# Emergent times in holographic duality

Sam Leutheusser and Hong Liu  Center for Theoretical Physics, Massachusetts
Institute of Technology,
77 Massachusetts Ave., Cambridge, MA 02139

###### Abstract

In holographic duality an eternal AdS black hole is described by two copies of
the boundary CFT in the thermal field double state. In this paper we provide
explicit constructions in the boundary theory of infalling time evolutions
which can take bulk observers behind the horizon. The constructions also help
to illuminate the boundary emergence of the black hole horizons, the
interiors, and the associated causal structure. A key element is the
emergence, in the large $N$ limit of the boundary theory, of a type III1 von
Neumann algebraic structure from the type I boundary operator algebra and the
half-sided modular translation structure associated with it.

††preprint: MIT-CTP/5382

###### Contents

  1. I Introduction
  2. II Emergent type III1 algebras at finite temperature
    1. II.1 Small excitations around the eternal black hole geometry
    2. II.2 Small excitations around thermal field double state on the boundary
    3. II.3 Complete spectrum and emergent type III1 structure
    4. II.4 Duality between the bulk and boundary from the algebraic perspective
  3. III Emergent type III1 algebras in boundary local regions
    1. III.1 GNS Hilbert space and bulk reconstruction
    2. III.2 Boundary theory in a Rindler region and AdS Rindler duality
    3. III.3 General boundary regions
  4. IV Physical implications
    1. IV.1 Role of the bifurcating horizon and RT surfaces
    2. IV.2 More general entanglement wedges
    3. IV.3 Emergent symmetries
  5. V Review of half-sided modular translations
    1. V.1 Review of half-sided modular translations
    2. V.2 Example I: null translations in Rindler spacetime
    3. V.3 Example II: Two copies of a large $N$ theory in the thermal field double state
  6. VI Positive extension of half-sided translations for generalized free fields
    1. VI.1 Unitary automorphism of the algebra
    2. VI.2 Determining the structure of $C_{kk^{\prime}}(s)$
  7. VII An example: generalized free fields in Rindler spacetime
    1. VII.1 The transformation
    2. VII.2 Crossing the Rindler horizon
  8. VIII Bulk reconstruction for AdS Rindler and BTZ revisited
    1. VIII.1 AdS Rindler and BTZ geometries
    2. VIII.2 Mode expansion in AdS-Rindler and BTZ
    3. VIII.3 Boundary support of a bulk operator
  9. IX Emergence of AdS Rindler horizons
  10. X Boundary emergence of an in-falling time in a black hole geometry
    1. X.1 Expressions for the transformations
    2. X.2 Near horizon transformation
    3. X.3 Left-right commutators and causal structure
    4. X.4 Transformation of a boundary operator
    5. X.5 Summary
  11. XI A point-wise transformation in the large mass limit
    1. XI.1 General setup and summary of results
    2. XI.2 Saddle-point equations
    3. XI.3 Transformed wave functions
      1. XI.3.1 Outside the horizon
      2. XI.3.2 Inside the horizon
    4. XI.4 Comments on initial operators localized in $\chi$
  12. XII Conclusions and discussions
  13. A Details on the GNS construction
  14. B Verification of $U(1)$ properties
    1. B.1 $s_{1}$ and $s_{2}$ of the same sign
    2. B.2 Opposite signs
  15. C Review of AdS Rindler and BTZ black hole
    1. C.1 AdS Rindler in $(2+1)$-dimension
    2. C.2 BTZ geometry
  16. D Analytic continuations of bulk mode functions
    1. D.1 AdS-Rindler Mode functions
    2. D.2 BTZ mode functions
  17. E Mode expansions in the boundary
  18. F Properties of hypergeometric functions
    1. F.1 Asymptotic behavior of hypergeometric function
      1. F.1.1 Case I
      2. F.1.2 Case II
    2. F.2 A derivation
    3. F.3 Some summation formulas for hypergeometric functions
  19. G Wavefunctions on the BTZ geometry in the large mass limit
    1. G.1 General $q$
    2. G.2 Expressions in the large mass limit at $q=0$

##  I Introduction

Figure 1: The Penrose diagram of an eternal black hole. The dashed lines are
event horizons and the wavy lines are the singularities.

Time is a baffling concept in quantum gravity. While it plays an absolute role
in the formulation of quantum mechanics, in gravity it can be arbitrarily
reparameterized by gauge diffeomorphisms and hence lacks a definite meaning.
In an asymptotic anti-de Sitter (AdS) spacetime, a sensible notion of boundary
time can be established in the asymptotic region as gauge transformations
generating time reparameterizations are required to vanish at spatial
infinities. For static spacetimes with a global timelike Killing vector, the
asymptotic time can be extended to the interior with the help of the symmetry.
But for spacetimes without such a symmetry, whether it is possible to describe
time flows in the interior in a diffeomorphism invariant way is a subtle
question whose understanding is important in many contexts.

For this purpose an eternal black hole in AdS, which is dual to two copies of
the boundary CFT in the thermal field double state Maldacena:2001kr  (see Fig.
1), offers perhaps a simplest nontrivial example. The black hole spacetime
possesses a time-like Killing vector in the exterior $R$ and $L$ regions. The
associated time ${\rm t}$, which can be considered as the extension of the
boundary time, however, ends at the event horizon, with no timelike Killing
vector inside the horizon. A natural question is whether the boundary theory
can describe an “infalling” time evolution, which we define as any evolution
which can take the Cauchy slice at ${\rm t}=0$ to Cauchy slices which go
inside the horizon. Such a time, if it exists, must be emergent, as the
evolutions of the usual boundary times do not probe the interior, see Fig. 2.

Figure 2: Left: Evolution of the ${\rm t}=0$ bulk slice under $H_{R}-H_{L}$,
where $H_{R,L}$ denote the Hamiltonians of the boundary theories. Center:
Evolution of the ${\rm t}=0$ slice under $H_{R}+H_{L}$, the action of which is
singular at the horizon. In fact any linear combination of $H_{R}$ and $H_{L}$
other than $H_{R}-H_{L}$ is expected to have a singular action at the horizon.
Right: A smooth Kruskal-like evolution. If such an evolution can be described
in a diffeomorphism invariant way, it must be emergent in the boundary theory.

There have been many different ways that boundary observables can probe
regions behind the horizon see e.g. Kraus:2002iv ; Fidkowski:2003nf ;
Festuccia:2005pi ; Hartman:2013qma ; Liu:2013iza ; Liu:2013qca ;
Susskind:2014rva ; Grinberg:2020fdj ; Zhao:2020gxq ; Haehl:2021emt ;
Haehl:2021prg ; Haehl:2021tft , but in these discussions neither an infalling
time evolution nor the casual structure of the horizon was visible from the
boundary, except in systems with symmetries Maldacena:2018lmt ; Lin:2019qwu .
Similarly, ER$=$EPR type arguments VanRaamsdonk:2010pw ; Maldacena:2013xja
are largely concerned with a single time slice. While it is possible to
express bulk operators in the black hole interior regions in terms of boundary
operators Hamilton:2005ju ; Hamilton:2006fh ; Papadodimas:2012aq ;
Papadodimas:2013jku ; Papadodimas:2015xma , such “bulk reconstructions”
require either evolving bulk equations of motion or analytic continuation
around the horizon, and thus are not intrinsically boundary constructions. See
also Jafferis:2020ora ; Gao:2021tzr  for an interesting recent discussion of
keeping track of the proper time of an in-falling observer using modular flows
and Nomura:2018kia ; Nomura:2019qps ; Nomura:2019dlz ; Langhoff:2020jqa ;
Nomura:2020ska  for a description of the black hole interior from the
perspective of coarse-graining.

In this paper we provide an explicit construction of infalling time evolutions
from the boundary theory.111A summary of the main idea and results has
appeared earlier in Leutheusser:2021qhd . It should be emphasized that our
goal is not to describe in-falling geodesic motion of some localized bulk
observers, which in general cannot be formulated in a diffeomorphism invariant
way. The goal is to construct “global” evolutions of a Cauchy slice as in Fig
2(c). Understanding such emergent evolutions also helps to illuminate the
emergence in the boundary theory of the bulk horizon and the associated causal
structure.

The key to our discussion is the emergence, in the large $N$ limit of the
boundary theory, of a type III1 von Neumann algebraic structure222For reviews
on the classification of von Neumann algebras see chapter III.2 of Haag:1992hx
or section 6 of Witten:2018zxz . from the type I boundary operator algebra and
the half-sided modular translation structure associated with it. A distinctive
property of the “evolution operators” $U(s)=e^{-iGs},\,s\in{\mathbb{{R}}},$
resulting from this construction is that the Hermitian generator $G$ has a
spectrum that is bounded from below,

$G\geq 0\ .$ (1)

The spectrum property is natural from the following perspectives: (i) It
distinguishes $G$, as a generator of “time” flow, from an operator generating
other unitary transformations, e.g. spacelike displacements or internal
symmetries, whose spectrum is not bounded from below. (ii) If we interpret the
eigenvalues of $G$ as energies associated with the “global” infalling time
$s$, they should be bounded from below to ensure stability. The existence of
the singularity means that such evolution may only have a finite “lifetime,”
but there should nevertheless exist a well-defined quantum mechanical
description before hitting the singularity. Also by construction $G$ involves
degrees of freedom from both CFTR and CFTL.333The necessity of left/right
couplings has previously been discussed. For example, see Mathur:2014dia .

Our discussion will be restricted to leading order in the $1/N$ expansion, but
we expect the structure uncovered should be present to any finite order in the
expansion. New structure from incorporating $1/N$ corrections to all orders is
discussed in WittenNew .

The plan for this paper is as follows. In section II we discuss the emergence
of a type III1 vN algebra in the boundary theory at finite temperature. In
section III we discuss the emergence of a new type III1 structure for local
boundary algebras in the large $N$ limit. In section IV we suggest several
physical implications of these emergent type III1 algebras. In section V we
review half-sided modular inclusion/translation. In section VI we show that
half-sided translations can be uniquely extended to all values of the
parameter and that the description of these evolution operators is completely
fixed, up to a phase, for algebras generated by generalized free fields. In
section VII we illustrate our construction of evolution operators in the
simple case of generalized free fields on Rindler spacetime. In section VIII
we review bulk reconstruction in the AdS-Rindler and BTZ spacetimes and then
provide new results on the boundary support of such bulk reconstructions. In
section IX we show how to cross the AdS-Rindler horizon and reconstruct the
bulk Poincaré time from Rindler patches of the boundary theory. In section X
we discuss boundary descriptions of Kruskal-like time evolution in the BTZ
geometry and sharp signatures of the black hole horizons and causal structure
in the boundary theory. In section XI we show that the emergent bulk evolution
becomes a point-wise transformation in the limit with the bulk field having a
very large mass. We then conclude in section XII with a discussion of our
results and we point out many future directions to be explored.

Conventions and notations:

In this paper we use $N^{2}\sim{1\over G_{N}}$ to denote the number of degrees
of freedom of the boundary theory, where $G_{N}$ is the bulk Newton constant.
For two-dimensional CFTs, $N^{2}$ should be understood as the central charge
$c$. The $1/N$ perturbative expansion of the CFT is dual to the perturbative
$G_{N}$ expansion around the corresponding classical geometry. In this regime,
the bulk gravity theory can be described by a weakly coupled quantum field
theory in a curved spacetime.

All operator algebras discussed in this paper should be understood as those of
bounded operators.444This is for mathematical convenience, but this constraint
does not sacrifice physical significance as essentially all observables can be
made to be bounded by putting restrictions on their spectra.

We will consider the boundary theory to be on ${\mathbb{{R}}}\times S^{d-1}$
or ${\mathbb{{R}}}^{1,d-1}$ and the discussion generalizes straightforwardly
to other boundary spatial manifolds such as hyperbolic space. A boundary point
is denoted by $x=({\rm t},{\vec{x}})$ with ${\vec{x}}$ denoting points on
either ${\mathbb{{R}}}^{d-1}$ or $S^{d-1}$. The corresponding Fourier space
will be denoted as $k=(\omega,q)$ with $q$ collectively denoting momentum on
${\mathbb{{R}}}^{d-1}$ or spherical harmonic labels on $S^{d-1}$. A bulk point
is denoted by $X=(r,x)$ with $r$ the bulk radial direction (later in the paper
we use $w$ as the radial variable).

${{\mathcal{A}}}^{\prime}$ denotes the commutant of an algebra
${{\mathcal{A}}}$, i.e. the algebra of operators commuting with the algebra
${{\mathcal{A}}}$. By type III1 algebras we mean a von Neumann (vN) algebra
which contains type III1 factor(s).

We use ${\rm t}$ to denote the boundary time whose translation is generated by
the Hamiltonian $H$, $\eta$ to denote the boundary time in units where the
inverse temperature is ${\beta}=2\pi$, i.e. $\eta={2\pi\over{\beta}}{\rm t}$,
and $t$ to denote the modular time.

##  II Emergent type III1 algebras at finite temperature

In this section we consider two copies of the boundary CFT in the thermal
field double state, which is dual to an eternal black hole in AdS. We argue
that there are emergent type III1 vN algebras in the large $N$ limit. We start
with a quick review of the bulk theory to set up the notation.

###  II.1 Small excitations around the eternal black hole geometry

Consider an eternal black hole in AdSd+1, whose metric can be written in a
form

$\displaystyle ds^{2}=-fd{\rm t}^{2}+{1\over
f}dr^{2}+{r^{2}}d\Sigma^{2}_{d-1}$ (2)

where $d\Sigma^{2}_{d-1}$ is the metric for the boundary spatial manifold
$\Sigma$ which we will take to be the unit sphere $S^{d-1}$ or
${\mathbb{{R}}}^{d-1}$, and $f$ is a function with a first order zero at event
horizon $r=r_{0}$. A bulk point is denoted by $X=({\rm t},r,{\vec{x}})$ where
${\vec{x}}$ denotes a point on $\Sigma$. The Schwarzschild coordinates $({\rm
t},r)$ can be used to cover any of the four regions of the fully extended
black hole geometry of Fig. 3, while the Kruskal coordinates $U,V$ cover all
the regions.

Figure 3: Kruskal diagram for an eternal black hole. The dashed lines are
event horizons, the solid red lines are the singularities, and the solid black
lines are the boundaries.

Small perturbations around the black hole geometry can be described using the
standard formalism of quantum field theory in a curved spacetime. Their
quantization results in a Fock space ${{\mathcal{H}}}_{\rm BH}^{\rm(Fock)}$.
We will use a real scalar field $\phi$ of mass $m$ as an illustration. The
restriction $\phi_{R}$ of $\phi$ to the $R$ region of the black hole geometry
can be expanded in terms of a complete set of properly normalized modes
$v_{\omega q}^{(R)}(X)$ in the $R$ region as

$\phi_{R}(X)=\sum_{q}\int_{0}^{\infty}{d\omega\over 2\pi}\,\left(v_{\omega
q}^{(R)}(X)a_{\omega q}^{(R)}+(v_{\omega q}^{(R)}(X))^{*}(a_{\omega
q}^{(R)})^{\dagger}\right)\ ,$ (3)

where $q$ collectively denotes quantum numbers associated with $\Sigma$,555The
sum over $q$ and ${{\delta}}_{qq^{\prime}}$ should be understood as integrals
and Dirac delta functions if there are continuous quantum numbers. and

$[a_{\omega
q}^{(R)},(a_{\omega^{\prime}q^{\prime}}^{(R)})^{\dagger}]=2\pi\delta(\omega-\omega^{\prime}){{\delta}}_{qq^{\prime}}\
.$ (4)

Below for notational simplicity we will write (3) as

$\phi_{R}(X)=\sum_{k}v_{k}^{(R)}(X)a_{k}^{(R)},\quad k=(\omega,q),\quad
v_{-k}^{(R)}=(v_{k}^{(R)})^{*},\quad a_{-k}^{(R)}=(a_{k}^{(R)})^{\dagger}\ .$
(5)

There is a similar expansion for the restriction of $\phi$ to the $L$ region,

$\phi_{L}(X)=\sum_{k}v_{k}^{(L)}(X)a_{k}^{(L)},\quad k=(\omega,q),\quad
v_{-k}^{(L)}=(v_{k}^{(L)})^{*},\quad a_{-k}^{(L)}=(a_{k}^{(L)})^{\dagger}\ .$
(6)

In the case of a Schwarzschild black hole, the $R$ and $L$ regions are related
by spacetime reflection symmetry $(U,V,{\vec{x}})$ to $(-U,-V,-{\vec{x}})$. It
is convenient to choose $v_{k}^{(L)}$ to be

$v_{k}^{(L)}({\rm t},r,{\vec{x}})=v_{k}^{(R)*}({\rm
t},r,{\vec{x}})=v_{-k}^{(R)}({\rm t},r,{\vec{x}})$ (7)

and the anti-unitary spacetime reflection operator $J$ then acts as

$J\phi_{R}({\rm t},r,{\vec{x}})J=\phi_{L}({\rm t},r,{\vec{x}}),\qquad
Ja_{k}^{(R)}J=a_{k}^{(L)}\ .$ (8)

Altogether

$(a_{k}^{({\alpha})})^{\dagger}=a_{-k}^{({\alpha})},\quad[a_{k}^{({\alpha})},a_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}},\quad{\alpha},{\beta}=R,L\
.$ (9)

The behavior of $\phi$ in the $F$ and $P$ regions can be determined from that
in the $R$ and $L$ regions by causal evolution or analytic continuation.

The Hartle-Hawking vacuum can be defined using the standard Unruh procedure by
first introducing modes $w_{k}$ which are analytic in the lower $U$ and $V$
planes for $\omega>0$,

$\displaystyle
w_{k}^{({\alpha})}={\mathfrak{b}}_{+}v^{({\alpha})}_{k}+{\mathfrak{b}}_{-}v^{(\bar{\alpha})}_{-k},\quad\bar{L}=R,\quad\bar{R}=L$
(10) $\displaystyle{\mathfrak{b}}_{\pm}={e^{\pm{{\beta}|\omega|\over
4}}\over\sqrt{2\sinh{{\beta}|\omega|\over
2}}},\quad{\mathfrak{b}}^{2}_{+}-{\mathfrak{b}}_{-}^{2}=1\ .$ (11)

Denoting the oscillators corresponding to the modes $w_{k}^{({\alpha})}$ as
$c^{({\alpha})}_{k}$ we then have on a Cauchy slice

$\phi=\sum_{{\alpha},k}v_{k}^{({\alpha})}a_{k}^{({\alpha})}=\sum_{{\alpha},k}w_{k}^{({\alpha})}c_{k}^{({\alpha})}\
,$ (12)

which implies the oscillators $c^{({\alpha})}_{k}$ and $a^{({\alpha})}_{k}$
are related by

$\displaystyle
c^{({\alpha})}_{k}={\mathfrak{b}}_{+}a^{({\alpha})}_{k}-{\mathfrak{b}}_{-}a^{(\bar{\alpha})}_{-k},\qquad
a^{({\alpha})}_{k}={\mathfrak{b}}_{+}c^{({\alpha})}_{k}+{\mathfrak{b}}_{-}c^{(\bar{\alpha})}_{-k},$
(13)
$\displaystyle[c_{k}^{({\alpha})},c_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}},\quad
Jc_{k}^{({\alpha})}J=c_{k}^{(\bar{\alpha})}\ .$ (14)

The Hartle-Hawking vacuum $\lvert{HH}\rangle$ is defined to satisfy

$\displaystyle
c_{k}^{({\alpha})}\lvert{HH}\rangle=0\quad\text{for}\quad\omega>0\ .$ (15)

The Fock space ${{\mathcal{H}}}_{\rm BH}^{\rm(Fock)}$ is built by acting
$c_{k}^{({\alpha})}$ with $\omega<0$ on $\lvert{HH}\rangle$. Note that

$a_{k}^{(R)}\lvert{HH}\rangle=e^{-{{\beta}\omega\over
2}}a_{-k}^{(L)}\lvert{HH}\rangle,\qquad\langle{HH}\rvert
a_{k}^{(R)}=e^{{{\beta}\omega\over 2}}\langle{HH}\rvert a_{-k}^{(L)}\ .$ (16)

We will denote the operator algebra generated by $\phi$ and other matter
fields (including metric perturbations) in the $R$ region as
$\widetilde{\mathcal{Y}}_{R}$ and similarly those generated by fields in the
$L$ region as $\widetilde{\mathcal{Y}}_{L}$. $\widetilde{\mathcal{Y}}_{R}$ and
$\widetilde{\mathcal{Y}}_{L}$ are commutants of each other, and are expected
to be type III1 von Neumann algebras Araki1964:2 ; Longo:1982zz ;
Fredenhagen:1984dc . Reflections of the type III1 structure include the non-
existence of the Schwarzschild vacuum state
$\lvert{0}\rangle_{R}\otimes\lvert{0}\rangle_{L}$ (which is defined to be
annihilated by $a_{k}^{({\alpha})}$ with $\omega>0$) in ${{\mathcal{H}}}_{\rm
BH}^{(\rm Fock)}$ and the entanglement entropy between $R$ and $L$ regions
being not well defined in the continuum limit.

###  II.2 Small excitations around thermal field double state on the boundary

We always consider the boundary CFT at a large but finite $N$. We denote the
Hilbert space of the boundary CFT as ${{\mathcal{H}}}$, its Hamiltonian as
$H$, the algebra of bounded operators as ${{\mathcal{B}}}$, and the algebra
generated by single-trace operators under multiplications and sums as
${{\mathcal{A}}}$, which is defined only perturbatively in the $1/N$
expansion. We use ${{\mathcal{O}}}$ to denote the single-trace operator dual
to the bulk field $\phi$. Now consider two copies of the boundary theory, to
which we refer respectively as CFTR and CFTL. Operators or states with
subscripts $R,L$ refer to those in the respective systems. The doubled system
has Hilbert space
$\hat{{\mathcal{H}}}={{\mathcal{H}}}_{R}\otimes{{\mathcal{H}}}_{L}$, operator
algebra $\hat{{\mathcal{B}}}={{\mathcal{B}}}_{R}\otimes{{\mathcal{B}}}_{L}$,
and single-trace operator algebra
$\hat{{\mathcal{A}}}={{\mathcal{A}}}_{R}\otimes{{\mathcal{A}}}_{L}$. Generic
operators in $\hat{{\mathcal{A}}}$ will be denoted as $a,b,\cdots$, those in
$\hat{{\mathcal{B}}}$ as $u,v,\cdots$, those in ${{\mathcal{A}}}_{R}$ as
$A_{R},B_{R},\cdots$.

The thermal field double state is defined as

$\displaystyle{\lvert{\Psi_{\beta}}\rangle}={1\over\sqrt{Z_{\beta}}}\sum_{n}e^{-{{\frac{1}{2}}}{\beta}E_{n}}\lvert{\tilde{n}}\rangle_{L}\lvert{n}\rangle_{R},$
(17)
$\displaystyle\lvert{\tilde{n}}\rangle={\theta}\lvert{n}\rangle,\quad{\left\langle{\tilde{m}|\tilde{n}}\right\rangle}={\left\langle{n|m}\right\rangle}={{\delta}}_{mn},\quad
Z_{\beta}=\sum_{n}e^{-{\beta}E_{n}}$ (18)

where $\lvert{n}\rangle$ denotes the full set of energy eigenstates of the
CFT, with eigenvalues $E_{n}$. $m,n$ here collectively denote all quantum
numbers including spatial momenta for the boundary theory on
${\mathbb{{R}}}^{1,d-1}$ or angular quantum numbers for the theory on
${\mathbb{{R}}}\times S^{d-1}$. ${\theta}$ is an anti-unitary operator and
will be taken to be the ${{\mathcal{C}}}{{\mathcal{P}}}{\mathcal{T}}$ operator
of the CFT. When tracing over degrees of freedom of one of the CFTs, we get
the thermal density operator at inverse temperature ${\beta}={1\over T}$ for
the remaining one

$\rho_{\beta}={1\over Z_{\beta}}e^{-{\beta}H}\ .$ (19)

Perturbatively in the $1/N$ expansion, excitations around
${\lvert{\Psi_{\beta}}\rangle}$ can be obtained by acting single-trace
boundary operators on it.666See also Witten:2021jzq  for a review of the
definition of the thermal field double state in the infinite volume and large
$N$ limits. In fact, the collection of excitations obtained this way has the
structure of a Hilbert space, which can be made precise mathematically using
the Gelfand-Naimark-Segal (GNS) construction. More explicitly, for each
operator $a\in\hat{{\mathcal{A}}}$ we associate a state $\lvert{a}\rangle$ and
define the inner product among them as

${\left\langle{a|b}\right\rangle}={\left\langle{\Psi_{\beta}|a^{\dagger}b|\Psi_{\beta}}\right\rangle},\quad
a,b\in\hat{{\mathcal{A}}}\ .$ (20)

In particular, for $A_{R},B_{R}\in{{\mathcal{A}}}_{R}$, we have

${\left\langle{A_{R}|B_{R}}\right\rangle}={\left\langle{\Psi_{\beta}|A_{R}^{\dagger}B_{R}|\Psi_{\beta}}\right\rangle}=\mathop{\rm
Tr}(\rho_{\beta}A^{\dagger}B)\ .$ (21)

Equation (20) does not yet define a Hilbert space as there can be operators
$y\in\hat{{\mathcal{A}}}$ satisfying
${\left\langle{\Psi_{\beta}|y^{\dagger}y|\Psi_{\beta}}\right\rangle}=0$ and
the corresponding $\lvert{y}\rangle$ should be set to zero. Denote the set of
such operators as ${\mathcal{J}}$. The GNS Hilbert space is the completion of
the set of equivalence classes $[a]$ which are defined by the equivalence
relations

$a\sim a+y,\qquad a\in\hat{{\mathcal{A}}},\quad y\in{\mathcal{J}}\ .$ (22)

The set ${\mathcal{J}}$ is non-empty, as from (17), for a Hermitian operator
$W\in{{\mathcal{B}}}$

$\displaystyle W_{R}({\rm
t},{\vec{x}})\lvert{\Psi_{\beta}}\rangle=W_{L}\left({\rm t}-i{{\beta}\over
2},{\vec{x}}\right)\lvert{\Psi_{\beta}}\rangle,\quad W_{L}({\rm
t},{\vec{x}})\lvert{\Psi_{\beta}}\rangle=W_{R}\left({\rm t}+i{{\beta}\over
2},{\vec{x}}\right)\lvert{\Psi_{\beta}}\rangle,$ (23)

where for simplicity we have assumed that
${\theta}^{\dagger}W(0){\theta}=W(0)$ and we have chosen the space and time
orientations of CFTL to be the opposite of those of CFTR.777That is,
$W_{R}({\rm t},{\vec{x}})=\mathbf{1}_{L}\otimes W({\rm t},{\vec{x}})$ while
$W_{L}({\rm t},{\vec{x}})=W(-{\rm t},-{\vec{x}})\otimes\mathbf{1}_{R}$. We
take that single-trace operators $W({\rm t},{\vec{x}})$ are analytically
continued to ${\rm Im}\,{\rm t}\in[0,{{\beta}\over 2}]$. Thus $W_{R}({\rm
t},{\vec{x}})$ are defined for ${\rm Im}\,{\rm t}\in[0,{{\beta}\over 2}]$
while $W_{L}({\rm t},{\vec{x}})$ for ${\rm Im}\,{\rm t}\in[-{{\beta}\over
2},0]$.  From (23) it can be shown that ${{\mathcal{A}}}_{R}$ or
${{\mathcal{A}}}_{L}$ alone can be used to generate the full GNS Hilbert
space, which we will denote as ${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$. See
Appendix A for details. In other words, any state in ${{\mathcal{H}}}_{\rm
TFD}^{\rm(GNS)}$ can be written as $\lvert{A_{R}}\rangle$ with
$A_{R}\in{{\mathcal{A}}}_{R}$ or as a limit of such states. The state in
${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ corresponding to the identity operator
is denoted as $\lvert{\Omega_{0}}\rangle$, which we sometimes refer to as the
GNS vacuum.

${{\mathcal{H}}}^{\rm(GNS)}_{{\rm TFD}}$ also provides a representation space
for $\hat{{\mathcal{A}}}$. The representation $\pi(a)$ of an operator
$a\in\hat{{\mathcal{A}}}$ acting on ${{\mathcal{H}}}^{\rm(GNS)}_{{\rm TFD}}$
can be defined as

$\pi(a)[b]=[ab],\qquad a,b\in\hat{{\mathcal{A}}}$ (24)

and as a result the inner product (20) can also be written as

${\left\langle{a|b}\right\rangle}={\left\langle{\Omega_{0}|(\pi(a))^{\dagger}\pi(b)|\Omega_{0}}\right\rangle}\
.$ (25)

We denote the representations of ${{\mathcal{A}}}_{R}$ and
${{\mathcal{A}}}_{L}$ in ${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ respectively
as ${\mathcal{Y}}_{R}$ and ${\mathcal{Y}}_{L}$. Given that
${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ can be generated by
${{\mathcal{A}}}_{R}$ or ${{\mathcal{A}}}_{L}$ alone, the GNS vacuum
$\lvert{\Omega_{0}}\rangle$ is cyclic and separating under both
${\mathcal{Y}}_{R}$ and ${\mathcal{Y}}_{L}$, and we have
${\mathcal{Y}}_{R}^{\prime}={\mathcal{Y}}_{L}$. We denote the operator algebra
on ${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ as ${\mathcal{Y}}$.

It can also be shown that ${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ is isomorphic
to ${{\mathcal{H}}}_{\rm\rho_{\beta}}^{\rm(GNS)}$, the GNS Hilbert space
corresponding to the thermal density operator $\rho_{\beta}$ over the single-
trace operator algebra ${{\mathcal{A}}}$.888For the construction of latter see
Sec. V.1.4 of Haag:1992hx  and Magan:2020iac .

To leading order in the $1/N$ expansion, the inner products (20) and thus (25)
can be written as sums of products of two-point functions of single-trace
operators. We can thus represent single-trace operators by generalized free
fields acting on ${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$ and the algebras
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are generated by generalized free
fields. More explicitly, for a single trace scalar operator ${{\mathcal{O}}}$,
we can expand its representations in terms of a complete set of functions on
the boundary manifold

$\displaystyle\pi({{\mathcal{O}}}_{R}(x))=\sum_{k}u_{k}^{(R)}(x)a_{k}^{(R)},\quad\pi({{\mathcal{O}}}_{L}(x))=\sum_{k}u_{k}^{(L)}(x)a_{k}^{(L)},$
(26) $\displaystyle u_{k}^{(R)}(x)=N_{k}e^{-i\omega{\rm
t}}h_{q}({\vec{x}}),\quad
u_{k}^{(L)}(x)=u_{-k}^{(R)}(x)=(u_{k}^{(R)}(x))^{*},\quad x=({\rm
t},{\vec{x}})$ (27)

where $N_{k}$ is some function of $k=(\omega,q)$, and $h_{q}({\vec{x}})$
denotes the complete set of functions on the boundary spatial manifold
$\Sigma$, and $a_{k}^{(R,L)}$ are operators acting on
${{\mathcal{H}}}^{\rm(GNS)}_{\rm TFD}$, normalized as999Note that this is
purely a boundary discussion. Even though we use the same notation,
$a_{k}^{({\alpha})}$ as in (5), at this stage these operators do not have
anything to do with each other.

$(a_{k}^{({\alpha})})^{\dagger}=a_{-k}^{({\alpha})},\quad[a_{k}^{({\alpha})},a_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}},\quad{\alpha},{\beta}=R,L\
.$ (28)

Using (21) and (25), $N_{k}$ can be deduced from the condition

$\mathop{\rm
Tr}(\rho_{\beta}{{\mathcal{O}}}(x_{1}){{\mathcal{O}}}(x_{2}))={\left\langle{\Omega_{0}|(\pi({{\mathcal{O}}}_{R}(x_{1}))\pi({{\mathcal{O}}}_{R}(x_{2}))|\Omega_{0}}\right\rangle}\
.$ (29)

Furthermore, applying (23) to $\pi({{\mathcal{O}}}_{R})$ and
$\pi({{\mathcal{O}}}_{L})$ we have

$\displaystyle a_{k}^{(R)}\lvert{\Omega_{0}}\rangle=e^{-{{\beta}\omega\over
2}}a_{-k}^{(L)}\lvert{\Omega_{0}}\rangle,\qquad\langle{\Omega_{0}}\rvert
a_{k}^{(R)}=e^{{{\beta}\omega\over 2}}\langle{\Omega_{0}}\rvert a_{-k}^{(L)}\
.$ (30)

We can introduce an anti-unitary “swap” operator $J$ which acts as

$J\lvert{\Omega_{0}}\rangle=\lvert{\Omega_{0}}\rangle,\quad
Ja_{k}^{({\alpha})}J=a_{k}^{(\bar{\alpha})},\quad
J\pi({{\mathcal{O}}}_{\alpha}(x))J=\pi({{\mathcal{O}}}_{\bar{\alpha}}(x)),\quad
J^{2}=1\ .$ (31)

Equations (30) motivate the introduction of (${\mathfrak{b}}_{\pm}$ were
introduced in (11))

$\displaystyle
c^{({\alpha})}_{k}={\mathfrak{b}}_{+}a^{({\alpha})}_{k}-{\mathfrak{b}}_{-}a^{(\bar{\alpha})}_{-k},\qquad
a^{({\alpha})}_{k}={\mathfrak{b}}_{+}c^{({\alpha})}_{k}+{\mathfrak{b}}_{-}c^{(\bar{\alpha})}_{-k},\quad\bar{L}=R$
(32)

which satisfy

$\displaystyle
c_{k}^{({\alpha})}\lvert{\Omega_{0}}\rangle=0\quad\text{for}\quad\omega>0,\quad\left(c_{k}^{({\alpha})}\right)^{\dagger}=c_{-k}^{({\alpha})},\quad[c_{k}^{({\alpha})},c_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}}\
.$ (33)

To conclude this subsection, we make some further general remarks:

  1. 1.

The algebras ${{\mathcal{A}}}_{R},{{\mathcal{A}}}_{L}$ are defined only in the
$1/N$ expansion. The algebras ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ act on
${{\mathcal{H}}}_{\rm TFD}^{\rm(GNS)}$, and are von Neumann algebras.

  2. 2.

$\pi({{\mathcal{O}}})$ is not the same as ${{\mathcal{O}}}$. The former is
defined only on ${{\mathcal{H}}}^{\rm(GNS)}_{\rm TFD}$ and is state-dependent
(i.e. it depends on the state we use to build the GNS representation), while
${{\mathcal{O}}}$ acts on the full CFT Hilbert space and is state-independent.
The algebras ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are thus also state-
dependent. For example, they depend on ${\beta}$.

  3. 3.

The operator algebras ${{\mathcal{B}}}_{R},{{\mathcal{B}}}_{L}$ are type I von
Neumann algebras, and ${\lvert{\Psi_{\beta}}\rangle}$ is cyclic and separating
with respect to them. The corresponding modular operator ${{\Delta}}$ is given
by $-\log{{\Delta}}={\beta}(H_{R}-H_{L})$. Note that the modular time $t$
defined by modular flow with ${{\Delta}}^{-it}$ is related to the usual CFT
time ${\rm t}$ by101010Recall that we take the time of CFTL to run in the
opposite direction to that CFTR.

$t={{\mathfrak{t}}\over{\beta}}\ .$ (34)

  4. 4.

Since $\lvert{\Omega_{0}}\rangle$ is cyclic and separating for
${\mathcal{Y}}_{R}$, there exists a modular operator ${{\Delta}}_{0}$ which
leaves $\lvert{\Omega_{0}}\rangle$ invariant and generates automorphisms of
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$. The modular flows generated by
${{\Delta}}_{0}$ again coincide with the time evolution of the respective
boundaries. More explicitly, combining with the previous item, we have

$\pi(A_{R}({\rm
t}={\beta}t))=\pi\left({{\Delta}}^{-it}A_{R}{{\Delta}}^{it}\right)={{\Delta}}^{-it}_{0}\pi(A_{R}){{\Delta}}_{0}^{it},\quad
A_{R}\in{{\mathcal{A}}}_{R}\ .$ (35)

###  II.3 Complete spectrum and emergent type III1 structure

For the boundary theory on ${\mathbb{{R}}}\times S^{d-1}$, we conjecture that
the algebras ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are type I below the
Hawking-Page temperature $T_{HP}$, but become type III1 above $T_{HP}$. Recall
that $T_{HP}$ is the temperature at which the boundary system exhibits a
first-order phase transition in the large $N$ limit, with $\log Z_{\beta}\sim
O(N^{0})$ for $T<T_{HP}$ but $\log Z_{\beta}\sim O(N^{2})$ for $T>T_{HP}$.
Below $T_{HP}$ thermal averages are dominated by contributions from states
with energies of $O(N^{0})$ while above $T_{HP}$ they are dominated by states
with energies of $O(N^{2})$. This change of dominance leads to dramatically
different behavior for thermal correlation functions. Since the inner products
(20)–(21) of ${{\mathcal{H}}}^{\rm(GNS)}_{\rm TFD}$ are determined by thermal
two-point functions of single-trace operators, the representations of elements
of ${{\mathcal{A}}}_{R},{{\mathcal{A}}}_{L}$, and thus the structure of the
algebras ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are sensitive to the behavior
of these two-point functions.

Consider thermal Wightman functions of a Hermitian scalar operator
${{\mathcal{O}}}$ of dimension ${{\Delta}}$

$G_{+}(x_{1}-x_{2})=\mathop{\rm
Tr}(\rho_{\beta}{{\mathcal{O}}}(x_{1}){{\mathcal{O}}}(x_{2}))={\left\langle{\Psi_{\beta}|{{\mathcal{O}}}_{R}(x_{1}){{\mathcal{O}}}_{R}(x_{2})|\Psi_{\beta}}\right\rangle}\
.$ (36)

Its Fourier transform has the Lehmann representation

$\displaystyle G_{+}(\omega,q)=\sum_{m,n}(2\pi)\delta(\omega-E_{nm})e^{-\beta
E_{m}}\rho_{mn}\equiv{1\over 1-e^{-\beta\omega}}\rho(\omega,q),$ (37)
$\displaystyle
E_{nm}=E_{n}-E_{m},\quad\rho_{mn}=|{\left\langle{m|{{\mathcal{O}}}(0)|n}\right\rangle}|^{2},$
(38)

where $\rho(\omega,q)$ is the (finite temperature) spectral function. In the
large $N$ limit and at strong coupling, $G_{+}$ and $\rho$ can be computed
using the standard procedure from gravity. Below $T_{HP}$, the finite
temperature Euclidean correlation function, $G_{E},$ of ${{\mathcal{O}}}$ is
determined by the Euclidean function $G_{E0}$ at zero temperature via
summation over images in the Euclidean time

$G_{E}(\tau,\vec{x})=\sum_{n}G_{E0}(\tau+n\beta,\vec{x})\ .$ (39)

When analytically continued back to the Lorentzian signature, this implies
that

$\rho(\omega,q)={\theta}(\omega)\rho_{0}(\omega,q)-{\theta}(-\omega)\rho_{0}(-\omega,q)$
(40)

where $\rho_{0}(\omega,q)$ is the spectral function at zero
temperature111111This can be defined by taking ${\beta}\to\infty$ in (37) and
can be found from the zero temperature momentum space Wightman function as
$G_{0+}(\omega,q)=\rho_{0}(\omega,q)$. and has the following form

$\rho_{0}(\omega,q)=\sum_{l=0}^{\infty}c_{l}\delta(\omega-\Delta-2l)\ ,$ (41)

In this case, $\rho$ is supported only at discrete points on the real
$\omega$-axis.

In contrast, for $T>T_{HP}$, $\rho(\omega,q)$ is smooth and supported on the
full real $\omega$-axis. For $d=2$, i.e. CFT on a circle, from the BTZ black
hole Banados:1992wn  it can be found that121212Now $q$ is the momentum on the
circle and $q_{\pm}={{\frac{1}{2}}}({{\Delta}}+i(\omega\pm
q)),\;\bar{q}_{\pm}={{\frac{1}{2}}}({{\Delta}}-i(\omega\pm q))$. $C$ is a
normalization constant. In (42) we have chosen units such that ${\beta}=2\pi$.

$\rho(\omega,q)=C\sinh{\pi\omega}{{\Gamma}}(q_{+}){{\Gamma}}(\bar{q}_{+}){{\Gamma}}(q_{-}){{\Gamma}}(\bar{q}_{-})\
.$ (42)

We will refer to such a $\rho$, a smooth function supported on the full real
$\omega$-axis for any $q$, as having a complete spectrum. For general $d$, the
explicit analytic expressions of $\rho(\omega,q)$ at strong coupling are not
known, but $\rho$ can be shown to always have a complete spectrum due to the
presence of the horizon in the black hole geometry (see e.g. Festuccia:2005pi
).

Conjecture: the algebras ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are type III1
if and only if the corresponding $\rho$ has a complete spectrum.

We now outline some motivational ideas for the conjecture, but will not
attempt a rigorous mathematical proof here. For any choice of a cyclic and
separating vector $\lvert{\phi}\rangle\in{{\mathcal{H}}}_{{\rm TFD}}^{(\rm
GNS)}$ for ${\mathcal{Y}}_{R}$ we have the corresponding modular operator
${{\Delta}}_{\phi}$. The spectrum $\sigma({{\Delta}}_{\phi})$ of
${{\Delta}}_{\phi}$ is $\lvert{\phi}\rangle$-dependent and consider the
spectral invariant

${\mathbb{{S}}}({\mathcal{Y}}_{R})=\cap_{\phi}\sigma({{\Delta}}_{\phi})\ .$
(43)

${\mathcal{Y}}_{R}$ is type III1 if
${\mathbb{{S}}}({\mathcal{Y}}_{R})={\mathbb{{R}}}_{+}$ Connes:73 ; Connes:74
(see also Sec. 5.2.4 of Haag:1992hx ). This implies that for all states
$\lvert{\phi}\rangle$ the spectrum of $\log{{\Delta}}_{\phi}$ should cover the
full real axis. In our construction of the GNS Hilbert space using generalized
free fields, $\omega$ can be interpreted as the eigenvalues of
$-\log{{\Delta}}_{0}$ (see (35)) for the GNS vacuum
$\lvert{\Omega_{0}}\rangle$. Furthermore, all states in the GNS Hilbert space
can be obtained by acting $a_{k}^{(R)}$ on $\lvert{\Omega_{0}}\rangle$ (or a
limit of such states). If there is a gap in the spectrum of $\rho(\omega,q)$
on the real $\omega$-axis, then we should be able to construct a state, whose
corresponding modular operator does not have a spectrum covering the full
${\mathbb{{R}}}_{+}$. Also note that when there is a complete spectrum, the
“vacuum” $\lvert{0}\rangle_{L}\otimes\lvert{0}\rangle_{R}$ for
$a_{k}^{({\alpha})}$, which is defined to be annihilated by
$a_{k}^{({\alpha})}$ with $\omega>0$, does not exist in ${{\mathcal{H}}}_{{\rm
TFD}}^{\rm(GNS)}$ and thus ${{\mathcal{H}}}_{{\rm TFD}}^{\rm(GNS)}$ cannot be
tensor factorized131313From (32), the normalization of
$\lvert{0}\rangle_{R}\otimes\lvert{0}\rangle_{L}$ is proportional to
$\prod_{\omega}(1-e^{-2\pi\omega})$, which is not well defined.. In Sec. VI we
show that when there is a complete spectrum it is possible to define a half-
sided modular translation structure (which is only possible for type III1
algebras).

We will discuss in the next subsection that the type III1 structure of
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ is also required by the duality of
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ with the bulk algebras
$\tilde{\mathcal{Y}}_{R},\tilde{\mathcal{Y}}_{L}$.

We emphasize that the complete spectrum is possible only in the large $N$
limit. CFT on $S^{d-1}$ has a discrete energy spectrum, i.e. the sums $m,n$ in
(37) are literally discrete. As a result, at finite $N$, the spectral function
$\rho$ is supported on only discrete values of $\omega=E_{mn}$. In the large
$N$ limit (for $T>T_{HP}$), the dominant contributions to the sums in (37)
come from states with energies of $O(N^{2})$, where the density of states is
$e^{O(N^{2})}$. If ${{\mathcal{O}}}$ has nonzero matrix elements between
generic states with energy differences $E_{mn}\sim O(N^{0})$, a complete
spectrum results in the large $N$ limit. In contrast, for $T<T_{HP}$, the
dominant contributions to the sums in (37) come from states with energies of
$O(N^{0})$, where the density of states is $O(N^{0})$, which leads to a
discrete spectrum for $\rho$. From the bulk perspective, the complete spectrum
can be attributed to the existence of an event horizon which results in a
continuum of modes for both signs of $\omega$. The emergent complete spectrum
in the large $N$ limit for $T>T_{HP}$ was emphasized before in
Festuccia:2005pi  as a possible reason for the emergence of a bulk horizon and
singularity in holography.

The complete spectrum of finite temperature spectral functions and thus the
emergent type III1 structure may not be restricted to strong coupling. In
Festuccia:2006sa  it was argued that a complete spectrum may arise generically
for a matrix-type theory in the large $N$ limit even at weak coupling (see
also Iizuka:2008hg ; Iizuka:2008eb ). A complete spectrum should also arise in
the SYK model.

The boundary theory on ${\mathbb{{R}}}^{1,d-1}$ is always in the high
temperature phase for any nonzero temperature and thus the corresponding
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ should be type III1 at any nonzero
temperature.

For theories defined on an uncompact space such as ${\mathbb{{R}}}^{d-1}$, the
spectral functions of a single-trace operator have a continuous spectrum at
both zero and finite temperature. Such a continuous spectrum alone is not
enough for the corresponding algebra to be type III1. For example, at $T=0$,
the spectral function for a scalar operator ${{\mathcal{O}}}$ with dimension
${{\Delta}}$ has the form ($C$ is a constant)

$\rho_{0}(k)=C{\theta}(\omega){\theta}(-k^{2})(-k^{2})^{\nu},\quad\nu={{\Delta}}-{d\over
2},\quad k^{2}=-\omega^{2}+\vec{q}^{2}\ .$ (44)

This has a continuous spectrum, but is not supported on the full real
$\omega$-axis for a given spatial momentum $\vec{q}$. In this case the
representation of the algebra generated by single-trace operators in the GNS
Hilbert space built from acting them on the vacuum should be type I. For a
free Yang-Mills theory at a finite temperature, the spectral function for a
typical gauge invariant operator is supported for $\omega^{2}\geq\vec{q}^{2}$
on the real $\omega$-axis for a given $\vec{q}$ (see Hartnoll:2005ju  for an
example), and thus also does not have a complete spectrum. The representation
of the algebra generated by this gauge invariant operator on the GNS Hilbert
space will therefore not be of type III1.

Our discussion of the emergent type III1 structure is at the generalized free
field theory level, which applies at leading order in the large $N$ limit. See
WittenNew  for a discussion on the deformation of this algebra when including
$1/N$ corrections.

###  II.4 Duality between the bulk and boundary from the algebraic perspective

Given that single-trace operators are dual to fundamental fields on the
gravity side, we can identify the Hilbert spaces of small excitations on both
sides and the corresponding operator algebras, i.e.

${{\mathcal{H}}}^{\rm(GNS)}_{{\rm TFD}}={{\mathcal{H}}}_{\rm
BH}^{\rm(Fock)},\quad\lvert{\Omega_{0}}\rangle=\lvert{HH}\rangle,\quad{\mathcal{Y}}_{R}=\widetilde{\mathcal{Y}}_{R},\quad{\mathcal{Y}}_{L}=\widetilde{\mathcal{Y}}_{L}\
.$ (45)

More explicitly, for a bulk scalar field $\phi$ dual to a boundary single-
trace operator ${{\mathcal{O}}}$, the last two equations of (45) imply that we
should identify oscillators, $a_{k}^{({\alpha})},$ constructed from the
generalized free field description of the boundary theory operators (26) with
those in the bulk mode expansions (5)–(6), which is the reason we have been
using the same notation for them. This identification is also reflected in the
standard extrapolate dictionary for the bulk and boundary operators ($C$ is a
normalization constant)

$\displaystyle\pi({{\mathcal{O}}}_{R}(x))=C\lim_{r\to\infty}r^{-{{\Delta}}}\phi_{R}(r,x),\qquad\pi({{\mathcal{O}}}_{L}(x))=C\lim_{r\to\infty}r^{-{{\Delta}}}\phi_{L}(r,x),$
(46) $\displaystyle
u_{k}^{(R)}(x)=C\lim_{r\to\infty}r^{-{{\Delta}}}v_{k}^{(R)}(r,x),\quad
u_{k}^{(L)}(x)=C\lim_{r\to\infty}r^{-{{\Delta}}}v_{k}^{(L)}(r,x)\ .$ (47)

We emphasize that it is the representations
$\pi({{\mathcal{O}}}_{R}),\pi({{\mathcal{O}}}_{L})$ of
${{\mathcal{O}}}_{R},{{\mathcal{O}}}_{L}$ in the GNS Hilbert space that appear
in the extrapolation formulas (46). This makes sense as the mode expansions of
$\phi_{R},\phi_{L}$ depend on the bulk geometry, which is reflected in the
state-dependence of $\pi({{\mathcal{O}}}_{R}),\pi({{\mathcal{O}}}_{L})$. The
identification of $\lvert{\Omega_{0}}\rangle$ with $\lvert{HH}\rangle$ then
follows from (16) and (30).

With the identifications of $a_{k}^{({\alpha})}$ in the boundary and bulk mode
expansions, $\phi_{R},\phi_{L}$ of equations (5)–(6) can now be directly
interpreted as boundary operators, which is the statement of bulk
reconstruction for the $R$ and $L$ regions of the black hole Hamilton:2005ju ;
Hamilton:2006fh ; Papadodimas:2012aq . We emphasize that the reconstruction
formula is in terms of operators in the GNS Hilbert space.

Since the algebras $\tilde{\mathcal{Y}}_{R},\tilde{\mathcal{Y}}_{L}$ of bulk
fields restricted to the $R$ and $L$ regions of the black hole are believed to
be type III1 von Neumann algebras, the duality can only hold if
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ are also type III1. The identifications
(45) should hold perturbatively in the $1/N$ expansion (or bulk $G_{N}$
expansion).

For the boundary theory on ${\mathbb{{R}}}\times S^{d-1}$, the above
discussion applies to $T={1\over{\beta}}>T_{HP}$. For $T<T_{HP}$, the bulk
dual for (17) is given by two disconnected copies of global AdS whose small
excitations are in the thermal field double state, see Fig. 4. In this case
${\mathcal{Y}}_{R}$ and ${\mathcal{Y}}_{L}$ are each dual respectively to the
algebra of bulk fields in the global AdS geometry and should be type I.

Figure 4: Below $T_{HP}$ the bulk theory is two separate global AdS spacetimes
whose small excitations are entangled in the thermal field double state.

##  III Emergent type III1 algebras in boundary local regions

The emergent type III1 structure discussed in the previous section concerned
the algebras generated by single-trace operators over the entire boundary
spacetime. We now would like to argue this phenomenon is more general,
applying to spacetime subregions, although in a more subtle way. The operator
algebra of a boundary CFT restricted to a subregion should be type III1
Araki1964:2 ; Longo:1982zz ; Fredenhagen:1984dc . We argue that there is a
further emergent type III1 structure in the large $N$ limit, and discuss its
manifestation in the bulk gravity dual.

Our discussion in this section will be for a single copy of the boundary CFT
at zero temperature. We again consider a finite but large $N$. For
definiteness, we will take the boundary spacetime to be
${\mathbb{{R}}}^{1,d-1}$. Recall that the Hilbert space of the boundary CFT is
${{\mathcal{H}}}$, with its full operator algebra ${{\mathcal{B}}}$. The
algebra generated by single-trace operators is ${{\mathcal{A}}}$, which is
only defined perturbatively in $1/N$ expansion. While ${{\mathcal{B}}}$ can be
defined on a single time slice, ${{\mathcal{A}}}$ is defined on the whole
spacetime as single-trace operators do not obey any equations of motion among
themselves, see Fig. 5.

Figure 5: The single-trace algebras ${{\mathcal{A}}}_{1}$ and
${{\mathcal{A}}}_{2}$ associated with the two different Cauchy slices shown
are inequivalent, even though they share a causal diamond, since single-trace
operators do not obey any equation of motion among themselves (standard
Heisenberg evolution takes a single-trace operator outside of the algebra).
The same statements apply to algebras generated by generalized free fields
(e.g. subalgebras of ${\mathcal{Y}}$) which do not obey any equations of
motion.

###  III.1 GNS Hilbert space and bulk reconstruction

For our discussion of the emergent type III1 structure for local boundary
subregions, it is again important to introduce the GNS Hilbert space of small
excitations, now around the vacuum state of the CFT. The procedure is similar
to our discussion of the GNS Hilbert space around the thermal field double
state in Sec. II.2, so we will not go into details.

Consider the GNS Hilbert space ${{\mathcal{H}}}^{\rm(GNS)}_{0}$ built from the
CFT vacuum state $\lvert{0}\rangle$ over the single-trace operator algebra
${{\mathcal{A}}}$. ${{\mathcal{H}}}^{\rm(GNS)}_{0}$ offers a representation
$\pi_{0}(A)$ for an operator $A\in{{\mathcal{A}}}$ and we denote the algebra
$\pi_{0}({{\mathcal{A}}})$ as ${\mathcal{Y}}$. As the entire operator algebra
on ${{\mathcal{H}}}^{\rm(GNS)}_{0}$, ${\mathcal{Y}}$ is a type I vN algebra.
We will denote the vector corresponding to the identity operator in
${{\mathcal{H}}}^{\rm(GNS)}_{0}$ as $\lvert{\Omega_{0}}\rangle_{\rm GNS}$. The
definition of ${\mathcal{Y}}$ is again only sensible perturbatively in the
$1/N$ expansion.

To leading order in the $1/N$ expansion, the algebra ${\mathcal{Y}}$ is again
generated by generalized free fields, with a mode expansion determined by
vacuum two-point functions of single-trace operators.

The boundary theory in the vacuum state $\lvert{0}\rangle$ is dual to the bulk
gravity theory in the empty AdS geometry (in our case the Poincaré patch as we
consider the boundary theory on ${\mathbb{{R}}}^{1,d-1}$). We can use the
standard procedure to build a Hilbert space of small excitations around the
Poincare vacuum $\lvert{0}\rangle_{\rm bulk}$, which we will denote as
${{\mathcal{H}}}_{0}^{(\rm Fock)}$. The algebra of bulk fields is denoted as
$\tilde{\mathcal{Y}}$. In terms of the algebraic language we are using, the
usual holographic dictionary can be written as

${{\mathcal{H}}}^{(\rm Fock)}_{0}={{\mathcal{H}}}^{(\rm
GNS)}_{0},\quad\lvert{0}\rangle_{\rm bulk}=\lvert{\Omega_{0}}\rangle_{\rm
GNS},\quad\tilde{\mathcal{Y}}={\mathcal{Y}}\ .$ (48)

In particular, the last equation in (48) identifies the bulk and boundary
creation/annihilation operators, and is equivalent to the statement of global
reconstruction.141414The HKLL global reconstruction Hamilton:2006az  is a
coordinate space version of the statement.

###  III.2 Boundary theory in a Rindler region and AdS Rindler duality

Figure 6: Left: Rindler regions of Minkowski spacetime. Right: AdS Rindler
regions of the bulk spacetime. The vertical lines denote the boundary and the
dashed lines are Rindler horizons. Each AdS Rindler region has the
corresponding Minkowski Rindler region as its boundary.

Now consider the boundary spacetime separated into Rindler regions, as on the
left of Fig. 6. We denote the algebra of operators restricted to the Rindler
$R$-region (or $L$-region) as ${{\mathcal{B}}}_{R}$ (or
${{\mathcal{B}}}_{L}$). These are type III1 vN algebras. The single-trace
operator algebras restricted to the $R$ and $L$ regions are denoted by
${{\mathcal{A}}}_{R}$ and ${{\mathcal{A}}}_{L}$. The restrictions of
${\mathcal{Y}}$ to the $R$ and $L$ regions are denoted by ${\mathcal{Y}}_{R}$
and ${\mathcal{Y}}_{L}$. They are von Neumann algebras and we have
${\mathcal{Y}}_{R}^{\prime}={\mathcal{Y}}_{L}$.

Our proposal is that ${\mathcal{Y}}_{R}$ and ${\mathcal{Y}}_{L}$ are type
III1. This new type III1 structure is only possible perturbatively in $1/N$
expansion, and is mathematically and physically distinct from the type III1
nature of ${{\mathcal{B}}}_{R}$ and ${{\mathcal{B}}}_{L}$. The support for our
proposal again comes from the complete spectrum of the spectral function of
single-trace operators restricted to a Rindler region and the half-sided
modular translation structure which we will study in detail in Sec. VI and
Sec. VII. It is also required by the duality with bulk gravity, which we now
elaborate.

The Poincaré patch of AdS can also be separated into four AdS Rindler regions
as on the right of Fig. 6. The standard procedures of the holographic
correspondence can be applied to an AdS Rindler region, leading to a duality
between the bulk gravity theory in the AdS Rindler ${\mathcal{R}}$
(${{\mathcal{L}}}$) region and the CFT in the boundary $R$ ($L$) region
Hamilton:2006az ; Czech:2012be ; Morrison:2014jha . Denoting the algebras of
bulk fields in the AdS ${\mathcal{R}}$ and ${{\mathcal{L}}}$ regions as
$\tilde{\mathcal{Y}}_{\mathcal{R}}$ and $\tilde{\mathcal{Y}}_{{\mathcal{L}}}$,
we have the identification

${\mathcal{Y}}_{R}=\widetilde{\mathcal{Y}}_{\mathcal{R}},\qquad{\mathcal{Y}}_{L}=\widetilde{\mathcal{Y}}_{{\mathcal{L}}}\
.$ (49)

As local operator algebras of the bulk low energy effective theory restricted
to a spacetime subregion, $\widetilde{\mathcal{Y}}_{\mathcal{R}}$ and
$\widetilde{\mathcal{Y}}_{{\mathcal{L}}}$ are type III1 vN algebras, thus so
are ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ due to the identifications (49).

The CFT vacuum $\lvert{0}\rangle$ is cyclic and separating for
${{\mathcal{B}}}_{R}$ and the corresponding modular Hamiltonian is
$-\log{{\Delta}}=K,$ where $K$ is the boost operator. Similarly,
$\lvert{\Omega_{0}}\rangle_{\rm GNS}$ is cyclic and separating for
${\mathcal{Y}}_{R}$, and the flows generated by the corresponding modular
operator ${{\Delta}}_{0}$ should again coincide with boosts. We thus have

$\pi_{0}(A_{R}(\eta=2\pi
t))=\pi_{0}\left({{\Delta}}^{-it}A_{R}{{\Delta}}^{it}\right)={{\Delta}}^{-it}_{0}\pi_{0}(A_{R}){{\Delta}}_{0}^{it},\quad
A_{R}\in{{\mathcal{A}}}_{R}$ (50)

where $\eta$ is the (dimensionless) Rindler time.

With ${{\mathcal{B}}}_{R}$ and ${\mathcal{Y}}_{R}$ being type III1, neither
${{\Delta}}$ nor ${{\Delta}}_{0}$ can be factorized into a product of
operators from the $R$ and $L$ regions, but their non-factorizations are
reflected very differently in the bulk. The non-factorization of ${{\Delta}}$
implies that the entanglement entropy $S_{R}$ between the $R$ and $L$ regions
in the full CFT can only be defined with a short-distance cutoff
$\epsilon_{\rm b}$ in the boundary, which corresponds to a bulk IR cutoff near
the intersection of the corresponding RT surface with the asymptotic boundary.
The non-factorization of ${{\Delta}}_{0}$ is reflected in the non-
factorization of the bulk field theory across the AdS Rindler horizon, which
implies that a bulk UV cutoff $\epsilon_{\rm UV}$ must be introduced in order
to define the bulk entanglement entropy ${\mathcal{S}}_{\mathcal{R}}$ between
the AdS Rindler ${\mathcal{R}}$ and ${{\mathcal{L}}}$ regions. See Fig. 6.

The above discussion of a boundary Rindler region can be straightforwardly
generalized to ball-shaped regions in the boundary which also have geometric
modular flow.

###  III.3 General boundary regions

We now generalize the above discussion of emergent type III1 algebras for
Rindler regions to general local boundary subregions. The story is similar, so
we will only emphasize those elements which are different.

We now use $R$ to denote a general spatial subregion in the boundary. Its
causal completion is denoted by $\hat{R}$. The restriction of
${{\mathcal{B}}}$ to $R$, ${{\mathcal{B}}}_{R},$ is the same as
${{\mathcal{B}}}_{\hat{R}}$, and is a type III1 vN algebra. Now consider the
restriction of ${\mathcal{Y}}$ to $R$, ${\mathcal{Y}}_{R},$ in the GNS Hilbert
space ${{\mathcal{H}}}_{0}^{\rm(GNS)}$. Note that
${\mathcal{Y}}_{R}\neq{\mathcal{Y}}_{\hat{R}}$ as ${\mathcal{Y}}$ is generated
by generalized free fields, which do not obey any equations of motion (see
Fig. 5). We now introduce

$\hat{\mathcal{Y}}_{R}\equiv({\mathcal{Y}}_{\hat{R}})^{\prime\prime}\ .$ (51)

From the definition, we have
$\hat{\mathcal{Y}}_{R}\supseteq{\mathcal{Y}}_{\hat{R}}$. For a half-space
(Rindler) or a ball-shaped region,
${\mathcal{Y}}_{\hat{R}}=\hat{\mathcal{Y}}_{R}$, as the modular flows are
geometric, but for general $R$ it may be that ${\mathcal{Y}}_{\hat{R}}$ is a
proper subset of $\hat{\mathcal{Y}}_{R}$. We propose that
$\hat{\mathcal{Y}}_{R}$ is type III1.

Subregion duality and bulk reconstruction associated with $R$ can be stated as

$\tilde{\mathcal{Y}}_{E_{R}}=\hat{\mathcal{Y}}_{R}\ ,$ (52)

where $E_{R}$ is the entanglement wedge of $R$ and
$\tilde{\mathcal{Y}}_{E_{R}}$ is the bulk field algebra restricted to $E_{R}$.
The containment of the causal wedge in the entanglement wedge can be
interpreted as a bulk reflection of
${\mathcal{Y}}_{\hat{R}}\subset\hat{\mathcal{Y}}_{R}$.151515 Denoting the
causal wedge for $R$ as $C_{R}$ and the corresponding bulk algebra as
$\tilde{\mathcal{Y}}_{C_{R}}$, we expect
$\tilde{\mathcal{Y}}_{C_{R}}\subseteq{\mathcal{Y}}_{\hat{R}}$. Note that since
$\tilde{\mathcal{Y}}_{C_{R}}$ is the algebra associated with a causally
complete bulk region, it should be a von Neumann algebra, but, in general,
${\mathcal{Y}}_{\hat{R}}$ does not have to be a von Neumann algebra.

We denote the modular operator of ${{\mathcal{B}}}_{R}$ with respect to the
CFT vacuum state $\lvert{0}\rangle$ as ${{\Delta}}$ and the modular operator
of $\hat{\mathcal{Y}}_{R}$ with respect to $\lvert{\Omega_{0}}\rangle_{\rm
GNS}$ as ${{\Delta}}_{0}$. It is tempting to postulate that modular flows
(with ${{\Delta}}^{-it}$) of elements of ${{\mathcal{A}}}_{\hat{R}}$ also have
a sensible $N\to\infty$ limit, in which case ${{\Delta}}_{0}$ may be viewed as
the representation of ${{\Delta}}$ in ${{\mathcal{H}}}_{0}^{(\rm GNS)}$, i.e.
we should have

$\pi_{0}\left({{\Delta}}^{-is}A{{\Delta}}^{is}\right)={{\Delta}}_{0}^{-is}\pi_{0}(A){{\Delta}}_{0}^{is},\quad
A\in{{\mathcal{A}}}_{R}$ (53)

which is the statement of the equivalence of bulk and boundary modular flows
Jafferis:2015del ; Faulkner:2017vdd  expressed in our language. Unlike those
in (35) and (50), the modular flow parameter $s$ in (53) does not have any
geometric interpretation.

The type III1 nature of ${{\mathcal{B}}}_{R}$ and $\hat{\mathcal{Y}}_{R}$ is
again reflected differently in the bulk, with the IR divergence of the area of
the Ryu-Takayanagi (RT) surface Ryu:2006bv  reflecting the type III1 nature of
${{\mathcal{B}}}_{R}$, while the divergence in the bulk entanglement entropy
${\mathcal{S}}_{E_{R}}$ for $E_{R}$ reflects the type III1 nature of
$\hat{\mathcal{Y}}_{R}$, see Fig. 7.

Figure 7: RT surface, $\gamma_{R}$, for a boundary spatial region $R$. $E_{R}$
denotes the entanglement wedge. Here we only draw a spatial section of the
bulk. The bulk IR divergence of the area of $\gamma_{R}$ comes from the part
near the boundary (circled red regions) and reflects type III1 nature of
boundary algebra ${{\mathcal{B}}}_{R}$. The type III1 nature of
$\hat{\mathcal{Y}}_{R}$ is reflected in the UV divergences of
${\mathcal{S}}_{E_{R}}$, which comes from UV degrees of freedom near
${{\gamma}}_{R}$ in the bulk (highlighted by orange wavy lines).

##  IV Physical implications

The emergent type III1 algebras potentially have many physical implications.
One such implication, which will be extensively explored in the rest of the
paper, is the emergent half-sided modular inclusion and translation structure,
which can be used to generate emergent in-falling flows in the bulk. Here we
discuss some other possible implications. Our discussion is somewhat vague,
but hopefully offers some pointers for future explorations.

###  IV.1 Role of the bifurcating horizon and RT surfaces

Consider first the case of the system in the thermal field double state. The
doubled system has a tensor product structure with
$\hat{{\mathcal{H}}}={{\mathcal{H}}}_{R}\otimes{{\mathcal{H}}}_{L}$,
$\hat{{\mathcal{B}}}={{\mathcal{B}}}_{R}\otimes{{\mathcal{B}}}_{L}$, and
$\hat{{\mathcal{A}}}={{\mathcal{A}}}_{R}\otimes{{\mathcal{A}}}_{L}$. The
emergent type III1 nature of ${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$ implies
that the GNS Hilbert space ${{\mathcal{H}}}_{{\rm TFD}}^{(\rm GNS)}$ does not
have a tensor product structure, i.e. it cannot be factorized into Hilbert
spaces associated with the $R$ and $L$ theories, and its operator algebra
${\mathcal{Y}}$ also lacks a tensor product structure in terms of
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$. This can have important implications
for describing the dynamics of low energy excitations around the thermal field
double state, including non-factorization of certain objects on the gravity
side. An immediate bulk example of such non-factorization comes from operators
inserted at the bifurcating horizon (suitably smeared), see the left of Fig.
8. The existence of conserved charges such as energy implies that
${\mathcal{Y}}_{R}$ and ${\mathcal{Y}}_{L}$ have a nontrivial center at the
leading order in the $1/N$ expansion WittenNew , i.e. they are not factors.
The presence of diffeomorphisms and other possible gauge symmetries on the
gravity side could also lead to a nontrivial center Casini:2013rba ;
Donnelly:2014fua ; Donnelly:2015hxa  for
$\tilde{\mathcal{Y}}_{R},\tilde{\mathcal{Y}}_{L}$ and thus for
${\mathcal{Y}}_{R},{\mathcal{Y}}_{L}$.

Figure 8: Left: operators inserted at the bifurcating horizon do not appear to
be factorizable. Right: a Wilson line in the eternal black hole geometry and
its factorization into left and right operators via the bifurcation surface.

The system can be factorized once we go beyond the $1/N$ expansion. From the
bulk perspective this requires going beyond the low energy approximation.
Interestingly, there are objects, which naively may be factorized only in the
full theory but turn out to be factorizable within the low energy description,
with the “help” of the bifurcating horizon. Here we will briefly comment on
two simple examples:

  1. 1.

${\mathcal{Y}}_{R}$ is type III1, so its modular operator ${{\Delta}}_{0}$
with respect to the GNS vacuum $\lvert{\Omega_{0}}\rangle$ cannot be
factorized, which translates to the bulk as the lack of a well-defined
entanglement entropy ${\mathcal{S}}_{R}$ between the $R$ and $L$ regions in
the continuum limit. Going beyond the $1/N$ expansion, the full theory can be
factorized, and there exists a well defined entanglement entropy $S_{R}$
between the $R$ and $L$ systems. It is a familiar fact that $S_{R}$ can
nevertheless be found using the low energy description on the gravity side by
the generalized entropy

$S_{R}={A_{\rm hor}\over 4G_{N}(\epsilon_{\rm
UV})}+{\mathcal{S}}_{R}(\epsilon_{\rm UV})$ (54)

where $A_{\rm hor}$ is the horizon area and $G_{N}(\epsilon_{\rm UV})$ is the
(bare) Newton constant at some bulk short-distance cutoff $\epsilon_{\rm UV}$.
The left-hand side is well defined mathematically but
${\mathcal{S}}_{R}(\epsilon_{\rm UV})$ cannot be defined in the $\epsilon_{\rm
UV}\to 0$ limit, and thus the two terms on the right hand side cannot be
individually defined in the continuum limit.

This emergent type III1 structure also provides a new perspective on the bulk
UV divergences and renormalization of the Newton constant $G_{N}$. Recall that
in the usual AdS/CFT dictionary, the bulk UV divergence is understood from the
boundary theory as coming from a truncation of operators dual to stringy modes
in the bulk. In particular, it is generally expected that the string theory
description of a physical quantity should be devoid of UV divergences at each
genus order. Here, however, the bulk UV divergences may be understood from the
boundary theory as arising from non-factorization of algebra ${\mathcal{Y}}$.
For this reason, even in string theory we expect that the two terms in (54)
which should come respectively from genus zero (the area term) and from higher
genera contributions cannot be individually finite.

  2. 2.

Another example was discussed by Harlow:2015lma , as indicated in the right of
Fig. 8, which is a Wilson line $W$ of a bulk gauge field going from the left
to the right boundary. To factorize the Wilson line into a product of left and
right operators requires breaking it up somewhere in the middle of the black
hole geometry, which cannot be done without introducing additional structure.
But there is an additional structure in the bulk: the bifurcating horizon. We
can break up the Wilson line in the low energy theory by taking advantage of
it, as indicated in Fig. 8, with

$W=\psi_{R}-\psi_{L},\quad\psi_{L,R}=\int_{\infty}^{r_{0}}A_{r}^{(L,R)}dr$
(55)

where $r_{0}$ is the location of the horizon. From discussions in
Nickel:2010pr ; Glorioso:2018mmw ; deBoer:2018qqm , $\psi_{R},\psi_{L}$ can be
identified respectively as effective fields describing diffusion in the right
and left theories at a finite temperature. These are collective dynamical
variables and cannot be expressed simply in terms of fundamental degrees of
freedom of the boundary theory. The bifurcating horizon can be described in a
diffeomorphism invariant way and thus $\psi_{R}$ and $\psi_{L}$ are also
diffeomorphism invariant and the factorization is well defined.

For both examples above, we see that the horizon plays the role of restoring
the factorization in the low energy description.

The above discussion can be generalized to the algebra $\hat{\mathcal{Y}}_{R}$
associated with a local boundary region $R$. We expect that
$\hat{\mathcal{Y}}_{R}$ and $(\hat{\mathcal{Y}}_{R})^{\prime}$ should share a
center whose “size” is characterized by the area of the RT surface. Similarly,
the RT surface can be used to restore factorization in the low energy
description. More explicitly, the entanglement entropy $S_{R}$ of a region $R$
in the full boundary theory can be obtained from the bulk by Faulkner:2013ana

$S_{R}={A_{{{\gamma}}_{R}}\over 4G_{N}(\epsilon_{\rm
UV})}+{\mathcal{S}}_{{\mathcal{R}}}(\epsilon_{\rm UV})$ (56)

where $A_{{{\gamma}}_{R}}$ is the area of the Ryu-Takayanagi (RT) surface
${{\gamma}}_{R}$. Recall that in this case $S_{R}$ is only defined with a UV
cutoff in the boundary which translates to a bulk IR cutoff. As remarked
earlier, due to the type III1 nature of $\hat{\mathcal{Y}}_{R}$,
${\mathcal{S}}_{{\mathcal{R}}}(\epsilon_{\rm UV})$ cannot be defined in the
continuum limit, and thus the two terms on the right hand side cannot be
defined separately in the limit $\epsilon_{\rm UV}\to 0$ as was the case in
(54).

The parallel with (54) and the thermal field double case can be made even
closer if we assume that the boundary theory has the split property, which is
believed to be satisfied by general quantum field theories Buchholz1 ;
Buchholz2 ; Fewster . Consider the situation in Fig. 9, where we separate the
two regions $R$ and $L$ by an infinitesimal distance $\epsilon_{\rm b}$. The
split property says that there exists a tensor product decomposition of the
global Hilbert space
${{\mathcal{H}}}={{\mathcal{H}}}_{N}\otimes{{\mathcal{H}}}_{\bar{N}}$, giving
rise to a type-I factor ${{\mathcal{N}}}$ corresponding to operators acting on
${{\mathcal{H}}}_{N}$, which satisfies

${{\mathcal{B}}}_{R}\subseteq{{\mathcal{N}}}\subseteq{{\mathcal{B}}}_{L}^{\prime}\
.$ (57)

The entanglement entropy associated with ${{\mathcal{H}}}_{N}$ is well defined
and in the limit $\epsilon_{\rm b}\to 0$ it can be used as a regularization of
$S_{R}$.161616Such a regularization was discussed earlier, for example, in
Dutta:2019gen . Under this regularization, the type III1 algebra
$\hat{\mathcal{Y}}_{R}$ can now be viewed as arising from the type I factor
${{\mathcal{N}}}$. In other words, in equation (53) we can treat the modular
operator ${{\Delta}}$ in the full theory as being factorizable. Similar to the
role played by the event horizon in (54), in (56) the RT surface restores the
factorization structure of
${{\mathcal{H}}}={{\mathcal{H}}}_{N}\otimes{{\mathcal{H}}}_{\bar{N}}$ in the
low energy description.

Figure 9: Slightly separated Rindler regions on a spatial slice. The split
property implies that there is a tensor factorization of the Hilbert space
with respect to an operator algebra ${{\mathcal{N}}}$ contained in the union
of the green and red regions above, even though no such tensor factorization
exists for the red and blue regions alone when $\epsilon_{\rm b}=0$

These discussions also imply that the bifurcating horizon of an eternal black
hole can be viewed as a special example of an RT surface from an algebraic
perspective. For a more general entangled state $\lvert{\Psi}\rangle$ between
the CFTR and CFTL, the RT surface which provides a signal of the factorization
of the full system in the low energy theory no longer coincides with the
horizon.

The role of the area terms in (54) and (56) in restoring the tensor product of
$\hat{{\mathcal{H}}}$ and ${{\mathcal{H}}}$ also provides a new perspective on
their physical origin and their universality. There are other ways to
understand the appearance of the area terms from the perspective of quantum
error correction Harlow:2016vwg ; Faulkner:2020hzi  and superselection sectors
Casini:2019kex . We believe that all these perspectives can be understood in a
unified way, which will be discussed elsewhere.

###  IV.2 More general entanglement wedges

The entanglement wedge of a boundary subregion $R$ can be defined to be the
maximal bulk subregion $E_{R}$ whose operator algebra can be reconstructed
from that of $R$. In the static situation and with a spatial region $R$ which
we are considering, the entanglement wedge $E_{R}$ for $R$ has been formulated
using the RT surface. It is the bulk causal completion of the region between
the RT surface ${{\gamma}}_{R}$ and $R$. This definition assumes that the
relevant operator algebra in the boundary for $R$ is ${{\mathcal{B}}}_{R}$
which is equivalent to ${{\mathcal{B}}}_{\hat{R}}$.

Our discussion in the previous sections suggests that entanglement wedges and
subregion duality can be formulated more generally, with the definition
associated to RT surfaces as a special case. As we emphasized, the bulk
reconstruction should be more precisely formulated in terms of operators in
the GNS Hilbert space, which are built from single-trace CFT operators. For
single-trace operators or their representations in the GNS Hilbert space, the
algebras associated with different Cauchy slices are inequivalent. This leads
to new ways of associating algebras with spacetime regions, which in turn
leads to new examples of entanglement wedges that are not related to RT
surfaces. We will discuss such examples in Sec. X. RT surfaces appear in
special situations where the operator algebra is associated with the causal
completion of a spatial (or null) region. In the more general case, these new
notions of entanglement wedges also raise the interesting question of whether
we can define more general notions of entropies that are associated with these
entanglement wedges. From our discussion of the close connections between bulk
area terms and emergent type III1 von Neumann algebras we expect the answer to
be yes. We leave this to future investigations.

###  IV.3 Emergent symmetries

There can be emergent symmetries associated with the emergent type III1
structure. In the example of two copies of CFT in the thermal field double
state, it can be shown that there are emergent null translation symmetries
along the past and future event horizons of the eternal black hole, which will
be discussed in detail in Sec. X.2.

There are likely many other examples where symmetries in the low energy
effective theory of gravity can be understood as being associated with
emergent type III1 algebras. Here we mention some possible candidates:

  1. 1.

In Goheer:2003tx  it has been argued that the compactification to
$(1+1)$-dimensional Rindler spacetime cannot exist in a quantum gravity, due
to incompatibility of an exact $SL(2,R)$ symmetry with a finite number of
states. It may be possible to understand the Rindler spacetime and associated
uncompact symmetries from an emergent type III1 algebra in the $G_{N}\to 0$
limit.

  2. 2.

An $SL(2,R)$ algebra in Jackiw-Teitelboim gravity was discussed in
Maldacena:2018lmt ; Lin:2019qwu  (see also Harlow:2021dfp ), which implements
AdS2 isometries on the matter fields. These symmetries may be understood from
emergent type III1 algebras in the SYK model.

  3. 3.

In DeBoer:2019kdj  local Poincare symmetry about a RT surface was discussed,
including its relevance for the modular properties of the boundary theory. As
with the near-horizon symmetries discussed in item 1 for a black hole, these
symmetries should be a consequence of the emergent type III1 structure
discussed in Sec. III.3.

##  V Review of half-sided modular translations

In this section we discuss how to generate new times in the boundary theory.
Our main tool is half-sided modular inclusion/translation Borchers:1991xk ;
Wiesbrock:1992mg , and an extension of it. This structure has played a role in
proofs of the CPT theorem Borchers:1991xk  and the construction of the
Poincaré group from wedge algebras Borchers1996 . There have also been
important applications of the half-sided modular inclusion structure to
understanding modular Hamiltonians of regions with boundaries on a null plane
for a quantum field theory in the vacuum, including average null energy
conditions Casini:2017roe ; Balakrishnan:2017bjg . See also Jefferson:2018ksk
for a discussion concerning black hole interiors.

###  V.1 Review of half-sided modular translations

Suppose ${{\mathcal{M}}}$ is a von Neumann algebra and the vector
$\lvert{\Omega_{0}}\rangle$ is cyclic and separating for ${{\mathcal{M}}}$.
The associated modular and conjugate operators are
${{\Delta}}_{{\mathcal{M}}}$ and $J_{{\mathcal{M}}}$. The commutant of
${{\mathcal{M}}}$ is denoted as ${{\mathcal{M}}}^{\prime}$.
${{\Delta}}_{{\mathcal{M}}}$ leaves $\lvert{\Omega_{0}}\rangle$ invariant and
can be used to generate flows within ${{\mathcal{M}}}$ or
${{\mathcal{M}}}^{\prime}$,

${{\Delta}}_{{\mathcal{M}}}^{-it}A{{\Delta}}_{{{\mathcal{M}}}}^{it}=e^{iK_{{\mathcal{M}}}t}Ae^{-iK_{{\mathcal{M}}}t}\in{{\mathcal{M}}},\quad
A\in{{\mathcal{M}}},\quad K_{{\mathcal{M}}}=-\log{{\Delta}}_{{\mathcal{M}}}\
,$ (58)

while the anti-unitary operator $J_{{\mathcal{M}}}$ takes ${{\mathcal{M}}}$ to
${{\mathcal{M}}}^{\prime}$ and vice versa

${{\mathcal{M}}}^{\prime}=J_{{\mathcal{M}}}{{\mathcal{M}}}J_{{\mathcal{M}}},\quad
J_{{\mathcal{M}}}^{2}=1\ .$ (59)

${{\Delta}}_{{\mathcal{M}}}$ acts on both ${{\mathcal{M}}}$ and
${{\mathcal{M}}}^{\prime}$, and in general cannot be factorized into operators
which act only on ${{\mathcal{M}}}$ or ${{\mathcal{M}}}^{\prime}$.

Now suppose there exists a von Neumann subalgebra ${{\mathcal{N}}}$ of
${{\mathcal{M}}}$ with the half-sided modular inclusion properties:

  1. 1.

$\lvert{\Omega_{0}}\rangle$ is cyclic for ${{\mathcal{N}}}$ (it is
automatically separating for ${{\mathcal{N}}}$ as
${{\mathcal{N}}}\subset{{\mathcal{M}}}$).

  2. 2.

The half-sided modular flow of ${{\mathcal{N}}}$ under
${{\Delta}}_{{\mathcal{M}}}$ lies within ${{\mathcal{N}}}$, i.e.

${{\Delta}}^{-it}_{{\mathcal{M}}}{{\mathcal{N}}}{{\Delta}}^{it}_{{\mathcal{M}}}\subset{{\mathcal{N}}},\quad
t\leq 0\ .$ (60)

We will denote the modular operator of ${{\mathcal{N}}}$ with respect to
$\lvert{\Omega_{0}}\rangle$ as ${{\Delta}}_{{\mathcal{N}}}$ with
$K_{{\mathcal{N}}}=-\log{{\Delta}}_{{\mathcal{N}}}$.

With these assumptions there are the following theorems Borchers:1991xk ;
Wiesbrock:1992mg ; Borchers:1998ye ; Borchers:1998 .

Theorem 1: There exists a unitary group $U(s),\,s\in{\mathbb{{R}}}$ with the
following properties:

  1. 1.

$U(s)$ has a positive generator, i.e.

$U(s)=e^{-iGs},\qquad G\geq 0$ (61)

  2. 2.

It leaves $\lvert{\Omega_{0}}\rangle$ invariant

$U(s)\lvert{\Omega_{0}}\rangle=\lvert{\Omega_{0}}\rangle,\quad\forall
s\in{\mathbb{{R}}}$ (62)

  3. 3.

Half-sided inclusion

$U^{\dagger}(s){{\mathcal{M}}}U(s)\subseteq{{\mathcal{M}}},\quad\forall s\leq
0\ .$ (63)

  4. 4.

${{\mathcal{N}}}$ can be obtained from ${{\mathcal{M}}}$ with an action of $U$

${{\mathcal{N}}}=U^{\dagger}(-1){{\mathcal{M}}}U(-1)\ .$ (64)

Theorem 2: Suppose $U(s)=e^{-iGs}$ is a continuous unitary group satisfying
(63), then any of the two conditions below imply the third:

$\displaystyle 1.$ $\displaystyle G\geq 0\ .$ (65) $\displaystyle 2.$
$\displaystyle U(s)\lvert{\Omega_{0}}\rangle=\lvert{\Omega_{0}}\rangle,\quad
s\in{\mathbb{{R}}}\ .$ (66) $\displaystyle 3.$
$\displaystyle\Delta^{-it}_{{\mathcal{M}}}U(s)\Delta^{it}_{{\mathcal{M}}}=U(e^{-2\pi
t}s),\;\;\text{and}\;\;J_{{\mathcal{M}}}U(s)J_{{\mathcal{M}}}=U(-s)\ .$ (67)

Theorem 3: Introducing

${{\mathcal{N}}}_{t}\equiv\Delta_{{{\mathcal{M}}}}^{-it}{{\mathcal{N}}}\Delta_{{{\mathcal{M}}}}^{it}$
(68)

we then have

  1. 1.

The family of algebras ${{\mathcal{N}}}_{t}$ with $t\in{\mathbb{{R}}}$ is
nested, i.e. ${{\mathcal{N}}}_{t_{1}}\subset{{\mathcal{N}}}_{t_{2}}$ for
$t_{1}<t_{2}$, with ${{\mathcal{N}}}_{\infty}={{\mathcal{M}}}$ and
${{\mathcal{N}}}_{0}={{\mathcal{N}}}$.

  2. 2.

The half-sided modular flow of any member of this family gives another algebra
in the family. In particular,

${{\Delta}}_{{\mathcal{N}}}^{-is}{{\mathcal{N}}}_{t}{{\Delta}}_{{{\mathcal{N}}}}^{is}={{\mathcal{N}}}_{f_{0}(s,t)},\qquad
f_{0}(s,t)=-{1\over 2\pi}\log\left(1+e^{-2\pi s}(e^{-2\pi t}-1)\right)$ (69)

valid for all $s,t$ such that the argument of the logarithm is positive. Note:

    1. (a)

For $t<0$, which means ${{\mathcal{N}}}_{t}\subset{{\mathcal{N}}}$, we always
have $f_{0}<0$ for any $s$ and $f_{0}<t$ for $s<0$. $f_{0}$ increases as $s$
increases and $f_{0}\to 0$ as $s\to\infty$.

    2. (b)

For $t>0$, which means that ${{\mathcal{N}}}\subset{{\mathcal{N}}}_{t}$, the
logarithm is defined only for $s\geq s_{t}\equiv{1\over 2\pi}\log(1-e^{-2\pi
t})<0$, and $f_{0}<t$ for $s>0$. As $s\to s_{t}$, $f_{0}\to+\infty$, while as
$s\to+\infty$, $f_{0}\to 0$. This can be intuitively understood as that the
part of ${{\mathcal{N}}}_{t}$ which is outside ${{\mathcal{N}}}$ is pushed
further away from (closer to) ${{\mathcal{N}}}$ for $s<0$ ($s>0$).

  3. 3.

The action of $U(s)$ on ${{\mathcal{N}}}_{t}$ has the structure

$e^{iGs}{{\mathcal{N}}}_{t}e^{-iGs}={{\mathcal{N}}}_{f_{1}(s,t)},\quad
f_{1}(s,t)=-{1\over 2\pi}\log(e^{-2\pi t}-s)$ (70)

valid for all $s,t$ such that the argument of the logarithm is positive. Note
that $f_{1}\to-\infty$ as $s\to-\infty$ and $f_{1}\to+\infty$ as $s\to
e^{-2\pi t}$.

  4. 4.

Modular operators of ${{\mathcal{M}}}$ and ${{\mathcal{N}}}$ satisfy the
algebra

$[K_{{\mathcal{M}}},K_{{\mathcal{N}}}]=-2\pi
i(K_{{\mathcal{M}}}-K_{{\mathcal{N}}})\ .$ (71)

  5. 5.

$U(s)$ can also be expressed in terms of modular flow operators of
${{\mathcal{M}}}$ and ${{\mathcal{N}}}$ as

${{\Delta}}_{{\mathcal{M}}}^{-it}{{\Delta}}_{{{\mathcal{N}}}}^{it}=U(\lambda(t)),\qquad\lambda(t)=e^{-2\pi
t}-1\ .$ (72)

Expanding both sides to linear order in $t$ we have

$K_{{\mathcal{M}}}-K_{{\mathcal{N}}}=2\pi G$ (73)

which gives the explicit form of $G$ in terms of modular Hamiltonians of
${{\mathcal{M}}}$ and ${{\mathcal{N}}}$.171717Note that the positivity of $G$
is manifest in this expression since
${{\mathcal{K}}}_{{\mathcal{M}}}-{{\mathcal{K}}}_{{\mathcal{N}}}=\log{{\Delta}}_{{\mathcal{N}}}-\log{{\Delta}}_{{\mathcal{M}}}\geq
0$

Theorem 4 Borchers:specCond : Suppose we have (i) nested von Neumann algebras
${{\mathcal{N}}}_{a},a\in{\mathbb{{R}}},\;{{\mathcal{N}}}_{a}\subset{{\mathcal{N}}}_{b},$
for $a<b$ with common cyclic and separating vector
$\lvert{\Omega_{0}}\rangle$; (ii) a one-parameter unitary group $T(a)$ with a
positive generator and
$T(a)\lvert{\Omega_{0}}\rangle=\lvert{\Omega_{0}}\rangle$; (iii) $T(a)$
translates the algebras

${{\mathcal{N}}}_{a}=T(a){{\mathcal{N}}}_{0}T(-a)\ .$ (74)

Then $T(a)$ is unique. Theorem 4 then says that given
${{\mathcal{M}}},{{\mathcal{N}}}$ and $\lvert{\Omega_{0}}\rangle$, $U(s)$ is
unique.

The above structure is called half-sided modular translation and exists only
if ${{\mathcal{M}}}$ is a type III1 von Neumann algebra Borchers:1998 .

Similarly, we can define half-sided modular inclusion (60) for $t\geq 0$ with
the corresponding half-sided modular translation for $s\geq 0$. All the
statements are parallel except with the following sign changes for equations
(67), (69), (70), (72)

$\displaystyle\Delta^{-it}_{{\mathcal{M}}}U(s)\Delta^{it}_{{\mathcal{M}}}=U(e^{2\pi
t}s),$ (75) $\displaystyle f_{0}(s,t)={1\over 2\pi}\log\left(1+e^{2\pi
s}(e^{2\pi t}-1)\right),$ (76) $\displaystyle f_{1}(s,t)={1\over
2\pi}\log(e^{2\pi t}+s),$ (77) $\displaystyle\lambda(t)=e^{2\pi t}-1\ .$ (78)

###  V.2 Example I: null translations in Rindler spacetime

Consider a quantum field theory in $(1+1)$-dimensional Minkowski spacetime
${\mathbb{{R}}}^{1.1}$ with coordinates $x^{\mu}=(x^{0},x^{1})$ and momentum
operators $P^{\mu}=(P^{0},P)$. Suppose the system is in the vacuum state
$\lvert{0}\rangle$ with respect to the Minkowski time $x^{0}$.

Consider the half space $A$ given by

$A=\\{x^{\mu}\in{\mathbb{{R}}}^{1,1}|x^{0}=0,x^{1}>0\\}$ (79)

whose domain of dependence is the Rindler $R$-region (see Fig. 6 Left). We
take ${{\mathcal{M}}}$ to be the operator algebra in the $R$-region, so
$\lvert{0}\rangle$ is cyclic and separating under ${{\mathcal{M}}}$. The
corresponding modular Hamiltonian $K_{{\mathcal{M}}}$ in this case is
proportional to the boost operator $K$

$K_{{\mathcal{M}}}=-\log{{\Delta}}_{{\mathcal{M}}}=2\pi K\
,\quad{{\Delta}}_{{\mathcal{M}}}^{-it}=e^{iK_{{\mathcal{M}}}t}$ (80)

and $J_{{\mathcal{M}}}$ is the ${{\mathcal{C}}}{{\mathcal{P}}}{\mathcal{T}}$
operator.

It is convenient to use light-cone coordinates

$x^{\pm}=x^{0}\pm x^{1},\quad P^{\pm}={{\frac{1}{2}}}(H\pm P)=-P_{\mp},\quad
H=P^{0}\ ,$ (81)

where the translation operator by a vector $a^{\mu}$ is given by

$e^{-iP_{\mu}\cdot a^{\mu}}=e^{iHa^{0}-iPa^{1}}=e^{iP^{-}a^{+}+iP^{+}a^{-}}\
.$ (82)

Note that

$\displaystyle[K,P^{\pm}]=\pm iP^{\pm},\qquad
e^{iK_{{\mathcal{M}}}s}P^{\pm}e^{-iK_{{\mathcal{M}}}s}=e^{\mp 2\pi s}P^{\pm},$
(83) $\displaystyle
e^{iK_{{\mathcal{M}}}s}\phi(x^{\mu})e^{-iK_{{\mathcal{M}}}s}=\phi(x^{\prime\mu}(s)),\qquad
x^{\prime\pm}(s)=e^{\pm 2\pi s}x^{\pm}$ (84)

where $\phi(x^{\mu})$ is a scalar operator.

${{\mathcal{M}}}$ is the operator algebra in the region $x^{+}>0,x^{-}<0$.
Below for simplicity we will simply use the spacetime region to denote the
operator algebra in that region. We take ${{\mathcal{N}}}$ to be the region
$\\{x^{+}>0,x^{-}<-1\\}$ (see Fig 10 Left), and then

${{\mathcal{N}}}_{t}\equiv
e^{iK_{{\mathcal{M}}}t}{{\mathcal{N}}}e^{-iK_{{\mathcal{M}}}t}=\\{x^{+}>0,x^{-}<-e^{-2\pi
t}\\}$ (85)

with ${{\mathcal{N}}}_{t}\subset{{\mathcal{N}}}$ for $t<0$. We thus have the
half-sided modular inclusion structure (60). In this case the modular operator
of ${{\mathcal{N}}}$ can be found explicitly and existence of the positive
generator $G$ can be directly verified. More explicitly, flows generated by
the modular operator of ${{\mathcal{N}}}$ correspond to boosts which which
leave the point $a^{\mu}=(a^{+},a^{-})=(0,-1)$ invariant. Thus
$K_{{\mathcal{N}}}$ should be given by

$e^{itK_{{\mathcal{N}}}}=e^{-ia^{\mu}P_{\mu}}e^{itK_{{\mathcal{M}}}}e^{ia^{\mu}P_{\mu}}=e^{-iP^{+}}e^{itK_{{\mathcal{M}}}}e^{iP^{+}},$
(86)

which gives

$K_{{\mathcal{N}}}=K_{{\mathcal{M}}}-2\pi P^{+}\ .$ (87)

From (73) we conclude that the corresponding $G$ is given by

$G=P^{+}$ (88)

and thus

$U(s)=e^{-isP^{+}}\ .$ (89)

We can now verify explicitly the statements of various theorems of last
subsection. For example,

$\displaystyle
U^{\dagger}(s){{\mathcal{M}}}U(s)=\\{x^{+}>0,x^{-}<s\\}\subset{{\mathcal{M}}},\quad
s<0,$ (90)
$\displaystyle{{\mathcal{N}}}=U^{\dagger}(-1){{\mathcal{M}}}U(-1),\quad{{\mathcal{N}}}_{t}=U^{\dagger}(-e^{-2\pi
t}){{\mathcal{M}}}U(-e^{-2\pi t}),$ (91)
$\displaystyle{{\Delta}}_{{\mathcal{M}}}^{-it}U(s){{\Delta}}_{{\mathcal{M}}}^{it}=e^{iK_{{\mathcal{M}}}t}e^{-iP^{+}s}e^{-iK_{{\mathcal{M}}}t}=U(e^{-2\pi
t}s)\ .$ (92)

By taking ${{\mathcal{N}}}$ to be the operator algebra associated with the
region in Fig. 10 (b), there is a half-sided modular inclusion structure with
$t\geq 0$, and the corresponding modular translation operator is given by
$G=P^{-}$.

Figure 10: Left: The algebra of the subregion ${{\mathcal{N}}}$ that leads to
the half-sided modular inclusion structure for $x^{-}$ translation. Right: The
algebra of the subregion ${{\mathcal{N}}}$ that leads to the half-sided
modular inclusion structure for $x^{+}$ translation.

###  V.3 Example II: Two copies of a large $N$ theory in the thermal field
double state

Figure 11: Left: ${{\mathcal{N}}}$ denotes the spacetime subregion with ${\rm
t}\leq 0$. The vertical axis is time and for simplicity we have taken the
spatial manifold to be a circle (vertical boundaries in the figure are
identified). Right: ${{\mathcal{N}}}$ denotes the spacetime subregion with
${\rm t}\geq 0$.

Now consider two CFTs in the thermal field double state in the large $N$
limit, as discussed in Sec. II.2. We now take
${{\mathcal{M}}}={\mathcal{Y}}_{R}$, which is the representation of the
single-trace algebra ${{\mathcal{A}}}_{R}$ in the GNS Hilbert space
${{\mathcal{H}}}_{\rm TFD}^{(\rm GNS)}$. The associated modular operator is
${{\Delta}}_{{\mathcal{M}}}={{\Delta}}_{0}$ with corresponding modular time
$t$ related to the usual time ${\rm t}$ by $t={{\rm t}\over{\beta}}$.

By choosing different subalgebras ${{\mathcal{N}}}$ we can construct different
generators $G$ whose spectra are bounded from below and thus generate new
“times”. As the simplest possibility we take ${{\mathcal{N}}}$ to be the
representation of the single-trace operator algebra associated with the region
indicated in the left plot of Fig. 11. Since generalized free fields do not
satisfy any Heisenberg equations, ${{\mathcal{N}}}$ is inequivalent to
${{\mathcal{M}}}$ (recall Fig. 5).

The GNS vacuum $\lvert{\Omega_{0}}\rangle$ is separating with respect to
${{\mathcal{N}}}$. While we do not have a rigorous mathematical proof, we will
assume that it is also cyclic with respect to ${{\mathcal{N}}}$. Since
${{\Delta}}_{0}^{-it}$ generates a time translation, clearly

${{\mathcal{N}}}_{t}\equiv{{\Delta}}_{0}^{-it}{{\mathcal{N}}}{{\Delta}}_{0}^{it}\subset{{\mathcal{N}}},\;\;\text{for}\;t<0\
.$ (93)

We thus have the half-sided modular inclusion structure of (60). In this case
${{\Delta}}_{{{\mathcal{N}}}}$ and $G$ are not explicitly known. The theorems
in Sec. V.1 can be used to anticipate the action of $U(s)=e^{-iGs}$, for
example as in (70). In Sec. X we will give the explicit action of $U(s)$ by
proposing the gravity description of it, which can be explicitly worked out.
Equation (70) then provides a nontrivial check of the proposal.

We can also consider choosing ${{\mathcal{N}}}$ to be associated with the
region in the right plot of Fig. 11, which gives a half-sided modular
inclusion structure for $t>0$.

For both plots in Fig. 11, instead of letting the region describing
${{\mathcal{N}}}$ be bounded by the ${\rm t}=0$ slice, we can choose an
arbitrary Cauchy slice (not necessarily with constant ${\rm t}$), see Fig. 12.
There is still a half-sided modular inclusion structure and the associated
modular translations. Thus there are an infinite number of emergent “times” in
the large $N$ limit.

Figure 12: For both plots in Fig. 11 instead of letting the region describing
${{\mathcal{N}}}$ be bounded by the ${\rm t}=0$ slice, we can choose a slice
${\rm t}=f(\chi)$ where $\chi$ is the boundary spatial coordinate and $f$ an
arbitrary periodic function.

##  VI Positive extension of half-sided translations for generalized free
fields

We now consider the general structure of half-sided modular translations for a
generalized free field theory. We will show that in this context it is
possible to determine the general structure of the action of $U(s)$ for all
values of $s$ without the need of specifying ${{\mathcal{N}}}$ or
${{\Delta}}_{{\mathcal{M}}}$.

As an illustration , we will use two CFTs in the thermal field double state in
the large $N$ limit, as discussed in Sec. II.2 and Sec. V.3, with
${{\mathcal{M}}}={\mathcal{Y}}_{R},\;{{\mathcal{M}}}^{\prime}={\mathcal{Y}}_{L}$,
and ${{\Delta}}_{{\mathcal{M}}}={{\Delta}}_{0}$. The generalized free fields
that generate the algebras are defined by (26)–(28). Below and for the rest of
the paper for notational simplicity we will write
$\pi({{\mathcal{O}}}_{R}(x))$ simply as ${{\mathcal{O}}}_{R}(x)$, but it
should be kept in mind that they are operators in the GNS Hilbert space. Also
for convenience for the rest of the paper we will rescale the CFT time such
that ${\beta}=2\pi$. The rescaled time will be denoted as $\eta$. From (34) we
thus have the relation between modular time $t$ and $\eta$

$t={\eta\over 2\pi}\ .$ (94)

From now on $\omega$ will be conjugate to $\eta$.

The discussion in this section also applies to ${\mathcal{Y}}_{R}$ for a local
subregion. For a Rindler region, $\eta$ is simply the Rindler time.

###  VI.1 Unitary automorphism of the algebra

For convenience we first copy here some relevant equations of Sec. II.2. In
the GNS Hilbert space ${{\mathcal{H}}}^{\rm(GNS)}_{\rm TFD}$, single-trace
operators ${{\mathcal{O}}}_{L},{{\mathcal{O}}}_{R}$ can be represented by
generalized free fields with mode expansions

$\displaystyle{{\mathcal{O}}}_{\alpha}(x)=\sum_{k}u_{k}^{({\alpha})}(x)a_{k}^{({\alpha})}=\sum_{k}({\mathfrak{b}}_{+}u_{k}^{({\alpha})}\,c_{k}^{({\alpha})}+{\mathfrak{b}}_{-}u_{-k}^{({\alpha})}\,c_{k}^{(\bar{\alpha})}),\quad{\alpha}=R,L,\quad\bar{R}=L\
,$ (95)

where $k=(\omega,q)$ denotes all quantum numbers including
$\omega\in(-\infty,\infty)$ with $\sum_{\omega}=\int{d\omega\over 2\pi}$. The
$R$ and $L$ systems are assumed to be symmetric with

$u_{-k}^{(R)}(x)=u_{k}^{(R)*}(x)=u_{k}^{(L)}(x)\ .$ (96)

The various oscillators satisfy the equations

$\displaystyle(a_{k}^{({\alpha})})^{\dagger}=a_{-k}^{({\alpha})},\quad[a_{k}^{({\alpha})},a_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}},\quad
a_{k}^{({\alpha})}\lvert{\Omega_{0}}\rangle=e^{-\pi\omega}a_{-k}^{(\bar{\alpha})}\lvert{\Omega_{0}}\rangle,$
(97)
$\displaystyle\left(c_{k}^{({\alpha})}\right)^{\dagger}=c_{-k}^{({\alpha})},\quad[c_{k}^{({\alpha})},c_{k^{\prime}}^{({\beta})}]=\epsilon(\omega){{\delta}}_{k+k^{\prime},0}{{\delta}}_{{\alpha}{\beta}},\quad
c_{k}^{({\alpha})}\lvert{\Omega_{0}}\rangle=0\;\;\text{for}\;\;\omega>0,$ (98)
$\displaystyle
c^{({\alpha})}_{k}={\mathfrak{b}}_{+}a^{({\alpha})}_{k}-{\mathfrak{b}}_{-}a^{(\bar{\alpha})}_{-k},\qquad
a^{({\alpha})}_{k}={\mathfrak{b}}_{+}c^{({\alpha})}_{k}+{\mathfrak{b}}_{-}c^{(\bar{\alpha})}_{-k},\quad{\mathfrak{b}}_{\pm}={e^{\pm{\pi|\omega|\over
2}}\over\sqrt{2\sinh{\pi|\omega|}}}\ .$ (99)

The anti-unitary modular conjugation operator $J$ takes ${{\mathcal{O}}}_{R}$
to ${{\mathcal{O}}}_{L}$ and vice versa, i.e.

$J{{\mathcal{O}}}_{\alpha}(x)J={{\mathcal{O}}}_{\bar{\alpha}}(x),\quad
Ja_{k}^{({\alpha})}J=a_{k}^{(\bar{\alpha})},\quad
Jc_{k}^{({\alpha})}J=c_{k}^{(\bar{\alpha})}\ .$ (100)

Now suppose there is a one-parameter unitary automorphism group
$U(s),s\in{\mathbb{{R}}}$,

${{\mathcal{O}}}_{R}(x;s)\equiv
U(s)^{\dagger}{{\mathcal{O}}}_{R}(x)U(s),\qquad{{\mathcal{O}}}_{L}(x;s)\equiv
U(s)^{\dagger}{{\mathcal{O}}}_{L}(x)U(s)$ (101)

which we require to satisfy the following properties:

  1. 1.

Half-sided inclusion, i.e.

${{\mathcal{O}}}_{R}(x;s)\in{\mathcal{Y}}_{R},\quad\text{for }s<0\ .$ (102)

  2. 2.

It leaves the state $\lvert{\Omega_{0}}\rangle$ invariant

$U(s)\lvert{\Omega_{0}}\rangle=\lvert{\Omega_{0}}\rangle,\quad\forall s\ .$
(103)

  3. 3.

$J$ acts on $U$ as

$JU(s)J=U(-s)\ .$ (104)

Acting on both sides of the first equation of (101) by $J$ we find

$J{{\mathcal{O}}}_{R}(x;s)J=JU(s)^{\dagger}JJ{{\mathcal{O}}}_{R}(x)JJU(s)J={{\mathcal{O}}}_{L}(x;-s)\
.$ (105)

  4. 4.

Under modular flow we require

${{\Delta}}^{-it}_{0}U(s){{\Delta}}_{0}^{it}=U(e^{-2\pi t}s)\ .$ (106)

  5. 5.

$U(1)$ group property

$U(s)^{\dagger}=U(-s),\qquad U(s_{1})U(s_{2})=U(s_{1}+s_{2}),\quad
s,s_{1},s_{2}\in{\mathbb{{R}}}\ .$ (107)

From Theorem 2 of Sec. V.1, $U(s)$ satisfying the above conditions is
generated by a Hermitian operator $G$ that is bounded from below.

We will now use the above properties to deduce the explicit transformations of
the oscillators under $U(s)$. For this purpose we denote

$\sigma_{s}(a_{k}^{({\alpha})})\equiv
U^{\dagger}(s)a_{k}^{({\alpha})}U(s)=\Lambda^{{\alpha}{\beta}}_{kk^{\prime}}(s)a_{k^{\prime}}^{({\beta})},\qquad\sigma_{s}(c_{k}^{({\alpha})})\equiv
U^{\dagger}(s)c_{k}^{({\alpha})}U(s)=\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)c_{k^{\prime}}^{({\beta})}\
.$ (108)

In the above equations and also subsequent discussions, repeated indices
$k^{\prime}$ and ${\beta}$ should be understood as being summed. The
transformation matrices $\Sigma$ and $\Lambda$ can be related to each other
using the basis changes (99). More explicitly, we have
(${\mathfrak{b}}^{\prime}_{\pm}$ denotes the corresponding expression as in
(99) for $\omega^{\prime}$)

$\displaystyle\Lambda^{{\alpha}{\beta}}_{kk^{\prime}}(s)={\mathfrak{b}}_{+}\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s){\mathfrak{b}}^{\prime}_{+}+{\mathfrak{b}}_{-}\Sigma^{\bar{\alpha}{\beta}}_{-kk^{\prime}}(s){\mathfrak{b}}^{\prime}_{+}-{\mathfrak{b}}_{+}\Sigma^{{\alpha}\bar{\beta}}_{k-k^{\prime}}(s){\mathfrak{b}}^{\prime}_{-}-{\mathfrak{b}}_{-}\Sigma^{\bar{\alpha}\bar{\beta}}_{-k-k^{\prime}}(s){\mathfrak{b}}^{\prime}_{-},$
(109)
$\displaystyle\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)={\mathfrak{b}}_{+}\Lambda^{{\alpha}{\beta}}_{kk^{\prime}}(s){\mathfrak{b}}^{\prime}_{+}-{\mathfrak{b}}_{-}\Lambda^{\bar{\alpha}{\beta}}_{-kk^{\prime}}(s){\mathfrak{b}}^{\prime}_{+}+{\mathfrak{b}}_{+}\Lambda^{{\alpha}\bar{\beta}}_{k-k^{\prime}}(s){\mathfrak{b}}^{\prime}_{-}-{\mathfrak{b}}_{-}\Lambda^{\bar{\alpha}\bar{\beta}}_{-k-k^{\prime}}(s){\mathfrak{b}}^{\prime}_{-}\
.$ (110)

Introducing

$\displaystyle
u^{({\alpha}{\beta})}_{k}(x;s)=u^{({\alpha})}_{k^{\prime}}(x)\Lambda^{{\alpha}{\beta}}_{k^{\prime}k}(s),\quad
w_{k}^{({\alpha}{\beta})}(x;s)={\mathfrak{b}}_{+}^{\prime}u_{k^{\prime}}^{({\alpha})}(x)\Sigma^{{\alpha}{\beta}}_{k^{\prime}k}(s)+{\mathfrak{b}}_{-}^{\prime}u_{-k^{\prime}}^{({\alpha})}(x)\Sigma^{\bar{\alpha}{\beta}}_{k^{\prime}k}(s),$
(111)

($\alpha$ is not summed) we have

$\displaystyle{{\mathcal{O}}}_{\alpha}(x;s)=u^{({\alpha}{\beta})}_{k^{\prime}}(x;s)a_{k^{\prime}}^{({\beta})}=w_{k^{\prime}}^{({\alpha}{\beta})}(x;s)\,c_{k^{\prime}}^{({\beta})},$
(112) $\displaystyle
w_{k}^{({\alpha}{\beta})}(x;s)=u^{({\alpha}{\beta})}_{k}(x;s){\mathfrak{b}}_{+}+u^{({\alpha}\bar{\beta})}_{-k}(x;s){\mathfrak{b}}_{-}\
.$ (113)

We now work out the conditions $\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}$ and
$\Lambda^{{\alpha}{\beta}}_{kk^{\prime}}$ should satisfy. Equation (102)
implies that

$\Lambda^{RL}_{kk^{\prime}}(s)=0,\quad\text{for }s<0$ (114)

and we denote

$\Lambda^{RR}_{kk^{\prime}}(s<0)=C_{kk^{\prime}}(s)\ .$ (115)

Taking the Hermitian conjugate of (108)

$(\sigma_{s}(c_{k}^{({\alpha})}))^{\dagger}=\sigma_{s}(c_{-k}^{({\alpha})})\quad\to\quad\Sigma^{{\alpha}{\beta}*}_{kk^{\prime}}(s)c_{-k^{\prime}}^{({\beta})}=\Sigma_{-kk^{\prime}}^{{\alpha}{\beta}}(s)c_{k^{\prime}}^{({\beta})}$
(116)

which requires that

$\Sigma^{{\alpha}{\beta}*}_{kk^{\prime}}(s)=\Sigma_{-k,-k^{\prime}}^{{\alpha}{\beta}}(s)\
.$ (117)

Similarly we have

$\Lambda^{{\alpha}{\beta}*}_{kk^{\prime}}(s)=\Lambda_{-k,-k^{\prime}}^{{\alpha}{\beta}}(s),\qquad
C_{kk^{\prime}}^{*}(s)=C_{-k-k^{\prime}}(s)\ .$ (118)

Acting $J$ on (108) we have

$J(\sigma_{s}(c_{k}^{({\alpha})}))J=JU^{\dagger}(s)c_{k}^{({\alpha})}U(s)J=\sigma_{-s}(c_{k}^{(\bar{\alpha})})\quad\to\quad\Sigma^{{\alpha}{\beta}*}_{kk^{\prime}}(s)c_{k^{\prime}}^{(\bar{\beta})}=\Sigma^{\bar{\alpha}\bar{\beta}}_{kk^{\prime}}(-s)c_{k^{\prime}}^{(\bar{\beta})}$
(119)

which implies that

$\Sigma^{{\alpha}{\beta}*}_{kk^{\prime}}(s)=\Sigma^{\bar{\alpha}\bar{\beta}}_{kk^{\prime}}(-s)$
(120)

and similarly

$\Lambda^{{\alpha}{\beta}*}_{kk^{\prime}}(s)=\Lambda^{\bar{\alpha}\bar{\beta}}_{kk^{\prime}}(-s)\
.$ (121)

From (103) and (98) we have

$\sigma_{s}(c_{k}^{({\alpha})})\lvert{\Omega_{0}}\rangle=0\quad\text{for
}\omega>0,\quad\langle{\Omega_{0}}\rvert\sigma_{s}(c_{k}^{({\alpha})})=0\quad\text{for
}\omega<0,$ (122)

i.e. the action of $\sigma_{s}$ does not mix $c-$type creation and
annihilation operators, which implies that $\Sigma$ should have the structure

$\Sigma_{kk^{\prime}}^{{\alpha}{\beta}}(s)={\theta}(\omega){\theta}(\omega^{\prime})A_{kk^{\prime}}^{{\alpha}{\beta}}(s)+{\theta}(-\omega){\theta}(-\omega^{\prime})B_{kk^{\prime}}^{{\alpha}{\beta}}(s)\
.$ (123)

Equations (120) and (117) imply that

$A_{kk^{\prime}}^{({\alpha}{\beta})*}(s)=A_{kk^{\prime}}^{(\bar{\alpha}\bar{\beta})}(-s)=B_{-k-k^{\prime}}^{({\alpha}{\beta})}(s),\quad
B_{kk^{\prime}}^{({\alpha}{\beta})*}(s)=B_{kk^{\prime}}^{(\bar{\alpha}\bar{\beta})}(-s)=A_{-k-k^{\prime}}^{({\alpha}{\beta})}(s)\
.$ (124)

We will now show that (123) further implies that the full
$\Lambda^{{\alpha}{\beta}}_{kk^{\prime}}(s)$ and
$\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)$ can be expressed in terms of
$C_{kk^{\prime}}(s)$ with $s<0$. From (111) and (113) we have for $s<0$

$\displaystyle
u_{k}^{(RR)}(x;s){\mathfrak{b}}_{+}={\mathfrak{b}}_{+}^{\prime}u_{k^{\prime}}^{(R)}(x)\Sigma^{RR}_{k^{\prime}k}+{\mathfrak{b}}_{-}^{\prime}u_{-k^{\prime}}^{(R)}(x)\Sigma^{LR}_{k^{\prime}k},$
(125) $\displaystyle
u_{-k}^{(RR)}(x;s){\mathfrak{b}}_{-}={\mathfrak{b}}_{+}^{\prime}u_{k^{\prime}}^{(R)}(x)\Sigma^{RL}_{k^{\prime}k}+{\mathfrak{b}}_{-}^{\prime}u_{-k^{\prime}}^{(R)}(x)\Sigma^{LL}_{k^{\prime}k}\
.$ (126)

From (123) we then have

$\displaystyle
u_{k^{\prime}}^{(R)}(x)C_{k^{\prime}k}(s){\mathfrak{b}}_{+}={\mathfrak{b}}_{+}^{\prime}u_{k^{\prime}}^{(R)}(x)\left({\theta}(\omega){\theta}(\omega^{\prime})A_{k^{\prime}k}^{RR}(s)+{\theta}(-\omega){\theta}(-\omega^{\prime})B_{k^{\prime}k}^{RR}(s)\right)$
(127)
$\displaystyle+{\mathfrak{b}}_{-}^{\prime}u_{k^{\prime}}^{(R)}(x)\left({\theta}(\omega){\theta}(-\omega^{\prime})A_{-k^{\prime}k}^{LR}(s)+{\theta}(-\omega){\theta}(\omega^{\prime})B_{-k^{\prime}k}^{LR}(s)\right)$
(128) $\displaystyle
u_{k^{\prime}}^{(R)}(x)C_{k^{\prime}-k}(s){\mathfrak{b}}_{-}={\mathfrak{b}}_{+}^{\prime}u_{k^{\prime}}^{(R)}(x)\left({\theta}(\omega){\theta}(\omega^{\prime})A_{k^{\prime}k}^{RL}(s)+{\theta}(-\omega){\theta}(-\omega^{\prime})B_{k^{\prime}k}^{RL}(s)\right)$
(129)
$\displaystyle+{\mathfrak{b}}_{-}^{\prime}u_{k^{\prime}}^{(R)}(x)\left({\theta}(\omega){\theta}(-\omega^{\prime})A_{-k^{\prime}k}^{LL}(s)+{\theta}(-\omega){\theta}(\omega^{\prime})B_{-k^{\prime}k}^{LL}(s)\right)\
.$ (130)

Considering respectively $\omega>0$ and $\omega<0$ on both sides we find that

$\displaystyle
A_{k^{\prime}k}^{RR}(s)={\theta}(\omega){\theta}(\omega^{\prime}){{\mathfrak{b}}_{+}\over{\mathfrak{b}}_{+}^{\prime}}C_{k^{\prime}k}(s),\quad
A_{k^{\prime}k}^{LR}(s)={\theta}(\omega){\theta}(\omega^{\prime}){{\mathfrak{b}}_{+}\over{\mathfrak{b}}_{-}^{\prime}}C_{-k^{\prime}k}(s),$
(131) $\displaystyle
A_{k^{\prime}k}^{RL}(s)={\theta}(\omega){\theta}(\omega^{\prime}){{\mathfrak{b}}_{-}\over{\mathfrak{b}}_{+}^{\prime}}C_{k^{\prime}-k}(s),\quad
A_{k^{\prime}k}^{LL}(s)={\theta}(\omega){\theta}(\omega^{\prime}){{\mathfrak{b}}_{-}\over{\mathfrak{b}}_{-}^{\prime}}C_{-k^{\prime}-k}(s),$
(132) $\displaystyle
B_{k^{\prime}k}^{RR}(s)={\theta}(-\omega){\theta}(-\omega^{\prime}){{\mathfrak{b}}_{+}\over{\mathfrak{b}}_{+}^{\prime}}C_{k^{\prime}k}(s),\quad
B_{k^{\prime}k}^{LR}(s)={\theta}(-\omega){\theta}(-\omega^{\prime}){{\mathfrak{b}}_{+}\over{\mathfrak{b}}_{-}^{\prime}}C_{-k^{\prime}k}(s),$
(133) $\displaystyle
B_{k^{\prime}k}^{RL}(s)={\theta}(-\omega){\theta}(-\omega^{\prime}){{\mathfrak{b}}_{-}\over{\mathfrak{b}}_{+}^{\prime}}C_{k^{\prime}-k}(s),\quad
B_{k^{\prime}k}^{LL}(s)={\theta}(-\omega){\theta}(-\omega^{\prime}){{\mathfrak{b}}_{-}\over{\mathfrak{b}}_{-}^{\prime}}C_{-k^{\prime}-k}(s)\
.$ (134)

The above equations can be written more compactly as

$A_{k^{\prime}k}^{{\alpha}{\beta}}(s)={{\mathfrak{b}}_{\beta}\over{\mathfrak{b}}_{\alpha}^{\prime}}{\theta}(\omega){\theta}(\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(s),\quad
B_{k^{\prime}k}^{{\alpha}{\beta}}(s)={{\mathfrak{b}}_{\beta}\over{\mathfrak{b}}_{\alpha}^{\prime}}{\theta}(-\omega){\theta}(-\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(s)$
(135)

where in subscripts for ${\mathfrak{b}}$ and $C$ it should be understood that
$R=+$ and $L=-$. From the above we also have the relations

$\displaystyle
A_{k^{\prime}k}^{({\alpha}{\beta})*}(s)=B_{-k^{\prime}-k}^{({\alpha}{\beta})}(s),\quad{{\mathfrak{b}}_{\alpha}^{\prime}\over{\mathfrak{b}}_{\beta}}A_{k^{\prime}k}^{{\alpha}{\beta}}(s)={{\mathfrak{b}}_{\bar{\alpha}}^{\prime}\over{\mathfrak{b}}_{\bar{\beta}}}B_{-k^{\prime}-k}^{\bar{\alpha}\bar{\beta}}(s),$
(136)

where the first relation follows from the second equation of (118). The above
expressions apply to $s<0$. We can now find the expressions for $A$ and $B$
for $s>0$ from (120)

$\displaystyle
A_{k^{\prime}k}^{({\alpha}{\beta})}(s)=A_{k^{\prime}k}^{(\bar{\alpha}\bar{\beta})*}(-s)={{\mathfrak{b}}_{\bar{\beta}}\over{\mathfrak{b}}^{\prime}_{\bar{\alpha}}}{\theta}(\omega){\theta}(\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(-s),$
(137) $\displaystyle
B_{k^{\prime}k}^{({\alpha}{\beta})}(s)=B_{k^{\prime}k}^{(\bar{\alpha}\bar{\beta})*}(-s)={{\mathfrak{b}}_{\bar{\beta}}\over{\mathfrak{b}}^{\prime}_{\bar{\alpha}}}{\theta}(-\omega){\theta}(-\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(-s)\
.$ (138)

Collecting everything together we thus have

$\displaystyle\Sigma_{k^{\prime}k}^{{\alpha}{\beta}}(s)=\begin{cases}{{\mathfrak{b}}_{\bar{\beta}}\over{\mathfrak{b}}^{\prime}_{\bar{\alpha}}}{\theta}(\omega\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(-s),&s>0\cr{{\mathfrak{b}}_{{\beta}}\over{\mathfrak{b}}^{\prime}_{{\alpha}}}{\theta}(\omega\omega^{\prime})C_{{\alpha}k^{\prime}{\beta}k}(s),&s<0\end{cases}$
(139)

and more explicitly

$\displaystyle\Sigma_{k^{\prime}k}^{RR}(s)={\sqrt{\sinh\pi|\omega^{\prime}|}\over\sqrt{\sinh\pi|\omega|}}\begin{cases}\left[e^{{\pi\over
2}(\omega-\omega^{\prime})}{\theta}(\omega){\theta}(\omega^{\prime})+e^{-{\pi\over
2}(\omega-\omega^{\prime})}{\theta}(-\omega){\theta}(-\omega^{\prime})\right]C_{k^{\prime}k}(s)&s<0\cr\left[e^{-{\pi\over
2}(\omega-\omega^{\prime})}{\theta}(\omega){\theta}(\omega^{\prime})+e^{{\pi\over
2}(\omega-\omega^{\prime})}{\theta}(-\omega){\theta}(-\omega^{\prime})\right]C_{k^{\prime}k}(-s)&s>0\end{cases},$
(140)
$\displaystyle\Sigma_{k^{\prime}k}^{RL}(s)={\sqrt{\sinh\pi|\omega^{\prime}|}\over\sqrt{\sinh\pi|\omega|}}\begin{cases}\left[e^{-{\pi\over
2}(\omega+\omega^{\prime})}{\theta}(\omega){\theta}(\omega^{\prime})+e^{{\pi\over
2}(\omega+\omega^{\prime})}{\theta}(-\omega){\theta}(-\omega^{\prime})\right]C_{k^{\prime}-k}(s)&s<0\cr\left[e^{{\pi\over
2}(\omega+\omega^{\prime})}{\theta}(\omega){\theta}(\omega^{\prime})+e^{-{\pi\over
2}(\omega+\omega^{\prime})}{\theta}(-\omega){\theta}(-\omega^{\prime})\right]C_{k^{\prime}-k}(-s)&s>0\end{cases},$
(141)

and similarly for $\Sigma^{LR}$ and $\Sigma^{LL}$. From (109) we then have

$\displaystyle\Lambda^{RR}_{k^{\prime}k}(s)=\begin{cases}C_{k^{\prime}k}(s)&s<0\cr{\sinh\pi\omega^{\prime}\over\sinh\pi\omega}C_{k^{\prime}k}(-s)&s>0\end{cases},\quad\Lambda^{RL}_{k^{\prime}k}(s)=\begin{cases}0&s<0\cr{\sinh\pi(\omega+\omega^{\prime})\over\sinh\pi\omega}C_{k^{\prime}-k}(-s)&s>0\end{cases},$
(142)
$\displaystyle\Lambda^{LL}_{k^{\prime}k}(s)=\begin{cases}{\sinh\pi\omega^{\prime}\over\sinh\pi\omega}C_{-k^{\prime}-k}(s)&s<0\cr
C_{-k^{\prime}-k}(-s)&s>0\end{cases},\quad\Lambda^{LR}_{k^{\prime}k}(s)=\begin{cases}{\sinh\pi(\omega+\omega^{\prime})\over\sinh\pi\omega}C_{-k^{\prime}k}(s)&s<0\cr
0&s>0\end{cases}\ .$ (143)

###  VI.2 Determining the structure of $C_{kk^{\prime}}(s)$

We first collect the conditions $C_{kk^{\prime}}(s)$ should satisfy and then
show that under certain assumptions it can be completely determined up to a
phase.

The $U(1)$ property implies that for $s_{1},s_{2}<0$

$C_{kk^{\prime}}(s_{1})C_{k^{\prime}k^{\prime\prime}}(s_{2})=C_{kk^{\prime\prime}}(s_{1}+s_{2})\
.$ (144)

Since the modular operator ${{\Delta}}_{0}$ generates a translation in time
$t$, it acts on $a_{k}^{({\alpha})}$ as181818Recall that $t={\eta\over 2\pi}$
and $\omega$ is the frequency for $\eta$.

${{\Delta}}_{0}^{-it}a^{({\alpha})}_{k}{{\Delta}}_{0}^{it}=e^{-2\pi
i{\alpha}\omega t}a^{({\alpha})}_{k}$ (145)

which also implies

${{\Delta}}^{-it}_{0}c^{({\alpha})}_{k}{{\Delta}}^{it}_{0}=e^{-2\pi
i{\alpha}\omega t}c^{({\alpha})}_{k}\ .$ (146)

Acting ${{\Delta}}^{-it}_{0}$ on (108) we find from (106)

${{\Delta}}^{-it}_{0}U(s)^{\dagger}c^{({\alpha})}_{k}U(s){{\Delta}}_{0}^{it}=e^{-2\pi
i{\alpha}\omega t}\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(e^{-2\pi
t}s)c^{({\beta})}_{k^{\prime}}=\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)c^{({\beta})}_{k^{\prime}}e^{-2\pi
i{\beta}\omega^{\prime}t}$ (147)

which implies that

$e^{-2\pi i{\alpha}\omega t}\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(e^{-2\pi
t}s)=\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)e^{-2\pi
i{\beta}\omega^{\prime}t}\ .$ (148)

The above equation implies that the $s$-dependence of $\Sigma$ must have the
form

$\Sigma^{{\alpha}{\beta}}_{kk^{\prime}}(s)\propto
s^{-i({\alpha}\omega-{\beta}\omega^{\prime})}$ (149)

which in turn implies that

$C_{kk^{\prime}}(s)\propto(-s)^{-i(\omega-\omega^{\prime})}\ .$ (150)

We will make a further assumption that $C_{kk^{\prime}}$ is diagonal in other
quantum numbers, i.e.

$C_{kk^{\prime}}(s)=(-s)^{-i(\omega-\omega^{\prime})}g(k,k^{\prime}){{\delta}}_{q,q^{\prime}}$
(151)

From (118) $g$ should satisfy

$g^{*}(k,k^{\prime})=g(-k,-k^{\prime})\ .$ (152)

Equation (144) requires

$\displaystyle\int{d\omega^{\prime}\over
2\pi}(-s_{1})^{-i(\omega-\omega^{\prime})}g(k,k^{\prime})(-s_{2})^{-i(\omega^{\prime}-\omega^{\prime\prime})}g(k^{\prime},k^{\prime\prime})=(-s_{1}-s_{2})^{-i(\omega-\omega^{\prime\prime})}g(k,k^{\prime\prime})\
.$ (153)

Without loss of generality, we take $|s_{2}|>|s_{1}|$. The above equation can
then be written as

$\displaystyle\int{d\omega^{\prime}\over
2\pi}z^{-i(\omega-\omega^{\prime})}g(k,k^{\prime})g(k^{\prime},k^{\prime\prime})=(1+z)^{-i(\omega-\omega^{\prime\prime})}g(k,k^{\prime\prime}),\quad
z={|s_{1}|\over|s_{2}|}<1\ .$ (154)

To compare with the LHS, let us expand the RHS in powers of $z$

$(1+z)^{-i(\omega-\omega^{\prime\prime})}={1\over{{\Gamma}}(i(\omega-\omega^{\prime\prime}))}\sum_{n=0}^{\infty}{(-1)^{n}z^{n}\over
n!}{{\Gamma}}(i(\omega-\omega^{\prime\prime})+n)$ (155)

and equation (154) can follow if the integral on the LHS can be evaluated
using Cauchy’s theorem with poles with $\omega-\omega^{\prime}=in$. This
motivates us to consider the function

$I_{\omega\omega^{\prime}}(x)=x^{-i(\omega-\omega^{\prime})}{{\Gamma}}(i(\omega-\omega^{\prime})+\epsilon)=\int_{0}^{\infty}{dp\over
p}p^{i(\omega-\omega^{\prime})+\epsilon}e^{-px},\quad x>0,$ (156)

which satisfies

$\displaystyle\int{d\omega^{\prime}\over
2\pi}I_{\omega\omega^{\prime}}(x_{1})I_{\omega^{\prime}\omega^{\prime\prime}}(x_{2})=I_{\omega\omega^{\prime\prime}}(x_{1}+x_{2}),\quad
I_{\omega\omega^{\prime}}(0)=2\pi{{\delta}}(\omega-\omega^{\prime})\equiv{{\delta}}_{\omega\omega^{\prime}}\
.$ (157)

Equation (153) can then be satisfied if $g(k,k^{\prime})$ has the form

$g(k,k^{\prime})={\lambda(k)\over\lambda(k^{\prime})}{{\Gamma}}(i(\omega-\omega^{\prime})+\epsilon)$
(158)

which gives $C$ of the form

$C_{kk^{\prime}}(s)={\lambda(k)\over\lambda(k^{\prime})}I_{\omega\omega^{\prime}}(-s){{\delta}}_{qq^{\prime}}\
.$ (159)

Equation (152) requires

${\lambda^{*}(k)\over\lambda^{*}(k^{\prime})}={\lambda(-k)\over\lambda(-k^{\prime})}\
.$ (160)

We still need to consider the full consequences of (103). The invariance of
$\lvert{\Omega_{0}}\rangle$ under $U(s)$ requires that

${\left\langle{\Omega_{0}|\sigma_{s}(c_{k_{1}}^{\alpha})\sigma_{s}(c_{k_{2}}^{\beta})|\Omega_{0}}\right\rangle}=\Sigma^{{\alpha}{{\gamma}}}_{k_{1}k_{1}^{\prime}}(s)\Sigma^{{\beta}{{\gamma}}}_{k_{2},-k_{1}^{\prime}}(s){\theta}(\omega_{1}^{\prime})$
(161)

is independent of $s$, which leads to191919Note
${\left\langle{\Omega_{0}|c_{k}^{\alpha}c_{k^{\prime}}^{\beta}|\Omega_{0}}\right\rangle}={{\delta}}_{{\alpha}{\beta}}{{\delta}}_{k,-k^{\prime}}{\theta}(\omega)\
.$ (162)

$\Sigma^{{\alpha}{{\gamma}}}_{k_{1}k_{1}^{\prime}}(s)\Sigma^{{\beta}{{\gamma}}}_{k_{2},-k_{1}^{\prime}}(s){\theta}(\omega_{1}^{\prime})={{\delta}}_{{\alpha}{\beta}}{{\delta}}_{k_{1},-k_{2}}{\theta}(\omega_{1})\
.$ (163)

The above equations are in turn equivalent to

$C_{k_{1}k^{\prime}}(s)C_{k_{2}-k^{\prime}}(s)({\mathfrak{b}}^{\prime
2}_{+}{\theta}(\omega^{\prime})+{\mathfrak{b}}^{\prime
2}_{-}{\theta}(-\omega^{\prime}))=({\mathfrak{b}}_{1+}^{2}{\theta}(\omega_{1})+{\mathfrak{b}}_{1-}^{2}{\theta}(-\omega_{1})){{\delta}}_{k_{1},-k_{2}}$
(164)

which can be written more explicitly as

$\sum_{k^{\prime}}C_{k_{1}k^{\prime}}(s)C_{k_{2}-k^{\prime}}(s){e^{\pi\omega^{\prime}}\over
2\sinh\pi|\omega^{\prime}|}=({\mathfrak{b}}_{1+}^{2}{\theta}(\omega_{1})+{\mathfrak{b}}_{1-}^{2}{\theta}(-\omega_{1})){{\delta}}_{k_{1},-k_{2}}\
.$ (165)

Inserting (159) into (165) leads to

$\sum_{k^{\prime}}I_{\omega_{1}\omega^{\prime}}(s){\lambda(k_{1})\over\lambda(k^{\prime})}{{\delta}}_{q_{1}q^{\prime}}I_{\omega_{2}-\omega^{\prime}}(s){\lambda(k_{2})\over\lambda(-k^{\prime})}{{\delta}}_{q_{2},-q^{\prime}}{e^{\pi\omega^{\prime}}\over
2\sinh\pi|\omega^{\prime}|}={e^{\pi\omega_{1}}\over
2\sinh\pi|\omega_{1}|}{{\delta}}_{k_{1},-k_{2}}\ .$ (166)

It can be checked that

$\sum_{\omega^{\prime}}I_{\omega_{1}\omega^{\prime}}(s)I_{\omega_{2}-\omega^{\prime}}(s)e^{\pi\omega^{\prime}}={{\delta}}_{\omega_{1},-\omega_{2}}e^{\pi\omega_{1}}$
(167)

and thus (166) is satisfied if

$\lambda(k^{\prime})\lambda(-k^{\prime})={1\over 2\sinh\pi|\omega^{\prime}|}\
.$ (168)

From (160) we then have

$\lambda(k)={e^{i{{\gamma}}_{k}}\over\sqrt{2\sinh\pi|\omega|}},\quad{{\gamma}}_{-k}=-{{\gamma}}_{k}$
(169)

and

$C_{kk^{\prime}}(s)=\sqrt{{\sinh\pi|\omega^{\prime}|\over\sinh\pi|\omega|}}e^{i{{\gamma}}_{k}-i{{\gamma}}_{k^{\prime}}}I_{\omega\omega^{\prime}}(s){{\delta}}_{qq^{\prime}}\
.$ (170)

With the above form $C_{kk^{\prime}}$ has the following “transpose” property

$C_{kk^{\prime}}(s)=C_{k^{\prime}k}^{*}(s){\sinh\pi|\omega^{\prime}|\over\sinh\pi|\omega|}\
.$ (171)

With (171), (165) can be rewritten as

$\sum_{k^{\prime}}C_{k_{1}k^{\prime}}(s)C_{k^{\prime}-k_{2}}(s)e^{\pi\omega^{\prime}}=e^{\pi\omega_{1}}{{\delta}}_{k_{1},-k_{2}}\
.$ (172)

From (107) we have

$\sigma_{s_{1}}(\sigma_{s_{2}}(a_{k}^{({\alpha})}))=\sigma_{s_{1}+s_{2}}(a_{k}^{({\alpha})})$
(173)

which implies that

$\Lambda_{kk^{\prime}}^{{\alpha}{\beta}}(s_{1})\Lambda_{k^{\prime}k^{\prime\prime}}^{{\beta}{{\gamma}}}(s_{2})=\Lambda_{kk^{\prime\prime}}^{{\alpha}{{\gamma}}}(s_{1}+s_{2})\
.$ (174)

For $s_{1},s_{2}$ with the same sign, it can be shown that the group
properties (174) follw from (144). For $s_{1},s_{2}$ of opposite signs, (144)
is not enough, but (174) can be shown to follow from the more explicit form
(170). See Appendix B for details.

To conclude, equations (170) and (142)–(143) give the explicit transformation
of $a_{k}^{({\alpha})}$ and thus ${{\mathcal{O}}}(x)$ under $U(s)$ (for all
$s\in{\mathbb{{R}}}$), which satisfies all the desired properties (102)–(107)
for half-sided modular translations. Without needing any explicit information
about ${{\mathcal{N}}}$ we have determined the action of $U(s)$ up to a phase
factor $e^{i{{\gamma}}_{k}}$. Information about different choices of
${{\mathcal{N}}}$ as well as the nature of emergent time $s$ is encoded in
this phase factor.

##  VII An example: generalized free fields in Rindler spacetime

In this section we use a simple example to illustrate the formalism developed
in Sec. VI. Consider a generalized free field ${{\mathcal{O}}}(x)$ in
Minkowski spacetime and the following question: given the restrictions
${{\mathcal{O}}}_{R},{{\mathcal{O}}}_{L}$ of ${{\mathcal{O}}}$ to the $R$ and
$L$ Rindler regions (see Fig. 6 Left), is it possible to recover the behavior
of the field in the full spacetime (i.e. also in the $F$ and $P$ regions)?
Intuitively the answer is no, since a generalized free field does not satisfy
any equation of motion, so we cannot obtain the behavior of ${{\mathcal{O}}}$
in the $F$ and $P$ regions by evolving ${{\mathcal{O}}}_{R}$ and
${{\mathcal{O}}}_{L}$ from a Cauchy slice as in a standard quantum field
theory. Here we show that by using the procedure of Sec. VI we in fact can
express ${{\mathcal{O}}}$ in the $F$ and $P$ regions in terms of those in $R$
and $L$ regions.

###  VII.1 The transformation

Consider $(1+1)$-dimensional Minkowski spacetime

$\displaystyle
ds^{2}=-(dx^{0})^{2}+(dx^{1})^{2}=-dx^{+}dx^{-}=e^{2\chi}\left(-d\eta^{2}+d\chi^{2}\right)=-e^{2\chi}d\xi^{+}d\xi^{-},$
(175) $\displaystyle x^{0}=e^{\chi}\sinh{\eta},\quad
x^{1}=e^{\chi}\cosh{\eta},\quad x^{\pm}=x^{0}\pm x^{1}=\pm
e^{\pm\xi^{\pm}},\quad\xi^{\pm}=\eta\pm\chi$ (176)

where the coordinates $\xi^{\pm}$ cover the $R$ patch. There is a similar
description for the $L$ patch.

For simplicity and also for the later connection with the AdS Rindler
discussion we take ${{\mathcal{O}}}$ to be given by an operator of dimension
${{\Delta}}$ in a CFT. The expression for the two-point function of
${{\mathcal{O}}}$ is fixed by conformal symmetry including the restriction to
the $R$ region. Accordingly the mode expansion for ${{\mathcal{O}}}_{R}(\xi)$
in the $R$ region can be written as (see Appendix E)

$\displaystyle{{\mathcal{O}}}_{R}(\xi)=\int\frac{d^{2}k}{(2\pi)^{2}}\,u_{k}^{(R)}(\xi)a_{k}^{(R)}\equiv\sum_{k}u_{k}^{(R)}(\xi)a_{k}^{(R)}$
(177) $\displaystyle u^{(R)}_{k}=N_{k}e^{{{{\Delta}}\over
2}(\xi^{-}-\xi^{+})}e^{-i\omega\eta+iq\chi}=N_{k}e^{\bar{q}_{+}\xi^{-}-q_{-}\xi^{+}},\quad
N_{k}={\sqrt{\sinh\pi|\omega|}\over\sqrt{2\pi}{{\Gamma}}({{\Delta}})}\left|\Gamma\left(q_{+}\right)\Gamma\left(q_{-}\right)\right|\
$ (178) $\displaystyle k=(\omega,q),\quad k^{\pm}={{\frac{1}{2}}}(\omega\pm
q),\quad q_{\pm}={{\frac{1}{2}}}({{\Delta}}+i(\omega\pm
q)),\quad\bar{q}_{\pm}={{\frac{1}{2}}}({{\Delta}}-i(\omega\pm q))\ .$ (179)

There is an analogous mode expansion for ${{\mathcal{O}}}_{L}$ with
$u^{(L)}_{k}=u^{(R)}_{-k}$.

Taking ${{\mathcal{M}}}$ to be the algebra generated by ${{\mathcal{O}}}_{R}$
and ${{\mathcal{N}}}$ the subalgebra associated with the region $\xi^{-}<0$
(Fig. 10 Left), as discussed in Sec. V.2 the generator $G$ for half-sided
modular translation simply corresponds to a null translation $x^{-}\to
x^{-}+s$, or in terms of $\xi^{\pm}$

$\xi^{-}\to\xi_{s}^{-}=\xi^{-}-\log\left(1-se^{\xi^{-}}\right),\quad\xi^{+}\to\xi^{+}_{s}=\xi^{+},\quad
s<0\ .$ (180)

We now show how to use the formalism developed in Sec. VI to extend the action
of $U(s)$ to positive $s$ and thereby extend
${{\mathcal{O}}}_{R},{{\mathcal{O}}}_{L}$ to the $F$ and $P$ regions.

For $s<0$, we have

${{\mathcal{O}}}_{R}(\xi;s)\equiv
U^{\dagger}(s){{\mathcal{O}}}_{R}(\xi)U(s)={{\mathcal{O}}}_{R}(\xi_{s})$ (181)

which implies that

$u_{k}^{(R)}(\xi_{s})=\ u_{k^{\prime}}^{(R)}(\xi)C_{k^{\prime}k}(s),\quad s<0\
.$ (182)

We then find that

$\displaystyle C_{k^{\prime}k}(s)$ $\displaystyle=$ $\displaystyle{N_{k}\over
N_{k^{\prime}}}\int
d^{2}\xi\,e^{-ik^{\prime}\cdot\xi+ik\cdot\xi_{s}+{{{\Delta}}\over
2}\xi_{s}^{-}}={{\frac{1}{2}}}{N_{k}\over
N_{k^{\prime}}}{{\delta}}_{k^{\prime-},k^{-}}{{{\Gamma}}(\bar{q}^{\prime}_{+})\over{{\Gamma}}(\bar{q}_{+})}I_{\omega^{\prime}\omega}(-s)$
(183) $\displaystyle=$
$\displaystyle{{\frac{1}{2}}}{{\delta}}_{k^{\prime-},k^{-}}{\sqrt{\sinh\pi|\omega|}\left|\Gamma\left(q_{+}\right)\right|\over\sqrt{\sinh\pi|\omega^{\prime}|}\left|\Gamma\left(q^{\prime}_{+}\right)\right|}{{{\Gamma}}(\bar{q}^{\prime}_{+})\over{{\Gamma}}(\bar{q}_{+})}I_{\omega^{\prime}\omega}(-s)$
(184)

where function $I$ was introduced previously in (156) and the last expression
has the form of (170) with

$e^{i{{\gamma}}_{k}}={{{\Gamma}}(\bar{q}_{+})\over|{{\Gamma}}(\bar{q}_{+})|}\
.$ (185)

###  VII.2 Crossing the Rindler horizon

We now consider ${{\mathcal{O}}}_{R}(\xi;s)$ with $s>0$,

$\displaystyle{{\mathcal{O}}}_{R}(\xi;s)=\sum_{k}u_{k}^{(R)}(\xi)\sigma_{s}(a_{k}^{(R)})=\sum_{k}u_{k}^{(R{\beta})}(\xi;s)a_{k}^{({\beta})},\quad
u_{k}^{(R{\beta})}(\xi;s)\equiv
u_{k^{\prime}}^{(R)}(\xi)\Lambda_{k^{\prime}k}^{R{\beta}}(s)\ .$ (186)

From (142) we have

$\displaystyle
u^{(RR)}_{k^{\prime}}(\xi;s)=N_{k^{\prime}}{e^{\bar{q}^{\prime}_{+}\xi^{-}-q^{\prime}_{-}\xi^{+}}\over{{\Gamma}}(\bar{q}^{\prime}_{+})\sinh(\pi\omega^{\prime})}J_{1},\quad
u^{(RL)}_{k^{\prime}}(\xi;s)=N_{k^{\prime}}{e^{q^{\prime}_{+}\xi^{-}-\bar{q}^{\prime}_{-}\xi^{+}}\over\sinh(\pi\omega^{\prime}){{\Gamma}}(q^{\prime}_{+})}J_{2},$
(187) $\displaystyle J_{1}=\int{d\omega\over
2\pi}\,\sinh\pi(\omega+\omega^{\prime})\,e^{-i\omega(\xi^{-}+\log
s)}{{\Gamma}}(i\omega+\epsilon){{\Gamma}}(-i\omega+\bar{q}_{+}^{\prime}),$
(188) $\displaystyle J_{2}=\int{d\omega\over 2\pi}\,e^{-i\omega(\xi^{-}+\log
s)}\Gamma\left(-i\omega+q^{\prime}_{+}\right){{\Gamma}}(i\omega+\epsilon)\sinh\pi\omega\
.$ (189)

The integrals (188)–(189) can be evaluated using contour integration. We have
the following situations:

  1. 1.

$\xi^{-}+\log s<0$, i.e. $s<s_{0}\equiv e^{-\xi^{-}}$. In this case we can
close the contour in the upper half complex-$\omega$ plane picking up poles at

$\omega=i\left(n+\epsilon\right),\quad n=0,1,\cdots\ $ (190)

which results in

$\displaystyle
J_{1}={{\Gamma}}(\bar{q}_{+}^{\prime})\sinh\pi\omega^{\prime}\left(1-se^{\xi^{-}}\right)^{-\bar{q}_{+}^{\prime}},\qquad
J_{2}=0$ (191)

and

$u^{(RR)}_{k^{\prime}}(\xi;s)=u^{(R)}_{k^{\prime}}(\xi^{-}_{s},\xi^{+}),\quad\hat{u}^{(RL)}_{k^{\prime}}(\xi;s)=0,\quad\to\quad{{\mathcal{O}}}_{R}(\xi;s)={{\mathcal{O}}}_{R}(\xi^{-}_{s},\xi^{+})\
.$ (192)

This is the situation where the transformation (180) is still well defined and
${{\mathcal{O}}}_{R}(\xi;s)$ remains in the $R$ region.

  2. 2.

$\xi^{-}+\log s>0$, i.e. $s>s_{0}$ ore equivalently $se^{\xi^{-}}>1$. In this
case we can close the contour in the lower half complex-$\omega$ plane picking
up poles for (188)–(189) respectively at

$\omega=-i\left(n+\bar{q}_{+}^{\prime}\right),\quad\omega=-i\left(n+q_{+}^{\prime}\right),\quad
n=0,1,\cdots\ $ (193)

which results in

$\displaystyle J_{1}=(-i)\sin\pi
q^{\prime}_{-}{{\Gamma}}(\bar{q}^{\prime}_{+})\left(se^{\xi^{-}}-1\right)^{-\bar{q}^{\prime}_{+}},$
(194) $\displaystyle J_{2}=(-i)\sin\pi
q_{+}^{\prime}{{\Gamma}}(q^{\prime}_{+})\left(se^{\xi^{-}}-1\right)^{-q^{\prime}_{+}},$
(195)

and

$\displaystyle u^{(RR)}_{k}(\xi;s)=-i{\sin\pi
q_{-}\over\sinh(\pi\omega)}N_{k}(x^{-}_{s})^{-\bar{q}_{+}}(x^{+})^{-q_{-}},\quad
x_{s}^{-}=x^{-}+s,$ (196) $\displaystyle u^{(RL)}_{k}(\xi;s)=-i{\sin\pi
q_{+}\over\sinh(\pi\omega)}N_{k}(x^{-}_{s})^{-q_{+}}(x^{+})^{-\bar{q}_{-}}$
(197)

In this range of $s$, (180) becomes complex and is no longer well-defined. But
the action of $U(s)$ leads to a well-defined new transformation described by
(196)–(197) if we use Minkowski $x^{\pm}$ coordinates of the initial point.
Note $x^{-}_{s}=x^{-}+s>0$, i.e. we are now in the future region. It can be
checked that the expressions (196)–(197) precisely agree with behavior of the
CFT in the $F$ region, see the second line of (446) and (447).

Thus we see that $s_{0}$ is the “critical” value for the half-sided modular
translation to take ${{\mathcal{O}}}_{R}(\xi)$ beyond the Rindler horizon and
into the $F$ region. Crossing the Rindler is signaled by the appearance of
$a_{k}^{(L)}$ in ${{\mathcal{O}}}_{R}(\xi;s)$.

By taking ${{\mathcal{N}}}$ to be given by the region indicated on the right
of Fig. 10, we can take ${{\mathcal{O}}}_{R}$ beyond the past Rindler horizon
and into the $P$ region.

##  VIII Bulk reconstruction for AdS Rindler and BTZ revisited

We will now use the formalism developed in earlier sections to study emergent
in-falling times in a black hole geometry. In particular, we will give an
explicit construction in the boundary theory of an evolution operator for a
family of bulk in-falling observers, making manifest the boundary emergence of
the black hole horizons, the interiors, and the associated causal structure.
As an illustration, we will work with the BTZ black hole in AdS3. For
contrast, it is also instructive to see how the AdS-Rindler horizon emerges in
the boundary theory in this framework.

In this section we first review the metrics of the BTZ black hole and an AdS3
Rindler region, as well as the mode expansions of a bulk scalar field in these
geometries. We then discuss the boundary support of a bulk field in the BTZ
black hole or AdS-Rindler spacetime. This part is new and will provide an
important preparation for our discussion in Sec. X.

We will set the AdS radius to be unity throughout the paper.

###  VIII.1 AdS Rindler and BTZ geometries

Consider the Poincaré patch of AdS3

$ds^{2}={1\over z^{2}}(-(dx^{0})^{2}+(dx^{1})^{2}+dz^{2})={1\over
z^{2}}(-dx^{+}dx^{-}+dz^{2}),\quad x^{\pm}=x^{0}\pm x^{1}$ (198)

which can be separated into four different AdS Rindler regions, labeled by
${\mathcal{R}},{{\mathcal{L}}},{{\mathcal{F}}},{{\mathcal{P}}}$ on the right
of Fig. 6, corresponding respectively to regions with $(x^{+},x^{-})$ having
signs $(+,-)$, $(-,+)$, $(+,+)$, $(-,-)$. They have respectively $R,L,F,P$
Rindler regions of Minkowski spacetime ${\mathbb{{R}}}^{1,1}$ (depicted on the
left of Fig. 6) as their boundaries (i.e. as $z\to 0$). It is also convenient
to introduce the so-called BTZ coordinates $(\eta,w,\chi)$, which for the
${\mathcal{R}}$ region have the form

$z=we^{\chi},\quad x^{+}=e^{\xi^{+}}\sqrt{1-w^{2}},\quad
x^{-}=-e^{-\xi^{-}}\sqrt{1-w^{2}},\quad\xi^{\pm}=\eta\pm\chi,$ (199)

and in terms of which the metric has a “black hole” form

$ds^{2}=\frac{1}{w^{2}}\left[-\left(1-w^{2}\right)d\eta^{2}+\left(1-w^{2}\right)^{-1}dw^{2}+d\chi^{2}\right]\
.$ (200)

The AdS Rindler horizon is at $w=1$ and the boundary is at $w=0$. When $w>1$,
the metric (200) covers the part of the ${{\mathcal{F}}}$ or ${{\mathcal{P}}}$
regions with $z^{2}-x^{+}x^{-}>0$. $w=\infty$ is a coordinate singularity
beyond which we have $z^{2}-x^{+}x^{-}<0$ and the BTZ coordinates
$(\eta,w,\chi)$ no longer apply. We will refer to the parts of
${{\mathcal{F}}}$ with $z^{2}-x^{+}x^{-}>0$ and $z^{2}-x^{+}x^{-}<0$
respectively as the ${{\mathcal{F}}}_{1}$ and ${{\mathcal{F}}}_{2}$ regions.
Similarly the ${{\mathcal{P}}}$ region is split into ${{\mathcal{P}}}_{1}$ and
${{\mathcal{P}}}_{2}$.

The BTZ black hole can be obtained by making $\chi$ compact Banados:1992wn ,
in which case $w=\infty$ becomes a genuine singularity where the spacetime
ends, and $w=1$ becomes an event horizon. The black hole has inverse
temperature ${\beta}=2\pi$ corresponding to the time $\eta$. For compact
$\chi$, the Poincaré coordinates (199) can no longer be used to connect
different regions. Instead, we can introduce the Kruskal coordinates in the
${\mathcal{R}}$ region

$\displaystyle U=-e^{\zeta-\eta}=-\sqrt{1-w\over 1+w}e^{-\eta},\quad
V=e^{\zeta+\eta}=\sqrt{1-w\over 1+w}e^{\eta}\ ,$ (201)

where $\zeta$ is the tortoise coordinate

$\zeta=-\int{dw\over 1-w^{2}}={{\frac{1}{2}}}\log{1-w\over 1+w}\ .$ (202)

The event horizons lie at $U,V=0$ and the boundary lies at $UV=-1$. See Fig.
3.

Note that the Kruskal coordinates $U,V$ can also be used for AdS-Rindler, with
$U,V=0$ corresponding to the AdS-Rindler horizons and $UV=1$ now a coordinate
singularity.202020Note that in terms of the range of Kruskal coordinates
$U,V$, the AdS-Rindler ${\mathcal{R}}$ and ${{\mathcal{L}}}$ regions coincide
respectively with the $R$ and $L$ regions of the BTZ black hole, but the $F$
and $P$ regions of the BTZ black hole only cover the ${{\mathcal{F}}}_{1}$ and
${{\mathcal{P}}}_{1}$ regions of AdS-Rindler.

For more extensive discussion of the AdS-Rindler and BTZ spacetimes, see
Appendix C.

###  VIII.2 Mode expansion in AdS-Rindler and BTZ

Consider a bulk scalar field $\phi$ dual to a boundary operator
${{\mathcal{O}}}$ of dimension ${{\Delta}}$. The restriction $\phi_{R}(X)$ of
$\phi$ to the AdS Rindler ${\mathcal{R}}$ region (with $X=(\eta,w,\chi)$) or
to the $R$ region of the BTZ black hole has the same mode expansion except
that the momentum $q$ along the $\chi$ direction is continuous for AdS-Rindler
and discrete for BTZ. Below we will use the same notation for both cases.

$\phi_{R}$ can be expanded in modes as

$\displaystyle\phi_{R}(X)=\sum_{k}v_{k}^{(R)}(X)a_{k}^{(R)},\quad
v_{k}^{(R)}(X)=N_{k}f_{k}(w)e^{ik\cdot x}$ (203) $\displaystyle
k=(\omega,q),\quad k\cdot x=-\omega\eta+q\chi\quad
q_{\pm}={{\frac{1}{2}}}({{\Delta}}+i(\omega\pm
q)),\quad\bar{q}_{\pm}={{\frac{1}{2}}}({{\Delta}}-i(\omega\pm q))$ (204)
$\displaystyle
N_{k}={\sqrt{\sinh\pi|\omega|}\over\sqrt{2\pi}{{\Gamma}}({{\Delta}})}\left|\Gamma\left(q_{+}\right)\Gamma\left(q_{-}\right)\right|,\quad
f_{k}(w)=w^{\Delta}(1-w^{2})^{-i\omega/2}{}_{2}F_{1}\left(\bar{q}_{-},\bar{q}_{+};\Delta;w^{2}\right)\
.$ (205)

The $a_{k}^{(R)}$ are creation (for $\omega<0$) and annihilation (for
$\omega>0$) operators of the boundary generalized free field theory in the $R$
region, and thus $\phi_{R}(X)$ can be interpreted as an operator in the
boundary theory. There is a similar “bulk reconstruction” equation for
$\phi_{L}$ in terms of $a_{k}^{(L)}$. Note that $f_{k}(w)$ is normalizable at
infinity

$\lim_{w\to 0}f_{k}(w)=w^{{\Delta}}+\cdots,$ (206)

and satisfies

$f_{k}(w)=f^{*}_{k}(w)=f_{-k}(w)=f_{-\omega,q}(w)=f_{\omega,-q}(w)\ .$ (207)

Near the horizon, $w\to 1$, we have

$v^{(R)}_{k}(X)={1\over\sqrt{2|\omega|}}e^{ik\cdot
x}\left(e^{-i\omega\xi+i{{\delta}}_{k}}+e^{i\omega\xi-i{{\delta}}_{k}}\right)\
,$ (208)

where the phase shift ${{\delta}}_{k}$ is given by

$e^{i{{\delta}}_{k}}={{{\Gamma}}(i\omega)|{{\Gamma}}(q_{-}){{\Gamma}}(q_{+})|\over|{{\Gamma}}(i\omega)|{{\Gamma}}(q_{-}){{\Gamma}}(q_{+})}e^{-i\omega\log
2}\ .$ (209)

We also note the asymptotic behavior for $|\omega|\to\infty$

${}_{2}F_{1}\left(\bar{q}_{-},\bar{q}_{+};\Delta;w^{2}\right)\approx
w^{{{\frac{1}{2}}}-{{\Delta}}}{{{\Gamma}}({{\Delta}})\over
2\sqrt{\pi}}\begin{cases}(1-w)^{i\omega}\left(-{i\omega\over
2}\right)^{{{\frac{1}{2}}}-{{\Delta}}}&{\rm
Im}\omega>0\cr(1+w)^{i\omega}\left({i\omega\over
2}\right)^{{{\frac{1}{2}}}-{{\Delta}}}&{\rm Im}\omega<0\end{cases}$ (210)

which can be obtained from the discussion of Appendix F. As $|q|\to\infty$ we
have Morrison:2014jha

$\displaystyle
f_{k}(w)=|q|^{{{\frac{1}{2}}}-\Delta}w^{{\frac{1}{2}}}(1-w)^{-{1\over
4}}\left(e^{q\arcsin{w}}+e^{-q\arcsin{w}}\right),$ (211)
$\displaystyle|N_{k}|^{2}={2^{1-2d}\over({{\Gamma}}({{\Delta}}))^{2}}\sinh(\pi|\omega|)|q|^{2({{\Delta}}-1)}e^{-\pi|q|}\left(1+O(|q|^{-2})\right),$
(212) $\displaystyle N_{k}f_{k}(w)\sim\left(e^{-|q|({\pi\over
2}-\arcsin{w})}+e^{-|q|({\pi\over 2}+\arcsin{w})}\right)\ ,$ (213)

and similarly for $\phi_{L}$. The expressions in the $F$ and $P$ regions of
BTZ, and in the ${{\mathcal{F}}}$ and ${{\mathcal{P}}}$ regions of Poincare
AdS can be obtained from analytic continuation which we discuss in Appendix D.

The corresponding boundary operators are obtained by the extrapolate
dictionary, i.e. removing $w^{{{\Delta}}}$ and taking $w\to 0$. We find

${{\mathcal{O}}}_{R}(x)=\sum_{k}u^{(R)}_{k}(x)a^{(R)}_{k},\qquad
u^{(R)}_{k}(x)=N_{k}e^{ik\cdot x}\ ,$ (214)

and similarly for ${{\mathcal{O}}}_{L}(x)$ with
$u^{(L)}_{k}(x)=(u^{(R)}_{k}(x))^{*}$. For the AdS Rindler the boundary limit
is taken by removing $z^{{\Delta}}$, so the corresponding boundary operator in
the Rindler patch has an extra factor $e^{-{{\Delta}}\chi}$ as in (178).

###  VIII.3 Boundary support of a bulk operator

The identification of bulk and boundary oscillators $a_{k}^{({\alpha})}$
implies that $\phi_{R}$ of (203) can be regarded as a boundary operator. This
is the statement of bulk reconstruction. We will now examine the support of
$\phi_{R}(X)$ on the boundary. We will use the notation for the BTZ spacetime
and exactly the same conclusion applies to AdS-Rindler.

Consider the smearing function $K(X,y)$ defined by Hamilton:2005ju ;
Hamilton:2006fh

$\displaystyle\phi_{R}(X)=\sum_{k}N_{k}e^{ik\cdot x}f_{k}(w)a_{k}^{(R)}=\int
d^{2}y\,K(X,y){{\mathcal{O}}}_{R}(y)$ (215)
$\displaystyle{{\mathcal{O}}}_{R}(x)=\sum_{k}N_{k}e^{ik\cdot
x}a_{k}^{(R)},\quad K(X,y)=\sum_{k}e^{ik\cdot(x-y)}f_{k}(w)$ (216)

where ${{\mathcal{O}}}_{R}(x)$ is obtained by taking the boundary limit of
$\phi_{R}(X)$ (see (214) or Appendix E). From the large $q$ behavior of
$f_{k}(w)$, see (211), the $q$-integral in (216) is divergent and thus
$K(X,y)$ cannot be consistently defined as a function Morrison:2014jha . The
origin of the divergence can be traced to the complete spectrum feature
emphasized in Sec. II.3: for any $\omega$, ${{\mathcal{O}}}$ has nonzero
support for arbitrary large values of $|q|$, but this support decays
exponentially at large $|q|$.212121We expect the amplitude for creating a mode
of large $q$ with a finite $\omega$ should be proportional to
$e^{-c{\beta}|q|}$ with $c$ an $O(1)$ number. The same statements apply to all
AdS-Rindler and black hole systems in all dimensions (see Bousso:2012mh  for
other arguments from the bulk).

The divergences can be avoided if we smear $\phi_{R}(X)$ in the $\chi$
direction by a function with sufficiently soft large $q$ behavior