Defects in low-dimensional TQFTs

In modern physics, quantum field theory (QFT) is arguably the most developed attempt at describing the dynamics of nature. At its core is the idea that the types of fundamental particles are in fact described by fields, e.g. there is an electron field, a quark field, etc. permeating every point in the universe. Individual particles are encoded into the ripples of these fields, for example a ripple pattern in the electron field can encode a state of a single electron moving with some velocity, another pattern can encode several electrons and so on. The field description of particles automatically incorporates the laws of quantum mechanics and special relativity, whereas the particle interactions depend on the QFT there are theories for electromagnetism, nuclear forces, etc. The mathematics of QFTs is notoriously difficult. In fact, most of the computations involving it can only be carried out approximately and can even be proven to be inconsistent at their foundations. Even though useful information can nonetheless be extracted from QFTs by introducing various clever workarounds, there is a great conceptual disadvantage in not having a formal grasp on what a QFT actually is. There are several schools of thought, whose aim is to remedy this issue. One of them, the so-called functorial field theories, utilises ideas from category theory a formal study on how concepts in different areas of mathematics are related. According to it, a QFT is a functor a dictionary between two sets of concepts converting the geometry of space and time into the algebraic tools needed to perform computations. This idea works best if the geometric side of this definition is easy enough the most prominent are the so- called topological quantum field theories (TQFTs), in which the only relevant properties of the spacetime are those that are invariant under deformations. For example, the distance between two points in a TQFT is irrelevant, while the presence of a wormhole is taken into account. The goal of this project is to push the current boundaries of understanding TQFTs using the aforementioned functorial method. In particular, we study the ways of adding defects to the spacetime of a TQFT. These are special regions which can be interpreted as particles, strings, branes, etc. A lot of attention will be dedicated to studying how new theories can emerge from TQFTs by filling their spacetime with a very fine defect foam. We will also examine the concept of extra dimensions and the role it plays in these emergent theories. Interestingly, the methods that we use in this study can lead beyond the purely theoretical considerations. For example, they can be applied to understanding the topological phases of matter, which have been proposed as a possible way to build a quantum computer.