# Deforming the ODE/IM correspondence with $\mathrm{T}\overline{\mathrm{T}}$

Fabrizio Aramini${}^{1\mbox{\musFontLarge\kern
0.70004pt\raisebox{1.50694pt}{\char 90\relax}\kern-0.1pt}}$, Nicolò
Brizio${}^{1\mbox{\musFont\kern 0.35002pt\raisebox{1.50694pt}{\char
7\relax\mbox{\musFont\kern 6.68501pt\raisebox{1.65764pt}{\char 16\relax}\kern
0.0pt}}\kern 1.39998pt}}$, Stefano Negro${}^{2\mbox{\musFontLarge\kern
0.70004pt\raisebox{1.50694pt}{\char 94\relax}\kern-0.1pt}}$ and Roberto
Tateo${}^{1\mbox{\musFontLarge\kern 0.70004pt\raisebox{1.50694pt}{\char
92\relax}\kern-0.1pt}}$

(1 Dipartimento di Fisica, Università di Torino and INFN, Sezione di Torino,
Via P. Giuria 1, 10125, Torino, Italy
2 Center for Cosmology and Particle Physics, New York University, New York,
NY 10003, U.S.A.
${}^{\mbox{\musFontLarge\kern 0.56003pt\raisebox{1.20555pt}{\char
<EMAIL_ADDRESS>${}^{\mbox{\musFont\kern
0.28001pt\raisebox{1.20555pt}{\char 7\relax\mbox{\musFont\kern
5.348pt\raisebox{1.32611pt}{\char 16\relax}\kern 0.0pt}}\kern
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0.56003pt\raisebox{1.20555pt}{\char
<EMAIL_ADDRESS>${}^{\mbox{\musFontLarge\kern
0.56003pt\raisebox{1.20555pt}{\char
<EMAIL_ADDRESS>
)

The ODE/IM correspondence is an exact link between classical and quantum
integrable models. The primary purpose of this work is to show that it remains
valid after $\mathrm{T}\overline{\mathrm{T}}$ perturbation on both sides of
the correspondence. In particular, we prove that the deformed Lax pair of the
sinh-Gordon model, obtained from the unperturbed one through a dynamical
change of coordinates, leads to the same Burgers-type equation governing the
quantum spectral flow induced by $\mathrm{T}\overline{\mathrm{T}}$. Our main
conclusions have general validity, as the analysis may be easily adapted to
all the known ODE/IM examples involving integrable quantum field theories.

###### Contents

  1. 1 Introduction
  2. 2 The quantum sine-Gordon model at finite volume
    1. 2.1 The non-linear integral equation
    2. 2.2 The $\mathrm{T}\overline{\mathrm{T}}$ deformation
  3. 3 The modified classical sinh-Gordon model
    1. 3.1 Second order linear differential equations
    2. 3.2 Integrals of motion
    3. 3.3 The integration countours $\Gamma_{1}$ and $\Gamma_{2}$
    4. 3.4 The circumference of the cylinder
  4. 4 Adding the $\mathrm{T}\overline{\mathrm{T}}$
    1. 4.1 The dynamical change of coordinates
    2. 4.2 The Burgers' equation
  5. 5 Conclusions
  6. A The conformal field theory limit
    1. A.1 WKB analysis and the monster potential

##  1 Introduction

The first instance of the ODE/IM correspondence dates back to 1998 [1] when a
surprising connection between works on spectral determinants for specific
Sturm-Liouville problems [2, 3] and the functional approach to conformal field
theories (CFTs) [4, 5, 6, 7, 8, 9, 10] was first established. The
correspondence holds its roots in the fact that seemingly different quantities
in the two contexts fulfil the same set of functional relations [1] with
identical analytic and asymptotic properties.

Though initially referring only to the CFT vacuum states, the ODE/IM
correspondence was later generalised to encompass excited states [11, 12] and
more recently extended to massive integrable quantum field theories based on
the $\mathfrak{sl}_{2}$ algebra [13, 14]. The correspondence was also extended
to systems based on higher-rank algebras, both critical [15, 8, 16, 17] and
off-critical [18, 19, 20]. In this work, we focus on systems based on the
$\mathfrak{sl}_{2}$ algebra. In the off-critical extension of the original
results [14], the Sturm-Liouville equation is replaced by the Lax equations of
the modified sinh-Gordon model, written as a pair of second-order differential
equations coupled through a field-dependent generalised potential (cf.
equation (3.7), below). In this perspective, we are dealing with two, a priori
unrelated systems: a classical integrable equation – the modified sinh-Gordon
one – and a quantum integrable model – such as the sine- and sinh-Gordon
models. The correspondence can then be viewed as an equality between the
conserved quantities of these two systems.[14, 21, 22].

The second main ingredient, relevant to the current purposes, is the recent
discovery that specific irrelevant perturbations of quantum field theories can
be studied using integrable models tools and hydrodynamics-type flow equations
[23, 24]. The observations made in [23, 24], triggered a considerable amount
of research activity, with applications ranging from simple systems in quantum
mechanics to AdS/CFT and nonlinear electrodynamics [25, 26, 27, 28, 29, 30,
31]. The perturbation involving Zamolodchikov's
$\mathrm{T}\overline{\mathrm{T}}$ operator [32] is arguably the most
interesting representative of an infinite tower of irrelevant deformations
[33, 34, 35] (see also [36] for some early results on irrelevant integrable
perturbations). It is related to the Nambu-Goto string [37, 38, 39], quantum
JT gravity [40, 41, 42, 43, 44, 45], and it possesses compelling
interpretation within the AdS/CFT [46, 47, 48, 49, 50, 51] and supersymmetry
[52, 53, 54, 55, 56, 57] frameworks. While at the quantum level, an inviscid
Burgers' equation governs the evolution of the
$\mathrm{T}\overline{\mathrm{T}}$-deformed spectrum, at the classical level,
the perturbation turns out to be equivalent to a dynamical change of the
space-time coordinates [58, 59, 60, 61].

The main objective of this work is to unify these two research strands by
proving the validity of the ODE/IM after $\mathrm{T}\overline{\mathrm{T}}$
perturbations of both the classical and the quantum sides of the
correspondence. In order to do so, we will prove that the conserved energy and
momentum of the theory on the $\mathrm{T}\overline{\mathrm{T}}$-deformed
classical side of the correspondence satisfy the same Burgers' equation as the
finite-size spectrum on the $\mathrm{T}\overline{\mathrm{T}}$-deformed quantum
side. This fact ensures that if the ODE/IM is valid for the undeformed
systems, it will hold for an arbitrary $\mathrm{T}\overline{\mathrm{T}}$
deformation. We will proceed in steps, first recalling in section 2 how a
$\mathrm{T}\overline{\mathrm{T}}$ deformation alters the finite-size spectrum
of a quantum integrable field theory and subsequently reviewing in section 3
the elements of the ODE/IM correspondence relevant for this work. Section 4
contains the key arguments and result: a proof that the energy and momentum of
a $\mathrm{T}\overline{\mathrm{T}}$-deformed classical field theory satisfy
the inviscid Burgers' equation.

In this work we choose to focus on the specific instance of the ODE/IM
correspondence analysed in [14], involving the quantum sine-Gordon and the
classical modified sinh-Gordon models. Consequently, the arguments of section
4 are adapted to the specificities of the latter. However, the final result
has a much broader validity. In particular, our derivation can be used – with
minimal modifications – to produce a proof of the Burgers' equation for the
energy and momentum of a generic, not necessarily integrable,
$\mathrm{T}\overline{\mathrm{T}}$-deformed classical field theory. To our
knowledge, the only previous appearance of the Burgers' equation in a
classical theory concerns a very specific class of zero-momentum solutions to
the sine-Gordon model [59].

Finally, our results should be considered as a first step toward the study of
irrelevant deformations using the ODE/IM correspondence as a powerful
quantisation tool.

##  2 The quantum sine-Gordon model at finite volume

Integrable models and their properties have been extensively studied during
past decades, and powerful methods developed to determine the finite-size
spectrum of an integrable quantum field theory. One of these is the non-linear
integral equation (NLIE) approach [62, 63, 64, 65, 66, 5, 67, 68], which we
briefly review in this section before discussing the effects of the
$\mathrm{T}\overline{\mathrm{T}}$ deformation.

###  2.1 The non-linear integral equation

Consider the sine-Gordon quantum field theory defined on a cylinder of radius
$R/2\pi$. The complete information on its spectrum can be extracted from the
_counting function_ $f_{\nu}(\theta)$, solution to the following NLIE [63, 64,
65, 67, 68]

$\displaystyle f_{\nu}(\theta)$
$\displaystyle=\nu(R,k\,|\,\theta)-\int_{\mathcal{C}_{1}}d\theta^{\prime}\,\mathcal{K}(\theta-\theta^{\prime})\,\log\left(1+e^{-f_{\nu}(\theta^{\prime})}\right)$
$\displaystyle+\int_{\mathcal{C}_{2}}d\theta^{\prime}\,\mathcal{K}(\theta-\theta^{\prime})\,\log\left(1+e^{f_{\nu}(\theta^{\prime})}\right)\,,$
(2.1)

where

$\nu(R,k\,|\,\theta)=2\pi\imath\,k-\imath\,mR\sinh(\theta)\;,$ (2.2)

is the so-called driving term. In equations (2.1) and (2.2), $m$ denotes the
sine-Gordon soliton mass and $k\in[-1/2,1/2]$ is the quasi-momentum, also
known as the twist, whose role is to select the vacuum [69, 14]. In (2.1), the
convolution kernel $\mathcal{K}(\theta)$ is

$\mathcal{K}(\theta)=\int_{\mathbb{R}}\frac{dp}{2\pi}\cos(p\,\theta)\,\frac{\sinh\left(\pi
p\frac{1-\alpha}{2\alpha}\right)}{2\cosh\left(\pi\frac{p}{2}\right)\,\sinh\left(\pi\frac{p}{2\alpha}\right)}\,,$
(2.3)

where $\alpha=\beta^{-2}-1$ and $\beta^{2}<1$ is the sine-Gordon coupling. For
technical reasons, as in [14], in the following we shall limit our attention
to the range $\beta^{2}<1/2$, which corresponds to the region $\alpha>1$.

The information on the specific energy eigenstate state under consideration is
encoded in the choice of the integration contours $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$. For the ground state $\mathcal{C}_{1}=\mathbb{R}+\imath
0^{+}=\mathcal{C}_{2}^{*}$, while for the excited states the contours
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ encircle a number of singularities
$\\{\theta_{i}\\}$ of $\log\left(1+e^{f_{\nu}(\theta_{i})}\right)$. See [70,
71, 67, 68] for more details.

Energy and momentum can be obtained from the counting function from the
following expression

$\displaystyle E$
$\displaystyle=m\left(\int_{\mathcal{C}_{1}}\frac{dy}{2\pi\imath}\,\sinh(y)\,\log\left(1+e^{-f_{\nu}(y)}\right)-\int_{\mathcal{C}_{2}}\frac{dy}{2\pi\imath}\,\sinh(y)\,\log\left(1+e^{f_{\nu}(y)}\right)\right)\,,$
(2.4) $\displaystyle P$
$\displaystyle=m\left(\int_{\mathcal{C}_{1}}\frac{dy}{2\pi\imath}\,\cosh(y)\,\log\left(1+e^{-f_{\nu}(y)}\right)-\int_{\mathcal{C}_{2}}\frac{dy}{2\pi\imath}\,\cosh(y)\,\log\left(1+e^{f_{\nu}(y)}\right)\right)\,.$

In particular, the momentum can be computed exactly via the so-called
_dilogarithm trick_. Using this, one easily checks that $P(R)=2\pi p/R$ with
$p\in\mathbb{Z}$ (see for example the Lemma in section 7 of [72]).

Finally, in the following we shall adopt the following specific
parametrization [14]

$mR=2\sqrt{\pi}\mathbf{s}^{\alpha+1}\frac{\Gamma\left(1+\frac{1}{2\alpha}\right)}{\Gamma\left(\frac{3}{2}+\frac{1}{2\alpha}\right)}\,,$
(2.5)

where $\mathbf{s}$ is a dimensionless scaling constant.

###  2.2 The $\mathrm{T}\overline{\mathrm{T}}$ deformation

At the level of the non-linear integral equation (2.1), the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation is introduced by implementing
the following modification of the convolution kernel [24]

$\mathcal{K}(\theta)\xmapsto{\mbox{$\mathrm{T}\overline{\mathrm{T}}$}}\mathcal{K}(\theta)-\tau\frac{m^{2}}{2\pi}\cosh(\theta)\,.$
(2.6)

Inserting (2.6) into equation (2.1), after simple manipulations, one finds
that $f_{\nu}^{(\tau)}(\theta)$, the counting function of the deformed theory,
still satisfies (2.1), up to a redefinition of the driving term

$\nu^{(\tau)}=\nu(R_{0},k\,|\,\theta-\theta_{0})\;,$ (2.7)

with

$\displaystyle R_{0}^{2}=\left(R+\tau
E(R,\tau)\right)^{2}-\tau^{2}P^{2}(R,\tau)\,,$ (2.8)
$\displaystyle\tanh{\theta_{0}}=\tau\frac{P(R)}{R+\tau E(R,\tau)}\,.$ (2.9)

The redefinition (2.7) implies the famous inviscid Burgers' flow equation for
the finite-volume quantum spectrum of
$\mathrm{T}\overline{\mathrm{T}}$-deformed field theories [24, 23]:

$\frac{\partial}{\partial\tau}E(R,\tau)=E(R,\tau)\frac{\partial}{\partial
R}E(R,\tau)+\frac{P(R)^{2}}{R}\,.$ (2.10)

##  3 The modified classical sinh-Gordon model

In this section, we will follow very closely [14] in the definition of the
quantities of interest. Let us consider the modified sinh-Gordon (mShG) model,
with equation of motion (EoM)

$\partial_{z}\partial_{\bar{z}}\eta-e^{2\eta}+p(z,s)p(\bar{z},s)e^{-2\eta}=0\,,$
(3.1)

where the complex coordinates $\mathbf{z}=(z,\bar{z})$ are dimensionless. The
function

$p(z,s)=z^{2\alpha}-s^{2\alpha}\,,$ (3.2)

is characterized by the pair of parameters $\alpha$ and $s\in\mathbb{R}_{>0}$.
Introducing polar coordinates,

$z=\rho\,e^{\imath\varphi}\,,\quad\bar{z}=\rho\,e^{-\imath\varphi}\,,$ (3.3)

we can describe the field configurations $\eta(\rho,\varphi)$ relevant to the
ODE/IM correspondence as those that satisfy the following requirements

  * –

$e^{-\eta(\rho,\varphi)}$ is a single-valued, non-zero complex function on the
cone $C_{\pi/\alpha}$ with apex angle $\pi/\alpha$ and $L+\bar{L}$ punctures;

  * –

the $\rho\rightarrow\infty$ asymptotic behaviour is
$e^{-\eta(\rho,\varphi)}\sim\rho^{-\alpha}$;

  * –

the $\rho\rightarrow 0$ asymptotic behaviour is
$e^{-\eta(\rho,\varphi)}\sim\rho^{-l}$, with $l\in\left[-1/2,1/2\right]$.

Through the ODE/IM correspondence, the conserved quantities of the _classical_
mShG model, evaluated on the above field configurations, are identified with
the _quantum_ ones of the sine-Gordon model, such as its energy and momentum
(2.4). The parameter $l$ is then related to the quasi-momentum in (2.2) as
$l=2|k|-1/2$. The number and positions of the $L+\bar{L}$ punctures determine
the specific energy eigenstate on the quantum side of the correspondence and
are constrained by a system of algebraic equations, the _monodromy-free
conditions_ of [12, 73], also described in [22]. In this section we will
describe more in detail the construction of the classical conserved charges of
the mShG model. These charges can be understood geometrically as integrals
along specific contours. We will leverage this perspective in the following
section to apply the $\mathrm{T}\overline{\mathrm{T}}$ deformation on the
classical side of the ODE/IM correspondence.

###  3.1 Second order linear differential equations

As it is well known, the mShG equation (3.1) can be interpreted as the
compatibility condition of the linear problem

$\partial_{z}\Psi=L_{1}\Psi\,,\quad\partial_{\bar{z}}\Psi=L_{2}\Psi\,,$ (3.4)

which involves the Lax pair

$\begin{aligned}
&L_{1}=-\frac{1}{2}\partial_{z}\eta\,\sigma^{3}+e^{\vartheta}\left(\sigma^{+}e^{\eta}-\sigma^{-}p(z,s)e^{-\eta}\right)\,,\\\
&L_{2}=\frac{1}{2}\partial_{\bar{z}}\eta\,\sigma^{3}+e^{-\vartheta}\left(\sigma^{-}e^{\eta}-\sigma^{+}p(\bar{z},s)e^{-\eta}\right)\,.\end{aligned}\,$
(3.5)

Here $\sigma^{3}$ and $\sigma^{\pm}$ are the Pauli matrices and $\vartheta$ is
the spectral parameter while $\Psi$ in (3.4) is a two-dimensional vector.

One can write the general solution of the linear problem (3.4) as

$\Psi=\begin{pmatrix}e^{\frac{\vartheta}{2}}e^{\frac{\eta}{2}}\psi\\\
e^{-\frac{\vartheta}{2}}e^{-\frac{\eta}{2}}(\partial_{z}+\partial_{z}\eta)\psi\end{pmatrix}=\begin{pmatrix}e^{\frac{\vartheta}{2}}e^{-\frac{\eta}{2}}(\partial_{\bar{z}}+\partial_{\bar{z}}\eta)\bar{\psi}\\\
e^{-\frac{\vartheta}{2}}e^{\frac{\eta}{2}}\bar{\psi}\end{pmatrix}\,,$ (3.6)

where the auxiliary fields $\psi$ and $\bar{\psi}$ solve the Schrödinger-type
equations

$\displaystyle\left(\partial_{z}^{2}-u(\mathbf{z})-e^{2\vartheta}p(z,s)\right)\psi(\mathbf{z})=0\,,$
(3.7)
$\displaystyle\left(\partial_{\bar{z}}^{2}-\bar{u}(\mathbf{z})-e^{-2\vartheta}p(\bar{z},s)\right)\bar{\psi}(\mathbf{z})=0\,.$

The function $u(\mathbf{z})$ depends on the field $\eta$ appearing in (3.1)
as:

$u({\bf z})=\left(\partial_{z}\eta({\bf
z})\right)^{2}-\partial_{z}^{2}\eta({\bf z})\,,\quad\bar{u}({\bf
z})=\left(\partial_{\bar{z}}\eta({\bf
z})\right)^{2}-\partial_{\bar{z}}^{2}\eta({\bf z})\,.$ (3.8)

###  3.2 Integrals of motion

The modified sinh-Gordon equation (3.1) is related to the unmodified one by a
simple change of variables. In terms of the new coordinates
$\mathbf{w}=(w,\bar{w})$,

$\displaystyle\left(\begin{array}[c]{cc}dw&\\\
d\bar{w}&\end{array}\\!\\!\\!\\!\\!\\!\right)=\left(\begin{array}[c]{cc}\sqrt{p(z,s)}dz&\\\
\sqrt{p(\bar{z},s)}d\bar{z}&\end{array}\\!\\!\\!\\!\\!\\!\right),$ (3.13)

the field

$\displaystyle\hat{\eta}({\bf w})=\eta({\bf z}({\bf
w}))-\frac{1}{4}\ln{p(z(w),s)p(\bar{z}(\bar{w}),s)}\,,$ (3.14)

satisfies the sinh-Gordon equation

$\partial_{w}\partial_{\bar{w}}\hat{\eta}-e^{2\hat{\eta}}+e^{-2\hat{\eta}}=0\,.$
(3.15)

From a WKB-type analysis of (3.7) [74, 75], one can find the expressions for
the integrals of motion (IMs) $\mathcal{J}_{2n-1}$ and
$\mathcal{\bar{J}}_{2n-1}$ with $n\in\mathbb{N}_{>0}$. Explicitly:

$\displaystyle\begin{aligned}
\mathcal{J}_{2n-1}&=\frac{1}{2(2n-1)\sin\left({\frac{\pi(2n-1)}{2\alpha}}\right)}\int\limits_{\Gamma_{1}}\left(dw\hat{P}_{2n}+d\bar{w}\hat{R}_{2n-2}\right)\,,\\\
\bar{\mathcal{J}}_{2n-1}&=\frac{1}{2(2n-1)\sin\left({\frac{\pi(2n-1)}{2\alpha}}\right)}\int\limits_{\bar{\Gamma}_{1}}\left(d\bar{w}\hat{\bar{P}}_{2n}+dw\hat{\bar{R}}_{2n-2}\right)\,.\end{aligned}$
(3.16)

The specification of the integration contours $\Gamma_{1}$ and
$\bar{\Gamma}_{1}$ is of primary importance and will be discussed in a moment.
The integrands appearing in (3.16) are closed 1-forms, meaning that they
satisfy a continuity equation

$\partial_{\bar{w}}{\hat{P}_{2n}}=\partial_{w}{\hat{R}_{2n-2}}\,,\quad\partial_{w}{\hat{\bar{P}}_{2n}}=\partial_{\bar{w}}{\hat{\bar{R}}_{2n-2}}\,.$
(3.17)

The expressions for $\hat{P}_{2n}$ and $\hat{R}_{2n-2}$, in terms of
$\hat{\eta}$, are known for generic $n$ and can be found in [14], however the
following analysis will only involve $\hat{P}_{2}$ and $\hat{R}_{0}$:

$\displaystyle\begin{aligned}
&\hat{P}_{2}=\frac{1}{2}\hat{u}\,,\quad\hat{R}_{0}=e^{-2\hat{\eta}}-1\,,\\\
&\hat{\bar{P}}_{2}=\frac{1}{2}\hat{\bar{u}}\,,\quad\hat{\bar{R}}_{0}=e^{-2\hat{\eta}}-1\,.\end{aligned}$
(3.18)

The potentials $\hat{u}$ and $\hat{\bar{u}}$ are defined as in (3.8) with
$\eta$ substituted for $\hat{\eta}$.

As a consequence of the continuity equation (3.17), these densities may be
written in terms of the stress-energy tensor components $T_{2}$, $\bar{T}_{2}$
and $\Theta_{0}$ of the sinh-Gordon model in dimensionless coordinates:

$\begin{split}&T_{2}(\mathbf{w})=\frac{1}{2}\hat{u}(\mathbf{w})=\hat{P}_{2}\,,\quad\bar{T}_{2}(\mathbf{w})=\frac{1}{2}\hat{\bar{u}}(\mathbf{w})=\hat{\bar{P}}_{2}\,,\\\
&\Theta_{0}(\mathbf{w})=\hat{R}_{0}=\hat{\bar{R}}_{0}=e^{-2\hat{\eta}}-1\,.\end{split}$
(3.19)

Finally, the total energy $E$ and the total momentum $P$ can be expressed as

$E=\mathcal{E}+\bar{\mathcal{E}}\,,\quad P=\mathcal{E}-\bar{\mathcal{E}}\,,$
(3.20)

where

$\displaystyle\mathcal{E}=\frac{\mathcal{M}}{4}\int\limits_{\Gamma_{1}}\left(dw\,T_{2}(\mathbf{w})+d\bar{w}\,\Theta_{0}(\mathbf{w})\right)\,,$
(3.21)
$\displaystyle\bar{\mathcal{E}}=\frac{\mathcal{M}}{4}\int\limits_{\bar{\Gamma}_{1}}\left(dw\,\Theta_{0}(\mathbf{w})+d\bar{w}\,\bar{T}_{2}(\mathbf{w})\right)\,.$

The dimensionful constant $\mathcal{M}$ is the equivalent of the soliton mass
$m$, appearing on the ODE/IM correspondence's quantum side.

###  3.3 The integration countours $\Gamma_{1}$ and $\Gamma_{2}$

While the analysis of the modified sinh-Gordon equation (3.1) is more easily
performed in the coordinates $\mathbf{w}$, we need to keep in mind that the
properties of the relevant field configurations are established in the
coordinates $\mathbf{z}$. Hence, we need to consider $\Gamma_{1}$ as the
image, through the map $\mathbf{w}(\mathbf{z})$ of a fundamental contour
$\Gamma_{2}$. Roughly speaking, this is a regularization of the contour
$\gamma_{2}$ introduced in [14] by a pair of arches around $z=\infty$ (in the
following denoted as $c_{2}^{\pm}$) starting just above and just below the
positive real axis and ending on the anti-Stokes lines with directions
$\varphi=\pi/(2\alpha+2)$ and $\varphi=-\pi/(2\alpha+2)$. The original contour
$\gamma_{2}$ in [14] started from $+\infty$ just below the real $z$ axis,
wound around the turning point $z=s$ and went back to $+\infty$ just above the
real axis, thus encircling the branch cut of $\sqrt{p(z,s)}$ clockwise. As we
will mention momentarily, the regularization provided by $\Gamma_{2}$ is
necessary to ensure the convergence of certain quantities relevant for the
ODE/IM correspondence. Notice that, this simple contour-type regularization
works well only in the regime $\alpha>1$ considered in this paper. We shall
return to this issue in section 3.4 below.

Figure 1: The $z$-plane with $\Gamma_{2}$ in blue. The solid rays
$\varphi=\pm\pi/(2\alpha)$ in black and the dashed rays
$\varphi=\pm\pi/(2\alpha+2)$ in gray.

The contour $\Gamma_{2}$ is represented in Figure 1, where the limit
$\rho\rightarrow+\infty$ for the arches' radius is implicitly assumed. Points
on the contour $\Gamma_{1}$ are defined as

$w(P)-w(A)=\int_{A}^{P\in\Gamma_{2}}dz\,\sqrt{p(z,s)}\,.$ (3.22)

The qualitative form of the contour $\Gamma_{1}$, obtained through the map
(3.22), is represented in Figure 2. Note that,
$\bar{\Gamma}_{2}=\Gamma_{2}^{*}$ and $\bar{\Gamma}_{1}=\Gamma_{1}^{*}$. It is
important to notice that, in the absence of singularities in the upper part of
the $w$-complex plane, $\Gamma_{1}$ can be straightened to a segment joining
the point $w(A)$ to $w(A^{\prime})$.

Finally, for integrands with a sufficiently fast vanishing asymptotics for
$\rho\rightarrow+\infty$ in the sector $|\arg(z)|\leq\pi/(2\alpha+2)$, the
contours $\Gamma_{2}$ ($\bar{\Gamma}_{2}$) and $\gamma_{2}$
($\bar{\gamma}_{2}$) become totally equivalent. In particular, given the
asymptotic requirements for the field $\eta(z,\bar{z})$, in the definition of
the IMs $\mathcal{J}_{2n-1}$, $\bar{\mathcal{J}}_{2n-1}$ one can safely trade
the contour $\Gamma_{1}$ with the image of $\gamma_{2}$ under the map
$\mathbf{w}(\mathbf{z})$.

Figure 2: The complex $w$-plane with (pictorial) $\bar{\Gamma}_{1}$ in orange.
$w(A^{\prime})-w(A)=\bar{w}(A)-\bar{w}(A^{\prime})>0$ is fixed at
$\rho\rightarrow+\infty$ (the points $w(B)$ and $w(B^{\prime})$ are pushed to
infinity).

###  3.4 The circumference of the cylinder

We shall now introduce the classical equivalent $\mathcal{M}\mathcal{R}$ of
the dimensionless quantity $mR$ appearing in (2.2). The former is written in
terms of simple integrals over $\Gamma_{2}$ and $\bar{\Gamma}_{2}$

$\mathcal{M}\mathcal{R}=\mathcal{M}\mathcal{R}(s)=\frac{2}{\tan\left(\frac{\pi}{2\alpha}\right)}\int_{\bar{\Gamma}_{2}}dz\,\sqrt{p(z,s)}=\frac{2}{\tan\left(\frac{\pi}{2\alpha}\right)}\int_{\Gamma_{2}}d\bar{z}\,\sqrt{p(\bar{z},s)}\,,$
(3.23)

where $\Gamma_{2}=c_{2}^{-}\cup\gamma_{2}\cup c_{2}^{+}$ is represented in
Figure 1. This object can be explicitly computed. First, let us consider the
integral appearing in [14]

$\displaystyle\int_{\bar{\gamma}_{2}}dz\,\sqrt{p(z,s)}$
$\displaystyle=2\lim_{\rho\rightarrow+\infty}\int_{\rho}^{s}dt\,\sqrt{p(t,s)}$
$\displaystyle=2\lim_{\rho\rightarrow+\infty}\left(\int_{\rho}^{s}dt\,\left(\sqrt{p(t,s)}-t^{\alpha}\right)+\frac{s^{\alpha+1}}{\alpha+1}-\frac{\rho^{\alpha+1}}{\alpha+1}\right)$
$\displaystyle=\tan{\left(\frac{\pi}{2\alpha}\right)}\sqrt{\pi}s^{\alpha+1}\frac{\Gamma\left(1+\frac{1}{2\alpha}\right)}{\Gamma\left(\frac{3}{2}+\frac{1}{2\alpha}\right)}-2\lim_{\rho\rightarrow+\infty}\frac{\rho^{\alpha+1}}{\alpha+1},$
(3.24)

where the factor two is due to the discontinuity on the square root type
branch cut. This integral is obviously divergent. We choose to regularize it
by altering the integration contour. In fact, we calculate the limiting
behaviour

$\lim_{\rho\rightarrow+\infty}\int_{\bar{c}^{+}_{2}(\rho)\,\cup\,\bar{c}^{-}_{2}(\rho)}dz\,\sqrt{p(z,s)}=\big{(}2-2\cos\left(\Omega(\alpha+1)\right)\big{)}\lim_{\rho\rightarrow+\infty}\frac{\rho^{\alpha+1}}{\alpha+1}.$
(3.25)

Now, if we set $\Omega=\pi/(2\alpha+2)$, the above quantity exactly cancels
the divergent term in (3.24) and we obtain the following finite expressions

$\displaystyle\mathcal{M}\mathcal{R}(s)=2\sqrt{\pi}s^{\alpha+1}\frac{\Gamma\left(1+\frac{1}{2\alpha}\right)}{\Gamma\left(\frac{3}{2}+\frac{1}{2\alpha}\right)}\,,$
(3.26)

establishing a match with the quantity defined in (2.5) on the quantum IM-side
of the correspondence: $\mathcal{M}\mathcal{R}=mR$ or, equivalently,
$s=\mathbf{s}$.

This same result could be obtained by more standard regularization schemes,
for example by subtracting the $\rho\rightarrow+\infty$ divergent part
appearing on the right-hand side of (3.24), or by replacing

$\sqrt{p(z,s)}\rightarrow\left(p(z,s)\right)^{a}\,,$ (3.27)

with $\operatorname{Re}(a)<-1/(2\alpha)$ and analytically continuing the
outcome to $a=1/2$. However, our contour type regularization is more natural,
especially in view of the introduction of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation as a field dependent change of
coordinates.

The same computation in the $\mathbf{w}$ complex plane yields the following
simple expression

$\displaystyle\mathcal{M}\mathcal{R}\tan\left(\frac{\pi}{2\alpha}\right)=2\left(\bar{w}(A)-\bar{w}(A^{\prime})\right)=2\left(w(A^{\prime})-w(A)\right)\,.$
(3.28)

According to the previous definitions (3.21) and to the above expression, the
total energy and momentum associated to a given field configuration should
depend on both $\mathcal{R}$ and $\mathcal{M}$. However, an explicit
calculation shows that the momentum is independent of $\mathcal{M}$:
$P=P(\mathcal{R})$ (see also the discussion in appendix A).

##  4 Adding the $\mathrm{T}\overline{\mathrm{T}}$

At the classical level, the simplest way to introduce the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation is through a dynamical change
of coordinates that involves the components of the stress-energy tensor [60].
Since we will repeatedly switch from one set of coordinates to the other, we
take a moment to fix the notations. The set of coordinates $\bf{w}$ will
always be that in which the EoMs (3.15) and, consequently the current
densities, look like the unperturbed ones. In these coordinates, the effect of
the $\mathrm{T}\overline{\mathrm{T}}$ perturbation is implicitly hidden in the
integration contour $\Gamma_{1}^{(t)}$, which is (equivalent to) the image of
$\Gamma_{1}$, under a dynamical change of coordinates [60]. On the other hand,
in the set of coordinates ${\bf x}$, the effect of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation will be visible as a
modification of both the EoMs and the current densities. One of the main
results of [60] is that these two alternative descriptions of a
$\mathrm{T}\overline{\mathrm{T}}$-perturbed theory are in fact equivalent.
This will allow us to prove that the energy of the
$\mathrm{T}\overline{\mathrm{T}}$-deformed, classical mShG model satisfies the
Burgers' equation (2.10), thus proving the validity of the ODE/IM
correspondence for $\mathrm{T}\overline{\mathrm{T}}$-deformed integrable
quantum field theories.

###  4.1 The dynamical change of coordinates

The dynamical change of space-time coordinates [60] is defined as the
following differential map:

$\left(\begin{array}[c]{cc}dw&\\\
d\bar{w}&\end{array}\\!\\!\\!\\!\\!\\!\right)=J^{T}\left(\begin{array}[c]{cc}dx&\\\
d\bar{x}&\end{array}\\!\\!\\!\\!\\!\\!\right)\,.$ (4.1)

The Jacobian and its inverse are

$\displaystyle\begin{aligned}
&J^{T}(\mathbf{x})=\frac{1}{\Delta(\mathbf{w}(\mathbf{x}),0)}\left(\begin{array}[c]{cc}1+2t\Theta_{0}(\mathbf{w}(\mathbf{x}),0)&-2t\bar{T}_{2}(\mathbf{w}(\mathbf{x}),0)\\\
-2tT_{2}(\mathbf{w}(\mathbf{x}),0)&1+2t\Theta_{0}(\mathbf{w}(\mathbf{x}),0)\end{array}\right)\,,\\\
&{\left(J^{T}\right)}^{-1}(\mathbf{w},0)=\left(\begin{array}[c]{cc}1+2t\Theta_{0}(\mathbf{w},0)&2t\bar{T}_{2}(\mathbf{w},0)\\\
2tT_{2}(\mathbf{w},0)&1+2t\Theta_{0}(\mathbf{w},0)\end{array}\right)\,,\end{aligned}$
(4.2)

where $t$ is a dimensionless coupling trivially related to $\tau$ in (2.6), as
we will see shortly, while

$\Delta(\mathbf{w},0)=(1+2t\Theta_{0}(\mathbf{w}),0)^{2}-4t^{2}T_{2}(\mathbf{w},0)\bar{T}_{2}(\mathbf{w},0)\,.$
(4.3)

The explicit form of (4.1) is

$\displaystyle
dw=\frac{1}{\Delta(\mathbf{w}(\mathbf{x}),0)}\left(dx+2t\Theta_{0}(\mathbf{w}(\mathbf{x}),0)dx-2t\bar{T}_{2}(\mathbf{w}(\mathbf{x}),0)d\bar{x}\right)\,,$
(4.4) $\displaystyle
d\bar{w}=\frac{1}{\Delta(\mathbf{w}(\mathbf{x}),0)}\left(d\bar{x}+2t\Theta_{0}(\mathbf{w}(\mathbf{x}),0)d\bar{x}-2tT_{2}(\mathbf{w}(\mathbf{x}),0)dx\right)\,.$

Now, after a $\mathrm{T}\overline{\mathrm{T}}$ deformation, all the quantities
introduced in the previous sections will depend on the two independent
parameters $\mathcal{R}$ and $t$. So for a generic value of $t$, and in
agreement with the previous discussions, we can write:

$\displaystyle\mathcal{E}(\mathcal{R},t)=\frac{\mathcal{M}}{4}\int\limits_{\Gamma_{1}^{(t)}}\left(dw\,T_{2}(\mathbf{w},0)+d\bar{w}\,\Theta_{0}(\mathbf{w},0)\right)\,,$
(4.5)
$\displaystyle\bar{\mathcal{E}}(\mathcal{R},t)=\frac{\mathcal{M}}{4}\int\limits_{\bar{\Gamma}_{1}^{(t)}}\left(dw\,\Theta_{0}(\mathbf{w},0)+d\bar{w}\,\bar{T}_{2}(\mathbf{w},0)\right)\,,$

where, according to (3.20):

$E(\mathcal{R},t)=\mathcal{E}(\mathcal{R},t)+\bar{\mathcal{E}}(\mathcal{R},t)\,,\quad
P(\mathcal{R},t)=\mathcal{E}(\mathcal{R},t)-\bar{\mathcal{E}}(\mathcal{R},t)\,.$
(4.6)

Let us stress again that in equations (LABEL:eqn:Perint) it is the deformation
of the integration contours which drives the evolution in $t$ of the conserved
charges. Notice also how the map (4.1) implies that the contours
$\bar{\Gamma}_{1}^{(t)}$ and $\Gamma_{1}^{(t)*}$ are not necessarily
equivalent, since in general

$(T_{2}(\mathbf{w}),0)^{*}-\bar{T}_{2}(\mathbf{w},0)\neq 0\,.$ (4.7)

We shall now shift to the alternative point of view, from which the current
densities evolve in $t$ while the contours remain untouched. Using the
invariance of $1$-forms under coordinate transformations, we can write

$\displaystyle\begin{aligned}
&T_{2}(\mathbf{w},0)\,dw+\Theta_{0}(\mathbf{w},0)\,d\bar{w}=T_{2}(\mathbf{x},t)\,dx+\Theta_{0}(\mathbf{x},t)\,d\bar{x}\,,\\\
&\Theta_{0}(\mathbf{w},0)\,dw+\bar{T}_{2}(\mathbf{w},0)\,d\bar{w}=\Theta_{0}(\mathbf{x},t)\,dx+\bar{T}_{2}(\mathbf{x},t)\,d\bar{x}\,,\end{aligned}$
(4.8)

so that

$\displaystyle\mathcal{E}(\mathcal{R},t)=\frac{\mathcal{M}}{4}\int\limits_{\Gamma_{1}}\left(dx\,T_{2}(\mathbf{x},t)+d\bar{x}\,\Theta_{0}(\mathbf{x},t)\right)\,,$
(4.9)
$\displaystyle\bar{\mathcal{E}}(\mathcal{R},t)=\frac{\mathcal{M}}{4}\int\limits_{\bar{\Gamma}_{1}}\left(dx\,\Theta_{0}(\mathbf{x},t)+d\bar{x}\,\bar{T}_{2}(\mathbf{x},t)\right)\,.$

Using the explicit form of the map (4.4) we obtain

$\displaystyle\begin{aligned}
&T_{2}(\mathbf{x},t)=\frac{T_{2}(\mathbf{w}(\mathbf{x}),0)}{\Delta(\mathbf{w}(\mathbf{x}),0)}\,,\quad\bar{T}_{2}(\mathbf{x},t)=\frac{\bar{T}_{2}(\mathbf{w}(\mathbf{x}),0)}{\Delta(\mathbf{w}(\mathbf{x}),0)}\,,\\\
&\Theta_{0}(\mathbf{x},t)=\frac{\Theta_{0}(\mathbf{w}(\mathbf{x}),0)+2t(\Theta_{0}(\mathbf{w}(\mathbf{x}),0)^{2}-T_{2}(\mathbf{w}(\mathbf{x}),0)\bar{T}_{2}(\mathbf{w}(\mathbf{x}),0))}{{\Delta(\mathbf{w}(\mathbf{x}),0})}\,,\end{aligned}$
(4.10)

from which we conclude that

$\displaystyle
dw=dx-2t\left(\Theta_{0}(\mathbf{x},t)\,dx+\bar{T}_{2}(\mathbf{x},t)\,d\bar{x}\right)\,,$
(4.11) $\displaystyle
d\bar{w}=d\bar{x}-2t\left(\Theta_{0}(\mathbf{x},t)\,d\bar{x}+T_{2}(\mathbf{x},t)\,dx\right)\,.$

The inverse map is

$\displaystyle
dx=dw+2t\left(\Theta_{0}(\mathbf{w},0)\,dw+\bar{T}_{2}(\mathbf{w},0)\,d\bar{w}\right)\,,$
(4.12) $\displaystyle
d\bar{x}=d\bar{w}+2t\left(\Theta_{0}(\mathbf{w},0)\,d\bar{w}+T_{2}(\mathbf{w},0)\,dw\right)\,.$

Next, we are going to consider the effect of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation on the `circumference'
$\mathcal{R}$ (3.28). It turns out to be useful to introduce the following
parametrization

$\mathcal{M}\mathcal{R}_{0}\,e^{-\uptheta_{0}}=\frac{2}{\tan{\left(\frac{\pi}{2\alpha}\right)}}\,\int\limits_{\bar{\Gamma}_{1}^{(t)}}dw\,,\quad\mathcal{M}\mathcal{R}_{0}\,e^{+\uptheta_{0}}=\frac{2}{\tan{\left(\frac{\pi}{2\alpha}\right)}}\,\int\limits_{\Gamma_{1}^{(t)}}d\bar{w}\,,$
(4.13)

where $\uptheta_{0}(t=0)=0$ and $\mathcal{R}_{0}(t=0)=\mathcal{R}$. These
expressions will be useful to make contact with the
$\mathrm{T}\overline{\mathrm{T}}$-deformed theory on quantum side of the
correspondence [24]. Using the map (4.11) we write

$\displaystyle\int\limits_{\bar{\Gamma}_{1}^{(t)}}dw$
$\displaystyle=\int\limits_{\bar{\Gamma}_{1}}dx-2t\int\limits_{\bar{\Gamma}_{1}}\Big{(}dx\,\Theta_{0}(\mathbf{x},t)+d\bar{x}\,\bar{T}_{2}(\mathbf{x},t)\Big{)}$
$\displaystyle=\frac{1}{2}\tan\left(\frac{\pi}{2\alpha}\right)\mathcal{M}\mathcal{R}-\frac{8}{\mathcal{M}}\,t\bar{\mathcal{E}}(\mathcal{R},t)\,,$
(4.14)

where relations (4.3) and (4.10) were used. Similarly, we obtain

$\int\limits_{\Gamma_{1}^{(t)}}d\bar{w}=\frac{1}{2}\tan{\left(\frac{\pi}{2\alpha}\right)}\mathcal{M}\mathcal{R}-\frac{8}{\mathcal{M}}\,t\mathcal{E}(\mathcal{R},t)\,.$
(4.15)

Setting

$\uptau=-\frac{8}{\mathcal{M}^{2}\tan{(\frac{\pi}{2\alpha})}}t\,,$ (4.16)

and using relations (4.6) we arrive at

$\displaystyle\begin{cases}\mathcal{R}_{0}\,e^{-\uptheta_{0}}=\mathcal{R}+\uptau\left(E(\mathcal{R},\tau)-P(\mathcal{R},\tau)\right)\\\
\mathcal{R}_{0}\,e^{+\uptheta_{0}}=\mathcal{R}+\uptau\left(E(\mathcal{R},\tau)+P(\mathcal{R},\tau)\right)\end{cases}\,.$
(4.17)

The previous analysis and the resulting equations (4.17) have the following
simple geometric interpretation: the net effect of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation is equivalent to a rescaling of
$\mathcal{R}$ plus a Minkowski rotation

$\displaystyle\mathcal{R}\rightarrow\mathcal{R}_{0}\,e^{+\uptheta_{0}},$
(4.18)

with rapidity

$\displaystyle\uptheta_{0}=\frac{1}{2}\log\left(\frac{\mathcal{R}+\uptau\left(E(\mathcal{R},\tau)+P(\mathcal{R},\tau)\right)}{\mathcal{R}+\uptau\left(E(\mathcal{R},\tau)-P(\mathcal{R},\tau)\right)}\right)\,,$
(4.19)

and

$\mathcal{R}_{0}^{2}=\left(\mathcal{R}+\uptau
E(\mathcal{R},\uptau)\right)^{2}-\uptau^{2}P^{2}(\mathcal{R},\uptau)\,,$
(4.20)

as pictorially represented in Figure 3(a). On the other hand, the dynamical
interpretation is encoded in the dependence of $\uptheta_{0}$ and
$\mathcal{R}_{0}$ on the energy and momentum of the state. Therefore, the net
effect of the $\mathrm{T}\overline{\mathrm{T}}$ perturbation corresponds to a
doubling/redefinition of the scaling parameter $s\rightarrow(s^{-},s^{+})$ of
the previous sections. For example, at fixed `mass-scale' $\mathcal{M}$, it
corresponds to

$\displaystyle\mathcal{R}(s)\rightarrow\left(\mathcal{R}^{+},\mathcal{R}^{-}\right)=\left(\mathcal{R}(s^{-}),\mathcal{R}(s^{+})\right)=\left(\mathcal{R}_{0}\,e^{-\uptheta_{0}},\mathcal{R}_{0}\,e^{+\uptheta_{0}}\right)\,.$
(4.21)

Notice also that, one might find the interpretation of equations (4.17) more
intuitively after switching to the Euclidean version of the theory, obtained
through the formal replacements: $P\rightarrow\imath P$ and
$\uptheta_{0}\rightarrow\imath\uptheta_{0}$. The interpretation using complex
variables is also helpful to guide the intuition in the following final steps
of our analysis.

(a)

(b)

Figure 3: Pictorial representation of the net effect of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation, as a rescaling of the `volume'
plus a Minkowski rotation. Figure 3(a) corresponds to equation (4.18), while
Figure 3(b) to the inverse relation (4.22).

We now need further, independent knowledge to close our system of equations
and make it equivalent to the Burgers' equation. Extra constraints can be
obtained by demanding the consistency between (4.12) and (4.11), encoding
exact information on the fundamental invertibility property of the Jacobian
matrix (4.1). Concretely, the idea is to derive the analogue of (4.17), for
the inverse transformation

$\displaystyle\mathcal{R}_{0}\rightarrow\mathcal{R}\,e^{-\uptheta_{0}},$
(4.22)

as depicted in Figure 3(b), by integrating the r.h.s of (4.12) along
appropriate contours $C_{1}$ and $\bar{C}_{1}=C^{*}_{1}$, of a form similar to
that of $\Gamma_{1}$ and $\bar{\Gamma}_{1}$ (see Figure 2). The endpoints of
$\bar{C}_{1}$ are $w(\tilde{A})$ and $w(\tilde{A}^{\prime})$, with relative
distance

$\displaystyle
w(\tilde{A}^{\prime})-w(\tilde{A})=\frac{1}{2}\,\mathcal{M}\tan\left(\frac{\pi}{2\alpha}\right)\mathcal{R}_{0}\,.$
(4.23)

Under the coordinate transformation $\mathbf{w}\rightarrow\mathbf{x}$, $C_{1}$
and $\bar{C}_{1}$ are mapped to $C_{1}^{(t)}$ and $\bar{C}_{1}^{(t)}$,
respectively, with

$\displaystyle\bar{C}_{1}^{(t)}$ $\displaystyle:\quad\quad
x(w(\tilde{A}^{\prime}))-x(w(\tilde{A}))=\frac{1}{2}\,\mathcal{M}\tan\left(\frac{\pi}{2\alpha}\right)\mathcal{R}e^{+\uptheta_{0}}\,,$
(4.24) $\displaystyle C_{1}^{(t)}$
$\displaystyle:\quad\quad\bar{x}(\bar{w}(\tilde{A}))-\bar{x}(\bar{w}(\tilde{A}^{\prime}))=\frac{1}{2}\,\mathcal{M}\tan\left(\frac{\pi}{2\alpha}\right)\mathcal{R}e^{-\uptheta_{0}}\,.$

Proceeding as above, we arrive at the following equations

$\displaystyle\begin{cases}\mathcal{R}\,e^{+\uptheta_{0}}=\mathcal{R}_{0}-\uptau\left(E(\mathcal{R}_{0},0)-P(\mathcal{R}_{0},0)\right)\\\
\mathcal{R}\,e^{-\uptheta_{0}}=\mathcal{R}_{0}-\uptau\left(E(\mathcal{R}_{0},0)+P(\mathcal{R}_{0},0)\right)\end{cases}\,.$
(4.25)

###  4.2 The Burgers' equation

As a final step in our analysis, we shall derive the Burgers' equation
involving only classical physical quantities.

From equations (4.17) and (4.25), we have

$\sinh{\uptheta_{0}}=\tau\frac{P(\mathcal{R},\uptau)}{\mathcal{R}_{0}}=\tau\frac{P(\mathcal{R}_{0},0)}{\mathcal{R}}\quad\Longrightarrow\quad\mathcal{R}P(\mathcal{R},\uptau)=\mathcal{R}_{0}P(\mathcal{R}_{0},0)\,,$
(4.26)

implying that $P(\mathcal{R},\uptau)=P(\mathcal{R})\propto\mathcal{R}^{-1}$.
Alternatively, the exactly-quantised expression

$P(\mathcal{R})=\frac{2\pi p}{\mathcal{R}}\,,\quad p\in\mathbb{Z}\,,$ (4.27)

can be obtained through the WKB type analysis reported in appendix A. Putting
(4.17) and (4.25) together, one get

$\displaystyle\begin{cases}e^{+\uptheta_{0}}\,\left(E(\mathcal{R},\uptau)-P(\mathcal{R},\tau)\right)=E(\mathcal{R}_{0},0)-P(\mathcal{R}_{0},0)\\\
e^{-\uptheta_{0}}\,\left(E(\mathcal{R},\tau)+P(\mathcal{R},\tau)\right)=E(\mathcal{R}_{0},0)+P(\mathcal{R}_{0},0)\\\
\end{cases}\,,$ (4.28)

that is

$E^{2}(\mathcal{R},\tau)-P^{2}(\mathcal{R},\tau)=E^{2}(\mathcal{R}_{0},0)-P^{2}(\mathcal{R}_{0},0)\,.$
(4.29)

Moreover, (4.17) and (4.25) imply the additional constraint [60]

$\frac{\partial}{\partial\uptau}\mathcal{R}=-E(\mathcal{R},\uptau)\,,$ (4.30)

at fixed $\mathcal{R}_{0}$. This result together with (4.29) is the solution,
in implicit form, of the Burgers' equation [58]:

$\frac{\partial}{\partial\uptau}E(\mathcal{R},\uptau)=E(\mathcal{R},\uptau)\frac{\partial}{\partial\mathcal{R}}E(\mathcal{R},\uptau)+\frac{P(\mathcal{R})^{2}}{\mathcal{R}}\,.$
(4.31)

In conclusion, assuming the validity of the ODE/IM at $\tau=0$, i.e.

$E(R,0)_{\textit{quantum}}=E(R,0)_{\textit{classical}}\,,\quad\quad
P(R)_{\textit{quantum}}=P(R)_{\textit{classical}}\,,$ (4.32)

the fact that the quantum and the classical
$\mathrm{T}\overline{\mathrm{T}}$-evolution equations coincide, implies

$\definecolor{cb@saved}{rgb}{0,0,0}\color[rgb]{.75,0,.25}\boxed{\color[rgb]{0,0,0}\definecolor{cb@saved}{rgb}{0,0,0}\color[rgb]{1,1,1}\boxed{\color[rgb]{0,0,0}E(R,\tau)_{\textit{quantum}}=E(R,\tau)_{\textit{classical}}}}\,.$
(4.33)

This equality constitutes the key result of the present work.

##  5 Conclusions

The main result of this paper is the extension to
$\mathrm{T}\overline{\mathrm{T}}$ perturbed models of the classical/quantum
duality associated with the off-critical variant of the ODE/IM correspondence.
Our work links properties of the $\mathrm{T}\overline{\mathrm{T}}$
perturbation on quantum and classical levels that previously stood on separate
grounds. The analysis highlights a deep connection between the classical
coordinate transformation, its invertibility, and the factorization property
of the operator $\mathrm{T}\overline{\mathrm{T}}$, which plays an essential
role at the quantum level. In this work, attention was restricted to the
modified sinh-Gordon case of [14], however, our result has a much wider
validity. First, we notice that the explicit form of the mShG potential (3.2)
does not appear anywhere in section 4 – except for a few constants.
Consequently, the validity of the result (4.33) can be immediately extended to
ODE/IM correspondence based on more general potentials, _e.g._ the polynomial
ones used in the context of AdS/CFT correspondence for the computation of
polygonal Wilson loops [76, 77]. Similarly, it is not difficult to see how the
arguments of section 4 remain valid for $\mathrm{T}\overline{\mathrm{T}}$
deformations of other known instances of the ODE/IM correspondence [19, 78,
18, 21, 22, 79] and are also easily adaptable to the study of the Lorentz
breaking deformations introduced in [60].

We wish to stress that the Burgers' equation (4.31) is actually a general
property of $\mathrm{T}\overline{\mathrm{T}}$-deformed classical field
theories. In fact, the argument used in section 4 teaches us that everything
we need to prove (4.31) are the differential maps (4.11, 4.12) and an
appropriate choice of Cauchy cycles for both the $\mathbf{w}$ and the
$\mathbf{x}$ coordinates, over which we integrate the closed $1$-forms (4.8)
in order to obtain the energy and the momentum. Integrating (4.11, 4.12) over
these cycles we arrive at the equivalent of (4.17, 4.25), from which the
Burgers' equation (4.31) immediately follows.

Another significant point emerging from our analysis is the link between the
dynamical change of coordinates and the factorization property of
$\mathrm{T}\overline{\mathrm{T}}$ [32]. This fact suggests the existence of
coordinate transformations also for deformations driven by operators built
using higher spin currents. However, preliminary inspections indicate that
spacetime maps exist, besides $\mathrm{T}\overline{\mathrm{T}}$, only for
$\mathrm{J}\overline{\mathrm{T}}$ [80, 81] and the
$\mathrm{T}\overline{\mathrm{T}}_{s}$ Lorentz-breaking models of [60], not for
the generic $\mathrm{T}_{s}\overline{\mathrm{T}}_{s^{\prime}}$ deformations.
Nevertheless, it may be worth investigating the relation between the latter
and maps between the spacetime and more complicated surfaces embedded in
higher dimensional spaces. The additional dimensions would correspond to the
higher times associated to the commuting Hamiltonian flows generated by the
higher conserved charges. A further, natural follow-up would be the extension
of the ODE/IM to the recently discovered
$\sqrt{\mbox{$\mathrm{T}\overline{\mathrm{T}}$}}$ deformations [61, 82, 83,
84, 85, 86, 87].

Finally, it would be nice to use the current framework to understand better
the geometric meaning of the Hagedorn singularity appearing in the deformed
quantum spectrum and explore at a deeper level the consequences of the
$\mathrm{T}\overline{\mathrm{T}}$ perturbation in the context of minimal
surfaces and polygonal Wilson loops [76, 77] (see also the recent review
[88]), and in the framework of quasi-normal modes of black holes [89].

## Acknowledgements

This project was partially supported by the INFN project SFT, grant
PHY-2210349, by the Simons Collaboration on Confinement and QCD Strings, and
by the FCT Project PTDC/MAT-PUR/30234/2017 ``Irregular connections on
algebraic curves and Quantum Field Theory". S.N. wishes to thank the
department of physics of the Università degli Studi di Torino for kind
hospitality.

##  Appendix A The conformal field theory limit

At the classical level, the ultraviolet limit yielding the conformal field
theory behaviour corresponds to the following (left-moving) scaling limit:

$z=y\,e^{-\frac{\vartheta}{1+\alpha}}\,,\quad\vartheta\rightarrow+\infty\,,\quad|z|\sim
s\rightarrow 0^{+}\,,$ (A.1)

with

$y=e^{\frac{\vartheta}{1+\alpha}}z\,\quad{\bf
E}=e^{2\alpha\vartheta/(1+\alpha)}s^{2\alpha}\,,$ (A.2)

kept finite. In this limit, the first equation in (3.7) reduced to the
Schrödinger equation of [1], with the addition of a finite number of regular
singularities $\left\\{y_{a}\right\\}$ [11, 12]

$\displaystyle\left(\frac{d^{2}}{dy^{2}}-V\left(y,\\{y_{a}\\}_{a=1}^{L}\right)-y^{2\alpha}+{\bf
E}\right)\psi_{\text{CFT}}(y)=0\,.$ (A.3)

Here $V$ is the so-called `monster potential' [11],

$V\left(y,\\{y_{a}\\}_{a=1}^{L}\right)=\frac{l(l+1)}{y}+2\frac{d^{2}}{dy^{2}}\sum_{a=1}^{L}\log\left(y^{2\alpha+2}-y_{a}\right)\,,$
(A.4)

and the additional singularities are defined on the cone $y_{a}\in
C_{\pi/\alpha}\setminus[{\bf E}^{1/(2\alpha)},+\infty)$, away from the apex:
$y_{a}\neq 0$.
In the right-moving sector, the opposite scaling limit

$\bar{z}=y\,e^{-\frac{\vartheta}{1+\alpha}}\,,\quad\vartheta\rightarrow-\infty\,,\quad|\bar{z}|\sim
s\rightarrow 0^{+},$ (A.5)

should be implemented. In this sector, a different number of singularities,
$\bar{L}$, will mark the monster potential.

###  A.1 WKB analysis and the monster potential

A naive, straightforward WKB analysis of (A.3) runs into problems, due to the
presence of two competing scales E and $\alpha$. A more refined analysis
involves the following change of variables [90]:

$y={\bf
E}^{\frac{1}{2\alpha}}\hat{y}^{\frac{1}{2l+1}}\,,\quad\psi_{\text{CFT}}(y)={\bf
E}^{-\frac{l}{2\alpha}}\hat{y}^{-\frac{l}{2l+1}}\hat{\psi}_{\text{CFT}}(\hat{y})\,.$
(A.6)

In this new set of coordinates, equation (A.3) reads

$\left(-\epsilon^{2}\frac{d^{2}}{d\hat{y}^{2}}+q(\hat{y})+\epsilon^{2}\xi_{L}(\hat{y},\epsilon)\right)\hat{\psi}_{\text{CFT}}(\hat{y})=0\,,$
(A.7)

with

$\displaystyle
q(\hat{y})=\frac{1}{4\lambda^{2}}\hat{y}^{\frac{1}{\lambda}}\left(\hat{y}^{\frac{\alpha}{\lambda}}-1\right)\,,$
(A.8)

The last term in the parenthesis, in equation (A.7), is

$\displaystyle\xi_{L}(\hat{y},\epsilon)$
$\displaystyle=-\frac{1}{\lambda}\frac{d^{2}}{d\hat{y}^{2}}\sum_{a=1}^{L}\log\left(\hat{y}^{\frac{\alpha+1}{\lambda}}-\epsilon^{2}y_{a}\right)$
$\displaystyle+\frac{2\lambda-1}{\lambda^{3}}(\alpha+1)^{2}\hat{y}^{\frac{\alpha+1}{\lambda}-2}\sum_{a=1}^{L}\frac{\epsilon^{2}y_{a}}{\left(\hat{y}^{\frac{\alpha+1}{\lambda}}-\epsilon^{2}y_{a}\right)^{2}}\,,$
(A.9)

with

$\lambda=l+\frac{1}{2}\,,\quad\epsilon={\bf E}^{-\frac{\alpha+1}{2\alpha}}\,.$
(A.10)

Now the WKB expansion of $\hat{\psi}_{\text{CFT}}(\hat{y})$ is naturally
organized by the single parameter $\epsilon$, which combines the two scales E
and $\alpha$

$\hat{\psi}_{\text{CFT}}(\hat{y})=\text{exp}\left(\frac{1}{\epsilon}\sum_{b\geq
0}\epsilon^{b}S_{b}(\hat{y})\right)\,,$ (A.11)

while

$\xi_{L}(\hat{y},\epsilon)=\xi_{L}(\hat{y},0)+\sum_{c\geq
1}\epsilon^{c}\xi_{c}^{(L)}(\hat{x})\,.$ (A.12)

The first few terms of the solution are easily computed by recursion,

$\displaystyle S_{0}^{\prime}(\hat{y})=-\sqrt{q(\hat{y})}\,,\quad
S_{1}^{\prime}(\hat{y})=-\frac{q^{\prime}(\hat{y})}{4q(\hat{y})}\,,$ (A.13)
$\displaystyle{S_{2}^{(L)}}^{\prime}(\hat{y})=-\frac{1}{48}\left(\frac{q^{\prime\prime}(\hat{y})}{q(\hat{y})^{\frac{3}{2}}}+5\frac{d}{d\hat{y}}\left(\frac{q^{\prime\prime}(\hat{y})}{q(\hat{y})^{\frac{3}{2}}}\right)\right)-\frac{\xi_{L}(\hat{y},0)}{2\sqrt{q(\hat{y})}}\,.$
(A.14)

From $S_{2}^{(L)}(\hat{y})$ we can extract the first IM [88] as

$\mathcal{E}=\frac{M}{4}\int_{\gamma_{\text{WKB}}}d\hat{y}\,{S_{2}^{(L)}}^{\prime}(\hat{y})\,.$
(A.15)

Here the WKB integration contour $\gamma_{\text{WKB}}$ starts from $+\infty$
just above the real axis, encircles the turning point $\hat{y}=1$ and goes
back to $+\infty$, just below the real axis. The mass $M$ sets the energy
scale. Similarly, for the right-moving sector

$\bar{\mathcal{E}}=\frac{M}{4}\int_{\gamma_{\text{WKB}}}d\hat{y}\,{S_{2}^{(\bar{L})}}^{\prime}(\hat{y})\,.$
(A.16)

Now, we can find the exact expression for the total momentum:

$\displaystyle P$
$\displaystyle=\mathcal{E}-\bar{\mathcal{E}}=-\frac{M}{4}\int_{\gamma_{\text{WKB}}}d\hat{y}\,\frac{\xi_{L}(\hat{y},0)}{2\sqrt{q(\hat{y})}}+\frac{M}{4}\int_{\gamma_{\text{WKB}}}d\hat{y}\,\frac{\xi_{\bar{L}}(\hat{y},0)}{2\sqrt{q(\hat{y})}}$
$\displaystyle=\frac{M}{\epsilon}L(\alpha+1)\,\int_{\mathbf{E}^{\frac{1}{2\alpha}}}^{+\infty}\frac{dy}{y^{2}}\frac{1}{\sqrt{y^{2\alpha}-\mathbf{E}}}-\frac{M}{\epsilon}\bar{L}(\alpha+1)\,\int_{\mathbf{E}^{\frac{1}{2\alpha}}}^{+\infty}\frac{dy}{y^{2}}\frac{1}{\sqrt{y^{2\alpha}-\mathbf{E}}}$
$\displaystyle=\frac{2\pi(L-\bar{L})}{\mathcal{R}}\,,$ (A.17)

where, in the last equality in (A.17), we used the explicit form of
$\mathcal{R}$

$\mathcal{R}=\frac{2\sqrt{\pi}}{M}\frac{\Gamma\left(1+\frac{1}{2\alpha}\right)}{\Gamma\left(\frac{3}{2}+\frac{1}{2\alpha}\right)}\,.$
(A.18)

Therefore, we arrived at the quantization rule for the total momentum

$P=\frac{2\pi\,p}{\mathcal{R}}\,,\quad p\in\mathbb{Z}\,,$ (A.19)

Similarly we can compute the total energy, arriving at the following
expression

$E=\mathcal{E}+\mathcal{\bar{E}}=-\frac{\pi}{6\mathcal{R}}c_{\text{eff}}\,,$
(A.20)

where $c_{\text{eff}}$ is the effective central charge:

$c_{\text{eff}}=1-\frac{6(l+\textstyle{\frac{1}{2}})^{2}}{\alpha+1}-12(L+\bar{L})\,.$
(A.21)

The results (A.19) and (A.20) can also be derived following other routes. As
an example, we have checked the scaling limit of [14] (in which
$s^{\alpha+1}/\mathcal{M}=M^{-1}$ is kept constant). From the general
assumptions in section 3, the WKB analysis reassures us that (A.19) is a
rather general result. See also [91] on recent interesting developments in the
study of excited-states in the ODE/IM framework.

Finally, it is worth stressing that the total contribution to the momentum $P$
comes from the region $(z,\bar{z})\sim(0,0)$, therefore the result (A.19) is
correct also for the full massive version, and after the perturbation with
$\mathrm{T}\overline{\mathrm{T}}$. This is an expected result as, at both the
classical and the quantum levels, the additions of the sin(h)-Gordon potential
and $\mathrm{T}\overline{\mathrm{T}}$ do not modify the original translational
properties along the space direction.

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