MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics String Theory 8 821 Prof J McGreevy Fall 2008 Problem Set 1 Reading 4 of d Hoker Freedman http arXiv org pdf hep th 0201253 Due Tuesday September 23 2008 1 Branes ending on branes The Dp brane effective action contains a term of the form Z S F Cp 1 Dp where Cp 1 is 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 Gauss law for the RR fields involved Interpret the boundary of the D p 2 brane in terms of the worldvolume theory of the Dp brane If you like focus on the case p 3 2 Timelike oscillators are evil Show that the commutation relation a a 1 which we found for the oscillators made from the time coordinates of the string implies that either a the energy H a a E0 1 is unbounded below if you treat a as the annihilation operator or b 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 extremal RN black hole works a Consider Einstein Maxwell theory in four dimensions with action Z 1 1 4 d x g R F F SEM 16 GN 4 1 The 1 comes from g 00 1 1 EM implies that Show that the Einstein equation 0 S g 1 2 R aGN 2F F g F 2 for some constant a b Consider the ansatz ds2 H 2 dt2 H 2 d 2 2 d 22 F bdt d H 1 EM where b is some constant Show that the Einstein equation 0 S and g SEM Maxwell s equation 0 A are solved by the ansatz if H is a harmonic function on the IR3 whose metric is ab dxa dxb d 2 2 d 22 Recall that H is harmonic iff 0 H 1 a ab b H c Find the form of the harmonic function which gives a spherically symmetric solution fix the two integration constants by demanding that i the spacetime is asymptotically flat and ii the black hole has charge Q meaning R F Q S 2 at f ixed d Take the near horizon limit Show that the geometry is AdS2 S 2 Determine the relationship between the size of the throat and the charge of the hole If you get stuck on this problem see Appendix F of Kiritsis book d If you re feeling brave add some magnetic charge to the black hole You will need to change the form of the gauge field to F bdt dH G 2 where 2 is 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 the equations of motion The action for type IIB supergravity when only the metric and the RR 5 form and possibly the dilaton are nontrivial can be written as Z 1 1 5 5 10 2 d x g e SIIB R 4 F F 16 GN 5 2 The self duality constraint F 5 F 5 must be imposed as a constraint and means that dF 5 0 implies the equations of motion for F 5 By the way this is the action for the string frame metric a Show that the equations of motion from this action imply 1 5 2 5 5 2 5F F g F R aGN e 2 for some constant a b Plug the following ansatz into the equations of motion ds2 p 1 H r dx dx p H r dy 2 F b 1 dt dx1 dx2 dx3 dH 1 0 b 0 are constants Determine the constant b and the condition on the function H for this to solve the equations of motion To do this there are two options some kind of symbolic algebra program like Mathematica or Maple or index shuffling by hand The latter is much more easily done using tetrad or vielbein methods I always forget these and have to relearn them every time For a lightning review of the vielbein method of computing curvatures I recommend d Hoker Freedman http arXiv org pdf hep th 0201253 pages 100 101 or Argurio http arXiv org pdf hep th 9807171 Appendix C To help with the former option I ve posted an example curvature calculation in Mathematica on the pset webpage Note by the way that for values of p other than 3 the dilaton is not constant With hindsight this specialness of p 3 is related to the fact that this is the critical dimension for YM theory where gY M is dimensionless 3