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Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Holographic entanglement entropy beyondAdS/CFT
Edgar Shaghoulian
Stanford Institute for Theoretical Physics
Kavli Institute for the Physics and Mathematics of the UniverseApril 28, 2014
Dionysios Anninos, Joshua Samani, and ES hep-th:1309.2579
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Contents
I Holographic entanglement entropy overview
I Warped AdS3 and warped CFT2 overview
I Holographic entanglement entropy for WAdS3
I Holographic entanglement entropy for locally AdS3 spacetime
I Perturbative holographic entanglement entropy for WAdS3
I Outlook
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Entanglement entropy
“Geometric” or entanglement entropy:
SA = −TrA(ρA log ρA), ρA = TrBρ .
CFT2 in ground state on plane [Holzhey, Larsen, Wilczek; Cardy, Calabrese]:
SA =c
3log
Lxε.
CFT2 in ground state on cylinder:
SA =c
3log
sin(Lθ/2)
ε
CFT2 at finite left-moving and right-moving temperatures:
SA =c
6log
(βLβRπ2ε2
sinh
(πLxβL
)sinh
(πLxβR
))
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Holographic entanglement entropy
For time-independent state in AdS/CFT, Ryu-Takayanagi (RT) proposed
SA =Area(γA)
4GN
for minimal surface γA on a given time slice.
Effectively proven by now [Casini, Huerta, Myers; Faulkner; Hartman; Lewkowycz,
Maldacena]. Extended to quantum corrections [Barella, Dong, Hartnoll, Martin;
Faulkner, Lewkowycz, Maldacena], higher spin theories [de Boer, Jottar; Ammon, Castro,
Iqbal], higher curvature theories [Hung, Myers, Smolkin; Dong].
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Time-dependent holographic entanglement entropy
Hubeny-Rangamani-Takayanagi (HRT) proposed generalization
SA =Area(γA)
4GN
for extremal surface γA not restricted to time slice. Formula unproven butsatisfies nontrivial checks, e.g. strong subadditivity [Callan, He, Headrick; Wall],and reproduces CFT2 formulae at finite TL and TR.
No extension of HRT proposal to non-AdS UV asymptotics.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Warped AdS3
Spacelike WAdS3 written as fibration over Lorentzian AdS2 base space:
ds2 =`2
4
(−(1 + r2)dτ2 +
dr2
1 + r2+ a2(du+ r dτ)2
)in global coordinates and
ds2 =1
4
(`2−dψ2 + dx2
x2+ a2
(dφ+ `
dψ
x
)2)
in Poincare-like coordinates.
I All coordinates valued in R; a ∈ [0, 2). R = 2(a2−4)
`2.
I Solution of Einstein gravity plus matter; exists in string theory.
I Isometry group SL(2,R)× U(1), unless a = 1.
I No conformal boundary (but there exists anisotropic conformal infinity[Horava, Melby-Thompson]).
I Discrete identification gives warped BTZ black hole.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Warped AdS3
Compactify fiber coordinate φ to get 3D part of NHEK geometry:
ds2 = 2JΩ(θ)2
(−dψ2 + dx2
x2+ dθ2 + a(θ)2
(dφ+
dψ
x
)2).
I Relevant for Kerr/CFT and understanding of astrophysical black holes.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Warped CFT2
Consider theories defined as having SL(2,R)× U(1) symmetry andproposed to be holographically dual to warped AdS3.
Symmetry automatically enhanced to infinite-dimensional V ir × U(1)Kac-Moody [Hofman, Strominger]; this case referred to as warped CFT2.
Cardy-like formula can be derived for density of states [Detournay, Hartman,
Hofman].
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Trivial warping: AdS3 spacetime
Set a = 1 to get AdS3 spacetime in fiberedPoincare-like coordinates:
ds2 =1
4
(−`2 dψ
2
x2+ `2
dx2
x2+
(dφ+ `
dψ
x
)2).
In fibered global coordinates we have
ds2 =`2
4
(−(1 + r2)dτ2 +
dr2
1 + r2+ (du+ r dτ)2
).
HRT proposal can be applied to these spacetimes!We stick to r = +∞. Coordinates on boundaryare null.
Figure : Adapted from0905.2612.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
HRT proposal for fibered Poincare-like AdS3 spacetime
Answer in terms of two null distances:
SEE =c
3log
(1
ε
√Lψ` sinh
(Lφ2`
))
=c
6log
Lψε
+c
6log
(`
εsinh
(Lφ2`
))ψ-movers in ground state and φ-movers at finite temperature `.Holographic renormalization shows
〈Tψψ〉 = 〈Tψφ〉 = 0; 〈Tφφ〉 6= 0 .
Compactifying fiber coordinate gives near-horizon limit of extremal BTZ,which has TL = 0 and TR 6= 0.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
HRT proposal for fibered global AdS3 spacetime
SEE =c
3log
(1
ε
√sin
(Lτ2
)sinh
(Lu2
))
=c
6log
(1
εsin
(Lτ2
))+c
6log
(1
εsinh
(Lu2
))τ -movers in ground state on cylinder and u-movers at finite temperature.Coordinates all dimensionless.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
WAdS3 Geodesics
Easy to solve for u(λ), τ(λ), and r(λ) in terms of conserved momenta cτ , cuand cv. For AdS3, limλ→∞ u(λ) = k for constant k, but for warped AdS3
solution for fiber coordinate has piece linear in λ. We find
Length ∼ λ∞ =log [r∞ f1(cu, cτ , a)]√
1 + (1− 1/a2)c2u
and
cτ = f2(cu, a) cot
(Lτ2
),
2
(−1 +
1
a2
)cuλ∞ + log
cu +
√1 + c2u −
c2ua2
cu −√
1 + c2u −c2ua2
= Lu .
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Perturbation theory
Troubling equation is
2
(−1 +
1
a2
)cuλ∞ + log
cu +
√1 + c2u −
c2ua2
cu −√
1 + c2u −c2ua2
= Lu ,
transcendental in cu. Solve perturbatively instead for a = 1 + δ:
cu = cu,0 + δ cu,1 + δ2cu,2 + · · · ,
with|δncu,n| |δn−1cu,n−1|
to assure convergence. Guaranteed if
Lu & 1, |λ∞δ| 1 .
Latter requirement interpreted as remaining in AdS3 part of geometry; thisis just AdS/CFT in presence of infinitesimal, irrelevant source! HRTproposal should apply.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Answer
Perturbative answer to all orders:
SEE =`
4GN
[(1 + δ coth2Lu
2
)log
(r∞ sin
Lτ2
sinhLu2
)]+
`
4GN
∞∑i=2
δi (−1)i+1coth2Lu2
csch2(i−1)Lu2
[log
(r∞ sin
Lτ2
sinhLu2
)]i
×
(i−2∑j=0
cij cosh(jLu)
).
Taking Lu 1 and δ > −1/2 lets us sum the entire series to get
SEE =`
2GN(1 + δ) log
(1
ε
√sin
Lτ2
exp
(Lu2
)).
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Reading off the central charge
Answer to all orders in δ in the large-Lu limit:
SEE =`
2GN(1 + δ) log
(1
ε
√sin
Lτ2
exp
(Lu2
)).
We have recovered universal CFT2 answer in large Lu limit, with
cL = cR =3`
2GN(1 + δ).
Peforming same perturbative expansion in Poincare-like coordinates againgives universal CFT2 answer:
SEE =`
2GN(1 + δ) log
(1
ε
√Lψ` exp
(Lφ2`
)).
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Warped BTZ black holeSpacelike warped BTZ black holes are locally spacelike warped AdS3
[Anninos, Li, Padi, Song, Strominger]:
ds2
`2=
3dt2
4− a2+
dr2
4(r − r+)(r − r−)+
6√
3
(4− a2)3/2(ar −√r+r−) dtdθ
+9r
(4− a2)2
((a2 − 1)r + r+ + r− − 2a
√r+r−
)dθ2, θ ∼ θ + 2π
Perturbative answer in large fiber-coordinate regime given by
SEE =`a
GNlog
(r+ − r−ε2
exp
(√3
a2(4− a2)∆t+
π∆θ
βL
)sinh
π∆θ
βR
),
with dimensionless temperatures
β−1L = TL =
3
2π(4− a2)
(r+ + r− −
2
a
√r+r−
),
β−1R = TR =
3(r+ − r−)
2π(4− a2).
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Nonperturbative (in δ) proposal
Cardy formula is nontrivial check (even for finite δ):
S =π2
3(cLTL + cRTR) =
(3π`
2GN (4− a2)(ar+ −
√r+r−)
)=
A
4GN.
Physically relevant range is a ∈ [0, 2). Our expansion converges fora ∈ (1/2, 2), i.e. δ > −1/2, so we propose it is valid in that range.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Open questions
I Full nonperturbative application of HEE proposal?I Nonlocality in the UV with volume law?
I Independent way to see WCFT2 reproduce CFT2 behavior; correlationfunctions?
〈φi(x−, x+)φj(y−, y+)〉 =
fij(x− − y−)
(x+ − y+)λi+λj
I Universal entanglement entropy formulae in WCFT2, withoutholography [ES, in progress (sort of)].
I Extension to TMG [Castro, Detournay, Iqbal, Perlmutter, in progress].
I Extend perturbative approach to spacetimes continuously connected toAdSd+2.
I Produce cL = 12J in NHEK.
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Take-away
I Sometimes useful to compute holographic entanglement entropyperturbatively.
I Warped CFT2 seems CFT2-like at finite temperature, as long as wetake an IR limit and large fiber coordinate separation.
I Central charge, TL, and TR predicted, with Cardy formula satisfied;first quantiatively successful application of (covariant) holographicentanglement entropy to non-asymptotically AdS spacetime!
I Many concrete directions for progress; outlook hopeful!
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Take-away
I Sometimes useful to compute holographic entanglement entropyperturbatively.
I Warped CFT2 seems CFT2-like at finite temperature, as long as wetake an IR limit and large fiber coordinate separation.
I Central charge, TL, and TR predicted, with Cardy formula satisfied;first quantiatively successful application of (covariant) holographicentanglement entropy to non-asymptotically AdS spacetime!
I Many concrete directions for progress; outlook hopeful!
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Take-away
I Sometimes useful to compute holographic entanglement entropyperturbatively.
I Warped CFT2 seems CFT2-like at finite temperature, as long as wetake an IR limit and large fiber coordinate separation.
I Central charge, TL, and TR predicted, with Cardy formula satisfied;first quantiatively successful application of (covariant) holographicentanglement entropy to non-asymptotically AdS spacetime!
I Many concrete directions for progress; outlook hopeful!
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Take-away
I Sometimes useful to compute holographic entanglement entropyperturbatively.
I Warped CFT2 seems CFT2-like at finite temperature, as long as wetake an IR limit and large fiber coordinate separation.
I Central charge, TL, and TR predicted, with Cardy formula satisfied;first quantiatively successful application of (covariant) holographicentanglement entropy to non-asymptotically AdS spacetime!
I Many concrete directions for progress; outlook hopeful!
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Overview AdS3 warmup Spacelike WAdS3 Warped BTZ Summary/Outlook
Take-away
I Sometimes useful to compute holographic entanglement entropyperturbatively.
I Warped CFT2 seems CFT2-like at finite temperature, as long as wetake an IR limit and large fiber coordinate separation.
I Central charge, TL, and TR predicted, with Cardy formula satisfied;first quantiatively successful application of (covariant) holographicentanglement entropy to non-asymptotically AdS spacetime!
I Many concrete directions for progress; outlook hopeful!
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
Additional Material
Compactification of common bosonic sector of IIA/B and heteroticSUGRAs (Einstein frame):
S =1
2κ210
∫d10x√−g(R10 −
1
2(∂φ)2 − 1
12e−ΦHMNPH
MNP
).
Compactify on S3 × T 3 × S1 and keep KK gauge field from S1, withbackground
ds210 = e−3Y/2
(eXds2
3 + e−X(dφ+A)2)
+ eY L2Sds
2(S3) + ds2(T 3)
H = hSL3SV ol(S
3) + H + F ∧ (dφ+A)
for H ≡ dB − F ∧A, LS the radius of S3, hS a constant, and ds2(S3) andVol(S3) the metric and volume forms on S3. We can thus reduce andconsistently truncate the resulting 3D action to
S3D =1
2κ23
∫d3x√−g3
(R3 −
1
8e3Y−ΦF 2 + Lkin(Φ, Y )− 2h2
3e2Φ−6Y
)+
1
2κ23
∫d3x√−g3
(12
L2S
e−4Y+Φ/2 − h2Se−6Y−Φ/2
)− h3
4κ23
∫A ∧ F
Holographic entanglement entropy beyond AdS/CFT Stanford Institute for Theoretical Physics
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