Transcript Static Quark Potential via Gauge

Selected Topics in AdS/CFT
lecture 1
Soo-Jong Rey
1st
Seoul National University
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
gauge dynamics
time
closed string channel
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 à 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 ( Fmn2
+ (Dm a)2 + [a, b]2
+ …. )
“||”
“?”
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 + …)





/2)V – E
NF
planar expansion: (N
= (1/N)
2
 : nonlinear interactions
1/N: quantum fluctuations
2h-2 2 l
observable: h l (1/N) ( ) Cl, h
2h-2
2 E-V
( )
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):
2
Stotal = S10d + sd4 x (-e- TrFmn + C4….)
Solution
 ds2 = Z-1/2 dx23+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

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) $ metric gmn
Tr (FmnFmn) $ dilaton 
Tr(FmpF*np) $ Ramond-Ramond C, Cmn
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 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

Why AdS Throat = D3-Brane?
D-brane absorption cross-section:
SUGRA computation = SYM computation

N D3-branes
AdS5
=
flat
flat
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
2
4
rotational symmetry implies L = Q gst ’ = 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 D3brane
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
2
YM
g N
V (r )   (1.254...)
r
Notice:
• Square-Root --- non-analyticity for 
• exact 1/r
--- conformal invariance
Notice:
R4=gstN’2
’2 cancels
out!
r=0
AdS radial scale
r=1
YM distance scale
Holography (boundary = bulk)
Anything that takes place HERE (AdS5) ----- is a result of that taking place HERE (R3+1)
Heavy Meson Configuration
r=0
AdS radial scale
bare anti-quark
r=oo
YM distance scale
bare 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: 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
Static Quark Potential
at Finite Temperature
1 1 
V (r ) ~  (1.254...) g N    (r  r* )
 r r* 
2
YM
Notice:
• Nonanalyticity in  persists
• exact 1/r persists
• potential vanishes beyond r*
UV-IR relation for T>0
L
T=0 relation
Maximum inter-quark distance!
T>0 relation
U*/M
Heavy Meson Configuration (T > 0)
r=M
r=0
AdS radial scale
bare anti-quark
r=oo
YM distance scale
bare 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
.
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]
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??)
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