Conformal Field Theory Techniques for Large N Group Theory

TL;DR

This paper introduces a novel approach using quantum mechanics on the U(N) group manifold to analyze large N group theory problems, connecting representation theory with free fermion models and collective field theory.

Contribution

It develops a formalism that maps U(N) representation theory problems to quantum mechanics and free fermions, enabling new analytical techniques for tensor product multiplicities and gauge theories.

Findings

Reduced complex problems to free fermions on a circle

Derived the collective field theory from bosonization

Provided solutions for large N Yang-Mills theories

Abstract

We show how to use quantum mechanics on the group manifold U(N) as a tool for problems in U(N) representation theory. The quantum mechanics reduces to free fermions on the circle, which in the large N limit become relativistic. The theory can be bosonized giving the Das-Jevicki-Sakita collective field theory. The formalism is particularly suited to problems involving tensor product multiplicity (Littlewood-Richardson) coefficients. As examples, we discuss the partition function of two-dimensional Yang-Mills theory on the sphere, and the zero magnetic field limit of D-dimensional Eguchi-Kawai Yang-Mills theory. We give the leading O(N^0) solution of the latter theory, using a method which allows computing corrections. Largely (but not completely) superseded by hep-th/9311130.