Masaki Shigemori. September 10, 2015 Int l Workshop on Strings, Black Holes and Quantum Info Tohoku Forum for Creativity, Tohoku U

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1 Masaki Shigemori September 10, 2015 Int l Workshop on Strings, Black Holes and Quantum Info Tohoku Forum for Creativity, Tohoku U

2 Plan BH microstates Microstate geometries Fuzzball conjecture & microstate geom program Microstate geom in 5D Microstate geom in 6D Conclusions 2

3 Black hole microstates

4 Black holes event horizon singularity Solution to Einstein equations Boundary of no return: event horizon Spacetime breaks down at spacetime singularity 5

5 BH entropy puzzle BH entropy: S BH = A 4G N A Stat mech: N micro = e S BH Where are the microstates? Uniqueness theorems Need quantum gravity? 6

6 AdS/CFT correspondence string theory / gravity in AdS space black hole quantum field theory (CFT) thermodyn. ensemble S BH = A 4G N! = S CFT = log N micro [Strominger-Vafa 96] Stat mech interpretation of BH put on firm ground 7

7 BH microstates string theory / gravity in AdS space black hole quantum field theory (CFT) thermodyn. ensemble? microstate in gravity (bulk) picture? individual microstate Ψ Must be a state of quantum gravity / string theory in general 8

8 Summary: We want gravity picture of BH microstates!

9 Microstate geometries

10 Are examples of gravity microstates known? Yes! We know examples of microstates called microstate geometries. Solution of classical gravity Has same mass & charge as the BH Smooth & horizonless

11 Example 1: LLM geometries [Lin-Lunin-Maldacena 2004]

12 = LLM geometries (1) Sugra in AdS 5 xs 5 ½ BPS BH: superstar (singular, A = 0) D=4, N=4 SYM ½ BPS states fermion gas in harmonic potential H = 1 2 (p2 + x 2 ) p LLM (bubbling) geometries x 13

13 LLM geometries (2) ds 2 = h 2 dt + V 2 + h 2 dy 2 + dx dx ye G dω ye G 2 d Ω 3 e 2G 1/2 + z h 2 = 2y cosh G = 1/2 z [ y y (y 1 y )]z x 1, x 2, y = 0 z S 3 S 3 LLM diagram encodes how S 3 s shrink Smooth horizonless geometries Non-trivial topology supported by flux no uniqueness thm in 10D 1-to-1 correspondence with coherent states in CFT 14

14 Classical limit How is naive singular geometry (superstar) recovered? Bubble area quantized area = 4π 2 l 4 p N, h = 4π 2 4 l p Classical limit: l p 0, N black & white black & white grayscale smooth smooth singular (superstar) 15

15 Example 2: LM geometries [Lunin-Mathur 2001] [Lunin-Maldacena-Maoz 2002]

16 = LM geometries (1) Sugra in AdS 3 xs 3 2-charge BH (singular, A = 0) N 1 D1-branes N 2 D5-branes LM geometries D=2, N=(4,4) CFT ½ BPS states free bosons in 2D Parametrized by integers n 1, n 2, n 3, k kn k = N 1 N 2 17

17 LM geometries (2) ds 2 = dv + β du + ω + Z 1 Z 2 dx Z 1 /Z 2 dx 6789 Z 1 Z 2 Z 1 ( x) = 1 + Q L 2 F 2 dλ L 0 x F λ, Z 2( x) = 1 + Q L 2 dλ 2 L 0 x F λ 2 arbitrary curve 4 x = F λ R 1234 λ v S 1 Fourier coeffs of F λ {n k } LM curve encodes how S 1 shrinks Smooth horizonless geometries supported by flux 1-to-1 correspondence with CFT states: F λ {n k } Entropy reproduced geometrically: S N 1 N 2 18

18 Classical limit How is naive singular geometry recovered? smooth classical limit singular R g s 1/3 ls 1/3 Q1 Q 2 1/6 g s 2/3 ls N 1 N 2 1/6 g s 0, l s 0, N 1,2 Fix Q 1,2 g s l s 2 N 1,2 19

19 Summary: Some BH microstates are represented by microstate geometries. Naive BH solutions are replaced by bubbling geometries with finite spread. R l P N α (but recall A = 0 so far)

20 Fuzzball conjecture & microstate geometry program

21 Maybe the same is true for genuine black holes? BH microstates are some stringy configurations spreading over a wide distance? R l P N α r H??

22 Fuzzball conjecture Mathur ~2001: QG effects r H horizon singularity conventional picture fuzzball picture BH microstates = QG/stringy fuzzballs No horizon, no singularity Spread over horizon scale 23

23 Sugra fuzzballs (1) Are fuzzballs describable in sugra? Unlikely in general General fuzzballs must involve all string modes Massive string modes are not in sugra Hope for supersymmetric states Massive strings break susy Only massless (sugra) modes allowed? Example : MSW (wiggling M5) [Maldacena+Strominger+Witten 1997] 24

24 Sugra fuzzballs (2) Are supersymmetric states any good? More tractable First order PDEs Can tell us about mechanism Mechanism for horizon-sized structure String theory objects are locally susy 25

25 Sugra fuzzballs (3) Caveats: Generic states have large curvature Higher derivative corrections nonnegligible But qualitative picture must be robust; DoF must be the same (cf. LLM) Non-geometries smooth, but curvature large Non-geometric microstates possible [Park+MS 2015] Need to extend framework (DFT, EFT) U-duality twist exotic supertube 26

26 Microstate geometry program: What portion of the BH entropy of (supersymmetric) BHs is accounted for by smooth, horizonless solutions of classical sugra? no horizon, no singularity

27 Comment: bottom-up vs. top-down [Mathur 09] O(1) deviation from flat space is needed for Hawking radiation to carry information Based on Q info (strong subadditivity) [AMPS 12] Firewall Same result, same Q info (monogamy etc.) 28 These arguments are bottom-up Mechanism to support finite size not explained Microstate geometry program is top-down Finite size supported by topology with fluxes

28 Microstate geometries in 5D

29 Setup D = 5, N = 1 sugra with 2 vector multiplets Action gauge fields: A μ I, I = 1,2,3. F I da I. scalars: X I, X 1 X 2 X 3 = 1 S bos = ( 5 R Q IJ dx I 5 dx I Q IJ F I 5 F J 1 6 C IJKF I F J A K ) Chern-Simons interaction C IJK = ε IJK, Q IJ = 1 2 diag(1/x1, 1/X 2, 1/X 3 ) 30

30 11D interpretation 6 M-theory on T 56789A A = 10 ds 2 11 = ds X 1 dx dx 6 + X 2 dx dx X 3 (dx dx A 2 ) A 3 = A 1 dx 5 dx 6 + A 2 dx 7 dx 8 + A 3 dx 9 dx A M2(56) M2(78) M2(9A) M5(λ789A) M5(λ569A) M5(λ5678) 31

31 BPS solutions [Gutowski-Reall 04] [Bena-Warner 04] Require susy 32 ds 5 2 = Z 2 dt + k 2 + Z ds 4 2 Linear system A I = Z I 1 dt + k + B I, elec Z = Z 1 Z 2 Z 3 1/3 ; X 1 = Z 2Z 3 Z 1 2 Θ I = 4 Θ I, 2 Z I = C IJK dk = Z I Θ I mag 4D base B 4 (hyperkähler) db I = Θ I All depends only on B 4 coordinates Θ J Θ K 1/3 and cyclic

32 Sol ns with U(1) sym [Gutowski-Gauntlett 04] Solving eqs in general is difficult. Assume U(1) symmetry in B 4 flat R 3 ds 2 4 = V 1 dψ + A 2 + V dy dy dy 2 3, (Gibbons-Hawking space) V is harmonic in R 3 : V = v 0 + p v p r r p 33

33 Complete solution All eqs solved in terms of harmonic functions in R 3 : H = V, K I, L I, M, H = h + p Q p r r p Θ I = d KI V (dψ + A) V 3 d KI V Z I = L I + 1 2V C IJKK J K K k = μ dψ + A + ω μ = M + 1 2V KI L I + 1 6V 2 C IJKK I K J K K 3 dω = VdM MdV KI dl I L I dk I 34

34 Multi-center solution H = V, K I, L I, M, H = h + p Q p r r p KK monopole mag (M5) elec (M2) KK momentum along ψ Multi-center config of BHs & BRs in 5D Positions r p satisfy bubbling eq (force balance) Reducing on ψ gives 4D BHs (same as Bates-Denef 2003) Q 1 Q 2 Q 3 R 3 r 1 r 2 r 3 35

35 Microstate geometries (1) Tune charges: l p I = C IJK 2 m p = C IJK 12 k p J k p K v p k p I k p J k p K v p 2 Smooth horizonless solutions [Bena-Warner 2006] [Berglund-Gimon-Levi 2006] S 2 S 2 flux R 3 r 1 r 2 r 3 Microstate geometries for 5D (and 4D) BHs Same asymptotic charges as BHs Topology & fluxes support the soliton Mechanism to support horizon-sized structure! 36

36 Microstate geometries (2) Various nice properties Scaling solutions [BW et al., 2006, 2007] Quantum effect cuts off the throat at finite depth Gap expected from CFT: ΔE 1 c 37

37 The real question: Are there enough? 3-chage sys (+ fluctuating supertube) Entropy enhancement mechanism [BW et al., 2008] Much more entropy? An estimate [BW et al., 2010] S Q 5 4 Q 3 2 Parametrically smaller supertube 4-chage sys [de Boer et al., ] Quantization of D6-D6-D0 config much less entropy 38

38 Summary: We found microstate geometries for genuine BHs, but they are too few. Possibilities: A) Sugra is not enough B) Need more general ansatz this talk

39 Microstate geometries in 6D

40 New hope 5D microstate geometries are not enough String theory and AdS/CFT suggest: There are solutions fluctuating along 6 th direction They are parametrized by functions of 2 variables superstratum [Bena, de Boer, Warner, MS ] Look for superstrata in 6D sugra! Can use AdS 3 /CFT 2 as guide: IIB on AdS 3 S 3 T 4 2D CFT (D1-D5 CFT) 43

41 6D sugra 6D N = 2 sugra with a vector multiplet Bosonic fields Metric g μν Dilaton φ 2-form B 2, field strength G 3 = db 2 4 IIB on T 6789 : D1 5 1-brane coupled to B 2 D brane coupled to B 2 44

42 Susy sol n (1): Base [Bena-Giutso-MS-Warner 11] 6D spacetime: u, v, x m u: isometry, v x 5 x m : 4D base 4D base B 4 (v) : almost hyper-kähler ds 4 2 = h mn (x, v)dx m dx n, m, n = 1,2,3,4 β x, v : 1-form ( KKM) J (A) x, v, A = 1,2,3 : almost HK 2-forms J A m B n J n p = ε ABC C J m p δ AB δ p m d 4 J (A) = v β J (A), D d 4 β v 45

43 Susy sol n (2): Fields Fields on B 4 Z 1 : scalar D1(v) Θ 1 : 2-form D1(λ) ω: 1-form J Z 2 : scalar D5(v6789) Θ 2 : 2-form D5(λ6789) F: scalar P(v) 6D fields ds 6 2 = 2 Z 1 Z 2 dv + β du + ω F dv + β Z 1Z 2 ds 4 2 G 3 = d[ 1 2 Z 1 1 du + ω dv + β ] DZ 2 + βz 2 + dv + β Θ 1 e 2φ = Z 1 /Z 2 46

44 Susy sol n (3): Linear structure First layer (Z, Θ) D 4 DZ I + βz I + 2Dβ Θ J = 0 DΘ J 1 β Θ J v 2 4 DZ I + βz I = 0 Second layer (F, ω) I, J = {1,2} v 4 D 4 L = Z 1 Z 2 + Z 1 Z 2 + Z 1 Z v Z 1 Z 2 h mn h mn + 1 Z 2 1Z 2 h mn h mn 1 2 hmn h np h pq h qm 2 β m L m 2 4 (Θ 1 Θ 2 ψ Dω) Dω = 2 Z 1 Θ 1 + Z 2 Θ 2 FDβ 4Z 1 Z 2 ψ L ω F β 1 2 DF ψ = 1 16 εabc A mn (B) C J J mnj Linear if solved in the right order 47

45 Superstratum around AdS 3 S 3 (1) Easiest to start from simplest background: AdS 3 S 3 [Bena-Giusto-Russo-MS-Warner 15] AdS/CFT dictionary for linear fluctuation known [Deger et al. 98] Correspond to descendants of chiral primaries in CFT Labeled by 3 quantum numbers (k, m, n) Supergraviton gas 48

46 Superstratum around AdS 3 S 3 (2) 49 Can use linear structure of 6D eqs to nonlinearly complete it Superposing multiple modes Sol s parametrized by funcs of 3 variables + = Correspond to non-chiral primaries in CFT most general microstate geom with CFT dual known!

47 What s missing? Does this class of superstrata reproduce S BH? Not yet These correspond to supergraviton gas = fluct around S 3. Entropy parametrically smaller. [de Boer 98] S 3 50

48 More general superstrata Next steps: Other backgrounds multiple S 3 s, Z k orbifolds CFT side: Need higher and fractional modes of SL(2, R) L SU 2 L S 3 S 3 S 3 multisuperstratum S 3 /Z k + J m 1 ψ + ψ J 1 J 2 + J 2 k k + ψ 51

49 Conclusions

50 Conclusions Microstate geometry program Interesting enterprise elucidating micro nature of BHs, whether answer turns out to be yes or no Microstate geom in 5D Have properties expected from CFT, but too few 6D: superstrata A new class of microstate geometries CFT duals precisely understood More general superstrata are crucial to reproduce S BH 53

51 Future directions 54 Superstratum More More general solution, multi-strata Count states, reproduce entropy (or not) Non-geometric microstates (exotic branes, DFT/EFT) Non-extremal BHs Information paradox Observational consequences? Early universe