aainstitutetext: David Rittenhouse Laboratory, University of Pennsylvania,
209 S.33rd Street, Philadelphia, PA 19104, USAbbinstitutetext: Santa Fe
Institute,
1399 Hyde Park Road, Santa Fe, NM 87501, USAccinstitutetext:  Theoretische
Natuurkunde, Vrije Universiteit Brussel,
Pleinlaan 2, B-1050 Brussels, Belgium

#  The entropy of finite gravitating regions

Vijay Balasubramanian a Charlie Cummings<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We develop a formalism for calculating the entanglement entropy of an
arbitrary spatial region of a gravitating spacetime at a moment of time
symmetry. The crucial ingredient is a path integral over embeddings of the
region into the overall spacetime, interpretable as a sum over the edge modes
associated with the region. We find that the entanglement entropy of a
gravitating region equals the minimal surface area among all regions that
enclose it. This suggests a notion of “terrestrial holography” where regions
of space can encode larger ones, in contrast to the standard form of
holography, in which degrees of freedom on the celestial sphere at the
boundary of the universe encode the interior.

##  1 Introduction

Gravity has a deep and mysterious connection to thermodynamics: the entropy of
some gravitating systems is geometrized as the area of a distinguished surface
in spacetime. For example, black holes appear to carry a coarse-grained
entropy of $S=A/4G_{N}$ where $A$ is the area of the horizon and $G_{N}$ is
Newton’s constant Bekenstein:1973ur ; hawkingBH . Cosmological Gibbons:1977mu
and Rindler Jacobson_2003 ; Unruh:1976db  horizons associated to accelerating
observers also obey a similar relation, although the statistical
interpretation of these entropies remains obscure. Associating an entropy to
causal horizons can even be used to _derive_ the Einstein equations
Jacobson_1995 . In addition, in the context of the holographic AdS/CFT
correspondence, certain extremal surfaces measure entanglement entropy between
the quantum degrees of freedom associated to opposite sides of the surface
Ryu_2006 ; Faulkner:2013ana ; Hubeny:2007xt ; Engelhardt:2014gca . These
relations lead us to ask whether there is a relation between entropy and area
for arbitrary subregions of a gravitating spacetime.

To answer this question, we propose a path integral for gravity in any finite
subregion of a global spacetime which is invariant under diffeomorphisms
inside that region. We use this path integral to argue that the entanglement
entropy of an open region $a$ at a moment of time symmetry is determined by a
(possibly larger) region $E(a)$ defined by: (1) $a\subseteq E(a)$, and (2)
$E(a)$ minimizes the generalized entropy among all regions satisfying (1).
This argument makes no assumption about the cosmological constant. For
AdS/flat spacetimes in the vacuum, and taking $a$ to have a single component,
$E(a)$ will simply be the region of minimal surface area containing $a$. The
result also implies the entropy of the de Sitter vacuum vanishes, at least in
the absence of external observers.

Three sections follow. In Sec. 2, we review and construct the technical tools
needed to define subregions of gravitating spacetimes. We then use these tools
to define and renormalize the entropy of such a subregion. In Sec. 3, we
propose a gravitational path integral for a finite region of space. This path
integral includes an integral over all embeddings of the region into
spacetime. We then take a saddlepoint approximation to compute the leading
order behavior of the entropy in the small coupling ($G_{N}\to 0$) limit. We
conclude in Sec. 4 with a discussion of the interpretations and implications
of our entropy formula.

##  2 The entropy of subregions in gravitating spacetimes

Consider a globally hyperbolic manifold $\mathcal{M}$. Let $\Sigma$ be a
Cauchy slice at a moment of time symmetry, and denote by $a$ any open region
of $\Sigma$ with $a^{\prime}=\text{int}(\Sigma\setminus a)$ the complement of
$a$ in $\Sigma$ minus the boundary of $a$. Also define the co-dimension 1 (co-
dimension 2 in spacetime) corner of $a$ as $\partial
a:=\bar{a}\cap\bar{a}^{\prime}$ where $\bar{a},\bar{a}^{\prime}$ are the
closures of $a,a^{\prime}$. Intuitively, $\partial a$ is the boundary between
$a,a^{\prime}$.

Let $\mathcal{H}_{\Sigma}$ be the Hilbert space of quantum gravity and matter
defined on $\Sigma$. This theory may or may not be known explicitly. In
AdS/CFT, $\mathcal{H}_{\Sigma}$ is isomorphic to the Hilbert space of a CFT in
one lower dimension Maldacena_1999 , but in flat or de Sitter spacetimes, less
is known about the structure of $\mathcal{H}_{\Sigma}$.111See
pasterski2021celestial ; Pasterski_2021 ; strominger2018lectures ;
raclariu2021lectures  for work towards understanding this Hilbert space in
flat space, and Strominger_2001 ; Balasubramanian:2001nb ; witten2001quantum ;
chakraborty2023hilbert  for the case of de Sitter spacetimes. In the small
Newton constant ($G_{N}$) limit, however, a great deal is known about
$\mathcal{H}_{\Sigma}$: it contains perturbative gravitons and matter
propagating on curved backgrounds. Take $\ket{\Psi}\in\mathcal{H}_{\Sigma}$ to
be a state with a well defined semi-classical description and a background
geometry $g^{0}_{\mu\nu}$ in the $G_{N}\to 0$ limit. In this limit, we can
define an (approximate) algebra of $\mathcal{A}_{\Sigma}$ of observables which
act on $\mathcal{H}_{\Sigma}$ Almheiri_2015 . $A_{\Sigma}$ includes both
matter fields and perturbative gravitons $h_{\mu\nu}$ such that the total
metric is $g_{\mu\nu}=g^{0}_{\mu\nu}+\sqrt{32\pi G_{N}}h_{\mu\nu}$, with the
prefactor chosen to normalize the graviton kinetic term. Furthermore,
$\mathcal{H}_{\Sigma}$ has a $\text{Diff}(\Sigma)$ diffeomorphism gauge
symmetry which corresponds to coordinate changes of the perturbative fields
defined on $\Sigma$. In the language of AdS/CFT, the subspace spanned by
$\mathcal{A}_{\Sigma}$ acting on $\ket{\Psi}$ defines a code subspace within
the full non-perturbative Hilbert space $\mathcal{H}_{\Sigma}$ Almheiri_2015 .
Our goal is to compute the entropy of the reduced density matrix
$\rho_{a}\sim\tr_{a^{\prime}}\outerproduct{\Psi}{\Psi}$, associated with a
bulk region $a$, in the weak coupling $G_{N}\to 0$ limit. A precondition for
this reduced state $\rho_{a}$ to be well defined is the existence of a bi-
partition of the Hilbert space into
$\mathcal{H}_{\Sigma}\sim\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}$.
However, this factorization is subtle in quantum gravity for two distinct
reasons.

First, there are UV divergence in the effective field theory (EFT) that we are
considering, associated with the sharp cut $\partial a$ between $a,a^{\prime}$
Haag:1992hx . We expect these divergences to be cured in the high energy
completion defining $\mathcal{H}_{\Sigma}$ in something like string theory.
RG-flow from this complete theory leads to our EFT below a cutoff. With this
is mind, we will assume that an appropriate regularization and renormalization
makes the path integral that we will consider well defined. Second, if there
is a gauge symmetry in the theory, constraints can correlate $a$ and
$a^{\prime}$ . This is even true for $U(1)$ lattice gauge theory Donnelly_2012
. We review this lack of factorization in the next two sections, following the
discussions in Casini_2014 ; Donnelly_2012 ; Donnelly_2016 .

###  2.1 Edge modes in lattice gauge theory

Consider a $U(1)$ lattice gauge theory, which manifestly has no UV
divergences, and let $\mathcal{H}_{\Sigma}$ be the gauge invariant Hilbert
space of the full lattice. As in the continuum, we separate the lattice into a
subset of nodes $a$ and its complement $a^{\prime}$, with $\partial a$ cutting
the links between them.

Suppose a Wilson line straddles the corner $\partial a$. Then tracing out
$a^{\prime}$ will leave behind an open Wilson line with endpoints on $\partial
a$. From the perspective of $a$, the endpoints at the cut will carry charges
transforming in some representation of the gauge group at $\partial a$. Since
the original Hilbert space was gauge invariant, these open Wilson lines
constitute new degrees of freedom called “edge modes”. The edge modes are
crucial in computations of entanglement entropy in gauge theories
Donnelly_2012 . Indeed, the total Hilbert space describing the degrees of
freedom on $a$ is the “extended Hilbert space”
$\widetilde{\mathcal{H}}_{a}=\mathcal{H}_{a}\otimes\mathcal{H}_{e.m.}$, where
$\mathcal{H}_{a}$ has support only inside $a$, and $\mathcal{H}_{e.m.}$ is the
edge mode Hilbert space supported on the corner.

Figure 1: The full Hilbert space $\mathcal{H}_{\Sigma}$ is defined on the
complete lattice. When restricting to a subset $a$ of the nodes on the
lattice, described by a reduced Hilbert space $\mathcal{H}_{a}$, one must also
include additional degrees of freedom to account for the endpoints of Wilson
lines not present in $\mathcal{H}_{\Sigma}$. These new degrees of freedom are
the edge modes in $\mathcal{H}_{e.m.}$.

To explain how to recover the original Hilbert space, we must first understand
the extended Hilbert space. The total gauge symmetry of the lattice $\Sigma$
is $U(1)_{\Sigma}$, i.e., $U(1)$ transformations at every link on the lattice.
This gauge group splits as

$\displaystyle U(1)_{\Sigma}=U(1)_{a}\oplus U(1)_{\partial a}\oplus
U(1)_{a^{\prime}}$ (1)

where $U(1)_{\partial a}$ are gauge transformations transforming links along
the cut, while $U(1)_{a}$ consists of gauge transformations of links with both
endpoints in $a$ (and similarly for $U(1)_{a^{\prime}}$). The extended Hilbert
space $\widetilde{\mathcal{H}}_{a}$ has a gauge symmetry $U(1)_{a}$ that it
inherits from $\mathcal{H}_{a}$, and a _global_ symmetry $U(1)_{\partial a}$
acting on $\mathcal{H}_{e.m.}$. The latter is a global symmetry because the
open Wilson lines transform in a non-trivial representation of $U(1)_{\partial
a}$. In fact, a more careful consideration shows that $\mathcal{H}_{e.m.}\cong
L^{2}(U(1)_{\partial a})$.222To see this, first note that the edge mode at
each link is simply a choice of phase Donnelly_2016 . Thus, if there are $n$
links on the cut $\partial a$, $\mathcal{H}_{e.m.}\cong L^{2}(U(1))^{\otimes
n}\cong L^{2}(U(1)^{\otimes n})\equiv L^{2}(U(1)_{\partial a})$ because
$L^{2}(A)\otimes L^{2}(B)\cong L^{2}(A\otimes B)$.  This makes the global
symmetry more transparent, it is simply the regular representation of
$U(1)_{\partial a}$ on $L^{2}(U(1)_{\partial a})$. We can reason similarly
about $\widetilde{\mathcal{H}}_{a^{\prime}}$

Because the extended Hilbert space has a global symmetry $U(1)_{\partial a}$,
it splits into superselection sectors of charge eigenstates. The
representations of $U(1)_{\partial a}$ are given by $\vec{Q}$, a choice of
$U(1)$ charge for every link on the cut, and so
$\widetilde{\mathcal{H}}_{a}=\oplus_{\vec{Q}}\widetilde{\mathcal{H}}_{\vec{Q},a}$.
Thus, the most general density matrix in $\widetilde{\mathcal{H}}_{a}$ takes
the form $\rho_{a}=\sum_{\vec{Q}}p_{\vec{Q}}\rho_{\vec{Q}}$, where
$\sum_{\vec{Q}}p_{\vec{Q}}=1$ and each charge-sector density matrix has unit
trace. $\rho_{\vec{Q}}$ is any state compatible with the boundary
representation. For example, $\rho_{0}$ is a possibly mixed state over Wilson
loops which do not touch the boundary. We will return to the implications of
this splitting below.

Let us now consider how to recover the original Hilbert space
$\mathcal{H}_{\Sigma}$. First, note that
$\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}\subset\mathcal{H}$ because
$\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}$ contains no Wilson lines
straddling the cut ${\partial a}$. This was why we had to introduce the edge
modes in the first place. On the other hand,
$\mathcal{H}_{\Sigma}\subset\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$
because
$\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$
contains states with open Wilson lines on the cut. We can recover
$\mathcal{H}_{\Sigma}$ if we restrict to states with a trivial charge under
$U(1)_{\partial a}$, that is,

$\displaystyle\mathcal{H}_{\Sigma}=\Pi_{0}[\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}]\Pi_{0}$
(2)

where $\Pi_{0}$ projects states onto the subspace satisfying
$\vec{Q}_{a}+\vec{Q}_{a^{\prime}}=0$. We now see why the total Hilbert space
is gauge invariant under the full $U(1)_{\Sigma}$ even if it is built from
states with global charges in
$\widetilde{\mathcal{H}}_{a},\widetilde{\mathcal{H}}_{a^{\prime}}$. While
$\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$ has a
global symmetry group $U(1)_{\partial a}\oplus U(1)_{\partial a}$, restricting
to the zero charge subspace identifies the proper links with each other and
gauges the remaining global symmetry. Alternatively, $\mathcal{H}_{\Sigma}$ is
not the same as $\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}$ because the
latter is missing the twisted sectors required for a consistent gauging of a
global symmetry Vafa:1989ih ; Dijkgraaf:1989hb .

The abelian criterion $\vec{Q}_{a}+\vec{Q}_{a^{\prime}}=0$ indicating that we
have gauged $U(1)_{\partial a}$ has a non-abelian generalization. Letting $G$
be a compact Lie group for a non-abelian lattice Yang-Mills theory, the full
gauge group splits in the same way as $U(1)$: $G_{a{a^{\prime}}}=G_{a}\oplus
G_{\partial a}\oplus G_{a^{\prime}}$. Acting on the extended Hilbert space
$\widetilde{\mathcal{H}}_{a}=\mathcal{H}_{a}\otimes\mathcal{H}_{e.m.}$, where
$\mathcal{H}_{e.m}$ is the Hilbert space of the non-abelian edge modes,
$G_{a}$ is a gauge symmetry and $G_{\partial a}$ is a global symmetry acting
on the endpoints of open Wilson lines ending on the cut. As above, let $g\in
G_{\partial a}$ be a non-abelian gauge transformation on the cut. Given a
state
$\ket{\psi_{a}}\otimes\ket{\psi_{a^{\prime}}}\in\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$,
we recover a gauge-invariant state on the full Cauchy slice by projecting onto
the equivalence classes under the diagonal action of $G_{\partial a}$. That
is, we identify $\ket{\psi_{a}}\otimes\ket{\psi_{a^{\prime}}}\sim
g\ket{\psi_{a}}\otimes g\ket{\psi_{a^{\prime}}}$ as being the same state. This
projection is sometimes referred to as the “entangling product” Donnelly_2016
, or the Hilbert space of co-invariants Chandrasekaran_2023 , and is denoted

$\displaystyle\mathcal{H}_{\Sigma}=\widetilde{\mathcal{H}}_{a}\otimes_{G_{\partial
a}}\widetilde{\mathcal{H}}_{a^{\prime}}$ (3)

The entangling product ensures continuity of the bulk fields across the cut.

Let $R$ be an irreducible representation (irrep) of $G_{\partial a}$. Since
$G_{\partial a}$ is a global symmetry for the extended Hilbert space, the most
general state in $\widetilde{\mathcal{H}}_{a}$ in the non-abelian case would
seem to be given by $\rho_{a}=\sum_{R}p_{R}\,\rho_{R}\otimes\sigma_{R}$, where
$\rho_{R}$ and $\sigma_{R}$ are density matrices on $\mathcal{H}_{a}$ and
$\mathcal{H}_{e.m.}$, respectively. Here $\rho_{R}$ is a possibly mixed state
over all bulk states compatible with a representation $R$ on the cut, similar
to the $U(1)$ case above. Even in the non-abelian case
$\mathcal{H}_{e.m.}\cong L^{2}(G_{\partial a})$333The edge mode corresponds to
a choice of trivialization of the gauge function at the boundary Donnelly_2016
. Thus, the proof works exactly the same as in the Abelian case. ; so we can
utilize the Peter-Weyl theorem PeterWeyl , a powerful result in non-abelian
harmonic analysis showing that

$\displaystyle L^{2}(G_{\partial
a})\cong\bigoplus_{R}\,\bigoplus_{n=1}^{d_{R}}V_{R,n}\,,$ (4)

where $V_{R,n}$ is the vector space of the irrep $R$ with dimension $d_{R}$,
and $n$ is a multiplicity index, ranging from $1,\cdots,d_{R}$. Note that the
latter is _not_ counting the number of states in a particular representation,
but the number of times the representation appears in the sum. The Peter-Weyl
theorem shows that each irrep $R$ has a degeneracy of $d_{R}$ in the
decomposition.444This is shown by computing the characters of the regular
representation on $L^{2}(G_{\partial a})$ and the orthogonality of characters
of irreps. One can view this theorem as a generalization of the Fourier
transform for compact groups or abelian locally compact groups (of which, the
usual Fourier transform is a special case). The existence of this
decomposition is equivalent to existence the $G_{\partial a}$ global symmetry
in the extended Hilbert space.

Naively, $\sigma_{R}$ seems to be unconstrained. However, let $\rho_{\Sigma}$
be a state in the gauge invariant Hilbert space (3). Because
$\mathcal{H}_{\Sigma}\subset\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$,
we can embed $\rho_{\Sigma}$ into
$\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$ and
define

$\displaystyle\rho_{a}=\tr_{\widetilde{\mathcal{H}}_{a^{\prime}}}(\rho_{\Sigma})$
(5)

where the trace is taken in the larger
$\widetilde{\mathcal{H}}_{a}\otimes\widetilde{\mathcal{H}}_{a^{\prime}}$
because this is the Hilbert space with a well defined biparititon. Also,
$\rho_{a}$ must commute with the generators of $G_{\partial a}$ because
$G_{\partial a}$ is a gauge symmetry of $\mathcal{H}$. By Schur’s lemma, this
implies that $\sigma_{R}$ is proportional to the identity for each $R$.
Therefore, the most general density matrix which arises from a globally gauge
invariant pure state takes the form Donnelly_2012

$\displaystyle\rho_{a}=\sum_{R}\sum_{n=1}^{d_{R}}p_{R,n}\,\rho_{R,n}\otimes\frac{1}{d_{R}}\text{Id}_{R,n}$
(6)

where $\text{Id}_{R,n}$ is the identity in representation $R$. This formula
will have a profound impact on the computation of the entropy of a subregion
in gravity.

###  2.2 Edge modes in gravity

Gravity is a gauge theory for the diffeomorphism group, and thus also has edge
modes Donnelly_2016 , but there are some key differences that will play a
crucial role in our entropy computation. As described above, given a static
slice $\Sigma$, there is a Hilbert space $\mathcal{H}_{\Sigma}$ with a gauge
group $\text{Diff}(\Sigma)$. Given subregions $a,a^{\prime}$, induced by the
corner $\partial a$, we would like to bipartition the Hilbert space into
$\mathcal{H}_{\Sigma}\sim\mathcal{H}_{a}\otimes\mathcal{H}_{a^{\prime}}$.
There are again two issues with this bipartition: UV divergences and
constraints from the $\text{Diff}(\Sigma)$ gauge symmetry. As discussed above,
we assume that the UV divergences have been regulated in some way, and later
discuss the removal of the regulator.

To deal with the constraints, we first decompose the gauge group as

$\text{Diff}(\Sigma)=\text{Diff}(a)\oplus\text{ECS}\oplus\text{Diff}(a^{\prime})$
(7)

Here, $\text{Diff}(a)$ and $\text{Diff}(a^{\prime})$ are diffeomorphisms with
support within $a,a^{\prime}$ respectively and vanishing on the corner. ECS is
the equivalence class of the remaining diffeomorphisms, called the “extended
corner symmetry” group555See Ciambelli_2023  for more information about the
importance of this group in gravity. Freidel_2021 . ECS has a Lie algebra
given by

$\mathfrak{ecs}=\mathfrak{diff}(\partial
a)\ltimes(\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^{2})_{\partial a}$ (8)

independently of the size and topology of $a$. The $\mathfrak{diff}(\partial
a)$ algebra describes coordinate reparameterizations of the corner. Now let
$p\in\partial a$ be a point on the corner and $x^{\pm}$ be coordinates on the
two dimensional plane normal to $\partial a$ that intersect $p$, such that $p$
is defined by $x^{\pm}=0$. Think of this as the plane spanned by two null rays
leaving $p$. The $\mathfrak{sl}(2,\mathbb{R})$ algebra describes
transformations of this plane which keep $p$ fixed and have unit determinant.
This includes boosts around $p$, but other transformations are possible as
well. The $\mathbb{R}^{2}$ part generates translations of the point $p$
itself:666This is the analog of supertranslations in the BMS group at infinity
in flat space. together, $\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$
form a kind of Euclidean group of the normal plane. The notation
$(\cdots)_{\partial a}$ means that we are actually considering a copy of this
Euclidean group at each point $p$.777More precisely, it is the symmetry group
of the normal bundle over $\partial a$. Remarkably, ECS has no central
extensions, which simplifies its representations Ciambelli_2022 . If gravity
were just like Yang-Mills theory, only $\mathfrak{diff}(\partial a)$ would
appear. The presence of the extra terms
$(\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^{2})_{\partial a}$ has no
analog in Yang-Mills, and correspond to diffeomorpisms which affect the time
and radial coordinates describing the slice $\Sigma$. In the lattice model,
$\mathfrak{diff}(\partial a)$ corresponds to gauge transformations on each
link, but $(\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^{2})_{\partial a}$
captures changes of the lattice itself. This is a manifestation of the famous
“problem of time” in quantum gravity, because $\text{Diff}(a)$,
$\text{Diff}(a^{\prime})$, and $\text{Diff}(\partial a)$ all keep $\Sigma$
fixed, while the extra
$(\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^{2})_{\partial a}$ factors
include time translations of the corner within the global spacetime. These
extra terms in ECS will play a crucial role in the computation of the entropy.

Next we should define more clearly what we mean by a subregion $a$. Since
diffeomorphisms are a gauge symmetry of gravity, simply giving the coordinate
values for the boundary of a region is not enough to define it, as this is not
a gauge invariant quantity without also fixing the coordinate system of the
spacetime, i.e., choosing a gauge. We will consider $a\subset\Sigma$ as an
open submanifold of $\Sigma$. Alternatively, we can view $a$ as a manifold in
its own right, independent of $\Sigma$. We will abuse notation and refer to
this abstract subregion also as $a$ from now on. We then view the original
region in $\Sigma$ as a particular embedding
$\phi_{e}:a\hookrightarrow\Sigma$. Taking this perspective, the full gauge
invariant specification of $a\hookrightarrow\Sigma$ is given by $(a,\phi)$,
where $\phi$ refers to an embedding of $a$ into $\Sigma$. This is because a
diffeomorphism of $\Sigma$ will also transform the map $\phi$ to keep the
embedding invariant. Explicitly keeping track of $\phi$ as a dynamical field,
essentially a Stuecklberg field for diffeomorphisms, as well as the fields on
the fixed model subregion $a$ is therefore sufficient for specifying
$a\hookrightarrow\Sigma$, as shown in Fig. 2. In fact, this $\phi$ is
precisely the edge mode of gravity Ciambelli_2021 ; Ciambelli_2022 . To see
this, let $\phi_{e}:a\hookrightarrow\Sigma$ be an embedding. Then, let
$g\in\text{ECS}$ be a diffeomorphism. Then $\phi_{g}:=g\cdot\phi_{e}$ is also
an embedding, and distinct from $\phi_{e}$ unless $g=e$. It is easy to see
from this definition that the set of $\phi_{g}$ form a representation of ECS,
as do the edge modes, so it is plausible that this is a correct way to view
them. A more careful analysis reveals this to indeed be the case
Ciambelli_2021 ; Ciambelli_2022 .

Figure 2: $a$ is a manifold with boundary, independent of $\Sigma$. By
specifying an embedding $\phi$, we can think of $a$ as a subset of $\Sigma$ in
a gauge invariant way.

The extended Hilbert space in gravity is explicitly given by
$\widetilde{\mathcal{H}}_{a}=\mathcal{H}_{a}\otimes L^{2}(ECS)$,888The edge
mode Hilbert space $L^{2}(ECS)$ has been considered previously in the
literature in the form of “quantum reference frames”
delahamette2021perspectiveneutral . It would be interesting to understand how
the results of this paper inform their constructions. where $\mathcal{H}_{a}$
represents the $\text{Diff}(a)$-invariant Hilbert space of bulk operators
(including gravitons) localized to $a$, and the $L^{2}(ECS)$ factor represents
wave functions over possible embeddings of the region $a$ into the spacetime.
Recall that we already cut off $\mathcal{H}_{a}$ to cure UV divergences and
make it well defined, essentially by excluding trans-Planckian fluctuations.
We will similarly regulate ECS so that we only keep the edge modes necessary
to match with the fluctuations below the cut off in the bulk of $a$.999More
specifically, it is the trans-Planckian fluctuations of the
$\mathfrak{diff}(\partial a)$ and $(\cdots)_{\partial a}$ factors within
$\mathfrak{ecs}$. This renders the regulated ECS locally compact so that we
can use its Haar measure to define an inner product on $L^{2}(ECS)$. We will
proceed with this regulator in place and remove it at the end.

Similarly to Yang-Mills theory, ECS acts as a global symmetry group for
$\widetilde{\mathcal{H}}_{a}$. We would like to then conclude that
$\widetilde{\mathcal{H}}_{a}\sim\oplus_{R}(\widetilde{\mathcal{H}}_{a})_{R}$,
where $R$ is an irrep of ECS (regulated to be locally compact). However,
mathematically, such a decomposition requires ECS to be “amenable” amenable .
This subtlety did not arise in Yang-Mills theory because all compact groups
are amenable amenable . Mathematically, amenability means that for all
$\epsilon>0$, there is a state $\ket{\alpha}$ such that
$|U(g)\ket{\alpha}-\ket{\alpha}|<\epsilon$ for all
$U(g)\in\mathcal{U}(L^{2}(ECS))$, the unitary operators on $L^{2}(ECS)$
amenable . This means the trivial representation is “weakly contained” within
$L^{2}(ECS)$, i.e., there is a state $\ket{\alpha}$ which ECS leaves
approximately invariant. If $G$ is a compact group with Haar measure
$d\mu_{g}$, then we could take

$\ket{\alpha}=\frac{1}{\text{Vol(G)}}\int d\mu_{g}\,U(g)\ket{\Psi}$

for any state $\ket{\Psi}$, but non-compact groups such as ECS can be more
complicated because there is no finite $\text{Vol}(ECS)$ to normalize the
group average. We therefore need a way of regulating the group, performing
calculations, and taking the limit at the end. As an example of such a
limiting definition, there is no normalizable constant function on
$L^{2}(\mathbb{R})$, but one can approximate this function arbitrarily well
with a gaussian by taking the limit of large variance. A physical way to
describe this requirement of amenability is that there exists an approximate
vacuum state for the wave functions over embeddings such that
$U(g)\ket{\alpha}\approx\ket{\alpha}$ for all $g\in ECS$. Such a vacuum state
manifestly satisfies the above condition for amenability. To this end, we
regulate the Hilbert space by truncating the dimension of the allowed
representations of ECS

$\widetilde{\mathcal{H}}^{\Lambda}_{a}\equiv\bigoplus^{\Lambda}_{R}\,\bigoplus_{n=1}^{d_{R}}\left(\widetilde{\mathcal{H}}_{a}\right)_{R,n}\,,$
(9)

where $\oplus_{R}^{\Lambda}$ means that we sum on irreps $R$ of ECS such that
$d_{R}\leq\Lambda$, and $n$ is a multiplicity index, ranging from
$1,\cdots,d_{R}$. Note that the latter is _not_ counting the number of states
in a particular representation, but the number of times the representation
appears in the sum. As explained above in the case of Yang-Mills, this
multiplicity actually equals the dimension of the representation $d_{R}$. We
refer to this regulated group as $ECS_{\Lambda}$, and take the limit
$\Lambda\to\infty$ at the end. This makes the group amenable because
$L^{2}(ECS_{\Lambda})$ manifestly decomposes into a direct sum over its
irreducible representations.

###  2.3 Renormalizing the entropy

Given this decomposition (9) of the regulated Hilbert space, as in Yang-Mills
theory, any density matrix $\rho_{a}$ which descends from a global state in
$\mathcal{H}_{\Sigma}$ can be written in the form

$\rho_{a}=\sum_{R}^{\Lambda}\sum_{n=1}^{d_{R}}p_{R,n}\,\rho_{R,n}\otimes\frac{1}{d_{R}}\text{Id}_{R,n}$
(10)

where $\sum_{R}^{\Lambda}$ means that we sum on irreps $R$ of ECS such that
$d_{R}\leq\Lambda$, and $n$ is a multiplicity index, ranging from
$1,\cdots,d_{R}$. $\rho_{R,n}$ is a state of the purely bulk degrees of
freedom, which is a possibly mixed state of fields whose boundary conditions
on $\partial a$ transform in a representation $R$ of ECS. Each $\rho_{R,n}$ is
a generalization of a “fixed area state” to the full ECS group
dong2023holographic , prepared with Dirichlet boundary conditions for the
fields at the boundary.101010Fixing the area of the boundary implies we fix
the value of the metric at the corner. This is Dirichlet boundary conditions
for the metric. The bulk fields transform non-trivially under ECS even though
these are diffeomorphisms of the full slice $\Sigma$ because ECS is a _global_
symmetry group for the subregion $a$, as described above. We would like to
compute the entropy of this state:

$\displaystyle S(\rho_{a})=-\tr_{ECS_{\Lambda}}\tr_{a}\rho_{a}\ln\rho_{a}$
(11)

Because of the superselection over representations, each term in the sum over
representations is orthogonal, allowing us to pull the sum out of the log.
Additionally, because $S(\rho_{1}\otimes\rho_{2})=S(\rho_{1})+S(\rho_{2})$, we
can compute the entropy of each factor seperately and add them together again.
All together,

$\displaystyle S(\rho_{a})=\sum^{\Lambda}_{R}\sum_{n=1}^{d_{R}}-p_{R,n}\ln
p_{R,n}+p_{R,n}\ln d_{R}-p_{R,n}\tr_{a}\rho_{R,n}\ln\rho_{R,n}.$ (12)

The sum on $R,n$ comes from the trace on $ECS_{\Lambda}$ in the representation
basis.

The first term in (12) is the Shannon entropy of the distribution over
representations. The second is the entropy of the maximally mixed state over
the edge modes $\frac{1}{d_{R}}\text{Id}_{R,n}$. The last term is the von
Neumann entropy of the bulk fields in the state $\rho_{R,n}$. Here, $\tr_{a}$
denotes the trace over purely bulk degrees of freedom $\mathcal{H}_{a}$, and
is well defined for the subregion $a$ because of our bulk regulator. Since the
symmetry group has a copy of the gauge group at each point on the corner (the
$(\cdots)_{\partial a}$ dependence), the sum over $\ln d_{R}$ will exhibit
area scaling (times a factor that diverges as we remove the regulators on
ECS). We may therefore wonder if it contributes directly to the area term in
the Bekenstein-Hawking entropy. However, this term exhibits area scaling in
any gauge theory, so this particular area scaling is not special to gravity.
Additionally, all graviton degrees of freedom are encoded within $\rho_{R,n}$,
suggesting that this area scaling is a red herring. Indeed, our calculations
below will reveal a different origin for the entanglement entropy that we are
computing.

As we take $\Lambda\to\infty$, the second sum in (12) diverges for a generic
distribution $p_{R,n}$ because of the $\ln d_{R}$ dependence of the entropy,
and we must renormalize it. Define

$\Omega:=\sum^{\Lambda}_{R}\sum_{n=1}^{d_{R}}d_{R}=\sum_{R}^{\Lambda}d_{R}^{2}=\tr_{L^{2}(ECS_{\Lambda})}(\text{Id}_{L^{2}(ECS_{\Lambda})})$
(13)

One can think of $\Omega$ as the infinite temperature partition function for
the microcanonical ensemble over the cutoff embeddings.111111This can be seen
by taking $p_{R,n,m}=\frac{1}{\Omega}$, where $m$ is an element of the
representation $R,n$ and noting that each representation $R$ is precisely
$d_{R}$ dimensional. To absorb the divergence as $\Lambda\to\infty$
(equivalently, $\Omega\to\infty$), we adopt the Susskind-Uglum
procedure121212The Susskind-Uglum procedure states that although each term in
the generalized entropy $S_{gen}=\frac{A}{4G_{N}}+S_{out}$ is divergent in the
$G_{N}\to 0$ limit, their sum should be made finite by renormalization of
$S_{out}$ and Newton’s constant. Here, we note that $\Omega$ is finite because
we have regulated the sub-planckian fluctuations in the embeddings, and
increasing $\Omega$ activates more and more fine grained modes. and
simultaneously renormalize $\tr_{a}$ over the bulk states as

$\displaystyle\tr_{a}$ $\displaystyle\mapsto\Omega^{-1}\tr_{a}$ (14)
$\displaystyle\rho_{R,n}$ $\displaystyle\mapsto\Omega\rho_{R,n}$ (15)

The simultaneous rescaling of $\rho_{R,n}$ is needed to ensure it remains
properly normalized under the new trace. Notice that this rescaling is state
independent and therefore well defined. In the language of von Neumann
algebras, the inverse of our $\Lambda\to\infty$ limit takes the form of a
“conditional expectation”.131313We thank Elliott Gesteau for noting that our
renormalization scheme can be viewed in this way. Given two von Neumann
algebras $N\subset M$, a conditional expectation is a map $\mathcal{E}:M\to N$
such that for $n_{1},n_{2}\in N$ and $m\in M$, we have that

$\displaystyle\mathcal{E}(\text{Id}_{M})$ $\displaystyle=\text{Id}_{N}$ (16)
$\displaystyle\mathcal{E}(n_{1}mn_{2})$
$\displaystyle=n_{1}\mathcal{E}(m)n_{2}$ (17)

In our case, we take $M=\mathcal{A}_{\Lambda^{\prime}}$ and
$N=\mathcal{A}_{\Lambda}$, the von Neumann algebras acting on the regulated
Hilbert spaces with cutoffs $\Lambda^{\prime}>\Lambda$. Crucially, the traces
in these algebras have been rescaled appropriately by equation (14). The map
$\mathcal{E}$ is then given explicitly by

$\mathcal{E}_{\Lambda^{\prime}\to\Lambda}(\cdot)=\Pi_{\Lambda}\,(\cdot)\,\Pi_{\Lambda}$
(18)

where $\Pi_{\Lambda}$ is the projector onto the
$\widetilde{\mathcal{H}}^{\Lambda}_{a}$ subspace of
$\widetilde{\mathcal{H}}^{\Lambda^{\prime}}_{a}$. One can see explicitly that
this map satisfies the properties of a conditional expectation, which has
recently been connected to the Susskind-Uglum procedure gesteau2023large .

Applying this rescaling to the entropy formula (12), and noting the
$\ln\rho_{R,n}$ dependence, we can rearrange the terms as

$\displaystyle S(\rho_{a})$
$\displaystyle=\sum^{\Lambda}_{R}\sum_{n=1}^{d_{R}}-p_{R,n}\ln
p_{R,n}+p_{R,n}\ln d_{R}-p_{R,n}\ln\Omega-
p_{R,n}\tr_{a}\rho_{R,n}\ln\rho_{R,n}$ (19)
$\displaystyle=\sum^{\Lambda}_{R}\sum_{n=1}^{d_{R}}-p_{R,n}\ln
p_{R,n}+p_{R,n}\ln\left(\frac{d_{R}}{\Omega}\right)-p_{R,n}\tr_{a}\rho_{R,n}\ln\rho_{R,n}$
(20)
$\displaystyle=-S_{rel}\left(p_{R,n}\left|\frac{d_{R}}{\Omega}\right)\right.-\sum^{\Lambda}_{R}\sum_{n=1}^{d_{R}}p_{R,n}\tr_{a}\rho_{R,n}\ln\rho_{R,n}$
(21)

where $S_{rel}$ is the relative entropy between the indicated distributions.

How does the value of the entropy depend on the specific renormalization
scheme we adopt here? If we instead picked a scheme where we rescaled the
trace over bulk states by $e^{S_{0}}\Omega$, we would simply shift the entropy
by a state independent constant $S_{0}$, implying a single ambiguity. This is
characteristic of states belonging to type II von Neumann algebras, consistent
with recent investigations of quantum gravity in subregions in
Leutheusser:2022bgi ; Witten_2022 ; Chandrasekaran_2023 ; Jensen:2023yxy ;
kudlerflam2023generalized .

We assumed that the global state $\ket{\Psi}$ was well described not just by a
gauge invariant state, but by a semi-classical state. Intuitively, a semi-
classical geometry should be represented by a coherent state over the
embeddings $\phi_{g}$. In other words, we expect the wave function in
$L^{2}(ECS)$ to be Gaussian. But we also expect this wave function to be
highly peaked around a particular embedding: semi-classically, the region $a$
has a definite coordinate location within the spacetime. This implies a large
uncertainty in the area of the region. To see this, note that fixing the boost
angle of $a$, i.e., the choice of time slice which is part of the data of the
embdedding, leads to a large variance in the area, as these are conjugate
variables. The area is determined by the boundary value of fields such as the
gravitons, which are a function of the state, explaining how a region could
have a definite embedding but indefinite area. In the end, when we take the
saddlepoint in $G_{N}\to 0$ limit, the region $a$ will have both a definite
embedding and area as the uncertainty principle relating area and boost angle
is $\sim G_{N}$ and so becomes trivial in the semiclassical limit.

As discussed above, each $\rho_{R,n}$ is prepared with Dirichlet boundary
conditions for the bulk fields. Thus, by the Heisenberg uncertainty principle,
a definite embedding demands a flat distribution over representations. The
microcanonical flatness of $p_{R,n}$ simply indicates that the embedding is
semiclassically well localized. The information about the particular embedding
is encoding in the phases appearing in the density matrices of each
representation $\rho_{R,n}$. Thus, the flat distribution over $R$ corresponds
to taking $\rho_{a}$ to have Neumann boundary conditions for bulk fields at
the corner.141414We thank Wayne Weng for helpful discussions on this point.

The flatness of $p_{R,n}$ for semi-classical states has appeared before in the
literature. For example, we can view this condition as a rephrasing of
Jacobson’s entanglement equilibrium hypothesis, which posits that the semi-
classical vacuum is the maximal entropy state Jacobson_2016 . For sufficiently
symmetric regions $a$, this condition is sufficent to recover the semi-
classical Einstein equations. It is also consistent with the “slowly-varying”
semi-classical assumptions on the observer wavefunctions in
Leutheusser:2022bgi ; Witten_2022 ; Chandrasekaran_2023 ; Jensen:2023yxy ;
kudlerflam2023generalized , suggesting their observer should be thought of as
a subset of the edge modes. This identification also makes intuitive sense,
since the observer in Leutheusser:2022bgi ; Witten_2022 ; Chandrasekaran_2023
; Jensen:2023yxy ; kudlerflam2023generalized  is an abstract model of a clock,
and embedding the region $a$ into partial time slices related by a boost
keeping $\partial a$ fixed within the spacetime is one way to measure time
within $a$. In fact, in the language of algebras, the mathematical description
of a gravitating observer and the edge modes takes a remarkably similar form
klinger2023crossed .

These arguments all suggest that we should take Neumann boundary conditions
for the bulk fields within $a$. At the level of the representations, this
implies we can at least approximately replace $p_{R,n}\to\frac{d_{R}}{\Omega}$
for large enough $\Omega$. Then the relative entropy in equation (21)
vanishes. Finally, we can substitute this form for $p_{R,n}$ in the average
over bulk entropies. To recap, we have argued that the entropy for an
arbitrary semi-classical state $\rho_{a}$ takes the form

$\displaystyle S(\rho_{a})$
$\displaystyle=\frac{1}{\Omega}\sum_{R}^{\Lambda}\sum_{n=1}^{d_{R}}d_{R}\left[-\tr_{a}\rho_{R,n}\ln\rho_{R,n}\right]$
(22)
$\displaystyle=\frac{1}{\Omega}\sum_{R}^{\Lambda}\sum_{n=1}^{d_{R}}\sum_{m=1}^{d_{R}}\left[-\tr_{a}\rho_{R,n}\ln\rho_{R,n}\right]$
(23)

We replaced the factor of $d_{R}$ with a sum over an auxiliary variable $m$
for later convenience. The fact that $\rho_{R,n}$ is independent of $m$, i.e.,
it is microcanonically flat within each representation, is expected from
AdS/CFT Akers_2019 ; Dong_2019 ; Harlow_2017  and plays a crucial role in the
quantum error correction underlying complementary recovery. In our case,
flatness can be traced back to the factorization
$\rho_{R,n}\otimes\text{Id}_{R,n}$ within each superselection sector, which in
turn was a consequence of diffeomorphism invariance in the global state. This
implies it is the bulk diffeomorphism invariance which leads to holographic
recovery.

We will now explain why the sum in equation (23) can be regarded as a trace
over $L^{2}(ECS_{\Lambda})$. First, as explained in Sec. 2.1, for compact
groups $G$, the regular representation $L^{2}(G)$ has a decomposition (4). In
particular, we can decompose the trace on $L^{2}(G)$ as

$\displaystyle\tr_{G}(\cdot)=\sum_{g\in
G}\bra{g}\cdot\ket{g}=\sum_{R}^{\Lambda}\sum_{n=1}^{d_{R}}\sum_{m=1}^{d_{R}}\bra{R,n,m}\cdot\ket{R,n,m}$
(24)

where $m$ is a label on the microstates within each representation $R,n$. This
expression for the trace also holds for the regulated $ECS_{\Lambda}$ by the
definition of our regulator. The first equality expresses the trace as a sum
over equivalence classes of embeddings $\phi_{g}$ of $a$ modulo the
diffeomorphisms $\text{Diff}(a)$, and the second equality expresses the trace
in the Fourier basis, i.e., the sum on representations of $ECS_{\Lambda}$. We
now recognize (23) as a regulated definition of the trace over $ECS_{\Lambda}$
in equation (11). Writing this trace in the basis of embeddings as in the
first equation in (24) we arrive at our final regulated formula

$\displaystyle S(\rho_{a})$
$\displaystyle=\frac{\tr_{ECS_{\Lambda}}}{\Omega}\left[-\tr_{a}\rho[\phi_{g}]\ln\rho[\phi_{g}]\right]$
(25)

Here, $\rho[\phi_{g}]$ is a bulk state with boundary conditions set by a
definite coordinate embedding $\phi_{g}$, as opposed to the definite charge of
$\rho_{R,n}$. In the limit $\Omega\to\infty$, the trace $\tr_{ECS_{\Lambda}}$
becomes a path integral,151515This is because in this limit, ECS is no longer
locally compact, but this is precisely what a path integral computes: a trace
over a configuration space, such as field configurations in spacetime, which
is not locally compact. as does $\tr_{a}$. Because
$\Omega=\tr_{ECS_{\Lambda}}\text{Id}_{ECS_{\Lambda}}$, this regularization
procedure has also recovered precisely the correct normalization for the path
integral. So we can now remove the cutoff on the edge modes and send
$\Omega\to\infty$, to arrive at a path integral for the the entropy:

$\displaystyle
S(\rho_{a})=-\int_{ECS}D\phi_{g}\,D(\phi_{g}^{*}\chi)\,\rho_{a}[\phi_{g},\phi^{*}_{g}\chi]\ln\rho_{a}[\phi_{g},\phi^{*}_{g}\chi]$
(26)

In this path integral, we integrate over the gauge fixed definitions of the
embeddings $\phi_{g}$. $\chi$ is a bulk field defined on $\phi_{g}(a)$, i.e.,
the embedding of $a$ into $\Sigma$, while $\phi^{*}_{g}$ is the pullback from
$\phi_{g}(a)\to a$. In other words, $\phi^{*}_{g}\chi$ are the matter fields
defined on the model region $a$. These are the fields prepared by $\tr_{a}$
because this trace was defined on the Hilbert space $\mathcal{H}_{a}$ of the
model region.

It is more convenient to instead integrate over the fields $\chi$ instead,
i.e., fields defined on the target space $\phi_{g}(a)$. Additionally, we wish
to sum over all embeddings $\phi$, not simply the gauge fixed equivalence
classes. In doing so, we must divide by a volume factor
$|\text{Diff}(\phi,\chi)|$ to remove the overcounting.
$|\text{Diff}(\phi,\chi)|$ divides out all diffeomorphisms with support on
$\phi(a)$, because this is the domain of the fields $\chi$. Making these
substitutions, we find that the entropy of a region $a$ is given by

$S(\rho_{a})=-\int\frac{D\phi
D\chi}{|\text{Diff}(\phi,\chi)|}\,\rho_{a}[\phi,\chi]\ln\rho_{a}[\phi,\chi]$
(27)

##  3 Computing the entropy

###  3.1 A path integral proposal

Above, we constructed a formula for the entropy of a bulk region $a$. We will
now explicitly evaluate this entropy via the replica trick, which proceeds as
follows. For an arbitrary quantum state $\rho$, we first define the Renyi
entropy as

$\displaystyle S_{n}(\rho)=\frac{1}{1-n}\ln\tr\rho^{n}\,.$ (28)

The von Neumann entropy is given by the limit

$\displaystyle S(\rho)=\lim_{n\to
1}S_{n}(\rho)=-\partial_{n}\tr(\rho^{n})|_{n=1}$ (29)

Thus, to calculate the entropy we must compute the trace of powers of
$\rho_{a}[\phi,\chi]$. We will use the gravitational path integral defined
below to perform this computation.

To recap, $\phi:a\hookrightarrow\Sigma$ is an embedding, as above. By letting
$\phi$ fluctuate, the region $\phi(a)$ itself moves around within $\Sigma$,
while $a$ is independent of $\phi$. To see that $\phi$ can really move the
region within the spacetime, consider two distinct embeddings $\phi_{1}$ and
$\phi_{2}$. Depending on their images, it is possible that
$\phi_{1}(a)\cap\phi_{2}(a)=\emptyset$. This would not be possible if
$\phi_{1,2}$ did not move the region $a$ within the slice $\Sigma$. This is an
active view of the role of the embeddings. Alternatively, we can view the
embedding as keeping $a$ fixed, while changing the coordinate definition of
$\partial a$.161616As well as the normal geometry at $\partial a$. Because
$\Sigma$ has a fixed background metric, these coordinate transformations move
the points within $\Sigma$ into and out of the region. This is a passive view
of the role of the embeddings $\phi_{1,2}$.

We now wish to define the path integral computing the (unnormalized) trace of
$\rho_{a}$, i.e., the partition sum ${\cal Z}(a)$, for a subregion $\phi(a)$.
Let $\chi$ be bulk fields, including perturbative gravitons, defined on
$\phi(a)$. It is important that the dynamical fields are $(\chi,\phi)$, not
$(\phi^{*}\chi,\phi)$, where $\phi^{*}$ is the pullback, i.e., $\chi$ is
defined in the target space $\phi(a)$ and not the model region $a$. Of course,
these two sets of variables are equivalent, so one could work with the other
choice instead Speranza_2018 , but our convention turns out to be the more
natural for future computations. By the same reasoning as in the previous
section, we impose Neumann boundary conditions at the corner for the bulk
fields $\chi$. We then propose that the partition sum for an arbitrary bulk
subregion $\phi(a)\subset\Sigma$ can be computed by the following
gravitational path integral that is consistent with the diffeomorphism
constraints:

$\displaystyle\tr(\rho_{a})$ $\displaystyle=\mathcal{Z}(a)=\int\frac{D\phi
D\chi}{|\text{Diff}(\phi,\chi)|}e^{-S[\phi,\chi]}$ (30) $\displaystyle
S[\phi,\chi]$
$\displaystyle=\int_{D[\phi(a)]}\mathcal{L}[\chi]=\int_{D[a]}\phi^{*}\mathcal{L}[\chi]$
(31)

$\mathcal{L}[\chi]$ is the Lagrangian for the effective field theory defined
on $\phi(a)$, in our case the Einstein-Hilbert Lagrangian plus matter terms,
but more generally we could take any covariant (i.e. diffeomorpism invariant)
Lagrangian. $\phi^{*}$ is the pullback of the Lagrangian to $a$. This is the
only way the edge modes $\phi$ enter into the action.
$|\text{Diff}(\phi,\chi)|$ is the volume of the group of diffeomorphisms with
compact support on $\phi(a)$. This is the gauge symmetry for $\phi(a)$. One
can think of $a$ as describing a gauge invariant notion of the region and
$\phi$ as characterizing the fluctuations of this region. Most importantly,
$D\phi$ is the flat measure on embeddings. This followed from the fact that
the measure $\frac{1}{\Omega}$ in our regulated sum over representations (23)
and (25) was independent of ECS. This measure (and therefore $\mathcal{Z}(a)$)
was specified by demanding that the background state be semiclassical and
gauge-invariant, which meant that it was micronanonically flat in the corner
symmetry representation basis for the embedding of $a$ into $\Sigma$.
Equivalently, $\mathcal{Z}(a)$ can be thought of as simply being
$\mathcal{Z}(\Sigma)$ with the fields on $a^{\prime}$ traced out. The fact
that the measure we arrived at is essentially unique suggests that the
particular regulator scheme we adopted above does not affect the final result.
Recall from equations (14) and (15) that the regulator on the bulk fields also
depends on the edge mode regulator in a way that cancels the divergences in
the edge mode sector. This suggests that our result is also independent of the
details of our bulk regulator. This novel proposal for the path integral of a
finite region is our main technical tool.

Returning to the Reyni entropy (29), we need to compute $\tr(\rho_{a}^{n})$.
Following our path integral expression in (30) for $\tr(\rho_{a})$, we can
prepare $\tr(\rho_{a}^{n})$ by computing
$\mathcal{Z}(a^{n})/\mathcal{Z}(a)^{n}$ on an n-fold replica of the model
region $a$ (equivalently, $\phi(a)$). In other words, we have to compute

$\displaystyle
S(\rho_{a})=-\partial_{n}\left(\frac{\mathcal{Z}(a^{n})}{\mathcal{Z}(a)^{n}}\right)_{n=1}$
(32)

Explicitly in terms of the path integral,

$\displaystyle S(\rho_{a})$ $\displaystyle=-\partial_{n}\left[\int
D\phi\int_{a^{n}}\frac{D\chi}{|\text{Diff}(\phi,\chi)|}\frac{e^{-S[\phi,\chi]}}{\mathcal{Z}(a)^{n}}\right]_{n=1}$
(33)

We will be interested in the value of the entropy in the semiclassical limit
$G_{N}\to 0$. So we can take a saddlepoint approximation for the $\chi$
integral. However, because the normalizing volume factor
$|\text{Diff}(\phi,\chi)|$ depends on $\chi$, we must include variations of
this factor in the saddle point equation. We will outline this procedure
explicitly in the next section, and find that the physical configurations of
the embeddings $[\phi]$ only include fluctuations which are strictly larger
than (or equal to) the original region $\phi_{e}(a)$. Letting $I_{n}[\phi]$
denote the on-shell effective action of the $n-$replicated manifold for a
fixed $\phi$, this gives

$\displaystyle S(\rho_{a})$ $\displaystyle=-\partial_{n}\left[\int
D[\phi]e^{-I_{n}[\phi]+nI_{1}}\right]_{n=1}\,.$ (34)

Here, $I_{1}$ is independent of $\phi$ because diffeomorphisms are represented
unitarily on the extended Hilbert space. We make no smoothness requirement on
the replicated (partial) saddles $\phi(a)^{n}$ at this time; at the present
stage $\phi$ can be off-shell and act as a source for possible singularities.

###  3.2 Why outward deformations

To characterize the classes of physical embeddings $[\phi]$, we expand the
partition sum (30) in the limit of small fluctuations around the original
embedding, which we refer to as $\phi_{e}(a)$. To clarify our notation, by
$\phi$, we mean all embeddings of $a\hookrightarrow\Sigma$ irrespective of
their relationship under small diffeomorphisms, i.e., they can be related by
gauge transformations. By $\phi_{g}$, we mean the equivalence classes of
$\phi$ under small diffeomorphisms, which are labeled by an element of ECS as
described in Sec. 2. Finally, by $[\phi]$, we mean the embeddings which
survive after accounting for the gauge fixing determinant in the $\chi$ saddle
point. We now argue that $[\phi]$ only contains fluctuations such that
$[\phi](a)$ strictly contains $\phi_{e}(a)$. As we will see, it is the
diffeomorphism invariance of the target space which breaks the symmetry
between inward and outward fluctuations. This is intuitively true because we
specified the problem by saying that we have complete information about the
state within $a$.

In the partition function (30), we are integrating over $\phi$ and $\chi$,
which are bulk fields defined on the target space $\phi(a)$. This means that
the divisor $|\text{Diff}(\phi,\chi)|$ is the volume of the group of
diffeomorphisms with support on $\phi(a)$. To simplify our analysis, we take
the limit of small fluctuations in $\phi$ around $\phi_{e}$, the embedding
defining the region $a$. In this limit, there are two distinct types of
diffeomorphisms: ones which do move the boundary $\partial\phi_{e}(a)$, and
ones which do not. Because we take the limit of small fluctuations, the volume
factor factorizes into

$|\text{Diff}(\phi,\chi)|\approx|\text{Diff}(\phi_{e},\chi)|\cdot|\text{Diff}(\phi,\chi_{e})|$
(35)

where $|\text{Diff}(\phi_{e},\chi)|$ is the volume of diffeomorphisms which do
not move the corner, but do move the fields within $\phi_{e}(a)$, and
$|\text{Diff}(\phi,\chi_{e})|$ is the volume of diffeomorphisms which do move
the corner but leave the bulk fields unchanged. The factorization follows
because in the strict $G_{N}=0$ limit the diffeomorphisms are irrelevant and
are therefore uncoupled. So this factorization holds approximately in the weak
coupling limit, and the path integral can be written as

$\mathcal{Z}(a)=\int\frac{D\chi
D\phi}{|\text{Diff}(\phi_{e},\chi)|\cdot|\text{Diff}(\phi,\chi_{e})|}e^{-S[\phi,\chi]}\,.$
(36)

Additionally, in this limit, we can split the measure for the embeddings into
the gauge orbits and the gauge degrees of freedom in the usual way via field
redefinitions. Because of the diffeomorphism invariance of the action, the
integral over small fluctuations of the boundary gives a volume factor, as
well as a determinant from the change of variables. This volume factor is
precisely $|\text{Diff}(\phi,\chi_{e})|$ because we are in the weak coupling
limit, so that the effect of moving $\chi$ at the same time as $\phi$ is a
higher order effect. This is analogous to non-abelian Yang-Mills. Explicitly,
the measure splits as

$D\phi=D\phi_{g}\,|\text{Diff}(\phi,\chi_{e})|\det(\cdots).$ (37)

where $\phi_{g}$ are the gauge orbits of the embeddings under diffeomorphisms
on $\phi_{0/g}(a)$, $|\text{Diff}(\phi,\chi_{e})|$ is the volume factor
discussed above, and $\det(\cdots)$ is the gauge fixing determinant. Plugging
this into (36),

$\displaystyle\mathcal{Z}(a)$ $\displaystyle\approx\int\frac{D\chi
D\phi_{g}}{|\text{Diff}(\phi_{e},\chi)|}\det(\cdots)e^{-S[\phi,\chi]}$ (38)
$\displaystyle=\int\frac{D\chi
D\phi_{g}D[\text{ghosts}]}{|\text{Diff}(\phi_{e},\chi)|}e^{-S[\phi,\chi]-S_{ghosts}}$
(39)

where we have used the ghost representation of the gauge fixing determinant in
the second line.

Now consider an analogy. In non-Abelian Yang-Mills theory, the role of the
ghosts is to eliminate the unphysical timelike and longitudinal degrees of
freedom from the path integral. These unphysical polarizations are a remnant
of the residual gauge symmetry left over after gauge fixing. If one could
explicitly separate the gauge orbits into those with physical (transverse) and
unphysical polarizations, then the ghost action would exactly cancel the
unphysical polarizations and leave an integral over just the physical degrees
of freedom. Thus, by identifying gauge orbits which are related by the
residual gauge symmetry, we can remove unphysical polarizations and can
eliminate ghosts. We will now do this for our gravitational path integral.

We have expanded the path integral around $\phi_{e}(a)$ as above. Here, the
ghosts will explicitly cancel particular configurations $\phi_{g}$ from the
path integral, and will leave behind only the physical embeddings $[\phi]$. We
now argue that these physical embeddings are strictly larger than $\phi_{e}$,
the original embedding we expand around.

Let $\delta_{g}$ be a diffeormorphism with support on the corner of the
original embedding $\partial\phi_{e}(a)$, such that
$\phi_{g}(a):=\delta_{g}\circ\phi_{e}(a)$, and furthermore assume that
$\phi_{g}(a)\supset\phi_{e}(a)$, i.e., an inward deformation. Because
$\phi_{g}(a)$ is a subset of the original embedding, there exists a
diffeomorphism with support on $\phi_{e}(a)$ which makes $\phi_{g}(a)$
arbitrarily close to $\phi_{e}(a)$. This means that they are gauge equivalent
under the residual gauge symmetry $\text{Diff}(\phi_{e},\chi)$, which is the
relevant group of diffeomorphisms to consider because the remaining divisor
term is precisely $|\text{Diff}(\phi_{e},\chi)|$. Therefore, the ghost degrees
of freedom will explicitly cancel this contribution, and we can discard it
from the path integral. On the other hand, consider
$\phi_{g^{-1}}:=\delta_{g}^{-1}\circ\phi_{e}(a)\supset\phi_{e}(a)$, i.e., an
outward deformation. Then, there is no diffeomorphism with support on
$\phi_{e}(a)$ which can equate them: they are in different equivalence classes
under the residual gauge symmetry. Therefore, the ghost degrees of freedom
will not cancel this contribution, and we must include it in the path
integral. See Fig. 3. Thus, to leading order in perturbation theory, the only
remaining embeddings to sum over strictly satisfy $[\phi]\supset\phi_{e}$,
namely, embeddings $[\phi]:a\hookrightarrow\Sigma$ that strictly include the
original region $\phi_{e}(a)$ to leading order in $G_{N}$. In Sec. 3.4, we
will argue that this strict containment persists to all orders in perturbation
theory.

Figure 3: Two examples of diffeomorphisms which can change the embedding map.
Left: The deformation moves the region (blue) strictly inside the original one
(blue and black). The residual gauge symmetry is diffeomorphisms with support
on the original embedded region (blue and black): such a diffeomorphism can
push the boundary back to the original configuration. Right: The deformation
moves the region (blue and red) strictly inside the original one (blue). The
residual gauge symmetry is diffeomorphisms with support on the original
embedded region (blue): such a diffeomorphism can never pull the boundary back
to the original configuration.

###  3.3 Imposing the saddlepoint equation

Returning to the entropy computation (34), because the effective action
$I_{n}=\mathcal{O}(G_{N}^{-1})$, we must also take a saddlepoint limit of the
remaining path integral over physical embeddings $[\phi]$ for consistency.
Letting $E$ be the embedding which solves this saddlepoint equation implies
that

$\displaystyle S(\rho_{a})$
$\displaystyle=-\partial_{n}\left[e^{-I_{n}[E]+nI_{1}}\right]_{n=1}$ (40)

For simplicity, we assume that $E(a)$ is homotopic to $\phi_{e}(a)$; we
consider the more general case at the end of this section. By definition of
$E$, the saddle $(E(a))^{n}$ must satisfy this final saddlepoint equation for
embeddings. Note that $E(a)\supset a$ by the argument in the previous section.
We now show this implies that $\partial E(a)$ must be extremal at points where
it does not coincide with the original corner, i.e.,
$\partial\phi_{e}(a)\neq\partial E(a)$, while the trace of the extrinsic
curvature is unconstrained wherever $\partial\phi_{e}(a)=\partial E(a)$.

The saddlepoint equation is equivalent to requiring $(E(a))^{n}$ be on shell.
As we showed in Sec. 2, when gluing two spacetime subregions together, one
must use the entangling product to ensure compatibility between the
Hamiltonian charges generating diffeomorphisms in the glued spacetime. This is
sufficient for the glued spacetime to be on-shell, because it will satisfy the
diffeomorphism constraints. Usually, this compatibility of charges is between
$a$ and $a^{\prime}$: but for the replicated space, the compatibility is
between $n$ copies of $E(a)$ itself. Recall that we viewed the corner symmetry
ECS in two different ways: one is as the group of diffeomorphisms with support
on $\partial a$, and the other was as parameterizing the space of embeddings
of $a\hookrightarrow\Sigma$. Therefore taking the saddle $\phi=E$ is
equivalent to picking a single vector field $\xi$ which generates the
diffeomorphism from $\phi_{e}(a)$ to $E(a)$, and imposing the constraint from
this particular diffeomorpism. We now compute this Hamiltonian charge, and
will see that its vanishing, which imposes the constraint, implies the
extremality condition we outlined above.

A simple way to explain the expression for this Hamiltonian charge is by using
the Hamilton-Jacobi equation, or Schwinger’s principle in the quantum case.
The Hamilton-Jacobi equation says that if one varies the on-shell action $I$
with respect to a generalized coordinate $x$, then
$\partialderivative{I}{x}=p$, where $p$ is the generalized momentum generating
a translation in $x$. The application of this principle to our setting can be
more rigorously justified using covariant phase space techniques Wald_1993 ;
Iyer_1994 ; Iyer_1995 , and relies on the fact we are using the extended phase
space of Ciambelli and Leigh Ciambelli_2021 ; Ciambelli_2022 ;
klinger2023crossed ; klinger2023extended .171717In their extended phase space
formalism, the charges are always integrable, so there are no flux terms which
also appear klinger2023extended . That is to say, the Poisson bracket between
any two Hamiltonian charges closes, implying they form a representation of ECS
independent of the boundary conditions for the bulk fields. In our setting,
the “momentum” is the Hamiltonian charge $H_{\xi}$ which generates the
diffeomorphism parameterized by $\xi$. The charge $H_{\xi}$ contains linear
combinations of the equations of motion, so its vanishing also implies that
$(E(a))^{n}$ is on shell.

Each copy of $a$ in the replica manifold has identical boundary conditions; so
the relevant total charge which must vanish is

$\sum_{i}H^{a_{i}}_{\xi}=nH_{\xi}=0\,.$ (41)

This implies that $H_{\xi}=0$ is the constraint for $E(a)$ to be on-
shell.181818This was recently noticed independently by the authors in
kastikainen2023gravitational . Explicitly, let $x_{\xi}$ be a coordinate of
the flow on $\Sigma$ generated by $\xi$. The Hamilton-Jacobi equation (or
Schwinger’s principle in the quantum case) says that the charge is given by

$H_{\xi}=\frac{1}{n}\partialderivative{I_{n}}{x_{\xi}}=\int_{\partial
E(a)}\xi\Theta=0$ (42)

where $\Theta$ is a local functional defined on the corner $\partial E(a)$
called the generalized expansion Wald_1993 . For example, we will see below
that when the matter contributions to the effective action can be neglected,
$\Theta=K$, where $K$ is the trace of the extrinsic curvature
freidel2023corner . More specifically, if $\xi=\partial_{r}$, where $r$ is a
radially outward coordinate, then $\xi\Theta=K^{r}$.

But crucially, $H_{\xi}$ is linear in $\xi$, and so the integrand vanishes
wherever $\xi=0$ along $\partial E(a)$. This implies that if $\xi=0$ at a
point $p$, then $\Theta$ can take any value at $p$ and we will still satisfy
the constraint $H_{\xi}=0$. By definition of $\xi$, at a point $p$,
$\xi|_{p}=0$ if and only if $\partial E(a)|_{p}=\partial\phi_{e}(a)|_{p}$. So,
by locally varying each point on the corner $\partial E(a)$, we can see that
the saddle point equation requires $\Theta=0$ when $\partial
E(a)\neq\partial\phi_{e}(a)$, and is otherwise unconstrained. This is
essentially the same as the result of Lewkowycz-Maldacena Lewkowycz_2013  when
they derived the Ryu-Takayanagi formula Ryu_2006 , except we have the
possibility of $\xi=0$ because $a$ can be a finite region.191919If $a$ is a
boundary subregion of an AdS spacetime, then $\partial E(a)$ will always be a
bulk geodesic in order to satisfy the minimality condition of the RT formula.
Finally, we take $E(a)$ to be the dominant saddle, and therefore the one which
minimizes the action $I_{n}$.

We can also justify as follows $E(a)$ can be finitely far from $a$. Recall
that in Sec. 3.2, we argued that to leading order in perturbation theory
around $\phi_{e}$, the physical embeddings $[\phi]$ in the path integral
strictly contain $\phi_{e}$, i.e., only contain outward deformations. To find
the saddlepoint $E$ connected to $\phi_{e}$, we can use gradient descent to
flow towards this solution. By repeating the arguments of Sec. 3.2 for the
perturbatively flowed solution, we can see that the outward-only property of
the physical embeddings persists to all orders in perturbation theory. This
gives an intuitive picture of the intermediate homotopy surfaces between
$\phi_{e}(a)$ and $E(a)$ as carrying the information from $a$ to $E(a)$ and
vice versa.

In this calculation, we assumed that $a$ was homotopic to $E(a)$ for
simplicity of discussion, which amounts to assuming that $a$ and $E(a)$ have
the same topology. This can be traced back to the assumption that $\phi$ was
an embedding. Embeddings are globally constrained to be one-to-one, and such
global constraints on field configurations are unnatural without the presence
of symmetries to impose them. A more natural specification for the maps
$\phi:a\hookrightarrow\Sigma$ is to demand that $\phi$ is only _locally_ an
embedding, i.e., it is an immersion. This relaxation from embeddings to
immersions amounts to summing over all topologies for $\phi(a)$. These non-
perturbatively different sectors were not apparent from the discussion about
edge modes because edge modes were defined perturbatively around the region
$a$. Because we defined the bulk fields $\chi$ in terms of the target space
$\phi(a)$, we can sum over the topologies of $\phi(a)$ in equation (30)
without ambiguity. The rest of analysis of in Sec. 3 then proceeds the same
way, modulo replacing $\phi_{e}(a)$ with topologically different
configurations $\phi_{e,n}(a)$ to expand around, where $n$ is an index
labeling the topological details of $\phi_{e,n}(a)$. The fact that we chose
$\phi_{e}(a)$ to be homeomorphic to $a$ instead of $\phi_{e,n}(a)$ is what
distinguishes the original region from these topologically distinct
configurations. Furthermore, considering the transitions from one topological
sector to another shows that it will still be true that $\phi\supset\phi_{e}$,
independent of the topology of $\phi(a)$. Thus, we remove the restriction that
$a$ was homotopic to $E(a)$ and merely require that $a$ is homologous to
$E(a)$, as this will remain true.

###  3.4 The final entropy formula

We can now finish the computation of the entropy. Let
$I_{n}=n\widetilde{I}_{n}$. One can interpret $\widetilde{I}_{n}$ as the on
shell action for the $\mathbb{Z}_{n}$ orbifolded version of the replicated
spacetime $(E(a))^{n}$. In terms of $\widetilde{I}_{n}$, the entropy (40) is

$\displaystyle S(\rho_{a})$
$\displaystyle=\partial_{n}\left[I_{n}[E]-nI_{1}\right]_{n=1}$ (43)
$\displaystyle=n\partial_{n}\left[\widetilde{I}_{n}[E]-I_{1}\right]_{n=1}$
(44)

To evaluate the on shell action, we can perform computations similar to the
ones done in Lewkowycz_2013 . One can show that the bulk contributions cancel
from the $I_{1}$ normalization, while the remaining boundary terms in
$\widetilde{I}_{n}[E]$ reduce to the Wald entropy Wald_1993  evaluated on
$\partial E(a)$ to leading order in $G_{N}$. Explicitly, if we neglect the
matter fields, the entropy evaluates to202020See Lewkowycz_2013 ;
kastikainen2023gravitational  for more details on the evaluation of the on-
shell action.

$S(\rho_{a})=\frac{1}{8G_{N}}\partial_{n}\int_{\partial
E(a)}\left(\nabla^{\mu}g_{\mu
r}-g^{\mu\nu}\nabla_{r}g_{\mu\nu}\right)=\frac{A(\partial E(a))}{4G_{N}}$ (45)

where $r$ is a local coordinate radially outward to $\partial E(a)$, which is
defined by $r=0$ in these coordinates. A generic surface $\partial E(a)$ would
have contributions from the equations of motion, unless the region $E(a)$ was
on shell; the fact that the equations of motion are satisfied for $E(a)$ was
argued in Sec. 3.3. By including in $I_{n}$ the effective action for the
matter fields, one also gets a contribution from the von Neumann entropy of
the bulk matter fields Ryu_2006 ; Faulkner:2013ana ; Hubeny:2007xt ;
Engelhardt:2014gca . In the end, we are left with

$\displaystyle S(\rho_{a})$
$\displaystyle=\min_{\phi\supset\phi_{e}}\left[\frac{A[\partial\phi(a)]}{4G_{N}}+S_{out}[\phi(a)]\right]$
(46)

This formula is the main result of our paper. The saddle $E(a)$ computing this
entropy is shown in Fig. 4 in the case of an AdS spacetime. The minimization
$\phi\supset\phi_{e}$ indicates we minimize $S_{gen}$ over all possible
embeddings which contain the original region $\phi_{e}(a)$. The similarity of
this formula to previous extremal surface calculations in holographic models
of gravity Ryu_2006 ; Faulkner:2013ana ; Hubeny:2007xt ; Engelhardt:2014gca
is striking, and will be discussed more in Sec. 4. We will especially comment
on the connection to Ryu_2006 , the most clear analogy between AdS/CFT and our
calculation due to the assumption that $\Sigma$ is static.

Figure 4: On a static slice $\Sigma$, a causally complete region  a and its
entanglement wedge  E(a), which includes the blue region as well.

###  3.5 Applications

Having determined our final entropy formula in (46), we now apply it to some
examples. Let $\Sigma$ be asymptotically AdS or flat, and let $a$ have a
single connected component. Additionally, assume the bulk matter fields are in
their ground state and that $\Sigma$ is two dimensional, i.e., is a Cauchy
slice for 2+1 dimensional spacetime. We will now show that $E(a)$ is the
convex hull of $a$.212121By convex hull, we mean the smallest region $E(a)$
containing the original region $a$ such that any two points in $E(a)$ are
connected by a geodesic. First, note that it is sufficient to prove that any
two points on $\partial E(a)$ are connected by a geodesic passing entirely
through $E(a)$. Letting $p,q$ be two points on $\partial E(a)$, and
$\gamma\subset E(a)$ the minimal length curve that connects $p$ to $q$ and is
contained within $E(a)$. Additionally, let $\gamma^{\prime}$ be the unique
geodesic connecting $p$ to $q$.222222If $p$ and $q$ are far enough separated,
there may not be a unique geodesic connecting them. However, the minimal
geodesic connecting them will suffice for this proof, and with generic initial
data for the matter fields, we expect this degeneracy to be resolved. If
$\gamma\neq\gamma^{\prime}$, then we could decrease the boundary length of the
generalized entanglement wedge by instead considering $E(a)^{\prime}$, defined
by replacing $\gamma\to\gamma^{\prime}$ and filling in the space. This
violates the minimality assumption of $E(a)$ unless $\gamma=\gamma^{\prime}$,
and thus $E(a)$ is convex. A similar proof holds in higher dimensions by
considering minimal area surfaces instead of curves. In higher dimensions,
$E(a)$ may no longer be convex,232323We thank Xin-Xiang Ju, Wen-Bin Pan, Ya-
Wen Sun and Yuan-Tai Wang for pointing out a subtlety about the convexity of
$E(a)$ in more than 2 spatial dimensions. but its boundary is still the
minimal area surface containing $a$. This result is perhaps most surprising
when $a$ is a thin spherical shell, as the region of minimal surface area
containing the shell simply fills in the missing sphere in its interior.

In the de Sitter vacuum, however, $E(a)$ will expand to cover all of $\Sigma$,
independent of the size of $a$ because the area will always be minimized by
covering the whole space, as $\Sigma$ is compact. This implies that the
entropy of de Sitter spacetimes is zero because the boundary of $E(a)$ is
trivial.242424In the renormalized sense of equation (21). However, if we put
an observer in the complementary patch of $a$, the constraints imposed by the
observer may change this result. For example, if we model the observer as an
excluded point from $a^{\prime}$, and provide sufficient entanglement between
this observer and the rest of the complementary patch $a^{\prime}$, then
$E(a)$ will minimize the generalized entropy by excluding $a^{\prime}$,
implying a non-zero answer. This qualitatively matches
balasubramanian2023sitter , and it would be interesting to see if making this
observer more precise in our model reproduces the calculations of the entropy
of de Sitter spacetimes in Chandrasekaran_2023 ; balasubramanian2023sitter .

##  4 Discussion

We proposed a path integral for finite gravitating regions and used it to
argue that the entropy of an open region $a\subset\Sigma$ of a spacetime at a
moment of time symmetry is determined by a (possibly larger) region
$E(a)\subset\Sigma$ defined by: (1) $a\subseteq E(a)$, and (2) $E(a)$
minimizes the generalized entropy among all regions satisfying (1). This
argument made no assumptions about the asymptotic structure of $\Sigma$ or the
cosmological constant. For AdS/flat spacetimes without matter fields, $E(a)$
will simply be the minimal area surface containing $a$. In dS spacetimes,
$E(a)$ will contain the whole Cauchy slice.

Several recent works have obtained results related to ours. In
Folkestad:2023cze , it is argued that extremal surfaces and non-perturbative
features of the background are sufficient structures to dress perturbations to
allow for independence of subregions in quantum gravity. Our findings are
consistent with this result. When $\partial a\neq\partial E(a)$, $\partial
E(a)$ is extremal and dressing to this portion of the corner is allowed. On
the other hand, if $\partial a=\partial E(a)$, then this portion of the corner
is fixed to all orders in perturbation theory, i.e. it is nonperturbatively
defined.

The authors of Dong_2020  define a notion of “effective entropy” for a
subregion in a gravitating spacetime. However, they arrive at a different
formula for the entropy, as they only consider area contributions when
$\phi(a)$ contains a different number of components than $a$, i.e. only the
area of islands. In contrast, we also have an area contribution from the
original region $a$ as well. They do this to avoid singularities and ensure
that the original region is the saddle in the $G_{N}\to 0$ limit.
Additionally, they did not fully specify how to define a subregion in a gauge
invariant way. We used the edge modes associated with $a$ to cure both of
these issues, as we used the algebraic feature of the entangling product to
ensure that the saddle $E(a)$ is on-shell, and the embedding map $\phi$
manifestly defined $a$ in a gauge invariant way.

Finally, in Jensen:2023yxy ; Chandrasekaran_2023 ; kudlerflam2023generalized
the authors compute the entropy of a subregion in a gravitating spacetime by
assuming the existence of an observer and dressing the fluctuations in $a$ to
this observer. Using algebraic techniques such as the crossed product, they
have all found that the algebra of observables undergoes a phase transition
from type III, as is the case with local QFT, to type II, which has a well
defined entropy up to an overall state-independent ambiguity. This is highly
consistent with our renormalization scheme. As we have emphasized in the main
text, there is a very suggestive analogy between the edge modes we consider
here and the observers they consider. Roughly, they impose the constraint
between the bulk fields and the edge modes for a subset of the edge modes with
all transform in some subalgebra of the $\mathfrak{sl}(2,\mathbb{R})$
component of $\mathfrak{ecs}$. On the other hand, our main result of the
larger region computing the entropy relied heavily on including the
$\mathbb{R}^{2}$ component of $\mathfrak{ecs}$. This suggests that it is the
$\mathfrak{sl}(2,\mathbb{R})$ part of $\mathfrak{ecs}$ which makes the entropy
UV finite, and the $\mathbb{R}^{2}$ part which determines its actual value.
The explicit construction of an algebraic picture connecting edge modes with
the crossed product has been investigated in klinger2023crossed , and it would
be interesting to make the connection of these results to our own more precise
in future work.

###  4.1 The meaning of our construction: terrestrial holography?

Our formula for the entropy recalls the Ryu-Takayanagi (RT) formula in the
AdS/CFT correspondence, which says that given a boundary subregion $R$ at a
moment of time symmetry, the entropy in the field theory is given by the
minimal area surface homologous to $R$ Ryu_2006 . In fact, in the case that
$\Sigma$ is asymptotically AdS and $a$ approaches a boundary subregion, our
prescription for computing the entropy in the $G_{N}\to 0$ limit precisely
matches the RT formula. In AdS/CFT, the RT formula was justified via the
gravitational path integral Lewkowycz_2013 , similarly to our approach. It is
surprising that the gravitational path integral, defined on inherently coarse
grained degrees of freedom, is sufficient to derive a microscopic entropy
formula, which seems like it would need non-perturbative control to compute.
For the entropy of black holes, recent work Penington:2019kki ;
Balasubramanian:2022gmo  has clarified why the gravitational path integral may
actually contain information about the microscopic density of states. It would
be useful to understand whether some similar insight applies to the the RT
formula and to our results.

We know that the RT formula is about more than just entropy: it also
determines the bulk information in the gravitating theory is localized within
the dual boundary theory. Specifically, given a boundary subregion $R$, and
its associated entanglement wedge $E(R)$, any bulk operator with support on
$E(R)$ can be reconstructed in the boundary using operators solely living in
$R$ Almheiri_2015 . This property is called entanglement wedge reconstruction
(EWR) and arises entirely from the fact that the RT formula computes
entanglement entropy, and the resulting quantum information constraints on
holographic encoding of information.252525The RT formula (more generally, the
QES formula) determines the entropy of regions, and therefore the relative
entropies as well. JLMS showed this implies the bulk and boundary modular
Hamiltonians agree Jafferis_2016 . Using the the tools of operator algebra
reconstruction Almheiri_2015 , or more generally the Petz map Chen_2020 , this
is sufficent to reconstruct the bulk degrees of freedom in the boundary
theory. The RT formula satisfies many non-trivial consistency checks, which
motivated EWR in the first place, such as no-cloning, nesting, and strong sub-
additivity. Does our proposal for the entropy of a bulk subregion pass the
same tests?

In fact, Bousso and Penington recently conjectured BP_static  that the entropy
of a bulk region $a$ in a static spacetime satisfies our equation (46), and
our work can be seen as evidence for their proposal. In fact, they
demonstrated that our definition of $E(a)$ _does_ satisfy no-cloning, nesting,
and strong sub-additivity. Their result is highly suggestive that the bulk
saddle $E(a)$ really is computing an entanglement entropy, as opposed to
merely computing a coarse-grained entropy. In AdS/CFT, these entangled degrees
of freedom are thought to be those of a CFT living on the spacetime boundary.
However, these boundary degrees of freedom can be _defined_ through the
extrapolate dictionary banks1998ads ; Balasubramanian:1999ri ; Hamilton:2006az
; harlow2011operator . Thus, not only is the bulk defined by the boundary
theory, but the boundary is defined by the bulk theory. It is therefore
natural to ask if one can view holographic encoding not as a map from the bulk
to the boundary, but also as a map from the bulk to other bulk subregions, the
scenario suggested by our entropy computation.

Sticking to AdS spacetimes, where the dual theory is better understood, let
$a_{\epsilon}$ be a spherical band containing the asymptotic boundary and with
radius $\rho=\epsilon$ in global coordinates, as in Fig. 5. Full knowledge of
the theory of quantum gravity within $a_{\epsilon}$ includes full knowledge of
the asymptotic boundary by the extrapolate dictionary. Because the
entanglement wedge of the entire boundary is the entire spacetime, this
suggests that in the full theory of gravity, $a_{\epsilon}$ encodes the full
slice $\Sigma$ as well. In fact, this argument generalizes to any bulk region
$a_{R}$ which contains a boundary subregion $R$, i.e. $a_{R}$ should encode
all of $E(R)$ as well, even though it is a bulk region. Furthermore, consider
a spacetime containing an evaporating black hole. If the Hawking radiation is
collected in an external reservoir, one can show that the interior of the
black hole is encoded in the Hawking radiation after the Page time Marolf_2021
; Penington:2019kki ; Almheiri:2019hni ; Hartman_2020 . Specifically, an
“island” behind the black hole horizon is holographically encoded in the
distant radiation. But now suppose that the Hawking radiation does not escape
to the reservoir, but instead is collected at a finite but large distance from
the black hole, say in the memory bank of a quantum computer. Performing the
same computations as if the radiation were in the reservoir, we expect that
the interior is still encoded within the Hawking radiation, i.e., that
“escaping past infinity” is not actually crucial to the physics of information
recovery through entanglement islands. But because the Hawking radiation is
still localized within the bulk, it would appear that we should view the
island as an entanglement wedge for the quantum computer, a finite region
within the gravitating system. Finally, this idea of a bulk-to-bulk encoding
has motivations in tensor network models of quantum gravity BP_static ;
Pastawski_2015 .

Figure 5: Subregion of AdS

Another check of our proposal in AdS comes from previous works
Balasubramanian:2013rqa ; Balasubramanian_2014 ; Myers_2014  which have
investigated the entropy of a finite region of spacetime by exploiting the
purity of the global state $\ket{\Psi}$ and computing the “differential
entropy” associated to a time band outside the region. A puzzle in these
calculations involved the treatment of non-convex regions Czech:2015qta ;
ju2023generalized ; ju2023squashed , for the time bands they construct will
always have a convex Rindler horizon. Our calculation suggests a potential
resolution to this puzzle in $2+1$ dimensions. Let $a$ be a non-convex
gravitating region in $2+1$d with entanglement wedge $E(a)$. If we assume that
the system satisfies complementary recovery,262626This is reasonable because
$\rho_{a}$ was microcanonically flat within each ECS representation $R$. In
the case when $a$ is a boundary region and so the $\mathbb{R}^{2}$ component
of $\mathfrak{ecs}$ is in the trivial representation, Donnelly_2016  show that
the area becomes central and therefore defines its eigenvalues the
superselection sectors that $\rho_{a}$ splits into. Akers_2019 ; Dong_2019 ;
Harlow_2017  show that in AdS/CFT, a fixed area state has a flat Reyni
spectrum, consistent with our result. They further argue that this flatness on
each sector allows one to interpret the JLMS formula Jafferis_2016  beyond the
code subspace, implying an exact (as opposed to approximate) equality between
the bulk and boundary modular Hamiltonians. then the complementary degrees of
freedom are captured by $E(a)^{\prime}$, namely the points in $\Sigma$ outside
$E(a)$. But $E(a)$ is the convex hull of $a$, and so the domain of dependence
$D[E(a)^{\prime}]$ has precisely of the form of the time bands studied in
Balasubramanian_2014 ; Myers_2014 . This implies that the entropy of $a$ is
equal to the entropy of the time band, a requirement for the global state to
be pure. If the entropy of $a$ was the area of $a$ and not $E(a)$, this would
not have been the case.

Finally, return to the case mentioned in Sec. 3.5 of a thin spherical shell.
Our entropy formula suggests that all the degrees of freedom in a compact
region of space $a$ live at an open neighborhood of the boundary of $E(a)$.
This is reminiscent of the original conception of the holographic principle by
’t Hooft and Susskind hooft2009dimensional ; Susskind_1995 . These examples
strongly motivate the existence of a more “terrestrial” formulation of
holography, where the dual degrees of freedom are located “closer to us”
within compact regions of the gravitating spacetime, as opposed to “the
heavens” at asymptotic infinity.272727Furthermore, the extended corner
symmetry group ECS, which played a fundamental role in our computation of the
entropy, can be thought of as a finite distance version of the BMS group,
which plays a key role in celestial holography. It would be very interesting
to more precisely define this “dual theory”.

#### Acknowledgments:

We thank Jon Sorce, Jonah Kudler-Flamm, Anthony Speranza, Sam Leutheusser, and
Gautam Satishchandran for detailed comments on an early version of this paper.
We are also grateful to Chitraang Murdia, Onkar Parrikar, Bartek Czech, Wayne
Weng, Roberto Emparan and Rob Myers for useful conversations. VB is supported
by the DOE through DE-SC0013528 and the QuantISED grant DE-SC0020360. CC is
supported by the National Science Foundation Graduate Research Fellowship
under Grant No. DGE-2236662. VB and CC thank the Yukawa Institute at the
University of Kyoto for hospitality during the YITP-T-23-01 Quantum
Information, Quantum Matter and Quantum Gravity workshop.

## References

  * (1) J.D. Bekenstein, _Black holes and entropy_ , _Phys. Rev. D_ 7 (1973) 2333.
  * (2) S.W. Hawking, _Particle creation by black holes_ , _Communications in Mathematical Physics_ 43 (1975) 199 .
  * (3) G.W. Gibbons and S.W. Hawking, _Cosmological Event Horizons, Thermodynamics, and Particle Creation_ , _Phys. Rev. D_ 15 (1977) 2738.
  * (4) T. Jacobson and R. Parentani, _Horizon entropy_ , _Foundations of Physics_ 33 (2003) 323–348.
  * (5) W.G. Unruh, _Notes on black hole evaporation_ , _Phys. Rev. D_ 14 (1976) 870.
  * (6) T. Jacobson, _Thermodynamics of spacetime: The einstein equation of state_ , _Physical Review Letters_ 75 (1995) 1260–1263.
  * (7) S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from the anti–de sitter space/conformal field theory correspondence_ , _Physical Review Letters_ 96 (2006) .
  * (8) T. Faulkner, A. Lewkowycz and J. Maldacena, _Quantum corrections to holographic entanglement entropy_ , _JHEP_ 11 (2013) 074 [1307.2892].
  * (9) V.E. Hubeny, M. Rangamani and T. Takayanagi, _A Covariant holographic entanglement entropy proposal_ , _JHEP_ 07 (2007) 062 [0705.0016].
  * (10) N. Engelhardt and A.C. Wall, _Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime_ , _JHEP_ 01 (2015) 073 [1408.3203].
  * (11) J. Maldacena, _The large n limit of superconformal field theories and supergravity_ , _International Journal of Theoretical Physics_ 38 (1999) 1113–1133.
  * (12) S. Pasterski, M. Pate and A.-M. Raclariu, _Celestial holography_ , 2111.11392.
  * (13) S. Pasterski, _Lectures on celestial amplitudes_ , _The European Physical Journal C_ 81 (2021) .
  * (14) A. Strominger, _Lectures on the infrared structure of gravity and gauge theory_ , 1703.05448.
  * (15) A.-M. Raclariu, _Lectures on celestial holography_ , 2107.02075.
  * (16) A. Strominger, _The ds/cft correspondence_ , _Journal of High Energy Physics_ 2001 (2001) 034–034.
  * (17) V. Balasubramanian, J. de Boer and D. Minic, _Mass, entropy and holography in asymptotically de Sitter spaces_ , _Phys. Rev. D_ 65 (2002) 123508 [hep-th/0110108].
  * (18) E. Witten, _Quantum gravity in de sitter space_ , hep-th/0106109.
  * (19) T. Chakraborty, J. Chakravarty, V. Godet, P. Paul and S. Raju, _The Hilbert space of de Sitter quantum gravity_ , 2303.16315.
  * (20) A. Almheiri, X. Dong and D. Harlow, _Bulk locality and quantum error correction in ads/cft_ , _Journal of High Energy Physics_ 2015 (2015) .
  * (21) R. Haag, _Local quantum physics: Fields, particles, algebras_ (1992).
  * (22) W. Donnelly, _Decomposition of entanglement entropy in lattice gauge theory_ , _Physical Review D_ 85 (2012) .
  * (23) H. Casini, M. Huerta and J.A. Rosabal, _Remarks on entanglement entropy for gauge fields_ , _Physical Review D_ 89 (2014) .
  * (24) W. Donnelly and L. Freidel, _Local subsystems in gauge theory and gravity_ , _Journal of High Energy Physics_ 2016 (2016) .
  * (25) C. Vafa, _Quantum Symmetries of String Vacua_ , _Mod. Phys. Lett. A_ 4 (1989) 1615.
  * (26) R. Dijkgraaf, C. Vafa, E.P. Verlinde and H.L. Verlinde, _The Operator Algebra of Orbifold Models_ , _Commun. Math. Phys._ 123 (1989) 485.
  * (27) V. Chandrasekaran, R. Longo, G. Penington and E. Witten, _An algebra of observables for de sitter space_ , _Journal of High Energy Physics_ 2023 (2023) .
  * (28) F. Peter and H. Weyl, _Die vollständigkeit der primitiven darstellungen einer geschlossenen kontinuierlichen gruppe_ , _Math. Ann. 97_ (1927) .
  * (29) L. Ciambelli and R. G. Leigh, _Universal corner symmetry and the orbit method for gravity_ , _Nuclear Physics B_ 986 (2023) 116053.
  * (30) L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, _Extended corner symmetry, charge bracket and einstein’s equations_ , _Journal of High Energy Physics_ 2021 (2021) .
  * (31) L. Ciambelli, R.G. Leigh and P.-C. Pai, _Embeddings and integrable charges for extended corner symmetry_ , _Physical Review Letters_ 128 (2022) .
  * (32) L. Ciambelli and R.G. Leigh, _Isolated surfaces and symmetries of gravity_ , _Physical Review D_ 104 (2021) .
  * (33) A.-C. de la Hamette, T.D. Galley, P.A. Hoehn, L. Loveridge and M.P. Mueller, _Perspective-neutral approach to quantum frame covariance for general symmetry groups_ , 2110.13824.
  * (34) J.-P. Pier, “Amenable locally compact groups.” Pure and Applied Mathematics. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. X, 418 p. £ 49.80 (1984)., 1984.
  * (35) X. Dong, S. McBride and W.W. Weng, _Holographic tensor networks with bulk gauge symmetries_ , 2309.06436.
  * (36) E. Gesteau, _Large $n$ von neumann algebras and the renormalization of newton’s constant_, 2302.01938.
  * (37) S. Leutheusser and H. Liu, _Subalgebra-subregion duality: emergence of space and time in holography_ , 2212.13266.
  * (38) E. Witten, _Gravity and the crossed product_ , _Journal of High Energy Physics_ 2022 (2022) .
  * (39) K. Jensen, J. Sorce and A. Speranza, _Generalized entropy for general subregions in quantum gravity_ , 2306.01837.
  * (40) J. Kudler-Flam, S. Leutheusser and G. Satishchandran, _Generalized Black Hole Entropy is von Neumann Entropy_ , 2309.15897.
  * (41) T. Jacobson, _Entanglement equilibrium and the einstein equation_ , _Physical Review Letters_ 116 (2016) .
  * (42) M.S. Klinger and R.G. Leigh, _Crossed products, extended phase spaces and the resolution of entanglement singularities_ , 2306.09314.
  * (43) C. Akers and P. Rath, _Holographic renyi entropy from quantum error correction_ , _Journal of High Energy Physics_ 2019 (2019) .
  * (44) X. Dong, D. Harlow and D. Marolf, _Flat entanglement spectra in fixed-area states of quantum gravity_ , _Journal of High Energy Physics_ 2019 (2019) .
  * (45) D. Harlow, _The ryu–takayanagi formula from quantum error correction_ , _Communications in Mathematical Physics_ 354 (2017) 865–912.
  * (46) A.J. Speranza, _Local phase space and edge modes for diffeomorphism-invariant theories_ , _Journal of High Energy Physics_ 2018 (2018) .
  * (47) R.M. Wald, _Black hole entropy is the noether charge_ , _Physical Review D_ 48 (1993) R3427–R3431.
  * (48) V. Iyer and R.M. Wald, _Some properties of the noether charge and a proposal for dynamical black hole entropy_ , _Physical Review D_ 50 (1994) 846–864.
  * (49) V. Iyer and R.M. Wald, _Comparison of the noether charge and euclidean methods for computing the entropy of stationary black holes_ , _Physical Review D_ 52 (1995) 4430–4439.
  * (50) M.S. Klinger, R.G. Leigh and P.-C. Pai, _Extended phase space in general gauge theories_ , 2303.06786.
  * (51) J. Kastikainen and A. Svesko, _Gravitational rényi entropy from corner terms_ , 2312.06765.
  * (52) L. Freidel, M. Geiller and W. Wieland, _Corner symmetry and quantum geometry_ , 2302.12799.
  * (53) A. Lewkowycz and J. Maldacena, _Generalized gravitational entropy_ , _Journal of High Energy Physics_ 2013 (2013) .
  * (54) V. Balasubramanian, Y. Nomura and T. Ugajin, _de sitter space is sometimes not empty_ , 2308.09748.
  * (55) A. Folkestad, _Subregion Independence in Gravity_ , 2311.09403.
  * (56) X. Dong, X.-L. Qi, Z. Shangnan and Z. Yang, _Effective entropy of quantum fields coupled with gravity_ , _Journal of High Energy Physics_ 2020 (2020) .
  * (57) G. Penington, S.H. Shenker, D. Stanford and Z. Yang, _Replica wormholes and the black hole interior_ , _JHEP_ 03 (2022) 205 [1911.11977].
  * (58) V. Balasubramanian, A. Lawrence, J.M. Magan and M. Sasieta, _Microscopic origin of the entropy of black holes in general relativity_ , 2212.02447.
  * (59) D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, _Relative entropy equals bulk relative entropy_ , _Journal of High Energy Physics_ 2016 (2016) .
  * (60) C.-F. Chen, G. Penington and G. Salton, _Entanglement wedge reconstruction using the petz map_ , _Journal of High Energy Physics_ 2020 (2020) .
  * (61) R. Bousso and G. Penington, _Entanglement wedges for gravitating regions_ , _Physical Review D_ 107 (2023) .
  * (62) T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, _AdS dynamics from conformal field theory_ , hep-th/9808016.
  * (63) V. Balasubramanian, S.B. Giddings and A.E. Lawrence, _What do CFTs tell us about Anti-de Sitter space-times?_ , _JHEP_ 03 (1999) 001 [hep-th/9902052].
  * (64) A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, _Holographic representation of local bulk operators_ , _Phys. Rev. D_ 74 (2006) 066009 [hep-th/0606141].
  * (65) D. Harlow and D. Stanford, _Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT_ , 1104.2621.
  * (66) D. Marolf and H. Maxfield, _Observations of hawking radiation: the page curve and baby universes_ , _Journal of High Energy Physics_ 2021 (2021) .
  * (67) A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, _The Page curve of Hawking radiation from semiclassical geometry_ , _JHEP_ 03 (2020) 149 [1908.10996].
  * (68) T. Hartman, Y. Jiang and E. Shaghoulian, _Islands in cosmology_ , _Journal of High Energy Physics_ 2020 (2020) .
  * (69) F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, _Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence_ , _Journal of High Energy Physics_ 2015 (2015) .
  * (70) V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, _The entropy of a hole in spacetime_ , _JHEP_ 10 (2013) 220 [1305.0856].
  * (71) V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, _Bulk curves from boundary data in holography_ , _Physical Review D_ 89 (2014) .
  * (72) R.C. Myers, J. Rao and S. Sugishita, _Holographic holes in higher dimensions_ , _Journal of High Energy Physics_ 2014 (2014) .
  * (73) B. Czech, L. Lamprou, S. McCandlish and J. Sully, _Integral Geometry and Holography_ , _JHEP_ 10 (2015) 175 [1505.05515].
  * (74) X.-X. Ju, W.-B. Pan, Y.-W. Sun and Y.-T. Wang, _Generalized rindler wedge and holographic observer concordance_ , 2302.03340.
  * (75) X.-X. Ju, B.-H. Liu, W.-B. Pan, Y.-W. Sun and Y.-T. Wang, _Squashed entanglement from generalized rindler wedge_ , 2310.09799.
  * (76) G. ’t Hooft, _Dimensional reduction in quantum gravity_ , gr-qc/9310026.
  * (77) L. Susskind, _The world as a hologram_ , _Journal of Mathematical Physics_ 36 (1995) 6377–6396.