Gravity in lower dimensions

Transcription

1 Gravity in lower dimensions Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology Uppsala University, December 2008

2 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions 2/42

3 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 3/42

4 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

5 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

6 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

7 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

8 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

9 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved There is a lot we still do not know about quantum gravity D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

10 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

11 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Non-perturbative understanding of quantum gravity? D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

12 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Non-perturbative understanding of quantum gravity? Microscopic understanding of non-extremal BH entropy? D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

13 Quantum gravity The Holy Grail of theoretical physics There is a lot we do know about quantum gravity already It should exist in some form String theory: (perturbative) theory of quantum gravity Microscopic understanding of extremal BH entropy Conceptual insight information loss problem resolved There is a lot we still do not know about quantum gravity Reasonable alternatives to string theory? Non-perturbative understanding of quantum gravity? Microscopic understanding of non-extremal BH entropy? Experimental signatures? Data? D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 4/42

14 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 5/42

15 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 5/42

16 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) 2D: lowest dimension exhibiting black holes (BHs) Simplest gravitational theories with BHs in 2D D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 5/42

17 Gravity in lower dimensions Riemann-tensor D2 (D 2 1) 12 components in D dimensions: 11D: 1210 (1144 Weyl and 66 Ricci) 10D: 825 (770 Weyl and 55 Ricci) 5D: 50 (35 Weyl and 15 Ricci) 4D: 20 (10 Weyl and 10 Ricci) 3D: 6 (Ricci) 2D: 1 (Ricci scalar) 2D: lowest dimension exhibiting black holes (BHs) Simplest gravitational theories with BHs in 2D 3D: lowest dimension exhibiting BHs and gravitons Simplest gravitational theories with BHs and gravitons in 3D D. Grumiller Gravity in lower dimensions Why lower-dimensional gravity? 5/42

18 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions Which 2D theory? 6/42

19 Attempt 1: Einstein Hilbert in and near two dimensions Let us start with the simplest attempt. Einstein-Hilbert action in 2 dimensions: I EH = 1 d 2 x g R = 1 (1 γ) 16π G 2G Action is topological No equations of motion Formal counting of number of gravitons: -1 D. Grumiller Gravity in lower dimensions Which 2D theory? 7/42

20 Attempt 1: Einstein Hilbert in and near two dimensions Let us continue with the next simplest attempt. Einstein-Hilbert action in 2+ɛ dimensions: I ɛ EH = 1 d 2+ɛ x g R 16π G Weinberg: theory is asymptotically safe Mann: limit ɛ 0 should be possible and lead to 2D dilaton gravity DG, Jackiw: limit ɛ 0 yields Liouville gravity lim I EH ɛ 1 = d 2 x g [ XR ( X) 2 + λe 2X] ɛ 0 16π G 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 7/42

21 Attempt 1: Einstein Hilbert in and near two dimensions Let us continue with the next simplest attempt. Einstein-Hilbert action in 2+ɛ dimensions: I ɛ EH = 1 d 2+ɛ x g R 16π G Weinberg: theory is asymptotically safe Mann: limit ɛ 0 should be possible and lead to 2D dilaton gravity DG, Jackiw: limit ɛ 0 yields Liouville gravity lim I EH ɛ 1 = d 2 x g [ XR ( X) 2 + λe 2X] ɛ 0 16π G 2 Result of attempt 1: A specific 2D dilaton gravity model D. Grumiller Gravity in lower dimensions Which 2D theory? 7/42

22 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ b a P b describes constant curvature gravity in 2D. Algorithm: D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

23 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ b a P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

24 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

25 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

26 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

27 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) Obtain the second order action 1 I = d 2 x g X [R Λ] 16π G 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

28 Attempt 2: Gravity as a gauge theory and the Jackiw-Teitelboim model Jackiw, Teitelboim (Bunster): (A)dS 2 gauge theory [P a, P b ] = Λ ɛ ab J [P a, J] = ɛ a b P b describes constant curvature gravity in 2D. Algorithm: Start with SO(1, 2) connection A = e a P a + ωj Take field strength F = da [A, A] and coadjoint scalar X Construct non-abelian BF theory [ ] I = X A F A = X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b ΛX Eliminate X a (Torsion constraint) and ω (Levi-Civita connection) Obtain the second order action 1 I = d 2 x g X [R Λ] 16π G 2 Result of attempt 2: A specific 2D dilaton gravity model D. Grumiller Gravity in lower dimensions Which 2D theory? 8/42

29 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area D. Grumiller Gravity in lower dimensions Which 2D theory? 9/42

30 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area Insert into D-dimensional EH action I EH = κ d D x g (D) R (D) : I EH = κ 2π(D 1)/2 Γ( D 1 2 ) d 2 x g φ D 2[ (D 2)(D 3) ( R + ( φ) 2 φ 2 1 ) ] D. Grumiller Gravity in lower dimensions Which 2D theory? 9/42

31 Attempt 3: Dimensional reduction For example: spherical reduction from D dimensions Line element adapted to spherical symmetry: ds 2 = g µν (D) }{{} dx µ dx ν = g αβ (x γ ) dx α dx β }{{} φ 2 (x α ) }{{} dω 2 S D 2, full metric 2D metric surface area Insert into D-dimensional EH action I EH = κ d D x g (D) R (D) : I EH = κ 2π(D 1)/2 Γ( D 1 2 ) d 2 x g φ D 2[ (D 2)(D 3) ( R + ( φ) 2 φ 2 1 ) ] Cosmetic redefinition X (λφ) D 2 : 1 I EH = d 2 x [ g XR + D 3 16π G 2 (D 2)X ( X)2 λ 2 X (D 4)/(D 2)] Result of attempt 3: A specific class of 2D dilaton gravity models D. Grumiller Gravity in lower dimensions Which 2D theory? 9/42

32 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity D. Grumiller Gravity in lower dimensions Which 2D theory? 10/42

33 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity Example: Katanaev-Volovich model (Poincare gauge theory) I KV d 2 x g [ αt 2 + βr 2] Kummer, Schwarz: bring into first order form: [ ] I KV X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b (αx a X a + βx 2 ) Use same algorithm as before to convert into second order action: 1 I KV = d 2 x g [XR + α( X) 2 + βx 2] 16π G 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 10/42

34 Attempt 4: Poincare gauge theory and higher power curvature theories Basic idea: since EH is trivial consider f(r) theories or/and theories with torsion or/and theories with non-metricity Example: Katanaev-Volovich model (Poincare gauge theory) I KV d 2 x g [ αt 2 + βr 2] Kummer, Schwarz: bring into first order form: [ ] I KV X a (de a + ɛ a bω e b ) + X dω + ɛ ab e a e b (αx a X a + βx 2 ) Use same algorithm as before to convert into second order action: 1 I KV = d 2 x g [XR + α( X) 2 + βx 2] 16π G 2 Result of attempt 4: A specific 2D dilaton gravity model D. Grumiller Gravity in lower dimensions Which 2D theory? 10/42

35 Attempt 5: Strings in two dimensions Conformal invariance of the σ model I σ d 2 ξ h [ g µν h ij i x µ j x ν + α φr +... ] requires vanishing of β-functions β φ 4b 2 4( φ) φ + R +... βµν g R µν + 2 µ ν φ +... Conditions β φ = βµν g = 0 follow from target space action 1 I target = d 2 x g [XR π G 2 X ( X)2 4b 2] where X = e 2φ D. Grumiller Gravity in lower dimensions Which 2D theory? 11/42

36 Attempt 5: Strings in two dimensions Conformal invariance of the σ model I σ d 2 ξ h [ g µν h ij i x µ j x ν + α φr +... ] requires vanishing of β-functions β φ 4b 2 4( φ) φ + R +... βµν g R µν + 2 µ ν φ +... Conditions β φ = βµν g = 0 follow from target space action 1 I target = d 2 x g [XR π G 2 X ( X)2 4b 2] where X = e 2φ Result of attempt 5: A specific 2D dilaton gravity model D. Grumiller Gravity in lower dimensions Which 2D theory? 11/42

37 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

38 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

39 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

40 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

41 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

42 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric Hamilton Jacobi counterterm contains superpotential S(X) S(X) 2 = e R X U(y) dy X V (y)er y U(z) dz dy and guarantees well-defined variational principle δi = 0 D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

43 Synthesis of all attempts: Dilaton gravity in two dimensions Second order action: 1 I = 16π G 2 1 8π G 2 M M d 2 x g [ XR U(X)( X) 2 V (X) ] dx γ [XK S(X)] + I (m) Dilaton X defined by its coupling to curvature R Kinetic term ( X) 2 contains coupling function U(X) Self-interaction potential V (X) leads to non-trivial geometries Gibbons Hawking York boundary term guarantees Dirichlet boundary problem for metric Hamilton Jacobi counterterm contains superpotential S(X) S(X) 2 = e R X U(y) dy X V (y)er y U(z) dz dy and guarantees well-defined variational principle δi = 0 Interesting option: couple 2D dilaton gravity to matter D. Grumiller Gravity in lower dimensions Which 2D theory? 12/42

44 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

45 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

46 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

47 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

48 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

49 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

50 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

51 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

52 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant δg EOM: complicated for non-constant dilaton... µ ν e 2φ g µν 2 e 2φ + 4 L 2 e 2φ g µν + L2 2 F µ λ F νλ L2 8 g µν F 2 = 0 D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

53 Recent example: AdS 2 holography Two dimensions supposed to be the simplest dimension with geometry, and yet... extremal black holes universally include AdS 2 factor funnily, AdS 3 holography more straightforward study charged Jackiw Teitelboim model as example I JT = α d 2 x g [e (R 2φ + 8L ) ] 2π 2 L2 4 F 2 Metric g has signature, + and Ricci-scalar R< 0 for AdS Maxwell field strength F µν = 2E ε µν dual to electric field E Dilaton φ has no kinetic term and no coupling to gauge field Cosmological constant Λ = 8 parameterized by AdS radius L L 2 Coupling constant α usually is positive δφ EOM: R = 8 AdS L 2 2! δa EOM: µ F µν = 0 E = constant δg EOM:...but simple for constant dilaton: e 2φ = L4 4 E2 µ ν e 2φ g µν 2 e 2φ + 4 L 2 e 2φ g µν + L2 2 F µ λ F νλ L2 8 g µν F 2 = 0 D. Grumiller Gravity in lower dimensions Which 2D theory? 13/42

54 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant D. Grumiller Gravity in lower dimensions Which 2D theory? 14/42

55 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) D. Grumiller Gravity in lower dimensions Which 2D theory? 14/42

56 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) Anomalous transformation above leads to central charge (HS, CGLM) c = 24αe 2φ = 3 G 2 = 3 2 ke2 L 2 D. Grumiller Gravity in lower dimensions Which 2D theory? 14/42

57 Some surprising results Hartman, Strominger = HS Castro, DG, Larsen, McNees = CGLM Holographic renormalization leads to boundary mass term (CGLM) I dx γ ma 2 Nevertheless, total action gauge invariant Boundary stress tensor transforms anomalously (HS) (δ ξ + δ λ ) T tt = 2T tt t ξ + ξ t T tt c 24π L 3 t ξ where δ ξ + δ λ is combination of diffeo- and gauge trafos that preserve the boundary conditions (similarly: δ λ J t = k 4π L tλ) Anomalous transformation above leads to central charge (HS, CGLM) c = 24αe 2φ = 3 G 2 = 3 2 ke2 L 2 Positive central charge only for negative coupling constant α (CGLM) α < 0 D. Grumiller Gravity in lower dimensions Which 2D theory? 14/42

58 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions Which 3D theory? 15/42

59 Attempt 1: Einstein Hilbert As simple as possible... but not simpler! Let us start with the simplest attempt. Einstein-Hilbert action: I EH = 1 d 3 x g R 16π G Equations of motion: R µν = 0 Ricci-flat and therefore Riemann-flat locally trivial! D. Grumiller Gravity in lower dimensions Which 3D theory? 16/42

60 Attempt 1: Einstein Hilbert As simple as possible... but not simpler! Let us start with the simplest attempt. Einstein-Hilbert action: I EH = 1 d 3 x g R 16π G Equations of motion: R µν = 0 Ricci-flat and therefore Riemann-flat locally trivial! Properties of Einstein-Hilbert No gravitons (recall: in D dimensions D(D 3)/2 gravitons) No BHs Einstein-Hilbert in 3D is too simple for us! D. Grumiller Gravity in lower dimensions Which 3D theory? 16/42

61 Attempt 2: Topologically massive gravity Deser, Jackiw and Templeton found a way to introduce gravitons! Let us now add a gravitational Chern Simons term. TMG action: I TMG = I EH + 1 d 3 x g 1 16π G 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) Equations of motion: with the Cotton tensor defined as R µν + 1 µ C µν = 0 C µν = 1 2 ε µ αβ α R βν + (µ ν) D. Grumiller Gravity in lower dimensions Which 3D theory? 17/42

62 Attempt 2: Topologically massive gravity Deser, Jackiw and Templeton found a way to introduce gravitons! Let us now add a gravitational Chern Simons term. TMG action: I TMG = I EH + 1 d 3 x g 1 16π G 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) Equations of motion: with the Cotton tensor defined as Properties of TMG R µν + 1 µ C µν = 0 C µν = 1 2 ε µ αβ α R βν + (µ ν) Gravitons! Reason: third derivatives in Cotton tensor! No BHs TMG is slightly too simple for us! D. Grumiller Gravity in lower dimensions Which 3D theory? 17/42

63 Attempt 3: Einstein Hilbert AdS Bañados, Teitelboim and Zanelli (and Henneaux) taught us how to get 3D BHs Add negative cosmological constant to Einstein-Hilbert action: I ΛEH = 1 d 3 x g ( R + 2 ) 16π G l 2 Equations of motion: G µν = R µν 1 2 g µνr 1 l 2 g µν = 0 Particular solutions: BTZ BH with line-element ds 2 BTZ = (r2 r+)(r 2 2 r ) 2 l 2 r 2 dt 2 l 2 r 2 + (r 2 r+)(r 2 2 r ) 2 dr2 + r 2 ( dφ r +r lr 2 dt ) 2 D. Grumiller Gravity in lower dimensions Which 3D theory? 18/42

64 Attempt 3: Einstein Hilbert AdS Bañados, Teitelboim and Zanelli (and Henneaux) taught us how to get 3D BHs Add negative cosmological constant to Einstein-Hilbert action: I ΛEH = 1 d 3 x g ( R + 2 ) 16π G l 2 Equations of motion: G µν = R µν 1 2 g µνr 1 l 2 g µν = 0 Particular solutions: BTZ BH with line-element ds 2 BTZ = (r2 r+)(r 2 2 r ) 2 l 2 r 2 dt 2 l 2 r 2 + (r 2 r+)(r 2 2 r ) 2 dr2 + r 2 ( dφ r +r lr 2 dt ) 2 Properties of Einstein-Hilbert-AdS No gravitons Rotating BH solutions that asymptote to AdS 3! Adding a negative cosmological constant produces BH solutions! D. Grumiller Gravity in lower dimensions Which 3D theory? 18/42

65 Cosmological topologically massive gravity CTMG is a 3D theory with BHs and gravitons! We want a 3D theory with gravitons and BHs and therefore take CTMG action I CTMG = 1 d 3 x [ g R π G l µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] Equations of motion: G µν + 1 µ C µν = 0 D. Grumiller Gravity in lower dimensions Which 3D theory? 19/42

66 Cosmological topologically massive gravity CTMG is a 3D theory with BHs and gravitons! We want a 3D theory with gravitons and BHs and therefore take CTMG action I CTMG = 1 d 3 x [ g R π G l µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] Equations of motion: G µν + 1 µ C µν = 0 Properties of CTMG Gravitons! BHs! CTMG is just perfect for us. Study this theory! D. Grumiller Gravity in lower dimensions Which 3D theory? 19/42

67 Einstein sector of the classical theory Solutions of Einstein s equations G µν = 0 R = 6 l 2 also have vanishing Cotton tensor C µν = 0 and therefore are solutions of CTMG. D. Grumiller Gravity in lower dimensions Which 3D theory? 20/42

68 Einstein sector of the classical theory Solutions of Einstein s equations also have vanishing Cotton tensor G µν = 0 R = 6 l 2 C µν = 0 and therefore are solutions of CTMG. This sector of solutions contains BTZ BH Pure AdS D. Grumiller Gravity in lower dimensions Which 3D theory? 20/42

69 Einstein sector of the classical theory Solutions of Einstein s equations also have vanishing Cotton tensor G µν = 0 R = 6 l 2 C µν = 0 and therefore are solutions of CTMG. This sector of solutions contains BTZ BH Pure AdS Line-element of pure AdS: ds 2 AdS = ḡ µν dx µ dx ν = l 2( cosh 2 ρ dτ 2 + sinh 2 ρ dφ 2 + dρ 2) Isometry group: SL(2, R) L SL(2, R) R Useful to introduce light-cone coordinates u = τ + φ, v = τ φ D. Grumiller Gravity in lower dimensions Which 3D theory? 20/42

70 AdS 3 -algebra of Killing vectors A technical reminder The SL(2, R) L generators obey the algebra L 0 = i u [ L ±1 = ie ±iu cosh 2ρ sinh 2ρ 1 u sinh 2ρ v i ] 2 ρ and have the quadratic Casimir [L 0, L ±1 ] = L ±1, [L 1, L 1 ] = 2L 0 L 2 = 1 2 (L 1L 1 + L 1 L 1 ) L 2 0 The SL(2, R) R generators L 0, L ±1 obey same algebra, but with u v, L L D. Grumiller Gravity in lower dimensions Which 3D theory? 21/42

71 Cotton sector of the classical theory Solutions of CTMG with G µν 0 necessarily have also non-vanishing Cotton tensor C µν 0 Few exact solutions of this type are known. D. Grumiller Gravity in lower dimensions Which 3D theory? 22/42

72 Cotton sector of the classical theory Solutions of CTMG with G µν 0 necessarily have also non-vanishing Cotton tensor C µν 0 Few exact solutions of this type are known. Perhaps most interesting solution: Warped AdS (stretched/squashed), see Bengtsson & Sandin Line-element of space-like warped AdS: ds 2 warped AdS = l2 ( cosh 2 ν 2 ρ dτ 2 + 4ν2 + 3 ν (du + sinh ρ dτ)2 + dρ 2) Sidenote: null-warped AdS in holographic duals of cold atoms: ( ds 2 null warped AdS = l 2 dy dx + dx y 2 ± (dx ) 2 ) y 4 D. Grumiller Gravity in lower dimensions Which 3D theory? 22/42

73 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

74 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

75 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions Like particle mechanics, but with up to three time derivatives D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

76 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions Like particle mechanics, but with up to three time derivatives Still surprisingly difficult to get exact solutions! D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

77 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions Like particle mechanics, but with up to three time derivatives Still surprisingly difficult to get exact solutions! Known solutions: AdS, BTZ, warped AdS D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

78 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions Like particle mechanics, but with up to three time derivatives Still surprisingly difficult to get exact solutions! Known solutions: AdS, BTZ, warped AdS Reduced action (Clement): [ ζ I C [ζ, X i ] dρ 2 Ẋi Ẋ j η ij 2 ζl 2 + ζ2 2µ ɛ ijk X i Ẋ j Ẍ k] Here ζ is a Lagrange-multiplier and X i = (T, X, Y ) a Lorentzian 3-vector D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

79 CTMG as particle mechanics problem Stationary and axi-symmetric solutions Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions Like particle mechanics, but with up to three time derivatives Still surprisingly difficult to get exact solutions! Known solutions: AdS, BTZ, warped AdS Reduced action (Clement): [ ζ I C [ζ, X i ] dρ 2 Ẋi Ẋ j η ij 2 ζl 2 + ζ2 2µ ɛ ijk X i Ẋ j Ẍ k] Here ζ is a Lagrange-multiplier and X i = (T, X, Y ) a Lorentzian 3-vector It could be rewarding to investigate this mechanical problem systematically and numerically! D. Grumiller Gravity in lower dimensions Which 3D theory? 23/42

80 CTMG at the chiral point...abbreviated as CCTMG Definition: CTMG at the chiral point is CTMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. D. Grumiller Gravity in lower dimensions Which 3D theory? 24/42

81 CTMG at the chiral point...abbreviated as CCTMG Definition: CTMG at the chiral point is CTMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? D. Grumiller Gravity in lower dimensions Which 3D theory? 24/42

82 CTMG at the chiral point...abbreviated as CCTMG Definition: CTMG at the chiral point is CTMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? Calculating the central charges of the dual boundary CFT yields c L = 3 ( 1 ) 1, cr = 3 ( 1 ) 1 + 2G µ l 2G µ l Thus, at the chiral point we get c L = 0, c R = 3 G D. Grumiller Gravity in lower dimensions Which 3D theory? 24/42

83 CTMG at the chiral point...abbreviated as CCTMG Definition: CTMG at the chiral point is CTMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? Calculating the central charges of the dual boundary CFT yields c L = 3 ( 1 ) 1, cr = 3 ( 1 ) 1 + 2G µ l 2G µ l Thus, at the chiral point we get c L = 0, c R = 3 G Notes: Abbreviate CTMG at the chiral point as CCTMG CCTMG is also known as chiral gravity D. Grumiller Gravity in lower dimensions Which 3D theory? 24/42

84 Gravitons around AdS 3 in CTMG Linearization around AdS background g µν = ḡ µν + h µν D. Grumiller Gravity in lower dimensions Which 3D theory? 25/42

85 Gravitons around AdS 3 in CTMG Linearization around AdS background g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α, (D M ) µ ν = δ ν µ + 1 µ ε µ αν α D. Grumiller Gravity in lower dimensions Which 3D theory? 25/42

86 Gravitons around AdS 3 in CTMG Linearization around AdS background g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α, (D M ) µ ν = δ ν µ + 1 µ ε µ αν α Three linearly independent solutions to (1): ( D L h L) µν = 0, ( D R h R) µν = 0, ( D M h M) µν = 0 D. Grumiller Gravity in lower dimensions Which 3D theory? 25/42

87 Gravitons around AdS 3 in CTMG Linearization around AdS background g µν = ḡ µν + h µν leads to linearized EOM that are third order PDE G (1) µν + 1 µ C(1) µν = (D R D L D M h) µν = 0 (1) with three mutually commuting first order operators (D L/R ) µ ν = δ ν µ ± l ε µ αν α, (D M ) µ ν = δ ν µ + 1 µ ε µ αν α Three linearly independent solutions to (1): ( D L h L) µν = 0, ( D R h R) µν = 0, ( D M h M) µν = 0 At chiral point left (L) and massive (M) branches coincide! D. Grumiller Gravity in lower dimensions Which 3D theory? 25/42

88 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all solutions of linearized EOM. Primaries: L 0, L 0 eigenstates ψ L/R/M with L 1 ψ R/L/M = L 1 ψ R/L/M = 0 D. Grumiller Gravity in lower dimensions Which 3D theory? 26/42

89 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all solutions of linearized EOM. Primaries: L 0, L 0 eigenstates ψ L/R/M with L 1 ψ R/L/M = L 1 ψ R/L/M = 0 Descendants: act with L 1 and L 1 on primaries D. Grumiller Gravity in lower dimensions Which 3D theory? 26/42

90 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all solutions of linearized EOM. Primaries: L 0, L 0 eigenstates ψ L/R/M with L 1 ψ R/L/M = L 1 ψ R/L/M = 0 Descendants: act with L 1 and L 1 on primaries General solution: linear combination of ψ R/L/M D. Grumiller Gravity in lower dimensions Which 3D theory? 26/42

91 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all solutions of linearized EOM. Primaries: L 0, L 0 eigenstates ψ L/R/M with L 1 ψ R/L/M = L 1 ψ R/L/M = 0 Descendants: act with L 1 and L 1 on primaries General solution: linear combination of ψ R/L/M Linearized metric is then the real part of the wavefunction h µν = Re ψ µν D. Grumiller Gravity in lower dimensions Which 3D theory? 26/42

92 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all solutions of linearized EOM. Primaries: L 0, L 0 eigenstates ψ L/R/M with L 1 ψ R/L/M = L 1 ψ R/L/M = 0 Descendants: act with L 1 and L 1 on primaries General solution: linear combination of ψ R/L/M Linearized metric is then the real part of the wavefunction h µν = Re ψ µν At chiral point: L and M branches degenerate. Get new solution (DG & Johansson) ψµν new ψµν(µl) M ψµν L = lim µl 1 µl 1 with property ( D L ψ new) µν = ( D M ψ new) µν 0, ( (D L ) 2 ψ new) µν = 0 D. Grumiller Gravity in lower dimensions Which 3D theory? 26/42

93 Sign oder nicht sign? That is the question. Choosing between Skylla and Charybdis. With signs defined as in this talk: BHs positive energy, gravitons negative energy D. Grumiller Gravity in lower dimensions Which 3D theory? 27/42

94 Sign oder nicht sign? That is the question. Choosing between Skylla and Charybdis. With signs defined as in this talk: BHs positive energy, gravitons negative energy With signs as defined in Deser-Jackiw-Templeton paper: BHs negative energy, gravitons positive energy D. Grumiller Gravity in lower dimensions Which 3D theory? 27/42

95 Sign oder nicht sign? That is the question. Choosing between Skylla and Charybdis. With signs defined as in this talk: BHs positive energy, gravitons negative energy With signs as defined in Deser-Jackiw-Templeton paper: BHs negative energy, gravitons positive energy Either way need a mechanism to eliminate unwanted negative energy objects either the gravitons or the BHs D. Grumiller Gravity in lower dimensions Which 3D theory? 27/42

96 Sign oder nicht sign? That is the question. Choosing between Skylla and Charybdis. With signs defined as in this talk: BHs positive energy, gravitons negative energy With signs as defined in Deser-Jackiw-Templeton paper: BHs negative energy, gravitons positive energy Either way need a mechanism to eliminate unwanted negative energy objects either the gravitons or the BHs Even at chiral point the problem persists because of the logarithmic mode. See Figure. (Figure: thanks to N. Johansson) Energy for all branches: D. Grumiller Gravity in lower dimensions Which 3D theory? 27/42

97 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 28/42

98 Witten s attempt Different approach (without gravitons!): D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

99 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

100 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

101 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

102 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

103 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

104 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! Therefore introduce negative cosmological constant D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

105 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! Therefore introduce negative cosmological constant Define quantum gravity by its dual CFT at the AdS boundary D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

106 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! Therefore introduce negative cosmological constant Define quantum gravity by its dual CFT at the AdS boundary Constructing this CFT still a monstrous effort... D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

107 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! Therefore introduce negative cosmological constant Define quantum gravity by its dual CFT at the AdS boundary Constructing this CFT still a monstrous effort... Maloney & Witten: taking into account all known contributions to path integral leads to non-sensible result for partition function Z. In particular, no holomorphic factorization: Z MW Z L Z R D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

108 Witten s attempt Different approach (without gravitons!): Naive remark 1: 3D gravity is trivial Naive remark 2: 3D gravity is non-renormalizable Synthesis of naive remarks: 3D quantum gravity may exist as non-trivial theory Positive cosmological constant: impossible? Vanishing cosmological constant: S-matrix, but no gravitons! Therefore introduce negative cosmological constant Define quantum gravity by its dual CFT at the AdS boundary Constructing this CFT still a monstrous effort... Maloney & Witten: taking into account all known contributions to path integral leads to non-sensible result for partition function Z. In particular, no holomorphic factorization: Z MW Z L Z R Various suggestions to interpret this problem: need cosmic strings, need sum over complex geometries, 3D quantum gravity does not exist by itself D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 29/42

109 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

110 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. 2. CCTMG has c L = 0 D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

111 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. 2. CCTMG has c L = 0 3. Massive graviton degenerates with left boundary graviton: ψ M = ψ L D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

112 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. 2. CCTMG has c L = 0 3. Massive graviton degenerates with left boundary graviton: ψ M = ψ L Thus, dual CFT chiral? If yes, we are done! D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

113 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. 2. CCTMG has c L = 0 3. Massive graviton degenerates with left boundary graviton: ψ M = ψ L Thus, dual CFT chiral? If yes, we are done! Suggestive to interpret LSS results as absence of gravitons D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

114 Li, Song & Strominger attempt Is CCTMG dual to a chiral CFT? Interesting observations: 1. If left-moving sector is trivial, Z L = 1, then problem of holomorphic factorization Z = Z L Z R = Z R is solved. 2. CCTMG has c L = 0 3. Massive graviton degenerates with left boundary graviton: ψ M = ψ L Thus, dual CFT chiral? If yes, we are done! Suggestive to interpret LSS results as absence of gravitons But: Disagrees with results by Carlip, Deser, Waldron & Wise! D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 30/42

115 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

116 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. Interesting property: L 0 ( ψ new ψ L L 0 ( ψ new ψ L ) = ) = ( ( ) ( ψ new ψ L ) ( ψ new ψ L ), ). D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

117 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. Interesting property: L 0 ( ψ new ψ L L 0 ( ψ new ψ L ) = ) = ( ( ) ( ) ψ new ψ L, ) ( ) ψ new. Such a Jordan form of L 0, L 0 is defining property of a logarithmic CFT! Note: called logarithmic CFT because some correlators take the form ψ L ψ new (z)ψ new (0) ln z +... D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

118 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. Interesting property: L 0 ( ψ new ψ L L 0 ( ψ new ψ L ) = ) = ( ( ) ( ) ψ new ψ L, ) ( ) ψ new. Such a Jordan form of L 0, L 0 is defining property of a logarithmic CFT! Note: called logarithmic CFT because some correlators take the form ψ L ψ new (z)ψ new (0) ln z +... Logarithmic CFT: not unitary and not chiral! D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

119 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. Interesting property: L 0 ( ψ new ψ L L 0 ( ψ new ψ L ) = ) = ( ( ) ( ) ψ new ψ L, ) ( ) ψ new. Such a Jordan form of L 0, L 0 is defining property of a logarithmic CFT! Note: called logarithmic CFT because some correlators take the form ψ L ψ new (z)ψ new (0) ln z +... Logarithmic CFT: not unitary and not chiral! Either logarithmic or chiral CFT dual (or none) D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

120 Gravitons in CCTMG Is CCTMG dual to a logarithmic CFT? New mode resolves apparent contradiction between LSS and CDWW. Interesting property: L 0 ( ψ new ψ L L 0 ( ψ new ψ L ) = ) = ( ( ) ( ) ψ new ψ L, ) ( ) ψ new. Such a Jordan form of L 0, L 0 is defining property of a logarithmic CFT! Note: called logarithmic CFT because some correlators take the form ψ L ψ new (z)ψ new (0) ln z +... Logarithmic CFT: not unitary and not chiral! Either logarithmic or chiral CFT dual (or none) Currently unknown which of these alternatives is realized! D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 31/42

121 Viability of the logarithmic mode, part 1 Explicit solution for logarithmic mode (DG & Johansson) Is the logarithmic mode really there? Collect in the following suggestions how the logarithmic mode could drop out of the physical spectrum and show that none of them is realized. D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 32/42

122 Viability of the logarithmic mode, part 1 Explicit solution for logarithmic mode (DG & Johansson) Is the logarithmic mode really there? Collect in the following suggestions how the logarithmic mode could drop out of the physical spectrum and show that none of them is realized. Before starting, here is the explicit form of the logarithmic mode: h new µν = sinh ρ ( ) c τ s ln cosh ρ cosh ρ with tanh 2 ρ ( s τ + c ln cosh ρ ) c = cos (2u), s = sin (2u), a = a 2 µν µν 1 sinh ρ cosh ρ (2) D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 32/42

123 Viability of the logarithmic mode, part 2 Physical mode with negative energy Suggestion 1 The logarithmic mode is pure gauge? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 33/42

124 Viability of the logarithmic mode, part 2 Physical mode with negative energy Suggestion 1 The logarithmic mode is pure gauge? No! h new does not solve linearized Einstein equations. Thus is not pure gauge. Note: confirmed by Sachs who considered logarithmic quasi-normal modes D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 33/42

125 Viability of the logarithmic mode, part 2 Physical mode with negative energy Suggestion 1 The logarithmic mode is pure gauge? No! h new does not solve linearized Einstein equations. Thus is not pure gauge. Note: confirmed by Sachs who considered logarithmic quasi-normal modes Suggestion 2 Logarithmic mode has infinite energy and thus must be discarded? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 33/42

126 Viability of the logarithmic mode, part 2 Physical mode with negative energy Suggestion 1 The logarithmic mode is pure gauge? No! h new does not solve linearized Einstein equations. Thus is not pure gauge. Note: confirmed by Sachs who considered logarithmic quasi-normal modes Suggestion 2 Logarithmic mode has infinite energy and thus must be discarded? No! E new = G l 3 Energy is finite and negative. Thus logarithmic mode leads to instability but cannot be discarded. D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 33/42

127 Viability of the logarithmic mode, part 3 Boundary conditions beyond Brown Henneaux Suggestion 3 New mode is not a small perturbation? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 34/42

128 Viability of the logarithmic mode, part 3 Boundary conditions beyond Brown Henneaux Suggestion 3 New mode is not a small perturbation? It is! h new diverges asymptotically like ρ, but AdS background diverges asymptotically like e 2ρ. Thus h new is really a small perturbation. D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 34/42

129 Viability of the logarithmic mode, part 3 Boundary conditions beyond Brown Henneaux Suggestion 3 New mode is not a small perturbation? It is! h new diverges asymptotically like ρ, but AdS background diverges asymptotically like e 2ρ. Thus h new is really a small perturbation. Suggestion 4 New mode is not asymptotically AdS? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 34/42

130 Viability of the logarithmic mode, part 3 Boundary conditions beyond Brown Henneaux Suggestion 3 New mode is not a small perturbation? It is! h new diverges asymptotically like ρ, but AdS background diverges asymptotically like e 2ρ. Thus h new is really a small perturbation. Suggestion 4 New mode is not asymptotically AdS? It is! Solution is asymptotically AdS ds 2 = dρ 2 + ( γ (0) ij e2ρ/l + γ (1) ij ρ + γ(0) ij + γ (2) ij e 2ρ/l +... ) dx i dx j but violates Brown-Henneaux boundary conditions! (γ (1) ij BH = 0) Henneaux et al. showed precedents where this may happen in 3D New boundary conditions replacing Brown-Henneaux (DG & Johansson) D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 34/42

131 Viability of the logarithmic mode, part 4 Brown York boundary stress tensor Suggestion 5 New mode leads to ill-defined Brown-York boundary stress tensor? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 35/42

132 Viability of the logarithmic mode, part 4 Brown York boundary stress tensor Suggestion 5 New mode leads to ill-defined Brown-York boundary stress tensor? No! Total action including boundary terms (Kraus & Larsen) I total = I CTMG + 1 d 2 x ( γ K 1 ) 8πG l Its first variation leads to Brown-York boundary stress-tensor: δi total = 1 d 2 x γ EOM 32πG (0) T ij δγ (0) ij DG & Johansson: T ij is finite, traceless and chiral: T ij = l ( ) π G 1 1 Note: coincides with Brown-York boundary stress-tensor of global AdS 3 D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 35/42 ij

133 Viability of the logarithmic mode, part 5 Artifact of linearization? Suggestion 6 Maybe some non-linear magic kills the new mode? D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 36/42

134 Viability of the logarithmic mode, part 5 Artifact of linearization? Suggestion 6 Maybe some non-linear magic kills the new mode? Unlikely! DG, Jackiw & Johansson: classical phase space analysis of CCTMG N = 1 2 ( ) 1 ( ) 2 D 2 N1 N 2 = = 1 2 N: number of physical degrees of freedom (per point) D: number of canonical pairs in full phase space N 1(2) : number of linearly independent first (second) class constraints confirmed in more general calculation by Carlip D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 36/42

135 Viability of the logarithmic mode, part 5 Artifact of linearization? Suggestion 6 Maybe some non-linear magic kills the new mode? Unlikely! DG, Jackiw & Johansson: classical phase space analysis of CCTMG N = 1 2 ( ) 1 ( ) 2 D 2 N1 N 2 = = 1 2 N: number of physical degrees of freedom (per point) D: number of canonical pairs in full phase space N 1(2) : number of linearly independent first (second) class constraints confirmed in more general calculation by Carlip Conclusion 1: logarithmic mode passed all tests so far Conclusion 2: CCTMG is unstable; dual CFT probably logarithmic D. Grumiller Gravity in lower dimensions How to quantize 3D gravity? 36/42

136 Outline Why lower-dimensional gravity? Which 2D theory? Which 3D theory? How to quantize 3D gravity? What next? D. Grumiller Gravity in lower dimensions What next? 37/42

137 Chiral vs. logarithmic Pivotal open question: does dual CFT exist? is it chiral or logarithmic? To Do Chiral route: must show consistency of truncation! Logarithmic: must show consistency of 2 nd order perturbations! D. Grumiller Gravity in lower dimensions What next? 38/42

138 Chiral vs. logarithmic Pivotal open question: does dual CFT exist? is it chiral or logarithmic? To Do Chiral route: must show consistency of truncation! Logarithmic: must show consistency of 2 nd order perturbations! ad chiral: restricting to Brown-Henneaux boundary conditions does not help Giribet, Kleban & Porrati showed that descendent of new mode L 1 ψ new µν = Y µν = X µν + L ξ ḡ µν after a diffeomorphism ξ obeys Brown-Henneaux boundary conditions Descendants of logarithmic mode are there even when boundary conditions are restricted beyond requiring variational principle! D. Grumiller Gravity in lower dimensions What next? 38/42

139 Chiral vs. logarithmic Pivotal open question: does dual CFT exist? is it chiral or logarithmic? To Do Chiral route: must show consistency of truncation! Logarithmic: must show consistency of 2 nd order perturbations! ad chiral: restricting to Brown-Henneaux boundary conditions does not help Giribet, Kleban & Porrati showed that descendent of new mode L 1 ψ new µν = Y µν = X µν + L ξ ḡ µν after a diffeomorphism ξ obeys Brown-Henneaux boundary conditions Descendants of logarithmic mode are there even when boundary conditions are restricted beyond requiring variational principle! Need different mechanism of truncation! D. Grumiller Gravity in lower dimensions What next? 38/42

140 Chiral vs. logarithmic Pivotal open question: does dual CFT exist? is it chiral or logarithmic? To Do Chiral route: must show consistency of truncation! Logarithmic: must show consistency of 2 nd order perturbations! ad logarithmic: straightforward but somewhat lengthy calculation expand metric around AdS background up to second order: EOM lead to linear PDE for h (2) µν : g µν = ḡ µν + h new µν + h (2) µν D (3) h (2) = f ( (h new µν ) 2) Check if h (2) really is smaller than h new µν D. Grumiller Gravity in lower dimensions What next? 38/42

141 Chiral vs. logarithmic Pivotal open question: does dual CFT exist? is it chiral or logarithmic? To Do Chiral route: must show consistency of truncation! Logarithmic: must show consistency of 2 nd order perturbations! ad logarithmic: straightforward but somewhat lengthy calculation expand metric around AdS background up to second order: EOM lead to linear PDE for h (2) µν : g µν = ḡ µν + h new µν + h (2) µν D (3) h (2) = f ( (h new µν ) 2) Check if h (2) really is smaller than h new µν Might be rewarding exercise for a student D. Grumiller Gravity in lower dimensions What next? 38/42

142 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G D. Grumiller Gravity in lower dimensions What next? 39/42

143 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G If AdS 3 is unstable in CCTMG because of logarithmic mode, where does it run to? D. Grumiller Gravity in lower dimensions What next? 39/42

144 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G If AdS 3 is unstable in (C)CTMG because of massive graviton mode, where does it run to? Both observations suggest that there might be a ground state different from pure AdS 3 in (C)CTMG. D. Grumiller Gravity in lower dimensions What next? 39/42

145 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G If AdS 3 is unstable in (C)CTMG because of massive graviton mode, where does it run to? Both observations suggest that there might be a ground state different from pure AdS 3 in (C)CTMG. Consider other possible ground states with less symmetry D. Grumiller Gravity in lower dimensions What next? 39/42

146 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G If AdS 3 is unstable in (C)CTMG because of massive graviton mode, where does it run to? Both observations suggest that there might be a ground state different from pure AdS 3 in (C)CTMG. Consider other possible ground states with less symmetry Example: warped AdS has four Killing vectors with U(1) L SL(2, R) R D. Grumiller Gravity in lower dimensions What next? 39/42

147 Which groundstate? Two observations: Global AdS 3 has mass and angular momentum in (C)CTMG M AdS 3 = µ J AdS3 = 1 8G If AdS 3 is unstable in (C)CTMG because of massive graviton mode, where does it run to? Both observations suggest that there might be a ground state different from pure AdS 3 in (C)CTMG. Consider other possible ground states with less symmetry Example: warped AdS has four Killing vectors with U(1) L SL(2, R) R Strominger et al. : Suggestive to consider warped AdS as possible groundstate of (C)CTMG D. Grumiller Gravity in lower dimensions What next? 39/42

148 Most crucial question we would like to answer Does 3D quantum gravity exist with no strings attached? D. Grumiller Gravity in lower dimensions What next? 40/42

149 Most crucial question we would like to answer Does 3D quantum gravity exist with no strings attached? Perhaps a win-win situation! Consider the possible outcomes to this question: D. Grumiller Gravity in lower dimensions What next? 40/42

150 Most crucial question we would like to answer Does 3D quantum gravity exist with no strings attached? Perhaps a win-win situation! Consider the possible outcomes to this question: If yes: we would have an interesting quantum theory of gravity with BHs and gravitons to get conceptual insight into quantum gravity D. Grumiller Gravity in lower dimensions What next? 40/42

151 Most crucial question we would like to answer Does 3D quantum gravity exist with no strings attached? Perhaps a win-win situation! Consider the possible outcomes to this question: If yes: we would have an interesting quantum theory of gravity with BHs and gravitons to get conceptual insight into quantum gravity if no: potentially exciting news for string theory D. Grumiller Gravity in lower dimensions What next? 40/42

152 Most crucial question we would like to answer Does 3D quantum gravity exist with no strings attached? Perhaps a win-win situation! Consider the possible outcomes to this question: If yes: we would have an interesting quantum theory of gravity with BHs and gravitons to get conceptual insight into quantum gravity if no: potentially exciting news for string theory Thank you for your attention! D. Grumiller Gravity in lower dimensions What next? 40/42

153 D. Grumiller Gravity in lower dimensions What next? 41/42