MIT OpenCourseWare http ocw mit edu 8 821 String Theory Fall 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 8 821 F2008 Lecture 19 Lecturer McGreevy November 17 2008 Pointlike probes of the bulk Geometric optics through D branes Baryons and Branes Baryon vertices in N 4 SU N gauge theory Giant gravitons Survey of other examples of the correspondence 1 Geometric Optics via D Branes Last time we observed that one could compute observables associated to k dimensional submanifolds C AdS by studying exp Extremum Vol C k 1 AdS in analogy with Wilson lines The simplest example is the case k 0 when C is a discrete point set This would lead us to consider one dimensional curves in AdS This one dimensional curve is the worldline of some point particle excitation in AdS Let C a a be the discrete point set and denote a curve parametrized as z with z 0 a and z 1 a as shown in Figure 1 The observable associated with C is clearly the two point function of some local operator O This local operator couples to a scalar eld of mass m in the gravity theory The usual prescription involves computing the two point function in the bulk and nding the on shell action Since we have a free scalar eld we can instead use a rst quantized approach The action for a point particle of mass m whose world line is is given by S z m ds The two point function for this particle z 0 z 1 is given by the Feynman path integral Z a dz exp S z 10 In the limit of large m we have Z a exp S z 1 where we have used the saddle point approximation z is the geodesic connecting the points a on the boundary 1 a a z Figure 1 The curve AdS connecting the two points in C The arrows on the curve indicate the orientation We now compute Z a in the saddle point approximation The metric restricted to is given by 2 ds2 Lz2 1 z 2 d 2 where z dz d This implies the action is L S z d 1 z 2 2 z The geodesic can be computed easily by noting that the action S z does not depend on explicitly This implies we have a conserved quantity L L 1 h z L z 1 z 2 z 2 L 1 z 2 z2 2 h z2 3 4 2 The above is a rst order di erential equation with solution zmax z 2 with z 0 zmax L h a This is the equation of a semi circle Substituting the solution back into the action gives a L dz a 2 S z d 1 z 2L 5 2 2 z a z z 2a 2L log terms 0 as 0 6 Recall that the expectation value of a circular Wilson loop of radius a was independent of a We argued that since we are computing this in a conformal eld theory the only scale in the problem is a and therefore the expectation value of the Wilson loop should be independent of a This argument fails in the case at hand because there are two scales a and The scale transformation is anomalous and this is manifested in the dependence of S z According to the GKPW prescription the generating functional Z a with the prescribed bound ary conditions corresponds to the two point function O a O a The saddle point approxima tion of large m corresponds to large in the CFT Therefore O a O a Z a exp S z 1 a2mL 7 This is exactly what we expect for the two point function since m 2 L2 D 2 in the large limit Without this anomaly the two point function of the operator O would be independent of 2 Figure 2 N F strings ending on a D5 brane at the origin 0 in global coordinates of AdS 5 The orientation of the strings is indicated with an arrow a forcing 0 which is impossible in a unitary CFT This is similar to the anomaly that gives a non zero scaling dimension of k 2 4 to the operator exp ikX for the 2d free boson CFT which has classical scaling dimension 0 Graham and Witten 1 showed that such a scale anomaly exists for surface observables for any even k This relation between correlators of large observables and geodesics in the bulk theory o ers a probe of the bulk spacetime 2 2 Baryons and branes In the previous section we considered a very heavy point particle excitation in the bulk theory The question that naturally arises is Who are these heavy objects A good candidate would be the D0 brane which is extremely heavy at weak coupling but in Type IIB there are no BPS D0 branes This is because the IIB theory has only even form RR potentials In the absence of a conserved charge this object would decay into a pu of closed string states The next best thing we could consider are Dq branes wrapping q cycles of S 5 The only closed cycles on S 5 are the 0 cycle and the 5 cycle This leaves us with only one choice the D5 brane wrapping the S 5 This would correspond to a point particle in AdS 5 The tension of the D5 brane is T5 1 gs ls6 Naively one would expect that the wrapped D5 brane would have a mass T 5 Vol S 5 L5 gs ls6 N 3 2 L However this turns out to be incorrect Recall from Problem Set 1 that the action of the D5 brane includes the term SD5 Fwv C 4 Awv F 5 M6 8 M6 where M 6 is the worldvolume of the D5 brane which happens to be S 5 worldline in AdS5 The subscript wv indicates that the elds live on the worldvolume of the D5 brane There are N units 3 B v Figure 3 Electric dipole moving in a magnetic eld of ve form ux threading the S 5 that stabilize its volume This implies that SD5 N Awv Worldline 9 The worldvolume gauge eld has N negative point charge sources induced by the ve form ux Since the space S 5 is compact this implies that we must have N positive point charge sources Recall that the fundamental string that ends on a D brane is electrically charged under the worldvolume gauge eld Thus we can cancel the net charge by attaching N F strings to the D5 brane Since these F strings can t end anywhere they extend all the way to in nity Figure 2 illustrates this state in global coordinates The attached strings have a mass N dz z 2 which is in nite and proportional to N This object is not a dynamical point particle in IIB gravity theory on AdS 5 since it is in nitely massive It does not couple to a nite operator in …