Graph planar algebra embeddings and infinite depth subfactors

TL;DR

This paper demonstrates that certain exotic subfactors' planar algebras embed into graph planar algebras, providing new obstructions and examples, including a hyperfinite subfactor with a specific non-integer index related to the Haagerup subfactor.

Contribution

It establishes an embedding of subfactor planar algebras into graph planar algebras and constructs a hyperfinite subfactor with a novel index, advancing understanding of subfactor invariants.

Findings

Subfactor planar algebras embed into Jones' graph planar algebra.

Constructed a hyperfinite subfactor with index (5+√13)/2.

Provided a new obstruction for the standard invariant.

Abstract

Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.