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 4 Fields in AdS 1 One more symmetry De ne xA z x so that the Poincare patch metric takes the form ds2 dz 2 d x2 dxA dxA z2 z2 Show that this metric is preserved by the following inversion symmetry xA xA xB xB i e z z 2 z x2 x z 2x x2 This is a conformal transformation which is not continuously connected to the identity Bonus question what does this do to the region of the global AdS space covered by the Poincare patch coordinates 2 Schro dinger description of AdS instabilities 1 a Consider the scalar wave equation in poincare AdS in momentum space 0 m2 with x z eik x z By rede ning the independent variable rewrite the scalar wave equation as a Scho dinger equation z2 V z z 2 z with 2 playing the role of energy Hint Rewrite z A z z and choose the function A to set to zero the term in the wave equation multiplying z 1 I learned about this trick from Sean Hartnoll If you get stuck or want to see a recent application of this technique see e g appendix A of 0810 1563 1 b A property of a eld con guration which generally determines whether it is allowed to participate in the physics i e whether it is xed by boundary con ditions or whether it can happen on its own is whether it is normalizable In Euclidean space this generally just means that the euclidean action evaluated on the con guration is nite since then it contributes to the path integral with a nite Boltzmann factor e S In Minkowski signature a eld con guration should be considered normalizible if it has nite energy for a scalar eld this energy is E TAB A nB where the integral is over some xed time slice n is a unit normal vector to is a time like killing vector and TAB is the stress tensor for the bulk eld TAB 2g g S AB Given the rede nition you found in part a what is the relationship between normalizable wave functions in the usual QM sense i e k k2 and nor malizable solutions of the AdS wave equation i e those with nite energy c Set k 0 Show that when m2 passes through the value m2BF from above this Schro dinger equation develops a single normalizable negative energy state 2 2 Recall that the BF bound is m2 m2BF 4DL2 This corresponds by the map above to a normalizable mode with imaginary frequency i e a linear instability 3 Dimensions of vector operators By studying the boundary behavior of solutions to the bulk Proca equations derived from the action m2 1 p 2 Sbulk A F F A A d x g 4 2 AdS compute the scaling dimension of the current coupled to a bulk vector eld A of arbitary mass Sbdy dp 1x A J AdS For purposes of this problem de ne to be the power of z of the subleading power law solution of the bulk equation near the boundary at z 0 2 If you want to learn more about potentials of this form see H Hammer and B Swingle Annals Phys 321 2006 306 317 arXiv quant ph 0503074v1 2 4 Saturating the unitarity bound 3 For this problem we will work in Euclidean AdS Feel free to redo the problem for the Lorenzian case using the notion of normalizability described in problem two a Consider the usual bulk action for a free scalar eld in AdSD 1 1 dD xdz g 2 m2 2 Susual 2 2 Show that for m2 L2 D4 1 Susual z That is the integral over the radial direction in this action diverges near z 0 when evaluated on a solution of the form z where is the smaller root of D L2 m2 such a solution is non normalizable On the other hand show that z gives a nite value and hence a solution with these asymptotics is normalizable With this action show that the smallest conformal dimension of a scalar operator in the dual theory is D2 b Show that the following modi ed action 1 D 1 2 d x g m Susual n SKW 2 AdS produces the same bulk equations but has the following property in the win dow of mass values we could call this the KW window D2 D2 L2 m2 1 4 4 both roots give normalizable solutions with respect to the action SKW i e SKW z What is the lowest conformal dimension you can obtain now Hint you can now consider to be the dimension of the source and to be the dimension of the operator 3 This result is from Klebanov Witten hep th 9905104 3