Non-BPS and wigglymulticenter solutionsand why should you like them

Iosif BenaIPhT, CEA Saclay

with Nick Warner, Gianguido Dall’Agata, Clement Ruef, Stefano Giusto

Strominger and Vafa (1996)+1000 other articles

Count BH Microstates Match B.H. entropy !!!

2 ways to understand:

Zero GravityFinite Gravity

FILL

String-QCD-BH

STATISTICAL

MECHANICS

BLACK HOLE

ENSEMBLE STATES CFT(Boundary)

ENTROPY MATCHINGSTROMINGER ! VAFA

WORKPRESENT

GEOMETRIESWITH NO HORIZON

Gravity(Bulk)

Figure 1: An illustrative description of this picture of black holes. My research focuses on con-structing more microstate geometries with no horizon, and on improving the dictionary between theexisting ones and the states of the CFT (the dashed vertical arrow).

paradox: microstates have unitary physics, and thus information is not lost. Second, sincethe maximal entropy in a region of space comes from microstates that have the same sizeas the would-be black hole, this would prove ’t Hooft’s holographic principle. Third, thesemicrostates will appear whenever we have a large-enough energy density, and it is quite likelythat their physics will be dominant in cosmological settings, like in the Big Bang and theBig Crunch singularities. To use an analogy from classical physics, one can say this picturecould revolutionize quantum gravity and the physics of black holes in the same way in whichstatistical physics revolutionized the understanding of thermodynamics.

Furthermore, this picture of black holes might also be experimentally testable with thegravity wave detector LISA or if black holes are found at the Large Hadron Collider. It istherefore 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 constructingand 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 agood 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 andthe solutions we have constructed. Per Kraus and I have been the first to describe black ringsin this CFT, and I believe I have some, and I can master the other tools needed to successfullyattack this problem. Once this map is obtained, I intend to find the bulk geometries thatcorrespond to the typical states of the CFT. These geometries would be then the typicalmicrostates of the black hole; if they are horizonless, this would give a proof that the blackhole is a thermodynamic description of an ensemble of horizonless configurations.

The second direction is to construct more generic three-charge solutions, that have blackhole charges and depend on several continuous functions. The solutions N. Warner and Ihave 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 constructedsolutions that depend on arbitrary functions, but do not have black hole charges. I think ourmethods can be combined to build these more generic solutions. We can then count them andsee if they can account for the entropy of the black hole. If they do, this would again establishthat black holes are ensembles of horizonless configurations.

Another direction that is important to pursue in longer term is the construction of mi-crostates of near-BPS and non-BPS black holes. Most of the effort in this field has so farconcentrated 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 thatBPS black holes are ensembles of horizonless microstates, then it will be rather unlikely thatnon-BPS black holes will not have the same description. However, if one is to use this picture

AdS-CFTCorrespondence

Strominger and Vafa (1996):Count Black Hole Microstates (branes + strings) Correctly match B.H. entropy !!!

Black hole regime of parameters:

Zero Gravity

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

SusskindHorowitz, Polchinski Damour, Veneziano

Black hole regime of parameters:

Identical to black hole far away. Horizon → Smooth cap

Giusto, Mathur, SaxenaBena, Warner

Berglund, Gimon, Levi

Strominger and Vafa (1996):Count Black Hole Microstates (branes + strings) Correctly match B.H. entropy !!!

Zero Gravity

BIG QUESTION: Are all black hole microstates becoming geometries with no horizon ?

Black hole = ensemble of horizonless microstates?

Mathur & friends

ThermodynamicsBlack Hole Solution

Statistical PhysicsMicrostate geometries

Thermodynamics(Air = ideal gas)P V = n R T

dE = T dS + P dV

Statistical Physics(Air -- molecules)eS microstatestypical atypical

Long distance physicsGravitational lensing

Physics at horizonInformation loss

Analogy with ideal gas

- Thermodynamics (LQF T) breaks down at horizon. Nonlocal effects take over.

- No spacetime inside black holes. Quantum superposition of microstate geometries.

A few corollaires

Can be proved by rigorous calculations:

1. Build most generic microstates + Count

2. Use AdS-CFTlots and lots of solutions∞ set of parametersblack hole charges

new low-mass degrees of freedom

Word of caution• To replace classical BH by BH-sized object

– Gravastar– Fuzzball– LQG muck – Quark-star, you name it …

satisfy very stringent test: Horowitz

– BH size grows with GN

– Size of objects in other theories becomes smaller

Same growth with GN !!!

- BH microstate geometries pass this test- Highly nontrivial mechanism

BPS Microstates geometriesLinear system

4 layers:

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

8 harmonic functions

Gauntlett, Gutowski, Bena, Kraus, Warner

Bena, Warner Gutowski, Reall

BPS Black Rings (in Taub-NUT) Elvang, Emparan, Mateos, Reall; Bena, Kraus, Warner; Gaiotto, Strominger, Yin

• Position of ring depends on charges and moduli• Ring can go to infinity and disappear from spectrum• Lines of marginal stability, wall crossing, and all that …

Examples: Multiple Black Rings• 5D BH on tip of Taub-NUT � 4D BH with D6 charge

• Black ring with BH in the middle � 2-centered 4D BH• 17 black rings + BH � 18-centered 4D BH Denef

• 4D D6,D4,D2,D0 BH � 5D black hole • 4D D4,D2,D0 BH � 5D black ring • 5D: ring supported by angular momentum • 4D: multicenter configuration supported by E x B

Compactified to 4D → multicenter configuration Denef

Abelian worldvolume flux Each: 16 supercharges 4 common supercharges

(D2,D2,D2)

Microstates geometriesMulti-center Taub-NUTmany 2-cycles + flux

• Where is the BH charge ? L = q A0

L = … + A0 F12 F34 + …• Where is the BH mass ? E = … + F12 F12 + …• BH angular momentum J = E x B = … + F01 F12 + …

magnetic

Microstates geometries2-cycles + magnetic flux

Charge disolved in fluxesKlebanov-Strassler

The deep microstates

• 4D perspective: points collapse on top of each other • 5D: throat deeper and deeper; cap remains similar !• Solution smooth throughout scaling !• Long throats → small mass gap → typical CFT sector • Scaling goes on forever !!! AdS-CFT unhappy

– Can it be stopped ? Quantum effects ? YES– Destroy huge chunk of a smooth horizonless solution !!!– Rethink when classical gravity breaks down !

• Put supertube in bubbling solution• Supertubes Mateos, Townsend; Emparan

– supersymmetric brane configs.– arbitrary shape:– smooth supergravity solutions

Lunin, Mathur; Lunin, Maldacena, Maoz

• Classical moduli space of microstates solutions has infinite dimension !– Much bigger than space of GH solutions (extra U(1))– Key ingredient in getting correct D1-D5 entropy– Wiggly supertubes do not descend to 4D sugra.

More general solutions

More general solutionsProblem: 2-charge supertubes have 2 charges

Marolf, Palmer; Rychkov

Solution:

• In deep scaling solutions: Bena, Bobev, Ruef, Warner

• Entropy enhancement !!! at typical depth

STUBE ~ SBH

STUBE << SBH

ENHANCED

TUBE

Backreacted wiggly solutions• Supertube in multicenter Taub-NUT -

arbitrary functions ! Bena, Bobev, Giusto Ruef, Warner

• Linear procedure but need Taub-NUT scalar + vector Green’s functions (painful)

• Supertube entropy ~ (d J)1/2 ~ limited by asymptotic charges. d~ Q1/2

– So if J ~ Q3/2, Sbackreacted ~ (Q2)1/2

– Extra supertube as J sink. J~Q2, S~(Q5/2)1/2

• Smooth backreacted solutions !• Not yet black hole-like, but getting there

BPS microstates – the story:• We have a huge number of them

– Arbitrary continuous functions– Infinite-dimensional moduli space– Supertube Entropy Enhancement– Black-Hole-like entropy in probe approx

Bena, Bobev, Giusto, Ruef and Warner

• Dual to CFT states in typical sector – This is where BH states live too – CFT perspective: highly weird if BH microstates were

anything but fuzzballs• Two non-backreacted calculations:

– BH entropy from horizon-less scaling multicenter configurations

Effective coupling ( gs )

Black HolesStrominger - Vafa

S = SBH

Multicenter Quiver QMDenef, Moore (2007)

S ~ SBH

Black Hole DeconstructionDenef, Gaiotto, Strominger, Van den Bleeken, Yin (2007)

S ~ SBH

Size grows

No Horizon

Smooth Horizonless Microstate Geometries

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

Always asked question:• Why are quantum effects affecting the 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

• It can be even worse – quantum effects significant even without horizon or singularity ! Bena, Wang, Warner; de Boer, El Showk, Messamah, van den Bleeken

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

Big deal ....

So what’s the big deal about fuzzball proposal ?

?Non-ExtremalResolution back in time

Extremal non-BPS black holes• Same singularity. • Same Penrose diagram.

– Why BPS different from extremal ?– some people in the room would strangle us...

• How can one show this ?– Construct multi-center extremal black holes – Easier said than done. No susy to help. – Second-order eqns

• Worse than Kerr ~ black ring– Could extremality make things easier ?

• Depends which kind of extremality Bossard’s talk

Extremal non-BPS multicenterAlmost – BPS:

Goldstein, Katmadas: -

-

• Still solves equations of motion !!!• Base needs just be Ricci-Flat

Bena, Giusto, Ruef, Warner

• R4 or R3 x S1 → BPS sols (absorb the – )• Nontrivial new solutions in Taub-NUT

Why non-BPS ? Bena, Dall’Agata, Giusto, Ruef, Warner

Objects are locally BPS M2 brane Killing Spinors Incompatible with base space:

Four-dimensional perspective D2-D2-D2 system has 4 supercharges Add D6 – susy preserved Add anti-D6 – susy broken Any 3 of the 4 branes preserve susy

Probe D2 still feels no force Built in the ansatzMutually BPS w.r.t. any of the background branes

-

Non-BPS Rings (in Taub-NUT) Bena, Dall’Agata, Giusto, Ruef, Warner

- Near-ring geometry same as BPS - Distance from center of Taub-NUT different- Almost-BPS equations more involved- But we solved them …

A flavor of the equations

• Non-BPS • BPS

Simple solution:

How to solve for k• Look at each term on the right hand side.• Find corresponding µ and ω e.g.

• Combine all pieces to find full k

Regularity

• Vanishing ω on axis → no Dirac strings– Implies “bubble” or “integrability equations”– Walls of marginal stability, wall crossing, etc– Bubble equations now cubic !!!

• No CTC’s at black ring horizons → free harmonic function in k has dipole part

• Add dipole part at Taub-NUT center → most general under-rotating non-BPS extremal 4D BH Bena, Dall’Agata, Giusto, Ruef, Warner

Multiple non-BPS Rings Bena, Giusto, Ruef, Warner

• 4D multicentered black holes • non-BPS BH at Taub-NUT tip: anti-D6 + D2 + rotation• BH’s coming from rings: D4-D2-D0 charges

• There exist scaling solutions (very long throats)• Scaling solutions can have nonzero angular momentum !• In scaling limit: non-BPS bubble eqs = BPS bubble eqs

Non-BPS horizonless microstates

• Impossible within almost-BPS ansatz• Anti-self-dual flux in GH spaces is

non-normalizable• Solution not asymptotically-flat• We have to be smarter !• 3 ways

– Paris– Italian– Potsdam Bossard, Nicolai, Ruef (Bossard’s talk)

1. Floating Branes - Paris method • Blast full force through equations

– Ansatz: M2 branes feel no force– Warp factors = electric potentials

• New class of 5D solutions: Bena, Giusto, Ruef, Warner

• 4D base = electrovac Euclidean solution • Any electrovac sol. → full solution • Linear procedure !• Two obvious families of base-spaces

– Israel-Wilson metrics– Euclidean Kerr-Newman Bobev, Ruef

Need good grad student + postdoc

• Israel-Wilson base

• Highly nontrivial non-BPS solutions: – D6-D4-D2-D0 BH in anti-D6 background– Can put normalizable flux on cycle between

anti-D6 and D6

1. Floating Branes - Paris method

• Sequence of dualities: – 4D supergravity electric-magnetic duality = 6 T-dualities– Do them Dall’Agata, Giusto, Ruef

• Reshuffles charges. BPS ⇔ BPS • Almost-BPS sols ⇔ Israel-Wilson ⇔ most general solution !

• Recipe: choose charges, solve almost-BPS eqs, reshuffle functions

• Smooth multicenter solutions, deep and scaling (as good as it gets)

2. Dualities - Italian method

Rotating non-BPS Ring in Taub-NUT Bena, Giusto, Ruef,

- Near-ring geometry intrinsically non-BPS - rotating D4-D4-D4 anti-D0

- No 5D decompactification (Taub-Nut → R4)- Base is R3 - this is not the extremal ring of Elvang-Emparan-Figueras

What about nonextremal ?• Given total energy budget: most entropy obtained by making

brane-antibrane pairs• S=2π (N1

1/2 + N11/2)(N2

1/2 + N21/2)(N3

1/2 + N31/2)

Horowitz, Maldacena, Strominger

• Mass gap = 1/N1N2N3

• Extend on long distances (horizon scale)• More mass – lower mass-gap – larger size• Only 3 solutions known. Work very nicely.

Jejjala Madden, Ross, Titchener (JMaRT); Cardoso, Dias, Hovdebo, Myers; Mathur & al.

• One multi-center solution (with horizon) Camps, Emparan, Figueras, Giusto, Saxena

The Running Bolt• Base space just needs be Ricci Flat• Take Euclidean Schwarzschild:

• Put fluxes on the Bolt Bena, Giusto, Ruef, Warner – Bolt starts running

• Smooth solution. • D4, D2, D0, mass of big fat nonextremal BH !

• Mass decreases with increasing |Q| !

Summary and Future Directions• Strong evidence that in string theory:

extremal black holes = ensembles of microstates – low-mass modes affect large (horizon) scales– Convergence of research directions see Spindel’s talk

• Classify all extremal solutions– R3 base: Dall’Agata, Giusto, Ruef; Bossard, Ruef

– non-R3 base: • overspinning Rashhed-Larsen BH • Elvang-Emparan-Figueras extremal dipole ring • JMaRT ?

• Extend to non-extremal black holes– Probably; at least near-extremal– More non-extremal microstates (~ multicenter JMaRT)

• Inverse scattering methods. • Underlying structure• Instabilities + CFT interpretation