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

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

3. 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

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

5. 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

6. 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

7. 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

8. 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”

9. 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

10. 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)

11. 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] “?”
+ …. )

12. 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.

13. 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

14. 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

15. 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?)

16. 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

17. 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

18. 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

19. 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

20. 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

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

22. 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

23. 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

24. 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!

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

27. 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?

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

29. Notice:
R4=gstN’2
’2 cancels out!

32. 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)

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

37. 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

38. 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

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

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

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

44. 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

45. 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]

46. 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??)

47. 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