dc.contributor.author Valera, Sachin Jayesh dc.date.accessioned 2021-08-19T09:14:19Z dc.date.available 2021-08-19T09:14:19Z dc.date.issued 2021-08-27 dc.date.submitted 2021-08-15T14:39:27.001Z dc.identifier container/1c/b0/23/eb/1cb023eb-1123-4dd7-8770-7e6a528b3837 dc.identifier.isbn 9788230869901 dc.identifier.isbn 9788230851586 dc.identifier.uri https://hdl.handle.net/11250/2770245 dc.description.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. en_US 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. dc.language.iso eng en_US dc.publisher The University of Bergen en_US dc.relation.haspart Paper I: Valera, S. J. (2021). “Fusion Structure from Exchange Symmetry in (2+1)-Dimensions”. Annals of Physics, 429, 168471. The article is available in the thesis file. The article is also available at: https://doi.org/10.1016/j.aop.2021.168471 en_US dc.relation.haspart Paper II: Valera, S. J. and Poudel, A. «Skein-Theoretic Methods for Unitary Fusion Categories». The article is available in the thesis file. The article is also available at: https://arxiv.org/abs/2008.07129 en_US dc.rights In copyright dc.rights.uri http://rightsstatements.org/page/InC/1.0/ dc.title Topological Quantum and Skein-Theoretic Aspects of Braided Fusion Categories en_US dc.type Doctoral thesis en_US dc.date.updated 2021-08-15T14:39:27.001Z dc.rights.holder Copyright the Author. All rights reserved en_US dc.description.degree Doktorgradsavhandling fs.unitcode 12-12-0
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