Virasoro hair on locally AdS 3 geometries

Transcription

1 Virasoro hair on locally AdS 3 geometries Kavli Institute for Theoretical Physics China Institute of Theoretical Physics ICTS (USTC) arxiv: , M. M. Sheikh-Jabbari and H. Y

2 Motivation

3 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

4 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

5 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

6 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

7 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

8 Introduction Solving the problem of quantisation of systems including gravity is notoriously hard. It does however simplify drastically in very low dimensions. (no local excitations ) The case d = 3 is noteworthy. In this case the quantum theory can be understood and is nevertheless non-trivial. AdS 3 gravity has been much more studied, mainly in connection with and through AdS 3 /CFT 2. AdS 3 gravity has black hole solutions [ M. Bañados, C. Teitelboim, and J. Zanelli] AdS 3 gravity appears as near horizon geometry of EVH black holes.

9 Killing vectors & causal structure of Bañados metrics

10 E.o.M is given by R µν = 2 l 2 g µν. Riemann curvature is determined through Ricci: R µνατ = 1 2 ɛ µνρɛ ατσ R ρσ. All solutions are hence locally AdS 3. All these solution have local SL(2, R) SL(2, R) isometry group.

11 E.o.M is given by R µν = 2 l 2 g µν. Riemann curvature is determined through Ricci: R µνατ = 1 2 ɛ µνρɛ ατσ R ρσ. All solutions are hence locally AdS 3. All these solution have local SL(2, R) SL(2, R) isometry group.

12 E.o.M is given by R µν = 2 l 2 g µν. Riemann curvature is determined through Ricci: R µνατ = 1 2 ɛ µνρɛ ατσ R ρσ. All solutions are hence locally AdS 3. All these solution have local SL(2, R) SL(2, R) isometry group.

13 E.o.M is given by R µν = 2 l 2 g µν. Riemann curvature is determined through Ricci: R µνατ = 1 2 ɛ µνρɛ ατσ R ρσ. All solutions are hence locally AdS 3. All these solution have local SL(2, R) SL(2, R) isometry group.

14 Introduction Classifications of the solutions are hence made through: global features, like global isometries or topology, asymptotic behaviour of the solution, the boundary conditions. Global AdS 3 is the ONLY geometry in this class which has global SL(2, R) SL(2, R) isometry. The other best known solutions are constructed through orbifolding of AdS 3 by a subgroup of its isometry group, with the condition that we do not get a closed time-like curve.

15 Introduction Classifications of the solutions are hence made through: global features, like global isometries or topology, asymptotic behaviour of the solution, the boundary conditions. Global AdS 3 is the ONLY geometry in this class which has global SL(2, R) SL(2, R) isometry. The other best known solutions are constructed through orbifolding of AdS 3 by a subgroup of its isometry group, with the condition that we do not get a closed time-like curve.

16 Introduction Classifications of the solutions are hence made through: global features, like global isometries or topology, asymptotic behaviour of the solution, the boundary conditions. Global AdS 3 is the ONLY geometry in this class which has global SL(2, R) SL(2, R) isometry. The other best known solutions are constructed through orbifolding of AdS 3 by a subgroup of its isometry group, with the condition that we do not get a closed time-like curve.

17 There are three well known such geometries: BTZ black holes: with global U (1) + U (1) + isometry. These U(1) s are compact. AdS 3 Self-Dual Orbifold with global SL(2, R) U(1) isometry. The U(1) is compact while U(1) SL(2, R) is non-compact. Conic space with global SL(2, R)/Z k SL(2, R)/Z l isometry, k, l N

18 The most general class of solutions with Brown-Henneaux boundary conditions ds 2 = l 2 dr 2 r 2 ) ) (rdx + l2 r f (x )dx (rdx l2 r f +(x + )dx + where x ± [0, 2π], or equivalently x ± = t l ± φ and φ [0, 2π] and f ± are two periodic holomorphic functions: f ± (x ± + 2π) = f ± (x ± )

19 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

20 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

21 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

22 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

23 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

24 ( ds 2 = l 2 dr 2 r 2 rdx + l2 r f (x )dx ) ( ) rdx l2 r f+(x + )dx + f + = f = 0 ds 2 = l2 dr 2 r 2 dx + dx ds 2 = l2 dr 2 r 2 r 2 f +, f = const. R + BTZ black hole ρ 2 = ( r 2 + l 2 f + ) ( r 2 + l 2 f ) r 2 dt2 l 2 + r 2 dϕ 2 r 2, t = l 2 (x + + x ), ϕ = 1 2 (x + x ), ( ds 2 = F (ρ)dt 2 + dρ2 F (ρ) + ρ2 dϕ ρ+ρ ) 2 dt lρ 2, F (ρ) = (ρ2 ρ 2 + )(ρ2 ρ 2 ) l 2 ρ 2 ρ ± = l( f + ± f ) Can be extended to negative values of r 2 CTC Allowed range of r 2 covers the geometry twice

25 Although all Bañados geometries are diffeomorphic to AdS 3 but these geometries are physically distinct, as there are NO everywhere smooth coordinate transformations which respect the periodicity in x ±. More precisely, these geometries are distinct because one can specify them with quasi-local conserved surface charges. In fact, one can distinguish two kinds of such conserved charges: those associated with exact symmetries (Killing symmetries), we denote the generators of the exact symmetries by J ±. Those which are in the family of symplectic symmetries, with generators L n, L n which form two (left and right) copies of Virasoro algebras at Brown-Henneaux central charge c. and that J ±.

26 Although all Bañados geometries are diffeomorphic to AdS 3 but these geometries are physically distinct, as there are NO everywhere smooth coordinate transformations which respect the periodicity in x ±. More precisely, these geometries are distinct because one can specify them with quasi-local conserved surface charges. In fact, one can distinguish two kinds of such conserved charges: those associated with exact symmetries (Killing symmetries), we denote the generators of the exact symmetries by J ±. Those which are in the family of symplectic symmetries, with generators L n, L n which form two (left and right) copies of Virasoro algebras at Brown-Henneaux central charge c. and that J ±.

27 Although all Bañados geometries are diffeomorphic to AdS 3 but these geometries are physically distinct, as there are NO everywhere smooth coordinate transformations which respect the periodicity in x ±. More precisely, these geometries are distinct because one can specify them with quasi-local conserved surface charges. In fact, one can distinguish two kinds of such conserved charges: those associated with exact symmetries (Killing symmetries), we denote the generators of the exact symmetries by J ±. Those which are in the family of symplectic symmetries, with generators L n, L n which form two (left and right) copies of Virasoro algebras at Brown-Henneaux central charge c. and that J ±.

28 Although all Bañados geometries are diffeomorphic to AdS 3 but these geometries are physically distinct, as there are NO everywhere smooth coordinate transformations which respect the periodicity in x ±. More precisely, these geometries are distinct because one can specify them with quasi-local conserved surface charges. In fact, one can distinguish two kinds of such conserved charges: those associated with exact symmetries (Killing symmetries), we denote the generators of the exact symmetries by J ±. Those which are in the family of symplectic symmetries, with generators L n, L n which form two (left and right) copies of Virasoro algebras at Brown-Henneaux central charge c. and that J ±.

29 L n = With the Brown-Henneaux boundary conditions, one can assign Virasoro charges to these geometries c 2π f + (x + )e inx + dx +, Ln = 12π 0 where c is the Brown-Henneaux central charge c 2π f (x )e inx dx 12π 0 c = 3l 2G N

30 Killing Vectors As local AdS 3 geometries, we have six Killing vectors: ζ = h r r + h h where h r = r 2 (K + + K ) h + = K + + l2 2 h = K + l2 2 r 2 K + l 2 f K + r 4 l 4 f + f r 2 K + + l 2 f + K r 4 l 4 f + f where K ± = K ± (x ± ) and prime denotes derivative with respect to the argument.

31 Functions K ± satisfy following equations K ± 4K ±f ± 2K ± f ± = 0 Two third-order differential equations with six independent solutions, giving rise to six local SL(2, R) SL(2, R) isometries. Global isometries iff we have real and periodic solutions. h a in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms There is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the class of our geometries.

32 Functions K ± satisfy following equations K ± 4K ±f ± 2K ± f ± = 0 Two third-order differential equations with six independent solutions, giving rise to six local SL(2, R) SL(2, R) isometries. Global isometries iff we have real and periodic solutions. h a in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms There is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the class of our geometries.

33 Functions K ± satisfy following equations K ± 4K ±f ± 2K ± f ± = 0 Two third-order differential equations with six independent solutions, giving rise to six local SL(2, R) SL(2, R) isometries. Global isometries iff we have real and periodic solutions. h a in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms There is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the class of our geometries.

34 Functions K ± satisfy following equations K ± 4K ±f ± 2K ± f ± = 0 Two third-order differential equations with six independent solutions, giving rise to six local SL(2, R) SL(2, R) isometries. Global isometries iff we have real and periodic solutions. h a in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms There is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the class of our geometries.

35 Functions K ± satisfy following equations K ± 4K ±f ± 2K ± f ± = 0 Two third-order differential equations with six independent solutions, giving rise to six local SL(2, R) SL(2, R) isometries. Global isometries iff we have real and periodic solutions. h a in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms There is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the class of our geometries.

36 Constructing explicit K ± solutions The three solutions to third order diff. eq. for K can be constructed through Schrodinger Eq. (Hill s Eq.) ψ f + ψ = 0, φ f φ = 0 with solutions ψ α, φ α, α = 1, 2, as K + = 1 2 ψ2 1, K 0 + = 1 2 ψ 1ψ 2, K + + = 1 2 ψ2 2, (1) K = 1 2 φ2 1, K 0 = 1 2 φ 1φ 2, K + = 1 2 φ2 2, where we used following normalisation ψ 1 ψ 2 ψ 1 ψ 2 = 1, φ 1 φ 2 φ 1 φ 2 = 1

37 Global vs. Local isometries Killing Vector analysis of isometry algebra gives local isometries. These would also be global isometry, iff K a ± are defined globally. That is, if the K s are periodic and smooth in x ±. Some (out of six) K s may be smooth and periodic, then generically global isometry is a subgroup of maximal global isometry group SL(2, R) SL(2, R). We will prove that regardless of what f ± are, we always have global U(1) U(1) isometry.

38 Global vs. Local isometries Killing Vector analysis of isometry algebra gives local isometries. These would also be global isometry, iff K a ± are defined globally. That is, if the K s are periodic and smooth in x ±. Some (out of six) K s may be smooth and periodic, then generically global isometry is a subgroup of maximal global isometry group SL(2, R) SL(2, R). We will prove that regardless of what f ± are, we always have global U(1) U(1) isometry.

39 Global vs. Local isometries Killing Vector analysis of isometry algebra gives local isometries. These would also be global isometry, iff K a ± are defined globally. That is, if the K s are periodic and smooth in x ±. Some (out of six) K s may be smooth and periodic, then generically global isometry is a subgroup of maximal global isometry group SL(2, R) SL(2, R). We will prove that regardless of what f ± are, we always have global U(1) U(1) isometry.

40 Global vs. Local isometries Killing Vector analysis of isometry algebra gives local isometries. These would also be global isometry, iff K a ± are defined globally. That is, if the K s are periodic and smooth in x ±. Some (out of six) K s may be smooth and periodic, then generically global isometry is a subgroup of maximal global isometry group SL(2, R) SL(2, R). We will prove that regardless of what f ± are, we always have global U(1) U(1) isometry.

41 Floquet theorem Solutions to the 1-d Sch s equation (with a periodic potential) ψ 1, ψ 2 are of the form: ψ 1 = e T x P 1 (x), ψ 2 = e T x P 2 (x), P α (x + 2π) = P α (x) the constant T is called the Floquet exponent. The Floquet theorem hence implies that: regardless of the form of f ± our geometries always have global U(1) + U(1) isometry, respectively associated with Killings ζ ±, and generated by ψ 1 ψ 2 and φ 1 φ 2.

42 Killing vectors have following structure ζ[k +,K ] = ζ r r + ζ ζ ζ r = r 2 (K+, +K ), ζ + = K +,+ l2 r 2 K + l 4 L K +, 2(r 4 l 4 L +L ), ζ = K + l2 r 2 K +, + l 4 L +K 2(r 4 l 4 L +L ) K ψ2 1, K ψ2 2, K ψ 1ψ 2, K φ2 1, K φ2 2, K φ 1φ 2, ψ f +ψ = 0, φ f φ = 0 where according to the Floquet theorem ψ 1 = e T+x+ P 1 (x + ), ψ 2 = e T x+ P 2 (x + ) φ 1 = e T x Q 1 (x ), φ 2 = e T x Q 2 (x ) P i and Q i are periodic functions with period 2π.

43 J ± are defined by δj ± = l 8πG 2π K± 0 δl ± dx ± 0 K 0 + = 2T + ψ 1 (x + ; T +)ψ 2 (x + ; T +), K 0 = 2T φ 1 (x ; T )φ 2 (x ; T ) J ± = δl ± = L ± δt ±. T ± l T± d T ± T ± = l 2G T ±0 4G (T ± 2 T±0 2 ) = c 6 (T ± 2 T±0 2 ) T ±0 is a reference point which has zero J ±. For BTZ black hole solution we get J ± = l 4G T 2 ± = lm BTZ ± J BTZ J ± commute with the Virasoro generators L n, L n, [J ±, L n] = [J ±, L n] = 0, n Z.

44 J ± are defined by δj ± = l 8πG 2π K± 0 δl ± dx ± 0 K 0 + = 2T + ψ 1 (x + ; T +)ψ 2 (x + ; T +), K 0 = 2T φ 1 (x ; T )φ 2 (x ; T ) J ± = δl ± = L ± δt ±. T ± l T± d T ± T ± = l 2G T ±0 4G (T ± 2 T±0 2 ) = c 6 (T ± 2 T±0 2 ) T ±0 is a reference point which has zero J ±. For BTZ black hole solution we get J ± = l 4G T 2 ± = lm BTZ ± J BTZ J ± commute with the Virasoro generators L n, L n, [J ±, L n] = [J ±, L n] = 0, n Z.

45 J ± are defined by δj ± = l 8πG 2π K± 0 δl ± dx ± 0 K 0 + = 2T + ψ 1 (x + ; T +)ψ 2 (x + ; T +), K 0 = 2T φ 1 (x ; T )φ 2 (x ; T ) J ± = δl ± = L ± δt ±. T ± l T± d T ± T ± = l 2G T ±0 4G (T ± 2 T±0 2 ) = c 6 (T ± 2 T±0 2 ) T ±0 is a reference point which has zero J ±. For BTZ black hole solution we get J ± = l 4G T 2 ± = lm BTZ ± J BTZ J ± commute with the Virasoro generators L n, L n, [J ±, L n] = [J ±, L n] = 0, n Z.

46 There is a certain combination of the two Killings which are normal to each other: ζ H± ζ + ± ζ. The norm of these vectors are given as where ζ H± 2 = ψ 1ψ 2 φ 1 φ 2 r 2 (r 2 r 2 1± )(r 2 r 2 2± ), r 2 1+ = l2 ψ 1 φ 1 ψ 1 φ 1, r 2 2+ = l2 ψ 2 φ 2 ψ 2 φ 2, r 2 1 = l2 ψ 1 φ 2 ψ 1 φ 2, r 2 2 = l2 ψ 2 φ 1 ψ 2 φ 1.

47 The charge associated with the outer (event) Killing horizon is S/2π. δs 2π = l 8πG K +δl + + K δl = β +δj + + β δj, β ± = 1 2T ± β ± is the inverse temperature associated with the left and right sectors. The above is nothing but the first law for a generic Bañados geometry. S = 2π S 2π = l 4G (T+ + T ) = 2(β +J + + β J ). c(j + + J +0 ) 6 + c(j + J 0 ), J ±0 = c 6 6 T ± 2. S inner 2π = l 4G (T+ T ), S S inner = π2 l G (J+ J )

48 The charge associated with the outer (event) Killing horizon is S/2π. δs 2π = l 8πG K +δl + + K δl = β +δj + + β δj, β ± = 1 2T ± β ± is the inverse temperature associated with the left and right sectors. The above is nothing but the first law for a generic Bañados geometry. S = 2π S 2π = l 4G (T+ + T ) = 2(β +J + + β J ). c(j + + J +0 ) 6 + c(j + J 0 ), J ±0 = c 6 6 T ± 2. S inner 2π = l 4G (T+ T ), S S inner = π2 l G (J+ J )

49 These sets are commuting, one may hence label the geometries by the J ± and the Virasoro charges L n, L n.

50 We classify Bañados geometries by the product of representations of the two, left and right, Virasoro groups, the Virasoro coadjoint orbits. Each Virasoro orbit is generically labelled by an integer n and a continuous real number T and then states in a given orbit are fully specified once we also give their Virasoro charges, the Virasoro hairs. There is a one-to-one relation between two copies of Virasoro orbits and Bañados geometries.

51 We classify Bañados geometries by the product of representations of the two, left and right, Virasoro groups, the Virasoro coadjoint orbits. Each Virasoro orbit is generically labelled by an integer n and a continuous real number T and then states in a given orbit are fully specified once we also give their Virasoro charges, the Virasoro hairs. There is a one-to-one relation between two copies of Virasoro orbits and Bañados geometries.

52 We classify Bañados geometries by the product of representations of the two, left and right, Virasoro groups, the Virasoro coadjoint orbits. Each Virasoro orbit is generically labelled by an integer n and a continuous real number T and then states in a given orbit are fully specified once we also give their Virasoro charges, the Virasoro hairs. There is a one-to-one relation between two copies of Virasoro orbits and Bañados geometries.

53 The picture we depict is: The information about Bañados geometries available to local observables of the usual classical GR, the geometric notions such as geodesic length, causal and boundary structure and the black hole (thermo)dynamics quantities like surface gravity and horizon angular velocity, entropy, are orbit invariant quantities. That is, all geometries which fall into the same orbit share these properties, regardless of their Virasoro hair.

54 On the other hand, the information about the Virasoro hair are semi- classical, in the sense that they are of the form of surface non-local ( quasi-local ) charges; the Virasoro charges may be viewed as the hair on classical geometries all sharing the same mass and angular momentum and, the causal structure. Given this picture, one may then hope to obtain a full quantum description upon quantisation of Virasoro coadjoint orbits.

55 Bañados Geometries & Coadjoint Representation

56 Rep. of Virasoro algebra Kirillov et. al , Witten 1988 Virasoro algebra is infinite dimensional and has central extension. Its representations are infinite dimensional. Witt algebra is the algebra of infinitesimal diffeomorphisms on a circle, diff (S 1 ). Virasoro algebra generators are ɛ(x) x, where x x + ɛ(x), ɛ(x + 2π) = ɛ(x). Virasoro group is then group of finite coordinate transformations on the circle S 1, Diff (S 1 ): x h(x), h(x + 2π) = h(x) + 2π

57 Virasoro group is infinite dimensional and its rep, the Virasoro coadjoint orbits, are also infinite dimensional. Virasoro coadjoint orbits, are associated with class of periodic functions (functions on S 1 ). Since f ± are periodic, the solutions of the Sch. equations are quasi-periodic with some monodromy, i.e., Ψ ± (x ± +2π) = Ψ ± (x ± )M Ψ±, M SL(2, R), Ψ ± = (ψ 1 ±, ψ 2 ±) Because the transformation Ψ ΨA with A SL(2, R) does not change K and transform the monodromy matrices by conjugation, one may assume without loss of generality that M Ψ belongs to a given set of representatives of the conjugacy classes SL(2, R) when writing the K.V solutions in terms of ψ s

58 the action of the conformal group (x h(x) with h > 0) on the set of Sch. equations f h 2 f (h(x)) 1 h 2 h + 3 h 2 4 h 2, ψ 1 ψ(h(x)) h For each chirality, this action is just the coadjoint action of the central extension of the conformal group Diff (S 1 ). Its orbits in the space of Virasoro densities f ± are known as the coadjoint orbits of the Virasoro algebrab. It is now clear that the classification of the Sch. equations under the conformal group is essentially the same as the classification of the Virasoro coadjoint orbits.

59 Given a periodic function f(x), Virasoro orbits of f(x) is generated by coordinate infinitesimal transformations x x + ξ(x) satisfying stabiliser equation δ ξ f = ξ 4f ξ 2f ξ = 0 Stabiliser equation is exactly the same equation which specifies the Killing vectors on Bañados geometries. Stabilizer equation has three solutions. Generically, only one is periodic and two are not.

60 Periodic solution is just a global coord. transf. on S 1 and is associated with a 1-dim. subgroup of Diff (S 1 ), T ξ. Non-periodic solutions move us up and down in the orbit (the Virasoro group multiplets ). The Virasoro coadjoint orbits would then be identified with the periodic solutions to the stabiliser equation. For a given f, with the subgroup T [f ], orbit of f is O f = Diff (S 1 )/T ξ[f ]. Elements in the same orbit O f are hence elements in Diff (S 1 ) up to T ξ[f ].

61 Upon a transformation in Diff (S 1 ), x h(x), f (x) f ( x) = h 2 f (h(x)) + S(h, x), ψ ψ = ψ((x)) h where S(h(x), x) = h 2h 3h 2 4h 2 is the Schwarz derivative. As it is seen ξ(x) x ξ(h(x)) h(x)

62 Classification of Virasoro coadjoint orbits Constant representative orbits: Exceptional orbits E n, with representative: f n = n2 2 4, ψ n = n sin nx 2 2, n cos nx 2, Elliptic orbits C(ν), with f ν = ν2 4, ψ ν = Zeroth Hyperbolic orbits B 0 (b), with f T = T 2, 2 ν sin νx 2 2, ν cos νx 2, ψ T = et x 2T, e T x 2T, T R + n N ν / N Zeroth order parabolic orbit P + 0 f + = 0, ψ = 2π, x 2π

63 Classification of Virasoro coadjoint orbits Non-Constant representative orbits: Hyperbolic orbits B n (T ), with f n,t = T 2 + T 2 + 4n 2 2F 3n2 4F 2, ψ n,t = et x ( 2 T F n n cos nx 2 + sin nx 2 where T R+, n N F = cos 2 nx 2 + (sin nx 2 + 2T n cos nx 2 )2 ), ψ n,t = e T x 2 T F n n cos nx 2

64 Classification of Virasoro coadjoint orbits Non-Constant representative orbits: Parabolic orbits P ± n, with f ± n 2H 3n2 (1 ± 1 2π ) 4H 2, n N = n2 ψ n = 1 sin nx H 2, ψ n = 1 ( ± x H 2π sin nx 2 2 n cos nx 2 where H = 1 ± 1 2π sin2 nx 2 ),

65 Geometry of Virasoro coadjoint orbits Exceptional orbits E n : Correspond to n-fold cover of AdS 3. In this case Tξ[f ] = PLS (n) (2, R). Elliptic orbits C(ν) ν < 1, correspond to conic singularities (particles on AdS 3 ). ν > 1 correspond to particles on n-fold cover of AdS 3 (n = [ν]) Zeroth Hyperbolic orbits B 0 (T ): correspond to BTZ black hole Hyperbolic orbits B n (T ) correspond to multi-btz black hole. Parabolic orbits P ± n correspond to selfdual orbifold of n-fold cover of AdS 3.

66 Causal Structure Horizon Structure of Bañados metrics Killing horizon is a co-dimension one null surface where a KV ζ H± vanishes. A linear combination of the two global Killings can be ζ H± = ζ + ± ζ ζ H± 2 = ψ 1ψ 2 φ 1 φ 2 r 2 (r 2 r 2 1± )(r 2 r 2 2± ) r 2 1+ = l2 ψ 1 φ 1 ψ 1 φ 1, r 2 2+ = l2 ψ 2 φ 2 ψ 2 φ 2, r 2 1 = l2 ψ 1 φ 2 ψ 1 φ 2, r 2 2 = l2 ψ 2 φ 1 ψ 2 φ 1

67 Sign structure of ψ and φ region function I 1,i I 2,i I 3,i I 4,i I 1,i+1 ψ 1 + ψ ψ 1 + ψ ψ 1 ψ 2 + ψ 1 /ψ ψ 2 /ψ I 1,1 = [0, x + 1,1 ), I 2,1 = (x + 1,1, x+ 2,1 ), I 3,1 = (x + 2,1, x+ 3,1 ), I 4,1 = (x + 3,1, x+ 4,1 ), I 1,2 = (x + 4,1, x+ 1,2 ), I 2,2 = (x + 1,2, x+ 2,2 ),, I 3,n + = (x + 2,n +, x + 3,n + ), I 4,n+ = (x + 3,n +, x + 4,n + ).

68 On the existence of the horizons: x + x I 1,j I 2,j I 3,j I 4,j I 1,j+1 I + 1,i I + 2,i I + 3,i + + I + 4,i + + I + 1,i x + x I 1,j I 2,j I 3,j I 4,j I 1,j+1 I + 1,i I + 2,i + + I + 3,i + + I + 4,i I + 1,i To distinguish which one is the inner horizon and which one the outer, we need to study the sign of rα± 2 functions. We can learn that r1± 2, r 2± 2 in the I+ and I regions, have the signs given in above and following tables. a,i b,j

69 x + x I 1,j I 2,j I 3,j I 4,j I 1,j+1 I + 1,i + + I + 2,i + + I + 3,i I + 4,i I + 1,i x + x I 1,j I 2,j I 3,j I 4,j I 1,j+1 I + 1,i + + I + 2,i I + 3,i I + 4,i + + I + 1,i+1 + +

70 Sign of ψ 1 ψ 2 φ 1 φ 2 x x + I 1,j I 2,j I 3,j I 4,j I 1,j+1 I + 1,i + + I + 2,i I + 3,i I + 4,i I + 1,i In summary: we have (n + + 1)(n + 1) causally disconnected regions in the range x ± [0, 2π] at the boundary. These regions are separated by the roots of ψ 1, ψ 2, φ 1 and φ 2.

71 One may show that 1 4 ζ H ± 2 ζh± 2 =0 = 1, Implying that the un-normalized surface gravity at the Killing horizons are equal (up to a sign). To read the physical surface gravity and determining its sign, we need to fix the normalization of the Killing vectors. To fix the normalization, we focus on the regions where ψ 1 ψ 2 and φ 1 φ 2 are both positive, and hence the event horizon is generated by ζ H+. At the large r ζ + K + +, ζ K.

72 One may show that 1 4 ζ H ± 2 ζh± 2 =0 = 1, Implying that the un-normalized surface gravity at the Killing horizons are equal (up to a sign). To read the physical surface gravity and determining its sign, we need to fix the normalization of the Killing vectors. To fix the normalization, we focus on the regions where ψ 1 ψ 2 and φ 1 φ 2 are both positive, and hence the event horizon is generated by ζ H+. At the large r ζ + K + +, ζ K.

73 One may show that 1 4 ζ H ± 2 ζh± 2 =0 = 1, Implying that the un-normalized surface gravity at the Killing horizons are equal (up to a sign). To read the physical surface gravity and determining its sign, we need to fix the normalization of the Killing vectors. To fix the normalization, we focus on the regions where ψ 1 ψ 2 and φ 1 φ 2 are both positive, and hence the event horizon is generated by ζ H+. At the large r ζ + K + +, ζ K.

74 The appropriate normalization is hence the one in which ζ ± are along the coordinates. Noting that where ζ + 1 2T + X +, ζ 1 2T X, X + = 1 2T + ln ψ 1 ψ 2, X = 1 2T ln φ 1 φ 2. The appropriate asymptotic time and angular variable, τ, ϕ are hence τ = l(x + + X )/2, ϕ = (X + X )/2. With the above normalization and recalling KVs, we learn that physical surface gravity κ is 1 κ = T + 2T

75 Concluding Remarks & Outlook

76 Considering Bañados geometries and studied their isometries, horizon and causal structure. Different probes can access different kind of information from the geometry. The "classical" probes, like geodesics, have only access to "classical, geometric" information. These geometric information are "orbit invariant". Classical observers are blind to Virasoro charges, "Virasoro hair". The Virasoro charges are semi-classical ones, they are given by "surface integrals". This information is not available to local classical probes.

77 Considering Bañados geometries and studied their isometries, horizon and causal structure. Different probes can access different kind of information from the geometry. The "classical" probes, like geodesics, have only access to "classical, geometric" information. These geometric information are "orbit invariant". Classical observers are blind to Virasoro charges, "Virasoro hair". The Virasoro charges are semi-classical ones, they are given by "surface integrals". This information is not available to local classical probes.

78 Considering Bañados geometries and studied their isometries, horizon and causal structure. Different probes can access different kind of information from the geometry. The "classical" probes, like geodesics, have only access to "classical, geometric" information. These geometric information are "orbit invariant". Classical observers are blind to Virasoro charges, "Virasoro hair". The Virasoro charges are semi-classical ones, they are given by "surface integrals". This information is not available to local classical probes.

79 There is a one-to-one relation between two copies of Virasoro coadjoint orbits and Bañados geometries. All geometries in the same orbit share the same "geometric" information, while they can be distinguished by their "Virasoro hairs". It is possible that at the level of the geometry we have extra requirements like absence of CTCs, which needs to be considered. Both the charges associated with exact symmetries J ± and the Virasoro hairs are symplectic charges [Covariant Phase-Space Formalism].

80 There is a one-to-one relation between two copies of Virasoro coadjoint orbits and Bañados geometries. All geometries in the same orbit share the same "geometric" information, while they can be distinguished by their "Virasoro hairs". It is possible that at the level of the geometry we have extra requirements like absence of CTCs, which needs to be considered. Both the charges associated with exact symmetries J ± and the Virasoro hairs are symplectic charges [Covariant Phase-Space Formalism].

81 There is a one-to-one relation between two copies of Virasoro coadjoint orbits and Bañados geometries. All geometries in the same orbit share the same "geometric" information, while they can be distinguished by their "Virasoro hairs". It is possible that at the level of the geometry we have extra requirements like absence of CTCs, which needs to be considered. Both the charges associated with exact symmetries J ± and the Virasoro hairs are symplectic charges [Covariant Phase-Space Formalism].

82 This analysis suggests the following general picture: different geometries which are diffeomorphic to each other share the same "geometric information". However, there could be a measure-zero set of diffeos producing semi-classically different geometries, they may be distinguished by their other surface charges, "semi-classical hairs". The states sharing the same classical geometric information fall into orbits of the semi-classical symmetry algebra. This symmetry algebra is a symplectic symmetry of the phase space constituted from diffeomorphic but distinguishable, geometries. If the geometry we are dealing with is a black hole, then the geometries which share the same geometric information may be viewed as hairs on this black hole.

83 This analysis suggests the following general picture: different geometries which are diffeomorphic to each other share the same "geometric information". However, there could be a measure-zero set of diffeos producing semi-classically different geometries, they may be distinguished by their other surface charges, "semi-classical hairs". The states sharing the same classical geometric information fall into orbits of the semi-classical symmetry algebra. This symmetry algebra is a symplectic symmetry of the phase space constituted from diffeomorphic but distinguishable, geometries. If the geometry we are dealing with is a black hole, then the geometries which share the same geometric information may be viewed as hairs on this black hole.

84 This analysis suggests the following general picture: different geometries which are diffeomorphic to each other share the same "geometric information". However, there could be a measure-zero set of diffeos producing semi-classically different geometries, they may be distinguished by their other surface charges, "semi-classical hairs". The states sharing the same classical geometric information fall into orbits of the semi-classical symmetry algebra. This symmetry algebra is a symplectic symmetry of the phase space constituted from diffeomorphic but distinguishable, geometries. If the geometry we are dealing with is a black hole, then the geometries which share the same geometric information may be viewed as hairs on this black hole.

85 This (hopefully) provides a handle on the BH microstate problem. Although we worked in a specific gauge, this above picture is gauge independent. It was shown that similar results hold in the Gaussian null coordinates. In this work have established this picture for AdS 3 case. Similar ideas have been worked through for the near horizon extremal geometries. We think this picture should be more general and applicable to any black hole. At a more technical level, Bañados geometries form a phase space. Elements in this phase space are classified by the Virasoro coadjoint orbits. One may hence use this picture to perform quantisation of AdS 3 gravity.

86 This (hopefully) provides a handle on the BH microstate problem. Although we worked in a specific gauge, this above picture is gauge independent. It was shown that similar results hold in the Gaussian null coordinates. In this work have established this picture for AdS 3 case. Similar ideas have been worked through for the near horizon extremal geometries. We think this picture should be more general and applicable to any black hole. At a more technical level, Bañados geometries form a phase space. Elements in this phase space are classified by the Virasoro coadjoint orbits. One may hence use this picture to perform quantisation of AdS 3 gravity.

87 This (hopefully) provides a handle on the BH microstate problem. Although we worked in a specific gauge, this above picture is gauge independent. It was shown that similar results hold in the Gaussian null coordinates. In this work have established this picture for AdS 3 case. Similar ideas have been worked through for the near horizon extremal geometries. We think this picture should be more general and applicable to any black hole. At a more technical level, Bañados geometries form a phase space. Elements in this phase space are classified by the Virasoro coadjoint orbits. One may hence use this picture to perform quantisation of AdS 3 gravity.

88 This (hopefully) provides a handle on the BH microstate problem. Although we worked in a specific gauge, this above picture is gauge independent. It was shown that similar results hold in the Gaussian null coordinates. In this work have established this picture for AdS 3 case. Similar ideas have been worked through for the near horizon extremal geometries. We think this picture should be more general and applicable to any black hole. At a more technical level, Bañados geometries form a phase space. Elements in this phase space are classified by the Virasoro coadjoint orbits. One may hence use this picture to perform quantisation of AdS 3 gravity.

89 Thank You For Your Attention