MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of PhysicsString Theory (8.821) – Prof. J. McGreevy – Fall 2008Problem Set 1Reading: §4 of d’Hoker-Freedman http://arXiv.org/pdf/hep-th/0201253Due: Tuesday, September 23, 2008.1. Branes ending on branes.The Dp-brane effective action contains a term of the formS ∋ZD pF ∧ Cp−1,where Cp−1is the RR p −1 form, which couples minimally to D(p −2)-branes.Show that a D(p −2) brane can end on a Dp brane without violating the Gausslaw for the RR fields involved. Interpret the boundary of the D(p-2)-brane interms of the worldvolume theory of the Dp brane. (If you like, focus on thecase p = 3.)2. Timelike oscillators are evil.Show that the commutation relation [a, a†] = −1 (which we found for theoscillators made from the time coordinates of the string) implies that eithera) the energy H = −a†a + E01is unbounded below (if you treat a†as theannihilation operator)orb) there are states with negative norms.3. Extremal Reissner-Nordstrom black hole.As a warmup for the 10-d RR soliton, let’s remind ourselves how the extremalRN black hole works.a) Consider Einstein-Maxwell theory in four dimensions, with actionSEM=116πGNZd4x√gR −14FµνFµν1The −1 comes from g00= −1.1Show that the Einstein equation 0 =δSEMδgµνimplies thatRµν= aGN2Fµ.F.ν−12gµνF2for some constant a.b) Consider the ansatzds2= H−2(ρ)−dt2+ H2(ρ)dρ2+ ρ2dΩ22,F = bdt ∧dH(ρ)−1where b is some constant. Show that the Einstein equation 0 =δSEMδgµνandMaxwell’s equation 0 =δSEMδAµare solved by the a nsatz if H is a harmonicfunction on the IR3whose metric isγabdxadxb:= dρ2+ ρ2dΩ22.Recall that H is harmonic iff 0 = H =1√γ∂a(√γγab∂bH).c) Find the form o f the harmonic function which gives a spherically sym-metric solution; fix the two integration constants by demanding that i) thespacetime is asymptotically flat and ii) the black hole has charge Q, meaningRS2at fixed ρ⋆F = Q.d) Take the near-horizon limit. Show that the geometry is AdS2× S2. De-termine the relationship between t he size of the throat and the charge of thehole.[If you get stuck on this problem, see Appendix F of Kiritsis’ book.]d) If you’re feeling brave, add some magnetic charge t o the black hole. Youwill need to change the form of the gauge field toF = bdt ∧ dH(ρ) + G(ρ)Ω2where Ω2is the area 2-form on the sphere, and G is some function.4. RR soliton.In this problem we’re going to check that the RR soliton is a solution of theequations of motion. The action for type IIB supergravity, when only the metricand the RR 5-form and possibly the dilaton are nontrivial can be written asSIIB=116πGNZd10x√ge−2Φ(R + 4∂µΦ∂µΦ) −15!F5.....F5 .....+ ...2(The self-duality constraint F5= ⋆F5must be imposed as a constraint, andmeans that dF5= 0 implies the equations of motion for F5.) By the way, thisis the action for the string frame metric.a) Show that the equations of motion from this action implyRµν= aGNe2Φ5F5µ....F5 ....ν−12gµν(F5)2for some constant a.b) Plug the following ansatz into the equations of motion:ds2=1pH(r)ηµνdxµdxν+pH(r)dy2F = b(1 + ⋆)dt ∧ dx1∧ dx2∧ dx3∧ dH−1Φ = φ0(b, φ0are constants.) Determine the constant b and the condition on the func-tion H for this to solve the equations of motion.To do this, there are two options – some kind o f symbolic algebra program likeMathematica or Maple, or index-shuffling by hand. The latter is much moreeasily done using ‘tetrad’ or ‘vielbein’ methods. I always forget these and haveto relearn them every time. For a lightning review of the vielbein method ofcomputing curvatures, I recommendd’Hoker-Freedman http://arXiv.org/pdf/hep-th/0201253, pages 100-101,or Argurio http://arXiv.org/pdf/hep-th/9807171, Appendix C. To helpwith the former option, I’ve posted an example curvature calculation in Math-ematica on the pset webpage.Note, by the way, that for values of p o ther than 3, the dilaton is not constant.With hindsight, this specialness of p = 3 is related to the fact that this is thecritical dimension for YM theory, where gY Mis