Field theories with defects and the centre functor

TL;DR

This paper introduces a functorial approach to quantum field theories with defects, focusing on two-dimensional models, and demonstrates how defect TFTs can define the algebraic centre using higher categories.

Contribution

It develops a functorial framework for 2D quantum field theories with defects, emphasizing the role of bicategories and applying it to define algebraic centres.

Findings

A functorial formulation of 2D TFTs with defects is established.

Defect TFTs can be used to define the centre of an algebra functorially.

Higher categories, especially bicategories, are crucial in describing field theories with defects.

Abstract

This note is intended as an introduction to the functorial formulation of quantum field theories with defects. After some remarks about models in general dimension, we restrict ourselves to two dimensions - the lowest dimension in which interesting field theories with defects exist. We study in some detail the simplest example of such a model, namely a topological field theory with defects which we describe via lattice TFT. Finally, we give an application in algebra, where the defect TFT provides us with a functorial definition of the centre of an algebra. This involves changing the target category of commutative algebras into a bicategory. Throughout this paper, we emphasise the role of higher categories - in our case bicategories - in the description of field theories with defects.