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MIT 8 821 - Study guide

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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 field theories, field theories in which all the fields 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 field Φ which we think of as potentially including scalars φ, gauge fields Aµ, and fermions ψα all of which are N x N matrices. The schematic Lagrangian is 1 L = Tr (∂Φ∂Φ − V (Φ)) (1)2gwhere V (Φ) ∼ Φ2 + Φ4 + ... 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 fields by conjugation: Φ UΦU−1 .→Second, that the fields are scaled so that the coupling g2 sits out in front of the entire Lagrangian. In particular, we have to very careful about field normalizations when doing large N counting for correlation functions. The t’Hooft limit is the limit where N → ∞ and g2 0 while λ = g2N remains fixed. λ is the t’Hooft coupling. → Note that we are adopting a different normalization condition than in the previous lecture on large N counting (there we used the field Φ˜= (1/g)Φ which had kinetic terms with the usual field theory normalization). With our current normalization the propagator �ΦΦ� goes like g2 = λ .N The propagator in the double line notation appears in Fig. 1. Remember that for the purposes of 1large N counting we are instructed to think only about counting powers of N, so we will suppress positions of fields, 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/g2 = N/λ from the overall 1 /g2multiplying the Lagrangian. A general diagram consists of propagators, interaction vertices, and index loops, and gives a contribution no. of prop. no. of int. vert. λdiagram ∼ � � �N �no. of index loopsN . (2)N λ For example, the planar diagram in Fig. 4 has 4 three point vertices, 6 propagators, and 4 index loops giving the final result N2λ2 . Figure 1: Propagator, λ/N Figure 2: Three point vertex, N/λ We learned last time how we to associate a triangulated surface with a Feynman diagram in two ways. The first 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 first 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 NF −E+V λE−V . The letters refer to the first way to triangulate Figure 3: Four point vertex, N/λ 2Figure 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−˜λF. It’s not a coincidence that the powers of N agree in both formulations. The exponent χ = 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 χ =2− 2h − b where h is the number of handles (the genus) and b is the number of boundaries. Note that the exponent of λ, 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 fixing λ = g2N 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 fix |g − gc|αN 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 χ) 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 classified 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


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