On Fuzzball conjecture Seiji Terashima (YITP, Kyoto) based on the work (PRD (2008), arXiv: ) in collaboration with Noriaki Ogawa (YITP) 2009 Feb 19 at Kinosaki

2 1. Introduction

3 In general relativity (or classical gravity): Black hole solutions exist and behave like blackbody: Thermodynamics d E = T d S δS ≥ 0 Blackbody radiation Black hole d M = κd A δA ≥ 0 Hawking radiation M: Mass of B.H. A: Area of horizon κ: surface gravity

4 Irreversibility ↔ horizon. No information inside the B.H. can escape outside the horizon. Information loss problem (quantum theory should be unitary)

5 Anyway, black hole is a classical solution, but thermodynamic object! General relativity will be just an effective action, and there will be micro states for the black hole, perhaps quantum mechanically.

6 String Theory - is well defined and understood perturbatively - is useful for Mathematics (ex. Mirror symmetry) - is applied to the QCD (ex. Holographic QCD) - can be applied to Particle Phenomenology and can be the Theory Of Everything. - includes Quantum Gravity Remember why String Theory is interesting:

7 Indeed, in string theory we can find the microstates of B.H. and show: log (# of microstates)= Area of B.H. for some BPS B.H. Strominger-Vafa using the Gauge/Gravity duality.

8 a Puzzle remains: Near the horizon, classical general relativity seems valid because curvature is small and the quantum effects would be confined within the plank length near the singularity Information loss? Inconsistency?

9 Fuzzball conjecture: “Black Hole” does not exist quantum mechanically (in string theory). (There is no horizon in quantum gravity.) Instead of black hole, somethings like fuzzball( 毛玉） exist. Mathur and his collaborators

10 Fuzzball (has complicated topololgy) Black hole Corse graining （粗視化）

11 B.H. is macroscopic object and has very large number of degree of freedom, N. Quantum effects will spread out by the large N effect. → 1. Macroscopic “hozizon” appear 2. Quantum effect near horizon ! Key point:

12 In supergravity, some classical solutions without horizon and singularity which are approximately same as the B.H. outside “horizon” were found (fuzzball solution). → “horizon” appear as approximate notion Note 1: the fuzzball in general are not represented as classical solution. Only for special cases. an evidence:

13 Note 2: BPS properties are essential to find the solutions. BPS equations is linear → reduced to one like Laplace equation → infinite sum of the solutions are also solutions →Fuzzball solutions No hair “theorem” is terribly violated

14 2. Half BPS Fuzzball in AdS5xS5

15 AdS5/CFT4 AdS5 x S5 ↔ N=4 SU(N) SYM restrict to ½ BPS sector Lin-Lunin-Maldacena ↔ N free fermions in (LLM)geometry harmonic oscillator potential

16 Other parts

17 AdS5x S5 solution In the coordinate, the metric becomes AdS5xS5 geometry with in the global coordinate

18 [Note] fermion distribution can be defined for special class of state in Hilbert space, i.e. semi-classical or “coherent states” spanning the base of Hilbert space

19 Pauli’s exclusion principle means for CFT side. LLM geometry have Closed Time like Curve for outside this region. Thus these solutions will be unphysical in quantum theory. LLM geometry is smooth for This phase space us quantized by

20 Thus, in this ½ BPS sectors of AdS5xS5 example, all microstates are represented by smooth supergravity solutions, which are the fuzzballs! More presicely, all states of a basis of microstates are represented by supergravity solutions.

21 Smooth geometry ↔ pure states in CFT Interpretation of the singular geometries in LLM is Mixed/coarse grained states in CFT. Singular geom. = “gray droplets” Superstar=simplest gray droplets =uniform disk Balasbramnian et.al.

22 Suparstar has naked singularity without horizon, but we expect it will have horizon after including quatum effects 5d N=2 supergravity reduced form is BPS “black hole”, but not extremal. If the horizon is at ζ,

23 Summary of our work: “Coarse-graining of bubbling geometries and the fuzzball conjecture” Consider charged “B.H.” called “Superstar” in AdS space-time Estimate the entropy of superstar, as log(# of microstates) Estimate the "horizon" size of the superstar, based on the fuzzball conjecture, from gravity side. We find the Bekenstein-Hawking entropy computed from this "horizon" agrees with log(# of microstates) Thus, this result supports the fuzzball conjecture.

24 Entropy = log of number of states which are not distinguishable from one another by macroscopic observations. So, what is “macroscopic observer”? We assume that One can only measure physical quantities up to If metric perturbation is bigger than We can detect two geometries are different

25 Consider two similar LLM geometries with fermion distributions and and their difference

26 Then, we can compute the entropy of the superstar (and other non-smooth geometries) by counting the number of possible microstates as

27 Now we observe region very close to the fuzzball and see where is “horizon”. Here the “horizon” is the surface of the region where the typical microstates are different each other. After some computation, we find “horizon” is at or in the coordinate for the superstar. Therefore, according to previous formula, we have We take with very small α

28 Conclusions

29 Summary –For some BPS black holes we find there are fuzzball solutions in super gravity, which are smooth and has no horizon. –These will correspond to the microstates of B.H. Future directions –Including higher derivative corrections –Non BPS, non extremal

30 Fin.