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 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics String Theory 8 821 Prof J McGreevy Fall 2008 Problem Set 5 Finally we can calculate 1 Bulk vector elds a Use the inversion trick to show that KAM w x cD w0D 1 J M w x w x 2 D 1 A where wA w0 w A labels a point in the bulk of AdSD 1 is the bulk to boundary propagator for a massless vector eld i e this object solves the bulk Maxwell equations and D w x as w0 0 Here JAM x AM 2 Show that xA x A A x with x 2 xM x is the Jacobian for the inversion transformation JAM x so this JAM xA xM x2 2 b Check that the gravity calculation of three point function O x1 O x2 J x3 J is the conserved current to which the bulk gauge eld AA couples has the form required by conformal symmetry given in lecture c extra bonus problem Show that the form of the three point function above is determined by conformal invariance up to a constant 2 Relation between AdS propagators Show that the bulk to boundary and boundary to boundary propagators for a scalar eld in AdS are related by G z x z x z K z x x lim 1 Hint use Green s second identity 2 2 g m m U n n U with G K The G appearing in this relation is de ned to be the normalizable solution to the wave equation with source 1 x m2 G z x z x D x x z z g which is regular in the interior K is de ned to be the solution to the homoge neous wave equation which is regular in the interior and approaches lim K z x x D x x z Note it is possible to show this using properties of the hypergeometric function appearing in the explicit expression for G this proof is not so illuminating and requires actually knowing the exact solution 3 Wilson line with cusp Use the AdS CFT duality to compute the strong coupling behavior of W v CF T i e the vev of a Wilson loop in the fundamental associated to a curve de scribed by a v of opening angle in a CFT with a AdS string dual Show that the renormalized expecation value behaves like ln W v ren CF T ln L 4 where is some function of the opening angle which vanishes as approaches this means that the discontinuity in the line is small and L are UV and IR cuto s on the radial coordinate in AdS i e z L in the coordinates we have been using 4 Surface gravity Compute the periodicity of y for which this metric is regular at z zm i e has no conical de cit dz 2 2 2 2 ds z f z dy dsother f z 2 where f z is a function with a rst order zero at z zm i e z f zm 0 and z is regular and non vanishing at z zm If we think of y as imaginary time this periodicity determines the temperature of the black hole since nite temperature means periodic Euclidean time a Specialize your answer to the case of the euclidean Schwarzchild black hole in at space for which f z 1 2GM 1 ds2other z 2 d x2 z b Specialize your answer to the case of the euclidean AdS black hole with planar horizon for which f z 1 z4 1 ds2other d x2 4 zm z2 This geometry is also relevant to the model of con nement obtained by com pactifying a supersymmetric gauge theory on a circle with supersymmetry breaking Scherk Schwarz boundary conditions c Show that you get the same answer by computing the surface gravity of the horizon the locus z zm which can be de ned by 1 2 a b c d gbd g ac z zm 2 where a is the tangent vector to the shrinking circle y 3
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