9 p-brane: in between p-brane: E = s g-2st (r )2 + (Gp+2)2 > g-1st s e- Gp+2M(p-brane) » (1/gst)p-brane = “quantum soliton”:more solitonic than F-stringbut less solitonic than NS5-braneF-string $ p-brane $ NS5-branequantum treatment of p-brane is imperative
10 Dp-brane (CFT description) strings can end on itopen stringstring endpoints labelledby Chan-Paton factors( ) = (i,j)mass set by disk diagramM ~ 1/gst---- identifiable with p-braneQL = QR (half BPS object)
12 Static source: heavy quarks semi-infinite string(M ~ 1)at rest or constant velocity(static source)labelled by a singleChan-Paton factor( ) = (i)heavy (anti)quarkin (anti)fundamental rep.
13 Perturbation theory (1) large N conformal gauge theory :double expansion in 1/N and 2 = gYM2 NSYM = (N /2) s d4 x Tr (Fmn2 + (Dm )2 + …)planar expansion: (N /2)V – E NF = (1/N)2h-2 (2)E-V2 : nonlinear interactions1/N: quantum fluctuationsobservable: 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 fluctuationSstring= (½’) s d2 X ¢ Xgst : string quantum loop fluctuationSSFT = * 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 structuregst $ (1/N)(’/R2) $ 1/ where R = characteristic scaleMaldacena’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….)Solutionds2 = Z-1/2 dx Z1/2 dy26G5 = (1+*) dVol4 Æ d Z whereZ = (1+R4/r4); r = |y| and R4 = 4 gst N ’2r ! 1: 10d flat spacetimer ! 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)=gstNNear D3-brane, spacetime= AdS5 x S54d D3-brane fluct. $ 10d spacetime fluct.(from coupling of D3-brane to 10d fields) Tr (FmpFnp) $ metric gmnTr (FmnFmn) $ dilaton Tr(FmpF*np) $ Ramond-Ramond C, Cmn
18 AdS/CFT correspondence Dirichlet problem in AdS5 or EAdS5=H5Zgravity = 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 hyperboloidX02 + X52 - a=14 Xa2 = R2SO(4,2) invariant, homogeneousAdS5 = induced geometry on hyperboloid<homework> derive the following coordinates
21 holographic scaling dimensions wave eqn for scalar field of mass m2 solns:normalizable vs. non-normalizable modes
22 Why AdS Throat = D3-Brane? D-brane absorption cross-section:SUGRA computation = SYM computationN D3-branes=flat flatAdS5AdS_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 regionSDinstanton = -(1/gst ’2) TrQ[1, 2]2 + ….< Tr(1)2 > » QL2, <Tr(2)2 > » Q2 gst ’2 / L2rotational 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]4Extensive quantities in 5d AdS gravity scales as [length]5So, 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-branenow strings are stretched and fluctuates inside AdS5Let’s compute interaction potential between quark and antiquarkDo we obtain physically reasonable answers?
28 Static Quark Potential at Zero Temperature Notice:Square-Root non-analyticity for exact 1/r conformal invariance
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 phasescalar field part = W-boson massunique N=4 supersymmetric structure withcontour 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)2r = 5th dim // 4d energy scale: < r < 1T>0 (finite-temperature 4d N=4 SYM):ds2 = r2 (- F dt2+dx2) + F-1 dr2/r2+(dS5)2F = (1 - (kT)4/r4): kT < r <15d AdS Schwarzschild BH = 4d heat bath
39 Static Quark Potential at Finite Temperature Notice:Nonanalyticity in persistsexact 1/r persistspotential vanishes beyond r*