Distortion for multifactor bimodules and representations of multifusion categories

Abstract

We call a von Neumann algebra with finite-dimensional center a multifactor. We introduce an invariant of bimodules over ${\mathrm{II}}_1$ multifactors that we call modular distortion, and use it to formulate two classification results.
We first classify finite depth finite index connected hyperfinite ${\mathrm{II}}_1$ 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 ${\mathrm{II}}_1$ 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 ${\mathrm{II}}_1$ multifactor inclusions.