Selected Topics in AdS/CFTlecture 1

Soo-Jong ReySeoul National University

1st Asian Winter School in String TheoryJanuary 2007

Review of AdS5/SYM4 Wilson/Polyakov loops Baryons, Defects Time-Dependent Phenomena

Topics: Loops, Defects and Plasma

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

open string channel time closed string channel

gauge dynamics gravity dynamics

Channel Duality

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

elementary excitations: open string + closed string solitons: D-branes (quantum) + NS-branes (classical) Mach’s principle: spacetime à elementary + soliton sources “holography”: gravity fluctuations = source fluctuations

Tool Kits for AdS/CFT

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 + H3

2] > g-2st sS3 e-2 H3

F-string:

E = s [g-2st (r )2 + g2

st K72] > g0

st 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-1

st 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

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)

Dp-brane (CFT description)

infinite tension limit (’ 0) open strings rigid rods (Mw ~ r/’) (N,N) string dynamics U(N) SYM(P+1)

L = g-2YM Tr ( Fmn

2 “||” + (Dm a)2 + [a, b]2 “?” + …. )

SYMp+1 at Low-Energy

semi-infinite string (M ~ 1) at rest or constant velocity (static source) labelled by a single Chan-Paton factor ( ) = (i) heavy (anti)quarkin (anti)fundamental rep.

Static source: heavy quarks

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 = h m gst

2h – 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 nonperturbativ

ely (evidence?)

D3-Brane Geometry 10d SUGRA(closed string)+4d SYM (D3-brane):

Stotal = S10d + sd4 x (-e- TrFmn2+ C4….)

Solution ds2 = Z-1/2 dx2

3+1 + Z1/2 dy26

G5 = (1+*) dVol4 Æ 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

D-brane stress tensor grows with (N/gst) At large N, curvedness grows with g2

st(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

Identifications

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

AdS5

In flat 4+2 dimensional space

ds2 = - dX02 – d X5

2 + a=14 d Xa

2 embed hyperboloid

X02 + X5

2 - a=14 Xa

2 = R2 SO(4,2) invariant, homogeneous AdS5 = induced geometry on hyperboloid <homework> derive the following coordinates

Global coordinates:

ds2 = R2(-cosh2 d2+d2+sinh2 d32)

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

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

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 fu

ndamental 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 Potentialat Zero Temperature

r

NgrV YM

2

...)254.1()(

Notice:• Square-Root --- non-analyticity for • exact 1/r --- conformal invariance

Notice:R4=gstN’2

’2 cancels out!

Holography (boundary = bulk)

r=0 AdS radial scale r=1

YM

dis

tanc

e sc

ale

Anything that takes place HERE (AdS5) ---

--- is a result of that taking place HERE (R3+1)

Heavy Meson Configuration

r=0 AdS radial scale r=oo

YM

dis

tanc

e sc

ale

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 (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

T=0 vs. T>0 SYM Theory

Static Quark Potential at Finite Temperature

)(11

...)254.1(~)( **

2 rrrr

NgrV YM

Notice:• Nonanalyticity in persists• exact 1/r persists• potential vanishes beyond r*

UV-IR relation for T>0

T>0 relation

L

U*/M

T=0 relation

Maximum inter-quark distance!

Heavy Meson Configuration (T > 0)

r=0 AdS radial scale r=oo

YM

dis

tanc

e sc

ale

bare quark

bare anti-quark

r=M

“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 modesNote:Note: 4d space-time, topology: gravity // gauge

Application:D3-branes on “thermal” S1

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]

YM2+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??)

YM3+1 glueball spectrum

The Story of Square-Root

branch cut from strong coupling? artifact of N1 limit heuristically, saddle-point of matrices

< Tr eM > = s [dM] (Tr eM) exp (--2 Tr M2) Recall modified Bessel function