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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