The classification of subfactors of index at most 5

TL;DR

This paper reviews the recent progress in classifying subfactors of von Neumann algebras with index up to 5, highlighting the finite classification results and the key ideas behind them.

Contribution

It summarizes the classification of subfactors with index up to 5, including the identification of only 10 such subfactors between indices 4 and 5, extending previous classifications.

Findings

Classification of subfactors up to index 4 is complete.

Only 10 subfactors exist between indices 4 and 5 outside infinite families.

Key ideas in the classification process are discussed.

Abstract

A subfactor is an inclusion $N \subset M$ of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action $M^G \subset M$, and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There's a measure of size of a subfactor, called the index. Remarkably the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the '80s and early '90s. The possible index values above 4 are not quantized, but once you exclude a certain family it turns out that again the possibilities are quantized. Recently the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.