seyed morteza hosseini - istituto nazionale di fisica nuclearehosseini/pdf/flat ee.pdf · 2016. 1....

Entanglement and flat space holography

Seyed Morteza Hosseini

University of Milano-Bicocca

November 17, 2015

Seyed Morteza Hosseini University of Milano-Bicocca

Entanglement and flat space holography 1 / 21

Outline

1. Motivations

2. The BMS3/GCA2 correspondence

3. Gravitational anomaly in CFT2

4. İnönü-Wigner contraction

5. Zero temperature entanglement entropy (EE) for GCFT2

6. Finite temperature EE for GCFT2

7. Holographic EE from BMS3/GCA2

8. Outlook

Based on

S. M. Hosseini and A. Veliz-Osorio, “Gravitational anomalies, entanglement

entropy, and flat-space holography,” arXiv:1507.06625 [hep-th].

Motivations

Main motivation

Holography beyond the AdS/CFT correspondence!

focus of this talk: Flat space holography!

1. Microstate counting from BMS3/GCA2?!

2. (Holographic) Weyl anomaly?!

3. (Holographic) c-function?!

4. (Holographic) entanglement entropy?!

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

Motivations

Main motivation

5. . . .

The BMS3/GCA2 correspondence

Galilean conformal algebra (GCA)

The GCA is the infinite dimensional algebra generated by theconformal isometries of Galilean spacetimes. In 1+1 dimensions thevector fields that generate these transformations are given by

Ln = −(n+ 1)xnt∂t − xn+1∂x , Mn = xn+1∂t ,

which satisfy the algebra

[Lm, Ln] = (m− n)Ln+m +cLL12

m(m2 − 1)δm+n,0 ,

[Lm,Mn] = (m− n)Mn+m +cLM12

m(m2 − 1)δm+n,0 ,

[Mm,Mn] = 0 . (1)

(Bagchi et al., Barnich et al.)

BMS3 algebra (Einstein gravity) (Barnich, Compere ‘06)

For (2+1)-dimensional asymptotically flat spacetimes, the symmetryalgebra at null infinity is given by the semi-direct sum of theinfinitesimal diffeomorphisms on the circle with an Abelian ideal ofsupertranslations.

[Jm,Jn] = (m− n)Jn+m , [Pm,Pn] = 0 , (2)

[Jm,Pn] = (m− n)Pn+m +1

4Gm(m2 − 1)δm+n,0 .

This algebra is isomorphic to a GCA2 with cLL = 0 and cLM = 3/G.

The BMS3/GCA2 correspondence!

BMS3 algebra (Einstein gravity) (Barnich, Compere ‘06)

For (2+1)-dimensional asymptotically flat spacetimes, the symmetryalgebra at null infinity is given by the semi-direct sum of theinfinitesimal diffeomorphisms on the circle with an Abelian ideal ofsupertranslations.

[Jm,Jn] = (m− n)Jn+m , [Pm,Pn] = 0 , (2)

[Jm,Pn] = (m− n)Pn+m +1

4Gm(m2 − 1)δm+n,0 .

This algebra is isomorphic to a GCA2 with cLL = 0 and cLM = 3/G.

The BMS3/GCA2 correspondence!

Gravitational anomaly in CFT2

EE in a CFT2 (T=0) for a single interval A = [z1, z2].

SEE(A) =[cL

6log(zδ

)+cR6

log( z̄δ

)],

where z2 − z1 = z. (Holzhey et al. ’94, Calabrese, Cardy ’04)

Let us rotate z in the complex plane to z = Reiθ. Under this rotation,the EE transforms into

SEE =cL + cR

6log

(R

δ

)+cL − cR

6iθ .

Notice that the second term vanishes for theories where the Lorentzsymmetry is non-anomalous (i.e., cL = cR), but is present otherwise!

EE in a CFT2 (T=0) for a single interval A = [z1, z2].

SEE(A) =[cL

6log(zδ

)+cR6

log( z̄δ

)],

where z2 − z1 = z. (Holzhey et al. ’94, Calabrese, Cardy ’04)

Let us rotate z in the complex plane to z = Reiθ. Under this rotation,the EE transforms into

SEE =cL + cR

6log

(R

δ

)+cL − cR

6iθ .

Notice that the second term vanishes for theories where the Lorentzsymmetry is non-anomalous (i.e., cL = cR), but is present otherwise!

In terms of spacetime variables, under the analytic continuationz = x− t, z̄ = x+ t, the angle θ corresponds to a boost parameter κas θ = iκ. Therefore,

EE for the boosted interval

SEE =cL + cR

6log

(R

δ

)− cL − cR

6κ . (3)

(Wall ’11, Castro, Detournay, Iqbal, Perlmutter ’14)

İnönü-Wigner contraction

Identify the GCA generators with the linear combinations

Ln = Ln + L̄n , Mn = −�(Ln − L̄n

),

of their Virasoro counterparts, where � is a small parameter at thelevel of algebra.

[Lm, Ln] = (m− n)Ln+m +cLL12

m(m2 − 1)δm+n,0 ,

[Lm,Mn] = (m− n)Mn+m +cLM12

m(m2 − 1)δm+n,0 ,

[Mm,Mn] = 0 .

From the spacetime point of view it corresponds to ∆t→ �∆t and∆x→ ∆x, in the �→ 0 limit.(Bagchi et al., Barnich et al.)

Zero temperature EE for GCFT2

Shortcut to EE in GCFTs

1. Decompose R into space and time as

R2 = (∆x)2 − (∆t)2 , κ = arctanh(

∆t

∆x

).

2. Make İnönü-Wigner contraction.

EE of GCFT2 at zero temperature

SGCFT2EE =cLL6

log

(∆x

δ

)+cLM

6

(∆t

∆x

). (4)

(Bagchi et al. ’14, SMH, Veliz-Osorio ’15)

Zero temperature EE for GCFT2

Shortcut to EE in GCFTs

1. Decompose R into space and time as

R2 = (∆x)2 − (∆t)2 , κ = arctanh(

∆t

∆x

).

2. Make İnönü-Wigner contraction.

EE of GCFT2 at zero temperature

SGCFT2EE =cLL6

log

(∆x

δ

)+cLM

6

(∆t

∆x

). (4)

(Bagchi et al. ’14, SMH, Veliz-Osorio ’15)

Finite temperature EE for GCFT2

The EE for a CFT2 with gravitational anomaly (cL 6= cR) can bewritten as

SEE =cL6

log

[βLπδ

sinh

(πz

βL

)]+cR6

log

[βRπδ

sinh

(πz̄

βR

)],

where βL (βR) is the left (right)-moving inverse temperature.(Holzhey et al. ’94, Calabrese, Cardy ’04)

Once again we want to break the Lorentz invariance of the theory,thus we boost the interval and follow the same strategy used for thezero temperature case.

The EE for a CFT2 with gravitational anomaly (cL 6= cR) can bewritten as

SEE =cL6

log

[βLπδ

sinh

(πz

βL

)]+cR6

log

[βRπδ

sinh

(πz̄

βR

)],

where βL (βR) is the left (right)-moving inverse temperature.(Holzhey et al. ’94, Calabrese, Cardy ’04)

Once again we want to break the Lorentz invariance of the theory,thus we boost the interval and follow the same strategy used for thezero temperature case.

SGCFT2EE =cLL6

log

[βLLπδ

sinh

(π∆x

βLL

)]− cLM

6

βLMβLL

+cLMπ

6βLL

(∆t+

βLMβLL

∆x

)coth

(π∆x

βLL

),

where we defined

βLL = lim�→0

1

2(βL + βR) , βLM = lim

�→0

1

2�(βL − βR) ,

which can be understood as the inverse temperatures of a GCFT2.(SMH, Veliz-Osorio ’15)

Cardy formula for GCFTs

SGCFT2thermal =π2

3β2LL(cLLβLL + cLMβLM ) .

SGCFT2EE =cLL6

log

[βLLπδ

sinh

(π∆x

βLL

)]− cLM

6

βLMβLL

+cLMπ

6βLL

(∆t+

βLMβLL

∆x

)coth

(π∆x

βLL

),

where we defined

βLL = lim�→0

1

2(βL + βR) , βLM = lim

�→0

1

2�(βL − βR) ,

which can be understood as the inverse temperatures of a GCFT2.(SMH, Veliz-Osorio ’15)

Cardy formula for GCFTs

SGCFT2thermal =π2

3β2LL(cLLβLL + cLMβLM ) .

Holographic EE from BMS3/GCA2

Topological massive gravity (TMG)

ITMG =1

16πG

∫d3x√−g[R− 2Λ + 1

2µCS(Γ)

],

where the Chern-Simons contribution is given by

CS(Γ) = εαβγΓρασ

(∂βΓ

σγρ +

2

3ΓσβηΓ

ηγρ

).

Canonical analysis of TMG yields the BMS3 algebra with

cLL =3

Gµ, cLM =

3

G.

The Einstein gravity limit corresponds to taking µ→∞.

Topological massive gravity (TMG)

ITMG =1

16πG

∫d3x√−g[R− 2Λ + 1

2µCS(Γ)

],

where the Chern-Simons contribution is given by

CS(Γ) = εαβγΓρασ

(∂βΓ

σγρ +

2

3ΓσβηΓ

ηγρ

).

Canonical analysis of TMG yields the BMS3 algebra with

cLL =3

Gµ, cLM =

3

G.

The Einstein gravity limit corresponds to taking µ→∞.

BTZ black hole

The equations of motion of TMG admit a rotating BTZ solution,

ds2BTZ`2

= −(r2 − r2+)(r2 − r2−)

r2dτ2 +

r2

(r2 − r2+)(r2 − r2−)dr2

+ r2(dφ+

r+r−r2

dτ)2

, (5)

where ` is the AdS radius of curvature. We define the quantities

m =r2+ − r2−

8G`2, j =

r+r−4G`

.

In pure Einstein gravity m and j correspond to the mass and angularmomentum of the black hole.

Due to the Chern-Simons term in the action there is a shift in theconserved quantities associated to the TMG-BTZ black hole

M = m− jµ`2

, J = j − mµ.

Holographic EE for the BTZ geometry in TMG

SBTZEE =cL + cR

12log

[βLβRπ2δ2

sinh

(πz

βL

)sinh

(πz̄

βR

)]

− cL − cR12

log

sinh(πz̄βR

)βR

sinh(πzβL

)βL

. (6)where

βL =2π`

r+ − r−, βR =

2π`

r+ + r−.

(Hubeny et al. ’07, Castro et al. ’14)Seyed Morteza Hosseini University of Milano-Bicocca

Flat space analogue of the rotating BTZ black hole solution!

Flat space cosmology (FSC)!

These geometries describe expanding (contracting) universes and canbe viewed as the flat space (Λ→ 0) limit of rotating BTZ black holes.

ds2FSC = r̂2+

(1− r

20

r2

)dτ2 − dr

2

r̂2+

(1− r

20

r2

) + r2(dφ− r̂+r0r2

dτ

)2,

r+ → `√

8Gm = `r̂+ , r− →√

2G

m|j| = r0 .

(Cornalba, Costa ’02)

Flat space analogue of the rotating BTZ black hole solution!

Flat space cosmology (FSC)!

These geometries describe expanding (contracting) universes and canbe viewed as the flat space (Λ→ 0) limit of rotating BTZ black holes.

ds2FSC = r̂2+

(1− r

20

r2

)dτ2 − dr

2

r̂2+

(1− r

20

r2

) + r2(dφ− r̂+r0r2

dτ

)2,

r+ → `√

8Gm = `r̂+ , r− →√

2G

m|j| = r0 .

(Cornalba, Costa ’02)

Claim

The holographic EE of FSC can be found by following ourprescription to implement an İnönü-Wigner contraction on theholographic EE of the rotating BTZ black hole!

Holographic EE of FCS

SFSCEE =cLL6

log

[β+πδ

sinh

(π∆x

β+

)]− cLM

6

β0β+

+cLMπ

6β+

(∆t+

β0β+

∆x

)coth

(π∆x

β+

), (7)

β+ =2π

r̂+, β0 =

2πr0r̂2+

, cLL =3

Gµ, cLM =

3

G.

Claim

The holographic EE of FSC can be found by following ourprescription to implement an İnönü-Wigner contraction on theholographic EE of the rotating BTZ black hole!

Holographic EE of FCS

SFSCEE =cLL6

log

[β+πδ

sinh

(π∆x

β+

)]− cLM

6

β0β+

+cLMπ

6β+

(∆t+

β0β+

∆x

)coth

(π∆x

β+

), (7)

β+ =2π

r̂+, β0 =

2πr0r̂2+

, cLL =3

Gµ, cLM =

3

G.

I Einstein gravity. We recover Einstein gravity by considering thelimit µ→∞.

cLM6β+

[π

(∆t+

β0β+

∆x

)coth

(π∆x

β+

)− β0

].

I Boost orbifold. This limit corresponds to taking the angularmomentum j to zero (β0 = 0).

cLL6

log

[β+πδ

sinh

(π∆x

β+

)]+cLMπ

6β+∆t coth

(π∆x

β+

).

(Bagchi et al. ’14)

I Einstein gravity. We recover Einstein gravity by considering thelimit µ→∞.

cLM6β+

[π

(∆t+

β0β+

∆x

)coth

(π∆x

β+

)− β0

].

I Boost orbifold. This limit corresponds to taking the angularmomentum j to zero (β0 = 0).

cLL6

log

[β+πδ

sinh

(π∆x

β+

)]+cLMπ

6β+∆t coth

(π∆x

β+

).

(Bagchi et al. ’14)

I Thermal entropy. The thermal entropy of the system can berecovered by considering a large entangling region.

SFSCthermal =2πr04G

+2πr̂+4Gµ

.

(Bagchi, Basu ‘13)

I Flat space chiral gravity. Finally, we consider the case where thebulk theory consists only of the gravitational Chern-Simonspiece.

Conformal Chern-Simons gravity (CSG)

G→∞,while keeping µG = 1/8k fixed.

cLL = 24k , cLM = 0 ,

(Bagchi, Detournay, Grumiller ’12)

+2πr̂+4Gµ

.

cLL = 24k , cLM = 0 ,

+2πr̂+4Gµ

.

cLL = 24k , cLM = 0 ,

Conformal Chern-Simons gravity action

ICSG =k

4π

∫d3x√−gCS(Γ) .

The asymptotic symmetry algebra for CSG reduces to a chiral copy ofthe Virasoro algebra.

SχEE =cLL6

log

[β+πδ

sinh

(π∆x

β+

)].

It has exactly the form of a chiral CFT at finite temperature!

Conformal Chern-Simons gravity action

ICSG =k

4π

∫d3x√−gCS(Γ) .

The asymptotic symmetry algebra for CSG reduces to a chiral copy ofthe Virasoro algebra.

SχEE =cLL6

log

[β+πδ

sinh

(π∆x

β+

)].

It has exactly the form of a chiral CFT at finite temperature!

Outlook

• Entanglement c-function?!

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

(SMH, work in progress.)

• Entanglement of localized excited states?!• Generalization to 4D?!• And many other open issues. . .

Longterm plan

A topologically twisted index for three-dimensional supersymmetric theories.

(Benini, Zaffaroni ’15)

Outlook

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

Longterm plan

Outlook

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

• Entanglement of localized excited states?!

• Generalization to 4D?!• And many other open issues. . .

Longterm plan

Outlook

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

• Entanglement of localized excited states?!• Generalization to 4D?!

• And many other open issues. . .

Longterm plan

Outlook

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

Longterm plan

Outlook

cLL(∆x,∆t) =

[∆x

∂

∂(∆x)+ ∆t

∂

∂(∆t)

]SEE(∆x,∆t) ,

cLM (∆x,∆t) = ∆x∂

∂(∆t)SEE(∆x,∆t) .

Longterm plan

seyed morteza hosseini - istituto nazionale di fisica nuclearehosseini/pdf/flat ee.pdf · 2016. 1....

Documents