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 2 N 4 SYM BPS property Reading D Hoker Freedman 2 4 Polchinski vol II appendix B 1 How to remember the N 4 action Show that ten dimensional N 1 SYM dimensionally reduces to 4d N 4 The 10d lagrangian density is L10 1 tr FM N F M N 2i M DM 2 2gY M M N are 10d indices here is a 10d Majorana Wel spinor 16 real com ponents By dimensionally reduce we mean consider the 10d theory on a 6 torus of volume V and restrict to eld con gurations which have no momen tum along the torus i e are independent of the coordinates on the torus The 4d N 4 lagrangian density is1 1 1 2 1 i 1 1 L4 2 tr F D X i D X i I D I X i X j 2 I iIJ X i J gY M 4 2 2 2 i j 2 Here I are four 4d Weyl spinors 4 real components are 4d gamma ma trices and iIJ are gamma matrices for SO 6 2 Show that the N 4 lagrangian explicitly breaks the diagonal abelian part U 1 U N 4 R of the R symmetry group 3 Show that the 1 loop function for N 4 SYM vanishes If you don t like arcane group theory do it for the case when the gauge group is SU N 1 This footnote corrects the expression I wrote in lecture the Yukawa terms were wrong 1 4 N 4 N 1 A 4d N 4 supersymmetric theory is a special case of a 4d N 1 supersym metric theory a Describe the eld content of N 4 SYM in terms of multiplets of some N 1 subgroup b Write the N 4 lagrangian in N 1 superspace c Convince yourself that this Lagrangian has SO 6 R symmetry e g by examining the Lagrangian written in components Is N 1 supersymmetry plus this R symmetry enough to convince you that this action is actually N 4 supersymmetric 5 Extremal BPS In this problem we will consider 4d N 2 supergravity which has eight real supercharges Along the lines of the discussion from lecture 5 one can see that the graviton multiplet for this theory must also contain a spin 3 2 gravitino eld and an abelian vector eld the graviphoton A So this is a simple supersymmetric completion of the system we studied in problem set 1 problem 3 The action is invariant under supersymmetry transformations which include the following variation for the gravitino 1 F 4 where 1 F F 2 are the self dual SD and anti self dual ASD parts of the eld strength of the graviphoton eld The covariant derivative acting on spinors is 1 ab ab 4 F F F where is the spin connection dea ab eb and ab 12 a b a Show that the extremal RN black hole from problem set 1 lies in a short BPS multiplet of the N 2 supersymmetry Do this by examining the supersymmetry variations of the fermionic elds given above and showing 2 that they vanish for some choice of spinors How many supersymmetries choices of which are called Killing spinors are left unbroken b Show that the Bertotti Robinson solution i e AdS2 S 2 which arises as the near horizon limit of the extremal RN black hole preserves eight Killing spinors c If you are feeling energetic show that AdS5 S 5 preserves 32 supercharges of 10 dimensional type IIB supergravity The supersymmetry variations of the fermion elds in that case are i F 1920 5 where F5 1 5 F5 1 5 is the dilatino eld and means terms that vanish when F5 is the only nontrivial eld other than the metric i e they depend on derivatives of the dilaton and the other uxes If you want to know what the full variations are see Kiritsis eqns H 26 28 6 W bosons from adjoint higgsing Using the N 4 lagrangian show that giving an expectation value to the scalars in the N 4 theory of the form hX i 1 6 i diag xi1 xiN gives a mass to the o diagonal gauge bosons of the form m2ab 6 xia xib 2 xa xb 2 i 1 this is meant to be the mass of the gauge boson with indices ab Show that the scalars with the same gauge charges get the same mass 3