An angle between intermediate subfactors and its rigidity

TL;DR

This paper introduces a new concept of angle between intermediate subfactors, explores its properties, and establishes bounds on the number of such subfactors, linking subfactor theory with geometric concepts like the kissing number.

Contribution

It defines a novel angle between intermediate subfactors, proves bounds on this angle, and relates these bounds to the enumeration of subfactors, answering a longstanding question.

Findings

Angles between minimal intermediate subfactors are bounded between 60 and 90 degrees.

Number of intermediate subfactors grows at most exponentially with the Jones index.

The results connect subfactor theory with geometric bounds like the kissing number.

Abstract

We introduce a new notion of angle between intermediate subfactors and prove various interesting properties of the angle and relate it with the Jones' index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo published in 2003.