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 17 More on 3 point functions the chiral anomaly and Wilson loops Lecturer McGreevy November 24 2008 1 Introduction 1 Finish OO J 2 Anomalies 3 Wilson loops 2 Calculation of a 3 point function Consider a CFT with a conserved U 1 current J The operator O is an operator in the CFT with scaling dimension that couples to a charged scalar eld in the bulk while the complex Suppose that there is a massless vector eld A in conjugate of this bulk scalar eld couples to O the bulk that couples to J The minimal coupling of this charged scalar eld to the vector eld leads to a vertex i g AB w which gives the leading contribution to the three point function G 123 O x1 O x2 J x3 D d wdw0 AB i g w K w x1 B K w x2 KA w x3 D 1 w w0 1 The double arrow means take the derivative acting to the right minus the derivative acting to the left K is the bulk to boundary propagator for the charged scalar eld and KA w x3 is the bulk to boundary propagator for the gauge boson The bulk to boundary propagator for the vector eld solves the bulk Maxwell equations and goes to a delta function at the boundary it goes to w x as w0 0 The Witten trick that we used to deduce the bulk to boundary propagator for the scalar eld can also be used to obtain the bulk to boundary propagator of the vector eld We claim that it implies KAB w x CD w0D 1 J B w x w x 2 D 1 A 1 2 were CD is some normalization constant and JAB is a Jacobian for inversion it is given by B JAB x A 2 xA xB x2 3 A xA A x which can be determined from JAB x 2 x This Jacobian arises because for the with x x 2 B Witten trick one rst nds the bulk to boundary propagator with the singularity at in nity and then performs an inversion to put the singularity at some nite value Using the inversion trick and doing the w integral yields G 123 S x1 x2 x3 D 2 D 2 D 2 D 2 4 S is the unique combination of x1 x2 and x3 allowed by conformal symmetry and is given by 1 x13 x 23 1 S x1 x2 x3 2 D 2 5 2 x2 D 2 D 2 x x12 x13 x23 13 23 Using the Ward identity G i x13 x23 OO x 3 123 6 we can determine the normalization of the two point function OO 2 D 1 2 D 2 D 2 x12 This is the k space calcualtion of this 3 point function it is 2 D 7 the position space answer Note the subtlety of the extra factor of 2 D which occurs only for two point functions because the dz is dicey For higher point functions this issue does not arise This factor comes from the following Recall our expression for the two point function Gz b0 z z z D 2 K kz a0 z b0 D 2 K k a 0 b0 diverging term a0 8 K is a Bessel function and we have used K k ab00 and D 2 The dependence on k is absorbed into b0 a0 The divergence corresponds to some kind of contact interaction that we don t care about Because of this divergence it is possible for the subleading term to actually matter This is where the coe cient comes from Witten s position space calculation takes the limit z 0 before taking this ratio so this factor does not appear in his position space calculation 3 Gravity dual of chiral anomaly Scale invariance and the SU 4 R symmetry of the N 4 Super Yang Mills theory are exact symmetries if the theory is de ned on at space with no sources But if we couple the SU 4 R 2 currents to some external gauge eld A which we can think of as the boundary value of some bulk gauge eld in AdS or if we have an external background metric which we may think of as the boundary value of the metric in AdS then there can be anomalies This is because the gauginos of SU 4 R So we may have anomalies in the trace of the stress tensor 4 the s are in the 4 T or in the R current The parity odd part of the three point function of three R currents in the N 4 SU N theory is N2 1 Tr 5 x 12 x 23 x 31 id 9 abc 32 6 x412 x423 x431 The subscript refers to the parity odd bit of the correlator and dabc 2T r Ta Tb Tc This three point correlation function implies that if we couple the R current to some background gauge eld then the divergence of the R current will be Ja x1 Jb x2 Jc x3 N2 1 b c idabc F F 384 2 This is one loop exact by the Adler Bardeen theorem D J a 10 Now we would like to see how to compute this anomaly at strong coupling using the gravity dual In AdS5 Ja couples to some SO 6 Kaluza Klein gauge eld Aa The statement of the anomaly is that the boundary symmetry generated by the R current J is broken by the non zero eld strength of the background gauge eld This global R symmetry is generated by gauge transformations in the bulk Aa Aa D a where the gauge parameter does not vanish at the boundary The e ective action for Type IIB supergravity includes SKK SY M A SCS A 11 where SCS is a Chern Simons term iN 2 SCS A 96 2 T rA F F 12 AdS5 Note that T rA F F d5 xdabc B1 B2 B3 B4 B5 AB1 FB2 B3 FB4 B5 The Chern Simons term arises because Type IIB supergravity breaks parity The basic reason for this parity breaking is that the ve form eld strength is self dual not anti self dual This Chern Simons term is gauge invariant in the bulk i e it is gauge invariant up to a total derivative 2p 1 SCS T rF F 13 A so under A A D the change in the action is iN 2 b c SCS d4 x dabc a F F 14 384 2 AdS Here we have assumed dF 0 or that there is no magnetic charge Alternatively since A couples to the R current J Sef f d4 xD a Ja d4 x a D Ja 15 so can read o the anomaly Du J a Notice that the coe cent matches at large N ignore 1 in N 2 1 The …