Holographic entanglement entropy beyond...

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


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



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

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].



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.



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.



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.



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].



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.



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.



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.



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 .



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.



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

)).



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`

)).



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).



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.



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.



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!


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


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