Topological Quantum and Skein-Theoretic Aspects of Braided Fusion Categories

Abstract

The first part of this thesis is dedicated to the study of anyons and exchange symmetry. We discuss the theory of identical particles and recap the standard algebraic framework for describing the exchange statistics of anyons. The novel component consists of a derivation of the fusion structure of anyons from exchange symmetry. In order to achieve this, we construct a precise notion of exchange symmetry that is compatible with the spatially localised nature of anyons. In particular, given a system of $n$ quasiparticles, we show that the action of a specific $n$-braid uniquely specifies its superselection sectors. This $n$-braid satisfies several internal symmetries corresponding to the decompositions of the $n$-quasiparticle Hilbert space, and its spectrum is related to the topological spins of the quasiparticles.

The second part of this thesis is primarily concerned with skein-theoretic aspects of unitary (braided) fusion categories. Specifically, we consider a fusion rule of the form $q\otimes q \cong {1}\oplus\bigoplus^k_{i=1}x_{i}$ in a unitary fusion category $\mathcal{C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a ``canonical basis''. Assuming self-duality of the summands $x_{i}$, some general observations are made e.g. the real-symmetricity of the $F$-matrix $F^{qqq}_q$. We then find explicit formulae for $F^{qqq}_q$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the (framed) Kauffman and Dubrovnik link polynomials.