Large N

TL;DR

This paper discusses the 1/N expansion technique in large-N gauge theories, emphasizing diagram planarity, and explores its implications for perturbative expansions, lattice theories, and renormalization, with exact solutions for zero-dimensional models.

Contribution

It provides an overview of the 1/N expansion in large-N gauge theories, including diagram classification, convergence properties, and exact solutions for counting planar diagrams.

Findings

1/N expansion sorts diagrams by planarity.

Zero-dimensional models allow exact counting of diagrams.

Insights into planar renormalization processes.

Abstract

In the first part of this lecture, the 1/N expansion technique is illustrated for the case of the large-N sigma model. In large-N gauge theories, the 1/N expansion is tantamount to sorting the Feynman diagrams according to their degree of planarity, that is, the minimal genus of the plane onto which the diagram can be mapped without any crossings. This holds both for the usual perturbative expansion with respect to powers of {tilde g}^2=g^2 N, as well as for the expansion of lattice theories in positive powers of 1/{tilde g}^2. If there were no renormalization effects, the tilde g expansion would have a finite radius of convergence. The zero-dimensional theory can be used for counting planar diagrams. It can be solved explicitly, so that the generating function for the number of diagrams with given 3-vertices and 4-vertices, can be derived exactly. This can be done for various kinds of Feynman diagrams. We end with some remarks about planar renormalization.