# Distortion for multifactor bimodules and representations of multifusion categories

October 02, 2020

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.

Keywords:

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