Flatspace Holography as a Limit of AdS/CFT

Flatspace Holography as a Limit of AdS/CFT
Reza Fareghbal
School of Particles and Accelerator,IPM,Tehran
Seventh Crete Regional Meeting in String Theory
June 2013
Based on:
A. Bagchi, R. F. (arXiv:1203.5795),
A. Bagchi, S. Detournay, R. F. , Joan Simon (arXiv:1208.4372)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Motivation
AdS/CFT provides a holographic picture for the gravitational
theory.
Is it possible to generalize holography for gravity in other
backgrounds?
Our universe is more like flat rather than AdS!
Does flat space have a holographic picture as a lower
dimensional field theory!
Our answer is yes! Starting from AdS/CFT, one can study
flat-space holography.
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

First step: Symmetries
AdS d isometry= SO(2, d − 1) = symmetries of CFT d−1 .
Dictionary of AdS/CFT → (Asymptotic symmetries of
gravity=symmetries of dual theory)
Often, ASG→ isometry of the vacuum:
ASG(AdS d+2 ) = SO(d + 1, 2). (for d > 1.)
Famous exception: ASG(AdS 3 ) = Virasoro ⊗ Virasoro.
[Brown, Henneaux 1986]
Another interesting example: Near Horizon Extreme Kerr.
[Bardeen, Horowitz 1999]; [Guica, Hartman, Song, Strominger 2008]
Non-trivial ASG for asymptotically Minkowski spacetimes at
null infinity in three and four dimensions.
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Asymptotic Symmetries and Flat-spacetimes
BMS 3 :
[J m , J n ] = (m − n)J m+n , [J m , P n ] = (m − n)P m+n , [P m , P n ] = 0.
(1)
J m = Global Conformal Generators of S 1 .
P m = ”Super” translations (depends on θ) → Infinite
dimensional.
[Ashtekar, Bicak, Schmidt 1996]
Further Extension: Can compute central extensions:
[Barnich-Compere 2006]
C JJ = 0, C JP = 3 G , C PP = 0 (2)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Asymptotic Symmetries and Flat-spacetimes .
BMS 4 :
[l m , l n ] = (m − n)l m+n , [¯l m ,¯l n ] = (m − n)¯l m+n , [l m ,¯l n ] = 0,
[l l , T m,n ] = ( l + 1 − m)T m+l,n , [¯l l , T m,n ] = ( l + 1 − n)T m,n+l .
2
2
l m ,¯l m = Global Conformal Generators of S 2 .
T m,n = ”Super” translations → Infinite dimensional.
(m, n = 0, ±1): Poincare group in 4d.
[ Bondi, van der Burg, Metzner; Sachs 1962]
Virasoros can have central extensions. [ Barnich, Troessaert 2009]
One may expect that holographic dual of flat space has BMS
symmetry.What is this theory?
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Flat space limit of AdS
AdS 3 space-time:
ds 2 = −(1 + r 2
l 2 )dt2 + (1 + r 2
l 2 )−1 dr 2 + r 2 dφ 2 (3)
In the limit l → ∞(zero cosmological constant), we find flat
space.
Asymptotic Symmetry is Vir ⊗ Vir:
[L m , L n ] = (m − n)L m+n + c 12 m(m2 − 1)δ m+n,0
[¯L m , ¯L n ] = (m − n)¯L m+n + ¯c 12 m(m2 − 1)δ m+n,0
where c = ¯c = 3l/2G.
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Flat space limit of AdS.
Define: L n = L n − ¯L −n ,
l → ∞ is well-defined and yields
where
M n = (L n + ¯L −n )/l.
[L m , L n ] = (m − n)L m+n + C 1
12 m(m2 − 1)δ m+n,0
[L m , M n ] = (m − n)M m+n + C 2
12 m(m2 − 1)δ m+n,0
[Barnich-Compere 2006]
C 1 = c − ¯c = 0, C 2 =
(¯c + c)
l
= 3 G
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Making sense of flat space limit in the CFT side
What does flat space limit mean in the CFT side?
Dual of AdS 3 in the global coordinate is a two dimensional
CFT on the cylinder with generators:
L n = −e nw ∂ w , ¯Ln = −e n ¯w ∂ ¯w (w = t + ix, ¯w = t − ix)
(4)
Define L n = L n − ¯L −n , M n = ɛ(L n + ¯L −n ).
Use a spacetime contraction: t → ɛt, x → x.
Final generators in the ɛ → 0 limit:
L n = −e inx (i∂ x + nt∂ t ), M n = −e inx ∂ t (5)
The resultant algebra in the ɛ → 0 limit is BMS 3 with central
charges: c LL = C 1 = c − ¯c, c LM = C 2 = ɛ(c + ¯c)
[A. Bagchi, R. F. (2012)]
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Making sense of flat space limit in the CFT side.
Proposal: Holographic dual of flat space-time is a
t-contracted CFT.
For the two-dimensional case, it is a non-relativistic field
theory which has Galilean Conformal symmetry.
For the higher-dimensional cases we can apply contraction of
time. The resultant algebra is higher-dimensional BMS.
Correlation function
The limit from the correlation functions of parent CFT yield
interesting structures.
Two point function of two primary operators is given by
G (2) (φ 1 , τ 1 , φ 2 , τ 2 ) = C 1 (1 − cos φ 12 ) −2h L
e ih Mτ 12 cot(φ 12 /2)
where h L = lim ɛ→0 (h − ¯h), h M = lim ɛ→0 ɛ(h + ¯h)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Next Step: Flat limit of BTZ
No asymptotically flat black hole solutions in the three
dimensional gravity. [D. Ida, 2000]
The flat limit of BTZ is well-defind!
BTZ black holes:
ds 2 = − (r 2 − r+)(r 2 2 − r−)
2
r 2 l 2 dt 2 r 2 l 2
+
(r 2 − r+)(r 2 2 − r−) 2 dr 2
l → ∞ :
+ r 2 ( dφ − r +r −
lr 2 dt ) 2
r − → r 0 =
√
2G
M |J|, r + → lˆr + = l √ 8GM (6)
Flat BTZ: a cosmiological solution of 3d Einstein gravity
ds 2 FBTZ = ˆr 2 +dt 2 −
r 2 dr 2
ˆr 2 +(r 2 − r 2 0 ) + r 2 dφ 2 − 2ˆr + r 0 dtdφ (7)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Next Step: Flat limit of BTZ.
FBTZ is a shifted- boost orbifold of R 1,2
[L. Cornalba, M. S. Costa(2002)]
r = r 0 is a cosmological horizon:
T = ˆr +
2 , S = πr 0
2πr 0 2G
(8)
Satisfy the inner horizon first law!
TdS = −dM + ΩdJ (9)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Dual field theory analysis: a Cardy-like formula
3d asymptotically flat spacetimes −→ states of field theory
with BMS symmetry.
The states are labelled by eigenvalues of L 0 and M 0 :
where
L 0 |h L , h M 〉 = h L |h L , h M 〉, M 0 |h L , h M 〉 = h M |h L , h M 〉,
h L = lim
ɛ→0
(h − ¯h) = J,
h M = lim
ɛ→0
ɛ(h + ¯h) = GM + 1 8 (10)
Degeneracy of states with (h L , h M ) −→ Entropy of
cosmological solution with (r 0 , ˆr + )
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Dual field theory analysis: a Cardy-like formula
Is there any Cardy-like formula? YES
Partition function of t-contracted CFT:
Z(η, ρ) = ∑ d(h L , h M )e 2π(ηh L+ρh M )
(11)
where
η = 1 2 (τ + ¯τ), ρ = 1 ɛ(τ − ¯τ) (12)
2
S-transformation:
(τ, ¯τ) → (− 1 τ , − 1¯τ ) ⇒ (η, ρ) → (− 1 η , ρ η 2 ) (13)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Dual field theory analysis: a Cardy-like formula .
Modular invariance of
Z 0 (η, ρ) = e −πiρC M
Z(η, ρ) (14)
results in
S = log d(h L , h M ) = 2πh L
√
C M
2h M
(15)
[A. Bagchi, S. Detournay, R. F. , Joan Simon (2012)]
In agreement with the entropy of cosmological solution!
At sufficiently high temperature hot flat space tunnels into a
universe described by flat space cosmology.[A. Bagchi, S.
Detournay, D. Grumiller, J. Simon ]
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Current Directions
Extension of BMS algebra to higher spin case [Work in progress
with Hamid Afshar, Daniel Grumiller and Jan Rosseel]
sl(3, R) algebra:
[L m , L n ] = (m − n)L m+n
[L m , W n ] = (2m − n)W m+n
[W m , W n ] = (m − n)(2m 2 + 2n 2 − mn − 8)L m+n (16)
where m, n are 0, ±1 in L m and 0, ±1, ±2 in W m .
New generators:
L m = L m − ¯L −m , M m = ɛ(L m + ¯L −m )
U m = (W m − ¯W −m ), V m = ɛ(W m + ¯W −m ) (17)
ɛ → 0 is well-defined:
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Current Directions
[L m , L n ] = (m − n)L m+n [L m , M n ] = (m − n)M m+n
[L m , U n ] = (2m − n)U m+n [L m , V n ] = (2m − n)V m+n
[M m , U n ] = (2m − n)V m+n
[U m , U n ] = (m − n)(2m 2 + 2n 2 − mn − 8)L m+n
[U m , V n ] = (m − n)(2m 2 + 2n 2 − mn − 8)M m+n (18)
Extension of BMS algebra−→ WBMS algebra!
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Current Directions
Question: Is WBMS a symmetry of higher spin gravity in the
flat spacetimes?
Spin 2 case: Chern-Simons formulation
A flat = A n M M n + A n L L n, n = 0, ±1 (19)
L n and M n are generators of BMS and
A M = e 1 , A 1 M = 1 2 (e0 + e 2 ), A −1
M = 1 2 (e0 − e 2 ),
A 0 L = ω1 , A 1 L = 1 2 (ω0 + ω 2 ), A −1
L
= 1 2 (ω0 − ω 2 ).
(20)
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Current Directions
Our observation:
L n = lim
l→∞
(L n − ¯L −n ),
1
M n = lim
l→∞ l (L n + ¯L −n ),
A n L = lim 1
l→∞ 2 (An + Ān ), A n M = lim l
l→∞ 2 (An − Ān ). (21)
A proposal: 3d spin three flat gravity with WBMS symmetry
where
A flat = A n M M n + A n L L n + A m U U m + A m V V m, (22)
U m = lim
l→∞
(W m − ¯W −m ),
n = 0, ±1, m = 0, ±1, ±2
1
V m = lim
l→∞ l (W m + ¯W −m )
A m U = lim 1
l→∞ 2 (Am W + Ām W ), Am V = lim
l→∞
Poisson-brackets of boundary preserving gauge
transformations results in WBMS algebra.
l
2 (Am W − Ām W ).
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Summary
Flat space limit in the bulk side is equivalent to t-contraction
in the boundary theory.
Equivalence of symmetries is an evidence.
Correlation functions are well-defined.
Counting of states supports this proposal.
One can talk about higher-spin gravity in the 3d flat
spacetime.
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Future Direction
Entanglement Entropy of BMS field theory
l → ∞ limit of holographic entanglement entropy in the
global AdS 3 is well-defined:
S = Λ ( ) ∆φ
2G sin = 1 ( ) ∆φ
2 6 c MΛ sin
2
(23)
Is it possible to derive it from BMS field theory?
Black Hole Entropy
Possible existence of BMS algebra in the NH geometries of
non-extremal Black holes (Branes).
Can we reproduce entropy of non-extremal black holes through
a microscopic description in terms of contracted CFT?
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT

Thank you
Reza Fareghbal
Flatspace Holography as a Limit of AdS/CFT