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.1 8.821 F2008 Lecture 10: CFTs in D > 2 Lecturer: McGreevy October 11, 2008 In today’s lecture we’ll elaborate on what conformal invariance really means for a QFT. Remember, we’ll keep working with CFTs in greater than 2 spacetime dimensions. A very good reference is Conformal Field Theory by Di Francesco et al, p95-107. Conformal Transformations To begin, we’ll give a few different perspectives on what conformal invariance is and what conformal transformations are. Ou r first perspective was emphasized by Brian Swingle after last lecture. Recall from last lecture that the group structure of the conformal group is the set of coordinate transformations: ′ x → x ′ gµν (x ′ ) = Ω(x)gµν (x) (1) However, we should point out that coordinate transformations don’t do anything at all in the sense that the metric is invariant, ds ′2 = ds2, and all we did was relabel points. To say something useful about physics, we really want to compare a theory at different distances. For example, we would like to compare the correlators of physical observables separated at different distances. C onformal symmetry relates these kind s of observables! To implement a conformal transformation, we need to change the metric via a Weyl rescaling x → x gµν → Ω(x)gµν (2) which changes physical distances ds2 → Ω(x)ds2 . Now follow this by a coordinate transformation such as the one in Eq.(1), so that x → x ′ and Ω(x)gµν → g ′ ′ ). This takes us to a coordinate µν (x system wh ere th e metric has the same form as the one we s tarted with, but the points have all been moved around and pushed closer together or farther apart. Therefore, we can view the conformal group as those coordinate transformations which can ‘undo’ a Weyl rescaling. 1� � � � � 1.1 Scale transformations Another useful perspective on conformal invariance, particularly for AdS/CFT, is to thin k about scale transformations. Any local (relativistic) QFT has a stress-energy tensor T µν which is conserved if the th eory is translation invariant. T µν encodes the linear response of the action to a small change in the metric: δS = T µν δgµν (3) If we put in gravity, then T µν = 0 by the equations of motion. But let’s return to QFT without gravity. Consider making a scale transformation, which changes the metric δgµν = 2λgµν → δS = Tµµ2λ (4) where λ is a constant. Therefore we see that if Tµµ = 0 then the theory is scale invariant. In fact, since nothing we said above depended on th e fact that λ is a constant, and since our th eory is a local th eory, we can actually make the following transformation: δgµν = δΩ(x)gµν → δS = TµµδΩ(x) (5) We conclude from the two equations above that, at least classically, • If Tµµ = 0 the theory has both scale invariance and conformal invariance. • If the theory is conformally invariant, it is also scale invariant, and Tµµ = 0 However scale invariance does not quite imply conformal invariance. For more details please see Polchinski’s paper, Scale and conformal invariance in quantum field theory. The conserved currents and charges of the transformations above are: Sµ = x ν Tµν → D ≡ S0dd x (6) Cµν = (2xµxλ − x 2 gµλ)Tνλ → Cµ ≡ C0µdd x (7) since both ∂µSµ and ∂µCµν are proportional to Tµµ . 1.2 Geometric Interpretation Lastly we’d like to give an alternative geometric interpretation of the conformal group. The con-formal group in Rd,1 is isomorphic to SO(d + 1, 2), the Lorentz group of a space with two extra dimensions. Suppose this space Rd+1,2 has metric ηab = diag(− + +... + −)ab (8) 22 where the last two dimensions are the ‘extra’ ones. A light ray in this space can be parameterized by d + 1 dimensional coordinates xµ in the following way: 1 1 ζa = κ(xµ, (1 − x 2), (1 + x 2)) (9) 2 2where κ is some arbitrary constant. The group SO(d + 1, 2) moves these light rays around. We can interpet these transformations as maps on xµ, and in fact these transformations are p recisely the conformal transformations, as you can check on your own. Invariants in Rd+1,2 should also be conformal invariants, for example ζ1 · ζ2 = ηabζ1aζ1 b =1 κ1κ2(x1 − x2)2 . (10) 2 This statement is not quite true because ζa and λζa are identified with the same xµ, so κ is a redundant variable. Conformal invariants actually are cross ratios of invariants in Rd+1,2, for example ζ1 · ζ2ζ3 · ζ4 . (11) ζ1 · ζ3ζ2 · ζ4 This perspective was lifted f rom a paper by Callan. It may be useful if you are in the business of relating CFTs to theories in higher dimensional spacetimes, but you are otherwise free to ignore it. Constraints on correlators in a CFT We will now attempt to say something concrete about constraints on a QFT from conformal in -variance, namely constraints on correlators. Consider a Green’s function G = �φ1(x1)...φn(xn)� (12) where φi is a conformal primary. R ecall from last lecture that this m eans φ satisfies [ ˆ=D, φ(0)] −iΔφ(0), where Δ is the weight and Dˆis the dilatation operator defined ab ove. How does φ(x) transform under Dˆ? Writing the field as φ(x) = e−iPˆ·xφ(0)e+iPˆ·x, and using the following result −iPˆ·x ˆP ·x ˆ ˆe De+i ˆ= D + x · P (13) which follows from the algebra, then [D, φ(x)] = −i(Δ + x · ∂)φ(x). The Green’s function thus transforms under the action of Dˆby: n δG = � (xiµ ∂x∂ µi + Δi)� � φi(xi)� + �0|Dˆ� φi(xi)|0� + �0| � φi(xi)Dˆ|0�. (14) i=1 Assuming that th e vacuum is conformally invariant, the last two terms above vanish. We will assume this in the