Distortion for multifactor bimodules and representations of multifusion categories

TL;DR

This paper introduces the modular distortion invariant for bimodules over II_1 multifactors and uses it to classify multifactor inclusions and representations of multifusion categories, advancing the understanding of their structure.

Contribution

It defines the modular distortion invariant and applies it to classify hyperfinite II_1 multifactor inclusions and representations of multifusion categories.

Findings

Classified finite depth hyperfinite II_1 multifactor inclusions using the standard invariant and modular distortion.

Characterized all distortions arising from representations of unitary multifusion categories.

Identified the subset of distortions corresponding to connected II_1 multifactor inclusions.

Abstract

We call a von Neumann algebra with finite dimensional center a multifactor. We introduce an invariant of bimodules over $\rm II_1$ multifactors that we call modular distortion, and use it to formulate two classification results. We first classify finite depth finite index connected hyperfinite $\rm II_1$ multifactor inclusions $A\subset B$ in terms of the standard invariant (a unitary planar algebra), together with the restriction to $A$ of the unique Markov trace on $B$. The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa's uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions. Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite $\rm II_1$ multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected $\rm II_1$ multifactor inclusions.