MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of PhysicsString Theory (8.821) – Prof. J. McGreevy – Fall 2008Problem Set 2N = 4 SYM, BPS propertyReading: 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 isL10= −12g2Y MtrFMNFMN− 2i¯λΓMDMλ;M, N are 10d indices; λ here is a 10d Majorana-Wel spinor (16 real com-ponents). By ‘dimensionally reduce,’ we mean consider the 10d theory on a6-torus of volume V , and restrict to field configurations which have no momen-tum along the torus (i.e. are independent of the coordinates on the torus).The 4d N = 4 lagrangian density is1L4= −1g2Y Mtr 14F2+12DµXiDµXi+i2¯λIγµDµλI−12Xi<j[Xi, Xj]2+12¯λIΓiIJ[Xi, λJ]!.Here λIare four 4d Weyl spinors (4 real components), γµare 4d gamma ma-trices, and ΓiIJare gamma matrices for SO(6 ) .2. Show that the N = 4 lagrangian explicitly breaks the diagonal abelian partU(1) ⊂ U(N = 4)Rof the R-symmetry group.3. Show that the 1-loop β-function for N = 4 SYM vanishes. If yo u don’t likearcane group theory, do it for the case when the gauge group is SU(N).1This footnote corrects the expression I wrote in lecture (the Yukawa terms were wrong).14. N = 4 ⊃ N = 1.A 4d N = 4 supersymmetric theory is a special case of a 4d N = 1 supersym-metric theory.a) D escribe the field content of N = 4 SYM in terms of multiplets of someN = 1 subgroup.b) Write the N = 4 lagra ng ia n in N = 1 superspace.c) Convince yourself that this Lagrangian has SO(6) R-symmetry (e.g. byexamining the Lagrangian written in components). Is N = 1 supersymmetryplus this R-symmetry enough to convince you that this action is actually N = 4sup ersymmetric?5. Extremal = BPS.In this problem we will consider 4d N = 2 supergravity (which has eight realsup ercharges). Along the lines of the discussion from lecture 5 one can see thatthe gravito n multiplet for this theory must also contain a spin-3/2 gravitinofield ψµ, and an abelian vector field (the ‘graviphoton’) Aµ. So this is a simplesup ersymmetric completion of the system we studied in problem set 1 problem3.The action is invariant under supersymmetry transformations which includethe following variation for the gravitino ψµ:δψµ=∇µ−14F−γµǫwhereF−≡ F−νργνγρ, F±νρ=12Fνρ± (⋆F )νρare the self-dual (SD) and anti-self-dual (ASD) parts of the field strength ofthe graviphoton field. The covariant derivative acting on spinors is∇µψ ≡∂µ+14ωabµγabψwhere ω is the spin connection, dea= −ωab∧ eband γab≡12[γa, γb].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 thesup ersymmetry variations of the fermionic fields (given above) and showing2that they vanish for some choice of spinors ǫ. How many sup ersymmetries(choices of ǫ, which are called Killing spinor s) are left unbroken?b) Show that the ‘Bertotti-Robinson’ so lution, i.e. AdS2× S2(which arises asthe near-horizon limit of the extremal RN black hole) preserves eight Killingspinors.c) If you are feeling energetic, show that AdS5× S5preserves 32 superchargesof 10-dimensional type IIB supergravity. The supersymmetry variations of thefermion fields in that case areδψµ=∇µ−i1920F−5Γµǫ + ...δλ = ...whereF+5≡ Γµ1...µ5(F5)µ1...µ5,λ is the ‘dilatino’ field and ... means terms that vanish when F5is the onlynontrivial field other than the metric (i.e. they depend on derivatives of thedilaton, and the other fluxes). If you want to know what the full variationsare, see Kiritsis eqns (H.26-28).6. W-bosons from adjoint higgsing. Using the N = 4 lagrangian, show thatgiving an expectation value to the scalars in the N = 4 theory of the for mhXi=1,...,6i = diag(xi1, . . . xiN)gives a mass to the off-diagonal gauge bosons of the formm2ab∝6Xi=1(xia− xib)2≡ |~xa− ~xb|2;this is meant to be the mass of the gauge boson with indices ab. Show that t hescalars with the same gauge charges get the same