Warped Black Holes in Lower-Spin Gravity

Transcription

1 Warped Black Holes in Lower-Spin Gravity Max Riegler Physique Theorique et Mathématique Université libre de Bruxelles Holography, Quantum Entanglement and Higher Spin Gravity Yukawa nstitute for Theoretical Physics, Kyoto University March 16 th, 2018 [T. Azeyanagi, S. Detournay, M.R; ]

2 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? 1

3 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? AdS3 deformation: 2 2 dswads = dsads 2Hξ ξ 3 3 1

4 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? Why? 1

5 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? Why? NHEK global sl(2, R) u(1) isometry. 1

6 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? Why? NHEK global sl(2, R) u(1) isometry. At fixed polar angle (selfdual spacelike) WAdS3. 1

7 ntroduction WAdS3, WCFT and Lower-Spin Gravity WAdS3 What? Why? NHEK global sl(2, R) u(1) isometry. At fixed polar angle (selfdual spacelike) WAdS3. study NHEK. 1

8 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What?

9 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What? QFT with global sl(2, R) u(1) symmetries.

10 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What? QFT with global sl(2, R) u(1) symmetries. Has conserved charges but not Lorentz invariant.

11 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What? Why?

12 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What? Why? Possible WAdS3 dual.

13 ntroduction WAdS3, WCFT and Lower-Spin Gravity 1 WCFT What? Why? Possible WAdS3 dual. better understanding of holography in a realistic setup!

14 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? 1

15 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? sl(2, R) u(1) Chern-Simons theory. 1

16 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? Why? 1

17 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? Why? Very simple model of WAdS3 without local DOF. 1

18 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? Why? Black Holes? 1

19 ntroduction WAdS3, WCFT and Lower-Spin Gravity Lower-Spin Gravity What? Why? Black Holes? Higher-Spin WAdS3 black holes? 1

20 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries.

21 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries. Determine thermal entropy.

22 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries. Determine thermal entropy. Consistency checks:

23 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries. Determine thermal entropy. Consistency checks: Compare with WCFT results.

24 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries. Determine thermal entropy. Consistency checks: Compare with WCFT results. Holographic entanglement entropy.

25 Outline Describing WAdS 3 BHs Using Lower-Spin Gravity 2 Boundary conditions asymptotic symmetries. Determine thermal entropy. Consistency checks: Compare with WCFT results. Holographic entanglement entropy. Metric interpretation.

26 WAdS 3 BHs in Lower-Spin Gravity Boundary Conditions and Asymptotic Symmetries 3 CS = k A da + 2 4π M 3 A A A + κ C dc 8π M

27 WAdS3 BHs in Lower-Spin Gravity Boundary Conditions and Asymptotic Symmetries CS k = 4π 3 Z 2 κ ha da + A A Ai + 3 8π M Z M hc dci Boundary Conditions A(ρ, t, ϕ) = b 1 (ρ) [a(t, ϕ) + d] b(ρ) C(ρ, t, ϕ) = c(t, ϕ) aϕ = L1 cϕ = 2π k 4π KS κ 2π 2 L K κ L 1 at = 0 ct = S

28 WAdS3 BHs in Lower-Spin Gravity Boundary Conditions and Asymptotic Symmetries CS = k 4π Z M ha da + 2 κ A A Ai + 3 8π 3 Z M hc dci Asymptotic Symmetries [Ln, Lm ] = (n m) Ln+m + [Ln, Km ] = m Kn+m κ [Kn, Km ] = n δn+m,0 2 [Hofman, Rollier 14] c n(n2 1)δn+m,0 12

29 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J.

30 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J.

31 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δs Th = β (δm ΩδJ) S Th.

32 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J δs Th = β (δm ΩδJ) S Th.

33 WAdS3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J δsth = β (δm ΩδJ) STh. Assumption ε = ξ µ Aµ, ε = ξ µ Cµ,

34 WAdS3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J δsth = β (δm ΩδJ) STh. Assumption ε = ξ µ Aµ, δm := δq[ε δj := δq[ε t ] + δq[ε ϕ ε = ξ µ Cµ, t ] + δq[ε ] = 2πδK ϕ ] = 2πδL

35 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δs Th = β (δm ΩδJ) S Th.

36 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δs Th = β (δm ΩδJ) S Th. δs Th = β (δm ΩδJ) = k h δa ϕ + κ 2 h δc ϕ where h = β 2π h = β 2π dϕ (a t + Ωa ϕ ) dϕ (c t + Ωc ϕ )

37 WAdS3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δsth = β (δm ΩδJ) STh. Assumption Eigen[h] = Eigen[2πL0 ] Eigen[h ] = Eigen[2πγS] δsth = β (δm ΩδJ) = k hh δaϕ i + where β h= dϕ (at + Ωaϕ ) 2π β h = dϕ (ct + Ωcϕ ) 2π κ hh δcϕ i 2

38 WAdS3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δsth = β (δm ΩδJ) STh. Assumption Eigen[h] = Eigen[2πL0 ] M β = 2π γ r κ k1 J Eigen[h ] = Eigen[2πγS] M2 κ 1 Ω= 2γ r 1 k J M2 κ 2M κ

39 WAdS 3 BHs in Lower-Spin Gravity First Law and Thermal Entropy 4 Determine mass M and angular momentum J. Relate β, Ω M, J. δs Th = β (δm ΩδJ) S Th.

40 WAdS3 BHs in Lower-Spin Gravity First Law and Thermal Entropy Determine mass M and angular momentum J. Relate β, Ω M, J. δsth = β (δm ΩδJ) STh. STh Thermal Entropy s! M2 c = 2π Mγ + J 6 κ 4

41 WAdS 3 BHs in Lower-Spin Gravity WCFT and the Vacuum 5 S Th = 4πiMMv κ [Detournay, Hartman, Hofman 12] ( + 4π J v (Mv ) 2 κ ) ( ) J M2 κ

42 WAdS3 BHs in Lower-Spin Gravity WCFT and the Vacuum 5 v! u v )2 u M2 4πiMM (M t v + 4π J J = κ κ κ v STh [Detournay, Hartman, Hofman 12] Assumption (Warped) vacuum: e H dϕ aϕ = 1l e H dϕ cϕ = e2πiγ

43 WAdS3 BHs in Lower-Spin Gravity WCFT and the Vacuum STh 5 v! u 2 u 4πiMM v M2 (M v ) t v = + 4π J J κ κ κ [Detournay, Hartman, Hofman 12] Assumption (Warped) vacuum: e H dϕ aϕ Jv = = 1l c γ2κ e H dϕ cϕ = e2πiγ Mv = iκγ 2

44 Consistency Checks HEE and Wilson Lines 6 B t f ϕ t A A C B t i B t B ϕ ρ

45 Consistency Checks HEE and Wilson Lines 6 [ ] [ ] S EE = log W sl(2,r) R (C; A) log W u(1) R (C; C) [Castro, Hofman, qbal 15]

46 Consistency Checks HEE and Wilson Lines 6 h i h i sl(2,r) u(1) SEE = log WR (C; A) log WR (C; C) [Castro, Hofman, qbal 15] SEE Holographic Entanglement Entropy κ c βϕ π ϕ =γ t + M ϕ + log sinh 2 6 π βϕ where βϕ = r 6 c π J M2 κ

47 Consistency Checks HEE and Wilson Lines SEE SEE Holographic Entanglement Entropy c κ βϕ π ϕ t + M ϕ + log sinh =γ 2 6 π βϕ βϕ π ϕ β δ δ v v = im t + ϕ + i M 4J log sinh βϕ π π βϕ v [Song, Wen, Xu 16] 6

48 Consistency Checks HEE and Wilson Lines SEE 6 Holographic Entanglement Entropy κ c βϕ π ϕ =γ t + M ϕ + log sinh 2 6 π βϕ β δ βϕ π ϕ δ SEE = im v t + ϕ + i M v 4J v log sinh βϕ π π βϕ [Song, Wen, Xu 16] δ = 2πγ

49 Consistency Checks HEE and Wilson Lines SEE 6 Holographic Entanglement Entropy κ c βϕ π ϕ =γ t + M ϕ + log sinh 2 6 π βϕ SEE ϕ 1 βϕ STh ϕ 2π

50 Consistency Checks Metric Formulation 7 Warped geometry lower-spin gravity.

51 Consistency Checks Metric Formulation 7 Warped geometry lower-spin gravity. Determine geometric variables: B = B (A, C; α, b, c)

52 Consistency Checks Metric Formulation 7 Warped geometry lower-spin gravity. Determine geometric variables: B = B (A, C; α, b, c) Metric: ds 2 = g µν dx µ dx ν = B, B

53 Consistency Checks Metric Formulation 7 Warped geometry lower-spin gravity. Determine geometric variables: B = B (A, C; α, b, c) Metric: ds2 = gµν dx µ dx ν = hb, Bi WAdS3 BH Parameters b2 = ν2 2`2 c= ν `2 α= 16 c2 k `2

54 Consistency Checks Metric Formulation 7 Warped geometry lower-spin gravity. Determine geometric variables: B = B (A, C; α, b, c) Metric: ds2 = gµν dx µ dx ν = hb, Bi WAdS3 BH Parameters b2 = ν2 2`2 c= ν `2 α= 16 c2 k `2 spacelike (stretched) warped AdS3 black hole!

55 Consistency Checks Metric Formulation 7 Killing vectors:

56 Consistency Checks Metric Formulation 7 nitial Assumption ε = ξ µ Aµ, ε = ξ µ Cµ, Killing vectors: Globally well defined Killing vectors X

57 Consistency Checks Metric Formulation 7 nitial Assumption (Warped) vacuum: e H dϕ aϕ = 1l e H dϕ cϕ Killing vectors: Globally well defined Killing vectors X 2 c J v = 24 + γ 4κ M v = iκγ X 2 = e2πiγ

58 Consistency Checks Metric Formulation 7 nitial Assumption Eigen[h] = Eigen[2πL0 ] Killing vectors: Eigen[h ] = Eigen[2πγS] Globally well defined Killing vectors X 2 c J v = 24 + γ 4κ M v = iκγ X 2 Thermodynamics: β(m, J), Ω(M, J) X

59 Consistency Checks Metric Formulation 7 Killing vectors: Globally well defined Killing vectors J v = c + γ2 κ M v = iκγ Thermodynamics: β(m, J), Ω(M, J) γ = b kα 2 2 = 2ν ν 2 +3

60 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X

61 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X Easiest toy model for WCFTS with T > 0.

62 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X Easiest toy model for WCFTS with T > 0. Straightforward extension to higher-spins.

63 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X Easiest toy model for WCFTS with T > 0. Straightforward extension to higher-spins. Outlook Higher-spin warped BHs.

64 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X Easiest toy model for WCFTS with T > 0. Straightforward extension to higher-spins. Outlook Higher-spin warped BHs. Full partition function as in [izuka, Tanaka, Terashima 15].

65 Conclusion and Outlook 8 Conclusion Warped BHs and lower-spin gravity. X Easiest toy model for WCFTS with T > 0. Straightforward extension to higher-spins. Outlook Higher-spin warped BHs. Full partition function as in [izuka, Tanaka, Terashima 15]. Near horizon soft hair?

66 Thank you for your attention!