8.821 F2008 Lecture 09: Preview of Strings in N = 4 SYM;Hierarchy of Scaling dimensions; Conformal Symmetry in QFTLecturer: McGreevy Scribe: Tarun GroverOctober 8, 20081 Emergence of Strings from Gauge TheoryContinuing from the previous lecture we substantiate the correspondence between the operators inN = 4 SYM theory with the excitations of some string theory. We saw that the primary chiraloperators (i.e. primary operators in the conformal field theory which also belong to the shortmultiplets of the supersymmetry) correspond to the SUGRA modes but we didn’t discuss whathappens when the operators in the gauge theory are in the long multiplet. Thu s consider a moregeneric operator (residin g in a long supersymmetry multiplet) of the matrix QFT in the large NlimitO(x) = tr(XXXXY XXY....Y X) (1)We consider the limit N ≫ J ≫ 1 where J is the number of entities in the above product. Cyclicityof the trace implies that the structure of the above operator has the symmetry of a closed loop.In fact the above operator corresponds to creation operator for excited string states with X′s andY′s the fields living on the worldsheet. Further, in the large J limit, J corresponds to the angularmomentum of the string excitations and in fact this relation could be used to reconstruct the stringtheory.Having sketched the correspondence between the various operators of th e SYM guage theory withSUGRA modes or string excitations, let us organize this knowledge to get a better perspective onwhen it could be useful.• The mass of the SUGRA mode is given by m2SU GRA= 1/L2AdS. Using the AdS-CFT corre-spondence m2SU GRAL2AdS= ∆(∆ −4) where ∆ is the scaling dimension of the correspondingchiral primary operator in the SYM theory. This implies that ∆SU GRA∼ N0λ0.• Moving on to the excited string states m2string states∼ 1/α′∼√λ/L2AdSwhere 1/α′is stringtension. T hus the corresponding scaling dimension equals ∆string states∼ N0λ1/4.• Finally, there are D-branes in the string theory which correspond to baryonic states in the1gauge theory. Their mass is given by1m2D −brane∼ 1/α′g2s∼√λ/g2sL2AdS⇒ ∆D −brane= Nλ1/4.Clearly, the scaling dimension ∆ for the different states (SUGRA, strings, D-branes) has distinctivedependence on N and λ. This hierarchy of ∆’s is what a QFT needs to have a weakly coupled gravitydescription without strings. For example the λ → ∞ limit removes all states except SUGRA mod es.2There are various forms of AdS-CFT conjecture. For example, one may believe that it only holdsin the limit λ → ∞, N → ∞ (yielding classical gravity) or perhaps also at finite λ and N → ∞limit (when one obtains classical strin gs on sm all AdS r ad ius ). Remarkably, all evidence till datepoints to a much stronger statement that it holds for all N and all λ.2 Conformal Symmetry in QFTThis is a worthy subject in itself and study of conformal invariance is relevant for understandingboth UV and IR limit of various QFT’s and also for the worldsheet theory of the strings in theconformal gauge. As you may already know (and we r ecapitulate it below) that conformal symmetryin two dimensions is very special due to existence of infinite number of conserved currents. Thereforeit pays to understand which aspects of a conformal field theory (CFT) are particular to D = 2and which apply to any dimension D > 2. We have already seen some of the constraints due tothe requirements of Lorentz invariance and SUSY on the QFT’s and expectedly, requirement ofconformal invariance makes it even more constrained. This is also the last s top in our ruthlessprogram to evade the loopholes in the Coleman-Mandula theorem.Some of the useful references are the book Conformal Field theory by Di Francesco, Mathieu andSenechal and the articles by Callan, Ginsparg and portions of MAGOO (links posted on the coursewebpage).2.1 The Conformal GroupWe define CFT by a list of operators and their Green functions which satisfy certain constraintsdescribed below. Succintly, a CFT is a theory with symmetries generated by conformal group .Obv iously, it behooves us to define conformal group which we do now. Please note that the routewe follow is not a good one for non-relativistic CFT’s.1The power of λ here depends on the specific geometry of the D-brane in question, i.e. which of the dimensionsof the bulk the D-brane is wrapping.2It is worth noting here that gauge theories in general also contain non-local operators such as Wilson loops:tr P eiRCAwhere C is some closed curve in the space on which the field theory lives. It can be thought of as the phase acquiredby a charged particle dragged along the specified path by an arbitrarily p owerful ex t ernal force. The connectionbetween these operators and strings turns out to be quite direct, as one might expect from the relationship betweencharges on D-branes and the ends of open strings.2Recall that isom etry group of a sp acetime with coordinates xµand metric gµνis the set of coordinatetransformations which leave the metric ds2= gµνdxµdxνunchanged. Conformal group correspondsto a bigger set of transformations which preserve the metric up to an overall (possibly position-dependent) rescaling: ds2→ Ω(x)ds2. Thus Poincare group (which is the group of isometries offlat spacetime) is a su bgroup of the conformal group with Ω(x) = 1. Conformal transformationscould also be seen as the successive app lication of a coordinate transformation x → x′, gµν→ g′µνwhich preserve ds2= gµνdxµdxνfollowed by a Weyl rescaling which takes x′→ x so that ds2is notpreserved.Conformal transformations preserve the angles between vectors, hence the name:cos(θ) =vµwµ√vµvµ√wµwµ→ cos(θ′) (2)Consider the space-time IRp,qwith a constant metric gµν= ηµνand let’s look at the piece ofinfinitesimal coordinate transformations connected to the identity:xµ→ xµ+ ǫµ(x) (3)gµν→ ηµν− (∂µǫν+ ∂νǫµ) (4)Note that above there are no terms with derivative(s) of metric since the metric is constant. Therequirement of conformal invariance implies that(∂µǫν+ ∂νǫµ) = f(x)ηµν(5)=2d∂ρǫρηµν(6)where the equality f (x) =2d∂ρǫρis obtained by taking trace of ∂µǫνon both sides of the eqn. 5.Applying an extra derivative ∂ρon 5 gives ǫν+ (1 −2d)∂ν(∂ρǫρ) = 0 (7)Pausing for a moment, we note that d = 2 is clearly s pecial since ǫν= 0 implies that