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
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
Daniel Grumiller Duality in 2D dilaton gravity
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)
5D: 50 (35 Weyl and 15 Ricci)
10D: 825 (770 Weyl and 55 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!
Daniel Grumiller Duality in 2D dilaton gravity
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)
5D: 50 (35 Weyl and 15 Ricci)
10D: 825 (770 Weyl and 55 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!
Daniel Grumiller Duality in 2D dilaton gravity
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)
5D: 50 (35 Weyl and 15 Ricci)
10D: 825 (770 Weyl and 55 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!
Daniel Grumiller Duality in 2D dilaton gravity
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)
5D: 50 (35 Weyl and 15 Ricci)
10D: 825 (770 Weyl and 55 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!
Daniel Grumiller Duality in 2D dilaton gravity
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
]
Daniel Grumiller Duality in 2D dilaton gravity
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
]
Daniel Grumiller Duality in 2D dilaton gravity
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
]
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
Daniel Grumiller Duality in 2D dilaton gravity
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)
Daniel Grumiller Duality in 2D dilaton gravity
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)
Daniel Grumiller Duality in 2D dilaton gravity
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)
Daniel Grumiller Duality in 2D dilaton gravity
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
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
Daniel Grumiller Duality in 2D dilaton gravity
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
Daniel Grumiller Duality in 2D dilaton gravity
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
Daniel Grumiller Duality in 2D dilaton gravity
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
Daniel Grumiller Duality in 2D dilaton gravity
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
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
Daniel Grumiller Duality in 2D dilaton gravity
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!)
Daniel Grumiller Duality in 2D dilaton gravity
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!)
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
Daniel Grumiller Duality in 2D dilaton gravity
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
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
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
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
Daniel Grumiller Duality in 2D dilaton gravity
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
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
Daniel Grumiller Duality in 2D dilaton gravity
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|
Daniel Grumiller Duality in 2D dilaton gravity
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|
Daniel Grumiller Duality in 2D dilaton gravity
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)
Daniel Grumiller Duality in 2D dilaton gravity
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)
Daniel Grumiller Duality in 2D dilaton gravity
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!
Daniel Grumiller Duality in 2D dilaton gravity
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!
Daniel Grumiller Duality in 2D dilaton gravity
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!
Daniel Grumiller Duality in 2D dilaton gravity
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 .
Daniel Grumiller Duality in 2D dilaton gravity