5 November 2012

15:45

Noah Snyder

Abstract

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.