The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})

TL;DR

This paper provides evidence that the landscape of subfactors with index in the interval (5, 3+√5) is nearly empty, establishing uniqueness results for known subfactors and showing the rarity of others through computational and theoretical methods.

Contribution

The paper demonstrates the uniqueness of known subfactors in the interval and shows that other potential subfactors would require very high rank, extending the classification boundary.

Findings

Only two known quantum-group subfactors exist in the interval.

The 1-supertransitive subfactor with index in this range is unique.

Any other subfactor would need rank at least 38.

Abstract

Progress on classifying small index subfactors has revealed an almost empty landscape. In this paper we give some evidence that this desert continues up to index 3+\sqrt{5}. There are two known quantum-group subfactors with index in this interval, and we show that these subfactors are the only way to realize the corresponding principal graphs. One of these subfactors is 1-supertransitive, and we demonstrate that it is the only 1-supertransitive subfactor with index between 5 and 3+\sqrt{5}. Computer evidence shows that any other subfactor in this interval would need to have rank at least 38. We prove our uniqueness results by showing that there is a unique flat connection on each graph. The result on 1-supertransitive subfactors is proved by an argument using intermediate subfactors, running the `odometer' from the FusionAtlas` Mathematica package and paying careful attention to dimensions.