Near Horizon Soft Hairs as Black Hole Microstates

Transcription

1 Near Horizon Soft Hairs as Black Hole Microstates Daniel Grumiller Institute for Theoretical Physics TU Wien Seminario-almuerzo Valdivia, November , ,

2 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of (higher spin) gravity theories Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates 2/27

3 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of (higher spin) gravity theories 2. Black hole microstates identified as specific soft hair descendants Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates 2/27

4 Two simple punchlines 1. Heisenberg algebra [X n, P m ] = i δ n, m fundamental not only in quantum mechanics but also in near horizon physics of (higher spin) gravity theories 2. Black hole microstates identified as specific soft hair descendants based on work with Hamid Afshar [IPM Teheran] Stephane Detournay [ULB] Wout Merbis [TU Wien] Blagoje Oblak [ULB] Alfredo Perez [CECS Valdivia] Stefan Prohazka [TU Wien] Shahin Sheikh-Jabbari [IPM Teheran] David Tempo [CECS Valdivia] Ricardo Troncoso [CECS Valdivia] Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates 2/27

5 Outline Motivation Near horizon boundary conditions Explicit construction of BTZ microstates Discussion Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates 3/27

6 Outline Motivation Near horizon boundary conditions Explicit construction of BTZ microstates Discussion Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 4/27

7 Black hole microstates Bekenstein Hawking S BH = A 4G N Motivation: microscopic understanding of generic black hole entropy Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 5/27

8 Black hole microstates Bekenstein Hawking S BH = A 4G N Motivation: microscopic understanding of generic black hole entropy Microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 5/27

9 Black hole microstates Bekenstein Hawking S BH = A 4G N Motivation: microscopic understanding of generic black hole entropy Microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) warped CFT: Detournay, Hartman, Hofman 12 Galilean CFT: Bagchi, Detournay, Fareghbal, Simon 13; Barnich 13 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 5/27

10 Black hole microstates Bekenstein Hawking S BH = A 4G N Motivation: microscopic understanding of generic black hole entropy Microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) Main idea: consider near horizon symmetries for non-extremal horizons Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 5/27

11 Black hole microstates Bekenstein Hawking S BH = A 4G N Motivation: microscopic understanding of generic black hole entropy Microstate counting from CFT 2 symmetries (Strominger, Carlip,...) using Cardy formula Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT, warped CFT,...) Main idea: consider near horizon symmetries for non-extremal horizons Near horizon line-element with Rindler acceleration a: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Meaning of coordinates: ρ: radial direction (ρ = 0 is horizon) ϕ ϕ + 2π: angular direction v: (advanced) time Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 5/27

12 Choices Rindler acceleration: state-dependent or chemical potential? Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

13 Choices Rindler acceleration: state-dependent or chemical potential? If state-dependent: need mechanism to fix scale Recall scale invariance a λa ρ λρ v v/λ of Rindler metric ds 2 = 2aρ dv dv dρ + γ 2 dϕ 2 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

14 Choices Rindler acceleration: state-dependent or chemical potential? If state-dependent: need mechanism to fix scale suggestion in : v v + 2πL Works technically but physical interpretation difficult Recall scale invariance of Rindler metric a λa ρ λρ v v/λ ds 2 = 2aρ dv dv dρ + γ 2 dϕ 2 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

15 Choices Rindler acceleration: state-dependent or chemical potential? If state-dependent: need mechanism to fix scale suggestion in : v v + 2πL Works technically but physical interpretation difficult If chemical potential: all states in theory have same (Unruh-)temperature T U = a 2π Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

16 Choices Rindler acceleration: state-dependent or chemical potential? If state-dependent: need mechanism to fix scale suggestion in : v v + 2πL Works technically but physical interpretation difficult If chemical potential: all states in theory have same (Unruh-)temperature T U = a 2π suggestion in We make this choice in this talk! Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

17 Choices Rindler acceleration: state-dependent or chemical potential? If state-dependent: need mechanism to fix scale suggestion in : v v + 2πL Works technically but physical interpretation difficult If chemical potential: all states in theory have same (Unruh-)temperature T U = a 2π Work in 3d Einstein gravity in Chern Simons formulation I CS = ± k A ± da ± + 2 4π 3 A± A ± A ± ± with sl(2) connections A ± and k = l/(4g N ) with AdS radius l = 1 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Motivation 6/27

18 Outline Motivation Near horizon boundary conditions Explicit construction of BTZ microstates Discussion Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 7/27

19 Diagonal gauge Standard trick: partially fix gauge A ± = b 1 ± (ρ) ( d+a ± (x 0, x 1 ) ) b ± (ρ) with some group element b SL(2) depending on radius ρ with δb = 0 Drop ± decorations in most of talk Manifold topologically a cylinder or torus, with radial coordinate ρ and boundary coordinates (x 0, x 1 ) (v, ϕ) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 8/27

20 Diagonal gauge Standard trick: partially fix gauge A = b 1 (ρ) ( d+a(x 0, x 1 ) ) b(ρ) with some group element b SL(2) depending on radius ρ with δb = 0 Standard AdS 3 approach: highest weight gauge a L + + L(x 0, x 1 )L b(ρ) = exp (ρl 0 ) sl(2): [L n, L m ] = (n m)l n+m, n, m = 1, 0, 1 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 8/27

21 Diagonal gauge Standard trick: partially fix gauge A = b 1 (ρ) ( d+a(x 0, x 1 ) ) b(ρ) with some group element b SL(2) depending on radius ρ with δb = 0 Standard AdS 3 approach: highest weight gauge a L + + L(x 0, x 1 )L b(ρ) = exp (ρl 0 ) sl(2): [L n, L m ] = (n m)l n+m, n, m = 1, 0, 1 For near horizon purposes diagonal gauge useful: a J (x 0, x 1 ) L 0 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 8/27

22 Diagonal gauge Standard trick: partially fix gauge A = b 1 (ρ) ( d+a(x 0, x 1 ) ) b(ρ) with some group element b SL(2) depending on radius ρ with δb = 0 Standard AdS 3 approach: highest weight gauge a L + + L(x 0, x 1 )L b(ρ) = exp (ρl 0 ) sl(2): [L n, L m ] = (n m)l n+m, n, m = 1, 0, 1 For near horizon purposes diagonal gauge useful: a J (x 0, x 1 ) L 0 Precise boundary conditions (ζ: chemical potential): a = (J dϕ + ζ dv) L 0 δa = δj dϕ L 0 and b = exp ( 1 ζ L +) exp ( ρ 2 L ). (assume constant ζ for simplicity) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 8/27

23 Near horizon metric Using g µν = 1 2 ( A + µ A ) ( µ A + ν A ) ν Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

24 Near horizon metric Using g µν = 1 2 yields (f := 1 + ρ/(2a)) ( A + µ A ) ( µ A + ν A ) ν ds 2 = 2aρf dv dv dρ 2ωa 1 dϕ dρ + 4ωρf dv dϕ + [ γ 2 + 2ρ a f(γ2 ω 2 ) ] dϕ 2 state-dependent functions J ± = γ ± ω, chemical potentials ζ ± = a ± Ω For simplicity set Ω = 0 and a = const. in metric above EOM imply v J ± = ± ϕ ζ ± ; in this case v J ± = 0 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

25 Near horizon metric Using g µν = 1 2 yields (f := 1 + ρ/(2a)) ( A + µ A ) ( µ A + ν A ) ν ds 2 = 2aρf dv dv dρ 2ωa 1 dϕ dρ + 4ωρf dv dϕ + [ γ 2 + 2ρ a f(γ2 ω 2 ) ] dϕ 2 state-dependent functions J ± = γ ± ω, chemical potentials ζ ± = a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: Comments: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Recover desired near horizon metric Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

26 Near horizon metric Using g µν = 1 2 yields (f := 1 + ρ/(2a)) ( A + µ A ) ( µ A + ν A ) ν ds 2 = 2aρf dv dv dρ 2ωa 1 dϕ dρ + 4ωρf dv dϕ + [ γ 2 + 2ρ a f(γ2 ω 2 ) ] dϕ 2 state-dependent functions J ± = γ ± ω, chemical potentials ζ ± = a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: Comments: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Recover desired near horizon metric Rindler acceleration a indeed state-independent Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

27 Near horizon metric Using g µν = 1 2 yields (f := 1 + ρ/(2a)) ( A + µ A ) ( µ A + ν A ) ν ds 2 = 2aρf dv dv dρ 2ωa 1 dϕ dρ + 4ωρf dv dϕ + [ γ 2 + 2ρ a f(γ2 ω 2 ) ] dϕ 2 state-dependent functions J ± = γ ± ω, chemical potentials ζ ± = a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: Comments: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Recover desired near horizon metric Rindler acceleration a indeed state-independent Two state-dependent functions (γ, ω) as usual in 3d gravity Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

28 Near horizon metric Using g µν = 1 2 yields (f := 1 + ρ/(2a)) ( A + µ A ) ( µ A + ν A ) ν ds 2 = 2aρf dv dv dρ 2ωa 1 dϕ dρ + 4ωρf dv dϕ + [ γ 2 + 2ρ a f(γ2 ω 2 ) ] dϕ 2 state-dependent functions J ± = γ ± ω, chemical potentials ζ ± = a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: Comments: ds 2 = 2aρ dv dv dρ + γ 2 dϕ Recover desired near horizon metric Rindler acceleration a indeed state-independent Two state-dependent functions (γ, ω) as usual in 3d gravity γ = γ(ϕ): black flower Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 9/27

29 Canonical boundary charges Canonical boundary charges non-zero for large trafos that preserve boundary conditions Zero mode charges: mass and angular momentum Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 10/27

30 Canonical boundary charges Canonical boundary charges non-zero for large trafos that preserve boundary conditions Zero mode charges: mass and angular momentum Background independent result for Chern Simons yields Q[η] = k dϕ η(ϕ) J (ϕ) 4π Finite Integrable Conserved Non-trivial Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 10/27

31 Canonical boundary charges Canonical boundary charges non-zero for large trafos that preserve boundary conditions Zero mode charges: mass and angular momentum Background independent result for Chern Simons yields Q[η] = k dϕ η(ϕ) J (ϕ) 4π Finite Integrable Conserved Non-trivial Meaningful near horizon boundary conditions and non-trivial theory! Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 10/27

32 Near horizon symmetry algebra Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Most general trafo δ ɛ a = dɛ + [a, ɛ] = O(δa) that preserves our boundary conditions for constant ζ given by with implying ɛ = ɛ + L + + ηl 0 + ɛ L v η = 0 δ ɛ J = ϕ η Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 11/27

33 Near horizon symmetry algebra Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Expand charges in Fourier modes J ± n What should we expect? = k 4π dϕ e inϕ J ± (ϕ) Virasoro? (spacetime is locally AdS3 ) BMS3? (Rindler boundary similar to scri) warped conformal algebra? (this is what we found for Rindleresque holography and what Donnay, Giribet, Gonzalez, Pino found in their near horizon analysis) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 11/27

34 Near horizon symmetry algebra Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Expand charges in Fourier modes J n ± = k dϕ e inϕ J ± (ϕ) 4π Near horizon symmetry algebra [ J ± n, J m ± ] = ± 1 2 knδ [ n+m, 0 J + n, Jm ] = 0 Two û(1) current algebras with non-zero levels Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 11/27

35 Near horizon symmetry algebra Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Expand charges in Fourier modes J n ± = k dϕ e inϕ J ± (ϕ) 4π Near horizon symmetry algebra [ J ± n, J m ± ] = ± 1 2 knδ n+m, 0 [ J + n, Jm ] = 0 Two û(1) current algebras with non-zero levels Much simpler than CFT 2, warped CFT 2, Galilean CFT 2, etc. Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 11/27

36 Near horizon symmetry algebra Near horizon symmetry algebra = all near horizon boundary conditions preserving trafos, modulo trivial gauge trafos Expand charges in Fourier modes J n ± = k dϕ e inϕ J ± (ϕ) 4π Near horizon symmetry algebra [ J ± n, J m ± ] = ± 1 2 knδ n+m, 0 [ J + n, Jm ] = 0 Two û(1) current algebras with non-zero levels Much simpler than CFT 2, warped CFT 2, Galilean CFT 2, etc. Map P 0 = J J 0 P n = i kn (J n + + J n ) if n 0 yields Heisenberg algebra (with Casimirs X 0, P 0 ) [X n, X m ] = [P n, P m ] = [X 0, P n ] = [P 0, X n ] = 0 [X n, P m ]= iδ n,m if n 0 X n = J + n J n Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 11/27

37 Map to asymptotic variables Usual asymptotic AdS 3 connection with chemical potential µ: Â = ˆb 1( d+â )ˆb âϕ = L L L ˆb = e ρl 0 â t = µl + µ L 0 + ( 1 2 µ 1 2 Lµ) L Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 12/27

38 Map to asymptotic variables Usual asymptotic AdS 3 connection with chemical potential µ: Â = ˆb 1( d+â )ˆb âϕ = L L L ˆb = e ρl 0 â t = µl + µ L 0 + ( 1 2 µ 1 2 Lµ) L Gauge trafo â = g 1 (d+a) g with g = exp (xl + ) exp ( 1 2 J L ) where v x ζx = µ and x J x = 1 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 12/27

39 Map to asymptotic variables Usual asymptotic AdS 3 connection with chemical potential µ: Â = ˆb 1( d+â )ˆb âϕ = L L L ˆb = e ρl 0 â t = µl + µ L 0 + ( 1 2 µ 1 2 Lµ) L Gauge trafo â = g 1 (d+a) g with g = exp (xl + ) exp ( 1 2 J L ) where v x ζx = µ and x J x = 1 Asymptotic chemical potential is combination of near horizon charge and chemical potential! µ J µ = ζ Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 12/27

40 Map to asymptotic variables Usual asymptotic AdS 3 connection with chemical potential µ: Â = ˆb 1( d+â )ˆb âϕ = L L L ˆb = e ρl 0 â t = µl + µ L 0 + ( 1 2 µ 1 2 Lµ) L Gauge trafo â = g 1 (d+a) g with g = exp (xl + ) exp ( 1 2 J L ) where v x ζx = µ and x J x = 1 Asymptotic chemical potential is combination of near horizon charge and chemical potential! µ J µ = ζ Asymptotic currents: twisted Sugawara construction with near horizon charges L = 1 2 J 2 + J Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 12/27

41 Map to asymptotic variables Usual asymptotic AdS 3 connection with chemical potential µ: Â = ˆb 1( d+â )ˆb âϕ = L L L ˆb = e ρl 0 â t = µl + µ L 0 + ( 1 2 µ 1 2 Lµ) L Gauge trafo â = g 1 (d+a) g with g = exp (xl + ) exp ( 1 2 J L ) where v x ζx = µ and x J x = 1 Asymptotic chemical potential is combination of near horizon charge and chemical potential! µ J µ = ζ Asymptotic currents: twisted Sugawara construction with near horizon charges L = 1 2 J 2 + J Get Virasoro with non-zero central charge δl = 2Lε + L ε ε Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 12/27

42 Remarks on asymptotic and near horizon variables Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey still Heisenberg algebra δq = k dϕ ε δl = k dϕ η δj 4π 4π Reason: asymptotic chemical potentials µ depend on near horizon charges J and chemical potentials ζ Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 13/27

43 Remarks on asymptotic and near horizon variables Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey still Heisenberg algebra δq = k dϕ ε δl = k dϕ η δj 4π 4π Reason: asymptotic chemical potentials µ depend on near horizon charges J and chemical potentials ζ Our boundary conditions singled out: (real) spectrum compatible with regularity [note: imaginary J 0 ± may lead to conical spaces] Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 13/27

44 Remarks on asymptotic and near horizon variables Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey still Heisenberg algebra δq = k dϕ ε δl = k dϕ η δj 4π 4π Reason: asymptotic chemical potentials µ depend on near horizon charges J and chemical potentials ζ Our boundary conditions singled out: (real) spectrum compatible with regularity [note: imaginary J 0 ± may lead to conical spaces] For constant chemical potential ζ: regularity = holonomy condition µµ 1 2 µ 2 µ 2 L = 2π 2 /β 2 Solved automatically from map to asymptotic observables; reminder: µ J µ = ζ L = 1 2 J 2 + J Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 13/27

45 Remarks on asymptotic and near horizon variables Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey still Heisenberg algebra δq = k dϕ ε δl = k dϕ η δj 4π 4π Reason: asymptotic chemical potentials µ depend on near horizon charges J and chemical potentials ζ Our boundary conditions singled out: (real) spectrum compatible with regularity [note: imaginary J 0 ± may lead to conical spaces] For constant chemical potential ζ: regularity = holonomy condition µµ 1 2 µ 2 µ 2 L = 2π 2 /β 2 Solved automatically from map to asymptotic observables; reminder: µ J µ = ζ L = 1 2 J 2 + J Near horizon boundary conditions natural for near horizon observer Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Near horizon boundary conditions 13/27

46 Outline Motivation Near horizon boundary conditions Explicit construction of BTZ microstates Discussion Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 14/27

47 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

48 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Near horizon algebra (conveniently rescaled) [J ± n, J ± m ] = 1 2 n δ n, m Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

49 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Near horizon algebra (conveniently rescaled) [J ± n, J ± m ] = 1 2 n δ n, m Near horizon Hilbert space: define vacuum by highest weight conditions J ± n 0 = 0 for all n 0. Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

50 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Near horizon algebra (conveniently rescaled) [J ± n, J ± m ] = 1 2 n δ n, m Near horizon Hilbert space: define vacuum by highest weight conditions J ± n 0 = 0 for all n 0. Construct near horizon Virasoro through standard Sugawara construction L ± n p Z : J ± n p J ± p : Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

51 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Near horizon algebra (conveniently rescaled) [J ± n, J ± m ] = 1 2 n δ n, m Near horizon Hilbert space: define vacuum by highest weight conditions J ± n 0 = 0 for all n 0. Construct near horizon Virasoro through standard Sugawara construction L ± n p Z : J ± n p J ± p : Get Virasoro algebra with central charge 1 [L ± n, L ± m] = (n m)l ± n+m (n3 n)δ n, m [L ± n, J ± m ] = mj ± n+m Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

52 Near horizon Hilbert space Denote near horizon generators with calligraphic letters Near horizon algebra (conveniently rescaled) [J ± n, J ± m ] = 1 2 n δ n, m Near horizon Hilbert space: define vacuum by highest weight conditions J ± n 0 = 0 for all n 0. Construct near horizon Virasoro through standard Sugawara construction L ± n p Z : J ± n p J ± p : Get Virasoro algebra with central charge 1 [L ± n, L ± m] = (n m)l ± n+m (n3 n)δ n, m [L ± n, J ± m ] = mj ± n+m Call this near horizon symmetry algebra (note: independent from l) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 15/27

53 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

54 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Near horizon Hamiltonian H J J 0 commutes with near horizon symmetry algebra Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

55 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Near horizon Hamiltonian H J J 0 commutes with near horizon symmetry algebra Descendants of vacuum have zero energy; dubbed soft hair (same true for descendants of black holes) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

56 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Near horizon Hamiltonian H J J 0 commutes with near horizon symmetry algebra Descendants of vacuum have zero energy; dubbed soft hair (same true for descendants of black holes) Immediate issue for entropy: infinite soft hair degeneracy! Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

57 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Near horizon Hamiltonian H J J 0 commutes with near horizon symmetry algebra Descendants of vacuum have zero energy; dubbed soft hair (same true for descendants of black holes) Immediate issue for entropy: infinite soft hair degeneracy! Note: descendants have positive eigenvalues of L ± 0 L ± 0 Ψ({n± i )} = i n ± i Ψ({n± i }) E± Ψ Ψ({n± i }) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

58 Soft hair Generic descendant of vacuum: Ψ({n ± i }) = {n ± i >0} ( J + J n + i n i ) 0 with set of positive integers {n ± i > 0} Near horizon Hamiltonian H J J 0 commutes with near horizon symmetry algebra Descendants of vacuum have zero energy; dubbed soft hair (same true for descendants of black holes) Immediate issue for entropy: infinite soft hair degeneracy! Note: descendants have positive eigenvalues of L ± 0 L ± 0 Ψ({n± i )} = i n ± i Ψ({n± i }) E± Ψ Ψ({n± i }) Will exploit this property to provide cut-off on soft hair spectrum! Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 16/27

59 Asymptotic Hilbert space Our algebra in original form [J ± n, J ± m] = k 2 n δ n, m Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 17/27

60 Asymptotic Hilbert space Our algebra in original form [J ± n, J ± m] = k 2 n δ n, m Map to asymptotic Virasoro algebra through twisted Sugawara construction [L ± n, J ± m] = m J ± n+m + i k 2 m2 δ n, m Note to experts: twist-term follows uniquely from mapping our connection into highest weight gauge Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 17/27

61 Asymptotic Hilbert space Our algebra in original form [J ± n, J ± m] = k 2 n δ n, m Map to asymptotic Virasoro algebra through twisted Sugawara construction [L ± n, J ± m] = m J ± n+m + i k 2 m2 δ n, m Note to experts: twist-term follows uniquely from mapping our connection into highest weight gauge Get asymptotic Virasoro with Brown Henneaux central charge [L ± n, L ± m] = (n m)l ± n+m + c 12 n3 δ n, m where c = 6k = 3l/(2G N ) in large k-limit Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 17/27

62 Asymptotic Hilbert space Our algebra in original form [J ± n, J ± m] = k 2 n δ n, m Map to asymptotic Virasoro algebra through twisted Sugawara construction [L ± n, J ± m] = m J ± n+m + i k 2 m2 δ n, m Note to experts: twist-term follows uniquely from mapping our connection into highest weight gauge Get asymptotic Virasoro with Brown Henneaux central charge [L ± n, L ± m] = (n m)l ± n+m + c 12 n3 δ n, m where c = 6k = 3l/(2G N ) in large k-limit Algebra gets twisted [L ± n, J ± m] = m J ± n+m + i k 2 m2 δ n, m Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 17/27

63 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Note: in whole Virasoro algebras are mapped, so even for n 0 cl ± n = L ± nc we use much weaker map above between zero modes as ansatz Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

64 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Consistency of algebras then implies J ± n = 1 6 J ± nc, n 0 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

65 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Consistency of algebras then implies J ± n = 1 6 J ± nc, n 0 Even though J n and J n algebras isomorphic, near horizon algebra has more generators than asymptotic one due to relation above, namely J m with m nc Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

66 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Consistency of algebras then implies J ± n = 1 6 J ± nc, n 0 Even though J n and J n algebras isomorphic, near horizon algebra has more generators than asymptotic one due to relation above, namely J m with m nc Relation between J 0 and J 0 also induced by above, but not needed Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

67 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Consistency of algebras then implies J ± n = 1 6 J ± nc, n 0 Even though J n and J n algebras isomorphic, near horizon algebra has more generators than asymptotic one due to relation above, namely J m with m nc Relation between J 0 and J 0 also induced by above, but not needed Main point: use eigenvalues of asymptotic generators L ± 0 (essentially mass and angular momentum of BTZ) to provide cut-off on soft hair spectrum L ± 0 BTZ = ± L ± n 0 BTZ = 0 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

68 Proposed map between near horizon and asymptotic generators Suggestive proposal (see Bañados ) cl ± 0 = L± Consistency of algebras then implies J ± n = 1 6 J ± nc, n 0 Even though J n and J n algebras isomorphic, near horizon algebra has more generators than asymptotic one due to relation above, namely J m with m nc Relation between J 0 and J 0 also induced by above, but not needed Main point: use eigenvalues of asymptotic generators L ± 0 (essentially mass and angular momentum of BTZ) to provide cut-off on soft hair spectrum L ± 0 BTZ = ± L ± n 0 BTZ = 0 Microstates = all states in near horizon Hilbert space obeying equations above Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 18/27

69 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

70 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B All BTZ microstates ( horizon fluffs ) are then of the form B({n ± i }) = N ( ) {n ± J + 0 i } {0<n ± i <c} n + i J n i Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

71 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B All BTZ microstates ( horizon fluffs ) are then of the form B({n ± i }) = N ( ) {n ± J + 0 i } This is the key result {0<n ± i <c} n + i J n i Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

72 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B All BTZ microstates ( horizon fluffs ) are then of the form B({n ± i }) = N ( ) {n ± J + 0 i } {0<n ± i <c} n + i J n i This is the key result Normalization constant N {n ± i } fixed by compatibility: B L ± 0 B = ± δ B,B Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

73 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B All BTZ microstates ( horizon fluffs ) are then of the form B({n ± i }) = N ( ) {n ± J + 0 i } {0<n ± i <c} n + i J n i This is the key result Normalization constant N {n ± i } fixed by compatibility: B L ± 0 B = ± δ B,B Useful observation 1: ± = B L ± 0 B 1 c B L± 0 B = 1 c i n ± i = 1 c E± B Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

74 Horizon fluffs as microstates We are now ready to identify all BTZ microstates Vector space V B of BTZ microstates defined by B L ± n 0 B = 0 B, B V B All BTZ microstates ( horizon fluffs ) are then of the form B({n ± i }) = N ( ) {n ± J + 0 i } {0<n ± i <c} n + i J n i This is the key result Normalization constant N {n ± i } fixed by compatibility: Useful observation 1: B L ± 0 B = ± δ B,B ± = B L ± 0 B 1 c B L± 0 B = 1 c i n ± i = 1 c E± B Useful observation 2: B J ± 2 0 B = 1 6 c ±, B J ± n 0 B = 0 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 19/27

75 Microstate counting Problem of counting microstates now reduced to combinatorics Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

76 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

77 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

78 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Work semi-classically, i.e., in limit of large N Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

79 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Work semi-classically, i.e., in limit of large N Problem above solved long ago by Hardy and Ramanujan p(n) N 1 1 ( N ) 4N 3 exp 2π 6 Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

80 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Work semi-classically, i.e., in limit of large N Problem above solved long ago by Hardy and Ramanujan p(n) N 1 1 ( N ) 4N 3 exp 2π 6 Number of BTZ microstates: p(e + B ) p(e B ) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

81 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Work semi-classically, i.e., in limit of large N Problem above solved long ago by Hardy and Ramanujan p(n) N 1 1 ( N ) 4N 3 exp 2π 6 Number of BTZ microstates: p(e + B ) p(e B ) Result for entropy: ( S = ln p(e + B ) + ln p(e B ) = 2π c + c ) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

82 Microstate counting Problem of counting microstates now reduced to combinatorics Count all horizon fluffs with same energy E ± B Mathematically, reduces to number p(n) of ways positive integer N partitioned into positive integers Work semi-classically, i.e., in limit of large N Problem above solved long ago by Hardy and Ramanujan p(n) N 1 1 ( N ) 4N 3 exp 2π 6 Number of BTZ microstates: p(e + B ) p(e B ) Result for entropy: ( S = ln p(e + B ) + ln p(e B ) = 2π c + c ) Agrees with Bekenstein Hawking and Cardy formula Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Explicit construction of BTZ microstates 20/27

83 Outline Motivation Near horizon boundary conditions Explicit construction of BTZ microstates Discussion Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 21/27

84 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

85 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

86 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

87 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Mathur 05: fuzzballs, like horizon fluffs, do not have horizon; we need only semi-classical near horizon description, not full quantum gravity Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

88 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Mathur 05: fuzzballs, like horizon fluffs, do not have horizon; we need only semi-classical near horizon description, not full quantum gravity Donnay, Giribet, Gonzalez, Pino 15: near horizon bc s (fixed Rindler) with centerless warped algebra Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

89 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Mathur 05: fuzzballs, like horizon fluffs, do not have horizon; we need only semi-classical near horizon description, not full quantum gravity Donnay, Giribet, Gonzalez, Pino 15: near horizon bc s (fixed Rindler) with centerless warped algebra Afshar, Detournay, DG, Oblak 15: near horizon bc s (varying Rindler) and null compactification with twisted warped algebra Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

90 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Mathur 05: fuzzballs, like horizon fluffs, do not have horizon; we need only semi-classical near horizon description, not full quantum gravity Donnay, Giribet, Gonzalez, Pino 15: near horizon bc s (fixed Rindler) with centerless warped algebra Afshar, Detournay, DG, Oblak 15: near horizon bc s (varying Rindler) and null compactification with twisted warped algebra Hawking, Perry, Strominger 16: introduced notion of soft hair, but did not attempt microstate counting Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

91 Relations to previous approaches Carlip 94: conceptually very close; main difference: Carlip stresses near horizon Virasoro, while we exploit near horizon Heisenberg Strominger, Vafa 96: string construction of microstates that relies on BPS/extremality Strominger 97: exploits Cardy formula for microstate counting, but does not identify microstates Mathur 05: fuzzballs, like horizon fluffs, do not have horizon; we need only semi-classical near horizon description, not full quantum gravity Donnay, Giribet, Gonzalez, Pino 15: near horizon bc s (fixed Rindler) with centerless warped algebra Afshar, Detournay, DG, Oblak 15: near horizon bc s (varying Rindler) and null compactification with twisted warped algebra Hawking, Perry, Strominger 16: introduced notion of soft hair, but did not attempt microstate counting Afshar, Detournay, DG, Merbis, Perez, Tempo, Troncoso 16: introduced near horizon bc s we use; did not attempt construction of microstates (but does Cardy-type of counting) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 22/27

92 Generalization to four dimensions Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 23/27

93 Generalization to four dimensions Compare with near horizon construction of Donnay, Giribet, Gonzalez, Pino 15 Near horizon algebra similar to but different from BT-BMS 4 : [Y ± n, Y ± m] = (n m) Y ± n+m [Y + l, T (n,m) ] = n T (n+l, m) [Y l, T (n,m) ] = m T (n, m+l) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 23/27

94 Generalization to four dimensions Compare with near horizon construction of Donnay, Giribet, Gonzalez, Pino 15 Near horizon algebra similar to but different from BT-BMS 4 : [Y ± n, Y ± m] = (n m) Y ± n+m [Y + l, T (n,m) ] = n T (n+l, m) [Y l, T (n,m) ] = m T (n, m+l) Intriguing algebraic observation: introducing again [J ± n, J ± m ] = 1 2 n δ n, m = [K ± n, K ± m] recovers 4d algebra above by Sugawara construction T (n, m) = ( J n + + K n + )( J m + Km ) Y n ± = ( J ± n p + K n p)( ± J ± p K ± ) p p Z Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 23/27

95 Generalization to four dimensions Compare with near horizon construction of Donnay, Giribet, Gonzalez, Pino 15 Near horizon algebra similar to but different from BT-BMS 4 : [Y ± n, Y ± m] = (n m) Y ± n+m [Y + l, T (n,m) ] = n T (n+l, m) [Y l, T (n,m) ] = m T (n, m+l) Intriguing algebraic observation: introducing again [J ± n, J ± m ] = 1 2 n δ n, m = [K ± n, K ± m] recovers 4d algebra above by Sugawara construction T (n, m) = ( J n + + K n + )( J m + Km ) Y n ± = ( J ± n p + K n p)( ± J ± p K ± ) p p Z Making AKVs in DGGP state-dependent to leading order relates their canonical boundary charges to Heisenberg boundary charges Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 23/27

96 Generalization to four dimensions Compare with near horizon construction of Donnay, Giribet, Gonzalez, Pino 15 Near horizon algebra similar to but different from BT-BMS 4 : [Y ± n, Y ± m] = (n m) Y ± n+m [Y + l, T (n,m) ] = n T (n+l, m) [Y l, T (n,m) ] = m T (n, m+l) Intriguing algebraic observation: introducing again [J ± n, J ± m ] = 1 2 n δ n, m = [K ± n, K ± m] recovers 4d algebra above by Sugawara construction T (n, m) = ( J n + + K n + )( J m + Km ) Y n ± = ( J ± n p + K n p)( ± J ± p K ± ) p p Z Making AKVs in DGGP state-dependent to leading order relates their canonical boundary charges to Heisenberg boundary charges Highly non-trivial indication for existence of soft Heisenberg hair in 4d Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 23/27

97 Microstates of non-extremal Kerr? Main challenge: how to provide (controlled) cut-off on soft hair spectrum in four dimensions? Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 24/27

98 Thanks for your attention! H. Afshar, D. Grumiller and M.M. Sheikh-Jabbari Near Horizon Soft Hairs as Microstates of Three Dimensional Black Holes, H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso Soft Heisenberg hair on black holes in three dimensions, Phys.Rev. D93 (2016) (R); Thanks to Bob McNees for providing the LATEX beamerclass! Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 25/27

99 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

100 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

101 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

102 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

103 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

104 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers H NH contained in H Vir Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

105 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers H NH contained in H Vir CFT side: expected to have discrete set of primaries Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

106 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers H NH contained in H Vir CFT side: expected to have discrete set of primaries H CFT contained in H Vir Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

107 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers H NH contained in H Vir CFT side: expected to have discrete set of primaries H CFT contained in H Vir Our proposal: H NH = H CFT Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27

108 On compatibility with AdS 3 /CFT 2 Punchline: our proposal is Bohr-type quantization of spectrum Unitary representations of Virasoro algebra span Hilbert space H Vir Gravity side, asymptotically: H Vir splits into H BTZ, H Conic and H AdS Only states in H BTZ captured by twisted Sugawara construction States in H BTZ labelled by two positive numbers (L ± 0 ) and set of integers (Virasoro excitations) Gravity side, near horizon: soft hair Hilbert space H NH includes all soft hair descendants of vacuum labeled by set of integers H NH contained in H Vir CFT side: expected to have discrete set of primaries H CFT contained in H Vir Our proposal: H NH = H CFT Example ([Lunin], Maldacena, Maoz 02): states in H CFT corresponding to conic spaces discrete family L ± 0 = rk/4 (r = 0: massless BTZ, r = k: global AdS, 0 < r < k: spectral flow) Daniel Grumiller Near Horizon Soft Hairs as Black Hole Microstates Discussion 26/27