###### Abstract

We investigate the asymptotic symmetries of three-dimensional AdS Einstein
gravity in the Weyl–Fefferman–Graham gauge, which is a generalization of the
Fefferman–Graham gauge inducing a Weyl connection as part of the boundary
structure. We show that this gauge arises as a natural intermediary step of
the gauge-fixing procedure in the Chern–Simons formulation. We prove that the
diffeomorphism required to go to the usual Fefferman–Graham gauge can be
charged, and thus its implementation has physical repercussions. We discuss
the holographic renormalization and the variational principle offering a new
holographic take on this gauge and its charges.

Luca Ciambellia, Arnaud Delfanteb111FRIA grantee of the Fund for Scientific
Research – FNRS, Belgium., Romain Ruzziconic, Céline Zwikela

aPerimeter Institute for Theoretical Physics,
31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5
bService de Physique de l’Univers, Champs et Gravitation,
Université de Mons – UMONS, 20 place du Parc, 7000 Mons, Belgium

cInstitute for Theoretical Physics, Technische Universität Wien,
Wiedner Hauptstrasse 8, A-1040 Vienna, Austria

<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Gauge fixing procedure
    1. 2.1 From FG to WFG
    2. 2.2 Why WFG
  3. 3 Asymptotic symmetries
    1. 3.1 Solution space
    2. 3.2 Charges
  4. 4 Holography and variational principle
    1. 4.1 Holographic renormalization
    2. 4.2 Boundary counterterm
    3. 4.3 Corner counterterm
  5. 5 Conclusions
  6. A Conventions
    1. A.1 Chern–Simons formulation
    2. A.2 Covariant phase space formalism
  7. B Second copy

##  1 Introduction

Three-dimensional Einstein gravity is a perfect playground to study
gravitational physics, due to its simple features [1, 2, 3]. Indeed, albeit
without propagating degrees of freedom, there are two important aspects of
this theory that makes it useful. The first one is the presence of black holes
[4, 5, 6], while the second is the asymptotic symmetry enhancement with
respect to the isometries of the vacuum [7, 8, 9, 10]. Since the precursory
work of Brown and Henneaux [7], it has been appreciated that AdS3 with
Dirichlet conditions at its conformal boundary has a centrally extended double
copy Virasoro algebra of charges. This has been later interpreted as the
algebra of modes of the stress tensor in the boundary conformal field theory
[11], a particularly rich instance of the AdS/CFT correspondence [12, 13, 14,
15, 16]. One of the advantages of three-dimensional gravity is that we can
exploit its topological nature and rephrase it as a Chern–Simons (CS) theory
[17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

The AdS/CFT correspondence is rooted in the geometric construction of the
ambient space formulated by Fefferman and Graham [27, 28]. These authors
introduced a gauge, called Fefferman–Graham (FG) gauge, that exploits all the
diffeomorphism freedom to completely lock the radial structure of the bulk
metric. It is a theorem that this gauge is always attainable and, for the
purposes of setting up the holographic dictionary, it has the virtue of
explicitly realize the boundary conformal multiplets as series in the
holographic coordinate, see e.g. [29]. It has already been noted in [30] that
restoring some of the diffeomorphism freedom allows one to realize the Weyl
connection as part of the induced boundary structure, and consequently, to
restore the boundary Weyl covariance. For this reason, this partial gauge has
been called the Weyl–Fefferman–Graham (WFG) gauge, which has received more
attention recently [31, 32].

While not often used in the AdS/CFT correspondence, the framework of charges
and the covariant phase space formalism have produced over the last decades a
lot of new understanding on the concept of observables in gravity and
generically in gauge theories. In particular, in the presence of boundaries of
interest, it is firmly established that Noether’s second theorem gives a
prescription to identify the physical asymptotic symmetries [33, 34, 35, 36,
37, 38, 8]. In gravity, the latter form the subgroup of the bulk
diffeomorphisms that are compatible with the falloffs and boundary conditions
and produce a non-vanishing Noether’s charge, which has support on
codimension-$2$ surfaces. Classifying the asymptotic symmetries is of
particular interest for holography since they correspond to the symmetries of
the dual theory through the holographic dictionary. Several attempts have been
made to find the most general boundary conditions in three-dimensional gravity
[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56] and
provide a physical interpretation from the boundary perspective. In this
spirit, the corner proposal has offered an interesting angle of attack to
address this problem by identifying some universal structures associated with
codimension-$2$ surfaces [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69].
Among these previous works, the results of Grumiller and Riegler [42] stand
out. In the latter, the most general solution space compatible with a well-
defined variational principle has been derived. In this paper, we consider a
field content which is encompassed in [42], but we refrain from constraining
the boundary geometry. Moreover, we compute the charges in the metric
formalism and show that they are finite approaching the boundary.

One of the main results in the theory of asymptotic symmetries is that a
charged diffeomorphism is a physical symmetry, mapping inequivalent physical
configurations. Utilizing a charged diffeomorphism to reach a particular gauge
is therefore a delicate procedure, as it restricts the physical fields of the
theory. The main result of this manuscript is to show that the diffeomorphism
mapping WFG to FG can be charged, and thus non-trivial. This generalizes to
WFG previous enhancements of boundary conditions in FG [7, 70, 50] and proves
that even though the FG gauge is always reachable, it could constrain the
physical solution space of our bulk theory.

This opens the fascinating prospect of a more general holographic dictionary,
geometrically formulated in the WFG gauge, with extra charges and observables
playing a role in the dual field theory. We explore the first steps towards
this endeavour by revisiting the holographic renormalization and variational
principle in WFG gauge, and identify the phase space variables with sources
and VEV in the dual theory. We show that the extra physical charges coming
from the partial gauge-fixing in the bulk can be modified exploiting different
boundary Lagrangians and prescriptions to compute the charges, such as [71,
72, 61, 73, 56]. In particular, we focus on three different symplectic
structures. In the first one, there are no extra charges with respect to the
FG gauge, and thus Weyl-covariantization can be attained for free. In the
second, the Weyl connection appears as a source in the boundary dual theory,
but the renormalization procedure is not covariant. Lastly, we propose a
symplectic structure that gives rise to a boundary Weyl-invariant extra source
(on top of the boundary metric), with a covariant renormalization procedure.
While all of them are mathematically sound, they lead to different physical
interpretations and consequences for the boundary dual theory.

The paper is organized as follows. We start in section 2 revisiting the gauge
fixing procedure in the CS formulation. We explain how WFG naturally arises as
an intermediate step in the gauge-fixing toward the FG gauge. We then comment
on why the WFG gauge is useful, reviewing the appearance of Weyl symmetry.
Section 3 contains the novelties of this paper. Specifically, we compute the
asymptotic symemtries and charges, showing explicitly that the diffeomorphism
bringing WFG to FG is charged. Section 4 discusses some holographic aspects of
the WFG gauge. The variational principle is presented and the holographic Ward
identities are derived. We then conclude offering possible outlooks in section
5. Two appendixes are offered, where we report conventions and some technical
details used in the main body of the manuscript.

##  2 Gauge fixing procedure

In this section, we introduce the Weyl–Fefferman–Graham (WFG) gauge [30]. In
subsection 2.1, we begin by studying a natural way to relax the
Fefferman–Graham (FG) gauge using the Chern–Simons (CS) formulation, realizing
the radially leading part of the WFG gauge. This will turn out to be
sufficient for the asymptotic analysis we are interested in. We then review in
subsection 2.2 the results of [30, 31, 32], where the WFG gauge was introduced
to restore the Weyl covariance of the boundary theory.

###  2.1 From FG to WFG

In this section, we relax the gauge conditions leading to the Fefferman–Graham
(FG) gauge. We show that there is a natural intermediate gauge obtained in
this way, which is the Weyl–Fefferman–Graham (WFG) gauge where only the Weyl
connection, appearing at the leading radial order of the bulk metric
expansion, is turned on. This relaxation was initially studied in [42, 30] and
further analyzed in [31, 32].

The Fefferman–Graham gauge of $3$-dimensional anti-de Sitter space consists in
choosing bulk coordinates $x^{\mu}=(\rho,x^{i})$, where $\rho\geq 0$ is a
radial coordinate and $x^{i}=(t,\theta)$ the boundary coordinates such that
the boundary is located at $\rho=0$, as well as the line element [74, 27, 28,
75, 76]222We follow conventions similar to [77, 48, 49].

$\text{d}s^{2}_{\text{FG}}=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\frac{\text{d}\rho^{2}}{\rho^{2}}+h_{ij}(\rho,x)\text{d}x^{i}\text{d}x^{j}\,,$
(2.1)

where we have fixed the AdS radius, $\ell=1$. The rewriting of this line
element in the Chern–Simons formulation leads to a natural relaxation of this
gauge. We refer to appendix A.1 for a review of the CS formalism. In this
context, the $\mathfrak{sl}(2,\mathbb{R})$ connection decomposes as

$A=A_{\mu}\text{d}x^{\mu}=A_{\rho}\text{d}\rho+A_{i}\text{d}x^{i}\,,$ (2.2)

and similarly for $\widetilde{A}$. The associated CS action is invariant under
SL(2,$\mathbb{R}$) gauge transformations

$A\to U^{-1}AU+U^{-1}\text{d}U\,,$ (2.3)

whose infinitesimal version is

$\delta_{\lambda}A=I_{V_{\lambda}}\delta A=\text{d}\lambda+[A,\lambda]\,,$
(2.4)

where $\lambda\in\mathfrak{sl}(2,\mathbb{R})$ and $U=\text{exp}(\lambda)$ is
its related group element. Depending on the holographic description we want to
study, we can exploit this gauge invariance to select a specific gauge for the
CS fields $A$. An arbitrary radial expansion is treated in subsection 2.2 and
in section 3. In this subsection we consider a gauge leading to a finite
$\rho$-expansion, as the Fefferman–Graham one. Indeed, if we impose that

$A_{\rho}=-\frac{1}{\rho}\,L_{0}\,,\qquad\widetilde{A}_{\rho}=\frac{1}{\rho}\,L_{0}\,,$
(2.5)

we get the following FG gauge-fixing condition

$g_{\rho\rho}=-\text{tr}\left(A_{\rho}-\widetilde{A}_{\rho}\right)^{2}=\frac{1}{\rho^{2}}\,,$
(2.6)

where we employed the following $\mathfrak{sl}(2,\mathbb{R})$ basis333A
possible matrix realization of this basis is
$L_{1}=-\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\,,\qquad
L_{-1}=-\frac{1}{\sqrt{2}}\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\,,\qquad
L_{0}=\frac{1}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\,.$ (2.7)

$[L_{1},L_{-1}]=-L_{0}\,,\qquad[L_{1},L_{0}]=L_{1}\,,\qquad[L_{-1},L_{0}]=-L_{-1}\,.$
(2.8)

In three spacetime dimensions, the on-shell FG line element has a truncated
radial expansion. This motivates the radial gauge choice for the CS
connections

$A=b^{-1}\,a\,b+b^{-1}\,\text{d}b\,,\qquad\widetilde{A}=\widetilde{b}^{-1}\,\widetilde{a}\,\widetilde{b}+\widetilde{b}^{-1}\,\text{d}\widetilde{b}\,.$
(2.9)

The one-forms $a=a_{i}\text{d}x^{i}$ and
$\widetilde{a}=\widetilde{a}_{i}\text{d}x^{i}$ encode then all the information
contained in the metric. The group elements $b$ and $\widetilde{b}$ are fixed
to satisfy

$A_{\rho}=b^{-1}\,\partial_{\rho}b\,,\qquad\widetilde{A}_{\rho}=\widetilde{b}^{-1}\,\partial_{\rho}\widetilde{b}\,,$
(2.10)

leading to

$b=\text{exp}\left(-\log\rho\,L_{0}\right)\,,\qquad\widetilde{b}=b^{-1}\,.$
(2.11)

We note that this gauge choice is always possible (at least locally): one can
always find such a group element $U$ that gives the identity at the boundary,
see [78]. Using one of the equations of motion, that is,

$F_{\rho i}=\partial_{\rho}A_{i}+A_{\rho}\,A_{i}-\left(\rho\leftrightarrow
i\right)=0\,,$ (2.12)

the $\mathfrak{sl}(2,\mathbb{R})$-valued one-forms $a$ and $\widetilde{a}$
depend only on the boundary coordinates. This fact constitutes a considerable
advantage in the calculation of surface charges, compared to the metric
formalism. Indeed, we will not need to recur to holographic renormalization,
because the radial dependence is completely captured by the gauge
transformation and factors out. We will comment on this last fact below eq.
(3.30). The remaining equations of motion simply tell us that these
connections are flat,

$\text{d}a+a\wedge a=0\,.$ (2.13)

If we expand them in the above basis (2.8) as

$a_{i}=a_{i}^{1}\,L_{1}+a_{i}^{0}\,L_{0}+a_{i}^{-1}\,L_{-1}\,,$ (2.14)

and similarly for $\widetilde{a}$, one can deduce that

$g_{\rho i}=-\frac{1}{2\rho}\left(a_{i}^{0}-\widetilde{a}_{i}^{0}\right).$
(2.15)

Therefore, without further restriction, this does not verify the FG gauge-
fixing condition $g_{\rho i}=0$. This then leads us to a natural relaxation of
the FG gauge (2.1), where the bulk metric takes the form

$g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\Big{(}\frac{\text{d}\rho}{\rho}-k_{i}(\rho,x)\text{d}x^{i}\Big{)}^{2}+h_{ij}(\rho,x)\text{d}x^{i}\text{d}x^{j}\,,$
(2.16)

such that

$k_{i}(\rho,x)=k_{i}^{(0)}(x)=\frac{1}{2}\left(a_{i}^{0}-\widetilde{a}_{i}^{0}\right),$
(2.17)

and

$\displaystyle
h_{ij}(\rho,x)=\frac{1}{\rho^{2}}h_{ij}^{(0)}(x)+h_{ij}^{(2)}(x)+\rho^{2}h_{ij}^{(4)}(x)\,,$
(2.18a) $\displaystyle
h_{ij}^{(0)}=-\frac{1}{4}\,a_{i}^{1}\,\widetilde{a}_{j}^{-1}\,,\qquad
h_{ij}^{(2)}=\frac{1}{4}\left(a_{i}^{1}\,a_{j}^{-1}+\widetilde{a}_{i}^{-1}\,\widetilde{a}_{j}^{1}\right),\qquad
h_{ij}^{(4)}=-\frac{1}{4}\,a_{i}^{-1}\,\widetilde{a}_{j}^{1}\,.$ (2.18b)

It is worth noting that the gauge choice (2.16) can be imposed off-shell as
the starting point, as we will do in section 3.

This gauge was first introduced in [19] in the Chern–Simons formulation, and
systematically analysed in [42]. It was studied geometrically in [30] in the
metric formalism for any dimension. In the latter, it was named
Weyl–Fefferman–Graham (WFG) since it was shown that it induces a Weyl
connection as part of the boundary geometry. We will review this in subsection
2.2. We note that (2.17) incorporates only the zero radial order of the new
structure $k_{i}(\rho,x)$. In reference [42], it has been proposed to choose
another group element instead of (2.11), i.e.,

$b(\rho)=\text{exp}(L_{-1})\text{exp}(-\log(\rho)L_{0})\,,$ (2.19)

leading to

$k_{i}(\rho,x)=\frac{1}{\rho}k_{i}^{(-1)}(x)+k_{i}^{(0)}(x)+\rho
k_{i}^{(1)}(x)\,,$ (2.20)

while in [30] the following infinite expansion was considered:

$k_{i}(\rho,x)=\sum_{n\geq 0}\rho^{2n}\,k_{i}^{(2n)}(x)\,.$ (2.21)

In the following, we will focus on the latter. In particular, in subsection
2.2 we will geometrically motivate the introduction of the WFG gauge via the
second order formalism. On the other hand, in section 3 we will come back to
the first order formalism and determine whether the new fields introduced have
associated non-vanishing charges or not. This is an important and delicate
step, because one has to determine whether restricting the radial expansion
(2.9) to be finite is a pure gauge choice, or instead it is a physical
restriction on the asymptotic phase space. Moreover, from a CS viewpoint, it
is always possible to go back to the FG metric (2.1) by performing a residual
gauge transformation which yields the following gauge fixing [79]:

$a_{i}^{0}=\widetilde{a}_{i}^{0}\qquad\Rightarrow\qquad g_{\rho
i}=-\frac{1}{\rho}k_{i}=0\,.$ (2.22)

This has been shown in [23] and is related to the FG theorem [28]. However, as
just stated, the radial and FG gauge fixing transformations can be physical,
which is determined via the study of surface charges in this theory.

###  2.2 Why WFG

In this subsection, we briefly review the WFG geometry [30, 31, 32]. Even
though the FG gauge (2.1) is convenient in the AdS/CFT correspondence, it
breaks the explicit Weyl covariance of the boundary by inducing the Levi-
Civita connection from the bulk. The WFG gauge was precisely introduced to
restore the Weyl covariance of the boundary.

In the FG gauge, a radial rescaling must be accompanied by a diffeomorphism in
the transverse space, so to preserve the gauge. Such a transformation is
called a Penrose–Brown–Henneaux (PBH) transformation [80, 7, 81, 22, 82, 83,
30], and has the form

$\rho\to\rho^{\prime}=\frac{\rho}{\mathfrak{B}(x)}\,,\qquad
x^{i}\to{x^{\prime}}^{i}=x^{i}+\xi^{i}(\rho,x)\,.$ (2.23)

Therefore, if we want to induce a Weyl transformation on the boundary in the
FG gauge we have to simultaneously introduce the diffeomorphism
$\xi^{i}(\rho,x)$, which vanishes at the boundary $\rho=0$. This
transformation impacts the subleading terms in the asymptotic radial FG
expansion so that they do not transform Weyl-covariantly under (2.23).

This issue can be overcome by relaxing the FG ansatz to the WFG gauge [30]

$\text{d}s^{2}_{\text{WFG}}=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\Big{(}\frac{\text{d}\rho}{\rho}-k_{i}(\rho,x)\text{d}x^{i}\Big{)}^{2}+h_{ij}(\rho,x)\text{d}x^{i}\text{d}x^{j}\,.$
(2.24)

In the last equation, we introduced the quantities $k_{i}$ and $h_{ij}$, which
can be asymptotically radially expanded as

$h_{ij}(\rho,x)=\frac{1}{\rho^{2}}\sum_{n\geq
0}\rho^{2n}\,h_{ij}^{(2n)}(x)\,,\qquad k_{i}(\rho,x)=\sum_{n\geq
0}\rho^{2n}\,k_{i}^{(2n)}(x)\,,$ (2.25)

where $h_{ij}^{(0)}$ is the boundary metric and $k_{i}^{(0)}$ is a boundary
Weyl connection. In the following, we will review why this is the right
interpretation. It is worth noting that (2.24) is the metric (2.16) we
obtained by a natural gauge relaxation in the CS formulation, but with an
arbitrary radial dependence of $k_{i}$. As we will explain later, this
arbitrary radial dependence can be fixed to recover exactly (2.16) by using
pure-gauge residual diffeomorphisms.

We start observing that the form of (2.24) is now preserved under the
following radial diffeomorphisms inducing boundary Weyl transformations;

$\rho\to\rho^{\prime}=\frac{\rho}{\mathfrak{B}(x)}\,,\qquad
x^{i}\to{x^{\prime}}^{i}=x^{i}\,,$ (2.26)

so that

$\displaystyle
k_{i}(\rho,x)\to{k^{\prime}}_{i}(\rho^{\prime},x^{\prime})=k_{i}(\mathfrak{B}(x)\rho^{\prime},x)-\partial_{i}\text{ln}\mathfrak{B}(x)\,,$
(2.27a) $\displaystyle
h_{ij}(\rho,x)\to{h^{\prime}}_{ij}(\rho^{\prime},x)=h_{ij}(\mathfrak{B}(x)\rho^{\prime},x)\,.$
(2.27b)

This then solves the above highlighted problem, because the subleading terms
in the radial expansion are now covariant under Weyl transformations. Thanks
to the last equations, we can also see that

$k_{i}^{(2n)}(x)\to
k_{i}^{(2n)}(x)\,\mathfrak{B}(x)^{2n}-\delta_{n,0}\,\partial_{i}\text{ln}\mathfrak{B}(x)\,,\qquad
h_{ij}^{(2n)}(x)\to h_{ij}^{(2n)}(x)\,\mathfrak{B}(x)^{2n-2}\,.$ (2.28)

Except for the leading term in $k_{i}$, all terms in the $\rho$ expansion
transform as Weyl tensors, with a definite Weyl weight given by the power of
$\mathfrak{B}(x)$. The leading term in $k_{i}$, on the other hand, transforms
inhomogeneously, i.e., as a Weyl connection,

$k_{i}^{(0)}\to k_{i}^{(0)}-\partial_{i}\text{ln}\mathfrak{B}\,.$ (2.29)

One also notices that the leading order of $h_{ij}$, the boundary metric,
varies as

$h_{ij}^{(0)}\to\mathfrak{B}^{-2}h_{ij}^{(0)}\,.$ (2.30)

In the following, we define a Weyl-covariant derivative in the boundary that
parallel transports the boundary metric $h^{(0)}_{ij}$. The idea is to choose
the following dual form basis444It should not be confused with the dreibein
used in the first order formulation as in, e.g., eq. (3.1). of the metric
(2.24)

$E^{\rho}=\frac{\text{d}\rho}{\rho}-k_{i}(\rho,x)\text{d}x^{i}\,,\qquad
E^{i}=\text{d}x^{i}\,,$ (2.31)

and its corresponding vector basis

$E_{\rho}=\rho\partial_{\rho}\equiv D_{\rho}\,,\qquad E_{i}=\partial_{i}+\rho
k_{i}(\rho,x)\partial_{\rho}\equiv D_{i}\,.$ (2.32)

The basis $\\{D_{\rho},D_{i}\\}$ spans the tangent space at any point
$(\rho,x^{i})$ of the bulk manifold $\mathcal{M}$, and the spatial vectors
$\\{D_{i}\\}$ span a $2$-dimensional distribution $\mathcal{D}$ on
$\mathcal{M}$. Their Lie brackets are given by

$[D_{\rho},D_{i}]=D_{\rho}k_{i}\,D_{\rho}\,,\qquad[D_{i},D_{j}]=f_{ij}\,D_{\rho}\,,$
(2.33)

where

$f_{ij}\equiv D_{i}k_{j}-D_{j}k_{i}$ (2.34)

is the curvature associated to $k_{i}$. Using the standard definition, the
coefficients of the bulk Levi-Civita (LC) connection $\nabla$ in the frame
$\\{D_{\rho},D_{i}\\}$ are

$\nabla_{D_{i}}D_{j}=\Gamma^{k}_{ij}D_{k}+\Gamma^{\rho}_{ij}D_{\rho}\,.$
(2.35)

Taking into account the radial expansions (2.25) to compute $\Gamma^{k}_{ij}$,
one obtains

$({\Gamma^{(0)}})^{k}_{ij}=\frac{1}{2}h^{kl}_{(0)}\left((\partial_{i}-2k^{(0)}_{i})h_{jl}^{(0)}+(\partial_{j}-2k^{(0)}_{j})h_{il}^{(0)}-(\partial_{l}-2k^{(0)}_{l})h_{ij}^{(0)}\right),$
(2.36)

at leading order. We emphasize that the coefficients $\Gamma^{k}_{ij}$ define
the induced connection coefficients on $\mathcal{D}$, see e.g. [84], while
their radial zero order (2.36) provides the coefficients of a torsion-free
connection with Weyl metricity [85, 86, 30].

Therefore, the WFG gauge has the novelty of being equipped with a Weyl
geometry and a metric $h_{ij}^{(0)}$ at the boundary. The induced connection
$\nabla^{(0)}$ acts as follows,

$\nabla_{i}^{(0)}h_{jk}^{(0)}=2k_{i}^{(0)}h_{jk}^{(0)}\,.$ (2.37)

For a generic Weyl-weight $\omega_{T}$ tensor $T$ of arbitrary type, we can
construct the Weyl covariant connection as

$\hat{\nabla}_{i}^{(0)}T\equiv\nabla_{i}^{(0)}T+\omega_{T}k_{i}^{(0)}T\,.$
(2.38)

So the connection $\hat{\nabla}^{(0)}$ is metric and $\hat{\nabla}_{i}^{(0)}T$
is Weyl covariant. Moreover, thanks to this, all geometric quantities built
with this connection are Weyl covariant.

We end this subsection with an important remark. If the Weyl curvature
$f_{ij}$ is zero, that is, $[D_{i},D_{j}]=0$, the distribution $\mathcal{D}$
is integrable according to the Frobenius theorem. In FG gauge $\mathcal{D}$
becomes $\\{\partial_{i}\\}$ and foliates $\mathcal{M}$ with $\rho$-constant
surfaces, which is not necessarily true for WFG.

##  3 Asymptotic symmetries

In this section, we come back to the Chern–Simons formulation of the WFG gauge
(2.24). More specifically, in subsection 3.1 we determine the bulk asymptotic
solution space in a conformal parametrization of the boundary, while we
compute the asymptotic symmetries and the associated surface charges in
subsection 3.2. We show that the transformation (2.22), which further gauge
fixes WFG down to FG, is charged, whereas the radial gauge fixing (2.9) is a
pure gauge diffeomorphism.

###  3.1 Solution space

We derive the asymptotic solution space in the WFG gauge (2.24) in the first
order formalism. Using $g_{\mu\nu}={e_{\mu}}^{B}\eta_{BC}\,{e_{\nu}}^{C}$,
with ($B=+1,-1,0$), we consider the following dreibein

$e^{0}=e_{\mu}{}^{0}\text{d}x^{\mu}=-\frac{\text{d}\rho}{\rho}+k_{i}(\rho,x)\text{d}x^{i}\,,\qquad
e^{\pm 1}=e_{\mu}{}^{\pm 1}\text{d}x^{\mu}=\frac{1}{\rho}{e_{i}}^{\pm
1}(\rho,x)\text{d}x^{i}\,,$ (3.1)

where we employed the basis (2.8) and the Minkowski metric

$\eta_{BC}=2\,\text{tr}\left(L_{B}\,L_{C}\right)=\begin{pmatrix}0&1&0\\\
1&0&0\\\ 0&0&1\end{pmatrix}.$ (3.2)

Using (2.25), we have the following radial expansion

$k_{i}(\rho,x)=\sum_{n\geq 0}\rho^{2n}\,k_{i}^{(2n)}(x)\,,\qquad{e_{i}}^{\pm
1}(\rho,x)=\sum_{n\geq 0}\rho^{2n}({e_{i}}^{\pm 1})^{(2n)}(x)\,.$ (3.3)

One can determine the associated spin connection and the CS forms using the
formulas collected in appendix A.1, in particular (A.6) and (A.11). It is then
possible to solve the flatness conditions reported in (A.16) order by order in
the radial expansion. Since the boundary is two-dimensional, we can always
express its metric in a conformally-flat parametrization

$h_{ij}^{(0)}(x)=\text{e}^{2\phi(x)}\eta_{ij}\,,$ (3.4)

where $\phi$ is an arbitrary conformal factor and $\eta_{ij}$ is the two-
dimensional Minkowski metric. We introduce the light-cone coordinates

$x^{\pm}=x^{1}\pm x^{2}=t\pm\theta\,,$ (3.5)

for which $\partial_{t}=\partial_{+}+\partial_{-}$ and
$\partial_{\theta}=\partial_{+}-\partial_{-}$. The boundary metric takes the
form

$\text{d}s^{2}_{\text{bdy}}=h_{ij}^{(0)}\text{d}x^{i}\text{d}x^{j}=\text{e}^{2\phi(x^{+},x^{-})}\text{d}x^{+}\text{d}x^{-}\,.$
(3.6)

The corresponding bulk metric is then

$\text{d}s^{2}_{\text{bulk}}=g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=\Big{(}\frac{\text{d}\rho}{\rho}-k_{i}(\rho,x)\text{d}x^{i}\Big{)}^{2}+h_{ij}(\rho,x)\text{d}x^{i}\text{d}x^{j}\,,$
(3.7)

where

$h_{ij}(\rho,x)=\frac{1}{\rho^{2}}h_{ij}^{(0)}(x)+h_{ij}^{(2)}(x)+\rho^{2}h_{ij}^{(4)}(x)+{\cal
O}(\rho^{4})$ (3.8)

such that

$\displaystyle h_{\pm\pm}^{(0)}$ $\displaystyle=0\,,$ (3.9a) $\displaystyle
h_{\pm\pm}^{(2)}$
$\displaystyle=\ell_{\pm}-(K_{\pm}^{(0)})^{2}-\partial_{\pm}K_{\pm}^{(0)},$
(3.9b)
$\displaystyle\begin{split}h_{\pm\pm}^{(4)}&=-\text{e}^{-2\phi}\,\partial_{\pm}K_{\mp}^{(0)}h_{\pm\pm}^{(2)}-k_{\pm}^{(2)}\left(\partial_{\pm}\phi+2K_{\pm}^{(0)}\right)-\frac{1}{2}\partial_{\pm}k_{\pm}^{(2)}\,,\end{split}$
(3.9c)

and

$\displaystyle h_{+-}^{(0)}$ $\displaystyle=\frac{1}{2}\text{e}^{2\phi}\,,$
(3.10a) $\displaystyle h_{+-}^{(2)}$
$\displaystyle=-\frac{1}{2}\Big{(}\partial_{-}K_{+}^{(0)}+\partial_{+}K_{-}^{(0)}\Big{)}\,,$
(3.10b)
$\displaystyle\begin{split}h_{+-}^{(4)}&=\frac{1}{4}\text{e}^{-2\phi}\Big{[}2\partial_{+}K_{-}^{(0)}\partial_{-}K_{+}^{(0)}-\text{e}^{2\phi}\Big{(}\partial_{-}k_{+}^{(2)}+\partial_{+}k_{-}^{(2)}+2k_{+}^{(0)}k_{-}^{(2)}+2k_{-}^{(0)}k_{+}^{(2)}\Big{)}+2h_{++}^{(2)}h_{--}^{(2)}\Big{]}\,.\end{split}$
(3.10c)

Here we defined

$K_{\pm}^{(0)}=k_{\pm}^{(0)}-\partial_{\pm}\phi\,,$ (3.11)

which is the Weyl gauge connection shifted by a pure gauge factor, see (2.29).
Moreover, the equation of motions impose

$\partial_{\pm}\ell_{\mp}=0\,.$ (3.12)

Note that the higher orders $h_{ij}^{(2n)}$ depend on $k_{i}^{(2n)}$. Just
like in [50] and as opposed to [40], we also notice that we do not perform a
chiral split of the solution space, because we consider here arbitrary
boundary functions $\phi$ and $k_{i}$. We can derive the following flat CS
connections

$\displaystyle A_{\rho}$
$\displaystyle=-\frac{1}{\rho}L_{0}+2\sqrt{2}\,\rho^{2}\text{e}^{-\phi}\left(k_{-}^{(2)}L_{1}-k_{+}^{(2)}L_{-1}\right)+{\cal
O}(\rho^{3})\,,$ (3.13a) $\displaystyle A_{+}$
$\displaystyle=\frac{\sqrt{2}}{\rho}\text{e}^{\phi}L_{1}+\left(2\,k_{+}^{(0)}-\partial_{+}\phi\right)L_{0}+\sqrt{2}\,\rho\,\text{e}^{-\phi}h_{++}^{(2)}L_{-1}+2\,\rho^{2}k_{+}^{(2)}L_{0}+{\cal
O}(\rho^{3})\,,$ (3.13b) $\displaystyle A_{-}$ $\displaystyle=\partial_{-}\phi
L_{0}-\sqrt{2}\,\rho\,\text{e}^{-\phi}\partial_{-}K_{+}^{(0)}L_{-1}+{\cal
O}(\rho^{3})\,.$ (3.13c)

A straightforward observation is that, unlike the FG gauge in three
dimensions, the WFG gauge has an infinite radial expansion. If we turn on only
the leading order of the Weyl connection, the expansion reduces to a finite
sum. As customary in the literature, we focus on the first
$\mathfrak{sl}(2,\mathbb{R})$ copy in the next steps. We collect the main
results on the second copy in appendix B.

###  3.2 Charges

We now turn to the identification of the residual symmetries of the solution
space and their associated surface charges.

#### Residual symmetries

We have to determine the gauge transformations (2.4) preserving the form of
the boundary values of $A$ in (3.13). We assume the following basis expansion
for the gauge parameters

$\lambda(\rho,x^{+},x^{-})=\epsilon^{B}(\rho,x^{+},x^{-})L_{B}\,,$ (3.14)

where the basis is introduced in (2.8). Then the components are given by

$\displaystyle\begin{split}\epsilon^{1}&=\frac{\sqrt{2}}{\rho}\text{e}^{\phi}Y^{+}+{\cal
O}(\rho^{3})\,,\end{split}$ (3.15a)
$\displaystyle\begin{split}\epsilon^{-1}&=\frac{\rho}{\sqrt{2}}\text{e}^{-\phi}\Big{[}\partial_{+}^{2}Y^{+}-2H_{+}^{(0)}+2\Big{(}\ell_{+}Y^{+}-(K_{+}^{(0)})^{2}Y^{+}+K_{+}^{(0)}\partial_{+}Y^{+}\Big{)}\Big{]}+{\cal
O}(\rho^{3})\,,\end{split}$ (3.15b) $\displaystyle\epsilon^{0}$
$\displaystyle=\sigma-\partial_{+}Y^{+}+2Y^{+}K_{+}^{(0)}\,,$ (3.15c)

where $\sigma$ and $H_{\pm}^{(0)}$ are arbitrary functions of the boundary
coordinates while $Y^{\pm}=Y^{\pm}(x^{\pm})$. In analogy with (3.11), we
introduce $h_{\pm}^{(0)}$ defined via

$H_{\pm}^{(0)}=h_{\pm}^{(0)}-\partial_{\pm}\sigma\,.$ (3.16)

Then, the asymptotic Killing vectors are given by555These are obtained using
$\xi^{\mu}=\tfrac{1}{2}{e_{B}}^{\mu}(\lambda^{B}-\widetilde{\lambda}^{B})$,
see [18].

$\xi^{\rho}=\rho\,\omega+\mathcal{O}(\rho^{3})\,,\qquad\xi^{\pm}=Y^{\pm}+\rho^{2}\zeta^{\pm}+\mathcal{O}(\rho^{4})\,,$
(3.17)

where we renamed

$\displaystyle\omega(x^{+},x^{-})$
$\displaystyle=-\sigma+\frac{1}{2}\partial_{i}Y^{i}+Y^{i}\partial_{i}\phi\,,$
(3.18a) $\displaystyle\zeta^{\pm}(x^{+},x^{-})$
$\displaystyle=\text{e}^{-2\phi}\left(K^{(0)}_{\mp}\partial_{\mp}Y^{\mp}-H^{(0)}_{\mp}-\frac{1}{2}\partial_{\mp}\partial_{i}Y^{i}+Y^{i}\partial_{i}K^{(0)}_{\mp}\right),$
(3.18b)

such that $\xi^{\rho}$ and $\xi^{\pm}$ have parameters $\omega$ and
$H^{(0)}_{\pm}$ that do not mix.

In particular, we see that the Weyl parameter $\omega$ is given by

$\omega-
k_{i}^{(0)}Y^{i}=-\frac{1}{2}\left(\epsilon^{0}-\widetilde{\epsilon}^{0}\right),$
(3.19)

which shows a link between radial rescalings and $\mathfrak{sl}(2,\mathbb{R})$
gauge transformations along the Cartan direction $L_{0}$. Indeed, the bulk
radial rescaling (2.23) acts on the boundary metric as [81]

$h_{ij}^{(0)}\to\text{e}^{2\omega}h_{ij}^{(0)}\,,$ (3.20)

that is, it induces a Weyl transformation with parameter $\omega(x)$. We note
that our expression (3.19) is Weyl covariant, since the standard relation
would rather be $\omega=-\tfrac{1}{2}(\epsilon^{0}-\widetilde{\epsilon}^{0})$,
see [79, 22, 87]. Once the residual gauge transformations are identified, we
can compute their commutation relations. Since these generators are field
dependent, one has to use a modified commutator666It is analogous to its
metric counterpart: the residual diffeomorphism algebra is closed under the
modified Lie bracket of two asymptotic Killing vectors [88, 89]
$[\xi_{1},\xi_{2}]_{M}\equiv[\xi_{1},\xi_{2}]-\delta_{\xi_{1}}\xi_{2}+\delta_{\xi_{2}}\xi_{1}\,,$
(3.21) where $[\xi_{1},\xi_{2}]$ is the standard Lie bracket. The modified
bracket takes into account the dependence of the vectors on fields that
transform themselves under the symmetry.

$[\lambda_{1},\lambda_{2}]_{M}\equiv[\lambda_{1},\lambda_{2}]-\delta_{\lambda_{1}}\lambda_{2}+\delta_{\lambda_{2}}\lambda_{1}\,,$
(3.22)

where, e.g., $\delta_{\lambda_{1}}\lambda_{2}$ denotes the variation of
$\lambda_{2}$ under $\lambda_{1}$. A similar relation applies for the second
copy, whose generators are called $\widetilde{\lambda}$. Taking
$\delta\sigma=\delta H_{\pm}^{(0)}=\delta Y^{\pm}=0$, the gauge parameters
close respectively the Lie algebras
$[\lambda_{1},\lambda_{2}]_{M}=\lambda_{12}$ and
$[\widetilde{\lambda}_{1},\widetilde{\lambda}_{2}]_{M}=\widetilde{\lambda}_{12}$,
where $\lambda_{12}$ and $\widetilde{\lambda}_{12}$ depend on

$Y^{\pm}_{12}={Y^{\pm}_{2}}\partial_{\pm}{Y^{\pm}_{1}}-{Y^{\pm}_{1}}\partial_{\pm}{Y^{\pm}_{2}}\,,\qquad\sigma_{12}=0\,,\qquad
H_{\pm\,12}^{(0)}=0\,.$ (3.23)

The residual symmetry algebra is given by the direct sum of two Witt algebras
generated by $Y^{i}$, an abelian sector generated by $\sigma$ and another one
generated by the boundary vector $H^{(0)}_{i}$.

If we introduce the following Fourier mode expansions of the symmetry
parameters

$Y^{\pm}\sim\text{e}^{\text{i}nx^{\pm}}\,,\quad\sigma\sim\text{e}^{\text{i}px^{+}}\text{e}^{\text{i}qx^{-}}\,,\quad{H_{\pm}^{(0)}}\sim\text{e}^{\text{i}px^{+}}\text{e}^{\text{i}qx^{-}}\,,$
(3.24)

with $n,p,q\in\mathbb{Z}$, we gather

$\displaystyle[\lambda_{n}^{Y^{\pm}},\lambda_{m}^{Y^{\pm}}]_{M}=\text{i}(n-m)\lambda_{n+m}^{Y^{\pm}}\,,\quad$
$\displaystyle[\lambda_{n}^{Y^{\pm}},\lambda_{m}^{Y^{\mp}}]_{M}=[\lambda_{n}^{Y^{\pm}},\lambda_{pq}^{\sigma}]_{M}=[\lambda_{n}^{Y^{\pm}},\lambda_{pq}^{H_{\pm}^{(0)}}]_{M}=0\,,$
(3.25)
$\displaystyle[\lambda_{pq}^{\sigma},\lambda_{rs}^{\sigma}]_{M}=0\,,\quad$
$\displaystyle[\lambda_{pq}^{\sigma},\lambda_{rs}^{H_{\pm}^{(0)}}]_{M}=0\,,$
$\displaystyle[\lambda_{pq}^{H_{\pm}^{(0)}},\lambda_{rs}^{H_{\pm}^{(0)}}]_{M}=0\,,\quad$
$\displaystyle[\lambda_{pq}^{H_{\pm}^{(0)}},\lambda_{rs}^{H_{\mp}^{(0)}}]_{M}=0\,.$

In these commutation relations, for example, we have denoted by
$\lambda_{n}^{Y^{\pm}}$ the gauge parameters where we turned on only $Y^{\pm}$
expanded as in (3.24) and set the other parameters $\sigma$ and
$H^{(0)}_{\pm}$ to zero.

#### Surface charges

Using (2.4), the variations of the boundary data under the above
transformations read

$\displaystyle\delta_{(\lambda,\widetilde{\lambda})}\ell_{\pm}$
$\displaystyle=Y^{\pm}\partial_{\pm}\ell_{\pm}+2\ell_{\pm}\partial_{\pm}Y^{\pm}-\frac{1}{2}\partial_{\pm}^{3}Y^{\pm}\,,\qquad\delta_{(\lambda,\widetilde{\lambda})}\phi=\sigma\,,\qquad\delta_{(\lambda,\widetilde{\lambda})}k_{\pm}^{(2n)}=h_{\pm}^{(2n)}\,,$
(3.26)

with $n\in\mathbb{N}$. In particular, we have that

$\delta_{(\lambda,\widetilde{\lambda})}K_{\pm}^{(0)}=H_{\pm}^{(0)}\,.$ (3.27)

We summarize the covariant phase space formalism applied to Chern–Simons
theories in appendix A.2. Following [19], the surface charges evaluated at
fixed value of the time coordinate $t$ are given by

$\delta
Q_{\lambda}=-\frac{\kappa}{2\pi}\int_{0}^{2\pi}\text{d}\theta\,\text{tr}\Big{(}\lambda\,\delta
A_{\theta}\Big{)}\,,\qquad\delta\widetilde{Q}_{\widetilde{\lambda}}=-\frac{\kappa}{2\pi}\int_{0}^{2\pi}\text{d}\theta\,\text{tr}\Big{(}\widetilde{\lambda}\,\delta\widetilde{A}_{\theta}\Big{)}\,,$
(3.28)

where $\kappa=\nicefrac{{1}}{{4\mathcal{G}}}$. We provide in (A.24) a
derivation of this result. In three-dimensional WFG gauge, the symplectic
structure does not diverge and the charges are thus finite in the limit
$\rho\to 0$. Substituting the gauge parameters, one obtains, up to integration
by parts,

$\delta Q_{\Lambda}=\lim_{\rho\to 0}\left(\delta
Q_{\lambda}-\delta\widetilde{Q}_{\widetilde{\lambda}}\right)=-\frac{\kappa}{2\pi}\int_{0}^{2\pi}\text{d}\theta\Big{[}\delta\ell_{+}Y^{+}+\delta\ell_{-}Y^{-}-\delta\phi
H^{(0)}_{t}+\sigma\delta K^{(0)}_{t}\Big{]}\,,$ (3.29)

where we denoted ${\Lambda}=(\lambda,\widetilde{\lambda})$. Note that we have
used $A_{\pm}=\tfrac{1}{2}(A_{t}\pm A_{\theta})$ in this computation. The
above variation is manifestly integrable for $\delta\sigma=\delta
H_{\pm}^{(0)}=\delta Y^{\pm}=0$, therefore the total surface charges are

$\boxed{Q_{\Lambda}=-\frac{\kappa}{2\pi}\int_{0}^{2\pi}\text{d}\theta\Big{[}\ell_{+}Y^{+}+\ell_{-}Y^{-}-\phi
H_{t}^{(0)}+\sigma K_{t}^{(0)}\Big{]}\,.}$ (3.30)

The charge associated to $Y$ is conserved. This is not the case for the
charges associated to $\sigma$ and $H_{t}^{(0)}$, as expected as we did not
impose any conditions on the conformal factor of the boundary metric and the
Weyl connection.

We would like to offer some remarks. Firstly, the fields associated with the
higher radial orders of the Weyl structure $k_{i}^{(2p)}$, for
$p\in\mathbb{N}_{0}$, and the component $K^{(0)}_{\phi}$ do not contribute to
the asymptotic charges. This means that they are pure gauge and so we can set
them to zero by a trivial diffeomorphism. Hence, we are allowed to reduce the
asymptotic expansion of the Weyl connection to the leading order only, which
in turns makes the metric expansion finite. In this way, we recover the gauge
(2.17) where one can perform a radial gauge fixing as in (2.9). On the other
hand, using (2.22) to further fix the gauge from WFG to FG would set to zero a
physical charge. This is one of the main results of this paper, that proves
how the complete FG gauge fixing is exploiting a charged diffeomorphism, and
thus it is restricting the set of physical states. Notice therefore that, in
the holographic setup where this charge is non-vanishing, the associated
current is physical. It was important to perform the asymptotic symmetry
analysis to reply to this question, raised in [30, 31].

Secondly, in the conformal parametrization (3.4), the the Ricci scalar
associated to the boundary metric reads as

$R^{(0)}=-8\text{e}^{-2\phi}\partial_{+}\partial_{-}\phi\,,$ (3.31)

while the Weyl curvature is given by

$f^{(0)}=\frac{1}{2}\mathcal{E}^{ij}f_{ij}^{(0)}=2\text{e}^{-2\phi}\left(\partial_{-}k_{+}^{(0)}-\partial_{+}k_{-}^{(0)}\right).$
(3.32)

In the last expression, we have used (2.34) and introduced
$\mathcal{E}_{ij}=\sqrt{-h^{(0)}}\varepsilon_{ij}$ with $\varepsilon_{01}=+1$
and $\mathcal{E}^{ik}\mathcal{E}_{kj}=\delta^{i}_{j}$. Then, imposing the
additional constraints

$k_{\pm}^{(0)}(x^{+},x^{-})=\partial_{\pm}\zeta(x^{+},x^{-})\,,\qquad\phi(x^{+},x^{-})\to\phi(x^{+},x^{-})+\zeta(x^{+},x^{-})\,,$
(3.33)

we set $f^{(0)}=0$, and we match the CS boundary conditions (3.13) studied in
[54]. Nonetheless, in the latter a different framework has been considered,
where the main focus was to obtain a smooth flat limit of the bulk metric,
attainable by focusing on a relaxed Bondi gauge. The main difference is that
in WFG the boundary Weyl scalar curvature contributes to the charges,777The
$2$-dimensional Weyl scalar curvature differs from the Levi-Civita scalar
curvature only by total derivatives [30]. while in the relaxed Bondi gauge it
is the boundary Weyl curvature that is charged, instead. We leave a better
understanding of the link between the different gauge relaxations in three-
dimensions for future work.

Finally, we notice that $\phi$ and $K^{(0)}_{t}$ are Heisenberg partners in
the sense that $\delta_{\xi}\phi=\sigma$ and
$\delta_{\xi}K^{(0)}_{t}=H_{t}^{(0)}$. This implies that they purely come from
a corner term in the symplectic potential, as they can be written as
$\text{d}C$, see (A.20). In section 4 we discuss different variational
principles switching on or off these charges. This is the manifestation that
$\phi$ and $K^{(0)}_{t}$ are kinematic charges, unlike the charges associated
to $\ell^{\pm}$ that are constrained due to Einstein equations.

#### Charge algebra

As a last step in this analysis, we show that, under the Poisson bracket, the
charges form a projective representation of the asymptotic symmetry algebra
(3.25)

$\\{{Q}_{{\Lambda}_{1}},{Q}_{{\Lambda}_{2}}\\}=\delta_{{\Lambda}_{2}}{Q}_{{\Lambda}_{1}}={Q}_{[{\Lambda}_{1},{\Lambda}_{2}]_{M}}+\mathcal{K}[{{\Lambda}_{1},{\Lambda}_{2}}]\,,$
(3.34)

where $\mathcal{K}[{{\Lambda}_{1},{\Lambda}_{2}}]$ is the central extension
satisfying the $2-$cocycle condition

$\mathcal{K}[[{{\Lambda}_{1},{\Lambda}_{2}}]_{M},\Lambda_{3}]+\mathcal{K}[[{{\Lambda}_{2},{\Lambda}_{3}}]_{M},\Lambda_{1}]+\mathcal{K}[[{{\Lambda}_{3},{\Lambda}_{1}}]_{M},\Lambda_{2}]=0\,.$
(3.35)

Indeed, we obtain that the $\text{Witt}\oplus\overline{\text{Witt}}$ part
gives rise to a double copy Virasoro algebra with

$\mathcal{K}[{\Lambda_{1}^{Y^{\pm}},\Lambda_{2}^{Y^{\pm}}}]=-\frac{\kappa}{4\pi}\int_{0}^{2\pi}\text{d}\theta\left(Y_{1}^{+}\partial_{+}^{3}Y_{2}^{+}+Y_{1}^{-}\partial_{-}^{3}Y_{2}^{-}\right),$
(3.36)

while the other parts are promoted to affine algebras with central extensions

$\mathcal{K}[{\Lambda_{1}^{\sigma},\Lambda_{2}^{H_{\pm}^{(0)}}}]=-\frac{\kappa}{2\pi}\int_{0}^{2\pi}\text{d}\theta\,\sigma_{1}H_{\pm\,2}^{(0)}\,.$
(3.37)

In terms of the mode expansions (3.24), the charge algebra (3.34) can be
written

$\\{Q_{\Lambda_{n}^{Y^{\pm}}},Q_{\Lambda_{m}^{Y^{\pm}}}\\}=\text{i}(n-m)Q_{\Lambda_{n+m}^{Y^{\pm}}}-\text{i}m^{3}\frac{c}{12}\delta_{n+m,0}\,,\qquad\\{Q_{\Lambda_{pq}^{\sigma}},Q_{\Lambda_{rs}}^{H_{\pm}^{(0)}}\\}=\frac{c}{3}\text{e}^{2\text{i}(q+s)t}\delta_{p+r,q+s}\,,$
(3.38)

where $c=\nicefrac{{3}}{{2\mathcal{G}}}$ is the Brown–Henneaux central charge
[7]. Note that we find the same time-dependency in the central charge as in
[50]. The charge algebra of the phase space is then a double copy of the
Virasoro algebra in direct sum with an Heisenberg algebra. This is precisely
the same algebra appearing in the Bondi-Weyl gauge [53] and in the analysis of
generic hypersurfaces [47, 90, 55].

##  4 Holography and variational principle

In this section, we discuss the holographic renormalization of the action
principle in second order formalism. We propose different prescriptions of
finite counterterms yielding different expressions for the holographic stress
tensor and the holographic Weyl current, as well as for the charges. In
particular, we show that the above discussion in first order formalism
corresponds to a specific choice of counterterms in the action.

###  4.1 Holographic renormalization

#### Variational principle

In previous works, the holographic renormalization in the WFG gauge has been
performed using dimensional regularization [30, 31]. In this section, we
revisit this problem using the cut-off regularization. Starting from the WFG
solution space (2.16), the renormalized action is

$\begin{split}S_{ren}=\frac{1}{16\pi\mathcal{G}}\int\text{d}^{3}x\,&(R+2)+\frac{1}{8\pi\mathcal{G}}\int\text{d}^{2}x\sqrt{-\gamma}\,(K-1)\\\
&+\frac{1}{16\pi\mathcal{G}}\int\text{d}^{2}x\sqrt{-\gamma}\,k_{i}\,\gamma^{ij}\,k_{j}+\frac{\rho^{2}\log\rho}{16\pi\mathcal{G}}\int\text{d}^{2}x\sqrt{-\gamma}\,\hat{R}^{(0)}\,,\end{split}$
(4.1)

where we denote $n_{\mu}$ the normal to constant-$\rho$ hypersurfaces
$n_{\mu}=-\sqrt{-\gamma}\delta_{\mu}^{\rho}$,
$\gamma_{\mu\nu}=g_{\mu\nu}-n_{\mu}n_{\nu}$ the induced metric, and
$K=g^{\mu\nu}\nabla_{\mu}n_{\nu}$ the extrinsic curvature. The first term in
the right-hand side of (4.1) is the Einstein–Hilbert bulk action, the second
is the Gibbons–Hawking–York boundary term, and the last two terms are boundary
counterterms ensuring the finiteness of the on-shell action in WFG gauge. The
variation of the on-shell action yields

$\delta S_{ren}\approx\int\text{d}^{2}x\,\Theta_{ren},$ (4.2)

with, following the prescription [71, 72, 61, 73] (see also appendix A.2), the
renormalized symplectic potential888In an abuse of vocabulary, we refer to
$\Theta$ as the symplectic potential, although at this stage it should be
called presymplectic, since it is degenerate.

$\Theta_{ren}=\lim_{\rho\to
0}\left[\Theta^{\rho}_{EH}+\delta\left(\frac{1}{8\pi\mathcal{G}}\sqrt{-\gamma}(K+1)+\frac{1}{16\pi\mathcal{G}}\sqrt{-\gamma}\,k_{i}\,\gamma^{ij}\,k_{j}\right)-\partial_{i}\vartheta_{GHY}^{i}\right].$
(4.3)

Here,
$\Theta^{\mu}_{EH}=\frac{\sqrt{-g}}{16\pi\mathcal{G}}[\nabla_{\nu}(\delta
g)^{\mu\nu}-\nabla^{\mu}{(\delta g)^{\nu}}_{\nu}]$ is the Einstein–Hilbert
symplectic potential, and $\lim_{\rho\to
0}\vartheta^{i}_{GHY}=\frac{1}{16\pi\mathcal{G}}\sqrt{-h^{(0)}}\delta(h_{(0)}^{ij}k^{(0)}_{j})$
is the Gibbons–Hawking–York symplectic potential. Explicitly, we have

$\displaystyle\Theta_{ren}=-\sqrt{-h^{(0)}}\left(\frac{1}{2}T^{ij}\delta
h_{ij}^{(0)}-J^{i}\delta k_{i}^{(0)}\right)$ (4.4)

where the holographic stress tensor $T^{ij}$ and the holographic Weyl current
$J^{i}$ read as999We denote by the round brackets the symmetrization on the
corresponding indices, $A_{(ij)}=\frac{1}{2}(A_{ij}+A_{ji})$.

$\displaystyle T^{ij}$ $\displaystyle=-\frac{2}{\sqrt{-h^{(0)}}}\frac{\delta
S_{ren}}{\delta
h_{ij}^{(0)}}\approx\frac{1}{8\pi\mathcal{G}}\left(h_{(2)}^{ij}+\frac{1}{2}h_{(0)}^{ij}R^{(0)}+\hat{\nabla}^{(i}_{(0)}k^{j)}_{(0)}\right),$
(4.5) $\displaystyle J^{i}$
$\displaystyle=\frac{1}{\sqrt{-h^{(0)}}}\frac{\delta S_{ren}}{\delta
k_{i}^{(0)}}\approx\frac{1}{8\pi\mathcal{G}}\,k_{(0)}^{i}\,.$ (4.6)

Notice the presence of the $\hat{\nabla}^{(i}_{{(0)}}k^{j)}_{(0)}$ term in the
holographic stress tensor compared to the usual Brown–York expression in FG
gauge. In the holographic dictionary, the renormalized symplectic potential
(4.4) has the standard form VEV $\times$ $\delta$(sources). The sources
correspond to the boundary geometry which, in WFG gauge, is provided by the
conformal class of boundary metric $h^{(0)}_{ij}$ and the Weyl connection
$k^{(0)}_{i}$. As a result, in addition to the VEV of the holographic stress
tensor associated with the source $h^{(0)}_{ij}$, there is a VEV for the Weyl
current operator $J^{i}$ associated with the source $k^{i}_{(0)}$. The unsual
feature, which appears in three bulk dimensions, is that the Weyl current is
given on shell by the Weyl connection (4.6).

#### Holographic Ward identities

As discussed in section 3.2, the residual gauge diffeomorphisms of the WFG
gauge are given by $\xi^{\rho}=\rho\omega+\mathcal{O}(\rho^{3})$ and
$\xi^{i}=Y^{i}+\rho^{2}\zeta^{i}+\mathcal{O}(\rho^{4})$ at leading order.
Under these residual diffeomorphisms, the boundary geometry transforms as

$\delta_{(Y,\omega,\zeta)}h_{ij}^{(0)}=\mathcal{L}_{Y}h_{ij}^{(0)}-2\,\omega\,h_{ij}^{(0)}\,,\qquad\delta_{(Y,\omega,\zeta)}k^{(0)}_{i}=\mathcal{L}_{Y}k^{(0)}_{i}-\partial_{i}\omega-2\,\zeta_{i}\,,$
(4.7)

while the holographic stress tensor and Weyl current transform as101010The
currents $J^{i}$ and $T^{i}{}_{j}$ are by construction of weight $+2$, the
dimension of the boundary. Moreover, one has
$\nabla_{i}^{(0)}J^{i}=\hat{\nabla}_{i}^{(0)}J^{i}$.

$\displaystyle\begin{split}\delta_{(Y,\omega,\zeta)}T^{ij}&=\mathcal{L}_{Y}T^{ij}+4\,\omega\,T^{ij}-\frac{1}{8\pi\mathcal{G}}\hat{\nabla}^{(i}_{(0)}\partial^{j)}\omega+\frac{1}{8\pi\mathcal{G}}h^{ij}_{(0)}\hat{\nabla}^{k}_{(0)}\partial_{k}\omega-\frac{1}{2\pi\mathcal{G}}k_{(0)}^{(i}\zeta^{i)}\\\
&\quad+\frac{1}{4\pi\mathcal{G}}h^{ij}_{(0)}k^{(0)}_{k}\zeta^{k}\,,\end{split}$
(4.8) $\displaystyle\delta_{(Y,\omega,\zeta)}J^{i}$
$\displaystyle=\mathcal{L}_{Y}J^{i}+2\,\omega\,J^{i}-\frac{1}{8\pi\mathcal{G}}\partial^{i}\omega-\frac{1}{4\pi\mathcal{G}}\zeta^{i}\,.$
(4.9)

The holographic Ward identities are obtained by evaluating the variation of
the on-shell action (4.2) on the above symmetries. Under the action of
boundary diffeomorphisms, we have

$\begin{split}\delta_{Y}S_{ren}&=-\int\text{d}^{2}x\sqrt{-h^{(0)}}\left(\frac{1}{2}T^{ij}\delta_{Y}h_{ij}^{(0)}-J^{i}\delta_{Y}k_{i}^{(0)}\right)\\\
&=-\int\text{d}^{2}x\sqrt{-h^{(0)}}\,Y^{j}\left(-\nabla_{i}^{(0)}{T^{i}}_{j}+J^{i}f_{ij}^{(0)}+\nabla_{i}^{(0)}J^{i}k^{(0)}_{j}\right)\end{split}$
(4.10)

where $f_{ij}^{(0)}=\nabla_{i}^{(0)}k^{(0)}_{j}-\nabla_{j}^{(0)}k^{(0)}_{i}$
is the Weyl curvature corresponding to the leading order of (2.34). To obtain
the second equality, we used the explicit transformation of the sources in
(4.7) under boundary diffeomorphisms and integrated by parts to isolate the
symmetry parameter $Y^{j}$. Einstein equations imply that this expression
vanishes

$\nabla_{i}^{(0)}{T^{i}}_{j}=J^{i}\,f_{ij}^{(0)}+\nabla_{i}^{(0)}J^{i}\,k^{(0)}_{j},$
(4.11)

which yields the invariance of the renormalized on-shell action under boundary
diffeomorphisms. Equation (4.11) can then be interpreted as the holographic
Ward identity associated with boundary diffeomorphisms. Notice the presence of
the additional terms in the right-hand side due to the Weyl connection
compared to the usual covariant conservation of the stress tensor.

The variation of the action under boundary Weyl transformations is

$\begin{split}\delta_{\omega}S_{ren}&=-\int\text{d}^{2}x\sqrt{-h^{(0)}}\left(\frac{1}{2}T^{ij}\delta_{\omega}h_{ij}^{(0)}-J^{i}\delta_{\omega}k_{i}^{(0)}\right)\\\
&=\int\text{d}^{2}x\sqrt{-h^{(0)}}\,\omega\left({T^{i}}_{i}+\hat{\nabla}_{i}^{(0)}J^{i}\right)\\\
&=\frac{c}{24\pi}\int\text{d}^{2}x\sqrt{-h^{(0)}}\,\omega\,\hat{R}^{(0)}\,.\end{split}$
(4.12)

In the last equality, we used Einstein equations to make the Weyl–Ricci scalar
appear. This shows that the renormalized on-shell action is not invariant
under Weyl, which unveils the presence of a holographic Weyl anomaly [91, 92].
Thus, the anomalous Ward identities associated with Weyl rescalings give

${T^{i}}_{i}+\hat{\nabla}_{i}^{(0)}J^{i}=\frac{c}{24\pi}\hat{R}^{(0)}\,.$
(4.13)

Plugging (4.13) into (4.11), we find a suggestive rewriting of the Ward
identity for boundary diffeomorphisms

$\hat{\nabla}_{i}^{(0)}{T^{i}}_{j}=J^{i}f_{ij}+\frac{c}{24\pi}\hat{R}^{(0)}\,k^{(0)}_{j}\,.$
(4.14)

As we shall now discuss, the symmetries $\zeta^{i}$ are pure gauge for the
phase space obtained from the holographic renormalization of the action (4.1),
and therefore do not lead to additional holographic Ward identities.

#### Charges

We compute the charges using the covariant phase space formalism [93, 94, 36,
38, 8] (see appendix A.2 and, e.g., [95, 96, 77, 97, 98] for recent reviews).
First, we contract the renormalized symplectic current

$\omega_{ren}=\delta\Theta_{ren}=-\frac{1}{2}\delta(\sqrt{-h^{(0)}}T^{ij})\wedge\delta
h_{ij}^{(0)}+\delta(\sqrt{-h^{(0)}}J^{i})\wedge\delta k_{i}^{(0)}$ (4.15)

with the symmetries encoded in the variations (4.7) and (4.8). Then, to
compare with the analysis of section 3.2 in Chern–Simons formulation, we
choose the conformal gauge (3.4) for the boundary metric. After the
redefinitions (3.11) and (3.16), assuming $\delta\sigma=\delta Y=0$, we find
that the canonical charge is finite and given by

$Q_{\xi}=\int
q_{\xi}=\frac{1}{8\pi\mathcal{G}}\int^{2\pi}_{0}\text{d}\theta\left[Y^{i}\,\ell_{i}+\partial_{t}\sigma\,\phi-\sigma\,\partial_{t}\phi\right].$
(4.16)

Notably, the charges (4.16) match precisely with the expressions found in the
FG gauge in [50]. In contrast with the analysis of section 3.2, the charge
associated with the symmetry $\zeta^{i}$ vanishes, which implies that this
residual gauge diffeomorphism is now pure gauge and can be quotiented out.
Note that this result holds for arbitrary boundary metric and not only in the
conformal gauge. We observe that the charge algebra is unaffected, and again
given by two copies of the Virasoro algebra in direct sum with a Heisenberg
algebra. However, here this Heisenberg factor is realized solely by the zero
mode of the conformal factor and its first derivative, while in (3.30) the
Heisenberg pair comes from different boundary fields.

Therefore, the phase space controlled by the action after holographic
renormalization (4.1) differs from the one in the Chern–Simons formulation
discussed in section 3.2. In this framework, the Weyl connection can be
consistently set to zero. The holographic interpretation is that there are no
observables that are sensitive to the latter, and thus the Weyl
covariantization of the boundary geometry can be achieved without enlarging
the physical phase space. On the other hand, this is not the case in section
3, where there is a non-vanishing charge, and we would like to understand how
to derive this result in second-order formulation, to gain a better
holographic perspective.

It is well known that the renormalized action (4.1) is not uniquely defined:
one can add finite counterterms. As we shall now explain, a judicious choice
of boundary or corner counterterms will allow to recover the phase space found
in the Chern–Simons formulation.

###  4.2 Boundary counterterm

We now connect the holographic discussion in the WFG gauge presented in
section 4.1 with the results of section 3.2. In the holographic
renormalization procedure, one can always add a finite boundary counterterm to
the action (4.1) as

$\bar{S}_{ren}=S_{ren}+S_{\circ},\qquad
S_{\circ}=\int\text{d}^{2}x\,L_{\circ}[h^{(0)}_{ij},k_{i}^{(0)}]$ (4.17)

where $L_{\circ}[h^{(0)}_{ij},k_{i}^{(0)}]$ is a boundary Lagrangian involving
the boundary geometry. Defining

$T^{ij}_{o}=-\frac{2}{\sqrt{-h^{(0)}}}\frac{\delta S_{o}}{\delta
h_{ij}^{(0)}},\qquad J^{i}_{o}=\frac{1}{\sqrt{-h^{(0)}}}\frac{\delta
S_{o}}{\delta k_{i}^{(0)}}$ (4.18)

and using the standard procedure to treat boundary terms [71, 72, 61, 73], the
variation of the renormalized on-shell action reads as

$\delta\bar{S}_{ren}\approx\int\text{d}^{2}x\,\bar{\Theta}_{ren}$ (4.19)

with

$\bar{\Theta}_{ren}=-\sqrt{-h^{(0)}}\left(\frac{1}{2}\bar{T}^{ij}\delta
h_{ij}^{(0)}-\bar{J}^{i}\delta k_{i}^{(0)}\right).$ (4.20)

The new currents are given by $\bar{T}^{ij}=T^{ij}+T^{ij}_{\circ}$ and
$\bar{J}^{i}=J^{i}+J^{i}_{\circ}$. Choosing the boundary term

$L_{\circ}=\lim_{\rho\to
0}\left[\frac{1}{16\pi\mathcal{G}}k_{i}\gamma^{ij}\partial_{j}\sqrt{-\gamma}\right]=\frac{1}{16\pi\mathcal{G}}k_{i}^{(0)}h^{ij}_{(0)}\partial_{j}\sqrt{-h^{(0)}}\,,$
(4.21)

we find explicitly

$\displaystyle\bar{T}^{ij}$
$\displaystyle=T^{ij}+J^{(i}\partial^{j)}\sqrt{-h^{(0)}}+\frac{1}{2}h^{ij}\nabla_{k}J^{k}\,,\qquad\bar{J}^{i}=J^{i}+\frac{1}{16\pi\mathcal{G}}\partial^{i}\log\sqrt{-h^{(0)}}\,.$
(4.22)

Notice that the boundary Lagrangian (4.21) is not covariant with respect to
boundary diffeomorphisms. However, the phase space associated with this choice
allows one to reproduce the results of the Chern–Simons formulation in section
3.2. Indeed, starting from (4.20), one reproduces exactly the charge
expressions (3.30) after imposing the conformal gauge condition (3.4) and
performing the redefinitions (3.11) and (3.16). In particular, with this
choice of counterterms one finds new symmetries compared to the FG gauge, as
explained in the CS section 3.

We see therefore that there is a trade-off: one can promote the Weyl
connection to a physical source at the expense of introducing a non-covariant
boundary Lagrangian. In this framework, we are reproducing the holographic
aspects explored in [30, 31], where it was proposed that the Weyl connection
should be treated as an independent source. In this setup, the extra current
should be added to the boundary partition function, and thus there will be
non-trivial correlation functions involving it. This is an open road to
explore further.

###  4.3 Corner counterterm

Interestingly, there is another procedure to extract the symplectic potential
from the variational principle [56], which yields the charges of section 3.2.
This procedure amounts to include corner contributions of the bulk symplectic
potential. Concretely the renormalized symplectic potential is

$\tilde{\Theta}_{ren}=\lim_{\rho\to
0}\left[\Theta^{\rho}_{EH}+\partial_{i}\int\text{d}\rho\,\Theta^{i}_{EH}+\delta
L_{b}\right]$ (4.23)

where $L_{b}$ are the boundary Lagrangians (that can include corner
Lagrangians). This differs from the prescriptions applied in sections 4.1 and
4.2 for which $C$ defined in eq. (A.20) is equal to minus the symplectic
potentials associated to the boundary Lagrangians.

Let us describe the choice of boundary and corner Lagrangians for the case of
interest. We add a finite corner term to the renormalized action (4.1):

$\tilde{S}_{ren}=S_{ren}+S_{C},\qquad
S_{C}=\int\text{d}^{2}x\,\partial_{i}L_{C}^{i}[h^{(0)}_{ij},k_{i}^{(0)}]$
(4.24)

where $L_{C}^{i}[h^{(0)}_{ij},k_{i}^{(0)}]$ is a corner Lagrangian involving
the boundary geometry. We take

$L_{C}^{i}=\lim_{\rho\to
0}\left[-\frac{1}{16\pi\mathcal{G}}\sqrt{-\gamma}\gamma^{ij}k_{j}\right]=-\frac{1}{16\pi\mathcal{G}}\sqrt{-h^{(0)}}h^{ij}_{(0)}k_{j}^{(0)}.$
(4.25)

This corner Lagrangian (4.25) has the advantage of being covariant with
respect to boundary diffeomorphisms. The corner contribution of the bulk
symplectic potential $\Theta_{EH}$ turns out to be $\delta$-exact,
$\lim_{\rho\to
0}\partial_{i}\Theta_{EH}^{i}=\delta\left(-\frac{1}{16\pi\mathcal{G}}\log\rho\sqrt{-h^{(0)}}\hat{R}^{(0)}\right)$,
precisely canceling the log-term contribution from the boundary Lagrangians.
Then the renormalized symplectic term is

$\begin{split}\tilde{\Theta}_{ren}&=\lim_{\rho\to
0}\bigg{[}\Theta^{\rho}_{EH}+\partial_{i}\int\text{d}\rho\,\Theta^{i}_{EH}+\frac{1}{8\pi\mathcal{G}}\delta\bigg{(}\sqrt{-\gamma}\,(K+1)+\frac{1}{2}\sqrt{-\gamma}\,k_{i}\gamma^{ij}k_{j}\\\
&\quad+\frac{\rho^{2}\log\rho}{2}\int\text{d}^{2}x\sqrt{-\gamma}\,\hat{R}^{(0)}-\frac{1}{2}\partial_{i}(\sqrt{-\gamma}\,\gamma^{ij}k_{i})\bigg{)}\bigg{]}\\\
&=\sqrt{-h^{(0)}}\left(-\frac{1}{2}\tilde{T}^{ij}\delta
h_{ij}^{(0)}+J^{i}\delta K_{i}^{(0)}\right),\end{split}$ (4.26)

where

$\displaystyle\tilde{T}^{ij}$
$\displaystyle=T^{ij}+\frac{1}{2}h^{ij}_{(0)}\hat{\nabla}_{k}^{(0)}J^{k}\,,\qquad
K_{i}^{(0)}=k^{(0)}_{i}-\frac{1}{2}\,\partial_{i}\ln\sqrt{-h^{(0)}}\,.$ (4.27)

These transform under the residual symmetries
$\xi^{\rho}=\rho\omega+\mathcal{O}(\rho^{3})$ and
$\xi^{i}=Y^{i}+\rho^{2}\zeta^{i}+\mathcal{O}(\rho^{4})$ as

$\displaystyle\delta_{(Y,\omega,\zeta)}K^{(0)}_{i}$
$\displaystyle=\mathcal{L}_{Y}K^{(0)}_{i}-2\,\zeta_{i}-\frac{1}{2}\partial_{i}\partial_{j}Y^{j}\,,$
(4.28)
$\displaystyle\begin{split}\delta_{(Y,\omega,\zeta)}\tilde{T}^{ij}&=\mathcal{L}_{Y}\tilde{T}^{ij}+4\,\omega\,\tilde{T}^{ij}-\frac{1}{8\pi\mathcal{G}}\bigg{(}\hat{\nabla}^{(i}_{(0)}\partial^{j)}\omega-\frac{1}{2}h^{ij}_{(0)}\hat{\nabla}^{k}_{(0)}\partial_{k}\omega+4k_{(0)}^{(i}\zeta^{i)}\\\
&\quad+h_{(0)}^{ij}\left(\hat{\nabla}_{k}^{(0)}\zeta^{k}-2k_{k}^{(0)}\zeta^{k}\right)\bigg{)}\,.\end{split}$
(4.29)

With this prescription, the holographic interpretation of (4.26) as VEV
$\times$ $\delta$(sources) leads to the identification of $h_{ij}^{(0)}$ and
$K_{i}^{(0)}$ (called Weyl source from now on) as sources. However, the latter
is not a Weyl connection. Rather, it has been dressed by a Weyl pure-gauge
shift, so that it is now Weyl invariant. It would be interesting to study the
holographic repercussions of this choice, where the two sources transform
independently of each other, similar to what happens when adding a $U(1)$
gauge field in the bulk – and indeed one could perhaps gain insights by
treating Einstein–Maxwell bulks with this analogy. In particular, the
transformation of the Weyl source is now reminiscent of a one-form symmetry,
and thus, from the dual perspective, there could be Wilson line associated to
it. Understanding this from the intrinsic boundary field-theoretically
perspective is a thrilling open question.

Repeating the same manipulations as above in conformal gauge, the phase space
associated with the symplectic potential (4.26) reproduces exactly the result
(3.30) of the charge in the first order formulation. Finally, since the
boundary terms in the action and the symplectic corner terms only contribute
to $\delta$-exact terms in the symplectic potential (4.26), the symplectic
current coincides precisely with the Einstein–Hilbert symplectic current.
Therefore, the charges obtained here and in the first order formulation
coincide with those obtained in the covariant phase space formalism, without
renormalization or any modification related to boundary terms in the action.

##  5 Conclusions

In this work, we have argued that the Weyl–Fefferman–Graham gauge is more
suitable than the standard Fefferman–Graham gauge for two reasons: $(i)$ the
Weyl rescalings at the boundary are induced by purely radial bulk
diffeomorphisms (2.26). This contrasts with the Weyl rescalings in
Fefferman–Graham gauge which necessarily involve a mixing with the transverse
components. As a consequence, each order in the radial expansion transforms
with a well-defined Weyl weight, see eq. (2.38). $(ii)$ The
Weyl–Fefferman–Graham gauge induces the full Weyl geometry (i.e. a conformal
class of metrics together with a Weyl connection) at the boundary. This allows
us to exploit the full Weyl covariance of the boundary theory. In particular,
the Weyl anomaly derived in (4.12) is Weyl covariant since it involves the
Weyl–Ricci scalar.

To obtain the WFG gauge, one has to relax the FG gauge and allow fluctuations
of $k_{i}(\rho,x)$ in (2.16) on the phase space. Relaxing this gauge condition
implies the appearance of a larger set of independent residual gauge
symmetries in the subleading order of the transverse bulk diffeomorphisms,
$\zeta^{\pm}$ in (3.17). The novelty of our work is to compute the charges
associated with the residual diffeomorphisms of the WFG gauge. We have shown
that, for a suitable choice of symplectic structure, these additional residual
symmetries are non vanishing and therefore carry physical information. These
results were discussed both in first order Chern–Simons formulation of
gravity, as well as in second order metric formulation. The new charge
associated to the Weyl connection combines with the Weyl charge to form a
Heisenberg algebra. These two charges are then purely corner charges
unconstrained by the equations of motion. The derivation of the symplectic
structure from the renormalized action has also been provided. These results
confirm previous works [42, 43, 46, 47, 48, 49, 52, 53, 54, 55] advocating
that, in presence of boundaries, the complete gauge-fixing might eliminate
potentially interesting physical degrees of freedom.

Investigating the implications of these additional physical symmetries for the
dual field theory is an interesting question for future endeavour. A first
glance revealed that the holographic interpretation turned out to be
polyvalent: either the Weyl connection is a new source, but there is a non-
covariant boundary Lagrangian, or the new source is a Weyl-invariant
combination of the Weyl connection and the metric, and the boundary Lagrangian
employed is covariant. We have commented both instances, and we in particular
noted that the Weyl connection and the Weyl-source transformation laws under
$H^{(0)}$ are reminiscent of a one-form symmetry. When the associated charge
is non-vanishing, then there are physical states at the boundary sensitive to
this operator, which can be interpreted as a Weyl Wilson line, and would be a
primer in the realization of higher form symmetries from bottom-up holography.
We refer e.g. to [48, 49, 54] for a holographic interpretation of symmetries
obtained beyond the complete gauge fixing.

While the analysis here has been performed in three dimensions of bulk, we
expect similar results to hold for higher dimensions. For instance, it would
be worth extending the systematic phase space analysis provided in [51] for
the WFG gauge. While in $3$-dimensional bulks the boundary is always
conformally flat and thus the full conformal isometries group is realized, in
higher dimensions there are situations in which the boundary is not
conformally flat (e.g. $3$-dimensional boundaries with non-vanishing Cotton
tensor), and thus the conformal isometries group is smaller. Nonetheless, the
dual theory enjoys Weyl covariance, because this symmetry arises solely from
the fact that the boundary sits at conformal infinity. This means that we are
dealing in these cases with a conformal field theory on curved spaces with
Weyl symmetry, i.e., a Weyl field theory. Extending our analysis to higher
dimensions could potentially shed light on these theories, of which little is
known.

As a general statement, we have shown that new charges might arise from
boundary Lagrangians. Therefore, this raises the important question of
classifying new charges associated to choices of symplectic spaces. As a
general guideline, one could argue that the more physical charges the better,
as this would lead to larger algebras that are more powerful to organize
observables of the theory. This specific example treated here is therefore
opening the door to a more fundamental problem, which is the classification of
charges steaming from partial gauge fixings. While this is an interesting
avenue in the theory of asymptotic symmetries, we foresee far-reaching
repercussions in both AdS and flat holography, yet to be unveiled.

#### Acknowledgements

We would like to thank Francesco Alessio, Glenn Barnich, Andrea Campoleoni,
Rob Leigh, Shahin Sheikh-Jabbari and James Weizhen for useful discussions.
Research at Perimeter Institute is supported in part by the Government of
Canada through the Department of Innovation, Science and Economic Development
Canada and by the Province of Ontario through the Ministry of Colleges and
Universities. AD is supported by the Fonds de la Recherche Scientifique - FNRS
under Grants No. FC.41161. RR is supported by the Austrian Science Fund (FWF),
project P 32581-N. RR thanks Perimeter Institute for its hospitality where the
last stages of this work were completed.

##  Appendix A Conventions

###  A.1 Chern–Simons formulation

In this appendix, we recall the salient features of the Chern–Simons (CS)
formulation of three-dimensional anti-de Sitter (AdS3) Einstein gravity. This
allows us to fix the notations used in main body of the paper. The isometry
algebra of AdS3 is the $\mathfrak{so}(2,2)$ algebra

$[M_{B},M_{C}]=\epsilon_{BCD}\,M^{D},\quad[M_{B},P_{C}]=\epsilon_{BCD}\,P^{D},\quad[P_{B},P_{C}]=\left(\frac{\mathcal{G}}{\ell}\right)^{2}\epsilon_{BCD}\,M^{D}\,,$
(A.1)

where $P_{B}$ and $M_{B}$ denote, respectively, the transvection and Lorentz
generators. The latter are related to the customary Lorentz generators as

$M_{B}=\frac{1}{2}\,\epsilon_{BCD}\,M^{CD}\,.$ (A.2)

In the above algebra, $\mathcal{G}$ is Newton’s constant and $\ell$ denotes
the AdS radius while we choose the convention $\epsilon^{012}=1$ for the Levi-
Civita symbol. Labelling by $\mu,\nu,\dots$ the bulk base-manifold indices and
introducing the $\mathfrak{so}(2,2)-$valued differential one-form

$\mathscr{A}=\left(\frac{1}{\mathcal{G}}{e_{\mu}}^{B}P_{B}+{\omega_{\mu}}^{B}M_{B}\right)\text{d}x^{\mu}\,,$
(A.3)

where ${e_{\mu}}^{B}$ is the bulk dreibein and ${\omega_{\mu}}^{B}$ is its
associated dualized spin connection, one can rewrite the $3-$dimensional
Einstein–Hilbert action as a CS action [17, 18]

$S_{EH}=\frac{1}{16\pi}\int_{\mathcal{M}}\text{Tr}\left(\mathscr{A}\wedge\text{d}\mathscr{A}+\frac{2}{3}\,\mathscr{A}\wedge\mathscr{A}\wedge\mathscr{A}\right).$
(A.4)

The operator d is the exterior derivative on the manifold
$\mathcal{M}=\text{AdS}_{3}$, such that $\text{d}^{2}=0$. In this expression
we introduced the Killing metric

$\text{Tr}\left(M_{B}M_{C}\right)=\text{Tr}\left(P_{B}P_{C}\right)=0\,,\qquad\text{Tr}\left(M_{B}P_{C}\right)=\eta_{BC}\,,$
(A.5)

with $\eta_{BC}$ the Minkowski metric whose signature is $(-,+,+)$, and used
the Cartan equation

$\text{d}e^{B}+\epsilon^{BCD}\,\omega_{C}\wedge e_{D}=0\,.$ (A.6)

For negative cosmological constant, $\Lambda=-\ell^{-2}<0$ (in the main body
of the paper we set $\ell=1$), one can take advantage of the isomorphism
$\mathfrak{so}(2,2)\cong\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,\mathbb{R})$
to rewrite (A.4) as

$S_{EH}=S_{CS}[A]-S_{CS}[\widetilde{A}]\,,$ (A.7)

with

$S_{CS}[A]=\frac{\kappa}{4\pi}\int_{\mathcal{M}}\text{tr}\left(A\wedge\text{d}A+\frac{2}{3}\,A\wedge
A\wedge A\right)$ (A.8)

where $\kappa=\nicefrac{{\ell}}{{4\mathcal{G}}}$, and a similar expression
holds for $\widetilde{A}$. In the latter we have introduced the
$\mathfrak{sl}(2,\mathbb{R})-$valued gauge connections $A=A^{B}J_{B}$, where
we introduced the $\mathfrak{sl}(2,\mathbb{R})$ generators

$[{J_{B}},{J_{C}}]={\epsilon_{BC}}^{D}\,J_{D}\,,\qquad\text{tr}(J_{B}J_{C})=\frac{1}{2}\,\eta_{BC}\,.$
(A.9)

In terms of the $\mathfrak{so}(2,2)$ generators, they can be written as

${J_{B}}=\frac{1}{2}\left(M_{B}+\frac{\ell}{\mathcal{G}}\,P_{B}\right).$
(A.10)

The dreibein and the spin connection are related to the CS forms $A$ and
$\widetilde{A}$ via

$A^{B}=\omega^{B}+\frac{1}{\ell}\,e^{B}\,,\qquad\widetilde{A}^{B}=\omega^{B}-\frac{1}{\ell}\,e^{B}\,.$
(A.11)

These are the main quantities that we scrupulously analyze in the paper.

###  A.2 Covariant phase space formalism

In section 3, we determined the solution space, the asymptotic symmetries and
the associated corner charges in the CS formulation. To do this, we could have
used methods from the Hamiltonian approach as in, e.g., [34, 99, 100, 19, 20,
101, 21, 102, 103, 95]. Inspired by the metric formulation, we instead used
the covariant phase space formalism. The latter was introduced in [104, 105,
106], refined in [94, 36, 38, 8]111111See, e.g., [96, 77, 98] for pedagogical
reviews., and it focuses on the Lagrangian approach. Its main idea is to put
together spacetime and phase space.

The differentiable manifold that we consider is the three-dimensional AdS
spacetime, $\mathcal{M}=\text{AdS}_{3}$. In appendix A.1, we have introduced
the exterior derivative, d, on this manifold. We denote $\iota$ its interior
product. The Lie derivative along the flow of a diffeomorphism $\xi\in
T\mathcal{M}$ is

$\mathcal{L}_{\xi}=\text{d}\iota_{\xi}+\iota_{\xi}\text{d}\,.$ (A.12)

One can put together this calculus with the one on field space $\Gamma$. We
denote by $\delta$ and $I$ the exterior derivative and the interior product on
the latter, respectively, with $\delta^{2}=0$. Given a vector field $V\in
T\Gamma$, we analogously have

$\mathfrak{L}_{V}=\delta I_{V}+I_{V}\delta\,.$ (A.13)

The CS Lagrangian form can be derived from its associated action principle
(A.8),

$S_{CS}=\int_{\mathcal{M}}L\,,\qquad
L=\frac{\kappa}{4\pi}\,\text{tr}\left(A\wedge\text{d}A+\frac{2}{3}\,A\wedge
A\wedge A\right).$ (A.14)

An arbitrary field variation $A\to A+\delta A$ of the latter yields, after
iterative applications of the inverse Leibniz rule,

$\delta L=(\text{eom})\delta A+\text{d}\Theta\,,$ (A.15)

where $(\text{eom})$ denotes the equations of motion of the theory

$(\text{eom})=\text{d}A+A\wedge A\approx 0\,,$ (A.16)

and the symbol $\approx$ indicates that we are on-shell of the equations of
motion. Here, $\Theta$ is the local symplectic potential form

$\Theta=-\frac{\kappa}{4\pi}\,\text{tr}\left(A\wedge\delta A\right)\,.$ (A.17)

We define the local symplectic two-form as

$\omega=\delta\Theta=-\frac{\kappa}{4\pi}\,\text{tr}\left(\delta A\wedge\delta
A\right)\,.$ (A.18)

This local expression can be integrated on an arbitrary Cauchy slice
$\Sigma\subset\mathcal{M}$ to give the symplectic two-form

$\Omega=\int_{\Sigma}\omega\,.$ (A.19)

In particular, we consider that the $3-$manifold
$\mathcal{M}=\mathbb{R}\times\Sigma$ has the topology of a cylinder, foliated
by the spatial slices, such that the boundary of the $2-$manifold $\Sigma$ is
a circle, $\partial\Sigma~{}=~{}S^{1}$.

Specifying a bulk Lagrangian is not enough to completely determine the theory.
Indeed, different choices of boundary Lagrangians give different associated
charges. In the following, we keep using the language of ambiguities, for
historical reasons, with the understanding that different choices of the
latter truly imply different physical theories, as indeed our holographic
analysis reveals. The symplectic potential possesses two types of ambiguities
in its definition (A.15), which do not affect the equations of motion,

$\Theta\to\Theta+\delta B-\text{d}C\,.$ (A.20)

Adding a boundary term to the bulk Lagrangian $L\to L+\text{d}B$ leads to the
first type of ambiguity ($B$) whose contribution to the symplectic form
$\omega$ vanishes since $\delta^{2}=0$. The nilpotent aspect and the
definition of $\Theta$ as a boundary term in $\delta L$ yields the second type
of ambiguity ($C$). It is worth noting that the latter alters $\omega$,

$\omega\to\omega-\delta\text{d}C=\omega+\text{d}\delta
C=:\omega+\text{d}\omega_{C}\,,$ (A.21)

reflecting our ignorance on how to select the boundary terms $\omega_{C}$,
called corner terms, in the symplectic form. This is related to the corner
proposal [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 56]. Ambiguities can
be used to renormalize the symplectic potential whenever the latter diverges
as one approaches the boundary [107, 92, 75, 108, 109, 71, 76, 110, 111, 112,
73, 113, 114, 56, 115], and can be further used to restore integrability [116,
53]. It turns out that such a renormalization procedure is always possible at
the level of the symplectic potential as recently shown in [111, 114, 56].

In the main text, we are using two different prescriptions to compute the
charges. The first one is due to Compère and Marolf [71]. It takes $B$ to be
the boundary Lagrangians that are used for the renormalization of the action
and $C$ to be minus the boundary symplectic potential associated to B. The
other prescription [56] also takes the same $B$ but $C$ to be the corner
contributions of the bulk symplectic potential, and hence solely relies on the
bulk symplectic potential to compute the charges.

In the presence of gauge symmetries, which is the case we are interested in,
Noether’s second theorem applies, which associates a codimension-2 conserved
quantity to a given local symmetry. The gauge symmetry of the CS field is

$\delta_{\lambda}A=I_{V_{\lambda}}\delta A=\text{d}\lambda+[A,\lambda]\,.$
(A.22)

Applying Noether’s second theorem to this symmetry

$I_{V_{\lambda}}\Omega=-\delta Q_{\lambda}\,,$ (A.23)

one can obtain the associated on-shell corner charges

$\delta Q_{\lambda}\approx\int_{S^{1}}\delta q_{\lambda}\,,\qquad\delta
q_{\lambda}=-\frac{\kappa}{2\pi}\,\text{tr}\left(\lambda\,\delta A\right).$
(A.24)

In three dimensions, we can always find integrable slicings for the charges
[47, 50, 117, 52, 53, 118]. This is related to the fact that, in this case,
there is no propagation of local degrees of freedom.

##  Appendix B Second copy

In this appendix, we collect the main results of the second copy of the CS
fields, since in section 3 we exclusively focus on the first copy in order to
compute the asymptotic symmetries and the charges. The bulk CS connections
using our boundary conditions are

$\displaystyle\widetilde{A}_{\rho}$
$\displaystyle=\frac{1}{\rho}L_{0}+2\sqrt{2}\rho^{2}\text{e}^{-\phi}\left(k_{-}^{(2)}L_{1}-k_{+}^{(2)}L_{-1}\right)+\mathcal{O}(\rho^{3})\,,$
(B.1a) $\displaystyle\widetilde{A}_{-}$
$\displaystyle=-\frac{1}{\rho}\sqrt{2}\text{e}^{\phi}L_{-1}-\left(2k_{-}^{(0)}-\partial_{-}\phi\right)L_{0}-\sqrt{2}\rho\text{e}^{-\phi}h_{--}^{(2)}L_{1}-2\rho^{2}k_{-}^{(2)}L_{0}+\mathcal{O}(\rho^{3})\,,$
(B.1b) $\displaystyle\widetilde{A}_{+}$ $\displaystyle=-\partial_{+}\phi
L_{0}+\sqrt{2}\rho\text{e}^{-\phi}\partial_{+}K_{-}^{(0)}L_{1}+\mathcal{O}(\rho^{3})\,.$
(B.1c)

If we expand the gauge parameters of the second copy as

$\widetilde{\lambda}(\rho,x^{+},x^{-})=\sum^{+1}_{B=-1}\widetilde{\epsilon}^{B}(\rho,x^{+},x^{-})L_{B}\,,$
(B.2)

one obtains that

$\displaystyle\widetilde{\epsilon}^{-1}$
$\displaystyle=-\frac{\sqrt{2}}{\rho}\text{e}^{\phi}Y^{-}+\mathcal{O}(\rho^{3})\,,$
(B.3a) $\displaystyle\widetilde{\epsilon}^{1}$
$\displaystyle=\frac{\rho}{\sqrt{2}}\text{e}^{-\phi}\Big{[}\partial_{-}^{2}Y^{-}+2H_{-}^{(0)}-2\Big{(}\ell_{-}Y^{-}-(K_{-}^{(0)})^{2}Y^{-}-K_{-}^{(0)}\partial_{-}Y^{-}\Big{)}\Big{]}+\mathcal{O}(\rho^{3})\,,$
(B.3b) $\displaystyle\widetilde{\epsilon}^{0}$
$\displaystyle=-\sigma+\partial_{-}Y^{-}-2Y^{-}K_{-}^{(0)}\,.$ (B.3c)

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