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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
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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].
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 2 / 21
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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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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. . . .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 3 / 21
-
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.)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 4 / 21
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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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 5 / 21
-
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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 5 / 21
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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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 6 / 21
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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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 6 / 21
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Gravitational anomaly in CFT2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 7 / 21
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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.)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 8 / 21
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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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 9 / 21
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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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 9 / 21
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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.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 10 / 21
-
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.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 10 / 21
-
Finite temperature EE for GCFT2
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 ) .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 11 / 21
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Finite temperature EE for GCFT2
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 ) .
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 11 / 21
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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→∞.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 12 / 21
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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→∞.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 12 / 21
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Holographic EE from BMS3/GCA2
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.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 13 / 21
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Holographic EE from BMS3/GCA2
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
Entanglement and flat space holography 14 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 15 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 15 / 21
-
Holographic EE from BMS3/GCA2
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.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 16 / 21
-
Holographic EE from BMS3/GCA2
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.
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 16 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 17 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 17 / 21
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Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 18 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 18 / 21
-
Holographic EE from BMS3/GCA2
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 18 / 21
-
Holographic EE from BMS3/GCA2
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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 19 / 21
-
Holographic EE from BMS3/GCA2
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!
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 19 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
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)
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 20 / 21
-
Seyed Morteza Hosseini University of Milano-Bicocca
Entanglement and flat space holography 21 / 21
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