On two dimensional black holes. and matrix models

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1 On two dimensional black holes and matrix models Based on: On Black Hole Thermodynamics of 2-D Type 0A, JHEP 0403 (04) 007, hep-th/ with J. L. Davis and D. Vaman Madison April, 2004

2 Motivation: Black Holes are the Hydrogen Atoms of Quantum Gravity Black Hole Thermodynamics: Effective, macroscopic Fundamental, microscopic 2-D Black Holes are the simplest gravity wise The dual theory is presumably manageable [Matrix Model]

3 Why take vacation in Flatland? Outline Black Holes in 2-d Type OA Thermodynamics of 2-d Black Holes Matrix Models dual of Black Holes Where is the Black Hole/Matrix Duality?

4 The Bosonic 2-d Black Hole [Witten] Background ds 2 = tanh 2 rdt 2 + dr 2, Φ = 2 ln cosh r + Φ 0. (1) R = 4/ cosh 2 r (2) Regular Null Coordinates ds 2 = dudv 1 uv (3) v uv=1 V u II r=0 IV I III r=0 uv=1 VI

5 2-D Type 0A Low energy Effective Action Takayanagi, Toumbas and Douglas,Klebanov,Kutasov,Maldacena,Martinec,Seiberg S = d 2 x g [ e 2Φ 2κ 2 8 α + R + 4( Φ)2 f 1 (T )( T ) 2 + f 2 (T ) πα 4 f 3(T )(F (+) ) 2 2πα 4 f 3( T )(F ( ) ) ], (4) f i (T ) functions of the tachyon field T. Dualize RR field Strength, q electric and magnetic D0 branes: S = d 2 x g [ e 2Φ c + R + 4( Φ) 2 ( T ) α T 2 +Λ(1+2T 2 )... ], c = 8/α, Λ = q 2 /(2πα ) units 2κ 2 = 1. (5)

6 The Black Hole Solution ds 2 = l(φ) dt 2 + dφ2 l(φ), (6) l(φ) = 1 4 c e cφ 1 4 Λ c φ + m, (7) Φ = c φ/2 (8) Λ < 0 the Reissner-Nordstrom [Two different roots for l(φ)] Singular at φ R = 2 φl(φ), (9)

7 This is Really a Brane-like Solution Flat space (with linear dilaton) φ ds 2 = dt 2 + dφ 2, Φ = c 4 φ. (10) The near horizon region of the extremal black hole is AdS 2 ds 2 = φ 2 dt 2 + dφ2, Φ = Constant (11) φ2 The extremal black hole interpolates between flat space and AdS 2

8 ADM Mass General Case [Mann] S[g, Φ] = d 2 x g [ D(Φ) R + H(Φ)g µν µ Φ ν Φ + V (Φ, Φ M ) ], (12) In our case S = d 2 x g [ e 2Φ ( R + 4( Φ) 2 + c ) + Λ ] (13) D(Φ) = e 2Φ, H(Φ) = 4e 2Φ, V = Λ + c e 2Φ. (14) a topologically conserved current, need no timelike Killing vector T µν the stress-energy tensor S µ = T µν ɛ νρ ρ F (15) Φ F = F 0 ds D s H(t) exp dt D. (16) (t)

9 Mass M: S µ = ɛ µν ν M [ Φ M = F 0 dsd s H(t) (s) V (s) exp dt D (t) ( D) 2 exp Φ H(t) dt D (t) (17) ]. M = 4F 0 e 2Φ[ ( Φ) 2 c Λ Φe2Φ] (18) For the solution M = 4 c m. (19)

10 Temperature Absence of conical singularity ds 2 = 4 l 2 dr l 2 r 2 dt 2. (20) T = 1 4π l (φ) φ=φh. (21) φ h = 4m Λ c + 1 c W Lambert function: W (z) exp(w (z)) = z. ( ) c Λ e4m/λ. (22) T = c 4π 1 + Λ ( c exp 4 m Λ + W ( c ) Λ e4m/λ ). (23)

11 Extremal and Near-extremal case φ 0 = 1 4 m 0 c Λ c = 1 ( ln c ), m 0 = 1 ( c Λ 4 Λ[ 1 + ln Λ) c ]. (24) Positivity of ADM mass [Park Strominger] Bound on the flux q 2 < 16πe. (25) m = m 0 + δm with δm/m 0 1 ( c ) δm Λ c Λ 1/2 4 3 δm c Λ + O φ h = 1 ln c δm 3/2. Λ (26) The horizon is pushed outward by adding a small amount of matter c T = δm 1/2. 2π Λ (27)

12 Dilaton Charge Conserved without equations of motion: topological µ j µ = ɛ µν [f (Φ) µ Φ ν Φ + f (Φ) µ ν Φ]. (28) D = Σ dσnµ j µ = φ 0 φ W dφ g φφ n t g tt ɛ tν ν f(φ) Canonical choice for f(φ): D = e 2Φ = φ 0 φ W dφ f (Φ). (29) D is the the function multiplying the Ricci scalar in the action.

13 Thermodynamical action Everything in term of observables at a Wall: T W and D W I onshell = M gλ + 2 M he 2Φ (K 2n a a Φ). (30) F = T W I, S = F T W, E = F + T W S, ψ = F D W. (31) I = β Λ(φ W φ h ) + βe c φ W ( 2l(φ W ) c + l (φ W )) = β Λ(φ W φ h ) β c e c φ W l(φ W ) Λ c e c φ W,(32) Tolmann Relation T W = T 1 l(φw ), (33) c T = 1 + Λ 4π c D 1 h, (34) m as a function of the position of the horizon m(φ h ) m(φ h ) = c D h + Λ ln D h, (35)

14 The On-Shell Action I W = 1 T Λ(φ W φ h ) D W T T W = T 1 + T 2 TW 2 + Λ. (36) c D W 1 l(φw ), (37) where T = c 4π 1 + Λ c D 1, (38) h The free energy: F = Λ c T W T ln D W D h c D W T W T 1 + T 2 TW 2 T and D h should be understood as functions of (T W, D W ) + Λ, (39) c D W

15 The zero RR flux limit[sanity Check] D h = D W Allows us to identify m(φ h ) as: 1 c 16π 2 T 2 W. (40) m(d W, T W ) = c D W c π 2 TW 2. (41) Thermodynamic properties: free energy, entropy and energy of the zero RR flux 2-d black hole [Gibbons, Perry]: F = 4π D W T W + S = 4π D W E = 8π D W 1 c 16π 2 T W c,, 16π 2 TW 2 c. (42) 16π 2 T W

16 The Extremal Black Hole δφ h = 4π T W c Evaluate the free energy F = T W T Λ ln( c D W c Λ φ h = 1 c ln 1 + Λ c D W ( c ) + δφ h. (43) Λ 1 + ln( cd W Λ ) 1/2. (44) ) c D W Λ Λδφ h T c D W c Independent of the temperature Robust entropy T W, (45) S = 4π Λ c = 1 4 q2. (46) The entropy of the extremal black hole with q units of electric and magnetic RR fluxes.

17 Energy and Mass E = 2 c D W Λ c ln D W + Λ c ln( Λ c ) 1. (47) Background Subtraction M = Λ c ln( Λ c ) 1 Coincides with ADM mass. Coincides with Matrix Model result q2 = 4π 2α ln q2 16π 1. (48)

18 Arbitrary Non-Extremal 0A Black Hole The general case: F = Λ c T W T ln D W D h c D W T W T 1 + T 2 TW 2 + Λ, (49) c D W D W, T W T, while keeping the temperature of the black hole T (and thus D h ) Free Energy F = 2 cd W αλ c ln( D W D h ) Λ c + O(D 1 W ). (50) DW (c(d W D h ) + Λ ln(d W /D h )) ( E = 2ΛcD W 4cΛD h + 2Λ 2 ln(d W /D h ) c 2 Dh 2 2ΛcDW + 3ΛcD Λ 2 h + Λ 2 (1 + α) ln(d h /D W ) αλcd h αλ 2). (51)

19 The limits D W E = 2 cd W + αλ c ln( D W D h ) α cd h (1+α) Λ c +O(D 1 W ) (52) M = cd h + Λ c ln D h = 4 c m (53) (T W, D W ) S = 4πD h (54) Thermodynamic Properties of 2-d Black Holes in Type 0A M T S F ψ Λ = 0 c cdh 4π 4πD h 0 2 c Λ 0 cd h + Λ 1 c ln D h 4π cd cd h + Λ 4πD h c Λ (ln D h 1) 2 c h

20 Matrix Models The matrix model for type 0A [DKKMMS]. The Jevicki-Yoneya model. The excitement E = V (λ) = λ 2 + q2 1/4 λ 2. (55) µ ɛρ(ɛ)dɛ 1 8π q2 ln q (56) Other quantities F = 1 8π q2 log q2 L + 1 [ 1 + (2πT ) 2 ] log q2 4 48π L... (57) 4 T temperature, L IR cut-off. The first non-vanishing contribution to the entropy is one-loop The Disagreement S = π 12 q2 T log L4. (58)

21 Vortex Condensation Another Candidate [Kazakov, Kostov, Kutasov] Summing over all possible SU(N) twists around the Euclidean time circle. Z N (β, λ) = r χ r (Ω ) Weyl character [DΩ]χr (Ω ) exp( n Z H r is the Hamiltonian in the representation r H r = P r N k=1 ( x k 1 ) 1 2 x2 k + 2 λ n tr(ω n ))T r r e βh r (59) i j τ r ijτ r ij (x i x j ) 2 (60) x i eigenvalues of the matrix with the inverted harmonic oscillator potential τ r ij SU(N) generators

22 Free Energy F = 1 1 2πR 4 (2 R)2 λ 4/(2 R) R + R 1 ln(λ 4/(2 R) ) + 48 h=2 (61) R radius of the compactified time circle, temperature 2πR = 1/T The genus zero contribution to the free energy is of the order f h (R)λ 4(1 h)/(2 R) 2 l (R 2) M 1 gs 2 (62) Nonsinglet sector gives a large entropy: S = β Hagedorn M +... (63) M 1 g 2 s Our calculation S = 4πD h with D h = 1/g 2 s Since the string coupling is related to the Order of magnitude agreement

23 AdS 2 in 2-d Type 0A AdS 2 Exist not only as a limit but also as a bona fide solution ds 2 = 1 φ 2 ( dt 2 + dφ 2) (64) Φ = Φ 0 (65) Apologies for effective action R = 8/α (66)

24 A Matrix Model for AdS 2 Strominger Type 0A potential V (λ) = λ 2 + q2 1/4 λ 2. (67) Flat Space AdS 2 Conjecture: Decoupling The AdS 2 solution is dual to the 1/λ 2 potential

25 The gravity side The GR calculation gives a vanishing result A candidate for black hole in global AdS 2 ds 2 = l 2 (1 br r 2 )dt 2 + l 2 (1 br + r 2 ) 1 dr 2 (68) Related to some identification of AdS 2 for some values of b [BTZ]

26 Outlook: The statistical entropy of 2-d black holes from matrix model. Implications for 4-d black holes How about the dynamics? Phase transitions?[gross-witten]