin to $x$. Poles in the integrand of (52) coming from the
expansion around $x=0$ ($x=\infty$) are associated to states in the ${\cal
D}_{j}^{+}$ (${\cal D}_{j}^{-}$) representations Eberhardt:2019ywk ;
Dei:2021xgh .

Spectral flow automorphisms of the current algebra (40) are defined as

$j^{\pm}(z)\to\tilde{j}^{\pm}(z)=z^{\pm w}j^{\pm}(z)\;,\quad\qquad
j^{3}(z)\to\tilde{j}^{3}(z)=j^{3}(z)-\frac{k\,\omega}{2}\frac{1}{z}\,,$ (55)

where the so-called spectral flow charge $\omega$ is an integer. Analogous
formulas hold for the antiholomorphic sector. We work with the universal cover
of SL$(2,\mathbb{R})$, which imposes that the holomorphic and antiholomorphic
spectral flow parameters must be equal, $\bar{\omega}=\omega$. The action of
(55) on the above representations defines in general inequivalent
representations that must be considered in order to generate a consistent
spectrum. This holds up to the so-called series identifications due to the
fact that the affine modules $\hat{{\cal{D}}}_{j}^{+,w}$ and
$\hat{{\cal{D}}}_{k/2-j}^{-,w+1}$ are isomorphic. Thus, as mentioned above,
the discrete series spectrum is constructed solely upon lowest weight
representations with $j$ restricted to the range (45).

At the level of vertex operators and for $w>0$, the spectral flow operation
introduced in (55) defines the so-called flowed primaries, whose OPEs with the
currents take the form

$\displaystyle j^{+}(z)V_{j,m}^{\omega}(w)$ $\displaystyle=$
$\displaystyle\frac{(m+1-j)V_{j,m+1}^{\omega}(w)}{(z-w)^{\omega+1}}+\sum_{n=1}^{\omega}\frac{(j^{+}_{n-1}V_{j,m}^{\omega})(w)}{(z-w)^{n}}+\dots\,,$
(56a) $\displaystyle j^{3}(z)V_{j,m}^{\omega}(w)$ $\displaystyle=$
$\displaystyle\frac{\left(m+\frac{k}{2}\omega\right)V_{j,m}^{\omega}(w)}{(z-w)}+\dots\,,$
(56b) $\displaystyle j^{-}(z)V_{j,m}^{\omega}(w)$ $\displaystyle=$
$\displaystyle(z-w)^{\omega-1}(m-1+j)V_{j,m-1}^{\omega}(w)+\dots\,,$ (56c)

where the ellipses indicate higher-order terms. Similar expressions hold for
$\omega<0$, with the roles of $j^{+}$ and $j^{-}$ inverted. The operators
$V_{j,m}^{\omega}(z)$ are not affine primaries. They are, however, Virasoro
primaries, with worldsheet conformal weight

$\hat{\Delta}\,=\,-\frac{j(j-1)}{k-2}-m\omega-\frac{k}{4}\omega^{2}\,.$ (57)

Note that for $\omega>0$ ($\omega<0$), independently of the characteristics of
the original state, these correspond to lowest (highest) weight states, with
SL(2,$\mathbb{R}$) spin

$h=m+\frac{k}{2}\omega\,,$ (58)

($h=-m-k\omega/2$, respectively). The notation $h$ anticipates that the
SL(2,$\mathbb{R}$) spin is identified with the holographic CFT conformal
weight Kutasov:1999xu  (see also e.g. Dabholkar:2007ey ), as we shall see in
Eq. (60).

The flowed affine modules alluded above are built by acting freely with the
currents on flowed primary states. In particular, the remaining states in the
zero-mode algebra, which are obtained by acting with $j_{0}^{-}$, are not
flowed primaries. Nevertheless, one can proceed as done for the unflowed
states, and combine them into a local operator, defined initially for
$\omega>0$ as

$V^{w}_{j,h}(x,z)\,=\,\sum_{n\in\mathbb{N}_{0}}x^{n}\bar{x}^{\bar{n}}\,V_{j,h+n,h+\bar{n}}^{w}(z)\,.$
(59)

Moreover, by inverting $x\to 1/x$ in the expansion, one also obtains the
states in the highest-weight representation with the same spin and opposite
$\omega$ and $m$. This shows that the resulting $x$-basis states are actually
defined in terms of the absolute value of $\omega$, its sign being irrelevant.
A direct $x$-basis definition for spectrally flowed vertex operators was
recently derived in Iguri:2022eat , extending the original proposal of
Maldacena:2001km  valid only for the singly flowed case.

The classical analog of the spectral flow operation (55) maps space-like
geodesics of point-like strings into solutions in which a long string wound
around the AdS3 angular direction at large radius comes in to the centre of
global AdS3, collapses to a point, and then re-expands to large radial
distance Maldacena:2000hw . The spectral flow parameter $\omega$ is thus
sometimes referred to in the literature as a “winding” number. Note that since
the AdS3 angular direction is contractible in the interior of global AdS3, the
parameter $\omega$ is not a conserved quantity. However, the $m$-basis two-
point functions are diagonal in $\omega$: it was shown in Maldacena:2001km
that the $m$-basis two-point function of flowed primaries is as in (53) with
an extra factor of $\delta_{\omega_{1},-\omega_{2}}$ and the worldsheet
conformal weight $\Delta_{1}$ replaced by $\hat{\Delta}_{1}$ given in Eq.
(57). On the other hand, in the $x$-basis one finds

$\langle
V_{j_{1},h_{1}}^{\omega_{1}}(x_{1},z_{1})V_{j_{2},h_{2}}^{\omega_{2}}(x_{2},z_{2})\rangle\,=\,\frac{1}{|x_{12}|^{4h_{1}}}\frac{\langle
V^{\omega_{1}}_{j_{1},m_{1}}V^{\omega_{2}}_{j_{2},m_{2}}\rangle}{V_{\mathrm{conf}}}\,.$
(60)

Thus, as mentioned above, the SL(2,$\mathbb{R}$) spin $h$ is identified with
the holographic CFT conformal weight Kutasov:1999xu , even though in the
flowed sectors the spin is independent of the value of $j$ of the
corresponding unflowed operator. The factor $V_{\mathrm{conf}}$ stands for the
divergent volume of the conformal group; it reflects the fact we are picking
up the contribution from a pole, and it will cancel in the relevant
computations that follow.

####  3.2.2 Bosonic WZW model for SU(2)

The bosonic WZW model based on the SU$(2)$ group manifold was studied in
Fateev:1985mm ; Zamolodchikov:1986bd . We denote the generators of the current
algebra by $k^{a}$, and for most quantities we use primes to distinguish them
from their SL$(2,\mathbb{R})$ counterparts. The currents satisfy the OPEs

$k^{a}(z)k^{b}(w)\,\sim\,\frac{k^{\prime}}{2}\frac{\delta^{ab}}{(z-w)^{2}}+\frac{f^{\prime
ab}_{\phantom{ab}c}\;\\!k^{c}(w)}{z-w}\;,$ (61)

where $k^{\prime}$ is the level of the affine Lie algebra, $\delta^{ab}$ is
the Killing form, and $f^{\prime abc}$ are the corresponding structure
constants,

$2\delta^{33}=\delta^{+-}=2\;,\quad\qquad f^{\prime+-}_{\phantom{+-}3}=2\
,\quad f^{\prime 3+}_{\phantom{3+}+}=-f^{\prime 3-}_{\phantom{3-}-}=1\,.$ (62)

The energy-momentum tensor and central charge are

$T_{\mathrm{su}}(z)=\frac{1}{k^{\prime}+2}\left[k^{3}(z)k^{3}(z)+\frac{1}{2}\,k^{+}(z)k^{-}(z)+\frac{1}{2}\,k^{-}(z)k^{+}(z)\right]\;,\qquad
c_{\text{su}}=\frac{3k^{\prime}}{k^{\prime}+2}.$ (63)

We denote SU$(2)$ vertex operators by
$V^{\prime}_{j^{\prime},m^{\prime},\bar{m}^{\prime}}(z,\bar{z})$. Again, their
zero-mode wavefunctions do not factorize into holomorphic and antiholomorphic
parts, however we shall mostly work holomorphically and suppress
antiholomorphic quantities ($\bar{m}^{\prime},\bar{z}$).

For SU$(2)$, the unitary representations of the zero-mode algebra are labeled
by

$\ 0\leq j^{\prime}\leq\frac{k^{\prime}}{2}\;,\quad\qquad
j^{\prime}\in\mathbb{Z}/2\ ,$ (64)

and their states are $|j^{\prime},m^{\prime}\rangle$ with
$m^{\prime}=-j^{\prime},-j^{\prime}+1,\dots,j^{\prime}-1,j^{\prime}$. Using
conventions that mimic those used above for SL$(2,\mathbb{R})$, we have

$\displaystyle k_{0}^{3}|j^{\prime},m^{\prime}\rangle$ $\displaystyle=$
$\displaystyle m^{\prime}|j^{\prime},m^{\prime}\rangle\,,$ (65a)
$\displaystyle k^{\pm}_{0}|j^{\prime},m^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\begin{cases}\,(j^{\prime}+1\pm
m^{\prime})|j^{\prime},m^{\prime}\pm 1\rangle&\text{if}~{}~{}m\neq\pm j\\\
~{}0&\text{if}~{}~{}m=\pm j\;,\end{cases}$ (65b) $\displaystyle
k_{n}^{a}|j^{\prime},m^{\prime}\rangle$ $\displaystyle=$ $\displaystyle 0\
\qquad\forall n>0\,,$ (65c)

and

$\Delta^{\prime}\,=\,\frac{j^{\prime}(j^{\prime}+1)}{k^{\prime}+2}\,.$ (66)

Unlike SL$(2,\mathbb{R})$, in the SU$(2)$ WZW model spectral flow is not
necessary for constructing a consistent spectrum due to the compactness of the
group manifold. Indeed, the spectral flow automorphisms merely reshuffle
primary and descendant fields, and they do not introduce new inequivalent
representations. Nevertheless, for superstring theory applications it is of
practical use to include it in the discussion Giribet:2007wp ;
Martinec:2018nco ; Bufalini:2021ndn . We will discuss this in more detail
shortly.

For SU$(2)$, spectral flow is defined as

$k^{\pm}(z)\to\tilde{k}^{\pm}(z)=z^{\mp w^{\prime}}k^{\pm}(z)\;,\quad\qquad
k^{3}(z)\to\tilde{k}^{3}(z)=k^{3}(z)-\frac{k^{\prime}\omega^{\prime}}{2}\frac{1}{z}\,.$
(67)

In this case, however, it is possible to have
$\bar{\omega}^{\prime}\neq\omega^{\prime}$. As before, spectrally flowed
primaries $V_{j^{\prime},m^{\prime}}^{\omega^{\prime}}(z)$ are Virasoro
primaries, with weight

$\hat{\Delta}^{\prime}\,=\,\frac{j^{\prime}(j^{\prime}+1)}{k^{\prime}-2}+m^{\prime}\omega^{\prime}+\frac{k^{\prime}}{4}\omega^{\prime
2}\,,$ (68)

but they are not affine primaries, and for $\omega^{\prime}>0$ they are
defined in terms of the OPEs

$\displaystyle k^{+}(z)V_{j^{\prime},m^{\prime}}^{\omega^{\prime}}(w)$
$\displaystyle=$
$\displaystyle(z-w)^{\omega^{\prime}-1}(j^{\prime}-m^{\prime})V_{j^{\prime},m^{\prime}+1}^{\omega^{\prime}}(w)+\dots\,,$
(69a) $\displaystyle k^{3}(z)V_{j^{\prime},m^{\prime}}^{\omega^{\prime}}(w)$
$\displaystyle=$
$\displaystyle\frac{\left(m^{\prime}+\frac{k}{2}\omega^{\prime}\right)V_{j^{\prime},m^{\prime}}^{\omega^{\prime}}(w)}{(z-w)}+\dots\,,$
(69b) $\displaystyle k^{-}(z)V_{j^{\prime},m^{\prime}}^{\omega^{\prime}}(w)$
$\displaystyle=$
$\displaystyle\frac{(j^{\prime}+m^{\prime})V_{j^{\prime},m^{\prime}-1}^{\omega^{\prime}}(w)}{(z-w)^{\omega^{\prime}+1}}+\sum_{n=0}^{\omega^{\prime}}\frac{(k^{-}_{n-1}V_{j^{\prime},m^{\prime}}^{\omega^{\prime}})(w)}{(z-w)^{n}}+\dots\,.$
(69c)

The corresponding two-point functions are, again, the unflowed ones times
$\delta_{\omega_{1},-\omega_{2}}$, with the appropriate powers of $z_{12}$.

####  3.2.3 Superstrings in
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$

We now review supersymmetric generalizations of the bosonic WZW models
discussed above. We introduce fermions $\psi^{a}$ and $\chi^{a}$ which are
superpartners of the SL$(2,\mathbb{R})$ and SU$(2)$ currents $J^{a}$ and
$K^{a}$ respectively. The appropriate ${\cal{N}}=1$ supersymmetric extensions
of the affine sl(2,$\mathbb{R}$)k and su(2)${}_{k^{\prime}}$ algebras are
generated by the supercurrents $\psi^{a}+\theta\>\\!J^{a}$ and
$\chi^{a}+\theta K^{a}$, where $\theta$ is a Grassmann variable. The currents
$J^{a}$ and $K^{a}$ satisfy the OPEs (40) and (61) respectively, with level
$n_{5}$ in both cases, and the OPEs involving the fermions $\psi^{a}$ and
$\chi^{a}$ are

$\displaystyle
J^{a}(z)\psi^{b}(w)\,\sim\,\frac{f^{ab}_{\phantom{ab}c}\;\\!\psi^{c}(w)}{(z-w)}\;,\quad\qquad$
$\displaystyle K^{a}(z)\chi^{b}(w)\,\sim\,\frac{f^{\prime
ab}_{\phantom{lab}c}\;\\!\chi^{c}(w)}{(z-w)},$ (70a)
$\displaystyle\psi^{a}(z)\psi^{b}(w)\,\sim\,\frac{n_{5}}{2}\frac{\eta^{ab}}{(z-w)}\;,\quad\qquad$
$\displaystyle\chi^{a}(z)\chi^{b}(w)\,\sim\,\frac{n_{5}}{2}\frac{\delta^{ab}}{(z-w)}\,.$
(70b)

One can split the currents into two independent contributions via

$J^{a}=j^{a}-\frac{1}{n_{5}}f^{a}_{\phantom{a}bc}\>\\!\psi^{b}\psi^{c}\;,\quad\qquad
K^{a}=k^{a}-\frac{1}{n_{5}}f^{\prime
a}_{\phantom{la}bc}\;\\!\chi^{b}\chi^{c}\,.$ (71)

The “bosonic” currents $j^{a}$ and $k^{a}$ commute with the free fermions, and
are currents of bosonic WZW models as described in Section 3.2, with levels
$k=n_{5}+2$ and $k^{\prime}=n_{5}-2$ respectively. In the fermionic sector,
the spectral flow automorphisms act as

$\tilde{\psi}^{\pm}(z)=z^{\mp}\psi^{\pm}(z)\,,\quad\tilde{\psi}^{3}(z)=\psi^{3}(z)\;,\quad\qquad\tilde{\chi}^{\pm}(z)=z^{\mp}\chi^{\pm}(z)\,,\quad\tilde{\chi}^{3}(z)=\chi^{3}(z)\,.$
(72)

The remaining flat compact directions are treated as usual. For the
$\mathbb{T}^{4}$, we simply have four (canonically normalized) free bosons
$Y^{i}$ and their fermionic partners $\lambda^{i}$ $(i=6,\dots,9)$, with OPEs

$Y^{i}(z)Y^{j}(w)\,\sim\,-\delta^{ij}\log(z-w)\;,\quad\qquad\lambda^{i}(z)\lambda^{j}(w)\,\sim\,\frac{\delta^{ij}}{(z-w)}\,.$
(73)

We can now write down the energy-momentum tensor $T$ and the supercurrent $G$
of the worldsheet theory for type II superstrings in
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$. The matter contributions
read

$\displaystyle T$ $\displaystyle\,=\,$
$\displaystyle\frac{1}{n_{5}}\Big{(}j^{a}j_{a}-\psi^{a}\partial\psi_{a}+k^{a}k_{a}-\chi^{a}\partial\chi_{a}\Big{)}+\frac{1}{2}\left(\partial
Y^{i}\partial Y_{i}-\lambda^{i}\partial\lambda_{i}\right),$ (74)
$\displaystyle G$ $\displaystyle\,=\,$
$\displaystyle\frac{2}{n_{5}}\left(\psi^{a}j_{a}-\frac{1}{3n_{5}}f_{abc}\psi^{a}\psi^{b}\psi^{c}+\chi^{a}k_{a}-\frac{1}{3n_{5}}f^{\prime}_{abc}\chi^{a}\chi^{b}\chi^{c}\right)+i\>\lambda^{j}\partial
Y_{j}\,,$ (75)

and the resulting central charge is compensated by the usual $bc$ and
$\beta\gamma$ ghost systems, leading to the BRST charge

${\cal{Q}}=\oint dz\left(c\left(T+T_{\beta\gamma}\right)-\gamma\,G+c(\partial
c)b-\frac{1}{4}b\gamma^{2}\right)\,.$ (76)

Here $T_{\beta\gamma}$ is the energy-momentum tensor of the $\beta\gamma$
system, which is bosonized as

$\displaystyle\beta=e^{-\varphi}\partial\xi\;,\quad\qquad\gamma=\eta\>\\!e^{\varphi}\,,$
(77)

where $\varphi(z)\varphi(w)\simeq-\ln(z-w)$ has background charge $2$, and
$\xi(z)\eta(w)\sim(z-w)^{-1}$. For computational purposes it is useful to also
bosonize the rest of the fermions Giveon:1998ns ; Dabholkar:2007ey . We thus
define (canonically normalized) bosonic fields $H_{I}$ with $I=1,\dots 5$, and
write

$\hat{H}_{I}=H_{I}+\pi\sum_{J<I}N_{J}\;,\quad\qquad N_{J}\equiv\oint i\partial
H_{J}\,,$ (78)

where the number operators $N_{I}$ are introduced in order to keep track of
the cocycle factors, namely

$e^{ia\hat{H}_{I}}e^{ib\hat{H}_{J}}=e^{ib\hat{H}_{J}}e^{ia\hat{H}_{I}}\,e^{i\pi
ab}\;,\quad\qquad\text{if}\quad I>J\,.$ (79)

We bosonize as

$\displaystyle\psi^{\pm}=\sqrt{n_{5}}\,e^{\pm
i\hat{H}_{1}}\,,\quad\chi^{\pm}=\sqrt{n_{5}}\,e^{\pm
i\hat{H}_{2}}\,,\quad\lambda^{6}\pm i\lambda^{7}=e^{\pm
i\hat{H}_{4}}\,,\quad\lambda^{8}\pm i\lambda^{9}=e^{\pm i\hat{H}_{5}}\,,$
(80a)
$\displaystyle\qquad\psi^{3}=\frac{\sqrt{n_{5}}}{2}\,\left(e^{i\hat{H}_{3}}-e^{-i\hat{H}_{3}}\right)\;,\quad\qquad\chi^{3}=\frac{\sqrt{n_{5}}}{2}\,\left(e^{i\hat{H}_{3}}+e^{-i\hat{H}_{3}}\right)\,,$
(80b)

where $\hat{H}_{I}^{\dagger}=\hat{H}_{I}$ for $I\neq 3$ and
$\hat{H}_{3}^{\dagger}=-\hat{H}_{3}$. Then we have

$\displaystyle
i\partial\hat{H}_{1}=\frac{1}{n_{5}}\,\psi^{+}\psi^{-}\;,\quad\qquad
i\partial\hat{H}_{2}=\frac{1}{n_{5}}\,\chi^{+}\chi^{-}\;,\quad\qquad
i\partial\hat{H}_{3}=\frac{2}{n_{5}}\,\psi^{3}\chi^{3}\,,$ (81a)
$\displaystyle i\partial\hat{H}_{4}=i\lambda^{6}\lambda^{7}\;,\quad\qquad
i\partial\hat{H}_{5}=i\lambda^{8}\lambda^{9}\,.$ (81b)

The phases in (79) ensure that bosonized fermions anticommute, and will be
important when working with states in the Ramond sector. From now on we will
simply omit the hats, and explicitly include the phase factors when they are
needed.

The spacetime supercharges can be written as:

$Q_{\varepsilon}=\oint dz\,e^{-\varphi/2}S_{\varepsilon}\;,\quad\qquad
S_{\varepsilon}=\exp\left(\frac{i}{2}\sum_{I=1}^{5}\varepsilon_{I}H_{I}\right),$
(82)

where $S_{\varepsilon}$ are spin fields and $\varepsilon_{I}=\pm 1$. Imposing
BRST invariance – where the relevant contributions come from the
$f_{abc}\psi^{a}\psi^{b}\psi^{c}$ and
$f_{abc}^{\prime}\chi^{a}\chi^{b}\chi^{c}$ pieces of $G$ in (75) – and mutual
locality (chiral GSO) leads to the conditions

$\prod_{I=1}^{3}\varepsilon_{I}=\prod_{I=1}^{5}\varepsilon_{I}=1\,.$ (83)

In the holomorphic sector this gives the expected four ‘ordinary’ supercharges
and four ‘superconformal’ supercharges. The same applies in the
antiholomorphic sector, giving the total 16 real supercharges of global
AdS${}_{3}\times\mathbb{S}^{3}$ Giveon:1998ns .

For later use, let us also recall that the R-symmetry of the boundary theory
is generated on the worldsheet by the SU(2) currents. More precisely, the zero
modes of the spacetime R-currents are given by the integrated worldsheet
currents Giveon:1998ns , i.e.

$\mathbb{J}^{a}_{0}=\oint dz\,K^{a}(z)\,.$ (84)

Consequently, the holomorphic R-charge in the holographic CFT is identified
with $m^{\prime}$, the eigenvalue of $K^{3}$. It is for this reason that we
used the notation $m^{\prime}$ in Sections 2.3 and 3.1.

###  3.3 Vertex operators and two-point functions

We now discuss physical vertex operators and their two-point functions, both
in NS and R sectors. This section is largely review, though we also give
explicit expressions for some R sector operators that to our knowledge have
not appeared before in the literature.

Our main interest is in worldsheet operators that correspond to chiral
primaries of the holographic CFT. We thus focus on states belonging to the
discrete representations of SL(2,$\mathbb{R}$). We also discuss the role of
SL(2,$\mathbb{R}$) and SU$(2)$ spectral flows in the string theoretical
construction. These vertex operators and their two-point functions will be
used as building blocks for constructing the vertex operators and two-point
functions of the null-gauged models in Section 4.

####  3.3.1 NS sector

The unflowed NS-NS sector was considered in Kutasov:1998zh , see also
Dabholkar:2007ey . We continue to suppress antiholomorphic parts of the
SL$(2,\mathbb{R})$ and SU$(2)$ vertex operators $V_{j,m,\bar{m}}$ and
$V^{\prime}_{j^{\prime},m^{\prime},\bar{m}^{\prime}}$. The complete NSNS
vertex is obtained by including the antiholomorphic fermions and ghosts.

We work in the canonical “$-1$” ghost picture, and consider only states with
vanishing momentum in the $\mathbb{T}^{4}$ directions. Then the (holomorphic
part of the) BRST invariant states with up to a single fermionic excitation
include the tachyon (which is projected out by GSO),

$\mathcal{T}_{j,m,j^{\prime},m^{\prime}}=e^{-\varphi}V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\,,$
(85)

and the spacetime vectors ($\varepsilon=\pm 1$, and recall $i=6,\dots,9$)

$\displaystyle\mathcal{V}^{i}_{j,m,j^{\prime},m^{\prime}}$ $\displaystyle=$
$\displaystyle
e^{-\varphi}\lambda^{i}V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\,,$ (86a)
$\displaystyle\mathcal{W}^{\varepsilon}_{j,m,j^{\prime},m^{\prime}}$
$\displaystyle=$ $\displaystyle e^{-\varphi}\left(\psi
V_{j}\right)_{j+\varepsilon,m}V^{\prime}_{j^{\prime},m^{\prime}}\,,$ (86b)
$\displaystyle\mathcal{X}^{\varepsilon}_{j,m,j^{\prime},m^{\prime}}$
$\displaystyle=$ $\displaystyle e^{-\varphi}V_{j,m}(\chi
V^{\prime}_{j^{\prime}})_{j^{\prime}+\varepsilon,m^{\prime}}\,,$ (86c)

where we have introduced the linear combinations

$(\psi
V_{j})_{j+\varepsilon,m}=c_{\varepsilon}^{r}\psi^{r}V_{j,m-r}\;,\quad\qquad(\chi
V^{\prime}_{j^{\prime}})_{j^{\prime}+\varepsilon,m^{\prime}}=d_{\varepsilon}^{r}\chi^{r}V^{\prime}_{j^{\prime},m^{\prime}-r}\,,$
(87)

where a summation over $r=+1,-1,0$ is implicit, “$0$” corresponding to the
“$3$” direction of the respective algebras. These combine the products of
bosonic primaries and free fermions into fields of total spins
$J=j+\varepsilon$ and $J^{\prime}=j^{\prime}+\varepsilon$ under the action of
the supersymmetric currents $J^{a}$ and $K^{a}$ Kutasov:1998zh . The Clebsh-
Gordan coefficients are given in our conventions by

$\displaystyle c^{r}_{-}$
$\displaystyle=\left(\frac{1}{2},\frac{1}{2},-1\right),\quad
d^{r}_{+}=\left(-\frac{1}{2},\frac{1}{2},1\right),$ $\displaystyle c_{+}^{r}$
$\displaystyle=\left(\frac{1}{2}(j+m)(j+m-1),\;\frac{1}{2}(j-m)(j-m-1),\;(j+m)(j-m)\right),$
(88) $\displaystyle d^{r}_{-}$
$\displaystyle=\left(\frac{1}{2}(j^{\prime}-m^{\prime})(j^{\prime}-m^{\prime}+1),\;-\frac{1}{2}(j^{\prime}+m^{\prime})(j^{\prime}+m^{\prime}+1),\;(j^{\prime}-m^{\prime})(j^{\prime}+m^{\prime})\right)\,.$

The Virasoro condition associated to all vertex operators in Eq. (86) reads

$\frac{1}{2}+\frac{1}{2}-\frac{j(j-1)}{n_{5}}+\frac{j^{\prime}(j^{\prime}+1)}{n_{5}}=1\,,$
(89)

and is solved by $j=j^{\prime}+1$ (or its reflection under $j\to 1-j$), thus
implying that we are dealing with bosonic primaries in the discrete
representations of SL(2,$\mathbb{R}$)k.

Let us briefly discuss the worldsheet two-point functions involving these
operators. The different bosonic sectors factorize and the fermions are free,
so we can express the results directly in terms of the non-trivial
contributions coming from the bosonic SL(2,$\mathbb{R}$) WZW model, namely Eq.
(53). By construction, the only non-vanishing two-point functions are the
diagonal ones. For the 6D scalars coming from the NS-NS sector polarizations
on the $\mathbb{T}^{4}$, (86a), we have

$\langle\mathcal{V}^{i\bar{i}}_{1}\mathcal{V}^{j\bar{j}}_{2}\rangle\,=\,\langle
V_{1}V_{2}\rangle\langle V_{1}^{\prime}V_{2}^{\prime}\rangle\left[\langle
e^{-\varphi_{1}}e^{-\varphi_{2}}\rangle\langle\lambda^{i}_{1}\lambda^{j}_{2}\rangle\times\mathrm{c.c.}\right]\,=\,\langle
V_{1}V_{2}\rangle\langle
V_{1}^{\prime}V_{2}^{\prime}\rangle\times\frac{\delta^{ij}\delta^{\bar{i}\bar{j}}}{|z_{12}|^{4}}\,.$
(90)

Since we are dealing with discrete representations, the contact term in (53)
vanishes, thus imposing $j_{1}=j_{2}\equiv j$. As discussed above Eq. (54),
the conformal weight in the holographic CFT is to be identified with the
SL(2,$\mathbb{R}$) spin, i.e. $h=j$. On the other hand, the R-charge is given
by $m^{\prime}$, with $|m^{\prime}|\leq j^{\prime}=j-1$. Thus
$h\neq|m^{\prime}|$, so $\mathcal{V}^{i}$ cannot correspond to a chiral
primary of the HCFT.

We now turn to the operators introduced in the second and third line of (86).
When computing correlators of two $\mathcal{W}$ states, we must deal with
expressions of the form

$\displaystyle\langle\left(\psi_{1}V_{j_{1}}\right)_{j_{1}+\varepsilon_{1},m_{1}}\left(\psi_{2}V_{j_{2}}\right)_{j_{2}+\varepsilon_{2},m_{2}}\rangle=\sum_{r_{1,2}}c^{r_{1}}_{\varepsilon_{1}}c^{r_{2}}_{\varepsilon_{2}}\braket{\psi^{r_{1}}\psi^{r_{2}}}\braket{V_{j_{1},m_{1}-r_{1}}V_{j_{2},m_{2}-r_{2}}}.$
(91)

We use the action of the bosonic currents (46) to express
$\braket{V_{j_{1},m_{1}-r_{1}}V_{j_{2},m_{2}-r_{2}}}$ in terms of
$\braket{V_{j_{1},m_{1}}V_{j_{2},m_{2}}}$, insert the coefficients (88), and
perform the sum. Recalling the shorthand $V_{i}\equiv V_{j_{i},m_{i}}$,
$V^{\prime}_{i}\equiv V^{\prime}_{j^{\prime}_{i},m^{\prime}_{i}}$, we obtain

$\langle\mathcal{W}^{\varepsilon_{1}}\mathcal{W}^{\varepsilon_{2}}\rangle=\frac{n_{5}^{2}}{4|z_{12}|^{4}}\langle
V_{1}V_{2}\rangle\langle
V_{1}^{\prime}V_{2}^{\prime}\rangle\times\left\\{\begin{array}[]{cc}j_{1}(1-2j_{1})(j_{1}^{2}-m_{1}^{2})\times\mathrm{c.c.}&\quad\varepsilon_{1}=\varepsilon_{2}=1\\\\[4.30554pt]
\dfrac{(j_{1}-1)(1-2j_{1})}{(j_{1}-1)^{2}-m_{1}^{2}}\times\mathrm{c.c.}&\quad\varepsilon_{1}=\varepsilon_{2}=-1{}{}\\\
0&\varepsilon_{1}=-\varepsilon_{2}\;.\end{array}\right.$ (92)

From Eq. (53) we find that the coefficients are exactly those needed to
produce the shift $j\to j+\varepsilon$ in the two-point function. Hence, Eq.
(52) shows that the weight of the corresponding holographic dual is
$h=j+\varepsilon$ Gaberdiel:2007vu . In particular, the operator
$\mathcal{W}^{-}$ with maximal SU(2) charge has $h=j-1=m^{\prime}$, and thus
corresponds to a chiral primary operator of the holographic CFT.

The computation of the $\langle\mathcal{X}\mathcal{X}\rangle$ correlators is
analogous; we obtain

$\langle\mathcal{X}^{\varepsilon_{1}}\mathcal{X}^{\varepsilon_{2}}\rangle=\frac{n_{5}^{2}}{4|z_{12}|^{4}}\langle
V_{1}V_{2}\rangle\langle
V_{1}^{\prime}V_{2}^{\prime}\rangle\times\left\\{\begin{array}[]{cc}\dfrac{(j_{1}^{\prime}+1)(1+2j_{1}^{\prime})}{(j_{1}^{\prime}+1)^{2}-{m_{1}^{\prime}}^{\\!2}}\times\mathrm{c.c.}&\varepsilon_{1}=\varepsilon_{2}=1\\\
j_{1}^{\prime}(1+2j_{1}^{\prime})({j_{1}^{\prime}}^{\\!2}-{m_{1}^{\prime}}^{\\!2})\times\mathrm{c.c.}{}&\varepsilon_{1}=\varepsilon_{2}=-1\\\
0&\varepsilon_{1}=-\varepsilon_{2}=\pm 1\end{array}\right.$ (93)

For $\mathcal{X}^{+}$ at highest SU(2) weight we have
$h=j=j^{\prime}+1=m^{\prime}$, leading to a second family of spacetime chiral
states. We will discuss the corresponding operators in the holographic CFT
theory and fix their normalization below.

So far, we have constructed chiral operators whose boundary weights $h=j-1$
and $h=j$ are bounded from above by $h<\frac{n_{5}+1}{2}$, see Eq. (45).
However, in the D1D5 CFT one can have chiral primaries in $n$-twisted sectors
with $n$ up to $n_{1}n_{5}$, and where $h$ grows linearly with $n$, as
discussed around Eq. (35). Thus, it seems that so far we are missing most of
the heavier chiral operators. However, as discussed in Giribet:2007wp , such
states lie in the sectors of the worldsheet theory with non-trivial spectral
flow charges, as we now review.

In the supersymmetric theory, spectrally flowed primary operators are built by
combining the bosonic flowed primaries introduced in Eqs. (56) and (69) with
fermionic excitations. The bosons $H_{I}$ allow us to express the spectral
flow operation in the fermionic sectors of SL(2,$\mathbb{R}$) and SU(2), Eq.
(72), in the following form,

$\psi^{\pm}_{\omega}\,=\,\psi^{\pm}e^{-i\omega
H_{1}}\;,\quad\qquad\chi^{\pm}_{\omega^{\prime}}\,=\,\chi^{\pm}e^{i\omega^{\prime}H_{2}},$
(94)

while the other fermions remain unchanged. Indeed, the OPEs between the
operators in (94) and the fermionic currents are analogous to those in (56)
and (69). Once factors of $e^{-i\omega H_{1}}$ and $e^{i\omega^{\prime}H_{2}}$
are included, the corresponding weights take the form in (57) and (68), with
$k-2=n_{5}=k^{\prime}+2$.

In principle, one could simply ignore the possibility of including spectral
flow in SU(2) since it does not give any new representations. However, for
discrete series states it is useful to do so in order to solve the modified
Virasoro condition, as discussed in Martinec:2018nco ; Bufalini:2021ndn . We
use the spectral flow operator with equal amount of spectral flow in
SL(2,$\mathbb{R}$) and SU(2),

$\exp\left(-i\omega H_{1}+\omega\sqrt{\frac{n_{5}+2}{2}}\phi+i\omega
H_{2}+i\omega\sqrt{\frac{n_{5}-2}{2}}\phi^{\prime}\right),$ (95)

which is mutually local with the supercharges, thus producing flowed singly-
excited states which will also survive the GSO projection. The corresponding
Virasoro condition for the flowed vertex operators is

$\frac{1}{2}+\frac{1}{2}-\frac{j(j-1)}{n_{5}}-m\omega-\frac{n_{5}}{4}\omega^{2}+\frac{j^{\prime}(j^{\prime}+1)}{n_{5}}+m^{\prime}\omega+\frac{n_{5}}{4}\omega^{2}\;=\;1\,.$
(96)

We seek to solve this for general $n_{5}$. We thus impose $j=j^{\prime}+1$ as
well as $m=m^{\prime}$. The latter constraint is quite restrictive, since by
definition we have

$\displaystyle{\cal{V}}\text{-type operators:}~{}~{}$ $\displaystyle\ \
|m|\geq j\ ,\ \quad~{}~{}|m^{\prime}|\leq j^{\prime}=j-1\,,$
$\displaystyle{\cal{W}}\text{-type operators:}~{}$ $\displaystyle\ \ |m|\geq
j-1\ ,\ ~{}~{}|m^{\prime}|\leq j^{\prime}=j-1\,,$
$\displaystyle{\cal{X}}\text{-type operators:}~{}~{}$ $\displaystyle\ \
|m|\geq j\ ,\ \quad~{}~{}|m^{\prime}|\leq j^{\prime}+1=j\,.$

Consequently, our only candidates are highest/lowest-weight
${\cal{W}}^{-}$-type operators with $m=m^{\prime}=j^{\prime}=j-1$ and
${\cal{X}}^{+}$-type operators with $m=m^{\prime}=j^{\prime}+1=j$. Their
explicit expressions are given by

$\displaystyle{\cal{W}}^{\omega}_{j}$ $\displaystyle=$ $\displaystyle
e^{-\varphi}\psi^{-}e^{-i\omega H_{1}}e^{i\omega
H_{2}}V^{\omega}_{j,j}V^{\prime\omega}_{j-1,j-1}\,,$ (97a)
$\displaystyle{\cal{X}}^{\omega}_{j}$ $\displaystyle=$ $\displaystyle
e^{-\varphi}e^{-i\omega H_{1}}\chi^{+}e^{i\omega
H_{2}}V^{\omega}_{j,j}V^{\prime\omega}_{j-1,j-1}\,.$ (97b)

These flowed states are also BRST-invariant Giribet:2007wp  since the
supercurrent $G$ can be written in the flowed frame as

$G(z)=\tilde{G}(z)+\frac{\omega}{z}\left(\chi^{3}-\psi^{3}\right),$ (98)

such that the extra terms on the RHS of this equation act trivially on
highest/lowest weight states.

The two-point functions of these spectrally flowed operators can be determined
straightforwardly from the corresponding bosonic ones. This is because the
latter impose $\omega_{2}=-\omega_{1}$, such that the charge conservation
rules for the $H_{I}$ exponentials are automatically satisfied. We define the
conjugate operators $\hat{\mathcal{W}}^{\omega}$,
$\hat{\mathcal{X}}^{\omega}$, and obtain

$\langle\mathcal{W}^{\omega_{1}}_{1}\hat{\mathcal{W}}^{\omega_{2}}_{2}\rangle\,=\,\langle\mathcal{X}^{\omega_{1}}_{1}\hat{\mathcal{X}}^{\omega_{2}}_{2}\rangle\,=\,\langle
V_{j_{1},j_{1}}^{\omega_{1}}V_{j_{1},-j_{1}}^{\omega_{2}}\rangle\langle
V_{j_{1}-1,j_{1}-1}^{{}^{\prime}\omega_{1}}V_{j_{1}-1,1-j_{1}}^{{}^{\prime}\omega_{2}}\rangle\frac{n_{5}^{2}}{|z_{12}|^{4(1+\omega_{1}+\omega_{1}^{2})}}\,.$
(99)

Spectral flowed primaries are always annihilated by $J_{0}^{-}$, and are thus
lowest-weight with respect to the SL(2,$\mathbb{R}$) zero mode algebra. After
the supersymmetric spectral flow (95), similarly to the bosonic
transformations (58), (69b), the spectral flowed primaries have quantum
numbers $h$ and $m^{\prime}$ that have increased by
$\,\frac{n_{5}}{2}\omega\,$ from their values for the unflowed vertex
operators. We thus conclude that these vertex operators correspond exactly to
the additional chiral operators we were looking for, with

${\cal{W}}^{\omega}_{j}~{}:~{}~{}h=m^{\prime}=j-1+\frac{n_{5}}{2}\omega\;,\quad\qquad{\cal{X}}^{\omega}_{j}~{}:~{}~{}h=m^{\prime}=j+\frac{n_{5}}{2}\omega\,.$
(100)

These quantum numbers extend to large values, by raising $\omega$. In the
holographic CFT there are states with conformal weight of order $n_{1}n_{5}$,
however in our worldsheet models $n_{1}$ is of order $g_{s}^{-2}$ (c.f. (19))
and we work in perturbation theory in $g_{s}$, so finite $n_{1}$ physics is
not accessible. Moreover, when considering holographic CFT operators with
conformal weight of order $n_{1}n_{5}$, the dual bulk configuration is not a
light probe on the original background, but rather a different background. The
rest of the modes associated to such boundary operators are obtained by acting
with the global current $J_{0}^{+}$ as in Eq. (59), and do not have simple
expressions in the $m$-basis since they are not flowed primaries.

####  3.3.2 Ramond sector

We now review the Ramond sector physical operators of the worldsheet theory,
in the $m$-basis. To our knowledge, this construction has only been carried
out explicitly in the literature for the case of highest/lowest-weight states
Dabholkar:2007ey ; Giribet:2007wp ; we shall present explicit expressions for
more general Ramond sector operators.

We will make use of the spin fields introduced in (82), and distinguish the
slightly more involved AdS${}_{3}\times\mathbb{S}^{3}$ sector, for which we
write the relevant factors as

$S_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}\,=\,e^{\frac{i}{2}(\varepsilon_{1}H_{1}+\varepsilon_{2}H_{2}+\varepsilon_{3}H_{3})}\,.$
(101)

We denote the AdS${}_{3}\times\mathbb{S}^{3}$ chirality by
$\varepsilon\equiv\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}$. We shall
implement this by considering $\varepsilon_{3}$ to be fixed to be
$\varepsilon_{3}=\varepsilon\varepsilon_{1}\varepsilon_{2}$. We impose the
chiral GSO projection via the mutual locality condition
$\prod_{I=1}^{5}\varepsilon_{I}=1$, and we implement this by fixing
$\varepsilon_{5}=\varepsilon\>\\!\varepsilon_{4}$. We introduce a generic
linear combination of bosonic primaries and spin fields of
AdS${}_{3}\times\mathbb{S}^{3}$ of fixed chirality,

$\left(SVV^{\prime}\right)^{\varepsilon}_{J,m,J^{\prime},m^{\prime}}=\sum_{\varepsilon_{1},\varepsilon_{2}=\pm
1}f^{\varepsilon}_{\varepsilon_{1}\,\varepsilon_{2}}S_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}V_{j,m-\frac{\varepsilon_{1}}{2}}V^{\prime}_{j^{\prime},m^{\prime}-\frac{\varepsilon_{2}}{2}}\;,\quad\qquad\varepsilon_{3}=\varepsilon\>\\!\varepsilon_{1}\varepsilon_{2}\,,$
(102)

where the total spins $(J,J^{\prime})$ will be related to $(j,j^{\prime})$ in
various ways momentarily. Note that for highest/lowest weight states, there
may be only one allowed choice of $\varepsilon_{1}$ and/or $\varepsilon_{2}$,
as we shall see in an example below. In the canonical “$-\frac{1}{2}$”
picture, the Ramond sector vertex operators then take the form

$\mathcal{Y}^{\varepsilon,\varepsilon_{4}}_{J,m,J^{\prime},m^{\prime}}=e^{-\frac{\varphi}{2}}\left(SVV^{\prime}\right)^{\varepsilon}_{J,m,J^{\prime},m^{\prime}}e^{\frac{i\varepsilon_{4}}{2}\left(H_{4}+\varepsilon
H_{5}\right)}\,.$ (103)

The Clebsch–Gordan coefficients
$f^{\varepsilon}_{\varepsilon_{1}\,\varepsilon_{2}}$ are computed by requiring
that the $\mathcal{Y}$ operators transform appropriately under the action of
the currents $J^{\pm}$, $K^{\pm}$. In our conventions, this gives four linear
combinations. In the equations below, the first bracket specifies how
$(J,J^{\prime})$ are related to $j$ and $j^{\prime}$; for instance, for case
$A$, $(J=j-1/2,J^{\prime}=j^{\prime}+1/2)$. For each case we write the
coefficients as a list,
$f^{\varepsilon}_{\varepsilon_{1}\,\varepsilon_{2}}=\left(f^{\varepsilon}_{++},f^{\varepsilon}_{+-},f^{\varepsilon}_{-+},f^{\varepsilon}_{--}\right)$.
We obtain

$\displaystyle
A\\!:\;(j-\tfrac{1}{2},\;\\!j^{\prime}+\tfrac{1}{2})\;\\!,\quad~{}f^{\varepsilon,A}_{\varepsilon_{1}\,\varepsilon_{2}}$
$\displaystyle\,=\,\big{(}1,\,i,\,\varepsilon,\,\varepsilon\;\\!i\big{)}\;\\!,$
(104a) $\displaystyle
B\\!:\;(j+\tfrac{1}{2},\;\\!j^{\prime}+\tfrac{1}{2})\;\\!,\quad~{}f^{\varepsilon,B}_{\varepsilon_{1}\,\varepsilon_{2}}$
$\displaystyle\,=\,\left(f^{B_{1}},\,i\;\\!f^{B_{1}},\,\varepsilon\;\\!f^{B_{2}},\,\varepsilon\;\\!i\;\\!f^{B_{2}}\right),$
(104b) $\displaystyle
C\\!:\;(j-\tfrac{1}{2},\;\\!j^{\prime}-\tfrac{1}{2})\;\\!,\quad~{}f^{\varepsilon,C}_{\varepsilon_{1}\,\varepsilon_{2}}$
$\displaystyle\,=\,\left(f^{C_{1}},\,-i\;\\!f^{C_{2}},\,\varepsilon\;\\!f^{C_{1}},\,\varepsilon(-i)f^{C_{2}}\right),$
(104c) $\displaystyle
D\\!:\;(j+\tfrac{1}{2},\;\\!j^{\prime}-\tfrac{1}{2})\;\\!,\quad~{}f^{\varepsilon,D}_{\varepsilon_{1}\,\varepsilon_{2}}$
$\displaystyle\,=\,\left(f^{B_{1}}f^{C_{1}},(-i)f^{B_{1}}f^{C_{2}},\,\varepsilon\;\\!f^{B_{2}}f^{C_{1}},\,\varepsilon(-i)f^{B_{2}}f^{C_{2}}\right),$
(104d)

where

$f^{B_{1}}=m+j-\frac{1}{2}\,,\quad f^{B_{2}}=m-j+\frac{1}{2}\,,\quad
f^{C_{1}}=j^{\prime}-m^{\prime}+\frac{1}{2}\,,\quad
f^{C_{2}}=j^{\prime}+m^{\prime}+\frac{1}{2}\,.$ (105)

We note that in all cases we have
$f^{+}_{\varepsilon_{1}\varepsilon_{2}}=\varepsilon_{1}\,f^{-}_{\varepsilon_{1}\varepsilon_{2}}$.
In addition, and using $j=j^{\prime}+1$, BRST-invariance gives four equations
for each chirality, out of which only two are linearly independent, namely

$\displaystyle f_{-+}^{\varepsilon}$
$\displaystyle\,=\,\frac{1}{\left(j+m-\frac{1}{2}\right)}\left[f_{++}^{\varepsilon}\left(\varepsilon\,m+m^{\prime}\right)-i\,f_{+-}^{\varepsilon}\left(j^{\prime}-m^{\prime}+\frac{1}{2}\right)\right],$
(106a) $\displaystyle f_{--}^{\varepsilon}$
$\displaystyle\,=\,\frac{1}{\left(j+m-\frac{1}{2}\right)}\left[i\,f_{++}^{\varepsilon}\left(j^{\prime}+m^{\prime}+\frac{1}{2}\right)+f_{+-}^{\varepsilon}\left(\varepsilon\,m-m^{\prime}\right)\right].$
(106b)

These are satisfied by only half of the states in Eq. (104). The physical
states in the “$-\frac{1}{2}$” picture are given by the $A$ and $D$ states
with $\varepsilon=1$ (and either choice of $\varepsilon_{4}=\pm 1$), plus the
$B$ and $C$ states with $\varepsilon=-1$ (and again either sign of
$\varepsilon_{4}$), making the correct eight physical polarizations.

The full list of expressions in (104) is useful in order to construct the
representatives of such operators in the “$-\frac{3}{2}$” ghost picture,
necessary for computing two-point functions. To obtain the “$-\frac{3}{2}$”
picture operators, we make an educated guess for their expressions, and then
apply the picture raising operator, i.e.

$\Phi^{(-\frac{1}{2})}(w)\,=\,\lim_{z\to
w}\left(e^{\varphi}G\right)(z)\;\Phi^{(-\frac{3}{2})}(w)\,.$ (107)

In order to get a non-trivial propagator, and up to an overall constant, the
appropriate guess is that they are given by the states with the same spins but
opposite chirality. Explicitly, we have

$\mathcal{Y}^{\varepsilon,\varepsilon_{4}\,(-\frac{3}{2})}_{J,m,J^{\prime},m^{\prime}}\,=\,\pm\frac{\sqrt{n_{5}}}{2j-1}e^{-\frac{3\varphi}{2}}\left(SVV^{\prime}\right)^{-\varepsilon}_{J,m,J^{\prime},m^{\prime}}e^{\frac{i\varepsilon_{4}}{2}\left(H_{4}+\varepsilon
H_{5}\right)}\,,$ (108)

where the negative (positive) sign holds for the cases $A$ and $B$ ($C$ and
$D$).

We can now compute the two-point functions in the unflowed Ramond sector. Only
diagonal pairings are non-zero, by construction. Denoting the antiholomorphic
sector contributions by “c.c.”, we obtain

$\displaystyle\langle\mathcal{Y}_{[A]}^{\varepsilon_{4},(-\frac{1}{2})}\mathcal{Y}_{[A]}^{-\varepsilon_{4},(-\frac{3}{2})}\rangle$
$\displaystyle\,=\,\frac{n_{5}}{|z_{12}|^{4}}\braket{V_{1}V_{2}}\braket{V^{\prime}_{1}V^{\prime}_{2}}\left(\dfrac{(2j-1)}{(j-m-\frac{1}{2})(j^{\prime}+m^{\prime}+\frac{1}{2})}\times\text{c.c.}\right),$
(109a)
$\displaystyle\langle\mathcal{Y}_{[B]}^{\varepsilon_{4},(-\frac{1}{2})}\mathcal{Y}_{[B]}^{-\varepsilon_{4},(-\frac{3}{2})}\rangle$
$\displaystyle\,=\,\frac{n_{5}}{|z_{12}|^{4}}\braket{V_{1}V_{2}}\braket{V^{\prime}_{1}V^{\prime}_{2}}\left(\dfrac{(2j-1)(j+m-\frac{1}{2})}{(j^{\prime}+m^{\prime}+\frac{1}{2})}\times\text{c.c.}\right),$
(109b)
$\displaystyle\langle\mathcal{Y}_{[C]}^{\varepsilon_{4},(-\frac{1}{2})}\mathcal{Y}_{[C]}^{-\varepsilon_{4},(-\frac{3}{2})}\rangle$
$\displaystyle\,=\,\frac{n_{5}}{|z_{12}|^{4}}\braket{V_{1}V_{2}}\braket{V^{\prime}_{1}V^{\prime}_{2}}\left(\dfrac{(2j-1)(j^{\prime}-m^{\prime}+\frac{1}{2})}{(m-j+\frac{1}{2})}\times\text{c.c.}\right),$
(109c)
$\displaystyle\langle\mathcal{Y}_{[D]}^{\varepsilon_{4},(-\frac{1}{2})}\mathcal{Y}_{[D]}^{-\varepsilon_{4},(-\frac{3}{2})}\rangle$
$\displaystyle\,=\,\frac{n_{5}}{|z_{12}|^{4}}\braket{V_{1}V_{2}}\braket{V^{\prime}_{1}V^{\prime}_{2}}\left((2j-1)(m+j-\frac{1}{2})(j^{\prime}-m^{\prime}+\frac{1}{2})\times\text{c.c.}\right),$
(109d)

where here $\braket{V_{1}V_{2}}=\braket{V_{m_{1}-1/2}V_{-m_{1}+1/2}}$ and
$\braket{V^{\prime}_{1}V^{\prime}_{2}}=\braket{V^{\prime}_{m_{1}^{\prime}-1/2}V^{\prime}_{-m_{1}^{\prime}+1/2}}$.
As expected, the coefficients resulting from the linear combinations
effectively shift the spins $j\to J$ and $j^{\prime}\to J^{\prime}$ in the
gamma functions coming from the bosonic correlators.

Among the states described above, the only chiral one corresponds to the SU(2)
highest-weight operator of type $A$. To simplify notation and for later
convenience, we suppress the SU(2) labels and use the label $j$ rather than
$J$ (here $J=j-1/2$). This operator has quantum numbers

$\mathcal{Y}_{j,m[A]}^{+,\varepsilon_{4}}~{}:~{}~{}~{}~{}h\>\\!=\>\\!J\>\\!=\,j-\frac{1}{2}\,=\,j^{\prime}+\frac{1}{2}\,=\>\\!J^{\prime}\>\\!=\>\\!m^{\prime}\,.$
(110)

The explicit form of this operator is simpler than the generic Ramond sector
operator, and is given by

$\mathcal{Y}_{j,m[A]}^{+,\varepsilon_{4}}\,=\,e^{-\frac{\varphi}{2}}\left(S_{+++}V_{j,m-\frac{1}{2}}+S_{-+-}V_{j,m+\frac{1}{2}}\right)_{j-\frac{1}{2},m}V^{\prime}_{j-1,j-1}e^{\frac{i\varepsilon_{4}}{2}\left(H_{4}+H_{5}\right)}\,.$
(111)

As in the NS sector, the rest of the chiral operators belong to the spectrally
flowed sectors. These are obtained by acting with the spectral flow operator
(95). From the flowed Virasoro condition, similar to Eq. (96), the resulting
operators must have $m=m^{\prime}$ (together with the relations in (110)), and
so only the second term in (111) is non-vanishing, giving rise to the flowed
Ramond operators (we now suppress also the label $m=J=j-1/2$)

$\mathcal{Y}^{+,\varepsilon_{4},\omega}_{j\>\\![A]}\,=\,e^{-\frac{\varphi}{2}}S_{-+-}^{\omega}\>\\!V_{j,j}^{\omega}\;\\!V_{j-1,j-1}^{{}^{\prime}\omega}\>\\!e^{\frac{i\varepsilon_{4}}{2}\left(H_{4}+H_{5}\right)}\,,$
(112)

where

$S_{-+-}^{\omega}\,\equiv\,e^{\frac{i}{2}\left[(1+2\omega)(-H_{1}+H_{2})-H_{3}\right]}\,.$
(113)

The corresponding two-point function is equivalent to the highest-weight case
of (109a), up to the usual additional $\delta_{\omega_{1},-\omega_{2}}$
factor.

####  3.3.3 Holographic dictionary for light chiral primaries

We have reviewed three sets of $m$-basis vertex operators corresponding to
chiral primaries of the holographic CFT. Two sets are in the NS sector:
$\mathcal{W}^{-}_{j,m}$ and $\mathcal{X}^{+}_{j,m}$, together with the
corresponding spectral flowed operators $\mathcal{W}^{\omega}_{j}$ and
$\mathcal{X}^{\omega}_{j}$. The third set is in the Ramond sector,
$\mathcal{Y}_{j,m[A]}^{+,\varepsilon_{4}}$ and its spectral flow,
$\mathcal{Y}^{+,\varepsilon_{4},\omega}_{j\>\\![A]}$. From now on we shall
omit the label $A$ and the AdS${}_{3}\times\mathbb{S}^{3}$ chirality
$\epsilon=+$, denoting this operator by
$\mathcal{Y}^{\omega,\varepsilon_{4}}_{j}$. Recall that in the spectral flowed
sectors, the remaining states in the zero-mode algebra are obtained by acting
with $J_{0}^{+}$, as discussed around Eqs. (59) and (100).

In order to reconstruct the corresponding local operators of the spacetime
CFT, we need to combine such modes by going to the $x$-basis, as done in Eqs.
(54) and (59) in the bosonic SL(2,$\mathbb{R}$) model.444In this paper we are
interested in operators of fixed R-charge. Hence, and in contrast to
Dabholkar:2007ey ; Giribet:2007wp , we do not introduce isospin variables in
the SU(2) sector. For the operators at hand, the sum over $m$ in the analog of
Eqs. (54) and (59) factorizes between fermionic and bosonic contributions,
leading to expressions of the following form:

$\displaystyle\mathcal{W}_{j}^{\omega}(x)$ $\displaystyle=$ $\displaystyle
e^{-\varphi}\psi^{\omega}(x)e^{i\omega
H_{2}}V_{j}^{\omega}(x)V^{\omega}_{j-1,j-1}\,,$ (114a)
$\displaystyle\mathcal{X}_{j}^{\omega}(x)$ $\displaystyle=$ $\displaystyle
e^{-\varphi}\psi^{\omega-1}(x)e^{i(\omega+1)H_{2}}V_{j}^{\omega}(x)V^{\omega}_{j-1,j-1}\,,$
(114b) $\displaystyle\mathcal{Y}_{j}^{\omega,\varepsilon_{4}}(x)$
$\displaystyle=$ $\displaystyle
e^{-\frac{\varphi}{2}}S^{\omega}(x)V_{j}^{\omega}(x)V^{\omega}_{j-1,j-1}e^{\frac{i\varepsilon_{4}}{2}\left(H_{4}+H_{5}\right)}\,.$
(114c)

Here $\psi^{\omega}(x)$ and $S^{\omega}(x)$ are defined as follows. First,
note that the fermions $\psi^{a}$ introduced in (70), which generate an affine
$\widehat{\text{sl}(2,\mathbb{R})}_{-2}$ algebra with level $k_{\psi}=-2$,
constitute affine primaries with spin $J_{\psi}=-1$, on which, however, the
zero-mode currents act as in (46) but with555This convention is perhaps more
natural from the SL(2,$\mathbb{R}$) point of view, but we have decided to
employ the conventions used in the most relevant literature for us, i.e.
Maldacena:2000hw ; Maldacena:2001km ; Dabholkar:2007ey ; Giribet:2007wp .
$J\to 1-J$, i.e. $J^{\pm}_{0}|J,m\pm 1\rangle=(m\pm J)|J,m\pm 1\rangle$. As a
consequence, and in contrast with what happens with bosonic primaries, the
action $J_{0}^{+}$ on $\psi^{-}$, the lowest-weight state, is truncated.
Identifying $\psi^{\omega=0}(0)=\psi^{-}$, the resulting $x$-basis operator
has only three terms:

$\psi^{\omega=0}(x)\,\equiv\,\psi^{-}(x)\,=\,e^{xJ_{0}^{+}}\psi^{-}e^{-xJ_{0}^{+}}\,=\,\psi^{-}-2x\>\\!\psi^{3}+x^{2}\psi^{+}\,.$
(115)

Of course, we already knew the action of the currents on $\psi^{a}$ from (70),
but the advantage of the above discussion is that it extends to the spectrally
flowed sectors. Indeed, $\psi^{\omega}(0)=\sqrt{n_{5}}e^{-i(1+\omega)H_{1}}$
is the lowest-weight component of a spin $J_{\psi}^{\omega}=-1-\omega$ field.
The corresponding $x$-basis operator is of the form

$\psi^{\omega}(x)\,\equiv\,\sqrt{n_{5}}\;e^{xJ_{0}^{+}}e^{-i(1+\omega)H_{1}}e^{-xJ_{0}^{+}}\,,$
(116)

and contains $1-2J_{\psi}^{\omega}=2\omega+3$ terms.

Similarly, the spin field (113) is the lowest-weight component of a
representation with SL(2,$\mathbb{R}$) and SU(2) spins
$(J^{\omega}_{S},J^{\prime\omega}_{S})=(-\frac{1}{2}-\omega,\frac{1}{2}+\omega)$,
such that the $x$-basis operator is

$S^{\omega}(x)\,\equiv\,e^{xJ_{0}^{+}}S_{-+-}^{\omega}e^{-xJ_{0}^{+}}\,,$
(117)

and contains $2(1+\omega)$ terms.

We thus have three types of vertices, $\mathcal{W}_{j}^{\omega}(x,z)$,
$\mathcal{X}_{j}^{\omega}(x,z)$ and $\mathcal{Y}_{j}^{\omega,\pm}(x,z)$, which
correspond to local chiral primary operators of the boundary theory. As
mentioned before, these should be completed with analogous antiholomorphic
excitations, which have been omitted in the presentation. As discussed around
Eqs. (100) and (110), their boundary weights are given by

$h\left[\mathcal{W}_{j}^{\omega}\right]=j_{\omega}-1\;,\quad\qquad
h\left[\mathcal{Y}_{j}^{\omega,\pm}\right]=j_{\omega}-\frac{1}{2}\;,\quad\qquad
h\left[\mathcal{X}_{j}^{\omega}\right]=j_{\omega}\,,$ (118)

where $j=j^{\prime}+1$ and so

$j_{\omega}=j+\frac{n_{5}}{2}\omega,\qquad
j=1,\frac{3}{2},\dots,\frac{n_{5}}{2},\qquad\omega=0,1,\dots\,.$ (119)

Up to normalization, which will be fixed shortly, these operators are
identified with the chiral primaries of the holographic CFT listed in Eq.
(39). In the $\mathcal{Y}_{j}^{\omega,\varepsilon_{4}}$ tower,
$\varepsilon_{4}=\pm$ is identified with the boundary quantum number $\dot{A}$
in (38). Note also that the $\mathcal{W}_{j}^{\omega}$ tower starts with the
identity operator of the boundary theory.666As discussed in Kutasov:1999xu ,
this is subtle, since the operator is actually a spectral-flow-sector
dependent constant. This subtlety is related to the fact that spectral flow
charge is not conserved in $n$-point functions with $n\geq 3$, and was
resolved in Kim:2015gak  by performing a Legendre transform to the
microcanonical ensemble, in which the total number of fundamental strings is
fixed. The dictionary is summarized in Table 1.

Worldsheet | Weight $h$ | Twist $n$ | Dual Operator
---|---|---|---
$\mathcal{W}^{\omega}_{j}$ | $j_{\omega}-1$ | $2j_{\omega}-1$ | $O^{-}_{n}$
$\mathcal{Y}^{\omega,\varepsilon_{4}}_{j}$ | $j_{\omega}-\frac{1}{2}$ | $2j_{\omega}-1$ | $O^{\dot{A}}_{n}$
$\mathcal{X}^{\omega}_{j}$ | $j_{\omega}$ | $2j_{\omega}-1$ | $O^{+}_{n}$
Table 1: Dictionary between worldsheet vertex operators and chiral primaries
of the holographically dual CFT. Here $j_{\omega}=j+\frac{n_{5}}{2}\omega$,
$j=1,\frac{3}{2},\dots,\frac{n_{5}}{2}$, and $\omega=0,1,\dots$

Although most of the chiral primaries of the holographic CFT are accounted for
by considering the ranges given in (119), it is known that those belonging to
the the $n$-twisted sectors with $n=pn_{5}$ with $p\in\mathbb{N}$ are still
missing Dabholkar:2007ey ; Giribet:2007wp . These would correspond to
operators sitting at the boundary of the allowed range of $j$ in Eq. (45), at
which the spectrum becomes degenerate and the continuous representations
appear Teschner:1997ft ; Giveon:2001up . The absence of these states in the
worldsheet spectrum has been related to the fact that the NS5-F1 model sits at
a singular point in the moduli space where all RR modes are turned off
Seiberg:1999xz .

The twist $n$ of the holographic CFT operators is identified as
Dabholkar:2007ey ; Giribet:2007wp

$n\,=\,2j-1+n_{5}\omega\,.$ (120)

Let us make a side comment regarding the limit in which there is only a single
NS5 brane sourcing the background, $n_{5}=1$. This model is special in that it
corresponds to the tensionless limit of the theory. It has to be treated with
care since the usual RNS formalism outlined above breaks down due to the fact
that the bosonic SU(2) level would become negative. It was shown in
Eberhardt:2018ouy ; Eberhardt:2020bgq  that for $n_{5}=1$ the worldsheet
theory is exactly dual to the supersymmetric symmetric orbifold
$(\mathbb{T}^{4})^{n_{1}}/S_{n_{1}}$. In this model, the discrete series is
absent, the spectrum truncates to $j=1/2$, physical states have $\omega>0$,
and the spectral flow charge is identified with $n$, i.e. $n=\omega$. Eq.
(120) is the known generalization of this relation for $n_{5}>1$.

In order to fix the normalization of the operators, we compute their two-point
functions. Making use of

$\displaystyle\langle\psi^{\omega_{1}}(x_{1})\psi^{\omega_{2}}(x_{2})\rangle\times{\rm
c.c.}\,=\,x_{12}^{2(\omega_{1}+1)}\langle
e^{-i(1+\omega_{1})H_{1}}\>\\!e^{i(1-\omega_{2})H_{1}}\rangle\times{\rm
c.c.}\,=\,\delta_{\omega_{1},-\omega_{2}}\frac{|x_{12}|^{4(\omega_{1}+1)}}{|z_{12}|^{2(\omega_{1}+1)^{2}}}\,,$
$\displaystyle\langle
S^{\omega_{1},+}(x_{1})S^{\omega_{2},-}(x_{2})\rangle\times{\rm
c.c.}\,=\,\frac{x_{12}^{2\omega_{1}+2}}{z_{12}}\langle
S_{-+-}^{\omega_{1}}S_{+-+}^{\omega_{2}}\rangle\times{\rm
c.c.}\,=\,\delta_{\omega_{1},-\omega_{2}}\frac{|x_{12}|^{4\omega_{1}+2}}{|z_{12}|^{4\omega_{1}(\omega_{1}+1)+\frac{5}{2}}}\,,$

we obtain

$\displaystyle\langle\mathcal{W}_{j_{1}}^{\omega_{1}}(x_{1},z_{1})\mathcal{W}_{j_{2}}^{\omega_{2}}(x_{2},z_{2})\rangle$
$\displaystyle\,=\,$
$\displaystyle\frac{n_{5}^{2}\>\\!B(j_{1})}{16}\frac{\delta(j_{1}-j_{2})\>\\!\delta_{\omega_{1},-\omega_{2}}}{|x_{12}|^{4(j_{1}-1)+2n_{5}\omega_{1}}|z_{12}|^{4}}\,,$
(121)
$\displaystyle\langle\mathcal{X}_{j_{1}}^{\omega_{1}}(x_{1},z_{1})\mathcal{X}_{j_{2}}^{\omega_{2}}(x_{2},z_{2})\rangle$
$\displaystyle\,=\,$
$\displaystyle\frac{n_{5}^{2}\>\\!B(j_{1})}{16}\frac{\delta(j_{1}-j_{2})\>\\!\delta_{\omega_{1},-\omega_{2}}}{|x_{12}|^{4j_{1}+2n_{5}\omega_{1}}|z_{12}|^{4}}\,,$
(122)
$\displaystyle\langle\mathcal{Y}_{j_{1}}^{\omega_{1},\pm}(x_{1},z_{1})\mathcal{Y}_{j_{2}}^{\omega_{2},\mp}(x_{2},z_{2})\rangle$
$\displaystyle\,=\,$
$\displaystyle\frac{n_{5}\>\\!B(j_{1})}{(2j_{1}-1+n_{5}\omega_{1})^{2}}\frac{\delta(j_{1}-j_{2})\>\\!\delta_{\omega_{1},-\omega_{2}}}{|x_{12}|^{4j_{1}+2(n_{5}\omega_{1}-1)}|z_{12}|^{4}}\,.$
(123)

Here we have used that in the spectrally flowed R sectors the denominator in
the extra factor of the corresponding vertex operator in the “$-\frac{3}{2}$”
picture is shifted as $2j-1\to 2j-1+n_{5}\omega$ as compared to (108).
Moreover, the $V_{\mathrm{conf}}$ factor in Eq. (60) is cancelled by the pole
appearing in (53) upon setting $m_{1}=j_{1}$.

The string two-point function is then obtained by including an extra factor
$g_{s}^{-2}\sim n_{1}/n_{5}$ as usual in string perturbation theory, fixing
$z_{1}=0$ and $z_{2}=1$, and dividing by a volume of the conformal group that
leaves such worldsheet insertions fixed. As discussed in Kutasov:1999xu ;
Maldacena:2001km , this cancels the divergence coming from
$\delta(j_{1}-j_{2})$, leaving a constant $j$-dependent factor of the form
$(2j-1+n_{5}\omega)$. As a consequence, the holographic dictionary reads

$\displaystyle O_{n}^{--}(x,\bar{x})$ $\displaystyle\leftrightarrow
A_{\mathrm{NS}}(j,\omega)\mathcal{W}_{j}^{\omega}(x,\bar{x}),\qquad\qquad
O_{n}^{++}(x)\leftrightarrow
A_{\mathrm{NS}}(j,\omega)\mathcal{X}_{j}^{\omega}(x,\bar{x}),$ (124)
$\displaystyle\qquad\qquad\qquad O_{n}^{\dot{A}\dot{B}}\leftrightarrow
A_{\mathrm{R}}(j,\omega)\mathcal{Y}_{j}^{\omega,\dot{A}\dot{B}}(x,\bar{x}),$
(125)

with $n$ related to the worldsheet quantum numbers as in (120) and where
Gaberdiel:2007vu ; Dabholkar:2007ey

$A_{\mathrm{NS}}(j,\omega)=\frac{4g_{s}}{\sqrt{n_{5}^{2}\,B(j)(2j-1+n_{5}\omega)}}\;,\quad\qquad
A_{\mathrm{R}}(j,\omega)=g_{s}\sqrt{\frac{(2j-1+n_{5}\omega)}{n_{5}\,B(j)}}\,.$
(126)

Of course, this identification is only expected to hold at small string
coupling, i.e. for $n_{1}\gg n_{5}$. Analysis and comparison of boundary and
worldsheet three-point functions was carried out in Pakman:2007hn ;
Dabholkar:2007ey ; Gaberdiel:2007vu .

##  4 Null-gauged description and worldsheet spectrum

We now proceed to describe massless excitations in the worldsheet theories
associated with the heavy backgrounds we study. We describe in detail how to
obtain the low-lying physical states via BRST quantization in these null-
gauged models, both in the NSNS and RR sectors. In the subset of the
backgrounds that preserve some supersymmetry, we discuss the BPS light
excitations.

In the full null-gauged models, these massless vertex operators describe
linearized fluctuations around the full asymptotically linear dilaton
solutions describing the heavy states, and so can be thought of as worldsheet
representatives of light states belonging to the Little String Theory living
on the NS5 branes.

Our main interest in this work will be computing correlators in the IR AdS3
limit, in which we have reviewed the fact that the backgrounds are related to
orbifolded AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$ via a spacetime
spectral flow large coordinate transformation. We describe how the AdS3 limit
can be taken on the worldsheet vertex operators. This leads to states that can
be understood holographically, in the spacetime (fractionally) spectrally
flowed frame defining the heavy background, as discussed around Eq. (27).

###  4.1 BRST quantization

We start by reviewing the quantization of the class of worldsheet coset models
introduced in Section 2, which describe the propagation of superstrings in the
JMaRT backgrounds and their (BPS and/or two-charge) limits Martinec:2017ztd ;
Martinec:2018nco ; Martinec:2020gkv ; Bufalini:2021ndn .

Before gauging, we have the WZW model associated to the (10+2)-dimensional
group manifold
SL(2,$\mathbb{R}$)$\,\times\,$SU(2)$\,\times\,\mathbb{R}_{t}\,\times\,$U(1)${}_{y}\,\times\,$U(1)4
as introduced in (1). This is described simply by adding the extra time
direction $t$ and spatial circle $y$ to the matter content employed in the
previous section, together with the corresponding fermionic partners
$\lambda^{t}$ and $\lambda^{y}$. The latter are bosonized using a canonically
normalised scalar $H_{6}$ as

$\displaystyle\lambda_{t}$
$\displaystyle=\frac{1}{2}\left(e^{iH_{6}}-e^{-iH_{6}}\right)\;,\quad\qquad\lambda_{y}=\frac{1}{2}\left(e^{iH_{6}}+e^{-iH_{6}}\right)\,,$
(127) $\displaystyle\hskip 28.45274pti\>\\!\partial
H_{6}=2\,\lambda^{t}\lambda^{y}\;,\quad\qquad H_{6}^{\dagger}=-H_{6}\,.$ (128)

The holomorphic parts of their OPEs are

$-t(z)t(w)\sim
y(z)y(w)\sim-\frac{1}{2}\log(z-w)\,,\quad~{}-\lambda^{t}(z)\lambda^{t}(w)\sim\lambda^{y}(z)\lambda^{y}(w)\sim\frac{1}{2}\frac{1}{(z-w)}\,,$
(129)

and they give additional free field contributions to the matter $T$ and $G$ in
(74) and (75), namely

$T_{(ty)}\,=\,\partial t\partial t-\partial y\partial y\;,\quad\qquad
G_{(ty)}\,=\,2i\left(-\lambda^{t}\partial t+\lambda^{y}\partial y\right).$
(130)

Introducing $P_{t,L}=i\>\\!\partial t\,$, $P_{t,R}=i\>\\!\bar{\partial}t\,$,
$P_{y,L}=i\>\\!\partial y\,$, and $P_{y,R}=i\>\\!\bar{\partial}y$, we gauge
the chiral null currents777The symbol $J$ for the current operators should not
be confused with the total SL(2,$\mathbb{R}$) spin that has appeared in
Section 3.3. The meaning should be clear from the context.

$J\,=\,i{\cal{J}}\,=\,J^{3}+l_{2}K^{3}+l_{3}P_{t,L}+l_{4}P_{y,L}\ ,\
\quad\bar{J}\,=\,i\bar{{\cal{J}}}\,=\,\bar{J}^{3}+r_{2}\bar{K}^{3}+r_{3}P_{t,R}+r_{4}P_{y,R}\,,$
(131)

which are the quantum operator versions of the classical currents in Eq. (4).
The supersymmetric partners of the currents $J$ and $\bar{J}$ are

$\boldsymbol{\lambda}=\psi^{3}+l_{2}\chi^{3}+l_{3}\lambda^{t}+l_{4}\lambda^{y}\;,\quad\qquad\bar{\boldsymbol{\lambda}}=\bar{\psi}^{3}+r_{2}\bar{\chi}^{3}+r_{3}\bar{\lambda}^{t}+r_{4}\bar{\lambda}^{y}\,.$
(132)

To perform the null gauging, one introduces additional fermionic and bosonic
first-order ghosts, denoted by $(\tilde{b},\tilde{c})$ and
$(\tilde{\beta},\tilde{\gamma})$, with conformal weights $\Delta[\tilde{c}]=0$
and $\Delta[\tilde{\gamma}]=1/2$ Martinec:2020gkv . The central charges
$c_{\tilde{b}\tilde{c}}=-2$ and $c_{\tilde{\beta}\tilde{\gamma}}=-1$ cancel
the additional matter contribution $c_{ty}=3$. The
$(\tilde{\beta},\tilde{\gamma})$ system has no background charge and is
bosonized via

$\tilde{\beta}=e^{-\tilde{\varphi}}\partial\tilde{\xi}\;,\quad\qquad\tilde{\gamma}=\tilde{\eta}\,e^{\tilde{\varphi}}\,.$
(133)

We shall momentarily introduce a modified BRST charge that imposes invariance
under the action of the null currents (131) and their supersymmetric partners
(132). Physical operators in 9+1 dimensions will be given by states of the
ungauged (10+2)-dimensional WZW model that survive the gauging procedure
Martinec:2020gkv . Of course, and as we shall see shortly, the Virasoro
conditions and the expressions of the BRST-exact states will be modified
accordingly. We consider a set of mutually local operators before the
gaugings, i.e. we perform the analog of the GSO projection in the
(10+2)-dimensional model. We thereby obtain get a tachyon-free spectrum in the
gauged models.

Underlying this procedure is the fact that for the case of chiral null
gaugings, the Polyakov-Wiegmann identity allows one to rewrite the gauged
action of the downstairs model into a form identical to that of the upstairs
model in terms of a new gauge-invariant variables Chung:1992mj ; Israel:2004ir
, see also Bufalini:2021ndn . This is achieved at the level of the path
integral by means of a field redefinition with a Jacobian that is almost
trivial except for a factor which, when exponentiated, gives rise to the
additional ghost fields described above.

Explicitly, physical operators in the coset model are defined by the
cohomology classes of the BRST charge Martinec:2020gkv

$\mathcal{Q}=\oint
dz\left[c\left(T+T_{\beta\gamma\bar{\beta}\bar{\gamma}}\right)+\gamma
G+\tilde{c}J+\tilde{\gamma}\boldsymbol{\lambda}+\text{ghosts}\right]\>\\!,$
(134)

where the last two terms implement the null-gauging procedure. Whether the
resulting spectrum is supersymmetric or not depends on whether some linear
combination(s) of the following supercharges are BRST-invariant
Martinec:2020gkv ,

$Q_{\varepsilon}=\oint
dz\,e^{-\left(\varphi-\tilde{\varphi}\right)/2}S_{\varepsilon}\;,\quad\qquad
S_{\varepsilon}=\exp\left(\frac{i}{2}\sum_{I=1}^{6}\varepsilon_{I}H_{I}\right).$
(135)

For these to be mutually local, we impose the analog of the GSO projection in
(10+2) dimensions,

$\prod_{I=1}^{6}\varepsilon_{I}=1\,.$ (136)

We shall discuss the conditions for spacetime supersymmetry after we have
analyzed more general Ramond sector vertex operators, around Eq. (156). For
now, we emphasize that only a subset of the backgrounds we consider preserve
some spacetime supersymmetry.

###  4.2 The unflowed NS sector

We now analyze physical NS sector states of the gauged models, focusing on
states with no spectral flow charges in SL(2,$\mathbb{R}$) or SU(2), and no
winding charge around the $y$-circle. As usual, the lightest physical
operators come with a single fermionic excitation on top of the tachyon state

$\mathcal{T}_{j,m,j^{\prime},m^{\prime}}\,=\,e^{-\varphi}V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}e^{i(-E\,t+P_{y}y)}\,.$
(137)

Note that since $t$ is a non-compact direction and $\omega_{y}=0$, both $E$
and $P_{y}$ are identical on the left and on the right sectors. For massless
states, the $L_{0}$ and $\bar{L}_{0}$ Virasoro constraints both read

$0\,=\,-\frac{j(j-1)}{n_{5}}+\frac{j^{\prime}(j^{\prime}+1)}{n_{5}}-\frac{1}{4}E^{2}+\frac{1}{4}P_{y}^{2}\,.$
(138)

Moreover, operators are uncharged with respect to the null-currents
$J,\bar{J}$ in (131) if and only if their quantum numbers are related by

$0=m+l_{2}\,m^{\prime}+\frac{l_{3}}{2}E+\frac{l_{4}}{2}P_{y}\;,\quad\qquad
0=\bar{m}+r_{2}\,\bar{m}^{\prime}+\frac{r_{3}}{2}E+\frac{r_{4}}{2}P_{y}\;.$
(139)

We will work in the canonical “$-1$” picture for the $\varphi$ ghost. On the
other hand, the fact that $\tilde{\varphi}$ has background charge
$Q_{\tilde{\varphi}}=0$ allows us to build NS states directly at
$\tilde{\varphi}$-picture zero. BRST-closed operators must then have a
vanishing second-order pole in their OPE with the supercurrent $G$, and
vanishing first-order pole in their OPE with the fermionic current
$\boldsymbol{\lambda}$ given in (132).

As can be expected from the fact that the $\mathbb{T}^{4}$ is untouched by the
gaugings, the simplest solutions are the 6D scalars

$\mathcal{V}^{i}_{j,m,j^{\prime},m^{\prime}}=e^{-\varphi}\lambda^{i}V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\,e^{i(-Et+P_{y}y)}\;,\quad\qquad
i=6,\dots,9.$ (140)

These are direct analogs of the global AdS3 states defined in Eq. (86a). They
were considered in detail in Martinec:2018nco , and their energies were
matched with those of the minimally coupled scalar perturbations on top of the
JMaRT background as computed in supergravity.

The remaining massless vertex operators will constitute the beginning of the
main new results of this work. They are slightly more involved to construct,
due to the fact that their polarization lies in a direction in which the null
currents act non-trivially. An important consequence is that the
raising/lowering operators $J^{\pm}_{0}$ and $K^{\pm}_{0}$ do not commute with
the BRST charge $\mathcal{Q}$ anymore. So, unlike in global
AdS${}_{3}\times\mathbb{S}^{3}\times\mathbb{T}^{4}$ as reviewed in Section
3.2, physical states need not have definite SL(2,$\mathbb{R}$) and SU(2)
spins. They will, however, have definite projections $m,\bar{m},m^{\prime}$
and $\bar{m}^{\prime}$, and also well-defined energy $E$ and momentum $P_{y}$.

This situation is a consequence of the fact that the
AdS${}_{3}\times\mathbb{S}^{3}$ isometries are absent in the asymptotically
linear dilaton geometry. Nevertheless, these isometries are restored in the
IR, by taking $R_{y}$ large while keeping $ER_{y}$ and $P_{y}R_{y}$ fixed, see
Eqs. (20)–(23). In this regime, the vertex operators of the gauged models will
reduce to the AdS${}_{3}\times\mathbb{S}^{3}$ expressions in Eqs. (86b) and
(86c).

Let us consider a generic linear combination of NS sector vertex operators,

$e^{-\varphi}\left[\left(c^{r}\psi^{r}V_{j,m-r}V^{\prime}_{j^{\prime},m^{\prime}}+d^{r}\psi^{r}V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}-r}\right)+\left(c^{t}\lambda^{t}+c^{y}\lambda^{y}\right)V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\right]e^{i(-Et+P_{y}y)},$
(141)

where the notation mirrors that of the AdS${}_{3}\times\mathbb{S}^{3}$
expressions in Eq. (87), in particular summation over $r=+1,-1,0$ is implicit,
with “$0$” corresponding to the “$3$” direction of the respective algebras. Of
these eight degrees of freedom, two are removed by the conditions arising from
the $G$ and $\boldsymbol{\lambda}$ terms in the BRST charge, which
respectively read

$0\,=\,mc^{3}+(m-j)c^{+}+(m+j)c^{-}+m^{\prime}d^{3}+(j^{\prime}+m^{\prime})d^{+}+(j^{\prime}-m^{\prime})d^{-}+c^{t}\frac{E}{2}+c^{y}\frac{P_{y}}{2},$
(142)

and

$0\,=\,n_{5}\left(-c^{3}+l_{2}d^{3}\right)-l_{3}c^{t}+l_{4}c^{y}.$ (143)

This leaves six states, out of which two turn out to be BRST-exact. The first
exact state comes, as usual, from the action of $G$ on the tachyon operator
(137), while the second one has no global AdS3 counterpart and appears due to
the action of $\boldsymbol{\lambda}$ on the same state. Their explicit
expressions are

$\displaystyle\Phi_{G}\,=\,e^{-\varphi}$
$\displaystyle\left[\frac{2}{n_{5}}\frac{1}{2}V^{\prime}_{j^{\prime},m^{\prime}}\left((m-j+1)\psi^{-}V_{j,m+1}+(m+j-1)\psi^{+}V_{j,m-1}-2m\psi^{3}V_{j,m}\right)\right.$
(144)
$\displaystyle\,\,\,+\frac{2}{n_{5}}\frac{1}{2}V_{j,m}\left((j^{\prime}+m^{\prime}+1)\chi^{-}V^{\prime}_{j^{\prime},m^{\prime}+1}+(j^{\prime}-m^{\prime}+1)\chi^{+}V^{\prime}_{j^{\prime},m^{\prime}-1}+2m^{\prime}\chi^{3}V^{\prime}_{j^{\prime},m^{\prime}}\right)$
$\displaystyle\,\,\,+\left.\frac{1}{2}\left(-E\lambda^{t}+P_{y}\lambda^{y}\right)V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\right]e^{i(-E\,t+P_{y}y)},$

and

$\Phi_{\boldsymbol{\lambda}}\,=\,e^{-\varphi}\left[\psi^{3}+l_{2}\chi^{3}+l_{3}\lambda^{t}+l_{4}\lambda^{y}\right]V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\>e^{i(-E\,t+P_{y}y)},$
(145)

respectively. Such states are trivially BRST invariant since $G$ and
$\boldsymbol{\lambda}$ square to the Virasoro constraint (138) and the null
condition (7), and the relevant term in their product is
$G(z)\,\boldsymbol{\lambda}(0)\sim\boldsymbol{\lambda}(z)\,G(0)\sim J(0)/z$,
whose action vanishes by means of the condition (139). In the end, we are left
with four physical vertex operators to add to the four from the
$\mathbb{T}^{4}$ directions to give the correct eight polarizations in the
holomorphic sector in 9+1 dimensions.

We choose a basis for these four physical vertex operators such that, in the
AdS3 limit, they reduce to the basis of global AdS3 vertex operators described
around Eq. (87). We thus obtain

$\displaystyle\mathcal{W}^{\varepsilon}$ $\displaystyle\,=\,$ $\displaystyle
e^{-\varphi}\left[(\psi
V_{j})_{j+\varepsilon,m}V^{\prime}_{j^{\prime},m^{\prime}}+\left(c^{t}_{\varepsilon}\,\lambda^{t}+c^{y}_{\varepsilon}\,\lambda^{y}\right)V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\right]e^{i\left(-E\,t+P_{y}y\right)},$
(146a) $\displaystyle\mathcal{X}^{\varepsilon}$ $\displaystyle\,=\,$
$\displaystyle e^{-\varphi}\left[V_{j,m}(\chi
V^{\prime}_{j^{\prime}})_{j^{\prime}+\varepsilon,m^{\prime}}+\left(d^{t}_{\varepsilon}\,\lambda^{t}+d^{y}_{\varepsilon}\,\lambda^{y}\right)V_{j,m}V^{\prime}_{j^{\prime},m^{\prime}}\right]e^{i\left(-E\,t+P_{y}y\right)},$
(146b)

where the SL(2,$\mathbb{R}$) and SU(2) coefficients are those given in
(87)–(88), while the novel ones are888The coefficients $c^{t,y}$ and $d^{t,y}$
were reported in the Letter Bufalini:2022wyp  with a slightly different
notation, related by
$c^{t,y}_{\text{there}}=c^{t,y}_{\varepsilon,\text{here}}/c^{3}_{\varepsilon}$,
and likewise for $d^{t,y}$.

$c^{t}_{\varepsilon}\,=\,-c^{3}_{\varepsilon}\frac{n_{5}P_{y}}{l_{4}E+l_{3}P_{y}}\;,\quad\qquad
c^{y}_{\varepsilon}\,=\,c^{3}_{\varepsilon}\frac{n_{5}E}{l_{4}E+l_{3}P_{y}}\,,$
(147)
$d^{t}_{\varepsilon}\,=\,d^{3}_{\varepsilon}\frac{n_{5}l_{2}P_{y}}{l_{4}E+l_{3}P_{y}}\;,\quad\qquad~{}~{}d^{y}_{\varepsilon}\,=\,-d^{3}_{\varepsilon}\frac{n_{5}l_{2}E}{l_{4}E+l_{3}P_{y}}\,.$
(148)

By construction, the resulting states are polarized transverse to the gauge
directions. As anticipated, they are built out of a linear combination of
terms of spin $j$ and $j+\varepsilon$ ($j^{\prime}$ and
$j^{\prime}+\varepsilon$). Moreover, at leading order in the large $R_{y}$
expansion, the coefficients in the $t,y$ directions go to zero, since
$E,P_{y}\sim{\cal{O}}(1/R_{y})$, $l_{3,4}\sim\mathcal{O}(R_{y})$, and
$l_{2}\sim\mathcal{O}(1)$.

###  4.3 The unflowed R sector

We now describe the physical states in the R sector of the null-gauged model.
The computation turns out to be more involved than in the NS sector, since the
spin fields necessarily involve all $\varepsilon$-chiralities. As a
consequence, we will not find a situation akin to (146) in which a subset of
coefficients are exactly those of the global AdS${}_{3}\times\mathbb{S}^{3}$
operators. However, we will again show that in the AdS3 limit the vertex
operators will reduce to their global AdS${}_{3}\times\mathbb{S}^{3}$
counterparts.

We introduce AdS${}_{3}\times\mathbb{S}^{3}$ and
$\mathbb{R}_{t}\times\mathbb{S}^{1}_{y}\times\mathbb{T}^{4}$ spin
fields,999The order of the spin fields in
$\mathcal{S}_{\varepsilon_{6}\varepsilon_{4}\varepsilon_{5}}$ has been chosen
for convenience in order to reduce clutter in computations involving cocycle
factors.

$S_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}=e^{\frac{i}{2}\left(\varepsilon_{1}H_{1}+\varepsilon_{2}H_{2}+\varepsilon_{3}H_{3}\right)}\;,\quad\qquad\mathcal{S}_{\varepsilon_{6}\varepsilon_{4}\varepsilon_{5}}=e^{\frac{i}{2}\left(\varepsilon_{6}H_{6}+\varepsilon_{4}H_{4}+\varepsilon_{5}H_{5}\right)}\,.$
(149)

Recalling the definition of the AdS3 chirality $\varepsilon$ and the mutual
locality / chiral GSO projection in (10+2) dimensions, (136), we substitute
away $\varepsilon_{3}$ and $\varepsilon_{6}$ via

$\varepsilon_{3}=\varepsilon\varepsilon_{1}\varepsilon_{2}\;,\quad\qquad\varepsilon_{6}=\varepsilon\varepsilon_{4}\varepsilon_{5}\,.$
(150)

The $H_{4,5}$ exponentials are spectators under the action of $\mathcal{Q}$,
so the parameters $\varepsilon_{4},\varepsilon_{5}$ will label the vertex
operators. For fixed $\varepsilon_{4},\varepsilon_{5}$, we consider
$\varepsilon_{6}$ to be controlled by $\varepsilon$ through the second
equation in (150), and we will form linear combinations of different values
over $\varepsilon_{1}$, $\varepsilon_{2}$, $\varepsilon$.

We work with vertex operators in ghost pictures
$(q_{\varphi},q_{\tilde{\varphi}})=(-\frac{1}{2},+\frac{1}{2})$, for which the
$\boldsymbol{\lambda}$-constraint is non-trival, while there is no need to
worry about BRST-exact states. We thus make an ansatz for R sector vertices of
the following form:

$\mathcal{Y}^{\varepsilon_{4},\varepsilon_{5}}\,=\,e^{-(\varphi-\tilde{\varphi})/2}\sum_{\varepsilon_{1},\varepsilon_{2},\varepsilon}F^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}S_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}}\mathcal{S}_{\varepsilon_{6}\varepsilon_{4}\varepsilon_{5}}V_{j,m-\frac{\varepsilon_{1}}{2}}V^{\prime}_{j^{\prime},m^{\prime}-\frac{\varepsilon_{2}}{2}}e^{i(-E\>\\!t+P_{y}\>\\!y)}\,,$
(151)

Note that the coefficients
$F^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}$
are not determined by the representation theory of
SL(2,$\,\mathbb{R}$)$\times$SU(2), since the states will not in general have
definite spin.

The $c\>\\!T$ and $\tilde{c}\>\\!J$ terms of the BRST operator $\mathcal{Q}$
(134) act as in the NS sector. Hence the unflowed, non-winding states of the R
sector also satisfy both the Virasoro condition (138) and the bosonic null-
gauge constraint (139).

Next, the $e^{\tilde{\varphi}}\boldsymbol{\lambda}$ term in $\mathcal{Q}$
leaves $\varepsilon_{1,2,4,5}$ unchanged, so for this term we can treat
$\varepsilon_{1,2,4,5}$ as fixed, and focus on the sum over $\varepsilon=\pm$.
The resulting constraints on
$F^{\pm}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}$ form
a two-dimensional homogeneous linear system, which is degenerate due to the
null condition on the gauge parameters, Eq. (7). For each choice of
$\varepsilon_{1,2,4,5}$, we have

$\displaystyle\begin{aligned}
(l_{3}+\varepsilon_{4}\varepsilon_{5}l_{4})F^{-}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}\,&-\,i\sqrt{n_{5}}\,(1-\varepsilon_{1}\varepsilon_{2}l_{2})F^{+}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}\,=\,0\,,\\\
i\sqrt{n_{5}}(1+\varepsilon_{1}\varepsilon_{2}l_{2})F^{-}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}\,&-\,(l_{3}-\varepsilon_{4}\varepsilon_{5}l_{4})F^{+}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}\,=\,0\,.\end{aligned}$
(152)

These constraints halve the degrees of freedom. When $|l_{2}|=1$ (and so
$|l_{3}|=|l_{4}|$), some of the
$F^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}$
get set to zero. For a given $\varepsilon_{1,2,4,5}$, when neither of
$F^{\pm}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}$ get
set to zero, their ratio
${F^{-}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}}/{F^{+}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}}$
becomes determined. So the 32 d.o.f. remaining after imposing GSO in (10+2)
dimensions have now become 16, corresponding to $\varepsilon_{1,2,4,5}$ in our
parameterization.

Let us pause to discuss how Eq. (152) behaves in the large $R_{y}$ limit. We
have $l_{2}\sim\mathcal{O}(1)$ while $l_{3}+l_{4}\sim\mathcal{O}(R_{y})$ and
$l_{3}-l_{4}\sim\mathcal{O}(1/R_{y})$, from (11)–(13). When
$\varepsilon_{4}\varepsilon_{5}=+1$, we obtain $F^{+}\sim\mathcal{O}(1)$ and
$F^{-}\sim\mathcal{O}(1/R_{y})$, so at leading order in large $R_{y}$ we
obtain a purely positive chirality operator. Similarly, when
$\varepsilon_{4}\varepsilon_{5}=-1$, at leading order in large $R_{y}$ we
obtain a purely negative chirality operator. So we obtain operators of
definite AdS${}_{3}\times\mathbb{S}^{3}$ chirality $\varepsilon$, with
$\varepsilon_{4}\varepsilon_{5}=\varepsilon$, exactly as in Section 3.2, see
Eq. (103). As before, one of $\varepsilon_{4}$ or $\varepsilon_{5}$ remains
unfixed, say $\varepsilon_{4}$.

We now examine the action of $e^{\varphi}G$ on the R vertex operator ansatz
(151). This will reduce the remaining 16 degrees of freedom to the correct 8
physical polarizations in the holomorphic sector. It leads to the following
set of equations (we suppress the $\varepsilon_{4},\varepsilon_{5}$ subscripts
on the RHS for ease of notation):

$\displaystyle\mathcal{B}^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}\varepsilon_{4}\varepsilon_{5}}$
$\displaystyle\,\equiv\,\left(m+\varepsilon_{1}j-\frac{\varepsilon_{1}}{2}\right)F^{\varepsilon}_{(-\varepsilon_{1})\varepsilon_{2}}+i\,\varepsilon_{1}\varepsilon_{2}\,\left(j^{\prime}-\varepsilon_{2}m^{\prime}+\frac{1}{2}\right)F^{\varepsilon}_{\varepsilon_{1}(-\varepsilon_{2})}$
$\displaystyle\qquad\qquad\quad~{}-(\varepsilon
m+\varepsilon_{1}\varepsilon_{2}m^{\prime})F^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}}+\frac{i\sqrt{n_{5}}}{2}\left(\varepsilon_{4}\varepsilon_{5}P-\varepsilon\,E\right)F^{(-\varepsilon)}_{\varepsilon_{1}\varepsilon_{2}}\;=\;0\,.$
(153)

Comparing to the AdS${}_{3}\times\mathbb{S}^{3}$ BRST condition, the only new
term is the fourth and final one, proportional to
$F^{(-\varepsilon)}_{\varepsilon_{1}\varepsilon_{2}}$, which has the effect of
mixing the $\varepsilon$ chiralities. The first three terms are unchanged from
the AdS${}_{3}\times\mathbb{S}^{3}$ BRST condition, so the
AdS${}_{3}\times\mathbb{S}^{3}$ limit of this condition is simply to drop the
fourth term. Note that Eq. (152) implies that half of these equations are
redundant, and allows us to decouple the $F^{+}$ from the $F^{-}$
coefficients. Moreover, by using the Virasoro constraint (138), the bosonic
null-gauge condition (139), and the null constraint on the gauge parameters
(7), one can show that for fixed $\varepsilon_{4}$ and $\varepsilon_{5}$,
actually only two equations are linearly independent. For generic values of
quantum numbers, such that all denominators appearing below are nonzero, the
linearly independent equations can be taken to be

$\displaystyle\begin{aligned}
F^{+}_{-+}&=-i\,\frac{j^{\prime}-m^{\prime}+\frac{1}{2}}{j+m-\frac{1}{2}}\,F^{+}_{+-}+\frac{-j(j-1)+j^{\prime}(j^{\prime}+1)+m^{2}-m^{\prime
2}}{(j+m-\frac{1}{2})\left[m-m^{\prime}+\frac{l_{4}-\varepsilon_{4}\varepsilon_{5}l_{3}}{2(l_{2}-1)}\left(\varepsilon_{4}\varepsilon_{5}E-P_{y}\right)\right]}F^{+}_{++}\,,\\\
F^{+}_{--}&=\frac{m-m^{\prime}+\frac{l_{4}-\varepsilon_{4}\varepsilon_{5}l_{3}}{2(l_{2}-1)}\left(\varepsilon_{4}\varepsilon_{5}E-P_{y}\right)}{j+m-\frac{1}{2}}F^{+}_{+-}+i\,\frac{j^{\prime}+m^{\prime}+\frac{1}{2}}{j+m-\frac{1}{2}}\,F^{+}_{++}\,.\end{aligned}$
(154)

Alternatively, the two linearly independent equations can generically be taken
to be

$\displaystyle F^{-}_{-+}$
$\displaystyle=-i\,\frac{j^{\prime}-m^{\prime}+\frac{1}{2}}{j+m-\frac{1}{2}}\,F^{-}_{+-}+\frac{-j(j-1)+j^{\prime}(j^{\prime}+1)+m^{2}-m^{\prime
2}}{(j+m-\frac{1}{2})\left[-m-m^{\prime}+\frac{n_{5}(l_{2}-1)}{2(l_{4}-\varepsilon_{4}\varepsilon_{5}l_{3})}\left(\varepsilon_{4}\varepsilon_{5}E+P_{y}\right)\right]}F^{-}_{++}\,,$
$\displaystyle F^{-}_{--}$
$\displaystyle=\frac{-m-m^{\prime}+\frac{n_{5}(l_{2}-1)}{2(l_{4}-\varepsilon_{4}\varepsilon_{5}l_{3})}\left(\varepsilon_{4}\varepsilon_{5}E+P_{y}\right)}{j+m-\frac{1}{2}}F^{-}_{+-}+i\,\frac{j^{\prime}+m^{\prime}+\frac{1}{2}}{j+m-\frac{1}{2}}\,F^{-}_{++}\,.$
(155)

Let us pause again to check consistency with the
AdS${}_{3}\times\mathbb{S}^{3}$ limit. Setting
$\varepsilon_{4}\varepsilon_{5}=\varepsilon$ and taking the large $R_{y}$
limit of Eqs. (154), we indeed find that a solution is given by setting (the
AdS${}_{3}\times\mathbb{S}^{3}$ limit of)
$F^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}}$ to be equal to the values
$f^{\varepsilon}_{\varepsilon_{1}\varepsilon_{2}}$ specified in Eqs.
(104)–(106).

Similarly to AdS${}_{3}\times\mathbb{S}^{3}$, Eqs. (154) are two equations for
four unknowns, so for each $\varepsilon_{4}$, $\varepsilon_{5}$ there is a
two-parameter family of solutions, which we take to be parameterized by the
values of $F^{+}_{+\pm}$. If working with Eqs. (155), we take the two-
parameter family of solutions to be parameterized by the values of
$F^{-}_{+\pm}$. Together with $\varepsilon_{4}$, $\varepsilon_{5}$, this gives
8 physical polarizations.

In Section 3.2, for AdS${}_{3}\times\mathbb{S}^{3}$ these unfixed coefficients
were chosen such that the vertex operators transform appropriately under the
action of the currents $J^{\pm}$ and $K^{\pm}$. However, in the null-gauged
worldsheet theory associated to the full asymptotically linear dilaton
geometry this need not necessarily be the case.

In the cosets we fix these coefficients by requiring a reasonable IR limit. We
treat the different $\varepsilon$ chiralities separately. For $\varepsilon=1$
we set the particular components $F^{+}_{+\pm}$ equal to their values in the
AdS limit, $F^{+}_{+\pm}=f^{+}_{+\pm}$. The rest of the coefficients are then
obtained using Eqs. (154) and (152). Alternatively, for $\varepsilon=-1$ we
set $F^{-}_{+\pm}=f^{-}_{+\pm}$ and again solve for the remaining coefficients
using Eqs. (155) and (152).

We now turn to the analysis of the spacetime supercharges preserved by the
null gauging, following on from the initial discussion around Eq. (135) (see
also Martinec:2020gkv ). The supercharge analysis corresponds to the limit of
the Ramond vertex operator analysis in which we take
$j=j^{\prime}=E=P_{y}=0,\;\\!m=\frac{\varepsilon_{1}}{2},\;\\!m^{\prime}=\frac{\varepsilon_{2}}{2}$,
as can be seen by comparing Eqs. (135) and (151). In this limit, the center-
of-mass wavefunction trivializes and we are left with integrated vertex
operators involving only the spin fields. As before, we parameterize the
$\varepsilon_{i}$ according to (150); $\varepsilon_{4}$ and $\varepsilon_{5}$
are spectators that will label the supercharges; and we will sum over
$\varepsilon,\varepsilon_{1},\varepsilon_{2}$ as in (151). The $J$ constraint
(139) reduces to

$\varepsilon_{1}+\varepsilon_{2}\,l_{2}=0\qquad\Rightarrow\qquad\varepsilon_{1}\varepsilon_{2}=-l_{2}\,,$
(156)

so supersymmetry is preserved in the holomorphic sector if and only if
$|l_{2}|=1$, and thus $|l_{3}|=|l_{4}|$. The $\gamma G$ constraint (153)
reduces directly to

$\varepsilon=-1\,,$ (157)

so all supercharges have negative AdS${}_{3}\times\mathbb{S}^{3}$ chirality.
Then the $\lambda$ constraint in the first line of (152), with $F^{+}=0$,
reduces to

$l_{3}+\varepsilon_{4}\varepsilon_{5}\,l_{4}=0\,.$ (158)

So when $|l_{2}|=1$ and $|l_{3}|=|l_{4}|\neq 0$, there are four holomorphic
supercharges, labelled by say $\varepsilon_{2}$ and $\varepsilon_{4}$.

Combining this analysis with the corresponding one in the antiholomorphic
sector, we observe consistency with the passage below Eq. (24) describing
which subset of the backgrounds are supersymmetric. In terms of the spacetime
spectral flow parameters $s$, $\bar{s}$ introduced in Eq. (27), we have
$l_{2}=2s+1$, $r_{2}=2\bar{s}+1$. The circular supertube backgrounds of
Maldacena:2000dr ; Balasubramanian:2000rt  have $s=\bar{s}=0$ and preserve
supersymmetry in both holomorphic and antiholomorphic sectors; the backgrounds
of Lunin:2004uu ; Giusto:2004id ; Giusto:2004ip ; Giusto:2012yz  have