# Quantum Entanglement in String Theory

Atish Dabholkar

International Centre for Theoretical Physics
Strada Costiera 11, Trieste 34151 Italy

###### Abstract

We define entanglement entropy in string perturbation theory using the
orbifold method – a stringy analog of the replica method in field theory. To
this end, we use the Newton series to analytically continue in $N$ the
partition functions for string orbifolds on $\mathbb{C}/\mathbb{Z}_{N}$
conical spaces, known for all odd integer $N$. In the concrete example of ten-
dimensional Type-IIB strings, the one-loop partition function can be computed
explicitly and the one-loop entropy can be expressed as a manifestly modular
invariant series in terms of the Weierstrass $\wp$ function. The convergence
of the series is not evident but, from physical arguments based on holography,
it is expected to yield a finite answer together with the tree level
contribution. This method has a natural generalization to other string
compactifications and to higher genus Riemann surfaces; it can provide a
modular invariant definition of generalized entropy in a given string vacuum
to all orders, of potential interest for the generalized second law of
thermodynamics.

###### Keywords:

quantum entanglement, black holes, superstrings, holography

##  1 Introduction

Our aim is to give a definition of entanglement entropy in Type-II superstring
theory in Minkowski spacetime using an analytic continuation of the partition
functions of the string orbifolds constructed in Dabholkar1995c  which can be
related to Euclidean Rindler spacetime. The entropy thus defined in Rindler
spacetime has two possible physical interpretations – either as quantum
entanglement entropy or as a quantum correction to black hole entropy. The two
are expected to be related since the near horizon geometry of the two-sided
eternal Schwarzschild black hole is approximated by Rindler spacetime. If a
proper definition of Rindler entropy exists within string theory, then it
would offer a way to talk about entanglement entropy in a consistent, ultra-
violet finite theory of quantum gravity. Alternatively, it can teach us
something about the quantum corrections to the Bekenstein-Hawking entropy for
generic nonsupersymmetric black holes in a generic string theory background.
Both these interpretations are very interesting.

In the context of $AdS/CFT$ holography, the entanglement entropy in the
boundary conformal field theory to leading order is related by the Ryu-
Takayanagi entropy formula to the area of the minimal surface in the quantum
gravity dual theory in the bulk. The quantum correction to the Ryu-Takayanagi
formula Ryu2006 ; Hubeny2007  in the bulk dual has been argued to be given by
the quantum contribution to the bulk entanglement entropy Lewkowycz2013a ;
Barrella2013 ; Faulkner2013 ; Engelhardt2015 ; Jafferis2016 . It is thus
desirable to compute the entanglement entropy in string theory also from this
perspective to better understand the relation between entanglement in the bulk
and the boundary.

Entanglement entropy and related notions such as relative entropy have come to
play a central role in quantum information theory in quantum mechanics and
quantum field theory Nielsen2002 ; Preskill1998 ; Witten2018b ; Calabrese2004
; Casini2007 ; Calabrese2009  and more recently in the context of black hole
physics and holograpy Rangamani2017  and in topics at the interface such as
the proof of the Bekenstein bound Casini2008 . Finiteness of entanglement
entropy in quantum gravity is at the heart of the information paradox and
related issues in black hole physics. For example, the formulation of the
strong sub-additivity paradox for Hawking emission Mathur2005 ; Mathur2009 ;
Almheiri2013  requires a notion of entanglement in a gravitational context.
Similarly, the proof of the generalized second law of thermodynamics to one
loop order Wall2012  requires one to separate the relative entropy into the
‘entropy term’ which captures the entanglement of the modes outside the black
hole and the ‘energy term’ which enters the Raychaudhuri equation. Each of
these terms is individually ultraviolet divergent in perturbative quantum
field theory but is expected to be finite in a consistent quantum theory of
gravity such as string theory. Thus, it is clearly desirable to arrive at an
appropriate notion of entanglement entropy within string theory that is
modular invariant and finite at one-loop order and possibly to all orders in
perturbation theory.

With these motivations, we review strings in Rindler spacetime in §2 and
outline the orbifold method for computing the entanglement entropy. For
concreteness, we consider ten-dimensional Type-II strings on a conical space
with opening angle $2\pi/N$ viewed as a $\mathbb{C}/\mathbb{Z}_{N}$ orbifold,
and construct the one-loop partition function in §3. After reviewing the
Newton series in §4, we use it in §5 to analytically continue in $N$ the
partition functions of the orbifolds to obtain a series expansion for the
quantum entanglement entropy. In §6 we discuss the classical contribution to
the entanglement entropy. At tree level, the worldsheet partition function
vanishes. However, there is boundary contribution to the spacetime free energy
which can be computed using the fact that the tree level dilaton equations of
motion are satisfied exactly for the string orbifolds Dabholkar2002 . This
allows one to compute both the classical and the quantum contributions to
entanglement entropy uniformly using the orbifold method.

At one-loop order, the formula (68) for quantum entanglement entropy can be
evaluated explicitly for the ten-dimensional Type-IIB strings. It yields an
intriguing modular invariant formula as a series involving the Weierstrass
$\wp$ function. It is not evident if this series can be summed in some way,
but one expects that the one-loop entropy should be finite by the following
physical reasoning in the context of $AdS/CFT$ holography. The near horizon
geometry of a Schwarzschild black hole in anti de Sitter spacetime is well
approximated by Rindler spacetime. In the two-sided boundary conformal field
theory living on the left and the right boundary, the black hole is
represented Maldacena2001a ; VanRaamsdonk2010 ; Maldacena2013  as the
thermofield double state $|\overline{\Omega}\rangle$ in the product Hilbert
space $\overline{\mathcal{H}}_{L}\otimes\overline{\mathcal{H}}_{R}$ where
$\overline{\mathcal{H}}_{L}$ and $\overline{\mathcal{H}}_{R}$ are the left and
the right boundary Hilbert spaces respectively111We use bar to distinguish all
boundary quantities from the bulk quantities.. Tracing the pure density matrix
for the ground state over the left Hilbert space gives the thermal density
matrix $\overline{\rho}_{R}$ on the right Hilbert space:

$\overline{\rho}_{R}={\raise-5.0pt\hbox{$\buildrel\textstyle{\rm
Tr}\over{\scriptscriptstyle\overline{\mathcal{H}}_{L}}$}~{}}|\overline{\Omega}\rangle\langle\overline{\Omega}|$
(1)

The von Neumann entropy $\overline{S}$ of this density matrix is finite at
each order in a large $N$ expansion. Hence, one expects that the corresponding
bulk quantity $S$ must also be finite order by order in string perturbation
theory. A detailed analysis of the finiteness of the entropy (69) will be
presented elsewhere DabholkarWiP .

These physical considerations have a more sophisticated formulation using von
Neumann algebras of local observables which are natural from the perspective
of the boundary conformal field theory Leutheusser2021 ; Witten2021b ;
Leutheusser2021a ; Witten2021a . The algebra of local observables in the right
conformal field theory is a Type-I von Neumann algebra. This is related to the
fact that $\overline{S}$ is finite. By contrast, the bulk entanglement entropy
in the naive field theory limit of string theory is divergent corresponding to
the fact that the von Neumann algebra of observables of a local quantum field
theory is Type-III. Holography implies that in a consistent quantum string
dual in the bulk, this situation must be ameliorated leading to finite
entanglement entropy, even though there is no natural notion of algebra of
local observables in quantum gravity. We discuss this perspective in §7.

##  2 Strings on Rindler Spacetime

Consider string theory on a ten-dimensional spacetime
$\mathbb{M}_{10}=\Sigma_{8}\times\mathbb{M}_{2}$ where $\Sigma_{8}$ is an
arbitrary Euclidean eight-manifold and $\mathbb{M}_{2}$ is two-dimensional
Minkowski spacetime with metric

$ds^{2}=-dT^{2}+dX^{2}\,,$ (2)

in the usual Minkowski coordinates $T,X$ with $-\infty<T<\infty$ and
$-\infty<X<\infty$. A uniformly accelerated observer in $\mathbb{M}_{2}$ uses
instead the Rindler coordinates $(t,r)$ related to the Minkowski coordinates
by

$T=r\cosh t\,,\quad X=r\sinh t\,$ (3)

with $-\infty<t<\infty$ and $0\leq r<\infty$, which cover only the right
diamond in Minkowski space. In these coordinates, the metric takes the form

$ds^{2}=-r^{2}dt^{2}+dr^{2}\,.$ (4)

Euclidean Rindler space is obtained by the Wick rotation $t=-i\theta$. The
Euclidean Rindler metric is thus identical to the flat Euclidean metric in
polar coordinates

$ds^{2}=r^{2}d\theta^{2}+dr^{2}\,\qquad 0\leq\theta\leq 2\pi\,.$ (5)

The metric is smooth at the origin if the angular coordinate has periodicity
$2\pi$.

As usual, periodicity of the Euclidean time $\theta$ is interpreted as the
inverse temperature. This implies that in the Minkowski vacuum, an observer
with unit proper acceleration sees a thermal bath of strings at inverse
Rindler temperature $2\pi$. The spacetime partition function is given by

$\hat{Z}=\mbox{Tr}\exp\left({{-2\pi H_{R}}}\right)\,.$ (6)

where $H_{R}$ is the Rindler Hamiltonian which generates the translations of
Rindler time $t$. The partition function can be represented formally as a
string field theory path integral on $\Sigma_{8}\times\mathbb{R}_{2}$ where
$\mathbb{R}_{2}$ is the flat two-dimensional Euclidean space.

The thermal entropy $S$ of the Rindler heat bath is given, as usual, by a
derivative of the partition function. Thus, to compute the Rindler entropy we
need to consider the partition function at inverse temperature $\beta$
slightly away from $2\pi$ to

$\displaystyle\beta:=2\pi-{\delta}:=\frac{2\pi}{N}\,.$ (7)

where $N\geq 1$ is continuous parameter and $N=1$ corresponds to flat space.
The partition function

$\hat{Z}(N):=\mbox{Tr}\left[\exp\left({-\frac{2\pi}{N}H_{R}}\right)\right]$
(8)

is then represented as a Euclidean path integral on
$\Sigma_{8}\times\mathbb{K}_{N}$ where $\mathbb{K}_{N}$ is a conical space
with Euclidean metric

$\displaystyle ds^{2}=r^{2}d\theta^{2}+dr^{2}\,\qquad
0\leq\theta\leq\frac{2\pi}{N}\,.$ (9)

The Rindler entropy is then given by

$\displaystyle
S=-\beta\frac{\partial(\log\hat{Z})}{\partial\beta}+\log\hat{Z}.$ (10)

evaluated at the Rindler temperature, or equivalently by

$\displaystyle S=\frac{d(N\,\log{\hat{Z}(N)})}{dN}\bigg{|}_{N=1}.$ (11)

The range of the angular coordinate in (9) implies a deficit angle
$\delta_{N}$

$\displaystyle\qquad\delta_{N}=2\pi(1-\frac{1}{N})\,$ (12)

and a conical curvature singularity at the origin. The Ricci scalar is given
by a delta function

$\displaystyle R(x)=4\pi(1-\frac{1}{N})\,\delta^{(2)}(x)\,.$ (13)

A conical space with a curvature singularity is not Ricci flat and Einstein
equations are not satisfied unless there is a delta-function energy source at
the tip of the cone. It would thus seem that string equations of motion would
also not be satisfied. If this were the case, then it would be necessary to
use an off-shell formulation of string theory to compute the partition
function and it would be a considerably difficult problem to make sense of
Rindler entropy in string theory.

One can instead take an indirect route Dabholkar1995c  using the fact that
string propagation on conical spaces $\mathbb{K}_{N}$ with deficit angle given
by (12) for arbitrary odd integer $N\geq 1$ is well defined. In this case, one
can regard $\mathbb{K}_{N}$ as a $\mathbb{Z}_{N}$ orbifold Dixon1985  of the
Euclidean space $\mathbb{R}_{2}$ or equivalently of complex plane
$\mathbb{C}$. String theory on these $\mathbb{Z}_{N}$ orbifolds can be
constructed using worldsheet orbifold techniques. Conformal invariance of the
worldsheet sigma model implies, somewhat surprisingly, that the classical
string equations of motion are satisfied Dabholkar2002 .

Given $\hat{Z}(N)$ for all integer values of $N\geq 1$, one can try to find an
analytic continuation valid in the complex $N$ plane. We will assume certain
analyticity conditions required by Carlson’s theorem Boas1954  so that the
resulting analytic continuation is unique. Given an analytic expression for
$\hat{Z}(N)$, using (69) one can then find the Rindler entropy. This is the
approach we will follow. For earlier work see He2015 ; Wittenb .

The orbifold method of analytically continuing from deficit opening angles
$2\pi/N$ is a variant of the replica method Mezard1986  of analytically
continuing from surplus opening angles $2\pi\mathcal{N}$ for an integer
$\mathcal{N}\geq 1$. See Wittenb  for a detailed discussion of the relation
betweeen the replica method and the orbifold method with $\mathcal{N}=1/N$. In
string theory, an obvious advantage of the orbifold method as mentioned above
is that the orbifolds are saddle points of the string field theory path
integral and the orbifold partition functions can be constructed relatively
easily.

It is convenient for later discussion to write the entropy as

$S=S^{(0)}+S_{q}$ (14)

where $S^{(0)}$ is the classical tree-level contribution and $S_{q}$ is the
quantum loop contribution from higher genus Riemann surfaces with $g\geq 1$.

At the quantum level, the spacetime partition function $\hat{Z}_{q}(N)$ of the
$\mathbb{Z}_{N}$ orbifold theory is related to the worldsheet partition
function $Z_{q}(N)$ by the relation

$\log(\hat{Z}_{q}(N))=Z_{q}(N)\,.$ (15)

with a genus expansion

$Z_{q}(N)=\sum_{g=1}^{\infty}Z^{(g)}(N)\,.$ (16)

The quantum entanglement entropy then has the corresponding expansion

$S_{q}=\frac{\partial}{\partial
N}\left(NZ_{q}(N)\right)\bigg{|}_{N=1}\,=\sum_{g=1}^{\infty}S^{(g)}.$ (17)

At the classical level, the relation (15) does not hold. The worldsheet
partition function vanishes on the conical backgrounds. On the other hand, one
can argue that the logarithm of the the classical partition function
$\hat{Z}^{(0)}(N)$ does not vanish and receives a nonzero boundary
contribution. One can show Dabholkar2002  by analytically continuing
$\log(\hat{Z}^{(0)}(N))$ in $N$ that $S^{(0)}$ is simply the Bekenstein-
Hawking entropy:

$S^{(0)}=\frac{A}{4G}$ (18)

Thus, the total entanglement entropy in string theory is given by

${S}=\sum_{g=0}^{\infty}S^{(g)}\,=\frac{A}{4G}+S_{q}\,.$ (19)

As mentioned earlier, the quantum entanglement entropy $S_{q}$ admits two
physical interpretations which a priori are distinct.

###  $\bullet$ Quantum Entanglement Entropy

One can imagine dividing space at $X=0$ into left and right sides separated by
the surface $\Sigma_{8}$. The Rindler Hamilotonian can be interpreted in the
Tomita-Takesaki theory Takesaki1970 ; Borchers2000  as the modular Hamiltonian
corresponding to the ‘cyclic and separating’ Minkowski vacuum state for the
algebra of local observables restricted to the right side. See Witten2018  for
a review and relevant references. The unnormalized density matrix $\rho$
obtained by tracing over the left wedge is then given formally by the thermal
density matrix for the Rindler Hamiltonian:

$\rho=\exp\left({{-2\pi H_{R}}}\right)\,.$ (20)

The partition function (8) can therefore be viewed as the Renyi entropy

$\mbox{Tr}(\rho^{\mathcal{N}})$ (21)

with $\mathcal{N}=1/N$ if it can be appropriately analytically continued to
integer $\mathcal{N}$. The Rindler entropy can thus be identified with the von
Neumann entropy of the thermal density matrix or equivalently with the
entanglement entropy associated with the surface $\Sigma_{8}$.

In quantum gravity one cannot define an algebra of local observables and it is
not clear how to define notions like modular Hamiltonian and entanglement
entropy. However, in perturbation theory around a fixed background metric, one
might hope to be able to give meaning to these concepts in the same way as in
a local quantum field theory. Moreover, given the ultra-violet finiteness of
the theory, Rindler entropy is expected to be better defined in string theory.

It has been argued that the field-theoretic divergences in the entanglement
entropy Susskind1994 ; Kabat1995  can be absorbed into the Einstein-Hilbert
action as the renormalization of Newton’s constant. The quantum effective
action for massless fields cannot be summarized entirely by local Wilsonian
terms and includes highly nonlocal terms in addition to the Einstein-Hilbert
term. One thus expects nonzero but finite contribution to the entropy even
after the renormalization of Newton’s constant has been taken into account. In
other words, there is no reason in general why finite piece in the
renormalization of Newton’s constant will equal the finite piece in the
entanglement entropy. Holographic considerations in §7 also lead one to expect
finite but nonzero contributions to the entanglement entropy order by order in
string perturbation theory.

###  $\bullet$ Quantum Black Hole Entropy

In black hole spacetimes with bifurcate horizons, the near horizon geometry of
the spacetime transverse to the horizon is a Rindler spacetime. Rindler
temperature is red-shifted to the inverse Hawking temperature $\beta_{H}={8\pi
GM}$ for the asymptotic observer. Rindler entropy can then be interpreted as
the quantum correction to the black hole entropy in the limit of infinite
horizon area Sorkin2014 ; Bombelli1986 ; Srednicki1993 ; Susskind1994 ;
Callan1994a ; Solodukhin1995 ; Fursaev1996 ; Solodukhin2011 ; Frolov1996 . To
be concrete, one can imagine computing the entropy of a four-dimensional
finite area black hole by performing the string field theory path integral in
the black hole background. In this case, one would expect an expansion for the
entropy of the form

$S=\frac{A}{l_{s}^{2}g^{2}}(\frac{1}{4}+a_{1}g^{2}+\ldots)+\log(A/l_{s}^{2})(b_{1}g^{2}+\ldots)+\frac{l_{s}^{2}}{A}(c_{0}+c_{1}g^{2}+\ldots)+\ldots$
(22)

where $l_{s}$ is the string length and $g^{2}$ is the four-dimensional string
coupling. The term logarithmic in $A$ arises from the loops of massless
particles Sen2012  and the terms suppressed by inverse powers of $A$
correspond to contributions to the Wald entropy from higher derivative terms.
In the Wilsonian effective action, such terms would arise from integrating out
massive string states including the string-loop corrections. In the large $A$
limit, the term proportional to $A$ will dominate. Our one-loop computation
could then be interpreted as a computation of the coefficient $a_{1}$. In
field theory, $a_{1}$ is ultraviolet divergent, but string theory is expected
to provide a natural UV cutoff to yield a finite answer.

##  3 Type II Strings on a Cone

As a concrete example of the considerations in the previous section, we
consider Type-IIB string on $\mathbb{M}_{8}\times\mathbb{K}_{N}$ where
$\mathbb{M}_{8}$ is $7+1$ dimensional Minkowski spacetime. We start with the
Green-Schwarz string on $\mathbb{M}_{8}\times\mathbb{C}$ and fix the light-
cone gauge by using two of the directions in $\mathbb{M}_{8}$ to obtain a
sigma model in flat space $\mathbb{R}_{6}\times\mathbb{C}$. The Green-Schwarz
Lagrangian is given by

${\cal
L}=-\frac{1}{\pi}\partial_{+}X^{i}\partial_{-}X^{i}-\frac{i}{\pi}S^{a}\partial_{+}S^{a}-\frac{i}{\pi}{\tilde{S}}^{\dot{a}}\partial_{-}{\tilde{S}}^{\dot{a}}\
,$ (23)

where the coordinates $X^{i}$ transform in the vector (${\bf 8}_{v}$)
representation of $SO(8)$ and $S^{a}$ and ${\tilde{S}}^{\dot{a}}$ transform
respectively in the spinor (${\bf 8}_{s}$) and conjugate spinor (${\bf 8}_{c}$
) representations of $Spin(8)$.

The orbifold group $\mathbb{Z}_{N}$ is a subgroup of planar rotations of the
complex plane $\mathbb{C}$, so we use the decomposition $Spin(8)\supset
Spin(6)\times Spin(2)$, or equivalently $Spin(8)\supset SU(4)\times U(1)$. The
Euclidean Rindler Hamiltonian is simply the generator $J$ of the $U(1)$
symmetry. The vector and the spinor representations then decompose as follows:

$\displaystyle{\bf 8}_{v}$ $\displaystyle\rightarrow$ $\displaystyle{\bf
6}(0)+{\bf 1}(1)+{\bf 1}(-1)$ (24) $\displaystyle{\bf 8}_{s}$
$\displaystyle\rightarrow$ $\displaystyle{\bf
4}(\frac{1}{2})+{\bf{\overline{4}}}(-\frac{1}{2})$ (25) $\displaystyle{\bf
8}_{c}$ $\displaystyle\rightarrow$ $\displaystyle{\bf
4}(-\frac{1}{2})+{\bf{\overline{4}}}(\frac{1}{2})$ (26)

Here the numbers in the parentheses are the $U(1)$ charges.

The fields charged under the $U(1)$ consist of one complex boson $X$ with
charge $1$, eight fermions $S^{m},{\tilde{S}}^{m}$ with charge $\frac{1}{2}$,
and their complex conjugates. The index $m$ transforms in the ${\bf 4}$ of
$SU(4)$. These are the only fields that get twisted by the orbifold group.
With the above conventions, the orbifold group
$\mathbb{Z}_{N}=\\{1,g,\ldots,g^{N-1}\\}$ is generated by222The $N$-th power
of the naive choice $g:=\exp{\frac{2\pi iJ}{N}}$ gives $(-1)^{F}$, where $F$
is the spacetime fermion number. We want $g^{N}=1$ even while acting on the
fermions. Otherwise, one would obtain a $\mathbb{Z}_{2N}$ orbifold which can
also be viewed as a $\mathbb{Z}_{N}$ orbifold of a $\mathbb{Z}_{2}$ orbifold.
The $\mathbb{Z}_{2}$ orbifold has ten-dimensional tachyons and corresponds to
a different vacuum of string theory denoted as Type-0 theory. We could
equivalently consider $g:=\exp{\frac{2\pi iJ}{N}}(-1)^{F}$ as the generator,
but it only relabels the sectors since $N$ is odd.

$g:=\exp{\frac{4\pi iJ}{N}}$ (27)

The orbifold partition function for these fields is of the form

$\mathcal{Z}(\tau,\overline{\tau};N)=\frac{1}{N}\sum_{k,l=0}^{N-1}\mathcal{Z}_{k,l}(\tau,\bar{\tau};N)\,.$
(28)

Each term $\mathcal{Z}_{k,l}$ is a partition function for the fields
$X,S^{m},{\tilde{S}}^{m}$ and their complex conjugates with twisted boundary
conditions

$\displaystyle S^{m}(\sigma_{1}+1\,,\,\sigma_{2})$
$\displaystyle=e^{\frac{2\pi ik}{N}}\,S^{m}(\sigma_{1}\,,\,\sigma_{2})\,,\quad
S^{m}(\sigma_{1}\,,\,\sigma_{2}+1)$ $\displaystyle=e^{\frac{2\pi
il}{N}}\,S^{m}(\sigma_{1}\,,\,\sigma_{2})\,$
$\displaystyle{\tilde{S}}^{m}(\sigma_{1}+1\,,\,\sigma_{2})$
$\displaystyle=e^{-\frac{2\pi
ik}{N}}\,{\tilde{S}}^{m}(\sigma_{1}\,,\,\sigma_{2})\,,\quad{\tilde{S}}^{m}(\sigma_{1}\,,\,\sigma_{2}+1)$
$\displaystyle=e^{-\frac{2\pi
il}{N}}\,{\tilde{S}}^{m}(\sigma_{1}\,,\,\sigma_{2})\,,$ $\displaystyle
X(\sigma_{1}+1\,,\,\sigma_{2})$ $\displaystyle=e^{\frac{4\pi
ik}{N}}\,X(\sigma_{1}\,,\,\sigma_{2})\,,\quad X(\sigma_{1}\,,\,\sigma_{2}+1)$
$\displaystyle=e^{\frac{4\pi il}{N}}\,X(\sigma_{1}\,,\,\sigma_{2})\,.$ (29)

Comparing these boundary conditions in Alvarez-Gaume1986  we see that the
fermions are twisted by

$a=\frac{k}{N}+\frac{1}{2}\quad and\quad b=\frac{l}{N}+\frac{1}{2}$ (30)

and the bosons are twisted by

$a=\frac{2k}{N}+\frac{1}{2}\quad and\quad b=\frac{2l}{N}+\frac{1}{2}\,.$ (31)

Given these boundary conditions, we can write down the one-loop partition
function for the charged coordinates by inspection or by explicit oscillator
computation Alvarez-Gaume1986 :

$\displaystyle\mathcal{Z}(\tau,\overline{\tau};N)=\sum_{k,l=0}^{N-1}\,\Bigg{|}\frac{\vartheta^{4}\bigg{[}{{\genfrac{}{}{0.0pt}{}{{\frac{k}{\scriptscriptstyle
N}+\frac{1}{2}}}{{\frac{l}{\scriptscriptstyle
N}+\frac{1}{2}}}}\bigg{]}}(0|\tau)}{\eta^{3}(\tau)\vartheta\bigg{[}{{\genfrac{}{}{0.0pt}{}{\frac{2k}{\scriptscriptstyle
N}+\frac{1}{2}}{\frac{2l}{\scriptscriptstyle
N}+\frac{1}{2}}}\bigg{]}}(0|\tau)}\Bigg{|}^{2}\,.$ (32)

The total one-loop partition function including the six uncharged bosonic
coordinates is

$Z^{(1)}(N)=\frac{A_{H}}{N}\int\frac{d^{2}\tau}{\tau_{2}^{5}}\big{|}\frac{1}{\eta^{6}(\tau)}\big{|}^{2}\mathcal{Z}(\tau,\bar{\tau};N)$
(33)

where $A_{H}$ is the regularized horizon area measured in string units

$A_{H}:=\frac{V_{8}}{(2\pi l_{s})^{8}}\,,\qquad\qquad
l_{s}^{2}=\alpha^{\prime}.$ (34)

The superscript indicates that we are computing the one-loop partition
function.

The theta functions with characteristics can be related to the more familiar
Jacobi theta function by the relation

$\vartheta\bigg{[}\genfrac{}{}{0.0pt}{}{a+\frac{1}{2}}{b+\frac{1}{2}}\bigg{]}(\tau)=e^{\pi
i\,a^{2}\tau+2\pi i\,a\,(b+\frac{1}{2})}\,\vartheta(a\tau+b|\tau)$ (35)

where the Jacobi theta function has the product representation

$\vartheta(z|\tau)=-2q^{\frac{1}{8}}\,\sin{\pi
z}\prod_{n=1}^{\infty}(1-q^{n})(1-q^{n}y)\,(1-q^{n}y^{-1})\ .$ (36)

Using (35) we can write

$\frac{1}{\eta^{9}(\tau)}\,\frac{\vartheta^{4}\bigg{[}{{\genfrac{}{}{0.0pt}{}{{\frac{k}{\scriptscriptstyle
N}+\frac{1}{2}}}{{\frac{l}{\scriptscriptstyle
N}+\frac{1}{2}}}}\bigg{]}}(0|\tau)}{\vartheta\bigg{[}{{\genfrac{}{}{0.0pt}{}{\frac{2k}{\scriptscriptstyle
N}+\frac{1}{2}}{\frac{2l}{\scriptscriptstyle
N}+\frac{1}{2}}}\bigg{]}}(0|\tau)}\,=\,\frac{1}{\eta^{9}(\tau)}\,\frac{\vartheta^{4}(\frac{k\tau}{N}+\frac{l}{N}|\tau)}{\vartheta(\frac{2k\tau}{N}+\frac{2l}{N}|\tau)}$
(37)

It is convenient to define

$G(z|\tau)\,:=\,\frac{1}{\eta^{9}(\tau)}\,\frac{\vartheta^{4}(z|\tau)}{\vartheta(2z|\tau)}$
(38)

The $G$-function so defined is a doubly periodic and odd function on the
torus:

$G(z|\tau)=G(z+1|\tau)=G(z+\tau|\tau)=-G(-z|\tau)\,$ (39)

and transforms like a modular form of weight $-3$:

$G(-\frac{z}{\tau}|-\frac{1}{\tau})=(-\tau)^{-3}\,G(z|\tau)\,\,.$ (40)

The partition function can then be written compactly as

$Z^{(1)}(N)=\frac{A_{H}}{N}\int\frac{d^{2}\tau}{\tau_{2}^{5}}\,\sum_{k,l=0}^{N-1}\,\bigg{|}G(\frac{k}{N}\tau+\frac{l}{N}\,|\tau)\bigg{|}^{2}$
(41)

The function $G(z|\tau)$ is a meromorphic function over the elliptic curve or
the genus-one Riemann surface $E(\tau)$ with the complex structure parameter
$\tau$. The elliptic curve can be thought of as the coset
$\mathbb{C}/\mathbb{Z}\tau+\mathbb{Z}$ of the complex plane $\mathbb{C}$ with
coordinate $z$, modded out by the lattice $\mathbb{Z}\tau+\mathbb{Z}$. Since
the Jacobi theta function has a simple zero at all lattice points, it is easy
to see from (38) that $G(z|\tau)$ has triple zeroes at all lattice points
$z=\lambda\tau+\mu$ for $\lambda,\tau\in\mathbb{Z}$ and simple poles at the
half lattice points. Knowing all its poles and zeroes, $G(z|\tau)$ can be seen
to be Wittenb  proportional to the inverse of the derivative of the
Weierstrass $\wp$-function:

$G(z|\tau)=-i\frac{(2\pi)^{3}}{\wp\textquoteright(z,\tau)}\,.$ (42)

Recall Chandrasekharan1985  that the $\wp$ function is defined by the lattice
sum

$\wp(z,\tau):=\sum_{n,m\in\mathbb{Z}}\frac{1}{\left(z-m\tau-n\right)^{2}}\,-\sideset{}{{}^{\prime}}{\sum}_{n,m\in\mathbb{Z}}\frac{1}{\left(m\tau+n\right)^{2}}\,.$
(43)

where the primed sum excludes the lattice point $n=m=0$. Its derivative

$\wp^{\prime}(z,\tau):=\frac{d}{dz}\wp(z,\tau)=-2\sum_{n,m\mathbb{Z}}\frac{1}{\left(z-m\tau-n\right)^{3}}$
(44)

is an odd function in $z$. For fixed $\tau$, it has triple poles at all
lattice points and single zeros at all half lattice points. Let
$x=\wp(z,\tau)$ and $y=\wp^{\prime}(z,\tau)$; then they satisfy the cubic
equation

$y^{2}=4x^{3}-g_{4}(\tau)x-g_{6}(\tau)\,,$ (45)

defining an elliptic curve with complex structure parameter $\tau$, where
$g_{4}$ and $g_{6}$ are proportional to the Eisenstein series:

$\displaystyle
g_{4}(\tau)=60\sum_{n,m\in\mathbb{Z}}\frac{1}{\left(m\tau+n\right)^{4}}\,,\qquad\qquad
g_{6}(\tau)=140\sum_{n,m\in\mathbb{Z}}\frac{1}{\left(m\tau+n\right)^{6}}\,.$
(46)

##  4 Newton Series

Newton series is an analog in finite calculus of the Taylor series in
infinitesimal calculus. For any polynomial function and many analytic
functions, it provides an exact series expansion. It is thus a natural
interpolation to consider for our purpose of analytically continuing in $N$.

Recall that for an analytic function $f(x)$, its Taylor series around $x=0$ is
given by

$f(x)=\sum_{r=0}^{\infty}\frac{D^{r}[f](0)}{r!}x^{r}\,,$ (47)

where

$D^{r}[f](0):=\frac{d^{r}f}{dx^{r}}(0)\equiv f^{(r)}(0)\,$ (48)

is the $r$-th derivative of the function evaluated at the origin. This
expansion is valid in the domain of analyticity around the origin. We are
interested in finding the function $f(x)$ for non-integer values of $x$ when
we do not know these derivatives at the origin but know only the values
$\\{f(n)\\}$ of the function for all nonnegative integers $\\{n\\}$. One would
like to construct an analytic function from this discrete data much as the
Taylor series constructs an analytic function from the discrete data
$\\{f^{(r)}(0)\\}$.

To find such an interpolation, the Newton series replaces the derivatives by a
natural definition of finite differences:

$\Delta^{r}[f](0)=\sum_{s=0}^{r}(-1)^{r-s}\frac{(r)_{s}}{s!}f(s)$ (49)

where $(r)_{s}$ is the $s$-th falling power of r:

$(r)_{s}:=r(r-1)(r-2)\ldots(r-s+1)=\frac{\Gamma(r+1)}{\Gamma(s+1)}$ (50)

The Newton series is then defined by

$f(x):=\sum_{r=o}^{\infty}\frac{\Delta^{r}[f](0)}{r!}\,(x)_{r}\,.$ (51)

with obvious formal analogy with the Taylor series. By construction, $f(x)$
thus defined takes values $f(n)$ at the non-negative integers $n$ but now
interpolates for non-integer $n$. Convergence of the Newton series is in
general not guaranteed. If the Newton series converges, then Carlson theorem
Boas1954  provides the necessary and sufficient conditions for the uniqueness
of the resulting function $f(x)$.

It is convenient to introduce an operator notation which makes various aspects
of the Newton Series more transparent, especially the appearance of binomial
coefficients in (49). Let $\mathbb{T}_{h}$ be an operator of step size $h$
acting on functions as

$\mathbb{T}_{h}[f](x)=f(x+h)\,.$ (52)

Let $\mathbb{I}$ be the identity operator. Then the difference operator of
step size $h$ can be defined by

$\Delta_{h}[f](x)=(\mathbb{T}-\mathbb{I})[f](x)=f(x+h)-f(x)$ (53)

In finite calculus, the difference operator above is a natural generalization
of the quantity $hD_{h}[f]$ in infinitesimal calculus where one takes the
limit $h\rightarrow 0$ to define the differential operator $D[f]$. Powers of
$\Delta_{h}$ can then be defined by the operator equation

$\Delta^{r}_{h}[f]=(\mathbb{T}_{h}-\mathbb{I})^{r}[f]$ (54)

For our purposes, the step size is unity, $h=1$, and in that case we drop the
subscript. The definition (49) then follows by binomial expansion. It is also
easier to see in this formalism that $f(x)$ defined by Newton series (51)
takes values $f(n)$ at $x=n$. If we substitute $x=n$ on the right hand side of
(51) then the infinite series terminates at $r=n$ and one obtains

$\sum_{r=0}^{n}\frac{\Delta^{r}[f](0)}{r!}\,(n)_{r}=\sum_{r=0}^{n}\frac{n!}{r!(n-r)!}\,\Delta^{r}[f](0)$
(55)

which we recognize as the binomial expansion of $(\mathbb{I}+\Delta)^{n}[f]$.
Hence the Newton series gives

$f(x)\big{|}_{n}:=(\mathbb{I}+\Delta)^{n}[f](0)=\mathbb{T}^{n}[f](0)=f(n)\,$
(56)

for all integer values $x=n$ as required. For more on calculus of finite
differences see Milne-Thomson .

This formulation also allows one to write the entropy formula in a more
succinct form. The expression of the entropy takes the form

$S_{q}=\frac{1}{2}\sum_{r=1}^{\infty}\frac{(-1)^{r+1}}{r}\Delta^{r}[f](0)\,.$
(57)

We recognize the infinite series as the Taylor expansion for
$\log(\mathbb{I}+\Delta)$. Therefore, one can write

$S_{q}=\frac{1}{2}\log(\mathbb{T})[f](0)$ (58)

with $f(n):=Z(2n+1)$.

To understand the relation between the differential operator $D$ and the
difference operator $\Delta$ more generally, we Taylor expand (49) in $h$ and
set $h=1$, to obtain

$\Delta[f]=\sum_{1}^{\infty}\frac{D^{r}}{r!}[f]=(e^{D}-\mathbb{I})[f]\,,$ (59)

or equivalently

$D[f]=\log(\mathbb{I}+\Delta)=\log(\mathbb{T})[f]\,.$ (60)

As an operator equation it holds for polynomials and for functions for which
the Newton series converges. The expression for the entropy above then
follows.

##  5 Quantum Entanglement Entropy in String Theory

To obtain the entanglement entropy we apply the Newton interpolation formula
to the function

$f(n):=Z(2n+1)\qquad n=0,1,2,\ldots\,.$ (61)

to obtain the function for real $x$ near the origin. This enables us to
express the derivatives at the origin in terms of the values $\\{f(n)\\}$.
Quantum entanglement entropy is then given by

$\displaystyle S_{q}$ $\displaystyle=$
$\displaystyle\frac{d(N\,Z(N)}{dN}\bigg{|}_{N=1}=\frac{d\left((2n+1)\,f(n)\right)}{d(2n+1)}\bigg{|}_{n=0}$
(62) $\displaystyle=$ $\displaystyle f(0)+\frac{1}{2}f\textquoteright(0)\,$
(63)

where $f\textquoteright(n)$ is the derivative with respect $x$ evaluated at
$n$. Assuming the convergence of the Newton series, the derivative of the
interpolated function is given by

$f\textquoteright(x):=\sum_{r=o}^{\infty}\frac{\Delta^{r}[f](0)}{r!}\,\frac{d(x)_{r}}{dx}\,.$
(64)

Since we are only interested in the derivative at the origin we note that

$\displaystyle\frac{d(x)_{r}}{dx}\big{|}_{x=0}$ $\displaystyle=$
$\displaystyle\frac{d}{dx}\left[x(x-1)(x-2)\ldots(x+1-r)\right]$ (65)
$\displaystyle=$ $\displaystyle(-1)(-2)(-3)\ldots(-(r-1))=(-1)^{r-1}(r-1)!\,.$
(66)

Using the fact that $f(0)=Z(1)=0$ we obtain

$S_{q}=\frac{1}{2}\sum_{r=1}^{\infty}\frac{(-1)^{r-1}}{r}\Delta^{r}[f](0)\,.$
(67)

Substituting for $\Delta^{r}[f]$, we obtain

$S_{q}=-\frac{1}{2}\,\sum_{r=1}^{\infty}\frac{1}{r}\sum_{s=0}^{r}\,(-1)^{s}\,\frac{(r)_{s}}{s!}\,Z(2s+1)\,.$
(68)

We have thus obtained an explicit expression for entanglement entropy as a sum
over the values of partition functions of orbifolds. This formula is valid for
any compactification and at any genus as long as the partition functions are
known. Since the partition functions are manifestly modular invariant, the
resulting entropy is also modular invariant.

For concreteness, one can apply these general considerations to the orbifolds
of ten-dimensional Type-IIB string constructed earlier. in this case, at one-
loop order one can write an explicit expression for the entropy using the
partition function (41) as an integral over the moduli space:

$S^{(1)}=A_{H}\int\frac{d^{2}\tau}{\tau_{2}^{2}}\,\mathcal{S}^{(1)}(\tau)$
(69)

where the entropy-density $\mathcal{S}^{(1)}(\tau)$ is a nonholomorphic
modular function over the moduli space:

$\mathcal{S}^{(1)}(\tau)=-\frac{(2\pi)^{6}}{2\tau_{2}^{3}}\,\sum_{r=1}^{\infty}\frac{1}{r}\,\sum_{s=0}^{r}\,(-1)^{s}\frac{(r)_{s}}{s!}\,\frac{1}{(2s+1)}\,\sum_{k,l=0}^{2s}\,\bigg{|}\frac{1}{\wp\textquoteright(\frac{k}{2s+1}\tau+\frac{l}{2s+1},\tau)}\bigg{|}^{2}\,.$
(70)

where we have substituted for $G(z|\tau)$ in terms of the Weierstrass $\wp$
function using (42).

It is not known _a priori_ if this series is convergent, divergent, or
asymptotic. It would be interesting to examine if it can be summed explicitly,
perhaps with appropriate regularization, or using other methods like Borel
summation to extract the physical implications. Since each of the partition
functions has a tachyonic divergence in the integral over moduli space, it
appears that the Rindler entropy would also inherit this divergence. However,
one may hope that, first performing the sum and then performing the modular
integral might ameliorate the divergence as is indicated in the closed string
channel in the analysis of open strings in Rindler spacetime Wittenb . As
mentioned in the introduction, there are physical reasons to expect that the
entanglement entropy in string theory should be finite. We explain the
reasoning in §7.

##  6 Classical Entanglement Entropy in String Theory

Its natural to ask if the tree-level classical contribution corresponding to
the Bekenstein-Hawking entropy should be added by hand to obtain the total
entanglement entropy (19) or if it can also be computed directly by the
orbifold method. This was investigated in Dabholkar2002  which we review
below. We will see that both the classical and quantum contributions to
entanglement can be treated uniformly using the orbifold method by analytic
continuation of spacetime free energy for the orbifolds including appropriate
boundary terms at tree level and using the fact that the orbifolds satisfy the
classical string equations of motion exactly.

If the naive relation (15) between the spacetime and worldsheet partition
functions were valid even at tree level, then the tree-level entropy would be
zero. However, as pointed out in Dabholkar2002  there is a potential subtlety.
For example, the Schwarzschild metric also satisfies string equations of
motion to leading order and defines a conformally invariant sigma model. The
bulk Einstein action for this solution is indeed zero but the total action is
nonzero because there is a boundary contribution coming from the Gibbons-
Hawking boundary term Gibbons1977  . If the naive identification (15) were
correct then one would conclude that the tree level free energy and
consequently the Gibbons-Hawking entropy of the Schwarzschild black hole is
zero. This is manifestly not correct. Thus the worldsheet partition function
could equal at most the bulk spacetime action and need not capture the
Gibbons-Hawking boundary contribution which one believes must exist from
spacetime considerations.

Motivated by the Gibbons-Hawking computation, we assume that the relation (15)
is valid only for the bulk spacetime action. For the $\mathbb{Z}_{N}$ orbifold
conformal field theory, the equations of motion for the dilaton and the
graviton are satisfied exactly with a constant dilaton. The tree level
spacetime action always comes multiplied by the factor $e^{-2\phi}$ where
$\phi$ is the dilaton and thus acts as a source for the dilaton. This implies
that the bulk action must vanish exactly because otherwise there would be a
dilaton tadpole. The vanishing of the bulk spacetime action would be
consistent with the vanishing of the worldsheet partition function.

We would now like to deduce the boundary contribution to the tree level action
following the Gibbons-Hawking procedure around these orbifold saddle points of
the string field theory action. We do not know of a worldsheet computation to
achieve this, so we will follow instead a spacetime approach as in
Dabholkar2002 . Consider the Lorentzian spacetime string effective action for
the orbifold, for concreteness, first to leading order in $\alpha^{\prime}$

$\displaystyle I=\frac{1}{16\pi
G}\int_{M}\sqrt{-g}\,e^{-2\phi}\,[R+4(\nabla\phi)^{2}-\delta^{2}(x)V(T)]+\frac{1}{8\pi
G}\int_{\partial{M}}\sqrt{-g}\,e^{-2\phi}\,K\,$ (71)

where $K$ is the extrinsic curvature and $\delta^{2}(x)V(T)$ denotes a
possible tachyon potential localized at the tip of the cone. A nontrivial
point here is that the twisted sector of string theory automatically furnishes
tachyons localized at the tip of the cone, and it is natural to consider their
dynamics localized at the tip of the cone. By contrast, in canonical gravity
there is no analogous reason why the classical Einstein equations would be
satisfied for any value of the opening angle of the cone. From the worldsheet
perspective, variation of this action generates the sigma model beta
functional equations. The beta functional vanishes at the critical points of
this action where the classical equations of motion are satisfied resulting in
a worldsheet conformal field theory.

The extrinsic curvature term in the action above is as usual necessary to
ensure that the effective action reproduces the string equations of motion for
variations $\delta\phi$ and $\delta g$ that vanish at the boundary. The
details of the tachyon potential are not important except that it supplies a
Euclidean 7-brane source term for gravity so that the Einstein equations are
satisfied even with a curvature singularity $R=\delta^{2}(x)V(T)$ at $x=0$.
The main point is that the above action correctly captures the fact that the
dilaton equations are satisfied with a constant dilaton, and the bulk
contribution to the action is zero.

To compute the boundary contribution, note that the boundary has topology
$\mathbb{R}^{8}\times{\bf S}^{1}$. For a cone, the circle ${\bf S}^{1}$ has
radius $r$ but the angular variable will go from $0$ to $\frac{2\pi}{N}$. The
extrinsic curvature for the circle equals $1/r$ and thus the contribution to
the tree level action from the boundary term is proportional to
$\frac{2\pi}{N}$. The total contribution can be written as

$\log(\hat{Z}^{(0)}(N))=\frac{A}{4G}(\frac{1}{N}-1)\,.$ (72)

where we have fixed an additive constant to $-1$ by requiring that the action
is zero for $N=1$. We have to also remember a factor of $-i$ in Euclidean
continuation of $\sqrt{-g}$. This formula can now be analytically continued
for complex $N$. Substituting in (69) we conclude that the tree level
contribution to the entropy is $A/4G$ equal to the Bekenstein-Hawking entropy.

Note that one would not be able to make a similar argument in canonical
gravity because a conical space with any opening angle does not solve the
equations of motion of canonical gravity and is an off-shell configuration
that does not correspond to a saddle point of the action.

In conformal field theory, we should worry about higher order
$\alpha^{\prime}$ corrections to the effective action. These corrections are
dependent on field redefinitions or equivalently on the renormalization scheme
of the world-sheet sigma model. However, the total contribution of these
corrections to bulk action must nevertheless vanish for the orbifold because
we know that the equations of motion of the dilaton are satisfied with a
constant dilaton _exactly_ to all orders in $\alpha^{\prime}$ which implies no
source terms for the dilaton in the bulk. Possible higher derivative
$\alpha^{\prime}$ corrections to the Gibbons-Hawking term are subleading by
dimensional analysis. Thus, the entire contribution to the action comes from
the leading Gibbons-Hawking boundary term above even when the
$\alpha^{\prime}$ corrections are taken into account Dabholkar2002 . This can
be calculated reliably in a scheme independent way by using the conical
geometry of the exact solution at the boundary to obtain the tree level
expression (72). It would be interesting to investigate if this boundary term
can be captured by a worldsheet computation in the conformal field theory
Kraus2002  or in string field theory DeLacroix2017 ; Erbin2021 .

##  7 Holography and Finiteness of Entanglement Entropy

Rindler spacetime arises as the near horizon geometry of any black hole with a
bifurcate horizon, in particular, the eternal Schwarzschild black hole in anti
de Sitter spacetime. The left region $U_{L}$ and the right region $U_{R}$ of
the two-sided black hole geometry correspond to the left Rindler wedge and the
right Rindler wedge respectively. In the limit of a very large $AdS$ and black
hole radius, this approximation is arbitrarily good. The Hartle-Hawking vacuum
$|\Omega\rangle$ of the black hole corresponds to the Minkowski vacuum for the
near horizon physics.

Tracing over the states localized in the left region of the black hole
geometry gives rise to the Rindler density matrix for the right region:

$\rho_{R}={\raise-5.0pt\hbox{$\buildrel\textstyle{\rm
Tr}\over{\scriptscriptstyle\mathcal{H}_{[}}$}~{}}L]|\Omega\rangle\langle\Omega|$
(73)

which is to be identified with the thermal density matrix considered earlier
(20). The associated von Neumann entropy is the entanglement entropy of our
interest.

In the field theory limit for the bulk modes, this entanglement entropy is
proportional to the area of the entanglement surface and has the well-known
ultra-violet divergence. This divergence is a reflection of the strong
correlations at short distances in a local quantum field theory, for example
through the derivatives of the field across the entangling surface. In taking
the trace (73) over the states in the left region, we have implicitly assumed
that the total Hilbert space $\mathcal{H}$ is a direct product
$\mathcal{H}_{L}\otimes\mathcal{H}_{R}$. This assumption is not quite correct
since the modes across the entangling surface on the two sides are strongly
correlated. Thus, one cannot really associate factorized Hilbert spaces
$\mathcal{H}_{L}$ and $\mathcal{H}_{R}$ with the regions $U_{L}$ and $U_{R}$.

In a local relativistic quantum field theory, there is no factorization of
Hilbert spaces but there is a factorization of algebras of local observables.
One can therefore frame the discussion in the language of local observables
within algebraic quantum field theory using the Tomita-Takesaki modular theory
Takesaki1970 ; Borchers2000 . With the regions $U_{L}$ and $U_{R}$ one can
associate von Neumann algebras of local observables $\mathcal{A}_{L}$ and
$\mathcal{A}_{R}$ respectively, rather than the factorized Hilbert spaces
$\mathcal{H}_{L}$ and $\mathcal{H}_{R}$. One can then proceed to define the
Rindler Hamiltonian $H_{R}$ as the modular Hamiltonian associated with the
state ‘cyclic and separating’ state $|\Omega\rangle$ and the algebra
$\mathcal{A}_{L}$. The entanglement entropy is then defined in terms of the
modular Hamiltonian without using factorized Hilbert spaces.

By causality, the two algebras of operators localized in the causally
separated regions $U_{L}$ and $U_{R}$ have to be mutually commuting:

$[\mathcal{A}_{L},\mathcal{A}_{R}]=0\,.$ (74)

Therefore, by Shur’s lemma, $\mathcal{A}_{R}$ cannot furnish an irreducible
representation on a Hilbert space unless $\mathcal{H}$ is factorized. This
implies that the algebra of observables restricted to the right rindler wedge
is a von Neumann algebra of Type III rather than Type I as would be the case
for a purely factorized quantum mechanical system. As emphasized in Witten2018
, the divergence in the entanglement entropy in a relativistic local quantum
field theory is not a property of the state $|\Omega\rangle$ but rather of the
von Neumann algebra $\mathcal{A}_{R}$ which is of Type III and not Type I, in
that the entanglement entropy in any typical state will manifest this
universal UV divergence.

In quantum gravity, there is no natural notion of algebra of local observables
and it is not clear how to pose these questions. However, within a holographic
context, one has the following expectation. The holographic boundary dual of
this bulk black hole geometry is the product of conformal quantum field
theories with Hilbert spaces $\overline{\mathcal{H}}_{L}$ and
$\overline{\mathcal{H}}_{R}$ of states living on the left and the right
boundaries respectively. The full boundary Hilbert space
$\overline{\mathcal{H}}$ is thus the direct product
$\overline{\mathcal{H}}_{L}\otimes\overline{\mathcal{H}}_{R}$. The black hole
Hartle-Hawking vacuum $|\Omega\rangle$ which appears like a thermal state to
the observers in the right region is represented as a maximally entangled
thermofield double state $|\overline{\Omega}\rangle$ in this product Hilbert
space Maldacena2001a ; VanRaamsdonk2010 ; Maldacena2013 . Tracing over the
left boundary states gives the thermal density matrix $\overline{\rho}_{R}$
for the CFT on the right boundary consistent with the fact that the black hole
in the bulk corresponds to a thermal state in the right boundary CFT.

One can consider the algebra of observables $\overline{\mathcal{A}}_{R}$ of
operators acting only on $\overline{\mathcal{H}}_{R}$. Since the boundary
Hilbert space $\overline{\mathcal{H}}$ is a direct product, the algebra of
boundary observables $\overline{\mathcal{A}}_{R}$ acting on
$\overline{\mathcal{H}}_{R}$ is a von Neumann algebra of Type I much like for
a quantum mechanical system and furnishes an irreducible representation on
$\overline{H}_{R}$. One can define the modular Hamiltonian $\overline{H}_{R}$
corresponding to this algebra and the cyclic separating state
$|\overline{\Omega}\rangle$. The modular density matrix $\overline{\rho}_{R}$
is then defined by

$\overline{\rho}_{R}:=e^{-2\pi\overline{H}_{R}}\,$ (75)

which coincides with the density matrix (1). The entanglement entropy
$\overline{S}$ of our interest is the von Neumann entropy of this density
matrix. As argued in Jafferis2016 ; Leutheusser2021a ; Witten2021a ,
$\overline{S}$ of the boundary theory should be identified with the total bulk
entropy $S$ including the tree level contribution from the classical
Bekenstein-Hawking entropy.

There is no divergence in the von Neumann entropy for the density matrix
$\overline{\rho}_{R}$ which is simply the thermal density matrix for one copy
of $\mathcal{N}=4$ super Yang-Mills theory. How can this be consistent with
the bulk? In particular, $\overline{\rho}_{R}$ cannot possibly be identified
with $\rho_{R}$ defined in (73) which has divergent von Neumann entropy. As
noted earlier, in the field theory limit in the bulk, there is no well-defined
factorization $\mathcal{H}_{L}\otimes\mathcal{H}_{R}$ and the divergence in
the entanglement entropy in the field theory limit is a reflection of this
lack of factorization. In particular, the algebra of observables on
$\mathcal{A}_{R}$ is a von Neumann algebra of Type III and cannot be
identified with the algebra of observables $\overline{\mathcal{A}}_{R}$.

Holography suggests that quantum gravity must improve the situation. The
divergence in the quantum field theory limit of the bulk should be ameliorated
in the full quantum string theory in the bulk so that the behavior in the bulk
correctly corresponds to the behavior in the boundary expected for a Type-I
von Neumann algebra of observables, even thought there is no natural notion of
local observables in quantum gravity. In particular,

$\overline{S}=S=\frac{A}{4G}+S_{q}\,.$ (76)

There is also a corresponding relation between the modular operators:

$\overline{H}_{R}=\frac{\hat{A}}{4G}+H_{R}$ (77)

where $\hat{A}$ is the horizon area operator Jafferis2016 .

Since the boundary entanglement entropy $\overline{S}$ is finite, the bulk
entanglement entropy $S$ should also be finite order by order in string
perturbation theory corresponding to the large $N$ expansion in the boundary.
In Type-II superstring theory with $32$ supercharges, there is no
renormalization of Newton’s constant to one-loop order and consequently there
are no UV or IR divergences in $S^{(0)}$. One therefore expects that the one-
loop entropy $S^{(1)}$ (69) in this theory would also not have any
divergences.

Recent investigations in Leutheusser2021 ; Witten2021b ; Leutheusser2021a ;
Witten2021a  support this reasoning. Instead of asking the harder question of
how the bulk quantum string theory improves the divergences of quantum field
theory, one could ask how the boundary mimics the divergence of the quantum
fields in the bulk. It was shown in Leutheusser2021 ; Leutheusser2021a  that a
Type-III von Neumann algebra of observables emerges from the Type-I algebra in
the boundary theory at large $N$ which can naturally correspond to the algebra
$\mathcal{A}_{R}$ of local fields in the bulk. This algebra was identified as
the ‘small algebra’ of generalized free fields in Papadodimas2012 ;
Papadodimas2013 ; Papadodimas2013c . For related work on the reconstruction of
bulk observables and the black hole inside see Hamilton2005 ; Hamilton2006 ;
Jafferis2020 ; Almheiri2015 ; Harlow2018 . The Type-III character of this
algebra is related to the ultra-violet divergence in the entropy in the field
theory limit. It was shown in Witten2021a  that corrections subleading in $N$
modify the character of this algebra rendering it Type-II to ‘improve’ the
situation.

These results indicate that the divergence of the entanglement entropy in the
bulk is an artifact of the large $N$ limit and should be ameliorated once
string loop corrections are taken into account. For example, in the context of
$AdS_{5}/CFT_{4}$, the tree-level Bekenstein entropy of Schwarzschild black
hole in the bulk Gubser1996  would correspond with the entropy of thermal gas
in the super Yang Mills theory on the boundary at large $N$ at large ’t Hooft
parameter. In the boundary theory, the correction to this entropy at
subleading order in $N^{2}$ is expected to be finite and nonzero and in
principle computable. Large ’t Hooft parameter is the limit of large $AdS$
radius. In the large $AdS$ radius limit, $AdS$ spacetime reduces to flat
Minkowski spacetime. Therefore, one expects a finite and nonzero value for the
quantity $S_{q}$ (68) of our interest in flat space after appropriate
renormalizations.

Given these arguments, one is encouraged to seek a summation of the series
(70) for the entropy density and a finite answer for the entanglement entropy
(69) to one-loop order. Modular invariance is a very stringent constraint in
string theory. It is thus nontrivial that we are able to obtain an expression
for the entropy density that is manifestly modular invariant. As we have seen,
the orbifold method naturally allows a definition to all orders in
perturbation theory including the tree level contribution. The generalized
entropy thus defined is the natural object to which the generalized second law
of thermodynamics can apply Wall2009 ; Wall2012 .

It remains to be seen if the resulting entropy is finite even after the
integral over the moduli space. While the considerations above rely on
properties of algebras of local observables in the boundary CFT, their
formulation is expected to involve very different physical concepts in the
bulk string theory, considering that there are no strictly local observables
in quantum gravity.

## Acknowledgements

I would like to thank Kyriakos Papadodimas, Upamanyu Moitra, Ashoke Sen,
Edward Witten, and Don Zagier for useful discussions.

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