8.821 F2008 Lecture 06: Supersymmetric Lagrangians and BasicChecks of AdS/CFTLecturer: McGreevy Scribe: David GuarreraSeptember 24, 2008We are on our way to talking about really awesome things about supercool stuff. Before we getthere, though, we n eed to develop some very powerful techn ology. To that end, today we will talkabout1. 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/CFTLooking past this lecture, we will be talking about strings from gauge theory next.1 N = 4 SYM and Other Supersymmetric LagrangiansRecall that the field content of N = 4 SYM is a vector Aµ, gaugini λI=1...4, and six scalars Xi, allin the adjoint of the gauge group. The Lagrangian density (which is completely determined by theamount of SUSY, up to two parameters, (gY M, ϑ)), isL =1g2Y MtrF2+ (DXi)2+ i¯λ /Dλ−6Xi<j[Xi, Xj]2− λ[X, λ] +¯λ[X,¯λ] ])+iϑ8π2tr(F ∧ F ) (1)The non-zero, N = 4 SUSY transformations are (very schematically!)1[Q, X] = λ{Q, λ} = F++ [X, X]{Q,¯λ} = DX[Q, A] = λ (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 theindices in, either leave that as a fun exercise, or check out Weinb erg, volume 3. As an example,F+≡ σµνFµν.1.1 A Superspace DetourThe N = 4 SYM Lagrangian is an example of a (highly) supersymmetric Lagrangian. So far, Ijust told you what it was and that it was SUSY invariant (something you could sit down in th eprivacy of your office and check, if you wanted). It’d be nice, though, if there were some sort of amachine that one could crank to generate supersymm etric lagrangians. That crankable machine issuperspace.To understand why superspace is u s eful, we should think about why fields are useful for representingtranslationally invariant Lagrangians in ordinary QFT. One reason is that the representations ofthe translation group on the fields are particularly simpleφ(x) = eiˆP ·xφ(0) (3)We’d like to introduce a superfield (that comes with its own supercapitalization) Φ(x, θ), which isnow a function of spacecoordinates x and “superspace” coordinates θ, that well represents transla-tional invariance AND supersymmetryΦ(x, θ) = eiˆP ·x+iˆQ·θΦ(0, 0) (4)where theˆQ′s are the operators that generate supersymmetry transformations.Now, in QFT, one can automatically get translationally invariant actionsS =Zddx L(φ, ∂φ) (5)as long as ∂xL = 0. Similarly the (here unproven) claim is thatZddx dN ·sθ L(Φ(x, θ)) (6)2is supersymmetric as long as ∂θL = 0. Here s denotes th e smallest (real) dimension of the spinorrepresentation in d dimensions and N , is, as usual the number of su persymmetries. Then N · s isjust the number of real supercharges. For example for N = 1, d = 4, N · s = 4, because either aWeyl or Majorana spinor (the minimum in four dimensions) has four real components.1.1.1 BPS or “Chiral” MultipletsIn lecture 5 we made a big deal about special representations of supersymmetry which are killed bysome of the supercharges. Such multiplets have correspondingly special pr operties in superspace.Consider a field which satisfies[¯Q, Φ] = 0[Q, Φ] 6= 0 (7)These multiplets, which are BPS (half of the supersymmetries annihilate them) are generally calledchiral mu ltiplets. Sometimes th ey are actually chiral (in the sense of the Lorentz group), but oftentimes they are not. This follows a long tr adition in physics of calling things other things whichthey are not.These mu ltiplets are functions of only half of superspace asΦ(x, θ,¯θ) = ei(Qθ+¯Q¯θ)Φ(x, 0, 0) = Φ(x, θ, 0) (8)Ok, well this equation is not exactly correct (as Senthil pointed out; r eally the RHS should beΦ(y ≡ x + i¯θθ, θ)), but it is morally correct: the Φ’s are functions of half of superspace. Becauseof this, it is possible to add terms to the Lagrangian density that are integrated over only half ofsuperspace and maintain supersymmetry:∆L =Zd2θ W (Φ) +Zd2¯θ¯W (¯Φ) (9)Here, we are explicitly working in d = 4, N = 1 superspace. These terms are supers ymmetric, aslong as W is a holomorphic function of Φ, ∂¯ΦW = 0. With this constraint, W is a function we arefree 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 ind = 4, N = 1 superspace (where we pick out a particular N = 1 subgroup from the N =4).In a certain sense, one of the λ’s and F can be thought of as comprising one N = 1 chiralmultiplet, while the remaining three λ’s and six λ’s can be thought of as another three chiralmultiplets. The second line of equation (1) is a superpotential for these th ree 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 theN = 1 superpotentialZd2θ τ tr(λαλα) (10)where the λ appearing is the superpartner of F (α is a s pinor index), and τ is a complexifiedcoupling constant τ ≡4πig2Y M+ϑ2π. Here λ should be thought of as a superfield w hose lowestcomponent is the gaugino. The superfield expansion contains a λ = . . . θF term. Multiplyingtwo together, one gets1g2Y MF2⊂Zd2θ τ θ2F2⊂Zd2θ τ tr(λαλα) (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 b e holomorphic in order for supersymmetry tobe preserved. We can take this line of reasoning one step further–we can think of promoting thecouplings to dynamical superfields (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, theform of r adiatively generated corrections to the superpotential are severely constrained–they mustbe holmorphic in the fields and the coup lings. For example, one could never generate a term in thesuperpotential that was a function of both τ and ¯τ, W 6= W (τ ¯τ).This leads to many non-renormalization theorems in supersymmetric field theories, one example