Claudia Scheimbauer on “Dualizabitility, higher categories, and topological field theories”

Claudia has given a series of seven lectures at the Copenhagen Center for Geometry and Topology.

Dualizabitility, higher categories, and topological field theories

In the past decade, using higher categories has proven to be an essential ingredient in the study of topological field theories (TFTs) from a mathematical perspective. The most prominent and seminal result is the Cobordism Hypothesis, which gives a beautiful classification of “fully extended” topological field theories. Here, fully extended means that our TFT can be evaluated at manifolds and bordisms of all dimensions below a given one; conversely, the mathematical language needed to describe the structure is that of higher categories and dualizability therein. In physics, we can interpret these values at all dimensions as (possibly higher) categories of boundary conditions, as point insertions/observables, line operators or higher dimensional operators, etc., depending on the stiuation.

The main goal of this master class will be to explain how to use the cobordism hypothesis to construct TFTs and variations thereof. One example we will look at in detail arises from factorization homology for E_n-algebras, which will also be a key tool in the parallel master class. We will discuss (∞,n)-categories and dualizability in detail, and, time permitting, some extensions and variations.