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 Due Thursday October 2 2007 roughly 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 components By dimensionally reduce we mean consider the 10d theory on a 6 torus of volume V and restrict to field configurations which have no momentum along the torus i e are independent of the coordinates on the torus The 4d N 4 lagrangian density is1 L4 1 gY2 M tr i 1 2 1 F D X i D X i I D I 4 2 2 2 Here I are four 4d Weyl spinors 4 real components are 4d gamma matrices 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 1 X i X j 2 I iIJ X i J 2 i j 1X 4 N 4 N 1 A 4d N 4 supersymmetric theory is a special case of a 4d N 1 supersymmetric theory a Describe the field 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 field and an abelian vector field 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 field strength of the graviphoton field 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 fields 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 fields in that case are i F 1920 5 where F5 1 5 F5 1 5 is the dilatino field and means terms that vanish when F5 is the only nontrivial field other than the metric i e they depend on derivatives of the dilaton and the other fluxes 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 off diagonal gauge bosons of the form m2ab 6 X 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