Lecture 25 8 251 Spring 2007 Last time considered e ects of Cal Baron eld Action S 1 S 2 1 x x 1 2 2 2 B x 2 d d X X X X d d dD xH H 2 6k First term String action Second term Interaction Third term Coupling to current Interaction term rewritten as x d d x x X dD xB x J x where current J x 1 H J k 2 x Antisymmetric in and J J J 0 x x Conservation index so we seem to have a collection of conserved currents J conservation index labels for various currents J 0k string charge densities to nd charge integrate over space B0k couples to J0k Conservation equation for 0 J 0 J 0 J 0k 0 0 0 x xk x 1 Lecture 25 8 251 Spring 2007 J o J 01 J 02 J 0d J 0 is a vector so J 0k J 0 k 0 0 J 0 0 k x xk String charge lives only on the string Magnetic elds help us but not a per fect analogy Perfect analogy string charge is like stationary electric currents Remember current conservation from E M J 0 t Stationary currents have t 0 J 0 e g a current on a closed loop or an in nite wire A current that e g ends at a capacitor to charge it is not stationary Open strings are problematic charge owing in string accumulates at ends 1 J t x 2 0k x0 xk xk x0 t x0 x x d d Static gauge x0 J 0k 1 2 1 2 J0 d x x t xk d x x t x If we have a closed string we ll have a string charge density vector everywhere along the string Now we know the direction of this vector along the direction of increasing String charge behaves like stationary electric current J ik for a static string is 0 It has to do with the velocity of the string Consider a static string in 3 1 dimensions could be an in nitely long string interesting and simple case Must solve 1 H J 2 x 2 Lecture 25 8 251 Spring 2007 Assume all H s are time independent with H ijk 0 Static string so really two equations are i j 1 H ij J ij 0 2 x X ij0 H ijk 0 0 x xk Expanding over index which can take values 0 and spatial 1 H 0 J 0 2 x0 Totally antisymmeteric so can t have other 0 indices so all other indices spatial 1 H 0kl J 0k 2 xl Let s recast this equation as something more familiar Let H 0kl 2 k0m B m Recall 123 1 Any reversal of index order changes its sign Plug in 1 2 k0m B m J 0k 2 xl klm l B m J 0k This is the familiar B k J 0 k B J 0 3 Lecture 25 8 251 Spring 2007 So nding Kalb Raman eld of a magnetic eld is mathematically equivalent to nding the electric current the magnetic eld B whose E M We have H whose E M analogue is B 0k vector potential J whose E M analogue is J current analogue is A Action for coupling to Kalb Raman eld x x d d B x 1 d d x x B x 2 SB where 1 2 and 12 1 21 1 and 1 2 and are coordinate indices on the worldsheet q A dx coupling of E M to a point charge Gauge invariant Yes Things made with A have a hard time being gauge invariant Reason above is gauge invariant q x d Let A x x q d x q x d q A dx q A x Gauge invariant if vanishes in past or future Good enough B B x x x x 4 Lecture 25 8 251 Spring 2007 SB d d x x x d d x d d x d d x SB d x x 0 Consider now a string ending on a D brane What kind of violation of the gauge invar will we get x m xm a xa 0 since xa at string ends is constant SB m d m xm m x 0 String cannot end on the D brane We ve accumulated so much evidence that this makes sense but then we get stuck Here s where we need inspiration 5 Lecture 25 8 251 Spring 2007 Possible hints String conservation charge conservation Maybe string ends not so innocent maybe they re electrically charged this would be good then string theory would include electric charge a very necessary element of a phys ical theory Approach Let s believe string endpoints are charged Say has a positive charge 0 has a negative charge S SB d Am x xm d Am x xm 0 B Bmm m n n m Am m Kalb Maron parameter is changing the magnetic eld Outrageous but neces sary Then get gauge invariant Such a strong gauge transformation might wonder if our cure is worse than the disease Not so S SB SEM is gauge invariant But what have we done to Maxwell Not so severe Fmn used to be gauge invar Now Fmn m An n Am m n n m Not gauge invariant but Fmn Bmn So Fmn Bmn is gauge invariant So in string theory with Kalb Raman elds can t just use F must use 6 Lecture 25 8 251 Spring 2007 mn Fmn Bmn gauge invar This looks good Bmn gravitational eld of closed string very small usually So practically mn Fmn most of the time Before 1 SEM F F 4 Now must write 1 1 1 1 SEM mn mn Fmn F mn Bmn B mn Fmn B mn 4 4 2 4 Recall string charge B 0k J0k So F0k couples to B 0k electric elds now have string charge Charge at string endpoints create electric elds in the brane that continue to carry string charge Maybe string is made of electric eld lines all bunched up together that can y in out on the brane But strings not just made of eld lines doesn t quite work This model implies that particle antiparticle annihilation are a closed string go ing o the brane Quarks and QCD Consider 3 D branes for 3 colors 7 Lecture 25 1 2 3 4 8 251 Spring 2007 Open string ending on the red brane is a red quark Blue quark Green quark Red anti quark going away Add in weak brane and leptonic branes Call all these branes D4 branes lling world 8 Lecture 25 8 251 Spring 2007 But string theory has extra dimensions But all those branes on a torus This model found to support all the particles we know and more we don t Not a complete story since we need symmetry breaking Maybe there s supersym metry so need some Higgs boson from tachyons from intersecting branes Not a 100 model but …