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IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
An example of non-AdS holography
Yang Lei
Institute of Theoretical physics, Chinese Academy of Science
Work with Simon Ross (1504.07252), Jelle Hartong, Niels Obersand Gerben Oling(1604.08054, 1712.05794)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
1 Introduction
2 Newton-Cartan gravityNon-relativistic symmtriesConstructing Newton-Cartan gravity
3 Novel non-AdS holography
4 Conclusion and future work
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Introduction: holography based on symmetry
It is believed the principle for quantum gravity theory isholographic principle. In the past decades, people are trying answerthe question: how general the holography principle can be? Thetheories dual to each other have the same symmetry group. Inabove case, SO(d , 2). In the special case
d = 2, we have SO(2, 2) ∼ SL(2,R)× SL(2,R). This isAdS3/CFT2 duality.
Higher spin generalization: SL(N,R) or more complicatedhs[λ];
p-adic holography, based on symmetry group SL(2,Qp)
Taking non-relativistic limit: this is found to beNewton-Cartan/NCFT duality
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Why non-relativistic holography is interesting?
Non-relativity means spacetime scales anisotropically
t → λz t, xi → λxi for z 6= 1
breaking Lorentz symmetry. This scaling is usually called Lifshitzsymmetry. Strongly correlated systems with this symmetry arecalled quantum Lifshitz fixed points.
Many condensed matter field theory exhibits non-relativisticscaling symmetry.Can one define conformal symmetry in non-relativistic sense?Non-relativistic quantum gravity (Horava-Lifshitz) can berenormalizable (Horava, 0812.4287)Does holography exist in general sense? (independent ofrelativity)Near horizon geometry of naked black hole (Horowitz, Ross,9704058)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
The theory cube
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic theory
Many body field theories describing anisotropic fixed points wereproposed to be holographically dual to gravity in the background ofLifshitz geometries, where time and space scale asymptoticallywith the same ratio z . Non-relativistic spacetime solutions arefound in Einstein gravity theory with gauge matter fields. Lifshitzspacetimes: (Kachru, Liu, Mulligan, 08)
ds2 = −r2zdt2 +dr2
r2+ r2dxidxi , At =
√2(z − 1)
zr z
Schrodinger spacetime (Son, 08)
ds2 = −r2zdt2 − 2r2dtdξ +dr2
r2+ r2dxidxi
z > 1 generically for Null Energy condition. Schrodinger algebracontains special conformal transformation at z = 2.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Not enough
The construction of non-relativistic holography by Einstein gravityis far beyond satisfaction.
We are embedding non-relativistic gravity into relativisticgravity
There are extra matter fields in the theory (massive gaugefields/higher curvature terms)
This does not appear to be non-relativistic limit of Einsteingravity.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
1 Introduction
2 Newton-Cartan gravityNon-relativistic symmtriesConstructing Newton-Cartan gravity
3 Novel non-AdS holography
4 Conclusion and future work
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Why should we consider Newton-Cartan gravity?
The ground breaking work to drag people’s attention toNewton-Cartan gravity is (Christensen, Hartong, Obers, Rollier,1311.6471). They found the Newton-Cartan gravity appears inLifshitz holography as one approaches the boundary of spacetime.
There is no need to discuss non-relativistic gravity by startingfrom relativistic gravity. Newton-Cartan gravity is built onnon-relativistic symmetry group.
Horava-Lifshitz gravity allows Lifshitz spacetime as vacuumsolution without matter to support.
Dynamical Newton-Cartan gravity is equivalent toHorava-Lifshitz gravity. (Hartong, Obers, 1504.07461)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Difficulty and debate
There are a lot of different arguments on Horava-Lifshitz gravityholgraphy, and some of relevant are
It has advantages: no need for extra massive gauge field;resolve non-renormalizable problem of gravity.
More negative arguments: it breaks diffeomorphism; as aquantum gravity, no Einstein limit; no well-defined black hole;no string embedding yet...
Main model: 3D gravity
We are going to study the problems in 3D gravity, which is alsotopological.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Non-relativistic symmetry group
The gravity theory is classified with respect to its isometry group.We are interested in several examples:
3D Galilean/Bargmann algebra contains H,Ga,Pa, J, (N)
[J,Pa] = εabPb, [J,Ga] = εabGb, [H,Ga] = Pa,
[Pa,Gb] = Nδab (Bargmann central extension)
a = 1, 2. Without non-relativistic boost, Galilean algebrareduces to Lifshitz-like algebra.
Newton-Hooke algebra, it includes Galilean algebra withcosmological constant
[H,Pa] = −ΛGa
Carrollian algebra (c → 0 limit)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Schrodinger algebra (introducing dilatation D) (Son...)
[H,D] = zH, [Pa,D] = Pa, [Ga,D] = (1− z)Ga
[D,N] = (z − 2)N
At z = 2, symmetry can be enhanced to include specialconformal transformation K , so that H,D,K form a SL(2,R)and [Pa,K ] = Ga
Galilean conformal algebra (Bagchi, Gopakumar) (too manycommutators...)
BMS algebra , with identification BMS3 is identical to GCA2
[Ln, Lm] = (n −m)Lm+n +cL12
n(n2 − 1)
[Ln,Mm] = (n −m)Mm+n +cM12
n(n2 − 1)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Einstein Gravity by gauging Poincare groups
In general dimensions, the Poincare algebra is
[MAB ,PC ] = ηACPB − ηBCPA
[MAB ,MCD ] = ηACMBD − ηADMBC − ηBCMAD + ηBDMAC
A = 0, 1, ..., d
Then gauge field
Aµ =1
2MABω
ABµ + PAe
Aµ
transforms under gauge transformation as δAµ = ∂µΛ + [Aµ,Λ]where
Λ =1
2MABσ
AB + PAζA
we can then derive
δeAµ = ∂µζA + eCµ σ
AC + ωAC
µ ζC
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
To identify gauge transformation ζa with diffeomorphism byζA = ξρeAρ , we need to study
Lξeaµ = ξρ(∂ρeAµ − ∂µeAρ ) + ∂µ(ξρeAρ )
= −ξρRρµ(PA) + ξρ(eρCωACµ + eµCω
ACρ ) + ∂µ(ξρeAρ )
whereF = dA + A ∧ A = PAR
A(P) + MABRAB(M)
we see we can make the identification as long as curvatureconstraint Rρµ(PA) = 0 is imposed. This is the torsion freecondition for Einstein gravity which is used to solvespin-connection uniquely from veilbein.
Invariant field
gµν = eAµ eBν ηAB is the geometric field invariant under local Lorentz
transformation.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Einstein gravity
Einstein equation is recovered by RAB(M) = 0. Note Einsteinequation is second order differential equation while gaugecurvature zero condition is first order differential equation. Theirequivalence is based on the fact spin-connection can be uniquelysolved by vielbein.
There are examples in which spin-connection cannot be uniquelysolved by vielbein. This can happen in higher spin non-relativisticsolutions (YL, Ross, 1504.07252). These solutions are said to bedegenerate. For these solutions, one cannot trust the descriptionby metric-like fields.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Gauging Galilean algebra
Galilean algebra in d + 1 dimensions contains H,Pa,Ga, Jab.Naively
A = Hτ + Paea + Gaω
a + Jabωab, a = 1, ..., d
But one immediately finds Ra(P) only contains d components ofequations. Recall Einstein gravity contains d + 1 components ofequations. This is because t component equation becomes
dτ = 0
imposing no constraints on spin-connection. As a result,spin-connection cannot be uniquely determined, making the gravitytheory ill-defined!
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Newton-Cartan gravity by gauging Bargmann algebra
First let me explain how to understand Newton-Cartan gravity ingeneral dimensions. By adding central extension [Pa.Gb] = Nδab,
A = Hτ + Paea + Gaω
a + Jabωab + Nm
m is the central extended gauge field, interpreted as mass current.Then it is easy to calculate
F = dA + A ∧ A
= HR(H) + Ra(P)Pa + Ra(G )Ga + Rab(J)Jab + R(N)N
We would like to impose the curvature constraints so thatspin-connection is uniquely solved in terms of smaller set of fields.In this case τ, ea and the gauge field m. Then we need to imposeR(H) = Ra(P) = R(N) = 0. (See Bergshoeff et. 1011.1145)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
After imposing curvature constraints, one can check
δτµ = ∂µ(ξρτρ)− ξρRρµ(H)
δeaµ = ∂µ(ξρeaρ)− ξρR aρµ (Pa) + (ξρωab
ρ )ebµ + (ξρωaρ)τµ
= ∂µξa + λabebµ + λaτµ
where ξρ = ξ0 = ξρτρ, ξa = ξρeaρ. Therefore, gauge
transformation is equivalent to diffeomorphism up to local Galileantransformation! The fields invariant under local Galileantransformations are τµ (clock one-form) and hµν = eµa eνb δ
ab
(spatial metric). One can always define the Lorentzian metric to be
gµν = −τµτν + hµν
Torsion constraints
One can show dτ is proportional to Γρ[µν], i.e. torsion tensor. SoBargmann theory with dτ = 0 is called torsionless Newton-Cartangravity.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Classifications
Based on the features of torsion, Newton-Cartan gravity isclassified into three kinds:
Torsionless Newton-Cartan gravity: dτ = 0. This correspondsto projectable Horava-Lifshitz gravity.
Twistless torsional Newton-Cartan gravity: τ ∧ dτ = 0. Thismeans there is a field b, such that dτ − zb ∧ τ = 0. Thismeans b is the field coupled to scaling generator D and[H,D] = zH. Therefore, we should expect once we introducescaling symmetry into Galilean algebra, (Schrodinger algebra),one should expect we obtain this kind of theory. This alsocorresponds to non-projectable Horava Lifshitz gravity
Torsional Newton-Cartan gravity: no constraints on τ (Neverseen examples yet..)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Chern-Simons theory
In d = 3 dimensional spacetime, one usually uses Ja = 12εabcM
bc
as the algebra generator. For
A = Paea + Jaω
a
we can write down Chern-Simons action
Scs = TrR
∫ (A ∧ dA +
2
3A ∧ A ∧ A
)Tr is actually an invariant bilinear product on Lie algebra, whichmaps two generators to a number. In case of ISO(2, 1), thisbilinear product is (Witten 88’)
< Pa, Jb >= ηab
resulting in Einstein-Hilbert action
S =
∫ea ∧ Ra(ω)
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Einstein AdS Chern-Simons
In terms of AdS gravity, gauge field A takes values in so(2, 2)algebra. This algebra allows a second parameters of bilinearproduct:
< Ja, Jb >= ηab, < Pa,Pb >= −Ληab
The presence of two parameters of bilinear products is the fact ofisomorphism
so(2, 2) = SL(2,R)× SL(2,R)
The bilinear product of such semisimple Lie algebra is given byCartan-Killing metric. This gives familiar AdS gravity
SEH = SCS [A]− SCS [A]
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Chern-Simons Newton-Cartan gravity
The Galilean/Bargmann algebra are not semi-simple. TheCartan-Killing metric is degenerate. It is pointed out by Wittenthat for non-semisimple Lie algebra, non-degenerate invariantbilinear product can exist sometimes. For Lie algebra[TA,TB ] = f C
AB TC , such invariant bilinear product is defined bysolving
f DABΩCD + f DACΩBD = 0
In (Witten, Nappi, 9310112), they consider algebraTA = J,P1,P2,T,
[J,Pa] = εabPb, [Pa,Pb] = εabT ,
which is known as central extended Poincare group Pc2 . They find
bilinear product< Pa,Pb >=< J,T > δab
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
However, even by solving the equations above, one cannot havenon-degenerate bilinear product (NDBP) for Bargmann algebra.This makes it hard to build Chern-Simons theory. A resolutionsuggested in (Papageorgiou, Schroers, 0907.2880) is to introducemore generators in the Lie algebra so that bilinear product can benon-degenerate. They find we only need to extend
[Ga,Gb] = Sεab
Then H,Pa,Ga, J,N,S form an NDBP, which is given by
< H,S >= − < J,N >= − < P1,G2 >=< P2,G1 >
Therefore, we can write down a Chern-Simons gravity action! Notethe whole algebra is Galilean with two u(1) extensions (Bergshoeff,Rosseel, 1604.08042).
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Chern-Simons action
Let’s take A = Hτ + Paea + GaΩa + JΩ + Nm + Sζ ,, then
Tr
(A ∧ dA +
2
3A ∧ A ∧ A
)=
−εabRa(G ) ∧ eb +1
2εabτ ∧ Ωa ∧ Ωb − Ω ∧ dm
+ζ ∧ dτ
where
Ra(G ) = dΩa − εabΩ ∧ Ωb
As one can see ζ works as Lagrangian multiplier, used to imposetorsionless condition dτ = 0! The extra generators are necessary.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Extended Bargmann gravity
The action above can be written in terms of gauge invariantmetric-like fields
L = e(hµρhνσKµνKρσ − (hµνKµν)2 − ΦR
),
where
Φ = −vµmµ +1
2hµνmµmν .
is the Newton Potential. Kµν is the extrinsic curvature, remindingus the decomposition of Einstein gravity in terms of extrinsiccurvature. R is the Ricci curvature on spatial directions.
u(1)2 extensions
N and S are two u(1) extensions. It has been proved the extendedBargmann algebra are non-relativistic limit of iso(2, 1)⊕ u(1)2
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Aside
For z = 2 Schrodinger algebra, we need to introduce three moregenerators to make Jacobi identity held and bilinear productnon-degenerate. (Niels Obers, Jelle Hartong, YL, 1604.08054)
A = Hτ + Paea + Gaω
a + Jω + Nm + Db + Kf
+Sζ + Yα + Zβ .
where [Pa,Pb] = εabZ and [Pa,Gb] = Nδab − Y εab. The action is
L = 2c1[R2(G ) ∧ e1 − R1(G ) ∧ e2 + τ ∧ ω1 ∧ ω2
−m ∧ dω − f ∧ e1 ∧ e2 + ζ ∧ (dτ − 2b ∧ τ)
+α ∧ (db − f ∧ τ) + β ∧ (df + 2b ∧ f )]
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
The extended Schrodinger Algebra
The z = 2 algebra we are considering is
[J,Pa] = εabPb, [J,Ga] = εabGb, [H,Ga] = Pa,
[Pa,Gb] = Nδab − εabY[Ga,Gb] = Sεab, [Pa,Pb] = εabZ
[H,D] = 2H, [Pa,D] = Pa, [Ga,D] = −Ga
[Pa,K ] = Ga, [D,K ] = 2K , [D,K ] = 2K
[H ,Y ] = −Z , [H , S ] = −2Y , [K ,Y ] = S ,
[K ,Z ] = 2Y , [D , S ] = 2S , [D ,Z ] = −2Z .
The bilinear product is
< H,S >= − < J,N >= − < P1,G2 >=< P2,G1 >
= < D,Y >=< K ,Z >
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Non-relativistic symmtriesConstructing Newton-Cartan gravity
Asymptotic symmetry
Take Lm = H,D,K, Mm = S ,Y ,Z, Y ir = Pa,Ga
[Lm, Ln] = (m − n)Lm+n +cL2
(m3 −m)δm+n,0
[Lm, Mn] = (m − n)Mm+n +c
2(m3 −m)δm+n,0
[Lm,Yir ] = (
m
2− r)Y i
m+r , [Lm, Jn] = −nJm+n,
[Lm,Nn] = −nNm+n, [Y ir ,Y
is ] = (r − s)Ns+r ,
[Y 1r ,Y
2s ] = −Mr+s + c
(s2 − 1
4
)δr+s,0, [Jn,Y
ir ] = Y j
r+nεij
[Jm,Nn] = cnδm+n,0 [Jn, Mm] = −2nNm+n,
[Jm, Jn] = cJnδm+n,0
Super GCA algebra (Bagchi, Gopakumar) with central extensions
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
1 Introduction
2 Newton-Cartan gravityNon-relativistic symmtriesConstructing Newton-Cartan gravity
3 Novel non-AdS holography
4 Conclusion and future work
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Newton-Hooke gravity
To get full analogy to AdS3/CFT2, we need cosmological constantto be introduced. Another feature we know of AdS3 in terms ofChern-Simons theory is that the Einstein-Hilbert action is differenceof two copies of Chern-Simons theory, as a result of isometry group
SO(2, 2) = SL(2,R)× SL(2,R)
What’s important to us is Newton-Hooke algebra can also bewritten in the similar product form.
pNH3 = Pc2 × Pc
2
Simple observation shows Pc2 is isomorphic to NH2.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
The Newton-Hooke Algebra
The algebra we are considering is
[J,Pa] = εabPb, [J,Ga] = εabGb, [H,Ga] = Pa,
[Pa,Gb] = Nδab
[Ga,Gb] = Sεab
[H,Pa] = −ΛGa, [Pa,Pb] = ΛSεab
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Bilinear product
H P1 P2 G1 G2 J N SH c5 0 0 0 0 c4 Λcc1 −c2P1 0 −Λcc1 0 0 −c2 0 0 0P2 0 0 −Λcc1 c2 0 0 0 0G1 0 0 c2 c1 0 0 0 0G2 0 −c2 0 0 c1 0 0 0J c4 0 0 0 0 c3 c2 c1N Λcc1 0 0 0 0 c2 0 0S −c2 0 0 0 0 c1 0 0
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
In fact, based on the extension in Bargmann case, we shouldconsider again the u(1)2 extension of AdS isometry algebra. Foreach copy of SL(2,R)× U(1), we consider their generators asLm,N0. Then
L−1 = cL−1 , L0 =1
2L0 +
c2
2N0 ,
L1 = cN1 , N0 = −1
2L0 +
c2
2N0 ,
by taking c →∞, we can see
[L−1 ,L0] = −iL−1 , [L−1 ,N1] = −iN0 , [L0 ,N1] = −iN1 .
which is exactly Pc2
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
One can write down Chern-Simons gravity action on Pc2 × Pc
2
Tr
(A ∧ dA +
2
3A ∧ A ∧ A
)=
2c1[− εabRa(G ) ∧ eb +
1
2εabτ ∧ Ωa ∧ Ωb − Ω ∧ dm
+ζ ∧ dτ + Λcτ ∧ e1 ∧ e2]
+ c2[Ωa ∧ Ra(G ) + 2ζ ∧ dΩ
+Λcea ∧ Ra(P)− 2Λcτ ∧ R(N) + Λce
a ∧ Ωa ∧ τ]
or equivalently
LNEH [c1] = τ ∧ dζ − ω ∧ dm + ω2 ∧ de1 − e2 ∧ dω1 − ω ∧ ea ∧ ωa
+Λτ ∧ e1 ∧ e2 + τ ∧ ω1 ∧ ω2
LTMG [c2] = 2ζ ∧ dω + ωa ∧ dωa + 2ω ∧ ω1 ∧ ω2
We can write LNEH [c1] = LCS [A]− LCS [A], for A takes values inPc2
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Reduction from Einstein gravity
We can obtain the theory above by starting from Einstein gravitywith two u(1) fields, and take non-relativistic limit.
IEH [c1] = −E 0 ∧ dΩ0 + E a ∧ dΩa + E 0 ∧ Ω1 ∧ Ω2 + E 1 ∧ Ω2 ∧ Ω0
+E 2 ∧ Ω0 ∧ Ω1 + ΛE 0 ∧ E 1 ∧ E 2 + Z1 ∧ dZ2
ITMG [c2] = −Ω0 ∧ dΩ0 + Ωa ∧ dΩa + 2Ω0 ∧ Ω1 ∧ Ω2 + Z2 ∧ dZ2
E 0 =1
2
(2cτ +
m
c
)Z1 =
1
2
(2cτ − m
c
)Ω1 =
1
cω2
Ω2 = −1
cω1, Ω0 =
1
2
(2ω − ζ
c2
), Z2 =
1
2
(2ω +
ζ
c2
)take c →∞ limit, one can obtain NH gravity.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Asymptotic symmetry
The story is even valid for Brown-Henneaux boundary condition.We can write down asymptotic solution:
a+ = N+ + F0L0 + F−L− + FNN0.
By calculating asymptotic symmetry algebra δa = dε+ [a, ε],
[Lm,Ln] = (m − n)Lm+n + 2πγ1m3δm+n,0,
[Lm,Nn] = −nNm+n − 2πi m2(
2γ2 −γ1c2
)δm+n,0,
[Nm,Nn] = −4πm
c2
(2γ2 −
γ1c2
)δm+n,0.
where γ1, γ2 are two parametres in non-degenerate bilinear product.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Asymptotic symmetry
For any finite c , the algebra is equivalent to Virasoro with u(1)affine by redefinition
L′n = Ln + aNn + b∑m
Nn−mNm
But for c →∞, which is case we are interested in, we should havetwisted u(1) Virasoro algebra
[Lm,Ln] = (m − n)Lm+n + 2πγ1m(m2 − 1)δm+n,
[Lm,Nn] = −nNm+n − 4πiγ2m(m + 1)δm+n.
up to a shift in zero mode.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Therefore, we have analogy with AdS/CFT.Relativistic Non-relativistic
AdS3 pNH 3D
isometry SL(2,R)× SL(2,R) isometry Pc2 × Pc
2
Vir × Vir Virt with U(1) × Virt with U(1)
Coset SO(2,2)SO(2,1) Coset
Pc2×Pc
2Pc2×U(1)
Topological massive gravity NH gravity with dynamics
CFT twisted WCFT (maybe)
higher dimension so(d+1,2) with extension so(d , 1)⊕ u(1)
WZW Exist, Liouville theory WZW Exist, (Nappi-Witten)
BTZ black hole ???
Higher spin generalization hs[λ] ???
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Higher dimensional generalizations
Consider the conformal algebra so(d + 1, 2) in d + 1 dimensions,
[D,Pa] = iPa, [D,Ka] = −iKa, [Pa,Kb] = −2iDηab − 2iMab
[Mab,Kc ] = i (ηacKb − ηbcKa) , [Mab,Pc ] = i (ηacPb − ηbcPa)
[Mab,Mcd ] = i (ηacMbd + ηbdMac − ηadMbc − ηbcMad) ,
We add to this a u(1) generator Q. We can also introduce anotherso(d , 1) algebra generated by Zab whose commutation relations are
[Zab,Zcd ] = i (ηacZbd + ηbdZac − ηadZbc − ηbcZad) .
By making combination
Pa = cPa, Ka = cKa, D =D2
+ c2N , Q = c2N − D2
Mab =Mab
2+ c2Sab, Zab =
Mab
2− c2Sab .
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Higher dimensional generalizations
The higher dimensional generalization of Galilean algebra withu(1)⊕ so(d , 1) extension is
[Pa,Kb] = −2iNηab − 2iSab, [D,Pa] = iPa, [D,Ka] = −iKa,
[Mab,Kc ] = i (ηacKb − ηbcKa) , [Mab,Pc ] = i (ηacPb − ηbcPa)
[Mab,Mcd ] = i (ηacMbd + ηbdMac − ηadMbc − ηbcMad)
[Mab,Scd ] = i (ηacSbd + ηbdSac − ηadSbc − ηbcSad) .
In d = 3, this theory is possibly related to spin matrix theory – thenon-relativistic limit of N = 4 super Yang-Mills theory (Harmark,Orselli, 1409.4417).
λ→ 0, E − ~Ω · ~J → 0,E − ~Ω · ~J
λfixed
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
String theory embedding
What is the string embedding of this gravity? The question isclosely related to the origin of two u(1) gauge fields. Onepossible interpretation is that the u(1) gauge fields areR-symmetry generators of supergravity. The enlightens us toconsider N = (2, 2) supergravity.
What happens to conformal algebra so(2, 2) if we takec →∞? The truth is by defining g = c−2,
E =D
c2= N +
g
2D, J = −Q
c2= N − g
2D
E is getting close to R-charge, so this is near BPS state.
Near BPS limit
We should think of this limit to be zooming in states near the BPSbound.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
1 Introduction
2 Newton-Cartan gravity
3 Novel non-AdS holography
4 Conclusion and future work
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Conclusion
We pave the way to build up a non-relativistic gravity theorywhich is completely analogous to AdS3 gravity. It isinteresting to talk about any generalizations about it.
We showed the whole phase space of WZW model of AdS canbe mapped to its non-relativistic limit.
We also derive the asymptotic symmetry of this non-AdSsolution, which is twisted u(1) Virasoro algebra.
We hope this work shed light upon a non-relativistic version ofholography, independent of Einstein gravity.
We hope this model could provide a simpler model ofholography, and the time is the holographic direction.Holographic direction decouples and is emergent.
Yang Lei
IntroductionNewton-Cartan gravity
Novel non-AdS holographyConclusion and future work
Future work
We have many future work can do:
NH supergravity/ NH torsional gravity
String embedding from AdS reduction
Higher spin
Generalization of conformal symmetry
BTZ black hole analogue
Applications to cosmology
Non-relativistic string theory and dualities
Generalization of entanglement entropy in non-relativistic CFT
Everything else you did in AdS3/CFT2
Yang Lei