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 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 1 Conformal Transformations To begin we ll give a few di erent perspectives on what conformal invariance is and what conformal transformations are Our rst 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 g x x g 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 di erent distances For example we would like to compare the correlators of physical observables separated at di erent distances Conformal symmetry relates these kinds of observables To implement a conformal transformation we need to change the metric via a Weyl rescaling x x g x g 2 which changes physical distances ds2 x ds2 Now follow this by a coordinate transformation x This takes us to a coordinate such as the one in Eq 1 so that x x and x g g system where the metric has the same form as the one we started 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 think about scale transformations Any local relativistic QFT has a stress energy tensor T which is conserved if the theory is translation invariant T encodes the linear response of the action to a small change in the metric S T g 3 If we put in gravity then T 0 by the equations of motion But let s return to QFT without gravity Consider making a scale transformation which changes the metric g 2 g S T 2 4 where is a constant Therefore we see that if T 0 then the theory is scale invariant In fact since nothing we said above depended on the fact that is a constant and since our theory is a local theory we can actually make the following transformation g x g S T x 5 We conclude from the two equations above that at least classically If T 0 the theory has both scale invariance and conformal invariance If the theory is conformally invariant it is also scale invariant and T 0 However scale invariance does not quite imply conformal invariance For more details please see Polchinski s paper Scale and conformal invariance in quantum eld theory The conserved currents and charges of the transformations above are S x T D C 2x x x2 g T C since both S and C are proportional to T 1 2 S0 dd x C0 dd x 6 7 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 ab diag ab 2 8 where the last two dimensions are the extra ones A light ray in this space can be parameterized by d 1 dimensional coordinates x in the following way 1 1 a x 1 x2 1 x2 2 2 9 where is some arbitrary constant The group SO d 1 2 moves these light rays around We can interpet these transformations as maps on x and in fact these transformations are precisely the conformal transformations as you can check on your own Invariants in Rd 1 2 should also be conformal invariants for example 1 1 2 ab 1a 1b 1 2 x1 x2 2 2 10 This statement is not quite true because a and a are identi ed with the same x so is a redundant variable Conformal invariants actually are cross ratios of invariants in Rd 1 2 for example 1 2 3 4 11 1 3 2 4 This perspective was lifted from 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 2 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 1 x1 n xn 12 0 where i is a conformal primary Recall from last lecture that this means satis es D i 0 where is the weight and D is the dilatation operator de ned above How does x transform under D Writing the eld as x e iP x 0 e iP x and using the following result iP x D x P e iP x De 13 which follows from the algebra then D x i x x The Green s function thus transforms under the action of D by G n x i i xi 0 D i xi 0 0 i xi D 0 i xi 14 i 1 Assuming that the vacuum is conformally invariant the last two terms above vanish We will assume this in the rest of the discussion N 4 is a good example where there is a conformally invariant vacuum 3 Now let s ignore what we just did with in nitesimal transformations and look at the nite trans formations of the green s function n n n x i d 2 G 15 i i i x i i x i x x xi i 1 where i 1 2 i 1 2b xi b2 x2i 2 i 1 scale transformation special conformal Conformal invariance tells us exactly how the Green s function must transform Therefore up to the i factors the Green s function can only depend on conformal invariants I made from x i What are the conformal invariants Assuming s are scalars let s constrain the conformal invariants by using the symmetries of the theory Translation invariance implies I depends only on di erences xi xj Therefore there are d n 1 numbers I can depend on Rotational invariance implies I depends only on magnitudes of di erences rij xi xj There are n n 1 2 of these numbers Scale invariance implies I depends only on ratios of magnitudes rij rkl Finally special conformal transformations transform rij by 2 r12 2 2 r12 r12 1 2 1 2 1 2b x1 b2 x21 1 2b x2 …