# Refining the Cutoff 3d Gravity / $T\overline{T}$ Correspondence

Per Kraus  Ruben Monten  Konstantinos Roumpedakis

###### Abstract

Pure gravity in AdS3 is a theory of boundary excitations, most simply
expressed as a constrained free scalar with an improved stress tensor that is
needed to match the Brown–Henneaux central charge. Excising a finite part of
AdS gives rise to a static gauge Nambu–Goto action for the boundary graviton.
We show that this is the $T\overline{T}$ deformation of the infinite volume
theory, as the effect of the improvement term on the deformed action can be
absorbed into a field redefinition. The classical gravitational stress tensor
is reproduced order by order by the $T\overline{T}$ trace equation. We
calculate the finite volume energy spectrum in static gauge and find that the
trace equation imposes sufficient constraints on the ordering ambiguities to
guarantee agreement with the light-cone gauge prediction. The correlation
functions, however, are not completely fixed by the trace equation. We show
how both the gravitational action and the $T\overline{T}$ deformation allow
for finite improvement terms, and we match these to the undetermined total
derivative terms in Zamolodchikov’s point splitting definition of the
$T\overline{T}$ operator.

###### Contents

  1. 1 Introduction
  2. 2 Boundary theory at zero cutoff
    1. 2.1 Gravity solutions
    2. 2.2 BTZ branch: $a$ and $\bar{a}$ negative
    3. 2.3 $a$ and $\bar{a}$ positive
    4. 2.4 Thermal partition function
  3. 3 The stress tensor in cutoff gravity and $T\overline{T}$-deformed CFT
    1. 3.1 $T\overline{T}$ deformation and stress tensor improvement
    2. 3.2 Nambu-Goto action from the trace equation
    3. 3.3 Computing the improved stress tensor
  4. 4 Static vs light-cone gauge
    1. 4.1 Static gauge
    2. 4.2 Light-cone gauge
    3. 4.3 Spectrum of static gauge Hamiltonian
  5. 5 Spectrum
    1. 5.1 Perturbation theory
  6. 6 Correlation functions
    1. 6.1 Gravity side
    2. 6.2 Field theory side
  7. A Deformed action from the trace equation
  8. B Conversion to Lorentzian signature

##  1 Introduction

The fact that General Relativity involves the dynamics of spacetime geometry
implies that the problem of defining boundary conditions in gravity is bound
to be more subtle than in non-gravitational theories. Gravitational boundary
conditions are typically imposed asymptotically, for example at spacelike or
null infinity in asymptotically flat spacetimes, or on the conformal boundary
of anti-de Sitter spacetime. It is of interest to ask whether imposing
boundary conditions at a “finite cutoff boundary” makes sense, for example by
imposing a Dirichlet condition on the metric on a non-asymptotic timelike
boundary. One setting where this does seem to make sense — though not without
subtleties — is the case of pure AdS3 gravity. This example is also of special
interest due to its holographic relation to $T\overline{T}$ deformed CFT2 [1,
2, 3].

This problem was studied at both the classical and quantum level in [4, 5].
The focus was on the dynamics of the boundary gravitons, since in 3d gravity
these are the only propagating degrees of freedom [6, 7, 8]. A main result was
that these boundary gravitons are governed by a static gauge Nambu-Goto action
defined on the cutoff boundary. Obtaining this result involved making a highly
nontrivial and nonlocal field redefinition between the variables describing
the embedding of the boundary surface and those appearing in the Nambu-Goto
action. Quantization of this theory was initiated, revealing the need for
renormalization counterterms, which is not surprising given the well-known
issues involved in quantizing the square root Nambu-Goto action.

In this paper we clarify, consolidate, and extend these results in a way that
provides a satisfying agreement between the gravity and $T\overline{T}$
deformed CFT sides of the story. We now summarize the main results of our
analysis. Other references on cutoff AdS3 gravity include [9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21].

The general problem involves the classical and quantum theory of boundary
gravitons living on top of some background geometry, which could be global
AdS, a conical defect, or a BTZ black hole. We begin by considering the
asymptotic AdS boundary case. The boundary action, which can be obtained
either in the Chern-Simons [22, 23] or metric [4, 5] formulation of 3d
gravity, is given by the Alekseev-Shatashvili theory [24, 25]. After a field
redefinition, this becomes the standard action for a free scalar, but with a
few special features. First, the scalar is not periodic around the boundary
cylinder, but obeys a fixed winding condition that encodes the mass and
angular momentum of the background gravity solution. Second, the theory is
subject to a gauge symmetry whose origin lies in the isometry group of the
background geometry. Finally, the stress tensor is not canonical but has an
improvement term, which is needed in order to realize the Brown-Henneaux
central charge. We carefully go though the quantization of this theory, which
has some subtleties depending on the form of the winding condition, showing
how to arrive at unitary representations of the Virasoro algebra.

It is well-known that the Nambu-Goto action is the $T\overline{T}$ deformed
action for an ordinary free scalar with canonical stress tensor [2]. As noted
above, the scalar arising from gravity has a non-canonical stress tensor, but
which does obey the trace equation $\Tr T=\frac{\lambda}{2\pi}\det T$ that
defines the $T\overline{T}$ flow. However, we show that adding an improvement
term to the stress tensor that preserves the trace equation does not change
the deformed action, possibly after performing a field redefinition, which
explains why the Nambu-Goto action also arises in the cutoff gravity context.

Assuming the standard Gibbons-Hawking-York plus cosmological constant boundary
term, the boundary stress tensor at finite cutoff obeys the trace equation
$\Tr T=\frac{\lambda}{2\pi}\det T$, where $\lambda$ is proportional to the
radial cutoff location $r_{c}$. This equation can be used to fix the explicit
form of the non-canonical stress tensor that appears in the Nambu-Goto action.
In particular, starting from the form of the stress tensor at $\lambda=0$,
including its improvement term, one can work out the stress tensor order by
order in $\lambda$ by imposing the trace equation. We carry this out, showing
how to reproduce the results obtained in [4, 5]. This stress tensor is
nonlocal, in the sense that each successive order in $\lambda$ involves terms
with more derivatives (we do not have a closed form expression to all orders).
This nonlocal property aligns with previous results, such as the “Dirac
string” picture developed in [26].

Having understood the emergence of the Nambu-Goto action from gravity at the
classical level, we turn to a discussion of its quantization.111 The general
problem of quantizing the Nambu-Goto action in general target space dimension
$D$ has received much attention in the context of “effective strings”, e.g.
[27, 28, 29, 30, 31, 32], and the works of Dubovsky et. al. developed the
connection to the $T\overline{T}$ deformation. One of the key facts about
$T\overline{T}$ deformed CFT is that the energy spectrum of the deformed
theory on the circle is related in a simple way to that of the undeformed CFT.
In particular, $E_{n}~{}\rightarrow~{}(1-\sqrt{1-2\lambda
E_{n}+\lambda^{2}P_{n}^{2}})/\lambda$ where $E_{n}$ and $P_{n}$ are the energy
and momentum of the state $n$ respectively. One approach to obtain this, which
we review, is to view the Nambu-Goto action as the gauge fixed version of the
coordinate invariant Nambu-Goto action for a string in a flat $d=3$ target
spacetime, which can alternatively be expressed in light cone gauge where the
action is quadratic and hence easily quantized. At the classical level this is
a canonical transformation [33]; quantum mechanically the equivalence is
somewhat murkier given that we are not in the critical dimension $d=26$.222The
usual Lorentz anomaly is trivially absent in $d=3$, so the procedure is
perhaps justified in this case as well. Regardless, this route does lead to
the correct spectrum. We also discuss the computation of the spectrum by
applying standard time-independent quantum mechanical perturbation theory.
Without additional input, this leads to divergent sums over intermediate
states, rendering the spectrum ambiguous, as one would expect for a non-
renormalizable theory. The additional ingredient needed to fix the spectrum is
the factorized form of the trace equation $\langle
T_{z{\bar{z}}}\rangle=-\frac{\lambda}{2\pi}(\langle T_{zz}\rangle\langle
T_{{\bar{z}}{\bar{z}}}\rangle-\langle T_{z{\bar{z}}}\rangle\langle
T_{z{\bar{z}}}\rangle)$, where the expectation value is taken in an
energy/momentum eigenstate. Of course, this is no surprise since this is the
key relation used in [1] to obtain the spectrum.

We finally turn to a discussion of boundary correlation functions, both of the
elementary fields and of the stress tensor. The Nambu-Goto action is non-
renormalizable, yet viewed as a $T\overline{T}$ deformed CFT its energy
spectrum is unambiguous. To what extent does this feature carry over to
correlation functions? Here the gravity picture provides a useful guide. Pure
3d gravity is renormalizable: using the equations of motion, all divergences
can be absorbed into the cosmological constant term [34]. This leads to the
expectation that on-shell correlators of elementary fields should be
unambiguous. Indeed, existing perturbative computations of correlators are
rendered finite by counterterms that vanish on-shell. The S-matrix is
therefore unambiguous, which is a well-known feature of $T\overline{T}$
deformed theories.[1, 2, 29]

For the stress tensor the story is a bit different due to the inherent
ambiguity in fixing the improvement terms. For the gravity theory with a
cutoff boundary nothing prevents us from adding a boundary term of the form
$\int\\!d^{2}x\sqrt{h}R^{(2)}(h)Z(K)$ where $R^{(2)}(h)$ is the Ricci
curvature of the boundary and $Z(K)$ some arbitrary function of the extrinsic
curvature.333This term vanishes in the asymptotically AdS limit, which is why
it is usually not considered. The effect of this term is to add an improvement
term to the stress tensor. In a general renormalizable QFT the stress tensor
is renormalized by such improvement terms [35], and the same is true here.
Although the stress tensor is ambiguous this does not affect the usefulness of
the equation $\langle T_{z{\bar{z}}}\rangle=-\frac{\lambda}{2\pi}(\langle
T_{zz}\rangle\langle T_{{\bar{z}}{\bar{z}}}\rangle-\langle
T_{z{\bar{z}}}\rangle\langle T_{z{\bar{z}}}\rangle$ in determining the energy
spectrum because the improvement terms have vanishing expectation value. One
might ask whether the form of the quantum stress tensor could be fixed by
imposing the operator equation $\Tr T=\frac{\lambda}{2\pi}\det T$, but we show
that this is not the case. The map to light-cone gauge also does not fix these
ambiguities; since it involves a state-dependent coordinate transformation, it
does not map correlation functions of local operators in light-cone gauge to
static gauge correlation functions of local operators. We conclude that
although off-shell correlators of the elementary fields and the stress tensor
can be computed in perturbation theory their definition is subject to
ambiguities. This might be taken as an indication that these are the wrong
observables to be looking at if the goal is to define the theory at a non-
perturbative level.

The rest of this paper is organized as follows. In section 2 we review the
boundary gravity theory in the asymptotically AdS case, and then discuss its
quantization and Hilbert space in detail. In section 3 we discuss the
construction of the boundary stress tensor, emphasizing how its form can be
fixed order by order by imposing the trace equation. The transformation to
light cone gauge is reviewed in section 4, and in section 5 we comment on the
use of ordinary perturbation theory. Correlation functions are discussed in
section 6 and two appendices contain some technical details.

##  2 Boundary theory at zero cutoff

In this section we review the boundary theory of pure three-dimensional
gravity without a cutoff. The boundary theory reduces to a free scalar field
with an improved stress tensor of linear dilaton (or background charge) type,
and a gauge symmetry. This will be particularly useful for later sections
where we study the $T\bar{T}$ deformation of this theory.

###  2.1 Gravity solutions

We start with the three-dimensional Euclidean action

$S=-\frac{1}{16\pi
G}\int\differential^{3}x\sqrt{g}\left(R+\frac{2}{\ell^{2}}\right)+S_{\rm
bndy}~{},$ (1)

where $S_{\rm bndy}$ is a boundary contribution that ensures a well-defined
variational principle . It is well-known that in three dimensions there are no
propagating gravitons in the bulk and the theory has only boundary degrees of
freedom. We consider solutions to Einstein equations with asymptotic AdS3
boundary conditions [7],

$\displaystyle
ds^{2}=\frac{dr^{2}}{4r^{2}}+\frac{1}{r}\left[dz-\frac{6}{c_{0}}rT_{{\bar{z}}{\bar{z}}}({\bar{z}})d{\bar{z}}\right]\left[d{\bar{z}}-\frac{6}{c_{0}}rT_{zz}(z)dz\right]~{},$
(2)

where

$\displaystyle c_{0}=\frac{3\ell}{2G}~{},$ (3)

is the Brown-Henneaux central charge [6]. We are working in Euclidean
signature with $z=x+it$, and $x\cong x+2\pi$ i.e., the boundary has the
topology of a cylinder. The free functions
$(T_{zz}(z),T_{{\bar{z}}{\bar{z}}}({\bar{z}}))$ can be identified as
components of the boundary stress tensor. The mass and angular momentum are

$\displaystyle M$
$\displaystyle=\int_{0}^{2\pi}\frac{dx}{2\pi}T_{tt}=-\int_{0}^{2\pi}\frac{dx}{2\pi}(T_{zz}+T_{{\bar{z}}{\bar{z}}})~{},$
(4) $\displaystyle J$
$\displaystyle=\int_{0}^{2\pi}\frac{dx}{2\pi}iT_{xt}=-\int_{0}^{2\pi}\frac{dx}{2\pi}(T_{zz}-T_{{\bar{z}}{\bar{z}}})~{}.$
(5)

The stationary and rotationally symmetric solutions are written in terms of
two constants $(a,\bar{a})$ as

$\displaystyle T_{zz}=\frac{c_{0}a}{24}~{},\quad
T_{{\bar{z}}{\bar{z}}}=\frac{c_{0}\bar{a}}{24}~{},$ (6)

and carry

$\displaystyle M=-\frac{c_{0}}{24}(a+\bar{a}),\quad
J=-\frac{c_{0}}{24}(a-\bar{a})~{}.$ (7)

Global AdS3 corresponds to $a=\bar{a}=1$. Upon continuation to Lorentzian
signature there are two braches of “healthy” solutions: rotating BTZ black
holes and conical defects; see e.g. [36]. Rotating BTZ black holes are given
by $a,\bar{a}<0$, with the extremal case $M=|J|$ occurring when one of
$(a,\bar{a})$ vanishes. Conical defects have $0<a=\bar{a}<1$. Taking
$a=\bar{a}>1$ yields “conical excess solutions” whose energy lie below global
AdS3 and correspond to non-unitary representations of Virasoro. Taking
$a\neq\bar{a}$ with one or both $(a,\bar{a})$ being positive can be seen to
yield naked closed timelike curves or singular horizons. More general
solutions that are “dressed with boundary gravitons” are obtained by taking

$\displaystyle
T_{zz}=\frac{c_{0}}{12}\left(\frac{a}{2}F^{\prime}(z)^{2}+\\{F(z),z\\}\right)~{},\quad
T_{{\bar{z}}{\bar{z}}}=\frac{c_{0}}{12}\left(\frac{\bar{a}}{2}\overline{F}^{\prime}({\bar{z}})^{2}+\\{\overline{F}({\bar{z}}),{\bar{z}}\\}\right)~{},$
(8)

where the Schwarzian derivative is

$\displaystyle\\{F(z),z\\}=\frac{F^{\prime\prime\prime}}{F^{\prime}}-\frac{3F^{\prime\prime
2}}{2F^{\prime 2}}~{}.$ (9)

The functions $(F,\overline{F})$ are each elements of diff$(S^{1})$, i.e.,
maps from the circle to itself. They are correspondingly monotonic and obey
the winding conditions

$F(x+2\pi,t)=F(x,t)+2\pi,\quad\bar{F}(x+2\pi,t)=\bar{F}(x,t)+2\pi~{}.$ (10)

Solutions with the same $(a,\bar{a})$ lie on a common diff orbit. These orbits
are symplectic manifolds, and the phase space action governing them is the so-
called Alekseev-Shatashvili action

$S=-\frac{c_{0}}{24\pi}\int\differential^{2}x\left[aF^{\prime}\partial_{\bar{z}}F-\left(\frac{1}{F^{\prime}}\right)^{{}^{\prime\prime}}\partial_{\bar{z}}F+\bar{a}\bar{F}^{\prime}\partial_{z}\bar{F}-\left(\frac{1}{\bar{F}^{\prime}}\right)^{{}^{\prime\prime}}\partial_{z}\bar{F}\right]~{}.$
(11)

This action has been obtained from gravity in both the Chern-Simons and metric
formulations. In this action $(F,\overline{F})=(F(x,t),\overline{F}(x,t))$ are
each arbitrary functions of $(x,t)$, subject to the winding and monotonicity
constraints. Primes denote $x$-derivatives and

$\partial_{z}=\frac{1}{2}(\partial_{x}-i\partial_{t}),\quad\partial_{\bar{z}}=\frac{1}{2}(\partial_{x}+i\partial_{t})~{}.$
(12)

In addition, (11) has the gauge redundancy444For the special value $a=1$ the
gauge redundancy is enhanced to $PSL(2,R)\times PSL(2,R)$ [23].

$F(x,t)\sim
F(x,t)+\epsilon(t),\quad\bar{F}(x,t)\sim\bar{F}(x,t)+\bar{\epsilon}(t)~{}.$
(13)

These gauge redundancies arise from the fact that they are invariances of the
stress tensor components. One can easily check that the Lagrangian transforms
by a total spatial derivative.

We now wish to quantize this theory. In the following we are going to treat
the cases of positive and negative $(a,\bar{a})$ separately. In both cases the
Hilbert space is a single unitary representation of the Vir $\times$ Vir
algebra, with lowest weight states determined by $(a,\bar{a})$. We refrain
from considering cases with opposite signs for $(a,\bar{a})$, since the
corresponding gravity solutions are pathological.

###  2.2 BTZ branch: $a$ and $\bar{a}$ negative

We write

$\displaystyle a=-b^{2}~{},\quad\bar{a}=-\bar{b}^{2}~{}.$ (14)

It was noted in [24, 25] that the field redefinition

$\left(e^{bF}\right)^{\prime}=b\;e^{f},\quad\left(e^{\bar{b}\bar{F}}\right)^{\prime}=\bar{b}\;e^{\bar{f}}~{},$
(15)

brings the action (11) to the simple form

$S=\frac{c_{0}}{24\pi}\int\differential^{2}x\left(f^{\prime}\partial_{\bar{z}}f+\bar{f}^{\prime}\partial_{z}\bar{f}\right)~{}.$
(16)

This is the action of a free scalar field in the first-order formalism. From
(15) we can read of the periodicity of the new variables

$f(x+2\pi,t)=f(x,t)+2\pi
b,\quad\bar{f}(x+2\pi,t)=\bar{f}(x,t)+2\pi\bar{b}~{},$ (17)

while the gauge redundancy is

$f(x,t)\sim
f(x,t)+b\epsilon(t),\quad\bar{f}(x,t)\sim\bar{f}(x,t)+\bar{b}\bar{\epsilon}(t)~{}.$
(18)

The final element needed to describe the classical boundary theory is the
boundary stress tensor. In [23] it was shown to be

$T=\frac{c_{0}}{12}\left(-\frac{1}{2}f^{{}^{\prime}2}+f^{\prime\prime}\right),\quad\bar{T}=\frac{c_{0}}{12}\left(-\frac{1}{2}\bar{f}^{{}^{\prime}2}+\bar{f}^{\prime\prime}\right)~{}.$
(19)

To make the connection with the free scalar theory more explicit let us define

$\phi=\sqrt{\frac{c_{0}}{48\pi}}(f+\bar{f}),\quad\Pi=\sqrt{\frac{c_{0}}{48\pi}}(f^{\prime}-\bar{f}^{\prime})~{}.$
(20)

The action (16) then takes the form

$S=\int\differential^{2}x\left(i\dot{\phi}\Pi+\frac{1}{2}\phi^{{}^{\prime}2}+\frac{1}{2}\Pi^{2}\right)~{}.$
(21)

Naively integrating out $\Pi$ leads to the usual free scalar Lagrangian.
However, this is not quite correct because (17) leads to the constraint

$\int_{0}^{2\pi}\differential x\;\Pi=\sqrt{\frac{\pi
c_{0}}{12}}(b-\bar{b})~{},$ (22)

which must be taken care of before employing the Euler-Lagrange equations.
This is the first-class constraint that generates the gauge symmetry (18):

$\phi(x,t)\rightarrow\phi(x,t)+\varepsilon(t),\quad\Pi(x,t)\rightarrow\Pi(x,t)~{},$
(23)

where
$\varepsilon(t)=\sqrt{\frac{c_{0}}{48\pi}}(b\epsilon(t)+\bar{b}\bar{\epsilon}(t))$.
The transformation of the Lagrangian (21) is a total time derivative by virtue
of the constraint (22). To proceed, we first need to pick a gauge, solve the
constraint, and then use the Euler-Lagrange equations. The net effect will be
that the zero modes of $\phi$ and $\Pi$ are fixed. Let us explain this
further. We start by expanding the fields into modes

$\displaystyle\phi$
$\displaystyle=\phi_{0}(t)+(b+\bar{b})\sqrt{\frac{c_{0}}{48\pi}}x+\frac{i}{2\sqrt{\pi}}\sum_{n\neq
0}\frac{\phi_{n}(t)}{n}e^{inx}~{},$ (24) $\displaystyle\Pi$
$\displaystyle=\pi_{0}(t)+\sum_{n\neq 0}\pi_{n}(t)e^{inx}~{},$ (25)

where we used (17) to fix the winding around the spatial circle. The
constraint (22) fixes $\pi_{0}=(b-\bar{b})\sqrt{c_{0}/48\pi}$. The gauge
symmetry can be used to choose $\dot{\phi}_{0}=i\pi_{0}$, i.e., the on-shell
value it would have in the absence of the constraint (22). We can use the
Euler-Lagrange equations for the remaining non-zero modes. Hence, we see that
we end up with a free scalar field with fixed winding and zero-modes. The
equations of motion then lead to the free field expansion

$\displaystyle\phi$
$\displaystyle=\sqrt{\frac{c_{0}}{48\pi}}[(b+\bar{b})x+i(b-\bar{b})t]+\frac{i}{2\sqrt{\pi}}\sum_{n\neq
0}\frac{\alpha_{n}}{n}e^{in(x+it)}-\frac{i}{2\sqrt{\pi}}\sum_{n\neq
0}\frac{\bar{\alpha}_{n}}{n}e^{-in(x-it)}~{},$ (26) $\displaystyle\Pi$
$\displaystyle=(b-\bar{b})\sqrt{\frac{c_{0}}{48\pi}}-\frac{1}{2\sqrt{\pi}}\sum_{n\neq
0}\alpha_{n}e^{in(x+it)}+\frac{1}{2\sqrt{\pi}}\sum_{n\neq
0}\bar{\alpha}_{n}e^{-in(x-it)}~{}.$ (27)

One can use (19) to write down a gauge-invariant expression for the stress
tensor in terms of spatial derivatives of $\phi$ and $\Pi$. However, the gauge
choice we made above $(\dot{\phi}_{0}=i\pi_{0})$ allows as to use the usual
form

$-\frac{1}{2\pi}T_{\mu\nu}=\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}\delta_{\mu\nu}(\partial\phi)^{2}-\sqrt{\frac{c_{0}}{12\pi}}(\partial_{\mu}\partial_{\nu}-\delta_{\mu\nu}\partial^{2})\phi~{}.$
(28)

keeping Lorentz symmetry manifest. Thus, we conclude that the boundary theory
includes also an improvement term for the stress tensor. As is well-known this
affects the representations of the boundary Virasoro algebra. The spectrum of
the theory is the same with or without the improvement terms but it organizes
itself differently into Virasoro modules.

The modes of the stress tensor in (19) are

$L_{n}=-\oint\frac{\differential
x}{2\pi}\;e^{-in(x+it)}T_{zz}=\frac{1}{2}\sum_{k\neq
0}\alpha_{n-k}\alpha_{k}+(in-b)\sqrt{\frac{c_{0}}{12}}\alpha_{n}+\frac{c_{0}b^{2}}{24}\delta_{n,0}~{},$
(29)

Conjugation acts as

$\alpha_{n}^{\dagger}=\alpha_{-n}^{\dagger}\Rightarrow
L_{n}^{\dagger}=L_{-n}^{\dagger}~{}.$ (30)

So far the discussion was classical. To quantize the theory we need to impose
the usual commutation relations

$\left[\phi(t,x),\Pi(t,x^{\prime})\right]=-i\delta(x-x^{\prime}),\quad\left[\Pi(t,x),\Pi(t,x^{\prime})\right]=0=\left[\phi(t,x),\phi(t,x^{\prime})\right]~{},$
(31)

which lead to

$\left[\alpha_{n},\alpha_{m}\right]=n\delta_{n+m}=\left[\bar{\alpha}_{n},\bar{\alpha}_{m}\right],\quad\left[\alpha_{n},\bar{\alpha}_{m}\right]=0~{}.$
(32)

We also choose normal ordering of the operators $\alpha_{-k}\alpha_{k}$ for
$L_{0}$ in (29). Using the above, one can show that the $L_{n}$’s obey the
algebra

$\left[L_{n},L_{m}\right]=(n-m)L_{n+m}+\frac{1}{12}(n^{3}-n)\delta_{n+m}+\frac{c_{0}}{12}n^{3}~{},$
(33)

while the operators

$L^{\prime}_{n}=L_{n}+\frac{c_{0}}{24}\delta_{n,0}~{},$ (34)

satisfy the usual Virasoro algebra with central charge $c=c_{0}+1$,

$\left[L^{\prime}_{n},L^{\prime}_{m}\right]=(n-m)L^{\prime}_{n+m}+\frac{c_{0}+1}{12}(n^{3}-n)\delta_{n+m}~{}.$
(35)

We see that the improvement term in (28) shifts the central charge by $c_{0}$.
To analyze the Hilbert space, note that all the $\alpha_{n}$ modes can be
written as combinations of the $L^{\prime}_{n}$ using (29) recursively. Hence
the theory has a single primary state, namely the “vacuum” defined by

$\alpha_{n}\ket{0}=0,\quad n>0.$ (36)

The $L_{n}$ operators do not obey the usual Virasoro algebra. Only
$L^{\prime}_{n}$ do, and we conclude that the Hilbert space of the theory
consists of a single Virasoro module with respect to the $L^{\prime}_{n}$’s.
The primary state satisfies

$L^{\prime}_{0}\ket{0}=\frac{c_{0}(1+b^{2})}{24}\ket{0}~{},\quad\bar{L}^{\prime}_{0}\ket{0}=\frac{c_{0}(1+\bar{b}^{2})}{24}\ket{0}~{}.$
(37)

Notice that this state is not annihilated by $L_{-1}=L^{\prime}_{-1}$. From
(34) we see that the energy of the primary state is

$(L_{0}+\bar{L}_{0})\ket{0}=\frac{c_{0}(b^{2}+\bar{b}^{2})}{24}\ket{0}.$ (38)

###  2.3 $a$ and $\bar{a}$ positive

In this case

$\displaystyle a=b^{2}~{},\quad\bar{a}=\bar{b}^{2}~{},$ (39)

and we modify (15) to

$\left(e^{ibF}\right)^{\prime}=ib\;e^{f},\quad\left(e^{-i\bar{b}\bar{F}}\right)^{\prime}=-i\bar{b}\;e^{\bar{f}}~{}.$
(40)

As can be seen after Wick rotating to Minkowski space, both $F$ and $\bar{F}$
are Hermitian. From the above definition it then follows that $f$ and
$\bar{f}$ obey complicated reality properties which we analyze below.

One can check that this again leads to the free scalar action (16). However,
in this case the periodicity of the new fields is

$f(x+2\pi,t)=f(x,t)+2\pi ib,\quad\bar{f}(x+2\pi,t)=\bar{f}(x,t)-2\pi
i\bar{b}~{},$ (41)

while the gauge redundancy is

$f(x,t)\sim
f(x,t)+ib\epsilon(t),\quad\bar{f}(x,t)\sim\bar{f}(x,t)-i\bar{b}\bar{\epsilon}(t)~{}.$
(42)

As a result, $\phi$ has imaginary winding

$\displaystyle\phi(x+2\pi,t)\to\phi(x,t)+i\sqrt{\frac{c_{0}\pi}{12}}(b-\bar{b})\
,$ (43)

Since $f$ and $\bar{f}$ are not hermitian, $\phi$ also does not have this
property, hence the appearance of the imaginary winding. However, this does
not imply the doubling of its modes as in the case of a complex $\phi$. On the
contrary, $\phi$ has more complicated reality properties and, as we will see
later, this translates to an unconventional conjugation for its modes.

In this case the momentum $\Pi$ obeys the constraint

$\int_{0}^{2\pi}dx\;\Pi=i(b+\bar{b})\sqrt{\frac{c_{0}\pi}{12}}~{},$ (44)

while the gauge symmetry acts as before (23) where now
$\varepsilon(t)=\sqrt{\frac{c_{0}}{48\pi}}ib(\epsilon(t)-\bar{\epsilon}(t))$.
We can proceed as before and write the free field expansion

$\displaystyle\phi$
$\displaystyle=\sqrt{\frac{c_{0}}{48\pi}}\left[i(b-\bar{b})x-(b+\bar{b})t\right]+\frac{i}{2\sqrt{\pi}}\sum_{n\neq
0}\frac{\alpha_{n}}{n}e^{in(x+it)}-\frac{i}{2\sqrt{\pi}}\sum_{n\neq
0}\frac{\bar{\alpha}_{n}}{n}e^{-in(x-it)}~{},$ (45) $\displaystyle\Pi$
$\displaystyle=i(b+\bar{b})\sqrt{\frac{c_{0}}{48\pi}}-\frac{1}{2\sqrt{\pi}}\sum_{n\neq
0}\alpha_{n}e^{in(x+it)}+\frac{1}{2\sqrt{\pi}}\sum_{n\neq
0}\bar{\alpha}_{n}e^{-in(x-it)}~{}.$ (46)

In this case the modes of the stress tensor are

$L_{n}=\frac{1}{2}\sum_{k\neq
0}\alpha_{n-k}\alpha_{k}+i(n-b)\sqrt{\frac{c_{0}}{12}}\alpha_{n}-\frac{c_{0}b^{2}}{24}\delta_{n,0}~{}.$
(47)

As before, to quantize the theory we impose the commutation relations (31)
which again lead to (32). However, conjugation in this case needs to be
modified. To preserve the action of conjugation on the $L_{n}$’s

$L_{n}^{\dagger}=L_{-n}~{},$ (48)

we need to define

$\alpha_{n}^{\dagger}=\frac{n+b}{n-b}\alpha_{-n}-i\frac{nb}{n-b}\sqrt{\frac{12}{c_{0}}}\sum_{k\neq
0}\frac{1}{(b+k)(b-k-n)}\alpha_{-n-k}\alpha_{k}+\dots~{},$ (49)

where the dots denotes terms with more $\alpha_{n}$’s. Higher order terms are
accompanied with higher powers of $1/c_{0}$. For large $c_{0}$ one can check
that for the first few orders, (49) is consistent with

$\left[\alpha_{n},\alpha_{m}\right]=n\delta_{n+m}\Rightarrow\left[\alpha_{n}^{\dagger},\alpha_{m}^{\dagger}\right]=-n\delta_{n+m}~{}.$
(50)

Equation (49) is valid only for $0<b<1$, but not for global AdS with
$b=\bar{b}=1$ in which case there is additional gauge symmetry.4 It is
surprising that one is forced to consider this peculiar definition of
conjugation. It would be interesting to understand its geometric
interpretation from the bulk 3d gravity point of view.

Let’s now consider the norm of the state $\alpha_{-n}\ket{0}$ for $n>0$. Using
the above expression for $\phi^{\dagger}_{n}$ one can show that only the first
term contributes

$||\alpha_{-n}\ket{0}||^{2}=\bra{0}\alpha_{-n}^{\dagger}\alpha_{-n}\ket{0}||=\frac{n(n-b)}{n+b}~{}.$
(51)

We see that the spectrum is unitary only for $-1\leq b\leq 1$. At the
endpoints of this interval null states appear. Combining with the results of
the previous section we see that the spectrum is unitary only for
$a<1$.555Solutions in gravity with $a,\bar{a}>1$ correspond to “conical
excesses”, and are indeed associated with non-unitary representations [37,
38]. Note the slightly curious fact that we obtain unitary representations of
Virasoro in this range for $a\neq\bar{a}$ even though the corresponding
gravity solutions have naked closed timelike curves. The only novel aspect of
this parameter regime is the fact that $\phi$ in (45) is not Hermitian.

As in the case with negative $a$, one can similarly define (34) which obey the
usual Virasoro algebra. We then conclude that the Hilbert space of the theory
contains a single Virasoro module with

$L^{\prime}_{0}\ket{0}=\frac{c_{0}(1-b^{2})}{24}\ket{0}\
,\quad\bar{L}^{\prime}_{0}\ket{0}=\frac{c_{0}(1-\bar{b}^{2})}{24}\ket{0}.$
(52)

For $b,\bar{b}<1$ this is a unitary representation whose primary state has
energy

$(L_{0}+\bar{L}_{0})\ket{0}=-\frac{c_{0}(b^{2}+\bar{b}^{2})}{24}\ket{0}.$ (53)

###  2.4 Thermal partition function

We end this section with a few comments. In the previous sections we showed
that the bulk theory with the boundary condition (2), where $T_{zz}$ and
$T_{{\bar{z}}{\bar{z}}}$ given by (6), leads to a single Virasoro primary on
the boundary that satisfies

$L^{\prime}_{0}\ket{0}=\frac{c_{0}(1-a)}{24}\ket{0},\quad\bar{L}^{\prime}_{0}\ket{0}=\frac{c_{0}(1-\bar{a})}{24}\ket{0}~{}.$
(54)

In our analysis the boundary was a cylinder. Let’s now compactify also the
time direction and consider the theory on the torus. If we define the
partition function as

$Z=(q\bar{q})^{-\frac{1}{24}}\Tr
q^{L_{0}}\bar{q}^{\bar{L}_{0}}=(q\bar{q})^{-\frac{c_{0}+1}{24}}\Tr
q^{L^{\prime}_{0}}\bar{q}^{\bar{L}^{\prime}_{0}}~{},$ (55)

we conclude that it is equal to

$Z=\chi_{\frac{c_{0}(1-a)}{24}}(q)\chi_{\frac{c_{0}(1-\bar{a})}{24}}(\bar{q})~{},$
(56)

where $\chi_{h}(q)$ is the character of a module of scaling dimension $h$.
This has been obtained in the literature before, either by computing the
partition function of the Alekseev-Shatashvili theory in the path integral
formulation [23] or by computing it directly in the bulk [8, 39].

Since the above expression (56) contains just a single Virasoro character it
cannot be the partition function of a modular invariant CFT. From the gravity
point of view we would expect that black holes of different masses and angular
momenta should be part of a common Hilbert space and so should be included in
the sum over states. However, gravity provides little guidance as to what
spectrum of masses to include. The spectrum should be discrete in order to
obtain a finite partition sum, and should also be in approximate agreement
with the Bekenstein-Hawking entropy formula. The alternative sum over
Euclidean geometries can produce a modular invariant result but suffers from
other pathologies [8, 40]. These facts have led to the suggestion that the
holographic dual of 3d gravity in AdS3 is actually an ensemble of CFTs [41,
42, 43].

##  3 The stress tensor in cutoff gravity and $T\overline{T}$-deformed CFT

###  3.1 $T\overline{T}$ deformation and stress tensor improvement

At the classical level the $T\overline{T}$ deformation corresponds to finding
an action $S_{\lambda}(\phi)$ that obeys the flow equation

$\displaystyle\frac{dS_{\lambda}}{d\lambda}=-\frac{1}{8\pi^{2}}\int\\!d^{2}x\det
T~{},$ (57)

where the factor is $\frac{1}{8\pi^{2}}$ is convention dependent. The stress
tensor $T_{\mu\nu}$ appearing on the right hand side should be conserved with
respect to the equations of motion of $S_{\lambda}$. However, given one
conserved stress tensor $T_{\mu\nu}$ one can always write down another
conserved stress tensor by adding an improvement term,666Although this is not
the most general improvement term for a generic QFT, it is for the case under
considerations which involves only a single scalar field.

$\displaystyle\tilde{T}_{\mu\nu}=T_{\mu\nu}+(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})Y(\phi)~{}.$
(58)

Different choices for the stress tensor might therefore seem to lead, via
(57), to different expressions for the deformed action. However, we now show
that these actions differ at most by a field redefinition.

We first note that in a deformed CFT, $\lambda$ is the only dimensionful
scale, and since $\Tr T$ is the generator of scale transformations we have
that $\lambda\frac{dS_{\lambda}}{d\lambda}=-\frac{1}{4\pi}\int\\!d^{2}x\Tr T$,
which implies that $\frac{\lambda}{2\pi}\det T=\Tr T$, up to a possible total
derivative. On the other hand, assume that we define the deformed action in
(57) using the improved stress tensor $\tilde{T}_{\mu\nu}$. For the reason we
just explained, $\tilde{T}_{\mu\nu}$ will also obey a trace equation
$\frac{\lambda}{2\pi}\det\tilde{T}=\Tr\tilde{T}$ up to a total derivative.
From (58) we have that $\Tr\tilde{T}=\Tr T-\partial^{2}Y$ from which it
follows that $\det T=\det\tilde{T}$ up to a total derivative. Therefore
$S_{\lambda}$ also obeys equation (57) but with $T_{\mu\nu}$ replaced by
$\tilde{T}_{\mu\nu}$.

This observation clarifies one of the main results of [5]. It is well-known
that the deformed action for a free scalar with canonical stress tensor is the
Nambu-Goto action. On the other hand, the undeformed action in [5] is a free
scalar but the stress tensor is not canonical, rather it has a linear dilaton
improvement term. The argument above explains why the deformed action starting
from this non-canonical stress tensor turns out to be the same Nambu-Goto
action as in the canonical case.

Actually, there is one important point in this argument that we glossed over.
The trace equation $\frac{\lambda}{2\pi}\det T=\Tr T$ is satisfied only on-
shell and therefore the statement $\det T=\det\tilde{T}$ holds only after we
use the equations of motion. If one uses two stress tensors that differ by
terms that vanish on-shell then the corresponding deformed actions may also
differ by such terms. However, on general grounds one knows that two actions
that differ in this way are related by a field redefinition; e.g [44]. So the
general statement is that the ambiguity regarding which stress tensor to use
in (57) does not translate into any ambiguity in the action, assuming one uses
the freedom to make field redefinitions. Indeed, in [5] a complicated field
redefinition, not known in closed form, was needed in order to bring the
action to the Nambu-Goto form.

###  3.2 Nambu-Goto action from the trace equation

In classical cutoff gravity with the standard boundary term (GHY plus boundary
cosmological term) the stress tensor obeys the trace equation

$\displaystyle T^{\mu}_{\mu}=\frac{\lambda}{2\pi}\det T^{\mu}_{~{}\nu}~{},$
(59)

We now quickly review how this deforms a free scalar field into the Nambu-Goto
action. We will use the canonical stress tensor, deferring the discussion of
the improvement term to the next section. From the argument of the previous
section we know that we will obtain the same action in either case. Since the
undeformed action and the canonical stress tensor contain only first
derivatives of the scalar field, (59) will not generate any dependence on
higher derivatives. To preserve this feature, we do not add a total derivative
term to (59).

To follow the above strategy it is more convenient to write the action in
terms of $\phi$ and its derivatives than in terms of $\phi$ and its canonical
momentum $\Pi$. In this approach Lorentz symmetry is manifest and we can
easily get a closed form expression for the action to all orders in the
deformation parameter $\lambda$. A subtlety in this formulation is that the
Lagrangian is not gauge invariant. However, once we convert back to canonical
variables the gauge symmetry is restored. The strategy we follow is to impose
that the Lagrangian and the stress tensor in terms of $\phi$ and its
derivatives are invariant under

$\phi\rightarrow\phi+\epsilon~{},$ (60)

for constant $\epsilon$. Equivalently we require that they do not depend on
$\phi$ itself but only on its derivatives. After obtaining the deformed
action, we replace time derivatives of $\phi$ with $\Pi$ to find expressions
that are gauge invariant under (23). It is possible to directly repeat the
derivations in this section using $\phi$ and $\Pi$ but it is not very
illuminating. We have included it in appendix A.

From dimensional analysis and Poincaré invariance, the action must take the
form

$\displaystyle S=\frac{1}{\lambda}\int d^{2}x\,F(s)~{},\quad
s\equiv\lambda\,\partial_{\mu}\phi\,\partial^{\mu}\phi\ .$ (61)

The canonical stress tensor defined as

$T^{\text{can}}_{\mu\nu}=2\pi\left(\delta_{\mu\nu}L-\frac{\partial
L}{\partial(\partial^{\mu}\phi)}\partial_{\nu}\phi\right)~{},$ (62)

is then equal to

$\displaystyle T_{\mu\nu}^{\text{can}}$
$\displaystyle=\frac{2\pi}{\lambda}(\delta_{\mu\nu}F-2\lambda\,\partial_{\mu}\phi\,\partial_{\nu}\phi\,F^{\prime})\
,$ (63)

and the trace equation (59) reduces to a differential equation for $F$,777A
convenient identity is  $\displaystyle\det T$
$\displaystyle=\frac{1}{2}(\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}-\delta_{\mu}^{\sigma}\delta_{\nu}^{\rho})T^{\mu}{}_{\rho}T^{\nu}{}_{\sigma}=\frac{1}{2}\big{(}(\Tr
T)^{2}-\Tr(T^{2})\big{)}\ .$ (64)

$\displaystyle\frac{2\pi}{\lambda}(2F-2sF^{\prime})$
$\displaystyle=\frac{2\pi}{\lambda}F(F-2sF^{\prime})\ .$ (65)

This ODE can be solved by integration. There are two branches and one
integration constant $n$: $F=1\pm\sqrt{1-ns}$. We want the action (61) to
reduce to the free boson action for $\lambda\to 0$, so we are forced to choose
the lower sign and fix $n=1$ to find

$\displaystyle S$ $\displaystyle=\frac{1}{\lambda}\int
d^{2}x\left(1-\sqrt{1-\lambda\partial_{\mu}\phi\,\partial^{\mu}\phi}\right)\
.$ (66)

From the deformed action (66) we can derive the canonical stress tensor

$\displaystyle T_{\mu\nu}^{\text{can}}$
$\displaystyle=\frac{2\pi}{\lambda}\left[(1-\sqrt{1-s})\delta_{\mu\nu}-\frac{\lambda\partial_{\mu}\phi\,\partial_{\nu}\phi}{\sqrt{1-s}}\right]~{},$
(67)
$\displaystyle=2\pi\left[\frac{1}{2}(\partial\phi)^{2}\delta_{\mu\nu}-\partial_{\mu}\phi\,\partial_{\nu}\phi\right]+\mathcal{O}(\lambda)~{},\
$ (68)

which indeed reduces to (28) with $c_{0}=0$ (corresponding to no improvement
term) in the limit $\lambda\rightarrow 0$.

###  3.3 Computing the improved stress tensor

We now explain how to obtain the stress tensor that arises from cutoff
gravity. This stress tensor takes the form

$\displaystyle T_{\mu\nu}$
$\displaystyle=T_{\mu\nu}^{\text{can}}+(\partial_{\mu}\,\partial_{\nu}-\delta_{\mu\nu}\partial^{2})Y\
,$ (69)

where $T_{\mu\nu}^{\text{can}}$ is the canonical part (67) derived from the
Nambu-Goto action. More precisely, starting from the Lagrangian in terms of
$\phi$ and $\Pi$, the stress tensor can be put in this form using the
equations of motion. In the asymptotically AdS case corresponding to
$\lambda=0$ we know that the improvement term is

$Y=\beta\phi+\mathcal{O}(\lambda)~{},\quad\beta=\sqrt{\frac{c_{0}\pi}{3}}~{}.$
(70)

Combining this with the fact that the stress tensor obeys the trace equation
(59), we can uniquely determine $Y$ order by order in $\lambda$, as we now
show. After using the equations of motion this procedure reproduces the stress
tensor computed in [5], and thereby establishes the precise sense in which the
cutoff gravity theory is the $T\overline{T}$-deformed version of the free
scalar with improved stress tensor.

At first order in $\lambda$ it is easy to see that the only Lorentz invariant
term allowed by dimensional analysis is

$Y=\beta\phi+\gamma\lambda(\partial\phi)^{2}+\mathcal{O}(\lambda^{2})~{}.$
(71)

Note that the gauge transformation (60) does not allow polynomials $\phi^{n}$
with $n\neq 1$. To this order the trace of the stress tensor becomes

$\Tr
T=-\beta\partial^{2}\phi-\frac{\pi\lambda}{2}(\partial\phi)^{4}-\gamma\lambda\;\partial^{2}(\partial\phi)^{2}+\mathcal{O}(\lambda^{2})~{},$
(72)

while the right-hand side of the trace equation (59) is equal to

$\frac{\lambda}{2\pi}\det T=-\frac{\lambda}{4\pi}\Tr
T^{2}+\mathcal{O}(\lambda^{2})=\beta\lambda\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi-\frac{\pi\lambda}{2}(\partial\phi)^{4}-\frac{\lambda\beta^{2}}{8\pi}\;\partial^{2}(\partial\phi)^{2}+\mathcal{O}(\lambda^{2})~{}.$
(73)

We can use the equations of motion

$\displaystyle\partial^{2}\phi$
$\displaystyle=-\lambda\frac{\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi}{1-\lambda(\partial\phi)^{2}}=-\lambda\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi+\mathcal{O}(\lambda^{2})\
.$ (74)

and demanding (59) we get $\gamma=\frac{\beta^{2}}{8\pi}$.

One can follow the same procedure and at each order in $\lambda$ determine the
improvement term $Y$ in (69). We carried out this calculation up to
$\lambda^{3}$ and we show that it leads the following unique $Y$

$\displaystyle Y$
$\displaystyle=\beta\phi+\frac{\lambda\beta^{2}}{8\pi}(\partial\phi)^{2}+\frac{\lambda^{2}\beta^{2}}{32\pi}(\partial\phi)^{4}+\frac{\lambda^{2}\beta^{3}}{32\pi^{2}}\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi$
$\displaystyle\quad+\frac{\lambda^{3}\beta^{2}}{64\pi}(\partial\phi)^{6}+\frac{5\lambda^{3}\beta^{3}}{192\pi^{2}}\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi(\partial\phi)^{2}$
$\displaystyle\quad+\frac{\lambda^{3}\beta^{4}}{384\pi^{3}}[\partial_{\mu}\phi\,\partial_{\nu}\phi\,\partial_{\rho}\phi\,\partial^{\mu}\partial^{\nu}\partial^{\rho}\phi+\frac{3}{2}(\partial\phi)^{2}\partial_{\mu}\partial_{\nu}\phi\,\partial^{\mu}\partial^{\nu}\phi]+\mathcal{O}(\lambda^{4})~{}.$
(75)

We believe that the trace equation determines $Y$ to all orders in $\lambda$
but unfortunately we were not able to get an answer in a closed form. All in
all we conclude that the full stress tensor is given by (69) with the
canonical part as in (67) and the improvement term as in (75).

To relate to results in [5] we should express $\dot{\phi}$ as a function of
$\phi^{\prime}$ and $\Pi$ using the equations of motion following from the
canonical form of the Nambu-Goto action,

$\displaystyle S$ $\displaystyle=\int
d^{2}x\left[i\Pi\dot{\phi}+\frac{1}{\lambda}\left(1-\sqrt{1-\lambda(\phi^{\prime
2}+\Pi^{2})+\lambda^{2}\phi^{\prime 2}\Pi^{2}}\right)\right]\ .$ (76)

This is indeed the $T\overline{T}$ deformed version of the action (21) (see
[33]). Note that the action is gauge invariant under (23). We further express
results in terms of $f$ and ${\overline{f}}$ using (20). The action in these
variables is given by

$\displaystyle S$
$\displaystyle=\frac{c_{0}}{48\pi}\int\differential^{2}x\;\left[if^{\prime}\dot{f}-i\bar{f}^{\prime}\dot{\bar{f}}+\frac{4}{r_{c}}\left(1-\sqrt{1-\frac{1}{2}r_{c}(f^{{}^{\prime}2}+\bar{f}^{{}^{\prime}2})+\frac{1}{16}r_{c}^{2}(f^{{}^{\prime}2}-\bar{f}^{{}^{\prime}2})^{2}}\right)\right]\
,$ (77)

where

$\displaystyle r_{c}=\frac{\lambda c_{0}}{12\pi}~{}.$ (78)

Carrying out the conversion, the canonical part of the stress tensor is given

$\displaystyle T^{\rm can}_{z{\bar{z}}}$
$\displaystyle=-\frac{c_{0}}{96}\left(r_{c}f^{\prime 2}{\overline{f}}^{\prime
2}+\frac{1}{2}r_{c}^{2}f^{\prime 2}{\overline{f}}^{\prime 2}(f^{\prime
2}+{\overline{f}}^{\prime 2})+\ldots\right)~{},$ (79) $\displaystyle T^{\rm
can}_{zz}$ $\displaystyle=-\frac{c_{0}}{24}\left(f^{\prime
2}+\frac{1}{2}r_{c}f^{\prime 2}{\overline{f}}^{\prime
2}+\frac{3}{16}r_{c}^{2}f^{\prime 2}{\overline{f}}^{\prime 2}(f^{\prime
2}+{\overline{f}}^{\prime 2})+\ldots\right)~{},$ (80) $\displaystyle T^{\rm
can}_{{\bar{z}}{\bar{z}}}$
$\displaystyle=-\frac{c_{0}}{24}\left({\overline{f}}^{\prime
2}+\frac{1}{2}r_{c}f^{\prime 2}{\overline{f}}^{\prime
2}+\frac{3}{16}r_{c}^{2}f^{\prime 2}{\overline{f}}^{\prime 2}(f^{\prime
2}+{\overline{f}}^{\prime 2})+\ldots\right)~{},$ (81)

while the improvement term is

$\displaystyle
Y=\frac{c_{0}}{6}\left[\frac{1}{2}(f+{\overline{f}})+\frac{r_{c}}{4}f^{\prime}{\overline{f}}^{\prime}+\frac{r_{c}^{2}}{16}(f^{\prime\prime}{\overline{f}}^{\prime
2}+f^{\prime 2}{\overline{f}}^{\prime\prime})+\ldots\right]~{}.$ (82)

Acting with the appropriate derivatives (69), the corresponding stress tensor
components are

$\displaystyle T_{z{\bar{z}}}$
$\displaystyle=\frac{c_{0}}{6}\left[-\frac{1}{4}r_{c}f^{\prime\prime}{\overline{f}}^{\prime\prime}+\frac{1}{8}r_{c}(f^{\prime\prime}{\overline{f}}^{\prime
2}+f^{\prime
2}{\overline{f}}^{\prime\prime})-\frac{1}{8}r_{c}^{2}(f^{\prime}f^{\prime\prime}{\overline{f}}^{\prime\prime\prime}+f^{\prime\prime\prime}{\overline{f}}^{\prime}{\overline{f}}^{\prime\prime})+\ldots\right]~{},$
(83) $\displaystyle T_{zz}$
$\displaystyle=\frac{c_{0}}{6}\left[\frac{1}{2}f^{\prime\prime}+\frac{1}{8}r_{c}(2f^{\prime\prime}{\overline{f}}^{\prime
2}+2f^{\prime}{\overline{f}}^{\prime}{\overline{f}}^{\prime\prime}-2f^{\prime}f^{\prime\prime}{\overline{f}}^{\prime})+\frac{1}{8}r_{c}^{2}(f^{\prime}f^{\prime\prime\prime}{\overline{f}}^{\prime\prime}+f^{\prime\prime
2}{\overline{f}}^{\prime\prime}+\frac{1}{2}f^{\prime\prime\prime\prime}{\overline{f}}^{\prime
2})+\ldots\right]~{},$ (84) $\displaystyle T_{{\bar{z}}{\bar{z}}}$
$\displaystyle=\frac{c_{0}}{6}\left[\frac{1}{2}{\overline{f}}^{\prime\prime}+\frac{1}{8}r_{c}(2{\overline{f}}^{\prime\prime}f^{\prime
2}+2{\overline{f}}^{\prime}f^{\prime}f^{\prime\prime}-2{\overline{f}}^{\prime}{\overline{f}}^{\prime\prime}f^{\prime})+\frac{1}{8}r_{c}^{2}({\overline{f}}^{\prime}{\overline{f}}^{\prime\prime\prime}f^{\prime\prime}+{\overline{f}}^{\prime\prime
2}f^{\prime\prime}+\frac{1}{2}{\overline{f}}^{\prime\prime\prime\prime}f^{\prime
2})+\ldots\right]~{}.$ (85)

Note that when expressed in terms of $f$ and $\bar{f}$ the constant in front
of the improvement term $c_{0}$ appears as an overall factor. Indeed, since
the gravity action is proportional to $\frac{1}{G}\sim c_{0}$ the classical
stress tensor must be proportional to $c_{0}$. The sum of (79) and (83)
reproduces the result in [5] obtained from the 3d bulk gravity.

##  4 Static vs light-cone gauge

As has been noted by various authors [45, 2, 3, 46, 33], the fact that the
$T\overline{T}$-deformed theory of a collection of free scalar fields turns
out to be the Nambu-Goto action in static gauge implies that one can solve the
theory by transforming to light-cone gauge where the action is quadratic. Of
course for a general number of scalar fields this is not justified at the
quantum level due to the target space Lorentz anomaly. However, the Lorentz
anomaly is absent in two special cases: for $24$ scalar fields one has the
usual cancellation of the anomaly, while for a single scalar field the anomaly
is trivially absent since it must be antisymmetric in transverse target space
indices. So there is a reason to believe that the procedure is justified in
the case relevant to 3d gravity, and indeed the resulting energy spectrum is
the correct one.

The canonical transformation between the static and light-cone gauge theories
was reviewed in the $T\overline{T}$ context in [33]. The only slight
difference we need to take into account is that our scalar field theory is a
gauged version of the one in [33]. Since this difference essentially goes
along for the ride, we simply state a few of the main results and add some
comments.

In this section we work in Lorentzian signature, taking our conventions as in
Appendix B, with action

$\displaystyle S=\int
d^{2}x\left[\Pi\dot{\phi}-\frac{1}{\lambda}\left(1-\sqrt{1-\lambda(\phi^{\prime
2}+\Pi^{2})+\lambda^{2}\phi^{\prime 2}\Pi^{2}}\right)\right]~{}.$ (86)

We will focus attention on the case
$(a=-b^{2},\overline{a}=-\overline{b}^{2})$ for which the winding conditions
are 888The other sign choice leads to imaginary winding conditions and
constraints, which clashes with the intepretation in terms of a string
embedded in a real target space.

$\displaystyle f(x+2\pi,t)=f(x,t)+2\pi
b~{},\quad{\overline{f}}(x+2\pi,t)={\overline{f}}(x,t)+2\pi\overline{b}~{}.$
(87)

This translate into the periodicities

$\displaystyle\phi(x+2\pi,t)=\phi(x,t)+\sqrt{\frac{c_{0}}{48\pi}}2\pi(b+\overline{b})~{},\quad\Pi(x+2\pi,t)=\Pi(x,t)~{},$
(88)

along with the constraint

$\displaystyle\int_{0}^{2\pi}\Pi(x,t)dx=\sqrt{\frac{\pi
c_{0}}{12}}(b-\overline{b}).$ (89)

We view this action as a gauge fixed version of the Hamiltonian form of the
Nambu-Goto action

$\displaystyle
S=\frac{1}{\lambda}\int\\!d^{2}x\sqrt{-\det\partial_{i}X^{\mu}\partial_{j}X_{\mu}}~{},$
(90)

which in Hamiltonian form reads

$\displaystyle
S=\int\\!d^{2}x\left[\Pi_{\mu}\dot{X}^{\mu}-\lambda_{1}C_{1}-\lambda_{2}C_{2}\right]~{},$
(91)

where the constraints are

$\displaystyle C_{1}=\Pi_{\mu}X^{\prime\mu}~{},\quad
C_{2}=\frac{1}{2}(\lambda^{2}\Pi_{\mu}\Pi^{\mu}+X^{\prime}_{\mu}X^{\prime\mu})~{}.$
(92)

Here $\mu=0,1,2$ with the Minkowski metric $\eta_{\mu\nu}=(-1,1,1).$ The
$X^{1}$ direction is taken to be compact and we consider the unit winding
sector, $X^{1}(x+2\pi,t)=X^{1}(x,t)+2\pi$ .

###  4.1 Static gauge

Static gauge is defined by

$\displaystyle X^{0}=t~{},\quad X^{1}=x~{}.$ (93)

Solving the constraints and plugging back into (91) reproduces (86) up to a
constant additive shift under the identification

$\displaystyle
X^{2}=\sqrt{\lambda}\phi~{},\quad\Pi_{2}=\frac{1}{\sqrt{\lambda}}\Pi~{}.$ (94)

The constraint (89) translates to a constraint on $\Pi_{2}$. In terms of the
string moving in a three-dimensional target spacetime, this constraint amounts
to fixing the spacetime momentum in the $X^{2}$ direction,
$P_{2}=\int_{0}^{2\pi}\Pi_{2}(x,t)dx$. The winding condition (88) on $\phi$
translates into

$\displaystyle X^{2}(x+2\pi,t)=X^{2}(x,t)+\sqrt{\frac{\lambda
c_{0}}{48\pi}}2\pi(b+\overline{b})~{}.$ (95)

The string therefore winds around both cycles of a target space torus in
$X^{1,2}$.

###  4.2 Light-cone gauge

Next we consider the light-cone gauge. Define

$\displaystyle X^{\pm}=X^{0}\pm
X^{1}~{},\quad\Pi_{\pm}=\frac{1}{2}(\Pi_{0}\pm\Pi_{1})~{},$ (96)

and fix the gauge as

$\displaystyle X^{+}(x,t)=-2\lambda p_{-}t+x~{},\quad\Pi_{-}=p_{-}~{},$ (97)

where $p_{-}$ is constant. The other light-cone component of the target space
momentum is $p_{+}=\int_{0}^{2\pi}\frac{dx}{2\pi}\Pi_{+}$. Integrating the
constraints allows us to relate $p_{\pm}$ to $(\phi,\Pi)$, defined in (94), as

$\displaystyle(p_{+}-p_{-})+P_{lc}=0~{},\quad 2\lambda
H_{lc}+(1-4\lambda^{2}p_{+}p_{-})=0~{},$ (98)

where

$\displaystyle P_{lc}=\int_{0}^{2\pi}\Pi\phi^{\prime}dx~{},\quad
H_{lc}=\int_{0}^{2\pi}\frac{1}{2}(\Pi^{2}+\phi^{\prime 2})dx~{}.$ (99)

The action is

$\displaystyle
S=\int\\!d^{2}x\left[\Pi\dot{\phi}-\frac{1}{2}(\Pi^{2}+\phi^{\prime
2})\right]~{},$ (100)

along with a decoupled part involving the zero modes that we refrain from
writing 999As shown in [33] there is also a decoupled term $\int
dtp_{-}\dot{q}^{-}$. This leads to an extra label for the states, namely the
eigenvalue value of $q_{-}$. However, in [33] it was also shown that because
of the constraints the value of $q_{-}$ is fixed in terms of the energy and
momentum and thus no additional degeneracy is introduced. . The free field
form of this action is of course the main virtue of the light-cone gauge. We
can view this free field theory as our undeformed theory. We have the spectra

$\displaystyle
H_{lc}=\frac{c_{0}}{24}(b^{2}+\overline{b}^{2})+N+\overline{N}~{},\quad
P_{lc}=\frac{c_{0}}{24}(b^{2}-\overline{b}^{2})+N-\overline{N}~{},$ (101)

where the ground state contributions come from the momentum and winding.

###  4.3 Spectrum of static gauge Hamiltonian

We are interested in the energy spectrum of the static gauge theory, since the
Hamiltonian of that theory is identified with that of the
$T\overline{T}$-deformed theory. Explicitly,

$\displaystyle H$
$\displaystyle=\int_{0}^{2\pi}dx\frac{1-\sqrt{1-\lambda(\phi^{\prime
2}+\Pi^{2})+\lambda^{2}\phi^{\prime 2}\Pi^{2}}}{\lambda}\ .$ (102)

$H$ generates $t$ translations, which in static gauge is the same as $X^{0}$
translations. In light-cone gauge the generator of $X^{0}$ translations is
$-(p_{+}+p_{-})$, so the static gauge Hamiltonian expressed in terms of light-
cone operators is $H=-(p_{+}+p_{-})$. The constraints (98) allow us write this
as

$\displaystyle H=\frac{1-\sqrt{1-2\lambda
H_{lc}+\lambda^{2}P^{2}_{lc}}}{\lambda}~{},$ (103)

where we added the constant piece by hand. The free field spectra of
$(H_{lc},P_{lc})$ then imply that the spectrum of $H$ is

$\displaystyle H$
$\displaystyle=\frac{1-\sqrt{1-2\lambda(\frac{c_{0}}{12}(b^{2}+\overline{b}^{2})+N+\overline{N})+\lambda^{2}(\frac{c_{0}}{12}(b^{2}-\overline{b}^{2})+N-\overline{N})^{2}}}{\lambda}\
.$ (104)

This reproduces the $T\overline{T}$ energy spectrum. We now make a few
comments. As shown in [33, 47] the phase space variables $(\phi,\Pi)$ in the
two gauges are related by a canonical transformation. While this establishes
the equivalence of the theories at the classical level, this does not
automatically extend to the quantum theory since classical canonical
transformations typically have no quantum counterpart.101010This fact is
sometimes stated by saying that there is no natural action of
symplectomorphisms on Hilbert space. Of course, the question of quantum
equivalence of the two theories is ill-defined unless one provides an
independent definition of the static gauge theory with its unwieldy square
root. The point to be emphasized here is that if we interpret the static gauge
action as being a $T\overline{T}$ deformed theory then this fixes the spectrum
of the theory and this spectrum coincides with that obtained from the light-
cone theory.

In [29] the quantum equivalence of the static and light-cone theories was
discussed in the context of the S-matrix. The static gauge theory has a
nontrivial $2\rightarrow 2$ S-matrix, which at first seems in conflict with
the fact that the light-cone theory is free. However this can be understood in
terms of the field dependent coordinate transformation that relates the
theories. Namely, the static gauge $2\rightarrow 2$ S-matrix is given by a
time delay, and under the coordinate transformation this time delay vanishes
in light-cone gauge, as it must in a free theory.

Since correlation functions are readily computed in the light-cone theory, one
might wonder whether correlators in static gauge could be computed by
transforming them to light-cone gauge. Unfortunately, the canonical
transformation maps simple static gauge operators to complicated light-cone
operators and vice versa, so this does not appear to be helpful. In slightly
more detail, consider a correlation function of local operators in static
gauge, each evaluated at a position $x_{n}^{s}$ with superscript $s$ for
static gauge. Because the coordinate transformation between static and light-
cone gauge is field-dependent, the same correlation function in light-cone
gauge involves operators evaluated at field-dependent locations
$x_{n}^{lc}[\phi,x_{n}^{s}]$. These are complicated objects. For example
calculating the correlation function as a path integral would involve
operators that move around (in light-cone coordinates) as one integrates over
the fundamental fields.

##  5 Spectrum

###  5.1 Perturbation theory

In this section we discuss the computation of the energy spectrum of our
theory by applying ordinary perturbation theory. This is useful in order to
clarify how the $T\overline{T}$ trace relation resolves the quantization
ambiguities. Our Hamiltonian is (102) where $\lambda=12\pi r_{c}/c_{0}$. For
simplicity we restrict here to the $b=\overline{b}=0$ case, where the mode
expansions (27) and (46) as well as the Hermiticity properties (30) and (49)
coincide. In terms of the fields $(f,{\overline{f}})$ (20), which are both
$2\pi$ periodic in $x$, we have the mode expansions

$\displaystyle f^{\prime}$ $\displaystyle=\sqrt{\frac{12}{c_{0}}}\sum_{m\neq
0}\alpha_{m}e^{imx}~{},$ (105) $\displaystyle{\overline{f}}^{\prime}$
$\displaystyle=\sqrt{\frac{12}{c_{0}}}\sum_{m\neq
0}\overline{\alpha}_{m}e^{-imx}~{},$ (106)

and impose the commutation relations (32). The Hamiltonian may be expanded to
the first few orders as

$\displaystyle
H=\frac{c_{0}}{24}\int_{0}^{2\pi}\frac{dx}{2\pi}\Big{[}f^{\prime
2}+{\overline{f}}^{\prime 2}+\frac{1}{2}r_{c}f^{\prime
2}{\overline{f}}^{\prime 2}+\frac{1}{8}r_{c}^{2}f^{\prime
2}{\overline{f}}^{\prime 2}(f^{\prime 2}+{\overline{f}}^{\prime
2})+\ldots\Big{]}~{}.$ (107)

The general arguments of Zamolodchikov and Smirnov imply that we expect the
energy spectrum to be

$\displaystyle E$
$\displaystyle=\frac{c_{0}}{6r_{c}}\left[1-\sqrt{1-\frac{12r_{c}}{c_{0}}(N+\overline{N})+\frac{36r_{c}^{2}}{c_{0}^{2}}(N-\overline{N})^{2}}\right]~{},$
(108)
$\displaystyle=N+\overline{N}+\frac{12r_{c}}{c_{0}}N\overline{N}+\frac{72r_{c}^{2}}{c_{0}^{2}}N\overline{N}(N+\overline{N})+O\left(\frac{r_{c}^{3}}{c_{0}^{3}}\right)~{},$
(109)

where $(N,\overline{N})$ are eigenvalues of the the level operators

$\displaystyle
N=\sum_{m>1}\alpha_{-m}\alpha_{m}~{},\quad\overline{N}=\sum_{m>1}\overline{\alpha}_{-m}\overline{\alpha}_{m}~{}.$
(110)

To define $H$ as an operator and compute its spectrum we of course need to
resolve ordering ambiguities among the mode operators. We think of expanding
out $H$ in mode operators, order by order in $1/c_{0}$ (equivalently $r_{c}$).
Before reordering, the order $c_{0}^{-p}$ contribution to $H$ is a homogeneous
degree $2p+2$ polynomial in the mode operators. The difference between two
choices of ordering will involve a polynomial of degree $q<2p+2$ in the mode
operators. Now imagine computing the matrix elements of $H$ in the unperturbed
basis where states are labelled by occupation numbers in each mode, again
expanding in $1/c_{0}$. Each squared matrix element at order $c_{0}^{-p}$ is a
polynomial of degree $2p+2$ in the occupation numbers. The point we wish to
emphasize here is that the order $2p+2$ piece of this polynomial is
unambiguous in the sense that it is independent of our choice of ordering and
is also UV finite. For example, the difference between
$\sum_{m>0}\alpha_{-m}\alpha_{m}$ and $\sum_{m>0}\alpha_{m}\alpha_{-m}$ is the
UV divergent quantity $\sum_{m>1}m$, whose matrix elements are independent of
the occupation numbers. From (108) we see that the terms that appear in the
energy spectrum involve only these unambiguous contributions. We conclude that
there is no ambiguity in reproducing these terms, and the only issue is how to
fix terms that are subleading in $(N,\overline{N})$ at a given order in the
$1/c_{0}$ expansion.

As we now discuss, these subleading ambiguities are fixed by the
$T\overline{T}$ trace relation. Classically, the relation

$\displaystyle
T_{z{\bar{z}}}=-\frac{6r_{c}}{c_{0}}(T_{zz}T_{{\bar{z}}{\bar{z}}}-T_{z{\bar{z}}}T_{z{\bar{z}}})~{},$
(111)

is simply an identity. However, we obtain a useful expression by taking the
expectation value of both sides in an eigenstate of $H$ and $P$. In
particular, Zamolodchikov showed that the right hand side factorizes, in the
sense that the following equation holds

$\displaystyle\langle T_{z{\bar{z}}}\rangle=-\frac{6r_{c}}{c_{0}}(\langle
T_{zz}\rangle\langle T_{{\bar{z}}{\bar{z}}}\rangle-\langle
T_{z{\bar{z}}}\rangle\langle T_{z{\bar{z}}}\rangle)~{}.$ (112)

Crucially, the expectation values in this equation are independent of whatever
improvement terms we wish to add to the stress tensor, as follows from the
fact that improvement terms are derivatives of local operators, but local
operators have constant expectation values in eigenstates of $H$ and $P$. We
are therefore free to impose (112) on the canonical stress tensor, which takes
the form (79) in terms of which the Hamiltonian (107) is

$\displaystyle
H=\int_{0}^{2\pi}\frac{dx}{2\pi}T_{tt}=\int_{0}^{2\pi}\frac{dx}{2\pi}(2T_{z{\bar{z}}}-T_{zz}-T_{{\bar{z}}{\bar{z}}})~{}.$
(113)

Our strategy is to carry out perturbation theory order by order in $1/c_{0}$,
or equivalently in $r_{c}$, using (112) to fix any ordering ambiguities at
each order, keeping in mind that both the states and the stress tensor receive
corrections at each order. We now illustrate this by working out the spectrum
to order $1/c_{0}^{2}$, verifying the prediction of (108).

We start with the theory at $r_{c}=0$, and define $(f^{\prime
2},{\overline{f}}^{\prime 2})$ by normal ordering so that $E=N+\overline{N}$.
States are labelled by their eigenvalues under the individual level numbers
(110). Now go to order $r_{c}$. The trace relation tells us that
$T_{z{\bar{z}}}=-\frac{c_{0}}{96}r_{c}f^{\prime 2}{\overline{f}}^{\prime 2}$
with $(f^{\prime 2},{\overline{f}}^{\prime 2})$ normal ordered.

The Hamiltonian at order $r_{c}$ involves the expressions $f^{\prime
2}{\overline{f}}^{\prime 2}$ in $T_{zz}$ and $T_{{\bar{z}}{\bar{z}}}$. It is
natural to guess that these operators are also normal ordered, and we will
indeed verify that is the correct prescription. Indeed, with this assumption
the correction to the energy is given by the standard formula (where
subscripts indicate the order in $r_{c}$)

$\displaystyle E$
$\displaystyle=E_{0}+\langle\psi_{0}|H_{1}|\psi_{0}\rangle+\ldots~{},$ (114)
$\displaystyle=N+\overline{N}+r_{c}\langle\psi_{0}|\frac{c_{0}}{48}\int_{0}^{2\pi}\frac{dx}{2\pi}f^{\prime
2}{\overline{f}}^{\prime 2}|\psi_{0}\rangle+\ldots~{},$ (115)
$\displaystyle=N+\overline{N}+\frac{12r_{c}}{c_{0}}N\overline{N}+\ldots~{},$
(116)

in agreement with (108).

At order $r_{c}^{2}$ the left hand side of (112) involves
$\langle\psi_{0}|f^{\prime 2}{\overline{f}}^{\prime 2}(f^{\prime
2}+{\overline{f}}^{\prime 2})|\psi_{0}\rangle$, which is UV divergent. We use
(112) to define this matrix element in terms of matrix elements with fewer
insertions of $(f,{\overline{f}})$. At this order we also need to include
corrections to the states according to the usual formula

$\displaystyle|E_{n}\rangle=|E_{n}^{0}\rangle+\sum_{k\neq n}\frac{\langle
E_{k}^{0}|H_{1}|E_{n}^{0}\rangle}{E_{n}^{0}-E_{k}^{0}}|E_{k}^{0}\rangle+\ldots~{},$
(117)

where we note that since $H_{1}$ is diagonal within a degenerate subspace only
non-degenerate unperturbed states contribute to the sum.

We can now work out the order $r_{c}^{2}$ correction to the energy from the
standard formula of second order perturbation theory

$\displaystyle
E_{2}=\langle\psi_{0}|H_{2}|\psi_{0}\rangle+\sum_{\psi_{0}^{\prime}\neq\psi_{0}}\frac{|\langle\psi_{0}^{\prime}|H_{1}|\psi_{0}\rangle|^{2}}{E_{0}-E_{0}^{\prime}}~{},$
(118)

with
$H_{2}=\frac{c_{0}}{24}\int_{0}^{2\pi}\frac{dx}{2\pi}\frac{1}{8}r_{c}^{2}f^{\prime
2}{\overline{f}}^{\prime 2}(f^{\prime 2}+{\overline{f}}^{\prime 2})$. We
explained above how to relate the expectation value of $H_{2}$ to lower order
matrix elements, but to see how the correct result comes out it is simpler to
compute directly. More precisely, in order to confirm (108) we will just
concentrate on the part cubic in occupation numbers, trusting that our general
arguments enforce the vanishing of the terms subleading in occupation numbers.
In terms of mode operators we have

$\displaystyle
H_{2}=\frac{9r_{c}^{2}}{c_{0}^{2}}\sum_{m_{i}}(\alpha_{m_{1}}\alpha_{m_{2}}\alpha_{m_{3}}\alpha_{m_{4}}\overline{\alpha}_{m_{5}}\overline{\alpha}_{m_{6}}+\alpha_{m_{1}}\alpha_{m_{2}}\overline{\alpha}_{m_{3}}\overline{\alpha}_{m_{4}}\overline{\alpha}_{m_{5}}\overline{\alpha}_{m_{6}})~{}.$
(119)

The sum should be restricted such that the sum of $\alpha$ mode numbers equals
the sum of $\overline{\alpha}$ mode numbers. The expectation value in a state
with level numbers $\\{N_{m},\overline{N}_{m}\\}$ works out to be

$\displaystyle\langle H_{2}\rangle$
$\displaystyle=\frac{108r_{c}^{2}}{c_{0}^{2}}\sum_{m_{1},m_{2}}\left[(N_{m_{1}})^{2}\overline{N}_{m_{2}}+N_{m_{1}}(\overline{N}_{m_{2}})^{2}\right]$
(120)
$\displaystyle\quad+\frac{432r_{c}^{2}}{c_{0}^{2}}\sum_{m_{1},m_{2},m_{3}}\left[N_{m_{1}}N_{m_{2}}\overline{N}_{m_{3}}+N_{m_{1}}\overline{N}_{m_{2}}\overline{N}_{m_{3}}\right]~{}.$
(121)

On the other hand, the second order contribution in $H_{1}$ works out to be

$\displaystyle\sum_{\psi_{0}^{\prime}\neq\psi_{0}}\frac{|\langle\psi_{0}^{\prime}|H_{1}|\psi_{0}\rangle|^{2}}{E_{0}-E_{0}^{\prime}}$
$\displaystyle=-\frac{36r_{c}^{2}}{c_{0}^{2}}\sum_{m_{1},m_{2}}\left[(N_{m_{1}})^{2}\overline{N}_{m_{2}}+N_{m_{1}}(\overline{N}_{m_{2}})^{2}\right]$
(122)
$\displaystyle\quad-\frac{288r_{c}^{2}}{c_{0}^{2}}\sum_{m_{1},m_{2},m_{3}}\left[N_{m_{1}}N_{m_{2}}\overline{N}_{m_{3}}+N_{m_{1}}\overline{N}_{m_{2}}\overline{N}_{m_{3}}\right]~{}.$
(123)

These nicely combine together in (118) to give

$\displaystyle
E_{2}=\frac{72r_{c}^{2}}{c_{0}^{2}}N\overline{N}(N+\overline{N})~{},$ (124)

in agreement with (108). This procedure can in principle be carried out to
higher orders. As long as one enforces (112) at each order the result is
guaranteed to reproduce (108), since the trace relation (112) implies a
differential equation for the energy levels that fixes their values, as shown
by Zamolodchikov and Smirnov. Our purpose in carrying out this exercise was to
emphasize that the contributions that actually appear in (108) are insensitive
to operator orderings. Operator ordering ambiguities only affect terms of
lower order in the level numbers, and these ambiguities are fixed by imposing
the trace relation.

##  6 Correlation functions

###  6.1 Gravity side

For a Dirichlet boundary condition on the metric the standard choice of
boundary term in (1) is [48]

$\displaystyle S_{\rm bndy}=-\frac{1}{8\pi G}\int_{\partial
M}\\!d^{2}x\sqrt{h}(K-1)~{},$ (125)

where $h_{\mu\nu}$ is the metric on the boundary and $K$ denotes the trace of
the extrinsic curvature. We are setting the AdS3 radius to unity, $\ell=1$,
for convenience. The term without $K$ is not required by the variational
principle but is added to yield a finite action in the limit that the boundary
is taken to infinity.111111Actually, finiteness also requires an additional
term associated with the Weyl anomaly [49], but since this term is (locally) a
total derivative it does not contribute to the stress tensor and will play no
role in our discussion. The boundary stress tensor is defined via the on-shell
variation of the action

$\displaystyle\delta S=\frac{1}{4\pi}\int_{\partial
M}\\!d^{2}x\sqrt{h}T^{\mu\nu}\delta h_{\mu\nu}~{},$ (126)

and works out to be

$\displaystyle T_{\mu\nu}=\frac{1}{4G}(K_{\mu\nu}-Kh_{\mu\nu}+h_{\mu\nu})~{}.$
(127)

Now consider the problem of computing correlation functions of $T_{\mu\nu}$ on
a flat boundary metric $h_{\mu\nu}$. This boundary is taken to be a finite
cutoff surface. This can in principle be carried out order by order in $G\sim
1/c_{0}$. The immediate question that arises is whether we will encounter UV
divergences that require counterterms. We first recall the standard argument
that the bulk Einstein-Hilbert action (1) is not renormalized apart from a
possible redefinition of $\ell$. This follows from the fact that any candidate
counterterm is proportional to $\int\\!d^{3}x\sqrt{g}$ upon using the lowest
order equation of motion; equivalently, a field redefinition may be used to
eliminate any counterterm involving curvatures.121212This statement is not
without subtleties. For example, if we employ dimensional regularization then
away from $d=3$ there are independent curvature dependent counterterms. These
can potentially contribute as “evanescent operators”, i.e., although the
operators formally vanish as $\epsilon\rightarrow 0$, if they are multiplied
by $1/\epsilon$ coefficients they can still contribute in the limit. We
implicitly assume that there exists a renormalization scheme where such
complications do not arise.

What about boundary terms? The Gibbons-Hawking-York (GHY) term is tied by the
variational principle to the Einstein-Hilbert term. The boundary cosmological
term can be fixed by the requirement that the stress tensor vanish on a flat
planar boundary. We should also consider boundary terms involving the
intrinsic and extrinsic curvature. A boundary term involving only intrinsic
curvatures cannot contribute either to the action or its first variation, and
hence not to the stress tensor; recall here that we are interested in
correlators on flat boundary. Regarding extrinsic curvature terms, we demand
that these do not spoil the variational principle. The only type of boundary
term which can contribute to the stress tensor without spoiling the
variational principle takes the form

$\displaystyle\Delta S_{\rm bndy}=-\frac{1}{4\pi}\int_{\partial
M}\\!d^{2}x\sqrt{h}R(h)Z(K_{\mu\nu})~{},$ (128)

where $Z(K_{\mu\nu})$ can depend on the extrinsic curvature and boundary
derivatives thereof. This term contributes to the stress tensor on a flat
boundary as an improvement term,

$\displaystyle\Delta
T_{\mu\nu}=(\partial_{\mu}\partial_{\nu}-h_{\mu\nu}\partial^{2})Z(K)~{}.$
(129)

In the case of an asymptotic AdS3 boundary (128) does not contribute since it
dies off as the boundary is taken to infinity, but it can contribute in the
case of a finite cutoff.

We conclude from this analysis that in the problem of computing stress tensor
correlators on a flat boundary we can encounter two types of divergences:
divergences in the action that vanish on-shell and hence can be removed by a
field redefinition and divergences in the stress tensor corresponding to
improvement terms. This conclusion from the gravity side nicely matches our
expectations on the 2d field theory side, as we now discuss.

###  6.2 Field theory side

The fact that on the gravity side all counterterms vanish for a flat boundary
surface leads to the expectation that in field theory all counterterms in the
action should vanish on-shell. This implies that the S-matrix should be
unambiguous, although this does not extend to off-shell correlators of the
elementary fields. This expectation is supported by previous analyses [50, 51,
28, 29, 27] of perturbation theory applied to the Nambu-Goto action131313In
[28, 29] the Lagrangian form of the Nambu-Goto action was considered, rather
than the Hamiltonian version that naturally arises in gravity, but we expect
the conclusions to be the same. and to $T\overline{T}$ theories more
generally, e.g [52]. For example, the one-loop computation of the four-point
function in [5] required the action counterterm
$\frac{r_{c}^{2}}{4\pi\varepsilon}\int
d^{2}x\partial_{z}(f^{\prime}\bar{f}^{\prime})\partial_{{\bar{z}}}(f^{\prime}\bar{f}^{\prime})$.
This expression can be seen to vanish after integration by parts and using the
lower order field equations
$\partial_{z}\bar{f}^{\prime}=\partial_{{\bar{z}}}f^{\prime}=0$.141414This
term is part of the evanescent 2d Ricci scalar that appears in dimensional
regularization of the low energy effective action for long strings [28]. We
thank Raphael Flauger for bringing this to our attention. A compelling story
regarding the finiteness of the $2\rightarrow 2$ S-matrix of the $d=3$ target
space Nambu-Goto action in static gauge was laid out in [28, 29, 30, 31, 32].

We next turn to correlators of the stress tensor computed on the plane.
Previous work on this problem includes [9, 53, 26, 54, 52, 55, 56, 5]. We
first recall the renormalization properties of the stress tensor in a general
QFT [35]. Let $S(\phi)$ be the renormalized action; that is $S(\phi)$ includes
counterterms that render finite all correlators of the elementary $\phi$
fields. Let $T_{\mu\nu}$ be the canonical stress tensor (62) obtained from
$S(\phi)$ by application of Noether’s theorem. The stress tensor Ward identity
relates correlators of $\partial^{\mu}T_{\mu\nu}$ with elementary fields to
correlators of elementary fields alone. Since the latter are assumed to be
finite, this implies that $\partial^{\mu}T_{\mu\nu}$ is a finite operator. The
only divergences that can show up in $T_{\mu\nu}$ therefore take the form of
identically conserved tensors151515By “identically conserved” we mean that the
conservation should hold without using the equations of motion, since an
equation of motion piece would spoil the contact terms in the Ward identity.,
which for a scalar field theory take the form of $\Delta
T_{\mu\nu}=(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})Y(\phi)$,
which we refer to as an improvement term. Besides the improvement terms needed
for finiteness there is also the freedom to add finite improvement terms, and
these must enter into the discussion of what one means by stress tensor
correlators.

In a CFT tracelessness of the stress tensor restricts the class of improvement
terms that can appear, and indeed Virasoro symmetry fixes the stress tensor
correlators uniquely in terms of a single number, the central charge $c$. In a
$T\overline{T}$ deformed CFT the stress tensor is not traceless; at the
classical level it obeys the trace equation $\Tr T=\frac{\lambda}{2\pi}\det
T$, and we have seen how this equation may be used to fix the form of the
stress tensor order by order in $\lambda$ starting from an initial seed. It is
natural to ask to what extent the quantum version of the trace equation fixes
the form of the quantum stress tensor and its correlators.

To address this we need to discuss the definition of the composite operator
$\det T$ in the quantum theory. Zamolodchikov famously showed that this
operator is well defined (UV finite) up to total derivatives of local
operators [57]. This has the important consequence that the expectation value
of $\det T$ is any energy-momentum eigenstate is finite and unambiguous, since
the total derivative pieces vanish in such states by translational invariance.
However, the total derivatives terms do contribute to off-diagonal matrix
elements, so in general we have to accept that $\det T$ requires such terms in
order to make sense as an operator.

Now suppose that we have defined a renormalized stress tensor that obeys the
relation

$\displaystyle\Tr T=\frac{\lambda}{2\pi}\det T+\partial_{\mu}W^{\mu}~{},$
(130)

where $\partial_{\mu}W^{\mu}$ denote operators added to $\det T$ to achieve UV
finiteness. Then, consider adding an infinitesimal improvement term to the
stress tensor,
$T^{\prime}_{\mu\nu}=T_{\mu\nu}+(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})Y$.
A simple computation shows that the new stress tensor obeys the equation $\Tr
T^{\prime}=\frac{\lambda}{2\pi}\det T^{\prime}+\partial_{\mu}W^{\prime\mu}$
with
$W^{\prime\mu}=W^{\mu}-2\partial^{\mu}Y-\frac{\lambda}{2\pi}T^{\mu\nu}\partial_{\nu}Y$,
where we used conservation of the stress tensor. The new stress tensor
therefore satisfies a trace equation that is equally valid as the original
one. We conclude that the quantum trace equation does not fix the form of the
quantum stress tensor, even though it does so at the classical level, the
reason being the appearance of the $\partial_{\mu}W^{\mu}$ term that is needed
for finiteness. It follows that stress tensor correlators are similarly
ambiguous. It is possible that there is some additional principle that may be
imposed on the stress tensor to render it unambiguous, but this would
presumably require a better understanding of how to define a $T\overline{T}$
deformed theory at the non-perturbative level.

We end this section by noting how these conclusions are borne out in
perturbation theory. In [5] the two-point function $\langle
T_{z\bar{z}}T_{z\bar{z}}\rangle$ was computed to two-loop order; all other
stress tensor two-point functions may be obtained from this by Ward
identities. In dimensional regularization the stress tensor was found to
require the counterterm $\Delta
T_{z\bar{z}}=-\frac{1}{2\varepsilon}r_{c}^{2}f^{\prime\prime\prime}\bar{f}^{\prime\prime\prime}=-\frac{1}{2\varepsilon}r_{c}^{2}\partial_{z}\partial_{\bar{z}}(f^{\prime\prime}\bar{f}^{\prime\prime})$,
where we used the equations of motion in the last line in order to define the
counterterm as an improvement term. The addition of this improvement term then
induces the appearance of a $W^{\mu}$ term in the trace equation. This pattern
is expected to persist at all orders in perturbation theory.

## Acknowledgments

The research of PK and RM is supported in part by the National Science
Foundation under research grant PHY-19-14412. The work of KR is supported by
the Mani L. Bhaumik Institute for Theoretical Physics

##  Appendix A Deformed action from the trace equation

In this appendix we show how one can determine the deformed action starting
from the trace equation (59) in the first order formalism using $\phi$ and its
canonical momentum $\Pi$. Although this approach might be most appropriate for
this work since we are mostly working in the first order formalism which makes
quantization easier, it is not very convenient for obtaining closed form
results mainly because it obscures Lorentz invariance.

We illustrate how one can construct the deformed action as a power in series
in $\lambda$ and we determine the first order correction. Let’s recall the
definition of the canonical stress tensor

$T_{\mu\nu}=2\pi\left(\delta_{\mu\nu}L-\frac{\partial
L}{\partial(\partial^{\mu}\phi)}\partial_{\nu}\phi\right)~{}.$ (131)

Using the definition of the canonical momentum

$\Pi=\frac{\partial L}{\partial\dot{\phi}}~{},$ (132)

we can express the components of the stress tensor as

$-\frac{1}{2\pi}T_{00}=H,\quad-\frac{1}{2\pi}T_{01}=\Pi\phi^{\prime},\quad-\frac{1}{2\pi}T_{10}=\frac{\partial
H}{\partial\Pi}\frac{\partial
H}{\partial\phi^{\prime}},\quad-\frac{1}{2\pi}T_{11}=H-\frac{\partial
H}{\partial\Pi}\Pi-\frac{\partial H}{\partial\phi^{\prime}}\phi^{\prime}~{}.$
(133)

To first order the most general gauge invariant (hence no $\phi$ polynomials)
expression for the Hamiltonian density is

$H=\frac{1}{2}(\phi^{{}^{\prime}2}+\Pi^{2})+\lambda(c_{1}\phi^{{}^{\prime}4}+c_{2}\phi^{{}^{\prime}3}\Pi+c_{3}\phi^{{}^{\prime}2}\Pi^{2}+c_{4}\phi^{{}^{\prime}}\Pi^{3}+c_{5}\Pi^{4})+\mathcal{O}(\lambda^{2})~{}.$
(134)

In this approach Lorentz invariance is encoded in the symmetry of the stress
tensor

$T_{01}=T_{10}\Rightarrow\Pi\phi^{\prime}=\frac{\partial
H}{\partial\Pi}\frac{\partial H}{\partial\phi^{\prime}~{}.}$ (135)

This fixes $c_{2}=c_{4}=0,c_{1}=c_{5}$ and $c_{3}=-2c_{5}$. H ence only one
coefficient remains by just requiring Lorentz invariance. Using (133) we then
find

$\displaystyle-\frac{1}{2\pi}T_{00}$
$\displaystyle=\frac{1}{2}(\phi^{{}^{\prime}2}+\Pi^{2})+c_{5}\lambda(\phi^{{}^{\prime}2}-\Pi^{2})^{2}+\mathcal{O}(\lambda^{2})~{},$
(136) $\displaystyle-\frac{1}{2\pi}T_{01}$
$\displaystyle=T_{10}=\Pi\phi^{\prime}~{},$ (137)
$\displaystyle-\frac{1}{2\pi}T_{11}$
$\displaystyle=-\frac{1}{2}(\phi^{{}^{\prime}2}+\Pi^{2})-3c_{5}\lambda(\phi^{{}^{\prime}2}-\Pi^{2})^{2}+\mathcal{O}(\lambda^{2})~{}.$
(138)

Plugging the above into the trace equation (59) we can fix the reaming
coefficient $c_{5}=\frac{1}{8}$.

##  Appendix B Conversion to Lorentzian signature

We relate the Euclidean and Lorentzian times as

$\displaystyle t_{E}=-it_{L}~{}.$ (139)

The (anti)-holomorphic coordinates then become light-cone coordinates

$\displaystyle z=x+it_{E}=x+t_{L}=x^{+}~{},$ (140)
$\displaystyle{\bar{z}}=x-it_{E}=x-t_{L}=x^{-}~{},$ (141)

so

$\displaystyle\partial_{+}=\frac{1}{2}(\partial_{x}+\partial_{t_{L}})~{},\quad\partial_{-}=\frac{1}{2}(\partial_{x}-\partial_{t_{L}})~{}.$
(142)

Since $dt_{E}=-idt_{L}$ the relation between the actions is $S_{E}=iS_{L}$,
i.e. so that

$\displaystyle S_{E}=\int dt_{E}dx((\partial_{t_{E}}\phi)^{2}+\phi^{\prime
2})=-i\int dt_{L}dx(-(\partial_{t_{L}}\phi)^{2}+\phi^{\prime 2})=iS_{L}~{}.$
(143)

The Euclidean action (16) then converts to the Lorentzian action

$\displaystyle
S_{L}=\frac{c_{0}}{24\pi}\int\\!dxdt_{L}(f^{\prime}\partial_{-}f+{\overline{f}}^{\prime}\partial_{+}{\overline{f}})~{}.$
(144)

We keep the relation between $f,{\overline{f}}$ and $\phi,\Pi$ as in (20) and
find

$\displaystyle
S_{L}=\int\\!dxdt_{L}\big{(}\Pi\dot{\phi}-\frac{1}{2}(\phi^{\prime
2}+\Pi^{2})\big{)}~{}.$ (145)

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