The SO(3) action on the space of finite tensor categories

The cobordism hypothesis gives a correspondence between the framed local topological field theories with values in $\mathcal C$ and a fully dualizable objects in $\mathcal C$. Changing framing gives an $O(n)$ action on the space of local TFTs, and hence by the cobordism hypothesis it gives a (homotopy coherent) action of $O(n)$ on the space of fully dualizable objects in $\mathcal C$. One example of this phenomenon is that $O(3)$ acts on the space of fusion categories. In fact, $O(3)$ acts on the larger space of finite tensorcategories.  I'll describe this action explicitly and discuss its relationship to the double dual, Radford's theorem, pivotal structures, and spherical structures.  This is part of work in progress joint with Chris Douglas and Chris Schommer-Pries.