Duality in 2D dilaton gravity based upon

Duality in 2D dilaton gravity

based uponhep-th/0609197with Roman Jackiw

Daniel Grumiller

CTP, LNS, MIT, Cambridge, Massachusetts

Supported by the European Commission, Project MC-OIF 021421

Brown University, December 2006

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

Geometry in 2D

As simple as possible but not simpler...

Riemann (Weyl+Ricci): ^N2(N₁₂²⁻¹⁾ 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!

Geometry in 2D

As simple as possible but not simpler...

Riemann (Weyl+Ricci): ^N2(N₁₂²⁻¹⁾ 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!

Geometry in 2D

As simple as possible but not simpler...

Riemann (Weyl+Ricci): ^N2(N₁₂²⁻¹⁾ 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!

Geometry in 2D

As simple as possible but not simpler...

Riemann (Weyl+Ricci): ^N2(N₁₂²⁻¹⁾ 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!

Spherical reduction

Line element adapted to spherical symmetry:

ds²= g_µν^(N)

|{z}

full metric

dx^µdx^ν =g_αβ(x^γ)

| {z }

2Dmetric

dx^αdx^β− φ²(x^α)

| {z }

surface area

dΩ²_S

N−2,

Insert into N-dimensional EH action I_EH =κR

d^Nxp

−g^(N)R^(N): I_EH =κ2π^(N−1)/2

Γ(^N−1₂ )

| {z }

N−2sphere

Z

d²x√

−gφ^N−2

| {z }

determinant

h

R+(N−2)(N−3) φ²

(∇φ)²−1

| {z }

Ricci scalar

i

Cosmetic redefinition X ∝√

λφN−2

:

I_EH ∝ Z

d²x√

−gh

XR+ N−3

(N−2)X(∇X)²−λX(N−4)/(N−2)

| {z }

Scalar−tensor theory a.k.a.dilaton gravity

i

Spherical reduction

Line element adapted to spherical symmetry:

ds²= g_µν^(N)

|{z}

full metric

dx^µdx^ν =g_αβ(x^γ)

| {z }

2Dmetric

dx^αdx^β− φ²(x^α)

| {z }

surface area

dΩ²_S

N−2,

Insert into N-dimensional EH action I_EH =κR

d^Nxp

−g^(N)R^(N): I_EH =κ2π^(N−1)/2

Γ(^N−1₂ )

| {z }

N−2sphere

Z

d²x√

−gφ^N−2

| {z }

determinant

h

R+(N−2)(N−3) φ²

(∇φ)²−1

| {z }

Ricci scalar

i

Cosmetic redefinition X ∝√

λφN−2

:

I_EH ∝ Z

d²x√

−gh

XR+ N−3

(N−2)X(∇X)²−λX(N−4)/(N−2)

| {z }

Scalar−tensor theory a.k.a.dilaton gravity

i

Spherical reduction

Line element adapted to spherical symmetry:

ds²= g_µν^(N)

|{z}

full metric

dx^µdx^ν =g_αβ(x^γ)

| {z }

2Dmetric

dx^αdx^β− φ²(x^α)

| {z }

surface area

dΩ²_S

N−2,

Insert into N-dimensional EH action I_EH =κR

d^Nxp

−g^(N)R^(N): I_EH =κ2π^(N−1)/2

Γ(^N−1₂ )

| {z }

N−2sphere

Z

d²x√

−gφ^N−2

| {z }

determinant

h

R+(N−2)(N−3) φ²

(∇φ)²−1

| {z }

Ricci scalar

i

Cosmetic redefinition X ∝√

λφN−2

:

I_EH ∝ Z

d²x√

−gh

XR+ N−3

(N−2)X(∇X)²−λX(N−4)/(N−2)

| {z }

Scalar−tensor theory a.k.a.dilaton gravity

i

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

Second order formulation

Similar action arises from string theory, from other kinds of dimensional reduction, from intrinsically 2D considerations, ...

Generic action:

I_2DG =κ Z

d²x√

−gh

XR+U(X)(∇X)²−λV(X) i

(1)

Special case U=0,V =X²: EOM R=2λX I∝

Z d²x√

−gR²

Similarly f(R)Lagrangians related to (1) with U =0 String context: X =e^−2φ, withφas string dilaton Conformal trafo to different model withU(X˜ ) =0:

V˜(X) = d

dXw(X) :=V(X)e^Q(X)

| {z }

conformally invariant

, withQ(X) :=RX

dyU(y)

Second order formulation

Similar action arises from string theory, from other kinds of dimensional reduction, from intrinsically 2D considerations, ...

Generic action:

I_2DG =κ Z

d²x√

−gh

XR+U(X)(∇X)²−λV(X) i

(1)

Special case U=0,V =X²: EOM R=2λX I∝

Z d²x√

−gR²

Similarly f(R)Lagrangians related to (1) with U =0 String context: X =e^−2φ, withφas string dilaton Conformal trafo to different model withU(X˜ ) =0:

V˜(X) = d

dXw(X) :=V(X)e^Q(X)

| {z }

conformally invariant

, withQ(X) :=RX

dyU(y)

Second order formulation

Similar action arises from string theory, from other kinds of dimensional reduction, from intrinsically 2D considerations, ...

Generic action:

I_2DG =κ Z

d²x√

−gh

XR+U(X)(∇X)²−λV(X) i

(1)

Special case U=0,V =X²: EOM R=2λX I∝

Z d²x√

−gR²

Similarly f(R)Lagrangians related to (1) with U =0 String context: X =e^−2φ, withφas string dilaton Conformal trafo to different model withU(X˜ ) =0:

V˜(X) = d

dXw(X) :=V(X)e^Q(X)

| {z }

conformally invariant

, withQ(X) :=RX

dyU(y)

Selected list of models

Model U(X) λV(X) λw(X)

1. Schwarzschild (1916) −¹

2X −λ −2λ√

X

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

3. Witten BH (1991) −¹

X −2b²X −2b²X

4. CGHS (1992) 0 −2b² −2b²X

5.(A)dS₂ground state (1994) −^a

X BX a6=2: _2−a^B X^2−a

6. Rindler ground state (1996) −^a

X BX^a BX

7. BH attractor (2003) 0 BX⁻¹ B ln X

8. SRG (N>3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2) −λ^{2 N−2}_N−3X(N−3)/(N−2) 9. All above: ab-family (1997) −^a

X BX^a+b b6=−1: _b+1^B X^b+1

10.Liouvillegravity a be^αX a6=−α: _a+α^b e^(a+α)X

11. Reissner-Nordström (1916) −_2X¹ −λ²+^Q2_X −2λ²√

X−2Q²/√ X

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X −2λ²√

X−²₃`X^3/2

13. Katanaev-Volovich (1986) α βX²−Λ RXe^αy(βy²−Λ)dy

14. Achucarro-Ortiz (1993) 0 ^Q2_X − ^J

4X 3−ΛX Q²ln X+ ^J 8X 2−¹₂ΛX²

15. Scattering trivial (2001) generic 0 const.

16. KK reduced CS (2003) 0 ¹₂X(c−X²) −¹

8(c−X²)²

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

1+γ²) 18. Symmetric kink (2005) generic −XΠⁿ_i=1(X²−X_i²) lengthy 19. KK red. conf. flat (2006) −¹₂tanh(X/2) A sinh X 4A cosh(X/2)

20. 2D type 0A −¹_X −2b²X+^{b2 q2}_8π −2b²X+^{b2 q2}_8π ln X

Red: relevant for strings Blue: pioneer models

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

First order formulation

Gravity 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,P_b] = Λε_abJ, [Pa,J] =εabP_b, Non-abelian BF theory:

I_BF = Z

X_AF^A= Z h

X_ade^a+X_aε^abω∧e^b+X dω+εabe^a∧e^bΛXi field strength F =dA+ [A,A]/2 contains SO(1,2)connection A=e^aP_a+ωJ, coadjoint Lagrange multipliers X_A

Generic first order action:

I_2DG∝ Z h

X_a T^a

|{z}

torsion

+X R

|{z}

curvature

+

|{z}

volume

(X^aX_aU(X) +λV(X))i (2)

T^a=de^a+ε^abω∧e^b, R=dω,=εabe^a∧e^b

First order formulation

Gravity 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,P_b] = Λε_abJ, [Pa,J] =εabP_b, Non-abelian BF theory:

I_BF = Z

X_AF^A= Z h

X_ade^a+X_aε^abω∧e^b+X dω+εabe^a∧e^bΛXi field strength F =dA+ [A,A]/2 contains SO(1,2)connection A=e^aP_a+ωJ, coadjoint Lagrange multipliers X_A

Generic first order action:

I_2DG∝ Z h

X_a T^a

|{z}

torsion

+X R

|{z}

curvature

+

|{z}

volume

(X^aX_aU(X) +λV(X))i (2)

T^a=de^a+ε^abω∧e^b, R=dω,=εabe^a∧e^b

Symmetries and equations of motion

Reinterpretation as Poisson-σmodel 94: Schaller, Strobl

I_PSM = Z

dX^I ∧A_I +1

2P^IJA_J∧A_I

gauge field 1-forms: A_I = (ω,ea), connection, Zweibeine target space coordinates: X^I = (X,X^a), dilaton, aux. fields P^IJ =−P^JI, P^Xa=ε^abX^b, P^ab =ε^ab(λV(X) +X^aX_aU(X)) Jacobi: P^IL∂_LP^JK +perm(IJK) =0

Equations of motion (first order):

dX^I+P^IJA_J =0 dA_I+ 1

2(∂IP^JK)A_K ∧A_J =0 Gauge symmetries (local Lorentz and diffeos):

δX^I =P^IJεJ

δA_I =−dε_I −

∂IP^JK εKA_J

Symmetries and equations of motion

Reinterpretation as Poisson-σmodel 94: Schaller, Strobl

I_PSM = Z

dX^I ∧A_I +1

2P^IJA_J∧A_I

gauge field 1-forms: A_I = (ω,ea), connection, Zweibeine target space coordinates: X^I = (X,X^a), dilaton, aux. fields P^IJ =−P^JI, P^Xa=ε^abX^b, P^ab =ε^ab(λV(X) +X^aX_aU(X)) Jacobi: P^IL∂_LP^JK +perm(IJK) =0

Equations of motion (first order):

dX^I+P^IJA_J =0 dA_I+ 1

2(∂IP^JK)A_K ∧A_J =0 Gauge symmetries (local Lorentz and diffeos):

δX^I =P^IJεJ

δA_I =−dε_I −

∂IP^JK εKA_J

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

Constant dilaton vacua and generic solutions

Light-cone components and Eddington-Finkelstein gauge (87: Polyakov, 92: Kummer, Schwarz, 96: Klösch, Strobl)

Constant dilaton vacua:

X =const. , V(X) =0, R=λV⁰(X) Minkowski, Rindler or (A)dS only

isolated solutions (no constant of motion) Generic solutions in EF gaugeω₀=e⁺₀ =0, e⁻₀ =1:

ds²=2e^Q(X)du dX+e^Q(X)(λw(X) +M)

| {z }

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 solutions

Light-cone components and Eddington-Finkelstein gauge (87: Polyakov, 92: Kummer, Schwarz, 96: Klösch, Strobl)

Constant dilaton vacua:

X =const. , V(X) =0, R=λV⁰(X) Minkowski, Rindler or (A)dS only

isolated solutions (no constant of motion) Generic solutions in EF gaugeω₀=e⁺₀ =0, e⁻₀ =1:

ds²=2e^Q(X)du dX+e^Q(X)(λw(X) +M)

| {z }

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

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

Exchange spacetime mass with reference mass

Recall

ds²=e^Q(X)h

2 du dX + (λw(X) +M)du²i Reformulate as

ds²=e^Q(X)w(X)

| {z }

e^{Q( ˜˜X)}

h

2 du dX w(X)

| {z }

dX˜

+(M 1

w(X)

| {z }

w( ˜˜ X)

+λ)du²i

Leads to dual potentials

U( ˜˜ X) =w(X)U(X)−e^Q(X)V(X) V˜( ˜X) =− V(X)

w²(X) and to dual action˜I= ˜κR

d²x√

−gh

X R˜ + ˜U(∇X˜)²−MV˜i

Exchange spacetime mass with reference mass

Recall

ds²=e^Q(X)h

2 du dX + (λw(X) +M)du²i Reformulate as

ds²=e^Q(X)w(X)

| {z }

e^{Q( ˜˜X)}

h

2 du dX w(X)

| {z }

dX˜

+(M 1

w(X)

| {z }

w( ˜˜ X)

+λ)du²i

Leads to dual potentials

U( ˜˜ X) =w(X)U(X)−e^Q(X)V(X) V˜( ˜X) =− V(X)

w²(X) and to dual action˜I= ˜κR

d²x√

−gh

X R˜ + ˜U(∇X˜)²−MV˜i

Exchange spacetime mass with reference mass

Recall

ds²=e^Q(X)h

2 du dX + (λw(X) +M)du²i Reformulate as

ds²=e^Q(X)w(X)

| {z }

e^{Q( ˜˜X)}

h

2 du dX w(X)

| {z }

dX˜

+(M 1

w(X)

| {z }

w( ˜˜ X)

+λ)du²i

Leads to dual potentials

U( ˜˜ X) =w(X)U(X)−e^Q(X)V(X) V˜( ˜X) =− V(X)

w²(X) and to dual action˜I= ˜κR

d²x√

−gh

X R˜ + ˜U(∇X˜)²−MV˜i

PSM perspective

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

Z d²x√

−g[XR−λ]

integrate in abelian gauge field (F =∗dA) Z

d²x√

−g[XR+YF−Y] on-shell: dY =0, so Y =λ

apply 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 motion Example (conformally transformed Witten BH, 92: Cangemi, Jackiw):

Z d²x√

−g[XR−λ]

integrate in abelian gauge field (F =∗dA) Z

d²x√

−g[XR+YF−Y] on-shell: dY =0, so Y =λ

apply trick to PSM: 4D Poisson manifold, 2 Casimirs (mass, charge)

duality: exchanges mass with charge

Outline

1 Gravity in 2D Models in 2D

Generic dilaton gravity action

2 Vienna School Approach Gravity as gauge theory All classical solutions

3 Duality

Casimir exchange Applications

The ab-family

Schwarzschild, Jackiw-Teitelboim, ...

Useful 2-parameter family of models:

U=−a

X , V =X^a+b Duality:˜a=1−(a−1)/b andb˜=1/b Global structure:

b>0: reflection at origin b<0: reflection atρ-axis

hereξ=lnp

|b|andρ= (a−1)/p

|b|

The ab-family

Schwarzschild, Jackiw-Teitelboim, ...

Useful 2-parameter family of models:

U=−a

X , V =X^a+b Duality:˜a=1−(a−1)/b andb˜=1/b Global structure:

b>0: reflection at origin b<0: reflection atρ-axis

hereξ=lnp

|b|andρ= (a−1)/p

|b|

Liouville gravity

cf. e.g. 04: Nakayama

Specific case (“almost Weyl invariant”):

Z d²x√

−gh

XR+1

2(∇X)²−m²e^Xi Dual model:

Z d²x√

−gh

X R˜ −λi (conformally transformed) Witten BH!

Note: on-shell R=0 (in both formulations)

Liouville gravity

cf. e.g. 04: Nakayama

Specific case (“almost Weyl invariant”):

Z d²x√

−gh

XR+1

2(∇X)²−m²e^Xi Dual model:

Z d²x√

−gh

X R˜ −λi (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+εto 2 dimensions:

U(X) =−1−ε

εX , V(X) =−ε(1−ε)X^1−2/ε

Limitε→0 not well-defined! No suitable rescaling of fields and coupling 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:

Z d²x√

−gh

XR+1

2(∇X)²−m²e^Xi .

This is the specific Liouville gravity model discussed previously!

Limiting action for gravity in 2 + ε dimensions (ε → 0)

79: Weinberg, 93: Mann, Ross

Spherical reduction from 2+εto 2 dimensions:

U(X) =−1−ε

εX , V(X) =−ε(1−ε)X^1−2/ε

Limitε→0 not well-defined! No suitable rescaling of fields and coupling 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:

Z d²x√

−gh

XR+1

2(∇X)²−m²e^Xi .

This is the specific Liouville gravity model discussed previously!

Limiting action for gravity in 2 + ε dimensions (ε → 0)

79: Weinberg, 93: Mann, Ross

Spherical reduction from 2+εto 2 dimensions:

U(X) =−1−ε

εX , V(X) =−ε(1−ε)X^1−2/ε

Limitε→0 not well-defined! No suitable rescaling of fields and coupling 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:

Z d²x√

−gh

XR+1

2(∇X)²−m²e^Xi .

This is the specific Liouville gravity model discussed previously!

Literature I

A. Jevicki, “Development in 2-d string theory,”

hep-th/9309115.

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

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