These lectures aim to be a gentle introduction to topological quantum field theory (TQFT) and the categorical structures it naturally gives rise to, with the aim to make connections to topological phases of matter and topological quantum computation. No prior knowledge of category theory will be assumed.
Lecture 1 will motivate the functorial definition of closed TQFTs, using desired properties of the path integral such as locality. We will introduce the categories of vector spaces and of n-dimensional bordisms, and then define a TQFT to be a structure-preserving map between the two. After a quick study of the case $n=1$, we will focus on the case $n=2$ and argue that 2-dimensional closed TQFTs are equivalent to commutative Frobenius algebras, which we will illustrate with examples coming from sigma models and Landau-Ginzburg models. Finally, we will give an overview of the case $n=3$ (to which we will return in Lecture 4)
Lectures 2 and 3 deal with refinements of 2-dimensional TQFTs where one studies a larger class of bordisms (with extra structure): "open-closed TQFTs" and "defect TQFTs". We will explain how the former naturally lead to "Calabi-Yau categories" (which will be defined along the way) that have the interpretation of describing certain D-branes and open strings in string theory. Similarly, 2-dimensional defect TQFTs give rise to three-layered structures called 2-categories. We will introduce them and explain how they neatly encode the totality of all bulk theories, interfaces and point defects of a given topological theory. After again discussing the examples of sigma models and Landau-Ginzburg models, we will illustrate the usefulness of the higher-categorical language by describing a generalisation of the orbifold construction which unifies state sum models, equivariant group actions, and provides new relations between 2-dimensional TQFTs.
Lecture 4 wants to lift all the previous constructions from 2 to 3 dimensions. In particular, we will define 3-dimensional defect TQFTs, sketch examples, explain what 3-categories are and how they arise naturally from TQFTs. Time permitting, we will discuss the role of such categories in the description of some (2+1)-dimensional topological phases of matter, and how 3-dimensional orbifolds can provide universal topological quantum computation.