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

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

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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 Starting Point

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

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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

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

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p-Brane (gravity description) string effective action S = s d 10 x [e -2 (R (10) +( r ) 2 +|H 3 | 2 + …) +( p |G p+2 | 2 + … ) ] = (1/g 2 st )(NS-NS sector) + (R-R sector) H 3 = d B 2, G p+2 = d C p+1 etc. R-R sector is “quantum” of NS-NS sector

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elementary and solitons fixed “magnetic” charge, energy-minimizing static configurations (necessary condition for BPS states) NS 5-brane: E = s g -2 st [( r ) 2 + H 3 2 ] > g -2 st s S3 e -2 H 3 F-string: E = s [g -2 st ( r ) 2 + g 2 st K 7 2 ] > g 0 st s S7 K 7 Mass: M(string) ~ 1 ; M(NS5) ~ (1/g 2 st ) “elementary” ; “soliton”

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p-brane: in between p-brane: E = s g -2 st (r ) 2 + (G p+2 ) 2 > g -1 st s e - G p+2 M(p-brane) » (1/g st ) 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

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strings can end on it open string string endpoints labelled by Chan-Paton factors ( ) = (i,j) mass set by disk diagram M ~ 1/g st ---- identifiable with p-brane Q L = Q R (half BPS object) Dp-brane (CFT description)

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infinite tension limit ( ’ 0) open strings rigid rods (Mw ~ r/ ’) (N,N) string dynamics U(N) SYM (P+1) L = g -2 YM Tr ( F mn 2 “||” + (D m a ) 2 + [ a, b ] 2 “ ? ” + …. ) SYM p+1 at Low-Energy

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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. Static source: heavy quarks

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Perturbation theory (1) large N conformal gauge theory : double expansion in 1/N and 2 = g YM 2 N S YM = (N / 2 ) s d 4 x Tr (F mn 2 + (D m ) 2 + …) planar expansion: (N / 2 ) V – E N F = (1/N) 2h-2 ( 2 ) E-V 2 : nonlinear interactions 1/N: quantum fluctuations observable: h l (1/N) 2h-2 ( 2 ) l C l, h

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Perturbation theory (2) Weak coupling closed string theory: double expansion in g st and ’ ’ : string worldsheet fluctuation S string = (½ ’) s d 2 X ¢ X g st : string quantum loop fluctuation S SFT = * Q B + g st * * observables = h m g st 2h – 2 (2 ’ p 2 ) m D m, h

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Identifying the Two Sides…. large-N YM and closed string theory have the same perturbation expansion structure g st $ (1/N) ( ’/R 2 ) $ 1/ where R = characteristic scale Maldacena’s AdS/CFT correspondence: “near-horizon”(R) geometry of D3-brane = large-N SYM 3+1 at large but fixed not only perturbative level but also nonperturbatively (evidence?)

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D3-Brane Geometry 10d SUGRA(closed string)+4d SYM (D3-brane): S total = S 10d + s d 4 x (-e - TrF mn 2 + C 4 ….) Solution ds 2 = Z -1/2 dx Z 1/2 dy 2 6 G 5 = (1+*) dVol 4 Æ d Z -1 where Z = (1+R 4 /r 4 ); r = |y| and R 4 = 4 g st N ’ 2 r ! 1 : 10d flat spacetime r ! 0 : characteristic curvature scale = R

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D-brane stress tensor grows with (N/g st ) At large N, curvedness grows with g 2 st (N/g st )=g st N Near D3-brane, spacetime= AdS 5 x S 5 4d D3-brane fluct. $ 10d spacetime fluct. (from coupling of D3-brane to 10d fields) Tr (F mp F np ) $ metric g mn Tr (F mn F mn ) $ dilaton Tr(F mp F * np ) $ Ramond-Ramond C, C mn Identifications

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AdS/CFT correspondence Dirichlet problem in AdS 5 or EAdS 5 =H 5 Z gravity = exp (-S AdS5 ( bulk,a, 1 a )) = Z SYM = s [dA] exp(-S SYM - s a 1 a O a ) AdS 5

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In flat 4+2 dimensional space ds 2 = - dX 0 2 – d X a=1 4 d X a 2 embed hyperboloid X X a=1 4 X a 2 = R 2 SO(4,2) invariant, homogeneous AdS 5 = induced geometry on hyperboloid derive the following coordinates

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Global coordinates: ds 2 = R 2 (-cosh 2 d 2 +d 2 +sinh 2 d 3 2 ) boundary = R t £ S 3 Poincare coordinates: ds 2 =R 2 [r 2 (-dt 2 + dx 2 + dy 2 + dz 2 )+ r -2 dr 2 ] boundary = R t £ R 3

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

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Why AdS Throat = D3-Brane? D-brane absorption cross-section: SUGRA computation = SYM computation N D3-branes = flat flat AdS 5 AdS_5 “interior” in gravity description = N D3-branes in gauge description

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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 S Dinstanton = -(1/g st ’ 2 ) Tr Q [ 1, 2 ] 2 + …. » QL 2, » Q 2 g st ’ 2 / L 2 rotational symmetry implies L 4 = Q g st ’ 2 = N g st ’ 2

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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 AdS d scales as X d-1 !

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Entropy Counting (3+1) SYM on V 3 with UV cutoff a AdS 5 gravity on V 3 with UV cutoff a

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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 AdS 5 Let’s compute interaction potential between quark and antiquark Do we obtain physically reasonable answers?

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Static Quark Potential at Zero Temperature Notice: Square-Root --- non-analyticity for exact 1/r --- conformal invariance

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Notice: R 4 =g st N ’ 2 ’ 2 cancels out!

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Holography (boundary = bulk) r=0 AdS radial scale r= 1 YM distance scale Anything that takes place HERE (AdS 5 ) is a result of that taking place HERE (R 3+1 )

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

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What have we evaluated? rectangular Wilson loop in N=4 SYM W[C] = Tr P exp s C (i A m dx m + a d y a ) gauge field part = Aharonov-Bohm phase scalar field part = W-boson mass unique N=4 supersymmetric structure with contour in 10-dimensions

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T=0 (zero-temperature 4d N=4 SYM): ds 2 = r 2 (-dt 2 +dx 2 ) + dr 2 /r 2 + (dS 5 ) 2 r = 5th dim // 4d energy scale: 0 < r < 1 T>0 (finite-temperature 4d N=4 SYM): ds 2 = r 2 (- F dt 2 +dx 2 ) + F -1 dr 2 /r 2 +(dS 5 ) 2 F = (1 - (kT) 4 /r 4 ): kT < r < 1 5d AdS Schwarzschild BH = 4d heat bath T=0 vs. T>0 SYM Theory

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

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

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

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“thermal S 1 ” 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 modesNote: 4d space-time, topology: gravity // gauge Application: D3-branes on “thermal” S 1

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

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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) * ** *** [M. Teper] other glueballs fit reasonably well (why??) YM 3+1 glueball spectrum

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The Story of Square-Root branch cut from strong coupling? artifact of N 1 limit heuristically, saddle-point of matrices = s [dM] (Tr e M ) exp (- -2 Tr M 2 ) Recall modified Bessel function

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