Massive gravity in three dimensions The AdS 3 /LCFT 2 correspondence

Transcription

1 Massive gravity in three dimensions The AdS 3 /LCFT 2 correspondence Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology CBPF, February 2011 with: Hamid Afshar, Mario Bertin, Branislav Cvetkovic, Sabine Ertl, Matthias Gaberdiel, Olaf Hohm, Roman Jackiw, Niklas Johansson, Ivo Sachs, Dima Vassilevich, Thomas Zojer

2 Outline Introduction to 3D gravity Topologically massive gravity Logarithmic CFT conjecture Consequences, Generalizations & Applications D. Grumiller Massive gravity in three dimensions 2/29

3 Outline Introduction to 3D gravity Topologically massive gravity Logarithmic CFT conjecture Consequences, Generalizations & Applications D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 3/29

4 Motivation Quantum gravity Address conceptual issues of quantum gravity Black hole evaporation, information loss, black hole microstate counting, virtual black hole production,... Technically much simpler than 4D or higher D gravity Integrable models: powerful tools in physics (Coulomb problem, Hydrogen atom, harmonic oscillator,...) Models should be as simple as possible, but not simpler D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 4/29

5 Motivation Quantum gravity Address conceptual issues of quantum gravity Black hole evaporation, information loss, black hole microstate counting, virtual black hole production,... Technically much simpler than 4D or higher D gravity Integrable models: powerful tools in physics (Coulomb problem, Hydrogen atom, harmonic oscillator,...) Models should be as simple as possible, but not simpler Gauge/gravity duality Deeper understanding of black hole holography AdS3 /CFT 2 correspondence best understood Quantum gravity via AdS/CFT? (Witten 07, Li, Song, Strominger 08) Applications to 2D condensed matter systems? Gauge gravity duality beyond standard AdS/CFT: warped AdS, asymptotic Lifshitz, non-relativistic CFTs, logarithmic CFTs,... D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 4/29

6 Motivation Quantum gravity Address conceptual issues of quantum gravity Black hole evaporation, information loss, black hole microstate counting, virtual black hole production,... Technically much simpler than 4D or higher D gravity Integrable models: powerful tools in physics (Coulomb problem, Hydrogen atom, harmonic oscillator,...) Models should be as simple as possible, but not simpler Gauge/gravity duality Deeper understanding of black hole holography AdS 3 /CFT 2 correspondence best understood Quantum gravity via AdS/CFT? (Witten 07, Li, Song, Strominger 08) Applications to 2D condensed matter systems? Gauge gravity duality beyond standard AdS/CFT: warped AdS, asymptotic Lifshitz, non-relativistic CFTs, logarithmic CFTs,... Physics Cosmic strings (Deser, Jackiw, t Hooft 84, 92) Black hole analog systems in condensed matter physics (graphene, BEC, fluids,...) D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 4/29

7 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 Massive gravity in three dimensions Introduction to 3D gravity 5/29

8 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 Massive gravity in three dimensions Introduction to 3D gravity 5/29

9 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 Massive gravity in three dimensions Introduction to 3D gravity 5/29

10 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 Massive gravity in three dimensions Introduction to 3D gravity 5/29

11 Pure gravity in 3D Let us switch off all matter fields and keep only the metric g. I 3DG = 1 d 3 x g L(g) 16π G D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 6/29

12 Pure gravity in 3D Let us switch off all matter fields and keep only the metric g. I 3DG = 1 d 3 x g L(g) 16π G Variation of L should lead to tensor equations Require absence of higher derivatives than fourth (for simplicity) Require absence of scalar ghosts D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 6/29

13 Pure gravity in 3D Let us switch off all matter fields and keep only the metric g. I 3DG = 1 d 3 x g L(g) 16π G Variation of L should lead to tensor equations Require absence of higher derivatives than fourth (for simplicity) Require absence of scalar ghosts The requirements above are fulfilled for L = L MG (R µν ) + L CS D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 6/29

14 Pure gravity in 3D Let us switch off all matter fields and keep only the metric g. I 3DG = 1 d 3 x g L(g) 16π G Variation of L should lead to tensor equations Require absence of higher derivatives than fourth (for simplicity) Require absence of scalar ghosts The requirements above are fulfilled for L = L MG (R µν ) + L CS with the possiblity for a gravitational Chern Simons term L CS = 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 6/29

15 Pure gravity in 3D Let us switch off all matter fields and keep only the metric g. I 3DG = 1 d 3 x g L(g) 16π G Variation of L should lead to tensor equations Require absence of higher derivatives than fourth (for simplicity) Require absence of scalar ghosts The requirements above are fulfilled for L = L MG (R µν ) + L CS with the possiblity for a gravitational Chern Simons term L CS = 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) and the higher derivative Lagrange density L MG (R µν ) = σr 2Λ + 1 m 2 ( Rµν R µν 3 8 R2) + O(R 3 µν) D. Grumiller Massive gravity in three dimensions Introduction to 3D gravity 6/29

16 Outline Introduction to 3D gravity Topologically massive gravity Logarithmic CFT conjecture Consequences, Generalizations & Applications D. Grumiller Massive gravity in three dimensions Topologically massive gravity 7/29

17 Action and equations of motion of topologically massive gravity (TMG) Consider the action (Deser, Jackiw & Templeton 82) I TMG = 1 d 3 x [ g R π G l µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] D. Grumiller Massive gravity in three dimensions Topologically massive gravity 8/29

18 Action and equations of motion of topologically massive gravity (TMG) Consider the action (Deser, Jackiw & Templeton 82) I TMG = 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: with the Cotton tensor defined as R µν 1 2 g µνr 1 l 2 g µν + 1 µ C µν = 0 C µν = 1 2 ε µ αβ α R βν + (µ ν) D. Grumiller Massive gravity in three dimensions Topologically massive gravity 8/29

19 Action and equations of motion of topologically massive gravity (TMG) Consider the action (Deser, Jackiw & Templeton 82) I TMG = 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: with the Cotton tensor defined as Some properties of TMG R µν 1 2 g µνr 1 l 2 g µν + 1 µ C µν = 0 C µν = 1 2 ε µ αβ α R βν + (µ ν) Massive gravitons and black holes AdS solutions and asymptotic AdS solutions warped AdS solutions and warped AdS black holes Schrödinger solutions and Schrödinger pp-waves D. Grumiller Massive gravity in three dimensions Topologically massive gravity 8/29

20 Classical solutions (exact) Stationarity plus axi-symmetry: Two commuting Killing vectors D. Grumiller Massive gravity in three dimensions Topologically massive gravity 9/29

21 Classical solutions (exact) Stationarity plus axi-symmetry: Two commuting Killing vectors Effectively reduce 2+1 dimensions to 1+0 dimensions D. Grumiller Massive gravity in three dimensions Topologically massive gravity 9/29

22 Classical solutions (exact) 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 Massive gravity in three dimensions Topologically massive gravity 9/29

23 Classical solutions (exact) 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 Massive gravity in three dimensions Topologically massive gravity 9/29

24 Classical solutions (exact) 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! Reduced action (Clement 94): [ 1 I C [e, X i ] dρ e 2 e 2 Ẋ i Ẋ j η ij 2 l µ e 3 ɛ ijk X i Ẋ j Ẍ k] Here e is the Einbein and X i = (T, X, Y ) a Lorentzian 3-vector D. Grumiller Massive gravity in three dimensions Topologically massive gravity 9/29

25 Classical solutions (exact) 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! Reduced action (Clement 94): [ 1 I C [e, X i ] dρ e 2 e 2 Ẋ i Ẋ j η ij 2 l µ e 3 ɛ ijk X i Ẋ j Ẍ k] Here e is the Einbein and X i = (T, X, Y ) a Lorentzian 3-vector Classification of solutions: Einstein solutions: AdS, BTZ warped solutions: warped AdS, warped black holes Schrödinger solutions: asymptotic Schrödinger spacetimes, pp-waves generic solutions (Ertl, Grumiller & Johansson, 10) D. Grumiller Massive gravity in three dimensions Topologically massive gravity 9/29

26 TMG at the chiral point Definition: TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. D. Grumiller Massive gravity in three dimensions Topologically massive gravity 10/29

27 TMG at the chiral point Definition: TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? (Li, Song & Strominger 08) D. Grumiller Massive gravity in three dimensions Topologically massive gravity 10/29

28 TMG at the chiral point Definition: TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? (Li, Song & Strominger 08) Calculating the central charges of the dual boundary CFT yields c L = 3l 2G ( 1 1 µ l Thus, at the chiral point we get c L = 0 ) c R = 3l 2G c R = 3l G ( µ l ) D. Grumiller Massive gravity in three dimensions Topologically massive gravity 10/29

29 TMG at the chiral point Definition: TMG at the chiral point is TMG with the tuning µ l = 1 between the cosmological constant and the Chern Simons coupling. Why special? (Li, Song & Strominger 08) Calculating the central charges of the dual boundary CFT yields c L = 3l 2G ( 1 1 µ l Thus, at the chiral point we get c L = 0 ) c R = 3l 2G c R = 3l G ( µ l Abbreviate Cosmological TMG at the chiral point as CTMG CTMG is also known as chiral gravity Dual CFT: chiral? (conjecture by Li, Song & Strominger 08) More adequate name for CTMG: logarithmic gravity D. Grumiller Massive gravity in three dimensions Topologically massive gravity 10/29 )

30 Gravitons around AdS 3 in CTMG Linearization around AdS background. Line-element ḡ µν of pure AdS: g µν = ḡ µν + h µν d s 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. The SL(2, R) L generators L 0 = i u [ L ±1 = ie ±iu cosh 2ρ sinh 2ρ 1 u sinh 2ρ v i ] 2 ρ obey the algebra [L 0, L ±1 ] = L ±1, [L 1, L 1 ] = 2L 0. The SL(2, R) R generators L 0, L ±1 obey same algebra, but with u v, L L D. Grumiller Massive gravity in three dimensions Topologically massive gravity 11/29

31 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 Massive gravity in three dimensions Topologically massive gravity 11/29

32 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 Massive gravity in three dimensions Topologically massive gravity 11/29

33 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 Massive gravity in three dimensions Topologically massive gravity 11/29

34 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all normalizable 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 Massive gravity in three dimensions Topologically massive gravity 12/29

35 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all normalizable 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 Massive gravity in three dimensions Topologically massive gravity 12/29

36 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all normalizable 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 Massive gravity in three dimensions Topologically massive gravity 12/29

37 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all normalizable 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 Massive gravity in three dimensions Topologically massive gravity 12/29

38 Degeneracy at the chiral point Will be quite important later! Li, Song & Strominger found all normalizable 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 log solution (Grumiller & Johansson 08) ψµν log ψµν(µl) M ψµν L = lim µl 1 µl 1 with property ( D L ψ log) µν = ( D M ψ log) µν 0, ( (D L ) 2 ψ log) µν = 0 D. Grumiller Massive gravity in three dimensions Topologically massive gravity 12/29

39 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 Massive gravity in three dimensions Topologically massive gravity 13/29

40 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 Carlip, Deser, Waldron, Wise 08: BHs negative energy, gravitons positive energy D. Grumiller Massive gravity in three dimensions Topologically massive gravity 13/29

41 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 Carlip, Deser, Waldron, Wise 08: 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 Massive gravity in three dimensions Topologically massive gravity 13/29

42 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 Carlip, Deser, Waldron, Wise 08: 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. (thanks to Niklas Johansson) Energy for all branches: D. Grumiller Massive gravity in three dimensions Topologically massive gravity 13/29

43 Outline Introduction to 3D gravity Topologically massive gravity Logarithmic CFT conjecture Consequences, Generalizations & Applications D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 14/29

44 Motivating the conjecture Log mode exhibits interesting property: ( ) ( ) ( ψ log 2 1 ψ log H ψ L = 0 2 ψ L ( ) ( ) ( ψ log 2 0 ψ log J = 0 2 ψ L ψ L ) ) Here H = L 0 + L 0 t is the Hamilton operator and J = L 0 L 0 φ the angular momentum operator. D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 15/29

45 Motivating the conjecture Log mode exhibits interesting property: ( ) ( ) ( ψ log 2 1 ψ log H ψ L = 0 2 ψ L ( ) ( ) ( ψ log 2 0 ψ log J = 0 2 ψ L ψ L ) ) Here H = L 0 + L 0 t is the Hamilton operator and J = L 0 L 0 φ the angular momentum operator. Such a Jordan form of H and J is defining property of a logarithmic CFT! D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 15/29

46 Motivating the conjecture Log mode exhibits interesting property: ( ) ( ) ( ψ log 2 1 ψ log H ψ L = 0 2 ψ L ( ) ( ) ( ψ log 2 0 ψ log J = 0 2 ψ L ψ L ) ) Here H = L 0 + L 0 t is the Hamilton operator and J = L 0 L 0 φ the angular momentum operator. Such a Jordan form of H and J is defining property of a logarithmic CFT! Logarithmic CFT conjecture CTMG dual to a logarithmic CFT (Grumiller, Johansson 08) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 15/29

47 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

48 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (Grumiller & Johansson 08) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

49 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (Grumiller & Johansson 08) Logarithmic mode 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) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

50 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (Grumiller & Johansson 08) Logarithmic mode 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) Consistent log boundary conditions replacing Brown Henneaux (Grumiller & Johansson 08, Martinez, Henneaux & Troncoso 09) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

51 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (Grumiller & Johansson 08) Logarithmic mode 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) Consistent log boundary conditions replacing Brown Henneaux (Grumiller & Johansson 08, Martinez, Henneaux & Troncoso 09) Brown York stress tensor is finite and traceless, but not chiral D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

52 Early hints for validity of conjecture Properties of logarithmic mode: Perturbative solution of linearized EOM, but not pure gauge Energy of logarithmic mode is finite E log = G l 3 and negative instability! (Grumiller & Johansson 08) Logarithmic mode 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) Consistent log boundary conditions replacing Brown Henneaux (Grumiller & Johansson 08, Martinez, Henneaux & Troncoso 09) Brown York stress tensor is finite and traceless, but not chiral Log mode persists non-perturbatively, as shown by Hamilton analysis (Grumiller, Jackiw & Johansson 08, Carlip 08) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 16/29

53 Correlators in logarithmic CFTs Any CFT has a conserved traceless energy momentum tensor. T z z = 0 T zz = O L (z) T z z = O R ( z) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 17/29

54 Correlators in logarithmic CFTs Any CFT has a conserved traceless energy momentum tensor. T z z = 0 T zz = O L (z) T z z = O R ( z) The 2- and 3-point correlators are fixed by conformal Ward identities. O R ( z) O R (0) = c R 2 z 4 O L (z) O L (0) = c L 2z 4 O L (z) O R (0) = 0 c R O R ( z) O R ( z ) O R (0) = z 2 z 2 ( z z ) 2 O L (z) O L (z ) O L c L (0) = z 2 z 2 (z z ) 2 O L (z) O R ( z ) O R (0) = 0 O L (z) O L (z ) O R (0) = 0 Central charges c L/R determine key properties of CFT. D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 17/29

55 Correlators in logarithmic CFTs Any CFT has a conserved traceless energy momentum tensor. T z z = 0 T zz = O L (z) T z z = O R ( z) The 2- and 3-point correlators are fixed by conformal Ward identities. Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε ˆB O M (z, z) O M (0, 0) = z4+2ε z 2ε which degenerates with O L in limit c L ε 0 D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 17/29

56 Correlators in logarithmic CFTs Any CFT has a conserved traceless energy momentum tensor. T z z = 0 T zz = O L (z) T z z = O R ( z) The 2- and 3-point correlators are fixed by conformal Ward identities. Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε ˆB O M (z, z) O M (0, 0) = z4+2ε z 2ε which degenerates with O L in limit c L ε 0 Then energy momentum tensor acquires logarithmic partner O log where O log = b L O L c L + b L 2 OM b L := lim c L 0 c L ε 0 D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 17/29

57 Correlators in logarithmic CFTs Any CFT has a conserved traceless energy momentum tensor. T z z = 0 T zz = O L (z) T z z = O R ( z) The 2- and 3-point correlators are fixed by conformal Ward identities. Central charges c L/R determine key properties of CFT. Suppose there is an additional operator O M with conformal weights h = 2 + ε, h = ε which degenerates with O L in limit c L ε 0 Then energy momentum tensor acquires logarithmic partner O log Some 2-point correlators: O L (z)o L (0, 0) = 0 O L (z)o log (0, 0) = b L 2z 4 O log (z, z)o log (0, 0) = b L ln (m 2 L z 2 ) z 4 New anomaly b L determines key properties of logarithmic CFT. D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 17/29

58 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

59 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

60 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

61 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

62 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

63 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT Except for value of new anomaly b L no freedom in this procedure. Either it works or it does not work. D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

64 Check of logarithmic CFT conjecture for 2- and 3-point correlators If LCFT conjecture is correct then following procedure must work: Calculate non-normalizable modes for left, right and logarithmic branches by solving linearized EOM on gravity side According to AdS 3 /LCFT 2 dictionary these non-normalizable modes are sources for corresponding operators in the dual CFT Calculate 2- and 3-point correlators on the gravity side, e.g. by plugging non-normalizable modes into second and third variation of the on-shell action These correlators must coinicde with the ones of a logarithmic CFT Except for value of new anomaly b L no freedom in this procedure. Either it works or it does not work. Works at level of 2-point correlators (Skenderis, Taylor & van Rees 09, Grumiller & Sachs 09) Works at level of 3-point correlators (Grumiller & Sachs 09) Value of new anomaly: b L = c R = 3l/G D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 18/29

65 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

66 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. Consider small but non-vanising central charge c L D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

67 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. Consider small but non-vanising central charge c L Then weights h = 2 + ε and h = ε of massive modes differ infinitesimally from weights 2 and 0 of left mode D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

68 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. Consider small but non-vanising central charge c L Then weights h = 2 + ε and h = ε of massive modes differ infinitesimally from weights 2 and 0 of left mode The new anomaly is given by the ratio of these two small quantities b L = lim ε 0 c L ε D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

69 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. Consider small but non-vanising central charge c L Then weights h = 2 + ε and h = ε of massive modes differ infinitesimally from weights 2 and 0 of left mode The new anomaly is given by the ratio of these two small quantities b L = lim ε 0 c L ε Result obtained in this way must coincide with result for b L from the 2- and 3-point correlators D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

70 Alternative calculation of new anomaly b L As another consistency check perform the following short-cut. Consider small but non-vanising central charge c L Then weights h = 2 + ε and h = ε of massive modes differ infinitesimally from weights 2 and 0 of left mode The new anomaly is given by the ratio of these two small quantities b L = lim ε 0 c L ε Result obtained in this way must coincide with result for b L from the 2- and 3-point correlators Recover the result (Grumiller & Hohm 09, Grumiller, Johansson & Zojer, 10) b L = 3l G D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 19/29

71 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

72 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

73 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral Calculate minimal part of partition function (Virasoro descendants of vacuum, descendants of log operator) on CFT side D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

74 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral Calculate minimal part of partition function (Virasoro descendants of vacuum, descendants of log operator) on CFT side Calculate the difference between these partition functions (corresponds to multiple log excitations) D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

75 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral Calculate minimal part of partition function (Virasoro descendants of vacuum, descendants of log operator) on CFT side Calculate the difference between these partition functions (corresponds to multiple log excitations) Check that all multi-log coefficients in this difference are non-negative D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

76 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral Calculate minimal part of partition function (Virasoro descendants of vacuum, descendants of log operator) on CFT side Calculate the difference between these partition functions (corresponds to multiple log excitations) Check that all multi-log coefficients in this difference are non-negative D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

77 1-loop partition function...yet another non-trivial check (Gaberdiel, Grumiller & Vassilevich 10) If LCFT conjecture is true, then the following procedure must work Calculate 1-loop partition function on gravity side Check that it is not chiral Calculate minimal part of partition function (Virasoro descendants of vacuum, descendants of log operator) on CFT side Calculate the difference between these partition functions (corresponds to multiple log excitations) Check that all multi-log coefficients in this difference are non-negative Conclusion: all consistency tests show validity of LCFT conjecture! D. Grumiller Massive gravity in three dimensions Logarithmic CFT conjecture 20/29

78 Outline Introduction to 3D gravity Topologically massive gravity Logarithmic CFT conjecture Consequences, Generalizations & Applications D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 21/29

79 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

80 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

81 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

82 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

83 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

84 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

85 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT Central charges: c L = 0, c R = 3l/G, new anomaly: b L = 3l/G D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

86 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT Central charges: c L = 0, c R = 3l/G, new anomaly: b L = 3l/G LCFTs non-unitary bulk gravitons negative energy D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

87 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT Central charges: c L = 0, c R = 3l/G, new anomaly: b L = 3l/G LCFTs non-unitary bulk gravitons negative energy LCFTs cannot be chiral Brown York stress tensor not chiral D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

88 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT Central charges: c L = 0, c R = 3l/G, new anomaly: b L = 3l/G LCFTs non-unitary bulk gravitons negative energy LCFTs cannot be chiral Brown York stress tensor not chiral Partition functions on gravity and LCFT sides appear to match D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

89 Summary and comments TMG at the chiral/logarithmic point µl = 1: 3D gravity theory with black holes and massive graviton excitations Conjectured to be dual to logarithmic CFT Conjecture passed several independent consistency tests Non-trivial Jordan cell structure on gravity side, like in LCFT Operator degenerates with energy-momentum tensor at the point where central charge vanishes good indication for a LCFT Correlators on gravity side match precisely those of LCFT Central charges: c L = 0, c R = 3l/G, new anomaly: b L = 3l/G LCFTs non-unitary bulk gravitons negative energy LCFTs cannot be chiral Brown York stress tensor not chiral Partition functions on gravity and LCFT sides appear to match If conjecture true: first example of AdS 3 /LCFT 2 correspondence! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 22/29

90 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

91 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

92 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G Partition function trivially factorizes holomorphically Z = Z L Z R = Z R Thus avoids problems with original approach by Witten 07 D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

93 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G Partition function trivially factorizes holomorphically Z = Z L Z R = Z R Thus avoids problems with original approach by Witten 07 Chiral gravity defined by truncation of the dual LCFT D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

94 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G Partition function trivially factorizes holomorphically Z = Z L Z R = Z R Thus avoids problems with original approach by Witten 07 Chiral gravity defined by truncation of the dual LCFT Truncation either by requiring periodicity in time or by imposing stricter fall-off conditions than ansymptotic AdS (Brown Henneaux) D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

95 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G Partition function trivially factorizes holomorphically Z = Z L Z R = Z R Thus avoids problems with original approach by Witten 07 Chiral gravity defined by truncation of the dual LCFT Truncation either by requiring periodicity in time or by imposing stricter fall-off conditions than ansymptotic AdS (Brown Henneaux) Not clear whether truncation consistent in full quantum theory D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

96 Consequences for chiral gravity Chiral gravity conjectured to exist as consistent quantum theory of gravity by Li, Song & Strominger 08 Dual CFT would be a chiral CFT with c L = 0 and c R = 3l/G Partition function trivially factorizes holomorphically Z = Z L Z R = Z R Thus avoids problems with original approach by Witten 07 Chiral gravity defined by truncation of the dual LCFT Truncation either by requiring periodicity in time or by imposing stricter fall-off conditions than ansymptotic AdS (Brown Henneaux) Not clear whether truncation consistent in full quantum theory Not clear yet if chiral gravity exists! If it exists: excellent toy model for quantum gravity! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 23/29

97 Generalizations to new massive gravity and generalized massive gravity Q: Is TMG the only gravity theory dual to a LCFT? D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 24/29

98 Generalizations to new massive gravity and generalized massive gravity Q: Is TMG the only gravity theory dual to a LCFT? A: No! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 24/29

99 Generalizations to new massive gravity and generalized massive gravity Q: Is TMG the only gravity theory dual to a LCFT? A: No! New massive gravity (Bergshoeff, Hohm & Townsend 09): I NMG = 1 d 3 x g [σr + 1 (R µν 16πG m 2 R µν 3 ) 8 R2 2λm 2] Similar story (Grumiller & Hohm 09, Alishahiha & Naseh 10): Linearized EOM around AdS 3 (g = ḡ + h) ( D R D L D M D Mh ) µν = 0 Logarithmic point for λ = 3: c L = c R = 0 Massive modes degenerate with left and right boundary gravitons 2-point correlators on gravity side match precisely those of a LCFT New anomalies: b L = b R = σ12l/g D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 24/29

100 Extended generalized massive gravity (Paulos 10) Reconsider higher curvature theories introduced in the beginning All actions of type L = L MG (R µν ) + L CS with gravitational Chern Simons term L CS = 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] and the specific higher derivative Lagrange density L MG (R µν ) = σr 2Λ + 1 m 2 ( Rµν R µν 3 8 R2) + O(R 3 µν) have an AdS solution (if Λ eff < 0) and linearized equations of motion ( D R D L D M D Mh ) µν = 0 Various degenerations of modes possible log excitations D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 25/29

101 Extended generalized massive gravity (Paulos 10) Reconsider higher curvature theories introduced in the beginning All actions of type L = L MG (R µν ) + L CS with gravitational Chern Simons term L CS = 1 2µ ελµν Γ ρ ( λσ µ Γ σ νρ Γσ µτ Γ τ νρ) ] and the specific higher derivative Lagrange density L MG (R µν ) = σr 2Λ + 1 m 2 ( Rµν R µν 3 8 R2) + O(R 3 µν) have an AdS solution (if Λ eff < 0) and linearized equations of motion ( D R D L D M D Mh ) µν = 0 Various degenerations of modes possible log excitations Thus, we have infinitely many gravity duals for LCFTs! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 25/29

102 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

103 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

104 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

105 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

106 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

107 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

108 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

109 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) Exploit LCFTs to compute correlators of quenched random systems D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

110 Potential applications in condensed matter physics LCFTs arise in systems with quenched disorder. Quenched disorder: systems with random variable that does not evolve in time Examples: spin glasses, quenched random magnets, dilute self-avoiding polymers, percolation For sufficient amount of disorder perturbation theory breaks down random critical point Infamous denominator in correlators: ( Dφ exp I[φ] d 2 z V (z )O(z ) ) O(z) O(0) O(z) O(0) = DV P [V ] ( Dφ exp I[φ] d 2 z V (z )O(z ) ) Different ways to deal with denominator (replica trick, SUSY) Result: operators degenerate and correlators acquire logarithmic behavior, exactly as in LCFT (Cardy 99) Exploit LCFTs to compute correlators of quenched random systems Apply AdS 3 /LCFT 2 to describe strongly coupled LCFTs! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 26/29

111 Next steps Quantum gravity Consistency of truncation to chiral gravity? Existence of (log) extremal CFTs for arbitrary level k? Unitary completion of dual logarithmic CFT? D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 27/29

112 Next steps Quantum gravity Consistency of truncation to chiral gravity? Existence of (log) extremal CFTs for arbitrary level k? Unitary completion of dual logarithmic CFT? Gauge/gravity duality Matching of 1-loop partition function in generalized massive gravity? (Bertin, Grumiller & Vassilevich, Zojer, in preparation) Higher dimensional generalization? (Lü & Pope, 11, Alishahiha & Fareghbal, 11, Bergshoeff, Hohm, Rosseel & Townsend, 11) Interesting fixed points in theory space? D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 27/29

113 Next steps Quantum gravity Consistency of truncation to chiral gravity? Existence of (log) extremal CFTs for arbitrary level k? Unitary completion of dual logarithmic CFT? Gauge/gravity duality Matching of 1-loop partition function in generalized massive gravity? (Bertin, Grumiller & Vassilevich, Zojer, in preparation) Higher dimensional generalization? (Lü & Pope, 11, Alishahiha & Fareghbal, 11, Bergshoeff, Hohm, Rosseel & Townsend, 11) Interesting fixed points in theory space? Physics Condensed matter physics applications? Identify relevant observables in strong coupling limit, like η/s in strongly coupled N = 4 SYM plasma! Compute relevant dual processes on gravity side and make predictions! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 27/29

114 Next steps Quantum gravity Consistency of truncation to chiral gravity? Existence of (log) extremal CFTs for arbitrary level k? Unitary completion of dual logarithmic CFT? Gauge/gravity duality Matching of 1-loop partition function in generalized massive gravity? (Bertin, Grumiller & Vassilevich, Zojer, in preparation) Higher dimensional generalization? (Lü & Pope, 11, Alishahiha & Fareghbal, 11, Bergshoeff, Hohm, Rosseel & Townsend, 11) Interesting fixed points in theory space? Physics Condensed matter physics applications? Identify relevant observables in strong coupling limit, like η/s in strongly coupled N = 4 SYM plasma! Compute relevant dual processes on gravity side and make predictions! Thanks for your attention! D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 27/29

115 D. Grumiller Massive gravity in three dimensions Consequences, Generalizations & Applications 28/29