Gravity in lower dimensions

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Gravity in lower dimensions

Daniel Grumiller

Institute for Theoretical Physics Vienna University of Technology

IPM, Teheran,January 2012

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Outline

Why lower-dimensional gravity?

Which 2D theory?

Holographic renormalization

Which 3D theory?

D. Grumiller — Gravity in lower dimensions 2/33

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Outline

Which 2D theory?

Which 3D theory?

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Motivation for studying gravity in 2 and 3 dimensions

I Quantum gravity

I Address conceptual issues of quantum gravity

I Black hole evaporation, information loss, black hole microstate counting, virtual black hole production, ...

I Technically much simpler than 4D or higher D gravity

I Integrable models: powerful tools in physics (Coulomb problem, Hydrogen atom, harmonic oscillator, ...)

I Models should be as simple as possible, but not simpler

I Gauge/gravity duality + indirect physics applications

I Deeper understanding of black hole holography

I AdS3/CFT2 correspondence best understood

I Quantum gravity via AdS/CFT? (Witten ’07, Li, Song, Strominger ’08)

I Applications to 2D condensed matter systems?

I Gauge gravity duality beyond standard AdS/CFT: warped AdS, asymptotic Lifshitz, non-relativistic CFTs,logarithmic CFTs, ...

I Direct physics applications

D. Grumiller — Gravity in lower dimensions Why lower-dimensional gravity? 4/33

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I Quantum gravity

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I Quantum gravity

I Cosmic strings (Deser, Jackiw, ’t Hooft ’84, ’92)

I Black hole analog systems in condensed matter physics (graphene, BEC, fluids, ...)

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Gravity in lower dimensions

Riemann-tensor ^D²^(D₁₂²⁻¹⁾ components inD dimensions:

I 11D: 1210 (1144 Weyl and 66 Ricci)

I 10D: 825 (770 Weyl and 55 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

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I 11D: 1210 (1144 Weyl and 66 Ricci)

I 3D: 6 (Ricci)

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I 11D: 1210 (1144 Weyl and 66 Ricci)

I 3D: 6 (Ricci)

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I 11D: 1210 (1144 Weyl and 66 Ricci)

I 3D: 6 (Ricci)

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Outline

Which 2D theory?

Which 3D theory?

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Attempt 1: Einstein–Hilbert in and near two dimensions

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

IEH= 1 16π G

Z

d²xp

|g|R= 1

2G(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:

D. Grumiller — Gravity in lower dimensions Which 2D theory? 7/33

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

IEH = 1 16π G

Z

d²⁺xp

|g|R

I Weinberg: theory is asymptotically safe

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

I DG, Jackiw: limit →0 yields Liouville gravity

→0limI_EH= 1 16π G₂

Z

d²xp

|g|

XR−(∇X)²+λe^−2X

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

IEH = 1 16π G

Z

d²⁺xp

|g|R

I Weinberg: theory is asymptotically safe

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

I DG, Jackiw: limit →0 yields Liouville gravity

→0limI_EH= 1 16π G₂

Z

d²xp

|g|

XR−(∇X)²+λe^−2X

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Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS2 gauge theory

[P_a, P_b] = Λ_abJ [P_a, J] =_a^bP_b describes constant curvature gravity in 2D. Algorithm:

I Start withSO(1,2)connection A=e^aP_a+ωJ

I Take field strengthF = dA+¹₂[A, A]and coadjoint scalarX

I Construct non-abelian BF theory I =

Z

X_AF^A= Z h

X_a(deâ+â_bω∧e^b) +Xdω+_abeâ∧e^bΛXi

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

I Obtain the second order action I = 1

16π G₂ Z

d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G₂ Z

d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G₂ Z

d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G₂ Z

d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G₂ Z

d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G2

Z d²x√

−g X[R−Λ]

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Z

X_AF^A= Z h

16π G2

Z d²x√

−g X[R−Λ]

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Attempt 3: Dimensional reduction

For example: spherical reduction fromDdimensions

Line element adapted to spherical symmetry:

ds² = g^(D)_µν

|{z}

full metric

dx^µdx^ν =g_αβ(x^γ)

| {z }

2Dmetric

dx^αdx^β− φ²(x^α)

| {z }

surface area

dΩ²_S_D−2,

Insert into D-dimensional EH actionI_EH =κR

d^Dxp

−g^(D)R^(D): IEH =κ2π^(D−1)/2

Γ(^D−1₂ ) Z

d²x√

−g φ^D−2h

R+(D−2)(D−3)

φ² (∇φ)²−1i

Cosmetic redefinition X∝(λφ)^D−2: IEH = 1

16π G2

Z d²x√

−gh

XR+ D−3

(D−2)X(∇X)²−λ²X(D−4)/(D−2)i

A specific class of 2D dilaton gravity models Result of attempt 3:

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ds² = g^(D)_µν

|{z}

full metric

| {z }

2Dmetric

| {z }

surface area

dΩ²_S_D−2,

d^Dxp

−g^(D)R^(D): IEH =κ2π^(D−1)/2

Γ(^D−1₂ ) Z

d²x√

−g φ^D−2h

R+(D−2)(D−3)

φ² (∇φ)²−1i

16π G2

Z d²x√

−gh

XR+ D−3

(D−2)X(∇X)²−λ²X(D−4)/(D−2)i

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ds² = g^(D)_µν

|{z}

full metric

| {z }

2Dmetric

| {z }

surface area

dΩ²_S_D−2,

d^Dxp

−g^(D)R^(D): IEH =κ2π^(D−1)/2

Γ(^D−1₂ ) Z

d²x√

−g φ^D−2h

R+(D−2)(D−3)

φ² (∇φ)²−1i

16π G2

Z d²x√

−gh

XR+ D−3

(D−2)X(∇X)²−λ²X(D−4)/(D−2)i

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Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial considerf(R) theories or/and theories with torsion or/and theories with non-metricity

I Example: Katanaev-Volovichmodel (Poincare gauge theory) IKV ∼

Z d²x√

−g

αT²+βR²

I Kummer, Schwarz: bring into first order form: IKV ∼

Z h

Xa(deâ+âbω∧e^b) +Xdω+abeâ∧e^b(αXâXa+βX²) i

I Use same algorithm as before to convert into second order action: IKV = 1

16π G₂ Z

d²x√

−gh

XR+α(∇X)²+βX² i

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Z d²x√

−g

αT²+βR²

I Kummer, Schwarz: bring into first order form:

IKV ∼ Z h

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

IKV = 1 16π G₂

Z d²x√

−gh

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Z d²x√

−g

αT²+βR²

I Kummer, Schwarz: bring into first order form:

IKV ∼ Z h

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

IKV = 1 16π G₂

Z d²x√

−gh

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Attempt 5: Strings in two dimensions

Conformal invariance of the σ model I_σ ∝

Z d²ξp

|h|

g_µνh^ij∂_ix^µ∂_jx^ν +α⁰φR+. . .

requires vanishing of β-functions

β^φ∝ −4b²−4(∇φ)²+ 4φ+R+. . . β_µν^g ∝R_µν+ 2∇_µ∇_νφ+. . .

Conditions β^φ=βµν^g = 0 follow from target space action Itarget= 1

16π G₂ Z

d²x√

−gh

XR+ 1

X(∇X)²−4b²i where X=e^−2φ

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Attempt 5: Strings in two dimensions

Conformal invariance of the σ model I_σ ∝

Z d²ξp

|h|

g_µνh^ij∂_ix^µ∂_jx^ν +α⁰φR+. . .

requires vanishing of β-functions

β^φ∝ −4b²−4(∇φ)²+ 4φ+R+. . . β_µν^g ∝R_µν+ 2∇_µ∇_νφ+. . .

Conditions β^φ=βµν^g = 0 follow from target space action Itarget= 1

16π G₂ Z

d²x√

−gh

XR+ 1

X(∇X)²−4b²i where X=e^−2φ

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Selected List of Models

Black holes in(A)dS,asymptotically flatorarbitrary spaces(Wheeler property)

Model U(X) V(X)

1. Schwarzschild (1916) −_2X¹ −λ²

2. Jackiw-Teitelboim (1984) 0 ΛX

3. Witten Black Hole (1991) −_X¹ −2b²X

4. CGHS (1992) 0 −2b²

5.(A)dS2 ground state (1994) −_X^a BX

6. Rindler ground state (1996) −_X^a BX^a

7. Black Hole attractor (2003) 0 BX⁻¹

8. Spherically reduced gravity (N >3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2)

9. All above: ab-family (1997) −_X^a BX^a+b

10. Liouville gravity a be^αX

11. Reissner-Nordstr¨om (1916) −_2X¹ −λ²+^Q_X²

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X

13. Katanaev-Volovich (1986) α βX²−Λ

14. BTZ/Achucarro-Ortiz (1993) 0 ^Q_X² −_4X^J₃−ΛX

15. KK reduced CS (2003) 0 ¹₂X(c−X²)

16. KK red. conf. flat (2006) −¹₂tanh (X/2) AsinhX 17. 2D type 0A string Black Hole −_X¹ −2b²X+^b²_8π^q² 18. exact string Black Hole (2005) lengthy lengthy

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Synthesis of all attempts: Dilaton gravity in two dimensions Second order action:

I = 1 16π G2

Z

M

d²xp

|g|

XR−U(X)(∇X)²−V(X)

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

I Dilaton X defined by its coupling to curvature R

I Kinetic term (∇X)² contains coupling functionU(X)

I Self-interaction potential V(X) leads to non-trivial geometries

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

I Hamilton–Jacobi counterterm contains superpotentialS(X) S(X)² =e⁻^R^X^{U(y) dy}

Z X

V(y)e^R^y^{U(z) dz}dy and guarantees well-defined variational principleδI = 0

I Interesting option: couple 2D dilaton gravity to matter

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

Z X

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

Z X

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

Z X

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

I Gibbons–Hawking–York boundary termguarantees Dirichlet boundary problem for metric

Z X

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

I Hamilton–Jacobi countertermcontains superpotentialS(X) S(X)² =e⁻^R^X^{U(y) dy}

Z X

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I = 1 16π G2

Z

M

d²xp

|g|

− 1 8π G2

Z

∂M

dxp

|γ| [XK−S(X)] +I^(m)

Z X

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Acknowledgments

List of collaborators on 2D classical and quantum gravity:

I Wolfgang Kummer (VUT, 1935–2007)

I Dima Vassilevich (ABC Sao Paulo)

I Luzi Bergamin

I Herbert Balasin (VUT)

I Rene Meyer (Crete U.)

I Alfredo Iorio (Charles U. Prague)

I Carlos Nu˜nez (Swansea U.)

I Roman Jackiw (MIT)

I Robert McNees (Loyola U. Chicago)

I Muzaffer Adak (Pamukkale U.)

I Alejandra Castro (McGill U.)

I Finn Larsen (Michigan U.)

I Peter van Nieuwenhuizen (YITP, Stony Brook)

I Steve Carlip (UC Davis)

I . . .

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Outline

Which 2D theory?

Which 3D theory?

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Why do we needholographic renormalization?

Holographic renormalizationis the subtraction of appropriate boundary terms from the action.

What is holographic renormalization?

Without holographic renormalization:

I Wrong black hole thermodynamics

I Wrong (typically divergent) boundary stress tensor

I Inconsistent theory (no classical limit)

I Unphysical divergences and finite parts of observables can be wrong

I Susskind, Witten ’98: in field theory: field theory UV divergences (which need to be renormalized) correspond to IR divergences on the gravity side if gauge/gravity duality exists

I DG, van Nieuwenhuizen ’09: SUSY at boundary requires unique holographic counterterm, at least in 2 and 3 dimensions

I Variational principle ill-defined

D. Grumiller — Gravity in lower dimensions Holographic renormalization 16/33

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AdS2

... the simplest gravity model where the need forholographic renormalizationarises!

Bulk action:

I_B =−1 2

Z

M

d²x√ gh

X R+ 2

`² i

Variation with respect to scalar fieldX yields R=−2

`²

This means curvature is constant and negative, i.e., AdS2. Variation with respect to metricg yields

∇_µ∇_νX−g_µνX+g_µνX

`² = 0 Equations of motion above solved by

X =r , gµνdx^µdx^ν = r²

`² −M

dt²+ dr²

r²

`² −M There is an important catch, however: Boundary terms tricky!

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AdS2

Bulk action:

I_B =−1 2

Z

M

d²x√ gh

X R+ 2

`² i

`²

This means curvature is constant and negative, i.e., AdS2.

Variation with respect to metricg yields

`² −M

dt²+ dr²

r²

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AdS2

Bulk action:

I_B =−1 2

Z

M

d²x√ gh

X R+ 2

`² i

`²

`² = 0

Equations of motion above solved by X =r , gµνdx^µdx^ν = r²

`² −M

dt²+ dr²

r²

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AdS2

Bulk action:

I_B =−1 2

Z

M

d²x√ gh

X R+ 2

`² i

`²

`² −M

dt²+ dr²

r²

`² −M

There is an important catch, however: Boundary terms tricky!

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AdS2

Bulk action:

I_B =−1 2

Z

M

d²x√ gh

X R+ 2

`² i

`²

`² −M

dt²+ dr²

r²

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Boundary terms, Part I

Gibbons–Hawking–York boundary terms: quantum mechanical toy model

Let us start with an bulk Hamiltonian action

IB=

tf

Z

ti

dt[−pq˙ −H(q, p)]

Want to set up a Dirichlet boundary value problemq= fixed atti, tf

Problem:

δI_B = 0 requiresq δp= 0 at boundary Solution: add“Gibbons–Hawking–York” boundary term IE =IB+IGHY , IGHY = pq|^t_t^f

i

As expectedIE =

tf

R

ti

[pq˙−H(q, p)] is standard Hamiltonian action

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IB=

tf

Z

ti

Problem:

δI_B = 0 requiresq δp= 0 at boundary Solution: add“Gibbons–Hawking–York” boundary term IE =IB+IGHY , IGHY = pq|^t_t^f

i

As expectedIE =

tf

R

ti

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IB=

tf

Z

ti

Problem:

δI_B = 0 requiresq δp= 0 at boundary

Solution: add“Gibbons–Hawking–York” boundary term IE =IB+IGHY , IGHY = pq|^t_t^f

i

As expectedIE =

tf

R

ti

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IB=

tf

Z

ti

Problem:

δI_B = 0 requiresq δp= 0 at boundary Solution: add“Gibbons–Hawking–York” boundary term I_E =I_B+I_GHY , I_GHY = pq|^t_t^f

i

As expectedIE =

tf

R

ti

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IB=

tf

Z

ti

Problem:

δI_B = 0 requiresq δp= 0 at boundary Solution: add“Gibbons–Hawking–York” boundary term I_E =I_B+I_GHY , I_GHY = pq|^t_t^f

i

As expectedIE =

tf

R

ti

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Boundary terms, Part II

Gibbons–Hawking–York boundary terms in gravity — something still missing!

That was easy! In gravity the result is I_GHY =−

Z

∂M

dx√ γ X K

where γ (K) is determinant (trace) of first (second) fundamental form.

Euclidean action with correct boundary value problem is I_E =I_B+I_GHY

The boundary lies at r =r0, with r0→ ∞. Are we done?

No! Serious Problem! Variation of IE yields

δIE ∼EOM+δX(boundary−term)− lim

r0→∞

Z

∂M

dt δγ

Asymptotic metric: γ =r²/`²+O(1). Thus,δγ may befinite! δIE 6= 0 for some variations that preserve boundary conditions!!!

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Z

∂M

dx√ γ X K

r0→∞

Z

∂M

dt δγ

Asymptotic metric: γ =r²/`²+O(1). Thus,δγ may befinite! δIE 6= 0 for some variations that preserve boundary conditions!!!

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Z

∂M

dx√ γ X K

r0→∞

Z

∂M

dtδγ Asymptotic metric: γ =r²/`²+O(1). Thus, δγ may befinite!

δIE 6= 0 for some variations that preserve boundary conditions!!!

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Z

∂M

dx√ γ X K

r0→∞

Z

∂M

dt δγ Asymptotic metric: γ =r²/`²+O(1). Thus, δγ may befinite!

δIE 6= 0 for some variations that preserve boundary conditions!!!

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Boundary terms, Part III

Holographic renormalization: quantum mechanical toy model

Key observation: Dirichlet boundary problem not changed under IE →Γ =IE−ICT =IEH+IGHY −ICT

with

I_CT = S(q, t)|^t^f

Improved action:

Γ =

tf

Z

ti

dt[−pq˙ −H(q, p)]+pq|^t_t^f

i −S(q, t)|^t^f

First variation (assuming p=∂H/∂p): δΓ =

p−∂S(q, t)

∂q

δq

tf

= 0? Works if S(q, t) is Hamilton’s principal function!