MIT OpenCourseWare http ocw mit edu 8 821 String Theory Fall 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 8 821 F2008 Lecture 06 Supersymmetric Lagrangians and Basic Checks of AdS CFT Lecturer McGreevy September 24 2008 We are on our way to talking about really awesome things about supercool stu Before we get there though we need to develop some very powerful technology To that end today we will talk about 1 SUSY Lagrangians and a whirlwind tour of the beauties of superspace 2 more on N 4 SYM 3 Back to the Big Picture Some basic checks of AdS CFT Looking past this lecture we will be talking about strings from gauge theory next 1 N 4 SYM and Other Supersymmetric Lagrangians Recall that the eld content of N 4 SYM is a vector A gaugini I 1 4 and six scalars X i all in the adjoint of the gauge group The Lagrangian density which is completely determined by the amount of SUSY up to two parameters gY M is L 1 gY2 M 6 D tr F 2 DX i 2 i X i X j 2 X X i j i tr F F 8 2 The non zero N 4 SUSY transformations are very schematically 1 1 Q X Q F X X Q DX Q A 2 A Note There are obviously indices and gamma sigma matrices suppressed ALL over the place Lorentz vector Lorentz spinor and SO 6 vector spinor supersymmetry If you want to put the indices in either leave that as a fun exercise or check out Weinberg volume 3 As an example F F 1 1 A Superspace Detour The N 4 SYM Lagrangian is an example of a highly supersymmetric Lagrangian So far I just told you what it was and that it was SUSY invariant something you could sit down in the privacy of your o ce and check if you wanted It d be nice though if there were some sort of a machine that one could crank to generate supersymmetric lagrangians That crankable machine is superspace To understand why superspace is useful we should think about why elds are useful for representing translationally invariant Lagrangians in ordinary QFT One reason is that the representations of the translation group on the elds are particularly simple x eiP x 0 3 We d like to introduce a super eld that comes with its own supercapitalization x which is now a function of spacecoordinates x and superspace coordinates that well represents transla tional invariance AND supersymmetry x eiP x iQ 0 0 4 where the Q s are the operators that generate supersymmetry transformations Now in QFT one can automatically get translationally invariant actions S dd x L 5 as long as x L 0 Similarly the here unproven claim is that dd x dN s L x 2 6 is supersymmetric as long as L 0 Here s denotes the smallest real dimension of the spinor representation in d dimensions and N is as usual the number of supersymmetries Then N s is just the number of real supercharges For example for N 1 d 4 N s 4 because either a Weyl or Majorana spinor the minimum in four dimensions has four real components 1 1 1 BPS or Chiral Multiplets In lecture 5 we made a big deal about special representations of supersymmetry which are killed by some of the supercharges Such multiplets have correspondingly special properties in superspace Consider a eld which satis es 0 Q Q 6 0 7 These multiplets which are BPS half of the supersymmetries annihilate them are generally called chiral multiplets Sometimes they are actually chiral in the sense of the Lorentz group but often times they are not This follows a long tradition in physics of calling things other things which they are not These multiplets are functions of only half of superspace as x ei Q Q x 0 0 x 0 8 Ok well this equation is not exactly correct as Senthil pointed out really the RHS should be but it is morally correct the s are functions of half of superspace Because y x i of this it is possible to add terms to the Lagrangian density that are integrated over only half of superspace and maintain supersymmetry L d2 W d2 W 9 Here we are explicitly working in d 4 N 1 superspace These terms are supersymmetric as long as W is a holomorphic function of W 0 With this constraint W is a function we are free to choose and is known as the superpotential Two examples of a superpotential are The second line of equation 1 Here we refer to the fact that this line can be written in d 4 N 1 superspace where we pick out a particular N 1 subgroup from the N 4 In a certain sense one of the s and F can be thought of as comprising one N 1 chiral multiplet while the remaining three s and six s can be thought of as another three chiral multiplets The second line of equation 1 is a superpotential for these three chiral multiplets On pset 2 you will have a chance to think about this more precisely 3 The gauge kinetic terms and third line of equation 1 can be thought of as coming from the N 1 superpotential d2 tr 10 where the appearing is the superpartner of F is a spinor index and is a complexi ed coupling constant g42 i 2 Here should be thought of as a super eld whose lowest YM component is the gaugino The super eld expansion contains a F term Multiplying two together one gets 1 gY2 M 2 F 2 2 2 d F d2 tr 11 The rest of the gauge kinetic terms and theta angle term can be understood similarly 1 2 Holomorphy and Non Renormalization aka Seibergology What s the big deal with this newfangled superpotential Well as we said before the superpotential has to be holomorphic in order for supersymmetry to be preserved We can take this line of reasoning one step further we can think of promoting the couplings to dynamical super elds whose lowest component vevs are just the constant couplings Then the superpotential must be holomorphic also in the couplings This statement is uncomfortably powerful For example it implies that if SUSY is not broken the form of radiatively generated corrections to the superpotential are severely constrained they must be holmorphic in the elds and the couplings For example one could never generate a term in the superpotential that was a function of both and W 6 W This leads to many non renormalization theorems in supersymmetric eld theories one example of which is for the function of supersymmetric gauge theories This says that Y M 1 loop non perturbative 12 This makes sense because we know the theta angle cannot appear in the beta function perturba tively However since the e ective gauge coupling function must be holomorphic in the only perturbative contribution to the beta function can …