Duality in 2D dilaton gravity based upon hep-th/0609197 with Roman Jackiw Daniel Grumiller CTP, LNS, MIT, Cambridge, Massachusetts Supported by the European Commission, Project MC-OIF 021421 Brown University, December 2006 Daniel Grumiller Duality in 2D dilaton gravity
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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
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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)
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
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 .