CMSA Colloquium: Unitary extended topological field theories and higher hermitian objects

Mathematically, topological field theories study the representation theory of manifolds under cutting and gluing. Extended topological field theories take this idea a step further by allowing one to cut manifolds in multiple directions, leading to a formulation in terms of higher category theory. It is an open problem to give a complete definition and description of unitary representations of manifolds as extended topological field theories.

In my talk, I will explain some recent progress towards the definition and classification of unitary extended topological field theories. This is based on a categorification of the concept of a hermitian object in a rigid symmetric monoidal category, introduced by Freed and Hopkins in the study of non-extended unitary topological field theories.

The talk is based on several joint works involving (in part) Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Nivedita, David Penneys, David Reutter, Claudia Scheimbauer, Luuk Stehouwer, and Chetan Vuppulury, as well as ongoing work with Theo Johnson-Freyd, Cameron Krulewski, and Luuk Stehouwer.