# Selected Topics in AdS/CFT lecture 1

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Selected Topics in AdS/CFT lecture 1
Soo-Jong Rey Seoul National University 1st 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 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

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

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

Tool Kits for AdS/CFT 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

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

elementary and solitons
fixed “magnetic” charge, energy-minimizing static configurations (necessary condition for BPS states) NS 5-brane: E = s g-2st [(r )2 + H32] > g-2st sS3 e-2 H3 F-string: E = s [g-2st (r )2 + g2st K72] > g0st sS7 K7 Mass: M(string) ~ 1 ; M(NS5) ~ (1/g2st) “elementary” ; “soliton”

p-brane: in between p-brane:
E = s g-2st (r )2 + (Gp+2)2 > g-1st s 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

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)

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 ( Fmn “||” + (Dm a)2 + [a, b] “?” + …. )

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.

Perturbation theory (1)
large N conformal gauge 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

Perturbation theory (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 m gst2h – 2 (2’ p2)m Dm, h

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?)

D3-Brane Geometry 10d SUGRA(closed string)+4d SYM (D3-brane):
Stotal = S10d + sd4 x (-e- TrFmn2+ C4….) Solution ds2 = Z-1/2 dx Z1/2 dy26 G5 = (1+*) dVol4 Æ d Z where Z = (1+R4/r4); r = |y| and R4 = 4 gst N ’2 r ! 1: 10d flat spacetime r ! 0 : characteristic curvature scale = R

Identifications At large N, curvedness grows with g2st(N/gst)=gstN
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) \$ metric gmn Tr (FmnFmn) \$ dilaton  Tr(FmpF*np) \$ Ramond-Ramond C, Cmn

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

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

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

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

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

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!

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

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?

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

Notice: R4=gstN’2 ’2 cancels out!

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

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

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

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: < 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

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

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

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

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

YM2+1 glueball spectrum 0++: solve dilaton eqn=2nd order linear ode
result: N=3 lattice N=oo lattice AdS/CFT (41) (55) (input) * (9) (13) ** (17) (22) *** [M. Teper]

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

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