Title: Distortion for II_1 multifactor bimodules
Speaker: David Penneys, OSU
Abstract: By a deep theorem of Popa, finite depth hyperfinite subfactors are completely determined by their standard invariants. As a consequence, every unitary fusion category admits a unique unitary tensor functor into Bim(R), where R is the hyperfinite II_1 factor. These results are no longer true when subfactors and fusion categories are replaced by inclusions of hyperfinite II_1 multifactors (finite direct sums of II_1 factors) and unitary multifusion categories. We introduce a new invariant of such inclusions called the distortion homomorphism and show that the standard invariant, together with the distortion, completely characterizes finite depth connected inclusions of hyperfinite multifactors. We also determine the possible values this distortion homomorphism can take for inclusions with a given standard invariant. We then show that the distortion homomorphism is a complete invariant of unitary tensor functors from an n\times n unitary multifusion category C into Bim(R^{\oplus n}), where all possible distortion homomorphisms are realized. As a corollary, in contrast to the subfactor/fusion setting, not every unitary tensor functor from C to Bim(R^{\oplus n}) arises from a multifactor inclusion. This is joint work with Bischoff, Charlesworth, Evington, Giorgetti, and Henriques which began at the 2018 AMS MRC on Quantum Symmetries: Subfactors and Fusion Categories.
