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April 24, 2020

Rank Finiteness for Braided Fusion Categories

Fusion categories arise as finite systems of bifinite bimodules of II1 factors, and play an important role in the theory of subfactors and algebraic quantum field theory. The number of (isomorphism classes) of simple objects in a fusion category is called its rank. It is natural to ask whether there are only finitely many fusion categories for each rank. While this question is wide open in general, we show that there are only finitely many braided fusion categories of each rank, interpolating the classical result for symmetric fusion categories and the more recent result for modular categories due to Bruillard, Ng, Rowell and Wang. Based on joint work with Scott Morrison, Dmitri Nikshych and Eric Rowell.