Subfactors of index exactly 5

TL;DR

This paper classifies subfactor planar algebras with index exactly 5, showing all examples originate from subgroup subfactors and establishing uniqueness and non-existence results.

Contribution

It provides a complete classification of subfactor planar algebras at index 5, extending previous work and including new non-existence proofs.

Findings

All subfactor planar algebras at index 5 are from subgroup subfactors

Uniqueness results are derived from Izumi's work

Certain subfactor planar algebras with index 5 do not exist

Abstract

We give the classification of subfactor planar algebras at index exactly 5. All the examples arise as standard invariants of subgroup subfactors. Some of the requisite uniqueness results come from work of Izumi in preparation. The non-existence results build upon the classification of subfactor planar algebras with index less than 5, with some additional analysis of special cases.