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 8 Large N Counting Lecturer McGreevy October 4 2008 1 Introduction Today we ll continue our discussion of large N scaling in the t Hooft limit of quantum matrix models In particular we ll review the N counting for vacuum diagrams and learn how to do N counting for correlators Our perspective on the large N limit continues to be one of convenience It is one of several assumptions that enable the simplest realization of holography and gauge gravity duality Next time we ll continue talking about simplifying assumptions and turn our attention to conformal invariance 2 Vacuum t Hooft counting Let s recall our starting point We were thinking about matrix eld theories eld theories in which all the elds are N x N matrices in addition to whatever other labels they may have In particular we write our theory schematically in terms of one big eld which we think of as potentially including scalars gauge elds A and fermions all of which are N x N matrices The schematic Lagrangian is 1 L 2 Tr V 1 g where V 2 4 and all products should be thought of as matrix products This schematic representation is meant to convey two pieces of information First that the Lagrangian is invariant under U N rotations which act on the matrix elds by conjugation U U 1 Second that the elds are scaled so that the coupling g 2 sits out in front of the entire Lagrangian In particular we have to very careful about eld normalizations when doing large N counting for correlation functions The t Hooft limit is the limit where N and g 2 0 while g 2 N remains xed is the t Hooft coupling Note that we are adopting a di erent normalization condition than in the previous lecture on large N counting there we used the eld 1 g which had kinetic terms with the usual eld theory normalization With our current normalization the propagator goes like g 2 N The propagator in the double line notation appears in Fig 1 Remember that for the purposes of 1 Figure 1 Propagator N Figure 2 Three point vertex N large N counting we are instructed to think only about counting powers of N so we will suppress positions of elds spin indices etc in our analysis In a generic matrix theory there will be three point Fig 2 and four point Fig 3 interaction vertices Both types of vertices come with factors of 1 g 2 N from the overall 1 g 2 multiplying the Lagrangian A general diagram consists of propagators interaction vertices and index loops and gives a contribution no of prop no of int vert N N no of index loops 2 diagram N For example the planar diagram in Fig 4 has 4 three point vertices 6 propagators and 4 index loops giving the nal result N 2 2 We learned last time how we to associate a triangulated surface with a Feynman diagram in two ways The rst way is to let the interaction vertices of the diagram be the vertices of the triangu lation In this case the index loops of the diagram bound pieces of surface The alternative is the dual triangulation in which we place vertices in the middle of the index loops of the Feynman dia gram Two vertices are connected if their respective index loops share a propagator in the Feynman diagram Don t be confused by multiple uses of the word vertex There are interaction vertices of various kinds in the Feynman diagrams and these correspond to vertices in the triangulation only in the rst formulation If E no of prop V no of int vert and F no of index loops then we showed last time that the diagram gives a contribution N F E V E V The letters refer to the rst way to triangulate Figure 3 Four point vertex N 2 Figure 4 This diagram consists of 4 three point vertices 6 propagators and 4 index loops the surface in which interaction vertices are triangulation vertices Then we interpret E as the number of edges F as the number of faces and V as the number of vertices in the triangulation In the dual triangulation there are dual faces F dual edges E and dual vertices V The relationship between the original and dual variables is E E V F and F V In the dual formulation we found that a diagram contributes N F E V E F It s not a coincidence that the powers of N agree in both formulations The exponent F E V F E V is the Euler character and it is a topological invariant of two dimensional surfaces In general it is given by 2 2h b where h is the number of handles the genus and b is the number of boundaries Note that the exponent of E V or E F is not a topological invariant and depends on the triangulation Feynman diagram Before we continue with the analysis its worth pointing out that there are other large N limits one might want to consider The motivation for the t Hooft limit comes in part from thinking about radiative corrections which grow with N and decrease with g This suggests some sort of balancing act might be possible which keeps the theory non trivial in the large N limit We encode this scaling by xing g 2 N as we make N large Other interesting possibilities are provided by critical points of the matrix model Suppose when g gc the matrix model is at a critical point Then there may be a sensible large N limit in which we x g gc N as we make N large One example of this is matrix quantum mechanics with an inverted oscillator potential Back to the t Hooft limit Because the N counting is topological depending only on we can sensibly organize the perturbation series for the free energy in terms of surfaces Because we re computing only vacuum diagrams for the moment the surfaces we re considering have no boundaries b 0 and are classi ed by their number of handles h h 0 is the two dimensional sphere h 1 is the torus and so on We may write the free energy as ln Z N h 0 2 2h c h 0 N 2 2h Fh 3 h 0 were the sum over surfaces is explicit Now we can see some similarities between this expansion and perturbative string expansions By writing N 2 2h 1 N 2h 2 we can identify the string coupling gs g N where we can allow some function of in addition to the pure N dependence Because of the proliferation of coupling constants let s rename the g 2 in our matrix model gY2 M in anticipation of the case of most interest to us when the matrix model is a gauge theory …