We introduce an invariant, called distortion, for inclusions of II_1 von Neumann algebras with finite index and finite-dimensional centers, thus called multifactors. Distortion allows to determine when the inclusion is extremal, when it admits an infinite tunnel and, if not, how many times one can perform the downward basic construction. We extend Popa’s classification of finite index finite depth hyperfinite subfactors to multifactors by means of the standard invariant, i.e. the 2-shaded unitary planar algebra associated with the inclusion, and the distortion.
As an application, we construct and classify, by means of their distortion, fully faithful representations of unitary multifusion categories into bimodules over hyperfinite II_1 multifactors.
Joint work with M. Bischoff, I. Charlesworth, S. Evington, D. Penneys,
arXiv:2010.01067, supported by ERC MSCA-IF beyondRCFT n. 795151.