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Selected Topics in AdS/CFT lecture 1

Selected Topics in AdS/CFT lecture 1. Soo-Jong Rey Seoul National University 1 st Asian Winter School in String Theory January 2007. Topics: Loops, Defects and Plasma. Review of AdS5/SYM4 Wilson/Polyakov loops Baryons, Defects Time-Dependent Phenomena. Starting Point.

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Selected Topics in AdS/CFT lecture 1

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  1. Selected Topics in AdS/CFTlecture 1 Soo-Jong Rey Seoul National University 1st Asian Winter School in String Theory January 2007

  2. Topics: Loops, Defects and Plasma • Review of AdS5/SYM4 • Wilson/Polyakov loops • Baryons, Defects • Time-Dependent Phenomena

  3. Starting Point • string theory:open + closed strings • low-energy limit of string theory: open string ! gauge theory (spin=1) closed string ! gravity theory (spin=2) • channel duality: open $ closed

  4. Channel Duality open string channel time closed string channel gauge dynamics gravity dynamics

  5. Gauge – Gravity Duality • keypoint: “gauge theory dynamics” (low energy limit of open string dynamics) is describable by “gravity dynamics” (low-energy limit of closed string dynamics) and vice versa

  6. Tool Kits for AdS/CFT • elementary excitations: open string + closed string • solitons: D-branes(quantum) +NS-branes (classical) • Mach’s principle: spacetimeà elementary + soliton sources • “holography”: gravity fluctuations = source fluctuations

  7. p-Brane (gravity description) • string effective action • S = s d10x [e-2(R(10)+(r)2+|H3|2 + …) +( p |Gp+2|2 + … ) ] = (1/g2st)(NS-NS sector) + (R-R sector) • H3 = d B2, Gp+2 = d Cp+1 etc. • R-R sector is “quantum” of NS-NS sector

  8. elementary and solitons • fixed “magnetic” charge, energy-minimizing static configurations (necessary condition for BPS states) • NS 5-brane: E = sg-2st [(r)2 + H32] > g-2stsS3 e-2 H3 • F-string: E = s [g-2st (r)2 + g2st K72] > g0stsS7 K7 • Mass: M(string) ~ 1 ; M(NS5) ~ (1/g2st) “elementary” ; “soliton”

  9. p-brane: in between • p-brane: E = sg-2st (r )2 + (Gp+2)2 > g-1sts e- Gp+2 • M(p-brane) » (1/gst) • p-brane = “quantum soliton”: more solitonic than F-string but less solitonic than NS5-brane • F-string $ p-brane $ NS5-brane • quantum treatment of p-brane is imperative

  10. Dp-brane (CFT description) • strings can end on it open string • string endpoints labelled by Chan-Paton factors ( ) = (i,j) • mass set by disk diagram M ~ 1/gst ---- identifiable with p-brane • QL = QR (half BPS object)

  11. SYMp+1 at Low-Energy • infinite tension limit (’ 0) • open strings  rigid rods (Mw ~  r/’) • (N,N) string dynamics U(N) SYM(P+1) L = g-2YM Tr ( Fmn2“||” + (Dma)2 + [a, b]2 “?” + …. )

  12. Static source: heavy quarks • semi-infinite string (M ~ 1) • at rest or constant velocity (static source) • labelled by a single Chan-Paton factor ( ) = (i) heavy (anti)quark in (anti)fundamental rep.

  13. Perturbation theory (1) • large N conformalgauge theory : double expansion in 1/N and 2 = gYM2 N • SYM = (N /2) s d4 x Tr (Fmn2 + (Dm)2 + …) • planar expansion: (N /2)V – E NF = (1/N)2h-2 (2)E-V • 2 : nonlinear interactions • 1/N: quantum fluctuations • observable: hl(1/N)2h-2(2)l Cl, h

  14. Perturbationtheory (2) • Weak coupling closed string theory: double expansion in gst and ’ • ’ : string worldsheet fluctuation Sstring= (½’)s d2 X ¢ X • gst : string quantum loop fluctuation • SSFT =  * QB + gst *  *  • observables = Sh mgst2h – 2(2’ p2)m Dm, h

  15. Identifying the Two Sides…. • large-N YM and closed string theory have the same perturbation expansion structure • gst $ (1/N) • (’/R2) $ 1/ where R = characteristic scale • Maldacena’s AdS/CFT correspondence: “near-horizon”(R) geometry of D3-brane = large-N SYM3+1 at large but fixed  • not only perturbative level but also nonperturbatively (evidence?)

  16. D3-Brane Geometry • 10d SUGRA(closed string)+4d SYM (D3-brane): Stotal = S10d + sd4 x (-e- TrFmn2+ C4….) • Solution • ds2 = Z-1/2 dx23+1 + Z1/2 dy26 • G5 = (1+*) dVol4Æ d Z-1 where Z = (1+R4/r4); r = |y| and R4 = 4 gst N ’2 • r !1: 10d flat spacetime • r ! 0 : characteristic curvature scale = R

  17. Identifications • D-brane stress tensor grows with (N/gst) • At large N, curvedness grows with g2st(N/gst)=gstN • Near D3-brane, spacetime= AdS5 x S5 • 4d D3-brane fluct. $10d spacetime fluct. (from coupling of D3-brane to 10d fields) Tr (FmpFnp) $metricgmn Tr (FmnFmn) $dilaton  Tr(FmpF*np) $Ramond-Ramond C, Cmn

  18. AdS/CFT correspondence • Dirichlet problem in AdS5 or EAdS5=H5 • Zgravity = exp (-SAdS5(bulk,a, 1a)) = ZSYM =s[dA] exp(-SSYM - s a1a Oa) AdS5

  19. AdS5 • In flat 4+2 dimensional space ds2 = - dX02 – d X52 + a=14 d Xa2 • embed hyperboloid X02 + X52 - a=14 Xa2 = R2 • SO(4,2) invariant, homogeneous • AdS5 = induced geometry on hyperboloid • <homework> derive the following coordinates

  20. Global coordinates: ds2 = R2(-cosh2 d2+d2+sinh2 d32) boundary = Rt£ S3 • Poincare coordinates: ds2=R2[r2(-dt2 + dx2 + dy2 + dz2)+ r-2dr2] boundary = Rt£ R3

  21. holographic scaling dimensions • wave eqn for scalar field of mass m • 2 solns: • normalizable vs. non-normalizable modes

  22. Why AdS Throat = D3-Brane? • D-brane absorption cross-section: SUGRA computation = SYM computation N D3-branes = flat flat AdS5 AdS_5 “interior” in gravity description = N D3-branes in gauge description

  23. Another argument • D-instantons probing (Euclidean) AdS5 • For U(N) gauge group, “homogeneous” instanton number < N (otherwise inhomogeneous) • Q D-instanton cluster in approx. flat region SDinstanton = -(1/gst’2) TrQ[1, 2]2 + …. • < Tr(1)2 > » QL2, <Tr(2)2 > » Q2 gst’2 / L2 • rotational symmetry implies L4 = Q gst’2 = N gst’2

  24. How can it be that 5d = 4d? • extensive quantities in 4d SYM theory scales as [length]4 • Extensive quantities in 5d AdS gravity scales as [length]5 • So, how can it be that quantities in 4d theory is describable by 5d theory?? • [Question] Show both area and volume of a ball of radius X in AdSd scales as Xd-1!

  25. Entropy Counting • (3+1) SYM on V3 with UV cutoff a • AdS5 gravity on V3 with UV cutoff a

  26. Shall we test AdS/CFT? • Recall that heavy quarks are represented by fundamental strings attached to D3-brane • now strings are stretched and fluctuates inside AdS5 • Let’s compute interaction potential between quark and antiquark • Do we obtain physically reasonable answers?

  27. Static Quark Potentialat Zero Temperature • Notice: • Square-Root --- non-analyticity for  • exact 1/r --- conformal invariance

  28. Notice: R4=gstN’2 ’2 cancels out!

  29. Holography (boundary = bulk) YM distance scale r=0 AdS radial scale r=1 Anything that takes place HERE (AdS5) --- --- is a result of that taking place HERE (R3+1)

  30. bare quark YM distance scale r=0 AdS radial scale r=oo bare anti-quark Heavy Meson Configuration

  31. What have we evaluated? • rectangular Wilson loop in N=4 SYM • W[C] = Tr P exp sC (i Am dxm + a d ya ) • gauge field part = Aharonov-Bohm phase • scalar field part = W-boson mass • unique N=4 supersymmetric structure with contour in 10-dimensions

  32. T=0 vs. T>0 SYM Theory T=0 (zero-temperature 4d N=4 SYM): ds2 = r2 (-dt2+dx2 ) + dr2/r2 + (dS5)2 r = 5th dim // 4d energy scale: 0< r <1 T>0 (finite-temperature 4d N=4 SYM): ds2 = r2 (- F dt2+dx2) + F-1 dr2/r2+(dS5)2 F = (1 - (kT)4/r4):kT< r <1 5d AdS Schwarzschild BH = 4d heat bath

  33. Static Quark Potential at Finite Temperature • Notice: • Nonanalyticity in  persists • exact 1/r persists • potential vanishes beyond r*

  34. L T=0 relation Maximum inter-quark distance! T>0 relation U*/M UV-IR relation for T>0

  35. bare quark r=M YM distance scale r=0 AdS radial scale r=oo bare anti-quark Heavy Meson Configuration (T > 0)

  36. Application:D3-branes on “thermal” S1 • “thermal S1” breaks N=4 susy completely • At low-energy, 3d Yang-Mills + (junks) • 5d AdS replaced by 5d Euclidean black hole (time $ space) • glueball spectrum is obtainable by studying bound-state spectrum of gravity modes Note: 4d space-time, topology: gravity // gauge

  37. YM2+1 glueball spectrum • 0++: solve dilaton eqn=2nd order linear ode • result: N=3 lattice N=oo lattice AdS/CFT . 4.329(41) 4.065(55) 4.07(input) * 6.52 (9) 6.18 (13) 7.02 ** 8.23 (17) 7.99 (22) 9.92 *** - - 12.80 [M. Teper]

  38. YM3+1 glueball spectrum • Use T>0 D4-brane instead • 0++: solve dilaton eqn=2nd-order linear ODE • result: N=3 lattice AdS/CFT . 1.61(15) 1.61(input) * 2.8 2.38 ** - 3.11 *** - 3.82 [M. Teper] other glueballs fit reasonably well (why??)

  39. The Story of Square-Root • branch cut from strong coupling? • artifact of N1 limit • heuristically, saddle-point of matrices < Tr eM > = s [dM] (Tr eM) exp (--2 Tr M2) Recall modified Bessel function

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