xxx

Root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
Deformed Boundary Conditions in Holography

Stephen Ebert,1 Christian Ferko2 and Zhengdi Sun3

1 Mani L. Bhaumik Institute for Theoretical Physics,
University of California, Los Angeles, CA 90095, USA

2 Center for Quantum Mathematics and Physics (QMAP),
Department of Physics & Astronomy, University of California, Davis, CA 95616,
USA

3 Department of Physics, University of California, San Diego, CA 92093, USA

We develop the holographic dictionary for pure $\mathrm{AdS}_{3}$ gravity
where the Lagrangian of the dual $2d$ conformal field theory has been deformed
by an arbitrary function of the energy-momentum tensor. In addition to the
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation,
examples of such functions include a class of marginal stress tensor
deformations which are special because they leave the generating functional of
connected correlators unchanged up to a redefinition of the source and
expectation value. Within this marginal class, we identify the unique
deformation that commutes with the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow, which is the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
operator, and write down the modified boundary conditions corresponding to
this root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation. We also identify the unique marginal stress tensor flow for the
cylinder spectrum of the dual CFT which commutes with the inviscid Burgers’
flow driven by ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$, and we propose this unique flow as a candidate root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of the
energy levels. We study BTZ black holes in $\mathrm{AdS}_{3}$ subject to
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed boundary conditions, and find that their masses flow in a way which
is identical to that of our candidate root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ energy flow equation,
which offers evidence that this flow is the correct one. Finally, we also
obtain the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformed boundary conditions for the gauge field in the Chern-Simons
formulation of $\mathrm{AdS}_{3}$ gravity.

###### Contents

  1. 1 Introduction
  2. 2 Holographic Dictionary for Stress Tensor Deformations
    1. 2.1 Multi-trace Deformations
    2. 2.2 Compatibility with Hubbard-Stratonovich
    3. 2.3 Stress Tensor Deformations of $\mathrm{AdS}_{3}/\mathrm{CFT}_{2}$
  3. 3 Root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ from Consistency Conditions
    1. 3.1 Derivation of Deformed Boundary Conditions
    2. 3.2 Derivation of Deformed Energy Levels
  4. 4 AdS3 Gravity with Root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ Deformed Boundary Conditions
    1. 4.1 Metric Formalism
    2. 4.2 Chern-Simons Formalism
  5. 5 Conclusion and Outlook
  6. A AdS3 Gravity with ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-Deformed Boundary Conditions
    1. A.1 Metric Formalism
    2. A.2 Chern-Simons Formalism

##  1 Introduction

A promising strategy for learning more about holography is to begin with a
relatively well-understood holographic correspondence and then to deform it in
some controlled way. We will focus on the case of an asymptotically
$\mathrm{AdS}_{3}$ bulk which is dual to a two-dimensional conformal field
theory. Given such a holographic boundary theory, we can view the
$\mathrm{CFT}_{2}$ as essentially defining the $3d$ gravitational theory. More
precisely, the $\mathrm{CFT}_{2}$ defines the boundary conditions which the
fields of the bulk gravity theory should obey at infinity.

To take a concrete example, we recall that every translation-invariant quantum
field theory admits a conserved stress tensor operator $T_{\alpha\beta}$. In
the holographic dictionary, this boundary stress tensor operator is dual to
the asymptotic bulk metric. One way to see this is to vary the action $S$ of
the $3d$ gravitational theory, including both the Einstein-Hilbert term and
appropriate boundary terms, and put this varied quantity on-shell using the
bulk equations of motion. The resulting expression can be written as a
boundary integral

$\displaystyle\delta S\Big{|}_{\text{on-
shell}}=\frac{1}{2}\int_{\partial\mathcal{M}}d^{2}x\,\sqrt{\gamma}\,T_{\alpha\beta}\,\delta\gamma^{\alpha\beta}\,,$
(1.1)

where $\mathcal{M}$ is the $3d$ spacetime manifold, $\partial\mathcal{M}$ is
its $2d$ boundary, and $\gamma^{\alpha\beta}$ is the metric on
$\partial\mathcal{M}$. In order for the on-shell variation of the action to
vanish, we require $\delta\gamma^{\alpha\beta}=0$ on $\partial\mathcal{M}$,
which means that we impose Dirichlet boundary conditions on the metric near
infinity. The quantity $T_{\alpha\beta}$ which appears in (1.1) is then
identified with the expectation value of the stress tensor operator of the
boundary theory; the procedure described above furnishes an explicit
expression for $T_{\alpha\beta}$ in terms of functions appearing in the
Fefferman-Graham expansion of the metric near infinity. We interpret this by
saying that the asymptotic metric $\gamma^{\alpha\beta}$ is a source for the
stress tensor operator of the dual $\mathrm{CFT}_{2}$.

Now consider a deformation of the boundary conformal field theory. One
familiar way to perform such a deformation is to add an integrated local
operator to the action defining the $2d$ theory, so that

$\displaystyle S_{0}\longrightarrow S_{0}+\delta S=S_{0}+\mu\int
d^{2}x\,\sqrt{\gamma}\,\mathcal{O}(x)\,,$ (1.2)

where $\mathcal{O}(x)$ is a local operator and $\mu$ is a real parameter.
Because the $\mathrm{CFT}_{2}$ defines the boundary conditions which the bulk
fields obey at infinity, it is natural to expect that such a deformation would
change these boundary conditions. This has been shown to be the case for many
such multi-trace deformations [1], at least subject to the usual caveats that
one should restrict attention to the effects on light single-trace operators
at large $N$.

For instance, one much-studied example is a double-trace deformation, where
the object $\mathcal{O}(x)$ appearing in (1.2) is the square of an operator
which is dual to a fundamental field in the gravity theory. In this work, we
will focus on deformations constructed from the stress-energy tensor
$T_{\alpha\beta}$; because this operator is present in _any_ translation-
invariant quantum field theory, such deformations are in a sense universal. An
operator $\mathcal{O}(x)$ which is constructed from products of components
$T_{\alpha\beta}$ is a double-trace operator, by the definition given above,
because the stress-energy tensor is dual to the bulk metric, which is a
fundamental field of the gravity theory. One particularly nice Lorentz-
invariant double-trace combination of components $T_{\alpha\beta}$ is

$\displaystyle\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}}=T^{\alpha\beta}T_{\alpha\beta}-\left(\tensor{T}{{}^{\alpha}_{\alpha}}\right)^{2}\,$

This combination defines the so-called ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator, which has
generated considerable research interest in recent years. For the moment, let
us focus on the properties of this operator purely as an object in the $2d$
boundary theory (and postponing its bulk interpretation). By infinitesimally
adding this operator $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ at each step along a
flow, one can define a one-parameter family of theories which obeys the
differential equation

$\displaystyle\frac{\partial S^{(\lambda)}}{\partial\lambda}=-\frac{1}{2}\int
d^{2}x\,\sqrt{\gamma}\,\mathcal{O}^{(\lambda)}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}(x)\,,$ (1.3)

where the superscript $\lambda$ is meant to emphasize that we must re-compute
the operator $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}^{(\lambda)}$ using the
deformed stress tensor $T_{\alpha\beta}^{(\lambda)}$ at each point along the
flow.111Throughout this work, we always use the symbol $\lambda$ to denote the
parameter of a ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ flow, while we use the symbol $\mu$ either for the parameter of a
generic deformation of a boundary field theory, or for the parameter of the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow,
which we introduce shortly. Note that $\mu$ is never a spacetime index.

We make three sets of observations.

  1. (I)

First note that $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ is a dimension-four
operator, which means that it is irrelevant in the Wilsonian sense. As a
consequence the flow equation (1.3) is quite unusual, from the perspective of
the renormalization group. Ordinarily one imagines beginning with a conformal
field theory and then adding an integrated _relevant_ operator in the spectrum
of the theory, which triggers a flow to the infrared. In a loose sense, the
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow is the
inverse of this familiar paradigm, as we add an integrated _irrelevant_
operator which modifies the definition of the theory in the ultraviolet.

  2. (II)

The quantity $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ defined in (1)
involves products of stress tensor operators. As products of coincident local
operators are generally divergent in quantum field theory, it is far from
obvious that the combination $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ actually defines a
local operator at all. However, it has been shown that one can begin with a
point-split quantity

$\displaystyle\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}}(x,y)=T^{\alpha\beta}(x)\,T_{\alpha\beta}(y)-\tensor{T}{{}^{\alpha}_{\alpha}}(x)\,\tensor{T}{{}^{\beta}_{\beta}}(y)\,,$
(1.4)

and then take a coincident point-limit $\lim\limits_{y\to
x}\mathcal{O}_{{T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}}(x,y)$. Surprisingly, this procedure _does_ define a sensible local
operator, up to certain total derivative ambiguities which can be ignored [2,
3].

  3. (III)

This deformation is “nice” in the sense that it preserves many desirable
properties of the undeformed theory, such as integrability [3, 4, 5, 6] and
supersymmetry [7, 8, 9, 10, 11, 12, 13]. Relatedly, observables in the
deformed theory can often be described with simple closed-form expressions; a
few examples include the finite-volume spectrum [3, 4], $S$-matrix [14], and
torus partition function [15, 16, 17].

Because the operator $\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ appears to be rather
special from the field theory perspective, one might suspect that this
deformation corresponds to some fairly natural modification of the asymptotic
boundary conditions for the bulk fields in the holographic dual. This turns
out to be the case [18]. To see this, one first defines the
$\lambda$-dependent quantities

$\displaystyle\begin{split}\gamma^{(\lambda)}_{\alpha\beta}&=\gamma^{(0)}_{\alpha\beta}-2\lambda\widehat{T}_{\alpha\beta}^{(0)}+\lambda^{2}\widehat{T}^{(0)}_{\alpha\rho}\,\widehat{T}^{(0)}_{\sigma\beta}\,\gamma^{(0)\rho\sigma}\,,\\\
\widehat{T}^{(\lambda)}_{\alpha\beta}&=\widehat{T}^{(0)}_{\alpha\beta}-\lambda\widehat{T}^{(0)}_{\alpha\rho}\,\widehat{T}^{(0)}_{\sigma\beta}\gamma^{(0)\rho\sigma}\,,\end{split}$
(1.5)

where
$\widehat{T}_{\alpha\beta}=T_{\alpha\beta}-\gamma_{\alpha\beta}\tensor{T}{{}^{\rho}_{\rho}}$
is the trace-reversed stress tensor. In terms of these quantities, the
boundary action which solves the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow equation (1.3)
has the property that its variation can be written as

$\displaystyle\delta S=\frac{1}{2}\int
d^{2}x\,\sqrt{\gamma^{(\lambda)}}\,T_{\alpha\beta}^{(\lambda)}\,\delta\gamma^{(\lambda)\alpha\beta}\,.$
(1.6)

This is _exactly_ of the same form as the usual on-shell bulk variation,
equation (1.1), except written in terms of the $\lambda$-dependent metric and
stress tensor. In order for the variation of the action to vanish, we now
require that $\delta\gamma^{(\lambda)\alpha\beta}=0$, which means that we
impose Dirichlet boundary conditions on the deformed metric
$\gamma^{(\lambda)\alpha\beta}$ at infinity. In terms of the original
variables, this looks like a certain choice of mixed boundary conditions on
the metric at infinity, since we now hold fixed a combination of the original
metric $\gamma^{(0)}_{\alpha\beta}$ and its radial derivative, which is
related to $T^{(0)}_{\alpha\beta}$.

One might ask whether there are other universal deformations constructed from
stress tensors which admit interpretations as particularly simple modified
boundary conditions. Another candidate is the recently-proposed root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator [19], which
is defined as

$\displaystyle\mathcal{R}=\sqrt{\frac{1}{2}T^{\alpha\beta}T_{\alpha\beta}-\frac{1}{4}\left(\tensor{T}{{}^{\alpha}_{\alpha}}\right)^{2}}\,.$
(1.7)

By way of comparison, let us revisit the three points (I) \- (III) which we
made concerning the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ operator and consider the analogous statements for root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$.

  1. ($\widetilde{\mathrm{I}}$)

Whereas ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ is
an irrelevant operator, the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator is
classically _marginal_. For instance, it has been checked in a large class of
examples that the stress tensor of a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed CFT still has
vanishing trace. As a consequence, the coupling constant $\mu$ parameterizing
the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
flow is dimensionless.

  2. ($\widetilde{\mathrm{II}}$)

Although ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ is
quantum-mechanically well-defined, it is not known whether the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator can be
defined at the quantum level by point-splitting. Understanding the quantum
properties of this operator remains an important open problem.

  3. ($\widetilde{\mathrm{III}}$)

The root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation shares some of the “niceness” properties of the ordinary ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. For
instance, flow equations for the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed Lagrangian
can often be solved in closed form [19], and the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation preserves
classical integrability in many examples [20]. However, formulas for
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed spectra, $S$-matrices, and partition functions have not been
obtained.

Although much less is known about the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator, there are
many hints that this deformation might lead to an interesting class of models.
One is the relation to the ModMax theory [21, 22, 23, 24] in four dimensions.
This theory and its Born-Infeld extension obey $4d$ analogues of the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow
equations, respectively [25, 5], and both flows can be supersymmetrized [26,
27]. The root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ operator also appears in a flow equation which generates the $3d$
Born-Infeld Lagrangian or its supersymmetric extension [28]. Further, the
dimensional reduction of the ModMax theory is identical to the theory obtained
by root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deforming a collection of $2d$ free scalars [29, 30]. A $(0+1)$-dimensional
version of the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation was studied in [31], which also preserves integrability.
This operator has been connected to ultra/non-relativistic limits and the BMS
group in three dimensions [32, 33], and to nonlinear automorphisms of the
conformal algebra [34]. See also [35] for an analysis of ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like deformations
using characteristic flows.

Given the interest in the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator from the
field theory perspective, it is natural to ask whether there are modified
boundary conditions for the bulk metric which implement this deformation, as
(1.5) do in the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ case. In this work, we will argue that the answer to this question is
yes, and the analogous expressions are

$\displaystyle\begin{split}\gamma_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}\widetilde{T}_{\alpha\beta}^{(0)}\,,\\\
\widetilde{T}_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\widetilde{T}_{\alpha\beta}^{(0)}+\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,,\end{split}$
(1.8)

where we have defined
$\widetilde{T}_{\alpha\beta}=T_{\alpha\beta}-\frac{1}{2}\gamma_{\alpha\beta}\tensor{T}{{}^{\rho}_{\rho}}$,
which is the traceless part of the stress tensor (not to be confused with the
_trace-reversed_ stress tensor $\widehat{T}_{\alpha\beta}$), and

$\displaystyle\mathcal{R}^{(0)}$
$\displaystyle=\sqrt{\frac{1}{2}T^{(0)\alpha\beta}T_{\alpha\beta}^{(0)}-\frac{1}{4}\left(\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}\right)^{2}}$
$\displaystyle=\sqrt{-\det\left(\widetilde{T}_{\alpha\beta}^{(0)}\right)}\,,$
(1.9)

is the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
operator as before.

This means that – from the viewpoint of holography – the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation plays a
similar role as the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation (or other $f(T)$ deformations), insofar as it imposes
certain mixed boundary conditions where some function of the metric
$\gamma_{\alpha\beta}$ and stress tensor $T_{\alpha\beta}$ is held fixed.
However, the mixed boundary conditions which appear in the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ case are considerably
more exotic because they involve the expression $\mathcal{R}^{(0)}$ which is
non-analytic in the stress tensor $T^{(0)}_{\alpha\beta}$. Despite this
unusual feature, we will show that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
conditions have several surprisingly nice properties: for instance, various
combinations of deformed quantities, like
$T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}$ and
$\det\left(\gamma_{\alpha\beta}^{(\mu)}\right)$, are equal to their undeformed
values, and the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformed boundary conditions commute with the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed boundary
conditions, in a sense which we will make precise below. These unexpectedly
simple relations, along with the pressing need to more deeply understand
theories of root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ type, motivates us to undertake a detailed study of the boundary
conditions (1.8) in the remainder of the present work.

The layout of this paper is as follows. In Section 2, we review the
holographic dictionary under general multi-trace deformations and apply these
results to stress tensor deformations of $\mathrm{AdS}_{3}/\mathrm{CFT}_{2}$.
In Section 3, we use consistency conditions, such as commutativity between
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$, to
identify the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformed boundary conditions and the flow equation for the finite-
volume spectrum of the field theory under a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. In
Section 4, we study $\mathrm{AdS}_{3}$ gravity with these root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
conditions in both the metric and Chern-Simons formalisms and perform a
holographic computation of the deformed spacetime mass which agrees with our
flow equation for the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed spectrum. In
Section 5, we conclude and identify directions for future research.

##  2 Holographic Dictionary for Stress Tensor Deformations

The connection between deformations of a field theory by local operators and
modified boundary conditions for the gravity dual was pointed out in the early
days of $\mathrm{AdS}$/$\mathrm{CFT}$. For double-trace deformations, the
effect on the CFT partition function was discussed in [36] and its relation to
modified boundary conditions was explored in [37, 1, 38, 39, 40, 41, 42]. A
generalization to multi-trace deformations, which we will follow in section
2.1, was laid out in [43]. Although earlier work focused on relevant or
marginal deformations, the analysis of irrelevant ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and $J\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu$ deformations is
described in [44, 18] and the lecture notes [45]. See also [46] for a recent
discussion of the generating functional of connected stress tensor correlators
in holography (without ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-like deformations).

In this section we will review some of this well-known material with the goal
of applying it to more general stress tensor deformations in
$\mathrm{AdS}_{3}$/$\mathrm{CFT}_{2}$. An arbitrary scalar constructed from
the stress tensor $T_{\alpha\beta}$ for a two dimensional field theory can be
written as a function of two independent invariants,

$\displaystyle
f\left(T_{\alpha\beta}\right)=f\left(\tensor{T}{{}^{\alpha}_{\alpha}},T^{\alpha\beta}T_{\alpha\beta}\right)\,,$
(2.1)

since all higher traces of $T_{\alpha\beta}$ are related to these two by trace
identities. At the classical level, any such function can be used to generate
a deformation of a quantum field theory. The usual ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation
corresponds to

$\displaystyle
f=T^{\alpha\beta}T_{\alpha\beta}-\left(\tensor{T}{{}^{\alpha}_{\alpha}}\right)^{2}=\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}\,,$ (2.2)

whereas the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation is

$\displaystyle
f=\sqrt{\frac{1}{2}T^{\alpha\beta}T_{\alpha\beta}-\frac{1}{4}\left(\tensor{T}{{}^{\alpha}_{\alpha}}\right)^{2}}=\mathcal{R}\,.$
(2.3)

As we will explain, for any operator $f$ which is chosen as a deformation of
the two-dimensional field theory, one can find the modified generating
functional in the large-$N$ limit by a path integral argument. For certain
choices of $f$, it is then possible to explicitly solve for the modified
boundary conditions in the $3d$ bulk gravity theory.

The surprising feature of a deformation by a _marginal_ combination of stress
tensors, such as the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator, is that the
additive shift in the generating functional of connected CFT correlators
vanishes to leading order in $\frac{1}{N}$. Although such a deformation still
has non-trivial effects on observables, this feature means that we will not be
able to find the corresponding modified boundary conditions in the usual way.
We will instead need to use a different argument which will be the subject of
Section 3.

###  2.1 Multi-trace Deformations

We first review the reasoning which is used to find the change in the
generating functional under a general multi-trace deformation of the
$\mathrm{CFT}$. We follow [43] except for the mild generalization that we
allow deformations by general scalar quantities constructed from operators
carrying arbitrary indices, which allows us to include the case of stress
tensor deformations. The analysis of this subsection applies in any dimension,
so we will temporarily work in general spacetime dimension $d$ before
specializing to $d=3$ in later subsections. In this section we will also
explicitly retain factors of $N$ in order to make the role of the large-$N$
limit more transparent. Although in later sections we will always implicitly
work in a large-$N$ or large-$c$ limit in order to have a classical bulk
gravity dual, we will typically not emphasize the central charge dependence of
quantities appearing in path integrals.

Consider a $\mathrm{CFT}_{d}$ dual to a bulk $\mathrm{AdS}_{d+1}$ gravity
theory. Let $\mathcal{O}_{A}$ be a collection of local operators in the
conformal field theory which are single trace in the sense that each is dual
to a fundamental field of the bulk gravity theory. For instance, one can
imagine each $\mathcal{O}_{A}$ as being dual to a light scalar field in the
$3d$ bulk, in which case $A$ is an internal index. When we specialize to
stress tensor deformations in $\mathrm{AdS}_{3}$/$\mathrm{CFT}_{2}$, we will
instead think of $\mathcal{O}_{A}$ as some component $T_{\alpha\beta}$ of the
energy-momentum tensor, in which case $A$ is a multi-index of spacetime
indices. For now, we will treat both cases uniformly by using an abstract
index $A$ which may transform under the action of some unspecified Lie group
$G$.

We will deform the action by adding $N^{2}\mu\int
d^{d}x\,\sqrt{\gamma}\,f(\mathcal{O})$. Here $f$ is a scalar function of
$\mathcal{O}_{A}$, in the sense that it is invariant under the action of $G$,
$\mu$ is a coupling constant with the appropriate dimension, and
$\gamma_{\alpha\beta}$ is the boundary metric. We will assume that $f(0)=0$
but make no further assumptions about the function $f$. For simplicity, in the
remainder of this subsection we will assume
$\gamma_{\alpha\beta}=\eta_{\alpha\beta}$ and thus omit factors of
$\sqrt{\gamma}$. Quantities in the deformed theory will be decorated by a
$\mu$ superscript or subscript, whereas quantities in the undeformed theory
will carry a $(0)$ label. Our goal will be to find a relationship between the
generating functionals of connected $\mathcal{O}_{A}$ correlators in the
deformed and undeformed theories, which we write as $W^{(\mu)}[J^{(\mu)}]$ and
$W^{(0)}[J^{(0)}]$, respectively, and which are defined by the path integrals

$\displaystyle\begin{split}e^{-W^{(0)}[J^{(0)}]}&=\int\mathcal{D}\psi\,\exp\left(-S_{0}-N^{2}\int
d^{d}x\,J^{(0)A}(x)\mathcal{O}_{A}(x)\right)\,,\\\
e^{-W^{(\mu)}[J^{(\mu)}]}&=\int\mathcal{D}\psi\,\exp\left(-S_{0}-N^{2}\int
d^{d}x\left(\mu
f(\mathcal{O})+J^{(\mu)A}(x)\mathcal{O}_{A}(x)\right)\right)\,.\end{split}$
(2.4)

Here $J^{(0)A}$ and $J^{(\mu)A}$ are sources which are linearly coupled to the
operators $\mathcal{O}_{A}$ in the undeformed and deformed theories,
respectively. For simplicity, we suppress the $A$ indices on the sources
$J^{(0)A}$, $J^{(\mu)A}$ when they appear as arguments in generating
functionals. Correlators of the operators $\mathcal{O}_{A}$ are obtained from
functional derivatives with respect to the source; for instance, the one-point
function in the undeformed theory is given by

$\displaystyle\langle\mathcal{O}_{A}\rangle_{0}=\frac{1}{N^{2}}\frac{\delta
W[J^{(0)}]}{\delta J^{(0)A}}\equiv\sigma_{A}(x)\,,$ (2.5)

where we introduce the shorthand $\sigma_{A}$ for convenience.

In the large-$N$ limit, all multi-point functions of operators
$\mathcal{O}_{A}$ factorize into products of one-point functions of the form
(2.5). This fact implies a simple relation between the two generating
functionals in equation (2.4). To see this, we begin by changing variables in
the path integral expression for $\exp\left(-W^{(\mu)}[J^{(\mu)}]\right)$,
defining

$\displaystyle\widetilde{J}_{A}=J^{(\mu)}_{A}+\mu\frac{\partial
f(\mathcal{O})}{\partial\mathcal{O}^{A}}\,.$ (2.6)

This shift is performed because a general function $f(\mathcal{O})$ will have
a term linear in $\mathcal{O}$ in its Taylor series expansion. Such a linear
term in the effective action obstructs us from directly applying the results
of large-$N$ factorization.222One way of understanding this, which is nicely
explained in chapter 8 of [47], is to consider diagrammatics. For an effective
action with a term linear in $\mathcal{O}$, there are infinitely many tree
graphs that can be constructed with two external lines, since any lines may
end on linear vertices. This complicates the large $N$ analysis, which usually
proceeds by noting that the leading contribution at large $N$ comes from tree
graphs with a minimal number of external lines (of which there should be
finitely many). Performing the shift (2.6) removes the linear vertex and
repairs this undesirable feature. After implementing this shift to remove the
linear term, the generating functional becomes

$\displaystyle\exp\left(-W^{(\mu)}\left(J^{(\mu)}\right)\right)=\int\mathcal{D}\psi\,\exp\left(-S_{0}-N^{2}\int
d^{d}x\,\left(\mu
f(\mathcal{O})+\widetilde{J}^{A}\mathcal{O}_{A}-\mu\mathcal{O}_{A}\partial^{A}f\right)\right)\,,$
(2.7)

where we introduce the shorthand $\partial^{A}f=\frac{\partial
f(\mathcal{O})}{\partial\mathcal{O}_{A}}$. We may now relate this expression
to the undeformed generating functional evaluated on $\widetilde{J}$ as

$\displaystyle\exp\left(-W^{(\mu)}\left[J^{(\mu)}\right]\right)$
$\displaystyle\quad=\int\mathcal{D}\psi\,\exp\left(-S_{0}-N^{2}\int
d^{d}x\,\left(\widetilde{J}^{A}\mathcal{O}_{A}\right)\right)\exp\left(-\mu
N^{2}\int
d^{2}x\,\left(f(\mathcal{O})-\mathcal{O}_{A}\partial^{A}f\right)\right)\,$
$\displaystyle\quad=e^{-W^{(0)}[\widetilde{J}]}\exp\left(-\mu N^{2}\int
d^{d}x\,\left(f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)\right)\right)+O\left(\frac{1}{N}\right)\,.$
(2.8)

The key observation is that the path integral on the second line of (2.1)
defines a certain expectation value, namely of the second exponential factor,
but in the large $N$ limit we may use factorization to evaluate this
expectation value by replacing all instances of $\mathcal{O}_{A}$ with its
one-point function $\sigma_{A}$. When $\mu=0$, the argument of the second
exponential factor vanishes and the two generating functions are equal, as
expected.

The upshot of this manipulation is that, by taking logarithms of the first and
last expressions of (2.1) and discarding subleading terms as $N\to\infty$, we
conclude

$\displaystyle-W^{(\mu)}\left[J^{(\mu)}\right]=-W^{(0)}\big{[}\,\widetilde{J}\,\big{]}-\mu
N^{2}\int d^{d}x\,\left(f(\sigma)-\sigma_{A}\partial^{a}f(\sigma)\right)\,,$
(2.9)

or in terms of the rescaled generating functionals $w[J]=\frac{1}{N^{2}}W[J]$,

$\displaystyle
w^{(\mu)}\left[J^{(\mu)}\right]=w^{(0)}\big{[}\,\widetilde{J}\,\big{]}+\mu\int
d^{d}x\,\left(f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)\right)\,,$ (2.10)

where now $\sigma(x)=\frac{\delta w^{(0)}[\tilde{J}]}{\delta\tilde{J}(x)}$.

Equation (2.10) is the main result which allows us to find the change in the
generating functional under an arbitrary multi-trace deformation, including by
non-analytic operators like root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$. The deformation by
any such operator has two separate effects. First, the generating functional
$w^{(\mu)}$ is shifted by a term involving an integral of
$f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)$. Second, the effective source
$J^{(\mu)A}$ which is used for computing one-point functions is shifted by a
term proportional to the derivative of $f$, as in equation (2.6).

It will sometimes be convenient to use a varied form of equation (2.10). The
variations of the two generating functionals are defined by varying the
sources and holding the corresponding one-point functions fixed:

$\displaystyle\begin{split}\delta W^{(0)}\left[J^{(0)}\right]&=\int
d^{d}x\,\langle O_{A}\rangle_{0}\,\delta J^{(0)A}\,,\\\ \delta
W^{(\mu)}\left[J^{(\mu)}\right]&=\int d^{d}x\,\langle
O_{A}\rangle_{\mu}\,\delta J^{(\mu)A}\,.\end{split}$ (2.11)

Varying equation (2.10) and substituting for $\delta w^{(0)}$ then gives

$\displaystyle\delta w^{(\mu)}\left[J^{(\mu)}\right]$ $\displaystyle=\delta
w^{(0)}\big{[}\,\widetilde{J}\,\big{]}+\mu\int
d^{d}x\,\delta\left(f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)\right)\,$
$\displaystyle=\int d^{d}x\,\left(\langle
O_{A}\rangle_{0}\,\delta\widetilde{J}^{A}+\mu\delta\left(f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)\right)\right)\,.$
(2.12)

Finally, equating this result with the expression for $\delta w^{(\mu)}$ in
terms of $\delta J^{(\mu)A}$ gives

$\displaystyle\int d^{d}x\,\langle O_{A}\rangle_{\mu}\,\delta J^{(\mu)A}=\int
d^{d}x\,\left(\langle
O_{A}\rangle_{0}\,\delta\widetilde{J}^{A}+\mu\delta\left(f(\sigma)-\sigma_{A}\partial^{A}f(\sigma)\right)\right)\,.$
(2.13)

Equation (2.13) will be useful for finding the modified boundary conditions
for bulk fields after deforming the boundary field theory by some operator
$f$. In particular, for a given deformation, one can match the coefficients of
independent variations in (2.13) to obtain differential equations whose
solution gives the deformed boundary conditions. This is especially helpful
for studying more general deforming operators $f$ which depend both on the
operators $\mathcal{O}_{A}$ and their sources $J^{A}$. Deformations by scalars
constructed from the stress tensor $T_{\alpha\beta}$ are of this more
complicated form, since they involve contractions with the boundary metric
$\gamma^{\alpha\beta}$ which plays the role of the source for
$T_{\alpha\beta}$. It is shown in [43] that an analysis of the varied equation
(2.13) yields the correct modification to the stress tensor $T_{\alpha\beta}$
after a multi-trace deformation, which is convenient because this analysis is
more straightforward than a direct computation from the deformed generating
functional.

As a sanity check, it is useful to consider the case of a double-trace
deformation,

$\displaystyle\mu f(\sigma)=\frac{1}{2}\mu^{AB}\sigma_{A}\sigma_{B}\,,$ (2.14)

where $\mu^{AB}$ is a field-independent symmetric tensor. In this case,

$\displaystyle\mu\partial^{A}f=\mu^{AB}\sigma_{B}\,,$ (2.15)

This means that the source $J_{\mu}^{(A)}$ satisfies

$\displaystyle J^{(\mu)A}=\widetilde{J}^{A}-\mu^{AB}\sigma_{B}\,,$ (2.16)

and thus the source has been shifted by a term which is linear in the
corresponding expectation value. The deformed and undeformed generating
functionals are related by

$\displaystyle
w^{(\mu)}\left[J^{(\mu)}\right]=w^{(0)}\big{[}\,\widetilde{J}\,\big{]}-\frac{1}{2}\int
d^{d}x\,\mu^{AB}\sigma_{A}\sigma_{B}\,.$ (2.17)

Therefore, we see that a double-trace deformation is especially simple:
although we deformed the action by _adding_ an integrated quantity
proportional to $\int d^{d}x\,\mu^{AB}\sigma_{A}\sigma_{B}$, the generating
functional has been deformed by _subtracting_ such a quantity.

###  2.2 Compatibility with Hubbard-Stratonovich

In the case of a double-trace deformation, the general analysis of Section 2.1
is equivalent to another common technique for deriving the modified
holographic dictionary, namely the Hubbard-Stratonovich transformation. This
method exploits the fact that a double-trace deformation is quadratic in
fields and therefore can be decoupled by integrating in an appropriate
auxiliary field. The Hubard-Stratonovich technique has a long history and was
already used in [36] to study the effect of a double-trace deformation on the
dual CFT, which is nicely reviewed in [44, 18]. We note that a similar
strategy was used in [15] in order to replace the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator with a
coupling to a metric-like field $h_{\alpha\beta}$ and interpret the
deformation as random geometry. However, this decoupling procedure does not
straightforwardly apply to more general multi-trace deformations, such as the
square-root-type deformation by $\mathcal{R}$. For completeness, we now
briefly review this alternative derivation and confirm that the resulting
modification to the generating functional is identical.

We again work in general spacetime dimension $d$ and focus on a deformation of
the CFT action which takes the form

$\displaystyle S_{0}\longrightarrow S_{0}+\frac{N^{2}}{2}\int
d^{d}x\,\mu^{AB}\mathcal{O}_{A}\mathcal{O}_{B}\,,$ (2.18)

where the $\mathcal{O}_{A}$ are single-trace operators as before, and consider
the deformed generating functional defined in equation (2.4),

$\displaystyle e^{-W^{(\mu)}[J^{(\mu)}]}$
$\displaystyle=\int\mathcal{D}\psi\,\exp\left(-S_{0}-N^{2}\int
d^{d}x\left(\frac{\mu^{AB}}{2}\mathcal{O}_{A}\mathcal{O}_{B}+J^{(\mu)A}(x)\mathcal{O}_{A}(x)\right)\right)\,.$
(2.19)

We now integrate in an auxiliary field. To emphasize the similarity with the
random geometry analysis of [15], we will use the notation $h_{A}$ for this
Hubbard-Stratonovich field. One has the path integral identity

$\displaystyle 1=\mathcal{N}\int\mathcal{D}h\,\exp\left(\frac{N^{2}}{2}\int
d^{2}x\,h^{A}\left(\mu^{-1}\right)_{AB}h^{B}\right)\,,$ (2.20)

Here the quantity $\mathcal{N}$ is a normalization factor which is defined by
the property that it normalizes the path integral on the right side of (2.20)
to one. It can also be formally written as
$\mathcal{N}=\frac{1}{\sqrt{\det\left(\frac{\mu}{N^{2}}\right)}}$, although we
will use the shorter expression $\mathcal{N}$ to avoid cluttering formulas.
Inserting this identity into the expression (2.19) for the generating
functional,

$\displaystyle\exp\left(-W^{(\mu)}\left(J^{(\mu)}\right)\right)$
$\displaystyle\quad=\mathcal{N}\int\mathcal{D}\psi\,\mathcal{D}h\,\exp[-S_{0}-N^{2}\int
d^{d}x\,\left(J^{(\mu)A}\mathcal{O}_{A}+\frac{\mu^{AB}}{2}\mathcal{O}_{A}\,\mathcal{O}_{B}-\frac{1}{2}\,h^{A}\left(\mu^{-1}\right)_{AB}h^{B}\right)\Bigg{]}\,$
$\displaystyle\quad=\mathcal{N}\int\mathcal{D}\psi\,\mathcal{D}h\,\exp[-S_{0}-N^{2}\int
d^{d}x\,\left(J^{(\mu)A}+\widehat{h}^{A}\right)\mathcal{O}_{A}-\frac{1}{2}\widehat{h}^{A}\mu^{-1}_{AB}\widehat{h}^{B}\Bigg{]}\,,$
(2.21)

where in the last step we have completed the square in the integrand by
writing quantities in terms of a shifted auxiliary field

$\displaystyle\widehat{h}^{A}=h^{A}+\mu^{AB}\mathcal{O}_{B}\,.$ (2.22)

Seeing that the combination $\widehat{h}^{A}+J^{(\mu)A}$ now acts as the
source for $\mathcal{O}_{A}$, we perform a second change of variables to

$\displaystyle\widetilde{h}^{A}=\widehat{h}^{A}+J^{(\mu)A}$ (2.23)

to find

$\displaystyle\exp\left(-W^{(\mu)}\left(J^{(\mu)}\right)\right)$
$\displaystyle\quad=\mathcal{N}\int\mathcal{D}\psi\,\mathcal{D}h\,\exp[-S_{0}-N^{2}\int
d^{d}x\,\left(\widetilde{h}^{A}\mathcal{O}_{A}+\frac{1}{2}\left(\widetilde{h}^{A}-J^{(\mu)A}\right)\mu^{-1}_{AB}\left(\widetilde{h}^{B}-J^{(\mu)B}\right)\right)\Bigg{]}\,$
$\displaystyle\quad=\mathcal{N}\int\mathcal{D}h\,e^{-W^{(0)}[\widetilde{h}]}\,\exp[-N^{2}\int
d^{d}x\,\left(\frac{1}{2}\left(\widetilde{h}^{A}-J^{(\mu)A}\right)\mu^{-1}_{AB}\left(\widetilde{h}^{B}-J^{(\mu)B}\right)\right)\Bigg{]}\,.$
(2.24)

In the second step we have noted that performing the path integral including
the first two terms in the exponential, $S_{0}$ and the coupling
$\widetilde{h}^{A}\mathcal{O}_{A}$, defines the undeformed generating
functional $\exp\left(-W^{(0)}\big{[}\,\widetilde{h}\,\big{]}\right)$, since
$\widetilde{h}^{A}$ acts as the source for $\mathcal{O}_{A}$. We have also
implicitly used large-$N$ factorization, since the third term in the
exponential also depends on the operators $\mathcal{O}_{A}$. In the last line
of (2.2), all implicit instances of such operators are understood to be
replaced with the corresponding one-point functions.

In the large $N$ limit, the remaining path integral over $h$ can be performed
using the saddle point approximation. The saddle occurs at the point
$\widetilde{h}^{A}$ which satisfies

$\displaystyle-\frac{\delta
W^{(0)}[\widetilde{h}]}{\delta\widetilde{h}^{A}}-\mu^{-1}_{AB}\left(\widetilde{h}^{B}-J^{(\mu)B}\right)=0\,.$
(2.25)

On the other hand, the quantity $-\frac{\delta W^{(0)}[\widetilde{h}]}{\delta
h^{A}}$ defines the one-point function
$\langle\mathcal{O}_{A}\rangle_{0}\equiv\sigma_{A}$, where we again introduce
the shorthand $\sigma_{A}$ for the undeformed expectation value of
$\mathcal{O}_{A}$.

Because we are modifying the field theory, the local operator
$\mathcal{O}_{A}^{(0)}$ in the undeformed theory could correspond to some
different operator $\mathcal{O}_{A}^{(\mu)}$ in the deformed theory.
Therefore, in principle we should distinguish between deformed and undeformed
operators, in addition to distinguishing between deformed and undeformed
_expectation values_ which we have written as $\langle\;\cdot\;\rangle_{\mu}$
and $\langle\;\cdot\;\rangle_{0}$, and which differ in that they are computed
using path integrals weighted by different actions. However, we will see that
for both the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation and the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, the
deformed and undeformed operators agree:

$\displaystyle\mathcal{O}_{A}^{(0)}=\mathcal{O}_{A}^{(\mu)}\,.$ (2.26)

More precisely, we will see that the derivatives of the operators
$\mathcal{O}^{(\lambda)}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$ and
$\mathcal{R}^{(\mu)}$ with respect to the appropriate flow parameters
$\lambda$ and $\mu$, respectively, both vanish. Thus we will simply assume
that (2.26) holds in the present analysis; one can view this as an extra
condition one might impose in order to single out a preferred class of
deforming operators.

Solving equation (2.25) then yields
$\widetilde{h}^{A}=J^{(\mu)B}+\mu^{AB}\sigma_{B}$. We therefore find that

$\displaystyle\exp\left(-W^{(\mu)}[J^{(\mu)}]\right)\sim\exp\left(-W^{(0)}\left[J^{(\mu)B}+\mu^{AB}\sigma_{B}\right]-\frac{N^{2}}{2}\int
d^{d}x\,\mu^{AB}\sigma_{A}\sigma_{B}\right)\,.$ (2.27)

Here we write $\sim$ to indicate both overall proportionality, since the
saddle point integral introduces an additional prefactor which we will not
track, and also the approximation to leading order in $\frac{1}{N}$. Taking
logarithms and discarding the normalization, we conclude

$\displaystyle W^{(\mu)}[J^{(\mu)}]$
$\displaystyle=W^{(0)}\big{[}\,\widetilde{J}\,\big{]}-\frac{N^{2}}{2}\int
d^{d}x\,\mu^{AB}\sigma_{A}\sigma_{B}\,,$ $\displaystyle\widetilde{J}^{A}$
$\displaystyle=J^{(\mu)A}+\mu^{AB}\sigma_{B}\,.$ (2.28)

We see that this exactly reproduces equations (2.16) and (2.17), after
shifting to the re-scaled generating functionals $w$ by dividing through by
$N^{2}$.

Therefore, the two approaches that we have described are equivalent for the
case of double-trace deformations. Both computations use the assumption of
large $N$ in a key way. In the first method we used large $N$ factorization in
equation (2.1), and in the Hubbard-Stratonovich approach we used both
large-$N$ factorization in equation (2.2) and a saddle-point approximation in
equation (2.27).

However, it is important to emphasize that the first method is more general
since it applies to arbitrary multi-trace deformations. The Hubbard-
Stratonovich technique crucially relies on the path integral identity (2.20),
which is a Gaussian integral and can therefore only introduce a quadratic
dependence on $h^{A}$. Such a quadratic auxiliary field term is sufficient to
decouple a double-trace deformation like the usual ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$, but for more general
operators such as root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$, we will instead
resort to the multi-trace analysis.

###  2.3 Stress Tensor Deformations of $\mathrm{AdS}_{3}/\mathrm{CFT}_{2}$

In the remainder of this work, we will focus on deformations which are
constructed from the energy-momentum tensor rather than from general operators
$\mathcal{O}_{A}$. It is worth pointing out that such deformations are
qualitatively different in three bulk spacetime dimensions, which is our
primary case of interest. In $\mathrm{AdS}_{3}$, the bulk metric has no local
propagating degrees of freedom. As a result, we do not need to impose the
usual restrictions that a deforming operator built from $\mathcal{O}_{A}$ be
relevant or marginal in order to retain analytic control.

An irrelevant deformation built from an operator $\mathcal{O}_{A}$ which is
dual to a dynamical field, such as a light scalar, would generically backreact
on the metric and therefore become difficult to study. In contrast, an
irrelevant deformation constructed from the $2d$ stress tensor
$T_{\alpha\beta}$, such as the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, does not
lead to any backreaction because the dual field is the (non-dynamical) bulk
metric $g_{\alpha\beta}$. This means that we are free to consider deformations
by _any_ scalar function $f(T)$ of the stress tensor, even those with
arbitrarily large dimension, and study the resulting mixed boundary conditions
in the $\mathrm{AdS}_{3}$ bulk.

As we mentioned around equation (2.1), the most general Lorentz scalar which
can be constructed from a $2d$ stress tensor $T_{\alpha\beta}$ is

$\displaystyle
f\left(T_{\alpha\beta}\right)=f\left(x_{1},x_{2}\right)\,,\qquad
x_{1}=\tensor{T}{{}^{\alpha}_{\alpha}}\,,\qquad
x_{2}=T^{\alpha\beta}T_{\alpha\beta}\,,$ (2.29)

where we introduce $x_{1}=\Tr(T)$ and $x_{2}=\Tr(T^{2})$.

In the notation of Section 2.1, this corresponds to
$\mathcal{O}_{A}=T_{\alpha\beta}$ and $\sigma_{A}=\langle
T_{\alpha\beta}\rangle_{0}$, where $A$ is a multi-index of two boundary
spacetime indices. We note that

$\displaystyle\frac{\partial f}{\partial T_{\alpha\beta}}=\frac{\partial
f}{\partial x_{1}}\gamma^{\alpha\beta}+2\frac{\partial f}{\partial
x_{2}}T^{\alpha\beta}\,,$ (2.30)

where $\gamma_{\alpha\beta}$ is the $2d$ metric.

We may now import the general results for the shift in the generating
functional under a multi-trace deformation defined by

$\displaystyle\frac{\partial S^{(\mu)}}{\partial\mu}=\int
d^{2}x\,\sqrt{\gamma}\,f(x_{1},x_{2})\,.$ (2.31)

Because the boundary metric $\gamma_{\alpha\beta}$ now plays a more important
role, we restore factors of $\sqrt{\gamma}$ in integrals, which were omitted
in Sections 2.1 and 2.2.

Using equation (2.10), we find

$\displaystyle
w^{(\mu)}\left[J^{(\mu)}\right]=w^{(0)}\big{[}\,\widetilde{J}\,\big{]}+\mu\int
d^{2}x\,\sqrt{\gamma}\,\left(f(x_{1},x_{2})-\left(x_{1}\frac{\partial
f}{\partial x_{1}}+2x_{2}\frac{\partial f}{\partial x_{2}}\right)\right)\,,$
(2.32)

where now we use $x_{1},x_{2}$ interchangeably for the operators
$\tensor{T}{{}^{\alpha}_{\alpha}}$, $T^{\alpha\beta}T_{\alpha\beta}$ and the
expectation values $\langle\tensor{T}{{}^{\alpha}_{\alpha}}\rangle$, $\langle
T^{\alpha\beta}\rangle\langle T_{\alpha\beta}\rangle$, as justified by
large-$N$ factorization.

In these formulas, the source $J^{(\mu)}$ which couples to the deformed stress
tensor $T_{\alpha\beta}^{(\mu)}$ is the deformed metric
$\gamma_{\alpha\beta}^{(\mu)}$. This means that the deformation by $f$
involves both single-trace operators and their sources, which makes the
behavior of this deformation more complicated. While a deformation by a
function which depends only on operators $\mathcal{O}_{A}$ (but not their
sources $J^{A}$) shifts the sources and leaves the expectation values
$\langle\mathcal{O}_{A}\rangle$ unchanged, a deformation which depends on both
$\mathcal{O}_{A}$ and $J^{A}$ will shift both the sources and the one-point
functions. In this case, as we discussed above, it is more convenient to use
the varied equation (2.13), which allows us (in principle, at least) to find
expressions for both the deformed sources and the deformed expectation values.
In this context, the appropriate varied equation for a stress tensor
deformation is

$\displaystyle\int
d^{d}x\,\sqrt{\gamma^{(\mu)}}\,T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}=\int
d^{d}x\,\sqrt{\gamma^{(\mu)}}\left[T^{(0)}_{\alpha\beta}\,\delta\gamma^{(0)\alpha\beta}+\mu\,\delta\left(f(x_{1},x_{2})-\left(x_{1}\frac{\partial
f}{\partial x_{1}}+2x_{2}\frac{\partial f}{\partial
x_{2}}\right)\right)\right]\,,$ (2.33)

or after taking the limit as $\mu\to 0$ to obtain a differential equation,

$\displaystyle\frac{\partial}{\partial\mu}\int
d^{d}x\,\sqrt{\gamma^{(\mu)}}\,T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}=\int
d^{d}x\,\sqrt{\gamma^{(\mu)}}\,\delta\left(f(x_{1},x_{2})-\left(x_{1}\frac{\partial
f}{\partial x_{1}}+2x_{2}\frac{\partial f}{\partial x_{2}}\right)\right)\,.$
(2.34)

The operator $\delta$ appearing on the right side acts on both
$T_{\alpha\beta}$ and $\gamma_{\alpha\beta}$. Given a particular choice of
deformation $f(x_{1},x_{2})$, one can then attempt to match the
$\delta\gamma_{\alpha\beta}$ and $\delta T_{\alpha\beta}$ terms on both sides
of (2.34) and solve the resulting coupled differential equations in $\mu$ to
obtain solutions for the deformed quantities $T_{\alpha\beta}^{(\mu)}$ and
$\gamma_{\alpha\beta}^{(\mu)}$.

The known results for the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation can be
recovered by setting

$\displaystyle
f(x_{1},x_{2})=-\frac{1}{2}\left(x_{2}-x_{1}^{2}\right)=-\frac{1}{2}\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}\,,$ (2.35)

where the factor of $-\frac{1}{2}$ is a choice of normalization for the
operator. Substituting this deformation $f$ into (2.34) and stripping off the
integrals gives the condition

$\displaystyle\partial_{\lambda}\left(\sqrt{\gamma^{(\lambda)}}T_{\alpha\beta}^{(\lambda)}\,\delta\gamma^{(\lambda)\alpha\beta}\right)=\delta\left(\sqrt{\gamma^{(\lambda)}}\left(T^{(\lambda)\alpha\beta}T_{\alpha\beta}^{(\lambda)}-\left(\tensor{T}{{}^{(\lambda)}{}^{\alpha}_{\alpha}}\right)^{2}\right)\right)\,,$
(2.36)

where we have changed the label for the deformation parameter from $\mu$ to
$\lambda$ to emphasize that this flow is associated to the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation (see
footnote 1). The indices in equation (2.36) are raised and lowered with the
deformed metric $\gamma^{(\lambda)}_{\alpha\beta}$. One can solve this
equation with the initial conditions
$\gamma^{(\lambda)}_{\alpha\beta}\to\gamma^{(0)}_{\alpha\beta}$,
$T_{\alpha\beta}^{(\lambda)}\to T_{\alpha\beta}^{(0)}$ as $\lambda\to 0$, as
described in [18] and reviewed in appendix A.1. The solution to this
differential equation can be expressed in terms of the trace-reversed stress
tensor,
$\widehat{T}_{\alpha\beta}=T_{\alpha\beta}-\gamma_{\alpha\beta}\tensor{T}{{}^{\rho}_{\rho}}$,
in terms of which one finds

$\displaystyle\begin{split}\gamma^{(\lambda)}_{\alpha\beta}&=\gamma^{(0)}_{\alpha\beta}-2\lambda\widehat{T}_{\alpha\beta}^{(0)}+\lambda^{2}\widehat{T}^{(0)}_{\alpha\rho}\,\widehat{T}^{(0)}_{\sigma\beta}\,\gamma^{(0)\rho\sigma}\,,\\\
\widehat{T}^{(\lambda)}_{\alpha\beta}&=\widehat{T}^{(0)}_{\alpha\beta}-\lambda\widehat{T}^{(0)}_{\alpha\rho}\,\widehat{T}^{(0)}_{\sigma\beta}\gamma^{(0)\rho\sigma}\,,\end{split}$
(2.37)

which reproduces equation (1.5) for the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed boundary
conditions which we quoted in the introduction.

One might ask whether there are other choices for the deforming operator $f$
which are distinguished in some sense. For instance, it is natural to ask
whether there is any choice of $f$ for which the shift in the generating
functional appearing in equation (2.32) vanishes. Such a function $f$
satisfies the differential equation

$\displaystyle f(x_{1},x_{2})=x_{1}\frac{\partial f}{\partial
x_{1}}+2x_{2}\frac{\partial f}{\partial x_{2}}\,,$ (2.38)

which has the general solution

$\displaystyle f(x_{1},x_{2})=x_{1}g\left(\frac{x_{2}}{x_{1}^{2}}\right)$
(2.39)

where $g$ is an arbitrary function. We demand that this deformation is well-
defined if the seed theory is a CFT, for which
$x_{1}=\tensor{T}{{}^{\alpha}_{\alpha}}=0$. The only way for the argument of
the function $g$ to remain finite when $x_{1}=0$ is if $g(y)=\sqrt{c_{1}y}$,
in which case

$\displaystyle
f(x_{1},x_{2})=x_{1}\sqrt{c_{1}\frac{x_{2}}{x_{1}^{2}}}=\sqrt{c_{1}x_{2}}\,.$
(2.40)

Choosing the normalization factor $c_{1}=\frac{1}{2}$, we find

$\displaystyle
f(x_{1},x_{2})=\sqrt{\frac{1}{2}x_{2}}=\sqrt{\frac{1}{2}T^{\alpha\beta}T_{\alpha\beta}}=\mathcal{R}\,\big{|}_{\tensor{T}{{}^{\alpha}_{\alpha}}=0}\,.$
(2.41)

Therefore, the only physical sensible stress tensor deformation of a CFT with
a vanishing shift in (2.32) is, up to proportionality, the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator $\mathcal{R}$
defined in (1.7). Note that this argument fixes the dependence of $f$ on
$x_{2}$ but not on $x_{1}$, since we have restricted to the case of a
conformal theory for which $x_{1}=0$. We will determine the dependence on
$x_{1}$ by demanding that this deformation commute with the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation in Section
3.

Suppose that we wish to identify the deformed metric
$\gamma_{\alpha\beta}^{(\mu)}$ and stress tensor $T_{\alpha\beta}^{(\mu)}$
associated with a deformation by this operator $\mathcal{R}$. One immediately
encounters the subtlety that the differential equation (2.34) reduces to

$\displaystyle\frac{\partial}{\partial\mu}\left(\sqrt{\gamma^{(\mu)}}T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}\right)=0\,.$
(2.42)

This means that the operator $\mathcal{R}$ is in the kernel of the map which
sends deformations to sources on the right side of the differential equation
(2.34). There are multiple solutions to equation (2.42). The most obvious one
is the trivial solution
$\gamma^{(\mu)}_{\alpha\beta}=\gamma^{(0)}_{\alpha\beta}$ and
$T^{(\mu)}_{\alpha\beta}=T^{(0)}_{\alpha\beta}$. Another, less obvious,
solution can be conveniently written in terms of the traceless part of the
stress tensor,
$\widetilde{T}_{\alpha\beta}=T_{\alpha\beta}-\frac{1}{2}\gamma_{\alpha\beta}\tensor{T}{{}^{\rho}_{\rho}}$.
That solution is

$\displaystyle\begin{split}\gamma_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}\widetilde{T}_{\alpha\beta}^{(0)}\,,\\\
\widetilde{T}_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\widetilde{T}_{\alpha\beta}^{(0)}+\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,,\end{split}$
(2.43)

where $\mathcal{R}^{(0)}$ is the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator constructed
from the undeformed metric and stress tensor. One can verify that the
expressions (2.43) solve the differential equation (2.42). In fact, several
quantities of interest remain individually undeformed along this flow:

$\displaystyle\det\left(\gamma_{\alpha\beta}^{(\mu)}\right)=\det\left(\gamma_{\alpha\beta}^{(0)}\right)\,,\qquad
T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}=T_{\alpha\beta}^{(0)}\,\delta\gamma^{(0)\alpha\beta}\,,\qquad\mathcal{R}^{(\mu)}=\mathcal{R}^{(0)}\,.$
(2.44)

If the only condition we impose is that our deformed metric and stress tensor
satisfy (2.42), then there is no way to distinguish between the trivial
solution and the $\mu$-dependent solution (2.43). Furthermore, it is not
immediately obvious whether there are other solutions
$\gamma^{(\mu)}_{\alpha\beta}$, $T^{(\mu)}_{\alpha\beta}$ which also satisfy
this flow. For this reason, from the perspective of the deformed generating
functional, we cannot uniquely identify a single solution for the deformed
metric and stress tensor which corresponds to the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation.

In order to circumvent this ambiguity, we will pursue a complementary analysis
which does not rely on the deformed generating functional. Instead, we will
stipulate a set of consistency conditions which we expect the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed metric and
stress tensor to satisfy, and demonstrate that (2.43) is the only solution
with these properties. This gives an independent piece of evidence that these
deformed boundary conditions are the correct ones which correspond to a
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation of the boundary theory. We turn to this argument in the next
section.

##  3 Root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
from Consistency Conditions

We have seen that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation is subtle
to treat holographically because it belongs to a class of deformations for
which the combination

$\displaystyle\int
d^{2}x\,\sqrt{\gamma^{(\mu)}}T_{\alpha\beta}^{(\mu)}\,\delta\gamma^{(\mu)\alpha\beta}$
(3.1)

is independent of $\mu$. This class also includes trivial deformations, such
as boundary diffeomorphisms or scale transformations, which leave the theory
unchanged.

However, we expect that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation is _not_
such a trivial deformation, and should modify the behavior of the theory in
some way. One piece of evidence for this is that the $2d$ root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of a
collection of bosons is the dimensional reduction of the $4d$ root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of the
free Maxwell theory [30], which gives rise to the ModMax theory. This ModMax
theory represents a genuine modification of the Maxwell theory, in that
physical observables are modified; one example is that the ModMax theory
exhibits birefringence whereas the Maxwell theory does not.

We would therefore like to distinguish the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed theory from
other deformations in the same class which obey (3.1). To do this, we will
enumerate a list of properties which we expect the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed theory to
obey and search for the most general deformation which satisfies these
properties. This will allow us to identify both a candidate set of deformed
boundary conditions $\gamma^{(\mu)}_{\alpha\beta}$, $T^{(\mu)}_{\alpha\beta}$
and a proposal for the deformed finite-volume spectrum of a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed CFT on a
cylinder.

An important ingredient in this analysis is the assumption that the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation commutes with the ordinary ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, in a
sense which we will make precise. This expectation is motivated by the
observation that classical ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flows for the
Lagrangian exhibit this property in many examples [19, 20]. The property of
commuting with ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ is _not_ shared by generic marginal stress tensor deformations. A
simple example is the marginal deformation generated by the trace of the
stress tensor,

$\displaystyle\frac{\partial S}{\partial\mu}=\int
d^{2}x\,\sqrt{\gamma}\,\tensor{T}{{}^{a}_{a}}\,.$ (3.2)

The flow (3.2) simply generates scale transformations, so a conformal field
theory is invariant under such a deformation. However, a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed field theory
is not scale-invariant because the theory has a dimensionful scale set by
$\lambda$. Thus, scale transformations do not commute with ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$. Deforming a CFT first
by (3.2) and then ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$-deforming with parameter $\lambda$ is the same as only performing the
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ step,
whereas first deforming the CFT by ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and then performing
the scale transformation (3.2) is _not_ the same as ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deforming by
$\lambda$.

###  3.1 Derivation of Deformed Boundary Conditions

We aim to find a one-parameter family of modified boundary conditions
$\gamma^{(\mu)}_{\alpha\beta}$, $T_{\alpha\beta}^{(\mu)}$ with the following
properties:

  1. (i)

The deformed boundary conditions should correspond to a classically marginal
deformation of the dual field theory. This means that the parameter $\mu$ is
dimensionless and that, if the undeformed stress tensor is traceless so that

$\displaystyle\gamma^{(0)\alpha\beta}T^{(0)}_{\alpha\beta}=0\,,$ (3.3)

then the deformed stress tensor is also traceless with respect to the deformed
metric,

$\displaystyle\gamma^{(\mu)\alpha\beta}T^{(\mu)}_{\alpha\beta}=0\,.$ (3.4)

  2. (ii)

The deformations of the metric and stress tensor form a group. In particular,
deformations compose. If we deform an initial configuration by $\mu_{1}$,

$\displaystyle\gamma^{(0)}_{\alpha\beta}\,,T^{(0)}_{\alpha\beta}\overset{\mu_{1}}{\longrightarrow}\gamma^{(\mu_{1})}_{\alpha\beta}\,,T^{(\mu_{1})}_{\alpha\beta}\,,$
(3.5)

and then use these quantities as the initial condition for a second
deformation by $\mu_{2}$,

$\displaystyle\gamma^{(\mu_{1})}_{\alpha\beta}\,,T^{(\mu_{1})}_{\alpha\beta}\overset{\mu_{2}}{\longrightarrow}\gamma^{(\mu_{1}+\mu_{2})}_{\alpha\beta}\,,T^{(\mu_{1}+\mu_{2})}_{\alpha\beta}\,,$
(3.6)

then the doubly-deformed quantities are identical to those obtained from doing
a single deformation by the total parameter $\mu_{1}+\mu_{2}$,

$\displaystyle\gamma^{(0)}_{\alpha\beta}\,,T^{(0)}_{\alpha\beta}\xrightarrow{\mu_{1}+\mu_{2}}\gamma^{(\mu_{1}+\mu_{2})}_{\alpha\beta}\,,T^{(\mu_{1}+\mu_{2})}_{\alpha\beta}\,.$
(3.7)

Here we assume that $\mu=0$ is the identity element, so that the deformed
boundary conditions reduce to the undeformed boundary conditions as the
deformation parameter is taken to zero. We further assume the group to be non-
trivial, so a deformation by $\mu\neq 0$ must be different from the identity.

  3. (iii)

The root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation commutes with the ordinary ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, in the
following sense. If we first deform the metric and stress tensor using the
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed
boundary conditions and flow by parameter $\lambda$, and then use these
deformed quantities as the initial condition for a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow by parameter
$\mu$, then the result is identical to first deforming by root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ with parameter $\mu$
and then by ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
with parameter $\lambda$.

${{\gamma^{(0)}_{\alpha\beta},T^{(0)}_{\alpha\beta}}}$${{\gamma^{(\lambda)}_{\alpha\beta},T^{(\lambda)}_{\alpha\beta}}}$${{\gamma^{(\mu)}_{\alpha\beta},T^{(\mu)}_{\alpha\beta}}}$${{\gamma^{(\lambda,\mu)}_{\alpha\beta},T^{(\lambda,\mu)}_{\alpha\beta}}}$$\scriptstyle{\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}}}$$\scriptstyle{\mathcal{R}}$$\scriptstyle{{\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}}}}$$\scriptstyle{\mathcal{R}}$

We will first use assumptions (i) and (ii) to determine the nature of the
modified boundary conditions when the seed theory is conformal, and then use
the third assumption to extend this procedure to the case when the undeformed
theory is non-conformal.

#### Conformal Seed Theory

For a conformal seed theory satisfying
$\gamma^{(0)\alpha\beta}T^{(0)}_{\alpha\beta}=0$, the only independent
dimensionful Lorentz scalar quantity in the problem is
$T^{(0)\alpha\beta}T^{(0)}_{\alpha\beta}$. Ordinarily, there are two
independent scalars that can be constructed from a general $2\times 2$ matrix
$M$ – for instance, $\tr(M)$ and $\tr(M^{2})$ – but we have assumed that the
trace of the stress tensor vanishes. We can equivalently say that any Lorentz
scalar built from a traceless stress tensor is a function of

$\displaystyle\mathcal{R}^{(0)}=\sqrt{\frac{1}{2}T^{(0)\alpha\beta}T_{\alpha\beta}^{(0)}}\,.$
(3.11)

On the other hand, there are also only two functionally independent symmetric
$2$-tensors available in this problem, namely $\gamma_{\alpha\beta}^{(0)}$ and
$T_{\alpha\beta}^{(0)}$. Again, one could attempt to form a new independent
$2$-tensor by taking products of the form

$\displaystyle\left(T^{2}\right)_{\alpha\beta}=T^{(0)}_{\alpha\gamma}\tensor{T}{{}^{(0)}{}^{\gamma}_{\beta}}\,,$
(3.12)

but because of the tracelessness condition, one has the identity

$\displaystyle\left(T^{2}\right)_{\alpha\beta}=\left(\mathcal{R}^{(0)}\right)^{2}\gamma_{\alpha\beta}\,,$
(3.13)

so this combination does not actually give an independent tensor structure.
Obviously, all higher powers of the stress tensor will also be proportional to
either $\gamma_{\alpha\beta}^{(0)}$ or $T_{\alpha\beta}^{(0)}$ with
coefficients that are functions of $\mathcal{R}^{(0)}$.

We therefore find that the most general ansatz for deformed symmetric tensors
$\gamma_{\alpha\beta}^{(\mu)}$ and $T_{\alpha\beta}^{(\mu)}$ with the correct
scaling dimensions is

$\displaystyle\gamma^{(\mu)}_{\alpha\beta}$
$\displaystyle=f_{1}(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{f_{2}(\mu)}{\mathcal{R}^{(0)}}T_{\alpha\beta}^{(0)}\,,$
(3.14) $\displaystyle T^{(\mu)}_{\alpha\beta}$
$\displaystyle=f_{3}(\mu)T_{\alpha\beta}^{(0)}+f_{4}(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,.$
(3.15)

All that remains is to fix the four functions $f_{i}(\mu)$. First we will use
the assumption that the deformed stress tensor remains traceless with respect
to the deformed metric, so that

$\displaystyle\gamma^{(\mu)\alpha\beta}T^{(\mu)}_{\alpha\beta}=0\,.$ (3.16)

This condition is satisfied if and only if

$\displaystyle f_{4}(\mu)=\frac{f_{2}(\mu)f_{3}(\mu)}{f_{1}(\mu)}\,,$ (3.17)

which fixes one of the functions.

Next we impose that subsequent deformations form a group, which is listed as
assumption (ii) above. One the one hand, we can first deform the metric and
stress tensor by parameter $\mu_{1}$ to obtain

$\displaystyle\gamma^{(\mu_{1})}_{\alpha\beta}$
$\displaystyle=f_{1}(\mu_{1})\gamma_{\alpha\beta}^{(0)}+\frac{f_{2}(\mu_{1})}{\mathcal{R}^{(0)}}T_{\alpha\beta}^{(0)}\,,$
(3.18) $\displaystyle T^{(\mu_{1})}_{\alpha\beta}$
$\displaystyle=f_{3}(\mu_{1})T_{\alpha\beta}^{(0)}+\frac{f_{2}(\mu_{1})f_{3}(\mu_{1})}{f_{1}(\mu_{1})}\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,,$
(3.19)

where we have used (3.17), and then use (3.18) as the initial condition for a
second deformation by parameter $\mu_{2}$. This gives one set of deformed
quantities, $\gamma^{(\mu_{1}+\mu_{2})}_{\alpha\beta}$ and
$T_{\alpha\beta}^{(\mu_{1}+\mu_{2})}$. On the other hand, we can deform all at
once by a combined parameter $\mu_{1}+\mu_{2}$, to yield

$\displaystyle\begin{split}\gamma^{\prime(\mu_{1}+\mu_{2})}_{\alpha\beta}&=f_{1}(\mu_{1}+\mu_{2})\gamma_{\alpha\beta}^{(0)}+\frac{f_{2}(\mu_{1}+\mu_{2})}{\mathcal{R}^{(0)}}T_{\alpha\beta}^{(0)}\,,\\\
T^{\prime(\mu_{1}+\mu_{2})}_{\alpha\beta}&=f_{3}(\mu_{1}+\mu_{2})T_{\alpha\beta}^{(0)}+\frac{f_{2}(\mu_{1}+\mu_{2})f_{3}(\mu_{1}+\mu_{2})}{f_{1}(\mu_{1}+\mu_{2})}\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,.\end{split}$
(3.20)

We have decorated the quantities in (3.20) with primes to emphasize that they
may differ from the results $\gamma^{(\mu_{1}+\mu_{2})}_{\alpha\beta}$ and
$T_{\alpha\beta}^{(\mu_{1}+\mu_{2})}$ of performing the two deformations
sequentially. We then impose the constraints

$\displaystyle\gamma^{(\mu_{1}+\mu_{2})}_{\alpha\beta}=\gamma^{\prime(\mu_{1}+\mu_{2})}_{\alpha\beta}\,,\qquad
T^{(\mu_{1}+\mu_{2})}_{\alpha\beta}=T_{\alpha\beta}^{\prime(\mu_{1}+\mu_{2})}\,.$
(3.21)

In performing the algebra to find the implications of equations (3.21), we
will assume that $f_{1}>0$ and $f_{3}>0$, which is convenient for simplifying
expressions like $\sqrt{f_{1}^{2}}$ which appear in intermediate steps. This
sign choice is reasonable because we are interested in deformations for which
$f_{1}(0)=f_{3}(0)=1$, so these functions should remain positive at least for
sufficiently small deformation parameter.

After making this assumption, one finds that these equations hold if and only
if

$\displaystyle\begin{split}f_{1}(\mu_{1}+\mu_{2})&=f_{1}(\mu_{1})f_{1}(\mu_{2})+f_{2}(\mu_{1})f_{2}(\mu_{2})\,,\\\
f_{2}(\mu_{1}+\mu_{2})&=f_{1}(\mu_{2})f_{2}(\mu_{1})+f_{1}(\mu_{1})f_{2}(\mu_{2})\,,\\\
f_{3}\left(\mu_{1}+\mu_{2}\right)&=f_{3}(\mu_{1})f_{3}(\mu_{2})\left(1+\frac{f_{2}(\mu_{1})f_{2}(\mu_{2})}{f_{1}(\mu_{1})f_{1}(\mu_{2})}\right)\,.\end{split}$
(3.22)

We can turn the first two conditions in (3.22) into differential equations for
$f_{1}$ and $f_{2}$ with the initial condition that $f_{1}(0)=1$ and
$f_{2}(0)=0$, which is required so that the deformation reproduces the
undeformed theory as $\mu\to 0$. For instance, taking a derivative of the
first line of (3.22) with respect to $\mu_{2}$ and then taking $\mu_{2}=0$
yields

$\displaystyle
f_{1}^{\prime}(\mu_{1})=f_{1}^{\prime}(0)f_{1}(\mu_{1})+f_{2}^{\prime}(0)f_{2}(\mu_{1})\,.$
(3.23)

To ease notation, let $f_{1}^{\prime}(0)=a$ and $f_{2}^{\prime}(0)=b$.
Differentiating the second line of (3.22) with respect to $\mu_{1}$ and then
taking $\mu_{1}$ to zero gives
$f_{2}^{\prime}(\mu_{2})=bf_{1}(\mu_{2})+af_{2}(\mu_{2})$. Thus we have a
system of differential equations

$\displaystyle f_{1}^{\prime}(x)=af_{1}(x)+bf_{2}(x)\,,\qquad
f_{2}^{\prime}(x)=bf_{1}(x)+af_{2}(x)\,,$ (3.24)

whose general solution with the initial conditions $f_{1}(0)=1$, $f_{2}(0)=0$
is

$\displaystyle f_{1}(x)=e^{ax}\cosh(bx)\,,\qquad f_{2}(x)=e^{ax}\sinh(bx)\,.$
(3.25)

For this class of solutions, the constraint in the third line of equation
(3.22) then imposes

$\displaystyle f_{3}\left(\mu_{1}+\mu_{2}\right)$
$\displaystyle=f_{3}(\mu_{1})f_{3}(\mu_{2})\left(1+\tanh(b\mu_{1})\tanh(b\mu_{2})\right)\,,$
(3.26)

which can be turned into a differential equation with the initial condition
$f_{3}(0)=1$ as above. The result of this procedure is
$f_{3}(x)=e^{cx}\cosh(bx)$, where $c$ is another constant.

Therefore, the most general $\mu$-dependent deformation of the metric and
stress tensor consistent with our assumptions is

$\displaystyle\begin{split}\gamma^{(\mu)}_{\alpha\beta}&=e^{a\mu}\left(\cosh(b\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(b\mu)}{\mathcal{R}^{(0)}}T_{\alpha\beta}^{(0)}\right)\,,\\\
T^{(\mu)}_{\alpha\beta}&=e^{c\mu}\left(\cosh(b\mu)T_{\alpha\beta}^{(0)}+\sinh(b\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\right)\,,\end{split}$
(3.27)

where $a$, $b$, $c$ are arbitrary constants.

Some comments are in order. First, the deformations associated with the
parameters $a$ and $c$ are simply the freedom to re-scale the metric or stress
tensor by a constant $\mu$-dependent factor, which is expected since such a
scaling respects conformal symmetry and forms a group. However, any such
change in coordinates can be un-done by a diffeomorphism along with a
redefinition of the stress tensor by a multiplicative factor (which does not
affect conservation). Therefore we will set $a=c=0$ in what follows.

Second, the choice of the parameter $b$ corresponds to the scaling of the
dimensionless flow parameter $\mu$, or equivalently to our choice of
normalization for the operator $\mathcal{R}$. If $b=0$, then there is no
change in the metric or stress tensor (up to diffeomorphisms) for any value of
$\mu$, and the group structure of our deformation is the trivial group. This
violates our assumption (ii), where we demand that the deformations form a
non-trivial group, so $b=0$ is forbidden. For simplicity, we will choose
$b=1$. With these choices, our modified boundary conditions for the case of a
conformal seed theory are

$\displaystyle\begin{split}\gamma^{(\mu)}_{\alpha\beta}&=\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}T_{\alpha\beta}^{(0)}\,,\\\
T^{(\mu)}_{\alpha\beta}&=\cosh(\mu)T_{\alpha\beta}^{(0)}+\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,.\end{split}$
(3.28)

We conclude that, up to diffeomorphisms and normalization, the unique choice
of modified $\mathrm{AdS}_{3}$ boundary conditions which implement a marginal
deformation of a $\mathrm{CFT}_{2}$ satisfying our assumptions are (3.28).
This is perhaps not too surprising, since there is only a single Lorentz
invariant that be constructed from a traceless stress tensor, so we had only
one choice of scalar $\mathcal{R}^{(0)}$ which could appear in the modified
boundary conditions. However, when both $T^{\alpha\beta}T_{\alpha\beta}$ and
$\tensor{T}{{}^{\alpha}_{\alpha}}$ are non-zero, there is more freedom in the
deformation and we will require additional input to uniquely identify the
appropriate analogue of (3.28).

#### Non-Conformal Seed Theory

Typically one would not be interested in seed theories for which
$\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}\neq 0$, since a generic non-
conformal $2d$ QFT will not have any $\mathrm{AdS}_{3}$ dual. An important
exception is if the seed theory _itself_ was obtained through applying an
irrelevant stress tensor deformation, such as the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, to a
conformal seed theory. Such a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed CFT has a
stress tensor with a non-vanishing trace,333 To wit, the trace satisfies the
trace flow equation
$\tensor{T}{{}^{\alpha}_{\alpha}}=-2\lambda\mathcal{O}_{{T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}}$. and yet it is dual
to an $\mathrm{AdS}_{3}$ bulk with modified boundary conditions as we have
described. One might therefore ask what happens if we use such a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed theory as the
input for a second deformation by root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$.

First we consider the most general expression for the modified boundary
conditions $\gamma^{(\mu)}_{\alpha\beta}$ and $T^{(\mu)}_{\alpha\beta}$, which
depend both on $\mu$ and on the undeformed quantities
$\gamma^{(0)}_{\alpha\beta}$ and $T^{(0)}_{\alpha\beta}$, with the property
that these expressions reduce to (3.28) in the special case where
$\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}=0$. Because the stress tensor is no
longer traceless, its square will not be proportional to the metric. As a
result, one might believe that there are now three independent tensor
structures in the problem, namely

$\displaystyle\gamma_{\alpha\beta}^{(0)}\,,\quad
T_{\alpha\beta}^{(0)}\,,\,\text{and}\quad\left(T^{(0)}\right)^{2}_{\alpha\beta}=T_{\alpha\rho}^{(0)}\gamma^{(0)\rho\sigma}T_{\sigma\beta}^{(0)}\,,$
(3.29)

and that the most general deformed metric $\gamma^{(\mu)}_{\alpha\beta}$ and
stress tensor $T^{(\mu)}_{\alpha\beta}$ will each be a linear combination of
three different tensor structures, with appropriate coefficients.

However, this is not the case and there are in fact still only two tensor
structures. One can see this by writing quantities in terms of the traceless
part of the stress tensor,

$\displaystyle\widetilde{T}_{\alpha\beta}=T_{\alpha\beta}-\frac{1}{2}\tensor{T}{{}^{\rho}_{\rho}}\gamma_{\alpha\beta}\,.$
(3.30)

Then
$T_{\alpha\beta}=\widetilde{T}_{\alpha\beta}+\frac{1}{2}\gamma_{\alpha\beta}T$,
where we write $T=\tensor{T}{{}^{\alpha}_{\alpha}}$ for the trace of the
stress tensor to lighten notation. We also suppress the $(0)$ superscripts for
the moment. Then the putative new tensor structure arising from the square of
the stress tensor is

$\displaystyle T^{2}_{\alpha\beta}$
$\displaystyle=T_{\alpha\sigma}\gamma^{\sigma\rho}T_{\rho\beta}$
$\displaystyle=\left(\widetilde{T}_{\alpha\sigma}+\frac{1}{2}\gamma_{\alpha\sigma}T\right)\gamma^{\sigma\rho}\left(\widetilde{T}_{\rho\beta}+\frac{1}{2}\gamma_{\rho\beta}T\right)$
$\displaystyle=\widetilde{T}^{2}_{\alpha\beta}+\widetilde{T}_{\alpha\beta}T+\frac{1}{4}T^{2}\gamma_{\alpha\beta}\,.$
(3.31)

We have already seen in equation (3.13) that
$\widetilde{T}^{2}_{\alpha\beta}=\mathcal{R}^{2}\gamma_{\alpha\beta}$. We
conclude that there are still only two independent tensor structures
$\gamma_{\alpha\beta}$ and $\widetilde{T}_{\alpha\beta}$, and that a generic
candidate expression for a deformed symmetric tensor like
$\gamma^{(\mu)}_{\alpha\beta}$ or $T^{(\mu)}_{\alpha\beta}$ must still be a
linear combination of these two structures with appropriate scalar
coefficients.

However, what _has_ changed is that there are now two Lorentz scalars that can
be constructed from $T^{(0)}_{\alpha\beta}$ rather than just one. One way of
parameterizing the two functionally independent scalars is
$x_{1}=\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}$,
$x_{2}=T^{(0)\alpha\beta}T^{(0)}_{\alpha\beta}$, as we have done above. It
will be more useful to instead use $x_{1}$ and
$\mathcal{R}^{(0)}=\sqrt{\frac{1}{2}x_{2}-\frac{1}{4}x_{1}^{2}}$. Clearly any
function of $x_{1}$ and $x_{2}$ can also be expressed as a function of $x_{1}$
and $\mathcal{R}^{(0)}$.

A convenient way to write most general deformed boundary conditions is

$\displaystyle\begin{split}\gamma^{(\mu)}_{\alpha\beta}&=f_{1}(\mu,y)\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+f_{2}(\mu,y)\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}\widetilde{T}_{\alpha\beta}^{(0)}\,,\\\
\widetilde{T}^{(\mu)}_{\alpha\beta}&=f_{3}(\mu,y)\cosh(\mu)\widetilde{T}_{\alpha\beta}^{(0)}+f_{4}(\mu,y)\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,.\end{split}$
(3.32)

The functions $f_{i}$ may depend on $\mu$ and on the dimensionless combination

$\displaystyle y\equiv\frac{x_{1}}{\mathcal{R}^{(0)}}\,.$ (3.33)

The expressions (3.32) give the most general deformed boundary conditions that
can be constructed from a non-conformal seed theory. In order to reduce to the
earlier results (3.28) in the case of a CFT seed, we impose

$\displaystyle f_{i}(\mu,0)=1\,.$ (3.34)

We now expect that there should be many solutions for the functions $f_{i}$
which correspond to boundary deformations by different marginal operators. For
instance, one could deform by some operator of the form

$\displaystyle\mathcal{O}=\sqrt{c_{1}\tensor{T}{{}^{(0)\alpha\beta}}T^{(0)}_{\alpha\beta}+c_{2}\left(\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}\right)^{2}}+c_{3}\tensor{T}{{}^{(0)}{}^{\alpha}_{\alpha}}\,,$
(3.35)

for any choice of $c_{i}$. There should exist some choice of modified boundary
conditions corresponding to any such operator, and we expect that any such
deformation will satisfy the group property described in assumption (ii)
above.

Rather than perform a systematic investigation of all such allowed deformed
boundary conditions, we will attempt to single out a unique deformation within
this family by imposing our assumption (iii), namely that this deformation
commute with ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$. More precisely, we can first substitute a metric
$\gamma_{\alpha\beta}^{(0)}$ and traceless stress tensor
$T_{\alpha\beta}^{(0)}$ into the expressions (2.37) to obtain the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed boundary
conditions $\gamma_{\alpha\beta}^{(\lambda)}$ and
$T_{\alpha\beta}^{(\lambda)}$, and then substitute these two expressions into
(3.32) to obtain

$\displaystyle\gamma_{\alpha\beta}^{(\lambda,\mu)}\,,\quad
T_{\alpha\beta}^{(\lambda,\mu)}\,.$ (3.36)

On the other hand, we could instead first substitute
$\gamma_{\alpha\beta}^{(0)}$, $T_{\alpha\beta}^{(0)}$ into (2.37) to obtain
$\gamma_{\alpha\beta}^{(\mu)}$ and $T_{\alpha\beta}^{(\mu)}$, and then plug
these into (2.37) to find

$\displaystyle\gamma_{\alpha\beta}^{(\mu,\lambda)}\,,\quad
T_{\alpha\beta}^{(\mu,\lambda)}\,.$ (3.37)

We then impose the two constraints

$\displaystyle\gamma_{\alpha\beta}^{(\mu,\lambda)}=\gamma_{\alpha\beta}^{(\lambda,\mu)}\,,\qquad
T_{\alpha\beta}^{(\mu,\lambda)}=T_{\alpha\beta}^{(\lambda,\mu)}\,.$ (3.38)

This equation can be analyzed explicitly in components, by beginning with a
general metric with entries $\gamma_{zz}$, $\gamma_{\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}$, $\gamma_{z\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}=\gamma_{\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5muz}$ and a general stress
tensor compatible with the tracelessness constraints, and then evaluating both
sides of (3.38). We will omit the general expressions resulting from this
procedure, which are not especially enlightening, and proceed to the
implications of (3.38). The constraint arising from demanding that
$\gamma^{(\mu,\lambda)}_{z\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=\gamma^{(\lambda,\mu)}_{z\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}$ is

$\displaystyle
f_{2}(\mu,y)=1+\frac{4+y^{2}}{4y}\coth(\mu)\left(1-f_{1}\right)\,.$ (3.39)

Substituting this result and demanding that
$\gamma^{(\mu,\lambda)}_{zz}=\gamma^{(\lambda,\mu)}_{zz}$ then yields

$\displaystyle f_{1}=1\,.$ (3.40)

Therefore $f_{1}=f_{2}=1$. Using these constraints and requiring that
$T^{(\mu,\lambda)}_{z\mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern
1.5mu}=T^{(\lambda,\mu)}_{z\mkern
1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu}$ gives

$\displaystyle f_{4}=1+\frac{4y\coth(\mu)}{4+y^{2}}\left(1-f_{3}\right)\,,$
(3.41)

and substituting this back into the equation
$T^{(\mu,\lambda)}_{zz}=T^{(\lambda,\mu)}_{zz}$ yields

$\displaystyle f_{3}=1\,.$ (3.42)

Therefore all four of the undetermined functions must satisfy $f_{i}=1$ in
order to commute with ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$. We conclude that the
only expressions for modified boundary conditions which are consistent with
our assumptions are

$\displaystyle\begin{split}\gamma_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}\widetilde{T}_{\alpha\beta}^{(0)}\,,\\\
\widetilde{T}_{\alpha\beta}^{(\mu)}&=\cosh(\mu)\widetilde{T}_{\alpha\beta}^{(0)}+\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,,\end{split}$
(3.43)

which are exactly the ones which we claim correspond to the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation.

###  3.2 Derivation of Deformed Energy Levels

The ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation of a QFT on a cylinder of radius $R$ has a well-known effect on
the spectrum of the theory [2, 3, 4]. For an energy eigenstate $|n\rangle$
with energy $E_{n}(R,\lambda)$ and momentum $P_{n}$, the deformed energy
satisfies the inviscid Burgers’ equation,

$\displaystyle\frac{\partial E_{n}}{\partial\lambda}=E_{n}\frac{\partial
E_{n}}{\partial R}+\frac{P_{n}^{2}}{R}\,,$ (3.44)

and the momentum $P_{n}$ remains unchanged. If the undeformed theory is a CFT,
so that all of the undeformed energy levels are of the form $E_{n}^{(0)}\equiv
E_{n}(R,0)=\frac{a_{n}}{R}$ for constants $a_{n}$, then (3.44) can be solved
in closed form to obtain

$\displaystyle E_{n}(R,\lambda)=\frac{R}{2\lambda}\left(\sqrt{1+\frac{4\lambda
E_{n}^{(0)}}{R}+\frac{4\lambda^{2}P_{n}^{2}}{R^{2}}}-1\right)\,.$ (3.45)

The flow equation (3.44) can be derived by using the point-splitting
definition of the local ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator in any
translation-invariant QFT and then expressing the components of the energy-
momentum tensor in terms of $E_{n}$, $R$, $\frac{\partial E_{n}}{\partial R}$,
and $P_{n}$.

In the case of the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, it is not
known whether one can define a local operator $\mathcal{R}$ by point-
splitting. Therefore we cannot give a rigorous derivation of a flow equation
like (3.44) for a quantum field theory deformed by root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$. However, in the
spirit of the preceding subsection, we can attempt to list the properties that
such a flow equation would necessarily possess and then see whether there
exists a unique differential equation satisfying these properties. We stress
that this type of argument does not constitute a proof that a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed QFT exists
and has a particular spectrum. It would merely show that, _assuming_ that the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation is well-defined quantum-mechanically and behaves in the expected
way, then there is only one possible flow equation that the spectrum could
satisfy.444Note that such a differential equation for the cylinder spectrum is
distinct from a flow equation for the classical Hamiltonian density, which has
been studied in [34].

Before enumerating the desired properties of such a flow equation, it is
useful to obtain a rough expectation for what the result might look like.
Suppose, for the sake of argument, that there exists a local operator
$\mathcal{R}(x)$ in the spectrum of a QFT with the property that

$\displaystyle\langle\mathcal{R}\rangle=\sqrt{\frac{1}{2}\langle
T^{\alpha\beta}\rangle\langle
T_{\alpha\beta}\rangle-\frac{1}{4}\langle\tensor{T}{{}^{\alpha}_{\alpha}}\rangle^{2}}\,,$
(3.46)

and consider a deformation of the action given by $\frac{\partial
S}{\partial\mu}=\int d^{2}x\,\mathcal{R}$. By expressing the components of the
stress tensor for the theory on a cylinder of radius $R$ in terms of energies
and momenta, exactly as in the derivation of the inviscid Burgers’ equation
for ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$, one
would arrive at a putative root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow equation

$\displaystyle\frac{\partial
E_{n}}{\partial\mu}=\sqrt{\frac{1}{4}\left(E_{n}-R\frac{\partial
E_{n}}{\partial R}\right)^{2}-P_{n}^{2}}\,,$ (3.47)

or equivalently

$\displaystyle\left(\frac{\partial
E_{n}}{\partial\mu}\right)^{2}-\frac{1}{4}\left(E_{n}-R\frac{\partial
E_{n}}{\partial R}\right)^{2}+P_{n}^{2}=0\,.$ (3.48)

If the initial condition for this flow is a CFT, so $E_{n}\sim\frac{1}{R}$ and
$P_{n}\sim\frac{1}{R}$, then the solution is

$\displaystyle
E_{n}(R,\mu)=\cosh(\mu)E_{n}(R,0)+\sinh(\mu)\sqrt{\left(E_{n}^{(0)}\right)^{2}-P_{n}^{2}}\,,$
(3.49)

where $E_{n}^{(0)}=E_{n}(R,0)$, and again the momenta $P_{n}$ are unaffected.

Much like the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ flow equation for the Lagrangian, this candidate deformation of the
energy levels forms a two-parameter family of commuting deformations along
with the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
flow. Beginning with a CFT, the solution for the doubly-deformed spectrum is

$\displaystyle
E_{n}(R,\mu,\lambda)=\frac{R}{2\lambda}\left(\sqrt{1+4\lambda\left(\cosh(\mu)E_{n}^{(0)}+\sinh(\mu)\sqrt{\left(E_{n}^{(0)}\right)^{2}-P_{n}^{2}}\right)+\frac{4\lambda^{2}}{R^{2}}P_{n}^{2}}-1\right)\,,$
(3.50)

where $E_{n}^{(0)}=E_{n}(R,0,0)$. The spectrum (3.50) satisfies the two
commuting flow equations

$\displaystyle\left(\frac{\partial
E_{n}}{\partial\mu}\right)^{2}-\frac{1}{4}\left(E_{n}-R\frac{\partial
E_{n}}{\partial R}\right)^{2}+P_{n}^{2}=0\,,\qquad\frac{\partial
E_{n}}{\partial\lambda}-E_{n}\frac{\partial E_{n}}{\partial
R}-\frac{P_{n}^{2}}{R}=0\,,$ (3.51)

corresponding to the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformations,
respectively.

In Section 4 we will give an argument that (3.48) is the correct flow equation
using holography. For now, we would like to argue that this partial
differential equation is the only reasonable possibility. To that end, we
would like to look for the most general flow equation for a spectrum with the
following properties:

  1. (a)

The flow is generated by a marginal stress tensor deformation. This means that
it is a partial differential equation for $\frac{\partial
E_{n}}{\partial\mu}$, where $\mu$ is a dimensionless parameter, which arises
from a deformation of the Euclidean action by a Lorentz scalar constructed
from the stress-energy tensor.

  2. (b)

The momentum $P_{n}$ is undeformed along the flow, so $P_{n}(\mu)=P_{n}(0)$.

  3. (c)

The flow equation forms a two-parameter family of commuting flows with the
inviscid Burgers’ equation associated with the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation.

We will show that the only flow equation consistent with (a) \- (c) is (3.48).
First, it will be useful to express the possible Lorentz scalars constructed
from $T_{\alpha\beta}$ in terms of energies and momenta. We work in Euclidean
signature with coordinates $(x,y)$, where $x\sim x+R$ is the compact direction
of the cylinder and $y$ is the Euclidean time direction. See figure 1.

$N$$\mathbb{R}^{+}$$y$$x$$R$ 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.

In an energy eigenstate $|n\rangle$, the components of the stress tensor are
related to the energy $E_{n}$ and momentum $P_{n}$ of the state as follows:

$\displaystyle T_{yy}=-\frac{1}{R}E_{n}(R)\,,\qquad T_{xx}=-\frac{\partial
E_{n}(R)}{\partial R}\,,\qquad T_{xy}=\frac{i}{R}P_{n}(R)\,.$ (3.52)

Furthermore, as we have described above, any Lorentz scalar constructed from
$T_{\alpha\beta}$ is a function of the two independent invariants

$\displaystyle
x_{1}=\tensor{T}{{}^{\alpha}_{\alpha}}=-\frac{2E_{n}}{R}-2\frac{\partial
E_{n}}{\partial R}\,,\qquad
x_{2}=T^{\alpha\beta}T_{\alpha\beta}=\left(\frac{\partial E_{n}}{\partial
R}\right)^{2}+\frac{E_{n}^{2}}{R^{2}}-\frac{2P_{n}^{2}}{R^{2}}\,.$ (3.53)

We are interested in a deformation of the form $\partial_{\mu}S_{E}=\int
d^{2}x\,f(x_{1},x_{2})$. The Euclidean Lagrangian density is the Hamiltonian
density, whose integral over a spatial slice gives the energy $E_{n}$ of a
state. Therefore this deformation of the Euclidean action can be written as

$\displaystyle\partial_{\mu}E_{n}=\int_{0}^{R}\,dx\,f(x_{1},x_{2})=Rf(x_{1},x_{2})\,.$
(3.54)

We have assumed that this flow is generated by a marginal deformation, which
means that $f(x_{1},x_{2})$ has mass dimension $2$. The function $f$ must
therefore be homogeneous of degree $\frac{1}{2}$ in $x_{2}$ and degree $1$ in
$x_{1}$. This allows us to scale the factor of $R$ into the arguments of $f$:

$\displaystyle\frac{\partial
E_{n}}{\partial\mu}=f\left(Rx_{1},R^{2}x_{2}\right)\,.$ (3.55)

Next we use the second assumption, that the momenta $P_{n}$ are undeformed
along the flow. This means that the theory is connected to some conformal
field theory by a flow along which the momenta are constant, and therefore the
dependence of momenta on the radius is fixed to be $P_{n}\sim\frac{1}{R}$ as
in a CFT. It is convenient to define dimensionless momenta $p_{n}=RP_{n}$. We
will also re-scale $x_{1}$ by a factor of $-\frac{1}{2}$ for convenience and
write

$\displaystyle\begin{split}\frac{\partial
E_{n}}{\partial\mu}&=f\left(\tilde{x}_{1},\tilde{x}_{2}\right)\,,\\\
\tilde{x}_{1}=E_{n}+R\frac{\partial E_{n}}{\partial
R}\,,&\qquad\tilde{x}_{2}=R^{2}\left(\frac{\partial E_{n}}{\partial
R^{2}}\right)^{2}+E_{n}^{2}-\frac{2p_{n}^{2}}{R^{2}}\,.\end{split}$ (3.56)

This is the most general ansatz for a flow equation consistent with our first
two assumptions. We will now fix the dependence of $f$ on
$\tilde{x}_{1},\tilde{x}_{2}$ using the third assumption.

Consider a two-parameter family of theories with energies
$E_{n}(R,\lambda,\mu)$ which satisfy the simultaneous partial differential
equations

$\displaystyle\frac{\partial
E_{n}}{\partial\mu}=f\left(\tilde{x}_{1},\tilde{x}_{2}\right)\,,\qquad\frac{\partial
E_{n}}{\partial\lambda}=E_{n}\frac{\partial E_{n}}{\partial
R}+\frac{p_{n}^{2}}{R^{3}}\,.$ (3.57)

Differentiating the $\mu$ flow equation with respect to $R$ gives

$\displaystyle\frac{\partial^{2}E_{n}}{\partial\mu\,\partial R}=\frac{\partial
f}{\partial\tilde{x}_{1}}\left(2\frac{\partial E_{n}}{\partial
R}+R\frac{\partial^{2}E_{n}}{\partial
R^{2}}\right)+\frac{2}{R^{3}}\frac{\partial
f}{\partial\tilde{x}_{2}}\left(2p_{n}^{2}+R^{3}\frac{\partial E_{n}}{\partial
R}\left(E_{n}+R\frac{\partial E_{n}}{\partial
R}+R^{2}\frac{\partial^{2}E_{n}}{\partial R^{2}}\right)\right)\,,$ (3.58)

while the derivative of the $\lambda$ flow equation with respect to $R$ is

$\displaystyle\frac{\partial^{2}E_{n}}{\partial\lambda\,\partial
R}=E_{n}\frac{\partial^{2}E_{n}}{\partial R^{2}}+\left(\frac{\partial
E_{n}}{\partial R}\right)^{2}-\frac{3p_{n}^{2}}{R^{4}}\,.$ (3.59)

We may compute the mixed second partial derivative
$\frac{\partial^{2}E_{n}}{\partial\mu\,\partial\lambda}$ in two ways. Taking a
$\mu$ derivative of the expression for $\frac{\partial
E_{n}}{\partial\lambda}$ and simplifying using (3.58) yields

$\displaystyle\frac{\partial^{2}E_{n}}{\partial\mu\,\partial\lambda}$
$\displaystyle=f(\tilde{x}_{1},\tilde{x}_{2})\frac{\partial E_{n}}{\partial
R}+E_{n}\frac{\partial f}{\partial\tilde{x}_{1}}\left(2\frac{\partial
E_{n}}{\partial R}+R\frac{\partial^{2}E_{n}}{\partial R^{2}}\right)$
$\displaystyle\quad+\frac{2E_{n}}{R^{3}}\frac{\partial
f}{\partial\tilde{x}_{2}}\left(2p_{n}^{2}+R^{3}\frac{\partial E_{n}}{\partial
R}\left(E_{n}+R\frac{\partial E_{n}}{\partial
R}+R^{2}\frac{\partial^{2}E_{n}}{\partial R^{2}}\right)\right)\,.$ (3.60)

On the other hand, the $\lambda$ derivative of $\frac{\partial
E_{n}}{\partial\mu}$ is

$\displaystyle\frac{\partial^{2}E_{n}}{\partial\lambda\,\partial\mu}$
$\displaystyle=\frac{1}{R^{3}}\frac{\partial
f}{\partial\tilde{x}_{1}}\left(-2p_{n}^{2}+R^{4}\left(\frac{\partial
E_{n}}{\partial R}\right)^{2}+R^{3}E_{n}\frac{\partial E_{n}}{\partial
R}+R^{4}\frac{\partial^{2}E_{n}}{\partial R^{2}}\right)$
$\displaystyle\qquad+\frac{\partial
f}{\partial\tilde{x}_{2}}\left(2E_{n}\left(\frac{p_{n}^{2}}{R^{3}}+E_{n}\frac{\partial
E_{n}}{\partial R}\right)+\frac{2}{R^{2}}\frac{\partial E_{n}}{\partial
R}\left(-3p_{n}^{2}+R^{4}\left(\frac{\partial E_{n}}{\partial
R}\right)^{2}+R^{4}E_{n}\frac{\partial^{2}E_{n}}{\partial
R^{2}}\right)\right)\,.$ (3.61)

By our third assumption, the $\lambda$-flow must commute with the $\mu$-flow
and hence the two mixed second partial derivatives must be equal. We set (3.2)
equal to (3.2) and eliminate the variables $p_{n}$ and $\frac{\partial
E_{n}}{\partial R}$ in favor of $y_{1},y_{2}$. This leads to the differential
equation

$\displaystyle 0=\tilde{x}_{1}f+\tilde{x}_{1}^{3}\frac{\partial
f}{\partial\tilde{x}_{2}}-3\tilde{x}_{1}\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{2}}-\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{1}}+E_{n}\left(\tilde{x}_{1}\frac{\partial
f}{\partial\tilde{x}_{1}}+2\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{2}}-f\right)\,.$ (3.62)

The function $f$ can depend on the variables $\tilde{x}_{1}$ and
$\tilde{x}_{2}$ but not on the function $E_{n}(R,\lambda,\mu)$ directly. Thus
in order for the equation (3.62) to be consistent, the $E_{n}$-dependent and
$E_{n}$-independent terms must vanish separately:

$\displaystyle\begin{split}0&=\tilde{x}_{1}\frac{\partial
f}{\partial\tilde{x}_{1}}+2\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{2}}-f\,,\\\
0&=\tilde{x}_{1}f+\tilde{x}_{1}^{3}\frac{\partial
f}{\partial\tilde{x}_{2}}-3\tilde{x}_{1}\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{2}}-\tilde{x}_{2}\frac{\partial
f}{\partial\tilde{x}_{1}}\,.\end{split}$ (3.63)

The solution to the first line of (3.63) is

$\displaystyle
f=\tilde{x}_{1}g\left(\frac{\tilde{x}_{2}}{\tilde{x}_{1}^{2}}\right)\,,$
(3.64)

where $g$ is an arbitrary function. Letting
$X=\frac{\tilde{x}_{2}}{\tilde{x}_{1}^{2}}$ and substituting this ansatz into
the second line of (3.63) gives

$\displaystyle\left(X-1\right)\left(g(X)+\left(1-2X\right)g^{\prime}(X)\right)=0\,.$
(3.65)

There are two possibilities. The first factor vanishes if $X=1$, which is the
case when

$\displaystyle
T^{\alpha\beta}T_{\alpha\beta}=\frac{1}{4}\left(\tensor{T}{{}^{\alpha}_{\alpha}}\right)^{2}\,.$
(3.66)

This is a trivial case in which the stress tensor for the theory is degenerate
and the two trace structures become dependent. We will discard this solution
and require that $X$ is not identically equal to $1$. This leaves us with the
second possibility, $g(X)+\left(1-2X\right)g^{\prime}(X)=0$, which has the
solution

$\displaystyle g(X)=c_{1}\sqrt{2X-1}\,,$ (3.67)

where $c_{1}$ is an arbitrary constant. Tracing back through the changes of
variables, this corresponds to a deformation of the form

$\displaystyle f(x_{1},x_{2})=c_{1}\sqrt{2x_{2}-x_{1}^{2}}\,.$ (3.68)

Choosing the normalization to be $c_{1}=\frac{1}{2}$, we conclude that the
function $f$ is

$\displaystyle
f\left(x_{1},x_{2}\right)=\sqrt{\frac{1}{2}x_{2}-\frac{1}{4}x_{1}^{2}}=\mathcal{R}\,,$
(3.69)

which is precisely the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator. The flow
equation for the energies is

$\displaystyle\frac{\partial
E_{n}}{\partial\mu}=\sqrt{\frac{1}{4}\left(E_{n}-R\frac{\partial
E_{n}}{\partial R}\right)^{2}-P_{n}^{2}}\,,$ (3.70)

and taking the square of this equation recovers (3.48).

Our conclusion is that there is only a single marginal deformation of the
cylinder spectrum for a $2d$ quantum field theory which is constructed from
the energy-momentum tensor and which commutes with the irrelevant ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow. This unique
deformation is the one which corresponds to the combination of stress tensors
which appears in the classical root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation. We
reiterate that this does not represent a proof that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ operator is
necessarily well-defined at the quantum level. However, if there exists _any_
deformation of the quantum theory with the properties that we listed, it must
lead to exactly the flow equation which one would have naïvely guessed would
correspond to the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, as we did
around equation (3.46).

##  4 AdS3 Gravity with Root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ Deformed Boundary
Conditions

In this section, we aim to show that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
conditions derived in section 3.1 are compatible with our proposed flow
equation for the spectrum in section 3.2. To do this, we will compute the mass
(or energy) of a spacetime with root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
conditions and compare this deformed mass to its undeformed value.

It is well-known that the notion of mass is subtle in a generally covariant
theory, and there are many definitions of the total mass of a spacetime which
are applicable in different contexts. In our case, since we are interested in
asymptotically $\mathrm{AdS}_{3}$ spacetimes, it will be most convenient to
define the spacetime mass as the integral of the quasi-local Brown-York stress
tensor [48]. In a $d$-dimensional spacetime $\mathcal{M}$, this mass integral
is given by

$\displaystyle
M=\int_{\Sigma}d^{d-1}x\,\sqrt{\sigma}\,u^{\mu}T_{\mu\nu}\xi^{\nu}\,,$ (4.1)

where $\Sigma$ is a spacelike surface in the boundary $\partial\mathcal{M}$
with metric $\sigma_{\alpha\beta}$, $u^{\mu}$ is a timelike unit normal, and
$\xi^{\nu}$ is the Killing vector associated with time translations. Our
strategy will be to compute the mass (4.1) by choosing a convenient coordinate
system generated by a field-dependent change of variables which implements our
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed boundary conditions. Such field-dependent diffeomorphisms have also
appeared in various works in the context of the ordinary ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation [49, 18,
30].

It would be very interesting to study the mass of $\mathrm{AdS}_{3}$
spacetimes subject to modified boundary conditions using a more general
prescription such as the covariant phase space formalism [50, 51]. The result
of a mass calculation in this formalism is guaranteed to agree with (4.1), but
because this machinery maintains covariance, it may be possible to obtain mass
flow equations associated with ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ and root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformations (or even
more general stress tensor deformations) without resorting to a field-
dependent diffeomorphism.

We will also obtain the corresponding root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
conditions in the Chern-Simons description of AdS3 gravity. In this formalism,
the definition of the deformed spacetime mass is not immediately obvious. As
we will review around equation (4.42), in the undeformed theory with
conventional boundary conditions, it is straightforward to show that the mass
(4.1) is equal to the value of the Chern-Simons boundary term which imposes
the appropriate boundary conditions. We will see by explicit computation that
this remains true when this boundary term is modified to the one which imposes
the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed boundary conditions. This provides evidence that the Chern-Simons
boundary term continues to yield the spacetime mass even with modified
boundary conditions, which one might attempt to prove more generally by a
computation in the canonical formulation.

###  4.1 Metric Formalism

First, we will briefly review the salient details in AdS3 gravity to set the
stage for the deformed energy spectrum computation.555For useful reviews on
the AdS3/CFT2 correspondence, see [52, 53, 54]. Pure three-dimensional general
relativity contains no local degrees of freedom, but is nontrivial enough to
have black hole solutions and is a useful arena to study interesting phenomena
in a controllable manner. A general solution of AdS3 gravity can be written in
the Fefferman-Graham expansion [55]

$ds^{2}=\frac{\ell^{2}d\rho^{2}}{4\rho^{2}}+\left(\frac{g^{(0)}_{\alpha\beta}(x^{\alpha})}{\rho}+g^{(2)}_{\alpha\beta}(x^{\alpha})+\rho
g^{(4)}_{\alpha\beta}(x^{\alpha})\right)dx^{\alpha}\,dx^{\beta}\,,$ (4.2)

which terminates at second order [56] and where $\rho=0$ is the AdS3 boundary.
The AdS3 radius is $\ell$ and the three-dimensional Einstein equations
determine $g^{(4)}_{\alpha\beta}$ in terms of the other two Fefferman-Graham
expansion coefficients as

$g^{(4)}_{\alpha\beta}=\frac{1}{4}g^{(2)}_{\alpha\gamma}g^{(0)\gamma\delta}g^{(2)}_{\delta\beta}\,.$
(4.3)

Asymptotically AdS3 solutions realize two copies of the Virasoro algebra,
which are generated by Brown-Henneaux diffeomorphisms [57] that preserve the
leading asymptotics of the metric (4.2). Such diffeomorphisms correspond to
conformal transformations in the $2d$ boundary theory. From the AdS/CFT
dictionary [58, 59], the Fefferman-Graham quantity $g^{(2)}_{\alpha\beta}$ is
proportional to the expectation value of the boundary CFT stress tensor

$g^{(2)}_{\alpha\beta}=8\pi
G\ell\left(T_{\alpha\beta}-g^{(0)}_{\alpha\beta}T^{\alpha}_{\alpha}\right)\equiv
8\pi G\ell\widehat{T}_{\alpha\beta}\,,$ (4.4)

and $g^{(0)}_{\alpha\beta}$ is the metric on the boundary where the dual CFT
lives. To derive the energy spectrum of this background (4.2) with the
root-$TT$ deformed boundary conditions, we borrow some of the key methods
developed to study holographic aspects of the double-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation in the
metric formalism [18] and Chern-Simons formalism [60] at large $N$. See
appendix A for a review of these methods. As a consequence of our analysis, we
will also find that the bulk spacetime exhibits superluminal propagation for
one sign of the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ deformation parameter $\mu$, which is also the case for the bad-sign
${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation.

#### Root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
Deformed Theory

In section 3.1, we argued that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed boundary
metric and stress tensor are

$\displaystyle\begin{split}\gamma_{\alpha\beta}^{(\mu)}=\cosh(\mu)\gamma_{\alpha\beta}^{(0)}+\frac{\sinh(\mu)}{\mathcal{R}^{(0)}}\widetilde{T}_{\alpha\beta}^{(0)}\,,\quad\widetilde{T}_{\alpha\beta}^{(\mu)}=\cosh(\mu)\widetilde{T}_{\alpha\beta}^{(0)}+\sinh(\mu)\mathcal{R}^{(0)}\gamma^{(0)}_{\alpha\beta}\,.\end{split}$
(4.5)

The strategy we follow here is motivated by the analysis of ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed AdS3/CFT2 in
[18]. As in the holographic analysis of the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, we will
make two assumptions about the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow. The first is
that this deformation is smooth and therefore preserves the boundary theory’s
degeneracy of states; for a black hole solution, this corresponds to the
statement that the black hole horizon area is unchanged. The second assumption
is that the deformation does not affect the momentum quantum number $P_{n}$ in
the boundary field theory, which is quantized in units of $\frac{1}{R}$, where
$R$ is the cylinder radius. Thus we will equate the deformed and undeformed
areas and angular momenta. These assumptions imply the root-deformed energy
spectrum which was derived by consistency conditions in section 3.2. Unlike
the ${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation, the trace of the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ theory does not flow,
as expected for a classically marginal deformation.

We will now focus on the case of a Bañados geometry [61] which is
parameterized by two quantities $\mathcal{L}(u)$ and $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu(v)$. A Bañados
geometry’s Fefferman-Graham quantities are defined

$\displaystyle g^{(0)}_{\alpha\beta}dx^{\alpha}\,dx^{\beta}=du\,dv\,,\quad
g^{(2)}_{\alpha\beta}dx^{\alpha}dx^{\beta}=\mathcal{L}(u)du^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)dv^{2}\,,\quad
g^{(4)}_{\alpha\beta}dx^{\alpha}dx^{\beta}=\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu(v)dudv$ (4.6)

implying that the metric (4.2) becomes

$\displaystyle ds^{2}$
$\displaystyle=\frac{\ell^{2}d\rho^{2}}{4\rho^{2}}+\frac{du\,dv}{\rho}+\mathcal{L}(u)du^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)dv^{2}+\rho\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu(v)\,du\,dv\,.$
(4.7)

The root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed boundary metric and stress tensor given in (4.5) are therefore

$\displaystyle\gamma^{(\mu)}_{\alpha\beta}$
$\displaystyle=(\cosh\mu)g_{\alpha\beta}^{(0)}+\frac{\sinh\mu}{2\sqrt{\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}}g^{(2)}_{\alpha\beta}=\frac{1}{2}\left(\begin{array}[]{cc}\sqrt{\frac{\mathcal{L}(u)}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}}\sinh\mu&\quad\cosh\mu\\\ \cosh\mu&\quad\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}{\mathcal{L}(u)}}\sinh\mu\end{array}\right)\,,$ (4.8)
$\displaystyle\widetilde{T}^{(\mu)}_{\alpha\beta}$
$\displaystyle=\frac{\cosh\mu}{2}g^{(2)}_{\alpha\beta}+\sqrt{\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}(\sinh\mu)g^{(0)}_{\alpha\beta}=\frac{1}{2}\left(\begin{array}[]{cc}\mathcal{L}(u)\cosh\mu&\quad\sqrt{\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu(v)}\sinh\mu\\\
\sqrt{\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}\sinh\mu&\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)\cosh\mu\end{array}\right)\,,$

where we work in conventions such that $4\pi G\ell=1$ and substituted (4.6)
into (4.5). We also used $g^{(2)}_{\alpha\beta}=2T_{\alpha\beta}$ and computed
the operator $\displaystyle{\mathcal{R}=\sqrt{\mathcal{L}(u)\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu(v)}}$. We now
identify a field-dependent diffeomorphism to new coordinates $U,V$ defined by

$dU=\left(\cosh\frac{\mu}{2}\right)du+\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}{\mathcal{L}(u)}}\left(\sinh\frac{\mu}{2}\right)dv\,,\quad
dV=\left(\cosh\frac{\mu}{2}\right)dv+\sqrt{\frac{\mathcal{L}(u)}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}}\left(\sinh\frac{\mu}{2}\right)du\,.$ (4.9)

which has the property that the metric, when written in these new variables,
returns to the standard form:

$\gamma^{(\mu)}_{\alpha\beta}dx^{\alpha}\,dx^{\beta}=dU\,dV\,.$ (4.10)

Expressing (4.9) in matrix notation, we may write the field dependent
coordinate transformation and its inverse as

$\displaystyle\left(\begin{array}[]{c}dU\\\ dV\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\cosh\frac{\mu}{2}&\quad\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}{\mathcal{L}(u)}}\sinh\frac{\mu}{2}\\\
\sqrt{\frac{\mathcal{L}(u)}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}}\sinh\frac{\mu}{2}&\quad\cosh\frac{\mu}{2}\end{array}\right)\left(\begin{array}[]{c}du\\\
dv\end{array}\right)\,,$ (4.11) $\displaystyle\left(\begin{array}[]{c}du\\\
dv\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{cc}\cosh\frac{\mu}{2}&\quad-\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}{\mathcal{L}(u)}}\sinh\frac{\mu}{2}\\\
-\sqrt{\frac{\mathcal{L}(u)}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v)}}\sinh\frac{\mu}{2}&\quad\cosh\frac{\mu}{2}\end{array}\right)\left(\begin{array}[]{c}dU\\\
dV\end{array}\right).$

For black hole solutions with constant $(\mathcal{L}(u),\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu(v))\equiv\left(\mathcal{L}_{\mu},\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu}\right)$,
the Fefferman-Graham quantities in the $(U,V)$ coordinates are

$\displaystyle g^{(0)}_{\alpha\beta}dx^{\alpha}\,dx^{\beta}$
$\displaystyle=du\,dv=-\frac{1}{2}\sinh\mu\left(\sqrt{\frac{\mathcal{L}_{\mu}}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}}dU^{2}+\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}{\mathcal{L}_{\mu}}}dV^{2}\right)+\cosh\mu dU\,dV\,,$ (4.12)
$\displaystyle g^{(2)}_{\alpha\beta}dx^{\alpha}\,dx^{\beta}$
$\displaystyle=\mathcal{L}_{\mu}du^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}dv^{2}=\cosh\mu\left(\mathcal{L}_{\mu}dU^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}dV^{2}\right)-2\sqrt{\mathcal{L}_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu}}\sinh\mu
dU\,dV\,,$ $\displaystyle g^{(4)}_{\alpha\beta}dx^{\alpha}\,dx^{\beta}$
$\displaystyle=\mathcal{L}_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}du\,dv=\mathcal{L}_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}\left(-\frac{1}{2}\sinh\mu\left(\sqrt{\frac{\mathcal{L}_{\mu}}{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}}dU^{2}+\sqrt{\frac{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}{\mathcal{L}_{\mu}}}dV^{2}\right)+\cosh\mu dU\,dV\right)\,.$

The metric (4.2) in terms of these Fefferman-Graham quantities (4.12) at the
event horizon $\rho_{h}=\left(\mathcal{L}_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}\right)^{-\frac{1}{2}}$ is

$ds^{2}\big{|}_{\rho=\rho_{h}}=\frac{\ell^{2}\mathcal{L}_{\mu}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}{4}d\rho^{2}+e^{-\mu}\left(\left(\sqrt{\mathcal{L}_{\mu}}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}\right)^{2}d\phi^{2}+\left(\sqrt{\mathcal{L}_{\mu}}-\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}\right)^{2}dT^{2}+2\left(\mathcal{L}_{\mu}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}\right)dT\,d\phi\right)\,,$ (4.13)

where $(U,V)=(\phi+T,\phi-T)$. The undeformed and deformed event horizon areas
are read off from (4.13)

$A^{(0)}=\int^{R}_{0}d\phi\sqrt{g_{\phi\phi}}=R\left(\sqrt{\mathcal{L}_{0}}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}}\right),\quad
A^{(\mu)}=\int^{R}_{0}d\phi\sqrt{g_{\phi\phi}}|_{\rho_{h}}=Re^{-\frac{\mu}{2}}\left(\sqrt{\mathcal{L}_{\mu}}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}\right)\,.$ (4.14)

Now to extract the deformed energy and angular momentum. Using (4.8) and
(4.12), we write the components of the stress tensor

$T^{(\mu)}_{\alpha\beta}\,dx^{\alpha}\,dx^{\beta}=\frac{1}{2}(\mathcal{L}_{\mu}dU^{2}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}dV^{2})=\frac{1}{2}(\mathcal{L}_{\mu}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu})(dT^{2}+d\phi^{2})+(\mathcal{L}_{\mu}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu})\,d\phi\,dT\,.$ (4.15)

Restoring factors of $4\pi G\ell$, the deformed energy and angular momentum
from (4.15) are

$E_{\mu}=\int^{R}_{0}d\phi\,T^{(\mu)}_{TT}=\frac{R}{8\pi
G\ell}(\mathcal{L}_{\mu}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu}),\quad
J_{\mu}=\int^{R}_{0}d\phi T^{(\mu)}_{T\phi}=\frac{R}{8\pi
G\ell}(\mathcal{L}_{\mu}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu})\,.$
(4.16)

The root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformed energy (4.16) is a simple sum $\mathcal{L}_{\mu}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu}$,
reminiscent of a CFT’s energy, which is a sign that the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed theory
remains a CFT. This simplicity of energy ceases to exist for the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation due to the
deformed theory being non-conformal. In the ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of
AdS3/CFT2, the energy is

$E_{\lambda}=\frac{R}{8\pi G\ell}\frac{\mathcal{L}_{\lambda}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\lambda}-2\rho_{c}\mathcal{L}_{\lambda}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\lambda}}{1-\rho_{c}^{2}\mathcal{L}_{\lambda}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\lambda}}\,,$
(4.17)

with $(\mathcal{L}_{\lambda},\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\lambda})$
defined in [18] and (A.30). The next ingredients are the areas and angular
momenta, which obey

$\sqrt{\mathcal{L}_{0}}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}}=e^{-\frac{\mu}{2}}\left(\sqrt{\mathcal{L}_{\mu}}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}}\right),\quad\mathcal{L}_{0}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}=\mathcal{L}_{\mu}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{\mu}\,,$
(4.18)

and the solutions of (4.18) are

$\mathcal{L}_{\mu}=\left(\sqrt{\mathcal{L}_{0}}\cosh\frac{\mu}{2}+\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}}\sinh\frac{\mu}{2}\right)^{2}\,,\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{\mu}=\left(\sqrt{\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}}\cosh\frac{\mu}{2}+\sqrt{\mathcal{L}_{0}}\sinh\frac{\mu}{2}\right)^{2}\,.$
(4.19)

Using the relations (4.19) to express the deformed energy in terms of
$\mathcal{L}_{0}$, $\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{0}$ then
yields

$\displaystyle E_{\mu}=\frac{R}{8\pi G\ell}\left((\mathcal{L}_{0}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0})\cosh\mu+2\sqrt{\mathcal{L}_{0}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}}\sinh\mu\right)\,.$ (4.20)

We can rewrite (4.20) in terms of the undeformed energy and angular momentum
by recalling that

$\displaystyle E_{0}$ $\displaystyle=\frac{R}{8\pi
G\ell}\left(\mathcal{L}_{0}+\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}\right)\,,\quad J_{0}=\frac{R}{8\pi
G\ell}\left(\mathcal{L}_{0}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{0}\right)\,,$
(4.21) $\displaystyle\mathcal{L}_{0}$ $\displaystyle=\frac{4\pi
G\ell}{R}\left(E_{0}+J_{0}\right)\,,\quad\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern 1.5mu_{0}=\frac{4\pi
G\ell}{R}\left(E_{0}-J_{0}\right)\,.$

After identifying the bulk angular momentum $J_{0}$ with the CFT momentum
$P_{0}$, this gives the same energy spectrum (3.49) we found from consistency
conditions, namely

$E_{\mu}=E_{0}\cosh\mu+\sqrt{E_{0}^{2}-P_{0}^{2}}\sinh\mu\,.$ (4.22)

#### Propagation Speed

In the case of the usual ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, there is
a sharp distinction between the two signs of the deformation parameter
$\lambda$. In our conventions, $\lambda>0$ corresponds to the “good sign” of
the flow. With this choice of sign, so long as $\lambda$ is not too large, all
of the energy eigenvalues in a ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformed CFT remain
real. For the “bad sign” $\lambda<0$, however, all but finitely many of the
energies in the deformed theory become complex.666In some cases, these complex
energies can be removed by performing multiple ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformations in a row.
For instance, one can deform a pair of CFTs by the bad sign of $\lambda$ and
then subsequently deform the tensor product of these theories by a good-sign
flow to cure the spectrum [62]. This signals a pathology in the bad-sign-
deformed theory which appears to be quite robust to the type of ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$-deformation one uses.
For instance, a single-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation with the
bad sign corresponds to a bulk dual with closed timelike curves [63]. The
conventional double-trace ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation, which is
the version considered in this work, with the bad choice of sign is dual to a
bulk spacetime which exhibits superluminal propagation [64, 65].

It is natural to ask whether the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation has a
similar pathology for one choice of the sign. Such a pathology would not be
visible at the level of the formula (4.22) for the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformed spectrum,
which appears to yield real energies for either sign of $\mu$. However, we
will now show that the sign choice $\mu<0$ leads to a bulk spacetime which
allows superluminal propagation. This suggests that, as with ${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$, only the positive
sign of the root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern
1.5mu}$ flow parameter may lead to a sensible deformed theory.

To demonstrate this superluminal propagation, we begin with a diagonal stress
tensor
$\widetilde{T}_{\alpha\beta}(0)=\operatorname{diag}(\widetilde{T}_{tt}(0),\widetilde{T}_{xx}(0))$
on a two-dimensional space equipped with $(t,x)$ coordinates and flat metric
$\eta_{\alpha\beta}=\operatorname{diag}(-1,1)$. The boundary deformed metric
(4.5) in this setting is

$\displaystyle ds^{2}$
$\displaystyle=\left(-dt^{2}+dx^{2}\right)\cosh\mu+\frac{\sinh\mu}{\mathcal{R}^{(0)}}\left(\widetilde{T}_{tt}(0)dt^{2}+\widetilde{T}_{xx}(0)dx^{2}\right)$
(4.23) $\displaystyle=-e^{-\mu}dt^{2}+e^{\mu}dx^{2}\,,$

where we have used $\widetilde{T}_{tt}=\widetilde{T}_{xx}=\mathcal{R}^{(0)}$.

Null geodesics obey $ds^{2}=0$ which have the following propagation speed

$v=e^{-\mu}\,,$ (4.24)

and in particular we see that $v>1$ if $\mu<0$. This confirms that the bulk
supports superluminal propagation for the negative sign of the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation parameter.

This result might have been anticipated because the root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation is closely
connected to the Modified Maxwell or ModMax theory of electrodynamics in four
spacetime dimensions. In particular, the $4d$ root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ deformation of the
free Maxwell theory yields the ModMax theory [25, 27], and the dimensional
reduction of this theory to two spacetime dimensions is the Modified Scalar
theory which is obtained from a root-${T\mkern
1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$ flow of free scalars
[30]. It was already pointed out in [21] that the $4d$ ModMax theory also
allows for superluminal propagation when $\gamma<0$, which corresponds to
$\mu<0$ in our notation. This gives another reason to suspect that the
root-${T\mkern 1.5mu\overline{\mkern-1.5muT\mkern-1.5mu}\mkern 1.5mu}$
deformation may be ill-behaved for $\mu<0$.

###  4.2 Chern-Simons Formalism

The three-dimensional Einstein-Hilbert action is expressible semi-classically
as the difference of two Chern-Simons actions [66, 67]

$S_{\text{EH}}[g_{\alpha\beta}]=S_{\text{CS}}[A]-S_{\text{CS}}[\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu]\,,$ (4.25)

where

$S_{\text{CS}}[A]=\frac{\ell}{16\pi G}\int\operatorname{Tr}\left(A\wedge
dA+\frac{2}{3}A\wedge A\wedge A\right)\,,$ (4.26)

and the bulk one-form Chern-Simons connections $(A,\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu)=(A_{\alpha}dx^{\alpha},\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu_{\alpha}dx^{\alpha})$
are expressed in terms of the vielbein $E^{a}=E^{a}{}_{\alpha}dx^{\alpha}$ and
spin connection $\Omega^{a}=\frac{1}{2}\epsilon^{abc}\Omega_{\alpha
bc}dx^{\alpha}$:

$A^{a}=\Omega^{a}+\frac{1}{\ell}E^{a},\quad\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu^{a}=\Omega^{a}-\frac{1}{\ell}E^{a}\,,$ (4.27)

where $a=-1,0,1$ are SL$(2,\mathbb{R})$ group indices. The equations of motion
imply flatness

$F=dA+A\wedge A=0,\quad\mkern 1.5mu\overline{\mkern-1.5muF\mkern-1.5mu}\mkern
1.5mu=d\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu+\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu\wedge\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu=0\,,$ (4.28)

and the bulk metric is related to the Chern-Simons gauge fields as

$g_{\alpha\beta}=\frac{\ell^{2}}{2}\operatorname{Tr}\left((A_{\alpha}-\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu_{\alpha})(A_{\beta}-\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu_{\beta})\right)\,,$
(4.29)

where the trace is over SL$(2,\mathbb{R})$ indices. The connections associated
to the Bañados geometry (4.7) are

$\displaystyle A$
$\displaystyle=-\frac{1}{2\rho}L_{0}d\rho+\frac{1}{\ell}\left(-\sqrt{\rho}\mathcal{L}_{0}L_{-1}+\frac{1}{\sqrt{\rho}}L_{1}\right)du=\left(\begin{array}[]{ll}-\frac{d\rho}{4\rho}&\quad-\frac{\sqrt{\rho}\mathcal{L}_{0}du}{\ell}\\\
-\frac{du}{\ell\sqrt{\rho}}&\quad\frac{d\rho}{4\rho}\end{array}\right)\,,$
(4.30) $\displaystyle\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}$
$\displaystyle=\frac{1}{2\rho}L_{0}d\rho+\frac{1}{\ell}\left(\frac{1}{\sqrt{\rho}}L_{-1}-\sqrt{\rho}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}L_{1}\right)dv=\left(\begin{array}[]{ll}\frac{d\rho}{4\rho}&\quad\frac{dv}{\ell\sqrt{\rho}}\\\
\frac{\sqrt{\rho}\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}dv}{\ell}&\quad-\frac{d\rho}{4\rho}\end{array}\right)\,,$

where the SL$(2,\mathbb{R})$ generators are

$L_{-1}=\left(\begin{array}[]{ll}0&1\\\ 0&0\end{array}\right),\quad
L_{0}=\frac{1}{2}\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
L_{1}=\left(\begin{array}[]{ll}0&0\\\ -1&0\end{array}\right)\,.$ (4.37)

These generators satisfy the standard commutation relations

$\displaystyle=(m-n)L_{m+n}\,.$ (4.38)

In radial gauge, we may extract the radial dependence from the bulk
connections as

$\displaystyle A(\rho,u)$
$\displaystyle=b^{-1}(\rho)(d+a(u))b(\rho),\quad\mkern
1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern 1.5mu(\rho,v)=b(d+\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern 1.5mu(v))b^{-1}(\rho)\,,$
(4.39) $\displaystyle b(\rho)$
$\displaystyle=e^{-\frac{1}{2}L_{0}\ln\rho}=\left(\begin{array}[]{cc}\rho^{-\frac{1}{4}}&\quad
0\\\ 0&\quad\rho^{\frac{1}{4}}\end{array}\right)\,,$

where the boundary connections are

$a(u)=\frac{1}{\ell}\left(-\mathcal{L}_{0}(u)L_{-1}+L_{1}\right)du,\quad\mkern
1.5mu\overline{\mkern-1.5mua\mkern-1.5mu}\mkern
1.5mu(v)=\frac{1}{\ell}\left(L_{-1}-\mkern
1.5mu\overline{\mkern-1.5mu\mathcal{L}\mkern-1.5mu}\mkern
1.5mu_{0}(v)L_{1}\right)dv\,.$ (4.40)

To compare more easily with our metric formalism analysis, we work in the same
temporal and periodic coordinates777We distinguish between the undeformed
coordinates $(t,\varphi)$ and the deformed coordinates $(T,\phi)$.

$t=\frac{1}{2}\left(u+v\right)\,,\quad\varphi=\frac{1}{2}\left(u-v\right)\,,\quad\varphi\sim\varphi+R\,.$
(4.41)

In these variables (4.41), the chiral boundary conditions are
$A_{t}=A_{\varphi}\,,\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern
1.5mu_{t}=-\mkern 1.5mu\overline{\mkern-1.5muA\mkern-1.5mu}\mkern