Title: Classifying small quantum symmetries
Speaker: David Penneys
Abstract: Classical symmetry is understood by group actions; an object has G-symmetry if G acts on it. Fixing a finite group G, the category of unitary representations on finite dimensional Hilbert spaces forms a unitary fusion category, which has direct sums, a tensor product, and a good notion of duals. More examples of such categories come from quantum groups, conformal field theory, and subfactor theory. Thus we think of unitary fusion categories as objects which encode quantum symmetry.
Through study of unitary fusion categories, we are naturally led to von Neumann algebras and subfactors. In fact, every unitary fusion category can be realized through the standard invariant of a finite index subfactor, and thus subfactors are universal hosts for quantum symmetries. In one sense, the simplest examples of quantum symmetry come from subfactors of small index. In this talk, I will discuss the ongoing small index subfactor classification program and the search for exotic examples. In recent joint work with Afzaly and Morrison, we give the complete classification of subfactor standard invariants to index 5+1/4, which includes 3+\sqrt{5}, the first interesting composite index.