11institutetext: Laurent Freidel 22institutetext: Perimeter Institute for
Theoretical Physics
31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5
22email<EMAIL_ADDRESS>33institutetext: Marc Geiller
44institutetext: ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France
44email<EMAIL_ADDRESS>55institutetext: Wolfgang Wieland
66institutetext: Institute for Quantum Gravity, Theoretical Physics III
Friedrich-Alexander Universität Erlangen-Nürnberg
Staudtstrasse 7/B2, 91058 Erlangen, Germany
Institute for Quantum Optics and Quantum Information (IQOQI)
Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna Austria
77institutetext: Vienna Center for Quantum Science and Technology (VCQ)
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna,
Austria
77email<EMAIL_ADDRESS>
88institutetext: This is a preprint of a chapter to appear in the “Handbook of
Quantum Gravity”, edited by Cosimo Bambi, Leonardo Modesto and Ilya Shapiro,
2023, Springer, reproduced with permission of Springer.

# Corner symmetry and quantum geometry

Laurent Freidel  Marc Geiller  Wolfgang Wieland

###### Abstract

By virtue of the Noether theorems, the vast gauge redundancy of general
relativity provides us with a rich algebra of boundary charges that generate
physical symmetries. These charges are located at codimension-2 entangling
surfaces called corners. The presence of non-trivial corner symmetries
associated with any entangling cut provides stringent constraints on the
theory’s mathematical structure and a guide through quantization. This report
reviews new and recent results for non-perturbative quantum gravity, which are
natural consequences of this structure. First, we establish that the corner
symmetry derived from the gauge principle encodes quantum entanglement across
internal boundaries. We also explain how the quantum representation of the
corner symmetry algebra provides us with a notion of quantum geometry. We then
focus our discussion on the first-order formulation of gravity and show how
many results obtained in the continuum connect naturally with previous results
in loop quantum gravity. In particular, we show that it is possible to get,
purely from quantization and without discretization, an area operator with
discrete spectrum, which is covariant under local Lorentz symmetry. We
emphasize that while loop gravity correctly captures some of the gravitational
quantum numbers, it does not capture all of them, which points towards
important directions for future developments. Finally, we discuss the
understanding of the gravitational dynamics along null surfaces as a
conservation of symmetry charges associated with a Carrollian fluid.

## Keywords

Boundary degrees of freedom, holography, Loop Quantum Gravity, Noether
charges, symmetry algebras, corner symmetries, Carrollian geometry,
representation theory of infinite-dimensional groups.

##  1 Introduction

Quantum gravity is one of our time’s most captivating and challenging
theoretical puzzles. Many different top-down approaches have been proposed to
address it, such as111This is by no means a complete list. See for example
deBoer:2022zka . String Theory, AdS/CFT Holography222Celestial holography can
also be understood as a bottom-up perspective on quantum gravity that is
perturbative in nature., and Loop Quantum Gravity (LQG hereafter). Despite
many successes, these approaches are based on fundamentally different
foundational principles and seem irreconcilable. What is missing, in our view,
is a bottom-up non-perturbative perspective that provides, from our
understanding of semi-classical gravity and its symmetries, solid foundations
and tests that any top-down approach to quantum gravity ought to pass. It
includes one of the foundational elements of holography, namely the fact that
the Hamiltonian of a gravitational system with fixed boundary conditions is
entirely contained on its boundary. It also respects one of the fundamental
tenants of LQG, namely background independence, and the foundations of the
equivalence principle, which implies that gravitational dynamics is the
expression of its symmetry. The local holography approach views both
holography and symmetry as two sides of the same coin. In this review, we
focus on presenting the connection of local holography with LQG and some of
its results such as the appearance of quantum geometry, leaving aside a more
general presentation of its link with traditional holography.

In particular, we show that gauge symmetry, which is usually understood as a
mere redundancy of our physical description, is much more than that. It
governs the quantum entanglement of subregions across entangling surfaces,
called corners. These entangling surfaces are shown to carry representations
of infinite-dimensional symmetry groups that are the physical consequence of
the underlying gauge invariance. Moreover, we show that these symmetries
represent, through their non-trivial commutation relations, the elements of
quantum geometry.

This perspective is very close to some of LQG’s most noticeable technical
achievements, in particular, the rigorous construction of a Hilbert space of
quantum Riemannian geometry based on a Hamiltonian formulation of general
relativity in terms of $\mathrm{SU}(2)$ Yang–Mills gauge connection variables.
In this context, symmetries such as spatial diffeomorphisms and local
$\mathrm{SU}(2)$ frame rotations were central for defining the physical
states. The emphasis on SU$(2)$ gauge symmetry led to the quantum numbers
labeling the kinematical spin network states Haggard:2023tnj .

Since it was initially conceived as a canonical quantization of general
relativity in the Hamiltonian formulation, LQG is often seen as a bulk
quantization. Little to no role is played by boundary structures. One goal of
the present review chapter is to explain why this is not so and how LQG
accommodates the description of some boundary degrees of freedom in its
foundations. For example, one of the key results of LQG is that spin networks
represent a notion of quantum geometry. Suppose we cut open a portion of
quantum geometry along a surface $S$. In that case, we obtain, for each
intersection between the spin network and the surface $S$, an $\mathrm{SU}(2)$
representation, which carries an elementary quantum of geometry. Moreover, the
$\mathrm{SU}(2)$ generators associated with each cut are the quantum flux
operators integrated along $S$. These quanta, which are fundamentally
entangled and represent quantum geometry, are controlled by gauge symmetry.
One limitation of the traditional LQG approach is that it assumes a
discretization of the fundamental geometrical excitations. Local holography
allows us to keep some of the key successes of LQG, such as the quantization
of the area spectra while being fully compatible with the standard
understanding of QFT in the continuum. The presentation will be done in four
main parts.

In section 2, we first explain how the notion of boundary degrees of freedom
arises in gauge theories, and highlight the fundamental role that it plays in
classical and quantum gravity. We develop these ideas further in section 3 by
explaining how boundary symmetries are related to entanglement and how their
quantization gives rise to a notion of quantum geometry. This then motivates
us to summarize the role played by boundary degrees of freedom in LQG. We
recall in section 4 various results establishing that spin network data can be
seen as boundary data and also summarize earlier results on boundaries and
entanglement in LQG. We then review recent results on the construction of
corner charges in the first-order formulation of gravity in the presence of
the Barbero–Immirzi parameter. And give the proof of a Lorentz covariant area
operator with discrete spectra. This section also reveals our shared viewpoint
on future developments. It shows how the non-perturbative quantum gravity
perspective started by LQG should be extended in order to fully provide a
representation of the boundary symmetries compatible with subsystem
decomposition and coarse-graining. Finally, in section 5, we review our
current understanding of the representation of supertranslation symmetry,
emphasizing the gravitational dynamics along null boundaries. We present our
understanding of such dynamics as Carrollian charge conservation equations. We
also construct a boundary action for finite null boundaries that encodes
radiation and is compatible with discrete area spectra.

##  2 Gauge theories in bounded regions

###  2.1 Generalized Noether charges and boundary symmetries

In the case of gauge symmetries, Noether’s first and second theorems together
imply that charges are associated to codimension-2 boundaries. More precisely,
to local gauge symmetries with parameter $\varepsilon$ one can assign the
object

$\displaystyle
Q(\varepsilon)=\int_{\Sigma}C(\varepsilon)+\oint_{S}q(\varepsilon),$ (1)

where $C(\varepsilon)\hat{\;=\;}0$ is the constraint and $q(\varepsilon)$ the
so-called _charge aspect_. This result implies that the charges of local
symmetries supported inside a region $\Sigma$ are given by an integral over
its boundary $S=\partial\Sigma$. By definition, what then distinguishes a
gauge transformation from a true physical symmetry is the possible non-
vanishing value of the charge. We therefore see that in the presence of a
boundary, whose intersection with $\Sigma$ is the corner $S$, gauge
transformations can potentially become physical symmetries associated with a
non-vanishing charge. Setting aside technical issue for the moment, one can
think of the covariant phase space formalism as a method for assigning the
charge (1) to an infinitesimal transformation $\delta_{\varepsilon}$.

Such surface charges are of fundamental importance as they describe essential
physical observables. This is the case of the ADM mass, the angular momentum,
the memory observables, the multipole moments, and for example the LQG fluxes.
Moreover, the charges come equipped with certain algebraic structures: they
form a local symmetry algebra, and their time development along the boundary
can give rise to so-called flux/balance laws, describing for example the
interplay between gravitational radiation and the failure of the charges to be
conserved. It is indeed important to appreciate that the physically meaningful
surface charges go beyond what one can be used to when studying Noether
charges in mechanics. The charges can be non-conserved, and also defined
without actually resorting to boundary conditions, i.e. by relaxing the need
for a variational principle. Moreover, the charges can also fail to be
integrable in the variational sense.

For example, in the case of a diffeomorphism with parameter $\xi$, the charge
takes the form

$\displaystyle I_{\xi}\Omega=\not{\delta}Q(\xi)=\delta
Q_{\text{N}}(\xi)-\oint_{S}i_{\xi}\theta,$ (2)

where $\Omega$ is the symplectic structure, obtained as the integral over
$\Sigma$ of the variation $\delta\theta$ of the symplectic potential $\theta$.
This latter appears as the boundary term of the variation of the Lagrangian,
i.e. $\delta L\hat{\;=\;}\mathrm{d}\theta$. Finally, here $Q_{\text{N}}(\xi)$
stands for the Noether charge associated with the Lie derivative
$\pounds_{\xi}$. Importantly, this formula shows that the full surface charge
is not only given by the Noether piece, but can contain a non-integrable flux
contribution $F_{N}(\xi)=\oint_{S}i_{\xi}\theta$. More generally, the charge
variations take the form

$\displaystyle I_{\varepsilon}\Omega=\delta Q(\varepsilon)+F(\varepsilon),$
(3)

where the right-hand side is a (non-unique) split between an integrable piece
$Q(\varepsilon)$ and a flux contribution $F(\varepsilon)$. We leave aside
technical discussions on how to select a preferred split of the charge between
integrable and non-integrable parts, but a canonical choice is the Noetherian
split, $Q(\varepsilon)=Q_{\text{N}}(\varepsilon)$, which is entirely
determined by the choice of boundary Lagrangian Freidel:2020xyx ;
Freidel:2021cjp . What is important is that a repeated symmetry action on a
charge leads to the equation

$\displaystyle\delta_{\varepsilon_{2}}Q(\varepsilon_{1})$
$\displaystyle=Q([\varepsilon_{1},\varepsilon_{2}])+I_{\varepsilon_{1}}F(\varepsilon_{2})$
(4) (5) evolution $\displaystyle=\text{conservation}+\text{dissipation}.$

As indicated, this can be thought of as an evolution generated by the
symmetry, which is composed of a rotation dictated by the symmetry algebra,
and a possible dissipation due to the flux term. This latter is typical of
open Hamiltonian systems, in which modes can escape e.g. through the null
boundary in the form of gravitational radiation. An important point of this
construction explained in details for the first time in Donnelly:2016auv  is
that once we take into account the presence of edge mode fields, it is
possible to show that the symmetry charges are gauge invariant. They commute
with the Hamiltonian constraints. In Gravity, the edge mode fields are given
by the choice of embedding maps from a reference ball to the spacetime
Freidel:2021dxw ; Ciambelli:2021nmv . For simplicity, we do not introduce them
in our presentation even if they play an important conceptual role.

This structure suggests that there is both kinematical and dynamical
information in the charges and their algebra. Kinematical charges are
integrable and associated with a vanishing flux, while the dynamics amounts to
restoring the flux and taking into account the radiative modes. The
kinematical charges belong to the so-called _corner symmetry group_ , which is
a subgroup $G_{0}^{S}\subset G^{S}_{\text{ext}}$ of the full symmetry group.
The corner symmetry charges form a closed algebra
$\\{Q_{0}(\varepsilon_{1}),Q_{0}(\varepsilon_{2})\\}=Q_{0}([\varepsilon_{1},\varepsilon_{2}])$
which can be readily quantized. The full symmetry group is called the
_extended corner symmetry group_. The extended corner symmetry charges can
only be understood as canonical symmetry charges after extending the
gravitational phase space with an edge mode field called the embedding field
Freidel:2021dxw ; Ciambelli:2021nmv . More recent work Kabel:2022efn  connects
this research to the geometry of open systems, dissipation and entropy
production via the framework of metriplectic geometry. At the quantum level,
physical observables should provide representations of $G_{0}^{S}$ and
$G^{S}_{\mathrm{ext}}$. This is a necessary condition in order to have a
quantization compatible with the symmetries of the underlying classical
theory. In general relativity, as we will show below, representing the corner
charge algebra amounts to a quantization of geometry, while representing the
full dynamical charge algebra is expected to provide a notion of quantum
dynamics. For a demonstration of this in the context of asymptotic infinity
see Freidel:2021qpz .

What we have left out of the discussion so far is an explanation of the
zoology of charges and charge algebras which can arise. This depends of course
on $i)$ the theory being considered (e.g. gravity or Maxwell), $ii)$ the
symmetry transformations being studied (e.g. diffeomorphisms or internal
Lorentz transformations), $iii)$ the type of boundary conditions (e.g.
conservative or radiative boundary conditions), $iv)$ the nature of the
boundary (e.g. an asymptotic boundary or an entangling surface at finite
distance), $iv)$ The choice of boundary Lagrangian or topological terms such
as theta terms. Recent years have seen a wealth of developments and new
results on the covariant phase space which has fueled lots of the developments
we present in this survey. In addition to the references Donnelly:2016auv ;
Freidel:2020xyx ; Freidel:2021cjp ; Freidel:2021dxw ; Ciambelli:2021nmv  cited
above the interested reader should consult Oliveri:2019gvm ;
Chandrasekaran:2020wwn ; Chandrasekaran:2021vyu ; Margalef-Bentabol:2020teu ;
Odak:2022ndm ; Wieland:2021eth . Moreover, the recent report view
Ciambelli:2022vot  contains a comprehensive review of the ideas we presented
here and many more references.

In order to make contact with LQG, we will focus below on gravity in tetrad
variables, and consider the corner symmetries and charges which can be derived
for any quasi-local region (i.e. a boundary at finite distance) and without
the need for boundary conditions. An immediate example of such a charge is
that obtained from (2) in the case of tangential vector fields, for which the
flux term vanishes.333One should distinguish between symplectic flux, which
can exist at a cut, from the flux due to time evolution. In the case of
tangential vector fields the former is absent, but of course there is still a
non-trivial flux of e.g. angular momentum due to its time evolution along
$\mathcal{I}$. There is therefore a translation flux but no rotation flux. On
equation (4), one can indeed see that even if $\varepsilon_{1}$ is tangent,
then there is still a non-trivial flux due to time evolution if
$\varepsilon_{2}$ is an infinitesimal translation in the $u$ direction.

###  2.2 Implications for classical and quantum gravity

The role of symmetries in the study of a classical and quantum theory cannot
be understated. Provided enough conserved quantities exist in a theory, they
can be used to solve the dynamics altogether. Although we do not expect this
to be the case in gravity, it is still the case that symmetries provide a
powerful organizational tool, and give the possibility of controlling the
quantization of geometry in terms of symmetry observables.

In the case of asymptotic symmetries for asymptotically-flat spacetimes, the
relationship between corner symmetries (described by the so-called BMS group
and its generalizations Bondi ; Sachs ), classical observables, and the
structure of the perturbative quantum theory, has already been demonstrated,
and is being used to build a notion of flat space holography (see
Strominger:2017zoo ; Raclariu:2021zjz ; Pasterski:2021rjz  and references
therein). This idea of building the quantum theory on the structure of the
classical symmetries is also what gave rise to the holographic AdS/CFT
correspondence. It is clear that the very existence of boundary charges and
symmetries in the classical theory should play a central role in any attempt
to quantize gravity and understand what form of holography holds in quantum
gravity.

##  3 Quantum geometry and corner symmetry

The viewpoint just presented helps understand the fundamental tension which
exists between gravity and local quantum field theory. In any gauge theory,
the presence of constraints implies that gauge-invariant observables are
necessarily non-local. This means, in turn, that the Hilbert space does not
factorize spatially, as factorization would leave out of the sub-factors some
of the non-local observables which cut across the entangling surface (see
Buividovich:2008gq ; Casini:2014aia ; Donnelly:2011hn ; Donnelly:2014gva ;
Donnelly:2015hxa ; Geiller:2019bti ; Gomes:2018dxs  for many discussions on
this issue in gauge theory).

###  3.1 Entanglement from symmetry

Let us now explain more in details, why corner symmetries are essential to
unravel the nature of spacetime entanglement and lead to a covariant
description of quantum spacetime that generalises the one given by LQG. The
key idea proposed in Donnelly:2016auv  is that corner symmetry allows us to
resolve the quantum entanglement of spacetime through symmetries derived from
the gauge principle. This correspondence between symmetry and entanglement was
first realised in the context of lattice gauge theory in Donnelly:2014gva .

Let us start with an example in standard field theory and let us focus on the
vacuum state $|0\rangle$. This state is an eigen state of the Hamiltonian $H$
but also of the boost operator $K$ that fixes the 2-dimensional plane
$S=\\{x^{3}=0\\}$ inside the $x^{0}=\mathrm{const}.$ Cauchy slice. In other
words we have

$\displaystyle H|0\rangle=K|0\rangle=0.$ (6)

The plane $S$ fixed by $K$ is called the _entangling surface_. It divides the
slice into a left and a right component $\Sigma=\Sigma_{L}\cup\Sigma_{R}$ with
$S=\partial\Sigma_{L/R}$. One can decompose the Hilbert space as a product of
left (resp. right) Hilbert spaces444Strictly speaking this decomposition is
only valid at the level of the observable algebras due to UV divergences. See
Witten:2018zxz  for the proper description. For simplicity, we keep the naive
language of Hilbert space factorisation, valid for a UV-regulated QFT.
obtained by acting on the vacuum by local operators supported on $\Sigma_{L}$
(resp. $\Sigma_{R}$). Tracing out the left Hilbert space one obtain the
density matrix $\rho_{S}=\mathrm{Tr}_{H_{L}}(|0\rangle\langle 0|)$.

Since the entangling cut is fixed by $K$, and the slice is defined at $t=0$,
we can decompose the total boost operator into a local sum $K=K_{R}-K_{L}$,
where $K_{R/L}=\int_{\Sigma_{R/L}}|x^{3}|T_{00}$ is the right/left boost
operator, also called the “twist operator”, which commutes with local
operators placed on the left side. $K_{R/L}$ is a positive operator, and it
acts non trivially on the vacua.This means that we can decompose the density
matrix in terms of eigenstates $|n\rangle$ of the “twist operator” $K_{R}$
with eigenvalue $K_{n}>0$.

We can now use these symmetry generators to construct the _symmetry-reduced_
state which are defined as the projection of $\rho_{S}$ onto the eigenspaces
of $K_{R}$. The main theorem of Unruh–Bisognano–Wichmann Bisognano:1976za
states that the symmetry-reduced vacuum is pure! The second statement of this
theorem is that this state is simply given by a complex boost associated with
an imaginary boost angle of $2i\pi$. Let us recall that
U$(\tau)=e^{\frac{i}{\hslash}\tau K_{R}}$ denotes the action of the boost
operator with boost angle $\tau$ on states supported on $\Sigma_{R}$. The
Unruh–Bisognano–Wichmann result therefore simply states that

$\displaystyle\rho_{S}=e^{-\frac{2\pi}{\hslash}K_{R}}.$ (7)

This means that the vacuum state can be decomposed as

$\displaystyle|0\rangle=\sum_{n}e^{-\frac{\pi}{\hslash}K_{n}}|n,\tilde{n}\rangle,$
(8)

where $|n,\tilde{m}\rangle$ denotes an eigen state of $K_{R}$ (resp. $K_{L}$)
with eigenvalue $K_{n}$ (resp. $K_{m}$). The lesson we learn from this example
is that the vacuum entanglement is entirely contained from the matching of
boost charges $K_{R}=K_{L}$ across the entangling surface. This is the
expression of boost invariance which implies that the state can be written as
$\sum_{n}\rho_{n}|n,\tilde{n}\rangle$ in term of a measure $\rho_{n}$. We also
learn that the form for this measure is determined by the demand of
translation invariance of the state.

So far we have reviewed a well-known result about quantum field theory (QFT)
in the presence of an entangling surface. How does this relate to quantum
gravity and corner symmetries? In QFT we saw that the only state to which we
can apply the previous symmetry argument to compute entanglement is the vacuum
state for which $K|0\rangle=0=H|0\rangle$. In gravity we know, as reviewed in
the previous section, that on a closed Cauchy surface the Hamiltonian
generator $Q(\xi)$ vanishes for _any_ vector field $\xi$.555A weaker statement
would be that $\Sigma$ has an asymptotic boundary but $\xi$ vanishes
sufficiently fast near infinity. At the quantum level this means that
$Q(\xi)|\Psi\rangle=0$ for _all_ quantum states of quantum gravity. Let us now
choose a 2-dimensional finite closed surface $S\in\Sigma$ which plays the role
of an entangling surface and denote $\Sigma_{L}$ the surface inside $S$ and
$\Sigma_{R}$ the slice outside $S$. Let us then consider the set of vector
fields $\xi$ which do not move $S$. We can then decompose the diffeomorphism
charge in terms of its left and right components
$Q(\xi)=Q_{R}(\xi)-Q_{L}(\xi)$. As we have seen in the previous section,
$Q(\xi)$ is the integral of the Hamiltonian constraint on $\Sigma$ while the
charges $Q_{R}(\xi)$ are given by corner integrals and the corner symmetry
algebra $\mathfrak{g}^{S}$ is the sub-algebra of infinitesimal diffeomorphisms
fixing $S$ that possess a non vanishing charge corner charge
$Q_{R}(\xi)=\oint_{S}q_{\xi}$. We are thus in a situation which is analogous
to the Unruh case. We expect that the physical Hilbert space splits into a sum
of representations $\rho$ of the corner symmetry group. In other words, given
the existence of a corner symmetry algebra, the quantum gravity states $\Psi$
can be decomposed into a sum

$\displaystyle\Psi=\sum_{\rho}\psi_{\rho}(S)\sum_{a}|\rho,a\rangle\otimes|\rho^{*},a\rangle.$
(9)

Here $\rho*$ denotes the dual representation and $a$ denotes a label of states
in the Hilbert space $V_{\rho}$. This decomposition can be extended to more
than one surfaces $S\to(S_{1},\cdots S_{n})$ and more generally to a
collections of cuts $\Gamma$ that represents a 2-dimensional cellular complex.
The previous calculation of charges as corner charges generalises to this
case. On the 2-dimensional cell of this complex we assign representations of
the corner symmetry group and on the 1-dimensional cells which glue the
2-cells we assign intertwiners of the corner symmetry algebra. At the end of
the day, we can therefore expect a decomposition of the quantum gravity state
as a sum

$\displaystyle|\Psi\rangle=\sum_{\rho_{\Gamma}}\psi_{\rho_{\Gamma}}(\Gamma)\sum_{a_{\Gamma}}|\rho_{\Gamma},a_{\Gamma}\rangle\otimes|\rho_{\Gamma}^{*},a_{\Gamma}\rangle.$
(10)

Here $\rho_{\Gamma}=(\otimes_{f}\rho_{f},\otimes_{e}I_{e})$, where $f$ labels
the faces of $\Gamma$ with representations $\rho_{f}$, and $e$ labels the
edges of the complex with intertwiners $I_{e}$. This decomposition must be
valid for _any_ quantum gravity state.

This decomposition is reminiscent of the spin network decomposition of quantum
gravity states that appears in loop quantum gravity666See Ashtekar:2004eh  or
the chapter Haggard:2023tnj  in this volume., but it is also more general. The
difference is that it appears in a continuum description and the group which
one needs to represent is the full corner symmetry group (and not only the
SU$(2)$ group of internal frame rotations as in LQG). As we are about to see,
in the corner symmetry proposal, the quantization of geometry appears through
the choice of representation and through quantization of the spectra of the
geometric operators that represents the corner symmetry charges. The central
questions of quantum gravity now become, in this context: what are the
representations which enter the decomposition of the quantum gravity states,
and what is the integration measure $\Psi_{\rho}$ which plays the role of the
Unruh measure $e^{\frac{\pi K_{n}}{\hslash}}$ for quantum gravitational
states?

###  3.2 Representation theory of corner symmetries

In metric gravity it has been shown that the corner symmetry group777The
extended corner symmetry group is obtained by adding the super-translations
along the two normal directions, and is given by Donnelly:2016auv ;
Ciambelli:2021vnn  $\displaystyle
G_{\mathrm{metric},\mathrm{ext}}^{S}=\left(\mathrm{Diff}(S)\ltimes\mathrm{SL}(2,\mathbb{R})^{S}\right)\ltimes\mathbb{R}^{2S}.$
(11) is simply the semi-direct product Donnelly:2016auv
$G_{\mathrm{metric}}^{S}=\mathrm{Diff}(S)\ltimes\mathrm{SL}(2,\mathbb{R})^{S},$
where $\mathrm{SL}(2,\mathbb{R})^{S}$ denotes the space of functions from the
corner $S$ to SL$(2,\mathbb{R})$. These represent the boost transformations
preserving $S$, while Diff$(S)$ are the sphere diffeomorphisms. In the first
order tetrad (or frame) formalism, the symmetry group at the corner is
extended by SL$(2,\mathbb{C})^{S}$, namely the group of internal local Lorentz
transformations supported on the sphere Freidel:2020svx . In the tetrad
formulation the corner symmetry group is therefore

$\displaystyle
G_{\mathrm{frame}}^{S}=\mathrm{Diff}(S)\ltimes\big{(}\mathrm{SL}(2,\mathbb{R})^{S}\times\mathrm{SL}(2,\mathbb{C})^{S}\big{)}.$
(12)

The study of the representations of these groups has been initiated in the
work Donnelly:2020xgu  and developed further in Freidel:2020svx ;
Freidel:2020ayo ; Donnelly:2022kfs ; Ciambelli:2022cfr . To connect this
general discussion to the traditional discussion in LQG, one can go back to
the work Freidel:2019ees , which proposed the study of the loop corner
symmetry group

$\displaystyle
G^{S}_{\mathrm{LQG}}=\mathrm{Diff}(S)\ltimes\mathrm{SU}(2)^{S},$ (13)

obtained in the study of the canonical formulation of gravity in the time
gauge. This is the subgroup of $G^{S}_{\mathrm{frame}}$ preserving the slice
normal at $S$. In this context, the gravitational phase space is described in
the bulk in terms of the SU$(2)$ valued Ashtekar–Barbero connection $A^{i}$,
and in terms of the 2-form flux element
$E_{i}=\frac{1}{2}\epsilon_{ijk}(e^{j}\wedge e^{k})$, where
$e^{i}=\mathrm{d}x^{a}e_{a}^{i}$ denotes the frame field of the induced metric
on $\Sigma$. Before imposing the constraints, these variables are canonically
conjugated as

$\displaystyle\big{\\{}A_{a}^{i}(x),E^{b}_{j}(y)\big{\\}}=\delta_{a}^{b}\delta^{i}_{j}\delta^{(3)}(x-y).$
(14)

The kinematical constraint equations associated with diffeomorphism and
SU$(2)$ gauge symmetry are simply, as shown in Freidel:2019ees , the Gauss
conservation laws

$\displaystyle\mathrm{d}_{A}E_{i}\hat{\;=\;}0,\qquad\qquad\mathrm{d}_{A}P_{i}\hat{\;=\;}0.$
(15)

Here $E_{i}$ is the geometrical 2-form flux, while
$P^{i}\coloneqq\mathrm{d}_{A}e^{i}$ is the torsion of the Ashtekar–Barbero
connection. On the reduced phase space $P^{i}$ computes components of the
extrinsic curvature tensor. One can also understand $P_{i}$ as the variable
conjugated to the frame field since

$\displaystyle\big{\\{}P^{a}_{i}(x),e_{b}^{j}(y)\big{\\}}=\delta_{b}^{a}\delta^{i}_{j}\delta^{(3)}(x-y)$
(16)

on the kinematical phase space.

It is important to appreciate that the commutation relations (14), or the one
just shown above, are only valid before imposition of the constraints. What
matters at the quantum level is to understand how these variables commute in
the _reduced phase space_ obtained after imposing the constraints. This is
where the corner symmetry enters as a key ingredient. It is direct to show
Freidel:2019ees ; Freidel:2019ofr  that the symmetry charge of infinitesimal
diffeomorphisms labelled by a vector $\xi^{a}\partial_{a}\in T\Sigma$ which is
tangent to $S$ and of infinitesimal SU$(2)$ gauge transformation labelled by
$\alpha^{i}\sigma_{i}\in\mathfrak{su}(2)$ are respectively given by the corner
charges

$\displaystyle P(\xi)\hat{\;=\;}\oint_{S}\xi^{a}e^{i}_{a}P_{i},\qquad\qquad
G(\alpha)\hat{\;=\;}\oint_{S}E_{i}\alpha^{i},$ (17)

where the hatted equality emphasizes that these equation are valid on the
reduced phase space.

In particular one sees that, after imposition of the Gauss constraints, the
pull-back on $S$ of the electric field operator satisfies ultralocal
commutation relations at the corner, is given by Cattaneo:2016zsq ;
Donnelly:2016auv

$\displaystyle\big{\\{}E^{i}(\sigma),E^{j}(\sigma^{\prime})\big{\\}}_{S}\;\hat{=}\;\epsilon^{ij}{}_{k}\delta^{(2)}(\sigma-\sigma^{\prime})E^{k}(\sigma),$
(18)

where $\sigma,\sigma^{\prime}\in S$. Usually such a non-trivial commutation
relation of the flux operator is only obtained in LQG after the introduction
of loops which discretize the support of the flux lines. What is remarkable is
that we can establish this non-commutation directly in the continuum and at
the classical level through the Noether theorem. Similarly, the momentum
operators $P_{A}=P_{i}e^{i}_{A}$ with $A=1,2$ indices tangent to $S$, are the
generators of diffeomorphisms along $S$. The commutation relations of these
momenta on the reduced phase space are

$\displaystyle\big{\\{}P_{A}(\sigma),P_{B}(\sigma^{\prime})\big{\\}}\;\hat{=}\;P_{A}(\sigma^{\prime})\partial_{B}\delta^{(2)}(\sigma,\sigma^{\prime})-P_{B}(\sigma)\partial^{\prime}_{A}\delta^{(2)}(\sigma,\sigma^{\prime}).$
(19)

At the quantum level the goal is to represent these commutation relations. The
commutations relations (18) for the flux fields define a 2-sphere
generalisation of an SU$(2)$ valued loop algebra. The representations are
classified by the SU$(2)^{S}$ whose Casimir is given by the corner area
element $\rho(\sigma)=|\det(e_{a}^{i})|(\sigma)$. At the classical level this
object defines a measure on the sphere which is strictly positive.

At the quantum level we have to chose a representation of the operators we
just described, that is we have to chose a unitary representation of
$G^{S}_{\mathrm{LQG}}$. These representations are classified by the choice of
sphere measure. We now focus our analysis to representations labelled by a
discrete measure on $S$ which now carry a new quantum number $N$: the number
of punctures on the sphere where this measure is non-vanishing. Representation
states of SU$(2)^{S}$ are given by functions
$\varphi:S^{N}\to\otimes_{i=1}^{N}V_{j_{i}}$ where $V_{j}$ denotes a spin-$j$
representations of SU$(2)$ and $N$ denotes the number of punctures activated
on the corner sphere. On these states, the action of the flux operator is then
simply given by

$\displaystyle
E_{i}(\sigma)\varphi(\sigma_{1},\cdots\sigma_{N})=\sum_{n=1}^{N}\delta^{(2)}(\sigma,\sigma_{n})\rho_{j_{n}}(X_{i})\varphi(\sigma_{1},\cdots\sigma_{N}),$
(20)

where $X_{i}$ is an SU$(2)$ generator and $\rho_{j}:\mathfrak{su}(2)\to V_{j}$
denote the spin-$j$ representations. This representation corresponds to a
choice of discrete density given by

$\displaystyle\hat{\rho}(\sigma)\varphi(\sigma_{1},\cdots,\sigma_{N})=\sum_{n=1}^{N}\rho_{n}\delta^{(2)}(\sigma,\sigma_{n})\varphi(\sigma_{1},\cdots,\sigma_{N}),$
(21)

where $\rho_{n}=\gamma\hslash\sqrt{j_{n}(j_{n}+1)}$ is the discrete area
spectrum of LQG.

An interesting analogy comes by noticing that
$G^{S}_{\mathrm{hydro}}=\mathrm{Diff}(S)\ltimes\mathbb{R}^{S}$, which is the
group generated by $(P_{A},\rho)$, is isomorphic to the symmetry group of a
2-dimensional barotropic fluid888Barotropic fluids are such that the fluid
pressure $P(\rho)$ is a function of the fluid density $\rho$ only. In general
the pressure can also depend on the entropy density. khesin1989invariants . In
this analogy $\rho$ is the fluid particle density, and it is well-known that
there exist two classes of fluid representations: the ones for which $\rho$ is
a strictly positive continuous measure, and the ones for which $\rho$ is a
discrete measure. In the second case the fluid is composed of molecules. Here
we see that the quantization of area is analog to having a corner quantum
fluid made of fundamental quanta.

This analogy also allows us to classify the representations of Diff$(S)$. It
is known in the mathematical physics litterature that these groups are non-
anomalous Khesin . This is why we can assume that the quantum theory provides
a representation for these groups.

As shown in Freidel:2021qpz , the representation of the diffeomorphism
symmetry group is then labelled by conformal dimensions
$(\Delta_{1},\cdots,\Delta_{N})$ and spins $(s_{1},\cdots,s_{N})$. These
representations are irreducible and also appears as representation labels for
BMSW . The Casimirs associated with diffeomorphism have been constructed in
Donnelly:2020xgu . The action of the diffeomorphism generator $P_{A}$ on the
discrete representation states is then given by

$\displaystyle\hat{P}_{A}(\sigma)\varphi(\sigma_{1},\cdots,\sigma_{N})=\;$
$\displaystyle\sum_{n=1}^{N}\delta^{(2)}(\sigma,\sigma_{i})\frac{\partial\varphi}{\partial\sigma_{n}^{A}}(\sigma_{1},\cdots,\sigma_{N})$
$\displaystyle-\sum_{n=1}^{N}\big{(}\Delta_{n}\delta_{A}{}^{B}+s_{n}\epsilon_{A}{}^{B}\big{)}\frac{\partial\delta^{(2)}(\sigma,\sigma_{n})}{\partial\sigma^{B}}\varphi(\sigma_{1},\cdots,\sigma_{N}).$
(22)

Here $\epsilon_{A}{}^{B}$ corresponds to the choice of a complex structure on
the sphere, i.e. it satisfies
$\epsilon_{A}{}^{B}\epsilon_{B}{}^{C}=-\delta_{A}{}^{C}$. We see that a
generic discrete representation of the LQG corner symmetry group assigns 3
quantum numbers $(j,\Delta,s)$ to each puncture. The states associated with
$(j,\Delta,s)$ can be expanded as a superposition of spin network states
purely labeled by SU$(2)$ indices. An explicit expansion is given in
Freidel:2019ees . It was shown in Wieland:2017cmf  that these discrete
representations labeled by $N$ appear naturally as a Fock space quantization
of the 2d corner symplectic fluid.

These preliminary considerations should give ample motivations to explore the
possibility of representing the corner symmetry group and connecting the
ensuring representations to an extension of the quantum geometric picture
arising from LQG, including in particular diffeomorphism symmetry. An
important point of clarification is that, although spin foam models do not
represent diffeomorphism symmetry, canonical LQG does by simply allowing the
embedded graph to be displaced by a diffeomorphism. What we can now clearly
see from (22) is that the LQG representation of diffeomorphism is trivial,
i.e. it correspond to choosing $(\Delta,s)=(0,0)$.

##  4 Boundaries in loop quantum gravity

We have seen in the previous section that corner symmetries allow us to
understand the entanglement of quantum spacetime from the representation
theory of corner symmetry groups. From the perspective of the corner symmetry
groups, the fundamental quantum discreteness of geometry is a simple
consequence of representation theory. These results suggests a form of local
holography, where the boundary data for local subregions plays a pivotal role
in the construction of the physical quantum states.

These ideas are profoundly connected to earlier LQG insights, for example
through the work on isolated horizons and black hole entropy BarberoG:2022ixy
, on the relationship between holography and quantum entanglement
Bianchi:2023avf , and on the geometrical understanding of spin networks and
spin foams (e.g. through twisted geometries) Freidel:2010aq ; Bianchi:2009tj ;
Freidel:2011ue . One element that the new corner perspective brings however is
the seamless connection with the continuum QFT, and the importance of
diffeomorphism symmetry and its representation labels such as $(\Delta,s)$. It
also opens up the possibility to connect with the vast literature on
holographic approaches to quantum gravity, and provides new avenues of
development for the core LQG results such as a Lorentz covariant formulation
of spin network states with discrete area spectra.

We now briefly summarize some results which highlight the role played by
boundaries in LQG. We refer the reader to the other review chapters
BarberoG:2022ixy  for black hole entropy, Bianchi:2023avf  for entanglement in
LQG, Haggard:2023tnj  for further details on quantum geometry and spin
networks, and Asante:2022dnj  for spin foams and renormalization.

Black hole entropy in LQG. One of the key ideas of LQG concerning black holes
was proposed by Krasnov Krasnov:1996tb , building up on previous work by
Smolin Smolin:1995vq  about edge mode degrees of freedom in gravity and TQFTs.
The key idea first developed was that black hole entropy really counts the
numbers of quantum degrees of freedom that live on the black hole horizon and
represent quantum geometrical states Rovelli:1996dv ; Ashtekar:1997yu . Early
on, a connection between these quantum states and horizon edge modes in the
gravity phase space was conjectured Ashtekar:1999wa . These results led on the
classical side to the study of boundary conditions for black holes and the
proposal of isolated horizons Hayward:1993wb ; Ashtekar:2000eq ;
Ashtekar:2004cn . It also led to a refined way of counting black hole
microstates Domagala:2004jt ; Meissner:2004ju ; Agullo:2010zz , and eventually
to important realization of the connection with U$(1)$ and SU$(2)$
Chern–Simons theory living on the horizon Engle:2010kt ; Engle:2011vf ;
Bodendorfer:2013jba ; Bodendorfer:2013sja . In these works, the central role
of the boundary symplectic structure was put forward, and it was finally
understood that horizon punctures could also support loop algebra symmetries
Ghosh:2014rra . These results from the black hole entropy counting led to a
direct connection with the corner symmetry in Freidel:2016bxd . Note also that
the notion of isolated horizons was recently generalized to the notion of
_non-expanding horizons_ Ashtekar:2021wld . These horizons correspond to null
surfaces where the corner symmetry charges are conserved, and the
corresponding Carrollian fluid is perfect, see section 5.1.

LQG and twisted geometries. We have seen that the entangling sphere carries
representations of the boundary symmetries. The traditional loop gravity
picture assigns, on the other hand, geometrical data to the vertex inside the
sphere. So there seems to be tension between the two interpretations. This
tension is resolved by the twisted geometry interpretation of spin networks,
developed in Freidel:2010aq ; Freidel:2010bw . In twisted geometries, one
assigns geometrical elements to each link that intertwines two vertices and
then proves that the matching of representations across the link allows one to
reconstruct the connection. In a twisted geometry, there are two discrete
geometries across the link: the one from the left and the one from the right.
If one tries to identify the group data in terms of polyhedral geometries, one
finds a mismatch: while the areas are matched, the shapes of the two adjacent
polyhedra might differ; hence the discrete geometry is _twisted_. Remarkably,
it was shown in Haggard:2012pm  that the difference in shapes can be encoded
into an SL$(2,\mathbb{R})$ connection. In Freidel:2018pvm  it was also shown,
following Freidel:2015gpa , that one has a natural generalization of spin
networks states that also carries representations of SL$(2,\mathbb{R})$
necessary to reconstruct the frame field. It was proven in Freidel:2020svx
that the generator of this SL$(2,\mathbb{R})$ symmetry is, in the continuum,
the induced geometry of the corner sphere transverse to the spin network link.
Finally, in Wieland:2021vef  the change of this SL$(2,\mathbb{R})$ was shown
to encode radiation.

Demanding that the geometry is not twisted projects twisted geometries, which
represent LQG states, onto Regge geometries Dittrich:2008ar . The fact that
LQG geometries are twisted and differ from the Regge geometries creates a
puzzle. This puzzle can be simply resolved if one accepts that the faces dual
to loop gravity states are not planar, and the edges of the dual polyhedra are
_spinning_ instead of being straight Freidel:2013bfa . Moreover, we know from
Freidel:2011ue  that there is an isomorphism between twisted geometries
representing LQG states and piecewise flat geometries which provides an exact
continuum description of the LQG phase space in agreement with the results
described in this review. This result also gives a geometrical understanding
of the dual LQG vacua constructed in Dittrich:2014wpa ; Dittrich:2014wda ;
Dittrich:2016typ . Another prescient paper by Bianchi Bianchi:2009tj  proposed
early on a similar understanding of classical LQG geometry compatible with the
action of diffeomorphisms. Now is a good time to go back to these results and
use them to connect LQG and corner symmetries more tightly.

Entanglement in LQG, spin foams and group field theories. Very early on it was
proposed that the area entanglement and its relation to boost symmetry could
be recovered from the spin network entanglement of links carrying
SL$(2,\mathbb{C})$ representations Bianchi:2012ui ; Bianchi:2012vp . Since
then there has been a recent interest in computing entanglement entropy of
spin network states and trying to relate it to the Ryu–Takayanagi formula
Chirco:2021chk ; Han:2016xmb . The central idea here is that quantum geometry
arises from a _network of entanglement_. The quantum entanglement in spin
networks is encoded through the representation link Donnelly:2008vx ;
Donnelly:2011hn ; Livine:2017fgq ; Baytas:2018wjd . Similar ideas about
entanglement have been explored in GFTs Chirco:2021chk . These results are a
discrete expression of the fundamental ideas presented earlier, namely that
entanglement can be derived from corner symmetry. It would be really
interesting to develop further the connection between entropy counting and the
renewed understanding of symmetry. In particular, in order to connect to the
Unruh calculation, one would need to construct, in quantum gravity, the
spacetime boost operator. Again, we refer the reader to the chapter
Bianchi:2023avf  for more details about entanglement in LQG.

Spin foams as quantization of boundary data. Spin foams are based on a lattice
truncation. The basic idea is to glue flat building blocks in such a way that
Einstein’s equations are satisfied at large scales compared to the typical
lattice scale. The resulting spin foam amplitudes are a sum over geometric
data in each fundamental four-simplex Perez:2012wv ; Wieland:2016exy ;
Asante:2022dnj . Each four-simplex is flat and contains no radiative data
inside. It is only through the non-trivial gluing between adjacent four-
simplices that curved geometries arise. Curvature is distributional and
concentrated at surfaces dual to the fundamental triangles. In fact, it goes
deeper than this: the traditional formulation of spin foam models focuses on
simplicity constraints at the vertices of the spin foam Engle:2007wy ;
Freidel:2007py , thereby restricting the geometry of boundary tetrahedra. One
can take an equivalent point of view where the simplicity constraints are
imposed by selecting appropriately the propagator going between the spin foam
vertices, hence by controlling the corner geometry matching. This point of
view adapted to corner and twisted geometry is already present in
Freidel:2007py , it was further developed in Kaminski:2009fm  and shown to be
extremely efficient in the proof of spin foam asymptotics Banburski:2014cwa .
This twisted geometry perspective on spin foams was first discovered in the
context of GFT models for quantum gravity Freidel:2005qe ; Oriti:2009nd  where
one witnessed that it is possible to encode quantum gravity constraints by the
right choice of propagator while keeping the interaction vertex simple.

What is missing? In asymptotically flat spacetimes, the simplest Dirac
observables are the ADM energy and momentum, angular momentum and centre of
mass together with an infinite-dimensional algebra of corner charges. The same
happens at finite distance. There are infinitely many corner charges that are
Dirac observables on the physical phase space of a bounded region. These
charges generate corner symmetries such as super-rotations (which represent
tangent diffeomorphisms), super-boosts and super-translations. We have seen in
section 3 that, even when we are in a vacuum state, some of the charges such
as the corner boost charges do not vanish. Therefore, a generic physical state
should carry a non-trivial representation of the corner symmetries. The
problem with the usual spin network representation is that the basis states do
not carry the corresponding representation labels. We know that these labels
characterize components of the 4d metric at the corner. For instance, the
generator of Diff$(S)$ symmetry knows about the off-diagonal components of the
metric in metric gravity Donnelly:2016auv  or knows about the torsion of the
Ashtekar–Barbero connection in loop gravity Freidel:2015gpa ; Freidel:2016bxd
; Freidel:2019ees . Another example concerns the geometry symmetry encoded
into an SL$(2,\mathbb{R})$ edge group, as described in the next section. Its
representation labels describe quantization of the tangential metric component
Freidel:2015gpa ; Freidel:2020svx . Importantly, we lack a proper
representation of asymptotically-flat spacetimes in terms of spin network
states (see Campiglia:2014xja ) which carries a representation of BMS
symmetry. By connecting the boundary charges in a finite region to asymptotic
charges at infinity, we obtain a new perspective to this long-standing problem
in the field (see Freidel:2021cjp ; Wieland:2020gno ; Campiglia:2020qvc  for
this connection at the semi-classical level).

By not representing these symmetries at the quantum level, we lose information
about the quantum geometry. In LQG, we have quantum numbers for internal
SU$(2)$ frame rotations and corresponding intertwiners, but there are no
representation labels for the rest of the corner symmetry group. For instance
as shown in Freidel:2016bxd , with SU$(2)$ labels we can represent the flux
but miss SL$(2,\mathbb{R})$ labels needed to represent the frame.

In Freidel:2016bxd  we have shown preliminary results on how to include the
conformal weights and spin labels of Diff$(S)$ representations, see (22), in
the spin network description. The main result is that these edge labels
control the gluing of states when we glue two elementary quantum subregions
along an edge. It forces specific spin network superpositions to enter, such
as $|\Delta,s\rangle\langle\Delta,s|=\sum_{j\geq
s}\rho_{j}(\Delta,s)|j,m\rangle\langle j,m|$. These superpositions arise from
the matching of diffeomorphism symmetry which determines the weights
$\rho_{j}(\Delta,s)$. Therefore, having extra representation labels does not
necessarily mean that spin networks are a wrong basis. It means that no
physical states can be supported on a single spin network state. Since the
nature of quantum geometry entanglement cannot simply be described by sharing
an SU$(2)$ spin. One needs extra symmetry labels control what superposition of
states is allowed when we glue two quantum subregions together. In other
words, the quantum geometry entanglement of subregions is considerably
reinforced by the presence of diffeomorphism symmetry at the quantum
level999As we have seen in section 3.2. The usual formulation of LQG contains
only trivial representations of the diffeomorphism group that do not create
any new entanglement.. Ultimately this is what we want: the demand of
invariance under super-rotation, super-boost and super-translation should
constrain the admissible spin network superposition allowed in the edge gluing
by requiring that these form representation states of the corner symmetry
group. The fact that after gluing the representation labels are matched is the
expression that the Hamiltonian and diffeomorphism constraints are
implemented. So it should be now clear that not including these representation
labels in the construction of spin network states in LQG is problematic. It is
therefore of utmost importance to revisit the investigations of quantum
geometry entanglement described in Bianchi:2023avf  in light of these results.

For the spin foam models, we can still use boundary states which are spin
networks. What the corner symmetry analysis provides, however, is the
possibility to constrain the value of spin foam amplitudes via symmetries. It
is well-known that in the presence of symmetries, path integral amplitudes
satisfy Ward identities which express the invariance of the quantum amplitude
under a symmetry transformation of the boundary state. We should therefore
expect spin foam models and GFT models to satisfy such Ward identities. These
identities represent the quantization of the infinite set of flux-balance laws
that arise from solving the Hamilton and vector constraints. At asymptotic
infinity, these balance laws turn into Ward identities, called soft theorems,
that put stringent constraints on the perturbative $S$-matrix
Strominger:2017zoo . This means that a new exciting possibility for spin foam
models is opening up: the possibility to identify the quasi-local generators
of symmetry charges acting on the spin network states and write down the Ward
identities that spin foam models ought to satisfy.

###  4.1 Classical theory and symplectic structure

We have already briefly discussed above in section 3.2 how LQG representation
of quantum geometry naturally arises from studying the corner symmetries group
appearing in tetrad gravity with a gauge-fixed internal normal. The choice of
this internal normal breaks Lorentz invariance and determines an internal
SU$(2)$ group. Our goal is now to show how a _generalization_ of the LQG
representation of quantum geometry, covariant under the internal Lorentz
group, naturally arises from studying the corner symmetries of tetrad gravity.
This result follows from a detailed canonical analysis of the first-order
Lagrangian for gravity.

Let us consider a tetrad 1-form $e^{I}$ and a connection 1-form $\omega^{IJ}$
with curvature $F^{IJ}$. These fields can be used to build the Lagrangian
4-form for Einstein–Cartan–Holst gravity

$\displaystyle L_{\text{ECH}}=\frac{1}{2}E_{IJ}\wedge F^{IJ},\qquad\qquad
E_{IJ}[e]=(\star+\beta)(e_{I}\wedge e_{J}).$ (23)

We are going to focus primarily on the symplectic structure of this
Lagrangian. In its analysis, a crucial role is played by the internal normal
$n^{I}=n^{\mu}e^{I}_{\mu}$ obtained from the time-like normal $n^{\mu}$ to a
spacelike hypersurface $\Sigma$. This normal is such that
$n^{I}n_{I}=n^{\mu}n_{\mu}=-1$. The importance of keeping track of this normal
in the phase space was already recognized in various contexts in
Bianchi:2012vp ; Bodendorfer:2013hla ; Alexandrov:2002br ; Wieland:2010ec ;
Wieland:2014nka ; Wieland:2017zkf ; Bodendorfer:2011nw ; Bodendorfer:2013jba .

Using the internal normal, we can decompose the Lorentz connection as

$\displaystyle\omega^{IJ}=\Gamma^{IJ}-2K^{[I}n^{J]},\qquad\mathrm{d}_{\Gamma}n^{I}=0,\qquad\mathrm{d}_{\omega}n^{I}=K^{I}.$
(24)

We also decompose the pull-back of the tensorial 2-form $E^{IJ}$ as

$\displaystyle E^{IJ}|_{\Sigma}=-2E^{[I}n^{J]}+\beta(e^{I}\wedge e^{J}).$ (25)

As explained at length in Freidel:2020svx , with these decomposition the
symplectic structure of the ECH Lagrangian becomes

$\displaystyle\Omega_{\text{ECH}}=\int_{\Sigma}\delta K^{I}\wedge\delta
E_{I}+\oint_{S}\left(\delta E_{I}\delta n^{I}-\frac{\beta}{2}\delta
e_{I}\wedge\delta e^{I}\right).$ (26)

One can see that the normal appears in the boundary contribution, and also
that the frame is conjugated to itself on the boundary due to the non-
vanishing Barbero–Immirzi parameter. To interpret this result, one should
notice that the bulk term is nothing but the symplectic structure of canonical
gravity. Indeed, with a slight rewriting, one can show that Freidel:2020svx

$\displaystyle\int_{\Sigma}\delta K^{I}\wedge\delta
E_{I}=\int_{\Sigma}\delta(K\times e)^{I}\wedge\delta
e_{I}=\frac{1}{2}\int_{\Sigma}\delta p^{ab}\delta h_{ab}=\Omega_{\text{GR}},$
(27)

where $p^{ab}=\sqrt{h}\big{(}Kh^{ab}-K^{ab}\big{)}$ is the usual momentum
density of the ADM formulation. In summary, we can write (26) in the form

$\displaystyle\Omega_{\text{ECH}}=\Omega_{\text{GR}}+\Omega^{S}_{\text{ECH/GR}},$
(28)

where $\Omega^{S}_{\text{ECH/GR}}$, which is the boundary contribution in
(26), is the relative boundary symplectic structure between the ECH and
canonical formulations of gravity. This is the main message of Freidel:2020xyx
, namely that any formulation $L_{\text{F}}$ of gravity has a symplectic
structure which is of the form
$\Omega_{\text{F}}=\Omega_{\text{GR}}+\Omega^{S}_{\text{F/GR}}$. The bulk
symplectic structure is universal and encodes the fact that the theories being
discussed are general relativity, while the boundary contribution is
formulation-dependent. Even if ECH gravity (23) is equivalent in the bulk to
general relativity, it carries a specific boundary symplectic structure.

Our goal is to understand the physical significance of the boundary symplectic
structure (26). This latter is responsible for the appearance of non-vanishing
charges for Lorentz transformations, and, as we will explain shortly, for the
non-commutativity of the tangential boundary metric (which in turn leads to
the discreteness of the boundary area). Note that studies of the relation
between the Einstein–Hilbert and Einstein–Cartan–Holst formulations through a
boundary term in the symplectic structure go back to Peldan:1993hi . The
emphasis on relative charges was made more explicit in DePaoli:2018erh ;
Oliveri:2019gvm .

###  4.2 Charges and discreteness

First, let us stress that the treatment presented here is in a sense
orthogonal to the usual approach to canonical LQG. Indeed, in standard
treatments of LQG one focuses on bulk variables, and in particular the
contribution of the Holst term proportional to $\beta$ is used to build a bulk
connection variable. This is made possible by the identity
$\beta\mathrm{d}(e_{I}\wedge\delta e^{I})\hat{\;=\;}\beta(e_{I}\wedge
e_{J})\wedge\delta\omega^{IJ}[e]$, which holds on-shell of the torsion
equation of motion. In the time gauge where $n^{I}=\delta^{I}_{0}$, this
reduces to $\beta\mathrm{d}(e_{i}\wedge\delta e^{i})\hat{\;=\;}\beta
E^{a}_{i}\delta\Gamma^{i}_{a}[e]$, where $i$ is the index in the
$\mathfrak{su}(2)$ subalgebra of $\mathfrak{so}(3,1)$ which survives the
fixing of the normal. This is the identity that enables, once the Holst term
is included, to define a bulk Ashtekar–Barbero connection variable conjugated
to the electric field $E$. Here we wish to shift emphasis from the bulk to the
boundary, and therefore keep the contribution of the Holst term in the
boundary symplectic structure.

The fact that the symplectic structure of ECH gravity differs from that of
canonical gravity $\Omega_{\text{GR}}$ by the presence of a surface term has
the important consequence that it gives rise to a new set of surface charges.
These are the charges of the internal Lorentz transformations acting as
$\delta_{\alpha}V^{I}=-{\alpha^{I}}_{J}V^{J}$ for $V^{I}=(E^{I},n^{I},e^{I})$
and $\delta_{\alpha}\omega^{IJ}=\mathrm{d}_{\omega}\alpha^{IJ}$. These charges
are obtained as usual by computing the contraction $I_{\alpha}\Omega$, and
their explicit expression is

$\displaystyle
G(\alpha)=\int_{\Sigma}E^{IJ}\wedge\mathrm{d}_{\omega}\alpha_{IJ}\hat{\;=\;}\oint_{S}\alpha_{IJ}E^{IJ}=\oint_{S}\alpha_{IJ}\Big{(}\beta(e^{I}\wedge
e^{J})-2E^{I}n^{J}\Big{)}.$ (29)

These boundary charges are the Lorentz covariant version of the SU$(2)$
charges (17). They have also been discussed extensively in the context of
black hole entropy counting in Bodendorfer:2013sja . It is important to stress
out that these charges are purely relative charges, in the sense that they are
produced entirely by the boundary symplectic structure in (26). This means
that, as expected, metric gravity with the symplectic structure
$\Omega_{\text{GR}}$ does not possess Lorentz charges. This is the key
property which differentiates, already at the classical level, LQG from any
metric based approach: it possesses extra surface charges which give rise to
the basic surface flux operators. We have here presented the covariant
formulation with an arbitrary internal normal $n^{I}$, but in LQG the normal
is fixed to $\delta^{I}_{0}$ and one recovers the SU$(2)$ fluxes (17) whose
action on states is (20).

We now turn to the main result obtained from the detailed analysis of the
boundary symplectic structure, which is the discreteness of area implied by
the presence of $\beta\neq 0$. In Freidel:2020ayo ; Freidel:2020svx  we have
decomposed the 6 components101010There are 6 components because
$e^{I}_{a}n_{I}=0$ and on $S$ we have $a=1,2$. of the tangential frame
$e^{I}_{a}$ at the surface $S$ into a so-called spin operator
$S^{I}=\frac{1}{2}\beta(e\times e)^{I}$ and the tangential metric
$q_{ab}=e^{I}_{a}e^{J}_{b}\eta_{IJ}$. The spin operator is the SU$(2)$ flux
$E^{i}$ written in internal Lorentz-covariant form. Equipped with this
decomposition, one can show that the tangential metric satisfies the
$\mathfrak{sl}(2,\mathbb{R})^{S}$ algebra

$\displaystyle\big{\\{}q_{ab}(x),q_{cd}(y)\big{\\}}=-\frac{1}{\beta}\big{(}q_{ac}\varepsilon_{bd}+q_{bc}\varepsilon_{ad}+q_{ad}\varepsilon_{bc}+q_{bd}\varepsilon_{ac}\big{)}(x)\delta^{2}(x-y).$
(30)

This is in line with earlier analysis carried out in Freidel:2018pvm ;
Freidel:2015gpa  in the SU$(2)$ gauge fixed formulation. This important result
shows that the corner metric becomes non-commutative in the presence of the
Barbero–Immirzi parameter! This shows the tangential metric component of $S$
are the $\mathfrak{sl}(2,\mathbb{R})^{S}$ generators. One can then show that
the quadratic Casimir is given by

$\displaystyle C_{\mathfrak{sl}(2,\mathbb{R})^{S}}=\beta^{2}\det(q).$ (31)

Since the metric is that of a 2-dimensional space-like surface, the
determinant on the right-hand side is positive, and we conclude from this
relation that the $\mathfrak{sl}(2,\mathbb{R})^{S}$ Casimir is positive as
well. This indicates that the surface area spectrum has to be labeled by the
discrete series of representations, if we were to label states by unitary
representations respecting the boundary symmetry structure.

Note that in order to make this construction rigorous a regularization of the
corner algebra is necessary, as outlined in Freidel:2018pvm . A treatment that
also includes diffeomorphism symmetry on the sphere is also required, along
the lines of Freidel:2019ees ; Freidel:2016bxd ; Donnelly:2020xgu ;
Wieland:2017cmf ; Donnelly:2022kfs . In particular, in Wieland:2017cmf  a
regularisation in terms of spherical harmonics with respect to a fiducial
metric at the sphere was introduced as an example of a regularisation which
does not involve any discretization. In Donnelly:2022kfs  such a
regularization was shown to be equivalent to a matrix model regularization and
arising by rendering the corner sphere non-commutative. This is ultimately
related to the understanding of quantization of diffeomorphisms in the corner
symmetry group and the role of the dynamics. We therefore now turn to a
description of the dynamics along null boundaries and show that in this
context we can also obtain the discreteness of area from the canonical
analysis of null surfaces.

##  5 Null boundaries, isolated horizons and fluid conservation law

In the previous sections, we have discussed the representation of the corner
symmetry group which contains the symmetry generators with vanishing flux. In
order to understand the dynamics one needs to describe the representation of
the extended corner symmetry group and include the super-translation
generators that move the corner transversally. It has proven invaluable to
focus on the representation of null super-translations and study the
gravitational dynamics projected along null surfaces and null horizons. There
has been a rich literature of the subject dating back to the membrane paradigm
price1986membrane , the study of isolated horizons Ashtekar:1999wa , and more
recently the dynamics of null surfaces, the construction of the symplectic
potential, of the symmetry charges, and their understanding in terms of
Carrollian geometry Donnay:2016ejv ; Hopfmuller:2016scf ; Hopfmuller:2018fni ;
Chandrasekaran:2018aop ; Donnay:2019jiz ; Wieland:2020gno ;
Chandrasekaran:2021hxc ; Adami:2021nnf ; Freidel:2022vjq .

###  5.1 Carrollian fluid conservation law

In this section we study the geometry and dynamics of a null boundary
$\mathcal{N}$. On a null boundary the pull-back of the four-dimensional metric
is degenerate. The degenerate direction is the direction of the null rays that
generate the surface. This null direction is in the kernel of the induced
metric

$q_{ab}=\varphi^{\ast}_{\mathcal{N}}g_{ab},\qquad\ell^{a}q_{ab}=0.$ (32)

Any such null direction is unique up to a local rescaling sending111111We
assume that $\ell^{a}$ is future pointing. $\ell^{a}$ into
$\mathrm{e}^{\lambda}\ell^{a}$. Mathematically this means that $\mathcal{N}$
is a fibered manifold with fibration $p:\mathcal{N}\to S$. The null direction
$\ell$ is in the kernel of $\mathrm{d}p$ (push-forward of $p:\mathcal{N}\to
S$) while the null metric $q_{ab}$ can be understood as the pull-back of a
time dependent metric $q^{S}_{AB}$ on the base. The Lie derivative of the
metric along $\ell$ decomposes in terms of the expansion and shear of the null
surface:

$\displaystyle\theta_{ab}\coloneqq\frac{1}{2}{\cal
L}_{\ell}q_{ab}=\frac{q_{ab}}{2}\theta+\sigma_{ab}.$ (33)

Note that the degenerate metric $q_{ab}$ determines a spatial area form
$\epsilon_{S}$ on $\mathcal{N}$ such that $i_{\ell}\epsilon_{S}=0$. Moreover
the choice of a null generator $\ell$ determine a volume form
$\epsilon_{\mathcal{N}}$ on $\mathcal{N}$ which is such that
$i_{\ell}\epsilon_{\mathcal{N}}=\epsilon_{S}$. The expansion relates the two
forms through the identity
$\mathrm{d}\epsilon_{S}=\theta\epsilon_{\mathcal{N}}$.

In order to construct a connection on $\mathcal{N}$ and describe the dynamics
of null surfaces, it is convenient to introduce a Ehresmann connection. This
is a 1-form $k_{a}$ dual to the null vector $\ell^{a}:k_{a}\ell^{a}=1$
Bekaert:2015xua ; Ciambelli:2019lap . The Ehresmann connection defines a
notion of horizontality, where $Y$ is an horizontal vector field on
$\mathcal{N}$ if $i_{Y}k=0$. A general vector $\xi\in T\mathcal{N}$ can be
decomposed as $\xi=T\ell+Y$, with $Y$ horizontal. The Ehresmann connection
also allows to decompose the volume form on $\mathcal{N}$ as
$\epsilon_{\mathcal{N}}=k\wedge\epsilon_{S}$. The data
${\mathcal{C}}_{\mathcal{N}}=(q_{ab},\ell^{a},k_{b})$ represents the
_Carrollian geometry_ of $\mathcal{N}$. This data defines a _rigging
structure_ Mars:1993mj , i.e. a projector
$\Pi_{a}{}^{b}=q_{a}{}^{b}+k_{a}\ell^{b}$ where $q_{a}{}^{b}$ is the
horizontal projector $\ell^{a}q_{a}{}^{b}=0=q_{a}{}^{b}k_{b}$. The presence of
a rigging structure allow us to uniquely raise indices of horizontal tensors.
For instance given the expansion tensor $\theta_{ab}$, the tensor
$\theta_{a}{}^{b}$ is defined as the unique tensor such that
$\theta_{a}{}^{c}q_{cb}=\theta_{ab}$ and such that $\theta_{a}{}^{b}k_{b}=0$.

The rigging structure allows us to introduce the notion of _Carrollian_
connections $D_{a}$ Chandrasekaran:2021hxc ; Freidel:2022vjq . These are
torsionless connections that preserve the rigging projector:
$D_{a}\Pi_{b}{}^{c}=0$. A Carrollian connection defines a 1-form $\omega_{a}$
on $\mathcal{N}$ given by the derivative of the volume form:

$\displaystyle D_{a}\epsilon_{\mathcal{N}}=-\omega_{a}\epsilon_{\mathcal{N}}.$
(34)

This 1-form can be decomposed into a transverse and horizontal component as
$\omega_{a}=\kappa k_{a}+\pi_{a}$, where $\pi_{a}$ is the Hajicek connection,
while $\kappa$ is the surface gravity measuring the inaffinity of $\ell$.
$\omega_{a}$ can be understood as a boost connection since, under the boost
rescaling $\ell\to e^{\lambda}\ell$ and $k\to e^{-\lambda}k$, it transforms as
$\omega_{a}\to\omega_{a}+\partial_{a}\lambda$. One can then prove that the
boost covariantized derivative of $\ell^{b}$ and $k_{b}$ are horizontal. They
define the expansion and transverse expansion tensors

$\displaystyle(D_{a}-\omega_{a})\ell^{b}=\theta_{a}{}^{b},\qquad
q_{a}{}^{c}(D_{c}+\omega_{c})k_{b}={\theta}^{(k)}_{ab}.$ (35)

The Carrollian connection preserves the metric when derivatives and tensor
indices are taken to be transverse directions. In general, one has

$\displaystyle D_{a}q_{bc}=-(k_{b}\theta_{ac}+k_{c}\theta_{ac}).$ (36)

We therefore see that the Carrollian connection compatible with the Carroll
structure $(q_{ab},\ell^{b},k_{b})$ is determined by the boost 1-form and the
transverse expansion $(\omega_{a},\theta^{(k)}_{ab})$. It is well-known that
the projection of Einstein’s equation along the null surface leads to two sets
of equations: the Raychaudhuri and Damour equations damour1978black . These
are given by

$\displaystyle(\ell+\theta)[\theta]$
$\displaystyle=\mu\theta-\sigma_{ab}\sigma^{ab},$ (37) $\displaystyle
q_{a}{}^{b}\left({\cal L}_{\ell}+\theta\right)[\pi_{b}]$
$\displaystyle=\overline{D}_{a}\mu-\overline{D}_{b}\sigma^{a}{}^{b},$ (38)

where we have introduced the surface tension
$\mu\coloneqq\left(\kappa+\frac{\theta}{2}\right)$ of $\mathcal{N}$ and
$\overline{D}_{a}=q_{a}{}^{b}D_{b}$ denotes the horizontal derivative.

As shown in Hopfmuller:2018fni , these equations can be understood as
conservation equations of corner symmetry charges associated with super-
translations along $S$ and diffeomorphism along $S$. The corresponding charge
aspects are $\theta\epsilon_{S}$ for super-translations and
$\pi_{A}\epsilon_{S}$ for super-rotations. These two equations can be
understood as conservation equations for Noether charges with non-trivial flux
as in (4), where the symplectic potential121212This expression is valid in the
boost frame where $k_{a}\delta\ell^{a}=0$. The boost transformation is pure
gauge in the metric formulation of gravity while it possesses a non-trivial
charge in the first order formulation as we are about to see. on $\mathcal{N}$
is given by Parattu:2015gga ; Hopfmuller:2016scf ; Chandrasekaran:2020wwn

$\displaystyle\Theta_{\mathcal{N}}=\int_{\mathcal{N}}\left(\frac{1}{2}\big{(}\sigma^{ab}-\mu
q^{ab}\big{)}\delta
q_{ab}-\pi_{a}\delta\ell^{a}\right)\epsilon_{\mathcal{N}}.$ (39)

From this we can read the canonical conjugate pairs of spin-0
$(\mu,\sqrt{q})$, spin-1 $(\pi_{a},\ell^{a})$ and spin-2
$(\sigma^{ab},q_{ab})$.

Quite remarkably, these equations are now understood as conservation equations
for a Carrollian fluid Donnay:2019jiz ; Chandrasekaran:2021hxc ;
Freidel:2022vjq . The Carrollian fluid stress tensor is given in terms of the
rigging structure and the Carrollian connection simply as

$\displaystyle T_{a}{}^{b}\coloneqq D_{a}\ell^{b}-\Pi_{a}{}^{b}D_{c}\ell^{c}.$
(40)

The tensor $D_{a}\ell^{b}=\omega_{a}\ell^{b}+\theta_{a}{}^{b}$ is called the
shape operator or Weingarten map. As first shown in Chandrasekaran:2021hxc
(see also Freidel:2022vjq ) the Einstein equations pulled back on
$\mathcal{N}$ can then simply be written as fluid conservation equations

$\displaystyle\Pi_{a}{}^{b}G_{\ell b}=D_{b}T_{a}{}^{b}=0.$ (41)

This generalizes for null surfaces the celebrated Brow–York result
brown1993quasilocal . In Freidel:2022vjq  it was shown that these equations
can be obtained from the gravitational symplectic potential and that the
symmetry charges associated with a vector field $\xi$ tangent to $\mathcal{N}$
are

$\displaystyle Q(\xi)=\int_{S}\xi^{a}T_{a}{}^{b}\epsilon_{b},$ (42)

where $\epsilon_{b}=i_{\partial_{b}}\epsilon_{\mathcal{N}}$. The Carrollian
fluid energy-momentum tensor can be decomposed in terms of the quantities
defined above as

$\displaystyle T_{a}{}^{b}=-\theta k_{a}\ell^{b}+\pi_{a}\ell^{b}-\mu
q_{a}{}^{b}+\sigma_{a}{}^{b}.$ (43)

This energy-momentum tensor can be interpreted as that of a Carrollian fluid
obtained from the $c\to 0$ limit of a relativistic energy-momentum tensor. In
this fluid analogy, $\theta$ plays the role of the fluid energy, $\pi_{a}$
that of the heat flux, $-\mu$ of the pressure and $\sigma_{ab}$ of the viscous
stress tensor. We therefore see that it is not only the representation theory
of the corner symmetry group that leads to a 2-dimensional fluid analogy, but
also the gravitational dynamics along null surfaces in agreement with the
membrane paradigm price1986membrane . The notion of non-expanding horizons
Ashtekar:2021wld ; Ashtekar:2021kqj  can now be simply understood in this
modern language as a perfect fluid where the energy $\theta$ and the
dissipation tensor $\sigma_{ab}$ vanishes.

In order to connect these results to the LQG literature we now translate the
analysis of dynamics on a null surface in terms of tetrads and connection
variables. This will allow us to understand the role of the Immirzi parameter
and show that a non-zero Immirzi parameter implies a discretization of the
spectra of the area operator which is the corner charge of internal boost
symmetry. We also show that the spinor variables are naturally quantized as
Fock space variables. These results are the null slice representation of the
results derived in section 3.2 and 4.1.

###  5.2 Boundary frame fields and boundary Lagrangian

Loop gravity is based on Einstein–Cartan geometry. Just like we have triads
and tetrads, we can also have orthonormal co-frames intrinsic to the null
boundary. The intrinsic metric is positive semi-definite and has one
degenerate null direction. We introduce co-dyads $m_{a}$ and
$\bar{m}_{a}\in\Omega^{1}(\mathcal{N}:\mathbb{C})$ that diagonalise the
signature $(0$$+$$+)$ metric of the null surface

$q_{ab}=2m_{(a}\bar{m}_{b)}.$ (44)

Given $q_{ab}$, the co-dyad $(m_{a},\bar{m}_{a})$ is unique up to a local
U$(1)$ transformation sending $m_{a}$ into
$\mathrm{e}^{\mathrm{i}\varphi}m_{a}$. This transformation will play an
important role below. Besides the metric $q_{ab}$, there is also a canonical
2-form at the boundary, namely the area-element

$\varepsilon_{ab}=-2\mathrm{i}\,m_{[a}\bar{m}_{b]}.$ (45)

The oriented area of any 2-dimensional cross-section $S$ of $\mathcal{N}$ is
then given by the integral $\mathrm{Ar}[S]=-\mathrm{i}\int_{S}m\wedge\bar{m}$.

The loop representation is based on quantum states that carry half-integer
spin labels. Rather than working with Lorentz vector-valued $p$-forms, it is
therefore more natural to work with a spinor representation of the frame
bundle. Using the soldering forms $\sigma^{AA^{\prime}}{}_{I}$ between Lorentz
vectors and spinors, we map Lorentz vectors into pairs of spinors,

$V^{AA^{\prime}}=\frac{\mathrm{i}}{\sqrt{2}}\sigma^{AA^{\prime}}{}_{I}V^{I},\qquad
V^{I}=\frac{\mathrm{i}}{\sqrt{2}}\bar{\sigma}_{AA^{\prime}}{}^{I}V^{AA^{\prime}}.$
(46)

Under this map, the Plebański 2-form $E^{IJ}=e^{I}\wedge e^{J}$ and the
SO$(1,3)$ connection $\omega^{IJ}$ split into left-handed and right-handed
parts. We can now move to the construction of the boundary Lagrangian.

In the spinor representation, the action neatly splits into left-handed and
right-handed parts,

$S_{M}[A,e]=\left[\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{M}\Sigma_{AB}\wedge F^{AB}\right]+\mathrm{cc}.$
(47)

We now need to explain how to couple this action to a boundary that is null.
First of all we note that the pull-back of the self-dual 2-form $\Sigma_{AB}$
onto the null surface

$\varphi_{\mathcal{N}}^{\ast}\Sigma_{AB}=e_{(A}\wedge\bar{m}\,\ell_{B)}.$ (48)

These boundary fields have a neat geometric interpretation. The spinor
$\ell_{A}$ is the square root of the null generators of $\mathcal{N}$. If
$e_{AA^{\prime}}$ is the bulk tetrad, we have, in fact,

$\varphi^{\ast}_{\mathcal{N}}\ell_{a}=0,\quad\ell_{a}=\mathrm{i}e_{AA^{\prime}a}\ell^{A}\bar{\ell}^{A^{\prime}}.$
(49)

The 2-form $e_{A}\wedge\bar{m}$, on the other hand, encodes the rest of the
intrinsic geometry of $\mathcal{N}$. If we contract, for example,
$e_{A}\wedge\bar{m}$ with $\ell^{A}$, we obtain the area 2-form,

$\varepsilon=\mathrm{i}\,e_{A}\wedge\bar{m}\,\ell^{A}.$ (50)

It is possible to show, see Wieland:2017zkf , that one can recover the entire
intrinsic geometry of the null surface from $\eta_{A}=e_{A}\wedge\bar{m}$ and
$\ell^{A}$ alone.

To introduce a basis in the spin bundle, we introduce a dual spinor $k_{A}$,
whose square returns the Ehresmann connection introduced earlier in section
5.1, i.e.

$k_{a}=-\mathrm{i}\,e^{AA^{\prime}}{}_{a}k_{A}\bar{k}_{A^{\prime}},\quad\text{such
that}\quad k_{A}\ell^{A}=1.$ (51)

In terms of this basis, the spinor-valued 2-form $e_{A}\wedge\bar{m}$ admits
the decomposition

$e_{A}\wedge\bar{m}=-(\ell_{A}k+k_{A}m)\wedge{\bar{m}},$ (52)

where $k$ is the Ehresmann connection on the null boundary (now viewed as a
1-form intrinsic to $\mathcal{N}$). The curvature of this connection
determines the Carrollian acceleration $\varphi$ and the Carrollian vorticity
$w$ Ciambelli:2019lap ; Freidel:2022vjq .

$\mathrm{d}k=-\varphi k\wedge\bar{m}-\bar{\varphi}k\wedge
m+\mathrm{i}\,w\,m\wedge\bar{m}.$ (53)

The exterior derivative of the U$(1)$ dyad $m_{a}$, on the other hand,
determines a U$(1)$ connection $\Gamma$ and the shear $\sigma$ and expansion
$\theta$ of the null boundary via

$\mathrm{d}m=-\mathrm{i}\,\Gamma\wedge m+\frac{1}{2}\theta k\wedge m+\sigma
k\wedge\bar{m}.$ (54)

Taking into account the Frobenius theorem, we thus see that for non-vanishing
shear, there is an obstruction to find a potential for $m_{a}$. If there is
shear, we can not find a holomorphic coordinate
$z:\mathcal{N}\rightarrow\mathbb{C}$ that would satisfy $m\propto\mathrm{d}z$.

Derivatives of the boundary spinors are important as well. Without assuming
special boundary conditions, the only available derivative is the pull-back of
the SL$(2,\mathbb{C})$ covariant (exterior) derivative from the bulk. We call
this derivative $D=\varphi^{\ast}_{\mathcal{N}}\nabla$ and it provides a
Carrollian connection as explained in the previous section. Given the spinor
basis $(k^{A},\ell^{A})$ at the null boundary, the spin coefficients are

$\displaystyle k_{A}D\ell^{A}$
$\displaystyle=\frac{1}{2\mathrm{i}}\Big{(}\Gamma+\mathrm{i}\omega\Big{)},$
(55) $\displaystyle\ell_{A}D\ell^{A}$ $\displaystyle=-\Big{(}\frac{1}{2}\theta
m+\sigma\bar{m}\Big{)},$ (56) $\displaystyle k_{A}Dk^{A}$
$\displaystyle=-\frac{1}{2}\big{(}\theta_{({k})}-\mathrm{i}\,w\big{)}\bar{m}-\bar{\sigma}_{(k)}\bar{m}+\big{(}\bar{\varphi}+\bar{\pi}\big{)}k,$
(57)

where the 1-form $\omega\in\Omega^{1}(\mathcal{N}:\mathbb{R})$ encodes the
non-affinity $\kappa$ of the null generators $\ell^{a}$ as well as the
momentum aspect $\pi_{a}$ of the Carrollian fluid, i.e.

$\omega=\kappa k+\bar{\pi}m+\pi\bar{m}.$ (58)

The 1-form $\omega\in T^{\ast}\mathcal{N}$ determines extrinsic data—it can
not be obtained from exterior derivatives of $m$ and $k$ intrinsic to
$\mathcal{N}$. In the same way, the transversal expansion
$\bar{\theta}=q^{ab}\nabla_{a}k_{b}$ and the transversal shear
$\sigma_{(k)}=\bar{m}^{a}\bar{m}^{b}\nabla_{a}k_{b}$ are extrinsic quantities
on $\mathcal{N}$.131313In here, we extended the boundary fields $\ell^{a}\in
T\mathcal{N}$ and $(k_{a},m_{a},\bar{m}_{a})\in T^{\ast}\mathcal{N}$ into a
Newman–Penrose null tetrad $(k^{a},\ell^{a},m^{a},\bar{m}^{a})$ in a
neighbourhood of $\mathcal{N}$ such that
$q^{ab}=m^{a}\bar{m}^{a}+\mathrm{cc}.$ In the following, it will be useful to
combine $\Gamma$ and $\omega$ into a single complexified boundary connection
$A\in\Omega^{1}(\mathcal{N}:\mathbb{C})$, which is given by

$A=-\mathrm{i}\,\Gamma+\omega-\theta k.$ (59)

The boundary fields carry a representation of the boundary symmetries. First
of all, they are clearly covariant under internal SL$(2,\mathbb{C})$ gauge
transformations and diffeomorphisms that preserve $\mathcal{N}$. This is
trivially so. Then they also admit a complexified U$(1)$ symmetry that acts on
boundary fields
$(\ell^{A},A,\sigma)\to(\mathrm{e}^{\frac{\mathrm{1}}{2}(\lambda+i\phi)}\ell^{A},A+d(\lambda+i\phi),e^{\lambda+2i\phi}\sigma)$,
where $\phi$ is a U$(1)$ gauge parameter intrinsic to $\mathcal{N}$ and
$\lambda$ generates internal boosts.

At the null boundary $\mathcal{N}$, we now have to choose specific boundary
conditions. At the kinematical level, the bulk fields $e_{AA^{\prime}}$ and
$\tensor{A}{{}^{A}_{B}}$ are completely arbitrary. What is kept fixed is a
gauge equivalence class of boundary fields,

$g=[m,A,\theta]/_{\sim},\qquad\delta g=0.$ (60)

The equivalence relations that define this equivalence class are
diffeomorphisms of the null boundary, complexified U$(1)$ transformations, in
addition to the following shift symmetries and conformal rescalings of the
boundary data:

$\displaystyle[m,A]\sim\big{[}m,A+\frac{\mathrm{i}}{2}\frac{\alpha}{\gamma+\mathrm{i}}\big{]},\sim[m,A+\zeta\bar{m}+\bar{\zeta}m],\sim[\mathrm{e}^{f}m,A],$
(61a)

where the 1-form $\alpha$ and the function $f$ are real, but $\zeta$ is
complex. It is easy to check that this equivalence class is determined by two
local degrees of freedom along $\mathcal{N}$, i.e. the two local degrees of
freedom of gravitational radiation along a null boundary.

Given the boundary conditions (61), the corresponding boundary action
Wieland:2021vef  is

$S_{\mathcal{N}}=\left[\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\int_{\mathcal{N}}\Big{(}e_{A}\wedge\bar{m}\wedge\Big{(}D-\frac{1}{2}A\Big{)}\ell^{A}-\frac{\theta}{4}e_{A}\wedge
e^{A}\wedge\bar{m}\Big{)}\right]+\mathrm{cc}.$ (62)

This action has simple structure. The first term is just a gauged kinetic term
for a configuration variable $\ell^{A}$, which is charged under
SL$(2,\mathbb{C})\times\mathrm{U}_{\mathbb{C}}(1)$, and $e_{A}\wedge\bar{m}$
is the corresponding chiral momentum variable. The second term is a simple
quadratic Hamiltonian. The coupled bulk plus boundary action is the sum of the
two terms

$S_{\text{bulk+boundary}}=S_{\mathcal{M}}+S_{\mathcal{N}}.$ (63)

Notice that the shift symmetry (61a) depends on the Barbero–Immirzi parameter
$\gamma$. In fact, variations of the boundary action with respect to the
boundary 1-form $\alpha\in\Omega^{1}(\mathcal{N}:\mathbb{R})$ impose a
constraint, namely the _reality condition_

$\varepsilon=\bar{\varepsilon}.$ (64)

From the variation of the boundary spinors $e_{A}$ and $\ell^{A}$, we obtain
additional boundary field equations that propagate the boundary spinors along
$\mathcal{N}$. The resulting boundary field equations can be written in the
following compact form

$\displaystyle\bar{m}\wedge\Big{(}D-\frac{1}{2}A\Big{)}\ell^{A}$
$\displaystyle=\frac{\theta}{2}e^{A}\wedge\bar{m},$ (65)
$\displaystyle\Big{(}D+\frac{1}{2}A\Big{)}\wedge(e_{A}\wedge\bar{m})$
$\displaystyle=0.$ (66)

If we take into account the decomposition of the spin connection with respect
to the spin frame $(k^{A},\ell^{A})$, it is easy to check that these boundary
field equations are satisfied on any embedded null surface. This is important
to note, because it means that the boundary field equations impose no
constraints on the free data along $\mathcal{N}$. In other words, the boundary
field equations (65) and (66) can be satisfied on any 3-dimensional null
boundary.

Given the boundary conditions and boundary field equations, we infer the
symplectic potential. On a partial Cauchy surface $M$ that intersects
$\mathcal{N}$ in a cross section $S$, there is a contribution from the bulk
and an additional boundary contribution

$\Theta_{\Sigma}=\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\left[\int_{\Sigma}\Sigma_{AB}\wedge\delta
A^{AB}-\oint_{\partial
M}e_{A}\wedge\bar{m}\,\delta\ell^{A}\right]+\mathrm{cc}.$ (67)

The expression for the symplectic structure on the null surface simplifies if
we restrict ourselves to only those variations $\delta[\cdot]$ that preserve
the ruling of the null surface, i.e. $\delta\ell^{a}\sim\ell^{a}$, see
Wieland:2021vef . Under this condition, we obtain

$\Theta_{\mathcal{N}}=-\frac{1}{8\pi\gamma
G}\int_{\mathcal{N}}\varepsilon\wedge\delta
A^{(\gamma)}-\frac{\mathrm{i}}{8\pi\gamma
G}\int_{\mathcal{N}}\Big{[}(\gamma+\mathrm{i})\ell_{A}D\ell^{A}\wedge\delta(k\wedge\bar{m})-\mathrm{cc}.\Big{]},$
(68)

where $k_{a}\in\Omega^{1}(\mathcal{N}:\mathbb{R})$ is the Ehresmann connection
such that $k_{a}\ell^{a}=1$. Notice that the expression for the symplectic
potential has the familiar structure from the loop gravity holonomy flux
algebra, with fluxes $\varepsilon$, $k\wedge m$, and $k\wedge\bar{m}$ that are
dual to 1-forms or connections. In fact, the area 2-form $\varepsilon$ is dual
to the U$(1)$ connection

$A^{(\gamma)}=\Gamma+\gamma\omega.$ (69)

The connection $A^{(\gamma)}$ is an abelian version of the SU$(2)$
Ashtekar–Barbero connection, namely the sum of the intrinsic U$(1)$ connection
$\Gamma$ and the extrinsic curvature $\omega$. In fact, given the U$(1)$ gauge
transformations, it is easy to check that $A^{(\gamma)}$ transforms as an
abelian U$(1)$ connection. If, on the other hand, we rescale $A^{(\gamma)}$ by
$\gamma^{-1}$, we obtain an abelian connection for the internal boost symmetry
$\ell^{a}\rightarrow\mathrm{e}^{\lambda}\ell^{a}$.

Let us now discuss two important limiting cases.

In the case of an isolated horizon
$\mathcal{N}\simeq\mathbb{S}^{2}\times\mathbb{R}$, we have no gravitational
radiation crossing the null boundary. Shear and expansion of the null
generators $\ell^{a}$ vanish. Therefore, the second term in the expression for
the pre-symplectic potential (68) disappears, and we are left with the first
term. Since the area element $\varepsilon_{ab}$ is conserved, the entire
geometry becomes effectively 2-dimensional. Given an arbitrary 2-dimensional
cross-section $S\simeq\mathbb{S}^{2}$ of $\mathcal{N}$, the symplectic
potential is now simply given by

$\Theta_{\,\mathrm{IH}}=\frac{1}{8\pi\gamma
G}\oint_{S}\varepsilon\,h^{-1}\delta h,$ (70)

where $h(x)=\mathrm{Pexp}(-i\int_{\gamma(\vec{x})}A^{(\gamma)}$ is the U$(1)$
holonomy along the null generator based at $\vec{x}$. This defines the phase
space $[T^{\ast}\mathrm{SU}(1)]^{S}$. Notice that the area 2-form is the
generator of U$(1)$ gauge transformations, hence area is quantized.

Another important limiting case is when we send $\mathcal{N}$ to null infinity
$\mathcal{I}^{\pm}$. In this case, the symplectic potential (68) returns the
usual radiative phase space Wieland:2020gno .

###  5.3 Spinor representation

From the previous analysis we see that at the corner of null surface, where
the $\Sigma$ intersects the null surface, we obtain the Poisson brackets

$\displaystyle\big{\\{}\tilde{\pi}_{A}(\vec{x}),\ell^{B}(\vec{y})\big{\\}}_{S}=\delta^{B}_{A}\tilde{\delta}_{S}(\vec{x},\vec{y}),\qquad\big{\\{}\tilde{\bar{\pi}}_{A^{\prime}}(\vec{x}),\bar{\ell}^{B^{\prime}}(\vec{y})\big{\\}}_{S}=\delta^{B^{\prime}}_{A^{\prime}}\tilde{\delta}_{S}(\vec{x},\vec{y}),$
(71)

where we introduced the momentum spinor $\tilde{\pi}_{A}$ as the pull-back of
the boundary field $e_{A}\wedge\bar{m}$ to the cross-section $S$, i.e.

$\tilde{\pi}_{A}=\frac{\mathrm{i}}{8\pi\gamma
G}(\gamma+\mathrm{i})\varphi^{\ast}_{S}(e_{A}\wedge\bar{m}).$ (72)

A number of Dirac observables can be written in terms of these boundary
fields. For example, the generator of infintesimal tangential diffeomorphism
associated with $\xi^{a}\in T\Sigma$, is given by

$P_{\xi}=\oint_{S}\xi^{a}[\tilde{\pi}_{A}D_{a}\ell^{A}]+\mathrm{cc}.$ (73)

In the same way, we find the generators of complexified U$(1)$
transformations,

$\displaystyle\tilde{\pi}_{A}\ell^{A}=\tilde{K}+\mathrm{i}\,\tilde{L}$
$\displaystyle=\frac{\gamma+\mathrm{i}}{8\pi\gamma G}\varepsilon,$ (74)

where $\tilde{L}$ is a $U(1)$ generator, and $\tilde{K}$ is a boost generator
and $\varepsilon$ is the area form. In terms of harmonic oscillators,
$\tilde{L}$ is the difference of two number operators, and $\tilde{K}$ is a
two-mode squeeze operator, see Wieland:2017cmf . The spectrum of $\tilde{K}$
is continuous, the spectrum of $\tilde{L}$ is discrete. The area 2-form (50)
is the contraction of the boundary spinors, At the kinematical level, this
surface density is complex. For the area to be real, we have to impose the
simplicity constraint $\tilde{K}-\gamma\tilde{L}=0$.

When acting on any physical state, the area density will be proportional to
$\tilde{L}$, which has a discrete spectrum. Hence area is quantized. This
selects a representation of the canonical commutation relations. In this way,
the quantization of area on the null surface boundary agrees with our analysis
in section 3.2. Both results only rely on the compactness of the orbits
generated by the area density $\varepsilon\propto\tilde{L}$. If we use a spin
network representation, the boundary spinors $(\tilde{\pi}_{A},\ell^{A})$ will
be excited at a number of punctures. We could also use, however, a more
standard Fock representation Wieland:2017cmf , where $\tilde{\pi}_{A}$ and
$\ell^{A}$ are continuous. In either case, the result is the same. To
summarise the the Barbero – Immirzi parameter, deforms the boundary symmetries
in such a way that the area spectrum becomes discrete.

##  6 Summary and perspectives

In this review, we have discussed three main ideas and related results. The
first result is about the classical phase space of general relativity in the
presence of inner boundaries of a spacelike or null slice. When there is a
boundary, an otherwise unphysical gauge redundancy in the bulk turns into a
physical degree of freedom intrinsic to the boundary. The corresponding
conjugate elements on phase space are the canonical generators of the boundary
symmetries. It is for this reason that the generators of diffeomorphisms turn
into surface charges that measure quasi-local observables of the gravitational
field. These observables generalise the ten Minkowskian Noether charges for
the Poincaré group into an infinite set of boundary charges that satisfy the
so-called corner symmetry algebra. We then reviewed a number of past results
from loop quantum gravity, where the boundary conditions and resulting
boundary symmetries have led to important results, from local entanglement
BarberoG:2022ixy  to the entropy counting of black hole horizons
Bianchi:2023avf , and to renormalization and the continuum limit of the theory
Asante:2022dnj . Then, we explained one of the key results of the corner
symmetry program thus far, namely the fact that the loop quantization of area
can be understood from the quantization of the boundary modes alone
Wieland:2021vef ; Wieland:2017cmf ; Freidel:2020ayo . Since there is a unique
area element on a null surface, the construction becomes particularly
transparent for such boundaries. The area 2-form at a 2-dimensional cross
section turns into a surface charge, whose spectrum is discrete. From the
perspective of the gravitational phase space in a finite region, this surface
charge is a complete Dirac observable. The discreteness of the spectrum of the
corresponding quantum operator can be traced back to the presence of the
Barbero–Immirzi parameter in the action. The Barbero–Immirzi parameter does
not affect the commutation relations for the radiative modes in the bulk, but
it deforms the algebra of boundary symmetries and their generators. This
discreteness has been proven to follows from representing the corner symmetry
as in (21). Finally, we have also reviewed a number of results on how the
Einstein equations induce a field theory for a Carrollian fluid on the null
surface boundary and how this can also be described naturally within the
tetrad formulation used in loop quantum gravity. This furthermore connects
seamlessly with the description of isolated horizons developed in the LQG
black hole entropy counting.

Some developments were not covered in our review. For example, there is a
growing body of research in quantum foundations and quantum information
science on quantum reference frames and their relation with edge modes
Goeller:2022rsx ; Vanrietvelde:2018pgb ; Carrozza:2022xut ; Kabel:2023jve .
Quantum reference frames are dual (in the phase space sense of the word) to
the generators of coordinate transformations. Recent results by Brukner and
collaborators gave an operational definition of _quantum reference frames_ ,
see e.g. Giacomini:2017zju . This definition includes a prescription for how
to _jump_ between quantum reference frames and identify properties of a
quantum state that remain invariant under such transformations. It turns out,
for example, that quantum entanglement is an observer-dependent notion and
depends on the frame of reference. Stenghthening further the connection
between corner symmetry and quantum reference frames is a promising area of
development.

Altogether, the idea we arrive at is that of local holography. While AdS
holography on asymptotic timelike boundaries is made possible by reflecting
boundary conditions, which prevent from including radiation Compere:2020lrt .
Celestial-carrollian Holography allows the inclusion of radiation, and is
adapted to the description of physically realistic setups Strominger:2017zoo ;
Raclariu:2021zjz ; Pasterski:2021rjz ; Fareghbal:2013ifa ; Donnay:2022aba ;
Donnay:2021wrk . Instead of considering null or timelike boundaries at
infinity, local holography aims at obtaining constraints on quantum gravity
from the local corner symmetry structure associated with any arbitrary
subregion defined by an entangling cut. This idea should not necessarily be
viewed as a new proposal for quantum gravity, but rather as a consistency
framework that brings a universal bottom-up perspective for all candidate
approaches. In this review, we have discussed the lessons from and for loop
quantum gravity, but it will be interesting and important to discuss local
holography in other contexts as well.

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