11institutetext: School of Biomedical Engineering & Imaging Sciences, King's
College London, UK. 22institutetext: Technical University Munich, Munich,
Germany 33institutetext: Helmholtz Center Munich, Germany 44institutetext:
Guy's and St Thomas' NHS Foundation Trust, London, UK 55institutetext:
Department of Cardiology, Heart and Lung Division, University Medical Center
Utrecht, Utrecht, The Netherlands. 66institutetext: Division of Cardiology,
Department of Medicine, Duke University, Durham, North Carolina, USA
77institutetext: British Heart Foundation Centre, King’s College London,
London, UK

# Automated Quality Controlled Analysis of 2D Phase Contrast Cardiovascular
Magnetic Resonance Imaging

Emily Chan 11 Ciaran O’Hanlon 11 Carlota Asegurado Marquez 11 Marwenie
Petalcorin 11 Jorge Mariscal-Harana 11 Haotian Gu Raymond J. Kim 7766 Robert
M. Judd 66 Phil Chowienczyk Julia A. Schnabel 11 7 7112233 Reza Razavi 1144
Andrew P. King 11 Bram Ruijsink 114455 Esther Puyol-Antón Joint last
authors.11

###### Abstract

Flow analysis carried out using phase contrast cardiac magnetic resonance
imaging (PC-CMR) enables the quantification of important parameters that are
used in the assessment of cardiovascular function. An essential part of this
analysis is the identification of the correct CMR views and quality control
(QC) to detect artefacts that could affect the flow quantification. We propose
a novel deep learning based framework for the fully-automated analysis of flow
from full CMR scans that first carries out these view selection and QC steps
using two sequential convolutional neural networks, followed by automatic
aorta and pulmonary artery segmentation to enable the quantification of key
flow parameters. Accuracy values of 0.958 and 0.914 were obtained for view
classification and QC, respectively. For segmentation, Dice scores were
$>$0.969 and the Bland-Altman plots indicated excellent agreement between
manual and automatic peak flow values. In addition, we tested our pipeline on
an external validation data set, with results indicating good robustness of
the pipeline. This work was carried out using multivendor clinical data
consisting of 986 cases, indicating the potential for the use of this pipeline
in a clinical setting.

Keywordscardiac magnetic resonance; deep learning; quality control; cardiac
function; view-selection; multivendor

##  1 Introduction

Cardiac Magnetic Resonance (CMR) is one of the most comprehensive modalities
for assessment of cardiovascular function, enabling the quantification of
cardiac volumes, myocardial motion and tissue characteristics. Aside from
these properties, CMR allows for the measurement of blood flow in the major
vessels. These measurements can be useful for quantifying a range of aspects
of cardiovascular function, including cardiac output, the shunting of blood
through intra-cardiac connections, the severity of valvular lesions (including
regurgitation), and vascular properties. While 2D flow can potentially be
measured in any blood vessel using CMR, pulmonary and aortic flow measurements
are the most important measures used in clinical assessments [7]. After
acquisition of 2D Phase Contrast (PC) images, flow measures are normally
obtained semi-automatically; a physician manually delineates the target blood
vessel and a semi-automatic (pixel threshold based) method propagates this
segmentation over the full cardiac cycle. After this, manual adjustment of the
target segmentation is performed and, importantly, the physician interrogates
the images and obtained flow data for quality. This quality control (QC)
includes checking for errors in imaging, such as fold-over and aliasing of the
blood flow signal, and checking the appearance of the flow curve for irregular
(unphysiological) behaviour [10].

Several works have proposed the semi-automatic or automatic quantification of
flow using PC-CMR images. For example, Bidhult et al. [2] incorporated shape
constraints to aid in active contour based semi-automatic segmentation of the
ascending aorta and pulmonary artery. Similarly, Bratt et al. [3] utilised a
modified U-Net to develop a method for fully-automatic aortic flow
quantification. However, these solutions ignore highly important steps in the
automation of flow assessment: image classification, to select the correct
images from a full CMR scan automatically, and the aforementioned quality
control, which is an integral part of CMR (flow) analysis.

In recent years, machine learning-enabled QC of CMR images has received
increasing interest, in particular for cine images. Tarroni et al. [14]
developed a decision forest-based solution for heart coverage estimation,
inter-slice motion detection and image contrast estimation in the cardiac
region, while Ruijsink et al. [12] proposed an end-to-end pipeline with
automated QC techniques to analyse cine CMR sequences. Other works focused on
detecting missing slices [18, 19] and motion artefacts [8], while some
addressed QC of automatic image segmentation [11, 1, 4, 16]. Additionally,
Vergani et al. [15] recently proposed the use of convolutional neural networks
for the automatic quality controlled selection of cine images for analysis. To
the best of our knowledge, the use of deep learning (DL) for view selection
and QC prior to flow quantification has not been explored with PC-CMR images.
Moreover, existing solutions for automated flow quantification have generally
been trained on single-centre/highly selective data sets, making their
generalisation to external data sets challenging.

In this paper, we propose a novel framework that takes into account these
important aspects to provide a fully-automated pipeline from image-to-report.
We develop a DL-based image classification algorithm to detect 2D PC flow
imaging, and classify/select the most important target sequences for analysis
(aortic flow and pulmonary flow); a QC algorithm to detect image quality
related errors; and a robust segmentation pipeline for 2D pulmonary and aortic
flow. Our pipeline calculates key parameters of interest in flow analysis:
peak, net, total forward and total backward flow. Furthermore, we utilise an
external validation set to assess the generalisability of our framework.

##  2 Methods

The framework we present is composed of four steps: 1) view selection aimed at
identifying conventional PC-CMR views, 2) QC to detect image quality related
artefacts, 3) automated segmentation of PC images, and 4) parameter extraction
(see Figure 1 for a summary). These steps are described below:

Figure 1: Overview of the proposed AI-based framework for automatic flow
quantification from a full CMR scan. The first block, view selection, aims to
identify conventional phase contrast views; the second block aims to perform
quality control to detect image quality related artefacts; the third block
aims to perform automatic segmentation of the quality controlled selected
images; and the fourth block aims to extract clinical parameters using the
resulting segmentations.

1\. View Selection: The first part of our framework aims to identify the
standard 2D PC views used for the analysis of cardiac function and
haemodynamics from a CMR acquisition. The manually classified data were
divided as follows: 80% was used for training, 10% was used for validation,
and 10% for testing. The training, validation and test data cohorts had a
mutually exclusive subject pool, i.e. acquisitions from the same subject could
only be used in one of the three cohorts. Prior to training, all images were
cropped to a standard size of 192x192 pixels. We used the AlexNet network [6]
to classify between the ascending aorta, pulmonary trunk and other views. The
network was trained with batch size 16 and learning rate 0.001 with stochastic
gradient descent for 200 epochs with cross entropy loss to classify images
into the three classes described. For the training data, data augmentation was
performed on-the-fly using random translations (±30 pixels), rotations (±90°),
flips (50% probability) and scaling (up to 20%) for each mini-batch of images
before feeding them to the network. The probability of augmentation for each
of the parameters was 50%.

2\. Quality Control: QC of image acquisition is of particular importance when
measuring flow because errors in the quantification of net flow can arise from
a range of different image acquisition or processing quality issues, including
tortuosity of the aorta, planning of the acquisition plane, arrhythmia and
breathing artefacts. These quality issues can lead to errors in flow analysis
and therefore need to be detected. We implemented the VGG-11 [13] network as a
binary classifier to detect such artefacts, utilising manual labels provided
by a cardiologist as ground truth. The same training, validation and test
splits of the data utilised for view selection were used here. Similarly, the
same pre-processing and data augmentation scheme were carried out for this
step in the framework, with the same network parameters used for training,
although binary cross entropy loss was used to classify the images as those
with and without artefacts.

3\. Automated Segmentation: We used the ‘nnU-Net’ framework [5] to segment all
frames through the cardiac cycle for the ascending aorta and pulmonary artery
classes, respectively. We used the 2D ‘nnU-Net’ implementation and the models
were trained for 1000 epochs. Training was performed with a batch size of 32
using stochastic gradient descent with Nesterov momentum ($\mu$=0.99) and an
initial learning rate of 0.01. The loss function was the sum of the cross
entropy and Dice losses. Data augmentation was performed on the fly and
included rotations, scaling, Gaussian noise, Gaussian blur, brightness,
contrast, simulation of low resolution, gamma correction and mirroring. An
ensemble of five different models trained on subsets of the training data was
used to achieve the final predicted segmentations on the test set. All deep
learning models used for this work were trained on an NVIDIA Quadro RTX 6000
GPU.

4\. Flow Curve Estimation & Parameter Estimation: pixel values were converted
to velocity using the established formula [17] as follows:

$\text{Velocity}=\frac{10\pi R}{\text{VENC}}*P*M$

where P and M represent the raw pixel values from the phase and magnitude
maps, VENC is an adjustable scanner parameter representing the maximum
measurable flow velocity extracted and R is a reconstruction scaling factor
specified from the PC-CMR DICOM files.

Flow was calculated from the automatic segmentation map of a given PC scan as:

$\text{Net Flow}=\sum_{n=0}^{N}\sum_{i=0}^{I}S_{n,i}V_{n,i}a\Delta_{t}$

where $n$ is the frame index, $N$ the number of temporal frames in the scan,
$I$ the number of pixels in each frame, $S$ the binary segmentation map, $V$
the velocity map calculated using the previous equation, $a$ is the pixel area
(in cm2), and $\Delta_{t}$ the time interval between frames.

From the flow curves, we quantify the peak, total forward, total backward and
net flow.

##  3 Materials

This is a retrospective multivendor study conducted on a large set of clinical
2D PC-CMR data from three centres that includes the full spectrum of cardiac
disease phenotypes. Details of the databases are provided below:
1\. UK Biobank (UKBB): This database contains a mixture of 80 patients and
healthy volunteers. CMR imaging was performed using a 1.5 T Siemens MAGNETOM
Aera (see [9] for further details of the image acquisition protocol).
2\. Guy’s and St Thomas’ NHS Foundation Trust (GSTFT): This database contains
830 patients acquired under routine clinical CMR practice with a range of
different cadiovascular diseases (CVDs). CMR imaging was performed using
different CMR scanners (Philips Achieva 1.5T/3.0T, Philips Ingenia 1.5T and
Siemens Aera 1.5T).
3\. Duke University Hospital (Duke): This database contains 102 patients
acquired in routine clinical CMR practice with a range of different CVDs. CMR
imaging was performed using different CMR scanners (Siemens Avanto 1.5T,
Siemens Sola 1.5T, Siemens Verio 3.0T and Siemens Vida 3.0T). This database is
used as an external validation data set.

For the three clinical databases, all data were manually classified and
annotated though the full cardiac cycle by multiple cardiologists, and an
additional review of all segmentations was performed to ensure the annotations
were of high quality. Table 1 summarises the cases used for view
classification, segmentation and QC.

Table 1: Number of images used in this study for view selection, automated segmentation and quality control (QC). Ao: ascending aorta, PA: pulmonary artery. | View Selection | QC | Segmentation
---|---|---|---
Database | Ao | PA | Other | Ao and PA | Ao | PA
UKBB | 80 | 0 | 0 | 80 | 79 | 0
GSTFT | 284 | 215 | 18 | 499 | 477 | 353
Duke | 54 | 48 | 0 | 100 | 40 | 36

##  4 Experiments and Results

We present the results in terms of the internal data set; this comprises the
UKBB and GSTFT data sets, which were merged and then randomly split into the
training/validation/test splits described in Section 2. External data set is
used to refer to the Anon Centre 2 data set, which was used for external
validation, where 20% of this data was combined with the original training
cohort and used to finetune the network before testing on the remaining
external validation data.

1\. View Selection: The performance of the view selection network was
evaluated in terms of precision, recall and the F1-score for each of the three
classes, as well as the accuracy achieved for all classes together. Table 2
summarises these results for the internal data set and the external data set.
High performance is shown on the internal data set, with an accuracy of 0.958,
while the accuracy of 0.833 on the external data set demonstrates good
generalisability for the view selection task.

2\. Quality Control: As with the view classification network, the performance
of the quality control network was measured using the precision, recall,
F1-score and accuracy, which are reported in Table 3. The results demonstrate
good performance on both the internal and external data sets, with accuracy
values of 0.914 and 0.850, respectively.

Table 2: View classification results, in terms of precision, recall and F1 score for each class, as well as the global accuracy. | Internal Data Set | External Data Set
---|---|---
| Precision | Recall | F1-score | Precision | Recall | F1-score
Ascending Aorta | 1.000 | 0.945 | 0.972 | 0.785 | 0.944 | 0.857
Pulmonary Artery | 0.913 | 0.977 | 0.944 | 0.912 | 0.708 | 0.800
Other | 0.800 | 1.000 | 0.889 | - | - | -
Accuracy | 0.958 | 0.833
Table 3: Quality control performance, in terms of the precision, recall, F1-score and accuracy. | Precision | Recall | F1-score | Accuracy
---|---|---|---|---
Internal Data Set | 0.914 | 0.913 | 0.907 | 0.914
External Data Set | 0.873 | 0.850 | 0.792 | 0.850

3\. Segmentation: The segmentation network for each view was evaluated in
terms of the average Dice coefficient between the manual and automatic
segmentations, and the results for the internal and external validation sets
can be seen in Table 4. Dice scores for the internal data set were excellent,
with good values also obtained for the external data set, where values of
greater than 0.969 and greater than 0.800 were achieved, respectively.

Table 4: Segmentation results in terms of the mean Dice coefficient for the two segmentation networks (ascending aorta and pulmonary artery) for the full cardiac cycle. | Internal Data Set | External Data Set
---|---|---
Ascending Aorta | 0.969 | 0.871
Pulmonary Artery | 0.971 | 0.800

4\. Parameter Estimation: The peak, net, total forward and total backward flow
values in the aorta and pulmonary artery estimated using the automatic
segmentations were compared to the manually obtained values using Pearson’s
correlation and Bland-Altman plots. Table 5 shows the correlation
coefficients; for all views, correlations are strong, with coefficients
$>$0.918 (p$<$0.001). Figure 2 shows the generated Bland-Altman plots; there
is good agreement between automated and manual analysis, without a significant
bias for any of the views and narrow limits of agreement.

Table 5: Pearson correlation coefficient values obtained when comparing the manual and automatic values for peak, net, total forward and total backward flow (p$<$0.001). | Internal Data Set | External Data Set
---|---|---
| Peak | Net | Forward | Backward | Peak | Net | Forward | Backward
Ascending Aorta | 0.997 | 0.998 | 0.988 | 0.980 | 0.998 | 0.970 | 0.953 | 0.918
Pulmonary Artery | 0.998 | 0.999 | 0.999 | 0.993 | 0.986 | 0.956 | 0.930 | 0.939

Figure 2: Bland-Altman plots for peak flow (first row), net flow (second row),
forward flow (third row) and backward flow(final row), for the ascending aorta
(left column) and the pulmonary artery (right column).

##  5 Discussion

In this work, we have proposed a novel end-to-end pipeline enabling automatic
flow analysis from full CMR scans, which encompasses four main steps: view
selection to detect conventional 2D PC-CMR scans; quality control to identify
images with artefacts; automatic segmentation of the ascending aorta and
pulmonary artery views; and the estimation of key flow parameters. To the best
of our knowledge, this is the first automatic flow analysis framework to
include the automatic quality-controlled selection of conventional 2D PC-CMR
views, which mimics crucial steps ordinarily carried out by physicians. We
utilised a large scale database from multiple centres, vendors and field
strengths, with one data set being used for external validation.

Our results show a good performance at each step in the pipeline, with
accuracy values of 0.958 and 0.914 for view classification and QC,
respectively, and Dice scores $>$0.969, indicating that our pipeline achieves
close to human-level performance and is comparable to other automatic flow
quantification frameworks [2, 3]. The obtained flow curves can be used to
calculate further metrics of interest in cardiovascular medicine, including
ejection times and first-phase ejection fraction (EF1). Additionally, we
obtained good results on the external validation set, indicating that our
pipeline has good generalisability. This is noteworthy, as the external data
set included images that were obtained with a different imaging strategy
(higher image-acceleration factors and breath-held imaging), leading to
significant differences in image characteristics.

A limitation of our pipeline is that the output automatic segmentations are
not checked for quality before being used for the computation of the flow
parameters. Future work will therefore include the addition of an extra
quality control step after the segmentation of the vessels to ensure the
quality of the parameter extraction. Furthermore, in this work, we excluded
views other than the ascending aorta and pulmonary artery from analysis. In
the future, we aim to extend our pipeline to these other views, such as
descending aorta, branch pulmonary artery and pulmonary vein flows.

## Acknowledgements

This work was supported by the UKRI London Medical Imaging & Artificial
Intelligence Centre for Value Based Healthcare, and the Wellcome EPSRC Centre
for Medical Engineering at the School of Biomedical Engineering and Imaging
Sciences, King's College London (WT 203148/Z/16/Z). The authors also
acknowledge financial support from the National Institute for Health Research
(NIHR) Cardiovascular MedTech Co-operative award to the Guy’s and St Thomas’
NHS Foundation Trust and the Department of Health National Institute for
Health Research (NIHR) comprehensive Biomedical Research Centre award to Guy’s
& St Thomas’ NHS Foundation Trust in partnership with King’s College London.

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2pf in single-trace $T\bar{T}$ holography

Amit Giveon

Racah Institute of Physics, The Hebrew University

Jerusalem, 91904, Israel

###### Abstract

The 2pf in single-trace $T\bar{T}$ holography was computed and analyzed for
operators in the $w=0$ sector in [1]. Here we present its immediate
generalization to non-zero winding $w$. For long strings, the result is
identical to the one obtained within the TsT/$T\bar{T}$ approach in [2] (up to
a factor of the reflection coefficient).

Single-trace $T\bar{T}$ holography [3] is obtained from the $AdS_{3}/CFT_{2}$
one by adding to the Lagrangian of the $CFT_{2}$ a certain dimension $(2,2)$
quasi-primary operator $D(x)$, $x\equiv(x^{0},x^{1})$, constructed in [4].
This irrelevant deformation can be controlled, since in the dual bulk string
theory on $AdS_{3}$, it corresponds to deforming the worldsheet theory, in
which $AdS_{3}$ is described by the $SL(2,R)$ WZW model, by a current-current
operator, $J^{-}\bar{J}^{-}$, [3].

A useful description of the deformed worldsheet theory is obtained by a null
gauging of the SCFT

$R^{1,1}\times AdS_{3}\times{\cal N}$ (1)

in the directions

$i\partial(y-t)+\alpha J^{-}~{},\qquad
i\bar{\partial}(y+t)\pm\alpha\bar{J}^{-}$ (2)

where $(t,y)$ are the canonically normalized bosonic fields of the $R^{1,1}$
SCFT. The resulting theory is obtained along the lines presented in [5, 1]; it
is referred to as the superstring on

${\cal M}_{3}\times{\cal N}$ (3)

in [3, 5, 1].

Following [1], a large class of observables in the superstring on (3) is given
by vertex operators in the (NS,NS) sector, which take the form (in the
$(-1,-1)$ picture)

$O^{w,j}(p)=\int
d^{2}ze^{-\varphi-\bar{\varphi}}\Phi_{h}^{w,j}(p)e^{-i\alpha(p_{0}t+p_{1}y)}O_{\cal
N}$ (4)

Here $\varphi,\bar{\varphi}$ are worldsheet fields associated with the
superconformal ghosts, that keep track of the picture, and $\Phi_{h}^{w,j}(p)$
are vertex operators on $AdS_{3}$ in the winding $w\geq 0$ sectors, obtained
by spectral flow from operators in the representation $j$ of $SL(2,R)$; they
are the Fourier transforms of the operators $\Phi_{h}^{w,j}(x;z)$ in [6] to
momentum space, $p\equiv(p_{0},p_{1})$. 111In [6], $h$ was denoted by $J$, and
we consider operators with $\bar{J}=J\equiv h$.

The operators in eq. (4) include in addition to the set of operators in the
$w=0$ sector (in which case $h=j$), considered in eq. (2.9) of [1] (in which
case the $AdS_{3}$ operator is $\Phi_{h}^{w=0,j=h}(x;z)\equiv\Phi_{h}(x;z)$),
also those with non-zero winding, $w>0$ (in which case the spacetime conformal
weight $h$ of the $AdS_{3}$ operator $\Phi_{h}^{w>0,j}(x;z)$ and its $j$ are
different quantum numbers).

When $w>0$, the mass-shell condition for the operators in eq. (4) is 222This
is obtained using e.g. eq. (4.6) in [7], which follows from eqs. (3.4),(3.5)
in [6]; we take $\bar{\Delta}_{\cal N}=\Delta_{\cal N}$, so that $\bar{h}=h$
below, as in [1].

$-{j(j-1)\over k}-w\left(h_{p^{2}}-{k\over 4}w\right)+{1\over 2}\lambda
p^{2}+\Delta_{\cal N}={1\over 2}$ (5)

where $\lambda=\pm\alpha^{2}$, with the $\pm$ amounting to axial and vector
gauging in (2), respectively. The subscript $p^{2}$ was added to $h$, to
emphasize that it depends on $p^{2}$ via the on-shell condition.

In the $w=0$ sector, $j=h_{p^{2}}$, and the on-shell condition is

$-{h_{p^{2}}(h_{p^{2}}-1)\over k}+{1\over 2}\lambda p^{2}+\Delta_{\cal
N}={1\over 2}$ (6)

This is eq. (3.5) in [1]. Equation (5) is the generalization of (6) to the
$w>0$ sectors.

Denoting by $h$ the value of $h_{p^{2}}$ in the undeformed theory,
$h\equiv\lim_{p\to 0}h_{p^{2}}$, from eq. (5) one finds that, for $w\neq 0$,

$h_{p^{2}}=h+{1\over 2}{\lambda\over w}p^{2}$ (7)

On the other hand, from eq. (6), one finds that, for $w=0$,

$2h_{p^{2}}-1=\sqrt{(2h+1)^{2}+2\lambda kp^{2}}$ (8)

instead. This is eq. (3.6) in [1].

Equation (7) is the generalization of eq. (8) to the non-zero winding cases,
$w\neq 0$.

The main result of this note is the following.

One finds that the 2pf that amounts to the worldsheet operators $O^{w>0,j}(p)$
in the target space is

$\langle\hat{O}^{w>0,j}(p)\hat{O}^{w>0,j}(-p)\rangle_{target}=R(w,j;h_{p^{2}})\gamma(1-2h_{p^{2}})\left(p^{2}\over
4\right)^{2h_{p^{2}}-1}$ (9)

with the $h_{p^{2}}$ in (7), and where

$R(w,j;h_{p^{2}})=k\nu^{1-2j}{\gamma\left(j-{k\over
2}w+h_{p^{2}}\right)\gamma\left(j+{k\over
2}w-h_{p^{2}}\right)\over\gamma\left({2j-1\over k}\right)\gamma(2j)}$ (10)

is the bulk reflection coefficient in the superstring on ${\cal
M}_{3}\times{\cal N}$, (1)–(3), 333For discrete states, the factor $R$ in (9)
should be replaced by a different factor; see below. with

$\gamma(x)\equiv{\Gamma(x)\over\Gamma(1-x)}$ (11)

A few comments are in order:

  1. 1.

Equation (9) has the same form as eq. (3.7) in [1], with $\pi D(h_{p^{2}})\to
R(w,j;h_{p^{2}})$, and with the $h_{p^{2}}$ given in eq. (7); it is obtained
following the discussion around eqs. (3.4)–(3.6) and section 5.1 in [6], and
applying it to the discussion around eqs. (3.4)–(3.7) in [1].

  2. 2.

Strictly speaking, the result (9) with (10) and (7) is derived using
(5.15),(5.16) of [6], which was obtained for $j={1\over 2}+is$. The radial
momentum $s$ is real for delta-function normalizable operators. In this case,
we have a continuous representation at $w=0$, prior to spectral flow in the
underlying $SL(2,R)$ of the null quotient of $R^{1,1}\times
AdS_{3}(\equiv{\cal M}_{3})$, (1)–(3). Non-normalizable operators can be
obtained by continuation to real positive $is$, as in e.g. [8].

  3. 3.

For delta-function normalizable operators, $j={1\over 2}+is$ and
$R(w,j;h_{p^{2}})$ is a phase:
$R(w,s;h_{p^{2}})=e^{i\delta(w,s;h_{p^{2}})}$(=(3.6) in [6] with $J=\bar{J}\to
h_{p^{2}}$); this is the phase shift in the scattering of long strings in
target space, ${\cal M}_{3}$. For non-normalizabe operators (${1\over 2}<j\in
R$), the discussion on p. 7,8 of [1], concerning the physics of such factors,
is relevant here as well.

  4. 4.

For discrete states, $R$ in (9) is not given by (10). Operators that amount to
discrete states in the null quotient of $R^{1,1}\times AdS_{3}(\equiv{\cal
M}_{3})$, (1)–(3), are obtained from normalizable operators in discrete
representations at $w=0$, a.k.a. with $m=\bar{m}=j+n$, $n=0,1,2,...$, ${1\over
2}<j<{k+1\over 2}$, in the underlying $SL(2,R)$, prior to spectral flow. It
thus follows from (5.18) in [6] 444With $J=\bar{J}\to h_{p^{2}}$ and $k-2\to
k$ in appropriate places, as required when going from the bosonic string in
[6] to the superstring, considered here. that in this case, one should replace
$R$ in (9) by 555A.k.a. by the residues of poles from one of the gamma
functions in (10), which amount to LSZ poles in the LST dual to the
superstring theory on the asymptotically linear-dilaton spacetime ${\cal
M}_{3}$, (1)–(3); see [9] and references therein.

$R\to\pi(2j-1+kw)B(j)\left({\Gamma(2j+n)\over
n!\Gamma(2j)}\right)^{2}~{},\qquad j=h_{p^{2}}-{k\over 2}w-n$ (12)

where

$B(j)={k\over\pi}{\nu^{1-2j}\over\gamma\left(2j-1\over k\right)}$ (13)

  5. 5.

Recall that in the $w=0$ sector, [1],

$\langle\hat{O}^{w=0,j=h_{p^{2}}}(p)\hat{O}^{w=0,j=h_{p^{2}}}(-p)\rangle_{target}=\pi
D(h_{p^{2}})\gamma(1-2h_{p^{2}})\left(p^{2}\over 4\right)^{2h_{p^{2}}-1}$ (14)

with the $h_{p^{2}}$ given by (8), and where

$D(j)=(2j-1)B(j)$ (15)

with the $B(j)$ in (13). Indeed, (12) collapses to $\pi D(h_{p^{2}})$ when
$w=0$ (in which case one should also set $n=0$ in eq. (12)).

  6. 6.

Equation (9), with (10) and (7), is (4.15) in [2], up to the factor of the
reflection coefficient, $R(w,j;h_{p^{2}})$, which is absent in [2], and is in
harmony with the dependence of the two-point correlator on the momentum at
large momentum in eq. (7) of [10], which was obtained using the JT-gravity
formulation of $T\bar{T}$-deformed CFTs.

To recapitulate, in the $w=1$ sector, the factor

$\gamma(1-2h-\lambda p^{2})\left(p^{2}\over 4\right)^{2h-1+\lambda p^{2}}$
(16)

in the 2pf (see (9), with (10) and (7), for delta-normalizable and non-
normalizable operators, and with (12) and (7), for discrete states) is the
same factor that appears in the 2pf of $\lambda T\bar{T}$ deformed $CFT_{2}$,
[2, 10].

For $w>1$, $\lambda\to\lambda/w$ in (16), which is in harmony with a symmetric
product of $\lambda T\bar{T}$ deformed $CFT_{2}$ (see e.g. the discussion
around eqs. (6.5) and (6.6) in [7] and references therein).

String theory also determines the prefactor of (16). For delta-normalizable
states, it is a phase – the phase shift in the scattering of long strings in
string theory on the null gauged $R^{1,1}\times AdS_{3}(\equiv{\cal
M}_{3})\times{\cal N}$, (1)–(3). For non-normalizable operators, it is the
continuation of the bulk reflection coefficient, (10), to imaginary radial
momentum in target space, ${\cal M}_{3}$. The poles from one of the gamma
functions in the numerator of (10) signal the existence of normalizable states
in the theory, whose on-shell two point functions are (9) with (12) and (13).
666Recall, [11] and references therein, that the discrete states are known not
to fit into a symmetric product. However, these states have measure zero in
the string spectrum, so one can say that the symmetric product describes
generic states in perturbative string theory on ${\cal M}_{3}$

On the other hand, the 2pf in the $w=0$ sector (see (14) with (8) and (15)
with (13)) is different, [1]. In particular, it has a factor

$\gamma\left(-\sqrt{(2h+1)^{2}+2\lambda kp^{2}}\right)\left(p^{2}\over
4\right)^{\sqrt{(2h+1)^{2}+2\lambda kp^{2}}}$ (17)

instead of (16), and its prefactor, $\pi D(h_{p^{2}})$, (15) with (13), is the
stringy contribution (a.k.a. non-perturbative in $\alpha^{\prime}$) to the
(continuation of the) reflection coefficient.

The above is in harmony with previous observations (see e.g. [3, 7, 12, 2, 11,
13] and references therein and thereof), and may shed more light regarding
intriguing open problems in single-trace $T\bar{T}$ holography and their
implications.

## Acknowledgments

This work was supported in part by the ISF (grant number 256/22), the BSF
(grant number 2018068) and the ISF center for excellence (grant number
2289/18).

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