# Holographic Renormalization Group and Stress Tensor Operators

Stephen Ebert

###### Abstract

The holographic duality conjectures a relation between strongly coupled
quantum systems and quantum gravity in higher-dimensional spacetimes.
Gravitational theories in two and three dimensions are meaningful examples for
classical and quantum exploration due to their unique characteristics, notably
the absence of propagating bulk degrees of freedom and the presence of only
boundary degrees of freedom, distinguishing them from higher-dimensional
counterparts. These gravitational theories exhibit complex interactions when
the bulk spacetime has a finite size, regulated by Zamolodchikov’s double-
trace irrelevant $T\overline{T}$ operator.

This thesis aims to gain a holographic understanding of $\mathrm{AdS}_{3}$ and
JT gravity under the influence of the $T\overline{T}$ deformation. Under a
finite radial cutoff, these theories exhibit perturbative behavior that
implies the emergence of the Nambu-Goto action for the corresponding boundary
graviton action. We also conducted semi-classical calculations of observables
related to finite-cutoff gravity and its dual $T\overline{T}$-deformed CFT
description, including correlation functions involving stress tensors and
gravitational Wilson lines, along with an analysis of their supersymmetric
extensions. Additionally, we explored the implications of general stress
tensor deformations within field-theoretic and holographic settings.

This thesis integrates previously adapted publications while also pioneering
new ground, notably exploring the definition of a quantum $T\overline{T}$
operator beyond two dimensions with $\frac{1}{N}$ corrections, investigating
quantum-corrected higher point correlators for a planar boundary, and offering
insights into a two-dimensional spherical boundary at a finite cutoff.
Furthermore, throughout the thesis, we show more details in calculations at
various points. [1]

Physics 2024 Zvi Bern Eric D’Hoker Michael Gutperle Per J. Kraus To my
parents.

###### Acknowledgements.

I am grateful to my advisor, Per Kraus, for his support and discussions
throughout my graduate school career at UCLA. Additional praise goes towards
the high energy theory faculty at UCLA: Zvi Bern, Eric D’Hoker, Thomas
Dumitrescu, and Michael Gutperle for facilitating the string theory journal
club, organizing the Southern California Strings Seminar at UCLA and Caltech,
and making my academic years at UCLA some of the most intellectually
stimulating. Besides UCLA’s high-energy theory faculty, I am also thankful to
our theory group’s graduate students and postdoctoral fellows from whom I have
learned several high-energy theory topics including Sriram Bharadwaj, Kevin
Chen, Shouvik Datta, Amey Gaikwad, John Gardiner, Nicholas Geiser, Enrico
Herrmann, Justin Kaidi, Seolhwa Kim, Nicholas Klein, Yanyan Li, Lukas
Lindwasser, Benjamin Michel, Emily Nardoni, Pierluigi Niro, Julio Parra-
Martinez, Konstantinos Roumpedakis, Trevor Scheopner, and Orr Sela. Despite
not working from my office frequently, I am happy to have two wonderful office
mates, Seolhwa Kim, and Jonah Hyman, to chat with when I was there. I am
incredibly thankful for my excellent collaborators’ camaraderie and from whom
I have learned a tremendous amount of physics related and unrelated to our
projects: Christian Ferko, Eliot Hijano, Per Kraus, Cian Luke Martin, Ruben
Monten, Richard Myers, Atul Sharma, Hao-Yu Sun, Zhengdi Sun, Gabriele
Tartaglino-Mazzucchelli, Diandian Wang, Ziqi Yan, and Mengyang Zhang.
Collaborating with these physicists has enriched my intellectual curiosity and
development as a researcher. In addition, many other physicists outside of
UCLA whom I had the privilege of learning and interacting: Chris Akers, Mykola
Dedushenko, Alexander Frenkel, Michael Green, Steven Gubser, Christian Jepsen,
Matthew Heydeman, Nathan Haouzi, Fernando Iniguez, Ken Intriligator, Jacob
Leedom, John Schwarz, Savdeep Sethi, Alessandro Sfondrini, Dmitri Sorokin,
Gabriele Veneziano, and Zhencheng Wang. I am also thankful to the professors
and advisors I had during my time as an undergraduate at UC Berkeley: Ashok
Ajoy, James Analytis, Richard Borcherds, Joel Fajans, William Holzapfel,
Lawrence Hall, Yury Kolomensky, Daniel Kasen, Alessandra Lanzara, Daniel
McKinsey, Joseph Orenstein, Matthias Reinsch, and Peter Sorenson. In
particular, I thank Ori Ganor and Petr Hořava for their in-depth courses in
string theory, quantum field theory, and general relativity. I especially
thank Ori for his mentoring during my numerous visits to the Berkeley Center
for Theoretical Physics in the Summer of 2019 when I rode Bay Area Rapid
Transit (BART) from Fremont to Berkeley. I am deeply grateful to my parents,
Stephen and Marsha Ebert, siblings Mark and Stephanie Ebert, and grandparents
Richard and Maxine Ebert, along with Raffi and Dorthy Krikorian, for their
lifelong love and support. Outside of research, I am grateful to the friends I
made in San Jose, other cities in the San Francisco Bay Area, and in my
academic career: Nijaid Arredondo, Eve Bodnia, Peter Boyle, Evan Bruce, Parker
Bruce, Steve Cao, Randall Clark, Andrew Christensen, Tahv Demayo, Jiashu Han,
Yifan Hong, Matthew Lee, Ruiyao Liu, and Dhruv Muley. I acknowledge funding
support throughout my Ph.D. from the Bhaumik Institute and the Dissertation
Year Fellowship from the UCLA graduate division. Contribution of authors
Chapter 1 of this thesis reports on work with Eliot Hijano, Per Kraus, Ruben
Monten, and Richard Myers [2]. Chapter 2 reports on work with Christian Ferko,
Hao-Yu Sun and Zhengdi Sun [3]. Chapter 3 reports on work with Christian
Ferko, Hao-Yu Sun, and Zhengdi Sun [4]. Chapter 4 reports on work with Hao-Yu
Sun, and Zhengdi Sun [5]. Chapter 5 reports on work with Christian Ferko and
Zhengdi Sun [6]. Chapter 6 reports on work with Christian Ferko, Cian Luke
Martin, and Gabriele Tartaglino-Mazzucchelli [7]. This thesis omits other
published papers [1, 8, 9, 10, 11], in which several authors and I
collaborated on. 2016 – 2018 B.A. Physics, University of California, Berkeley.
2018 – 2020M.Sc. Physics, University of California, Los Angeles. 2018 – 2024
Ph.D. Physics, University of California, Los Angeles (Expected). 9 S. Ebert,
C. Ferko, C. L. Martin, and G. Tartaglino-Mazzucchelli, “Flows in the Space of
Interacting Chiral Boson Theories,” arXiv: 2403.18242. S. Ebert, Z. Yan,
“Anistropic Compactification of Nonrelativistic M-theory,”
JHEP 11 (2023) 135. S. Ebert, C. Ferko, and Z. Sun, “Root-$T\overline{T}$
Deformed Boundary Conditions in Holography,” Phys. Rev. D 107, 126022.S.
Ebert, C. Ferko, H.-Y Sun and Z. Sun, “$T\overline{T}$ in JT Gravity and BF
Gauge Theory,” SciPost Phys. 13 (2022) 096.S. Ebert, C. Ferko, H.-Y. Sun, and
Z. Sun, “$T\overline{T}$ Deformations of Supersymmetric Quantum Mechanics,”
JHEP 08 (2022) 121.S. Ebert, H.-Y. Sun, and Z. Sun, “$T\overline{T}$-deformed
free energy of the Airy model,” JHEP 08 (2022) 026.S. Ebert, E. Hijano, P.
Kraus, R. Monten, and R. M. Myers, “Field Theory of Interacting Boundary
Gravitons,” SciPost 13 (2022) 038.S. Ebert, H.-Y. Sun, and Z. Yan, “Dual
D-brane actions in nonrelativistic string theory,” JHEP 04 (2022) 161.S.
Ebert, H.-Y. Sun, and Z. Sun, “$T\overline{T}$ deformation in SCFTs and
integrable supersymmetric theories,” JHEP 09 (2021) 082.S. Ebert, A. Sharma,
and D. Wang, “Descendants in celestial CFT and emergent multi-collinear
factorization,” JHEP 03 (2021) 030.S. Ebert, H.-Y. Sun, and M.-Y. Zhang,
“Probing holography in $p$-adic CFT,” Phys. Rev. D 107, 126011.

##  Chapter 0 Introduction

In contemporary perspectives, the holographic principle provides the most
comprehensive framework for understanding quantum gravity, sometimes called
gauge/gravity duality or the Maldacena conjecture, which, loosely speaking,
states that quantum gravity in $d+1$ dimensions is equivalent to a
$d$-dimensional field theory. The AdS/CFT correspondence stands out as a solid
and early foundation for this phenomenon, providing what is arguably our most
modern advanced understanding of quantum gravity. AdS stands for “anti-de
Sitter,” which is the maximally symmetric solution of Einstein’s equations
with a negative cosmological constant. CFT stands for “conformal field
theory,” which is a type of quantum field theory characterized by its
invariance under conformal transformations. The AdS/CFT correspondence is one
of the most significant results that string theory produced during the second
superstring revolution.

String theory, ambitiously aiming to unify quantum mechanics and general
relativity, gained widespread acceptance in the first superstring revolution
(1984-1985). Key developments include the Green-Schwarz anomaly mechanism
[12], validating the consistency of supersymmetric gauge theories in ten
dimensions when coupled to supergravity with gauge groups $SO(32)$ or
$E_{8}\times E_{8}$. The discovery of heterotic string theories with these two
gauge groups [13] and the realization that $E_{8}\times E_{8}$ heterotic
string theory allows solutions with a six-dimensional Calabi-Yau space,
resulting in a realistic four-dimensional field theory at low energies [14],
marked significant progress. Eventually, five superstring theories emerged in
ten dimensions: type I, type IIA, type IIB, $SO(32)$ heterotic, and
$E_{8}\times E_{8}$ heterotic [15, 13]. The understanding of non-perturbative
effects awaited the second superstring revolution in the mid-1990s.

The second superstring revolution brought groundbreaking advancements,
unveiling perturbative and non-perturbative dualities (i.e. T-duality and
S-duality) that connect the five distinct ten-dimensional superstring
theories. These theories emerge from a single theory called M-theory found by
Edward Witten [16], which lacks description in terms of quantized fundamental
strings but instead described by non-perturbative solitonic membranes living
in eleven dimensions. There were other developments, including the Matrix-
theory interpretation of M-theory [17, 18, 19, 20, 21, 22, 23, 24, 25], the
black hole entropy derived from string theory [26], and the establishment of
the AdS/CFT correspondence [27, 28, 29, 30] were key developments.111For
reviews and textbooks on string/M-theory, see [31, 32, 33, 34, 35, 36, 37, 38,
39, 40, 41, 42].

To truly grasp the significance of the AdS/CFT correspondence, it is essential
to delve deeper. On one side, we have a theory of quantum gravity, operating
in spacetime with $d+1$ dimensions, encompassing $d$ spatial dimensions and
one temporal dimension. On the other side, there is what we call a “CFT” also
inhabiting $d$ spacetime dimensions. This CFT does not incorporate gravity. In
essence, the AdS/CFT correspondence reveals the emergence of an additional
spatial dimension as we transition between the gravity-free realm of the CFT
and the gravity-filled AdS space.

In 1997, Juan Maldacena proposed groundbreaking dualities that connect
specific string and M-theory solutions in AdSd+1 times a compact space to a
CFTd [27]. The shared $SO(d,2)$ symmetry between the conformal symmetry group
and AdSd+1’s isometry group supports these connections. Gubser, Klebanov,
Polyakov, and Witten established the GKPW dictionary, precisely relating
amplitudes and correlation functions in these theories [28, 30].

Maldacena explored three maximally symmetric backgrounds, with the most
studied in the literature being type IIB superstring theory in
$\operatorname{AdS}_{5}\times S^{5}$ dual to four-dimensional $\mathcal{N}=4$
super Yang-Mills theory with an $SU(N)$ gauge group, based on $N$ coincident
D3-branes. Two other examples linked M-theory in $\operatorname{AdS}_{4}\times
S^{7}$ and $\operatorname{AdS}_{7}\times S^{4}$ to three-dimensional and six-
dimensional superconformal field theories. A decade later from [27], the
understanding of the $\operatorname{AdS}_{4}\times S^{7}$ example was expanded
by Aharony, Bergman, Jafferis, and Maldacena (ABJM) [43]. They considered a
more general setting: M-theory on $\operatorname{AdS}_{4}\times
S^{7}/\mathbb{Z}_{k}$ with $k$ as the level of the Chern-Simons terms, leading
to a dual gauge theory, a three-dimensional $\mathcal{N}=6$ superconformal
Chern-Simons theory with gauge group $U(N)_{k}\times U(N)_{-k}$ and $N$
coincident M2-branes. The $\operatorname{AdS}_{7}\times S^{4}$ example,
involving $N$ coincident M5-branes, remains challenging and less understood.

The foundation of the AdS/CFT correspondence comes from a strategy developed
by ’t Hooft in 1974 for the large-$N$ expansion of quantum chromodynamics
(QCD) and similar theories [44]. This strategy posits that a gauge theory with
large degrees of freedom $N$ can be equated with a string theory. In this
context, the primary variables for the gauge theory are $N\times N$ matrices,
where $N$ corresponds to the rank of a specific gauge group.

However, while this universal argument provides a starting point for the
large-$N$ duality, it doesn’t inherently indicate which specific string theory
is the dual description of a given gauge theory. Determining the dynamics of
the string worldsheet requires separate, independent methods, a task that is
feasible only in select instances. One notable example is found in maximally
supersymmetric Yang-Mills theories, which possess distinct characteristics
that enable the unique identification of their dual string theory, as
elucidated by Maldacena [27].

This thesis explores the AdS/CFT correspondence in low dimensions under stress
tensor deformations.222For more in-depth background and applications of the
holographic duality, refer to [45, 46, 47, 48, 49, 50, 51, 52, 53].

###  1 AdS/CFT in low dimensions

Despite the rigorous string-theoretic formulation and motivation of the
AdS/CFT correspondence from the previous section, the earliest precursor of
the AdS/CFT correspondence comes from the study of pure AdS3 gravity by Brown
and Henneaux [54]. Brown and Henneaux famously discovered the asymptotic
spacetime algebra of AdS3 given by two copies of the Virasoro algebra with
central charge $c=\frac{3\ell}{2G}$. The CFT quantity $c$ is related to two
bulk quantities: $\ell$ being the AdS3 length scale and Newton’s gravitational
coupling $G$. The large central charge limit corresponds to semi-classical
behavior, whereas finite central charge leads to fully quantized AdS3 gravity.

The perturbative expansion around an AdS3 background is well understood: one
obtains a theory of boundary gravitons governed by Virasoro symmetry [54, 55].
The quantum theory of these boundary gravitons is perfectly sensible and self-
contained, with a well-defined Hilbert space and spectrum of local operators.
Indeed, the boundary graviton theory is simple as the action and stress tensor
are rendered quadratic in appropriate field variables [56, 57, 58]. In CFT
parlance, this theory describes the Virasoro vacuum block of some putative CFT
with some spectrum of primary operators. A much-studied problem is how to
reconcile the desired modular invariance of such a spectrum with a sum over
geometries interpretation in gravity, e.g., [59, 55, 60].

There are two salient features of three-dimensional gravity motivating this
thesis.

The first feature is that the three-dimensional Einstein-Hilbert action

$I[g_{\mu\nu}]=-\frac{1}{16\pi
G}\int_{M_{3}}d^{3}x\sqrt{g}\left(R-2\Lambda\right)-\frac{1}{8\pi
G}\int_{\partial M_{3}}d^{2}x\sqrt{h}\left(K-1\right)\,.$ (1)

in the first-order formulation of general relativity; may be written semi-
classically as a Chern-Simons gauge theory, as observed by [61, 62]. One can
then calculate fundamental observables in the gravitational Chern-Simons
theory, such as Wilson lines and loops obtained from path-ordered exponential
integrals of the one-form connection $A_{\mu}(x)$ along an open interval and a
closed contour, respectively.

In the following discussion, we will use the term “gravitational Wilson lines”
to refer to Wilson lines associated with the Chern-Simons gauge field
$A_{\mu}$. When considering two endpoints, denoted as $Z_{1}=(r_{1},z_{1})$
and $Z_{2}=(r_{2},z_{2})$ located on the AdS3 boundary, the gravitational
Wilson line linking these points can be expressed as:

$W[Z_{2},Z_{1}]=P\exp\left(\int^{Z_{2}}_{Z_{1}}A_{\mu}(x)\,dx^{\mu}\right)\,.$
(2)

In the classical (i.e. large-$c$) limit, the object $W[Z_{2},Z_{1}]$ has the
peculiar property that it transforms as a bi-local primary operator at its
endpoints. In [63, 64], at least in perturbative terms with respect to
$\frac{1}{c}$, the _quantum_ Wilson line appears to undergo a transformation
resembling that of a bi-local primary operator at its endpoints. As argued in
[65, 66, 67, 68, 69, 70, 63, 64], another physical feature of the Wilson line
is that it serves as a convenient repackaging of the Virasoro vacuum OPE block

$\langle T_{zz}(w_{1})\cdots T_{zz}(w_{n})W[z_{2},z_{1}]\rangle_{0}=\langle
T_{zz}(w_{1})\cdots T_{zz}(w_{n})O(z_{2})O(z_{1})\rangle_{0}\,,$ (3)

where $\langle W[z_{2},z_{1}]\rangle_{0}$ will be later defined by (209) in
terms of a path-ordered exponential integral involving only the stress tensor
operator’s holomorphic component $T_{zz}$.

The Virasoro vacuum OPE block – whose characteristics are similar to those of
the gravitational Wilson line – precisely captures all operators built out of
the stress tensor operator appearing in the OPE of two primary operators
$O(z_{1})$ and $O(z_{2})$. Schematically (suppressing numerical factors,
coordinate dependence, and derivatives in the OPE coefficient $C_{OOO_{i}}$,
as well as omitting the $T_{\bar{z}\bar{z}}$ piece), the first term in the
following OPE of two primary operators

$O(z_{2})O(z_{1})=\left(1+T_{zz}+T_{zz}T_{zz}+\cdots\right)+\sum_{i}C_{OOO_{i}}\left(O_{i}+O_{i}T_{zz}+O_{i}T_{zz}T_{zz}+\cdots\right)\,,$
(4)

corresponds to the Virasoro vacuum OPE block.

From the bulk perspective, the (open) gravitational Wilson line calculates the
exponential of the worldline action for a massive point particle, accounting
for gravitational self-interaction effects that renormalize its mass [71, 63,
72, 64]. Conversely, the closed gravitational Wilson line, or the Wilson loop,
quantifies the holonomy of the gauge connection. In the context of a BTZ black
hole, when the Wilson loop encircles the horizon, its value corresponds to the
Bekenstein-Hawking entropy [73, 74].

The second feature is that upon dimensionally reducing $3d$ gravity (1) on a
circle with a radius equal to the dilaton $\Phi$, one obtains the JT gravity
action

$I[g^{(2)}_{\mu\nu},\Phi]=-\frac{1}{16\pi
G}\int_{M_{2}}d^{2}x\sqrt{g^{(2)}}\Phi\left(R-2\Lambda\right)-\frac{1}{8\pi
G}\int_{\partial M_{2}}d\tau\sqrt{\gamma}\Phi\left(K-1\right)\,,$ (5)

where $g^{(2)}_{\mu\nu}$ is the $2d$ bulk metric and $\gamma_{\tau\tau}$ is
the $1d$ boundary metric. The holographic dual description of JT gravity is
the $1d$ Schwarzian theory [75, 76, 77, 78, 79].

Just as one can reduce $3d$ gravity on a circle in the metric formalism, one
may also perform the dimensional reduction for the Chern-Simons description of
$3d$ gravity and obtain the so-called BF gauge theory, which enjoys similar
features to $2d$ Yang-Mills gauge theory with a non-compact group. BF theory
with gauge group $G$ is holographically dual to the worldline theory of a
particle with a free kinetic Lagrangian and moves on target space $G$; we call
this the “particle-on-a-group” theory.

Given these complementary perspectives and technical advantages of lower
dimensional gravity, it is desirable to understand these corners of the
following diagram below in different settings.

(6)

An area of particular relevance for this thesis involves the examination of
diagram (6) in the presence of Zamolodchikov’s double-trace $T\overline{T}$
operator, which is considered an irrelevant operator [80]. Let us now review
the $T\overline{T}$ deformation.

###  2 Zamolodchikov’s $T\overline{T}$ deformation

Understanding irrelevant deformations are notoriously difficult compared to
marginal and relevant deformations. Turning on an irrelevant operator will
generically turn on infinitely many additional operators at high energies,
which modifies the theory in the UV and leads to a loss of predictive power.
However, the $T\overline{T}$ deformation is one irrelevant operator333The
stress tensor has mass dimension $d$ in $d$-dimensions, so the $T\overline{T}$
operator has dimension $2d$ and is irrelevant in any number of spacetime
dimensions since $2d>d$. Marginal operators have scaling dimension $\Delta=d$
while relevant operators have scaling dimension $\Delta>d$. which circumvents
these technical difficulties: a $T\overline{T}$-deformed quantum field theory
remains under some analytic control and is “solvable.” Since this operator is
irrelevant, a $T\overline{T}$ flow may seem to be the opposite of a
conventional renormalization group flow triggered by the addition of a
relevant operator. However, a conventional RG flow connects a family of local
QFTs controlled by an RG fixed point in the UV. It is known that a
$T\overline{T}$-deformed field theory is not a local QFT and thus not
controlled by a CFT in the UV, so this flow is not like a conventional RG flow
even in reverse.

A remarkable consequence arising from the behavior of $T\overline{T}$-deformed
one-point functions is the existence of a differential equation governing the
finite-volume spectrum of a $T\overline{T}$-deformed theory. This equation
establishes a relationship between energy levels in the
$T\overline{T}$-deformed theory and those in the undeformed seed theory [81,
82]. This relationship between the energy levels exemplifies what we mean by
characterizing the deformation as “solvable.” The $T\overline{T}$ deformation
is solvable because various quantities within the deformed theory, including
the torus partition function [83, 84, 85], flat space S-matrix [86, 87], and
correlation functions [88, 89, 90, 91, 92, 93, 94, 9, 5, 95, 2, 96], can be
expressed in terms of corresponding data from the undeformed theory.

It is worth noting that the $T\overline{T}$ operator in two spacetime
dimensions retains several symmetries of the original seed theory. These
preserved symmetries encompass integrability [81, 87] and supersymmetry [97,
98, 99, 100, 101]. For more information on manifestly supersymmetric
$T\overline{T}$-like flows, refer to [102, 103, 104, 105, 106, 4]. However,
it’s important to note that conformal symmetry is not preserved because the
flow parameter $\lambda$ introduces a dimensionful scale to the deformed
theory.

The $T\overline{T}$ operator is a $2d$ bilinear operator constructed of the
stress tensor $T_{\mu\nu}$ which can be expressed as the determinant

$\det\left(T_{\mu\nu}\right)=\frac{1}{2}\left(\left(T^{\mu}{}_{\mu}\right)^{2}-T^{\mu\nu}T_{\mu\nu}\right)\,,$
(7)

and is unambiguously defined by point-splitting up to total derivatives of
local operators that can be neglected [80, 81]. Zamolodchikov assumed the seed
QFT has the following properties:

  1. 1.

Local translation and rotation symmetry

The stress tensor is conserved $\partial^{\mu}T_{\mu\nu}=0$ and symmetric
$T_{\mu\nu}=T_{\nu\mu}$.

  2. 2.

Global translation symmetry

One-point functions are independent of the position
$\langle\mathcal{O}_{i}(z)\rangle=\langle\mathcal{O}_{i}(0)\rangle$ and any
two-point functions
$\langle\mathcal{O}_{i}(z)\mathcal{O}_{j}(z^{\prime})\rangle=F_{ij}(z-z^{\prime})$.

  3. 3.

Clustering

There exists some direction $r\rightarrow\infty$ such that
$\lim_{r\rightarrow\infty}\langle\mathcal{O}_{i}(r)\mathcal{O}_{j}(0)\rangle=\langle\mathcal{O}_{i}\rangle\langle\mathcal{O}_{j}\rangle$.

  4. 4.

UV CFT

The seed QFT is a CFT at short distances.

With these four conditions, we show the coincident point limit defines a local
operator:

$T\overline{T}=\lim_{z^{\prime}\rightarrow
z}\left(T_{zz}(z^{\prime})T_{\bar{z}\bar{z}}(z)-T_{z\bar{z}}(z^{\prime})T_{z\bar{z}}(z^{\prime})\right)\,.$
(8)

Furthermore, the conservation of the stress tensor gives

$\displaystyle\partial_{\bar{z}}\left(T_{zz}(z)T_{\bar{z}\bar{z}}\left(z^{\prime}\right)-T_{z\bar{z}}(z)T_{z\bar{z}}\left(z^{\prime}\right)\right)=\left(\partial_{z}+\partial_{z^{\prime}}\right)T_{z\bar{z}}(z)T_{\bar{z}\bar{z}}\left(z^{\prime}\right)-\left(\partial_{\bar{z}}+\partial_{\bar{z}^{\prime}}\right)T_{z\bar{z}}(z)T_{z\bar{z}}\left(z^{\prime}\right)$
(9)
$\displaystyle\partial_{z}\left(T_{zz}(z)T_{\bar{z}\bar{z}}\left(z^{\prime}\right)-T_{z\bar{z}}(z)T_{z\bar{z}}\left(z^{\prime}\right)\right)=\left(\partial_{z}+\partial_{z^{\prime}}\right)T_{zz}(z)T_{\bar{z}\bar{z}}\left(z^{\prime}\right)-\left(\partial_{\bar{z}}+\partial_{\bar{z}^{\prime}}\right)T_{zz}(z)T_{z\bar{z}}\left(z^{\prime}\right)\,.$

Next, recall the stress tensor’s OPEs

$\displaystyle T_{zz}(z)T_{z\bar{z}}(z^{\prime})$
$\displaystyle=\sum_{i}A_{i}(z-z^{\prime})\mathcal{O}_{i}(z^{\prime})\,,\quad
T_{z\bar{z}}(z)T_{z\bar{z}}(z^{\prime})=\sum_{i}C_{i}(z-z^{\prime})\mathcal{O}_{i}(z^{\prime})\,,$
(10) $\displaystyle T_{z\bar{z}}(z)T_{\bar{z}\bar{z}}(z^{\prime})$
$\displaystyle=\sum_{i}B_{i}(z-z^{\prime})\mathcal{O}_{i}(z^{\prime})\,,\quad
T_{zz}(z)T_{\bar{z}\bar{z}}(z^{\prime})=\sum_{i}D_{i}(z-z^{\prime})\mathcal{O}_{i}(z^{\prime})\,.$

The stress tensor conservation equations (LABEL:eq:f0e23ersdfs) yield

$\displaystyle\sum_{i}\partial_{\bar{z}}\left(D_{i}(z-z^{\prime})-C_{i}(z-z^{\prime})\right)\mathcal{O}_{i}\left(z^{\prime}\right)$
(11)
$\displaystyle=\sum_{i}\left(B_{i}\left(z-z^{\prime}\right)\partial_{z^{\prime}}\mathcal{O}_{i}(z^{\prime})-C_{i}\left(z-z^{\prime}\right)\partial_{\bar{z}^{\prime}}\mathcal{O}_{i}(z^{\prime})\right)$

and

$\displaystyle\sum_{i}\partial_{z}\left(D_{i}(z-z^{\prime})-C_{i}(z-z^{\prime})\right)\mathcal{O}_{i}\left(z^{\prime}\right)$
(12)
$\displaystyle=\sum_{i}\left(D_{i}\left(z-z^{\prime}\right)\partial_{z^{\prime}}\mathcal{O}_{i}\left(z^{\prime}\right)-A_{i}\left(z-z^{\prime}\right)\partial_{\bar{z}^{\prime}}\mathcal{O}_{i}\left(z^{\prime}\right)\right)\,.$

Consequently, this implies that any operator arising in the following OPE

$T_{zz}(z)T_{\bar{z}\bar{z}}(z^{\prime})-T_{z\bar{z}}(z)T_{z\bar{z}}(z^{\prime})=\sum_{i}\left(D_{i}(z-z^{\prime})-C_{i}(z-z^{\prime})\right)\mathcal{O}_{i}(z^{\prime})$
(13)

must have a coordinate independent coefficient function
$D_{i}(z-z^{\prime})-C_{i}(z-z^{\prime})$ or itself is the derivative of
another local operator

$T_{zz}(z)T_{\bar{z}\bar{z}}(z^{\prime})-T_{z\bar{z}}(z)T_{z\bar{z}}(z^{\prime})=\mathcal{O}_{T\overline{T}}(z^{\prime})+\sum_{\alpha}A_{\alpha}(z-z^{\prime})\partial_{z^{\prime}}O_{\alpha}(z^{\prime})\,.$
(14)

Therefore, we arrive at the composite operator Zamolodchikov [80] found

$T\overline{T}=\mathcal{O}_{T\overline{T}}(z)$ (15)

up to derivative terms, but these vanish due to global translation symmetry
where a one-point function of a total derivative is zero.

At the classical level, given a seed theory’s Lagrangian $\mathcal{L}^{(0)}$,
the $T\overline{T}$ flow is captured by the differential equation

$\frac{\partial\mathcal{L}^{(\lambda)}}{\partial\lambda}=-\det\left(T^{(\lambda)}_{\mu\nu}\right)\,,$
(16)

where the notation $T^{(\lambda)}_{\mu\nu}$ emphasizes that at each step along
the flow, the stress tensor is recomputed from the deformed Lagrangian
$\mathcal{L}^{(\lambda)}$. We give an example of $N$ free massless scalars in
the subsequent subsection.

####  1 $T\overline{T}$-deformed free scalars

In this subsection, we review the solution for the $T\overline{T}$ flow
equation for the deformed Lagrangian of multiple free scalar fields444Appendix
9.C considers the case for a scalar theory with a potential $V(\phi)$.

$S(0)=\frac{1}{4\pi}\sum^{N}_{n=1}\int
d^{2}x\sqrt{g}g^{\mu\nu}\partial_{\mu}\phi_{n}\partial_{\nu}\phi_{n}\,.$ (17)

We first consider a single scalar field and write the ansatz for the deformed
action

$S(\lambda)=\int
d^{2}x\sqrt{g}\frac{F(\lambda\partial^{\mu}\phi\partial_{\mu}\phi)}{\lambda}$
(18)

and substitute (18) into the flow equation

$\frac{dS(\lambda)}{d\lambda}=\int d^{2}x\sqrt{g}T\overline{T}(x)$ (19)

to obtain a differential equation for $F(z)$ with solution

$S(\lambda)=\int
d^{2}x\sqrt{g}\left(\frac{1-\sqrt{1-\pi\lambda\partial^{\mu}\phi\partial_{\mu}\phi}}{2\pi^{2}\lambda}\right)\,.$
(20)

A more sophisticated ansatz is required when dealing with multiple scalars

$\mathcal{L}(\lambda)=F(\lambda
g^{\mu\nu}\partial_{\mu}\phi_{n}\partial_{\nu}\phi_{n},\lambda^{2}g^{\mu\sigma}g^{\nu\rho}\partial_{\rho}\phi_{m}\partial_{\sigma}\phi_{m}\partial_{\mu}\phi_{n}\partial_{\nu}\phi_{n})$
(21)

and the solution to the partial differential equation for $F$ is now a
determinant

$S(\lambda)=\int
d^{2}x\left(\frac{\sqrt{g}-\sqrt{\det\left(g_{\mu\nu}-\pi\lambda\partial_{\mu}\phi_{n}\partial_{\nu}\phi_{n}\right)}}{2\pi^{2}\lambda}\right)\,.$
(22)

The action (22) is the Nambu-Goto action written in static gauge

$X^{0}=x^{0},\quad X^{n}=\sqrt{-\pi\lambda}\phi_{n},\quad X^{N+1}=x^{1}$ (23)

implying

$S(\lambda)=\frac{1}{2\pi^{2}\lambda}\int
d^{2}x\sqrt{\det\left(\partial_{\mu}X^{A}\partial_{\nu}X^{A}\right)}+\operatorname{constant}\,,\quad
A=0,1,\cdots,N+1\,.$ (24)

The Nambu-Goto action is invariant under an $SO(N+2)$ global symmetry and
reparametrization invariance. The nonlinear realization of the $SO(2)$
symmetry in the gauge-fixed form (22) arises due to the necessity of
implementing a compensatory reparametrization to preserve the static gauge.

As we will see in later chapters of this thesis, the same kind of Nambu-Goto
action for the deformed scalar theory also appears in the context of low-
dimensional $T\overline{T}$ deformed gravitational theories. For example, in
Euclidean signature, the resulting boundary graviton action in cutoff AdS3
with planar boundary is

$\displaystyle I(r_{c})=\frac{1}{32\pi G}\int d^{2}x\bigg{[}$ $\displaystyle
i\left(f^{\prime}\dot{f}-\bar{f}^{\prime}\dot{\bar{f}}\right)$ (25)
$\displaystyle-\frac{4}{r_{c}}\left(\sqrt{1-\frac{r_{c}}{2}\left(f^{\prime
2}+\bar{f}^{\prime 2}\right)+\frac{r_{c}^{2}}{16}\left(f^{\prime
2}-\bar{f}^{\prime 2}\right)^{2}}-1\right)\bigg{]}$

and its dimensional reduction yields the deformed Schwarzian theory in JT
gravity. To be explained in greater depth, $r_{c}$ is the radial location of
the boundary such that $r_{c}\rightarrow 0$ (or $\lambda\rightarrow 0$) is the
conformal boundary and the fields $(f,\bar{f})$ nonlinearly realize Lorentz
symmetry.

Holographically, the $T\overline{T}$ deformation has the interpretation of
gravity with Dirichlet boundary conditions for the metric at a finite radial
cutoff $r_{c}$ which is related to $\lambda$ via the $T\overline{T}$ flow
equation (16).

In the forthcoming chapters, we delve into $T\overline{T}$-deformed low-
dimensional holography to understand (6) in greater detail. Nevertheless,
before delving into that discussion, we will briefly touch upon some
distinctive features of the $T\overline{T}$ deformation. Following this, we
will provide the rationale for exploring the AdS/CFT correspondence with a
finite radial cutoff in the subsequent two sections.

###  3 Factorizaton and flow equation

We derive the $T\overline{T}$ flow equation for the energy levels of a QFT on
a cylinder with radius $R$. We demonstrate that in the case of a conformal
seed theory, the energy spectrum of the $T\overline{T}$-deformed system
exhibits square root behavior. Throughout this thesis, we exclusively examine
seed theories characterized by conformal symmetry, and we anticipate observing
analogous square root-like behaviors in these cases.

It is imperative to establish the independence of the expectation value of the
$T\overline{T}$ operator with respect to the distance $D(z,w)$ between
insertion points

$D(z,w)=\langle T_{zz}(z)T_{\bar{z}\bar{z}}(w)\rangle-\langle
T_{z\bar{z}}(z)T_{z\bar{z}}(w)\rangle\,.$ (26)

Next, use (LABEL:eq:f0e23ersdfs) and differentiate with respect to $\bar{z}$

$\partial_{\bar{z}}D(z,w)=\partial_{z}\langle
T_{z\bar{z}}(z)T_{\bar{z}\bar{z}}(w)\rangle-\partial_{\bar{z}}\langle
T_{z\bar{z}}(z)T_{z\bar{z}}(w)\rangle\,.$ (27)

From the assumption of global translation invariance,

$\partial_{\bar{z}}\langle
T_{z\bar{z}}(z)T_{z\bar{z}}(w)\rangle=-\partial_{\bar{w}}\langle
T_{z\bar{z}}(z)T_{z\bar{z}}(w)\rangle$ (28)

and

$\partial_{z}\langle
T_{z\bar{z}}(z)T_{\bar{z}\bar{z}}(w)\rangle=-\partial_{w}\langle
T_{z\bar{z}}(w)T_{\bar{z}\bar{z}}(w)\rangle\,.$ (29)

Hence, by conservation of the stress tensor,

$\partial_{\bar{z}}D(z,w)=-\langle
T_{z\bar{z}}(z)\partial_{w}T_{\bar{z}\bar{z}}(w)\rangle+\langle
T_{z\bar{z}}(z)\partial_{\bar{w}}T_{z\bar{z}}(w)\rangle=0\,.$ (30)

Therefore, we have shown that $D(z,w)$ is a constant. When $z\rightarrow w$

$\lim_{z\rightarrow w}D(z,w)=\langle T_{zz}T_{\bar{z}\bar{z}}\rangle\,.$ (31)

In the context of a QFT on a cylinder, we consider the points $z$ and $w$
infinitely separated along the non-compact dimension of the cylinder. We use
the fact that the vacuum two-point functions cluster decompose into products
of one-point functions at infinite separation to find

$\lim_{|z-w|\rightarrow\infty}D(z,w)=\langle T_{zz}\rangle\langle
T_{\bar{z}\bar{z}}\rangle-\langle T_{z\bar{z}}\rangle^{2}\,.$ (32)

Hence, with the above discussion, we conclude

$\langle T_{zz}T_{zz}\rangle=\langle T_{zz}\rangle\langle
T_{\bar{z}\bar{z}}\rangle-\langle T_{z\bar{z}}\rangle^{2}$ (33)

proves the factorization property of the $T\overline{T}$ operator. This
derivation assumes we are in the vacuum state $|0\rangle$, but the
$T\overline{T}$ operator factorizes for any energy eigenstate $|n\rangle$.

Proving that factorization holds for any energy eigenstate $|n\rangle$ is
straightforward. First, from a spectral expansion for a complete set of
states, we have

$\displaystyle\langle n|T_{zz}(z)T_{\bar{z}\bar{z}}(w)|n\rangle$ (34)
$\displaystyle=\sum_{m}\langle n|T_{zz}(z)|m\rangle\langle
m|T_{\bar{z}\bar{z}}(w)|n\rangle
e^{(E_{n}-E_{m})|\operatorname{Im}(z)-\operatorname{Im}(w)|+i(P_{n}-P_{m})|\operatorname{Re}(z)-\operatorname{Re}(w)|}\,.$

The exponential factors contain dependence on coordinates $(z,w)$. However, we
know that the function $D(z,w)$ is constant for correlators of any energy
eigenstate. Hence, the sum expressed in equation (LABEL:eq:erie) is zero for
$m\neq n$, rendering the exponential factor equal to one. Consequently, we
attain a meaningful limit as the points coincide, i.e. $z\rightarrow w$:

$\langle n|T_{zz}T_{zz}|n\rangle=\langle n|T_{zz}|n\rangle\langle
n|T_{\bar{z}\bar{z}}|n\rangle-\langle n|T_{z\bar{z}}|n\rangle^{2}$ (35)

implying that factorization holds for _all_ energy eigenstates.

Now that we have proven factorization, we are prepared to derive the flow
equation for a field theory on a cylinder $S^{1}\times\mathbb{R}$ with line
element

$ds^{2}=dy^{2}+R^{2}dx^{2}\,.$ (36)

Figure 1: We denote the compact direction by $x\sim x+R$, where $R$ is the
radius of the spatial $S^{1}$, and write $y$ for the non-compact Euclidean
time direction.

The components of the stress tensor are555The momentum is quantized in units
of $\frac{1}{R}$ and so it does not change with $\lambda$ because $\lambda$ is
continuous.

$\displaystyle\langle n|T_{yy}|n\rangle$
$\displaystyle=-\frac{E_{n}(R,\lambda)}{R}\,,$ (37) $\displaystyle\langle
n|T_{xx}|n\rangle$ $\displaystyle=-\frac{\partial}{\partial
R}E_{n}(R,\lambda)\,,$ $\displaystyle\langle n|T_{xy}|n\rangle$
$\displaystyle=\frac{i}{R}P_{n}(R,0)\,.$

The governing flow equation (16) for the energy levels are

$\displaystyle\frac{\partial}{\partial\lambda}E_{n}(R,\lambda)$
$\displaystyle=-R\left(\langle n|T_{xx}|n\rangle\langle
n|T_{yy}|n\rangle-\langle n|T_{xy}|n\rangle^{2}\right)\,.$ (38)

Substituting (37) into (38), we arrive at the same one-dimensional inviscid
Burgers’ equation commonly used to describe the dynamics of a diffusionless
fluid

$\frac{\partial}{\partial\lambda}E_{n}(R,\lambda)=E_{n}(R,\lambda)\frac{\partial}{\partial
R}E_{n}(R,\lambda)+\frac{P_{n}(R,0)^{2}}{R}\,.$ (39)

Before addressing the deformed energy levels, it is essential to establish the
initial conditions. While solving the differential equation in a closed form
may not be feasible for general seed theories, the situation significantly
simplifies when the seed theory conforms to CFT principles. The initial
conditions for any CFT on a cylinder of radius $R$ are the following:

$\displaystyle E_{n}(R,0)$
$\displaystyle=\frac{1}{R}\left(n+\bar{n}-\frac{c}{12}\right)\,,$ (40)
$\displaystyle P_{n}(R,0)$
$\displaystyle=\frac{1}{R}\left(n-\bar{n}\right)\,,$

where $n$ and $\bar{n}$ are the eigenvalues of the Virasoro generators $L_{0}$
and $\bar{L}_{0}$ respectively.

With the initial conditions (40), the solution to (39) is given by

$\displaystyle E_{n}(R,\lambda)$
$\displaystyle=\frac{R}{2\lambda}\left(\sqrt{1+\frac{4\lambda
E_{n}(R,0)}{R}+\frac{4\lambda^{2}P_{n}^{2}}{R^{2}}}-1\right)$ (41)
$\displaystyle=\frac{R}{2\lambda}\left(\sqrt{1+\frac{4\lambda}{R^{2}}\left(n+\bar{n}-\frac{c}{12}\right)+\frac{4\lambda^{2}}{R^{4}}\left(n-\bar{n}\right)^{2}}-1\right)\,.$

###  4 $T\overline{T}$ and holography

Besides introducing new states, another route to enriching and extending the
theory of boundary gravitons is to move the AdS3 boundary radially inwards,
and there are several motivations for doing so. One is a way to access
observables that are “more local” than the usual asymptotically defined
quantities, namely the S-matrix in Minkowski space and boundary correlators in
AdS. The need to develop such observables has long been appreciated,
particularly in a cosmological context where there may not exist any “far away
spatial region” that an observer at a fixed time can appeal to. In general
dimensions, the complications of defining quantum gravity in a finite spatial
region are hard to disentangle from the usual UV problems,666See [107] for a
review of the boundary value problem in $D>3$ Euclidean gravity. but the
situation is better in AdS3 since the renormalizability argument of (1)
applies to the case of a finite boundary.777At least if the boundary is flat,
as will be the main case of interest in this thesis. More generally, we might
need boundary counterterms involving boundary curvature. See appendix 8.B for
more discussion. The problem is also interesting due to its proposed
description [108] as a $T\overline{T}$-deformed CFT [80, 81]. These are
theories described in the IR as CFTs perturbed by irrelevant operators; their
UV description is not well understood, but they conceivably represent a new
type of quantum theory in which locality breaks down in a controlled manner.
We take the perspective that these two descriptions – cutoff AdS3 and
$T\overline{T}$-deformed CFT2 – are mutually illuminating.

Another holographic proposal is for $\lambda<0$. This sign choice alters the
asymptotic boundary conditions for the metric as $r\to\infty$, yet it is
crucial to emphasize that it does not impose a finite cutoff on the spacetime,
as discussed in [109]. While this modification enforces the Dirichlet
condition at a finite radial cutoff for $\lambda>0$, it’s noteworthy that the
same prescription applied in the case of $\lambda<0$, where the bulk theory
extends to $r\to\infty$. In this context, the entire spectrum of the boundary
field theory remains real, at least for sufficiently small values of
$\lambda$.

These two holographic interpretations of the $T\overline{T}$ deformation in
the metric formalism for $3d$ gravity have been studied in the Chern-Simons
formalism by [110, 111, 112, 113, 2, 114]. In particular, the Chern-Simons
analysis of [110] studies $3d$ gravity in Lorentzian signature for $\lambda<0$
and shows that a $T\overline{T}$ deformation of the dual CFT corresponds to a
modification of the boundary conditions for the gauge field $A_{\mu}$. This
interpretation of the modified boundary conditions introduces a novel
variational principle wherein specific linear combinations of the undeformed
source and expectation value are held constant. Consequently, this combination
is a newly defined deformed source.

To complete the corners of low-dimensional gravity in the diagram (6), it is
desirable to understand the bottom half under the $T\overline{T}$ deformation.
The process of dimensional reduction from 3D to JT gravity, along with its
dual Schwarzian description and the associated partition functions in the
metric formalism, was initially explored by [115, 116] in the case of
$\lambda<0$.888The case for $\lambda>0$ was studied by [117]. One immediately
encounters a complex-valued energy spectrum and partition function when
$E>\frac{1}{8\lambda}$. To cure this problem and obtain a real-valued
partition function, [117] included a non-perturbative contribution. However,
unlike the extensive literature on $T\overline{T}$ deformations in the metric
and Chern-Simons descriptions of $3d$ gravity, such $T\overline{T}$-like
deformations of the BF theory description of JT gravity (and its dual
“particle-on-a-group” theory) have received less attention, which is one of
the motives for part of this thesis.

###  5 Effective field theory interpretation and higher dimensions

A different perspective to understand the deformation is from the observation
that the partition function of the effective field theory (EFT) on a radial
slice must be a solution to the radial Wheeler-DeWitt equation to describe
gravitational physics. In other words, there is a holographic dictionary for
the EFT dual to a sharp radial cutoff in AdS. This interpretation was
thoroughly studied in [118], and a few more intermediate calculations are
provided in appendix 7. The difference is that the $T^{2}$ operator is a
$d$-dimensional analog of the two-dimensional $T\overline{T}$ operator in the
large-$N$ limit so we have factorization. We start with Euclidean Einstein
gravity coupled to matter with the appropriate gravitational counterterm

$\displaystyle I$ $\displaystyle=-\frac{1}{16\pi
G}\int_{M_{d}}d^{d}x\sqrt{g}\left(R-2\Lambda\right)-\frac{1}{8\pi
G}\int_{\partial M_{d}}d^{d-1}x\sqrt{g^{0}}K$ (42)
$\displaystyle+\frac{1}{8\pi G}\int_{\partial
M_{d}}d^{d-1}x\sqrt{g^{0}}(d-1+\mathcal{L}_{\text{Curvature}})+I_{\text{Matter}}$

and the renormalized Brown-York stress tensor of (42) is defined by

$\delta I=\frac{1}{2}\int_{\partial M_{d}}\sqrt{g^{0}}\widetilde{T}^{ij}\delta
g^{0}_{ij}$ (43)

which is determined in [119]

$\widetilde{T}_{ij}=\frac{1}{8\pi
G}\left(K_{ij}-Kg^{0}_{ij}+(d-1)g^{0}_{ij}\right)-a_{d}\widetilde{C}_{ij}\,,$
(44)

where $K$ is the trace of the extrinsic curvature $K_{ij}=2\nabla_{(i}n_{j)}$,
$n_{j}$ is the outward-pointing normal vector, $a_{d}$ is a constant,
$\widetilde{C}_{ij}$ are curvature-dependent counterterms,
$g^{0}_{ij}(x)\equiv g_{ij}(r_{c},x)$ is the metric on the cutoff surface, and
the tildes are for bulk quantities are related to their undeformed
counterparts up to a multiplicative factor of the cutoff. The boundary stress
tensor (44) obeys

$\widetilde{T}_{i}^{i}+a_{d}\widetilde{C}_{i}^{i}=-4\pi
G\left[\left(\widetilde{T}_{ij}+a_{d}\widetilde{C}_{ij}\right)^{2}-\frac{1}{d-1}\left(\widetilde{T}_{i}^{i}+a_{d}\widetilde{C}_{i}^{i}\right)^{2}\right]-\frac{\widetilde{R}}{16\pi
G}+\widetilde{t}_{r}^{r}\,,$ (45)

where $t_{ij}$ represents the stress tensor associated with matter, and one
can find the proof provided in appendix 7.A. Here, (45) is the $T^{2}$ flow
equation999We derive this explicit form of $\mathbb{X}$ in appendix 7 and the
explicit form of the holographic cutoff dictionary.

$\frac{\partial S_{\text{EFT}}}{\partial\lambda}=\int
d^{d}x\sqrt{\gamma}\mathbb{X}\,,$ (46)

where

$\mathbb{X}=\left[\left(T_{ij}+a_{d}\widetilde{C}_{ij}\right)^{2}-\frac{1}{d-1}\left(T_{i}^{i}+a_{d}\widetilde{C}_{i}^{i}\right)^{2}\right]-\frac{r_{c}^{d}}{d\lambda}\left(-\frac{\widetilde{R}}{16\pi
G}+\widetilde{t}_{r}^{r}-a_{d}\widetilde{C}_{i}^{i}\right)$ (47)

and

$\lambda=\frac{4\pi G}{dr_{c}^{d}}\,.$ (48)

_Torus energy spectrum and quantum corrections_

We can derive the energy spectrum under the $T^{2}$ deformation in the
large-$N$ limit where factorization holds following [118]. An example to
consider is a CFT on a manifold $\mathbb{R}\times\mathcal{M}^{d-1}$ with
metric

$ds^{2}=d\tau^{2}+h_{ab}dx^{a}dx^{b}$ (49)

and the flow for the energy levels are

$\frac{\partial E}{\partial\lambda}=\int d^{d-1}x\sqrt{\gamma}\mathbb{X}\,.$
(50)

Let’s specialize $\mathcal{M}^{d-1}$ to be $(d-1)$-dimensional rectangular
torus so the line element is

$ds^{2}=d\tau^{2}+\sum^{d-1}_{i=1}L^{2}_{i}dx_{i}^{2}\,,$ (51)

where $\tau$ is valued on $\mathbb{R}$, $L_{i}$ is the radius of the
$i^{\text{th}}$ cycle on the torus and $x_{i}$ are compact coordinates on the
torus. For this background, the Einstein tensor vanishes, and there are no
trace anomalies on the torus. We furthermore set $\tilde{t}^{r}_{r}=0$.
Therefore, the operator $\mathbb{X}$ simplifies to

$\mathbb{X}=T_{ij}T^{ij}-\frac{1}{d-1}\left(T_{i}^{i}\right)^{2}$ (52)

and the flow equation for the energy becomes101010Here we take the torus to be
square $L_{1}=L_{2}=\cdots=L_{d-1}=L$. See appendix 7 for more details.

$\frac{\partial\varepsilon}{\partial\lambda}=\frac{d-2}{d-1}\varepsilon^{2}-\frac{2\varepsilon}{(d-1)L^{d-2}}\frac{\partial}{\partial
L}\left(L^{d-1}\varepsilon\right),$ (53)

where the energy density is $\varepsilon=E/L^{d-1}$. The solution to this
differential equation (53) is

$E=\frac{(d-1)L^{d-1}}{2d\lambda}\left(1-\sqrt{1-\frac{4d\lambda}{d-1}E_{0}}\right),$
(54)

where the undeformed energy is $E_{0}=\frac{M}{L}$.

Related to this discussion, an unanswered question one could ask is if it is
possible a higher dimensional $T\overline{T}$-like an operator exists that one
can reliably compute $\mathcal{O}(1/N)$ corrections to the energy, such as for
a QFT on a torus. For large-$N$ and $d>2$, we have factorization in the
operator $\mathbb{X}$ defined in (52), but can one easily create an operator
up to $\mathcal{O}(1/N)$ corrections111111I thank Per Kraus for discussions on
this.

$\mathbb{X}(x,y)=T_{ij}(x)T^{ij}(y)-\frac{1}{d-1}T_{i}^{i}(x)T_{j}^{j}(y)+\frac{1}{N}\mathcal{O}^{(1)}_{\mathbb{X}}(x,y)+\sum^{\infty}_{p=2}\frac{1}{N^{p}}\mathcal{O}^{(p)}_{\mathbb{X}}(x,y)$
(55)

so that $\langle E,P|\mathbb{X}(x,y)|E,P\rangle$ is independent of $(x,y)$?
For a straightforward example, consider a theory with $T^{i}_{i}=0$, such as
$N$ free massless scalars, then

$\langle E,P|T_{ij}(x)T^{ij}(y)|E,P\rangle=\langle
E,P|T_{ij}(x)|E,P\rangle\langle E,P|T^{ij}(y)|E,P\rangle+Nf_{E,P}(x-y)\,.$
(56)

where $f_{E,P}(x-y)$ is a nontrivial energy-momentum state dependent function
of $x-y$. The first piece is factorized (i.e. connected) proportional to
$N^{2}$, while the second piece is unfactorized (i.e. disconnected). The
unwanted contribution is the connected piece is removed via

$\langle
E,P|\mathcal{O}^{(1)}_{\mathbb{X}}(x,y)|E,P\rangle=-N^{2}f_{E,P}(x-y)\,.$ (57)

Now the candidate operator with the normal ordering of stress tensors up to
$\mathcal{O}(1/N)$ is

$\displaystyle\mathbb{X}(x,y)$
$\displaystyle=T_{ij}(x)T^{ij}(y)-\frac{1}{d-1}T_{i}^{i}(x)T_{j}^{j}(y)$ (58)
$\displaystyle-\left[:T_{ij}(x)::T^{ij}(y):-\frac{1}{d-1}:T^{i}_{i}(x)::T^{j}_{j}(y):\right]+\sum^{\infty}_{p=2}\frac{1}{N^{p}}\mathcal{O}^{(p)}_{\mathbb{X}}(x,y)\,.$

This operator exhibits the desired properties within the free boson theory’s
vacuum state as all connected diagrams up to $\mathcal{O}(1/N)$ are removed.
However, it appears to extend differently for excited states because the
normally ordered operators do not annihilate them.

###  6 Outline

The outline of this thesis is as follows.

In chapter 1, we determine the boundary graviton action of AdS3 at a finite
planar boundary cutoff in the metric and Chern-Simons formalism to calculate
the stress tensor correlators at the two-loop order. This analysis extends the
previous tree-level discussion. We make new comments on higher point quantum
corrected planar correlators. Correlators of the deformed gravitational AdS3
Wilson line are computed semi-classically via conformal perturbation theory.

In chapter 2, we study JT gravity in the BF formalism under the
$T\overline{T}$ deformation to determine the dual $T\overline{T}$-deformed
Schwarzian theory. Finally, we calculate deformed correlators involving
gravitational BF Wilson lines defined up to the Hagedorn temperature.

In chapter 3, we define a manifestly supersymmetric $T\overline{T}$
deformation for $(0+1)d$ QFTs with $\mathcal{N}=(0,1)$ and $\mathcal{N}=(1,1)$
supersymmetry. We determine the $T\overline{T}$-deformed super-Schwarzian
theory dual to JT supergravity. These deformations, denoted as
$f(\mathcal{Q})$, are explicitly expressed within the Noether currents
associated with supertranslations. We show three seemingly different
procedures to determine the same deformed supersymmetric quantum mechanics
theory.

In chapter 4, we consider the $T\overline{T}$ deformation of JT gravity and
its ‘t Hooft limit, the Airy model. Various aspects of the $T\overline{T}$
deformation are considered, such as complexification of the spectrum at the
disk level, negativities in the deformed spectral density, and, in particular,
the free energy for $\lambda>0$ and $\lambda<0$.

In chapter 5, we investigate properties of the newly defined
root-$T\overline{T}$ operator using holographic properties. In contrast to the
better-known $T\overline{T}$ operator, the root-$T\overline{T}$ operator is
much harder to make precise in the quantum theory but is also classically
marginal, which makes it particularly interesting as a possible tool in CFT.
We assume a large-$N$ limit corresponds to a large central charge in the CFT.
This limit is needed to perform the path integral to the lowest order in a
$1/N$ expansion using large-$N$ factorization. Using this method, a suggestive
form for the deformed metric and stress tensor of the CFT in terms of the
undeformed quantities is derived. The same results are then derived using the
trace of the stress tensor staying zero in the deformed theory. The results
require the root-$T\overline{T}$ deformation to commute with the ordinary
$T\overline{T}$ deformation. Similar ideas lead one to conjecture the deformed
energy levels in the root-$T\overline{T}$ theory. This conjecture is supported
by calculating the energy of a class of AdS spacetimes with the
root-$T\overline{T}$ boundary conditions derived in the previous sections and
showing that it is consistent with the flow of the energy levels.

In chapter 6, we conclude this thesis by continuing our previous studies of
the root-$T\overline{T}$ deformation from chapter 5 to the context of
interacting chiral boson theories and perturbatively study their quantum
aspects for particular backgrounds in two dimensions. We extend our
interacting chiral boson analysis to the three-dimensional abelian Chern-
Simons context.

Details of various calculations, applications, and additional observations are
relegated in the appendices.

##  Chapter 1 The Field Theory of Interacting Boundary Gravitons

We study pure three-dimensional gravity at a finite radial cutoff in this
chapter. Pure three-dimensional gravity is a renormalizable theory with two
free parameters labeled by Newton’s constant $G$ and the cosmological constant
$\Lambda$. As a result, correlation functions of the boundary stress tensor in
AdS3 are unambiguously determined solely by the central charge of the Virasoro
algebra. The same argument implies that AdS3 gravity at a finite radial cutoff
is a renormalizable theory, but now with one additional parameter
corresponding to the cutoff location. This theory is conjecturally dual to a
$T\overline{T}$-deformed CFT, assuming these theories exist. To elucidate
this, we study the quantum theory of boundary gravitons living on a cutoff
planar boundary and the associated correlation functions of the boundary
stress tensor. We compute stress tensor correlation functions to two-loop
order (Newton’s gravitational coupling $G$ being the loop counting parameter),
extending existing tree-level results. This computation is feasible since the
boundary graviton action simplifies greatly upon making a judicious field
redefinition, turning into the Nambu-Goto action. After imposing Lorentz
invariance, the correlators in this order are found to be unambiguous up to a
single undetermined renormalization parameter.

###  1 Introduction

We develop the quantum theory of boundary gravitons on a cutoff planar
surface, focusing on obtaining the optimal form of the action and using it to
compute correlation functions of boundary operators.

We work in the framework of the covariant phase space formalism [120, 121],
and in both the metric and Chern-Simons formulation [61, 62] of $3d$ gravity,
since they offer complementary perspectives. This cutoff phase space is
established by considering an AdS3 background and applying all coordinate and
gauge transformations that maintain a Dirichlet boundary condition. These
phase space coordinates consist of two functions defined at some initial time
on the boundary; these are the coordinate transformations
$(x,t)\rightarrow\big{(}x+A(x,t),t+B(x,t)\big{)}$ evaluated at $t=0$. We need
a symplectic form and a Hamiltonian on this phase space to construct the
canonical formulation, and we develop efficient methods for computing these.
In the asymptotically AdS3 case, this procedure is simple to carry out
exactly, and we readily arrive at the Alekseev-Shatashvili action [56, 57], as
was obtained via the Chern-Simons formulation in [58]. At finite cutoff, life
is more complicated; we work order-by-order in the $(A,B)$ variables, but the
resulting expressions quickly become complicated because the phase space
action contains an ever-growing number of higher derivatives acting on these
fields.

A pleasant surprise is that a field redefinition $(A,B)\rightarrow(f,\bar{f})$
can be used to remove all higher derivatives from the action, at least to the
order we have checked (eighth-order in the fields). The resulting (imaginary
time) action is the Nambu-Goto action written in Hamiltonian form (25). We
obtain further evidence for this action by deriving it to all orders in the
special case of linearly varying $(f,\bar{f})$.

However, the stress tensor is not the canonical stress tensor of the Nambu-
Goto theory due to the non-linear action of the Poincaré group on the fields.
The deformed stress tensor includes a series of higher derivative correction
terms reflecting the nonlocal nature of the theory, e.g.111See (92) for all
three components.

$4GT_{zz}=\frac{1}{2}f^{\prime\prime}-\frac{1}{4}f^{\prime
2}+\frac{1}{4}r_{c}f^{\prime\prime\prime}\bar{f}^{\prime}-\frac{1}{8}r_{c}\big{(}f^{\prime
2}-2f^{\prime}\bar{f}^{\prime}\big{)}^{\prime}~{}\bar{f}^{\prime}+\frac{1}{16}r_{c}^{2}\big{(}f^{\prime\prime\prime\prime}\bar{f}^{\prime
2}+(f^{\prime 2})^{\prime\prime}\bar{f}^{\prime\prime}\big{)}+\cdots\,.$ (1)

With the expressions for the action and stress tensor, we seek its
quantization. The main interest here is in computing two-point functions of
the stress tensor order-by-order in the loop counting parameter
$G$.222Previous works [122, 123] on this problem in the gravitational
formulation stopped at tree level.  There is some tension coming from two
perspectives on this problem: on the one hand, the Nambu-Goto action with its
square root is problematic to quantize directly without ambiguity; on the
other hand, the underlying theory is pure $3d$ gravity, which one expects to
be renormalizable.

The subtlety in reconciling these perspectives has to do with the complicated
(nonlinear and nonlocal) manner in which the symmetries of the gravitational
description are realized once we pass to the reduced phase space description,
and in particular with preserving these symmetries in the quantum theory. In
this chapter, we compute the stress tensor correlators to two-loop order using
dimensional regularization. At tree level and one-loop, the results are finite
and unambiguous. At two loops, we find that a single renormalization of the
stress tensor is required, and the divergent part comes as usual with an
associated undetermined finite part parametrized here by $\mu$. For example,
we find the $\langle T_{zz}T_{zz}\rangle$ correlator at the two-loop order to
be333 The full set of two-point functions is written in (190).

$\displaystyle\langle
T_{zz}(x)T_{zz}(0)\rangle=\leavevmode\resizebox{366.40831pt}{}{$\frac{1}{z^{4}}\left[\frac{c}{2}+10(3+4G)\left(\frac{r_{c}}{z\overline{z}}\right)^{2}+96G\left(8+60\ln(\mu^{2}z\overline{z})\right)\left(\frac{r_{c}}{z\overline{z}}\right)^{3}+2520G\left(\frac{r_{c}}{z\overline{z}}\right)^{4}\right]\,.$}$
(2)

Here $c=c_{0}+1=\frac{3\ell}{2G}+1$ is the one-loop corrected Brown-Henneaux
central charge [54, 58] of the $r_{c}=0$ theory.444In this thesis, the symbol
$c$ will be used, and its meaning – whether it represents the tree-level
central charge ($c_{0}$) or the loop-corrected central charge – will be
inferred from the context in which it is mentioned. Regarding
renormalizability, the main result in this chapter is inconclusive: we suspect
that the free parameter reflects that dimensional regularization is not
preserving all symmetries, but further work is required to substantiate this
by imposing the relevant Ward identities.

While our primary focus centers on a flat planar boundary, it is also valuable
to explore the case of a curved boundary metric. In preparation for the curved
boundary metric, we meticulously derive the Chern-Simons formulation that
accommodates a general boundary metric. As an illustrative example, we
demonstrate the procedure for calculating the action for Euclidean AdS3 with a
finite $S^{2}$ boundary, taking into account the substantial radius-related
divergence linked to the Weyl anomaly of the boundary theory, as presented in
appendix 8.B. This action is straightforward to obtain in the metric
description. However, it is somewhat subtle in the Chern-Simons formulation
due to the need to introduce two overlapping patches for the gauge potentials.

Furthermore, we focus on Wilson lines in $2d$ theories deformed by the double-
trace version of the $T\overline{T}$ operator. A conceptually related analysis
involving deformed Wilson loops in the single-trace setting was presented in
[124]. However, in that work, the Wilson line was computed for the DBI gauge
field living on a D1-brane, rather than the gauge field arising from a gauge
theory presentation of a gravity theory. In this chapter, we content ourselves
with the double-trace version of the $T\overline{T}$ deformation and Wilson
lines for gauge fields associated with gravitational theories. One reason for
doing this is that the bulk gravity dual to a single-trace
$T\overline{T}$-deformed CFT also involves the dilaton and the three-form flux
$H_{3}$, and writing the kinetic terms for these fields in Chern-Simons
variables is somewhat unwieldy. A second reason is that, although the gravity
solution relevant for single-trace $T\overline{T}$ is local AdS3 in the deep
interior, the solution approaches a linear dilaton spacetime. The linear
dilaton region is qualitatively different from AdS3 (in fact, its causal
structure is, in some sense, more similar to that of Minkowski space), and the
Chern-Simons formulation of $3d$ gravity is not straightforwardly applicable
in this regime.

We highlight some prior research in a related vein. In [122], the authors
investigated AdS3 gravity with a finite cylindrical boundary. A result from
[122] was that the asymptotic Virasoro $\times$ Virasoro algebra was deformed
by breaking the conformal invariance associated with the finite boundary.
Another result was that the free boundary graviton spectrum was deformed in a
manner compatible with $T\overline{T}$ considerations. In this chapter, the
main focus is on a planar boundary; this is simpler, and we make other
technical advances that allow us to go further than before. Stability and
causality for gravity with cutoff boundary conditions are discussed in [125,
126, 127]. The covariant phase space in the presence of boundaries is reviewed
in [128]. Jackiw-Teitelboim gravity [129, 130] at a finite radial boundary
cutoff was studied in [131, 117, 132], with results relating to the spectrum
of $T\overline{T}$-deformed quantum mechanics obtained in [115, 116, 3]. An
important subtlety that arises, discussed in [131], is the distinction between
microscopic versus effective theories of the JT gravity path integral, and the
resulting non-trivial relations among the parameters and couplings; presumably
these issues are also present in this chapter’s context.

Correlation functions in the $2d$ field theory or $3d$ bulk were studied in
[123, 88, 89, 90, 133, 91, 92, 93, 134, 94, 9]. Results in those papers were
found either at low order in the $T\overline{T}$ coupling555In this chapter,
we denote the $T\overline{T}$ coupling by $\lambda_{T\overline{T}}$ instead of
$\lambda$ to avoid a clash of notation with the AdS3 Gauss parameters
$(\lambda,\Psi,F,\bar{\lambda},\bar{\Psi},\bar{F})$ defined in (LABEL:zm). In
later chapters, we resort to $\lambda$ being the $T\overline{T}$ coupling.
$\lambda_{T\overline{T}}\sim\frac{r_{c}}{c}$, or lowest order in the
$\frac{1}{c}$ expansion (tree level in this chapter’s language). An exception
is [89] that proposed some results of all orders in $\lambda_{T\overline{T}}$.
The results in this chapter hold to all orders in $r_{c}$ but are perturbative
in $\frac{1}{c}$ (we go to two-loop order, extending the previous tree-level
results). In the context of a massive scalar [133] and a Dirac fermion [135],
integrability fixed renormalization ambiguities. In [110, 111, 112, 113], the
$T\overline{T}$-deformed CS formulation of $3d$ gravity was discussed.

The remaining portion of this chapter unfolds subsequently: In section 2, we
lay out some general principles in computing the action for boundary gravitons
common to the metric and Chern-Simons formulations. In section 3, we discuss
the metric formulation, developing a streamlined approach to computing the
boundary action for a flat radial cutoff boundary. We show how to obtain the
Alekseev-Shatashvili action straightforwardly in the $r_{c}=0$ limit and
acquire the all-order action at finite $r_{c}$ in the special case of constant
$(f^{\prime},\bar{f}^{\prime})$: the Nambu-Goto action. In section 4, we turn
to the Chern-Simons formulation. Since it is of interest beyond the immediate
concerns of this work, we carefully develop the variational principle for CS
gravity with a general curved cutoff boundary. We carry out a perturbative
computation of the action for gravitons on a cutoff planar boundary, obtaining
results to eight order in $(f,\bar{f})$; the results turn out to match the
expansion of the Nambu-Goto action to this order, leading us to conjecture
that this extends to all orders. In section 5, we turn to correlation
functions. We compute correlators of both elementary fields and the stress
tensor. The one-loop four-point function of elementary fields is found to
require one counterterm in the gravitational action, and the two-loop stress
tensor correlators require a single renormalization of the stress tensor.
Finally, in section 3, we calculate the deformed gravitational AdS3 Wilson
line correlators semi-classically. We conclude with a brief discussion in
section 7. In appendix 8, we provide further details on a general trace flow
equation, the Chern-Simons formulation, including the comparison to the metric
formulation, relating Chern-Simons theory at finite cutoff and coupling to
topological gravity, details regarding the evaluation of Feynman diagrams and
a discussion of how to compute the action in the case of a spherical boundary.

###  2 Generalities on the phase space formulation of boundary graviton
theories

In the upcoming two sections, we obtain the action and stress tensor for
boundary gravitons localized on a finite cutoff surface by working in the
metric and Chern-Simons formulations. The formalisms provide complementary
viewpoints and bring distinct technical advantages to the table, yet it is
crucial to note that their outcomes are in complete agreement. To pave the way
for the in-depth analysis that ensues, we delve into several fundamental
facets of the issue.

The Einstein-Hilbert action (1), or its Chern-Simons equivalent, contains a
mixture of “physical modes” and “pure gauge modes,” and the goal in this
chapter is to arrive at a reduced action that omits the latter as much as
possible. In general, one pays a price by reducing the degrees in the form of
a loss of manifest symmetry, for example, in the light cone gauge treatment of
Yang-Mills theory or string/M-theory. In Yang-Mills perturbation theory, this
price is typically too high, and a Lorentz invariant formulation with
unphysical modes is usually adopted. However, in a topological theory, like
pure three-dimensional gravity, the reduction of degrees of freedom is so
dramatic (removing all but the boundary modes) that the cost of losing some
manifest symmetry is more than repaid.

We will construct a reduced action living on a flat boundary surface with
coordinates $(x,t)$. The action is of the phase space variety, built out of a
Hamiltonian $H$ and a “canonical one-form” $\Upsilon$.666Note that $\Upsilon$
is a one-form on phase space, not on spacetime. We use $\delta$ to denote the
exterior derivative on phase space, reserving $d$ for the exterior derivative
on spacetime.  The phase space action takes the form

$I=-\int dt\big{(}i_{V_{\eta}}\Upsilon-H\big{)}\,,$ (3)

where $t$ is Euclidean time and $i_{V_{\eta}}$ denotes contraction with the
phase space vector field $V_{\eta}$ that implements (Lorentzian) time
translation. For example, for a particle moving in one dimension, we might
take $\Upsilon=p\delta q$ and $H(p,q)=\frac{p^{2}}{2m}+V(q)$. We have
$i_{V_{\eta}}\Upsilon=-ip\dot{q}$ implying that
$I=\int\\!dt\big{(}ip\dot{q}+H(p,q)\big{)}$.

The symplectic form $\Omega$ is given by $\Omega=\delta\Upsilon$. On the true
phase space of the theory, $\Omega$ should be nondegenerate, meaning that
$i_{V}\Omega=0$ if and only if $V=0$. In gauge theory or gravity, one begins
with a larger “pre-phase space” with a degenerate, closed two-form $\Omega$.
The null directions of $\Omega$ on pre-phase space correspond to small gauge
transformations. Part of the task in this chapter will be to remove the pure
gauge modes corresponding to these null directions.

In the case of $3d$ gravity, the dynamical variables appearing in the phase
space action will be fields $\big{(}f(x,t),\bar{f}(x,t)\big{)}$ on the
boundary which therefore comprises the physical degrees of freedom.777More
precisely, these fields are subject to residual gauge equivalences associated
with isometries of AdS3. The route to obtaining the action for these fields
differs in the metric versus Chern-Simons descriptions.

In the metric formulation, the idea is to start with some reference solution
and then apply boundary-condition-preserving coordinate transformations to
construct a space of solutions. The symplectic form for gravity on pre-phase
space was calculated in [120] and implies that coordinate transformations that
vanish at the boundary correspond to degenerate modes. All that matters is the
form of the coordinate transformation near the boundary, and this information
is specified by the fields $(f,\bar{f})$. The coordinate transformations
preserve the metric on the boundary but change the value of the boundary
stress tensor $T_{ij}(x)$. As we discuss in detail in the next section, the
phase space action follows immediately from the expressions for the boundary
momentum density $\mathcal{P}=\frac{i}{2\pi}T_{tx}$ and energy density
$\mathcal{H}=\frac{1}{2\pi}T_{tt}$.

Turning to the Chern-Simons formulation, in this approach, one can pass
directly from the Chern-Simons action to the reduced phase space action once
one has been sufficiently careful in defining boundary conditions and adding
the associated boundary terms in the gravitational action. In the case of an
asymptotic AdS3 boundary, previous work on this problem includes [136, 137,
58, 138]. The general approach in this chapter follows [58]. Nonetheless, in
our context, we are dealing with a spacetime subject to a finite cutoff.
Consequently, our initial task is to establish a rigorously defined
variational principle. Additionally, we must incorporate the relevant boundary
terms into the action, as they deviate from the conventions typically employed
in most of the existing literature. We do this for a general curved boundary
geometry, although our primary focus here is the case of a flat boundary. As
usual in gauge theory, the time components of the gauge fields $A_{t}$ act as
Lagrange multipliers imposing constraints [136]. Essentially, all one needs to
do is to solve these constraint equations in a manner compatible with the
boundary conditions, and then substitute back into the action. The Lagrangian
density is observed to be a total derivative, and the resulting boundary term
is the desired phase space action. In this approach, the fields $(f,\bar{f})$
appear as free functions that parameterize solutions to the constraint
equations and boundary conditions.

In either approach, obtaining the action (using perturbation theory if
necessary) is rather mechanical since the resulting expression may be unwieldy
due to a suboptimal choice of coordinates on phase space. Selecting
coordinates such that the kinetic term in the gravitational action (the terms
involving time derivatives) is purely quadratic in fields is convenient for
performing quantum mechanical perturbation theory. This corresponds to
choosing “Darboux coordinates” such that the components of the symplectic form
are constant, which is always possible locally.888In general coordinates, the
complication is that the path integral measure is proportional to the
nontrivial Pfaffian of the symplectic form. The $(f,\bar{f})$ are such Darboux
coordinates, and part of our analysis will be to identify them. We also find
that it is possible999To at least eighth order in fields – we do not yet have
a general proof. to choose these coordinates such that the Hamiltonian takes a
simple form, namely that of the Nambu-Goto action (25). The Nambu-Goto action
is well-known to be the $T\overline{T}$-deformed action of a free scalar with
canonical stress tensor [82]; the new features here are that the stress tensor
derived from the gravity theory is not the canonical stress tensor and the
existence of a highly nontrivial field redefinition that relates the natural
gravitational variables to those appearing in the Nambu-Goto action.

###  3 Metric formulation of boundary graviton action on the plane

####  1 Preliminaries

We start from the Euclidean signature action of $3d$ gravity with cosmological
constant $\Lambda$,

$I=-\frac{1}{16\pi
G}\int_{M_{3}}d^{3}x\sqrt{g}\left(R-2\Lambda\right)+I_{\operatorname{bndy}}\,,$
(4)

where the boundary terms are written below. Einstein’s equations are

$R_{\mu\nu}-{1\over 2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=0\,.$ (5)

For $\Lambda<0$, we define the AdS3 radius as $\Lambda=-\frac{1}{\ell^{2}}$,
and henceforth choose units such that $\ell=1$. We impose Dirichlet boundary
conditions on the metric of a two-dimensional boundary surface and pick
coordinates $(r,x^{i})$ such that this surface lives at $r=r_{c}$. The
interior of the surface is selected to be the region $r>r_{c}$, so that the
vector $\partial_{r}$ is inward pointing. It is convenient to choose Gaussian
normal coordinates in the vicinity of the boundary so that the metric is

$ds^{2}=\frac{dr^{2}}{4r^{2}}+g_{ij}(r,x)dx^{i}dx^{j}\,.$ (6)

These coordinates may or may not fail away from the boundary, but this is
largely immaterial to study the boundary graviton theory. The Dirichlet
boundary condition means fixing $g_{ij}(r_{c},x^{i})$ as well as the form (6)
(we could, in principle, fix only the induced metric on the boundary, but it
is convenient to also put “gauge” conditions on the radial coordinate). The
appropriate boundary action appearing in (4) is then

$I_{\text{bndy}}=-\frac{1}{8\pi G}\int_{\partial M_{3}}\\!d^{2}x\sqrt{\det
g_{ij}}(K-1)-\frac{\log r_{c}}{32\pi G}\int_{\partial
M_{3}}\\!d^{2}x\sqrt{\det g_{ij}}R(g_{ij})\,,$ (7)

where the extrinsic curvature and its trace are

$K_{ij}=-r\partial_{r}g_{ij}\,,\quad K=g^{ij}K_{ij}\,.$ (8)

The terms in $I_{\text{bndy}}$ not involving $K$ depend only on the Dirichlet
boundary data and are unnecessary for a proper variational principle; however,
one adds them to ensure the finiteness of the action in the asymptotic AdS3
limit $r_{c}\rightarrow 0$.

The boundary stress tensor $T_{ij}$ is defined in terms of the on-shell
variation of the action [139, 119],

$\delta I=\frac{1}{4\pi}\int_{\partial M_{3}}\\!d^{2}x\sqrt{\det
g_{ij}}T^{ij}\delta g_{ij}\,,$ (9)

and works out to be

$T_{ij}=\frac{1}{4G}(K_{ij}-Kg_{ij}+g_{ij})\,.$ (10)

One can easily confirm the boundary stress tensor for three-dimensional
gravity (10) is conserved

$\displaystyle\nabla^{i}T_{ij}$
$\displaystyle=\frac{1}{4G}\left[\nabla^{i}\left(K_{ij}-Kg_{ij}\right)+\nabla^{i}g_{ij}\right]=0\,,$
(11)

where the first term $\nabla^{i}\left(K_{ij}-Kg_{ij}\right)$ vanishes from the
vacuum Einstein equations (5) for metric (6) and the second term always
vanishes since $\nabla^{i}g_{ij}=0$.

To compare to a dual (deformed) CFT, we think of the latter as living on a
rescaled metric $\gamma_{ij}$, defined as
$\gamma_{ij}=r_{c}g_{ij}(r_{c},x^{i})$, which is in particular finite in the
asymptotically AdS3 case. The Einstein equations can be used to show that the
stress tensor obeys the $T\overline{T}$ trace relation (16) which was shown by
[88]101010We emphasize (12) holds only for AdS3. When $\Lambda>0$, the trace
relation, for example in dS3, differs from AdS3 (12) by an additional constant
$\propto\frac{1}{\lambda_{T\overline{T}}}$. See [140, 141, 142, 143, 144] and
appendix 8.A.

$\gamma^{ij}T_{ij}=\pi\lambda_{T\overline{T}}\det(\gamma^{ik}T_{kj})-\frac{1}{8G}R(\gamma)$
(12)

with [108]

$\lambda_{T\overline{T}}=\frac{4Gr_{c}}{\pi}\,.$ (13)

On a flat surface, the $T\overline{T}$ deformation of the action is defined as

$\partial_{\lambda_{T\overline{T}}}I_{\lambda_{T\overline{T}}}=-\frac{1}{4}\int\\!d^{2}x\sqrt{\gamma}\det(\gamma^{ik}T_{kj})\,$
(14)

We now specialize in the case of a flat boundary metric,

$g_{ij}(r_{c})dx^{i}dx^{j}=\frac{dt^{2}+dx^{2}}{r_{c}}\,,$ (15)

where $t$ lives on the real line, and at this stage $x$ is allowed to live on
either the line or the circle.

Now, let $\xi^{\mu}\partial_{\mu}$ generate a diffeomorphism vector field that
preserves the boundary conditions; namely,
$(\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu})\big{|}_{r_{c}}=0$. At this
stage, we should emphasize that we do not restrict to vector fields $\xi$ that
are tangent to the boundary; we allow for $\xi$ with a nonzero normal
component $\xi^{r}$, which in an active sense corresponds to moving the
location of the boundary. To clarify this, note that a more geometrical
characterization of our setup consists of finding flat surfaces embedded in an
ambient AdS3 background. To translate this picture into the one we use, we
note that near each surface, we can construct a Gaussian normal coordinate
system, with the surface at $r=r_{c}$ in these coordinates. The coordinate
transformation needed to relate two such surfaces requires diffeomorphisms
that are not tangent to the surfaces. Alternatively, one could take the more
algebraic view that the vector field $\xi$ is a computationally efficient way
of representing a transformation on field space. Since this canonical
formalism only cares that our transformations preserve the boundary
conditions, there is no difficulty posed by nonzero $\xi^{r}$. Associated to
each boundary condition preserving vector field is a (not necessarily
conserved) boundary charge [139, 122]111111These charges are only conserved
and only generate symmetries if $\xi$ is tangent to the boundary. It is then
also a boundary Killing vector. We should also note that the notation $Q[\xi]$
is not to be confused with Wald’s $Q_{\xi}$ [145] as explained in footnote 20
of [128].

$Q[\xi]=\frac{i}{2\pi}\int\\!T_{ti}\xi^{i}dx\,,$ (16)

where the integral evaluates on a constant $t$ slice of the boundary, and the
appearance of $i$ is due to our choice of Euclidean signature. Translation
invariance of the boundary metric implies the existence of conserved energy
and momentum charges,

$\displaystyle H$ $\displaystyle=Q[-i\partial_{t}]=\frac{1}{2\pi}\int
T_{tt}dx\,,$ (17) $\displaystyle P$
$\displaystyle=Q[\partial_{x}]=\frac{i}{2\pi}\int T_{tx}dx\,.$

####  2 Phase space

Adopting the framework of covariant phase space [120, 121], we think of phase
space as the space of classical solutions that obey the boundary conditions
(15) where we identify solutions related by “small gauge transformations,”
made precise momentarily. By definition, a phase space equips to a symplectic
form $\Omega$, which is a non-degenerate, closed two-form. For pure gravity
described by the Einstein-Hilbert action in arbitrary dimension, a closed (and
conserved) two-form was found in [120] as an integral over a Cauchy surface
$\Sigma$ of an expression involving the metric and its derivatives

$\Omega=\frac{1}{16\pi
G}\int_{\Sigma}d\Sigma_{\alpha}\sqrt{g}\left[\delta\Gamma^{\alpha}{}_{\mu\nu}\wedge\left(\delta
g^{\mu\nu}+\frac{1}{2}g^{\mu\nu}\delta\ln
g\right)-\delta\Gamma^{\nu}{}_{\mu\nu}\wedge\left(\delta
g^{\alpha\mu}+\frac{1}{2}g^{\alpha\mu}\delta\ln g\right)\right]\,.$ (18)

This object is degenerate on the space of all classical solutions since it
gives zero when contracted against an infinitesimal displacement in solution
space corresponding to a coordinate transformation that vanishes at the
boundary. To obtain the symplectic form, we must mod out by such coordinate
transformations so that the two-form becomes non-degenerate on the quotient
space. In the context of AdS3 gravity, this procedure is straightforward.

We denote $\delta_{\xi}g_{\mu\nu}$ as the change of the metric under an
infinitesimal coordinate transformation,
$\delta_{\xi}g_{\mu\nu}=\nabla_{\mu}\xi_{\mu}+\nabla_{\nu}\xi_{\mu}$. We let
$V_{\xi}$ denote the corresponding vector field on the space of solutions; in
terms of the Lie derivative, this corresponds to the statement
$\delta_{\xi}g_{\mu\nu}=\mathcal{L}_{V_{\xi}}g_{\mu\nu}$. A key relation,
verified by direct computation in [122], is121212Note that this result is
valid even when $\xi$ has a component normal to the boundary, unlike what is
often assumed, e.g., in [128]. This relation holds in pure gravity, where one
can choose a gauge so that the quantity $C$ defined in [128] vanishes.

$i_{V_{\xi}}\Omega=-\delta Q[\xi]\,,$ (19)

where $i$ denotes the contraction operation and $Q[\xi]$ is given by (16).
Since $Q[\xi]$ takes the form of a boundary integral, this makes explicit the
statement that diffeomorphisms that vanish at the boundary correspond to null
directions of $\Omega$.

To provide an explicit expression for the symplectic form, we need a
description for the phase space with small diffeomorphisms properly quotient
out. Given that (19) establishes that diffeomorphisms persisting at the
boundary correspond to non-degenerate directions, it becomes a logical choice
to employ these diffeomorphisms as our coordinate system on phase space. We
then begin with some chosen reference solutions and perform all possible
diffeomorphisms that preserve the boundary conditions. This construction gives
us the “boundary graviton phase space” associated with the chosen reference
solution. It may not coincide with the complete phase space if the latter
encompasses distinct solutions that cannot be connected through finite
diffeomorphisms.

Within the realm of pure AdS3 gravity, our point of reference will be the AdS3
solution in global or Poincaré coordinates. As for potential alternative
solutions, we consider BTZ black holes and conical defects with varying masses
and angular momenta. However, BTZ black holes obey different boundary
conditions than vacuum AdS3 — i.e. the BTZ black hole has two asymptotic
boundaries in Lorentzian signature and a periodic time direction in Euclidean
signature — while conical defects have singular metrics. We are interested in
the phase space of boundary gravitons living on the boundary of vacuum AdS3.

We start by writing down the reference metric, which, in the vicinity of the
boundary, takes the form (6). We then change coordinates,
$r=r(r^{\prime},x^{\prime i})$, $x^{i}=x^{i}(r^{\prime},x^{\prime j})$ and
demand that the metric at the boundary is unchanged,

$ds^{2}\big{|}_{r^{\prime}=r_{c}}=\frac{dr^{\prime
2}}{4r_{c}^{2}}+g_{ij}(r_{c},x^{\prime i})dx^{\prime i}dx^{\prime j}\,.$ (20)

We demand that the metric components at the boundary take the same form in the
primed coordinates as in the original coordinates. Note, in particular, that
in the new coordinates, the location of the boundary is at $r^{\prime}=r_{c}$;
since this, in general, differs from $r=r_{c}$ and can think of the coordinate
transformation as actively changing the location of the boundary. Imposing the
boundary conditions in the new coordinates amounts to solving a system of PDEs
for $r(r^{\prime},x^{\prime i})$ and $x^{i}(r^{\prime},x^{\prime j})$. Given
the nature of the problem, it is natural to expand the coordinates
transformation near the boundary, and so we write

$\displaystyle x$
$\displaystyle=x^{\prime}+A(x^{\prime},t^{\prime})+(r^{\prime}-r_{c})U(x^{\prime},t^{\prime})+\cdots\,,$
(21) $\displaystyle t$
$\displaystyle=t^{\prime}+B(x^{\prime},t^{\prime})+(r^{\prime}-r_{c})V(x^{\prime},t^{\prime})+\cdots\,,$
$\displaystyle r$
$\displaystyle=r^{\prime}+r^{\prime}C(x^{\prime},t^{\prime})+r^{\prime
2}W(x^{\prime},t^{\prime})+\cdots\,.$

We then further expand around an initial time surface (taken to be $t=0$) on
the boundary,

$\displaystyle x$
$\displaystyle=x^{\prime}+A(x^{\prime})+\sum_{n=1}^{\infty}A_{n}(x^{\prime})t^{\prime
n}+(r^{\prime}-r_{c})\sum_{n=0}^{\infty}U_{n}(x^{\prime})t^{\prime
n}+\cdots\,,$ (22) $\displaystyle t$
$\displaystyle=t^{\prime}+B(x^{\prime})+\sum_{n=1}^{\infty}B_{n}(x^{\prime})t^{\prime
n}+(r^{\prime}-r_{c})\sum_{n=0}^{\infty}V_{n}(x^{\prime})t^{\prime
n}+\cdots\,,$ $\displaystyle r$
$\displaystyle=r^{\prime}+r^{\prime}\sum_{n=0}^{\infty}C_{n}(x^{\prime})t^{\prime
n}+r^{\prime 2}\sum_{n=0}^{\infty}W_{n}(r^{\prime})t^{\prime n}+\cdots\,.$

The reason for writing things in this way is that an inspection of the PDEs
reveals that one can take as “initial data” any freely chosen functions
$(A(x^{\prime}),B(x^{\prime}))$ (which respect periodicity conditions, if any,
on $x$) and then determine the remaining functions in terms of these. At
finite $r_{c}$, it seems rather arduous to solve this problem in closed form,
so we will work perturbatively and treat the amplitudes of $(A,B)$ small. This
analysis is equivalent to a small $G$ expansion. Perturbation theory is
straightforward to carry out since the functions $(U_{n},V_{n},C_{n},W_{n})$
are determined algebraically in terms of $(A,B)$ — no differential equations
need to be solved — and one only needs a finite number of these functions at
any given order.

The functions $(A(x^{\prime}),B(x^{\prime}))$ will (modulo the gauge
invariances discussed below) serve as coordinates on phase space. These
functions determine the location of the boundary in the original reference
spacetime. The new boundary at $r^{\prime}=r_{c}$ is located at
$r=r_{c}+r_{c}C(x^{\prime},t^{\prime})+r_{c}^{2}W(x^{\prime},t^{\prime})+\cdots$
with $(C,W,\cdots)$ are functions of $(A,B)$.

####  3 Boundary stress tensor

Proceeding, the output of this analysis that we will need is an expression for
the boundary stress tensor $T_{ij}$, evaluated at $t^{\prime}=0$, in terms of
$(A,B)$. For this procedure to work, we need the radial derivatives of the
metric components evaluated at the boundary. One can straightforwardly work
this out to any desired order. We focus on the case that the reference
solution is Poincaré AdS3 given by (15) with $x$ non-compact. Up to the
quadratic order in fields, the stress tensor works out to be

$\displaystyle 4GT_{tt}$
$\displaystyle=-A^{\prime\prime\prime}+\frac{1}{2}\big{(}(A^{\prime
2})^{\prime\prime}+A^{\prime\prime 2}\big{)}-\frac{1}{2}\big{(}(B^{\prime
2})^{\prime\prime}+B^{\prime\prime 2}\big{)}-\frac{1}{2}(A^{\prime\prime
2})^{\prime\prime}r_{c}+\cdots\,,$ (23) $\displaystyle 4GT_{xt}$
$\displaystyle=-B^{\prime\prime\prime}+(A^{\prime}B^{\prime})^{\prime\prime}+A^{\prime\prime}B^{\prime\prime}+(A^{\prime\prime\prime}B^{\prime\prime})^{\prime}r_{c}+\cdots\,,$
$\displaystyle 4GT_{xx}$
$\displaystyle=A^{\prime\prime\prime}-\frac{1}{2}\big{(}(A^{\prime
2})^{\prime\prime}+A^{\prime\prime 2}\big{)}+\frac{1}{2}\big{(}(B^{\prime
2})^{\prime\prime}+B^{\prime\prime
2}\big{)}+(A^{\prime\prime}A^{\prime\prime\prime\prime}-B^{\prime\prime
2})r_{c}+\cdots\,.$

At higher orders in the fields, the stress tensor components are more
complicated but, as we discuss below, are greatly simplified after making an
appropriate field redefinition.

The general structure of the stress tensor is fixed by gauge invariance (see
section 6 for more detail). Consider the four-parameter subgroup of the entire
six-dimensional isometry group corresponding to translations, rotations, and
dilatations of $(x,t)$ (with the dilatation accompanied by a rescaling of
$r$). Since the stress tensor vanishes for the vacuum solution at $A=B=0$,
this means it vanishes for any $(A,B)=(a+bx,c+dx)$ because any such coordinate
transformation is expressed as some combination of these isometries. These
choices of $(A,B)$ are “pure gauge.” From this, we deduce that $(A,B)$ can
only appear with at least one derivative and that every term in the stress
tensor must have at least one factor with at least two derivatives.

####  4 Symplectic form

We now discuss how to compute the symplectic form $\Omega$. We can use the
relation $i_{V_{\chi}}\Omega=-\delta P$, where $\chi$ acts as an
$x$-translation on the boundary, to efficiently deduce $\Omega$ given $P$. We
first note that the gauge invariance of the preceding paragraph implies that
$P$ can always be put in the form, possibly after integrating by parts,

$P=\int\\!dx\Big{(}P_{A}(A,B)A^{\prime}+P_{B}(A,B)B^{\prime}\Big{)}\,,$ (24)

where $P_{A,B}(A,B)$ are local functions of $(A,B)$ and their derivatives
(regular as $A,B\rightarrow 0$), and are furthermore total derivatives of such
local functions. The latter statement follows from the fact that each term in
$P$ contains at least one factor with at least two derivatives. We now argue
that the symplectic form is given by $\Omega=\delta\Upsilon$ with

$\Upsilon=\int\\!dx\Big{(}P_{A}(A,B)\delta A+P_{B}(A,B)\delta B\Big{)}\,.$
(25)

To compute $i_{V_{\chi}}\Omega$ and verify equality with $\delta P$, we need
expressions for $i_{V_{\chi}}\delta A$ and $i_{V_{\chi}}\delta B$. To this
end, we start from (21) and perform a subsequent infinitesimal translation
$(x^{\prime},t^{\prime})=(x^{\prime\prime}+\chi^{x},t^{\prime\prime})$ and
evaluate the effect at $t^{\prime\prime}=0$. For an $x$-translation, take
$\chi=\chi^{i}\partial_{i}=\partial_{x}$. This $x$-translation acts as

$\displaystyle A(x)$ $\displaystyle\rightarrow
A(x)+\chi^{x}+A^{\prime}(x)\chi^{x}\,,$ (26) $\displaystyle B(x)$
$\displaystyle\rightarrow B(x)+B^{\prime}(x)\chi^{x}\,,$

so that

$i_{V_{\chi}}\delta A=\chi^{x}+A^{\prime}(x)\chi^{x}\,,\quad
i_{V_{\chi}}\delta B=B^{\prime}(x)\chi^{x}\,.$ (27)

Using this to evaluate $i_{V_{\chi}}\Omega$, the first term in
$i_{V_{\chi}}\delta A$ gives zero since $P_{A}$ is a total derivative. The
remaining terms give

$\displaystyle i_{V_{\chi}}\Omega$ $\displaystyle=-\int\\!dx\Big{(}P_{A}\delta
A^{\prime}+P_{B}\delta B^{\prime}+\delta P_{A}A^{\prime}+\delta
P_{B}B^{\prime}\Big{)}$ (28) $\displaystyle=-\delta P\,,$

as desired.

We also need to establish that $\Omega$ defined above is the unique object
obeying all requirements. Consider replacing
$\Omega\rightarrow\Omega+\Delta\Omega$. We can always write $\Delta\Omega$ in
the form

$\Delta\Omega=\int\\!dx\Big{(}\delta X_{A}(A,B)\wedge\delta A+\delta
X_{B}(A,B)\wedge\delta B\Big{)}$ (29)

with $X_{A,B}(A,B)$ being local functions with a perturbative expansion by the
following argument. First, $\Delta\Omega$ is closed. Second, by gauge
invariance each $A$ or $B$ in $X_{A,B}$ appears with at least one derivative,
and $X_{A,B}$ must be total derivatives of local functions. Using these facts,
we contract with the $x$-translation vector field and find

$\displaystyle i_{V_{\chi}}\Delta\Omega$
$\displaystyle=\delta\int\\!dx\Big{(}X_{A}A^{\prime}+X_{B}B^{\prime}\Big{)}$
(30)
$\displaystyle=-\delta\int\\!dx\Big{(}X_{A}^{\prime}A+X_{B}^{\prime}B\Big{)}\,.$

We proceed to assert that this quantity must equal zero to maintain
$i_{V_{\chi}}\Omega=-\delta P$. Using that $X_{A,B}$ admit a perturbative
expansion and does not involve undifferentiated $A$ or $B$, it is easy to see
that $i_{V_{\chi}}\Delta\Omega=0$ requires $X^{\prime}_{A}=X^{\prime}_{B}=0$.
Since $X_{A}$ and $X_{B}$ are constants, they obey $\delta X_{A}=\delta
X_{B}=0$, which then implies $\Delta\Omega=0$.

####  5 Action and equations of motion

From (21), the canonical equations of motion are

$\dot{A}=A_{1}\,,\quad\dot{B}=B_{1}\,,$ (31)

where we recall that $(A_{1},B_{1})$ are functions of $(A,B)$ and their
derivatives. Related to this, if we consider a diffeomorphism by the time
translation vector field $\eta^{i}\partial_{i}=\eta^{t}\partial_{t}$, we have

$i_{V_{\eta}}\delta A=A_{1}\eta^{t}\,,\quad i_{V_{\eta}}\delta
B=\eta^{t}+B_{1}\eta^{t}\,.$ (32)

The equations of motion (31) are equivalent to the statement that $V_{\eta}$
is the Hamiltonian vector field corresponding to $Q[\eta]$, i.e.
$i_{V_{\eta}}\Omega=-\delta Q[\eta]$, as this is a special case of (19). In
particular, taking $\eta^{t}=-i$ gives $Q[\eta]=H$.

We now seek an action whose Euler-Lagrange equations coincide with these
equations of motion, $i_{V_{\eta}}\Omega=-\delta H$. The answer is

$\displaystyle I$
$\displaystyle=-\int\\!dt\big{(}i_{V_{\eta}}\Upsilon-H\big{)}$ (33)
$\displaystyle=\int\\!d^{2}x\big{(}iP_{A}\dot{A}+iP_{B}\dot{B}+\mathcal{H}\big{)}\,,$

where $\mathcal{H}={1\over 2\pi}T_{tt}$ so that $H=\int\\!dx\mathcal{H}$. The
equality between the two lines makes use of the fact that $P_{A,B}$ are total
derivatives so that the $\eta^{t}$ term in $i_{V_{\eta}}\delta B$ does not
contribute. To see that this is the correct action, we evaluate
$i_{V_{\eta}}\Omega$ as we did in (28) in the case of an $x$-translation. We
now get

$\displaystyle i_{V_{\eta}}\Omega$
$\displaystyle=-i\int\\!dx\big{(}\dot{P}_{A}\delta A-\delta
P_{A}\dot{A}+\dot{P}_{B}\delta B-\delta P_{B}\dot{B}\big{)}$ (34)
$\displaystyle=i\int\\!dx\delta\big{(}P_{A}\dot{A}+P_{B}\dot{B}\big{)}\,,$

which implies

$i_{V_{\eta}}\Omega+\delta
H=i\int\\!dx\delta\big{(}P_{A}\dot{A}+P_{B}\dot{B}-i\mathcal{H}\big{)}\,.$
(35)

The vanishing of $i_{V_{\eta}}\Omega+\delta H$ is equivalent to the Euler-
Lagrange equations for (33). From (23), the action in quadratic order is

$I={1\over 16\pi
G}\int\\!d^{2}x\Big{(}A^{\prime\prime\prime}(\dot{B}-A^{\prime})+B^{\prime\prime\prime}(\dot{A}+B^{\prime})+\operatorname{cubic}+\cdots\Big{)}\,.$
(36)

The cubic and higher order terms are complicated when expressed in terms of
$(A,B)$, but we will later find a field redefinition that significantly
simplifies the action.

####  6 Symmetries

The action (33) is invariant under certain gauge and global symmetries. We
begin with the former, which originates due to the isometries of the reference
solution. AdS3 has a six-parameter group of coordinate transformations that
leave the metric invariant, which in Lorentzian signature is
$\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})$. Applying one of these
followed by a general coordinate transformation has the same effect as
applying only the latter, and this statement implies an equivalence relation
between distinct $(A,B)$.

We will adopt more compact notation, with $x^{i}=(x,t)$, writing (21) as

$\displaystyle x^{i}$ $\displaystyle=x^{\prime i}+A^{i}(x^{\prime
j})+(r^{\prime}-r_{c})U^{i}(x^{\prime j})+\cdots\,,$ (37) $\displaystyle r$
$\displaystyle=r^{\prime}+r^{\prime}C(x^{\prime j})+r^{\prime 2}W(x^{\prime
j})+\cdots\,.$

Let $\xi^{\mu}$ obey $\nabla^{0}_{\mu}\xi_{\nu}+\nabla^{0}_{\nu}\xi_{\mu}=0$,
where the derivatives are defined with respect to the reference solution
$g^{0}_{\mu\nu}$. Writing the coordinate transformation
$x^{\mu}=x^{\prime\mu}+\xi^{\mu}(x^{\prime})$ in the form (37) defines
$(A^{i}_{\xi},U^{i}_{\xi},C_{\xi},F_{\xi})$. We compose this with the
subsequent transformation

$\displaystyle x^{\prime i}$ $\displaystyle=x^{\prime\prime
i}+A^{i}(x^{\prime\prime i})+(r^{\prime\prime}-r_{c})U^{i}(x^{\prime\prime
i})+\cdots\,,$ (38) $\displaystyle r^{\prime}$
$\displaystyle=r^{\prime\prime}+r^{\prime\prime}C(x^{\prime\prime
i})+r^{\prime\prime 2}W(x^{\prime\prime i})+\cdots\,,$

and evaluate at $(t^{\prime\prime},r^{\prime\prime})=(0,r_{c})$. This defines
a transformation between $x^{\mu}$ and $x^{\prime\prime\mu}$ labeled by some
modified $A^{i}$ functions and is equivalent to the transformation (37) in
which $(x^{i},r)$ appears on the left-hand side. Explicitly, the gauge
equivalence is

$A^{i}(x)\cong
A^{i}(x)+\left[A^{i}_{\xi}\big{(}x^{j}+A^{j}(x^{k})\big{)}+\big{(}r_{c}C(x^{j})+r_{c}^{2}W(x^{j})\big{)}U^{i}_{\xi}\big{(}x^{j}+A^{j}(x^{k})\big{)}\right]\Big{|}_{t=0}\,.$
(39)

Recalling that $C$ and $W$ are nonlinear functions of $A^{i}$ and their
derivatives, we see that gauge transformations act in a complicated nonlinear
way. The stress tensor is invariant under these transformations. The
symplectic form is also invariant; this is not entirely obvious from our
somewhat indirect method for extracting the symplectic form, but it is
manifest when one expresses (as in [120]) the symplectic form as an integral
over a spacelike surface since that expression is expressed in terms of the
metric, which is by definition invariant under the isometries. Furthermore,
both the stress tensor and the symplectic form do not depend on time
derivatives of $(A,B)$, so the invariance extends to transformations in which
the parameters are allowed to have arbitrary dependence on time. This type of
gauge symmetry was referred to as “quasi-local” in [58]. The phase space
action is determined from the symplectic form and Hamiltonian, which implies
that the action is also gauge invariant. As noted, the
SL($2,\mathbb{R})\times$SL$(2,\mathbb{R})$ gauge transformations act on
$(A,B)$ in a complicated nonlinear fashion. The formulas of course simplify
markedly for $r_{c}=0$, and we write out the corresponding transformations
explicitly in the next section.

Global symmetries correspond to isometries of the metric on the boundary,
which are simply translations and rotations (i.e. Poincaré transformations in
real-time). In this case, we first apply (37) followed by the infinitesimal
transformation

$x^{\prime i}=x^{\prime\prime i}+\epsilon^{i}+\theta^{ij}x^{\prime\prime
j}\,,\quad r^{\prime}=r^{\prime\prime}\,,$ (40)

with $\theta^{ij}=-\theta^{ji}$.

Composing these transformations, we find

$\displaystyle A(x)$ $\displaystyle\rightarrow
A(x)+(1+A^{\prime})\epsilon^{x}+A_{1}\epsilon^{t}+A_{1}x\theta^{tx}\,,$ (41)
$\displaystyle B(x)$ $\displaystyle\rightarrow
B(x)+(1+B_{1})\epsilon^{t}+B^{\prime}\epsilon^{x}+B_{1}x\theta^{tx}\,,$

where we used $\dot{A}|_{t=0}=A_{1}$ and $\dot{B}|_{t=0}=B_{1}$. These
transformations are again highly nonlinear due to the appearance of
$(A_{1},B_{1})$. The stress tensor transforms into a symmetric tensor under
these translations and rotations.

####  7 Asymptotic AdS3 case: Alekseev-Shatashvili action

For illustrative purposes, we consider the asymptotic case of $r_{c}=0$ where
it is simple to carry out our general procedure in closed form. Starting from

$ds^{2}={dr^{2}\over 4r^{2}}+{1\over r}dzd\bar{z}\,,$ (42)

the coordinate transformation [146]

$\displaystyle z$ $\displaystyle=F(z^{\prime})-{2r^{\prime}F^{\prime
2}\bar{F}^{\prime\prime}\over
4F^{\prime}\bar{F}^{\prime}+r^{\prime}F^{\prime\prime}\bar{F}^{\prime\prime}}\,,$
(43) $\displaystyle\bar{z}$
$\displaystyle=\bar{F}(\bar{z}^{\prime})-{2r^{\prime}\bar{F}^{\prime
2}F^{\prime\prime}\over
4F^{\prime}\bar{F}^{\prime}+r^{\prime}F^{\prime\prime}\bar{F}^{\prime\prime}}\,,$
$\displaystyle r$ $\displaystyle={16r^{\prime}F^{\prime 3}\bar{F}^{\prime
3}\over(4F^{\prime}\bar{F}^{\prime}+r^{\prime}F^{\prime\prime}\bar{F}^{\prime\prime})^{2}}\,,$

gives

$ds^{2}={dr^{\prime 2}\over 4r^{\prime 2}}+{1\over
r^{\prime}}dz^{\prime}d\bar{z}^{\prime}-{1\over 2}\\{F,z^{\prime}\\}dz^{\prime
2}-{1\over 2}\\{\bar{F},\bar{z}^{\prime}\\}d\bar{z}^{\prime 2}+{1\over
4}r^{\prime}\\{F,z^{\prime}\\}\\{\bar{F},\bar{z}^{\prime}\\}dz^{\prime}d\bar{z}^{\prime}\,.$
(44)

Here $F=F(z^{\prime})$, $\bar{F}=\bar{F}(\bar{z}^{\prime})$, primes on
$(F,\bar{F})$ denote derivatives, and the Schwarzian derivative is

$\\{F,z^{\prime}\\}={F^{\prime\prime\prime}\over F^{\prime}}-{3\over
2}{F^{\prime\prime 2}\over F^{\prime 2}}\,.$ (45)

Writing $z=x+it$, and comparing (43) to (21) we read off

$\displaystyle A+iB=F-z^{\prime}\,,$ (46) $\displaystyle
A-iB=\bar{F}-\bar{z}^{\prime}\,,$ $\displaystyle U+iV=-{1\over
2}{F^{\prime}\bar{F}^{\prime\prime}\over\bar{F}^{\prime}}\,,$ $\displaystyle
U-iV=-{1\over 2}{F^{\prime\prime}\bar{F}^{\prime}\over F^{\prime}}\,,$
$\displaystyle C=F^{\prime}\bar{F}^{\prime}\,,$ $\displaystyle W=-{1\over
4}F^{\prime\prime}\bar{F}^{\prime\prime}\,.$

The stress tensor is

$\displaystyle T_{zz}$ $\displaystyle={c_{0}\over
12}\\{F,z^{\prime}\\}\,,\quad T_{\bar{z}\bar{z}}={c_{0}\over
12}\\{\bar{F},\bar{z}^{\prime}\\}\,,\quad T_{z\bar{z}}=0\,,$ (47)

where $c_{0}={3\over 2G}$ is the Brown-Henneaux central charge.

In this case, we, of course, could have written down (47) directly from
knowledge of the asymptotic Virasoro symmetry of AdS3 with $r_{c}=0$, but,
here, we are emulating the procedure we carry out for the general cutoff case.
Expressing $T_{ij}$ in terms of $(A,B)$ gives the all-order version of the
expressions (23) at $r_{c}=0$.

From here, it is straightforward to calculate the Alekseev-Shatashvili action
for $(F,\bar{F})$. It is useful to generalize a bit by taking the stress
tensor to be

$\displaystyle T_{zz}$ $\displaystyle={c_{0}\over 12}\left({a\over 2}F^{\prime
2}+\\{F,z^{\prime}\\}\right)\,,\quad T_{\bar{z}\bar{z}}={c_{0}\over
12}\left({a\over 2}\overline{F}^{\prime
2}+\\{\bar{F},\bar{z}^{\prime}\\}\right)\,,\quad T_{z\bar{z}}=0\,,$ (48)

where $a$ is a parameter that can be thought of as the stress tensor of a more
general reference solution with $T_{zz}=T_{\bar{z}\bar{z}}={ac_{0}\over 24}$.
For example, $a=1$ corresponds to global AdS3 provided $x\cong x+2\pi$, while
values $0<a<1$ correspond to conical defect solutions. The energy and momentum
from (17) are

$\displaystyle H$ $\displaystyle=-{1\over
2\pi}\int\\!dx(T_{zz}+T_{\bar{z}\bar{z}})=-{c_{0}\over
24\pi}\int\\!dx\left({a\over 2}F^{\prime 2}+\\{F,z^{\prime}\\}+{a\over
2}\overline{F}^{\prime 2}+\\{\bar{F},\bar{z}^{\prime}\\}\right)\,,$ (49)
$\displaystyle P$ $\displaystyle=-{1\over
2\pi}\int\\!dx(T_{zz}-T_{\bar{z}\bar{z}})=-{c_{0}\over
24\pi}\int\\!dx\left({a\over 2}F^{\prime 2}+\\{F,z^{\prime}\\}-{a\over
2}\overline{F}^{\prime 2}-\\{\bar{F},\bar{z}^{\prime}\\}\right)\,.$

We now apply our general procedure to compute the symplectic form. This
procedure was previously stated in terms of $(A,B)$ but applies equally in
terms of $(F,\bar{F})$. We write $P$ in the form

$P=\int\\!dx\left(P_{F}F^{\prime}+P_{\bar{F}}\bar{F}^{\prime}\right)\,,$ (50)

where $P_{F}$ and $P_{\bar{F}}$ are total derivatives. This is easily achieved
using

$\displaystyle\\{F,z\\}=-{1\over 2}\left({1\over
F^{\prime}}\right)^{\prime\prime}F^{\prime}+\operatorname{total~{}derivative}$
(51)

yielding

$P_{F}=-{c_{0}\over 48\pi}\left(aF^{\prime}-\left({1\over
F^{\prime}}\right)^{\prime\prime}\right)\,,\quad P_{\bar{F}}={c_{0}\over
48\pi}\left(a\bar{F}^{\prime}-\left({1\over\bar{F}^{\prime}}\right)^{\prime\prime}\right)\,.$
(52)

The value of $\Upsilon$ (25) in this case gives

$\Upsilon=-{c_{0}\over 48\pi}\int\\!dx\left[\left(aF^{\prime}-\left({1\over
F^{\prime}}\right)^{\prime\prime}\right)\delta
F-\left(a\bar{F}^{\prime}-\left({1\over\bar{F}^{\prime}}\right)^{\prime\prime}\right)\delta\bar{F}\right]\,.$
(53)

The general formula (33) then yields the Alekseev-Shatashvili action [56]

$I=-{c_{0}\over
24\pi}\int\\!d^{2}x\left[aF^{\prime}\partial_{\bar{z}}F-\left({1\over
F^{\prime}}\right)^{\prime\prime}\partial_{\bar{z}}F+a\bar{F}^{\prime}\partial_{z}\bar{F}-\left({1\over\bar{F}^{\prime}}\right)^{\prime\prime}\partial_{z}\bar{F}\right]\,,$
(54)

with

$\partial_{z}={1\over
2}(\partial_{x}-i\partial_{t})\,,\quad\partial_{\bar{z}}={1\over
2}(\partial_{x}+i\partial_{t})\,.$ (55)

The theory (54) describes a single (left and right moving) boson with variable
central charge. As such, one expects it to be equivalent to the standard
action for a free boson with a linear dilaton (or background charge)
contribution to its stress tensor. Indeed, as noted in [56, 57] the field
redefinition

$\left(e^{i\sqrt{a}F}\right)^{\prime}=\sqrt{a}e^{f}\,,\quad\left(e^{-i\sqrt{a}\bar{F}}\right)^{\prime}=\sqrt{a}e^{\bar{f}}\,.$
(56)

yields

$\displaystyle T_{zz}$ $\displaystyle=-{c_{0}\over 12}\left({1\over
2}f^{\prime 2}-f^{\prime\prime}\right)\,,$ (57) $\displaystyle
T_{\bar{z}\bar{z}}$ $\displaystyle=-{c_{0}\over 12}\left({1\over
2}\bar{f}^{\prime 2}-\bar{f}^{\prime\prime}\right)\,,$ $\displaystyle\Upsilon$
$\displaystyle={c_{0}\over 48\pi}\int\\!dx\left(f^{\prime}\delta
f-\bar{f}^{\prime}\delta\bar{f}\right)\,,$ $\displaystyle I$
$\displaystyle={c_{0}\over
96\pi}\int\\!d^{2}x\left(f^{\prime}\partial_{\bar{z}}f+\bar{f}^{\prime}\partial_{z}\bar{f}\right)\,.$

As $a\rightarrow 0$, the field redefinition reads

$F^{\prime}=e^{f}\,,\quad\bar{F}^{\prime}=e^{\bar{f}}$ (58)

and relations (57) continue to hold. Each chiral half of the action $I$ in
(57) is the Floreanini-Jackiw action [147]; the two halves combined give a
standard free scalar action in Hamiltonian form. On the other hand, the stress
tensor in (57) coming from gravity includes an improvement term (unlike
considered in [111]). The improvement term is crucial since without it, the
central charge would be fixed at $c=1$.

Restricting now to $a=0$, the gauge transformations act as

$\delta_{\epsilon}F=\sum_{n=0}^{2}\epsilon_{n}(t)F^{n}\,,\quad\delta_{\epsilon}\bar{F}=\sum_{n=0}^{2}\overline{\epsilon}_{n}(t)\bar{F}^{n}\,.$
(59)

To compute the gauge variation of the stress tensor, symplectic form, and the
action, we only need the transformations of $(f^{\prime},\bar{f}^{\prime})$
which are

$\delta_{\epsilon}f^{\prime}=2\epsilon_{2}(t)e^{f}\,,\quad\delta_{\epsilon}\bar{f}^{\prime}=2\overline{\epsilon}_{2}(t)e^{\bar{f}}\,.$
(60)

Gauge invariance of the stress tensor fixes the relative coefficient of the
two terms in $T_{zz}$ and $T_{\bar{z}\bar{z}}$. Using standard formulas from
free boson CFT, it follows that correlators of stress tensors are those of a
CFT with central charge $c=c_{0}+1$; we elaborate on this more in the course
of our $r_{c}\neq 0$ discussion below.

It is also instructive to see how the stress tensor in (57) arises from
Noether’s theorem applied to the quadratic action in (57). An infinitesimal
rigid translation of the boundary coordinates, $x^{i}\rightarrow
x^{i}+\epsilon^{i}$ acts on $(F,\bar{F})$ as

$\delta_{\epsilon}F=\partial_{i}F\epsilon^{i}\,,\quad\delta_{\epsilon}\bar{F}=\partial_{i}\bar{F}\epsilon^{i}\,,$
(61)

derived using the same reasoning that led to (41). We want a transformation on
phase space, and so we use the equations of motion
$\partial_{\bar{z}}F=\partial_{z}\bar{F}=0$ to trade away time derivatives and
obtain

$\delta_{\epsilon}F=F^{\prime}\epsilon^{z}\,,\quad\delta_{\epsilon}=\bar{F}^{\prime}\epsilon^{\bar{z}}\,.$
(62)

As usual in the derivation of Noether’s theorem, we now consider the
transformation (62) that $\epsilon^{i}$ depends arbitrarily on $x^{i}$. We
then work out the transformation of $(f,\bar{f})$ as

$\delta_{\epsilon}f=f^{\prime}\epsilon^{z}+(\epsilon^{z})^{\prime}\,,\quad\delta_{\epsilon}\bar{f}=\bar{f}^{\prime}\epsilon^{\bar{z}}+(\epsilon^{\bar{z}})^{\prime}\,.$
(63)

We finally compute the variation of the action and write this in the form

$\delta I=-{1\over 2\pi}\int\\!d^{2}xT_{ij}\partial^{i}\epsilon^{j}\,,$ (64)

yielding the stress tensor in (57).

####  8 Exact action for constant $(f^{\prime},\bar{f}^{\prime})$

Besides the asymptotic $r_{c}=0$ limit, there is another special case in which
we can derive the action to all orders. This is a consequence of the fact that
we can find exact solutions to the boundary value problem in the case that
second and higher derivatives of $f$ and $\bar{f}$ vanish. This leads to a
result for the action which captures all dependence on
$(f^{\prime},\bar{f}^{\prime})$, but not on higher derivatives of these
fields.

We start from

$ds^{2}={d\rho^{2}\over 4\rho^{2}}+{1\over\rho}dwd\bar{w}$ (65)

and first, perform the coordinate change

$\displaystyle w$ $\displaystyle={1-r\over 1+r}e^{2\left({x\over
1+r_{c}}+{it\over 1-r_{c}}\right)}\,,$ (66) $\displaystyle\bar{w}$
$\displaystyle={1-r\over 1+r}e^{2\left({x\over 1+r_{c}}-{it\over
1-r_{c}}\right)}\,.$ $\displaystyle\rho$
$\displaystyle={4r\over(1+r)^{2}}e^{{4x\over 1+r_{c}}}\,,$

which gives the line element

$ds^{2}={dr^{2}\over 4r^{2}}+{1\over r}\left[\left(1-r\over
1-r_{c}\right)^{2}dt^{2}+\left(1+r\over 1+r_{c}\right)^{2}dx^{2}\right]\,.$
(67)

We now perform a further two-parameter coordinate redefinition that preserves
the form of the metric at $r=r_{c}$. This corresponds to a rescaling by a
parameter $a$,

$r\rightarrow ar\,,\quad t\rightarrow{\sqrt{a}(1-r_{c})\over
1-ar_{c}}t\,,\quad x\rightarrow{\sqrt{a}(1+r_{c})\over 1+ar_{c}}x\,,$ (68)

followed by a rotation by angle $\theta$,

$t\rightarrow t\cos\theta+x\sin\theta\,,\quad x\rightarrow
x\cos\theta-t\sin\theta\,.$ (69)

Writing the combined transformation in the form $w=A(r)e^{f(x,t)}$ and
$\bar{w}=\overline{A}(r)e^{\bar{f}(x,t)}$, we compute

$\displaystyle f^{\prime}$ $\displaystyle={2\sqrt{a}\over
1-a^{2}r_{c}^{2}}(e^{i\theta}-ar_{c}e^{-i\theta})\,,$ (70)
$\displaystyle\bar{f}^{\prime}$ $\displaystyle={2\sqrt{a}\over
1-a^{2}r_{c}^{2}}(e^{-i\theta}-ar_{c}e^{i\theta})\,.$

On the other hand, it is straightforward to compute the stress tensor in terms
of $(a,\theta)$ and re-express the results in terms of
$(f^{\prime},\bar{f}^{\prime})$. Writing $P=\int\\!dxp={i\over
2\pi}\int\\!T_{tx}dx$, we find

$\displaystyle p={i\over 2\pi}T_{tx}={1\over 32\pi G}(f^{\prime
2}-\bar{f}^{\prime 2})$ (71)

as in (84). Further, writing $H=\int\\!dxh={1\over 2\pi}\int\\!T_{tt}dx$ we
find that $h$ obeys

$h-4\pi Gr_{c}(h^{2}-p^{2})=h_{0}\,,$ (72)

where

$\displaystyle h_{0}={1\over 32\pi G}(f^{\prime 2}+\bar{f}^{\prime 2})\,.$
(73)

Solving (72) for $h$ gives,

$\displaystyle h$ $\displaystyle=-{1\over 8\pi Gr_{c}}\left(\sqrt{1-16\pi
Gr_{c}h_{0}+(8\pi Gr_{c}p)^{2}}-1\right)$ (74) $\displaystyle=-{1\over 8\pi
Gr_{c}}\left(\sqrt{1-{1\over 2}r_{c}(f^{\prime 2}+\bar{f}^{\prime 2})+{1\over
16}r_{c}^{2}(f^{\prime 2}-\bar{f}^{\prime 2})^{2}}-1\right)\,,$

where we chose the root which obeys $\lim_{r_{c}\rightarrow 0}h=h_{0}$.

This result implies that the full action takes the form

$\displaystyle I=$ $\displaystyle{1\over 32\pi
G}\int\\!d^{2}x\left[if^{\prime}\dot{f}-i\bar{f}^{\prime}\dot{\bar{f}}-4{\sqrt{1-{1\over
2}r_{c}(f^{\prime 2}+\bar{f}^{\prime 2})+{1\over 16}r_{c}^{2}(f^{\prime
2}-\bar{f}^{\prime 2})^{2}}-1\over r_{c}}\right]$ (75)
$\displaystyle+{\text{higher~{}derivatives}}\,,$

where the higher derivative terms vanish when $f^{\prime\prime}=0$ and
$\bar{f}^{\prime\prime}=0$.

From [114], we define

$\phi=\frac{1}{\sqrt{32\pi
G}}\left(f+\bar{f}\right),\qquad\Pi=\frac{1}{\sqrt{32\pi
G}}\left(f^{\prime}-\bar{f}^{\prime}\right)$ (76)

so that the full action (75) becomes the Nambu-Goto action in static
gauge131313The static gauge is the gauge choice that one identifies two of the
target space coordinates with the worldsheet coordinates. More specifically:
$X^{0}=t,~{}X^{1}=x,~{}X^{2}=\sqrt{\lambda}\phi$ and the direction $X^{1}$ is
taken to be compact $X^{1}(x+2\pi,t)=X^{1}(x,t)+2\pi$ for unit winding sector.

$\displaystyle I$ $\displaystyle=\int
d^{2}x\left[i\Pi\dot{\phi}-\frac{1}{\lambda}\left(1-\sqrt{1-\lambda\left(\phi^{\prime
2}+\Pi^{2}\right)+\lambda^{2}\phi^{\prime 2}\Pi^{2}}\right)\right]$ (77)
$\displaystyle=\int
d^{2}x\left[i\Pi\dot{\phi}-\frac{1}{\lambda}+\mathcal{L}_{\operatorname{Nambu-
Goto}}\right]$

and

$\mathcal{L}_{\operatorname{Nambu-
Goto}}=\frac{1}{\lambda}\sqrt{-\det\left(\eta^{\mu\nu}\partial_{i}X_{\mu}\partial_{j}X_{\nu}\right)}\,,$
(78)

with target space indices being $\mu,\nu=0,1,2$, worldsheet indices being
$i,j=0,1$ and Minkowski target space metric
$\eta_{\mu\nu}=\operatorname{diag}\left(-1,1,1\right)$.

In the $\lambda\rightarrow 0$ limit, or going from the “string” to the
particle $\alpha^{\prime}\rightarrow 0$ limit, of (77), we obtain141414The
quotation marks are to remind us from section 3 that the
$T\overline{T}$-deformed theory is not a full-fledged string theory due to the
absence of dynamical gravity.

$I=\int d^{2}x\left(i\dot{\phi}\Pi+\frac{1}{2}\left(\phi^{\prime
2}+\Pi^{2}\right)\right)\,.$ (79)

In the next section, we will provide strong evidence that the higher
derivative terms are absent in general provided we define $(f,\bar{f})$
appropriately.

In this context, we can address the fact that the square root can become
imaginary in some region of the $(f^{\prime},\bar{f}^{\prime})$ plane. From
(70), we have

$1-{1\over 2}r_{c}(f^{\prime 2}+\bar{f}^{\prime 2})+{1\over
16}r_{c}^{2}(f^{\prime 2}-\bar{f}^{\prime
2})^{2}=\left({1-2ar_{c}\cos(2\theta)+a^{2}r_{c}^{2}\over
1-a^{2}r_{c}^{2}}\right)^{2}$ (80)

or in terms of $(\phi,\Pi)$

$1-4\pi Gr_{c}\left(\phi^{\prime 2}+\Pi^{2}\right)+(8\pi
Gr_{c})^{2}\phi^{\prime
2}\Pi^{2}=\left({1-2ar_{c}\cos(2\theta)+a^{2}r_{c}^{2}\over
1-a^{2}r_{c}^{2}}\right)^{2}\,.$ (81)

Therefore, the square root is real for any real value of $(a,\theta)$. It is
natural to expect that in general (i.e. dropping the linearity assumption) the
domain of $(f,\bar{f})$ is bounded such that the integrand in (75) is real,
but this remains to be shown.

####  9 Boundary gravity action on planar cutoff

We now consider the general case, a planar boundary at $r=r_{c}$ with
arbitrary functions $f$ and $\bar{f}$. As explained, the strategy is to start
from the reference solution

$ds^{2}={dr^{2}\over 4r^{2}}+{dx^{2}+dt^{2}\over r}$ (82)

and look for coordinate transformations, expressed in the form (21)-(22), such
that

$g_{r^{\prime}r^{\prime}}|_{r^{\prime}=r_{c}}={1\over 4r_{c}^{2}}\,,\quad
g_{r^{\prime}i^{\prime}}|_{r^{\prime}=r_{c}}=0\,,\quad
g_{i^{\prime}j^{\prime}}|_{r^{\prime}=r_{c}}={\delta_{ij}\over r_{c}}\,,$ (83)

where $i$ and $j$ run over $(x,t)$. This problem can be solved order-by-order
as an expansion in the freely specifiable functions
$(A(x^{\prime}),B(x^{\prime}))$. Only algebraic equations need to be solved in
each order, and the procedure is easily automated on the computer. What we
need from this procedure are expressions for the components of the boundary
stress tensor evaluated at $t^{\prime}=0$, which in turn depend on
$\partial_{r^{\prime}}g_{i^{\prime}j^{\prime}}|_{r^{\prime}=r_{c}}$. The
resulting expressions to quadratic order were written in (23).

At $r_{c}=0$, we saw that the expressions for the stress tensor simplify
dramatically under the field redefinition (58), and so we seek a version of
this at nonzero $r_{c}$. As a criterion for what constitutes an optimal field
redefinition, we note that the symplectic form $\Omega=\delta\Upsilon$ will
have a complicated expansion in $(A,B)$. Quantization of the phase space
action uses the natural measure Pf$(\Omega)$. A nontrivial measure is
incorporated by expressing the Pfaffian as a fermionic path integral, but life
is much simpler if the Pfaffian is constant, which is indeed the case at
$r_{c}=0$ after making the field redefinition. We therefore try to generalize
this feature to $r_{c}\neq 0$. Recall that the symplectic form is obtained
from the momentum $P$ via the formulas (24)-(25).151515We argued for this
using the $(A,B)$ fields, but we will see below that the argument also holds
after the field redefinition to new fields $(f,\bar{f})$. We look to define
new fields $(f,\bar{f})$ such that

$P={i\over 2\pi}\int\\!T_{tx}dx={1\over 32\pi G}\int\\!dx(f^{\prime
2}-\bar{f}^{\prime 2})\,,$ (84)

which implies

$\Upsilon={1\over 32\pi G}\int\\!dx(f^{\prime}\delta
f-\bar{f}^{\prime}\delta\bar{f})\,.$ (85)

In particular, we want

$T_{tx}=-{i\over 16G}(f^{\prime 2}-\bar{f}^{\prime
2})+{\text{total~{}derivative}}\,.$ (86)

By explicit computation, we find that this is achieved by taking

$\displaystyle A^{\prime}+iB^{\prime}$ $\displaystyle=\exp\left[f-{1\over
4}r_{c}\bar{f}^{\prime 2}-{1\over
8}r_{c}^{2}\bar{f}^{\prime}(f^{\prime}\bar{f}^{\prime})^{\prime}-{1\over
16}r_{c}^{2}f^{\prime}(\bar{f}^{\prime 2})^{\prime}+\cdots\right]-1\,,$ (87)
$\displaystyle A^{\prime}-iB^{\prime}$
$\displaystyle=\exp\left[\bar{f}-{1\over 4}r_{c}f^{\prime 2}-{1\over
8}r_{c}^{2}f^{\prime}(f^{\prime}\bar{f}^{\prime})^{\prime}-{1\over
16}r_{c}^{2}\bar{f}^{\prime}(f^{\prime 2})^{\prime}+\cdots\right]-1\,.$

This redefinition involves the spatial derivatives of $(A,B)$, so $(A,B)$ are
non-locally related to $(f,\bar{f})$. However, no undifferentiated $(A,B)$
will ever appear in the stress tensor, and so the stress tensor and action
will be local in terms of $(f,\bar{f})$. The terms written in (87) are
sufficient to work out the stress tensor and action up to quartic order in the
new fields, which is sufficient for the two-loop computations we perform in
this work. For $T_{xt}$, we have

$\displaystyle 4GT_{xt}$ $\displaystyle=-{i\over 4}(f^{\prime
2}-\bar{f}^{\prime
2})+\frac{i}{16}r_{c}^{2}\big{(}2f^{\prime}f^{\prime\prime}\bar{f}^{\prime\prime}-f^{\prime
2}\bar{f}^{\prime\prime\prime}-2f^{\prime\prime}\bar{f}^{\prime}\bar{f}^{\prime\prime}+f^{\prime\prime\prime}\bar{f}^{\prime
2}\big{)}^{\prime}$ (88)
$\displaystyle+\text{quartic~{}total~{}derivatives}\,.$

$T_{tt}$ is found to be

$\displaystyle 4GT_{tt}$ $\displaystyle={1\over 4}(f^{\prime
2}+\bar{f}^{\prime 2})-{1\over
16}r_{c}^{2}\left(f^{\prime\prime}\bar{f}^{\prime 2}+f^{\prime
2}\bar{f}^{\prime\prime}\right)^{\prime\prime}+{1\over 8}r_{c}f^{\prime
2}\bar{f}^{\prime 2}$ (89) $\displaystyle+\text{quartic~{}total~{}
derivatives}\,.$

The Hamiltonian to quartic order is

$H=\int\\!dx\mathcal{H}={1\over 2\pi}\int\\!dxT_{tt}={1\over 16\pi
G}\int\\!dx\left({1\over 2}f^{\prime 2}+{1\over 2}\bar{f}^{\prime 2}+{1\over
4}r_{c}f^{\prime 2}\bar{f}^{\prime 2}+\cdots\right)\,,$ (90)

leading to the simple action

$\displaystyle I$
$\displaystyle=-\int\\!dt\big{(}i_{V_{\eta}}\Upsilon-H\big{)}$ (91)
$\displaystyle={1\over 16\pi
G}\int\\!d^{2}x\Big{(}f^{\prime}\partial_{\bar{z}}f+\bar{f}^{\prime}\partial_{z}\bar{f}+{1\over
4}r_{c}f^{\prime 2}\bar{f}^{\prime 2}+\cdots\Big{)}\,,$

where as usual $(z,\bar{z})=(x+it,x-it)$. This agrees with the expansion of
(75) to this order, with no higher derivative terms present.161616In the next
section, we use the Chern-Simons formulation to verify (75) to eighth order.
Coming back to our choice of the $1/16$ in (87), for any other choice of
coefficient the Hamiltonian includes a term proportional to
$r_{c}^{2}f^{\prime}\bar{f}^{\prime}f^{\prime\prime}\bar{f}^{\prime\prime}$.
The full expressions for the stress tensor components to cubic order are

$\displaystyle 4GT_{zz}$ $\displaystyle={1\over 2}f^{\prime\prime}-{1\over
4}f^{\prime 2}+{1\over 4}r_{c}f^{\prime\prime\prime}\bar{f}^{\prime}-{1\over
8}r_{c}\big{(}f^{\prime
2}-2f^{\prime}\bar{f}^{\prime}\big{)}^{\prime}\,\bar{f}^{\prime}$ (92)
$\displaystyle+{1\over
16}r_{c}^{2}\big{(}f^{\prime\prime\prime\prime}\bar{f}^{\prime 2}+(f^{\prime
2})^{\prime\prime}\bar{f}^{\prime\prime}\big{)}+{\text{quartic}}\,,$
$\displaystyle 4GT_{\bar{z}\bar{z}}$ $\displaystyle={1\over
2}\bar{f}^{\prime\prime}-{1\over 4}\bar{f}^{\prime 2}+{1\over
4}r_{c}\bar{f}^{\prime\prime\prime}f^{\prime}-{1\over
8}r_{c}\big{(}\bar{f}^{\prime
2}-2\bar{f}^{\prime}f^{\prime}\big{)}^{\prime}\,f^{\prime}$
$\displaystyle+{1\over
16}r_{c}^{2}\big{(}\bar{f}^{\prime\prime\prime\prime}f^{\prime
2}+(\bar{f}^{\prime
2})^{\prime\prime}f^{\prime\prime}\big{)}+{\text{quartic}}\,,$ $\displaystyle
4GT_{z\bar{z}}$ $\displaystyle=-{1\over
4}r_{c}f^{\prime\prime}\bar{f}^{\prime\prime}+{1\over
8}r_{c}(f^{\prime\prime}\bar{f}^{\prime 2}+f^{\prime
2}\bar{f}^{\prime\prime})-{1\over
8}r_{c}^{2}(f^{\prime\prime\prime}\bar{f}^{\prime}\bar{f}^{\prime\prime}+f^{\prime}f^{\prime\prime}\bar{f}^{\prime\prime\prime})+{\text{quartic}}\,.$

The action and stress tensor are invariant under the gauge transformation
discussed in section 6. The complicated form of these transformations, along
with the need to re-express them in terms of $(f,\bar{f})$, make these
symmetries difficult to use in practice. However, we expect that the full
expressions for the stress tensor are fixed by gauge invariance in terms of
their leading terms. Let us also comment that the stress tensor in principle
can be derived via Noether’s theorem using the transformations (41), as was
done at $r_{c}=0$ at the end of section 7.

Finally, we justify why we can pass from the momentum (84) to the canonical
one-form $\Upsilon$ in (85). As before, the question is whether the equation
$i_{V_{\chi}}\Omega=-\delta P$ fixes the symplectic form $\Omega$ according to
our rule, or whether there is an ambiguity of the form
$\Delta\Omega=\int\\!dx(\delta X_{f}\wedge\delta
f+X_{\bar{f}}\wedge\delta\bar{f})$. To this end, we note that we can invert
(87) order-by-order to obtain local expressions for $(f,\bar{f})$ in terms of
$(A,B)$. $(A,B)$ will always appear with a least one derivative. Given this,
it follows that any candidate $\Delta\Omega$ of the form just noted will,
under the field redefinition to $(A,B)$ turn into a $\Delta\Omega$ of the sort
that we previously excluded. This justifies the procedure in the $(f,\bar{f})$
frame.

###  4 Chern-Simons formulation of boundary graviton action

Classically or in quantum perturbation theory, $3d$ Einstein gravity can be
formulated as a gauge theory, namely a Chern-Simons theory whose connections
are constructed from the spin connection and vielbein of the first-order
formulation of general relativity. The relation between Chern-Simons theory
and Einstein gravity at the non-perturbative level is unclear. One of the main
ingredients in a non-perturbative theory of gravity is a sum over topologies,
while such a sum is not natural from the perspective of gauge theory. These
issues are, however, beyond the scope of this work. In this section, we are
interested in understanding the perturbative theory of boundary gravitons from
the Chern-Simons perspective.

The general strategy will be similar to section 3: to identify the boundary
phase space, we consider all gauge transformations of a chosen solution that
preserve the boundary conditions and then quotient out small gauge
transformations. Having identified the phase space, we evaluate the action.
This will be done for the case of boundary conditions imposed at the
asymptotic boundary of AdS3, as well as the case of a finite cutoff boundary.

We start with a quick review of Chern-Simons gravity in three dimensions. In
this section, we work in Lorentzian signature, and only Wick rotate to
Euclidean signature at the end of the computation to connect to the results of
section 3.171717To be precise, we will relate Lorentzian to Euclidean time by
$t_{L}=it_{E}$ and Lorentzian actions $S$ to Euclidean actions $I$ through
$I=iS|_{t_{L}\to it_{E}}$.

####  1 Action and boundary conditions

As mentioned above, Einstein gravity in $3d$ is classically equivalent to a
gauge theory. For negative cosmological constant, the gauge group is
$SO(2,2)=\operatorname{SL}(2,\mathbb{R})\times\operatorname{SL}(2,\mathbb{R})/\mathbb{Z}_{2}$.

We denote the generators of $\operatorname{sl}(2,\mathbb{R})$ by $L_{0,\pm 1}$
and take them to obey

$[L_{m},L_{n}]=(m-n)L_{m+n}\,.$ (93)

An explicit representation is

$L_{0}=\left(\begin{array}[]{cc}\frac{1}{2}&0\\\ 0&-\frac{1}{2}\\\
\end{array}\right)\,,\quad L_{1}=\left(\begin{array}[]{cc}0&0\\\ -1&0\\\
\end{array}\right)\,,\quad L_{-1}=\left(\begin{array}[]{cc}0&1\\\ 0&0\\\
\end{array}\right)\,,$ (94)

which obey

$\operatorname{Tr}(L_{0}^{2})={1\over
2}\,,\quad\operatorname{Tr}(L_{1}L_{-1})=-1\,,$ (95)

with other traces vanishing.

The Chern-Simons connections are related to the dreibein and the spin
connection of the first-order formulation of gravity as follows

$A=L_{a}\left(\omega_{\mu}^{a}+e^{a}_{\mu}\right)dx^{\mu}\,,\quad\bar{A}=L_{a}\left(\omega_{\mu}^{a}-e^{a}_{\mu}\right)dx^{\mu}\,.$
(96)

The base manifold $M_{3}$ equips with coordinates $x^{\mu}=\\{r,t,x\\}$, where
$r$ is the holographic coordinate for which $r\to 0$ at the conformal
boundary. Greek indices are reserved for the boundary $M_{3}$, which is
equipped with coordinates $x^{i}=\\{t,x\\}$. The metric extracted from the
Chern-Simons connections is

$g_{\mu\nu}=2\operatorname{Tr}\left(e_{\mu}e_{\nu}\right)={1\over
2}\operatorname{Tr}\left[(A-\bar{A})_{\mu}(A-\bar{A})_{\nu}\right]$ (97)

The gravitational action is written in terms of the connections as

$S_{\text{grav}}=S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]+S_{\text{bndy}}\,,$
(98)

where the Chern-Simons action at level $k$ reads

$S_{\text{CS}}[A]={k\over 4\pi}\int_{M_{3}}\operatorname{Tr}\left(A\wedge
dA+{2\over 3}A\wedge A\wedge A\right)\,.$ (99)

Here $k$ is related to Newton’s constant $G$ and the AdS3 length scale
as181818We are again working with $\ell=1$ in this section.

$k=\frac{\ell}{4G}\,.$ (100)

The equations of motion imply the flatness of the Chern-Simons connections,
which correspond to the Einstein equations and the vanishing of torsion.

We now turn to the choice of boundary conditions and the associated boundary
term in the action. In complete generality, we write the connections as

$\displaystyle A$ $\displaystyle=E^{+}L_{1}+\Omega L_{0}+f^{-}L_{-1}\,,$ (101)
$\displaystyle\bar{A}$
$\displaystyle=f^{+}L_{1}+\bar{\Omega}L_{0}+E^{-}L_{-1}\,,$

where at this stage, all functions depend arbitrarily on all three
coordinates. The corresponding metric is

$ds^{2}={1\over 4}(\Omega-\bar{\Omega})^{2}+(E^{+}-f^{+})(E^{-}-f^{-})\,.$
(102)

We choose boundary conditions that mimic our construction in the metric
formulation, where we choose to fix all metric components at $r=r_{c}$. We
write

$\left(\Omega-\bar{\Omega}\right)|_{r_{c}}={1\over
r_{c}}dr\,,\quad(E-f)^{\pm}|_{r_{c}}=\pm 2e^{\pm}_{i}dx^{i}\,,$ (103)

so that

$ds^{2}\big{|}_{r_{c}}={dr^{2}\over
4r_{c}^{2}}-4e^{+}_{i}e^{-}_{j}dx^{i}dx^{j}\,.$ (104)

Here, $e^{\pm}_{i}$ is the fixed boundary zweibein. A boundary term compatible
with these boundary conditions is191919This boundary term appears in a
different form in [110].

$S_{\text{bndy}}={k\over 4\pi}\int_{\partial
M_{3}}\operatorname{Tr}A\wedge\bar{A}-{k\over 4\pi}\int_{\partial
M_{3}}\operatorname{Tr}\left[L_{0}(A-\bar{A})\wedge(A-\bar{A})\right]\,.$
(105)

In particular, a straightforward computation yields the following on-shell
variation of the action

$\delta S_{\text{bulk}}+\delta S_{\text{bndy}}={k\over 2\pi}\int_{\partial
M_{3}}\Big{(}f^{-}\wedge\delta e^{+}+f^{+}\wedge\delta e^{-}\Big{)}\,.$ (106)

Since only variations of the fixed quantities $e^{\pm}$ appear, our
variational principle is consistent.

In practice, it is convenient to impose additional boundary conditions that
are compatible with the variational principle and incorporate all solutions of
interest. The boundary spin connection $\omega$, given by

${1\over 2}\left(\Omega+\bar{\Omega}\right)|_{r_{c}}=\omega\,,$ (107)

is so far unfixed. However, the vanishing of the Chern-Simons field strength
implies

$de^{+}-\omega\wedge e^{+}=de^{-}+\omega\wedge e^{-}=0\,,$ (108)

which are the usual torsionless conditions that uniquely fix the boundary spin
connection $\omega$ in terms of the vielbein $e^{\pm}$. We therefore impose
(108) as a boundary condition. The remaining flatness conditions evaluated at
the boundary impose conservation of the stress tensor (defined below) and also
fix its trace.

####  2 Stress tensor

The boundary stress tensor is defined in terms of the on-shell variation of
the action as202020In this formula $\sqrt{g}$ and $\det e$ refer to boundary
quantities, related by $\sqrt{g}=2\det e$ according to our convention (96).

$\displaystyle\delta S$ $\displaystyle={1\over
4\pi}\int\\!d^{2}x\sqrt{g}\,T^{ij}\delta g_{ij}$ (109)
$\displaystyle={1\over\pi}\int\\!d^{2}x\det e\,T^{i}_{a}\delta e^{a}_{i}\,.$

Comparing to (106), we read off

$T_{+}^{j}={k}\epsilon^{ij}f^{-}_{i}\,,\quad
T_{-}^{j}={k}\epsilon^{ij}f^{+}_{i}\,,$ (110)

where our boundary orientation is defined by

$\epsilon^{tx}=-\epsilon^{xt}={1\over\det e}\,.$ (111)

####  3 Relation to metric formulation

We write the relation between the bulk and boundary terms in the metric versus
Chern-Simons descriptions. For the bulk Einstein-Hilbert action, we have

${1\over 16\pi
G}\int_{M_{3}}\sqrt{g}(R+2)=S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]-{k\over
4\pi}\int_{\partial M_{3}}d\operatorname{Tr}(A\wedge\bar{A})\,.$ (112)

For the Gibbons-Hawking terms, where $h_{\mu\nu}$ is the boundary metric, we
explain in appendix 8.E how it can be rewritten as

${1\over 8\pi G}\int_{\partial M_{3}}\\!d^{2}x\sqrt{h}K={k\over
2\pi}\int_{\partial M_{3}}\operatorname{Tr}(A\wedge\bar{A})\,,$ (113)

under the condition $\partial_{a}n^{a}=0$, which is satisfied given our choice
of gauge (103). We finally have the boundary area counterterm,

$-{1\over 8\pi G}\int_{\partial M_{3}}\sqrt{h}=-{k\over 4\pi}\int_{\partial
M_{3}}\operatorname{Tr}[L_{0}(A-\bar{A})\wedge(A-\bar{A})]\,.$ (114)

The relationship between the complete actions is therefore212121Note that we
are defining the action with an overall sign flip compared (1); this has to do
with the fact that (1) defined in Euclidean signature.

$\displaystyle{1\over 16\pi G}\int_{M_{3}}\sqrt{g}(R+2)+{1\over 8\pi
G}\int_{\partial M_{3}}\\!d^{2}x\sqrt{h}K-{1\over 8\pi G}\int_{\partial
M_{3}}\sqrt{h}$ (115)
$\displaystyle=S_{\text{CS}}[A]-S_{\text{CS}}[\bar{A}]+{k\over
4\pi}\int_{\partial M_{3}}\operatorname{Tr}(A\wedge\bar{A})~{}-{k\over
4\pi}\int_{\partial
M_{3}}\operatorname{Tr}[L_{0}(A-\bar{A})\wedge(A-\bar{A})].$

So, our Chern-Simons action agrees with the standard gravity action.

####  4 Boundary action

We now reduce the bulk theory to the boundary by solving the constraints of
the theory and substituting it back in. The Chern-Simons connections can be
written in a space-time split as

$A=A_{t}dt+\tilde{A}\,,\quad\bar{A}=\bar{A}_{t}dt+\tilde{\bar{A}}$ (116)

and we similarly write the exterior derivative on spacetime as
$d=dt\,\partial_{t}+\tilde{d}$. The components $A_{t}$ and $\bar{A}_{t}$
appear in the action as Lagrange multipliers,

$S_{\text{CS}}[A]={k\over
4\pi}\int_{M_{3}}\operatorname{Tr}\left[2A_{t}dt\wedge\tilde{\mathcal{F}}+\tilde{A}\wedge
dt\wedge\partial_{t}\tilde{A}-\tilde{d}\left(\tilde{A}\wedge
A_{t}dt\right)\right]\,,$ (117)

where the spatial field strength is
$\tilde{\mathcal{F}}=\tilde{d}\tilde{A}+\tilde{A}\wedge\tilde{A}$. The $A_{t}$