An example of non-AdS holographyicts.ustc.edu.cn/chinese/seminar/transparencies/Yang Lei/Newton-Cartan.pdf · AB is the geometric eld invariant under local Lorentz transformation.

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



1 Introduction

2 Newton-Cartan gravityNon-relativistic symmtriesConstructing Newton-Cartan gravity

3 Novel non-AdS holography

4 Conclusion and future work

Yang Lei



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



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



The theory cube

Yang Lei



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



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



Non-relativistic symmtriesConstructing Newton-Cartan gravity

1 Introduction




Yang Lei




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




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




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




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




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




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




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




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




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 + 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




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




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




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




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




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




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




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




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




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 + 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




The extended Schrodinger Algebra

The z = 2 algebra we are considering is


[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




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



1 Introduction




Yang Lei



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



The Newton-Hooke Algebra

The algebra we are considering is


[Pa,Gb] = Nδab

[Ga,Gb] = Sεab

[H,Pa] = −ΛGa, [Pa,Pb] = ΛSεab

Yang Lei



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



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



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



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



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



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



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



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



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



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



1 Introduction

2 Newton-Cartan gravity



Yang Lei



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



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

An example of non-AdS holographyicts.ustc.edu.cn/chinese/seminar/transparencies/Yang Lei/Newton-Cartan.pdf · AB is the geometric eld invariant under local Lorentz transformation.

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