Renormalization group approach to matrix models via noncommutative space

Abstract

We develop a new renormalization group approach to the large-N limit of matrix models. It has been proposed that a procedure, in which a matrix model of size (N − 1) × (N − 1) is obtained by integrating out one row and column of an N × N matrix model, can be regarded as a renormalization group and that its fixed point reveals critical behavior in the large-N limit. We instead utilize the fuzzy sphere structure based on which we construct a new map (renormalization group) from N × N matrix model to that of rank N − 1. Our renormalization group has great advantage of being a nice analog of the standard renormalization group in field theory. It is naturally endowed with the concept of high/low energy, and consequently it is in a sense local and admits derivative expansions in the space of matrices. In construction we also find that our renormalization in general generates multi-trace operators, and that nonplanar diagrams yield a nonlocal operation on a matrix, whose action is to transport the matrix to the antipode on the sphere. Furthermore the noncommutativity of the fuzzy sphere is renormalized in our formalism. We then analyze our renormalization group equation, and Gaussian and nontrivial fixed points are found. We further clarify how to read off scaling dimensions from our renormalization group equation. Finally the critical exponent of the model of two-dimensional gravity based on our formalism is examined.