8 821 F2008 Lecture 09 Preview of Strings in N 4 SYM Hierarchy of Scaling dimensions Conformal Symmetry in QFT Lecturer McGreevy Scribe Tarun Grover October 8 2008 1 Emergence of Strings from Gauge Theory Continuing from the previous lecture we substantiate 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 correspond to the SUGRA modes but we didn t discuss what happens when 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 product 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 m2SU GRA 1 L2AdS Using the AdS CFT correspondence m2SU GRA L2AdS 4 where is the scaling dimension of the corresponding chiral primary operator in the SYM theory This implies that SU GRA N 0 0 Moving on to the excited string states m2string states 1 L2AdS where 1 is string tension Thus the corresponding scaling dimension equals string states N 0 1 4 Finally there are D branes in the string theory which correspond to baryonic states in the 1 gauge theory Their mass is given by 1 m2D brane 1 gs2 gs2 L2AdS D brane N 1 4 Clearly the scaling dimension for the different states SUGRA strings D branes has distinctive dependence on N and This hierarchy of s is what a QFT needs to have a weakly coupled gravity description without strings For example the limit removes all states except SUGRA modes 2 There are various forms of AdS CFT conjecture For example one may believe that it only holds in the limit N yielding classical gravity or perhaps also at finite and N limit when one obtains classical strings on small AdS radius Remarkably all evidence till date points to a much stronger statement that it holds for all N and all 2 Conformal Symmetry in QFT This is a worthy subject in itself and study of conformal invariance is relevant for understanding 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 Lorentz 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 1 The power of 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 2 It is worth noting here that gauge theories in general also contain non local operators such as Wilson loops tr P ei R C A 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 turns out to be quite direct as one might expect from the relationship between charges on D branes and the ends of open strings 2 Recall that isometry group of a spacetime with coordinates x and metric g is the set of coordinate transformations which leave the metric ds2 g dx dx unchanged Conformal group corresponds to a bigger set of transformations which preserve the metric up to an overall possibly positiondependent rescaling ds2 x ds2 Thus Poincare group which is the group of isometries of flat spacetime is a subgroup of the conformal group with x 1 Conformal transformations could also be seen as the successive application of a coordinate transformation x x g g which preserve ds2 g dx dx followed by a Weyl rescaling which takes x x so that ds2 is not preserved Conformal transformations preserve the angles between vectors hence the name cos v w cos v v w w 2 Consider the space time IRp q with a constant metric g and let s look at the piece of infinitesimal coordinate transformations connected to the identity x x x g 3 4 Note that above there are no terms with derivative s of metric since the metric is constant The requirement of conformal invariance implies that f x 2 d 5 6 where the equality f x d2 is obtained by taking trace of on both sides of the eqn 5 Applying an extra derivative on 5 gives 2 1 0 d 7 Pausing for a moment we note that d 2 is clearly special since 0 implies that any holomorphic antiholomorphic function z z of complex coordinates z x1 ix2 would correspond to a conformal transformation A little more thought leads to the result that conformal algebra is infinite dimensional in two dimensions Applying one more partial derivative on this equation and applying on eqn 5 and using the both equations thus obtained together gives 2 d 3 8 Contracting with implies 0 and thus x is at most a quadratic function of x Let s organize x by its degree in x Degree