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MIT 8 821 - Supersymmetric Lagrangians and Basic Checks of AdS/CFT

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MIT OpenCourseWare 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit:� � � 1 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 stuff. 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. N = 4 SYM and Other Supersymmetric Lagrangians Recall that the field content of N = 4 SYM is a vector Aµ, gaugini λI=1...4, and six scalars Xi, all in the adjoint of the gauge group. T he Lagrangian density (which is completely determined by the amount of SUSY, up to two parameters, (gY M, ϑ)), is L = 2 1 tr F2 + (DXi)2 λ /+ i¯Dλ gY M 6 − [Xi, Xj]2 − λ[X, λ] + λ¯[X, λ¯] ]) i<j iϑ + tr(F ∧ F ) (1) 8π2The 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, L orentz 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+ ≡ σµνFµν. 1.1 A Superspace Detour The N = 4 SYM Lagrangian is an example of a (h ighly) 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 office 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 sh ould think about why fields are useful for representing translationally invariant Lagrangians in ordinary QFT. One reason is that the representations of the translation group on the fields are particularly simple φ(x) = e iPˆ·xφ(0) (3) We’d like to introduce a superfield (that comes with its own supercapitalization) Φ(x, θ), which is now a function of spacecoordin ates x and “superspace” coordinates θ, th at well represents transla-tional invariance AND supersymmetry Φ(x, θ) = e iPˆ·x+iQˆ·θΦ(0, 0) (4) where the Qˆ′ s are the operators that generate supersymmetry transformations. Now, in QFT, one can automatically get translationally invariant actions S = dd x L(φ, ∂φ) (5) as long as ∂xL = 0. Similarly the (here unproven) claim is that ddx dN ·sθ L(Φ(x, θ)) (6) 2� � is supersymmetric as long as ∂θL = 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 m inimum 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 s upercharges. Such multiplets have correspondingly sp ecial properties 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 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 p hysics of calling things other things which they are not. These multiplets are fu nctions of on ly half of superspace as Φ(x, θ, θ¯) = e i(Qθ+Q¯θ¯)Φ(x, 0, 0) = Φ(x, θ, 0) (8) Ok, well this equation is not exactly correct (as Senthil pointed out; really the RHS should b e Φ(y ≡ x + i¯correct: the Φ’s are functions of half of superspace. Because θθ, θ)), but it is morally of this, it is possible to add terms to the Lagrangian dens ity that are integrated over only half of superspace and maintain supersymmetry: d2θ¯W¯(¯ΔL = d2θ W (Φ) + Φ) (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 Φ, ∂Φ¯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 th at this line can be written in d = 4, N = 1 superspace (where we pick out a particular N = 1 subgroup f rom the N =4). In a certain sense, one of the λ’s and F can be thought of as comprising one N = 1 chiral multiplet, while the remaining three λ’s and six λ’s can be thought of as another three chiral multiplets. The second line of equation (1) is a superpotential for these three chir al multiplets. On pset 2 you will h ave 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 d2θ τ tr(λαλα) (10) where the λ appearing is the superpartner of F (α is a spinor index), and τ is a complexified coupling constant τ ≡ g42 πi + 2ϑπ. Here λ should be thought of as a superfield whose lowest Y M component is the gaugino. The s uperfield expansion contains a λ = . . . θF term. Multiplying two together, one gets 1 F2 ⊂ d2θ τ θ2F2 ⊂ d2θ τ tr(λαλα) (11) 2gY M The rest of the gauge kinetic terms and theta angle term can be understood

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