Rigid C∗ tensor categories of bimodules over interpolated free group factors
The notion of a fantastic (or factor) planar algebra will be presented and some examples will be given. I will then show how such an object can be used to diagrammatically describe a rigid, countably generated C∗ tensor category C. Following in the steps of Guionnet, Jones, and Shlyakhtenko, I will present a diagrammatic construction of a II1 factor M and a category of bimodules over M which is equivalent to C. Finally, I will show that the factor M is an interpolated free group factor and can always be made to be isomorphic to L(F∞). Therefore we will deduce that every rigid, countably generated C∗ tensor category is equivalent to a category of bimodules over L(F∞).
This is joint work with Arnaud Brothier and David Penneys.