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AdS / CFT aka Anti de Sitter (space) / Conformal Field Theory W.A. Zajc Columbia University

AdS / CFT aka Anti de Sitter (space) / Conformal Field Theory W.A. Zajc Columbia University. Explaining the Connection. Maldacena’s extraordinary conjecture. 1) Weakly Coupled (classical) gravity in Anti-deSitter Space (AdS). 3) Strongly Coupled (Conformal) gauge Field Theories (CFT).

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AdS / CFT aka Anti de Sitter (space) / Conformal Field Theory W.A. Zajc Columbia University

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  1. AdS / CFT aka Anti de Sitter (space) / Conformal Field Theory W.A. Zajc Columbia University Journal Club

  2. Explaining the Connection • Maldacena’s extraordinary conjecture 1) Weakly Coupled (classical) gravity in Anti-deSitter Space (AdS) 3) Strongly Coupled (Conformal) gauge Field Theories (CFT) Journal Club

  3. All You Need To Know About Strings Journal Club

  4. All You Need To Know About D-branes • ‘D’ = Dirichlet  an extended object that imposes boundary conditions on (open) string endpoints • D-branes characterized by • Their dimensionality; Dp-brane lives in p spatial dimensions • Their tension Tp , defined such that • Required, e.g., to open closed strings upon brane contact • D-branes are essential dynamical objects in string theory String explores the full space  “the bulk” String endpointsconstrained to live on “the brane” Journal Club

  5. “Stack” of N D3-branes These shown as 2-d slices of 3-volumes This direction has no meaning, branes are really coincident • D3-brane properties: • Mass ~ 1/gS • Source gauge quantum number • Open strings end on them Journal Club

  6. String Interactions on D3-branes D3-branes shown as ~1-d slices of 3-volumes This direction has no meaning, branes are really coincident String world One string “indexed” on green + anti-red Gauge world SU(N) gauge theory of gluon interactions Journal Club

  7. Gauge  Gravity These shown as 2-d slices of coincident3-volumes • Mass ~ N/gS • Sources gravity • Curves space • Generates (sort of) anAnti de Sitterspacetime • D3-brane properties: • Mass ~ 1/gS • Source gauge quantum number • Open strings end on them Journal Club

  8. The Gravity Solution • “Towards a gravity dual of RHIC Collision”, Sang-Jan Sin, http://him.phys.pusan.ac.kr/PDS_HIM/HIM/2005-11/3_shin.pdf Where’s my AdS ? There it is! Journal Club

  9. The Correspondence • Q. Where do the N D3-branes live? • A. On the boundary of an Anti de Sitter space (that they create!) ~ Essentially flat space Curvature matters ! This direction ( r ) has meaning; ~ energy scale Journal Club

  10. So What’s the CFT Part ? • “Real” AdS in n spacetime dimensions • The D-brane induced “almost AdS” • Their limits (which are also called AdS): • “Real” AdS : • D-brane “almost AdS”: • The scaling form of the limit (which is also called AdS) Journal Club

  11. The Conformal Part • Note that this metric has no scale, that is, is invariant under (xm,z)  (lxm, lz) • Potential must scale as 1/r • AdS interpretation: Still an area law for Wilson lines, but the warp factor 1/z makes the“area” fall as 1/r Journal Club

  12. The Icky Part Horizon • Icky, that is, if you want to use this correspondence to study QCD • Conformal • no scale • “It’s 1/r all the way down” • No confinement ! • One way out (Witten, hep-th/9803002) • Modify space to have a horizon: • More recently: “More on a holographic dual of QCD”, T. Sakai and S. Sugimoto, http://arxiv.org/abs/hep-th/0507073 Journal Club

  13. We Don’t Care About Confinement • The duality, as described, applies to • More accurately: Q. How to thermalize the theory? A. Shine a “black” hole on it (!) T=0 CFT in flat 3+1 spacetime Gravity in curved 4+1 AdS spacetime  (Strongly coupled)T=0 CFT in flat 3+1 spacetime (~Classical)Gravity in curved 4+1 AdS spacetime  Journal Club

  14. Black Hole Thermodynamics • ~1970, Bekenstein: • Black hole area law “feels like” 2nd law of thermodynamics: • AMERGED ≥ A1 + A2 • Charge for black hole contributes to energy as dM = F dQ,feels like chemical potential • So why not dM = T dSBH + F dQ , with SBH ~Black Hole Area ?? • Counter-arguments: • “Black holes have no hair”  no internal d.o.f  no entropy • Entropy  temperature  radiation, but black holes are black • ~1974, Hawking: • Black holes do radiate ! • Semi-classical computation allowed determination of entropy: Journal Club

  15. BH Radiation  BH’s are Unstable • Starting from this: it’s easy to compute • Black Hole entropy: • Black Hole temperature: • Black Hole lifetime(assuming Stefan-Boltzmann) Journal Club

  16. Black Holes in Higher Dimensions • Apply same basic formalism starting from D-dimensional result for Schwarzschild radius: • Show that higher-dimensional BH’s • Have a temperature • And therefore radiate • And therefore have finite lifetime • Unless the background spacetime is curved ! Journal Club

  17. Black Holes in AdS • The metric becomes • The spacetime curvature R introduces a new scale in the problem • Especially because light reaches the boundary in time T = p R and is “reflected” • Black hole is in a “box”: • Small black holes: rbh << R  rbh ~ M1/2  Unstable • Large black holes: rbh ~ R  rbh ~ M1/4  STABLE ! • In addition, for large black holes: • In 5-d spacetime, BH “area” ~ Length3  S ~ M3/4 • T ~ M1/4  S ~ T 3 , that is, just like a QGP Journal Club

  18. This is Your Brane • This is your brane on AdS • Negative curvature R • Finite time ~R for light to reach boundary and return • Black holes of lifetime > ~ R are STABLE ! Journal Club

  19. Viscosity Primer • Remove your organic prejudices • Don’t equate viscous with “sticky” ! • Think instead of a not-quite-ideal fluid: • “not-quite-ideal”  “supports a shear stress” • Viscosity hthen defined as • Dimensional estimate: • Viscosityincreases withtemperature • Largecross sections  small viscosity • The gauge/string duality is one that mapsstrongly coupled gauge fields  Weak (semi-classical) gravity Journal Club

  20. Ideal Hydrodynamics • Why the interest in viscosity? A.) Its vanishing is associated with the applicability of ideal hydrodynamics (Landau, 1955): B.) Successes of ideal hydrodynamics applied to RHIC data suggest that the fluid is “as perfect as it can be”, that is, it approaches the (conjectured) quantum mechanical limit See “A Viscosity Bound Conjecture”, P. Kovtun, D.T. Son, A.O. Starinets, hep-th/0405231 Journal Club

  21. Why Does This Work?? • The easy part: • Recall • that is, viscosity ~ x-momentum transport in y-direction ~ Txy • There are standard methods (Kubo relations) to calculate such dissipative quantities • The hard part: • This calculation is difficult in a strongly-coupled gauge theory • The weird part: • A (supersymmetric) pseudo-QCD theory can be mapped to a 10-dimensional classical gravity theory on the background of black 3-branes • The calculation can be performed there as the absorption of gravitons by the brane • THE SHEAR VISCOSITY OF STRONGLY COUPLED N=4 SUPERSYMMETRIC YANG-MILLS PLASMA., G. Policastro, D.T. Son , A.O. Starinets, Phys.Rev.Lett.87:081601,2001 hep-th/0104066 hmn Am An Journal Club

  22. The Result • Viscosity h = “Area”/16pG • Normalize by entropy (density) S = “Area”/4G • Dividing out the infinite “areas” : • Conjectured to be a lower bound “for all relativistic quantum field theories at finite temperature and zero chemical potential”. • See “Viscosity in strongly interacting quantum field theories from black hole physics”, P. Kovtun, D.T. Son, A.O. Starinets, Phys.Rev.Lett.94:111601, 2005, hep-th/0405231 Infinite “Area” ! Journal Club

  23. Isn’t This Result “Just” Quantum Mechanics? • Recall from previous discussion: • e = energy density • t = lifetime of quasiparticle • Entropy density s ~ kB n  • where last step • follows from requirement that lifetime of quasiparticle must exceed ~h/Energy • establishes that the bound is from below Journal Club

  24. How Perfect is “Perfect” • All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity • h = 0 “perfect fluid” • But there is a (conjectured) quantum limit: • Where do “ordinary” fluids sit wrt this limit? • RHIC “fluid” mightbe at ~2-3 on this scale (!) T=1012 K Journal Club

  25. Water  RHIC  Water  RHIC • The search for QCD phase transition of course was informed by analogy to ordinary matter • Results from RHIC are now “flowing” back to ordinary matter h / s “On the Strongly-Interacting Low-Viscosity Matter Created in Relativistic Nuclear Collisions”,L.P. Csernai, J.I. Kapusta and L.D. McLerran, Phys.Rev.Lett.97:152303,2006, nucl-th/0604032 Journal Club

  26. QCD Critical Point Journal Club

  27. A Loophole To The Bound? • Kovtun, Son and Starinets also note • Cohen seeks to exploit this loophole: • “Is there a 'most perfect fluid' consistent with quantum field theory?”, Thomas D. Cohen, hep-th/0702136 Journal Club

  28. Entropy of Mixing V/NA V/NA V/NA 2V/2NA V/NB 2V/NA+2V/NB • It’s “in” the Sackur-Tetrode equation:   Journal Club

  29. Entropy For Distinguishable Particles Journal Club

  30. Incorporating Indistinguishability Journal Club

  31. Incorporating Multiple Species Journal Club

  32. Cohen’s Scaling Parameter Journal Club

  33. The Scaling Regime Journal Club

  34. How Low Can It Go? Journal Club

  35. Not Discussed • Counter-counter arguments: • Bousso’s entropy bound on spacetime regions? • Counter-counter-counter arguments: • Residual entropy ? Journal Club

  36. Suggested Reading • November, 2005 issue of Scientific American • “The Illusion of Gravity” • J. Maldacena • A test of this prediction comes from the Relativistic Heavy Ion Collider (RHIC) at BrookhavenNational Laboratory, which has been colliding gold nuclei at very high energies. A preliminary analysis of these experiments indicates the collisions are creating a fluid with very low viscosity. Even though Son and his co-workers studied a simplified version of chromodynamics, they seem to have come up with a property that is shared by the real world. Does this mean that RHIC is creating small five-dimensional black holes? It is really too early to tell, both experimentally and theoretically. (Even if so, there is nothing to fear from these tiny black holes-they evaporate almost as fast as they are formed, and they "live" in five dimensions, not in our own four-dimensional world.) Journal Club

  37. A Spooky Connection • RHIC physics clearly relies on • The quantum nature of matter (Einstein, 1905) • The relativistic nature of matter (Einstein, 1905) but presumably has no connection to • General relativity (Einstein, 1912-7) • Wait ! Both sides of this equation were calculated using black hole physics (in 10 dimensions) MULTIPLICITY Entropy  Black Hole Area DISSIPATION Viscosity  Graviton Absorption Journal Club

  38. Spooky Connection at a Distance • We’ve yet to understand the discrepancy between lattice results and Stefan-Boltzmann limit: • The success of naïve hydrodynamics requires very low viscosities • Both are predicted from ~gravitational phenomena in N = 4 supersymmetric theories: Journal Club

  39. New Dimensions in RHIC Physics • “The stress tensor of a quark moving through N=4 thermal plasma”, J.J. Friess et al., hep-th/0607022 Jet modifications from wake field Our 4-d world The stuff formerly known as QGP Heavy quark moving through the medium String theorist’s 5-d world Energy loss from string drag Journal Club

  40. The Way Forward • Recall • “ We need to learn to expand in powers of 1 / g(T) ” • For example, the mean free path lmfp • Limit lmfp 0 is hydrodynamics Journal Club

  41. Landau Knew It • Landau (1955) significant extension of Fermi’s approach • Considers fundamental roles of • hydrodynamic evolution • entropy • “The defects of Fermi’s theory arise mainly because the expansion of the compound system is not correctly taken into account…(The) expansion of the system can be considered on the basis of relativistic hydrodynamics.” • (Emphasis added by WAZ) Journal Club

  42. But We’re Not Quite Done Making Mistakes • Recall our argument for short mean free paths: • But this relies on the number density n , which is not well-defined for a relativistic field theory at strong coupling(!) • But wait, it get worse… • Even the classical coupling parameter is not well-defined relativistically(!) Journal Club

  43. A Way Out • How can we quantify the coupling properties of our “plasma” ? • A solution was provided by Dam Son: • n( T ) is not well-defined … but s(T) is • mean free path not well-defined… but viscosity h is • coupling G is not well defined… but s / h is • Note: • Short mean free paths small viscosity Journal Club

  44. This is Your Brane • This is your brane on AdS • More seriously: • Negative curvature R • Finite time ~R for light to reach boundary and return • Black holes of lifetime > ~ R are STABLE ! Journal Club

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