So far we have constructed topological defects and their relevant mathematical structures from the bottom up: starting from a d-dimensional QFT and looking at its unitary (d-1)-fusion category of topological defects. We have seen this approach in detail in the context of 2d QFTs.

One can proceed in a similar way and analyze the case of unitary 2-fusion categories (Douglas-Reutter 2018) and their lattice realizations (Imamura-Ohmori, 2023). The logic is very similar to what we have outlined in the 2d case.

In the final part of the course we will introduce a different approach to characterize topological defects, which exploits an auxiliary bulk-boundary system in one dimension higher, the Symmetry TQFT.

Aspects of TQFTs:

Bordisms groups and bordism category
Symmetric monoidal functors
Atyiah-Segal axiom: a TQFT is a symmetric monoidal functor from the bordism category to Vect
Partition function and Hilbert space of a TQFT
Dualizability and Hilbert space finite dimension
Example of classification result: Frobenius algebras and 2d TQFTs
Examples: Finite gauge theories as TQFTs

Aspects of extended TQFTs:

A rough sketch of higher categories: the infinity grupoid category
The bordism n-category and extended TQFT as symmetric monoidal functors of n-categories
Extended TQFTs and locality: the cobordism hypothesis
Topological defects in extended TQFTs and Topological Symmetry Theory: calculus of defects
Example: finite gauge theories as extended TQFTs and boundary conditions
Electromagnetic Duality and Duality Defects: Topological Symmetry Theory for the Ising Model

References:

Dan Freed, Lectures on Field Theory and Topology, CBMS Number 133, AMS 2019

Davi Costa et al., Simons Lectures on Categorical Symmetries, https://arxiv.org/abs/2411.09082

D. Freed and C. Teleman, Topological Dualities in the Ising Model, https://arxiv.org/abs/1806.00008

Lecture notes:

To appear