Radford's theorem and the belt trick

5 November 2012
Noah Snyder
Topological field theories give a connection between topology and algebra. This connection can be exploited in both directions: using algebra to construct topological invariants, or using topology to prove algebraic theorems. In this talk, I will explain an interesting example of the latter phenomena. Radford's theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a finite tensor category the quadruple dual functor is easy to understand. It's somewhat mysterious that the double dual is hard to understand but the quadruple dual is easy. Using topological field theory, we show that Radford's theorem is exactly the consequence of the Dirac belt trick in topology. That is, the double dual corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the quadruple dual is trivial in an appropriate sense exactly because $\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large project, joint with Chris Douglas and Chris Schommer-Pries, to understand local field theories with values in the 3-category of tensor categories via the cobordism hypothesis.