Covariant Holographic Entanglement: A Derivation
Mukund Rangamani
QMAP & Dept of Physics @ UC Davis
Quantum Matter, Spacetime, and Information
Yukawa Institute for theoretical physics, Kyoto
June 17, 2016
Xi Dong, Aitor Lewkowycz, MR
to appear….

Introduction
The holographic entanglement entropy prescription of Ryu-Takayanagi
relates geometric data in the bulk with quantum features on the boundary
Ryu, Takayanagi ‘06✦ Motivated by analogy between entanglement entropy and Bekenstein-

Hawking formula.
✦ Derived by invoking the basic entry in the AdS/CFT dictionary: mapping ofboundary and bulk path integrals and evaluate the latter in the semiclassicalsaddle point approximation.
Lewkowycz, Maldacena ‘13✦ Derivation makes clear how the bulk gravity dynamics picks out the minimal

surface prescription.
✦ Gives us insight into stringy/quantum corrections.
✦ Relates nicely bulk and boundary relative entropies.

Faulkner, Lewkowycz, Maldacena ‘ 13; Jafferis, Lewkowycz, Maldacena, Suh ‘16

Just as for other causal sets, we can extend these definitions to any region R, namely
S[R] := \p2R S(p) is the set of points which are causally disconnected from the entire
region R, etc.
Having established our notation for general causal relations, let us now specify thenotation relevant for holographic entanglement entropy. As before we will fix a region
A on the boundary. The HRT proposal [3] states that the entanglement entropy SA is✦ Given the boundary region A the prescription to compute entanglement
holographically computed by the area of a bulk codimension-two
extremal surface E
holographically involves finding a bulk extremal surface EA which is anchored A
that isanchored on @A; specifically,on @A and is homologous to A .
Area(EA )
SA =
.
(2.4)
4GN

Covariant Holographic Entanglement Entropy✦
surfaceis a codimension-2
surfacein isthebulk to be homolInThe
the extremal
static (RT)case, it isA known
that the extremal
surfacerequiredE

ogous
to A, meaning
there exists
a bulkregion@RMR,
A [ EA‘07.
M (nb:
asymptotically
AdSthatspacetime@M= B )RA such thatA =Hubeny,Takayanagi
So far, it has not been entirely clear what the correct covariant generalization of this✦ Motivated by using bulk covariance as a guiding principle.
condition is. In particular, should it merely be a topological condition, or should one
imposegeometricalcausalRA , for example,
that it be spacelike?
✦
Covariancedoesornot
pick requirementsout a uniqueon
prescription,
but supplemented(A critical discussion of the issues involved can be found in [32].) In this paper, wewith intuition arising from covariant entropy bounds we land on the
9These aresurface
necessaryprescription.
but not sufficient conditions for the spacetime to be asymptotically AdS.
extremal✦ Various consistency checks; an attempt at a derivation using Lorentzian

AdS/CFT.

–9–

Entanglement in QFT✦ Consider a QFT in a density matrix, living on a backgroundB which isglobally hyperbolic spacetime with a nice time foliation (Cauchy slices ⌃ ).
✦A is a subregion of the Cauchy slice, with an entangling surface @A .
⌃⇢A

reduced density
matrix

Ac
A

@A

SA =

Tr (⇢A log ⇢A )

Causality and Entanglement✦ Entanglement entropy in QFT is a wedge observable.

J + [@A]
D+ [A]

A

D[Ac ]D[A]

D [A]
J [@A]
D[A] = D+ [A] [ D [A]✦ The entanglement entropy can only be influenced by changing conditionsin the past domain J [@A] .

Implementing causality in the path integral✦ The real time path integral which is cognizant of the causality constraints

invokes the Schwinger-Keldysh formalism.

of a time dependent density matrix ⇢A (t). Let us first start with the entire system andconstruct the density matrix (which may be pure) on A [ Ac at time t. Consider the purecase ⇢ =| ih | for simplicity (if it is mixed, we can always purify it). Since a density matrix⇢ is an operator on the Hilbert space, we are required to evolve both the state vector and its
dual from the initial state up to the time of interest ⇢(t) = Ut, 1 ⇢i U 1,t , with Ut1 ,t2 being
the unitary operator that evolves the state forward from time t2 to t1 , and ⇢i being the initialstate prepared in some manner at t = 14 . As has been appreciated by many authors in
✦ The
path
integralwhich oneis cognizantof ledthetocausality
constraints
thepast,real
fromtimea pathintegralperspective,
is necessarily
doubling thedegrees of
freedom[15–17].
invokesthe Schwinger-Keldysh formalism.

Implementing causality in the path integral✦ We compute the matrix elements of thetotal density matrix at a given time by
evolving forward from the initial state, tothe instant of interest and thence retrace
our footsteps to the far past.
✦ This ensures that sources inserted in the

future do not affect the operator of interest.
✦ The forward and backward evolutions are

Fig. 1: A schematic representation of the Schwinger-Keldysh
contourson
necessarythe computation
oft .
glued together
someforCauchy
slice ⌃
the (a) density matrix and (b) its powers. We have explicitly shown the computation of ⇢3 in (b).
The dots and lines in red correspond to an entangled initial state prepared in some manner. This
picture does not carry the spatial information necessary to ascertain the reduced density matrices
themselves, which is better understood from Figs. 3. Note that in contrast to the usual depiction

of a time dependent density matrix ⇢A (t). Let us first start with the entire system andconstruct the density matrix (which may be pure) on A [ Ac at time t. Consider the purecase ⇢ =| ih | for simplicity (if it is mixed, we can always purify it). Since a density matrix⇢ is an operator on the Hilbert space, we are required to evolve both the state vector and its
dual from the initial state up to the time of interest ⇢(t) = Ut, 1 ⇢i U 1,t , with Ut1 ,t2 being
the unitary operator that evolves the state forward from time t2 to t1 , and ⇢i being the initialstate prepared in some manner at t = 14 . As has been appreciated by many authors in
✦ The
path
integralwhich oneis cognizantof ledthetocausality
constraints
thepast,real
fromtimea pathintegralperspective,
is necessarily
doubling thedegrees of
freedom[15–17].
invokesthe Schwinger-Keldysh formalism.

Implementing causality in the path integral✦ We compute the matrix elements of thetotal density matrix at a given time by
evolving forward from the initial state, tothe instant of interest and thence retrace
our footsteps to the far past.
✦ This ensures that sources inserted in the

future do not affect the operator of interest.
⇢ =| ih |! e iHt | ih | eiHt

✦ The forward and backward evolutions are

Fig. 1: A schematic representation of the Schwinger-Keldysh
contourson
necessarythe computation
oft .
glued together
someforCauchy
slice ⌃
the (a) density matrix and (b) its powers. We have explicitly shown the computation of ⇢3 in (b).
The dots and lines in red correspond to an entangled initial state prepared in some manner. This
picture does not carry the spatial information necessary to ascertain the reduced density matrices
themselves, which is better understood from Figs. 3. Note that in contrast to the usual depiction

Implementing causality in the path integral✦ The real time path integral which is cognizant of the causality constraints

invokes the Schwinger-Keldysh formalism.
✦ We compute the matrix elements of theA
Ac matrix
total density
at a given
time by⌃t
evolving forward@Afrom the initial state, tothe instant of interestand thence retracet
our footsteps to the far past.

⌃tt✦ This ensures that sources inserted in the

future do not affect the operator of interest.
⇢ =| ih |! e iHt | ih | eiHt

✦ The forward and backward evolutions are
Ac

A
glued
togetheron some Cauchy slice ⌃t .
@A
⌧ + 2⇡i⌧ + ⇡i⌧

The reduced density matrix elements✦ This general prescription can be minimally modified to obtain the reduced

density matrix elements.
✦ We cut open the path integral along the
Ac

A

⌃t

@A

region A on the Cauchy slice ⌃t .
✦ Imposing suitable boundary conditions int

the future/past segments leads to thematrix elements of the reduced density
matrix (⇢A )± .

✦ Knowing the reduced density matrix elements we can then attempt tocompute the von Neumann entropy by the replica trick.
@AA⌧ + 2⇡i⌧t

Local Rindler structure✦ In what follows, it will be useful to pay attention to the local geometry near

the entangling surface.
A

c

@A

A

⌧ + 2⇡i⌧ + ⇡i⌧✦ We can adapt local Rindler like coordinates in the neighbourhood, which

also allows for a suitable complexification.

Out-of-time-ordered Schwinger-Keldysh✦ To compute powers of the reduced density matrix we need a slight

generalization of the Schwinger-Keldysh contour.
✦ Consider computing not time-ordered, but out-of-time ordered correlators.
✦ These can be generated by suitably stringing together Schwinger-Keldysh

contours, with multiple switchbacks or timefolds.

Out-of-time-ordered Schwinger-Keldysh

To appreciate this point let us first ask what is involved in computing the matrix elof a time dependent density matrix ⇢A (t). Let us first start with the entire systemc at time t. Consider thconstructthedensitymatrix(which
may
be
pure)
on
A
[
A
✦ To compute powers of the reduced density matrix we need a slightcase ⇢ =| ih | for simplicity (if it is mixed, we can always purify it). Since a density mgeneralization⇢oftheoperator
Schwinger-Keldysh
contour.
is anon the Hilbert space,
we are required to evolve both the state vector adual from the initial state up to the time of interest ⇢(t) = Ut, 1 ⇢i U 1,t , with Ut1 ,t2✦ Consider computing not time-ordered, but out-of-time ordered correlators.
the unitary operator that evolves the state forward from time t2 to t1 , and ⇢i being thestate prepared in some manner at t = 14 . As has been appreciated by many auth✦ These can be generated by suitably stringing together Schwinger-Keldysh
the past, from a path integral perspective, one is necessarily led to doubling the deg
contours, withfreedom
multipleswitchbacks or timefolds.
[15–17].

✦ For instance computing a three-fold out-of-time ordered correlation function will
involve some contour of the form:

Replicated reduced density matrix✦ In this timefold picture we can straightforwardly compute powers of the

reduced density matrix.
✦ We string together copies of the path integral with the pieces identified

cyclically as required for multiplying out the matrix elements@A@A⌧3 + 2⇡i⌧3⌧2 + 2⇡i⌧2@A⌧1 + 2⇡i⌧1

Zq symmetric gluing conditions

:coordinate.
We bundle
would liketo admitsreiterateLorentzian
what it means
for fields
be neighbourhood
periodic in ⌧ withis⌧ that
the normal
to @A
sections.
In an toopenof theperiod 2⇡iq. In the QFT path integral we integrate over all allowed field configurations.
zero
section i.e., at @A we can use the Rindler coordinates described above, which provides a
The ⌧ domain of QFT fields consists of 4q disconnected horizontal lines in the complex ⌧convenient way to keep track of theproperties of the replicated geometry.
⇡

From replicas to entanglement

plane, each with imaginary part i 2 m, m = 0, 1, · · · , 4q 1. We then impose boundary conditions
at the asymptotic
infinitiesof thesehorizontal
lines. entropies
It is easiest to do this in the
2.3
Productsof reduced
densitymatrixand Rényi
language of asymptotically incoming/outgoing modes. The coefficients of these mode mustAtthe end
day,
the
isand
a and
means
for
us⌧ tocomputethe1entangle1
1✦ Our
aimofisthetocomputeentropies
thenanalyticallycontinue
tomatch
between⌧=m⇡i+ reduced
1theandRenyi⌧ density= (m+matrix)⇡i+1,
between
=
(m
+
)⇡i
and
2
2
ment
entropythereplicatheNotebasic
definitions:⌧ =(m+ 1)⇡i1,
forall m trick.
=entropy
0, 1,We· · · recall, 2q 1.
thatin saying this, we are identifying
obtaintheviavon
Neumann⌧ = 2⇡iq 1 with ⌧ = 1 – this is what we mean by the periodicity⌧ ⇠ ⌧ + 2⇡iq. To(q)SA = TrA (⇢A log ⇢A ) = lim SA ,recap: we are gluing q copies of the reduced density matrix,
q!1 with regions A identified across(2.5)
the copies. This can be just as(q)well statedin the {xJ , qtJ } coordinates, but the ⌧ coordinate
1
SA =log TrA (⇢A ) .
is more useful for delineating the analogous
1 q boundary conditions in gravity. [to clean up] [Inow think this is ok?]
✦ It isWhile
the togeneralfocusahereis on thecomputationof therelatedRényi entropies
themselves,usefulconsiderparticular
auxiliaryquantity
to the Renyi
it entropies,
will transpirethat thecomputation
nicer for theavatar.
derivative of the Rényi
whichwill gravitational
turn out to havea cleanisgeometric– 11–
entropy with respect to its index. Define thus
a related
quantity:
1(q)S̃A = q 2 @q
log TrA (⇢A )q(2.6) eq:q
Dong ‘16

In writing this expression we have already assumed that we can analytically continue theRényi entropies away from the integer values of the index q. To our knowledge, this object
has not been considered before in the quantum information literature, but it is rather natural.

Time reflection symmetric case
✦ If the quantum state is at a moment of time symmetry, then we can eschew

various complications of the Schwinger-Keldysh construction.
✦ Essentially we unwrap the switchbacks and work with regular path integral

contours; cutting them open to obtain (⇢A )± which may then be glued
together cyclically.
✦ Useful to view this Euclidean path integral in terms of QFT on a new

background: q-fold branched cover of the original background, branched at
the entangling surface.

Review: Lewkowycz-Maldacena✦ AdS/CFT relates holographic field theories on some background B to a

gravitational theory on a bulk spacetime M subject to boundary conditions
which demand that @M = B.

✦ The bulk spacetime can be obtained in the semiclassical limit using a saddle

point solution of the quantum gravity path integral.
✦ The branched cover construction gives us a boundary manifold Bq

which

we use as boundary conditions to determine Mq .

✦ This sets up the gravitational problem. Our job is to find the appropriate

solution, determine the boundary partition function, which is related to theon-shell action, and thence analytically continue.
✦ LM argue that the analytic continuation is simpler in gravity and employ it todirectly derive the RT prescription.

+ terms in the Taylor expansion about theq
We have retained only the leading
main
point
to noteisovertheB explicit
qatdependence.
Itsprovides
presence
impliesthata q-foldbranchedcover(branched
@A).
Per
se
this
a
clean
boundaryx = 0, tE = 0. One can equivalently parameterize the normal directions in
the gravity
problem
as 0,
described
above.
However, wesome
canexploitthat fact that
cndition
to beforsmooth
near
r
=
we
must
encounternon-trivial
backr
±i
⌧
16
x ± i tE = r e 52 , where
⌧
⇠
⌧
+
2⇡
for
regularity.
CHAPTER 5. DERIVING HOLOGRAPHIC ENTANGLEe partition function has a Zq symmetry of Bq that exchanges the di↵erent replicas. This is
mply
identify⇠ introduce⌧ + 2⇡q suchin (3.1).
If⌧ welocal coordinates in the vicinity of eq as in (3.1)
LM:
Kinematics
symmetry owing to the cyclicityof thetrace
intheindefinition
of Rényi entropies.
problem.
We
will
work
a
singlefundamental
domainquoti
symmetry
implies wethatcanthe restrict
action is ourinvariantwith respect
toofatheglobalg Assuming
the replica
symmetry
attentiontoasinglefund
as in [6] that thisreplica
Zis
extends
to thebulk,
we cantakethe
q symmetry
we
have
seen
topologicallyisomorphic
totheoriginal
bulk
spacetim
coordinate in the normal plane ⌧ , viz., ⌧ ! ⌧ + 2⇡; see Fig. 6 for an illustra
mooth
dual
M
and
considerthequotientspaceM̂di↵erent/Z
quotient
geometry
eplica)oftheinthe havebulk.
Thence,thetotalactionof the✦bulk
Assumebulk
saddles
replicaqZ
q = Mqgeometries
q . Thisq action
twoarespacetimes
vastlyowing
to theboundary
replicacoordinate
has atocodimension-2
be identified fixedunderpoint⌧ ⇠ locus⌧ + 2⇡
q.
We
can
now
15 u
not
smoothandgenericallycontainsoftheZ
action.
symmetricandconstruct
thecanorbifolddi↵erencebeaccounted
for
@ M̂
Bqthe
/Zqsingular= B locus eq .q This codimen
will beq timesthatofasingledomain,
viz.,
q =by
17
of theMq , tothatbethepartlocal
geometrynear eq inasthe
e will call this
fixedcovering
point treatedsetspaceof theeqofinfer– energy-momentum
it willofwhichthe kinematic
as a bulk
sourcebackreactsdata
on the spa
18
be
of
the
form
e build up an ansatz for construction.
Apart
from
being
invariantunder
the geometry
Zq symmetry
it to M̂q . To
ensurethatthe wehave thecorrect
we mus
2 I[M̂
2 q ]2
2
2
2
I[M
]
=
q
q
B
ds
=
q
dr
+
r
d⌧
+
ds
...
conditions
at thefixed
point
for e+
changing the replicas, eq is boundary
the naturalextension
of @A
into
the
bulk.itself,
transverse
q is not a gener
locus
of locus
orbifold
singularities

that arises from a smooth spacetime M by an orbifold construction.

q
We have left implicit here the M̂transverse
part of qthe geometry
which we w
2⇡
openingangleotient space has a conicaldefect angle, thecovering
taking singularity
an orbifold of a with(d + 1)-dimensional
spacetimeto geta codimeqcourse. The main
point
tothe
note
is thelocus
explicit
q dependence.
Its
presenceimsuggeststhatsingularshouldbetreatedasacosmicbranew@AA
observation
will
play
a
crucialroleinsettinguptheboundary
conditio
B metric to be smooth near r = cosmic
braneof tensionsome non-tri
for the0, we mustencounter1 becomes
q 1
antage of thinking
orbifolded
quotient
space
manifeone cannotabout
simply theidentify⌧ ⇠ ⌧ + 2⇡qin (3.1).
T
=
q
r
(d+1)
q
M̂
q
4G
N
Exploitingthereplicasymmetry
wecanrestrictourattentiontoa s
ut computing the entanglement
entropywhichrequiresanalyticconteq⌧
the reinstated
Zq actionallinthethe
Thence,the totalappro
act
where
weofhaveof Newton’sconstant
to q = 1.domain
From(orthereplica)
geometric
perspective
qfactors
isbulk.
simplya parametert@A
A domain,surface
computation will
bedensity
q timesthat ofinaasingle
viz., of the spacetime.
energylocalized
codimension-2

th of the opening angle at the conical defect in M̂q . Working in the or
The claim then is that we can compute the geometry of M̂q and t
I[MOn-shell actioncontinue
we want isqsimply
related the
mply ✦analyticallyby dialing
strengthofq ]theq ] = q I[M̂
with M withthe codimension-2
cosmic branewithsingularity.
the above value TrmatterEinstein’s
equations(4.2.3)withTarising from the cosmic2⇡brane✦AB
Subtleties
arisingfromglobaltopology…
While
the
quotient
space
has
a
conicalsingularity
withdefectangle
ience we are going to use the same notation for theHartman,normalbundlecoordinatesin, thethqHaehl,
Marolf,Maxfield,MR‘14termined the solution⌧for M̂q we computeeq the on-shell action of thisved quantity @q I[M̂q ] for any value of q. This is desirable for one not only
to figure out
what
configurations
dominateandthencecomputeon-shellaction.
To
2
2 2
2
2
q
t q
3 xtheirialbackreaction;one
cannotsimplyidentify
⌧
⇠
⌧
+
2⇡q
in
(5.1.7).
ds
=
q
dr
+
r
d⌧
+
+
2
K
r
cos
⌧
+
2
K
r
sin
⌧
tion about he entanglement entropy but also we do not have to set q = 1 ij
in
ij
ij
h
enforcetheboundary
conditions
inthegravitational
solution,letusexaminethemetricclose
cussion.
fq (q 1)

ave an toansatzshouldsimplyxcomputediscern when+ ther action
eq inwepolarcoordinate± i tE = rthee±i ⌧field. Theequations
replica Zqtosymmetry
implies thatea
involves viewing
the derivativewith respect
to q,geometry@ , as a changeof we
the find divergent
satisfied.
Evaluatingthecurvatures
for the(5.1.8)is invariant
withrespect
to a global
shift qof the polar
coordinate
in the normal plane ⌧ , viz.,nd its boundary conditions).K aStandard
variationalsays thatfunctionanya fq is acalculus
ij
where
some
analyticof
qhand
thatastakes
nonnegativ
proportionalto(q This1) r feature
whereisKillustrated⌘ Kij in Fig.
is thetraceof theextrinsic
⌧
!
⌧
+
2⇡.
??.
On
the
other
we
approacheq
assical action can be written as a combination of the equations of motion
is a positive
integer.
[AddedXi’ssuggestion][excellent]amining
potential
higherorder
terms,
onelearns
that
none
of
these
can
help
the
coordinate⌧hastotraverse
throughallthereplicacopiesbeforerevertingback
to itself,
erms (using integration by parts where necessary). In gravity this takes the✦ Gravitational
symmetric
integerq. Evaluating
the brane!
Ricci
for theganalytic
continuation:dial
thetensioni.e., Zit should
identifiedunder⇠ ⌧for+ 2⇡
q. motion
Using
globalsmoothness
the saddlepoint
his contribution.
Thebe
only
way
for
the ⌧equationof
to of
becosmicsatisfiedbyof tensor
Ka
⇥ AB
⇤
find
proportional(q 1) r where
covering E
space(ggeometry
M
we
infer
spacetimenear eto
M̂K1.8)I[is✦determined;
set eq:var0of admissible
q in the quotient
q
M̂for
)qABe
,qdivergent@to)ABthat) contributions. the localthe(5.1.10)q ] =the extrinsicq )geometry
AB + d⇥((gqofq (gqbeLocalManalysis
of eomin extrinsic
the vicinityof singular
locus
givescontribution
RT prescription.
2
qcurvature.
This
divergent
has
be
of
the
form
surfaces are required to have vanishing trace of the extrinsic curvature in the cannot be com
dary terms have be collected into a pre-symplectic
form⇥. metric.
For a typicalt We are thence led to conclude thatcomponents
of
the
ons!
Since
we
have
a
t
!
t
symmetry,
wehavetrivallyK=t 0q and one ithus
2
2
2
2
2
x
q
j
ppears in a standard
AdS/CFT
calculation,
this
would
evaluatetoaterm
ds = q dr + r d⌧ us+a constraint
⌧
+
2
K
r
sin
⌧
dy
dy
+ · codimension-2·· ,(5.1.8)
ij + 2 Kij roncos
ij e . The allowed
theallowedqnimal
surfaceof [9]:
ic boundary@Mconditionq = Bq . However, we wish to consider the variation of q,
vanishingtrace
theextrinsic
curvaturein the normal
directelidingover condition
higher order
again.
We
attention
to the explicit
q dependence.
hanges theboundarynearterms
the
fixed
point
setwisheof
Fordraw
the choiceq . toxt = 0. (5.1.9)
lim
e
!
E
,
E
2
M
with
t
=
0,
K
=
0
.
eq:lmrtbysymmetrysymmetry,
wehavetriviallyK
The
constraint
is equivqAA
he variation
satisfies
@
(g
)
=
0,
@
(g
)
=
6
0.
So
we
see
that
the
Its
presence
implies
that
in
order
for
the
metrictobesmooth
near
r
=
0,
we
must
encounter
q
q
AB
q
q
AB
q!1
Bq
eq
conditionof[1]:point
ed by thesome
replicaindex variation
is localized
at thefixed
locus
and has⌧ ⇠ ⌧ + 2⇡q in (5.1.7).3
non-trivial
backreaction;one
cannotsimplyidentify
B
from
the PRESCRIPTION
asymptotic boundary of the spacetime. One may therefore
lim write
eq !
EA55
, can
EAbe
2 M with t = 0, K
E
HRT
consider higher
derivative
gravitational
theories,
thegeneralanalysisOnce we Zhave an ansatz we should simply computethe field equations to discern whenq!1M̂qh in a similar
fashionas⇥((gdiscussed
in
[71,
72].
What
becomes
clear
is
that
the
@q I[wouldM̂, @q (g))(5.1.11)
eq:var1
they
Evaluatingthecurvaturesforthegeometry(5.1.8)wefinddivergentq ] =be satisfied.
q )AB
q
AB
19
The astute
may
worry
that
as =
a consequence we will have somee(✏)
q in an open neighbourhood
Kqareadercalcoordinates(5.1.8)
of
e
,
one
finds
a but
a Kij✏doesqsuffices
to
pin
down
the
singular
locus
in
the
!
1
limit,
this
nottrace
in of the extrinsic
contributions proportionalsingular
to (q near1) rr = whereK⌘ K
is the0 for q 2
/ Z. This,ijwhile true,
turns
out
@A
A to be tamable –chosentoregulatetheresultbyblowingup
thesingularlocus
toatubularmine thefunctional
which wepotential
minimizeto obtain
curvature.
Examininghigherorder
terms,
in a suitablesense,
asthe
we surface.
shallonesee. learns that none of these can help
In other words the fix point set eq which was
20 at r = 0 is now being regulated
Greek(lowercase)indices of
refer
tothe
thetofullcompensate
thiscontribution.
Theof only
way forthe equation
motionbespacetime,satisfied mibyArea(eqfibration) Notation:
risticasthegeometryisanontrivial
thenormalbundleparameterizedbyr
n-1 surface eq (✏) @at
r M̂
= ✏.
the correct answer when ✏ ! 0.(5.1.13) eq:renderq= will2obtainq I[q ] Wei,j,· · · refer to the tangent space of the fixed point set eq , and early-alpha
the ansatz
theq G
dimension-2
base. (5.1.8) is4for
N extrinsic geometry of eq to be determined; the set of admissible

LM: Dynamics

eq
to the(which
bundleof solutions
eqtrace.
nt case weactually
evaluate
thisreferintegral
can
be
done
given
±! ⌧
derstandthewon’tfactors
of r e±i⌧by looking
atwhich
ofnormal
the
local
mode
(rqtheei ⌧ )extrinsic
and curvature
codimension-2
surfacesarerequiredtohavevanishingofin thetheformula.
, us
butwillRTfollowequivalentroute. In the presence of a boundary for thehevicinityofr=an
0directions!
areadmissible.
normalSince we have a t ! t symmetry, we have trivally K t = 0 and one thus
ulus to be well-defined and give the correct equations of motion, we would
re allows
us to theanalyticallycontinue
the on-shellderivesminimal surface
conditionof [9]:action I[M ] to nonthe correct boundary terms. While in our case the surface eq (✏) isq notLorentzian AdS/CFT✦ To compute entanglement entropy from the bulk we need to figure out

how to set up the bulk quantum gravity path integral.
✦ Gravity dual of the Schwinger-Keldysh path integral contour?

Lorentzian AdS/CFT✦ To compute entanglement entropy from the bulk we need to figure out

how to set up the bulk quantum gravity path integral.
✦ Gravity dual of the Schwinger-Keldysh path integral contour?

✦ Assumption: The boundary Schwinger-Keldysh contour is piecewise

extended into the bulk gravity theory.
✦ If the global state admits a semiclassical dual, then each segment of thebulk path integral is dominated by the the corresponding geometry.
✦ We have to respect the time ordering constraint in the bulk, which now has

some extra redundancy…
Skenderis, van Rees ‘08

A bulk redundancy✦ A boundary time slice does notuniquely extend into the bulk.
✦ The redundancy is captured bybulk Cauchy surfaces that areanchored on and spacelike to ourboundary slice.
✦ This ambiguity maps out a causal

domain in the bulk, the FRW
wedge.

The bulk ansatz✦ Prescription: Pick some bulk

˜ t within the FRW
Cauchy slice ⌃wedge.

˜t⌃
RA
A✦ We will glue copies of the

e

RAc

Ac

˜ t to obtaingeometry past of ⌃the dual of the SK contour.
˜ t is irrelevant for✦ The choice of ⌃@A

computing time-orderedcorrelation functions.
✦ For entanglement entropy we will(b)

˜ t is forced to containfind that ⌃the extremal surface.

Bulk density matrix elements
identify

˜t
⌃

˜t
⌃
RA

A

RA
A

c

A

@A

B

A

c

@A

B
M

Replicating the bulk
identify

RA
Ac

Ac

A

@A

identify

MqRA

identify

RA

Ac

A
@A

identify

AcRA

identify

RA
Ac

@A

RA

A

Ac

@A

A

@A

A

A
@A

Replicating the bulk
identify

RA
Ac

Ac

A

@A

A
@A

identify

MqRA

identify

RA

Ac

A
@A

@A

A

identify

AcRA

identify

˜t⌃identify

˜t
⌃
RA
A

Ac
RA
Ac

@A

@A

RA

A

Ac

@A

RA
Ac

A
@A

A

M̂q = Mq /ZqKinematics for covariant construction✦ Construct the boundary Schwinger-Keldysh contour which gives a suitably

doubled spacetime B.
✦ Filling in this contour with appropriate bulk pieces we have the bulk

geometry which serves as the seed ansatz M for computing the reduced
density matrix elements.
✦ We string together copies of M sewing together successive segments

cyclically to obtain the covering space Mq .

✦ Gravitational dynamics is supposed to pin down Mq .
✦ We again exploit the relative simplicity of the gravitational analytic

continuation, by working in the quotient spacetime M̂q = Mq /Zq which
now features a spacelike cosmic brane eq .

Local boundary conditions
B

@A

A

e

looks like Rindler spacetime
with the entangling surface
extended out into the bulk asa Rindler horizon.
✦ The replica boundary˜t⌃⌧ ⇠ ⌧ + 2⇡i✦ The q=1 geometry locallyconditions for M̂q canequivalently be stated interms of modulating this local
structure; the Rindler
2⇡
temperature is lowered to.
qof the bulk under the replica Zq symmetry, namely a Schwinger-Keldysh double geometry
constructing the dual of the trace of the total density matrix Tr⇢(t) with a Zq symmetric˜ t . The remaining task at hand is tofixed point set, eq , localized on the Cauchy surface ⌃employ the bulk equations of motion, see what they imply for eq , and compute the on-shellactionWiththereafter.
this ansatz for the geometry, we can now analyze the consequences of the equations of motion. This is in fact quite easy, since the local geometry resembles the Euclidean
1. The extremality condition: We have described the boundary conditions that we needdiscussion. We have a deviation away from flat space (in Rindler coordinates) owing to an
to satisfy in §3.2.1. As in §3.1 it is useful to switch to Rindler-like coordinates {r, ⌧ } for the
instantonic
brane sourcewith tension
set replicaby q. Theboundary
gravitational conditions,
equations of motionaway✦ Oncewe have
and
the
all
that
normalbundletheof eansatz
in
the
bulk.
In
the
following
discussion,
we
will
focus
on
the
forwardq
from eq however do not care for this.
segmentof theSchwinger-Keldysh
contour (⌧of< motion.
0).29 Analogous
toin
(3.4)
in §3.1,
the metricremains
is
to
solve
the
bulk
equations
Work
local
coordinatesIndeed, evaluating the terms in the equations of motion for Einstein-Hilbert dynamics in
is constrained by the Zq symmetry, boundary conditions, and regularity for integer q to haveadaptedtheproportional
singular locus:
the bulk,towethefind normal
potentiallybundle
divergentof
terms
to [I just put indices]
the following expansion in the vicinity of eq :
q 1 a
a
a
EOM /x q K + regular(3.12) eq:eomL
2
2 2
2
2
t
q
i
j
r
ds = q dr
r d⌧ + ij + 2 Kij r cosh ⌧ + 2 Kij r sinh ⌧ dy dyWith this ansatz for the geometry,
we can now analyze
the consequencesof the equahifor small q 1 [Removed g in previous equation].
Basically,thepresenceoffq (q 1)
µ
⌫
+
r
1
g
dx
dx
+
·
·
·
(3.11)µ⌫
tions of motion. This is in fact quite easy, since the local geometry resembles
the Euclidean eq:lmL
the extrinsic curvature terms in (3.11) leads to potentially singular behaviour of the Ricci
discussion.
have
athe
deviation
from
space
Rindlerincoordinates)owing
ana because
whereinWe
wecoefficientsrq flat
terms
as be
K(in
the
1 limittheytogive
tensorthedenoteneighbourhoodof awayr =of0.theThesecannotfor qby!any
correction
ijcompensatedinstantonic
branethatsourcewith tension
set theby q.geometry
Theboundary
gravitationalmotionaway
thecurvature.
✦ Bulktoequations
theextrinsic
metricrespects
theZq symmetry
andOneoflocus.
thenlearnsof motionthen
fix
of conditions.
the equations
singularTo
fromthateq 29however
donotisextrinsic
care
for
this.
˜ t , we do in
the
trace
of thecurvatureof worrythe normal
directions
must vanish,
i.e.,
Since
the
analysis
local
below
⌃
noteachneed toabout thekink. [Commented
discussion,leading
order
in
q-1
we
fix
the
geometry.
In
Einstein-Hilbert
theory
this
gives
t
x
t
Indeed,thetermsin the refers
equationsmotionfor timelike
Einstein-Hilbertremoveif=agree]K= K evaluating
0. While
thisstatement
to theoftracein the(K ) and dynamics
spacelike inx ) directions
theextremalsurfacecondition
respectively,
we can bytakingsuitable linear
combinations
express this in
the (Kbulk,
we findpotentiallydivergentterms
proportional
to [Ijust put indices]
terms of the null expansions which are qmore
for codimension-2 spacelike surfaces
1 naturala±aa0±
1 we thus
EOM x / = p1 K
+xregular(3.12) eq:eom
in Lorentzian manifolds. Defining
x
have
the
extremalsurfacer 2 – 25 –30
condition postulated in [3], viz.,

Gravitational dynamics

for small q 1 [Removed g in previous equation]. Basically, the presence of1
the extrinsic curvatureK atermsin (3.11)
to0 ±potentiallyp K= 0 =)✓± =leads
K 1 = 0 , singular behaviour of the Ricci
2(3.13) eq:extrema
tensor in the neighbourhood of r = 0. Thesecannot be compensated for by any correction=) lim eq = EA ,EA 2 M is extremal.
to the metric that respects the Zqq!1
symmetry and boundary
conditions. One then learns
thatHaving
the traceof the extrinsic
curvaturein eachthe normal directions must vanish, i.e.,
ascertained
the dynamical
constraint
on eofq in the limit q ! 1, let us return to ourThe on-shell gravitational action✦ Since the singular locus is a spacelike source, it influences the geometry

along the past light cones. This complicates the analysis of the on-shellaction, which requires a suitable regulating procedure.
✦ The non-trivial part is to argue that the Renyi entropies work out correctly;

the limit q 7! 1 turns out to be simpler. In any event we can show thatS̃q =i @q I[M̂q ] =

i
@q
8⇡GN

Area(eq )
K✏ =
2G
4q
N
eq (✏)

ˆ

Farhi, Guth, Guven ’90, Neiman ’12, ‘13

Comments…
✦ The construction builds in the homology constraint.
✦ The bulk Cauchy surface we pick is forced to admit the extremal surface.
✦ The construction explicitly ensures that the proposal satisfies boundary

causality; the extremal surface lies in the causal shadow of the boundary
region’s domain of dependence .
Wall’ 12; Headrick, Hubeny, Lawrence, MR ‘14✦ In spirit the construction has elements of the maximin reformulation of the

HRT prescription.

Wall’ 12

✴ Pick a bulk slice and find a minimal surface on the slice.
✴ Maximize the area of the surface across all bulk slices in the FRW wedge.

Open Questions✦ Can we rule out the occurrence of complex saddle points?

Fischetti, Marolf + Wall ’14; Maxfield ‘14✦ Should they be relevant, how does one reconcile their presence withcausality restrictions?

✦ A cleaner derivation of the bulk dual to the Schwinger-Keldysh

prescription?
✦ Can we put the topological symmetries inherent in Schwinger-Keldysh touse efficiently for this purpose?

Haehl, Loganayagam, MR ’15