# Subalgebra-subregion duality: emergence of space and time in holography

Sam Leutheusser  Princeton Gravity Initiative, Princeton University,
Princeton, NJ 08544, USA  Hong Liu  Center for Theoretical Physics,
Massachusetts Institute of Technology,
77 Massachusetts Ave., Cambridge, MA 02139

###### Abstract

In holographic duality, a higher dimensional quantum gravity system emerges
from a lower dimensional conformal field theory (CFT) with a large number of
degrees of freedom. We propose a formulation of duality for a general causally
complete bulk spacetime region, called subalgebra-subregion duality, which
provides a framework to describe how geometric notions in the gravity system,
such as spacetime subregions, different notions of times, and causal
structure, emerge from the dual CFT. Subalgebra-subregion duality generalizes
and brings new insights into subregion-subregion duality (or equivalently
entanglement wedge reconstruction). It provides a mathematically precise
definition of subregion-subregion duality and gives an independent definition
of entanglement wedges without using entropy. Geometric properties of
entanglement wedges, including those that play a crucial role in interpreting
the bulk as a quantum error correcting code, can be understood from the
duality as the geometrization of the additivity anomaly of certain algebras.
Using general boundary subalgebras rather than those associated with geometric
subregions makes it possible to find duals for general bulk spacetime regions,
including those not touching the boundary. Applying subalgebra-subregion
duality to a boundary state describing a single-sided black hole also provides
a precise way to define mirror operators.

††preprint: MIT-CTP/5510

###### Contents

  1. I Introduction
  2. II AdS/CFT duality in the large $N$ limit
    1. II.1 Boundary theory
    2. II.2 Bulk theory
    3. II.3 AdS/CFT duality at large $N$
  3. III Formulation of subalgebra-subregion duality
  4. IV Reformulation of subregion-Subregion duality
    1. IV.1 Formulation
    2. IV.2 Entanglement wedge algebra
    3. IV.3 RT surfaces without entropy
      1. IV.3.1 ${\mathcal{X}}_{R}$ and ${\mathcal{Y}}_{\hat{R}}$ in various examples
      2. IV.3.2 Entanglement and causal wedges
    4. IV.4 Additivity anomaly: union and intersection of wedge algebras
    5. IV.5 Summary
  5. V Duality for general bulk regions
    1. V.1 Bulk dual regions of time bands in the boundary
    2. V.2 Vacuum
      1. V.2.1 Temporally separated time bands
      2. V.2.2 More general boundary spacetime regions
    3. V.3 An interior causal diamond
    4. V.4 Finite temperature
    5. V.5 Single-Sided Black Holes
  6. VI Discussion
  7. A Rindler decomposition of the generalized free field
    1. A.1 Global and local mode expansions
    2. A.2 Deriving the change of basis
    3. A.3 Shifted Rindler wedges
    4. A.4 Equivalence of ${\mathcal{X}}_{S}$ and ${\mathcal{Y}}_{\hat{S}}$
  8. B Interval decomposition of the generalized free field
    1. B.1 Local mode expansion
    2. B.2 Derivation of the basis change
    3. B.3 Relations between ${\mathcal{X}}_{I,O}$ and ${\mathcal{Y}}_{\hat{I},\hat{O}}$
      1. B.3.1 Lack of invertibility and the additivity anomaly
  9. C RT surface without entropy: Rindler
  10. D RT surface without entropy: interval and its complement
    1. D.1 Calculation of $I_{\pm}(k)$
  11. E Showing the additivity and intersection anomalies
    1. E.1 Showing the additivity anomaly
    2. E.2 Showing the intersection anomaly
  12. F Boundary Geometric Superadditivity
    1. F.1 Bulk dual of ${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}$
    2. F.2 Causal wedge of $(\hat{R}_{+}\cup\hat{R}_{-})_{\text{ext}}$
    3. F.3 A boundary only approach to superadditivity
      1. F.3.1 Double commutant is not the causal completion
  13. G Proof of geodesic convexity

##  I Introduction

Understanding the precise manner in which space, time, and the associated
causal structure of a bulk gravity system arise in its boundary description
has been an outstanding question in holography. Geometric notions of a bulk
gravity system such as local spacetime regions, event horizons, and causal
structure can be sharply defined only in the $G_{N}\to 0$ limit,111$G_{N}$ is
Newton’s constant. In this paper we take $\hbar=1$, and by $G_{N}\to 0$ we
mean the perturbative $G_{N}$ expansion to any finite order. At finite
$G_{N}$, quantum spacetime fluctuations will make geometric concepts fuzzy.
which means that their emergence can be described in a rigorous manner only in
the $N\to\infty$ limit of the boundary theory.222We use $N$ to characterize
the number of degrees of freedom of the boundary theory, which is related to
the bulk Newton constant $G_{N}$ as $G_{N}\sim{1\over N^{2}}$. The
$N\to\infty$ limit refers to perturbative expansions in $1/N^{2}$ to any
finite order. An important step to understand the emergence of the bulk
geometry is to pinpoint the underlying mathematical structure in the boundary
theory that is responsible. As $N\to\infty$, many states and operators of a
finite $N$ theory do not have a sensible limit, and drop out of the $N=\infty$
theory. As a result, the structures of the Hilbert space and operator algebras
undergo dramatic changes shortPaper ; longPaper ; Witten:2021jzq ;
Witten:2021unn ; Schlenker:2022dyo ; Chandrasekaran:2022cip ;
Chandrasekaran:2022eqq ; Faulkner:2022ada  (see also Gao:2021tzr ;
Chandrasekaran:2022qmq ; Dabholkar:2022mxo ; Sugishita:2022ldv ; Gomez:2022eui
; Bahiru:2022mwh ; Verlinde:2022xkw ; deBoer:2022zps ; Seo:2022pqj ;
Donnelly:2022kfs ; Bzowski:2022kgf ). In particular, it was argued in
shortPaper ; longPaper  that there is ubiquitous emergence of type III1 von
Neumann (vN) algebras, which in the example of the thermal field double state
was used to explain the emergence of event horizons, Kruskal-like times, and
the associated causal structure in an eternal black hole geometry.

In this paper we further elaborate on the mathematical structure of the
$N\to\infty$ limit, and provide a general picture of how a local bulk
spacetime region emerges from the boundary theory. We will argue that there is
the following one-to-one correspondence

$\text{\parbox{142.26378pt}{emergent type III${}_{1}$ \\\ boundary
subalgebra}}\Leftrightarrow\quad\quad\text{\parbox{142.26378pt}{causally
complete local \\\ bulk spacetime region}}.$ (1)

Namely, for any causally complete local bulk spacetime region
${\mathfrak{b}}$, there exists an emergent boundary type III1 vN algebra that
is equivalent to the bulk operator algebra in ${\mathfrak{b}}$, i.e.

$\displaystyle\text{an emergent type III${}_{1}$ vN {(boundary)}
subalgebra}=\text{bulk operator algebra in ${\mathfrak{b}}$}\ .$ (2)

Furthermore, bulk geometric notions, such as interior times, lightcones, and
horizons, can be viewed as geometrizations of algebraic properties of emergent
type III1 vN subalgebras. Our main message can be summarized with a slogan:

Bulk locality is a geometrization of emergent boundary type III1 subalgebras.

The correspondence (1)–(2) gives a formulation of the duality for a general
local bulk spacetime subregion, to which we will refer as subalgebra-subregion
duality. See Fig. 1 for some examples. We stress that this duality for a bulk
subregion can be precisely formulated only in the $N\to\infty$ limit.
Nevertheless, it should have important implications for understanding the bulk
theory at a small but finite $G_{N}$ (dual to a boundary theory at a large
finite $N$). An analogy is provided by a phase transition in statistical
physics, which can be sharply defined only in the thermodynamic limit, but
does control essential physics at a large but finite volume.

Subalgebra-Subregion duality generalizes the previously formulated subregion-
subregion duality VanRaamsdonk:2009ar ; Czech:2012bh ; Czech:2012be ;
Wall:2012uf ; Lewkowycz:2013nqa ; Faulkner:2013ana ; Headrick:2014cta ;
Almheiri:2014lwa ; Jafferis:2014lza ; Pastawski:2015qua ; Jafferis:2015del ;
Hayden:2016cfa ; Dong:2016eik ; Harlow:2016vwg ; Faulkner:2017vdd ;
Cotler:2017erl . The Ryu-Takayanagi (RT) formula and its covariant
generalization Ryu:2006bv ; Hubeny:2007xt  say that

$S(\rho_{R})=\frac{{\rm Area}({{\gamma}}_{R})}{4\hbar G_{N}}$ (3)

where $\rho_{R}$ is the reduced density matrix associated with a spatial
boundary region $R$ with $S(\rho_{R})$ the corresponding von Neumann entropy.
${{\gamma}}_{R}$ is the minimal surface which ends on $\partial R$ on the
boundary, see Fig. 1(b). Various arguments have suggested that bulk physics in
the corresponding entanglement wedge $\hat{E}_{R}$, i.e. the causal completion
of the region $E_{R}$ between ${{\gamma}}_{R}$ and $R$, is equivalent to
boundary physics in $R$.

(a)

(b)

(c)

(d)

(e)

Figure 1: Some examples of subalgebra-subregion duality. (a) The right region
of an eternal black hole (shaded) is dual to the single-trace operator algebra
${{\mathcal{M}}}_{R}$ of CFTR (i.e. the one defined on the right boundary) in
a thermal field double state. (b) Duality between the entanglement wedge of a
boundary spatial region $R$ and an algebra defined in ${\mathfrak{b}}_{R}$
(only a time slice of the bulk is shown). (c) A wedge region (shaded) in an
eternal black hole geometry dual to the boundary subalgebra associated with a
time band indicated by $T$ in the CFTR in a thermal field double state. (d) A
radial wedge region (between the red surfaces and the conformal boundary) in
global AdS dual to the boundary subalgebra ${{\mathcal{M}}}_{T}$ associated
with a time band $T$ for the CFT in the vacuum state. (e) A causal diamond in
global AdS that does not touch the boundary is dual to the commutant of
${{\mathcal{M}}}_{T}$ where $T$ is a time band.

Even for cases like Fig. 1 (b), the prototypical situation of subregion-
subregion duality, the subalgebra-subregion formulation (1)–(2) brings new
insights. It provides a mathematically precise definition of subregion-
subregion duality by identifying the bulk operator algebra in $\hat{E}_{R}$
with a boundary type III1 subalgebra, which we will denote as
${\mathcal{X}}_{R}$. We propose that ${\mathcal{X}}_{R}$ can be defined as the
large $N$ limit of ${{\mathcal{B}}}_{R}^{(N)}$, the operator algebra of the
region $R$ in the boundary CFT at a finite $N$, i.e.

${\mathcal{X}}_{R}=\pi_{\Psi}(\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{R}^{(N)})$
(4)

where here $\Psi$ is the boundary state dual to the bulk geometry. The precise
definition of the limit as well as the notation $\pi_{\Psi}$ will be explained
in detail later in Sec. IV. Here it is enough to note that the large $N$ limit
depends on the state $\Psi$ under consideration. Conversely, one can use
${\mathcal{X}}_{R}$ of (4) to provide an alternative definition of the
entanglement wedge and the associated RT surface without using entropy, by
instead directly identifying the bulk region whose operator algebra is
equivalent to ${\mathcal{X}}_{R}$. As a nontrivial explicit example, we are
able to reconstruct the bulk entanglement wedge in an example where the
entanglement wedge exceeds the causal wedge. See Fig. 2.

(a)

(b)

Figure 2: Starting with the algebra ${\mathcal{X}}_{R}$ for the boundary
region $R=R_{1}\cup R_{2}$ (where $R_{1},~{}R_{2}$ are half-lines separated
from each other by an interval) in the vacuum state, it is possible (see Sec.
IV.3) to reconstruct the bulk spacetime region dual to ${\mathcal{X}}_{R}$,
i.e. the entanglement wedge of $R,$ using equivalence of algebras alone. The
shaded regions are time-slices of the reconstructed entanglement wedges, and
the union of the regions bounded by dashed lines gives the causal domain. (a):
a single time slice in the Poincare patch. The region outside the dashed lines
is the causal wedge. (b) a single time slice in global AdS. The union of
regions between the dashed lines and the boundary is the causal domain.

A very interesting consequence of the limit (4) is the additivity anomaly:
while we expect ${{\mathcal{B}}}_{R}^{(N)}$ to obey additivity, i.e. (with
$R_{1,2}$ subregions of the same Cauchy slice)

${{\mathcal{B}}}_{R_{1}}^{(N)}\lor{{\mathcal{B}}}_{R_{2}}^{(N)}={{\mathcal{B}}}_{R_{1}\cup
R_{2}}^{(N)},\qquad{{\mathcal{B}}}_{R_{1}}^{(N)}\land{{\mathcal{B}}}_{R_{2}}^{(N)}={{\mathcal{B}}}_{R_{1}\cap
R_{2}}^{(N)},$ (5)

${\mathcal{X}}_{R}$ in general does not, i.e.333The fact that the algebras
dual to entanglement wedges should not be additive was pointed out earlier in
Casini:2019kex , though the interpretation there in terms of superselection
sectors differs from our interpretation in terms of the large $N$ limit. (See
also Benedetti:2022aiw  for a recent discussion)

${\mathcal{X}}_{R_{1}}\lor{\mathcal{X}}_{R_{2}}\subseteq{\mathcal{X}}_{R_{1}\cup
R_{2}},\qquad{\mathcal{X}}_{R_{1}\cap
R_{2}}\subseteq{\mathcal{X}}_{R_{1}}\land{\mathcal{X}}_{R_{2}}\ .$ (6)

The additivity anomaly (6) underlies geometric properties of the corresponding
entanglement wedges. See Fig. 3 for an illustration. Fig. 2 is another example
where $R_{1}\cap R_{2}=\emptyset$.

Subregion-subregion duality has been interpreted in terms of quantum error
correction since information about certain bulk operators can be recovered
with only partial information in the boundary theory. Furthermore, the
geometric properties of entanglement wedges in Fig. 3 have been interpreted in
terms of error correcting properties Almheiri:2014lwa ; Harlow:2016vwg .
Subalgebra-subregion duality and the additivity anomaly give an explanation of
their origin (see also Mintun:2015qda ; Freivogel:2016zsb  for previous
discussions).444For example, the theorems discussed in Almheiri:2014lwa ;
Harlow:2016vwg  have the form of an equivalence of various statements, but it
is not clear from those discussions that any of the statements actually
applies to holographic systems. The additivity anomaly implies that, in the
large $N$ limit, ${\mathcal{X}}_{R}$ is not locally generated on the
boundary.555For an interesting discussion of addivity violation in the context
of generalized symmetries in QFT see Casini:2020rgj .  This delocalization of
quantum information is a crucial ingredient in the quantum error correction
interpretation, and underlies various other quantum informational aspects of
the holographic duality, which will be discussed elsewhere.

Subregion-subregion duality covers the situations in Fig. 1 (a) and (b), but
not those in (c)–(e). Using type III1 subalgebras on the boundary rather than
causally complete geometric subregions makes it possible to find duals for
general bulk spacetime regions. A particularly interesting example is the
causal diamond region of Fig. 1(e), which does not touch the boundary. The
corresponding dual subalgebra has no direct geometric description in the
boundary theory. Applying subalgebra-subregion duality to a boundary state
describing a single-sided black hole also provides a precise way to define the
mirror operators postulated in Papadodimas:2012aq ; Papadodimas:2013jku . See
Fig. 4.

Figure 3: The additivity anomaly (6) is realized on the gravity side through
geometric properties of entanglement wedges, i.e. $E_{R_{1}}\cup
E_{R_{2}}\subseteq E_{R_{1}\cup R_{2}}$ and $E_{R_{1}\cap R_{2}}\subseteq
E_{R_{1}}\cap E_{R_{2}}$. These properties played important roles in
interpreting the bulk gravity system as a quantum error correcting code.
Figure 4: For the boundary CFT in a state dual to a bulk single-sided black
hole. The operator algebra ${{\mathcal{M}}}_{T}$ associated with a half-
infinite time band $T$ should be dual to the shaded bulk wedge region
${\mathfrak{c}}_{T}$. The commutant of ${{\mathcal{M}}}_{T}$ gives a precise
definition of mirror operators advocated for in Papadodimas:2012aq ;
Papadodimas:2013jku , and is dual to the shaded bulk region labeled by
$({\mathfrak{c}}_{T})^{\prime}$.

In this paper we work to leading order in the $N\to\infty$ limit, when the
boundary theory is described by a generalized free field theory. The
qualitative picture should survive to any finite order in the $1/N$
perturbative expansion. When including $1/N$ corrections to all orders, the
type III1 algebras become type II∞ Witten:2021unn , which leads to new ways of
understanding black hole and de Sitter entropies Witten:2021unn ;
Chandrasekaran:2022cip .666The black hole entropy can also be understood by
working with the microcanonical thermal field double Chandrasekaran:2022eqq .
There is also a very interesting recent paper Faulkner:2022ada  which
discusses the embedding of bulk subalgebras in the $G_{N}\to 0$ limit to the
boundary theory at a finite $N$, which in our language corresponds to the
embedding of ${\mathcal{X}}_{R}$ defined in (4) into
${{\mathcal{B}}}_{R}^{(N)}$ at a finite $N$. Since the bulk and boundary
theories are not in the same parameter regime, the embedding is not part of
the duality, but provides an important alternative probe of the limit in (4).

The plan of the paper is as follows. In Sec. II we highlight various features
of the AdS/CFT duality in the large $N$ limit that are important for
subsequent discussions. In Sec. III we describe our general setup and give an
explicit formulation of (1)–(2). In Sec. IV subregion-subregion duality is
discussed in detail from the new algebraic perspective. In particular, we
provide a boundary derivation of the additivity anomaly (6), and derive
entanglement wedges for some choices of boundary regions—including that of
Fig. 2—from equivalence of algebras. In Sec. V we consider some more general
examples of subalgebra-subregion duality for local bulk spacetime regions. We
conclude with a discussion of some future directions in Sec. VI.

Notational Conventions
We will denote boundary spatial subregions by captial Latin letters, i.e.
$I,O,R,S,...$ We only discuss subregions of bulk Cauchy slices in connection
to the entanglement wedge or causal domain of a boundary subregion. Such bulk
spatial subregions are denoted by capital Latin letters with subscripts
identifying the corresponding boundary subregion. We denote a homology
hypersurface associated to the entanglement wedge of a boundary spatial
subregion $R$ by $E_{R}.$ $E_{R}$ is a piece of a bulk Cauchy slice. We denote
general bulk spacetime subregions by gothic letters
${\mathfrak{a}},{\mathfrak{b}},...$ The entanglement wedge of a boundary
spatial subregion $R$ is denoted by ${\mathfrak{b}}_{R}$ and the causal domain
by ${\mathfrak{c}}_{R}.$

We will use apostrophes to denote causal complements, taken in the same
spacetime in which the original region is defined. For example, $R^{\prime}$
is the boundary causal complement of the boundary subregion $R,$ while
${\mathfrak{a}}^{\prime}$ is the bulk causal complement of the bulk subregion
${\mathfrak{a}}.$ We use a bar to denote complements on a spatial slice. For
example, $\bar{R}$ is the spatial subregion that is the complment to $R$ on a
boundary Cauchy slice, while $\overline{E_{R}}$ is bulk spatial subregion that
is the complement to $E_{R}$ on a bulk Cauchy slice.

We use hats to denote causal completions. For example,
$\hat{R}=R^{\prime\prime}$ denotes the boundary causal completion of $R,$
while $\hat{\mathfrak{a}}={\mathfrak{a}}^{\prime\prime}$ denotes the bulk
causal completion of ${\mathfrak{a}}.$

##  II AdS/CFT duality in the large $N$ limit

To set the stage for formulating the duality (1)–(2), we first elaborate on
various aspects of the AdS/CFT duality in the large $N$ limit that will be
important later.

In the AdS/CFT duality we have a bulk quantum gravity system in AdSd+1 dual to
a CFTd on ${\mathbb{{R}}}\times S_{d-1}$. The number of degrees of freedom of
the CFTd is characterized by a parameter $N$, related to the bulk Newton
constant $G_{N}$ as $G_{N}\sim{1\over N^{2}}$. While our discussion is
general, it is useful to keep in mind some explicit examples, such as the
duality between the ${{\mathcal{N}}}=4$ Super-Yang-Mills (SYM) theory with
gauge group $SU(N)$ on ${\mathbb{{R}}}\times S_{3}$ and the IIB superstring in
AdS${}_{5}\times S_{5}$. In our explicit calculations, for technical
simplicity, we will mostly work with $d=2$, with the boundary CFT2 of central
charge $c\propto N^{2}$ defined on a cylinder ${\mathbb{{R}}}\times S_{1}$.

At a finite (but large) $N$, the boundary theory has a Hilbert space
${{\mathcal{H}}}_{\rm CFT}$, which is identified with that of the bulk quantum
gravity theory

${{\mathcal{H}}}_{\rm bulk}={{\mathcal{H}}}_{\rm CFT},$ (7)

in the sense that there is a one-to-one mapping between quantum states of the
bulk and boundary theories. Accordingly, there is also a one-to-one
correspondence between boundary and bulk operator subalgebras acting on the
above Hilbert space, and the correspondence is state-independent. For example,
consider a local boundary subregion $R$, with an associated type III1 local
operator subalgebra ${{\mathcal{B}}}_{R}^{(N)}$. There must be a corresponding
type III1 algebra $\tilde{{\mathcal{B}}}_{R}^{(N)}$ on the gravity side. Being
state-independent, $\tilde{{\mathcal{B}}}_{R}^{(N)}$ cannot be associated with
any bulk geometry. Furthermore, ${{\mathcal{B}}}_{R}^{(N)}$ contains very
heavy operators with dimension ${{\Delta}}\sim N^{2}$ that can create “black
holes.”777We put black holes in quotes as at a finite $G_{N}$, we do not
really know how to define a black hole in precise terms. Without a full
quantum theory of gravity, there is little we could say in a precise manner
about the mapping of states and subalgebras at a finite $N$.

In the large $N$ limit, only some of the states and operators of the finite
$N$ theory survive, which makes the structures of the Hilbert space and
operator algebras very different shortPaper ; longPaper ; Witten:2021jzq ;
Witten:2021unn ; Schlenker:2022dyo ; Chandrasekaran:2022cip ;
Chandrasekaran:2022eqq ; Faulkner:2022ada . In particular, the ubiquitous
emergence of type III1 subalgebras in the large $N$ limit will be a main theme
in this paper. These emergent type III1 structures are not related to the type
III1 nature of ${{\mathcal{B}}}_{R}^{(N)}$ at a finite $N$. In fact, to make
the story sharper we could put the boundary theory on a spatial lattice, so
that ${{\mathcal{B}}}_{R}^{(N)}$ is type I, and the type III1 structure would
still emerge.

For the convenience of later discussions, below we elaborate in detail the
structures of the boundary and bulk theories in this limit, and the duality
between them.

###  II.1 Boundary theory

We denote the full operator algebra of the CFT at a finite $N$ as
${{\mathcal{B}}}^{(N)}$. We say an operator has a sensible large $N$ limit if:
(i) it can be defined for sufficiently large $N$; (ii) its vacuum correlation
functions have a well-defined $N\to\infty$ limit. For the ${{\mathcal{N}}}=4$
SYM theory with the Hamiltonian normalized as $H=N\mathop{\rm Tr}(\cdots)$,
from the standard large $N$ scaling, these are operators generated by finite
products of single-trace operators of the form ${{\mathcal{O}}}=\mathop{\rm
Tr}(\cdots)$. We denote the vector space of all finite products of single-
trace operators by ${{\mathcal{A}}}$. We can introduce an algebraic structure
on ${{\mathcal{A}}}$ by truncating the OPEs of single-trace operators to the
identity operator. More explicitly, consider two single-trace operators
${{\mathcal{O}}}_{1}(x_{1}),{{\mathcal{O}}}_{2}(x_{2})$, then their OPE has
the form

${{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2})=:{{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2}):+f_{12}(x_{1}-x_{2})\mathbf{1}+\cdots$
(8)

where $:{{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2}):$ denotes the
double-trace operators associated with product and $\cdots$ denote terms whose
OPE coefficients are suppressed by $1/N$. We define the algebra obtained from
keeping the first two terms in (8) as ${{\mathcal{A}}}_{\Omega}$. It can be
viewed as an abstract $C^{*}$-algebra, with norm inherited from a finite $N$
theory. We note that ${{\mathcal{A}}}_{\Omega}$ is not a subalgebra of
${{\mathcal{B}}}^{(N)}$ as ${{\mathcal{A}}}_{\Omega}$ is only strictly defined
in the large $N$ limit.888Previous attempts to understand low energy
excitations about a fixed bulk background have defined a set of low-energy
observables that can be defined at finite $N$ Papadodimas:2012aq ;
Papadodimas:2013jku . In that case, the low-energy observables do not form a
closed algebra.

An elementary point is that single-trace operators at different times are
independent, in the sense that they cannot be expressed in terms of one
another. Consider, for example, a single-trace operator $V=\mathop{\rm
Tr}F_{\mu\nu}F^{\mu\nu}$ in the ${{\mathcal{N}}}=4$ SYM theory. At finite $N$,
we can express it in terms of operators at $t=0$,

$\displaystyle V(t)$ $\displaystyle=$ $\displaystyle e^{iHt}V(0)e^{-iHt}$ (9)
$\displaystyle=$ $\displaystyle\sum_{i}c_{i}(t)O_{i}(0)$ (10)

where the second line can (in principle) be obtained by solving the Heisenberg
equation, with $\\{O_{i}\\}$ denoting the full set of operators and
$\\{c_{i}(t)\\}$ being some numbers independent of $N$. As $N\to\infty$, while
$V(t)$ is well-defined through the limit of its correlation functions, neither
equation (9) nor (10) survives the limit. This is because the Hamiltonian $H$
has an explicit $N$ dependence in its definition, and the set of $O_{i}$’s
appearing in (10) contains not just single-trace operators, but also more
complicated operators that do not survive the large $N$ limit.

The loss of the relations (9)–(10) in the large $N$ limit has some immediate
implications, which have played crucial roles in the discussion of shortPaper
; longPaper  and will continue to do so in this paper:

  1. 1.

Single-trace operator algebras associated with different Cauchy slices cannot
be expressed in terms of one another, and thus are inequivalent. See Fig. 5
(a).

  2. 2.

The algebra of single-trace operators includes operators at all boundary
times, i.e. they already contain some information about the boundary time
evolution for all times.

  3. 3.

There are (infinitely) many more subalgebras at large $N$ that are not present
at a finite $N$. For example, there are distinct vN subalgebras associated to
almost all distinct spacetime subregions, even though the algebras of many of
these regions would be equivalent in the finite $N$ theory. See Fig. 5 (b).
There are also subalgebras which do not have boundary geometric definitions.
For example, the commutant of the subalgebra associated with the time band in
Fig. 5 (b) does not have any direct geometric definition.

(a)

(b)

Figure 5: (a) Algebras ${{\mathcal{M}}}_{1}$ and ${{\mathcal{M}}}_{2}$ of
single-trace operators on two different Cauchy slices are inequivalent. (b):
At finite $N$, the algebra associated with the time band indicated in the plot
is equivalent to that on a single Cauchy slice. But in the large $N$ limit, it
is inequivalent and is a nontrivial subalgebra of the full algebra.

Now we turn to the large $N$ limit of ${{\mathcal{H}}}_{\rm CFT}$. We will
first consider pure states. We say a state $\lvert{\Psi}\rangle$ has a
sensible large $N$ limit, if there exists a sequence of states,
$\\{\lvert{\Psi^{(N)}}\rangle\\},$ such that correlation functions of single-
trace operators (with their expectation values subtracted) in the states have
a well-defined $N\to\infty$ limit, e.g.

$\lim_{N\to\infty}{\left\langle{\Psi^{(N)}|\hat{{\mathcal{O}}}_{1}\hat{{\mathcal{O}}}_{2}\cdots|\Psi^{(N)}}\right\rangle}={\rm
finite},\quad\hat{{\mathcal{O}}}_{i}\equiv{{\mathcal{O}}}_{i}-{\left\langle{\Psi^{(N)}|{{\mathcal{O}}}_{i}|\Psi^{(N)}}\right\rangle}.$
(11)

Above each $\hat{{\mathcal{O}}}_{i}$ should be understood to depend on $N$,
which we have suppressed. $\lvert{\Psi}\rangle$ is defined as the large $N$
limit of $\\{\lvert{\Psi^{(N)}}\rangle\\}$ through the limits of correlation
functions (11). By definition the vacuum $\lvert{\Omega}\rangle$ has a
sensible limit, but a typical state with energy $E-E_{\Omega}\sim
O(N^{2})\to\infty$ may not. We further define a semi-classical state as one in
which correlation functions (11) factorize into products of two-point
functions of single-trace operators at the leading order in the large $N$
expansion. For a semi-classical state $\lvert{\Psi}\rangle$, we use
${{\mathcal{O}}}_{\Psi}$ to denote the $N\to\infty$ limit of
$\hat{{\mathcal{O}}}$,999Note that from the standard large $N$ scaling we
expect ${\left\langle{\Psi^{(N)}|{{\mathcal{O}}}|\Psi^{(N)}}\right\rangle}\sim
O(N)$ as $N\to\infty$. which by definition satisfies
${\left\langle{\Psi|{{\mathcal{O}}}_{\Psi}|\Psi}\right\rangle}=0$. For
notational simplicity, we will often suppress the subscript $\Psi$ in
${{\mathcal{O}}}_{\Psi}$, which should be understood from the
context.101010For example, below ${{\mathcal{O}}}_{1}$ should be understood as
${{\mathcal{O}}}_{1\Psi}$. We now define products of
${{\mathcal{O}}}_{1}(x_{1}),{{\mathcal{O}}}_{2}(x_{2})$ by

${{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2})=:{{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2}):+f_{12}^{\Psi}(x_{1},x_{2})\mathbf{1},\quad
f_{12}^{\Psi}(x_{1},x_{2})\equiv{\left\langle{\Psi|{{\mathcal{O}}}_{1}(x_{1}){{\mathcal{O}}}_{2}(x_{2})|\Psi}\right\rangle}$
(12)

We denote the algebra defined using (12) as ${{\mathcal{A}}}_{\Psi}$. Clearly
for $\lvert{\Psi}\rangle=\lvert{\Omega}\rangle$, ${{\mathcal{A}}}_{\Psi}$
coincides with ${{\mathcal{A}}}_{\Omega}$ defined earlier using (8).111111In
the vacuum, expectation values of the $\cdots$ terms in (8) are all zero. But
in a semi-classical state $\lvert{\Psi}\rangle$, the $\cdots$ terms can sum to
give a finite contribution, which leads to the difference between
$f_{12}^{\Psi}$ in (12) and $f_{12}$ in (8). While we have been using the
notation for a pure state, the discussion can also be applied to a mixed
state.

We can build a Hilbert space ${{\mathcal{H}}}_{\Psi}^{\rm GNS}$ “around” a
semi-classical state $\lvert{\Psi}\rangle$ by acting products of single-trace
operators on $\lvert{\Psi}\rangle$, via a procedure called the Gelfand-
Naimark-Segal (GNS) construction gelfandNaimark ; segal . More explicitly, we
introduce a vector $\lvert{A}\rangle$ for each operator $A$ in the algebra
${{\mathcal{A}}}_{\Psi}$ of single-trace operators, and define an inner
product between these vectors as

${\left\langle{A|B}\right\rangle}={\left\langle{\Psi|A^{\dagger}B|\Psi}\right\rangle},\qquad
A,B\in{{\mathcal{A}}}_{\Psi}\ .$ (13)

Equation (13) does not yet define a Hilbert space as there can be zero-norm
vectors from operators $X\in{{\mathcal{A}}}_{\Psi}$ that satisfy
${\left\langle{\Psi|X^{\dagger}X|\Psi}\right\rangle}=0$. The set of such
operators will be denoted as ${\mathcal{J}}$. To eliminate the zero-norm
vectors, we introduce equivalence classes $[A]$ by the equivalence relation

$A\sim A+X,\quad A\in{{\mathcal{A}}}_{\Psi},\quad X\in{\mathcal{J}}$ (14)

and associate $\lvert{A}\rangle$ instead with each $[A]$. The state
$\lvert{\mathbf{1}}\rangle\equiv\lvert{0}\rangle_{\Psi,{\rm GNS}}$ associated
with the equivalence class $[\mathbf{1}]$ of the identity operator is also
referred to as the GNS vacuum. The GNS Hilbert space
${{\mathcal{H}}}_{\Psi}^{\rm GNS}$ is then the completion of the set
$\\{\lvert{A}\rangle\\}$. The representation $\pi_{\Psi}(A)$ of an operator
$A\in{{\mathcal{A}}}_{\Psi}$ in ${{\mathcal{H}}}^{\rm GNS}_{\Psi}$ is defined
as

$\pi_{\Psi}(A)[B]=[AB],\qquad A,B\in{{\mathcal{A}}}_{\Psi}\ .$ (15)

We will denote the representation of ${{\mathcal{A}}}_{\Psi}$ on
${{\mathcal{H}}}_{\Psi}^{\rm GNS}$ by
${{\mathcal{M}}}_{\Psi}=\pi_{\Psi}({{\mathcal{A}}}_{\Psi}).$

Given the factorization property of (11) in a semi-classical state, any inner
product (13) reduces to sums of products of two-point functions of single-
trace operators at the leading order in $1/N$ expansion, which implies that
${{\mathcal{H}}}^{\rm GNS}_{\Psi}$ can be associated with a Gaussian theory of
single-trace operators. Since single-trace operators at different times are
independent, it is a generalized free field theory with each single-trace
operator being a generalized free field.121212For earlier discussion of
generalized free fields in the large $N$ limit, see e.g. Duetsch:2002hc ; El-
Showk:2011yvt . More explicitly, we can expand the representation of a single-
trace operator ${{\mathcal{O}}}_{\Psi}$ as

$\displaystyle\pi_{\Psi}({{\mathcal{O}}}_{\Psi}(x))=\sum_{n}\left(v_{n}(x)a_{n}+v_{n}^{*}(x)a_{n}^{\dagger}\right)$
(16)
$\displaystyle[a_{m},a_{n}^{\dagger}]={{\delta}}_{mn},\quad[a_{m},a_{n}]=[a_{m}^{\dagger},a_{n}^{\dagger}]=0,\quad
a_{n}\lvert{0}\rangle_{\Psi,{\rm GNS}}=0\ ,$ (17)

where $\\{v_{n}(x)\\}$ is a complete set of mode functions in the boundary
spacetime chosen so that the Wightman function of ${{\mathcal{O}}}_{\Psi}$ is
recovered

${\left\langle{\Psi|{{\mathcal{O}}}_{\Psi}(x){{\mathcal{O}}}_{\Psi}(0)|\Psi}\right\rangle}={{}_{\Psi,{\rm
GNS}}}{\left\langle{0|\pi_{\Psi}({{\mathcal{O}}}_{\Psi}(x))\pi_{\Psi}({{\mathcal{O}}}_{\Psi}(0))|0}\right\rangle}_{\Psi,{\rm
GNS}}\ .$ (18)

We stress that $\pi_{\Psi}({{\mathcal{O}}}_{\Psi})$ is defined only in
${{\mathcal{H}}}^{\rm GNS}_{\Psi}$ and is state-dependent, while
${{\mathcal{O}}}$ is state-independent. ${{\mathcal{H}}}^{\rm GNS}_{\Psi}$ is
generated by acting $\\{a_{n}^{\dagger}\\}$ (and creation operators from other
single-trace operators) on $\lvert{0}\rangle_{\Psi,{\rm GNS}}$.

By definition, any product of single-trace operators has dimension $O(N^{0})$
and as a result a state
$\lvert{\Psi^{\prime}}\rangle\in{{\mathcal{H}}}_{\Psi}^{\rm GNS}$ has energy
$E_{\Psi^{\prime}}-E_{\Psi}\sim O(N^{0})$. ${{\mathcal{H}}}_{\Psi}^{\rm GNS}$
may be interpreted as the space of “small” excitations around
$\lvert{\Psi}\rangle$. Furthermore, given the factorization property of
$\lvert{\Psi}\rangle$, $\lvert{\Psi^{\prime}}\rangle$, which is obtained by
acting $a^{\dagger}_{n}$’s on $\lvert{\Psi}\rangle$, does not satisfy the
factorization property. Thus two semi-classical states
$\lvert{\Psi_{1}}\rangle$, $\lvert{\Psi_{2}}\rangle$ cannot be in each other’s
GNS Hilbert space, i.e. their respective GNS Hilbert spaces
${{\mathcal{H}}}_{\Psi_{1}}^{\rm GNS}$, ${{\mathcal{H}}}_{\Psi_{2}}^{\rm GNS}$
do not overlap. In the large $N$ limit, the full space of states then has the
structure of “disjoint continents” around semi-classical states. See Fig. 6.
There is no operator that can take states from one GNS Hilbert space to
another.131313 While in a finite $N$ theory, there exists an operator that
takes $\lvert{\Psi_{1}^{(N)}}\rangle$ to $\lvert{\Psi_{2}^{(N)}}\rangle$, this
operator does not survive the $N\to\infty$ limit, as the operators that
survive (finite products of single-trace operators) act within each GNS
Hilbert space. We also note that ${{\mathcal{H}}}^{(\rm GNS)}_{\Psi}$ cannot
be viewed as a subspace of the ${{\mathcal{H}}}_{\rm CFT}$ as
${{\mathcal{H}}}^{(\rm GNS)}_{\Psi}$ is precisely defined only in the large
$N$ limit.

Figure 6: In the large $N$ limit, the full Hilbert space of the boundary CFT
separates into disjoint “continents” of GNS Hilbert spaces around semi-
classical states. As we will describe below the same structure exists on the
gravity side in the $G_{N}\to 0$ limit, and there is a one-to-one
correspondence between the continents.

In the discussion above we considered pure states. For a mixed state, we can
purify it first and consider the resulting pure state in the doubled theory,
as in the case of thermal ensemble (which becomes the thermal field double
state). In fact, the GNS construction on a mixed state $\rho$ automatically
provides such a purification.

While the set of single-trace operators can be defined abstractly without
referring to any state (except the vacuum), their algebra is state-dependent.
Since the full state space splits into disjoint GNS Hilbert spaces in the
large $N$ limit, the structure of a large $N$ operator algebra depends on the
specific GNS Hilbert space it acts on. As an example, consider two copies of a
boundary CFT in the thermal field double state discussed in detail in
longPaper . The structure of the subalgebra ${{\mathcal{M}}}_{R}$
(${{\mathcal{M}}}_{L}$) associated with the set of single-trace operators
${{\mathcal{A}}}_{R}$ (${{\mathcal{A}}}_{L}$) for the CFTR (CFTL) depends on
the temperature $T$. For $T<T_{HP}$ (with $T_{HP}$ the Hawking-Page
temperature), ${{\mathcal{M}}}_{R}$ is type I, but it becomes type III1 for
$T>T_{HP}$.

###  II.2 Bulk theory

In the $G_{N}\to 0$ limit, quantum states for a gravity system can be obtained
by quantizing excitations around classical solutions of bulk equations of
motion. More explicitly, consider a bulk classical solution $\chi_{\Psi}$
corresponding to some state $\lvert{\Psi}\rangle$ of the quantum gravity
theory, where $\chi$ collectively denotes all bulk fields including the
metric. Quantizing small metric and matter perturbations around $\chi_{\Psi}$,
with an appropriately chosen “vacuum” state $\lvert{0}\rangle_{\chi_{\Psi}}$,
results in a Fock space ${{\mathcal{H}}}_{\Psi}^{\rm Fock}$. At the leading
order in the $G_{N}$ expansion we have a free quantum field theory in the
curved spacetime specified by $\chi_{\Psi}$. For example, a bulk field
$\Phi(X)$ (with $X$ denoting a bulk point) can be expanded in terms of a
complete set of mode functions $\\{u_{n}(X)\\}$ as

$\displaystyle\Phi(X)=\Phi_{0}(X)+\sum_{n}\left(u_{n}(X)a_{n}+u_{n}^{*}(X)a_{n}^{\dagger}\right),$
(19)
$\displaystyle[a_{m},a_{n}^{\dagger}]={{\delta}}_{mn},\qquad[a_{m},a_{n}]=[a_{m}^{\dagger},a_{n}^{\dagger}]=0,\quad
a_{n}\lvert{0}\rangle_{\chi_{\Psi}}=0,$ (20)

where $\Phi_{0}$ denotes the background value of $\Phi$ we are expanding
around, and $n$ collectively denotes all quantum numbers including continuous
ones. ${{\mathcal{H}}}_{\Psi}^{\rm Fock}$ is generated by acting
$a_{n}^{\dagger}$’s on the vacuum state $\lvert{0}\rangle_{\chi_{\Psi}}$.

The classical solution $\chi_{\Psi}$ together with the associated vacuum
$\lvert{0}\rangle_{\chi_{\Psi}}$ can be interpreted as giving the low energy
description of the quantum state $\lvert{\Psi}\rangle$ in the full theory.
Similarly, an excited state
$\lvert{\psi^{\prime}}\rangle\in{{\mathcal{H}}}_{\Psi}^{\rm Fock}$ along with
$\chi_{\Psi}$ corresponds to some quantum state $\lvert{\Psi^{\prime}}\rangle$
that differs from $\lvert{\Psi}\rangle$ by some low energy excitations.
Denoting the energy expectation value of $\lvert{\Psi}\rangle$ as $E_{\Psi}$,
we have

$E_{\Psi^{\prime}}=E_{\Psi}+{{\delta}}E,\quad{{\delta}}E\sim O(G_{N}^{0})\sim
O(N^{0}),\quad G_{N}\to 0\ ,$ (21)

since the canonically normalized quadratic action of small perturbations used
to construct ${{\mathcal{H}}}_{\Psi}^{\rm Fock}$ is independent of $G_{N}$.
For two distinct classical solutions $\chi_{\Psi_{1}}$ and $\chi_{\Psi_{2}}$,
their Fock spaces ${{\mathcal{H}}}_{\Psi_{1}}^{\rm Fock}$ and
${{\mathcal{H}}}_{\Psi_{2}}^{\rm Fock}$ do not overlap, as $\chi_{\Psi_{1}}$
cannot be obtained from $\chi_{\Psi_{2}}$ by small quantum fluctuations. The
space of states thus separates into disjoint “continents” around different
classical solutions, see Fig. 6.

###  II.3 AdS/CFT duality at large $N$

Under the AdS/CFT duality, a semi-classical state $\lvert{\Psi}\rangle$ in the
CFT can be mapped to a bulk quantum state which has a low energy description
in terms of a classical geometry $\chi_{\Psi}$ together with a vacuum state
$\lvert{0}\rangle_{\chi_{\Psi}}$ for matter and perturbative metric
excitations. Single-trace operators are in one-to-one correspondence with
elementary fields on the gravity side. We should then identify (see Fig. 6)

${{\mathcal{H}}}_{\Psi}^{\rm Fock}={{\mathcal{H}}}_{\Psi}^{\rm
GNS},\quad\lvert{0}\rangle_{\chi_{\Psi}}=\lvert{0}\rangle_{\Psi,{\rm GNS}}\ ,$
(22)

which also imply that the set of creation/annihilation operators in (16) and
(19) must be equivalent and is the reason that we have used the same notation
for them. With ${{\mathcal{M}}}_{\Psi}$ the algebra of the single-trace
operators on ${{\mathcal{H}}}_{\Psi}^{\rm GNS}$, and
$\widetilde{{{\mathcal{M}}}}_{\chi_{\Psi}}$ the operator algebra of bulk
fields on ${{\mathcal{H}}}_{\Psi}^{\rm Fock}$, we should then have

${{\mathcal{M}}}_{\Psi}=\widetilde{{{\mathcal{M}}}}_{\chi_{\Psi}}\ .$ (23)

In particular, there should a one-to-one correspondence between subalgebras of
${{\mathcal{M}}}_{\Psi}$ and $\widetilde{{{\mathcal{M}}}}_{\chi_{\Psi}}$.

The extrapolate dictionary between bulk and boundary operators can be viewed
as a manifestation of (23), which also underlies global bulk reconstruction
Banks:1998dd ; Bena:1999jv ; Hamilton:2006az . We stress that the boundary
operators in global reconstruction should be understood as those acting on
${{\mathcal{H}}}^{\rm GNS}_{\Psi}$, i.e. in ${{\mathcal{M}}}_{\Psi}$, not in
${{\mathcal{B}}}$ or ${{\mathcal{A}}}$.

An example of $\chi_{\Psi}$ is the empty AdS, with
$\lvert{\Psi}\rangle=\lvert{\Omega}\rangle$ being the vacuum state of the CFT,
and $\lvert{0}\rangle_{\rm AdS}\in{{\mathcal{H}}}_{\Omega}^{\rm Fock}$ being
the vacuum of bulk perturbations around empty AdS. Other examples include the
thermal field double state or any state which has a classical gravity
description.141414Since here we are mainly concerned with the large $N$ limit,
by classical gravity description we also include those geometries which may
not be describable by the Einstein gravity, but still have a classical string
theory description.

To leading order in the $G_{N}\to 0$ limit, the bulk theory is that of free
quantum fields on a (fixed) curved spacetime. Bulk fields obey bulk equations
of motion and thus the algebra $\widetilde{{{\mathcal{M}}}}_{\chi_{\Psi}}$ of
bulk operators restricted to a single time slice is the full algebra of
operators in the bulk. In contrast, ${{\mathcal{M}}}_{\Psi}$ is defined on the
full boundary spacetime. See Fig. 7. This implies that bulk time evolution
should be viewed as an automorphism of ${{\mathcal{M}}}_{\Psi}$, which can be
used to study the boundary emergence of bulk times shortPaper ; longPaper .

Bulk causality associates distinct bulk subalgebras to distinct causally
complete subregions of the bulk. In particular, we expect that the bulk
effective field theory obeys additivity and Haag duality Araki:1964 . Namely,
for any bulk causally complete subregions
${\mathfrak{r}}_{1},~{}{\mathfrak{r}}_{2}$ we should have

$\displaystyle\widetilde{{{\mathcal{M}}}}_{({\mathfrak{r}}_{1}\cup{\mathfrak{r}}_{2})^{\prime\prime}}$
$\displaystyle=\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}_{1}}\vee\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}_{2}}$
(24) $\displaystyle\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}_{1}^{\prime}}$
$\displaystyle=\left(\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}_{1}}\right)^{\prime}\
,$ (25)

with ${\mathfrak{r}}^{\prime}$ denoting the bulk causal complement of
${\mathfrak{r}}$ and
$\left(\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}}\right)^{\prime}$ denoting
the commutant of $\widetilde{{{\mathcal{M}}}}_{{\mathfrak{r}}}.$ We will see
these properties impose nontrivial constraints on the identifications of bulk
subregions with boundary subalgebras.

Figure 7: The algebra of a bulk Cauchy slice is mapped to the algebra of
single-trace operators on the entire boundary spacetime.

##  III Formulation of subalgebra-subregion duality

In this section we introduce the formulation of subalgebra-subregion duality
(1)–(2). The discussion of this section will be somewhat abstract; we leave
more detailed discussions in specific cases to Sec. IV–Sec. V.

Consider a semi-classical state $\lvert{\Psi}\rangle$ dual to some bulk
geometry $\chi_{\Psi}$. The single-trace operator algebra
${{\mathcal{M}}}_{\Psi}$ in the GNS Hilbert space ${{\mathcal{H}}}_{\Psi}^{\rm
GNS}$ is identified with the bulk operator algebra
$\widetilde{{\mathcal{M}}}_{\chi_{\Psi}}$. This identification implies that
there is a one-to-one correspondence between subalgebras on two sides. Now
consider a causally complete spacetime subregion ${\mathfrak{a}}$ in the bulk.
The bulk operator algebra $\tilde{\mathcal{Y}}_{\mathfrak{a}}$ associated with
${\mathfrak{a}}$ is a type III1 algebra in the $G_{N}\to 0$ limit. There must
therefore be a corresponding type III1 boundary subalgebra
${\mathcal{Y}}\in{{\mathcal{M}}}_{\Psi}$ that can be identified with
$\tilde{\mathcal{Y}}_{\mathfrak{a}}$, i.e.

${\mathcal{Y}}=\tilde{\mathcal{Y}}_{\mathfrak{a}}\ .$ (26)

This implies that, in the large $N$ limit, ${{\mathcal{M}}}_{\Psi}$ must be
rich enough to contain all the subalgebras associated to local bulk
subregions.

Conversely, for any type III1 subalgebra
${\mathcal{Y}}\in{{\mathcal{M}}}_{\Psi}$ there is a corresponding bulk
subalgebra $\tilde{\mathcal{Y}}\in\widetilde{{\mathcal{M}}}_{\chi_{\Psi}}$.
Since, in the $G_{N}\to 0$ limit, the bulk theory can be viewed as a quantum
field theory on a curved spacetime, it is natural to expect that any type III1
subalgebra is associated with a causally complete local subregion.151515In all
known QFTs, type III1 algebras are associated with local subregions. We thus
propose a one-to-one correspondence between type III1 subalgebras of the
boundary theory and causally complete bulk spacetime subregions, i.e.
equations (1)–(2).

In Fig. 1 we gave some examples, which include subregion-subregion duality
when the boundary type III1 algebra ${\mathcal{X}}_{R}$ is associated with a
boundary spatial subregion $R$. Other examples include the bulk regions that
are dual to the boundary subalgebra associated with a time band, and also bulk
regions that do not touch the boundary, which we will discuss in more details
in Sec. V.

Here we make some general comments on how properties of a type III1 boundary
subalgebra translate into geometric properties of the corresponding bulk
region, including local time evolutions and causal structure. For this
purpose, we first briefly review the story of the example of Fig. 1(a) where
the $R$ region of an eternal black hole is dual to the single-trace operator
subalgebra ${{\mathcal{M}}}_{R}$ of CFTR in the thermal field double state:

  1. 1.

Modular flow of ${{\mathcal{M}}}_{R}$ can be identified as the Schwarzschild
time evolution of the $R$ region, which is an internal time flow, i.e. a time
evolution which takes $\tilde{{\mathcal{M}}}_{R}={{\mathcal{M}}}_{R}$ to
itself.

  2. 2.

Kruskal-like times which can take an element of ${{\mathcal{M}}}_{R}$ into the
$F$ or $P$ regions of the black hole spacetime are generated by half-sided
modular translations. They are obtained by identifying a subalgebra of
${{\mathcal{M}}}_{R}$ that satisfies the half-sided modular inclusion
properties.

  3. 3.

The event horizon can be “seen” in the boundary theory from non-analytic
behavior of an operator in ${{\mathcal{M}}}_{R}$ under half-sided modular
translations.

We believe that this example provides a general paradigm for understanding
emergent local properties in the bulk. More explicitly, consider a causally
complete bulk subregion ${\mathfrak{b}}$, whose algebra
$\tilde{\mathcal{Y}}_{\mathfrak{b}}$ is equivalent to some boundary subalgebra
${\mathcal{Y}}$. Then we should have (see Fig. 8):

  1. 1.

The identification of the bulk operator algebra
$\tilde{\mathcal{Y}}_{{\mathfrak{b}}}$ that is equivalent to ${\mathcal{Y}}$
completely determines the region ${\mathfrak{b}}$.

  2. 2.

Modular flows of ${\mathcal{Y}}$ can be used to describe internal time flows
in ${\mathfrak{b}}$ (though they may not be local).

  3. 3.

Half-sided modular translations can be used to describe “global” time flows
that can take one outside of ${\mathfrak{b}}$.

  4. 4.

The light-cone boundary of ${\mathfrak{b}}$ and the corresponding causal
structure can be “seen” in the boundary theory from non-analytic properties of
the flow of an operator under half-sided modular translations.

We mention that, except in some simple cases, it is technically challenging to
construct the dual given ${\mathfrak{b}}$ or ${\mathcal{Y}}$, as is finding
the corresponding modular flows and half-sided modular translations. The above
discussion provides a conceptual framework where such questions can be
precisely formulated.

Figure 8: Internal and global flows of a causally complete bulk spacetime
region ${\mathfrak{b}}$ in the global AdS.

##  IV Reformulation of subregion-Subregion duality

In this section we give a reformulation of subregion-subregion duality
VanRaamsdonk:2009ar ; Czech:2012bh ; Czech:2012be ; Wall:2012uf ;
Headrick:2014cta ; Almheiri:2014lwa ; Jafferis:2014lza ; Pastawski:2015qua ;
Jafferis:2015del ; Hayden:2016cfa ; Dong:2016eik ; Harlow:2016vwg ;
Faulkner:2017vdd ; Cotler:2017erl  based on the more general subalgebra-
subregion duality outlined in Sec. III.

###  IV.1 Formulation

Consider a local boundary spatial subregion $R$, with $\hat{R}$ being its
(boundary) causal completion. At finite $N$, the boundary operator algebra
${{\mathcal{B}}}_{R}^{(N)}={{\mathcal{B}}}_{\hat{R}}^{(N)}$ associated with
$R$ is state-independent and type III1, but not all operators in
${{\mathcal{B}}}_{R}^{(N)}$ have a sensible large $N$ limit.

Now consider the system in a semi-classical state $\lvert{\Psi}\rangle$ dual
to some bulk geometry $\chi_{\Psi}$. Subregion-subregion duality says that
bulk physics in the entanglement wedge ${\mathfrak{b}}_{R}$ can be obtained
from boundary physics in $\hat{R}$. The entanglement wedge
${\mathfrak{b}}_{R}$ is defined as the domain of dependence of the region
$E_{R}$ that satisfies $\partial E_{R}={{\gamma}}_{R}\cup R$ where
${{\gamma}}_{R}$ is the RT surface for $R$. In particular, it has been
proposed that any bulk operator $\phi$ in ${\mathfrak{b}}_{R}$ acting on
$\lvert{\Psi}\rangle$ can be represented by some operator
${{\mathcal{O}}}\in{{\mathcal{B}}}_{R}^{(N)}$, which is called entanglement
wedge reconstruction Almheiri:2014lwa ; Pastawski:2015qua ; Hayden:2016cfa ;
Dong:2016eik ; Harlow:2016vwg ; Faulkner:2017vdd ; Cotler:2017erl .

From the perspective of subalgebra-subregion duality of Sec. III, in the large
$N$ limit there should be a boundary subalgebra
${\mathcal{X}}_{R}\subset{{\mathcal{M}}}_{\Psi}$ that is identified with the
bulk operator algebra
$\widetilde{{\mathcal{M}}}_{{\mathfrak{b}}_{R}}=\widetilde{{\mathcal{M}}}_{E_{R}}\subset\widetilde{{\mathcal{M}}}_{\chi_{\Psi}}$
in the entanglement wedge ${\mathfrak{b}}_{R}$. Below we will give a proposal
for ${\mathcal{X}}_{R}$, discuss its properties, and comment on the connection
of our formulation with entanglement wedge reconstruction formulated in
Almheiri:2014lwa ; Dong:2016eik ; Harlow:2016vwg ; Faulkner:2017vdd  toward
the end in Sec. IV.5.

Another bulk region often discussed in connection with the entanglement wedge
is the causal wedge Hubeny:2012wa , which is defined as the intersection of
bulk causal future and causal past of $\hat{R}$. The causal wedge as defined,
however, may not be causally complete.161616We thank Netta Engelhardt for
discussions on this. We define the causal domain ${\mathfrak{c}}_{\hat{R}}$ of
$\hat{R}$, as the smallest bulk domain of dependence enclosing the causal
wedge. There should also be a boundary subalgebra ${\mathcal{Y}}_{\hat{R}}$
that is identified with the bulk operator subalgebra
$\widetilde{{\mathcal{M}}}_{{\mathfrak{c}}_{\hat{R}}}\subset\widetilde{{\mathcal{M}}}_{\chi_{\Psi}}$
for the causal domain ${\mathfrak{c}}_{\hat{R}}$. Since the entanglement wedge
always encloses the causal wedge Hubeny:2012wa ; Wall:2012uf ; Hubeny:2013gba
, we have ${\mathfrak{c}}_{\hat{R}}\subseteq{\mathfrak{b}}_{R}$, and thus
$\widetilde{{\mathcal{M}}}_{{\mathfrak{c}}_{\hat{R}}}\subseteq\widetilde{{\mathcal{M}}}_{{\mathfrak{b}}_{R}}$.

We propose that, in the boundary theory, ${\mathcal{X}}_{R}$ and
${\mathcal{Y}}_{\hat{R}}$ can be defined as follows:

$\displaystyle{\mathcal{X}}_{R}=\pi_{\Psi}(\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{R}^{(N)}),$
(27)
$\displaystyle{\mathcal{Y}}_{\hat{R}}=\left({{\mathcal{M}}}_{\hat{R}}\right)^{\prime\prime}\
.$ (28)

Here $\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{R}^{(N)}$ is the subalgebra of
${{\mathcal{A}}}_{\Psi}$ obtained from ${{\mathcal{B}}}_{R}^{(N)}$ in the
large $N$ limit. It is defined in the same manner as ${{\mathcal{A}}}_{\Psi}$
in the discussion around (12). It depends on $\lvert{\Psi}\rangle$.
${{\mathcal{M}}}_{\hat{R}}$ denotes the restriction of
${{\mathcal{M}}}_{\Psi}$ to $\hat{R}$ and ${\mathcal{Y}}_{\hat{R}}$ is its
completion under the weak operator topology. By definition, both
${\mathcal{X}}_{R}$ and ${\mathcal{Y}}_{\hat{R}}$ are subalgebras of
${{\mathcal{M}}}_{\Psi}$. Introducing a restricting operation
${{\mathcal{P}}}_{R}$ to region $R$, we can also write (27)–(28) as

${\mathcal{X}}_{R}=\pi_{\Psi}(\lim_{N\to\infty,\Psi}{{\mathcal{P}}}_{R}{{\mathcal{B}}}^{(N)}),\quad{\mathcal{Y}}_{\hat{R}}={{\mathcal{P}}}_{\hat{R}}\pi_{\Psi}(\lim_{N\to\infty,\Psi}{{\mathcal{B}}}^{(N)})\
.$ (29)

Since
${{\mathcal{P}}}_{R}{{\mathcal{B}}}^{(N)}={{\mathcal{P}}}_{\hat{R}}{{\mathcal{B}}}^{(N)}$,
the difference in the definitions of ${\mathcal{X}}_{R}$ and
${\mathcal{Y}}_{\hat{R}}$ lies only in the order of taking the restriction to
$R$ and the large $N$ limit. The two procedures in general do not commute, as
we will see in an explicit example in Sec. IV.3.

An underlying assumption of the proposal is that both ${\mathcal{X}}_{R}$ and
${\mathcal{Y}}_{\hat{R}}$ should be type III1. We also expect
$\lvert{0}\rangle_{\Psi,{\rm GNS}}$ to be cyclic and separating with respect
to both of them. Both algebras depend on $\lvert{\Psi}\rangle$, and can in
principle have very different properties for different $\lvert{\Psi}\rangle$.

###  IV.2 Entanglement wedge algebra

We now list properties of the entanglement wedge algebra ${\mathcal{X}}_{R}$,
most of which are immediate:

  1. 1.

Since $\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{R}^{(N)}$ includes all the
single-trace operators in $\hat{R}$, we have
${\mathcal{Y}}_{\hat{R}}\subseteq{\mathcal{X}}_{R}$.

  2. 2.

${\mathcal{X}}_{R}$ depends only on the spacetime subregion $\hat{R}$;
however, it can be understood as being associated with $R$, which is a
subregion of a Cauchy slice. In particular, if $\tilde{R}$ is a subregion of
another Cauchy slice such that $\hat{\tilde{R}}=\hat{R},$ then we can equally
well think of ${\mathcal{X}}_{R}$ as being associated to $\tilde{R}$. At
finite $N$, we have
${{\mathcal{B}}}_{R}^{(N)}={{\mathcal{B}}}_{\tilde{R}}^{(N)}$, which implies
${\mathcal{X}}_{R}={\mathcal{X}}_{\tilde{R}}$. In contrast,
${\mathcal{Y}}_{\hat{R}}$ is only defined for $\hat{R}$ and cannot be
associated to any spatial subregion.

  3. 3.

Nesting: for $R_{1}\subset R_{2}$, we have
${\mathcal{X}}_{R_{1}}\subset{\mathcal{X}}_{R_{2}}$, as
${{\mathcal{B}}}_{R_{1}}^{(N)}\subset{{\mathcal{B}}}_{R_{2}}^{(N)}$.

  4. 4.

Suppose ${{\mathcal{B}}}_{R}^{(N)}$ satisfies Haag duality, i.e.
${{\mathcal{B}}}_{R}^{(N)}=({{\mathcal{B}}}_{\bar{R}}^{(N)})^{\prime}$, where
$\bar{R}$ denotes the complement of $R$ on a Cauchy slice. Assuming that the
procedure of taking $N\to\infty$ and the commutant can be exchanged, i.e.

$\lim_{N\to\infty,\Psi}({{\mathcal{B}}}_{\bar{R}}^{(N)})^{\prime}=(\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{\bar{R}}^{(N)})^{\prime},$
(30)

where the commutant on the right hand side should be understood as being taken
within ${{\mathcal{A}}}_{\Psi}$, we then have

${\mathcal{X}}_{R}={\mathcal{X}}_{\bar{R}}^{\prime}\ .$ (31)

i.e. ${\mathcal{X}}_{R}$ satisfies Haag duality. Whether (30) is indeed
correct should clearly be carefully understood. Since in the supergravity
limit, the bulk effective field theory is expected to satisfy Haag duality
(see Fig. 9), for our proposal to work, (30) should at least hold in the
strong coupling limit. In longPaper  it was found (31) holds for a two-
dimensional CFT when $R$ is given by a half space,171717That discussion is
independent of couplings and should work even in the free theory. which we
will discuss further in Sec. IV.3.

Figure 9: Since bulk effective field theory obeys Haag duality, we should have
$\overline{E_{R}}=E_{\bar{R}}$.

Since ${\mathcal{Y}}_{\hat{R}}$ is defined in a generalized free field theory,
in general it does not satisfy Haag duality, i.e. in general
${\mathcal{Y}}_{\hat{R}}\neq{\mathcal{Y}}_{\hat{R}^{\prime}}^{\prime}$.

  5. 5.

The definition (27) is not explicit, as we currently do not know how to
precisely describe $\lim_{N\to\infty,\Psi}{{\mathcal{B}}}_{R}^{(N)}$.
${\mathcal{X}}_{R}$ can be given an alternative definition in terms of modular
flows using the results of Jafferis:2015del ; Faulkner:2017vdd . Denote the
modular operator for ${\mathcal{X}}_{R}$ with respect to
$\lvert{0}\rangle_{\Psi,{\rm GNS}}$ by ${{\Delta}}_{0}$. Now consider

${{\mathcal{O}}}(s;{\vec{x}})\equiv{{\Delta}}^{is}_{0}\pi_{\Psi}({{\mathcal{O}}}({\vec{x}})){{\Delta}}^{-is}_{0}\in{\mathcal{X}}_{R},\quad
s\in(-\infty,\infty),\quad{\vec{x}}\in R,$ (32)

where ${{\mathcal{O}}}({\vec{x}})$ is a single-trace operator. We should
emphasize that ${{\mathcal{O}}}(s;{\vec{x}})$, is, by definition, part of the
single-trace operator algebra; however, in general it may not be expressible
in terms of local single-trace operators in $\hat{R}$. We will discuss an
explicit example in Sec. IV.3. Denote the algebra generated by
${{\mathcal{O}}}(s;{\vec{x}})$ as ${\mathcal{Z}}_{R}$. The discussion of
Jafferis:2015del ; Faulkner:2017vdd  can be translated to our language as

${\mathcal{X}}_{R}={\mathcal{Z}}_{R}\ .$ (33)

Denoting the modular operator of ${{\mathcal{B}}}^{(N)}_{R}$ with respect to
$\lvert{\Psi}\rangle$ at finite $N$ by ${{\Delta}}_{\Psi}$, the statement of
Jafferis:2015del  of the identification of the boundary and bulk modular
operators can be written in our language as

$\pi_{\Psi}\left(\lim_{N\to\infty,\Psi}{{\Delta}}^{-is}_{\Psi}{{\mathcal{O}}}({\vec{x}}){{\Delta}}^{is}_{\Psi}\right)={{\Delta}}_{0}^{-is}\pi_{\Psi}({{\mathcal{O}}}({\vec{x}})){{\Delta}}_{0}^{is}\
.$ (34)

Equation (34) is a striking statement, as it implies that the modular flow
${{\Delta}}^{-is}_{\Psi}{{\mathcal{O}}}({\vec{x}}){{\Delta}}^{is}_{\Psi}$ of a
single-trace operator has a well-defined large $N$ limit, i.e. remains part of
the single-trace operator algebra. This is the property that connects the two
definitions.

The identification (33) implies that for a CFT in the vacuum state, and for
$R$ given by a half-space or a sphere

${\mathcal{X}}_{R}={\mathcal{Y}}_{\hat{R}}$ (35)

as in these cases, modular flows are geometric, and thus
${{\mathcal{O}}}(s;{\vec{x}})$ can be expressed in terms of single-trace
operators in $\hat{R}$.

  6. 6.

Let $R_{1}$ and $R_{2}$ be subregions on the same boundary Cauchy slice. While
${{\mathcal{B}}}_{R}^{(N)}$ obeys additivity, i.e.

${{\mathcal{B}}}_{R_{1}}^{(N)}\lor{{\mathcal{B}}}_{R_{2}}^{(N)}={{\mathcal{B}}}_{R_{1}\cup
R_{2}}^{(N)},$ (36)

it can be checked that ${\mathcal{X}}_{R}$ in general does not, i.e.

${\mathcal{X}}_{R_{1}}\lor{\mathcal{X}}_{R_{2}}\subseteq{\mathcal{X}}_{R_{1}\cup
R_{2}}\ .$ (37)

The inequality in (37) should then be a consequence of the large $N$ limit,
and will be referred as the additivity anomaly. The anomaly has its origin
again in Fig. 5, i.e. single-trace operators at different times are
independent. We will give some explicit examples in Sec. IV.3 and IV.4.

The additivity anomaly also leads to an anomaly in the intersection of
${\mathcal{X}}_{R_{1}}$ and ${\mathcal{X}}_{R_{2}}$

${\mathcal{X}}_{R_{1}\cap
R_{2}}\subseteq{\mathcal{X}}_{R_{1}}\land{\mathcal{X}}_{R_{2}}\ .$ (38)

while finite-$N$ algebras ${{\mathcal{B}}}_{R}^{(N)}$ should satisfy

${{\mathcal{B}}}_{R_{1}}^{(N)}\land{{\mathcal{B}}}_{R_{2}}^{(N)}={{\mathcal{B}}}_{R_{1}\cap
R_{2}}^{(N)}\ .$ (39)

To see (38), we start with (37) for $\bar{R}_{1}$ and $\bar{R}_{2}$

${\mathcal{X}}_{\bar{R_{1}}}\lor{\mathcal{X}}_{\bar{R_{2}}}\subseteq{\mathcal{X}}_{\bar{R_{1}}\cup\bar{R_{2}}}$
(40)

and take the commutant on both sides of (40). Equation (38) follows by using
(31). Note $\overline{\bar{R_{1}}\cup\bar{R_{2}}}=R_{1}\cap R_{2}$, and
$({\mathcal{X}}_{\bar{R_{1}}}\lor{\mathcal{X}}_{\bar{R_{2}}})^{\prime}={\mathcal{X}}_{\bar{R_{1}}}^{\prime}\land{\mathcal{X}}_{\bar{R_{2}}}^{\prime}={\mathcal{X}}_{R_{1}}\land{\mathcal{X}}_{R_{2}}$.

The above properties are consistent with the bulk dual of ${\mathcal{X}}_{R}$
being the entanglement wedge. For example, the additivity anomaly (37)–(38)
describes precisely what was observed geometrically in the bulk for
entanglement wedges Casini:2019kex , see Fig. 3. Haag duality, (31), also
implies that there is no additivity anomaly when $R_{1}=\overline{R_{2}}$,
i.e.

${\mathcal{X}}_{R}\lor{\mathcal{X}}_{\bar{R}}={{\mathcal{M}}}_{\Psi}\ .$ (41)

###  IV.3 RT surfaces without entropy

Given the identifications (27)–(28), we can in principle derive the
entanglement wedge ${\mathfrak{b}}_{R}$ and the causal domain
${\mathfrak{c}}_{\hat{R}}$ from the boundary theory. In particular, in the
case of (27), subalgebra-subregion duality can be used to provide an
independent definition of the RT (or HRT) surface without using entropy. It
should be possible to derive the minimal surface prescription of the RT (or
HRT) surface, and the quantum extremal surface prescription (including
islands), from equivalence of the bulk and boundary subalgebras. These are
ambitious goals and they will not be attempted here.

Here we discuss three explicit examples in which the entanglement wedges (and
the associated RT surface) and causal wedges can be found by brute force from
the identification of algebras.

We will consider a boundary CFT2 on ${\mathbb{{R}}}^{1,1}$ in the vacuum state
$\lvert{\Omega}\rangle$, with boundary coordinates $x^{\mu}=(x^{0},x^{1})$.
The bulk theory is defined on a Poincare patch of AdS3, with coordinates
$X^{M}=(z,x^{\mu})$ and boundary at $z=0$. At leading order in the large $N$
limit, ${{\mathcal{M}}}_{\Omega}$ factorizes into a product of subalgebras
generated by different single-trace operators. An important and highly
nontrivial self-consistency condition is that the entanglement and causal
wedges obtained from the subalgebra generated by any single-trace operator
should be the same. We will consider the subalgebra
${{\mathcal{M}}}_{{\mathcal{O}}}\subset{{\mathcal{M}}}_{\Omega}$ generated by
a scalar operator ${{\mathcal{O}}}$ of dimension ${{\Delta}}$, dual to a bulk
field $\phi$. For notational simplicity, we will simply refer to it as
${{\mathcal{M}}}_{\Omega}$, and similarly for ${\mathcal{X}}_{R}$ and
${\mathcal{Y}}_{\hat{R}}$. We will use the definition of ${\mathcal{X}}_{R}$
based on (33) and (32). We show that the resulting entanglement and causal
wedges from ${\mathcal{X}}_{R}$ and ${\mathcal{Y}}_{\hat{R}}$ for various
examples indeed agree with their standard bulk definitions. By a conformal
transformation (or isometric transformation in the bulk), the examples
considered can be converted to those in the global AdS. Below when talking
about a boundary spatial region $R$, we always refer to the time slice
$x^{0}=0$.

The examples we will consider are:

  1. E1:

$R$ is the half space $S:x^{1}\in(0,\infty)$ with $\hat{S}$ the right Rindler
wedge of ${\mathbb{{R}}}^{1,1}$. From discussion around (35),
${\mathcal{X}}_{S}$ and ${\mathcal{Y}}_{\hat{S}}$ coincide. We show that
${\mathcal{X}}_{S}$ is equivalent to the bulk subalgebra in the domain of
dependence of the bulk region $x^{1}>0$ on the $x^{0}=0$ slice. See Fig. 10
(a).

  2. E2:

$R$ is an interval $I:x^{1}\in(-a,a)$. In this ${\mathcal{X}}_{I}$ and
${\mathcal{Y}}_{\hat{I}}$ again coincide, and can be shown to be equivalent to
the bulk subalgebra in the domain of dependence of the bulk region inside the
half circle $z^{2}+(x^{1})^{2}=a^{2}$ on the $x^{0}=0$ slice. See Fig. 10 (b).

  3. E3:

Take $R=O=O_{L}\cup O_{R}=\bar{I}$, where $O_{L}$ and $O_{R}$ are two half
spaces separated by the finite interval $I:x\in(-a,a)$. In this case,
${\mathcal{X}}_{O}$ and ${\mathcal{Y}}_{\hat{O}}$ are different, with

${\mathcal{Y}}_{\hat{O}}={\mathcal{X}}_{O_{L}}\lor{\mathcal{X}}_{O_{R}}\subset{\mathcal{X}}_{O}\
,$ (42)

and the corresponding bulk regions whose algebras are equivalent to
${\mathcal{X}}_{O}$ and ${\mathcal{Y}}_{\hat{O}}$ can be shown to be those in
Fig. 10 (c). This is also an example of the additivity anomaly (37).

(a)

(b)

(c)

Figure 10: Algebraic entanglement wedges in three examples. (a) $x^{0}=0$
slice of the entanglement wedge of a half-space. (b) $x^{0}=0$ slice of the
entanglement wedge of an interval. (c) $x^{0}=0$ slice of the entanglement
wedge of two-finitely separated half-spaces. Note that the causal domain (the
region outside of the dashed lines) is properly contained in the entanglement
wedge.

We now outline the calculations behind the above examples, leaving details to
various appendices. We first describe the boundary side of the story, i.e. how
to define ${\mathcal{X}}_{R}$ and ${\mathcal{Y}}_{\hat{R}}$ for each example,
and their relations. We then discuss how to obtain the bulk regions whose
algebras are identified with them.

####  IV.3.1 ${\mathcal{X}}_{R}$ and ${\mathcal{Y}}_{\hat{R}}$ in various
examples

In the large $N$ limit, a scalar single-trace operator ${{\mathcal{O}}}$ can
be described by a generalized free field, with a mode expansion181818Below
${{\mathcal{O}}}(x)$ should be more precisely understood as
$\pi_{\Omega}({{\mathcal{O}}}(x))$. We suppress $\pi_{\Omega}$ for notational
simplicity.

$\displaystyle{{\mathcal{O}}}(x)=\int{d^{2}p\over(2\pi)^{2}}\,f_{p}e^{ip\cdot
x}b_{p},\quad f_{p}=C\theta(-p^{2})\kappa^{{{\Delta}}-1},\quad p\cdot
x=-p^{0}x^{0}+p^{1}x^{1},$ (43) $\displaystyle\kappa=\sqrt{-p^{2}},\quad
b_{p}^{\dagger}=b_{-p},\quad[b_{p},b_{p^{\prime}}]=(2\pi)^{2}\epsilon(p^{0}){{\delta}}^{(2)}(p+p^{\prime}),$
(44)

where $C$ is a constant depending on the normalization of ${{\mathcal{O}}}$.
It is convenient to choose
$C=2^{1-{{\Delta}}}\sqrt{\pi}{{\Gamma}}({{\Delta}})^{-1}$ as this puts the
extrapolate dictionary for a canonically normalized bulk field in the simplest
form. We make this choice of normalization from here on. The above equations
follow from two-point correlation functions of ${{\mathcal{O}}}(x)$, which are
fully determined by conformal symmetry. ${{\mathcal{M}}}_{\Omega}$ can also be
regarded as being generated by $\\{b_{p}\\}$. ${\mathcal{Y}}_{\hat{R}}$ for a
spacetime region $\hat{R}$ is generated from ${{\mathcal{O}}}(x)$ in (43) with
the restriction $x\in\hat{R}$.

Now take $R$ to be the half space $S:x^{1}>0$, and $\hat{S}$ is given by the
right Rindler region with metric

$ds^{2}=e^{2\chi}(-d\eta^{2}+d\chi^{2})=-e^{2\chi}d\xi^{+}d\xi^{-},\quad\xi^{\pm}=\eta\pm\chi\
.$ (45)

We will use $\xi=(\eta,\chi)=(\xi^{+},\xi^{-})$ to denote the Rindler
coordinates. From two-point functions of ${{\mathcal{O}}}$ in the Rindler
region we can obtain a mode expansion for the restriction of ${{\mathcal{O}}}$
to $\hat{S}$ as

$\displaystyle{{\mathcal{O}}}_{S}(\xi)=\int\frac{d^{2}k}{(2\pi)^{2}}\,u_{k}^{(S)}a_{k}^{(S)},\quad
u^{(S)}_{k}=N_{k}e^{\bar{q}_{+}\xi^{-}-q_{-}\xi^{+}}$ (46) $\displaystyle
q_{+}={{{\Delta}}\over 2}+ik^{+},\quad q_{-}={{{\Delta}}\over
2}+ik^{-},\quad\bar{q}_{+}={{{\Delta}}\over
2}-ik^{+},\quad\bar{q}_{-}={{{\Delta}}\over 2}-ik^{-}$ (47) $\displaystyle
k^{+}={\omega+q\over 2},\quad k^{-}={\omega-q\over 2},\quad
N_{k}={\sqrt{\sinh\pi|\omega|}\over\sqrt{2\pi}{{\Gamma}}({{\Delta}})}\left|\Gamma\left(q_{+}\right)\Gamma\left(q_{-}\right)\right|$
(48)
$\displaystyle(a_{k}^{(S)})^{\dagger}=a_{-k}^{(S)},\qquad[a_{k}^{(S)},a_{k^{\prime}}^{(S)}]=\epsilon(\omega)(2\pi)^{2}{{\delta}}^{(2)}(k+k^{\prime})\
.$ (49)

Here $\omega$ is the frequency for $\eta$, and $q$ is the momentum for $\chi$.
The subalgebra ${\mathcal{X}}_{S}={\mathcal{Y}}_{\hat{S}}$ can be regarded as
being generated by $\\{a_{k}^{(S)}\\}$.

$\bar{S}$ is the half space $x^{1}<0$ at $x^{0}=0$, and we can similarly write
down the mode expansion for ${{\mathcal{O}}}_{{\bar{S}}}(\xi)$, where now
$\xi$ refers to the Rindler coordinates in the left Rindler region,

${{\mathcal{O}}}_{{\bar{S}}}(\xi)=\int\frac{d^{2}k}{(2\pi)^{2}}\,u_{k}^{(\bar{S})}a_{k}^{(\bar{S})},\quad
u^{(\bar{S})}_{k}=N_{k}e^{q_{+}\xi^{-}-\bar{q}_{-}\xi^{+}}\ .$ (50)

$\\{a_{k}^{(\bar{S})}\\}$ can be used to generate
${\mathcal{X}}_{\bar{S}}={\mathcal{Y}}_{\hat{\bar{S}}}$.

Since a modular flow for operators in $S$ and $\bar{S}$ corresponds to a
translation in $\eta$, $\omega$ can be interpreted as the modular frequency,
i.e. conjugate to the modular time.

It can be shown that ${{\mathcal{O}}}(x)$ for any $x\in{\mathbb{{R}}}^{1,1}$
can be expressed in terms of $\\{a_{k}^{(S)}\\}$ and $\\{a_{k}^{(\bar{S})}\\}$
longPaper ,

${{\mathcal{O}}}(x)=\int\frac{d^{2}k}{(2\pi)^{2}}\,\sum_{{\alpha}=S,\bar{S}}f^{({\alpha})}(x;k)a^{({\alpha})}_{k},$
(51)

which is the statement

${\mathcal{X}}_{S}\lor{\mathcal{X}}_{\bar{S}}={{\mathcal{M}}}_{\Omega}\ .$
(52)

More explicitly, there exists an (invertible) transformation matrix
$A_{kp}^{({\alpha})}$ between two sets of oscillators $\\{b_{p}\\}$ and
$\\{a^{(S)}_{k},a^{(\bar{S})}_{k}\\}$,

$b_{p}=\sum_{k,{\alpha}}A_{kp}^{({\alpha})}a_{k}^{({\alpha})},\quad{\alpha}=S,\bar{S}\
.$ (53)

The explicit form of $A_{kp}^{({\alpha})}$ as well as its inverse are given in
Appendix A.2.

(a) The decomposition of the Minkowski $x-$plane with respect to the causal
structure of the interval and its complement. The solid orange lines are the
image of the $y-$plane Rindler horizon, while the dashed lines are the
singularities of the inverse of the conformal transformation.

(b) The decomposition of the Minkowski $y$-plane into the nine regions that
map to the nine regions of the $x$-plane shown in the left plot, with $b=c=1$.
The solid orange lines are the Rindler horizons in $y,$ while the dashed lines
are $y^{-}=b$ and $y^{+}=-c$, where the transformation is singular.

Figure 11: The decomposition of the Minkowski $x-$ and $y-$planes with respect
to the causal structure of the $x-$plane decomposition. Note that while in the
right plot, the region $\hat{I}$ is spacelike separated from $P_{M},F_{M}$,
they are timelike separated in the left plot. This is due to the singularity
in the transformation.

Now consider examples E2 and E3. Since the regions in E2 and E3 are
complements on a single time slice, we will treat them together.
${\mathcal{Y}}_{\hat{I}}$ and ${\mathcal{Y}}_{\hat{O}}$ are defined by
restricting (43) to $\hat{I}$, which is the causal diamond for $I$, and
$\hat{O}=\hat{O}_{R}\cup\hat{O}_{L}$, respectively. In particular, since
$O_{R}$ and $O_{L}$ are disconnected, we have

${\mathcal{Y}}_{\hat{O}}={\mathcal{Y}}_{\hat{O}_{R}}\lor{\mathcal{Y}}_{\hat{O}_{L}}={\mathcal{X}}_{O_{R}}\lor{\mathcal{X}}_{O_{L}}\
.$ (54)

We can obtain a mode expansion for an operator in $\hat{I}$ or $\hat{O}$ by
conformally mapping the mode expansions discussed above for the right and left
Rindler regions to $\hat{I}$ and $\hat{O}$. Consider another spacetime spanned
by $y^{\mu}=(y^{0},y^{1})$ with $y^{\pm}=y^{0}\pm y^{1}$, and the following
conformal transformation that maps smoothly from the right Rindler wedge in
the $y$-plane to $\hat{I}$ in $x$-spacetime

$x^{-}=a{y^{-}+b\over b-y^{-}},\qquad x^{+}=a{y^{+}-c\over y^{+}+c}\ ,$ (55)

where $b,c$ are arbitrary real positive constants. The above transformation is
singular at null lines $y^{-}=b$ and $y^{+}=-c$, which lie outside the right
Rindler quadrant in the $y$-plane. Below we will take $b=c=1$. Depending on
their causal relations with $\hat{I}$ and $\hat{O}_{R},\hat{O}_{L}$, the
$x$-plane can be separated into nine different regions. How various regions in
the $y$-plane are mapped to these nine regions is shown in Fig. 11. Due to the
singular behavior at $y^{-}=b$ and $y^{+}=-c$, the mapping preserves casual
structure only for the regions lying within the quadrant to the right of the
dashed lines in Fig. 11(b) (the shaded region). For example, while the region
$\hat{I}$ is causally disconnected from regions $F_{M}$ and $P_{M}$ in plot
(b), it is causally connected with them in plot (a). We can then reliably
conformally transform the mode expansions in the $y$-plane to the $x$-plane
only for the shaded region in the plot (b).

Using the mode expansions (46) and (50) for ${{\mathcal{O}}}(y)$ in the right
and left Rindler regions of the $y$-plane, we can use the conformal
transformation to obtain the mode expansion for ${{\mathcal{O}}}(x)$ in the
$\hat{I}$ and $\hat{O}_{L}$ regions (after expressing $\xi^{\pm}$ in terms of
$y^{\pm}$ and setting $b=c=1$)

$\displaystyle{{\mathcal{O}}}(x\in\hat{I})=\int{d^{2}k\over(2\pi)^{2}}u^{(I)}_{k}(x)~{}a^{(I)}_{k}\
,$ (56)
$\displaystyle{{\mathcal{O}}}(x\in\hat{O}_{L})=\int{d^{2}k\over(2\pi)^{2}}u^{(O)}_{k}(x)~{}a^{(O)}_{k}\
,$ (57) $\displaystyle
u^{(I)}_{k}(x)=N_{k}(2a)^{{{\Delta}}}(a-x^{-})^{-\bar{q}_{+}}(x^{-}+a)^{-q_{+}}(a-x^{+})^{-\bar{q}_{-}}(x^{+}+a)^{-q_{-}},$
(58) $\displaystyle
u^{(O)}_{k}(x)=N_{k}(2a)^{{{\Delta}}}(x^{-}-a)^{-q_{+}}(x^{-}+a)^{-\bar{q}_{+}}(a-x^{+})^{-q_{-}}(-x^{+}-a)^{-\bar{q}_{-}}\
.$ (59)

We can obtain the expressions for ${{\mathcal{O}}}$ in other regions of the
$x$-plane by analytic continuations of (56)–(59). The analytic continuation
procedure is a generalization of that for a generalized field in Rindler
introduced in longPaper  (which is in turn a generalization of the standard
Unruh procedure). The details are given in Appendix B.1. We find that
equations (57) and (59) (negating the arguments of all complex powers) in fact
also apply to $\hat{O}_{R}$ region and thus the full $\hat{O}$ region, i.e.

${{\mathcal{O}}}(x)=\int{d^{2}k\over(2\pi)^{2}}u^{(O)}_{k}(x)~{}a^{(O)}_{k},\quad
x\in\hat{O}=\hat{O}_{L}\cup\hat{O}_{R}\ .$ (60)

In other regions of the $x$-plane, ${{\mathcal{O}}}(x)$ can be expressed in
terms of both $a_{k}^{(I)}$ and $a_{k}^{(O)}$. Again there is an invertible
basis transformation between $\\{b_{p}\\}$ and
$\\{a_{k}^{(I)},a_{k}^{(O)}\\}$,

$b_{p}=\int{d^{2}k\over(2\pi)^{2}}\,\left(P^{(I)}_{kp}a^{(I)}_{k}+P^{(O)}_{kp}a^{(O)}_{k}\right)\
.$ (61)

The explicit expressions for $P^{(I)}_{kp}$ and $P^{(I)}_{kp}$ are given in
appendix B.2.

Since (56) is obtained from conformal transformation of the right Rindler
region $\hat{S}$, whose associated algebra
${\mathcal{X}}_{S}={\mathcal{Y}}_{\hat{S}}$, we should have
${\mathcal{X}}_{I}={\mathcal{Y}}_{\hat{I}}$, which can also be considered as
being generated by $\\{a^{(I)}_{k}\\}$.

We will now interpret ${\mathcal{X}}_{O}$ as being generated by
$\\{a^{(O)}_{k}\\}$. We note that the quantum numbers $k=(\omega,q)$ have no
obvious geometric meaning in the $O$ region; they went along for the ride with
the conformal transformation. Since before the conformal transformation,
$\omega$ corresponds to the modular frequency of Rindler regions, $\omega$ in
(56) and (60) can be interpreted as the modular frequency for the $I$ and $O$
regions. From (60), we have
${\mathcal{X}}_{O_{L}}\lor{\mathcal{X}}_{O_{R}}\subseteq{\mathcal{X}}_{O}$,
but now an important difference from the $I$ region is that equation (60) is
not invertible, i.e. it cannot be inverted to solve for $a_{k}^{(O)}$ in terms
of ${{\mathcal{O}}}(x)$ in $\hat{O}$. More explicitly, it is not possible to
find a set of functions $\\{f_{k}(x),k\in\mathbb{R}^{2}\\}$ supported for
$x\in\hat{O}$ such that

$a^{(O)}_{k}=\int_{x\in\hat{O}}d^{2}x~{}f_{k}(x){{\mathcal{O}}}(x),\quad\forall
k\ .$ (62)

See Appendix B.3 for a proof. This non-invertibility implies that

${\mathcal{X}}_{O_{R}}\lor{\mathcal{X}}_{O_{L}}\subset{\mathcal{X}}_{O}\quad{\Longrightarrow}\quad{\mathcal{Y}}_{\hat{O}}\subset{\mathcal{X}}_{O}\
.$ (63)

We can also describe ${\mathcal{X}}_{O}$ in terms of a “coordinate” basis by
introducing operators defined in terms of the modular time $s$ conjugate to
$\omega$

${{\mathcal{O}}}(s;x^{1})=\int{d^{2}k\over(2\pi)^{2}}\,u^{(O)}_{k}(x^{1},x^{0}=0)e^{-i\omega
s}a^{(O)}_{k},\qquad x^{1}\in O,$ (64)

which is invertible

$\displaystyle a^{(O)}_{k}=\int_{x^{1}\in
O}dx^{1}ds~{}f_{k}(x^{1},s){{\mathcal{O}}}(s;x^{1}),$ (65) $\displaystyle
f_{k}(x^{1},s)={1\over N_{k}}\left({(x^{1})^{2}-a^{2}\over
2a}\right)^{{{\Delta}}-1}\left({x^{1}-a\over x^{1}+a}\right)^{-iq}e^{i\omega
s}\ .$ (66)

Thus ${\mathcal{X}}_{O}$ can also be viewed as being generated by the set
$\\{{{\mathcal{O}}}(s;x^{1})\\}$. We also note that there does not exist a
conformal transformation that can smoothly map a Rindler region in the
$y$-plane to $\hat{O}$. Otherwise, we would have ${\mathcal{X}}_{O}$
equivalent to ${\mathcal{Y}}_{\hat{O}}$.

####  IV.3.2 Entanglement and causal wedges

We now turn to finding the bulk regions corresponding to ${\mathcal{X}}_{R}$
and ${\mathcal{Y}}_{\hat{R}}$ for the examples discussed above.

A bulk scalar field $\phi(X)$ in AdS3 can be expanded in modes as

$\displaystyle\phi(X)=\int{d^{2}p\over(2\pi)^{2}}v_{p}(X)b_{p},\quad
v_{p}(X)=\theta(-p^{2})\sqrt{\pi}z^{d\over 2}J_{{{\Delta}}-1}(\kappa
z)e^{ip\cdot x},$ (67)

where $X=(x^{\mu},z)$ are the Poincare coordinates of a bulk point, and the
rest of the notation is the same as in (43). Note that $\phi(X)$ is expanded
in terms of the same set of $\\{b_{p}\\}$ as
$\pi_{\Omega}({{\mathcal{O}}}(x))$. This follows from the extrapolate
dictionary and gives the identification (23). With the identification,
$\phi(X)$ as written in (67) can be regarded as a boundary operator.

Consider first the case the boundary region is given by half space
$R=S:x^{1}>0$. Using the basis change (53) we can write (67) as

$\phi(X)=\sum_{k}\left(\tilde{v}^{(S)}_{k}(X)a^{(S)}_{k}+\tilde{v}^{(\bar{S})}_{k}(X)a^{(\bar{S})}_{k}\right),\quad\tilde{v}^{({\alpha})}_{k}(X)=\int{d^{2}p\over(2\pi)^{2}}A_{kp}^{({\alpha})}v_{p}(X)\
.$ (68)

The entanglement wedge ${\mathfrak{b}}_{R}$ for $R$ is then obtained by
finding the collection of $X$ such that $\tilde{v}^{(\bar{S})}_{k}(X)=0$ for
all $k$. We then find the entanglement wedge (which coincides with the causal
domain as ${\mathcal{X}}_{S}={\mathcal{Y}}_{\hat{S}}$) is given by that in
Fig. 10(a). For calculation details see Appendix C.

The same discussion applies to the $I$ and $O$ regions. Using (61) we can
write (67) as

$\phi(X)=\sum_{k}\left(\tilde{v}^{(I)}_{k}(X)a^{(I)}_{k}+\tilde{v}^{(O)}_{k}(X)a^{(O)}_{k}\right),\quad\tilde{v}^{({\alpha})}_{k}(X)=\int{d^{2}p\over(2\pi)^{2}}P_{kp}^{({\alpha})}v_{p}(X)\
.$ (69)

The entanglement wedge of $I$ is obtained by identifying the collection of $X$
such that $\tilde{v}^{(O)}_{k}(X)=0$ for all $k$, which gives the region in
Fig. 10(b) (the causal domain is the same as
${\mathcal{X}}_{I}={\mathcal{Y}}_{\hat{I}}$). Similarly, the entanglement
wedge of $O$ is obtained by identifying the collection of $X$ such that
$\tilde{v}^{(I)}_{k}(X)=0$ for all $k$, which gives the region in Fig. 10(c).
In this case the entanglement wedge is larger than the causal domain. For
calculation details see Appendix D.

###  IV.4 Additivity anomaly: union and intersection of wedge algebras

We have already seen an example of the additivity anomaly in (63). We will now
explore it further by examining unions and intersections of
${\mathcal{X}}_{R}$’s in a few situations. We start by considering a simple
example purely from the boundary perspective using the generalized free field
theory description of a scalar operator, which already illustrates well
various surprising and unintuitive features associated with such unions and
intersections. We then discuss these features from the bulk perspective using
the duality with entanglement wedges. We see that the bulk description can be
interpreted as providing a geometrization of these unintuitive properties.

Consider a boundary CFT2 in the vacuum state, and two intersecting intervals
$R_{-}:x^{1}\in(-b-a,-b+a)$ and $R_{+}:x^{1}\in(b-a,b+a)$ (with $a>b>0$) on
the $x^{0}=0$ slice. $R_{U}=R_{-}\cup R_{+}$ is an interval of width $2(a+b)$
while $R_{I}=R_{-}\cap R_{+}$ is an interval of width $2(a-b)$. These regions
and their causal completions are shown in Fig. 12. Note that
$\hat{R}_{-}\cup\hat{R}_{+}$ is not causally complete. We would like to
understand the relations among
${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}$,
${\mathcal{X}}_{R_{-}}\land{\mathcal{X}}_{R_{+}}$, ${\mathcal{X}}_{R_{U}}$,
and ${\mathcal{X}}_{R_{I}}$. If we were at finite $N$, i.e. dealing with
${{\mathcal{B}}}_{R}$ (suppressing $N$ below) we would have

${{\mathcal{B}}}_{R_{-}}\lor{{\mathcal{B}}}_{R_{+}}={{\mathcal{B}}}_{R_{U}},\quad{{\mathcal{B}}}_{R_{-}}\land{{\mathcal{B}}}_{R_{+}}={{\mathcal{B}}}_{R_{I}}\
.$ (70)

Recall that in the large $N$ limit and the vacuum state
$\lvert{\Omega}\rangle$, ${\mathcal{X}}_{R}$ for $R$ given by an interval is
generated by single-trace operators in $\hat{R}$, i.e.
${\mathcal{X}}_{R}={{\mathcal{M}}}_{\hat{R}}$. Since $R_{-},R_{+},R_{U},R_{I}$
are all intervals we have
${\mathcal{X}}_{R_{i}}={{\mathcal{M}}}_{\hat{R}_{i}},$ for $i=+,-,U,I$. Recall
that ${{\mathcal{M}}}_{A}$ denotes the restriction of single-trace algebra
${{\mathcal{M}}}_{\Omega}$ to a spacetime subregion $A$.

(a)

(b)

Figure 12: Causal completions of various intervals on the boundary. (a) The
regions $\hat{R}_{-},~{}\hat{R}_{+}$ and $\hat{R}_{I}.$ (b) The region
$\hat{R}_{U}.$

Since single-trace operators at different times are independent, naively we
may expect that

${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}={{\mathcal{M}}}_{\hat{R}_{-}}\lor{{\mathcal{M}}}_{\hat{R}_{+}}={{\mathcal{M}}}_{\hat{R}_{-}\cup\hat{R}_{+}}\subset{{\mathcal{M}}}_{\hat{R}_{U}}={\mathcal{X}}_{R_{U}},\quad{\mathcal{X}}_{R_{-}}\land{\mathcal{X}}_{R_{+}}={{\mathcal{M}}}_{\hat{R}_{I}}={\mathcal{X}}_{R_{I}}\
.$ (71)

This expectation turns out to be incorrect. Instead, we find

$\displaystyle{{\mathcal{M}}}_{\hat{R}_{-}\cup\hat{R}_{+}}\subset{\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}\subset{{\mathcal{M}}}_{\hat{R}_{U}}={\mathcal{X}}_{R_{U}},$
(72)
$\displaystyle{\mathcal{X}}_{R_{I}}={{\mathcal{M}}}_{\hat{R}_{I}}\subset{\mathcal{X}}_{R_{-}}\land{\mathcal{X}}_{R_{+}}\
.$ (73)

To see (72), recall that ${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}$ is
defined as

${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}\equiv({\mathcal{X}}_{R_{-}}\cup{\mathcal{X}}_{R_{+}})^{\prime\prime}\
.$ (74)

The first relation in (72) implies that there are operators outside
${{\mathcal{M}}}_{\hat{R}_{-}\cup\hat{R}_{+}}$ that commute with
$({\mathcal{X}}_{R_{-}}\cup{\mathcal{X}}_{R_{+}})^{\prime}$, while the second
relation says that there are operators in ${{\mathcal{M}}}_{\hat{R}_{U}}$ that
do not commute with
$({\mathcal{X}}_{R_{-}}\cup{\mathcal{X}}_{R_{+}})^{\prime}$. Equation (73)
says that there exist operators which can be expressed separately in terms of
single-trace operators in $\hat{R}_{-}$ and $\hat{R}_{+}$, but not those in
$\hat{R}_{I}$. See Fig 12 (a). The second and third properties can be
demonstrated explicitly by using the mode expansion of ${{\mathcal{O}}}$ in
terms of an interval and its complement. For details, see Appendix E. Their
bulk geometric descriptions are given in Fig. 3 and Fig. 13 (b).

The first relation of (72) is more subtle, and so far we can only show it
explicitly with the help of the bulk description (see Appendix F.3 for a pure
boundary discussion). In the bulk,
${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}$ is given by operators lying
inside the domain of dependence of $E_{R_{-}}\cup E_{R_{+}}$ in Fig. 13 (a).
From the duality, ${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}$ is given by
${{\mathcal{M}}}_{R_{e}}$ where $R_{e}$ is the largest boundary spacetime
subregion that has $\widehat{E_{R_{-}}\cup E_{R_{+}}}$ as its causal domain.
$R_{e}$ is found explicitly in Appendix F, and is shown in Fig. 14. It
encloses $\hat{R}_{-}\cup\hat{R}_{+}$.

(a) The $x^{0}=0$ slices of ${\mathfrak{b}}_{R_{-}}$ and
${\mathfrak{b}}_{R_{+}}.$

(b) The $x^{0}=0$ slice of ${\mathfrak{b}}_{R_{-}\cup R_{+}}$.

Figure 13: Bulk time slices of the entanglement wedges of intervals.  Figure
14: The region $R_{e}$ defined by
${\mathcal{X}}_{R_{+}}\lor{\mathcal{X}}_{R_{-}}={{\mathcal{M}}}_{R_{e}}$. It
is given by the largest boundary subregion whose causal domain is
$\widehat{E_{R_{-}}\cup E_{R_{+}}}$. The points between the black and blue
curves lie in $R_{e}$ but not in $\hat{R}_{-}\cup\hat{R}_{+}.$ The blue curves
are given by $x^{+}x^{-}=a^{2}-b^{2}$ for $x^{1}\in(-b,b)$.

We note that when $R_{-}$ and $R_{+}$ are non-intersecting (i.e. if $b>a>0$),
we have

${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}={{\mathcal{M}}}_{\hat{R}_{-}\cup\hat{R}_{+}}\
.$ (75)

Depending on the sizes of $R_{-},R_{+}$ and the distance between them, we can
have the additivity anomaly

${\mathcal{X}}_{R_{-}}\lor{\mathcal{X}}_{R_{+}}\subset{\mathcal{X}}_{R_{-}\cup
R_{+}}$ (76)

when the entanglement wedge is connected, of which (63) is a special example.
Thus the “phase transition” of the entanglement wedge for two intervals can be
considered as the onset of violation of the additivity property.

In contrast to the intersecting interval case, the anomaly does not have a
geometric description in the boundary theory, as $\widehat{R_{-}\cup
R_{+}}=\hat{R}_{-}\cup\hat{R}_{+}$. We also note that the violation of the
intersection property is never geometrically realized on the boundary. This is
because intersections of causally complete regions are also causally complete
themselves.

A “phase transition” in the entanglement wedge is also encountered for a
single interval at finite temperature, with the bulk geometry described by a
BTZ black hole. See Fig. 15. This transition can again be understood as the
onset of violation of the additivity property, as indicated in figures 15 and
16. For boundary diamonds of half-widths $w<w_{c}<\pi,$ the entanglement wedge
and causal domain coincide and are given in Fig. 15 (a). When $w>w_{c}$ (but
still $w<\pi$) the entanglement wedge undergoes a “phase transition” to
suddenly include points all the way up to the black hole horizon, see Fig. 15
(b). The entanglement wedge now exceeds the causal domain which is given by
the red region in Fig. 15 (c) and does not include points near the horizon.

(a)

(b)

(c)

Figure 15: The “phase transition” in the entanglement wedge for an interval of
angular half-width $w$ at finite temperature. (a) the $t=0$ slice of the
entanglement wedge for $w<w_{c}$. (b) the $t=0$ slice of the entanglement
wedge for $w>w_{c}$. (c) The red shaded region gives the $t=0$ slice of the
causal domain for $w>w_{c}$. The blue shaded region is the causal domain for
its complement (which coincides with its entanglement wedge).

We can understand this “phase transition” in the entanglement wedge for an
interval at finite temperature from the addivity anomaly. Going to the
covering space of the boundary cylinder, the domains of dependence of the
interval and its complement are mapped to the repeated diamond regions,
$\cup_{i}D_{i},~{}\cup_{i}\tilde{D}_{i},$ shown in red and blue respectively
in Fig. 16 (a). We can consider two possible algebras for the union of red
diamonds in Fig. 16 (a): $\lor_{i}{\mathcal{X}}_{D_{i}}$ or
${\mathcal{X}}_{\cup_{i}D_{i}}$ (with respect to the thermal state). The phase
transition of the entanglement wedge at $w=w_{c}$ can then be interpreted as
the onset of the additivity anomaly at $w=w_{c}$: these two algebras become
different for $w>w_{c}$. The algebra $\lor_{i}{\mathcal{X}}_{D_{i}}$ has bulk
dual shown in red in Fig. 16 (b) while ${\mathcal{X}}_{\cup_{i}D_{i}}$ has
bulk dual shown in red in Fig. 16 (c). With $\lor_{i}{\mathcal{X}}_{D_{i}}$ in
the covering space, upon mapping back to the cylinder we obtain the bulk dual
shown in red in Fig. 15 (c), the causal domain of the boundary causal
completion of the interval. With ${\mathcal{X}}_{\cup_{i}D_{i}}$ in the
covering space, upon mapping back to the cylinder we instead obtain the bulk
dual shown in red in Fig. 15 (b), the entanglement wedge of the interval.

The additivity anomaly occurs when the algebra associated to a subregion, $R,$
is larger than the algebra that is ‘built up’ from smaller subregions whose
union is $R.$ The existence of such an anomaly reflects some degree of non-
locality in the field theory. Unlike a local field theory where we expect that
the algebras of a subregion $R$ can be constructed from those associated to
infinitesimal diamond-shaped subregions whose union covers $R,$ this is not
possible in a theory with an additivity anomaly. In such a theory, the algebra
of a region cannot necessarily be decomposed into those of subregions covering
it. Physical implications of the additivity anomaly will be further discussed
elsewhere addPaper .

(a)

(b)

(c)

Figure 16: Causal domains and entanglement wedges of diamonds on the covering
space (the Minkowski plane) of a Lorentzian cylinder. (a) Image of the causal
completion of an interval (red) and its causal complement (blue) on the
cylinder in the covering space. (b) Causal domain of the unions of red or blue
diamonds. (c) Entanglement wedge of the union of red or blue diamonds.

###  IV.5 Summary

To conclude this section, we give a summary of the main insights into
subregion-subregion duality from the lens of equivalence of bulk and boundary
operator algebras:

  1. 1.

Subalgebra-subregion duality provides a mathematically precise definition of
subregion-subregion duality. It highlights that subregion-subregion duality
can be precisely formulated only in the large $N$ limit. Previous formulations
Almheiri:2014lwa ; Pastawski:2015qua ; Hayden:2016cfa ; Dong:2016eik ;
Harlow:2016vwg ; Cotler:2017erl  assumed a bulk Hilbert space embedded into
the boundary Hilbert space and that both the code subspace corresponding to
the bulk Hilbert space and the boundary Hilbert space could be factorized into
Hilbert spaces corresponding to subregions. These assumptions are not realized
in (field theoretic) examples of the duality.191919Some of these assumptions
have been relaxed in Kang:2018xqy ; Faulkner:2020hzi , but the formulations
are still incompatible with the Reeh-Schlieder theorem Kelly:2016edc ;
Faulkner:2020hzi . Properly generalized, such formulations may be interpreted
at a finite $N$. But note that at a finite $N$, the bulk geometric picture
becomes murky (for example geometric concepts such as the RT surface and the
entanglement wedge cannot be precisely defined), and such formulations can at
most be approximate. Recently, Faulkner:2022ada  discussed embeddings of bulk
subalgebras in the $G_{N}\to 0$ limit to boundary theory at a finite $N$,
which helps to clarify how subregion-subregion duality emerges in the large
$N$ limit. Note that since the boundary and bulk theories are not defined in
the same regime of $N$, the embedding cannot be viewed as a duality map.

  2. 2.

The new formulation provides an alternative way to define the entanglement
wedge and the associated RT or HRT surfaces without using entropy associated
to a boundary subregion. It also provides a boundary explanation of why the
entanglement wedge always encloses causal domain.

Since the boundary operator algebra in a local subregion is type III1,
entanglement entropy is not well defined: a regularization is needed and the
answer depends on the regularization. Furthermore, in the $N\to\infty$ limit,
even the regularized entropies go to infinity, which means that they are not
intrinsically defined objects in the large $N$ limit. In contrast,
entanglement wedges and RT or HRT surface are well-defined geometric objects
in the $G_{N}\to 0$ limit—there is no ambiguity in their definitions. The new
algebraic formulation resolves this tension: the objects on both sides of the
duality are well-defined in the same limit.

  3. 3.

The equivalence of algebras immediately implies that, in the large $N$ limit,
the relative entropies of the bulk and boundary theory in the corresponding
regions must agree Jafferis:2015del . The conclusion not only applies to
entanglement wedges, but also causal domains.202020Note that the notions of
relative entropies associated to causal domains have no analogue at finite
$N.$ Such quantities can thus only be used to probe properties of states in
the large $N$ limit.

  4. 4.

A surprising result following from Jafferis:2015del  and global reconstruction
Banks:1998dd ; Bena:1999jv ; Hamilton:2006az  is that modular flows of single-
trace operators in a spatial subregion $R$ are part of the associated single-
trace algebra in the large $N$ limit, although they may not be written in
terms of local single-trace operators in $\hat{R}$. This underlies our
proposal (33). We have now been able to see this in an explicit example in
Sec. IV.3. It may be expected that operators generated from non-geometric
modular flows can be exponentially complex when viewed at a large, but finite
$N$. This means that certain exponentially complex computational tasks at a
finite but large $N$, may be achieved “easily” using operators in the algebra
of single-trace operators. An example is the reconstruction of bulk operators
in the region lying outside the causal domain, but still in the entanglement
wedge, i.e. what is called Python’s lunch Brown:2019rox ; Engelhardt:2021mue ;
Engelhardt:2021qjs .

##  V Duality for general bulk regions

Entanglement wedge reconstruction and the RT or HRT prescription allow us to
study bulk subregions bounded by extremal surfaces that contain near-boundary
points. In order to study more general bulk subregions, subregion-subregion
duality is not sufficient, and we must pass to the more general subalgebra-
subregion duality. In this section we discuss some explicit examples.

###  V.1 Bulk dual regions of time bands in the boundary

Consider the boundary with topology $\mathbb{R}\times X,$ for some spatial
manifold $X.$ A time-band212121For other discussions of the algebras
associated to time-bands, see Banerjee:2016mhh ; Bahiru:2022oas  on the
boundary is a spacetime subregion that contains an entire Cauchy slice. It can
be specified as the region between two non-intersecting Cauchy slices,
$\Sigma_{p}$ and $\Sigma_{f}.$ See figure 17. We will denote the time band
bounded by $\Sigma_{p}$ to the past and $\Sigma_{f}$ to future as $T_{p,f}.$

Figure 17: A general time band for $X=S^{1}$. $T_{p,f}$ is the spacetime
region on the cylinder between the two Cauchy slices $\Sigma_{f}$ and
$\Sigma_{p}$.

While $T_{p,f}$ is not a causally complete boundary spacetime subregion (its
causal completion is the entire boundary) we can still associate to it a
subalgebra, ${{\mathcal{M}}}_{T_{p,f}}$, by restriction of
${{\mathcal{O}}}(x)$ to $x\in T_{p,f}$. Any $T_{p,f}$ can be constructed from
a union of an infinite number of diamond shaped regions $\\{D_{i}\\}$, with
$\cup_{i}D_{i}=T_{p,f}$, see Fig. 18 for an example.

Figure 18: A boundary time-band (blue) can be constructed from the union of an
infinite number of diamond-shaped subregions (yellow). The left and right
dashed lines in the figure are identified. Here the time band is assumed to be
not too long and only show two diamonds.

Since ${{\mathcal{M}}}_{A}$ is defined as the restriction of
${{\mathcal{O}}}(x)$ to $x\in A$ and the union of the diamonds exactly covers
the time-band, i.e. $\cup_{i}D_{i}=T_{p,f},$ we have

$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}=\lor_{i}{{\mathcal{M}}}_{D_{i}}\
.$ (77)

Notice that the resulting algebra
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}$ is independent of the
choice of diamonds $\\{D_{i}\\}$ used to build $T_{p,f}$.

From the duality for diamond-shaped boundary subregions
${{\mathcal{M}}}_{D_{i}}=\widetilde{{{\mathcal{M}}}}_{{\mathfrak{c}}_{D_{i}}}$
and additivity of the bulk field theory we have

$\lor_{i}{{\mathcal{M}}}_{D_{i}}=\lor_{i}\widetilde{{{\mathcal{M}}}}_{{\mathfrak{c}}_{D_{i}}}=\widetilde{{{\mathcal{M}}}}_{\left(\cup_{i}{\mathfrak{c}}_{D_{i}}\right)^{\prime\prime}}\
.$ (78)

The causal domain of the time band ${\mathfrak{c}}_{T_{p,f}}$ is defined by

${\mathfrak{c}}_{T_{p,f}}=\left(J^{-}(T_{p,f})\cap
J^{+}(T_{p,f})\right)^{\prime\prime}\ ,$ (79)

i.e. the bulk causal completion of the intersection of the bulk past and
future of $T_{p,f}.$ Since $\cup_{i}D_{i}=T_{p,f},$ from the bulk causal
structure we have

${\mathfrak{c}}_{T_{p,f}}=\left(\cup_{i}{\mathfrak{c}}_{D_{i}}\right)^{\prime\prime}\Rightarrow\widetilde{{{\mathcal{M}}}}_{{\mathfrak{c}}_{T_{p,f}}}=\widetilde{{{\mathcal{M}}}}_{\left(\cup_{i}{\mathfrak{c}}_{D_{i}}\right)^{\prime\prime}}\
.$ (80)

Thus, from (80), (78), and (77) we have

$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}=\widetilde{{{\mathcal{M}}}}_{{\mathfrak{c}}_{T_{p,f}}}\
,$ (81)

showing that ${\mathfrak{c}}_{T_{p,f}}$ defined in (79) is indeed the bulk
dual of $\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}.$

Since the bulk theory obeys Haag duality, the commutant,
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime},$ of
${{\mathcal{M}}}_{T_{p,f}}$ should then be dual to the bulk spacetime region
$({\mathfrak{c}}_{T_{p,f}})^{\prime}$. The bulk dual of the time-band is a
causally complete bulk subregion and thus is the domain of dependence of a
spatial bulk subregion on some Cauchy slice. Consider that bulk Cauchy slice
with a subregion $C_{p,f}$ and its complement $D_{p,f}=\overline{C_{p,f}}$
such that $\widehat{C_{p,f}}={\mathfrak{c}}_{T_{p,f}}$ and
$\widehat{D_{p,f}}=({\mathfrak{c}}_{T_{p,f}})^{\prime}$, see Fig. 19.
$D_{p,f}$ and its boundary $b_{p,f}=\partial D_{p,f}$ lie completely in the
bulk, with no near-boundary points, but we can still identify the boundary
algebra dual to $\widehat{D_{p,f}}$ as
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime}.$

Figure 19: The bulk dual of
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}$ intersects a bulk
Cauchy slice in the yellow region, $C_{p,f}.$ By bulk Haag duality the
complementary region $D_{p,f}=\overline{C_{p,f}}$ shown in pink must be
described by the commutant algebra,
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime}$.

In Balasubramanian:2013lsa ; Headrick:2014eia , there was a very interesting
proposal to associate an entropy to bulk surfaces like $b_{p,f}$ by connecting
it to a collection of boundary diamonds analogous to the set $\\{D_{i}\\}$
discussed above. The discussion there was in the reverse direction: given a
co-dimension two bulk surface, they used co-dimension two extremal surfaces
tangent to the surface at each point to give a collection of diamond-shaped
boundary regions, $\\{\tilde{D}_{i}\\}$. In our language, their procedure
describes the bulk dual222222In the case that the co-dimension two bulk
surface is nowhere-concave. of $\lor_{i}{\mathcal{X}}_{\tilde{D}_{i}},$ which
differs from $\lor_{i}{{\mathcal{M}}}_{\tilde{D}_{i}}$ for general non-vacuum
states.232323The bulk dual of
$\lor_{i}{{\mathcal{M}}}_{\tilde{D}_{i}}=\left({{\mathcal{M}}}_{\tilde{T}_{p,f}}\right)^{\prime\prime}$
with $\tilde{T}_{p,f}\equiv\cup_{i}\tilde{D}_{i},$ will be a bulk subregion
contained within the causal completion of the exterior of the co-dimension two
surface studied in Balasubramanian:2013lsa ; Headrick:2014eia . Despite these
differences, it is of great interest to see whether the entropy discussed
there can be considered as being associated with the algebra
$\left({{\mathcal{M}}}_{T_{p,f}}\right)^{\prime\prime}$ or
$({{\mathcal{M}}}_{T_{p,f}})^{\prime}$ when working in the vacuum.

We will now study some examples in which an explicit description of the causal
domain of the time-band can be given.

###  V.2 Vacuum

Take the boundary spatial slice to be a circle, i.e. $X=S^{1}.$ In the vacuum
state, the bulk dual is pure AdS${}_{3},$ which can be described by
coordinates $(t,\phi,\rho)$ with metric

$ds^{2}={l^{2}\over\cos^{2}\rho}\left(-dt^{2}+d\rho^{2}+\sin^{2}\rho~{}d\phi^{2}\right)\
,$ (82)

and coordinate ranges
$t\in\mathbb{R},~{}\phi\in(-\pi,\pi],~{}\rho\in(0,{\pi\over 2}).$ The boundary
is at $\rho={\pi\over 2}$.242424Note that we have compactified the radial
coordinate in this discussion.

We consider a uniform time band of total width $2w,$ i.e. we take $\Sigma_{p}$
to be described by $t=t_{0}-w$ and $\Sigma_{f}$ by $t=t_{0}+w$ for some fixed
$w>0.$ When $w<\pi,$ for each $\phi_{0}\in(-\pi,\pi],$ the diamond of angular
(and temporal) width $w$ centered at $(t=t_{0},\phi=\phi_{0})$ is contained in
the time-band $T_{p,f}$ and the union of such diamonds over all $\phi_{0}$ is
$T_{p,f}$. When $w<{\pi\over 2},$ the intersection of the entanglement wedge
(or, equivalently in this case, the causal domain) of such a diamond with the
$t=t_{0}$ bulk slice is bounded by a bulk spacelike geodesic,
$\rho=\rho_{\phi_{0},w}(\phi),$ reaching a minimum value of the (compactified)
radial coordinate $\rho_{\min}(w)={\pi\over 2}-w$ at $\phi=\phi_{0}$. See
figure 20 (a). The union of the entanglement wedges of all such diamonds then
intersects the $t=t_{0}$ slice in the subregion
$H_{t_{0},w}=\\{(\phi,\rho)~{}|~{}\rho>\rho_{\min}(w)\\}$. See figure 20 (b).
The bulk dual of the uniform time-band of half-width $w<{\pi\over 2}$ is then
the bulk causal completion of $H_{t_{0},w}.$

(a)

(b)

Figure 20: Bulk time slice of the causal domains for diamond-shaped regions
and the time-band. (a) Intersection of the causal domain (and entanglement
wedge) of a diamond-shaped region of half-width $w={\pi\over 4}$ with the
$t=t_{0}$ bulk slice. (b) Intersection of the union of causal domains of
sixteen diamond-shaped regions, each of half-width $w={\pi\over 4},$ with the
$t=t_{0}$ bulk slice. In the limit of the continuous union of diamonds around
the boundary circle, the causal wedge covers $H_{t_{0},w}$ on the $t=t_{0}$
slice.

When ${\pi\over 2}<w<\pi,$ the time-band can still be covered by a one-
parameter family of diamond-shaped regions centered on the $t=t_{0}$ slice.
However, the entanglement wedge of such a diamond now covers more than half of
the bulk $t=t_{0}$ slice, including the ‘centre’ of AdS, $\rho=0.$ The union
of entanglement wedges of such diamond-shaped regions thus covers the entire
$t=t_{0}$ Cauchy slice of the bulk, and therefore the bulk dual of a time band
of total width between $\pi$ and $2\pi$ is the entire bulk spacetime. When
$w>\pi,$ there are no longer diamond-shaped regions of half-width $w$ on the
boundary, however, since such a time-band clearly contains another time-band
of half-width larger than ${\pi\over 2},$ the bulk dual of this larger time
band is also the entire bulk spacetime.

This is precisely as one would expect purely from the boundary GFF theory.
Around the vacuum, the GFF $\pi_{\Omega}({{\mathcal{O}}}(x^{\mu}))$ can be
expanded as

$\displaystyle\pi_{\Omega}({{\mathcal{O}}}(t,\phi))=\sum_{nl}C_{nl}\left[a_{nl}e^{-i\omega_{nl}t+il\phi}+a_{nl}^{\dagger}e^{i\omega_{nl}t-il\phi}\right],$
(83) $\displaystyle\omega_{nl}={{\Delta}}+l+2n,\quad n=0,1,\cdots,\quad
l\in{\mathbb{{Z}}},$ (84)

where $C_{nl}$ are some constants. Due to quantized nature of $\omega_{nl}$ we
can express $a_{nl}$ in terms of $\pi_{\Omega}({{\mathcal{O}}}(t,\phi))$
within any time band with half width $w\geq{\pi\over 2}$. Explicitly, we have
(as in Hamilton:2006az )

$a_{nl}={1\over C_{nl}}\int_{t_{0}-{\pi\over 2}}^{t_{0}+{\pi\over
2}}{dt\over\pi}\int{d\phi\over
2\pi}e^{i\omega_{nl}t-il\phi}~{}\pi_{\Omega}({{\mathcal{O}}}(t,\phi))_{+}\ ,$
(85)

where $\pi_{\Omega}({{\mathcal{O}}}(t,\phi))_{+}$ denotes the positive
frequency part $\pi_{\Omega}({{\mathcal{O}}}(t,\phi)).$

####  V.2.1 Temporally separated time bands

Now consider two temporally separated time-bands,
$T_{1}:t\in(t_{1}-w_{1},t_{1}+w_{1})$ and
$T_{2}:t\in(t_{2}-w_{2},t_{2}+w_{2}).$252525We thank Ahmed Almheiri, Venkatesa
Chandrasekaran, and Henry Lin for discussions on this. Without loss of
generality we take $t_{1}>t_{2}.$ If $w_{1}>{\pi\over 2}$ or $w_{2}>{\pi\over
2}$ then the corresponding time-band algebra is equal to the algebra on the
entire boundary spacetime. In this case the algebra of the union of time-bands
$\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}\equiv{{\mathcal{M}}}_{T_{1}}\lor{{\mathcal{M}}}_{T_{2}}$
is equal to the full algebra, ${{\mathcal{B}}}({{\mathcal{H}}}^{\rm
GNS}_{\Omega})$ as well. We will thus restrict to $w_{1},w_{2}<{\pi\over 2}$.
While we can in principle deduce $\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}$ by working with boundary GFF, we will do so by
using the bulk duals of $T_{1}$ and $T_{2}$, which can be directly read from
the bulk geometry. From Fig. 21, we find that
$\left({{\mathcal{M}}}_{T_{1}\cup T_{2}}\right)^{\prime\prime}$ is given by
the time band formed by joining the two time bands and the spacetime region
between them.

(a)

(b)

Figure 21: (a) Bulk duals of ${{\mathcal{M}}}_{T_{1}}$ and
${{\mathcal{M}}}_{T_{1}}$. (b) Bulk dual of $\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}$ (with $t_{1}+w_{1}-(t_{2}-w_{2})<\pi$). Both
figures are a constant $\phi$ slice of global AdS3.

Explicitly, we find that $\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}={{\mathcal{M}}}_{T_{12}},$ where

$T_{12}=\\{(t,\phi)~{}|~{}t_{m}-w_{m}<t<t_{m}+w_{m}\\}\ ,$ (86)

with $t_{m}={{\frac{1}{2}}}\left(t_{1}+w_{1}+t_{2}-w_{2}\right)$ and
$w_{m}\equiv{t_{1}+w_{1}-(t_{2}-w_{2})\over 2}.$ i.e. The algebra of the union
of two temporally separated time-bands is equal to the algebra of the smallest
single (connected) time-band that contains the original time-bands.

####  V.2.2 More general boundary spacetime regions

Consider the case of the boundary spatial manifold $X=\mathbb{R},$ so that the
boundary spacetime is $(1+1)-$dimensional Minkowski space. With the boundary
theory in the vacuum state, the bulk dual is Poincaré AdS3. Consider a time-
reflection spacetime subregion. Such time-reflection symmetric regions are of
interest since their bulk duals are domains of dependence of bulk subregions
contained within a time-reflection symmetric Cauchy slice of the bulk. For
simplicity, we consider a connected subregion with axis of time reflection
symmetry being the $t=0$ boundary slice. Such a subregion can then be
described by $T_{w}=\\{(t,x)~{}|~{}-w(x)<t<w(x),~{}x\in I\\},$ where $I$ is an
interval on the $x-$axis and $w(x)>0$ on $I$.

In this case, the causal domain of $T_{w}$ is the domain of dependence of a
bulk spatial subregion $C_{w}$ on the bulk $t=0$ slice. Since Poincaré AdS3 is
conformally flat, a point $(t,x,z)$ is in the past of $T_{w}$ only if
$(t-w(x_{0}))^{2}>(x-x_{0})^{2}+z^{2},~{}t<w(x_{0})$ for some $x_{0}\in I.$
Focusing on points on the bulk $t=0$ slice, one finds that the only points in
the bulk past of $T_{w}$ are

$\\{(t,x,z)~{}|~{}t=0,~{}0<z<z_{w}(x)\\}\ ,$ (87)

where

$z_{w}(x)=\max_{x_{0}\in I}\sqrt{w(x_{0})^{2}-(x-x_{0})^{2}}\ ,$ (88)

and for each $x,$ the maximum is over all $x_{0}$ such that the argument of
the square root is positive. If no such $x_{0}$ exists for a given $x,$ we
have $z_{w}(x)=0.$ By time-reflection symmetry, the exact same set of points
on the $t=0$ bulk slice are in the bulk future of $T_{w},$ and thus the
intersection of the causal domain of $T_{w}$ with the $t=0$ slice is exactly
this set,

$C_{w}=\\{(t,x,z)~{}|~{}t=0,~{}0<z<z_{w}(x)\\}\ .$ (89)

For example, the choice of a ‘square’ boundary subregion, see figure 22 (a),
of half-width $w>0,$ i.e.

$w(x)=w,~{}x\in I=[-w,w]$ (90)

leads to a causal domain whose intersection with the bulk $t=0$ slice,
$C_{w}$, is shown in figure 22 (b). The causal domain of this square region
turns out to be identical to the causal domain of the larger boundary
subregion shown in figure 22 (c).262626We thank Ahmed Almheiri, Venkatesa
Chandrasekaran, and Henry Lin for discussions on this.

(a)

(b)

(c)

Figure 22: (a) A ‘square’ boundary spacetime subregion. (b) $t=0$ slice of
causal domain of the ‘square’ boundary subregion. (c) A boundary subregion
with the same bulk causal domain.

An interesting feature of the bulk spatial subregions found in this procedure
is that their complements on the $t=0$ bulk slice,
$D_{w}\equiv\overline{C_{w}},$ are geodesically convex, i.e. for any two
points $p_{1},~{}p_{2}\in D_{w},$ there is a unique minimizing geodesic,
${{\gamma}}_{12},$ on the $t=0$ bulk slice that connects them and, moreover,
this geodesic lies entirely in $D_{w}.$ For a proof of this, see appendix G.
It would be interesting to study if the geodesic convexity observed here can
be understood in terms of the entanglement wedges for gravitating regions
recently discussed by Bousso and Penington Bousso:2022hlz .

###  V.3 An interior causal diamond

Using Haag duality of the bulk field theory we may use the results of the
previous section to understand which boundary algebras describe bulk operators
within an interior causal diamond of global AdS. In particular, the algebra of
operators in the bulk domain of dependence of a spherically symmetric bulk
spatial subregion
$H_{t_{0},\rho_{m}}=\\{(t,\phi,\rho)~{}|~{}t=t_{0},\rho<\rho_{m}(<{\pi\over
2})\\}$ in AdS3 is the commutant of the algebra of operators on the domain of
dependence of
$\overline{H_{t_{0},\rho_{m}}}=\\{(t,\phi,\rho)~{}|~{}t=t_{0},\rho>\rho_{m}\\}.$
By the discussion of the previous section, this algebra of operators is dual
to the algebra of boundary operators in the uniform time-band, $T,$ of width
$\pi-2\rho_{m}$ centered at $t_{0}.$ See figure 23.

Figure 23: An interior causal diamond is shown in blue. Its causal complement
is the region between the red surfaces and the conformal boundary. The algebra
in the causal complement region is dual to a boundary time-band.

From bulk Haag duality, we then obtain an explicit boundary characterization
of the algebra of bulk operators in the interior causal diamond that is the
domain of dependence of the spatial subregion $H_{t_{0},\rho_{m}}.$ The
boundary dual of this interior diamond is
$\left({{\mathcal{M}}}_{T}\right)^{\prime},$ the commutant of a time-band
algebra.

Notice that the algebra of such an interior causal diamond, which contains no
near-boundary points, is the commutant of the algebra of a time-band.
Recalling that time-band algebras are only non-trivial subalgebras in the
large $N$ limit of the boundary CFT, we see that there is no finite $N$
analogue of the algebra dual to bulk operators in an interior causal diamond.
At finite $N$ the commutant of a “time-band algebra” is trivial. Thus, bulk
subregions with no near-boundary points are purely emergent in the large $N$
limit. The finite $N$ theory can at most describe such bulk subregions
approximately, and at the moment we do not have the proper language for such
an approximate description.

For more general interior causal diamonds in the bulk the story should be the
same. Bulk operators in those patches are described by the commutant of a
time-band algebra, though the corresponding time-bands will no longer be
spherically symmetric. One interesting subtlety is that, in the examples we
have studied, time-reflection symmetric boundary time-bands always describe a
subregion of a time-reflection symmetric bulk Cauchy slice that is the
complement of a geodesically convex subregion on that slice. Thus, it appears
that only interior causal diamonds associated to geodesically convex
subregions of a bulk time-reflection symmetric slice can be described by the
commutant of the algebra of some boundary time-reflection symmetric time-band.
It would be very interesting to understand the boundary dual of interior
causal diamonds associated to non-geodesically convex spatial subregions,
though there may not be a simple geometric understanding of the boundary
subalgebra in this case.

###  V.4 Finite temperature

Consider again the case of boundary spatial manifold $X=S^{1},$ but now with
the boundary in a thermal state above the Hawking-Page temperature. The bulk
dual is then the BTZ black hole, whose right exterior can be described by
coordinates $(t,\phi,r)$ with metric

$ds^{2}=-l^{2}f(r)dt^{2}+{dr^{2}\over f(r)}+r^{2}d\phi^{2},\qquad
f(r)={r^{2}-r_{+}^{2}\over l^{2}}\ ,$ (91)

and coordinate ranges
$t\in\mathbb{R},~{}\phi\in(-\pi,\pi],~{}r\in(r_{+},\infty).$ The boundary is
at $r=\infty$ and the horizon at $r=r_{+}.$ The AdS length scale is denoted by
$l$. The temperature with respect to $t$ (which is dimensionless) is then
$T={r_{+}\over 2\pi l}.$

Consider a uniform time-band, $T_{w}:\\{(t,\phi)~{}|~{}-w<t<w\\},$ of half-
width $w$ centered at $t=0$ on the boundary. When $w<\pi,$ we may decompose
the time-band into diamond-shaped subregions of angular (and temporal) half-
width $w$ centered at a point $(t=0,\phi=\phi_{0}),$ which we denote by
$D_{\phi_{0},w}.$ We then have $\cup_{\phi_{0}}D_{\phi_{0},w}=T_{w}.$ The
causal and entanglement wedges of such diamond-shaped boundary subregions in
the BTZ black hole geometry were discussed in Fig. 15. The subalgebra of the
time-band is then
$\left({{\mathcal{M}}}_{T_{w}}\right)^{\prime\prime}=\lor_{\phi_{0}}{{\mathcal{M}}}_{D_{\phi_{0},w}}.$

The bulk region dual to $\left({{\mathcal{M}}}_{T_{w}}\right)^{\prime\prime}$
can then be obtained by the union of causal domains for $D_{\phi_{0},w}$, i.e.

${\mathfrak{c}}_{T_{w}}=\left(\cup_{\phi_{0}}{\mathfrak{c}}_{{\phi_{0},w}}\right)^{\prime\prime},$
(92)

where ${\mathfrak{c}}_{{\phi_{0},w}}$ is the causal completion of the region
shown in Fig. 24 (a). The intersection of ${\mathfrak{c}}_{T_{w}}$ with the
$t=0$ slice is then

$H_{r_{m}(w)}=\\{(t,\phi,r)~{}|~{}t=0,r>r_{m}(w)\\}\ ,$ (93)

with

$r_{m}(w)=r_{+}~{}\coth{wr_{+}\over l}\ .$ (94)

The bulk dual of such a time-band is then the bulk domain of dependence of
$H_{r_{m}(w)}$. See figure 24 (b). $r_{m}(w)$ can be obtained from the
intersection with the $t=0$ slice of the radial null geodesics from $t=w$
given in Cruz:1994ir .

(a)

(b)

Figure 24: The $t=0$ slice of the right exterior of the BTZ black hole. The
black hole is the black region $r\in(0,r_{+}).$ (a) Bulk dual of a diamond
boundary subegion ($\phi_{0}={\pi\over 4},~{}w={\pi\over 2}$). (b) Bulk dual
of the boundary time band of half-width $w=\pi.$

When $w>\pi,$ we can no longer build the time-band from diamond-shaped
boundary subregions centered on a single boundary time-slice. We can, however,
construct such a time-band from the union of time-bands each with half-width
$w<\pi.$ Consider for example, two time bands $T_{1}$ (centered around $t_{1}$
with half width $w_{1}$) and $T_{2}$ (centered around $t_{2}$ with half width
$w_{2}$) as indicated in Fig. 25. From the bulk geometry we find that the dual
bulk region for $\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}\equiv{{\mathcal{M}}}_{T_{1}}\lor{{\mathcal{M}}}_{T_{2}}$
is given by the causal completion of the region $H_{r_{m}(w_{12})}$ where
$w_{12}$ is the half width of the time band $T_{12}$ obtained by including
both $T_{1}$ and $T_{2}$, and the spacetime region between them. Here
$H_{r_{m}(w_{12})}$ lies on the bulk time-slice at the value of $t$ about
which $T_{12}$ is time-reversal symmetric. It is thus natural to propose that
$\left({{\mathcal{M}}}_{T_{1}\cup
T_{2}}\right)^{\prime\prime}=\left({{\mathcal{M}}}_{T_{12}}\right)^{\prime\prime}$.
Thus we conclude that the dual region of the boundary time-band $T_{w}$ is
always given by (94) for any $w>0$.

(a) Bulk duals of time-bands, $T_{1},~{}T_{2}$.

(b) Bulk dual of the union $T_{1}\cup T_{2}$.

Figure 25: Bulk duals of spherically symmetric time-bands at finite
temperature.

It is convenient to describe the bulk subregion dual to a time of width $w$ in
Kruskal coordinates, defined in the right exterior of the BTZ black hole by

$U=-\sqrt{r-r_{+}\over r+r_{+}}e^{-{r_{+}\over l}t},~{}V=\sqrt{r-r_{+}\over
r+r_{+}}e^{{r_{+}\over l}t}\ .$ (95)

The ‘innermost’ points of $H_{t_{0},r_{m}(w)}$ are then described by Kruskal
coordinates

$U_{m}=-e^{-{wr_{+}\over l}}e^{-{r_{+}\over l}t_{0}},~{}V_{m}=e^{-{wr_{+}\over
l}}e^{{r_{+}\over l}t_{0}}\ .$ (96)

Considering a semi-infinite time-band obtained by the limit
$w,(-t_{0})\to\infty$ with $t_{0}+w$ held fixed, i.e. the time-band
$t\in(-\infty,t_{0}+w)$. From (96) we then have the bulk dual to be

$U<U_{m}=-e^{-{r_{+}\over l}(t_{0}+w)},~{}V>V_{m}=0,~{}UV<-1\ ,$ (97)

which is exactly the bulk dual argued for in longPaper , see Fig. 26.

Figure 26: Bulk dual of the algebra of the semi-infinite time-band
$T_{-\infty,t_{0}+w}$ at finite temperature.

The commutant $({{\mathcal{M}}}_{T_{w}})^{\prime}$ of the time-band subalgebra
$\left({{\mathcal{M}}}_{T_{w}}\right)^{\prime\prime}$ is dual to the blue bulk
region in Fig. 27 (a). This is a bulk subregion that contains points in the
right and left exterior regions as well as the future and past interior
regions. In particular, it contains the entire left exterior region as well as
near singularity points. Since the corresponding algebra is defined as the
commutant of an algebra that has no non-trivial analogue at finite $N,$ this
algebra is necessarily emergent in the large $N$ limit.

(a)

(b)

Figure 27: (a) Bulk dual of the algebra of a finite time-band at finite
temperature (red) and its commutant (blue). (b) Bulk dual of the addition of
the algebras of finite time-bands on the left and right in the thermal field
double (red) and the commutant (blue).

Interestingly though, in the case that we work with the thermal field double
on two copies of the system, this equivalence of the algebra of two disjoint
time-bands with that of a larger time-band only occurs when those time-bands
are on the same copy of the boundary. The addition of the algebras of two
time-bands on different copies of the boundary is not associated with any
larger boundary subregion, i.e. for left/right time bands $T_{L/R}$ we have
${\mathfrak{c}}_{T_{L}\cup
T_{R}}={\mathfrak{c}}_{T_{L}}\cup{\mathfrak{c}}_{T_{R}}$. The commutant of
this algebra is interesting though. See the blue region of Fig. 27 (b). This
bulk subregion includes the bifurcation surface ($U=V=0$) and may include
points near the singularity. It does not contain any near boundary points, and
exactly as for such subregions in the vacuum case, it therefore is emergent in
the large $N$ limit as there is no finite $N$ analogue of its associated
algebra.

###  V.5 Single-Sided Black Holes

In the case of a high-energy272727Note that $\lvert{\Psi}\rangle$ should not
be an energy eigenstate but we should have $\langle{\Psi}\rvert
H\lvert{\Psi}\rangle=O(N^{2}).$ semiclassical pure state,
$\lvert{\Psi}\rangle,$ on the boundary, the bulk dual should be a single-sided
black hole. Since we are working with a pure state on the boundary, the GNS
vacuum $\lvert{0}\rangle_{\Psi,{\rm GNS}}$ should not be separating for
${{\mathcal{M}}}_{\Psi},$ the single-trace algebra on the entire boundary, and
moreover ${{\mathcal{M}}}_{\Psi}$ should have a trivial commutant on
${{\mathcal{H}}}^{\rm GNS}_{\Psi}.$ This implies that ${{\mathcal{M}}}_{\Psi}$
should be of type $I.$

In the large $N$ limit, the boundary theory should still be described by a GFF
theory where operators on different time-slices are independent. In
particular, this implies that we can assign non-trivial subalgebras to time-
bands. We expect that the algebra of a semi-infinite time-band from some late
time to infinity should be dual to bulk field operators supported at late
times outside the horizon of the black hole.282828See Chandrasekaran:2022eqq
for an interesting discussion of the relationship between these late time
algebras and the generalised second law of black hole entropy. See the blue
region of fig. 4. On the field theory side, two-point functions of these
operators should look approximately thermal. On the bulk side, the two-point
functions look like those in the exterior of the eternal black hole. Such late
time boundary operators should not be able to annihilate the GNS vacuum and
thus their algebra ${{\mathcal{M}}}_{\rm late},$ should have a non-trivial
commutant on ${{\mathcal{H}}}^{\rm GNS}_{\Psi}.$ This commutant is precisely
the algebra of mirror operators whose existence was argued for in
Papadodimas:2012aq ; Papadodimas:2013jku . In particular,
$\left({{\mathcal{M}}}_{\rm late}\right)^{\prime}$ should describe excitations
of the bulk field in the causal complement of the dual of
${{\mathcal{M}}}_{\rm late},$ i.e. $\left({{\mathcal{M}}}_{\rm
late}\right)^{\prime}$ should describe bulk excitations in the red region of
figure 4.

Similar to what was found in the case of the interior causal diamond in global
AdS, these mirror operators are emergent in the large $N$ limit. They do not
have any analogue at finite $N.$ At finite $N,$ the algebra of late time
operators is equal to the full boundary algebra and its commutant is trivial.
Nevertheless, we expect some approximate version of such “mirror” operators to
play important roles at a large but finite $N$

##  VI Discussion

Subalgebra-subregion duality opens up new conceptual perspectives and powerful
technical tools for understanding the emergence of space and time in
holography. There are many further questions to investigate, which include:
(i) working out an explicit construction for a bulk diamond region (with no
near-boundary points) in the vacuum state of bulk gravity, (ii) deriving the
Wheeler-de Witt equation for such a local bulk region from the boundary
theory, (iii) understanding how the construction works for a black hole formed
from gravitational collapse such as the Vaidya geometry, and (iv)
understanding the algebraic structures of boundary theories dual to AdS2 and
Jackiw-Teitelboim (JT) gravity theories Lin:2022zxd ; Lin:2022rzw ;
Lin:2022rbf , where the boundary is dynamical and leads to new elements beyond
those considered in shortPaper ; longPaper . JT gravity also provides a
laboratory where we may be able to probe the operator algebraic structure in
the full quantum gravity regime.

Since physics in a sufficiently small local spacetime region (which does not
touch the boundary) of an AdS spacetime is essentially the same as that in an
asymptotically flat or de Sitter spacetime, the boundary description of such a
region can potentially give important hints on how to formulate holographic
duality for asymptotically flat and cosmological spacetimes. See
Chandrasekaran:2022cip  for recent progress for understanding operator
algebras in de Sitter spacetime.

Subalgebra-subregion duality also leads to new insights into and perspectives
on entanglement wedge reconstruction and its interpretation in terms of
quantum error correction Almheiri:2014lwa ; Dong:2016eik . It can also be used
to obtain entanglement wedges directly from equivalence of algebras, rather
than using entropies as in Ryu:2006bv ; Hubeny:2007xt ; Engelhardt:2014gca .
It should be possible to use the duality to give a first principle derivation
of the “island” prescription used in the calculation of the Page curve for an
evaporating black hole Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni
.

There have also been exciting developments Witten:2021unn ;
Chandrasekaran:2022eqq ; Chandrasekaran:2022cip  in understanding operator
algebras in quantum gravity when $\hbar G_{N}$ corrections are included, which
have shed new light onto the nature of the de Sitter and black hole entropies.
It is of great interest to understand whether that construction can be applied
for general spacetime regions to define entropies for general spacelike
surfaces.

It is also important to understand the role played by couplings in the nature
of emergent operator algebras. Our boundary discussion has been concerned with
only the large $N$ limit. But the explicit examples we have considered are all
in the vacuum state, where the behavior of two-point functions follows from
conformal symmetry and is independet of couplings. When considering general
states, the effects of couplings become important, and the bulk gravity
descriptions are only available in the strong coupling limit. It is of immense
importance to understand whether various bulk geometric concepts persist away
from strong couplings, which corresponds to the stringy regime in the bulk.

Acknowledgements

We would like to thank Netta Engelhardt, Thomas Faulkner, Daniel Harlow,
Stefan Hollands, Finn Larsen, Nima Lashkari, Javier Magan, Jonathan Sorce, and
Edward Witten for discussions. This work is supported by the Office of High
Energy Physics of U.S. Department of Energy under grant Contract Number DE-
SC0012567 and DE-SC0020360 (MIT contract # 578218). SL acknowledges the
support of NSF grant No. PHY-2209997.

##  Appendix A Rindler decomposition of the generalized free field

In this appendix we derive the change of basis from oscillators associated to
a Rindler decomposition, $\\{a^{(S)}_{k},~{}a^{(\bar{S})}_{k}\\},$ to those of
the Minkowski mode expansion, $\\{b_{p}\\}.$ We first recall the global and
local mode expansions of the generalized free field and then use equivalence
of these mode expansions to derive the change of basis. We then briefly
mention how the change of basis for Rindler decompositions about an arbitrary
point $x_{0}$ can be derived. Finally, we discuss the equivalence of the
entanglement wedge and causal domain algebras,
${\mathcal{X}}_{S}={\mathcal{Y}}_{\hat{S}}$ in this case.

###  A.1 Global and local mode expansions

Recall that the generalized free field can be written in terms of a Minkowski
mode expansion

$\displaystyle{{\mathcal{O}}}(x)=\int{d^{2}p\over(2\pi)^{2}}\,f_{p}e^{ip\cdot
x}b_{p},\quad
f_{p}={\sqrt{\pi}\over{{\Gamma}}({{\Delta}})2^{{{\Delta}}-1}}\theta(-p^{2})\kappa^{{{\Delta}}-1},\quad
p\cdot x=-p^{0}x^{0}+p^{1}x^{1},$ (98)
$\displaystyle\kappa=\sqrt{-p^{2}},\quad
b_{p}^{\dagger}=b_{-p},\quad[b_{p},b_{p^{\prime}}]=(2\pi)^{2}\epsilon(p^{0}){{\delta}}^{(2)}(p+p^{\prime}),$
(99)

on all of $\mathbb{R}^{1,1}.$

Within the right (left) Rindler wedge, ${\hat{S}}:x^{1}>|x^{0}|$
(${\hat{\bar{S}}}:x^{1}<|x^{0}|$), we have (with $x^{\pm}=x^{0}\pm x^{1}$) a
Rindler mode expansion

$\displaystyle{{\mathcal{O}}}(x\in{\hat{S}})=\int\frac{d^{2}k}{(2\pi)^{2}}u_{k}^{(S)}(x)a_{k}^{(S)},\qquad
u_{k}^{(S)}(x)=N_{k}\left(-x^{-}\right)^{-\bar{q}_{+}}\left(x^{+}\right)^{-q^{-}}$
(100)
$\displaystyle{{\mathcal{O}}}(x\in{\hat{\bar{S}}})=\int\frac{d^{2}k}{(2\pi)^{2}}u_{k}^{(\bar{S})}(x)a_{k}^{(\bar{S})},\qquad
u_{k}^{(\bar{S})}(x)=N_{k}\left(x^{-}\right)^{-q_{+}}\left(-x^{+}\right)^{-\bar{q}^{-}}$
$\displaystyle q_{+}={{{\Delta}}\over 2}+ik^{+},\quad q_{-}={{{\Delta}}\over
2}+ik^{-},\quad\bar{q}_{+}={{{\Delta}}\over
2}-ik^{+},\quad\bar{q}_{-}={{{\Delta}}\over 2}-ik^{-}$ $\displaystyle
k^{+}={\omega+q\over 2},\quad k^{-}={\omega-q\over 2},\quad
N_{k}={\sqrt{\sinh\pi|\omega|}\over\sqrt{2\pi}{{\Gamma}}({{\Delta}})}\left|\Gamma\left(q_{+}\right)\Gamma\left(q_{-}\right)\right|$
$\displaystyle(a_{k}^{({\alpha})})^{\dagger}=a_{-k}^{({\alpha})},\qquad[a_{k}^{({\alpha})},a_{k^{\prime}}^{({\beta})}]=\epsilon(\omega)(2\pi)^{2}{{\delta}}(k+k^{\prime}){{\delta}}_{{\alpha}{\beta}}\
,$

where $\omega,~{}q$ are the quantum numbers associated respectively to the
boost and dilatation symmetries of the Rindler wedge.

The mode functions $u_{k}^{(S,{\bar{S}})}$ can be analytically continued to
the entire Minkowski plane in the lower (upper) half $x^{\pm}$ planes for
positive (negative) $\omega$ to the (normalized) functions

$\displaystyle w_{k}^{(S)}(x)$ $\displaystyle={e^{\pi|\omega|\over
2}\over\sqrt{2\sinh\pi|\omega|}}N_{k}\left(-x^{-}+i\epsilon(\omega)\epsilon\right)^{-\bar{q}_{+}}\left(x^{+}-i\epsilon(\omega)\epsilon\right)^{-q_{-}}$
(101) $\displaystyle w_{k}^{(\bar{S})}(x)$ $\displaystyle={e^{\pi|\omega|\over
2}\over\sqrt{2\sinh\pi|\omega|}}N_{k}\left(x^{-}-i\epsilon(\omega)\epsilon\right)^{-q_{+}}\left(-x^{+}+i\epsilon(\omega)\epsilon\right)^{-\bar{q}_{-}}\
,$

which are supported on purely positive (negative) frequency Minkowski modes
for positive (negative) $\omega.$ One can associate oscillators
$\\{c_{k}^{({\alpha})}\\}$ to these mode functions with