# Canonical analysis of the gravitational description of the $T\bar{T}$
deformation

Florencia<EMAIL_ADDRESS>Guzmán Hernández-
<EMAIL_ADDRESS>and Esteban<EMAIL_ADDRESS>Instituto
de Física, Facultad de Ingeniería (Universidad de la República), Julio Herrera
y Reissig 565, 11300 Montevideo, Uruguay.

###### Abstract

The description of the $T\bar{T}$ deformation in terms of two-dimensional
gravity is analyzed from the Hamiltonian point of view, in a manner analogous
to the ADM description of general relativity. We find that the Hamiltonian
constraints of the theory imply relations between target-space momentum at
finite volume which are equivalent to the $T\bar{T}$ finite volume flow
equations. This fully-quantum $T\bar{T}$ result emerges already at the
classical level within the gravitational theory. We exemplify the analysis for
the case when the undeformed sector is a collection of $D-2$ free massless
scalars, where it is shown that –somewhat non-trivially– the target-space two-
dimensional Poincaré symmetry is extended to $D$ dimensions. The connection
between canonical quantization of this constrained Hamiltonian system and
previous path integral quantizations is also discussed. We extend our analysis
to the “gravitational” description of $J\bar{T}$-type deformations, where it
is found that the flow equations obtained involve deformations that twist the
spatial boundary conditions.

##  I Introduction

The so-called $T\bar{T}$ deformation has gathered considerable attention in
recent years due to its connection to a variety of different aspects of
quantum field theory, ranging from integrability to holography and 3D gravity.
This deformation, originally introduced in Smirnov:2016lqw ; Cavaglia:2016oda
, consists in a procedure to define a one-parameter family of quantum field
theories starting from a seed or “undeformed” theory. The procedure may be
defined as follows Smirnov:2016lqw . Any 2D QFT may be infinitesimally
deformed by the $T\bar{T}$ operator, defined as

$T\bar{T}\equiv\det\left(T_{\mu\nu}\right)$ (1)

As shown in Smirnov:2016lqw , this deformation is well defined due to the fact
that, when regularizing this product of operators by point splitting, any
singularities in the coincident limit appear only as total derivatives, and
thus are absent from the infinitesimal deformation, which is built from the
spacetime integral of this composite operator. This argument relies on the
conservation of the stress energy tensor, which holds for any relativistic 2D
QFT. Thus any relativistic 2D QFT may be infinitesimally deformed in this
manner, so that the deformation defines a vector field in the space of
relativistic 2D QFTs. Integral curves of this vector field define one-
parameter families of theories, indexed by some parameter $\lambda$. Following
some such curve starting from some particular theory $S_{0}$ –the ”seed” or
”undeformed” theory– one obtains at finite $\lambda$ the theory $S_{\lambda}$,
which is often referred to as the $T\bar{T}$ deformation of $S_{0}$444In this
paragraph the letter $S$ doesn’t nececesarily refer to the action but rather
more abstractly to some 2d quantum field theory $S$..
Interest in these one-parameter families of theories has been due to the fact
that they satisfy some remarkable properties. For instance, as noticed first
in Smirnov:2016lqw ; Cavaglia:2016oda , the finite volume spectrum of a given
one-parameter family satisfies the exact equation

$\partial_{\lambda}E=E\partial_{R}E+\frac{P^{2}}{R}$ (2)

which in hydrodynamics is known as Burgers’ equation. Here $R$ denotes the
size of the compact spatial direction and $P$ is the total spatial momentum,
which is conserved under the deformation. This is a remarkable formula: while
finite volume spectra of QFTs are in general very difficult to calculate, for
$T\bar{T}$-deformed theories these can be obtained by solving the PDE in (2),
assuming the spectrum of the seed theory is known.
In this work we elaborate on the idea put forward in dubovsky2017asymptotic ;
dubovsky2018tt  that the $T\bar{T}$ deformation may be realized by coupling
the undeformed theory to a two dimensional gravitational sector. In
dubovsky2017asymptotic  it was shown that this reproduces the $T\bar{T}$
deformed S-matrix, whereas dubovsky2018tt  showed how the $T\bar{T}$-deformed
finite-volume spectrum emerges in the gravitational theory.
The proposal of dubovsky2018tt  suggests that555This is a slight reformulation
of the action presented in dubovsky2018tt  to contemplate the possibility of a
non flat geometry in target-space (see for instance Mazenc:2019cfg  for a
discussion on this). at finite deformation parameter $\lambda$ the deformed
theory may be described by the action

$S_{\lambda}=S_{0}[g,\phi]+S_{grav}[X,e]$ (3)

where

$S_{grav}=\frac{-1}{2\lambda}\int\epsilon_{ab}(X^{*}E-e)^{a}\wedge(X^{*}E-e)^{b}$
(4)

and $S_{0}[g,\phi]$ is the action of the undeformed theory coupled to the
dynamical metric $g$ of the gravitational sector, with $\phi$ standing for the
fields in the undeformed theory. The formulation of the gravitational part of
the action requires the addition of an auxiliary spacetime with fixed geometry
which we will refer to as “target space”. This auxiliary spacetime is taken to
have the same topology as the spacetime in which we are formulating the
undeformed theory, which we will refer to as “worldsheet”, although in general
no stringy interpretation will be implied. The fixed geometry of the target
space is described by the dyad $E^{a}$. In the gravitational action the
dynamical metric on the worldsheet appears through the dyad $e^{a}$, although
the coupling to matter - that is, to the undeformed action $S_{0}$\- is
accomplished through the coupling to the two-dimensional metric $g$ built out
of $e^{a}$ in the standard way. The gravitational action also includes an
auxiliary field $X$ which is a map from the worldsheet to target space in a
particular homotopy class, on which we will elaborate below for the cases of
interest, and $X^{*}E$ stands for the pullback to the worldhseet of the
target-space form $E^{a}$. Thus the dynamical fields in the gravitational
sector are the worldsheet dyad $e^{a}$ and the map $X$, whereas the dyad
$E^{a}$ is to be thought of as a background field.
In this paper we analyse this gravitational theory from the Hamiltonian point
of view, in analogy of the Arnowitt-Deser-Misner (ADM) Arnowitt:1962hi
analysis of classical general relativity (see for instance Bojowald:2010qpa ).
From the canonical point of view, the theory behaves as a constrained
Hamiltonian system, with the constraints generating the gauge transformations
(diffeomorphisms) of the system when acting on canonical variables. The
dynamics is restricted to the constraint surface inside of the phase space of
the theory. One of the main results of this paper is to show that, when
considering the gravitational theory in finite volume, imposing the
aforementioned constraints necessarily requires the observables associated to
target-space time translations (the physical energy and momentum) to satisfy
the $T\bar{T}$-flow equation (2). It is interesting that this happens already
when imposing the constraints at the classical level. 666A similar analysis to
that done here can be found in Tolley:2019nmm , although here the focus is
more on how the burgers equation for target-space observables emerges from the
constraints, and we consider a somewhat different class of undeformed theories
to analyze their spectrum, perhaps more general in a certain sense.
We also analyze the gravitational description of the $T\bar{T}$ at the quantum
level. In dubovsky2018tt , a path integral quantization of this theory was
introduced, and the torus partition function

$Z=\frac{\int DeDX}{V_{Diffs}}e^{S_{grav}}Z_{0}[g(e)]$ (5)

was calculated, showing that this theory indeed reproduces the
$T\bar{T}$-deformed spectrum given by (2). Here $Z_{0}[g]$ is the torus
partition function of the undeformed theory with background metric $g(e)$
(built out of $e$) and $V_{Diffs}$ is the volume of the group of spacetime
diffeomorphisms, which must be factored out of the integration measure via a
Faddeev-Popov type procedure.
Our objective here will be to compare with the results of dubovsky2018tt
starting from the canonical perspective. The starting point of the approach
considered here is the construction of the so-called “reduced” phase space:
the phase space obtained by quotienting the constraint surface by the gauge
transformations generated by them. In practice, this phase space is
constructed by picking a representative of each gauge orbit by a suitable
gauge-fixing condition. By choosing a specific gauge, it is shown that the
reduced phase space is identical to that of the undeformed theory, for which
we assume a canonical quantization prescription exists. We consider the
partition function over the quantized reduced phase space built taking the
generators of spacetime translations in the gravitational theory as energy and
momentum. We then compare this canonical partition function to that given by
5, and attempt to rederive the $T\bar{T}$ deformed spectrum. The procedure we
follow is straightforward and similar to that followed in many textbooks
Itzykson:1980rh ; weinberg1995quantum  to go from the canonical quantization
of gauge-fixed gauge theories to the covariant Faddeev-Popov path integral in
terms of the action functional. We present this calculation in some detail to
focus on the analysis of the path integral measure obtained from this
procedure and compare it with that of dubovsky2018tt . Since in that work the
path integral measure played a crucial role in getting the $T\bar{T}$ result,
it is interesting to compare with the measure obtained from the canonical
analysis.
Regarding the general presentation of our results, many of the ideas applied
here are well-known textbook level concepts about the canonical analysis of
constrained Hamiltonian systems (e.g. henneaux ). We spell these out in detail
to keep the presentation self-contained and because we believe that this
system is simple yet interesting enough to warrant a detailed exposition
exemplifying this type of analyses.
Before we proceed however, some further comments are warranted regarding the
sense in which the theory defined by (3) may be regarded as a “gravitational”
theory. To start with, even though the gravitational dynamics of this theory
is formulated in terms of the frame field $e^{a}$, similar to what is done in
the first order formalism for gravity, in contrast to what happens in that
case the spin connection does not appear in this theory. This implies that the
gravitational sector of (3) can only couple to bosonic matter, through the
metric obtained from the frame field in the standard way. Coupling the
gravitational part to fermionic matter does not appear to be possible in
principle, given that it would involve the spin connection which is absent
from this theory. Another unusual aspect of the theory formulated in (3) is
that it is not invariant under local Lorentz transformations (or in the
euclidean case, of local rotations) of the internal indices of the frame field
$e^{a}$. This makes the previously mentioned problem of coupling to spinors
seemingly insurmountable: even if one were to write the spin connection in the
coupling to fermions in terms of the frame field by solving for the spin
connection from the zero torsion condition, this is inconsistent in a theory
not invariant under $SO(1,1)$ (or $SO(2)$) gauge transformations in the
internal indices of the frame field.
The lack of invariance under local Lorenz transformations also implies that
the theory may be ill-defined (as a gravitational theory) in manifolds of
arbitrary topology. Indeed, for compact manifolds of non-zero Euler character,
given a non-singular metric there is no globally defined frame field $e^{a}$
describing it. For these manifolds, given a non-singular metric, the
corresponding frame field must be defined patch-wise and in such a way that on
patch overlaps the corresponding frame fields are related by an internal
rotation. However, since the action is not invariant under these rotations it
is thus not single valued on the overlaps of the different patches. A possible
way out of this problem would be to consider only field configurations where
$e^{a}$ is globally defined, although this necessarily would involve
configurations of $e$ that vanish at certain points of the manifold.
The aforementioned problems do not appear however when one considers the
theory on the manifolds $\mathbb{R}^{2}$, $T^{2}$ or $S\times\mathbb{R}$,
which are indeed the cases of clearer physical interest. The first two cases
were considered in dubovsky2018tt  and the latter will be considered in this
paper. In all of these cases any non-singular metric may be described in terms
of a single globally defined frame field, and thus there is no problem with
the well-definedness of the action. Furthermore, if in these cases the target-
space is taken to be flat -notice all these manifolds all admit metrics with
everywhere-vanishing curvature- then the gravitational part of (3) is
equivalent to the first order formulation of Jackiw-Teitelboim gravity, with
the extra condition that one restricts to the sector where the spin connection
has zero holonomy around all noncontractible cycles, if any777Within JT
gravity, this condition may be imposed by compactifying the field range of the
lagrange multiplier associated to the zero curvature condition as explained
for instance in appendix A of Witten:2018xfj .. Indeed, this is the way that
the gravitational description of the $T\bar{T}$ deformation given by (3) was
originally derived in dubovsky2018tt . Given that the theory with a spin
connection may be coupled to fermions, it would appear that under these
conditions the theory given by (3) may also be consistently coupled to
fermionic matter. We will however not deal with fermionic matter for the rest
of our analysis and leave it for possible future work.
The rest of the paper proceeds as follows. In Section II, we develop the
canonical formalism for the $T\bar{T}$-gravitational theory at the classical
level and obtain our main result regarding the deformed spectrum. In section
III we discuss the free massless scalar field case and its connection to non-
critical strings. Section IV is devoted to exploring the coupling to more
general matter content. In section V we apply similar ideas to a
$J\bar{T}$-type of deformation. Section VI discusses the emergence of the
covariant path integral quantization of dubovsky2018tt  from canonical
quantization. Finally, we present some conclusions drawn from our analysis and
outline future work.

##  II Canonical analysis of $T\bar{T}$ gravity - classical equations of
motion

We now proceed to study the theory specified in (3) in the canonical formalism
at the level of the classical equations of motion. As explained in the
introduction, this theory is expected to describe the $T\bar{T}$ deformation
of the seed theory $S_{0}$ at deformation parameter $\lambda$. From the point
of view of the gravitational interpretation, the undeformed theory $S_{0}$ is
the “matter” sector to which the gravitational field $e^{a}$ couples to. Our
objective will be to see how the formula (2) for the finite volume spectrum
arises from the point of view of the canonical formalism.
We will proceed as follows. We first foliate spacetime into spatial surfaces
as required by the canonical formalism and identify the canonical variables
and the gravitational and matter hamiltonians. These are seen to be of the
“pure-constraint” form –as usual in the canonical formalism for gravity– the
constraints being imposed by the equations of motion of the non-dynamical
parts of the metric field, which act as Lagrange multipliers. We then identify
the target-space observables and via a partial gauge fixing, we will show that
the zero modes of the constraints yield relations between these observables
equivalent to the $T\bar{T}$ flow equations.
As mentioned before, we begin by foliating spacetime by spatial surfaces, as
required by the canonical formalis. Since we are interested in placing the
theory in finite volume, this implies that we need to consider the theory in a
spacetime with the topology of a cylinder $S\times\mathbb{R}$. Notice that the
prescription for the gravitational description of the $T\bar{T}$ deformation
requires both worldsheet and target-space to be of this form. The worldsheet
cylinder will be coordinatized by the pair $(t,x)$, where the compact spatial
coordinate $x$ satisfies $x\sim x+1$. We take the target space cylinder to
have circumference $R$: this will be the “physical” volume our finite-volume
spectra will eventually depend on. Target-space recoordinatizations are
realized as field redefinitions of the field $X^{a}$. We choose a
coordinatization where we take the target-space dyad to be constant, which
without loss of generality we can take to be of the form 888In our notation,
greek indices are used for tangent space indices both of the worldsheet and
the target space spacetimes. Latin letters are reserved for the frame field
internal indices. When specializing the indices to particular values, the
spacetime indices will turn into $t$ or $x$, whereas the internal indices take
the values $0$ and $1$

$E_{\mu}^{a}=\delta^{a}_{\mu}.$ (6)

With $E$ constant in our coordinatization of target spcace, it is useful to
define

$X^{a}\equiv X^{\mu}E_{\mu}^{a}$ (7)

In these terms the action (3) becomes

$S=S_{0}+S_{grav}$ (8)

with

$S_{grav}=-\frac{1}{2\lambda}\int
d^{2}x\epsilon_{ab}\epsilon^{\alpha\beta}\left(\partial_{\alpha}X^{a}-e^{a}_{\alpha}\right)\left(\partial_{\beta}X^{b}-e^{b}_{\beta}\right)$
(9)

As mentioned in the introduction, the field $X$ in our gravitational sector
must be restricted to a particular homotopy class. In this case, where $X$
describes a mapping between the worldsheet and target space cylinders, we fix
$X$ so that the mapping has winding $1$. The target space is coordinatized in
such a way that this implies

$\displaystyle X^{1}(t,x+1)=X^{1}(t,x)+R$ (10)

In order to implement this it is useful to perform the change of variables

$X^{a}=X^{a}_{w}+Y^{a},$ (11)

where $X^{a}_{w}$ is some fixed function implementing the winding and $Y^{a}$,
which satisfies periodic boundary conditions on the worldsheet spatial circle,
is our new dynamical field.
We now proceed to analyze the theory specified by the action (8) in the
canonical formalism. The first step is to identify what the phase space
variables are and the symplectic structure in said phase space. After
performing the aforementioned split of spacetime, we notice that the action
$S_{grav}$ is first order in time derivatives, and that there are no time
derivatives acting upon $e^{a}_{t}$:

$S_{grav}=\frac{1}{\lambda}\int
dtdx\left(\epsilon_{ab}X^{a}\partial_{t}e^{b}_{x}-e^{a}_{t}\epsilon_{ab}\left(\partial_{x}X^{b}+e^{b}_{x}\right)\right)$
(12)

from which we would conclude that the phase space variables in the
gravitational sector are $Y^{a}$ and $e_{x}^{a}$, the corresponding symplectic
form being given by

$\Omega_{grav}=-\frac{1}{\lambda}\int dx\epsilon_{ab}\delta X^{a}\wedge\delta
e^{b}_{x}$ (13)

From $S_{grav}$ we can conclude that $e^{a}_{t}$ is not be a phase space
variable but rather plays the role of a Lagrange multiplier. These conclusions
may however be altered depending on the nature of the matter action $S_{0}$.
Indeed, if there are time derivatives acting on the metric in $S_{0}$, in
principle this could turn $e^{a}_{t}$ into a dynamical variable and alter the
symplectic structure of the theory so that it is not described by
$\Omega_{grav}$ in the gravitational sector. In order to avoid these
complications, we consider a somewhat less general undeformed theory and take
$S_{0}$ to be the action of a single scalar field with only one derivative per
field terms. In cases with either scalars with higher derivative terms or
fields in other representations of the Lorentz group the symplectic structure
is more complicated. In fact, in these cases –in general– even the symplectic
structure of the theory without coupling to gravity has its complications.
Therefore, we will restrict to this simplified situation for now and will
comment further on the general case in section .
The matter action $S_{0}$ thus may be written in terms of an arbitrary scalar
function $K$ of the scalars $\phi$ and $u\equiv
g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi$

$S_{0}=\int d^{2}x\sqrt{-g}K(u,\phi)$ (14)

Under these circumstances the only phase space variables in the gravitational
sector are indeed $e^{a}_{x}$ and $X^{a}$, with the symplectic form on the
full phase space being given by

$\Omega=\Omega_{grav}+\int dx\delta\phi\wedge\delta\pi.$ (15)

Notice that in our conventions the field $\pi$ –the conjugate momentum to
$\phi$–is a spatial one-form. The variables $e^{a}_{t}$ are indeed non-
dynamical in this case and act as Lagrange multipliers, as will be seen
explicitly below.
The Hamiltonian of the full deformed theory is of the form

$H=H_{grav}+H_{mat}$ (16)

where $H_{grav}$ is the contribution from the gravitational sector and
$H_{mat}$ denotes the matter sector contribution. From (12) we see that
$H_{grav}$ is given by

$H_{grav}=\frac{1}{\lambda}\int
dxe^{a}_{t}\epsilon_{ab}\left(\partial_{x}X^{b}+e^{b}_{x}\right)$ (17)

The matter contribution $H_{mat}$ is obtained from the Legendre transform of
(14) in the standard way, although with the extra complication that the action
(14) for the scalar is somewhat general and is in the presence of an arbitrary
metric. Details on the procedure to obtain $H_{mat}$ are provided in Appendix
B. The result is that $H_{mat}$ is of the form

$H_{mat}=\int dx\frac{-e_{t}^{a}\eta_{ab}e_{x}^{b}}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{P}_{0}+\int
dx\frac{|e_{t}^{a}\epsilon_{ab}e_{x}^{b}|}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{H}_{0}$ (18)

where

$\mathcal{P}_{0}=-\frac{\pi\partial_{x}\phi}{\sqrt{|e_{x}\cdot e_{x}|}}$ (19)

and the function $\mathcal{H}_{0}$ is of the form

$\mathcal{H}_{0}=\sqrt{|e_{x}\cdot
e_{x}|}\tilde{f}\left(\frac{\pi}{\sqrt{|e_{x}\cdot
e_{x}|}},\frac{\partial_{x}\phi}{\sqrt{|e_{x}\cdot e_{x}|}},\phi\right).$ (20)

Here $\tilde{f}$ is some function whose details depend on the particular form
of $K$ (see Appendix B for more details) and whose only importance for us here
is to specify the dependence of $\mathcal{H}_{0}$ on the canonical variables.
Crucially for what follows –as explained in more detail in Appendix B– it
should be noted that

$\int dx\mathcal{H}_{0}=E_{0}(L)$ (21)

is the energy of the undeformed theory in flat space when placed on a spatial
circle of size

$L=\int dx\sqrt{|e_{x}\cdot e_{x}|}.$ (22)

Similarly

$\int dx\mathcal{P}_{0}=P_{0}(L)$ (23)

is the spatial momentum density of the undeformed scalar theory when placed on
a spatial circle of size $L$.
We therefore find, as expected for a diffeomorphism-invariant theory, that the
Hamiltonian is a linear combination of the constraints:

$H=\int e_{t}^{a}F_{a}$ (24)

where the $e_{t}^{a}$ play the role of Lagrange multipliers as anticipated and
the $F_{a}=0$ are the constraints of the theory, with999Here we have lifted
the absolute value of $|e_{t}^{a}\epsilon_{ab}e_{x}^{b}|$ under the assumption
that $e_{t}^{a}\epsilon_{ab}e_{x}^{b}>0$. This is merely a convention that
somewhat simplifies what follows, as changing the sign of this quantity is
equivalent to changing the sign of $\lambda$ and of the definition of
$\mathcal{P}_{0}$.

$F_{a}\equiv\epsilon_{ab}\frac{\left(\partial_{x}X^{b}+e^{b}_{x}\right)}{\lambda}+\frac{-\eta_{ab}e_{x}^{b}}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{P}_{0}+\frac{\epsilon_{ab}e_{x}^{b}}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{H}_{0}.$ (25)

To summarize our discussion thus far, from the canonical point of view the
gravitational $T\bar{T}$ theory consists of a constrained Hamiltonian system.
The unconstrained phase space, which we will denote $M_{kin}$, is described by
the canonical variables $\\{e_{x},X,\phi,\pi\\}$ and is imbued with the
symplectic form $\Omega$ given by (15). It is over this phase space that the
constraints $F_{a}=0$ are imposed.
We now turn to the analysis of this constrained Hamiltonian system. The first
important aspect of this is the fact that, as expected for a gravitational
theory, the constraints are first class. Given the generality of the matter
action we consider, it is worthwhile to discuss this in some detail, even if
this is a completely expected result.
It turns out that in order to analyze their structure, it is convenient to
consider a redefined form of the constraints which we take to be

$\displaystyle C_{1}$ $\displaystyle\equiv-
F_{b}\epsilon^{b}_{a}e^{a}_{x}=\left(e_{x}^{a}\eta_{ab}(e^{b}_{x}+\partial_{x}X^{b})+\mathcal{H}_{0}\right)$
(26) $\displaystyle C_{2}$ $\displaystyle\equiv
e^{a}_{x}F_{b}=\left(e^{a}_{x}\epsilon_{ab}\partial_{x}X^{b}+\mathcal{P}_{0}\right).$
(27)

Notice that, since we are assuming that the $e_{x}$, seen as a vector in the
Minkowski-signature auxiliary space, is non-lightlike everywhere, the
conditions $C_{i}=0$ are equivalent to the original constraints $F_{a}=0$
(there is an invertible relation between the $C$s and $F$s under this
assumption). Observe that if $e_{x}$ were lightlike at a given point, the
induced metric on the spatial slice would be singular at that point. In fact,
many of the previous formulas are only sensible within the assumption that
this does not happen.
The advantage of the formulation in terms of the $C_{i}$ over the $F_{a}$ is
that, the $C_{i}$ being (at least) quadratic in the fields, their action via
Poisson brackets on the canonical variables starts with a linear term instead
of a constant (as happens for the $F_{a}$), and thus their Poisson brackets
have a better chance of closing on a more manageable algebraic structure.
Furthermore, in order to check their brackets it is more practical to define
the smeared constraints

$\displaystyle\mathcal{C}[N]$ $\displaystyle=\int dxNC_{1}$ (28)
$\displaystyle\mathcal{C}_{x}[N^{x}]$ $\displaystyle=\int dxN^{x}C_{2}.$ (29)

defined for some arbitrary functions $N$ and $N_{x}$. Notice that this is
equivalent to performing a redefinition of the Lagrange multipliers

$\displaystyle e^{a}_{t}\longrightarrow
N^{x}e_{x}^{a}+Ne_{x}^{b}\epsilon^{a}_{b}.$ (30)

where $N$ and $N^{x}$ are taken to be independent from the canonical
variables. With this redefinition, (24) becomes:

$\displaystyle H=\mathcal{C}[N]+\mathcal{C}_{x}[N^{x}].$ (31)

Although this redefinition does not look too covariant, this is not an issue
since in the canonical approach explicit covariance is broken by picking a
particular foliation of spacetime.
Let us examine the Poisson brackets between the smeared constraints. A simple
calculation shows that the smeared constraint $\mathcal{C}_{x}[N^{x}]$
generates, via its Poisson bracket, spatial diffeomorphisms parametrized by
the spatial vector field $N^{x}$. Therefore, all Poisson brackets involving
$\mathcal{C}_{x}[N^{x}]$ are obtained trivially. The only bracket that is left
is that of $\mathcal{C}[N]$ with itself, which is somewhat subtle due to the
degree of generality we are assuming for the matter action. The details are
presented in appendix C. The end result is:

$\displaystyle\left\\{\mathcal{C}[N],\mathcal{C}[M]\right\\}$
$\displaystyle=\mathcal{C}_{x}[N\partial_{x}M-M\partial_{x}N]$ (32)
$\displaystyle\left\\{\mathcal{C}_{x}[N^{x}],\mathcal{C}[N]\right\\}$
$\displaystyle=\mathcal{C}[N^{x}\partial_{x}N-N\partial_{x}N^{x}]$ (33)
$\displaystyle\left\\{\mathcal{C}_{x}[N^{x}],\mathcal{C}_{x}[M^{x}]\right\\}$
$\displaystyle=\mathcal{C}_{x}[M^{x}\partial_{x}N^{x}-N^{x}\partial_{x}M^{x}]$
(34)

We see then that the $C_{i}$ constraints are indeed first class. Since the
$F_{a}$ can be obtained from the $C_{i}$ by an invertible linear
transformation then this implies that the $F_{a}$ are also first
class.101010In the terminology of constrained Hamiltonian systems, the $F_{a}$
would be secondary constraints. The primary constraints are those
corresponding to the vanishing of the conjugate momenta to the $e^{a}_{t}$
variables. The $F_{a}$ are secondary constraints in that they result from the
requirement that the primary constraints be conserved. The fact that these are
first class and that the Hamiltonian is of the form (31) ensures that no
tertiary constraints arise. The distinction between primary, secondary and
tertiary constraints is, however, irrelevant at the end of the day (see, e.g.
henneaux )..
As it usually happens for constrained Hamiltonian systems, first class
constraints generate gauge transformations on functions of the canonical
variables via their Poisson brackets. In this case, the
$\mathcal{C}_{x}[N^{x}]$ generate spatial diffeomorphisms and the
$\mathcal{C}[N]$ generate the evolution along the time-like
direction111111Modulo a spatial diffeomorphism.
Regarding the constraint algebra obtained, a few final comments are in order.
First, the constraint algebra obtained here is the usual one present in the
canonical formulation of string theory and $1+1$ dimensional gravity theories
(see for example Brink:1988nh  or jackiw1995two ). Second, there are no
structure functions present in the right side of the Poisson Brackets, only
structure constants (so the algebra obtained here is thus a true Lie Algebra);
the presence of structure functions in the constraint algebra is one of the
main obstacles in the canonical quantization of gravity, since they imply that
the group associated to the action of the constraints is non compact thus
rendering the group averaging procedure to find physical states non-viable
(See e.g. rovelli2004quantum ; thiemann2008modern ). Finally, it is worth
noting that the previous result was obtained considering matter consisting of
a scalar field with a fairly general form of the action, having assumed only
that the Lagrangian density depends on Lorentz invariant combinations of the
scalar field and its derivatives up to second order. We haven’t been able to
find analyses of the constraint structure of gravitational systems with matter
in the degree of generality considered here in the literature, and for this
reason we present this calculation in detail in appendix C.
We now turn to the description of the physical observables of the theory. From
the point of view of the canonical formalism these will be what are known as
Dirac observables: functions of the phase space variables that commute with
the constraints of the theory. In general, these will to a certain degree
depend on the particularities of the matter sector. However, in the case of
$T\bar{T}$ gravity there are observables associated to the symmetries of the
target space (which is the “physical” space of the theory) that are present
independently of the nature of the matter sector. Indeed, examining (3) it is
clear that the theory is symmetric under the action on $X$ of the isometries
of the target-space metric. In the case of constant $E_{a}$ which we are
restricting to here, these will be Poincarè transformations of target-space.
The conserved quantities associated to these symmetries are given
by121212Notice that the $X$ equations of motion imply $de^{a}=0$, and so
$\int_{\mathcal{C}}e^{a}$ depends only topologically on the curve
$\mathcal{C}$.

$\displaystyle P^{a}$ $\displaystyle\equiv\int
dx\frac{\epsilon^{a}_{b}e^{b}}{\lambda}$ (35) $\displaystyle J$
$\displaystyle\equiv-\int dx\frac{Y^{a}e^{b}_{x}\eta_{ab}}{\lambda}$ (36)

It is easy to check that the Poisson algebra of these quantities represents
the Poincaré algebra

$\displaystyle\left\\{J,P^{a}\right\\}$ $\displaystyle=\epsilon^{a}_{b}P^{b}$
(37) $\displaystyle\left\\{P^{a},P^{b}\right\\}$ $\displaystyle=0$ (38)

and that these generate Poincaré transformations of the $X^{\mu}$ through the
Poisson brackets

$\displaystyle\left\\{X^{a},J\right\\}$ $\displaystyle=\epsilon^{a}_{b}X^{b}$
(39) $\displaystyle\left\\{X^{a},P^{b}\right\\}$ $\displaystyle=\eta^{ab}$
(40)

On the $e^{a}$ these act as

$\displaystyle\left\\{e^{a}_{x},J\right\\}=\epsilon^{a}_{b}e^{a}_{x}$ (41)
$\displaystyle\left\\{e^{a}_{x},P^{a}\right\\}=0.$ (42)

and they Poisson commute with the matter fields.
From the canonical action of the $P^{a}$ on the fields it is clear that they
are indeed Dirac observables of the theory, as they generate constant shifts
of $X^{a}$, which appear with derivatives in the constraints. On the other
hand, $J$ is only a Dirac observable when periodic boundary conditions are
taken for the $X^{a}$, but not in the winding sector, as expected, given that
compactifying the target-space spatial slice breaks target-space boosts.
An important observation for what follows is that $P^{0}$ and $P^{1}$ generate
target-space translations and are thus the target-space energy $E$ and
momentum $P$ respectively, which we crucially interpret as the
$T\bar{T}$-deformed energy and momentum. The key point that we will show is
that the constraints of the gravitational theory imply nontrivial relations
between these observables that are equivalent to the finite volume $T\bar{T}$
flow equation (2).
In order to do this, it will be useful to perform a gauge-fixing procedure,
that will allow to carve out the reduced phase space (in a sense, the
“physical” phase space) out of the full phase space of the theory $M_{kin}$.
As we have recalled earlier, in a Hamiltonian system with first-class
constraints these are interpreted to generate via their Poisson action
infinitesimal gauge transformations of the system. Thus to remove ambiguities
coming from gauge evuivalences, the reduced phase space of the system is
obtained by quotienting the full phase space by the action of these gauge
transformations. One way to accomplish this is to supplement the constraints
of the theory with a set of extra constraints –the gauge-fixing conditions– so
that imposing the full set of constraints picks out a single representative of
each gauge orbit. Notice that the specific way these gauge-fixing conditions
are imposed does not affect Dirac observables since they –by definition–
Poisson-commute with the first class constraints and are therefore gauge
invariant. Indeed we will see that the $T\bar{T}$ finite volume flow equations
arise from the constraints as gauge-invariant algebraic relations between
gauge-invariant observables and thus are independent of the particular gauge-
fixing procedure will follow.
We will perform the gauge-fixing procedure in two steps and start by imposing
first the partial gauge-fixing condition

$\partial_{x}e^{a}_{x}=0$ (43)

Notice that indeed this does not fully fix the gauge freedom of the system
given that the zero modes of the constraints Poisson-commute with this
condition

$\left\\{\int dxF_{a},\partial_{x}e^{b}_{x}\right\\}=0.$ (44)

However, condition (43) does fully fix the gauge transformations generated by
the non-zero modes. To see this, let us define the expansion of the
constraints and gauge-fixing conditions over the non-zero modes

$\displaystyle G^{a}_{(n)}\equiv\int dxG^{a}(x)e^{ix2\pi n}\hskip 20.0ptn\neq
0$ (45) $\displaystyle F_{a(m)}\equiv\int dxF_{a}(x)e^{ix2\pi m}\hskip
20.0ptm\neq 0$ (46)

Then the matrix $G^{b}_{amn}$ of Poisson brackets between $G^{a}_{(n)}$ and
$F^{a}_{(m)}$ is given by

$G^{b}_{amn}\equiv\left\\{F_{a(m)},G^{b}_{(n)}\right\\}=-\lambda
m^{2}\delta^{a}_{b}\delta_{mn}.$ (47)

Let us now define the space $\tilde{M}_{kin}$ as that obtained from imposing
the conditions $F^{a}_{(n)}=0$ and $G^{b}_{(m)}=0$ on the phase space
$M_{kin}$. Then the fact that $G^{b}_{amn}$ is invertible implies that
condition (43) does fix the gauge transformations generated by $F^{a}_{(m)}$
and that the closed differential form $\tilde{\Omega}$ induced on
$\tilde{M}_{kin}$ is invertible and thus symplectic. Indeed, this symplectic
form is given by

$\tilde{\Omega}=-\frac{1}{\lambda}\int
dx\epsilon_{ab}\delta\tilde{X}^{a}\wedge\delta\tilde{e}^{b}_{x}+\int
dx\delta\phi\wedge\delta\pi$ (48)

where the $\tilde{X}^{a}$ and $\tilde{e}^{b}_{x}$ denote the zero modes for
the corresponding variables. The partial gauge fixing procedure finally leaves
us then with the phase space $\tilde{M}_{kin}$ on which we have two remaining
constraints

$\tilde{F}_{a}\equiv\int dxF_{a}|_{kin}=0$ (49)

where the $|_{kin}$ implies that this phase space function is to be evaluated
at $F_{a(m)}=G^{b}_{(n)}=0$.
We will now see how these left-over constraints imply non-trivial relations
between the Dirac observables of the theory $P^{a}$ that are equivalent to the
$T\bar{T}$ flow equation for the finite-volume spectrum.
Notice first that on $\tilde{M}_{kin}$ we have

$e_{x}^{a}|_{kin}=\tilde{e}_{x}^{a}=\lambda\epsilon^{a}_{b}P^{b}.$ (50)

If we now consider, as explained in the introduction, winding boundary
conditions for $X^{a}$ where

$\int dx\partial_{x}X^{a}=R\delta^{a}_{1}$ (51)

then the $\tilde{F}_{a}=0$ constraints can be written as

$\epsilon_{a1}\frac{R}{\lambda}+P_{b}+\operatorname{sgn}(\lambda)\frac{P_{a}}{|P^{2}|^{1/2}}E_{0}(L)-\operatorname{sgn}{\lambda}\frac{\epsilon_{ab}P^{b}}{|P^{2}|^{1/2}}P_{0}(L)=0$
(52)

with $L$, $E_{0}(L)$ and $P_{0}(L)$ defined as in (21),(22) and (23). We now
identify $P^{a}$ with the $T\bar{T}$ deformed energy $E_{\lambda}$ and the
momentum $P_{\lambda}$ as follows

$\displaystyle P^{0}=-R/|\lambda|+\operatorname{sgn}(\lambda)E_{\lambda}(R)$
(53) $\displaystyle P^{1}=-\operatorname{sgn}(\lambda)P_{\lambda}(R)$ (54)

and define $\theta_{0}$ so that

$\displaystyle\cosh\theta_{0}=-\frac{P^{0}}{|P^{2}|^{1/2}}=\frac{R-\lambda
E_{\lambda}(R)}{L}$ (55)
$\displaystyle\sinh\theta_{0}=-\frac{P^{1}}{|P^{2}|^{1/2}}=\lambda\frac{P_{\lambda}(R)}{L}$
(56)

where combining (22) with (50) we have

$L=|\lambda||P^{2}|^{1/2}=|(R-\lambda
E_{\lambda}(R))^{2}-P_{\lambda}(R)^{2}|^{1/2}$ (57)

(notice that the “physical” sign for $\lambda$ in our conventions is
negative). Then the $\tilde{F}_{a}=0$ equations can be written as

$\displaystyle E_{\lambda}(R)$
$\displaystyle=\cosh\theta_{0}E_{0}(L)-\sinh\theta_{0}P_{0}(L)$ (58)
$\displaystyle P_{\lambda}(R)$
$\displaystyle=-\sinh\theta_{0}E_{0}(L)+\cosh\theta_{0}P_{0}(L).$ (59)

These are precisely the equations that define the implicit form of the
solution to the $T\bar{T}$ finite-volume energy flow equation (2), as can be
found in Cavaglia:2016oda ; Conti:2018jho . These is one of the main results
of this paper: that the constraints arising from the gravitational system
imply relations between the target-space energy and momentum that are
equivalent to the $T\bar{T}$-deformed spectrum. Even though it was necessary
to fix (partially) the gauge to arrive at these equations, these are algebraic
relations between gauge-invariant observables and are thus gauge-invariant.
Following Cavaglia:2016oda ; Conti:2018jho , let us further analyze the flow
equations (58) and (59) to clarify their meaning. Noticing that they are of
the form of a Lorentz transformation of rapidity $\theta_{0}$ between
$(E_{0}(L),P_{0}(L))$ and $(E_{\lambda}(R),P_{\lambda}(R))$, it is
straightforward to invert these equations into

$\displaystyle E_{0}(L)$
$\displaystyle=\cosh\theta_{0}E_{\lambda}(R)+\sinh\theta_{0}P_{\lambda}(R)$
(60) $\displaystyle P_{0}(L)$
$\displaystyle=\sinh\theta_{0}E_{\lambda}(R)+\cosh\theta_{0}P_{\lambda}(R).$
(61)

The last of these equations can be written as

$LP_{0}(L)=RP_{\lambda}(R).$ (62)

Since in finite spatial volume spatial momentum is quantized, then for any
given state there are integers $N_{0}$ and $N_{\lambda}$ such that
$P_{0}(L)=2\pi N_{0}/L$ and $P_{\lambda}(R)=2\pi N_{\lambda}/R$. Equation (62)
then immediately implies that $N_{0}=N_{\lambda}$, so that the integer
describing the quanta of total spatial momentum is undeformed by the flow, as
expected for an integer-quantized quantity in the presence of a continuous
deformation. In the zero momentum sector, the deformation takes a particularly
simple form

$E_{\lambda}(R)=E_{0}(R-\lambda E_{\lambda}(R))$ (63)

which is the well-known implicit solution of (2) when $P=0$.
It is interesting to note that the $T\bar{T}$ flow equations between
$E_{\lambda}(R)$ and $E_{0}(R)$, which are supposed to be exact quantum
mechanically, already arise at the classical level within the gravitational
description we have presented. This hints at the fact that perhaps quantum
effects in this gravitational theory are such that to take them into account
one needs only account for them at the classical theory. That is, that to
account for quantum corrections to $E_{\lambda}$ one merely needs to include
quantum effects in the calculation of $E_{0}$, and then the quantum mechanical
$E_{\lambda}$ emerges from taking this (quantum) $E_{0}$ as input into the
classical flow equations. We discuss this point a bit further in section VI.
To finish our analysis of the canonical structure of the theory we will
construct the fully gauged-fixed “reduced” phase-space. As discussed above,
the gauge-fixing condition (43) does not fully fix the gauge: transformations
infinitesimally generated by the zero modes of the constraints remained
unfixed. To obtain the fully gauge-fixed Hilbert space we have to supplement
(43) with extra conditions that remove this leftover gauge freedom. we will
now briefly discuss two possible ways in which this can be achieved.
One possibility is to fix the gauge in order to simplify the on-shell behavior
of the metric. The $X^{a}$ equations of motion in the spacetime split we have
chosen are

$\partial_{t}e^{a}_{x}-\partial_{x}e^{a}_{t}=0.$ (64)

Combined with the gauge-fixing condition (43) these EOMs imply that
$\partial_{x}e^{a}_{t}$ is only a function of $t$ only, meaning that
$e^{a}_{t}$ is of the form

$e^{a}_{t}=x\partial_{x}e^{a}_{t}(t)+g^{a}(t)$ (65)

for some functions $g^{a}(t)$. If we impose that $e^{a}_{t}$ is periodic on
the spatial circle then this periodicity requires the first term in the
previous expression to vanish, and thus $e^{a}_{t}$ is independent of $x$ and
only a function of $t$ in the gauge we have chosen. The remaining gauge
freedom is generated by two functions (the zero modes of the $F_{a}$) and thus
can be used to fully fix to constants the two functions $g^{a}(t)$ that
determine $e^{a}_{t}$ on-shell. This is another way of seeing that
$e^{a}_{\mu}$ can be gauge-fixed to be on-shell flat (indeed constant on the
cylinder), which was one of the key points used in dubovsky2018tt  to
calculate the quantum partition function and spectrum of this theory. Notice
that the matter equations of motion in this gauge are the undeformed equations
of motion in the presence of a constant metric on the cylinder, and thus the
quantities $E_{0}(L)$ and $P_{0}(L)$ that appear in the $T\bar{T}$ flow
equations are obviously constant in this gauge.
From the canonical point of view, the gauge-fixing outlined in the previous
paragraph is somewhat counter intuitive, as one is gauge-fixing Lagrange
multipliers and not canonical variables, and so the fully-constrained phase
space and induced symplectic structure are not easy to construct from this.
Another perhaps more natural possibility from the canonical point of view is
to take the gauge-fixing condition

$\tilde{X}^{a}=0$ (66)

From the point of view of the matter equations of motion, this condition has
the disadvantage that these will involve a time dependent metric. However, in
this gauge the structure of the fully-fixed phase space $M_{red}$ is far more
evident. Indeed, in this gauge $M_{red}$ is identical to the phase space of
the unconstrained matter theory –in this case a simple real scalar– and the
induced symplectic form is just the symplectic form of the matter sector. The
gravitational sector is completely removed by the gauge fixing conditions and
equations of motion.

##  III Free massless scalars and non-critical strings

As a particular example of the procedure we have discussed, it is instructive
to consider the simplest possible case, namely, that of a system of $N$ free
massless scalars. This is particularly interesting because, as it is well-
known, the $T\bar{T}$ deformation of free massless scalars appears to be
related to non-critical strings. A number of previous studies have discussed
this (see Smirnov:2016lqw ; Cavaglia:2016oda ; Dubovsky:2012wk ;
Tolley:2019nmm ; Callebaut:2019omt ; Frolov:2019nrr  for a sample of various
perspectives on this) and we will discuss this mostly to give a simple
concrete example of the ideas discussed in the previous section.
The matter sector of the action we consider is therefore simply131313In our
normalization whatever factor in front of the matter action may be reabsorbed
into $\lambda$ without altering the classical equations of motion.

$\displaystyle
S_{0}=\int\mathrm{d}t\mathrm{d}x\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{I}$
(67)

where $I=1\dots N$. The matter hamiltonian and momentum densities are

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\frac{\left(\pi^{I}\pi^{I}+\partial_{x}\phi^{I}\partial_{x}\phi^{I}\right)}{2|e_{x}\cdot
e_{x}|^{1/2}}=\frac{h_{0}}{|e_{x}\cdot e_{x}|^{1/2}}$ (68)
$\displaystyle\mathcal{P}_{0}$
$\displaystyle=\frac{\pi\partial_{x}\phi}{|e_{x}\cdot
e_{x}|^{1/2}}=\frac{p_{0}}{|e_{x}\cdot e_{x}|^{1/2}}$ (69)

(we have defined $h_{0}$ and $p_{0}$ for later convenience) so that the total
Hamiltonian becomes:

$\displaystyle
H=\int\mathrm{d}x\left\\{N\left[\frac{\left(e_{x}^{a}\eta_{ab}(e^{b}_{x}+\partial_{x}X^{b})\right)}{\lambda}+\left(\frac{1}{2}\pi^{2}+\frac{1}{2}\left(\partial_{x}\phi\right)^{2}\right)\right]+N^{x}\left(\frac{e^{a}_{x}\epsilon_{ab}\partial_{x}X^{b}}{\lambda}+\pi\partial_{x}\phi\right)\right\\}$
(70)

It is interesting to note that the $ISO(1,1)$ algebra of target-space
isometries described in the previous section is now extended to $ISO(1,N+1)$.
This is as not surprising from the fact that at the classical level this
theory is expected to describe a bosonic string propagating in flat
$(N+2)$-dimensional spacetime, with the fields $\phi^{I}$ realizing the
embedding into the extra $N$ coordinates. The extra generators allowing to go
from $ISO(1,1)$ to $ISO(1,N+1)$ are given by

$\displaystyle P^{I}$ $\displaystyle=\int\mathrm{d}x~{}\pi^{I}$ (71)
$\displaystyle Q^{IJ}$ $\displaystyle=\int
dx\left(\pi^{I}\phi^{J}-\pi^{J}\phi^{I}\right)$ (72) $\displaystyle Q^{aI}$
$\displaystyle=\int
dx\left(\phi^{I}~{}\left(e_{x}^{a}+\frac{\partial_{x}X^{a}}{2}\right)-\pi^{I}\epsilon^{a}_{b}\frac{X^{b}}{2}\right)$
(73)

Then $P^{I}$ and $Q^{IJ}$ generates translations and rotations in the
$\phi^{I}$ sector, whereas $Q^{aI}$ has the following action on the canonical
variables

$\displaystyle\left\\{X^{a},Q^{bI}\right\\}$
$\displaystyle=\epsilon^{ab}\phi^{I}$ (74)
$\displaystyle\left\\{e_{x}^{a},Q^{bI}\right\\}$
$\displaystyle=\eta^{ab}\frac{\pi^{I}}{2}-\epsilon^{ab}\left(\frac{\partial_{x}\phi^{I}}{2}\right)$
(75) $\displaystyle\left\\{\phi^{I},Q^{aJ}\right\\}$
$\displaystyle=\delta^{IJ}\epsilon^{a}_{b}\frac{X^{b}}{2}$ (76)
$\displaystyle\left\\{\pi^{I},Q^{aJ}\right\\}$
$\displaystyle=-\delta^{IJ}\eta^{a}_{b}\left(e_{x}^{b}+\frac{1}{2}\partial_{x}X^{b}\right)$
(77)

so that the action of $Q^{aI}$ is a rotation/boost between the $X^{a}$
variables and the $\phi^{I}$. A straighforward calculation shows that indeed
the poisson brackets of the generators
$\left\\{P^{a},P^{I},Q,Q^{IJ},Q^{aI}\right\\}$ close into the $ISO(1,N+1)$
algebra. This shows that indeed at the classical level the theory realizes the
isometries of flat $N+2$-dimensional spacetime. We interpret this theory to
give a kind of “Polyakov-like” formulation of a non-critical bosonic string,
where $X^{\mu}=\left\\{X^{a},\phi^{I}\right\\}$ give the embedding coordinates
and $e^{a}_{x}$ plays the role of a “Polyakov metric”. Interestingly, in
contrast to what happens in the usual Polyakov formulation of the bosonic
string, in this formulation the dynamical geometry on the worldsheet
$e^{a}_{\mu}$ is also charged under target-space transformations.
Let us now analyze the spectrum of this theory according to the analysis
outlined in the previous section. We will consider that we are as before in
the winding sector in $X^{1}$ so that we have an $N+2$-dimensional closed
string winding around a compact target-space dimension of size $R$, with
$\phi^{I}$ being the transverse coordinates. from our previous arguments, we
start from the constraint equations, which formulated in terms of (70) appear
naturally in the $C_{i}=0$ form

$\displaystyle\lambda P^{2}-P^{a}\epsilon_{ab}\partial_{x}X^{b}+h_{0}=0$ (78)
$\displaystyle P^{a}\eta_{ab}\partial_{x}X^{b}-p_{0}=0$ (79)

where we have already formulated the constraints in the
$\partial_{x}e^{a}_{x}=0$ gauge and identified the $P^{a}$ as in (50). The
zero mode equations in the winding sector then yield

$\displaystyle\lambda P^{2}-P^{0}R+\int dxh_{0}=0$ (80) $\displaystyle
P^{1}=\frac{\int dxp_{0}}{R}$ (81)

which combine to give $P^{0}$ of the form

$P^{0}=\frac{R}{2\lambda}\pm\frac{1}{2\lambda}\sqrt{R^{2}-4\lambda\left(\int
dxh_{0}-\lambda\left(\frac{\int dxp_{0}}{R}\right)^{2}\right)}$ (82)

which yields the target-space energy spectrum of the string in terms of the
spectrum of the operators $\int dxh_{0}$ and $\int dxp_{0}$, which are the
flat-space energy and momentum of a system of $N$ free two-dimensional
massless bosons on a compactified spatial circle of size $1$. Notice that in
our formulation from section II the constraint equations giving rise to the
$T\bar{T}$ deformed spectrum were naturally written in terms of the
$E_{0}(L)\equiv\int dx\mathcal{H}_{0}$ and $P_{0}(L)\equiv\int
dx\mathcal{P}_{0}$. For the simple system we consider in this section the
relation between $E_{0}(L)$ and $P_{0}(L)$ and $\int dxh_{0}$ and $\int
dxp_{0}$ is trivial at constant $e_{x}$ and the equation for the spectrum is
readily obtained in terms of the latter.
The undeformed theory of free massless bosons is readily solved at the full
quantum level and its spectra is expressed in terms of two non-negative
integers $N_{L}$ and $N_{R}$ describing the total number of left- and right-
moving momentum

$\displaystyle\int dxh_{0}$ $\displaystyle=2\pi N_{R}+2\pi N_{L}-2\pi N/12$
(83) $\displaystyle\int dxp_{0}$ $\displaystyle=2\pi N_{R}-2\pi N_{L}.$ (84)

Plugging this into expression (82) and taking $\lambda=-l_{s}^{2}/2$

$P^{0}=-\frac{R}{l_{s}^{2}}\mp\frac{1}{l_{s}}\sqrt{\frac{R^{2}}{l_{s}^{2}}+4\pi\left(N_{L}+N_{R}-N/12\right)+\frac{4\pi^{2}l_{s}^{2}(N_{L}-N_{R})^{2}}{R^{2}}}.$
(85)

This is precisely the lightcone-quantized bosonic string spectrum in $D=N+2$
in the unit-winding sector (for the positive energy branch, and up to
subtraction of some linear in $R$ terms). Notice that we have obtained this by
plugging in the full quantum result for the undeformed free theory describing
the transverse modes into the “classical” $T\bar{T}$ flow equations. As
mentioned before, the fact that the $T\bar{T}$ deformation of $N$ free
massless bosons yields this spectrum is well known and can be seen directly by
solving Burgers’ equation (2) with free theory spectra as initial condition.
We have merely presented this in order to exemplify our more general
procedure.
Notice that even from our point of view it is hardly surprising that we have
obtained the light-cone string spectrum, given that the zero modes of our
gauge-fixed constraints, equations (80) and (81), coincide with the zero modes
of the Virasoro constraints

$\displaystyle\int
dx\left(\partial_{x}X^{\mu}\partial_{x}X_{\mu}+\partial_{t}X^{\mu}\partial_{t}X_{\mu}\right)=0$
(86) $\displaystyle\int dx\partial_{t}X^{\mu}\partial_{x}X_{\mu}=0$ (87)

in lightcone gauge where in the winding sector one has141414Here we are using
the somewhat unconventional notation of $t$ and $x$ for the worldsheet
coordinates.

$\displaystyle X^{0}$
$\displaystyle=\left(l_{s}^{2}P^{0}+R\right)t+\displaystyle\sum_{n\neq
0}\left(\alpha^{-}_{n}e^{in(x-t)}+\tilde{\alpha}^{-}_{n}e^{in(x+t)}\right)$
(88) $\displaystyle X^{1}$
$\displaystyle=l_{s}^{2}P^{1}t+Rx-\displaystyle\sum_{n\neq
0}\left(\alpha^{-}_{n}e^{in(x-t)}+\tilde{\alpha}^{-}_{n}e^{in(x+t)}\right)$
(89)

where we have introduced the slightly unconventional parametrization of
target-space momentum in order to reproduce equation (80) on the nose.
A very important point to mention before moving on is that from our previous
construction one may be misled into thinking that the spectrum we have
obtained is that of a well-defined quantum bosonic string theory propagating
in $N+2$ dimensions. Of course this is not the case, and it is well-known that
this spectrum is incompatible with target-space Poincaré invariance except for
$N=24$ or $N=1$. In lightcone quantization, the Poincaré algebra is not
realized quantum mechanically outside those dimensions, and indeed our
comments about $ISO(1,N+1)$ generators were only about classical Poisson
brakets and are not expected to carry on into the full quantum theory.

##  IV Coupling to other matter sectors

So far we have shown that for matter consisting of scalars with actions with
only one-derivative terms but otherwise completely arbitrary, the classical
equations of motion imply the $T\bar{T}$-flow equations for the (classical)
spectra on the cylinder. This result is immediately generalized to actions
with multiple scalar fields under the aforementioned conditions. The question
is then whether the arguments presented here can be extended to more general
matter content, given that the $T\bar{T}$-flow equation for the finite volume
spectrum is expected to hold for the deformation of an arbitrary relativistic
two-dimensional QFT. We briefly address this point in this section.
From the point of view of the analysis performed in the previous section, it
appears that for any matter content for which the gravitational constraints
(the $e_{t}^{a}$ equations of motion) are of the form $F_{a}=0$ with $F_{a}$
as in (25), then the $T{\bar{T}}$-flow equation for the energy spectra would
follow from identical considerations as before. In principle this would seem
to require that the symplectic structure of the matter+gravity system remain
unchanged, for all of the previous discussion to hold exactly as presented
here. Let us then examine to what extent beyond scalars with single-derivative
terms and examine what happens for other representations of the Lorentz group
and higher derivative interactions.
Let us start considering for instance a $U(1)$ gauge field with one-derivative
interactions. A similar analysis can be performed for the non-abelian case but
the extra complication is irrelevant for our present discussion151515The
$T\bar{T}$ deformation of YM theories has been explored in many works, see for
instance Ireland:2019vvj ; Griguolo:2022hek ; Griguolo:2022vdx ;
Santilli:2018xux . The most general “matter” action in this case is of the
form

$S_{0}=\int\sqrt{-g}K(F^{2})$ (90)

for some arbitrary function $K$. The resulting “matter” Hamiltonian $H_{0}$ in
the presence of the gravitational field is then of the form

$H_{mat}=\int dx\sqrt{-g}f(\pi)+\int dxA_{t}\partial_{x}\pi$ (91)

where $f$ is some function whose details depend on $K$, $\pi$ is the conjugate
momentum to $A_{x}$ and $A_{t}$ ends up a non-dynamical variable enforcing the
$\partial_{x}\pi=0$ constraint. For this theory the phase space is spanned by
$\left\\{X^{a},e^{a}_{x},A_{x},\pi\right\\}$ and the symplectic form is

$\Omega=\Omega_{grav}+\int dx\delta\pi\wedge\delta A_{x}.$ (92)

From equation (91) it is clear that the gravitational constraints will be of
the form (25) and thus from our previous argument the $T\bar{T}$-deformed
spectrum follows.
However, when a more general action for a field in the vector representation
is considered, difficulties immediately arise, even at the one-derivative per
field level. Before we expand on this, notice that this also illustrates the
kind of problems that arise in this setup when considering higher derivative
interactions for scalars for instance. This is because a second derivative
term of the form $\nabla_{\mu}\partial_{\mu}\phi$ can be written as a one-
derivative term in terms of an auxiliary (co)vector field $V_{\mu}$ by
imposing the constraint

$V_{\mu}=\partial_{\mu}\phi.$

Let us then consider an arbitrary action in terms of a scalar $\phi$ and a
vector $V_{\mu}$ with up to one-derivative terms

$S_{0}=\int\sqrt{-g}K(\phi,\partial_{\mu}\phi,V_{\mu},\nabla_{\mu}V_{\nu})$

where the arguments appear only in Lorentz-invariant combinations and the
action may involve also Lagrange multipliers involving the aforementioned
constraints which are in any case irrelevant for the present discussion. The
crucial difference between this action and the ones previously considered is
that the metric enters now as well in $\nabla_{\mu}$. In this case, two of the
important properties used to analyze the previous examples may fail: (1) There
is no reason for the matter Hamiltonian to be linear in $e_{t}^{x}$, and thus
it is no longer evidently a Lagrange multiplier nor is the constraint
structure of the theory as evident as before. (2) The phase space and
symplectic structures may be radically more complicated than the simple
$\Omega_{grav}+\Omega_{m}$ found in the previous examples. See for instance
the discussion surrounding equation (24) in Iyer:1994ys , where the symplectic
form generically mixes the matter and gravitational sectors, even in the zero
curvature limit (to which the equations of motion localize to in our case).
Thus, without the properties associated to (1) and (2) it is not at all clear
how an argument of the form presented so far can be extended beyond the type
of theories we have examined so far.
On the other hand, the analysis in dubovsky2018tt  of the fully-quantum torus
partition function of the gravitational theory is completely agnostic about
the nature of the undeformed theory. In fact, given that the gravitational
path integral localizes to the space of flat metrics, one only ever needs the
matter partition function evaluated over a flat-background metric on the
torus. For this object a general expression can be given in terms of the
spectrum of energy and momentum which is expected to be valid for any
reasonable theory, and only this is used in the derivation of dubovsky2018tt .
It is somewhat unsatisfactory that this result appears so complicated from the
canonical point of view.

##  V $J\bar{T}$-type deformations

The type of classical flow equations for the spectra obtained in the previous
sections for the $T\bar{T}$ deformation from its gravitational description can
be extended to $J\bar{T}$-type deformations. These are deformations of
theories with global $U(1)$ symmetries, and the deformations are built out of
Lorentz-breaking bilinears built out of the $U(1)$-current and the stress-
energy tensor. Deformations of this type have been the subject of a several
studies (see for instance Guica:2017lia ; Bzowski:2018pcy ;
Chakraborty:2018vja ; Aharony:2018ics ; Apolo:2018qpq  for a small sample of
these). As explained in Aguilera-Damia:2019tpe , these deformations can be
described in a “gravitational” manner analogous to that of the $T\bar{T}$
deformation by coupling the undeformed theories $S_{0}$ not only to the
dynamical geometry $e^{a}_{\mu}$ but also to a dynamical $U(1)$ gauge field
$A_{\mu}$. Again we consider a set-up where the undeformed theory consists of
a scalar field, in this case a complex scalar charged under the aforementioned
$U(1)$ symmetry. The action of the deformed theory is then of the form

$S^{J\bar{T}}=S_{grav}^{J\bar{T}}+S_{0}[D_{\mu}\bar{\phi}D_{\nu}\phi
g^{\mu\nu},\bar{\phi}\phi],$ (93)

where the coupling of the undeformed theory to the gravitational sector is as
before and the coupling to the $U(1)$ gauge fields is through the covariant
derivatives

$\displaystyle D_{\mu}\phi$
$\displaystyle\equiv\partial_{\mu}\phi+iA_{\mu}\phi$ (94) $\displaystyle
D_{\mu}\bar{\phi}$
$\displaystyle\equiv\partial_{\mu}\bar{\phi}-iA_{\mu}\bar{\phi}.$ (95)

Similarly to what happens with the $T\bar{T}$ case, the deformation is
implemented by the “gravitational” action governing the dynamics of the $e$
and $A$ fields which we take to be form

$S_{grav}^{J\bar{T}}=S_{grav}^{T\bar{T}}+\frac{1}{\lambda}\int\partial_{\mu}X^{a}\epsilon_{ab}n^{b}\left(\partial_{\nu}\alpha-
A_{\nu}\right)\epsilon^{\mu\nu}+\frac{1}{\lambda}\int
e_{\mu}^{a}\epsilon_{ab}n^{\prime b}\left(\partial_{\nu}\alpha-
A_{\nu}\right)\epsilon^{\mu\nu}$ (96)

where $\alpha$ is an auxiliary scalar playing a similar role to the $X^{a}$ in
this framework and $n^{a}$ and $n^{\prime a}$ are fixed vectors in target-
space. Notice that this action breaks target-space Lorentz invariance, as
expected for these types of Lorentz-breaking deformations. Notice also that in
Aguilera-Damia:2019tpe  the deformation involves two $U(1)$ gauge fields, as
the authors consider the undeformed theory to be a CFT and the possibility of
coupling separately to the right-moving and left-moving currents. Here we
consider a somewhat different situation where the undeformed theory is an
arbitrary theory with a $U(1)$ symmetry and attempt to build a Lorentz-
breaking deformation out of a procedure inspired by the “gravitational” action
suggested in Aguilera-Damia:2019tpe  but with only one gauge field.
Taking this formulation as a starting point, we may proceed as in section II
and extract from the constraints flow equations for the finite-volume
observables of the theory. To the observables available before, associated to
the generators of target-space translations, now is added a generator of the
symmetry

$\alpha\longrightarrow\alpha+\text{const},$ (97)

which we interpret as the deformed $U(1)$ charge. We proceed in an analogous
way to section II and sketch the relevant steps, commenting on where they
differ to the earlier procedure.
The symplectic form of the theory is extracted from the part of the action
that is linear in time derivatives

$S_{t}^{J\bar{T}}=\frac{1}{\lambda}\int\epsilon_{ab}X^{a}\partial_{t}e_{x}^{b}+\frac{1}{\lambda}\int\partial_{t}X^{a}\epsilon_{ab}n^{b}A_{X}+\frac{1}{\lambda}\int
e_{x}^{a}\epsilon_{ab}n^{\prime b}\partial_{t}\alpha.$ (98)

It is useful to bring this expression to Darboux form by the redefinition

$\tilde{X}^{a}:=X^{a}+n^{\prime a}\alpha,$ (99)

and

$\tilde{e}^{a}:=e^{b}+n^{b}A_{X}.$ (100)

in terms of which the symplectic form becomes161616We introduce the notation
$n^{\prime}\cdot\epsilon\cdot n\equiv n^{\prime a}\epsilon_{ab}\dot{n}^{b}$

$\Omega^{J\bar{T}}=-\frac{1}{\lambda}\int\delta\tilde{X}^{a}\wedge\tilde{e}_{X}^{b}\epsilon_{ab}-\frac{1}{\lambda}\int\delta\alpha\wedge\delta
A_{X}(n^{\prime}\cdot\epsilon\cdot n),$ (101)

The Poisson brackets are then the following

$\displaystyle\\{\tilde{X}^{a},\tilde{e}_{X}^{b}\\}$
$\displaystyle=-\lambda\epsilon^{ab}$ (102) $\displaystyle\\{\alpha,A_{X}\\}$
$\displaystyle=-\frac{\lambda}{n^{\prime}\cdot\epsilon\cdot n}$ (103)

We can express the Hamiltonian of the full deformed theory in the same way we
did in section II

$H^{J\bar{T}}=H_{grav}^{J\bar{T}}+H_{mat}^{J\bar{T}}$ (104)

with

$\displaystyle H_{mat}^{J\bar{T}}$
$\displaystyle=\sqrt{-g}\tilde{F}\left(\frac{\pi_{\phi}}{\sqrt{-gg^{00}}},\frac{\pi_{\bar{\phi}}}{\sqrt{-gg^{00}}},\frac{D_{x}\bar{\phi}D_{x}\phi}{\sqrt{-gg^{00}}}\right)+A_{t}\left(i\bar{\phi}\pi_{\bar{\phi}}-i\phi\pi_{\phi}\right)$
(105)
$\displaystyle-\frac{g^{01}}{g^{00}}(\partial_{X}\phi+iA_{x}\phi)\pi_{\phi}-\frac{g^{01}}{g^{00}}(\partial_{X}\bar{\phi}-iA_{x}\bar{\phi})\pi_{\bar{\phi}}$
(106)

being the matter contribution, which is derived through the Legendre transform
of the action using the standard approach.
From (96) we can see that the gravitational part of (104) is given by

$H_{grav}^{J\bar{T}}=H_{grav}^{T\bar{T}}+A_{t}\left(\frac{\partial_{X}X^{a}}{\lambda}\epsilon_{ab}n^{b}+\frac{e_{X}^{a}\epsilon_{ab}n^{\prime
b}}{\lambda}\right)+e_{t}^{a}\frac{\epsilon_{ab}n^{\prime
b}}{\lambda}\left(\partial_{X}\alpha-A_{X}\right).$ (107)

(superindices $T\bar{T}$ indicate terms that are identical to the pure
$T\bar{T}$ case). Again, the total Hamiltonian is a linear combination of
constraints of the form

$\displaystyle H^{J\bar{T}}=e_{t}^{a}F^{J\bar{T}}_{a}+A_{t}G^{J\bar{T}}$ (108)

where

$G\equiv\left(\frac{\partial_{x}X^{a}}{\lambda}\epsilon_{ab}n^{b}+\frac{e_{x}^{a}\epsilon_{ab}n^{\prime
b}}{\lambda}\right)-i(\pi_{\phi}\phi-\pi_{\bar{\phi}}\bar{\phi})$ (109)

and

$F_{a}\equiv\frac{\epsilon_{ab}}{\lambda}\left(\partial_{x}X^{b}+e^{b}_{x}+n^{\prime
b}[\partial_{x}\alpha-
A_{X}]\right)-\frac{\eta_{ab}e_{x}^{b}}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{P}_{0}^{J\bar{T}}-\frac{\epsilon_{ab}e_{x}^{b}}{\sqrt{|e_{x}\cdot
e_{x}|}}\mathcal{H}_{0}^{J\bar{T}}.$ (110)

Note that $\mathcal{P}_{0}$ and $\mathcal{H}_{0}$ have the same form as in
section II, but they are evaluated in the background of the gauge field
$A_{X}$, so

$\mathcal{P}_{0}^{J\bar{T}}=-\frac{\pi D_{x}\phi}{\sqrt{|e_{x}\cdot e_{x}|}}$
(111)

and

$\mathcal{H}_{0}^{J\bar{T}}=\sqrt{|e_{x}\cdot
e_{x}|}\tilde{f}\left(\frac{\pi_{\phi}\pi_{\bar{\phi}}}{\sqrt{|e_{x}\cdot
e_{x}|}},\frac{D_{x}\phi D_{x}\bar{\phi}}{\sqrt{|e_{x}\cdot
e_{x}|}},\bar{\phi}\phi\right).$ (112)

To summarize, as for the case of the $T\bar{T}$ deformation theory, the
$J\bar{T}$ theory is also a constrained Hamiltonian system. The imposition of
the constraints $F_{a}=0$ and $G=0$ occurs over the phase space described by
the canonical variables
$\\{e_{x},X,A,\alpha,\phi,\pi_{\phi},\bar{\phi},\pi_{\bar{\phi}}\\}$ equiped
with the symplectic form (101). In an analogous manner to section II, it can
be verified that constraints are indeed first class, as expected.

Let us now identify the Dirac observables of the theory. As previously
mentioned, the theory is symmetric under the action of isometries of the
target-space metric on $X$ and the shift (97). The conserved quantities
associated with these symmetries are

$P^{a}=\int
dx\frac{\epsilon^{a}_{b}e^{b}_{x}-\epsilon^{a}_{b}n^{b}A_{x}}{\lambda}$ (113)

and

$Q_{\lambda}=-\int dx\frac{n^{\prime}\epsilon n}{\lambda}A_{x},$ (114)

which are the generators of translations in $X^{a}$ and $\alpha$,
respectively.

As before, we use diffeomorphisms and $U(1)$ gauge transformations to fix the
gauge where $e^{a}_{x}$ and $A_{x}$ are constants (denoted by tildes), so

$\displaystyle\tilde{A}_{x}$ $\displaystyle=-\frac{\lambda
Q_{\lambda}}{(n^{\prime}\cdot\epsilon\cdot n)}$ (115)
$\displaystyle\tilde{e}^{a}_{x}$
$\displaystyle=\epsilon^{a}_{b}P^{b}\lambda-\frac{\lambda
Q_{\lambda}n^{a}}{(n^{\prime}\cdot\epsilon\cdot n)}$ (116)

Proceeding similarly to section II, we find that the zero modes of the
constraints lead to the following equations, which represent a deformation
mixing the $P^{a}$ with the deformed charge $Q_{\lambda}$

$\displaystyle Q_{\lambda}$ $\displaystyle=Q_{0}+P\cdot
n^{\prime}+\frac{Rn^{1}}{\lambda}$ (117)
$\displaystyle\frac{R}{\lambda}+P^{0}-Q_{\lambda}\frac{n^{1}-n^{\prime
1}}{n^{\prime}\cdot\epsilon\cdot n}$
$\displaystyle=\cosh\theta_{0}E_{0}(L,\mu)+\sinh\theta_{0}P_{0}(L,\mu)$ (118)
$\displaystyle P^{1}-Q_{\lambda}\frac{n^{0}-n^{\prime
0}}{n^{\prime}\cdot\epsilon\cdot n}$
$\displaystyle=\sinh\theta_{0}E_{0}(L,\mu)+\cosh\theta_{0}P_{0}(L,\mu)$ (119)

where we have defined

$\displaystyle
L=|\tilde{e}_{x}^{2}|^{1/2}=|\lambda|\left\lvert\left(P^{0}-Q_{\lambda}\frac{n^{1}}{n^{\prime}\cdot\epsilon\cdot
n}\right)^{2}-\left(P^{1}-Q_{\lambda}\frac{n^{0}}{n^{\prime}\cdot\epsilon\cdot
n}\right)^{2}\right\rvert^{1/2}$ (120)
$\displaystyle\cosh\theta_{0}=\frac{\tilde{e}^{1}_{x}}{L}=\left(P^{0}-Q_{\lambda}\frac{n^{1}}{n^{\prime}\cdot\epsilon\cdot
n}\right)\frac{\lambda}{L}$ (121)
$\displaystyle\sinh\theta_{0}=\frac{\tilde{e}^{0}_{x}}{L}=\left(P^{1}-Q_{\lambda}\frac{n^{0}}{n^{\prime}\cdot\epsilon\cdot
n}\right)\frac{\lambda}{L}.$ (122)

Also, in the previous equations $Q_{0}$ stands for the $U(1)$ charge generator
in the undeformed theory

$Q_{0}=i\int dx\left(\pi_{\phi}\phi-\pi_{\bar{\phi}}\bar{\phi}\right)$ (123)

and

$\displaystyle E_{0}(L,\mu)=\int
dx\mathcal{H}_{0}^{J\bar{T}}\big{\rvert}_{e_{x}=\tilde{e}_{x}}$ (124)
$\displaystyle P_{0}(L,\mu)=\int
dx\mathcal{P}_{0}^{J\bar{T}}\big{\rvert}_{e_{x}=\tilde{e}_{x}}$ (125)

are the energy and momentum of the undeformed theory when placed on a spatial
circle of length $L$ and in the presence of a background constant background
$U(1)$ gauge field of the form

$A_{\alpha}=\delta_{\alpha x}\mu=-\delta_{\alpha
x}\lambda\frac{Q_{\lambda}}{n^{\prime}\cdot\epsilon\cdot n},$ (126)

which can be equivalently thought as a $U(1)$ twist of the boundary conditions
for the scalars along the circle

$\phi(t,x+L)=e^{i\mu}\phi(t,x)$ (127)

with $\mu$ defined in (126).
In principle it is not evident how to relate the flow equations we have
obtained to those found for the $J\bar{T}$-deformations in the literature.
This is because our expressions involve the undeformed charges evaluated at
twisted boundary conditions, with the twist parameter itself depending on the
deformed spectra, in an analogous way as happens for the radius in the usual
$T\bar{T}$ flow. This feature –to our knowledge– is absent from usual
formulations of $J\bar{T}$-type flows.
It is interesting to point our however that having the flow depend on the
undeformed theory at twisted boundary conditions with twists depending on the
deformed conserved quantities is precisely what was found in Hernandez-
Chifflet:2019sua  for the flows induced by higher-spin currents in integrable
theories. Even though these results were specifically about these types of
deformations, it seems likely that this twist dependence is a general feature
of deformations of integrable theories by CDD factors. In any case, we have
presented this result mostly as an application of the “gravitational” approach
to a different type of deformation and relegate further study of the flow
equation we have obtained to future work.

##  VI From canonical quantization to the path-integral formulation

Let us see now how the covariant path integral quantization (5) discussed in
dubovsky2018tt  emerges from the canonical point of view. We will follow the
standard procedure (as found in many textbook treatments of QED
Itzykson:1980rh ; weinberg1995quantum ; henneaux ) which starts from canonical
quantization of a fully constrained system and its canonical-variables path
integral and builds from there the covariant path integral over the
unconstrained configuration space.
We will start from the assumption that we have managed to quantize the theory
in a “constrain first” type of approach, meaning that we have built a fully
constrained “reduced” or “physical” phase space $M_{phys}$ by imposing at the
classical level all of the constraints of the theory and have performed the
canonical quantization procedure of this phase space. To be concrete, we will
assume that this phase space is constructed out of the gauge-fixing procedure
discussed in the last paragraphs of section II by imposing the gauge fixing
conditions (43) and (66). As explained before, the phase space thus
constructed is equal to the phase space of the undeformed theory imbued with
the symplectic form of the undeformed theory. Therefore, we may well assume
that a quantization of this phase space is available, that is, that the
undeformed theory is defined at the quantum level. In particular the canonical
quantization procedure applied to the undeformed theory would yield a Hilbert
space $\mathcal{H}_{phys}$ and a way to promote phase space variables in
$M_{phys}$ to operators (modulo ordering issues as usual) and our preceding
argument implies that this quantizes the fully constrained “gravitational”
theory as well.
From the point of view of the constrained gravitational theory the $P^{a}$ are
natural (commuting) observables realizing physical spacetime translations and
it is natural then to write down the partition function171717We are keeping
the Minkowskian $i$ in the exponent whereas for the partition function
euclidean signature would be more natural (as considered in dubovsky2018tt ).
This is irrelevant for our discussion which is completely formal in any case.

$Z(L,R)=\text{Tr}_{\mathcal{H}_{phys}}\left\\{e^{-iL_{a}\hat{P}^{a}}\right\\}.$
(128)

This function is theoretically well-defined, given that on the constraint
surface, $P^{a}$ can be written in terms of phase space variables of the
undeformed theory for which we assume a canonical quantization procedure makes
sense. The $\hat{P}^{a}$ denote the operators on $\mathcal{H}_{phys}$ that
would be obtained from this procedure. The $R$ dependence appears because the
equations $\int dxF_{a}=0$ that are used to write $P^{a}$ in terms of the
reduced phase space canonical variables depend on $R$ via the winding of $X$
along the spatial cycle, which we take to be non-zero.
Standard manipulations allow to write the $Z(L,R)$ partition function as a
path integral over the fully-constrained phase space

$Z(L,R)=\int_{z^{i}(t=1)=z^{i}(t=0)}Dze^{i\int
dt(\alpha^{(const)}_{i}\dot{z}^{i}-L_{a}P^{a})}$ (129)

where the $z^{i}$ stand for the phase-space variables of the fully-constrained
phase space in a Darboux coordinatization and $\alpha^{(const)}_{i}$ for the
corresponding symplectic potential. Of course, as we have mentioned, these
coincide with the undeformed phase-space variables and in our scalar field
example thy would be represented by the $\left\\{\phi,\pi\right\\}$ canonical
pair. It is important to point out that in the previous expression $P^{a}$
stand for phase space functions built out of the $\hat{P}^{a}$ operators
according to some ordering procedure (for instance, following
weinberg1995quantum , by pushing all canonical momentum to the right). Notice
that since the $P^{a}$ on the fully-constrained phase space are obtained by
solving the constraint equations, these are very nontrivial functions over
this phase space and ordering issues may be very complicated to resolve in
practice. Keeping this in mind, we can follow the usual procedure
(Itzykson:1980rh ; henneaux ) and express this path integral as one over the
unconstrained gravitational phase space by adding the suitable delta functions
and measure factors

$Z(L,R)=\int De_{x}^{a}DY^{a}D\phi
D\pi\delta(F_{a})\delta(G^{b})\det\left\\{F_{a},G\right\\}e^{i\int\alpha^{(unconst)}-i\int
dtL_{a}P^{a}}$ (130)

where all of the path-integral integration variables are subject to periodic
boundary conditions along $t\in(0,1)$. Notice that our canonical variable is
the periodic $Y$ and not $X$ as we are assuming we are in the sector with
spatial winding so that $X=X^{wind}+Y$. Here $F_{a}$ are the constraints as
previously discussed and $G^{b}=0$ stand for our gauge fixing conditions (43)
and (66). In the previous formula $\alpha^{(unconst)}$ stands for (the
pullback to the circle of) the unconstrained symplectic potential which is
given by

$\int\alpha^{(unconst)}=\frac{1}{\lambda}\int
dtdx\epsilon_{ab}e^{a}_{x}\partial_{t}Y^{b}+\int dtdx\pi\partial_{t}\phi.$
(131)

We can now write the delta function enforcing the constraints of the theory as
an exponential by integrating-in a Lagrange multiplier which is immediately
identified with $e^{a}_{t}$

$\delta(F_{a})=\int De^{a}_{t}e^{-i\int dtdxe^{a}_{t}F_{a}}=\int
De^{a}_{t}e^{-i\int dt(H_{grav}+H_{0})}.$ (132)

The path integral therefore becomes

$\displaystyle Z(L,R)$ $\displaystyle=$ $\displaystyle\int
DY^{a}De_{x}^{a}De_{t}^{a}\delta(G^{b})\det\left\\{F_{a},G\right\\}\times$
$\displaystyle\exp\left\\{\frac{i}{\lambda}\int
dtdx\epsilon_{ab}e^{a}_{x}\partial_{t}Y^{b}-\frac{i}{\lambda}\int
dtdxe^{a}_{t}\epsilon_{ab}\left(\partial_{x}X^{b}+e^{b}_{x}\right)-i\int
dtL_{a}P^{a}\right\\}Z_{0}[e]$ (133)

where $X$ includes the part realizing the spatial winding and we have used
expression (17) for $H_{grav}$. Here we have identified the undeformed
partition function in the presence of the gravitational field

$Z_{0}[e]\equiv\int D\phi D\pi e^{i\int dtdx\pi\partial_{t}\phi-i\int
dtH_{0}}$ (134)

where the dependence on the dynamical geometry enters through $H_{0}$ as given
in (18). Notice that we have been using the notations for a scalar field,
although we expect these rather formal manipulations to be valid for some more
general cases under the assumptions discussed in section IV. Finally, we can
use (35) to write $P^{a}$ in the exponent in terms of the unconstrained
canonical variables

$\exp\left\\{-i\int dtL_{a}P^{a}\right\\}=\exp\left\\{-i\int dtL_{a}\int
dx\frac{\epsilon^{a}_{b}e^{b}_{x}}{\lambda}\right\\}=\exp\left\\{i\int dt\int
dxe^{a}_{x}\epsilon_{ab}\frac{\partial_{t}(tL^{a})}{\lambda}\right\\},$ (135)

given that the path integral is actually non-zero only in the support of the
delta function $\delta(F_{a})$ that enforces the constraints. Notice that this
last point is actually somewhat subtle: the $P^{a}$ that appears in the
constrained phase space path integral is really some ordered version of an
operator whose expression in terms of the constrained canonical variables is
expected to be complicated. The fact that after the ordering procedure this
function of the constrained canonical variables can still be written as in
(135) is not at all evident. Under the assumption that this is so, if we
redefine the field $X$ so that now it also includes winding along the time
cycle

$X^{a}\longrightarrow X^{a}+tL^{a}$ (136)

we then recover (up to a constant term coming from total derivatives) the
covariant path integral on the torus

$Z(L,R)=\int
DX^{a}De^{a}\delta(G^{b})\det\left\\{F_{a},G\right\\}e^{iS_{grav}[X,e]}Z_{0}[e].$
(137)

This is precisely a gauge-fixed form of the covariant path integral on the
torus considered in dubovsky2018tt , where the $X$ windings around the two
cycles of the torus encode the $L$ and $R$ dependence. Crucially, this path-
integral expression is in fact independent of the gauge-fixing conditions
used, see for instance Itzykson:1980rh  for arguments directly within the
canonical approach181818This can also be seen by the usual BRST arguments..
Notice that usually (e.g. in actions that are second-order in time
derivatives) to get from the canonical path integral to the covariant path
integral it is necessary to integrate out some conjugate momenta, whereas here
the phase space integral already yields a covariant path integral after
exponentiation of the constraints (which is natural from the fact that actions
that are first order in time derivatives –such as this one– naturally weight
paths in phase space, not in configuration space).
It is interesting to compare the path integral expressions obtained here with
that used in dubovsky2018tt . In (137), the integration measures obtained from
the canonical procedure and implied in $De$ and $DX$ are in a sense trivial,
in that they simply run over all phase space values of $X^{a}$ and $e_{x}^{a}$
and all real values of $e_{t}^{a}$ and there is no extra measure factor
attached except what would come from the
$\delta(G^{b})\det\left\\{F_{a},G^{b}\right\\}$ factor. From the canonical
point of view this is natural, as the only requirement for our measure over
the fully-constrained phase space is that it is invariant under canonical
transformations. Indeed the path integral measure in (129) can be thought of,
at every time slice, as the symplectic measure of the induced symplectic form
on the gauge-fixed phase space.
This is in contrast with the situation in the gauge-fixed path integral
considered in dubovsky2018tt , where even aside from the determinant factors
coming from the Faddeev-Popov procedure the path integral measures over $e$
and $X$ are non-trivial and defined via diff-invariant inner products on the
space of fields (as is usual in path integral formulations of string theory
Polchinski:1985zf ; DHoker:1988pdl ). Indeed, in the treatment of
dubovsky2018tt  these measure factors were crucial to get the “correct”
$T\bar{T}$ answer. Furthermore, the range of integration of the fields
considered in the path integral formulation of dubovsky2018tt  ran only over
explicitly invertible geometries by construction191919One could argue however
that since that path integral ended up being one-loop exact in a sense, the
localization on the saddle point makes questions regarding the region of
integration somewhat subtle. in the canonical formulation the langrange
multiplier variable $e_{t}$ runs over all real values, including values
corresponding to non-invertible metrics for instance.
This discrepancy of the covariant and canonical approaches in the range of
integration for metric variables imposing constraints is often present in
quantization of gravitational theories202020See for instance Witten:2022xxp
for recent comments on this. In fact, this problem is already present for the
simplest possible diffeomorphism invariant action, that of the free
relativistic massive particle. As discussed for instance in Hartle:1986yu ;
Cohen:1985sm  the worldline “lapse” function emerges in the canonical
formalism as a Lagrange multiplier, whose range is naturally unrestricted.
However, for the path integral to correctly yield the Klein-Gordon propagator,
its range of integration must be restricted ultimately to positive
values212121It is also interesting to compare statement in Hartle:1986yu  and
Cohen:1985sm  regarding to what extent locality fixes the path integral
measure.
It is also interesting to compare the two path integral formulations for the
$T\bar{T}$ gravity theory regarding the one-loop exactness property found in
dubovsky2018tt . There, after the path integral has localized into the space
of constant metrics (constant dyads, strictly speaking) the remaing finite-
dimensional integral is also found to be one-loop exact. From the point of
view of that procedure this one-loop exactness was somewhat surprising,
although crucial to obtain the correct $T\bar{T}$ deformed spectrum. As we
have pointed out before, considering this perspective, the emergence of the
equation for the $T\bar{T}$ spectrum at the classical level t is not
surprising, given that at the level of the quantum partition function it
arises from some saddle-point equation.
In the context of the analysis presented in this paper, the one-loop exactness
of the path integral and the resulting $T\bar{T}$ spectrum follow from our
canonical analysis and the expression for the canonical partition function we
start with (128). It is instructive to see how this one-loop exactness comes
about, and as a sanity check of this whole procedure, to rederive the
$T\bar{T}$ spectrum from (137) working backwards. From this point of view, the
step where one would find the localization to constant geometries would be
when arriving at (130). In order to analyze this expression it is convenient
to separate the constraints $F_{a}$ and the gauge fixing conditions $G^{a}$
into zero modes (denoted with tildes) and non-zero modes (denoted with
primes). For the gauge-fixing conditions, the non-zero mode part $G^{\prime}a$
is the $\partial_{x}e^{a}_{x}=0$ constraint, while the zero-mode part
$\tilde{G}^{a}$ is given by $\tilde{X}^{a}=0$ (as in (66)). Notice that since
the Poisson bracket between the zero mode of $F_{a}$ and the non-zero mode of
$G$ vanish, then the determinant factor in (130) factors into

$\det\left\\{F_{a},G\right\\}=\det\left\\{\tilde{F}_{a},\tilde{G}\right\\}\det\left\\{F^{\prime}_{a},G^{\prime
b}\right\\}$ (138)

where the second factor is a functional determinant given by

$\det\left\\{F^{\prime}_{a},G^{\prime
b}_{(n)}\right\\}=\det(-\partial^{2}_{x})^{2}.$ (139)

To perform the path integral in (130) it is useful to factor the integration
into zero and non-zero modes as

$\displaystyle DY^{a}=d\tilde{Y}^{a}DY^{\prime a}$ (140) $\displaystyle
De_{x}^{a}=d\tilde{e}^{a}_{x}De^{\prime a}_{x}$ (141)

where the primed variables again refer to the non-zero modes. Notice that the
integral over the non-zero modes have no extra measure factor coming from a
non-trivial path integral measure as in dubovsky2018tt . The integrals over
the nonzero modes of $Y^{a}$ and $e^{a}_{x}$ are immediately taken care of by
the non-zero-mode part of the constraints and gauge-fixing conditions
respectively. The fact that this sets to zero the non-zero mode part of
$e^{a}_{x}$ means that these variables (that is, their non-zero mode parts)
disappear from the exponent in (130) since this exponent is linear in
$e^{a}_{x}$ and depends on $Y$ via its product with $e^{a}_{x}$. Then these
integrations yield only left-over functional determinants which are given by

$\displaystyle\int DY^{\prime a}\delta(F^{\prime}_{a})=\int DY^{\prime
a}\delta(\partial_{x}Y^{a}+\dots)=\det(\partial_{x})^{-2}$ (142)
$\displaystyle\int De^{\prime
a}_{x}\delta(\partial_{x}e^{a}_{x})=\det(\partial_{x})^{-2}$ (143)

These functional determinants can be turned into determinants of hermitian
operators by using

$\det(Q)=\det(Q^{\dagger}Q)^{1/2}$ (144)

If we assume, as mentioned before, that the path integral measures of
integration for the $Y$ and $e_{x}$ are defined by the trivial inner-product
for functions on the circle

$(f,g)\equiv\int dxf^{*}g$ (145)

(notice that, as mentioned before, there is no need to use a diffeomorphism
invariant inner product on the circle to define a path integral over the
canonical variables) then, using this inner-product to define the adjoint, the
aforementioned manipulations of functional determinants (144) yield

$\det(\partial_{x})=\det(-\partial_{x}^{2})^{1/2}.$ (146)

Thus, the functional determinants coming from the integration over the non-
zero modes cancel with the functional determinant factor from
$\det\left\\{F_{a},G\right\\}$. The integral over the zero mode of $Y$
trivially sets it to zero because of the zero mode of the gauge fixing
condition and we are left with a finite-dimensional integral over constant
$e_{x}^{a}$ and a path integral over the canonical variables of the undeformed
theory

$Z(L,R)=\int D\phi D\pi\int d\tilde{e}^{a}_{x}e^{i\int
dtdx\pi\partial_{t}\phi}\delta(\tilde{F_{a}})\det\left(\frac{\partial\tilde{F_{a}}}{\partial\tilde{e}^{b}_{x}}\right)e^{-iL_{a}\epsilon^{a}_{b}\tilde{e}^{b}_{x}/\lambda}$
(147)

where we have used that at constant $e^{a}_{x}$

$\det\left\\{\tilde{F}_{a},\tilde{G}\right\\}=\det\left(\frac{\partial\tilde{F_{a}}}{\partial\tilde{e}^{b}_{x}}\right)$
(148)

and $P^{a}=\epsilon^{a}_{b}\tilde{e}^{b}_{x}/\lambda$. The delta function that
imposes $\tilde{F}_{a}$ fixes $\tilde{e}^{a}_{x}$ to be such that $P^{a}$
satisfy the $T\bar{T}$ flow equation, as explained in section II. This of
course gives back the expression (129) for the partition function, modulo the
fact that in (129) $P^{a}$ requires some ordering procedure whereas the
expression we have just obtained $P^{a}$ appears to be just the phase space
function obtained from solving $\tilde{F}_{a}=0$, this subtlety being another
avatar of the ordering issues we have pointed out earlier.
However, in order to make contact with the procedure performed in
dubovsky2018tt , some different manipulations are useful. First, the finite-
dimensional delta function can be exponentiated by integrating-in a constant
variable $\tilde{e}_{t}^{a}$

$\delta(\tilde{F}^{a})=\int
d\tilde{e}_{t}^{a}e^{-i\tilde{e}_{t}^{a}F_{a}}=\int
d\tilde{e}_{t}^{a}e^{-i\tilde{H}_{grav}-i\tilde{H}_{0}}$ (149)

where $\tilde{H}_{grav}$ and $\tilde{H}_{0}$ are the zero modes of the gravity
and matter Hamiltonians respectively evaluated at constant geometry
$\tilde{e}^{a}_{\mu}$. Then the partition function can be written as

$Z(L,R)=\int
d\tilde{e}^{a}_{x}d\tilde{e}^{a}_{t}e^{-i\tilde{H}_{grav}-iL_{a}P^{a}\tilde{e}^{b}_{x}/\lambda}W\left[\tilde{e}\right]$
(150)

where

$W\left[\tilde{e}\right]=\int D\pi D\phi e^{i\int
dtdx\pi\partial_{t}\phi}\det\left(\frac{\partial\tilde{F_{a}}}{\partial\tilde{e}^{b}_{x}}\right)e^{-i\tilde{H}_{0}}$
(151)

This last quantity $W\left[\tilde{e}\right]$ is calculated purely in terms of
the matter theory (the undeformed theory) at fixed constant geometry. As we
have mentioned for our starting point, we may assume that it is understood how
to quantize the undeformed matter theory, in the sense that when placing the
theory in a cylinder with constant geometry of size $L$ we have a basis of
energy and momentum eigenstates of the undeformed theory $\ket{k}$ with
eigenvalues $E_{0}^{(k)}(L)$ and $P_{0}^{(k)}(L)$ respectively (here $k$
stands for some multi-index labelling these states). Then, modulo ordering
subtleties, we can write

$W\left[\tilde{e}\right]=\sum_{k}\bra{k}\det\left(\frac{\partial\hat{\tilde{F_{a}}}}{\partial\tilde{e}^{b}_{x}}\right)e^{-i\hat{\tilde{H}}_{0}}\ket{k}$
(152)

where the hat denotes that we have promoted these functions of the phase space
of the undeformed theory into operators in the Hilbert space of the undeformed
theory. Both $\tilde{F}_{a}$ and $\tilde{H}_{0}$ are functions of the
undeformed energy $E_{0}(L)$ and momentum $P_{0}(L)$ on a circle of size $L$,
where $L$ in these expressions depends on $\tilde{e}^{a}_{x}$ via
$L|e_{x}^{2}|^{1/2}$. Then naively we would want to write

$W\left[\tilde{e}\right]=\sum_{k}\det\left(\frac{\partial\tilde{F}^{(k)}_{a}}{\partial\tilde{e}^{b}_{x}}\right)e^{-i\tilde{H}^{(k)}_{0}}$
(153)

where $\tilde{F}^{(k)}_{a}$ and $\tilde{H}^{(k)}_{0}$ denote these quantities
with $E_{0}(L)$ and $P_{0}(L)$ evaluated at $E_{0}^{(k)}(L)$ and
$P_{0}^{(k)}(L)$.
What may obstruct the passage from (152) to (153) is the fact that the
expectation values summed over also include insertions of
$\partial_{L}E_{0}(L)$ and $\partial_{L}P_{0}(L)$ coming from the derivatives
of $\tilde{F}_{a}$ with respect to $\tilde{e}_{x}^{a}$. Because of the
quantization of total momentum at fixed size $L$, $P_{0}(L)$ is necessarily of
the form $P_{0}(L)=N/L$ for some integer $N$. Therefore $\ket{k}$ are also
eigenvectors of $\partial_{L}P_{0}(L)$ and insertions of this operator do not
introduce complications in the passage from (152) to (153).
Let us focus on terms in (152) involving insertions of $\partial_{L}E_{0}(L)$.
Because of the structure of $\tilde{F}_{a}$, given by (52) with
$P^{a}=\epsilon^{a}_{b}\tilde{e}^{a}_{x}/\lambda$, these insertions can appear
at the most at quadratic order in (152). However, because of the structure of
the determinant it turns out that quadratic insertions of these operator
vanish from (152). Terms linear in $\partial_{L}E_{0}(L)$ are easily taken
care of by the fact that
$\bra{k}\partial_{L}E_{0}(L)\ket{k}=\partial_{L}E_{0}^{(k)}(L)$ by standard
perturbation theory arguments and thus expression (153) appears to be valid.
Plugging in then (153) into our expression (150) for $Z(L,R)$ yields

$Z(L,R)=\sum_{k}\int
d\tilde{e}^{a}_{x}d\tilde{e}^{a}_{t}e^{-i\tilde{H}_{grav}-iL_{a}P^{a}\tilde{e}^{b}_{x}/\lambda}\det\left(\frac{\partial\tilde{F}^{(k)}_{a}}{\partial\tilde{e}^{b}_{x}}\right)e^{-i\tilde{H}^{(k)}_{0}}.$
(154)

This is –if not precisely the same– of the form found for the gravitational
$T\bar{T}$ partition function found in dubovsky2018tt  (see for instance
equations (43) to (50) there): an integral over constant dyads on the torus
and a sum over the eigenstates of the undeformed theory. However, for the
integral found in dubovsky2018tt , the fact it was one-loop exact seemed quite
surprising, a conspiracy of factors coming from the non-trivial path integral
measures, coming from inner products in the space of fields fixed by demanding
diffeomorphism invariance.
Here however this saddle-point exactness property is quite simple. Indeed,
undoing in these last expression the steps going from (147) to (150) by
integrating over $\tilde{e}^{a}_{t}$ we get

$Z(L,R)=\sum_{k}\int
d\tilde{e}^{a}_{x}\delta(\tilde{F}^{(k)}_{a})\det\left(\frac{\partial\tilde{F}_{a}^{(k)}}{\partial\tilde{e}^{b}_{x}}\right)e^{-iL_{a}P^{a}(\tilde{e}_{x})}$
(155)

Then the delta function fixes the integral to values of $\tilde{e}^{a}_{x}$
such that $P^{a}(\tilde{e}_{x})$ coincide with the $T\bar{T}$ deformed
spectrum, (as discussed in detail in section (II)). The determinant factor
gives the right measure so that

$Z(L,R)=\sum_{k}e^{-iL_{0}(E^{(k)}_{\lambda}(R)+R/\lambda)-iL_{1}P^{(k)}_{\lambda}(R)}$
(156)

without there being any extra factors. Here $E^{(k)}_{\lambda}(R)$ and
$P^{(k)}_{\lambda}(R)$ are the solutions to the $T\bar{T}$ flow equations with
$E^{(k)}_{0}(R)$ and $P^{(k)}_{0}(R)$ as intial ($\lambda=0$) conditions. We
have arrived thus at the $T\bar{T}$-deformed partition function.
Let us briefly recap what we have accomplished. We started from the
observation that in a particular gauge, the gauge-fixed phase space is
identical to the undeformed phase space, thus we have assumed that the gauge-
fixed phase space admits a canonical quantization procedure identical to that
of the undeformed theory. The physical spacetime translation operators of the
gravitational theory are the $P^{a}$, which are made into functions
exclusively of this phase space after one solves the constraint equations.
Since these generate physical spacetime translations, it makes sense to
consider the partition function defined by (128) over the gauge-fixed
–equivalently, the undeformed– phase space. Via an expression of this
partition function as a path integral over canonical variables in the
constrained phase space and a type of canonical Faddeev-Popov trick we managed
to write this as a gauge-fixed form (in a particular gauge) of the covariant
path-integral discussed in dubovsky2018tt . Following a slightly different
version of this procedure in reverse we have shown that this reproduces the
$T\bar{T}$-deformed partition function.
Many of the steps we have followed are dependent on how one handles ordering
ambiguities that may arise, which we have mentioned at every relevant step,
yet not analyzed in detail. It would seem to be difficult to do this at the
level of generality about the undeformed theory we are assuming here, and
perhaps this analysis is better performed in a case-by-case basis.
It is interesting to point out however that in going from the gauge-fixed
covariant expression (137) to the $T\bar{T}$-deformed partition function (156)
the only ordering issues that arose were those within the quantization of the
undeformed theory, related to the operators appearing in the expectation
values taken in (152). This lends some weight to the intuitive idea mentioned
in section II that perhaps to get the fully quantum $T\bar{T}$ deformed
spectrum in a sense only quantum corrections at the level of the undeformed
theory need to be considered. It appears that further analysis is required
from this point of view, perhaps in a case-by-case basis considering specific
undeformed theories, to put these types of argument in more solid footing.

##  VII Conclusions and future directions

In this paper we have analyzed the gravitational description of the $T\bar{T}$
deformation from the Hamiltonian point of view, where it is described as a
constrained canonical system. We have found that the constraints imply non-
trivial relations between the target-space observables of the theory (the
physical energy and momentum) and the energy and momentum of the undeformed
theory. These relations, given by (58) and (59) coincide with the implicit
form of the solutions of the $T\bar{T}$ finite-volume flow equation (2) once
one identifies the target space energy and momentum with the
$T\bar{T}$-deformed energy and momentum. We have found that this result holds
for a quite general undeformed matter content. It is interesting that these
equations emerge within the gravitational description at the level of the
classical equations of motion, whereas the $T\bar{T}$ deformed spectrum is
supposed to hold fully quantum-mecanically. As it is well known, this already
happens at the level of the lightcone-quantized critical bosonic string, and
we have pointed out the similarity between the two cases.
We have followed a similar approach to study a similar deformation, inspired
by “gravitational” formulations of the Lorentz-breaking $J\bar{T}$ deformation
proposed in the literature. Here we have found that the deformed spectrum
depends on the undeformed quantities evaluated at twisted boundary conditions.
Although this feature is expected if one considers $S$-matrix deformations of
integrable theories, it makes comparison with previous results somewhat
difficult and further work is warranted in this direction.
Perhaps the most interesting avenue for future work in terms of possible
applications is to extend the $T\bar{T}$-gravity formulation of non-critical
string theories discussed in section III to the sector without winding,
particularly for a three-dimensional target space, where the spectrum in the
winding sector is known to be compatible with target-space Lorentz symmetries
at the quantum level. Here instead of the finite-volume energy spectrum
$E(R)$, one would be interested in the energies as function of the angular
momentum $J$. A possible approach would be to consider the path integral on
the Euclidean torus with boundary conditions twisted by $J$ along some cycle.
A first step towards attempting this calculation is to identify how rotations
act on the gravitational variables (in contrast to what happens in the
Polyakov formulation, this action is non-trivial in our case), which we have
done in this paper. Very likely, this path integral calculation will involve
considering singular metric configurations, which adds an extra complication
with respect to the non-zero winding case. In any case, it will be interesting
to see whether an exact formula emerges from this procedure, as happens in the
winding sector.

## Acknowledgements

We thank Miguel Campiglia, Rodolfo Gambini and Sergei Dubovsky for many
helpful discussions regarding this work. GHC acknowledges the support of CSIC
I+D grant number 583. This work is also supported by PEDECIBA.

##  Appendix A Constrained Hamiltonian Systems

In this section, we will introduce the most general aspects of constrained
Hamiltonian systems (which include gauge systems). Part of the importance of
having an appropriate Hamiltonian formulation for constrained systems lies in
the need to develop a valid canonical quantization procedure.

It can be shown (see henneaux ) that the total Hamiltonian in these cases is
of the form

$H_{T}=H+u^{m}c_{m}$ (157)

where the $u^{m}$ are Lagrange multipliers of the constraints $c_{m}$, and
$H=\dot{q}^{n}p_{n}-L$.

In order to appreciate the relation between quantities that appear in the
Hamiltonian formalism, a distinction between first and second class functions
is usually made. This relies only on the fundamental structure of the
Hamiltonian theory, the Poisson bracket. Any quantity that has a weakly null
bracket with all the constraints is called first class. Quantities that do not
have this property are called second class. The Total Hamiltonian (157) is
expressed as the sum of a first-class Hamiltonian plus a linear combination of
first-class constraints.

All the first-class constraints generate gauge transformations that are
independent if and only if the constraints are irreducible.The transformation
generated by a first-class constraint preserves all first and second-class
constraints, so it maps an allowed state onto an allowed state. Also, the
Poisson brackets of any two first-class constraints is a first-class
constraint and generates a gauge transformation.

On the other hand, second-class constraints are present if the matrix
$C_{jj^{\prime}}=[c_{j},c_{j^{\prime}}]$ does not vanish on the constraint
surface. This kind of constraints do not generate any transformation of
physical significance, including gauge transformations, because the
transformation generated by a second-class constraint does not preserve all
the constraints of the system, so it maps an allowed state onto a nonallowed
state. To treat second-class constraints we must define the Dirac bracket:

$[F,G]^{*}=[F,G]-[F,c_{i}]C^{ij}[c_{j},G]$ (158)

If the theory is formulated in terms of the Dirac brackets, the second-class
constraints become strong identities expressing canonical variables in terms
of others, and in some cases they can be used to eliminate some degrees of
freedom.

##  Appendix B Hamiltonian for scalar fields with one derivative interactions
in gravitational background

In order to obtain the Hamiltonian for the scalar matter theory in the
gravitational background we must first find the canonical momentum $\pi$ of
the matter field in terms of the time derivatives of $\phi$. This is
accomplished via the defining relation

$\pi=\sqrt{-g}\partial_{u}K(u,\phi)\frac{\partial
u}{\partial\partial_{t}\phi}$ (159)

Using that $u$ can be written as

$u=r\left(\sqrt{|g^{00}|}\partial_{t}\phi+r\frac{g^{01}}{\sqrt{|g^{00}|}}\partial_{x}\phi\right)^{2}+\frac{1}{g^{00}g}(\partial_{x}\phi)^{2}$
(160)

where $r\equiv\operatorname{sgn}(g^{00})$. The equation to invert
$\partial_{t}\phi$ in terms of $\pi$ is thus of the form

$\sqrt{|g^{00}|}\partial_{t}\phi+r\frac{g^{01}}{\sqrt{|g^{00}|}}\partial_{x}\phi=f\left(\frac{\pi}{\sqrt{|g^{00}g|}},\frac{(\partial_{x}\phi)^{2}}{g^{00}g},\phi\right)$
(161)

for some function $f$ of the variables made explicit above, the specific
details of which depend on the particular form of $K$. Using this we get that
the Hamiltonian can be written as

$\displaystyle H_{mat}$ $\displaystyle\equiv\int dx\pi\partial_{t}\phi-\int
dx\sqrt{-g}K(u,\phi)$ $\displaystyle=\int
dx\left(-\frac{g^{01}}{g^{00}}\pi\partial_{x}\phi+\frac{\pi}{\sqrt{|g^{00}|}}f-\sqrt{-g}K\left(f^{2}+\frac{(\partial_{x}\phi)^{2}}{g^{00}g},\phi\right)\right)$
$\displaystyle=\int
dx\left(-\frac{g^{01}}{g^{00}}\pi\partial_{x}\phi+\sqrt{-g}\tilde{f}\left(\frac{\pi}{\sqrt{|g^{00}g|}},\frac{(\partial_{x}\phi)^{2}}{g^{00}g},\phi\right)\right)$
(162)

from which (18) immediately follows. Notice also that by evaluating the
previous expression in flat space $g_{\mu\nu}=\eta_{\mu\nu}$ it is clear that
$\tilde{f}$ is the flat-space matter Hamiltonian density, so that by a change
of variables it is seen that the interpretation of (21) given in the main text
follows.

##  Appendix C Constraint algebra

In this section we are going to show some details of the calculation of the
constraint algebra and compute the Poisson brackets (32). Defining the
gravitational part of (28) as

$\mathcal{C}_{grav}[N]=\int
N~{}e_{x}^{a}\eta_{ab}(e_{x}^{b}+\partial_{x}X^{b}),$ (163)

and the material part as

$\mathcal{C}_{mat}[N]=\int N\mathcal{H}_{0}(x)$ (164)

we have

$\displaystyle\left\\{\mathcal{C}[N],\mathcal{C}[M]\right\\}$
$\displaystyle=\left\\{\mathcal{C}_{grav}[N],\mathcal{C}_{grav}[M]\right\\}+\left\\{\mathcal{C}_{mat}[N],\mathcal{C}_{mat}[M]\right\\}$
.
$\displaystyle+\left\\{\mathcal{C}_{grav}[N],\mathcal{C}_{mat}[M]\right\\}+\left\\{\mathcal{C}_{mat}[N],\mathcal{C}_{grav}[M]\right\\}$
(165)

The first term in (165) corresponds to the Poisson brackets between the
gravitational part. This is

$\displaystyle\left\\{\int\mathrm{d}xN~{}e_{x}^{a}\eta_{ab}(e_{x}^{b}+\partial_{x}X^{b}),\int\mathrm{d}x^{\prime}M~{}e_{x}^{a^{\prime}}\eta_{a^{\prime}b^{\prime}}(e_{x}^{b^{\prime}}+\partial_{x^{\prime}}X^{b^{\prime}})\right\\}$
(166)

Before continuing, we first note that the only terms that contribute to the
final result are those involving the Poisson brackets
$\left\\{\partial_{x}X^{b},e^{a^{\prime}}_{x}\right\\}$ and
$\left\\{e^{a}_{x},\partial_{x}X^{b^{\prime}}\right\\}$, while the other two
terms cancel due to the anti symmetry of the Levi-Civita (which appears when
taking the Poisson bracket of the canonical variables). With this in mind, we
can write (166) as

$\displaystyle\int
N(x)M(x^{\prime})e^{x}_{a}(x)\eta_{ab}\left\\{\partial_{x}X^{b}(x),e^{a^{\prime}}_{x}(x^{\prime})\right\\}\eta_{a^{\prime}b^{\prime}}(e^{b^{\prime}}_{x}(x^{\prime})+\partial_{x^{\prime}}X^{b^{\prime}}(x^{\prime}))$
(167) $\displaystyle+\int
N(x)M(x^{\prime})\left\\{e^{a}_{x}(x),\partial_{x^{\prime}}X^{b^{\prime}}(x^{\prime})\right\\}\eta_{ab}(e^{b}_{x}(x)+\partial_{x}X^{b}(x))e^{a^{\prime}}_{x}(x^{\prime})\eta_{a^{\prime}b^{\prime}}.$

It is a simple task to verify that only each second term in the curved
parenthesis contributes to the final result. We are then left with

$\displaystyle\int
N(x)M(x^{\prime})e^{x}_{a}(x)\eta_{ab}\left\\{\partial_{x}X^{b}(x),e^{a^{\prime}}_{x}(x^{\prime})\right\\}\eta_{a^{\prime}b^{\prime}}\partial_{x^{\prime}}X^{b^{\prime}}(x^{\prime})$
(168) $\displaystyle+\int
N(x)M(x^{\prime})\left\\{e^{a}_{x}(x),\partial_{x^{\prime}}X^{b^{\prime}}(x^{\prime})\right\\}\eta_{ab}\partial_{x}X^{b}(x)e^{a^{\prime}}_{x}(x^{\prime})\eta_{a^{\prime}b^{\prime}}$
$\displaystyle=\int
N(x)M(x^{\prime})e^{x}_{a}(x)\partial_{x^{\prime}}X^{b^{\prime}}(x^{\prime})\eta_{ab}\varepsilon^{ba^{\prime}}\eta_{a^{\prime}b^{\prime}}\partial_{x}\delta(x-x^{\prime})$
$\displaystyle+\int
N(x)M(x^{\prime})e^{a^{\prime}}_{x}(x^{\prime})\partial_{x}X^{b}(x)\eta_{ab}\varepsilon^{ab^{\prime}}\eta_{a^{\prime}b^{\prime}}\partial_{x^{\prime}}\delta(x-x^{\prime})$
$\displaystyle=\int
N(x)M(x^{\prime})e^{x}_{a}(x)\varepsilon_{ab}\partial_{x^{\prime}}X^{b}(x^{\prime})\partial_{x}\delta(x-x^{\prime})$
$\displaystyle+\int
N(x)M(x^{\prime})e^{x}_{a}(x^{\prime})\varepsilon_{ab}\partial_{x}X^{b}(x)\partial_{x^{\prime}}\delta(x-x^{\prime}).$

Integrating over the Dirac delta, and noticing that the only non vanishing
terms are those involving the derivative of the Lagrange multipliers, we
finally get

$\displaystyle\left\\{\mathcal{C}_{grav}[N],\mathcal{C}_{grav}[M]\right\\}$
$\displaystyle=\int\mathrm{d}x(N\partial_{x}M-M\partial_{x}N)e^{x}_{a}\varepsilon_{ab}\partial_{x}X^{b}$
(169) $\displaystyle=\mathcal{C}_{x,grav}[N\partial_{x}M-M\partial_{x}N],$

where $C_{x,grav}$ stands for the gravitational part of the diffeomorphism
constraint.
We now proceed to calculate the second term in (165), which will be the most
complicated because of the degree of generality we are assuming for the matter
action. We start by remembering that the matter Hamiltonian density is
obtained after performing a Legendre transform

$\displaystyle\mathcal{H}_{mat}=\pi\partial_{t}\phi-\sqrt{-g}K,$ (170)

where $\sqrt{-g}K$ is the Lagrangian density and we have assumed it to be of
the form

$\displaystyle
K=K(\phi,u),~{}~{}u=\partial_{\mu}\phi~{}\partial^{\mu}\phi=g^{\mu\nu}\partial_{\mu}\phi~{}\partial_{\nu}\phi,$
(171)

so that $K$ depends exclusively on Lorentz invariant combinations of the field
and its derivatives up to second order. $\pi$ is the conjugate momentum
associated to the field $\phi$ and is defined as

$\displaystyle\pi=\sqrt{-g}\frac{\partial K}{\partial(\partial_{t}\phi)}$
(172)

Let us now perform some preliminary calculations:

$\displaystyle\left\\{\pi(x)\partial_{t}\phi(x),\phi(x^{\prime})\right\\}$
$\displaystyle=-\partial_{t}\phi(x)\delta(x-x^{\prime})-\pi(x)\frac{\delta\partial_{t}\phi(x)}{\delta\pi(x^{\prime})}$
(173)
$\displaystyle\left\\{\pi(x)\partial_{t}\phi(x),\pi(x^{\prime})\right\\}$
$\displaystyle=\pi(x)\frac{\delta\partial_{t}\phi(x)}{\delta\phi(x^{\prime})}$
$\displaystyle\left\\{\sqrt{-g}(x)K(x),\phi(x^{\prime})\right\\}$
$\displaystyle=-\sqrt{-g}\pi(x)\frac{\delta\partial_{t}\phi(x)}{\delta\pi(y)}$
$\displaystyle\left\\{\sqrt{-g}(x)K(x),\pi(x^{\prime})\right\\}$
$\displaystyle=\sqrt{-g}\left.\frac{\delta
K(x)}{\delta\phi(x^{\prime})}\right|_{\partial_{t}\phi}+\pi(x)\frac{\delta\partial_{t}\phi(x)}{\delta\phi(x^{\prime})},$

where we used the shorthand notation $K(x)=K(\phi(x),u(x))$ and
$\partial_{t}\phi(x)$ is determined by inverting equation (172), the vertical
bar on the first term on the RHS denoting that the functional derivative is
taken at constant $\partial_{t}\phi(x)$. With these results we have

$\displaystyle\left\\{\mathcal{H}_{mat}(x),\phi(x^{\prime})\right\\}=-\partial_{t}\phi(x)\delta(x-x^{\prime})$
(174)
$\displaystyle\left\\{\mathcal{H}_{mat}(x),\pi(x^{\prime})\right\\}=-\sqrt{-g}\left.\frac{\delta
K}{\delta\phi(x^{\prime})}\right|_{\partial_{t}\phi}.$

We now come to the key trick that is used to perform the calculation. Notice
that the matter part of the $\mathcal{C}[N]$ constraints, that is
$\mathcal{C}_{mat}[N]$, can be obtained from $\int dx\mathcal{H}_{mat}(x)$ by
the substitution

$\displaystyle e_{t}^{a}\rightarrow Ne_{x}^{b}\varepsilon^{a}_{b}.$ (175)

However, due to the arguments in appendix B, $\mathcal{H}_{mat}$ is linear in
$e_{t}^{a}$ and therefore

$\mathcal{C}_{mat}[N]=\int dx\mathcal{H}_{mat}\rvert_{e_{t}^{a}\rightarrow
Ne_{x}^{b}\varepsilon^{a}_{b}}=\int
dxN(x)\mathcal{H}_{mat}\rvert_{e_{t}^{a}\rightarrow
e_{x}^{b}\varepsilon^{a}_{b}}.$ (176)

Thus, in order to calculate
$\left\\{\mathcal{C}_{matter}[N],\mathcal{C}_{matter}[M]\right\\}$ we start by
calculating

$\displaystyle\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yM(y)\mathcal{H}_{mat}(y)\right\\}=\int\mathrm{d}x\mathrm{d}yN(x)M(y)\left\\{\mathcal{H}_{mat}(x),\mathcal{H}_{mat}(y)\right\\}$
$\displaystyle=\int\mathrm{d}x\mathrm{d}yN(x)M(y)\mathrm{d}z\left\\{\frac{\delta\mathcal{H}_{mat}(x)}{\delta\phi(z)}\frac{\delta\mathcal{H}_{mat}(y)}{\delta\pi(z)}-\frac{\delta\mathcal{H}_{mat}(x)}{\delta\pi(z)}\frac{\delta\mathcal{H}_{mat}(y)}{\delta\phi(z)}\right\\}$
(177)

and we will later on evaluate this at $e_{t}^{a}\rightarrow
e_{x}^{b}\varepsilon^{a}_{b}$. Using now the results (174), we get

$\displaystyle\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yN(y)\mathcal{H}_{mat}(y)\right\\}$
$\displaystyle=\int
N(x)N(y)\left[\delta(x-z)\partial_{t}\phi(x)\sqrt{-g}(y)\frac{\delta
K(y)}{\delta\phi(z)}-(x\leftrightarrow y)\right],$ (178)

where we have now omitted the measures of integration. The functional
derivative appearing in the previous equation can be calculated as

$\displaystyle\frac{\delta
K(x)}{\delta\phi(y)}=\partial_{\phi}K(x)\delta(x-y)+\partial_{u}K(x)\left.\frac{\delta
u(x)}{\delta\phi(y)}\right|_{\partial_{t}\phi}$ (179)

When inserting this expression into (C), we can see that the contribution
proportional to the first term vanishes due to the antisymmetrization, so we
are left with

$\displaystyle\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yM(y)\mathcal{H}_{mat}(y)\right\\}$
$\displaystyle=\int
N(x)M(y)\left[\delta(x-z)\partial_{t}\phi(x)\sqrt{-g}(y)\partial_{u}K(y)\left.\frac{\delta
u(y)}{\delta\phi(z)}\right|_{\partial_{t}\phi}-(x\leftrightarrow y)\right].$
(180)

Since $u=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$, we have

$\displaystyle\left.\frac{\delta
u(y)}{\delta\phi(z)}\right|_{\partial_{t}\phi}=2g^{01}\partial_{t}\phi(y)\partial_{y}\delta(y-z)+2g^{11}\partial_{y}\phi(y)\partial_{y}\delta(y-z)$
(181)

From the expression for the metric in terms of the dyad
$g_{\mu\nu}=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b}$, we then have

$\displaystyle
g^{\mu\nu}=\frac{1}{g}\begin{pmatrix}e_{x}^{a}e_{xa}&-e_{t}^{a}e_{xa}\\\
-e_{t}^{a}e_{xa}&e_{t}^{a}e_{ta}\end{pmatrix}$ (182)

We now evaluate at $e_{t}^{a}=e^{b}_{x}\varepsilon^{a}_{b}$. With this choice
of $e_{t}^{a}$, (181) becomes

$\displaystyle\left.\frac{\delta
u(y)}{\delta\phi(z)}\right|_{\partial_{t}\phi}=2g^{11}\partial_{y}\phi(y)\partial_{y}\delta(y-z)=2g^{00}\partial_{y}\phi(y)\partial_{y}\delta(y-z).$
(183)

Then, we have

$\displaystyle\left.\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yM(y)\mathcal{H}_{mat}(y)\right\\}\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon^{a}_{b}}=$
(184) $\displaystyle\int
N(x)M(y)\left[\delta(x-z)\partial_{t}\phi(x)\sqrt{-g}(y)\partial_{u}K(y)\left.\frac{\delta
u(y)}{\delta\phi(z)}\right|_{\partial_{t}\phi}-(x\leftrightarrow y)\right]$
$\displaystyle\int
N(x)M(y)\left[\delta(x-z)\partial_{t}\phi(x)\sqrt{-g}(y)\partial_{u}K(y)2g^{00}\partial_{y}\phi(y)\partial_{y}\delta(y-z)-(x\leftrightarrow
y)\right]$

Integrating over the Dirac delta (keeping in mind that only the terms
involving the derivatives of $N(x)$ or $M(y)$ will survive due to the
antisymmetrizacion) and then integrating on $z$ and on one of the other
spatial variables, we obtain

$\displaystyle\left.\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yM(y)\mathcal{H}_{mat}(y)\right\\}\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon^{a}_{b}}=$
(185)
$\displaystyle\int(N\partial_{x}M-M\partial_{x}N)\left[\sqrt{-g}~{}\partial_{u}K~{}\partial_{t}\phi~{}2~{}g^{00}\partial_{x}\phi\right].$

Let us now recall the definition of $\pi$:

$\displaystyle\pi=\sqrt{-g}\frac{\partial
K}{\partial(\partial_{t}\phi)}=\sqrt{-g}~{}\partial_{u}K\frac{\partial
u}{\partial(\partial_{t}\phi)}=\sqrt{-g}\partial_{u}K(2g^{00}\partial_{t}\phi+2g^{01}\partial_{x}\phi).$
(186)

With the choice of $e_{t}^{a}=e_{x}^{b}\varepsilon^{a}_{b}$, we have
$g^{01}=0$, so (185) becomes

$\displaystyle\left.\left\\{\int\mathrm{d}xN(x)\mathcal{H}_{mat}(x),\int\mathrm{d}yM(y)\mathcal{H}_{mat}(y)\right\\}\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon^{a}_{b}}$
$\displaystyle=\int(N\partial_{x}M-M\partial_{x}N)\pi\partial_{x}\phi$ (187)
$\displaystyle=\mathcal{C}_{x,mat}[N\partial_{x}M-M\partial_{x}N]$

where in the last equality we have identified the matter contribution to the
$\mathcal{C}_{x}$ constraint.
Finally, it is possible to verify that the last two terms in (165) vanish. In
order to see this, it is enough to note that

$\displaystyle\left\\{\mathcal{C}_{grav}[N],\mathcal{C}_{mat}[M]\right\\}$

will be proportional to

$\displaystyle\left\\{e_{x}^{a}\varepsilon^{b}_{a}\varepsilon_{bc}\partial_{x}X^{c}(x),\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)(y)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}\right\\}$
(188)

Since the second term depends on the gravitational variables only through the
dyad, we can write the previous term as

$\displaystyle
e_{x}^{a}\varepsilon^{b}_{a}\varepsilon_{bc}\left\\{\partial_{x}X^{c}(x),\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)(y)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}\right\\}=\partial_{x}\delta(x-y)e_{x}^{a}\varepsilon^{b}_{a}\frac{\delta\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\delta
e^{b}_{x}}$ (189)

The function $\pi\partial_{t}-\sqrt{-g}K$ depends on $e^{b}_{x}$ both
explicitly and through $\partial_{t}\phi$, therefore

$\displaystyle\frac{\delta\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\delta
e^{b}_{x}}$
$\displaystyle=\left.\left(\frac{\partial\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\partial
e^{b}_{x}}\right)\right|_{\partial_{t}\phi}$ (190)
$\displaystyle+\left.\left(\frac{\partial\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\partial\left(\partial_{t}\phi\right)}\right)\right|_{\pi}\frac{\partial\left.(\partial_{t}\phi)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\partial
e^{b}_{x}}$

The second term vanishes when taking the derivative with respect to
$\partial_{t}\phi$ while holding $\pi$ fixed by virtue of the properties of
the Legendre transform. We are then left with

$\displaystyle\left.\left(\frac{\partial\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\partial
e^{b}_{x}}\right)\right|_{\partial_{t}\phi}$ (191)

The first term in (191) vanishes since the derivative is taken while holding
$\partial_{t}\phi$ fixed. Moreover, keeping in mind that we are also
evaluating on $e^{a}_{t}=e^{b}_{x}\varepsilon^{a}_{b}$, using the chain rule,
we have that

$\displaystyle\left.\left(\frac{\partial\left.\left(-\sqrt{-g}K\right)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}}{\partial
e^{b}_{x}}\right)\right|_{\partial_{t}\phi}\propto\frac{\partial(\eta_{ac}e_{x}^{a}e_{x}^{c})}{\partial
e^{b}_{x}}$ (192)

When holding $\partial_{t}\phi$ fixed and evaluating on
$e^{a}_{t}=e^{b}_{x}\varepsilon^{a}_{b}$, the term $-\sqrt{-g}K$ can be
regarded as a function of $e_{x}^{a}e_{x}^{b}\eta_{ab}$. This allows us to
obtain the following result for (188)

$\displaystyle\left\\{e_{x}^{a}\varepsilon^{b}_{a}\varepsilon_{bc}\partial_{x}X^{c}(x),\left.\left(\pi\partial_{t}\phi-\sqrt{-g}K\right)(y)\right|_{e_{t}^{a}=e_{x}^{b}\varepsilon_{b}^{a}}\right\\}$
$\displaystyle\propto
e^{a}_{x}\varepsilon^{b}_{a}\frac{\partial(\eta_{cd})e^{c}_{x}e^{d}_{x}}{\partial
e^{b}_{x}}$ (193) $\displaystyle\propto
e^{a}_{x}\varepsilon^{b}_{a}\eta_{bc}e^{c}_{x}=e^{a}_{x}e^{b}_{x}\varepsilon_{ab}=0$

So that the last two terms in (165) vanish, giving us the desired result

$\displaystyle\left\\{\mathcal{C}[N],\mathcal{C}[M]\right\\}=\mathcal{C}_{x}[N\partial_{x}M-M\partial_{x}N]$
(194)

## References

  * (1) F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B 915 (2017) 363–383,  1608.05499.
  * (2) A. Cavaglià, S. Negro, I. M. Szécsényi, and R. Tateo, “$T\bar{T}$-deformed 2D Quantum Field Theories,” JHEP 10 (2016) 112,  1608.05534.
  * (3) S. Dubovsky, V. Gorbenko, and M. Mirbabayi, “Asymptotic fragility, near AdS 2 holography and $T\bar{T}$,” Journal of High Energy Physics 2017 (2017), no. 9, 1–37.
  * (4) S. Dubovsky, V. Gorbenko, and G. Hernández-Chifflet, “$T\bar{T}$ partition function from topological gravity,” Journal of High Energy Physics 2018 (2018), no. 9, 1–23.
  * (5) E. A. Mazenc, V. Shyam, and R. M. Soni, “A $T\bar{T}$ Deformation for Curved Spacetimes from 3d Gravity,”  1912.09179.
  * (6) R. L. Arnowitt, S. Deser, and C. W. Misner, “The Dynamics of general relativity,” Gen. Rel. Grav. 40 (2008) 1997–2027,  gr-qc/0405109.
  * (7) M. Bojowald, Canonical Gravity and ApplicationsCosmology, Black Holes, and Quantum Gravity.  Cambridge University Press, 12, 2010.
  * (8) A. J. Tolley, “$T\overline{T}$ deformations, massive gravity and non-critical strings,” JHEP 06 (2020) 050,  1911.06142.
  * (9) C. Itzykson and J. B. Zuber, Quantum Field Theory.  International Series In Pure and Applied Physics. McGraw-Hill, New York, 1980.
  * (10) S. Weinberg, The quantum theory of fields.  Cambridge university press, 1995.
  * (11) C. T. Marc Henneaux, Quantization of gauge systems.  Princeton University press, 1991.
  * (12) E. Witten, “Open Strings On The Rindler Horizon,” JHEP 01 (2019) 126,  1810.11912.
  * (13) L. Brink and M. Henneaux, PRINCIPLES OF STRING THEORY.  1988\.
  * (14) R. Jackiw, “Two lectures on two-dimensional gravity,” arXiv preprint gr-qc/9511048 (1995).
  * (15) C. Rovelli, Quantum gravity.  Cambridge university press, 2004.
  * (16) T. Thiemann, Modern canonical quantum general relativity.  Cambridge University Press, 2008.
  * (17) R. Conti, L. Iannella, S. Negro, and R. Tateo, “Generalised Born-Infeld models, Lax operators and the $\mathrm{T}\overline{\mathrm{T}}$ perturbation,” JHEP 11 (2018) 007,  1806.11515.
  * (18) S. Dubovsky, R. Flauger, and V. Gorbenko, “Solving the Simplest Theory of Quantum Gravity,” JHEP 09 (2012) 133,  1205.6805.
  * (19) N. Callebaut, J. Kruthoff, and H. Verlinde, “$T\overline{T}$ deformed CFT as a non-critical string,” JHEP 04 (2020) 084,  1910.13578.
  * (20) S. Frolov, “$T\overline{T}$ Deformation and the Light-Cone Gauge,” Proc. Steklov Inst. Math. 309 (2020) 107–126,  1905.07946.
  * (21) A. Ireland and V. Shyam, “$T\overline{T}$ deformed YM2 on general backgrounds from an integral transformation,” JHEP 07 (2020) 058,  1912.04686.
  * (22) L. Griguolo, R. Panerai, J. Papalini, and D. Seminara, “Exact $T\overline{T}$ deformation of two-dimensional Yang-Mills theory on the sphere,” JHEP 10 (2022) 134,  2207.05095.
  * (23) L. Griguolo, R. Panerai, J. Papalini, and D. Seminara, “The phase diagram of $T\overline{T}$-deformed Yang-Mills theory on the sphere,” JHEP 11 (2022) 078,  2209.06222.
  * (24) L. Santilli and M. Tierz, “Large N phase transition in $T\overline{T}$ -deformed 2d Yang-Mills theory on the sphere,” JHEP 01 (2019) 054,  1810.05404.
  * (25) V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50 (1994) 846–864,  gr-qc/9403028.
  * (26) M. Guica, “An integrable Lorentz-breaking deformation of two-dimensional CFTs,” SciPost Phys. 5 (2018), no. 5, 048,  1710.08415.
  * (27) A. Bzowski and M. Guica, “The holographic interpretation of $J\bar{T}$-deformed CFTs,” JHEP 01 (2019) 198,  1803.09753.
  * (28) S. Chakraborty, A. Giveon, and D. Kutasov, “$J\overline{T}$ deformed CFT2 and string theory,” JHEP 10 (2018) 057,  1806.09667.
  * (29) O. Aharony, S. Datta, A. Giveon, Y. Jiang, and D. Kutasov, “Modular covariance and uniqueness of $J\bar{T}$ deformed CFTs,” JHEP 01 (2019) 085,  1808.08978.
  * (30) L. Apolo and W. Song, “Strings on warped AdS3 via $\mathrm{T}\bar{\mathrm{J}}$ deformations,” JHEP 10 (2018) 165,  1806.10127.
  * (31) J. Aguilera-Damia, V. I. Giraldo-Rivera, E. A. Mazenc, I. Salazar Landea, and R. M. Soni, “A path integral realization of joint $J\overline{T}$, $T\overline{J}$ and $T\overline{T}$ flows,” JHEP 07 (2020), no. 07, 085,  1910.06675.
  * (32) G. Hernández-Chifflet, S. Negro, and A. Sfondrini, “Flow Equations for Generalized $T\overline{T}$ Deformations,” Phys. Rev. Lett. 124 (2020), no. 20, 200601,  1911.12233.
  * (33) J. Polchinski, “Evaluation of the One Loop String Path Integral,” Commun. Math. Phys. 104 (1986) 37.
  * (34) E. D’Hoker and D. H. Phong, “The Geometry of String Perturbation Theory,” Rev. Mod. Phys. 60 (1988) 917.
  * (35) E. Witten, “A Note On The Canonical Formalism for Gravity,”  2212.08270.
  * (36) J. B. Hartle and K. V. Kuchar, “Path integrals in parametrized theories: The Free relativistic particle,” Phys. Rev. D 34 (1986) 2323–2331.
  * (37) A. G. Cohen, G. W. Moore, P. C. Nelson, and J. Polchinski, “An Off-Shell Propagator for String Theory,” Nucl. Phys. B 267 (1986) 143–157.