Gravity in lower dimensions - quark.itp.tuwien.ac.at

Gravity in lower dimensions

Daniel Grumiller

Institute for Theoretical PhysicsVienna University of Technology

Center for Theoretical Physics, Massachusetts Institute of Technology,December 2008

Outline

Why lower-dimensional gravity?

Which 2D theory?

Which 3D theory?

How to quantize 3D gravity?

What next?

D. Grumiller — Gravity in lower dimensions 2/42

Outline


Which 2D theory?

Which 3D theory?


What next?

D. Grumiller — Gravity in lower dimensions Why lower-dimensional gravity? 3/42

Quantum gravityThe Holy Grail of theoretical physics

There is a lot we do know about quantum gravity already

I It should exist in some form

I String theory: (perturbative) theory of quantum gravity

I Microscopic understanding of extremal BH entropy

I Conceptual insight — information loss problem resolved

There is a lot we still do not know about quantum gravity

I Reasonable alternatives to string theory?

I Non-perturbative understanding of quantum gravity?

I Microscopic understanding of non-extremal BH entropy?

I Experimental signatures? Data?















































































































Riemann-tensor D2(D2−1)12 components in D dimensions:

I 11D: 1210 (1144 Weyl and 66 Ricci)I 10D: 825 (770 Weyl and 55 Ricci)I 5D: 50 (35 Weyl and 15 Ricci)I 4D: 20 (10 Weyl and 10 Ricci)

I 3D: 6 (Ricci)I 2D: 1 (Ricci scalar)

I 2D: lowest dimension exhibiting black holes (BHs)

I Simplest gravitational theories with BHs in 2D

I 3D: lowest dimension exhibiting BHs and gravitons

I Simplest gravitational theories with BHs and gravitons in 3D




I 11D: 1210 (1144 Weyl and 66 Ricci)I 10D: 825 (770 Weyl and 55 Ricci)I 5D: 50 (35 Weyl and 15 Ricci)I 4D: 20 (10 Weyl and 10 Ricci)I 3D: 6 (Ricci)I 2D: 1 (Ricci scalar)






















Outline


Which 2D theory?

Which 3D theory?


What next?

D. Grumiller — Gravity in lower dimensions Which 2D theory? 6/42

Attempt 1: Einstein–Hilbert in and near two dimensions

Let us start with the simplest attempt. Einstein-Hilbert action in 2dimensions:

IEH =1

16πG

∫d2x

√|g|R =

12G

(1− γ)

I Action is topological

I No equations of motion

I Formal counting of number of gravitons: -1

A specific 2D dilaton gravity model

Result of attempt 1:



Let us continue with the next simplest attempt. Einstein-Hilbertaction in 2+ε dimensions:

IEHε =

116πG

∫d2+εx

√|g|R

I Weinberg: theory is asymptotically safe

I Mann: limit ε→ 0 should be possible and lead to 2D dilaton gravity

I DG, Jackiw: limit ε→ 0 yields Liouville gravity

limε→0

IEHε =

116πG2

∫d2x

√|g|

[XR− (∇X)2 + λe−2X

]





Let us continue with the next simplest attempt. Einstein-Hilbertaction in 2+ε dimensions:

IEHε =

116πG

∫d2+εx

√|g|R

I Weinberg: theory is asymptotically safe

I Mann: limit ε→ 0 should be possible and lead to 2D dilaton gravity

I DG, Jackiw: limit ε→ 0 yields Liouville gravity

limε→0

IEHε =

116πG2

∫d2x

√|g|

[XR− (∇X)2 + λe−2X

]




Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model

Jackiw, Teitelboim (Bunster): (A)dS2 gauge theory

[Pa, Pb] = Λ εabJ [Pa, J ] = εabPb

describes constant curvature gravity in 2D. Algorithm:

I Start with SO(1, 2) connection A = eaPa + ωJI Take field strength F = dA+ 1

2 [A,A] and coadjoint scalar XI Construct non-abelian BF theory

I =∫XAF

A =∫ [

Xa(dea + εabω ∧ eb) +X dω + εabea ∧ eb ΛX

]I Eliminate Xa (Torsion constraint) and ω (Levi-Civita connection)I Obtain the second order action

I =1

16πG2

∫d2x

√−g X [R− Λ]







describes constant curvature gravity in 2D. Algorithm:I Start with SO(1, 2) connection A = eaPa + ωJ

I Take field strength F = dA+ 12 [A,A] and coadjoint scalar X

I Construct non-abelian BF theory

I =∫XAF

A =∫ [



I =1

16πG2

∫d2x

√−g X [R− Λ]







describes constant curvature gravity in 2D. Algorithm:I Start with SO(1, 2) connection A = eaPa + ωJI Take field strength F = dA+ 1

2 [A,A] and coadjoint scalar X

I Construct non-abelian BF theory

I =∫XAF

A =∫ [



I =1

16πG2

∫d2x

√−g X [R− Λ]









I =∫XAF

A =∫ [


]

I Eliminate Xa (Torsion constraint) and ω (Levi-Civita connection)I Obtain the second order action

I =1

16πG2

∫d2x

√−g X [R− Λ]









I =∫XAF

A =∫ [


]I Eliminate Xa (Torsion constraint) and ω (Levi-Civita connection)

I Obtain the second order action

I =1

16πG2

∫d2x

√−g X [R− Λ]









I =∫XAF

A =∫ [



I =1

16πG2

∫d2x

√−g X [R− Λ]









I =∫XAF

A =∫ [



I =1

16πG2

∫d2x

√−g X [R− Λ]




Attempt 3: Dimensional reductionFor example: spherical reduction from D dimensions

Line element adapted to spherical symmetry:

ds2 = g(D)µν︸︷︷︸

full metric

dxµ dxν = gαβ(xγ)︸︷︷︸2D metric

dxα dxβ − φ2(xα)︸︷︷︸surface area

dΩ2SD−2

,

Insert into D-dimensional EH action IEH = κ∫

dDx√−g(D)R(D):

IEH = κ2π(D−1)/2

Γ(D−12 )

∫d2x

√−g φD−2

[R+

(D − 2)(D − 3)φ2

((∇φ)2 − 1

) ]Cosmetic redefinition X ∝ (λφ)D−2:

IEH =1

16πG2

∫d2x

√−g

[XR+

D − 3(D − 2)X

(∇X)2 − λ2X(D−4)/(D−2)]

A specific class of 2D dilaton gravity models





ds2 = g(D)µν︸︷︷︸

full metric



dΩ2SD−2

,


dDx√−g(D)R(D):

IEH = κ2π(D−1)/2

Γ(D−12 )

∫d2x

√−g φD−2

[R+

(D − 2)(D − 3)φ2

((∇φ)2 − 1

) ]

Cosmetic redefinition X ∝ (λφ)D−2:

IEH =1

16πG2

∫d2x

√−g

[XR+

D − 3(D − 2)X

(∇X)2 − λ2X(D−4)/(D−2)]






ds2 = g(D)µν︸︷︷︸

full metric



dΩ2SD−2

,


dDx√−g(D)R(D):

IEH = κ2π(D−1)/2

Γ(D−12 )

∫d2x

√−g φD−2

[R+

(D − 2)(D − 3)φ2

((∇φ)2 − 1

) ]Cosmetic redefinition X ∝ (λφ)D−2:

IEH =1

16πG2

∫d2x

√−g

[XR+

D − 3(D − 2)X

(∇X)2 − λ2X(D−4)/(D−2)]




Attempt 4: Poincare gauge theory and higher power curvature theories

Basic idea: since EH is trivial consider f(R) theories or/and theories withtorsion or/and theories with non-metricity

I Example: Katanaev-Volovich model (Poincare gauge theory)

IKV ∼∫

d2x√−g

[αT 2 + βR2

]I Kummer, Schwarz: bring into first order form:

IKV ∼∫ [

Xa(dea + εabω ∧ eb) +X dω + εabea ∧ eb (αXaXa + βX2)

]I Use same algorithm as before to convert into second order action:

IKV =1

16πG2

∫d2x

√−g

[XR+ α(∇X)2 + βX2

]







IKV ∼∫

d2x√−g

[αT 2 + βR2


IKV ∼∫ [



IKV =1

16πG2

∫d2x

√−g

[XR+ α(∇X)2 + βX2

]







IKV ∼∫

d2x√−g

[αT 2 + βR2


IKV ∼∫ [



IKV =1

16πG2

∫d2x

√−g

[XR+ α(∇X)2 + βX2

]




Attempt 5: Strings in two dimensions

Conformal invariance of the σ model

Iσ ∝∫

d2ξ√|h|

[gµνh

ij∂ixµ∂jx

ν + α′φR+ . . .]

requires vanishing of β-functions

βφ ∝ −4b2 − 4(∇φ)2 + 4φ+R+ . . .

βgµν ∝ Rµν + 2∇µ∇νφ+ . . .

Conditions βφ = βgµν = 0 follow from target space action

Itarget =1

16πG2

∫d2x

√−g

[XR+

1X

(∇X)2 − 4b2]

where X = e−2φ




Attempt 5: Strings in two dimensions

Conformal invariance of the σ model

Iσ ∝∫

d2ξ√|h|

[gµνh

ij∂ixµ∂jx

ν + α′φR+ . . .]

requires vanishing of β-functions

βφ ∝ −4b2 − 4(∇φ)2 + 4φ+R+ . . .

βgµν ∝ Rµν + 2∇µ∇νφ+ . . .

Conditions βφ = βgµν = 0 follow from target space action

Itarget =1

16πG2

∫d2x

√−g

[XR+

1X

(∇X)2 − 4b2]

where X = e−2φ




Synthesis of all attempts: Dilaton gravity in two dimensions

Second order action:

I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)

I Dilaton X defined by its coupling to curvature RI Kinetic term (∇X)2 contains coupling function U(X)I Self-interaction potential V (X) leads to non-trivial geometriesI Gibbons–Hawking–York boundary term guarantees Dirichlet boundary

problem for metricI Hamilton–Jacobi counterterm contains superpotential S(X)

S(X)2 = e−R X U(y) dy

∫ X

V (y)eR y U(z) dz dy

and guarantees well-defined variational principle δI = 0I Interesting option: couple 2D dilaton gravity to matter




I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)

I Dilaton X defined by its coupling to curvature R

I Kinetic term (∇X)2 contains coupling function U(X)I Self-interaction potential V (X) leads to non-trivial geometriesI Gibbons–Hawking–York boundary term guarantees Dirichlet boundary



∫ X






I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)

I Dilaton X defined by its coupling to curvature RI Kinetic term (∇X)2 contains coupling function U(X)

I Self-interaction potential V (X) leads to non-trivial geometriesI Gibbons–Hawking–York boundary term guarantees Dirichlet boundary



∫ X






I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)

I Dilaton X defined by its coupling to curvature RI Kinetic term (∇X)2 contains coupling function U(X)I Self-interaction potential V (X) leads to non-trivial geometries

I Gibbons–Hawking–York boundary term guarantees Dirichlet boundaryproblem for metric

I Hamilton–Jacobi counterterm contains superpotential S(X)


∫ X






I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)


problem for metric

I Hamilton–Jacobi counterterm contains superpotential S(X)


∫ X






I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)




∫ X


and guarantees well-defined variational principle δI = 0

I Interesting option: couple 2D dilaton gravity to matter




I =1

16πG2

∫M

d2x√|g|

[XR− U(X)(∇X)2 − V (X)

]− 1

8πG2

∫∂M

dx√|γ| [XK − S(X)] + I(m)




∫ X




Recent example: AdS2 holographyTwo dimensions supposed to be the simplest dimension with geometry, and yet...

I extremal black holes universally include AdS2 factorI funnily, AdS3 holography more straightforwardI study charged Jackiw–Teitelboim model as example

IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2

]

I Metric g has signature −,+ and Ricci-scalar R< 0 for AdSI Maxwell field strength Fµν = 2E εµν dual to electric field EI Dilaton φ has no kinetic term and no coupling to gauge fieldI Cosmological constant Λ = − 8

L2 parameterized by AdS radius LI Coupling constant α usually is positiveI δφ EOM: R = − 8

L2 ⇒ AdS2!I δA EOM: ∇µF

µν = 0 ⇒ E = constantI δg EOM:

∇µ∇νe−2φ − gµν ∇2e−2φ+

4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2

]I Metric g has signature −,+ and Ricci-scalar R< 0 for AdS

I Maxwell field strength Fµν = 2E εµν dual to electric field EI Dilaton φ has no kinetic term and no coupling to gauge fieldI Cosmological constant Λ = − 8





4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2

]I Metric g has signature −,+ and Ricci-scalar R< 0 for AdSI Maxwell field strength Fµν = 2E εµν dual to electric field E

I Dilaton φ has no kinetic term and no coupling to gauge fieldI Cosmological constant Λ = − 8





4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2

]I Metric g has signature −,+ and Ricci-scalar R< 0 for AdSI Maxwell field strength Fµν = 2E εµν dual to electric field EI Dilaton φ has no kinetic term and no coupling to gauge field

I Cosmological constant Λ = − 8L2 parameterized by AdS radius L

I Coupling constant α usually is positiveI δφ EOM: R = − 8




4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2

]I Metric g has signature −,+ and Ricci-scalar R< 0 for AdSI Maxwell field strength Fµν = 2E εµν dual to electric field EI Dilaton φ has no kinetic term and no coupling to gauge fieldI Cosmological constant Λ = − 8

L2 parameterized by AdS radius L

I Coupling constant α usually is positiveI δφ EOM: R = − 8




4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2


L2 parameterized by AdS radius LI Coupling constant α usually is positive

I δφ EOM: R = − 8L2 ⇒ AdS2!

I δA EOM: ∇µFµν = 0 ⇒ E = constant

I δg EOM:


4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2



L2 ⇒ AdS2!

I δA EOM: ∇µFµν = 0 ⇒ E = constant

I δg EOM:


4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2




µν = 0 ⇒ E = constant

I δg EOM:


4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2




µν = 0 ⇒ E = constantI δg EOM: complicated for non-constant dilaton...


4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0




IJT =α

2π

∫d2x

√−g

[e−2φ

(R+

8L2

)− L2

4F 2




µν = 0 ⇒ E = constantI δg EOM: ...but simple for constant dilaton: e−2φ = L4

4 E2


4L2

e−2φ gµν+L2

2Fµ

λ Fνλ−L2

8gµν F

2 = 0


Some surprising resultsHartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM

I Holographic renormalization leads to boundary mass term (CGLM)

I ∼∫

dx√|γ|mA2

Nevertheless, total action gauge invariant

I Boundary stress tensor transforms anomalously (HS)

(δξ + δλ)Ttt = 2Ttt∂tξ + ξ∂tTtt −c

24πL∂3

t ξ

where δξ + δλ is combination of diffeo- and gauge trafos that preservethe boundary conditions (similarly: δλJt = − k

4πL∂tλ)I Anomalous transformation above leads to central charge (HS, CGLM)

c = −24αe−2φ =3G2

=32kE2L2

I Positive central charge only for negative coupling constant α (CGLM)

α < 0




I ∼∫

dx√|γ|mA2

Nevertheless, total action gauge invariantI Boundary stress tensor transforms anomalously (HS)


24πL∂3

t ξ


4πL∂tλ)

I Anomalous transformation above leads to central charge (HS, CGLM)

c = −24αe−2φ =3G2

=32kE2L2


α < 0




I ∼∫

dx√|γ|mA2



24πL∂3

t ξ



c = −24αe−2φ =3G2

=32kE2L2


α < 0




I ∼∫

dx√|γ|mA2



24πL∂3

t ξ



c = −24αe−2φ =3G2

=32kE2L2


α < 0


Outline


Which 2D theory?

Which 3D theory?


What next?


Attempt 1: Einstein–HilbertAs simple as possible... but not simpler!

Let us start with the simplest attempt. Einstein-Hilbert action:

IEH =1

16πG

∫d3x

√−g R

Equations of motion:Rµν = 0

Ricci-flat and therefore Riemann-flat – locally trivial!

I No gravitons (recall: in D dimensions D(D − 3)/2 gravitons)

I No BHs

I Einstein-Hilbert in 3D is too simple for us!

Properties of Einstein-Hilbert


Attempt 1: Einstein–HilbertAs simple as possible... but not simpler!

Let us start with the simplest attempt. Einstein-Hilbert action:

IEH =1

16πG

∫d3x

√−g R

Equations of motion:Rµν = 0

Ricci-flat and therefore Riemann-flat – locally trivial!

I No gravitons (recall: in D dimensions D(D − 3)/2 gravitons)

I No BHs

I Einstein-Hilbert in 3D is too simple for us!

Properties of Einstein-Hilbert


Attempt 2: Topologically massive gravityDeser, Jackiw and Templeton found a way to introduce gravitons!

Let us now add a gravitational Chern–Simons term. TMG action:

ITMG = IEH +1

16πG

∫d3x

√−g 1

2µελµν Γρ

λσ

(∂µΓσ

νρ +23

ΓσµτΓτ

νρ

)Equations of motion:

Rµν +1µCµν = 0

with the Cotton tensor defined as

Cµν =12εµ

αβ∇αRβν + (µ↔ ν)

I Gravitons! Reason: third derivatives in Cotton tensor!

I No BHs

I TMG is slightly too simple for us!

Properties of TMG


Attempt 2: Topologically massive gravityDeser, Jackiw and Templeton found a way to introduce gravitons!

Let us now add a gravitational Chern–Simons term. TMG action:

ITMG = IEH +1

16πG

∫d3x

√−g 1

2µελµν Γρ

λσ

(∂µΓσ

νρ +23

ΓσµτΓτ

νρ


Rµν +1µCµν = 0

with the Cotton tensor defined as

Cµν =12εµ

αβ∇αRβν + (µ↔ ν)

I Gravitons! Reason: third derivatives in Cotton tensor!

I No BHs

I TMG is slightly too simple for us!

Properties of TMG


Attempt 3: Einstein–Hilbert–AdSBanados, Teitelboim and Zanelli (and Henneaux) taught us how to get 3D BHs

Add negative cosmological constant to Einstein-Hilbert action:

IΛEH =1

16πG

∫d3x

√−g

(R+

2`2


Gµν = Rµν −12gµνR−

1`2gµν = 0

Particular solutions: BTZ BH with line-element

ds2BTZ = −(r2 − r2+)(r2 − r2−)

`2r2dt2 +

`2r2

(r2 − r2+)(r2 − r2−)dr2 + r2

(dφ− r+r−

`r2dt

)2

I No gravitons

I Rotating BH solutions that asymptote to AdS3!

I Adding a negative cosmological constant produces BH solutions!

Properties of Einstein-Hilbert-AdS


Attempt 3: Einstein–Hilbert–AdSBanados, Teitelboim and Zanelli (and Henneaux) taught us how to get 3D BHs

Add negative cosmological constant to Einstein-Hilbert action:

IΛEH =1

16πG

∫d3x

√−g

(R+

2`2


Gµν = Rµν −12gµνR−

1`2gµν = 0

Particular solutions: BTZ BH with line-element

ds2BTZ = −(r2 − r2+)(r2 − r2−)

`2r2dt2 +

`2r2

(r2 − r2+)(r2 − r2−)dr2 + r2

(dφ− r+r−

`r2dt

)2

I No gravitons

I Rotating BH solutions that asymptote to AdS3!

I Adding a negative cosmological constant produces BH solutions!

Properties of Einstein-Hilbert-AdS


Cosmological topologically massive gravityCTMG is a 3D theory with BHs and gravitons!

We want a 3D theory with gravitons and BHs and therefore take CTMGaction

ICTMG =1

16πG

∫d3x

√−g

[R+

2`2

+12µ

ελµν Γρλσ

(∂µΓσ

νρ +23

ΓσµτΓτ

νρ

)]Equations of motion:

Gµν +1µCµν = 0

I Gravitons!

I BHs!

I CTMG is just perfect for us. Study this theory!

Properties of CTMG


Cosmological topologically massive gravityCTMG is a 3D theory with BHs and gravitons!

We want a 3D theory with gravitons and BHs and therefore take CTMGaction

ICTMG =1

16πG

∫d3x

√−g

[R+

2`2

+12µ

ελµν Γρλσ

(∂µΓσ

νρ +23

ΓσµτΓτ

νρ

)]Equations of motion:

Gµν +1µCµν = 0

I Gravitons!

I BHs!

I CTMG is just perfect for us. Study this theory!

Properties of CTMG


Einstein sector of the classical theory

Solutions of Einstein’s equations

Gµν = 0 ↔ R = − 6`2

also have vanishing Cotton tensor

Cµν = 0

and therefore are solutions of CTMG.

This sector of solutions contains

I BTZ BH

I Pure AdS

Line-element of pure AdS:

ds2AdS = gµν dxµ dxν = `2(− cosh2 ρdτ2 + sinh2 ρdφ2 + dρ2

)Isometry group: SL(2,R)L × SL(2,R)R

Useful to introduce light-cone coordinates u = τ + φ, v = τ − φ




Gµν = 0 ↔ R = − 6`2


Cµν = 0

and therefore are solutions of CTMG.This sector of solutions contains

I BTZ BH

I Pure AdS








Gµν = 0 ↔ R = − 6`2


Cµν = 0

and therefore are solutions of CTMG.This sector of solutions contains

I BTZ BH

I Pure AdS






AdS3-algebra of Killing vectorsA technical reminder

The SL(2,R)L generators

L0= i∂u

L±1 = ie±iu[cosh 2ρsinh 2ρ

∂u −1

sinh 2ρ∂v ∓

i

2∂ρ

]obey the algebra

[L0, L±1] = ∓L±1 , [L1, L−1] = 2L0

and have the quadratic Casimir

L2 =12(L1L−1 + L−1L1)− L2

0

The SL(2,R)R generators L0, L±1 obey same algebra, but with

u↔ v , L↔ L


Cotton sector of the classical theory

Solutions of CTMG withGµν 6= 0

necessarily have also non-vanishing Cotton tensor

Cµν 6= 0

Few exact solutions of this type are known.

Perhaps most interesting solution:

I Warped AdS (stretched/squashed), see Bengtsson & Sandin

Line-element of space-like warped AdS:

ds2warped AdS =`2

ν2 + 3(− cosh2 ρdτ2 +

4ν2

ν2 + 3(du+ sinh ρdτ)2 + dρ2

)Sidenote: null-warped AdS in holographic duals of cold atoms:

ds2null warped AdS = `2(dy2 + 2 dx+ dx−

y2± (dx−)2

y4

)


Cotton sector of the classical theory

Solutions of CTMG withGµν 6= 0

necessarily have also non-vanishing Cotton tensor

Cµν 6= 0

Few exact solutions of this type are known.Perhaps most interesting solution:

I Warped AdS (stretched/squashed), see Bengtsson & Sandin

Line-element of space-like warped AdS:

ds2warped AdS =`2

ν2 + 3(− cosh2 ρdτ2 +

4ν2

ν2 + 3(du+ sinh ρdτ)2 + dρ2

)Sidenote: null-warped AdS in holographic duals of cold atoms:

ds2null warped AdS = `2(dy2 + 2 dx+ dx−

y2± (dx−)2

y4

)D. Grumiller — Gravity in lower dimensions Which 3D theory? 22/42

CTMG as particle mechanics problemStationary and axi-symmetric solutions

Stationarity plus axi-symmetry:

I Two commuting Killing vectors

I Effectively reduce 2+1 dimensions to 1+0 dimensions

I Like particle mechanics, but with up to three time derivatives

I Still surprisingly difficult to get exact solutions!

I Known solutions: AdS, BTZ, warped AdS

Reduced action (Clement):

IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]

Here ζ is a Lagrange-multiplier and Xi = (T,X, Y ) a Lorentzian 3-vector

It could be rewarding to investigate this mechanicalproblem systematically and numerically!










IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]












IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]












IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]












IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]












IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]












IC[ζ,Xi] ∼∫

dρ[ζ2XiXjηij −

2ζ`2

+ζ2

2µεijk X

iXjXk]




CTMG at the chiral point...abbreviated as CCTMG

Definition: CTMG at the chiral point is CTMG with the tuning

µ ` = 1

between the cosmological constant and the Chern–Simons coupling.

Why special?Calculating the central charges of the dual boundary CFT yields

cL =3

2G(1− 1

µ `

), cR =

32G

(1 +

1µ `

)Thus, at the chiral point we get

cL = 0 , cR =3G

Notes:

I Abbreviate “CTMG at the chiral point” as CCTMGI CCTMG is also known as “chiral gravity”




µ ` = 1

between the cosmological constant and the Chern–Simons coupling.Why special?

Calculating the central charges of the dual boundary CFT yields

cL =3

2G(1− 1

µ `

), cR =

32G

(1 +

1µ `


cL = 0 , cR =3G

Notes:





µ ` = 1

between the cosmological constant and the Chern–Simons coupling.Why special?Calculating the central charges of the dual boundary CFT yields

cL =3

2G(1− 1

µ `

), cR =

32G

(1 +

1µ `


cL = 0 , cR =3G

Notes:





µ ` = 1

between the cosmological constant and the Chern–Simons coupling.Why special?Calculating the central charges of the dual boundary CFT yields

cL =3

2G(1− 1

µ `

), cR =

32G

(1 +

1µ `


cL = 0 , cR =3G

Notes:



Gravitons around AdS3 in CTMG

Linearization around AdS background

gµν = gµν + hµν

leads to linearized EOM that are third order PDE

G(1)µν +

1µC(1)

µν = (DRDLDMh)µν = 0 (1)

with three mutually commuting first order operators

(DL/R)µν = δν

µ ± ` εµαν∇α , (DM )µ

ν = δνµ +

1µεµ

αν∇α

Three linearly independent solutions to (1):(DLhL

)µν

= 0 ,(DRhR

)µν

= 0 ,(DMhM

)µν

= 0

At chiral point left (L) and massive (M) branches coincide!






G(1)µν +

1µC(1)



(DL/R)µν = δν


ν = δνµ +

1µεµ

αν∇α


)µν

= 0 ,(DRhR

)µν

= 0 ,(DMhM

)µν

= 0







G(1)µν +

1µC(1)



(DL/R)µν = δν


ν = δνµ +

1µεµ

αν∇α


)µν

= 0 ,(DRhR

)µν

= 0 ,(DMhM

)µν

= 0







G(1)µν +

1µC(1)



(DL/R)µν = δν


ν = δνµ +

1µεµ

αν∇α


)µν

= 0 ,(DRhR

)µν

= 0 ,(DMhM

)µν

= 0



Degeneracy at the chiral pointWill be quite important later!

Li, Song & Strominger found all solutions of linearized EOM.I Primaries: L0, L0 eigenstates ψL/R/M with

L1ψR/L/M = L1ψ

R/L/M = 0

I Descendants: act with L−1 and L−1 on primariesI General solution: linear combination of ψR/L/M

I Linearized metric is then the real part of the wavefunction

hµν = Reψµν

I At chiral point: L and M branches degenerate. Get new solution(DG & Johansson)

ψnewµν = lim

µ`→1

ψMµν(µ`)− ψL

µν

µ`− 1with property(

DLψnew)µν

=(DMψnew

)µν6= 0 ,

((DL)2ψnew

)µν

= 0




L1ψR/L/M = L1ψ

R/L/M = 0

I Descendants: act with L−1 and L−1 on primaries

I General solution: linear combination of ψR/L/M


hµν = Reψµν


ψnewµν = lim

µ`→1

ψMµν(µ`)− ψL

µν


DLψnew)µν

=(DMψnew

)µν6= 0 ,

((DL)2ψnew

)µν

= 0




L1ψR/L/M = L1ψ

R/L/M = 0



hµν = Reψµν


ψnewµν = lim

µ`→1

ψMµν(µ`)− ψL

µν


DLψnew)µν

=(DMψnew

)µν6= 0 ,

((DL)2ψnew

)µν

= 0




L1ψR/L/M = L1ψ

R/L/M = 0



hµν = Reψµν


ψnewµν = lim

µ`→1

ψMµν(µ`)− ψL

µν


DLψnew)µν

=(DMψnew

)µν6= 0 ,

((DL)2ψnew

)µν

= 0




L1ψR/L/M = L1ψ

R/L/M = 0



hµν = Reψµν


ψnewµν = lim

µ`→1

ψMµν(µ`)− ψL

µν


DLψnew)µν

=(DMψnew

)µν6= 0 ,

((DL)2ψnew

)µν

= 0


Sign oder nicht sign?That is the question. Choosing between Skylla and Charybdis.

I With signs defined as in thistalk: BHs positive energy,gravitons negative energy

I With signs as defined inDeser-Jackiw-Templetonpaper: BHs negative energy,gravitons positive energy

I Either way need a mechanism toeliminate unwanted negativeenergy objects – either thegravitons or the BHs

I Even at chiral point the problempersists because of thelogarithmic mode. See Figure.(Figure: thanks to N. Johansson)

Energy for all branches:























Outline


Which 2D theory?

Which 3D theory?


What next?

D. Grumiller — Gravity in lower dimensions How to quantize 3D gravity? 28/42

Witten’s attempt

Different approach (without gravitons!):

I Naive remark 1: 3D gravity is trivialI Naive remark 2: 3D gravity is non-renormalizableI Synthesis of naive remarks: 3D quantum gravity may exist as

non-trivial theoryI Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...

Maloney & Witten: taking into account all known contributions to pathintegral leads to non-sensible result for partition function Z.In particular, no holomorphic factorization:

ZMW 6= ZL · ZR

Various suggestions to interpret this problem: need cosmic strings, needsum over complex geometries, 3D quantum gravity does not exist by itself


Witten’s attempt

Different approach (without gravitons!):I Naive remark 1: 3D gravity is trivial

I Naive remark 2: 3D gravity is non-renormalizableI Synthesis of naive remarks: 3D quantum gravity may exist as



ZMW 6= ZL · ZR



Witten’s attempt

Different approach (without gravitons!):I Naive remark 1: 3D gravity is trivialI Naive remark 2: 3D gravity is non-renormalizable

I Synthesis of naive remarks: 3D quantum gravity may exist asnon-trivial theory

I Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt

Different approach (without gravitons!):I Naive remark 1: 3D gravity is trivialI Naive remark 2: 3D gravity is non-renormalizableI Synthesis of naive remarks: 3D quantum gravity may exist as

non-trivial theory

I Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt


non-trivial theoryI Positive cosmological constant: impossible?

I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt


non-trivial theoryI Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!

I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt


non-trivial theoryI Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constant

I Define quantum gravity by its dual CFT at the AdS boundaryI Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt


non-trivial theoryI Positive cosmological constant: impossible?I Vanishing cosmological constant: S-matrix, but no gravitons!I Therefore introduce negative cosmological constantI Define quantum gravity by its dual CFT at the AdS boundary

I Constructing this CFT still a “monstrous” effort...


ZMW 6= ZL · ZR



Witten’s attempt




ZMW 6= ZL · ZR



Witten’s attempt




ZMW 6= ZL · ZR



Witten’s attempt




ZMW 6= ZL · ZR

Various suggestions to interpret this problem: need cosmic strings, needsum over complex geometries, 3D quantum gravity does not exist by itselfD. Grumiller — Gravity in lower dimensions How to quantize 3D gravity? 29/42

Li, Song & Strominger attemptIs CCTMG dual to a chiral CFT?

Interesting observations:

1. If left-moving sector is trivial, ZL = 1, then problem of holomorphicfactorization

Z = ZL · ZR = ZR

is solved.

2. CCTMG hascL = 0

3. Massive graviton degenerates with left boundary graviton: ψM = ψL

Thus, dual CFT chiral? If yes, we are done!

Suggestive to interpret LSS results as absence of gravitons

Disagrees with results by Carlip, Deser, Waldron & Wise!

But:





Z = ZL · ZR = ZR

is solved.2. CCTMG has

cL = 0





But:





Z = ZL · ZR = ZR


cL = 0





But:





Z = ZL · ZR = ZR


cL = 0





But:





Z = ZL · ZR = ZR


cL = 0





But:





Z = ZL · ZR = ZR


cL = 0





But:


Gravitons in CCTMGIs CCTMG dual to a logarithmic CFT?

New mode resolves apparent contradiction between LSS and CDWW.

Interesting property:

L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).

Such a Jordan form of L0, L0 is defining property of a logarithmic CFT!Note: called “logarithmic CFT” because some correlators take the form

〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .

I Logarithmic CFT: not unitary and not chiral!

I Either logarithmic or chiral CFT dual (or none)

I Currently unknown which of these alternatives is realized!





L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).


〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .








L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).


〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .








L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).


〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .








L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).


〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .








L0

(ψnew

ψL

)=

(2 1

20 2

) (ψnew

ψL

),

L0

(ψnew

ψL

)=

(0 1

20 0

) (ψnew

ψL

).


〈ψnew(z)ψnew(0)〉 ∼ ln z + . . .





Viability of the logarithmic mode, part 1Explicit solution for logarithmic mode (DG & Johansson)

Collect in the following suggestions how the logarithmic mode could dropout of the physical spectrum and show that none of them is realized.

Is the logarithmic mode really there?

Before starting, here is the explicit form of the logarithmic mode:

hnewµν =

sinh ρcosh3ρ

(c τ − s ln cosh ρ

) 0 0 10 0 11 1 0

µν

− tanh2ρ(s τ + c ln cosh ρ

) 1 1 01 1 00 0 −a2

µν

(2)

with

c = cos (2u) , s = sin (2u) , a =1

sinh ρ cosh ρ


Viability of the logarithmic mode, part 1Explicit solution for logarithmic mode (DG & Johansson)

Collect in the following suggestions how the logarithmic mode could dropout of the physical spectrum and show that none of them is realized.

Is the logarithmic mode really there?

Before starting, here is the explicit form of the logarithmic mode:

hnewµν =

sinh ρcosh3ρ

(c τ − s ln cosh ρ

) 0 0 10 0 11 1 0

µν

− tanh2ρ(s τ + c ln cosh ρ

) 1 1 01 1 00 0 −a2

µν

(2)

with

c = cos (2u) , s = sin (2u) , a =1

sinh ρ cosh ρD. Grumiller — Gravity in lower dimensions How to quantize 3D gravity? 32/42

Viability of the logarithmic mode, part 2Physical mode with negative energy

The logarithmic mode is pure gauge?

No!

Suggestion 1

hnew does not solve linearized Einstein equations. Thus is not pure gauge.Note: confirmed by Sachs who considered logarithmic quasi-normal modes

Logarithmic mode has infinite energy and thus must be discarded?

No!

Suggestion 2

Enew = − 471152G`3

Energy is finite and negative.Thus logarithmic mode leads to instability but cannot be discarded.



The logarithmic mode is pure gauge? No!

Suggestion 1



No!

Suggestion 2

Enew = − 471152G`3





Suggestion 1



No!

Suggestion 2

Enew = − 471152G`3





Suggestion 1


Logarithmic mode has infinite energy and thus must be discarded? No!

Suggestion 2

Enew = − 471152G`3

Energy is finite and negative.Thus logarithmic mode leads to instability but cannot be discarded.D. Grumiller — Gravity in lower dimensions How to quantize 3D gravity? 33/42

Viability of the logarithmic mode, part 3Boundary conditions beyond Brown–Henneaux

New mode is not a small perturbation?

It is!

Suggestion 3

hnew diverges asymptotically like ρ, but AdS background divergesasymptotically like e2ρ. Thus hnew is really a small perturbation.

New mode is not asymptotically AdS?

It is!

Suggestion 4

Solution is asymptotically AdS

ds2 = dρ2 +(γ

(0)ij e

2ρ/` + γ(1)ij ρ+ γ

(0)ij + γ

(2)ij e

−2ρ/` + . . .)

dxi dxj

but violates Brown-Henneaux boundary conditions! (γ(1)ij

∣∣BH

= 0)Henneaux et al. showed precedents where this may happen in 3DNew boundary conditions replacing Brown-Henneaux (DG & Johansson)



New mode is not a small perturbation? It is!

Suggestion 3



It is!

Suggestion 4


ds2 = dρ2 +(γ

(0)ij e

2ρ/` + γ(1)ij ρ+ γ

(0)ij + γ

(2)ij e

−2ρ/` + . . .)

dxi dxj


∣∣BH





Suggestion 3



It is!

Suggestion 4


ds2 = dρ2 +(γ

(0)ij e

2ρ/` + γ(1)ij ρ+ γ

(0)ij + γ

(2)ij e

−2ρ/` + . . .)

dxi dxj


∣∣BH





Suggestion 3


New mode is not asymptotically AdS? It is!

Suggestion 4


ds2 = dρ2 +(γ

(0)ij e

2ρ/` + γ(1)ij ρ+ γ

(0)ij + γ

(2)ij e

−2ρ/` + . . .)

dxi dxj


∣∣BH



Viability of the logarithmic mode, part 4Brown–York boundary stress tensor

New mode leads to ill-defined Brown-York boundary stress tensor?

No!

Suggestion 5

Total action including boundary terms (Kraus & Larsen)

Itotal = ICTMG +1

8πG

∫d2x

√−γ

(K − 1

`

)Its first variation leads to Brown-York boundary stress-tensor:

δItotal∣∣EOM

=1

32πG

∫d2x

√−γ(0) T ij δγ

(0)ij

DG & Johansson: Tij is finite, traceless and chiral:

Tij = − `

16πG

(1 11 1

)ij

Note: coincides with Brown-York boundary stress-tensor of global AdS3


Viability of the logarithmic mode, part 4Brown–York boundary stress tensor

New mode leads to ill-defined Brown-York boundary stress tensor? No!

Suggestion 5

Total action including boundary terms (Kraus & Larsen)

Itotal = ICTMG +1

8πG

∫d2x

√−γ

(K − 1

`

)Its first variation leads to Brown-York boundary stress-tensor:

δItotal∣∣EOM

=1

32πG

∫d2x

√−γ(0) T ij δγ

(0)ij

DG & Johansson: Tij is finite, traceless and chiral:

Tij = − `

16πG

(1 11 1

)ij

Note: coincides with Brown-York boundary stress-tensor of global AdS3


Viability of the logarithmic mode, part 5Artifact of linearization?

Maybe some non-linear “magic” kills the new mode?

Unlikely!

Suggestion 6

DG, Jackiw & Johansson: classical phase space analysis of CCTMG

N =12

(2×D − 2×N1 −N2

)=

12

(2× 18− 2× 14− 6

)= 1

I N : number of physical degrees of freedom (per point)I D: number of canonical pairs in full phase spaceI N1(2): number of linearly independent first (second) class constraints

confirmed in more general calculation by Carlip

I Conclusion 1: logarithmic mode passed all tests so far

I Conclusion 2: CCTMG is unstable; dual CFT probably logarithmic



Maybe some non-linear “magic” kills the new mode? Unlikely!

Suggestion 6


N =12

(2×D − 2×N1 −N2

)=

12

(2× 18− 2× 14− 6

)= 1







Maybe some non-linear “magic” kills the new mode? Unlikely!

Suggestion 6


N =12

(2×D − 2×N1 −N2

)=

12

(2× 18− 2× 14− 6

)= 1






Outline


Which 2D theory?

Which 3D theory?


What next?

D. Grumiller — Gravity in lower dimensions What next? 37/42

Chiral vs. logarithmicPivotal open question: does dual CFT exist? is it chiral or logarithmic?

I Chiral route: must show consistency of truncation!

I Logarithmic: must show consistency of 2nd order perturbations!

To Do





To Do

ad chiral:

I restricting to Brown-Henneaux boundary conditions does not help

I Giribet, Kleban & Porrati showed that descendent of new mode

L−1ψnewµν = Yµν = Xµν + Lξ gµν

after a diffeomorphism ξ obeys Brown-Henneaux boundary conditions

I Descendants of logarithmic mode are there even when boundaryconditions are restricted beyond requiring variational principle!

I Need different mechanism of truncation!





To Do

ad chiral:

I restricting to Brown-Henneaux boundary conditions does not help

I Giribet, Kleban & Porrati showed that descendent of new mode

L−1ψnewµν = Yµν = Xµν + Lξ gµν

after a diffeomorphism ξ obeys Brown-Henneaux boundary conditions

I Descendants of logarithmic mode are there even when boundaryconditions are restricted beyond requiring variational principle!

I Need different mechanism of truncation!D. Grumiller — Gravity in lower dimensions What next? 38/42




To Do

ad logarithmic:I straightforward but somewhat lengthy calculationI expand metric around AdS background up to second order:

gµν = gµν + hnewµν + h(2)

µν

EOM lead to linear PDE for h(2)µν :

D(3) h(2) = f((hnew

µν )2)

I Check if h(2) really is smaller than hnewµν

I Might be rewarding exercise for a student





To Do

ad logarithmic:I straightforward but somewhat lengthy calculationI expand metric around AdS background up to second order:

gµν = gµν + hnewµν + h(2)

µν

EOM lead to linear PDE for h(2)µν :

D(3) h(2) = f((hnew

µν )2)

I Check if h(2) really is smaller than hnewµν

I Might be rewarding exercise for a student


Which groundstate?

Two observations:I Global AdS3 has mass and angular momentum in (C)CTMG

MAdS3 = µJAdS3 = − 18G

I If AdS3 is unstable in CCTMG because of mode,where does it run to?

Both observations suggest that there might be a ground state differentfrom pure AdS3 in (C)CTMG.

Consider other possible ground states with less symmetry

Example: warped AdS has four Killing vectors with U(1)L × SL(2,R)R

Suggestive to consider warped AdS as possible groundstate of (C)CTMG

Strominger et al. :


Which groundstate?



I If AdS3 is unstable in CCTMG because of logarithmic mode,where does it run to?





Strominger et al. :


Which groundstate?



I If AdS3 is unstable in (C)CTMG because of massive graviton mode,where does it run to?





Strominger et al. :


Which groundstate?








Strominger et al. :


Which groundstate?








Strominger et al. :


Which groundstate?








Strominger et al. :


Most crucial question we would like to answer

Does 3D quantum gravity existwith no strings attached?

I Consider the possible outcomes to this question:

I If yes: we would have an interesting quantum theory ofgravity with BHs and gravitons to get conceptualinsight into quantum gravity

I if no: potentially exciting news for string theory

Perhaps a win-win situation!

Thank you for your attention!



































Some literature

D. Grumiller, W. Kummer, and D. Vassilevich, “Dilaton gravity in twodimensions,” Phys. Rept. 369 (2002) 327–429, hep-th/0204253.

T. Hartman and A. Strominger, “Central charge for AdS2 quantum gravity,”[arXiv:0803.3621 [hep-th]].

A. Castro, D. Grumiller, F. Larsen, and R. McNees, “Holographic Descriptionof AdS2 Black Holes,” [arXiv:0809.4264 [hep-th]].

W. Li, W. Song and A. Strominger, JHEP 0804 (2008) 082, 0801.4566.

S. Carlip, S. Deser, A. Waldron and D. Wise, Phys.Lett. B666 (2008) 272,0807.0486, 0803.3998

D. Grumiller and N. Johansson, JHEP 0807 (2008) 134, 0805.2610.

G. Giribet, M. Kleban and M. Porrati, JHEP 0810 (2008) 045, 0807.4703.

D. Grumiller, R. Jackiw and N. Johansson, 0806.4185.

Thanks to Bob McNees for providing the LATEX beamerclass!


http://arXiv.org/abs/hep-th/0204253

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