Duality in 2D dilaton gravity - TU Wienquark.itp.tuwien.ac.at/~grumil/pdf/brown06.pdfDuality in 2D dilaton gravity ... 10D: 825 (770 Weyl and 55 Ricci) 11D: ... pioneer models Daniel

Duality in 2D dilaton gravitybased upon hep-th/0609197 with Roman Jackiw

Daniel Grumiller

CTP, LNS, MIT, Cambridge, MassachusettsSupported by the European Commission, Project MC-OIF 021421

Brown University, December 2006

Daniel Grumiller Duality in 2D dilaton gravity

Outline

1 Gravity in 2DModels in 2DGeneric dilaton gravity action

2 Vienna School ApproachGravity as gauge theoryAll classical solutions

3 DualityCasimir exchangeApplications


Outline





Geometry in 2DAs simple as possible but not simpler...

Riemann (Weyl+Ricci): N2(N2−1)12 components in N dimensions

4D: 20 (10 Weyl and 10 Ricci)



11D: 1210 (1144 Weyl and 66 Ricci)

3D: 6 (Ricci)

2D: 1 (Ricci scalar) → Lowest dimension with curvature

1D: 0

But: 2D Einstein-Hilbert: no equations of motion!Number of graviton modes: N(N−3)

2

Stuck already in the formulation of the model?

Have to go beyond Einstein-Hilbert in 2D!







11D: 1210 (1144 Weyl and 66 Ricci)

3D: 6 (Ricci)


1D: 0


2









11D: 1210 (1144 Weyl and 66 Ricci)

3D: 6 (Ricci)


1D: 0


2









11D: 1210 (1144 Weyl and 66 Ricci)

3D: 6 (Ricci)


1D: 0


2




Spherical reduction

Line element adapted to spherical symmetry:

ds2 = g(N)µν︸︷︷︸

full metric

dxµ dxν = gαβ(xγ)︸︷︷︸2D metric

dxα dxβ − φ2(xα)︸︷︷︸surface area

dΩ2SN−2

,

Insert into N-dimensional EH action IEH = κ∫

dNx√−g(N)R(N):

IEH = κ2π(N−1)/2

Γ(N−12 )︸︷︷︸

N−2 sphere

∫d2x

√−gφN−2︸︷︷︸

determinant

[R +

(N − 2)(N − 3)

φ2

((∇φ)2 − 1

)︸︷︷︸

Ricci scalar

]

Cosmetic redefinition X ∝(√

λφ)N−2

:

IEH ∝∫

d2x√−g

[XR +

N − 3(N − 2)X

(∇X )2 − λX (N−4)/(N−2)︸︷︷︸Scalar−tensor theory a.k.a. dilaton gravity

]


Spherical reduction


ds2 = g(N)µν︸︷︷︸

full metric



dΩ2SN−2

,


dNx√−g(N)R(N):

IEH = κ2π(N−1)/2

Γ(N−12 )︸︷︷︸

N−2 sphere

∫d2x

√−gφN−2︸︷︷︸

determinant

[R +

(N − 2)(N − 3)

φ2

((∇φ)2 − 1

)︸︷︷︸

Ricci scalar

]


λφ)N−2

:

IEH ∝∫

d2x√−g

[XR +

N − 3(N − 2)X


]


Spherical reduction


ds2 = g(N)µν︸︷︷︸

full metric



dΩ2SN−2

,


dNx√−g(N)R(N):

IEH = κ2π(N−1)/2

Γ(N−12 )︸︷︷︸

N−2 sphere

∫d2x

√−gφN−2︸︷︷︸

determinant

[R +

(N − 2)(N − 3)

φ2

((∇φ)2 − 1

)︸︷︷︸

Ricci scalar

]


λφ)N−2

:

IEH ∝∫

d2x√−g

[XR +

N − 3(N − 2)X


]


Outline





Second order formulation

Similar action arises from string theory, from other kinds ofdimensional reduction, from intrinsically 2D considerations, ...Generic action:

I2DG = κ

∫d2x

√−g

[XR + U(X )(∇X )2 − λV (X )

](1)

Special case U = 0, V = X 2: EOM R = 2λX

I ∝∫

d2x√−gR2

Similarly f (R) Lagrangians related to (1) with U = 0String context: X = e−2φ, with φ as string dilatonConformal trafo to different model with U(X ) = 0:

V (X ) =d

dXw(X ) := V (X )eQ(X)︸︷︷︸

conformally invariant

, with Q(X ) :=∫ X dyU(y)




I2DG = κ

∫d2x

√−g

[XR + U(X )(∇X )2 − λV (X )

](1)


I ∝∫

d2x√−gR2


V (X ) =d

dXw(X ) := V (X )eQ(X)︸︷︷︸






I2DG = κ

∫d2x

√−g

[XR + U(X )(∇X )2 − λV (X )

](1)


I ∝∫

d2x√−gR2


V (X ) =d

dXw(X ) := V (X )eQ(X)︸︷︷︸




Selected list of models

Model U(X) λV (X) λw(X)

1. Schwarzschild (1916) − 12X −λ −2λ

√X

2. Jackiw-Teitelboim (1984) 0 ΛX 12 ΛX2

3. Witten BH (1991) − 1X −2b2X −2b2X

4. CGHS (1992) 0 −2b2 −2b2X5. (A)dS2 ground state (1994) − a

X BX a 6= 2 : B2−a X2−a

6. Rindler ground state (1996) − aX BXa BX

7. BH attractor (2003) 0 BX−1 B ln X8. SRG (N > 3) − N−3

(N−2)X −λ2X (N−4)/(N−2) −λ2 N−2N−3 X (N−3)/(N−2)

9. All above: ab-family (1997) − aX BXa+b b 6= −1 : B

b+1 Xb+1

10. Liouville gravity a beαX a 6= −α : ba+α

e(a+α)X

11. Reissner-Nordström (1916) − 12X −λ2 + Q2

X −2λ2√X − 2Q2/√

X

12. Schwarzschild-(A)dS − 12X −λ2 − `X −2λ2√X − 2

3 `X3/2

13. Katanaev-Volovich (1986) α βX2 − ΛR X eαy (βy2 − Λ) dy

14. Achucarro-Ortiz (1993) 0 Q2X − J

4X3 − ΛX Q2 ln X + J8X2 −

12 ΛX2

15. Scattering trivial (2001) generic 0 const.16. KK reduced CS (2003) 0 1

2 X(c − X2) − 18 (c − X2)2

17. exact string BH (2005) lengthy −γ −(1 +p

1 + γ2)

18. Symmetric kink (2005) generic −XΠni=1(X2 − X2

i ) lengthy19. KK red. conf. flat (2006) − 1

2 tanh (X/2) A sinh X 4A cosh (X/2)

20. 2D type 0A − 1X −2b2X + b2q2

8π−2b2X + b2q2

8πln X

Red: relevant for strings Blue: pioneer models


Outline





First order formulationGravity as gauge theory (89: Isler, Trugenberger, Chamseddine, Wyler, 91: Verlinde,92: Cangemi, Jackiw, Achucarro, 93: Ikeda, Izawa, 94: Schaller, Strobl)

Example: Jackiw-Teitelboim model (U = 0, λV = ΛX )

[Pa, Pb] = ΛεabJ , [Pa, J] = εabPb ,

Non-abelian BF theory:

IBF =

∫XAF A =

∫ [Xa dea+Xaε

abω∧eb+X dω+εabea∧ebΛX

]field strength F = dA + [A, A]/2 contains SO(1, 2) connectionA = eaPa + ωJ, coadjoint Lagrange multipliers XA

Generic first order action:

I2DG ∝∫ [

Xa T a︸︷︷︸torsion

+X R︸︷︷︸curvature

+ ε︸︷︷︸volume

(X aXaU(X ) + λV (X ))]

(2)

T a = dea + εabω ∧ eb, R = dω, ε = εabea ∧ eb


First order formulationGravity as gauge theory (89: Isler, Trugenberger, Chamseddine, Wyler, 91: Verlinde,92: Cangemi, Jackiw, Achucarro, 93: Ikeda, Izawa, 94: Schaller, Strobl)

Example: Jackiw-Teitelboim model (U = 0, λV = ΛX )

[Pa, Pb] = ΛεabJ , [Pa, J] = εabPb ,

Non-abelian BF theory:

IBF =

∫XAF A =

∫ [Xa dea+Xaε

abω∧eb+X dω+εabea∧ebΛX

]field strength F = dA + [A, A]/2 contains SO(1, 2) connectionA = eaPa + ωJ, coadjoint Lagrange multipliers XA

Generic first order action:

I2DG ∝∫ [

Xa T a︸︷︷︸torsion

+X R︸︷︷︸curvature

+ ε︸︷︷︸volume

(X aXaU(X ) + λV (X ))]

(2)

T a = dea + εabω ∧ eb, R = dω, ε = εabea ∧ eb


Symmetries and equations of motionReinterpretation as Poisson-σ model 94: Schaller, Strobl

IPSM =

∫ [dX I ∧ AI +

12

P IJAJ ∧ AI

]gauge field 1-forms: AI = (ω, ea), connection, Zweibeinetarget space coordinates: X I = (X , X a), dilaton, aux. fieldsP IJ = −PJI , PXa = εa

bX b, Pab = εab(λV (X ) + X aXaU(X ))Jacobi: P IL∂LPJK + perm (IJK ) = 0

Equations of motion (first order):

dX I + P IJAJ = 0

dAI +12(∂IP

JK )AK ∧ AJ = 0

Gauge symmetries (local Lorentz and diffeos):

δX I = P IJεJ

δAI = −dεI −(∂IP

JK)

εK AJ


Symmetries and equations of motionReinterpretation as Poisson-σ model 94: Schaller, Strobl

IPSM =

∫ [dX I ∧ AI +

12

P IJAJ ∧ AI

]gauge field 1-forms: AI = (ω, ea), connection, Zweibeinetarget space coordinates: X I = (X , X a), dilaton, aux. fieldsP IJ = −PJI , PXa = εa

bX b, Pab = εab(λV (X ) + X aXaU(X ))Jacobi: P IL∂LPJK + perm (IJK ) = 0

Equations of motion (first order):

dX I + P IJAJ = 0

dAI +12(∂IP

JK )AK ∧ AJ = 0

Gauge symmetries (local Lorentz and diffeos):

δX I = P IJεJ

δAI = −dεI −(∂IP

JK)

εK AJ


Outline





Constant dilaton vacua and generic solutionsLight-cone components and Eddington-Finkelstein gauge (87: Polyakov, 92: Kummer,Schwarz, 96: Klösch, Strobl)

Constant dilaton vacua:

X = const. , V (X ) = 0 , R = λV ′(X )

Minkowski, Rindler or (A)dS only

isolated solutions (no constant of motion)

Generic solutions in EF gauge ω0 = e+0 = 0, e−0 = 1:

ds2 = 2eQ(X) du dX + eQ(X)(λw(X ) + M)︸︷︷︸Killing norm

du2 (3)

Birkhoff theorem: at least one Killing vector ∂u

one constant of motion: mass M

one parameter in action: λ

dilaton is coordinate x0 (residual gauge trafos!)


Constant dilaton vacua and generic solutionsLight-cone components and Eddington-Finkelstein gauge (87: Polyakov, 92: Kummer,Schwarz, 96: Klösch, Strobl)

Constant dilaton vacua:

X = const. , V (X ) = 0 , R = λV ′(X )

Minkowski, Rindler or (A)dS only

isolated solutions (no constant of motion)

Generic solutions in EF gauge ω0 = e+0 = 0, e−0 = 1:

ds2 = 2eQ(X) du dX + eQ(X)(λw(X ) + M)︸︷︷︸Killing norm

du2 (3)

Birkhoff theorem: at least one Killing vector ∂u

one constant of motion: mass M

one parameter in action: λ

dilaton is coordinate x0 (residual gauge trafos!)


Outline





Exchange spacetime mass with reference mass

Recallds2 = eQ(X)

[2 du dX + (λw(X ) + M) du2

]Reformulate as

ds2 = eQ(X)w(X )︸︷︷︸eQ(X)

[2 du

dXw(X )︸︷︷︸

dX

+(M1

w(X )︸︷︷︸w(X)

+λ) du2]

Leads to dual potentials

U(X ) = w(X )U(X )− eQ(X)V (X )

V (X ) = − V (X )

w2(X )

and to dual action I = κ∫

d2x√−g

[XR + U(∇X )2 −MV

]Daniel Grumiller Duality in 2D dilaton gravity


Recallds2 = eQ(X)

[2 du dX + (λw(X ) + M) du2

]Reformulate as


[2 du

dXw(X )︸︷︷︸

dX

+(M1

w(X )︸︷︷︸w(X)

+λ) du2]


U(X ) = w(X )U(X )− eQ(X)V (X )

V (X ) = − V (X )

w2(X )


d2x√−g

[XR + U(∇X )2 −MV



Recallds2 = eQ(X)

[2 du dX + (λw(X ) + M) du2

]Reformulate as


[2 du

dXw(X )︸︷︷︸

dX

+(M1

w(X )︸︷︷︸w(X)

+λ) du2]


U(X ) = w(X )U(X )− eQ(X)V (X )

V (X ) = − V (X )

w2(X )


d2x√−g

[XR + U(∇X )2 −MV


PSM perspective

Trick: convert parameter in action to constant of motionExample (conformally transformed Witten BH, 92: Cangemi,Jackiw): ∫

d2x√−g[XR − λ]

integrate in abelian gauge field (F = ∗dA)∫d2x

√−g[XR + YF − Y ]

on-shell: dY = 0, so Y = λapply trick to PSM: 4D Poisson manifold, 2 Casimirs (mass,charge)duality: exchanges mass with charge


PSM perspective

Trick: convert parameter in action to constant of motionExample (conformally transformed Witten BH, 92: Cangemi,Jackiw): ∫

d2x√−g[XR − λ]

integrate in abelian gauge field (F = ∗dA)∫d2x

√−g[XR + YF − Y ]

on-shell: dY = 0, so Y = λapply trick to PSM: 4D Poisson manifold, 2 Casimirs (mass,charge)duality: exchanges mass with charge


Outline





The ab-familySchwarzschild, Jackiw-Teitelboim, ...

Useful 2-parameter family of models:

U = − aX

, V = X a+b

Duality: a = 1− (a− 1)/b and b = 1/bGlobal structure:

b>0: reflection at origin b<0: reflection at ρ-axishere ξ = ln

p|b| and ρ = (a − 1)/

p|b|


The ab-familySchwarzschild, Jackiw-Teitelboim, ...

Useful 2-parameter family of models:

U = − aX

, V = X a+b

Duality: a = 1− (a− 1)/b and b = 1/bGlobal structure:

b>0: reflection at origin b<0: reflection at ρ-axishere ξ = ln

p|b| and ρ = (a − 1)/

p|b|


Liouville gravitycf. e.g. 04: Nakayama

Specific case (“almost Weyl invariant”):∫d2x

√−g

[XR +

12(∇X )2 −m2eX

]Dual model: ∫

d2x√−g

[XR − λ

](conformally transformed) Witten BH!Note: on-shell R = 0 (in both formulations)


Liouville gravitycf. e.g. 04: Nakayama

Specific case (“almost Weyl invariant”):∫d2x

√−g

[XR +

12(∇X )2 −m2eX

]Dual model: ∫

d2x√−g

[XR − λ

](conformally transformed) Witten BH!Note: on-shell R = 0 (in both formulations)


Limiting action for gravity in 2 + ε dimensions (ε → 0)79: Weinberg, 93: Mann, Ross

Spherical reduction from 2 + ε to 2 dimensions:

U(X ) = −1− ε

εX, V (X ) = −ε(1− ε)X 1−2/ε

Limit ε → 0 not well-defined! No suitable rescaling of fields andcoupling constants possible! [recall: in action XR + U(X )(∇X )2]Solution: dualize, take limit in dual formulation, dualize back[why it works? MGS ' model with U = 0]Result: ∫

d2x√−g

[XR +

12(∇X )2 −m2eX

].

This is the specific Liouville gravity model discussed previously!




U(X ) = −1− ε

εX, V (X ) = −ε(1− ε)X 1−2/ε


d2x√−g

[XR +

12(∇X )2 −m2eX

].





U(X ) = −1− ε

εX, V (X ) = −ε(1− ε)X 1−2/ε


d2x√−g

[XR +

12(∇X )2 −m2eX

].



Literature I

A. Jevicki, “Development in 2-d string theory,”hep-th/9309115 .

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

D. Grumiller and R. Jackiw, “Duality in 2-dimensional dilatongravity,” Phys. Lett. B642 (2006) 530, hep-th/0609197 .


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

Documents

Duality in 2D dilaton gravity - TU Wienquark.itp.tuwien.ac.at/~grumil/pdf/brown06.pdfDuality in 2D dilaton gravity ... 10D: 825 (770 Weyl and 55 Ricci) 11D: ... pioneer models Daniel