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 $\mathcal{C}$. Following in the steps of Guionnet, Jones, and Shlyakhtenko, I will present a diagrammatic construction of a $II_{1}$ factor $M$ and a category of bimodules over $M$ which is equivalent to $\mathcal{C}$. Finally, I will show that the factor $M$ is an interpolated free group factor and can always be made to be isomorphic to $L(\mathbb{F}_{\infty})$. Therefore we will deduce that every rigid, countably generated $C^{*}$ tensor category is equivalent to a category of bimodules over $L(\mathbb{F}_{\infty})$.
This is joint work with Arnaud Brothier and David Penneys.