Invariants and Topological Quantum Field Theories of Four-Manifolds

There are very, very many four-dimensional smooth manifolds. One way to study them is by defining invariants for them. Many invariants come from Topological Quantum Field Theories (TQFTs), which are very interesting from the physical perspective, since it is generally assumed that we all live on a four-dimensional manifold. In the 90s, the Crane-Yetter TQFT was defined in an attempt to understand quantum gravity as a TQFT. It was not very successful, and the resulting Crane-Yetter invariant was assumed to be quite trivial. Recently, I have worked out that it actually isn't as trivial as expected, since it contains at least finite gauge theory.

In this talk, I will explain how to make your own 4-manifold from simple building blocks, called "handles". I will mostly draw lots of nice pictures. Then, I'll give these pictures a semantics in category theory, more specifically, in ribbon fusion categories. At the end, I will bring everything together and show you how you can easily evaluate invariants of 4-manifolds yourself.