Masaki Shigemori. November 16 & 18, th Taiwan String Workshop

Transcription

1 Masaki Shigemori November 16 & 18, th Taiwan String Workshop

2 Plan 1. BH microstates 2. Microstate geom 3. Fuzzball conjecture & microstate geom program 4. Microstate geom in 5D 5. Double bubbling 6. Superstratum 2

3 Black hole microstates black hole = thermodyn. ensemble individual microstate?? ρ = i e βe i ψ i ψ i ψ i 3

4 Microstate geometries singularity horizon finite spread!! no horizon, no singularity In some cases, black hole microstates are described by smooth horizonless solutions of classical gravity 4

5 Fuzzball conjecture QG effects r H horizon singularity conventional picture fuzzball picture Can that be generally true? BH microstates are some stringy configurations spreading over a wide distance? 5

6 Microstate Geometry Program How much of black hole entropy can be accounted for by smooth, horizonless solutions of classical gravity? singularity horizon no horizon, no singularity 6

7 Superstrata A new class of microstate geometries for D1-D5-P 3-charge BH. Most general class with known CFT dual Explicitly constructed basic ones Need more general ones to reproduce BH entropy 7

8 1. Black hole microstates

9 Black holes event horizon singularity Solution to Einstein equations Boundary of no return: event horizon Spacetime breaks down at spacetime singularity Classical (macro) physics: Well understood Quantum (micro) physics: Various puzzles 9

10 BH entropy puzzle BH entropy: S BH = A 4G N A Stat mech: N micro = e S BH BH with solar mass: S BH Uniqueness theorem: only one BH solution N gravity = 1 10

11 BH entropy puzzle N gravity N micro = : huge discrepancy Cf. Cosmo. const. problem: Λ expected Λ observed = Where are the microstates? Uniqueness theorems Avoided in higher dimensions Need quantum gravity (string theory)? We don t really know QG. What to do? 11

12 AdS/CFT correspondence A way out : AdS/CFT string theory / gravity in AdS space quantum field theory (CFT) Defines string / QG Can in principle study BHs in full string / QG 12

13 BH microstate counting in AdS/CFT 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 Actually, SV predates AdS/CFT. Valid for all couplings for 2D CFT 13

14 BH microstates string theory / gravity in AdS space black hole quantum field theory (CFT) thermodyn. ensemble? microstate in gravity picture? Must be a state of quantum gravity / string theory in general individual microstate Ψ By gravity picture, I mean the bulk side of AdS/CFT and not classical gravity. Can be fully stringy / QG 14

15 Summary: We want to know the gravity picture of BH microstates! Again, by gravity picture, I don t mean classical gravity; it can be fully stringy / QG

16 2. Microstate geometries

17 Are any examples of gravity microstates known? They generally require full string theory Involves all string oscillators g μν, B μν, Φ, C are massless truncations

18 Yes! We know examples of microstates called microstate geometries. Solution of classical gravity (no massive string modes) Has same mass & charge as the BH Smooth & horizonless Many are susy, but some are non-susy

19 Example 1: LLM geometries [Lin-Lunin-Maldacena 2004] N D3-branes, 16 supersymmetries

20 = AdS 5 /SYM 4 in ½ BPS sector Sugra in AdS 5 xs 5 ½ BPS BH: superstar (singular, A = 0) D=4, N=4 SYM ½ BPS states (16 susy s) N D3-branes LLM (bubbling) geometries no stringy modes! N free fermions in harmonic potential H = 1 2 (p2 + x 2 ) p x At large N, the state is represented by droplets on the phase space 20

21 CFT side: free fermions ½ BPS states = N free fermions in harmonic potential H 1 particle = 1 2 (p2 + x 2 ) [Berenstein 04] [Corley+Jevicki+Ramgoolam 02] p p x x 1-particle coherent state large number (N) particles: droplets in 1-particle phase space 21

22 LLM geometries (1) ds 2 = h 2 dt + V 2 + h 2 dy 2 + dx dx ye G dω ye G 2 d Ω 3 h 2 = 2y cosh G e 2G 1/2 + z = 1/2 z [ y y (y 1 y )]z x 1, x 2, y = 0 y black: z white: z 1 2 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 22

23 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 y y V i = ε ij j z y i V j = ε ij y z F (5) = F 2 dω 3 + F 2 d Ω 3 F 2 = db t dt + V + B t dv + d B F 2 = d B t dt + V + B t dv + d B B t = 1 4 y2 e 2G B t = 1 4 y2 e 2G d B = 1 z + 1/2 4 y3 3 d( y 2 ) d B = 1 z 1/2 4 y3 3 d( y 2 ) 23

24 Smoothness (1) 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 x 2 [ y y (y 1 y )]z x 1, x 2, y = 0 Laplace eq explicitly solved: z x, y = y2 π z x, 0 d 2 x x x 2 + y 2 2 Behavior near the y=0 plane z = 1 2 z = 1 2 x 1 z = 1 2 at y = 0 z = 1 2 y2 c + x 2 +, e G yc + x, h 2 c + (x) y 0 behavior of S 3, S 3 : h 2 dy 2 + ye G dω ye G d Ω 3 3 c + x dy 2 + y 2 d Ω dω 3 2 c + (x) 24 S 3 shrinks smoothly S 3 remains finite (radius: 1/c + (x))

25 Smoothness (2) Behavior near the black/white boundary z = 1 2 x 2 z x, y = 1 2 x 2 x y 2 = ± 1 2 y2 4x z = 1 2 Exact expression: x 1 c ± x = 1 2 x 2 h 2 dy 2 + ye G dω ye G 3 d Ω 3 dy 2 = 2 x y + x y 2 x 2 d Ω x y x 2 dω 3 x 2 0: S 3 shrinks smoothly, S 3 remains finite On the black-white boundary, S 3 shrinks 25

26 Smoothness (3) Non-trivial S 5 S 5 S 5 F 5 S 3 shrinks as one approaches the boundary from inside S 3 S 3 S 3 Flux supports non-trivial S 5 S 3 over a disk is S 5 Flux through S 5 is proportional to the area of black region E.g. pure AdS 5 xs 5 x 2 x 1 N = area 4π 2 l p 4 l p = g s 1/4 ls 26

27 Flux quantization F (5) = F 2 dω 3 + F 2 d Ω 3 F 2 = db t dt + V + B t dv + d B F 2 = d B t dt + V + B t dv + d B B t = 1 4 y2 e 2G B t = 1 4 y2 e 2G d B = 1 z + 1/2 4 y3 3 d( y 2 ) d B = 1 z 1/2 4 y3 3 d( y 2 ) N = 1 2π 2 l p 4 d B = 1 8π 2 l p 4 Σ 2 y 3 3 d z + = Area z=1/2 4π 2 l p 4 y

28 Classical limit How is naive singular geometry (superstar) recovered? Bubble area quantized area = Nh, h = 4π 2 l p 4 = 4π 2 g s l s 4 cf. what enters metric: H D3 = 1 + 4πNg sl s 4 r 4 Classical limit: l p 0, N with fixed Q = Ng s l s 4 = Nh black & white black & white grayscale smooth smooth singular (superstar) 28

29 Quantization Why are we talking about classical sol ns and h at the same time? LLM solutions: basis on which quantization is carried out (solution space in gravity) = (phase space) quantum Hilbert space Each LLM solution is like x of a free particle q, p = iħ (Or, it s like classical orbit of electron in old quantum theory) [Grant+Maoz+Marsano+Papadodimas+Rychkov 05] 29

30 Comments Generic states have large curvature Higher derivative corrections non-negligible LLM sol s are soln s of two-derivative gravity and are not reliable But higher derivative corrections should not change qualitative picture; DoF must be the same (no massive stringy modes needed) smooth, but curvature large Smooth, no singularity to be resolved 30

31 Example 2: LM geometries [Lunin-Mathur 2001] [Lunin-Maldacena-Maoz 2002] N 1 D1-branes + N 5 D5-branes, 8 supersymmetries

32 = 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 X Σ n e in(σ τ) α n Parametrized by integers n 1, n 2, n 3, (8 susy s) k kn k = N 1 N 2 32

33 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 LM curve encodes how S 1 shrinks Smooth horizonless geometries supported by flux 1-to-1 correspondence with CFT states: F λ {n k } Fourier coeffs of F λ {n k } 33 Entropy reproduced geometrically: S N 1 N 2 [Rychkov 05] [Krishnan+Raju 15]

34 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 Q: what enters sugra solution N: quantized charge 34

35 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) The more branes you put, the larger the spread (this is against standard intuition!)

36 3. Fuzzball conjecture & microstate geometry program

37 Maybe the same is true for genuine black holes? BH microstates are some stringy configurations spreading over a wide distance? I m talking about the size, not about whether it s describable in classical gravity. R l P N α r H?? Again, this goes against standard intuition in gravity

38 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 There may be singularities allowed in string theory, but scattering is unitary 38

39 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 Sugra = all fields in massless sector of string theory. g μν, B μν, Φ, C, fermions Hope for susy (BPS) states Massive strings break susy Only massless (sugra) modes allowed? Example : MSW (wiggling M5) [Maldacena+Strominger+Witten 1997] LLM and LM are supporting evidence, although A = 0 39

40 Sugra fuzzballs (2) Are supersymmetric states any good? More tractable First order PDEs Can tell us about mechanism Mechanism for horizon-sized structure Basic string theory objects are locally susy 40

41 Sugra fuzzballs (3) Absolutely no hope for non-susy states? If something is described in sugra or not is a tricky question Cf. Macroscopic string Large # of quanta more relevant for classicality? The effect of massive modes captured by massless sector? Need all orders in α expansion? Some non-susy microstates known [Gubser+Klebanov+Polyakov 02] e.g. [Jejjala-Madden-Ross-Titchener 05] 41

42 Sugra fuzzballs (3) Caveats: Generic states have large curvature Higher derivative corrections nonnegligible But should not change qualitative picture; DoF must be the same Non-geometries Example in LLM: smooth, but curvature large Non-geometric microstates possible [Park+MS 2015] Need to extend framework (DFT, EFT) U-duality twist exotic supertube 42

43 Microstate geometry program: What portion of the BH entropy of supersymmetric BHs is accounted for by smooth, horizonless solutions of classical sugra? The answer may turn out to be 0, or 1. This is my definition here. Some other people want to construct microstates for non-susy ones too.

44 Comment: bottom up approach [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 More arguments based on Q info (monogamy, etc.) 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 44 I m not saying that bottom-up approach is bad because it can t explain the size. It s a feature of the approach.

45 Summary of the last lecture MGP works for certain systems: D3 LLM geom 2-charge sys (D1-D5 sys) LM geom However, they are not real BHs (A=0). Need to go to systems with a finite horizon to carry out MGP 45

46 Let s review a class of Finite horizon real BH microstate geometries, including their pros & cons. 5D microstate geometries: circa

47 4. Microstate geometries in 5D

48 3-charge system Susy BH in 5D (4 supercharges) Canonical rep [Strominger-Vafa 1996] IIB on S T N 1 D1 O ~ ~ ~ ~ N 2 D5 O O O O O N 3 P O ~ ~ ~ ~ Decoupling AdS 3 S 3 T 4 / D1-D5 CFT Macroscopic entropy: S N 1 N 2 N 3 48

49 4-charge system Susy BH in 4D / BS in 5D (4 supercharges) Canonical rep [Maldacena-Strominger-Witten 1997] M on 6 T A N 1 M5 ~ ~ O O O O O N 2 M5 O O ~ ~ O O O N 3 M5 O O O O ~ ~ O N 4 P ~ ~ ~ ~ ~ ~ O Decoupling AdS 3 S 2 T 6 / MSW CFT Macroscopic entropy: S N 1 N 2 N 3 N 4 49

50 M-theory frame Want to find gravity microstates for 3- & 4-charge systems Start from 3-charge system 5 IIB / T D1 5, D , P(5) T 5, T 6, T 7 5 IIA / T M / T 56789A D2 67, D2 89, F1(5) lift along x A M2 67, M2 89, M2(5A) Nicely symmetric 50 Take M-theory on T 6 and go to 5D 4-charge MSW system is already in M-theory frame

51 Ansatz 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) 51

52 5D theory 8 susys 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 X 1 X 2 X 3 is a hyper scalar 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 ) 52

53 No solitons without topology (1) Komar mass/smarr formula V μ = t : Killing [Gibbons-Warner 13] [Haas 14] used 2 K μ = R μν K ν M S 3 5 dv Σ 4 5 (V μ R μν dx ν ), if there is no internal boundary. S 3 Σ 4 t = const If there is a horizon, it will be an internal boundary and we get Smarr s formula relation M,S,T (for vacuum grav, R μν = 0 and no bulk contribution. 53

54 No solitons without topology (2) EOMs / Bianchi: df I = 0 dg I = 0, G I 5 Q IJ F J + C IJK F J A K R μν = Q IJ μ X I ν X J + Q IJ FI J μρ F ρ ν + Q IJ ρσ G I μρσ G J ν Assume time-independent config: (ignoring numerical factors) (*) L V X I = L V F I = L V G I = 0 d ι V F I = d ι V G I = 0 (used L V = dι V + ι V d) ι V F I = f I + (exact), ι V G I = g I + exact. H 1 (Σ 4 ): elec flux H 2 (Σ 4 ): mag flux Now contract (*) with K μ, and plug it into Komar integral 54

55 No solitons without topology (3) elec mag elec mag M Σ 4 f I G I + g I F I M can be topologically supported by crossing of elec & mag fluxes in the cohomology H Σ 4. No spatial topology M = 0 Spacetime is flat 55

56 BPS solutions Require susy [Gutowski-Reall 04] [Bena-Warner 04] 4D base B 4 (hyperkähler) ds 5 2 = Z 2 dt + k 2 + Z ds 4 2 A I = Z I 1 dt + k + B I, db I = Θ I * timelike class F g Θ f 0 elec mag Z = Z 1 Z 2 Z 3 1/3 ; X 1 = Z 2Z 3 Z 1 2 1/3 and cyclic All depend only on B 4 coordinates BPS eqs: linear system Θ I = 4 Θ I, 2 Z I = C IJK 4 Θ J Θ K dk = Z I Θ I 56

57 Sol ns with U(1) sym [Gutowski-Gauntlett 04] Solving BPS eqs in general is difficult. Assume U(1) symmetry in B 4 * tri-holomorphic U(1) 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 : da = 3 dv V = 0 V = v 0 + Multi-center KK monopole / Taub-NUT p v p r r p 57

58 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 58

59 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 59

60 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! 60

61 Komar mass/smarr relation Θ S 2 S 2 Θ F g Θ f 0 R 3 r 1 r 2 r 3 M F g Θ Θ Q elec Non-trivial flux supported by magnetic fluxes d F F F Electric flux sourced by crossing of magnetic fluxes 61

62 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 62

63 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., ] S Q 4 3 Q 2 Quantization of D6-D6-D0 config much less entropy 63

64 Further issues (1) Lifting [Dabholkar, Giuca, Murthy, Nampuri 09] moduli space free CFT point sugra point with B 2 = 0 Single-center BH exists everywhere Single-ctr BH exists everywhere and contributes to index (elliptic genus). Microstates must also exist everywhere and contribute to index. But >2 center solns do not contribute to index! They disappear when generic moduli are turned on? They are irrelevant for microstates? Cf. Moulting BH [Bena, Chowdhury, de Boer, El-Showk, MS 2011] Expect correspondence even for states that generically lift? 64

65 Further issues (2) Pure Higgs branch [Bena, Berkooz, de Boer, El-Showk, Van den Bleeken 12] Vacua of Quiver QM (scaling regime) Coulomb branch Corresponds to multi-center solutions Small entropy Generally J 0 Pure Higgs branch Corresponding sugra solution unclear Large entropy J = 0 Cf. supereggs 65 [Denef+Gaiotto+Stroinger+van den Bleeken+Yin 07] [Martinec 15] [Martinec+Niehoff 15] [Reaymaekers+van den Bleeken 15]

66 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

67 5. Double bubbling 2010

68 What are we missing? A guiding principle for constructing microstate geometries. Revisit better understood example: 2-charge system (LM geometries) 68

69 Supertube transition [Mateos+Townsend 2001] D0 + F1(1) D2 1λ F1(1) D0 polarize E B D2 1λ dipole charge x 1 x 1 λ Spontaneous polarization phenomenon Produces new dipole charge Represents genuine bound state Cross section = arbitrary curve (cf. Myers effect) 69

70 F1-P frame F1 9 + P(9) F1 λ x 9 F1(9) P polarize wiggly F1 x 1 x 8 λ To carry momentum, F1 must wiggle in transverse R 8 Projection onto transverse R 8 is an arbitrary curve 70

71 D1-D5 frame D1(5) + D KKM(λ6789,5) D1 D5 polarize KKM λ arbitrary curve 4 x = F λ R 1234 This is LM geometry Arbitrary curve large entropy S N 1 N 2 Explains origin of 2-charge microstate geometries 71

72 3-charge case Double bubbling D1 5 D P(5) KKM λ6789,5 D5 λ6789 D1(λ) θ, λ56789 KKM λ6789, θ (θ6789, λ) θ D1 D5 P arbitrary curve: supertube λ arbitrary surface: superstratum?? λ Multiple transitions can happen in principle Arbitrary surface larger entropy? Non-geometric in general [de Boer+MS 2010, 2012] [Bena+de Boer +Warner+MS 2011] 72

73 A geometric channel D1 5 D P(5) D5 λ6789 D1(λ) KKM λ6789, θ x 5 D1 D5 wiggly D1 wiggly D5 wiggly KKM = P geometric x 1 x 4 superstratum Dependence on x 5 is crucial Must live in 6D Possibility to recover S N 1 N 2 N 3 [Bena+ +Warner+MS 2014] 73

74 Two routes to superstratum D1 D5 polarize LM geom = straight superstratum add P add P wiggly D1 wiggly D5 polarize wiggly KKM = geometric superstratum 74

75 Summary: Existence of superstrata depending on functions of two variables is a necessary condition for S BH S geom

76 6. Microstate geometries in 6D (sugra superstratum) 2011

77 Sugra side

78 Goal: Explicitly construct superstrata or wiggly KKM in 6D They must depend on functions of two variables: F(v, w)

79 Susy solutions in 6D 4 IIB sugra on T 6789 No dependence on T 4 coordinates Require same susy as preserved by D1-D5-P Expected charges / dipole charges: D1(v) D5(v6789) P(v) D1(λ) D5(λ6789) u = t x5 2, v = t+x5 2 x 5 : compact KKM(λ6789, v) [Gutowski+Martelli+Reall 2003] [Cariglia+Mac Conamhna 2004] [Bena+Giusto+MS+Warner 2011] [Giusto+Martucci+Petrini+Russo 2013] 79

80 The sol n is characterized by scalars Z 1 D1(v) Z 2 D5(v6789) F P(v) Z 4 NS5(v6789)+F1(v) 2-forms Θ 1 D1(λ) Θ 2 D5(λ6789) Θ 4 NS5(λ6789)+F1(λ) 1-forms β KKM(λ6789, v) ω P(λ) 80

81 Explicit form of BPS solution ds 2 10 = 2α dv + β du + ω + 1 F dv + β Z Z 1 Z 2 2 1Z 2 ds 2 (B 4 ) + Z 1 ds 2 (T 4 ) Z 2 e 2Φ = αz 1 Z 2 α Z 1Z 2 Z 1 Z 2 Z 4 2 D d 4 β v v H 3 = du + ω dv + β D αz 4 Z 1 Z 2 αz 4 Z 1 Z 2 β + dv + β Θ 4 αz 4 Z 1 Z 2 Dω + αz 4 Z 1 Z 2 du + β Dβ + 4 (DZ 4 + Z 4 β) F 1 = D Z 4 Z 1 + dv + β v Z 4 Z 1 F 3 = du + ω dv + β D 1 Z 1 1 Z 1 β + αz 4 Z 1 Z 2 D Z 4 Z 1 + dv + β Θ 1 Z 4 Z 1 Θ 4 1 Z 1 Dω + 1 Z 1 du + β Dβ + 4 (DZ 2 + Z 2 β) Z 4 Z 1 4 (DZ 4 + Z 4 β) 81

82 0 th layer: 4D base 6D spacetime: u, v, x m 4D base B 4 (v) : almost hyper-kähler ds 2 (B 4 ) = h mn (x, v)dx m dx n, m, n = 1,2,3,4 β x, v : 1-form ( KKM) u: isometry v x 5 (compact) x m : 4D base 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 82

83 BPS equations First layer (Z, Θ) D 4 DZ 1 + βz 1 = Dβ Θ 2 DΘ 2 β Θ 2 = v 4 DZ 1 + βz 1 Θ 2 Z 1 ψ = 4 Θ 2 Z 1 ψ ψ 1 8 εabc J A mn (B) C J mnj Second layer (ω, F) Dω + FDβ = Z 1 4 Θ 1 + Z 2 Θ 2 Z Θ 4 4 D 4 L + 2 β i L i = Z 1 Z 2 + Z 1 Z 2 + Z 1 Z 2 Z 2 4 2Z 4 Z v Z 1 Z 2 Z 2 4 h mn h mn + 1 Z 2 2 1Z 2 Z 4 h mn h mn 1 2 hmn h np h pq h qm ( Θ 1 Z 2 ψ Θ 2 Z 1 ψ Θ 4 Z 4 ψ Θ 4 Z 4 ψ + Z 1Z2ψ ψ 2ψ Dω) α L ω + F β DF Linear if solved in the right order 83

84 Very complicated! Hard to find general superstrata Strategy: To prove concept, construct simple superstrata depending on functions of two variables [Bena-Giusto-Russo-MS-Warner 15]

85 Background (1) Starting point: simplest D1-D5 configuration (no P yet): circular LM geom = pure AdS 3 S 3 = round superstratum with no wiggle (yet) Circular LM profile v S 1 S 3 at the center of AdS 3 (previous naive picture) 85

86 Background (2) Circular profile: F 1 + if 2 = a exp 2πiλ/L S 3 Explicit solution: 86 Flat base (B 4 = R 4 ) ds 2 (R 4 ) = Σ Σ r 2 + a 2 cos 2 θ Other data: Z 1 = 1 + Q 1 Σ dr 2 r 2 + a 2 + dθ2 + r 2 + a 2 sin 2 θ dφ 2 + r 2 cos 2 θ dψ 2 Z 2 = 1 + Q 2 Σ Z 4 = F = Θ 1 = Θ 2 = Θ 4 = 0 β = R 5a 2 2Σ (sin2 θdφ cos 2 θdψ) ω = R 5a 2 2Σ (sin2 θdφ + cos 2 θdψ)

87 Putting momentum Now we want to add P Putting momentum deforms the round superstratum = S 3 by putting wiggles on it S 3 87

88 Linear fluctuation Certain linear solutions can be found by solution generating technique [Mathur+Saxena+Srivastava 2003] Z 4 = b R 5Δ km Σ cos v km Θ 4 = 2bmΔ km r sinθ Ω 1 sin v km + Ω 2 cos v km Δ km a r 2 + a 2 k sin k m θ cos m θ v km m 2 R 5 ds 2 B 4, Z 1,2,, β, ω, Θ 1,2 : unchanged at O(b) v + k m φ mψ v dependence (P) Depends on two params (k,m) CFT dual: descendants of chiral primary 88

89 How to get function of two variables Regard solution with (k, m) as Fourier modes on S 3 f S 3 = b k,m Y k,m k,m b km independent + S 3 SU 2 L SU 2 R BPS function of two variables! = Non-linearly complete to get genuine geometric superstratum + 89

90 Non-linear completion Use linear structure of BPS eqs to nonlinearly complete Assume 0 th data B 4, β are unchanged Regard Z 4, Θ 4 as non-linear sol n of 1 st layer D 4 DZ 4 = Dβ Θ 4 DΘ 4 = v 4 DZ 4 Find ω, F as non-linear sol n of 2 nd layer NL completion dω + Fdβ = Z 1 Θ 1 + Z 2 Θ 2 2Z 4 Θ 4 4 D 4 ω 1 2 df = Z 1 Z 2 + Z 1 Z 2 + Z 1 Z 2 Z 2 4 2Z 4 Z 4 Enough to do it for each pair of modes Regularity determines solution It also determines Z 1,2, Θ 1,2 90

91 Ex 1: k 1, m 1 = k 2, m 2 Z 4 b Δ k 1 m 1 Σ cos v k1 m 1, Z 2 : unchanged Z 1 b 2 Δ 2k1,2m 1 cos v 2k1,2m 1 F = 2m 2 (0,0) 1 F 2k1,2m 1 : needed to make ω regular ω = μ dψ + dφ + ζ(dψ dφ) undetermined 91 μ = R Δ 2k 1,2m 1 Σ ζ = + 2k 1 2m 2 0,0 1 F 2k1,2m m r2 + a 2 sin 2 θ 1 Σ Regularity ω = 0 at r = θ = 0 x = R ,0 F 2k1,2m 1 NL completed, with coeff fixed by regularity k 1 m x Σ

92 Ex 2: k 1, m 1 : any, k 2, m 2 = (1,0) Z 4 b 1 Δ k 1m1 Σ Z 1 b 1 b 2 F = 0 cos v k1 m 1 + b 2 Δ 10 Σ cos v 10, Δ k 1+1,m1 Σ ω = cω (1) + ω (2) ω (1) = R 5 2 Δ k 1 1,m 1 dr r r 2 +a 2 cos v k1 +1,m 1 + c Δ k1 1,m1 Σ undetermined cos v k1 1,m 1, sin v k 1 1,m 1 + sin2 θdφ+cos 2 θdψ cos v Σ k1 1,m 1 ω (2) = R 5 Δ k 1 1,m1 m 1 k 1 dr m 1 tanθdθ sin v 2 r 2 +a 2 k 1 r k k1 1,m r2 +a 2 Σ sin 2 θdφ + r2 +a 2 Σ cos 2 θ m 1 k 1 dψ cos v k1 1,m 1 Z 2 : unchanged Regularity ω = 0 at r = θ = 0 c = k 1 m 1 k 1 NL completed, with coeff fixed by regularity 92

93 CFT side

94 BOUNDARY CFT D1-D5 CFT d = 2, N = (4,4) SCFT Bosonic sym: SL 2, R L SU 2 L SL 2, R R SU 2 R Bosonic currents: T(z), J i z, T z, J i z L n Ji n L n Ji n Orbifold CFT There are component strings with total length N = N 1 N N 94

95 2-CHARGE STATES (1) Component strings come with flavors related to SU 2 L SU 2 R charge Round LM geom NS vacuum 1 all strings of length 1 95

96 2-CHARGE STATES (2) Linear fluct around circular LM single-trace chiral primary k 1 General LM geom general chiral primary various component strings 96

97 3-CHARGE STATES (1) P-carrying linear fluct around circular LM descendant of chiral primary + m J 1 k 1 + Single chiral primary acted on with J 1 Labeled by (k, m) State of a single supergraviton with quantum numbers (k, m) 97

98 3-CHARGE STATES (2) General P-carrying fluct around circular LM descendant of non-chiral primary n k1 m 1 strings n k2 m 2 strings + m J 1 1 k 1 + m J 1 + m J k 1 k 2 + J 1 m 2 k Various modes k, m excited with arbitrary amp. The most general microstate geometry with known CFT dual Individual J + 1 act independently State of supergraviton gas

99 Where are we?

100 Summary Superstratum depending of two variables Having modes with different k, m NL completion for pair of modes Succeeded in NL completion for various pairs of modes Constructive proof of existence of superstrata! Big step toward general 3-charge microstate geometries Correspond to non-chiral primaries in CFT Most general microstate geom with known CFT dual 100

101 Toward more general superstrata Does this class of superstrata reproduce S BH? No. These correspond to coherent states of graviton gas. Entropy is parametrically smaller. S geom N 3 4 N 1 S BH (N 1 N 5 N p J N) Need more general superstrata In CFT language, we only considered rigid generators of SU 1,1 2 L SU 1,1 2 R e.g. L 0, L 1, L 1, J 0 Need higher and fractional modes e.g. J 1 k They probably correspond to multiple superstrata 101

102 Multiple superstrata More generally, one has multiple S 3 s Can fluctuate each S 3 multi-superstratum Can use AdS 3 S 3 as local model Large redshift in scaling geometries entropy enhancement? S N 1? 102

103 Comment on issues Lifting Not directly applicable to 6D configuration Pure Higgs branch Superstratum reminiscent of Higgs branch Maybe only states that have J = 0 survive when moduli are turned on? 103

104 Conclusions

105 Conclusions Microstate geometry program Interesting enterprise elucidating micro nature of BHs, whether answer turns out to be yes or no Microstate geometries in 5D sugra Have properties expected from CFT, but too few Superstratum A new class of microstate geometries CFT duals precisely understood More general superstrata are crucial to reproduce S BH 105

106 Future directions 106 Superstratum More general solution, multi-strata Clarify issues (lifting, pure Higgs) Count states, reproduce entropy (or not) Non-geometric microstates Exotic branes, DFT Novel ways to store information More Non-extremal BHs Information paradox Observational consequences? Early universe

107 Thanks!

108 Appendix

109 Some formulas 109