Scott Morrison, Miller Institute & UC Berkeley
Connections and planar algebras
With Emily Peters, I've been exploring subfactors with index in the
interval $(5, 3+\sqrt{5})$. We've recently obtained a classification
of 1-supertransitive subfactors in this range, and performed an
extensive computer search in higher supertransitivities. I'll describe
the examples of subfactors we've found. We have two main new
techniques. First, even when we only know a fragment of a principal
graph, we can extract certain inequalities by considering the norms of
the entries of a connection. This allows the new classification
result. Second, we extend the theory of bi-unitary connections to the
bi-invertible case, and find we can then work over a fixed number
field. This allows effective use of the "hybrid method" of
constructing subfactors: given a not-necessarily flat bi-invertible
connection, we can efficiently solve the equations for a flat element
in the graph planar algebra. This lets us completely analyse the
examples.