The Classification of SU(3) Modular Invariants Revisited

TL;DR

This paper revisits the classification of SU(3) modular invariants, simplifying previous proofs, clarifying arguments, and highlighting independent results, all based on fundamental principles like modular invariance and positivity.

Contribution

The paper provides a simplified, clearer classification of SU(3) modular invariants using only basic assumptions, improving upon previous complex methods.

Findings

Complete classification of SU(3) modular invariants achieved.

Simplified proofs and clearer explanations provided.

Identification of smaller independent results within the classification.

Abstract

The SU(3) modular invariant partition functions were first completely classified in Ref.\ \SU. The purpose of these notes is four-fold: \item{(i)} Here we accomplish the SU(3) classification using only the most basic facts: modular invariance; $M_{\la\mu}\in{\bf Z}_{\ge}$; and $M_{00}=1$. In \SU{} we made use of less elementary results from Moore-Seiberg, in addition to these 3 basic facts. \item{(ii)} Ref.\ \SU{} was completed well over a year ago. Since then I have found a number of significant simplifications to the general argument. They are all included here. \item{(iii)} A number of people have complained that some of the arguments in \SU{} were hard to follow. I have tried here to be as explicit and as clear as possible. \item{(iv)} Hidden in \SU{} were a number of smaller results which should be of independent value. These are explicitly mentioned here.