Non-BPS and wiggly multicenter solutions and why should you like them
Iosif Bena IPhT, CEA Saclay
with Nick Warner, Gianguido Dall’Agata, Clement Ruef, Stefano Giusto Strominger and Vafa (1996) +1000 other articles Count BH Microstates 2 ways to understand: Match B.H. entropy !!!
Finite Gravity Zero Gravity String-QCD-BH
STATISTICAL CFT ENSEMBLE STATES MECHANICS (Boundary)
STROMINGER ! VAFA FILLPRESENT ENTROPY MATCHING WORK AdS-CFT GEOMETRIES Gravity BLACK HOLE Correspondence WITH NO HORIZON (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 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 mi- crostates 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 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 BIG QUESTION: Are all black hole microstates becoming geometries with no horizon ? ? Black hole = ensemble of horizonless microstates Mathur & friends 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 new low-mass A few corollaires degrees of freedom
- Thermodynamics (LQF T) breaks down at horizon. Nonlocal effects take over. - No spacetime inside black holes. Quantum superposition of microstate geometries.
Can be proved by rigorous calculations:
1. Build most generic microstates + Count 2. Use AdS-CFT lots and lots of solutions ∞ set of parameters black hole charges 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
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 BPS Microstates geometries Linear system 4 layers: Bena, Warner Gutowski, Reall
Focus on Gibbons-Hawking (Taub-NUT) base:
8 harmonic functions
Gauntlett, Gutowski, Bena, Kraus, Warner 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 Microstates geometries
Multi-center Taub-NUT many 2-cycles + flux
Compactified to 4D → multicenter configuration Denef
Abelian worldvolume flux Each: 16 supercharges 4 common supercharges (D2,D2,D2) 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 + … Charge disolved in fluxes Klebanov-Strassler • BH angular momentum
J = E x B = … + F01 F12 + … 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 ! More general solutions
• 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 Problem: 2-charge supertubes have 2 charges
TUBE STUBE << SBH Marolf, Palmer; Rychkov Solution:
• In deep scaling solutions: Bena, Bobev, Ruef, Warner
• Entropy enhancement !!! at typical depth S ENHANCED ~ S TUBE BH 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 3/2 2 1/2 – So if J ~ Q , Sbackreacted ~ (Q ) – 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 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 S ~ SBH
Size grows
No Horizon
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-Extremal Resolution 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 multicenter
Almost – 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 ansatz Mutually 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 Need good grad • Blast full force through equations student + postdoc – 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 1. Floating Branes - Paris method • 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 2. Dualities - Italian 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) 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 1/2 1/2 1/2 1/2 1/2 1/2 • S=2π (N1 + N1 )(N2 + N2 )(N3 + N3 ) 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