String Theory, QCD and Black Holes

Black Hole Microstate Structure Fuzzballs and Firewalls Iosif Bena

IPhT, CEAEA Saclay

with Nick Warner, Jan deBoer, Micha Berkooz, Simon Ross, Gianguido Dall’Agata, Stefano Giusto, Masaki Shigemori, Dieter van den Bleeken, Monica Guica, Sheer El-Showk, Stanislav Kuperstein, Hagen Triendl, Bert Vercnocke, Andrea Puhm,

Ref. Ares(2012)790792 - 29/06/2012 ! Department C "Grant Management"

! Mittwoch, 12. März 2014

Subject: Aggressive marketing from publishing houses

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Mechtild May Head of Department C a.i. Black Holes Bekenstein & Hawking: Behave like thermodynamic objects Temperature, entropy:

S ~ 10 77 solar mass BH S ~ 10 90 BH at center of Milky Way

Mittwoch, 12. März 2014 Black Holes Bekenstein & Hawking: Behave like thermodynamic objects Temperature, entropy:

S ~ 10 77 solar mass BH S ~ 10 90 BH at center of Milky Way

Quantum 10 90 10000000 … 00000 Mechanics: e = e states

Mittwoch, 12. März 2014 Black Holes Bekenstein & Hawking: Behave like thermodynamic objects Temperature, entropy:

S ~ 10 77 solar mass BH S ~ 10 90 BH at center of Milky Way

Quantum 10 90 10000000 … 00000 Mechanics: e = e states

General HAIR Relativity: 1 big fat state

Information paradox

Mittwoch, 12. März 2014 WHERE ARE THE STATES ??? HOW DO THEY LOOK ??? • Very hard question – Quantum Gravity • Simpler question: – Count black hole states in any other way ? Strominger and Vafa (1996) + 1792 other articles

Strings and Black Hole Branes Finite Gravity Zero Gravity

Mittwoch, 12. März 2014 Ø Count quantum states at zero gravity Ø Entropy matches black hole classical horizon area !!! Ø 2 absolutely different calculations (Cardy Formula vs. classical area) Ø Amazing success Ø Modular forms, hypergeometric, other beasts Ø Unmatched in other theories of gravity String-QCD-BH

STATISTICAL CFT ENSEMBLE STATES MECHANICS (Boundary)

another way STROMINGER ! VAFA FILLPRESENT to understand: ENTROPY MATCHING WORK AdS-CFT GEOMETRIES Gravity BLACK HOLE Correspondence WITH NO HORIZON (Bulk)

FigureMittwoch, 1: 12. MärzAn 2014 illustrative description of this picture of black holes. My research focuses on constructing more microstate geometries with no horizon, and on improving the dictionary between the existing ones and the states of the CFT (the dashed vertical arrow). paradox: microstates have unitary physics, and thus information is not lost. Second, since the maximal entropy in a region of space comes from microstates that have the same size as the would-be black hole, this would prove ’t Hooft’s holographic principle. Third, these microstates will appear whenever we have a large-enough energy density, and it is quite likely that their physics will be dominant in cosmological settings, like in the Big Bang and the Big Crunch singularities. To use an analogy from classical physics, one can say this picture could revolutionize quantum gravity and the physics of black holes in the same way in which statistical physics revolutionized the understanding of thermodynamics. Furthermore, this picture of black holes might also be experimentally testable with the gravity wave detector LISA or if black holes are found at the Large Hadron Collider. It is therefore a crucial problem in quantum gravity to establish whether this picture is correct. In previous work I have taken quite a few important steps in this direction by constructing and analyzing huge families of black hole microstates, both in string theory and in supergravity. In the future, I believe there are two directions in this research programme that both have a good shot at proving or disproving this revolutionary picture of black holes. The first is to construct a precise map between the states of the dual boundary theory and the solutions we have constructed. Per Kraus and I have been the first to describe black rings in this CFT, and I believe I have some, and I can master the other tools needed to successfully attack this problem. Once this map is obtained, I intend to find the bulk geometries that correspond to the typical states of the CFT. These geometries would be then the typical microstates of the black hole; if they are horizonless, this would give a proof that the black hole is a thermodynamic description of an ensemble of horizonless configurations. The second direction is to construct more generic three-charge solutions, that have black hole charges and depend on several continuous functions. The solutions N. Warner and I have constructed have the appropriate charges, but do not depend on arbitrary functions. Other groups (including my present postdoc, A. Saxena) have on the other hand constructed solutions that depend on arbitrary functions, but do not have black hole charges. I think our methods can be combined to build these more generic solutions. We can then count them and see if they can account for the entropy of the black hole. If they do, this would again establish that black holes are ensembles of horizonless configurations. Another direction that is important to pursue in longer term is the construction of microstates of near-BPS and non-BPS black holes. Most of the effort in this field has so far concentrated on constructing and analyzing microstates of BPS (supersymmetric) black holes. This is enough for the purpose of establishing this picture of black holes: if one can prove that BPS black holes are ensembles of horizonless microstates, then it will be rather unlikely that non-BPS black holes will not have the same description. However, if one is to use this picture Strominger and Vafa (1996): Count Black Hole Microstates (branes + strings) Correctly match B.H. entropy !!! Zero Gravity

Black hole regime of parameters:

Standard lore: As gravity becomes stronger, - brane configuration becomes smaller - horizon develops and engulfs it Susskind Horowitz, Polchinski - recover standard black hole Damour, Veneziano

Mittwoch, 12. März 2014 Strominger and Vafa (1996): Count Black Hole Microstates (branes + strings) Correctly match B.H. entropy !!! Zero Gravity

Black hole regime of parameters:

Identical to black hole far away. Horizon → Smooth cap

Giusto, Mathur, Saxena Bena, Warner Berglund, Gimon, Levi

Mittwoch, 12. März 2014 BIG QUESTION: Are all black hole microstates becoming geometries with no horizon ? ? Black hole = ensemble of horizonless microstates Mathur & friends

Mittwoch, 12. März 2014 Analogy with ideal gas

Thermodynamics Statistical Physics (Air = ideal gas) (Air -- molecules) S P V = n R T e microstates dE = T dS + P dV typical atypical

Thermodynamics Statistical Physics Black Hole Solution Microstate geometries

Long distance physics Physics at horizon Gravitational lensing Information loss

Mittwoch, 12. März 2014 Other formulations: e.g. Bena, Warner, 2007

- Thermodynamics (EF T) breaks down at horizon. New low-mass d.o.f. kick in. - No spacetime inside black holes. Quantum superposition of microstate geometries.

Not some hand-waving idea - provable by rigorous calculations in String Theory

Mittwoch, 12. März 2014 Other formulations: e.g. Bena, Warner, 2007

- Thermodynamics (EF T) breaks down at horizon. New low-mass d.o.f. kick in. Highly Unusual - No spacetime inside black holes. Quantumin this field superposition of microstate geometries. ☺

Not some hand-waving idea - provable by rigorous calculations in String Theory

Mittwoch, 12. März 2014 Word of caution • To replace classical BH by BH-sized object – Gravastar – Infinite density firewall hovering above horizon – LQG configuration – Quark-star, you name it … – satisfy 2 very stringent tests: Horowitz

1. Same growth with GN !!!

• BH size grows with GN • Size of objects in other theories becomes smaller

1. Same growth with GN !!!

• BH size grows with GN • Size of objects in other theories becomes smaller - BH microstate geometries pass this test - Highly nontrivial mechanism: - D-brane tension ~ 1/gs ➙ lighter and fluffier as GN increases

Mittwoch, 12. März 2014 2. Mechanism not to fall into BH

Mittwoch, 12. März 2014 2. Mechanism not to fall into BH Very difficult !!!

Mittwoch, 12. März 2014 2. Mechanism not to fall into BH Very difficult !!! General Relativity Dogma: Thou shalt not put anything at the horizon !!!

- Horizon is null - Must go at speed of light. - If massive: ∞ boost ➙ ∞ energy - If massless: dilutes with time - Nothing can live there ! (nor carry degrees of freedom) - No membrane - No (fire)wall, wave sent by Bob

Mittwoch, 12. März 2014 2. Mechanism not to fall into BH Very difficult !!! General Relativity Dogma: Thou shalt not put anything at the horizon !!!

Mittwoch, 12. März 2014 Microstate geometries

3-charge 5D black hole Strominger, Vafa; BMPV

Want solutions with same asymptotics, but no horizon

Mittwoch, 12. März 2014 Microstate geometries

Bena, Warner Gutowski, Reall

Mittwoch, 12. März 2014 BPS Microstates geometries - 11D SUGRA / T6

5 D 3-charge BH (Strominger-Vafa)

Linear system 4 layers: Bena, Warner Gutowski, Reall

Focus on Gibbons-Hawking (Taub-NUT) base:

8 harmonic functions

Gauntlett, Gutowski, Bena, Kraus, Warner

Mittwoch, 12. März 2014 Simplest Microstate Geometries

Multi-center Taub-NUT: many 2-cycles + flux

Completely smooth (@ Taub-NUT centers geometry ~ R4) No singular sources or horizons Same mass, charge, size as BH with large horizon area Lots and lots of solutions !

Mittwoch, 12. März 2014 Microstates geometries

• Where is the BH charge ? 2-cycles + magnetic flux

L = q A0 magnetic

L = … + A0 F12 F34 + … • Where is the BH mass ? 12 E = … + F12 F + … Bubbling Geometries • BH angular momentum Black Hole Solitons GR story behind Gibbons, Warner J = E x B = … + F01 F12 + …

The charge is dissolved in magnetic fluxes. No singular sources. Klebanov-Strassler

Mittwoch, 12. März 2014 Four Scales

• Classical BH has 2 scales: – Mass / Horizon Size – Planck Length

Mittwoch, 12. März 2014 The important conclusion here is that besides Chern-Simons terms and cohomological fluxes, there is no other mechanism that can support a stationary microstate geometry. In a generic black hole this will require a mixture of self-dual and anti-self dual fluxes and so if one is to construct a stationary Schwarzschild microstate this would require new classes of long-lived “flux-anti-flux” solutions. This could present a very interesting set of new challenges to the solution generating tech- niques that come from inverse scattering methods. Based upon the structure of BPS solutions, we expect to find very large families of axisymmetric, non-extremal microstate geometries, which depend on several functions of two variables. Their construction can be thus reduced to an effec- tively two-dimensional problem that should be amenable to inverse scattering methods applied to the corresponding equivalent scalar coset models. [?, ?]. However, as we have seen, the construction of these geometries critically depends upon the existence of topological terms, whose incorporation in this technology has proven strenuous so far. Once this barrier is overcome, we expect finding these rich families of axisymmetric, non-extremal microstate geometries to be within reach. Finally, independent of the entire fuzzball proposal, this supergravity analysis suggests another possible end-point of stellar collapse: Topological Stars, supported by higher-dimensional magnetic fluxes. It is important to realize that the topological cycles necessarily require extra, compactified dimensions of space-time and so the structure of such a star can only appear at the scale of the extra dimensions. Through the effects of warp factors, it is possible that the scale of (some of) these extra dimensions could become much larger near the star than they are at infinity. The fact that these stars are made of flux and anti-flux suggests that they will slowly decay. In the fuzzball program we suspect that this will be how fuzzballs generate Hawking radiation but, more generally, it will mean that topological stars will still emit radiation. We will discuss this further in Section 8.

3.2 Geometric transitions and new black-hole scales In order to characterize the microstate solutions, and to understand the features one expects from them if they are to describe the typicalFour Scales microstates of the black hole, it is useful to discuss several scales one encounters in their physics. First, one can define the scale λT of the geometric transition that is responsible• Classical for replacing BH has certain2 scales: singular brane sources with smooth geometry, or otherwise the typical size– ofMass the / Horizon homology Size cycles, or bubbles in a microstate solution. This – Planck Length scale is determined by a balance between gravity that tends to collapse the bubbles and the flux • Microstate geometries have 2 more zmax that tends to drive their expansion.– Redshift from For the BPS bottom and of the extremal bubbles we typically find that throat, zmax – Size of bubbles: λ k (3.4) T ∼ P 4 where k is the number of flux quanta threading the typical cycle30 and P is the Planck length . After the geometric transition, the entire region where the black hole would have formed is now populated by a clusterMittwoch, 12. März of 2014 homology cycles and fluxes whose overall size is approximately that of what would have been the black hole. Thus one would expect that the number of cycles,

4In a BPS microstate geometry with a long throat that corresponds to a four-dimensional inter-center distance of order d, the warp factors scale as Z k2/d, V 1/d and hence λ2 d2(Z Z Z )1/3V . ∼ ∼ T ∼ 1 2 3

10 The important conclusion here is that besides Chern-Simons terms and cohomological fluxes, there is no other mechanism that can support a stationary microstate geometry. In a generic black hole this will require a mixture of self-dual and anti-self dual fluxes and so if one is to construct a stationary Schwarzschild microstate this would require new classes of long-lived “flux-anti-flux” solutions. This could present a very interesting set of new challenges to the solution generating tech- niques that come from inverse scattering methods. Based upon the structure of BPS solutions, we expect to find very large families of axisymmetric, non-extremal microstate geometries, which depend on several functions of two variables. Their construction can be thus reduced to an effec- tively two-dimensional problem that should be amenable to inverse scattering methods applied to the corresponding equivalent scalar coset models. [?, ?]. However, as we have seen, the construction of these geometries critically depends upon the existence of topological terms, whose incorporation in this technology has proven strenuous so far. Once this barrier is overcome, we expect finding these rich families of axisymmetric, non-extremal microstate geometries to be within reach. Finally, independent of the entire fuzzball proposal, this supergravity analysis suggests another possible end-point of stellar collapse: Topological Stars, supported by higher-dimensional magnetic fluxes. It is important to realize that the topological cycles necessarily require extra, compactified dimensions of space-time and so the structure of such a star can only appear at the scale of the extra dimensions. Through the effects of warp factors, it is possible that the scale of (some of) these extra dimensions could become much larger near the star than they are at infinity. The fact that these stars are made of flux and anti-flux suggests that they will slowly decay. In the fuzzball program we suspect that this will be how fuzzballs generate Hawking radiation but, more generally, it will mean that topological stars will still emit radiation. We will discuss this further in Section 8.

3.2 Geometric transitions and new black-hole scales In order to characterize the microstate solutions, and to understand the features one expects from them if they are to describe the typicalFour Scales microstates of the black hole, it is useful to discuss several scales one encounters in their physics. First, one can define the scale λT of the geometric transition that is responsible• Classical for replacing BH has certain2 scales: singular brane sources with smooth geometry, or otherwise the typical size– ofMass the / Horizon homology Size cycles, or bubbles in a microstate solution. This – Planck Length scale is determined by a balance between gravity that tends to collapse the bubbles and the flux • Microstate geometries have 2 more zmax that tends to drive their expansion.– Redshift from For the BPS bottom and of the extremal bubbles we typically find that throat, zmax – Size of bubbles: λ k (3.4) T ∼ P Can be traded for gap in 4 where k is the number of flux quanta threading the typical cycle30 and P is the Planck length . energy spectrum Egap After the geometric transition, the entire region where the black hole would have formed is now populated by a clusterMittwoch, 12. März of 2014 homology cycles and fluxes whose overall size is approximately that of what would have been the black hole. Thus one would expect that the number of cycles,

10 More general bubbling solutions • Add supertubes – supersymmetric brane configs – arbitrary shape: • Construct backreacted solution – Taub-NUT Green’s functions (painful) • Smooth ! – exactly as in flat space Mathur Lunin2 Maldacena Maoz • Entropy: S~(Q5/2)1/2 • 5D, 6D SUGRA - evade bounds of entropy of 4D multicenter solutions de Boer, El Showk, Messamah, van den Bleeken • Not yet black-hole-like (Q3/2); getting there J

Mittwoch, 12. März 2014 30 Even more general solutions Bena, deBoer, Shigemori, Warner

• Supertubes (locally 16 susy) - 8 functions of one variable (c = 8) • Superstrata (locally 16 susy) - 4 functions of two variables (c= ∞) • Double supertube transition: Should be Smooth !!!

Figure 2: The doubleMittwoch, bubbling 12. März of the 2014 D1-D5-P system. There are two ways to obtain a superstratum: The D1 and P can fuse into a D1-P supertube spiral (red dotted line), and the D5 and P can fuse into a D5-P spiral (blue continuous line). The spirals can then fuse into a superstratum. Alternatively the D1-D5 can fuse into a D1-D5-KKM tube (violet straight supertube), which upon adding momentum can start shaking and become a superstratum. superstratum

16 supersymmetries: One applies a second supertube transition that involves adding a KKM dipole charge and angular momentum. Locally, this is the same as the standard supertube transition of the D1-D5 system. It is important to remember that this transition decreases the codimension of the system, and because the KKM shrinks to zero the D1-D5 common direction the resulting configuration is smooth [5, 6]. Hence, the puff-up into a codimension-three object completely resolves the singularity of the D1-D5 system. To be more specific, letz ˆ denote the common direction within of D1 and D5 branes before puffing up and recall that there is, locally, a patch, ,ofR4 transverse to the branes (see Fig. 1). The smooth solution is obtained by introducing aU KKM dipole charge along a closed path,γ ˆ, in and smearing the D1 and D5 charge along this path. We will parametrize the curve,γ ˆ, byU an angle, ψ, so the puffed up brane is a codimension 3 object that sweeps out the (ˆz,ψ)- plane. The resulting object is now described by the curve,γ ˆ,in and the three-dimensional transverse geometry in in the neighborhood of a point onγ ˆ, appears,U at first sight, to be singular. However, it is aU Kaluza-Klein monopole and if thez ˆ direction is compactified with the proper periodicity then the KKM fiber shrinks to zero at a certain profile in R4 in such a way that the resulting geometry is smooth.

6 Even more general solutions Bena, deBoer, Shigemori, Warner

• Supertubes (locally 16 susy) - 8 functions of one variable (c = 8) • Superstrata (locally 16 susy) - 4 functions of two variables (c= ∞) • Double supertube transition: Should be Smooth !!!

6 Even more general solutions Bena, deBoer, Shigemori, Warner

• Supertubes (locally 16 susy) - 8 functions of one variable (c = 8) • Superstrata (locally 16 susy) - 4 functions of two variables (c= ∞) • Double supertube transition: Should be Smooth !!!

6 Superstrata • Want smooth solution depending on arbitrary function of 2 variables F(ψ,v) • ψ = GH fiber, v = D1-D5 common direction • ψ-dependent solutions Mathur Lunin2 Maldacena Maoz • interchange fibers: v-dependent solutions • more general: f(ψ) and g(v) Niehoff, Warner

Mittwoch, 12. März 2014 Superstrata • Want smooth solution depending on arbitrary function of 2 variables F(ψ,v) • ψ = GH fiber, v = D1-D5 common direction • ψ-dependent solutions Mathur Lunin2 Maldacena Maoz • interchange fibers: v-dependent solutions • more general: f(ψ) and g(v) Niehoff, Warner

• Superstrata entropy: • D1-D5 supertube - dimension of moduli space – classically: ∞

– quantize: 4 N1 N5 = number of momentum carriers • Counting (+ fermions) (à la Maldacena Strominger Witten) 1/2 S=2 π (N1 N5 Np) !!! Bena, Shigemori, Warner (to appear)

Mittwoch, 12. März 2014 SUSY microstates – the story: • We have a huge number of them – Arbitrary continuous functions – Smooth solutions. 4 scales ! – Superstrata reproduce black hole entropy J Bena, Shigemori, Warner • Dual to CFT states in typical sector – This is where BH states live too J – AdS-CFT perspective: highly weird if BH microstates had horizon Bena, Wang, Warner • Two non-backreacted calculations: – BH entropy - scaling multicenter config J Denef, Moore; Denef, Gaiotto, Strominger, Van den Bleeken, Yin – Higgs-Coulomb map. Bena, Berkooz, de Boer, El Showk, Van den Bleeken; Lee, Wang, Yi

Mittwoch, 12. März 2014 Black Hole Deconstruction Black Strominger - Vafa Denef, Gaiotto, Strominger, Holes Van den Bleeken, Yin (2007) S = SBH S ~ SBH

Effective coupling ( gs )

Multicenter Quiver QM Smooth Horizonless Denef, Moore (2007) Microstate Geometries Bena, Berkooz, de Boer, El Showk, Van den Bleeken. S ~ SBH Size grows

No Horizon

Punchline: Typical states grow as GN increases. Horizon never forms. Quantum effects from singularity extend to horizon

Mittwoch, 12. März 2014 Black Hole Deconstruction Black Strominger - Vafa Denef, Gaiotto, Strominger, Holes Van den Bleeken, Yin (2007) S = SBH S ~ SBH

Effective coupling ( gs )

Multicenter Quiver QM Smooth Horizonless Denef, Moore (2007) Microstate Geometries Bena, Berkooz, de Boer, El Showk, Van den Bleeken. S ~ SBH Size grows

No Horizon

Punchline: Typical states grow as GN increases. Horizon never forms. Quantum effects from singularity extend to horizon Similar story for non-SUSY extremal black holes

Mittwoch, 12. März 2014 Why destroy horizon ? Low curvature !

Mittwoch, 12. März 2014 Why destroy horizon ? Low curvature ! • Answer: space-time has singularity: – low-mass degrees of freedom – change physics on long distances

Mittwoch, 12. März 2014 Why destroy horizon ? Low curvature ! • Answer: space-time has singularity: – low-mass degrees of freedom – change physics on long distances • Very common in string theory !!! – Polchinski-Strassler – Klebanov-Strassler – Giant Gravitons + LLM – D1-D5 system • Non-Abelian ⇔ brane polarization ⇔ bubbling • Nothing holy about singularity behind horizon Bena, Kuperstein, Warner

Mittwoch, 12. März 2014 BPS Black Hole = Extremal • This is not so strange • Horizon in causal future of singularity • Time-like singularity resolved by stringy low- mass modes extending to horizon

Penrose Poisson, Israel Dafermos Marolf

Big deal .... Penrose Poisson, Israel Dafermos Marolf

Mittwoch, 12. März 2014 Mittwoch, 12. März 2014 The really big deal fuzzball, firewall

Mittwoch, 12. März 2014 The really big deal fuzzball, firewall

? Non-Extremal Resolution back in time

Mittwoch, 12. März 2014 The really big deal fuzzball, firewall

? Non-Extremal Build lots and Resolution back in time lots of such solutions !!!

Mittwoch, 12. März 2014 Very few known. Extremely hard to build... – Coupled nonlinear 2‘nd order PDE’s do not factorize

Mittwoch, 12. März 2014 Very few known. Extremely hard to build... – Coupled nonlinear 2‘nd order PDE’s do not factorize Do not pray to the saint who does not help you ! Romanian proverb

Mittwoch, 12. März 2014 Very few known. Extremely hard to build... – Coupled nonlinear 2‘nd order PDE’s do not factorize Do not pray to the saint who does not help you ! Romanian proverb • Idea: perturbative construction - near-BPS • antibranes in backgrounds with charge dissolved in fluxes Kachru, Pearson, Verlinde • Add supertubes to BPS bubbling sols.

• Metastable minima Bena, Puhm, Vercnocke • Decay to susy minima: brane-flux annihilation - Hawking radiation • Microstates of near-extremal BH

Mittwoch, 12. März 2014 30

Mittwoch, 12. März 2014 Near-Extremal BH Microstates • Microstate geometries: BH:

• Some longer, some shorter

• Force on branes (à la KKLMMT) wild fluctuations !!! 30 • Incoming observer cooked ? Definitely feel it !

Mittwoch, 12. März 2014 Mittwoch, 12. März 2014 The really big deal

At lest for Near-Extremal Resolution “backwards in time!”

!!!

Mittwoch, 12. März 2014 What is the mechanism ?

• Topological cycles • Opposite fluxes • (+) and (-) charges dissolved in fluxes.

• No Solitons without Topology Gibbons, Warner – Only way to build stationary solutions with BH charges • One mechanism to hold stuff at horizon three hypostases: Bubbling ⇔ Brane polarization ⇔ NonAbelian • Same physics - different regimes of parameters • Similar to flux compactifications Aganagic, Beem, Seo, Vafa

Mittwoch, 12. März 2014 What about other black holes?