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MIT 8 821 - Hierarchy of Scaling dimensions

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MIT OpenCourseWarehttp://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 09: Preview of Strings in N = 4 SYM; Hierarchy of Scaling dimensions; Conformal Symmetry in QFT Lecturer: McGreevy October 8, 2008 Emergence of Strings from Gauge Theory Continuing from the previous lecture we subs tantiate the correspondence between the operators in N = 4 SYM theory with the excitations of some string theory. We saw that the primary chiral operators (i.e. primary operators in the conformal field theory which also belong to the short multiplets of the supersymmetry) corresp ond to the SUGRA modes but we didn’t discuss what happens w hen the operators in the gauge theory are in the long multiplet. Thus consider a more generic operator (residing in a long supersymmetry multiplet) of the matrix QFT in the large N limit O(x) = tr(XXXXY XXY....Y X) (1) We consider the limit N ≫ J ≫ 1 where J is the number of entities in the above pr oduct. Cyclicity of 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 and Y′s the fields living on the worldsheet. Further, in the large J limit, J corresponds to the angular momentum of the string excitations and in fact this relation could be used to reconstruct the string theory. Having sketched the correspondence between the various operators of the SYM guage theory with SUGRA modes or string excitations, let us organize this knowledge to get a better perspective on when it could be useful. The mass of the SUGRA mode is given by m2 SU GRA = 1/L2 AdS. Using the AdS-CFT corre-• spondence mSU GRA2 LAdS 2 = Δ(Δ − 4) where Δ is the scaling dimension of the corresponding chiral primary operator in the SYM theory. This implies that ΔSU GRA ∼ N0λ0 . 2Moving on to the excited string states mstring states ∼ ∼√λ/L2 where 1/α′ is string 1/α′ AdS • tension. Thus the corresponding scaling dimension equals Δstring states ∼ N0λ1/4 . • Finally, there are D-branes in the string theory which corresp ond to baryonic states in the 1gauge theory. Their mass is given by 1 2 2 m g √λ/g2L2 = Nλ1/4 .D −brane ∼ 1/α ′ s ∼sAdS ⇒ ΔD −brane Clearly, the scaling dimension Δ for the different states (SUGRA, strings, D-branes) has distinctive dependen ce on N and λ. This hierarchy of Δ’s is what a QFT needs to have a weakly coupled gravity description without strings. For example the λ → ∞ limit removes all states except SUGRA modes.2 There are various forms of AdS-C FT conjecture. For example, one may believe that it only holds in the limit λ → ∞, N → ∞ (yielding classical gravity) or perhaps also at finite λ and N → ∞ limit (when one obtains classical strings on small AdS radius). Remarkably, all evidence till date points to a much s tron ger statement that it holds for all N and all λ. 2 Conformal Symmetry in QFT This is a worthy subject in itself and study of conformal invariance is relevant for understandin g both UV and IR limit of various QFT’s and also for the worldsheet theory of the strings in the conformal gauge. As you may already know (and we recapitulate it below) that conformal symmetry in two dimensions is very special due to existence of infinite number of conserved currents. Therefore it pays to understand which aspects of a conformal field theory (CFT) are particular to D = 2 and which apply to any dimension D > 2. We have already seen some of the constraints due to the requirements of L orentz invariance and SUSY on the QFT’s and expectedly, requirement of conformal invariance makes it even more constrained. This is also the last stop in our ruthless program to evade the loopholes in the Coleman-Mandula theorem. Some of the useful references are the book Conformal Field theory by Di Francesco, Mathieu and Senechal and the articles by Callan, Ginsparg and portions of MAGOO (links posted on the course webpage). 2.1 The Conformal Group We define CFT by a list of operators and their Green functions which satisfy certain constraints described below. Succintly, a CFT is a theory with symmetries generated by conformal group . Obviously, it behooves us to define conformal group which we do now. Please note that the route we 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 dimensions of 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: R i A tr P eC where C is some closed curve in the space on which the field theory lives. It can be thought of as the phase acquired by a charged particle dragged along the specified path by an arbitrarily powerful external force. The connection between these operators and strings turn s out to be quite direct, as one might expect from the relationship between charges on D-branes and th e ends of open strings. 2Recall that isometry group of a spacetime with coordinates xµ and metric gµν is th e set of coordinate transformations which leave the metric ds2 = gµνdxµdxν unchanged. Conformal group corresponds to 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 of →flat spacetime) is a subgroup of the conformal group with Ω(x) = 1. Conformal transformations ′ ′could also be seen as th e successive application of a coordinate transformation x x , gµν gµν ′ → →which preserve ds2 = gµνdxµdxν followed


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