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The braid group, representations and non-abelian anyons
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2015 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

This bachelor’s thesis concerns unitary linear representations of the braid group, motivated by their connection to two dimensional particle statistics and certain quasi-particles called non-abelian anyons. Particle statistics in two dimensions is related to the braid group via the fundamental group of the configuration space of indistinguishable particles in two dimensions, and non-abelian anyons correspond to non-commutative unitary representations of the braid group. In the aim of understanding these connections and studying the mathematical possibilities for non-abelian anyons, algebraic and topological definitions as well as results about braids and their representations are presented and investigated. The focus is on representations of low dimension, and a characterisation of low-dimensional irreducible complex representations is analysed. The unitarisability of such representations and the consequences for non-abelian anyons are then considered.

Abstract [sv]

Detta kandidatexamensarbete berör unitära linjära representationer av flätgruppen, med motivering av deras koppling till två dimensionell partikelstatistik och de kvasipartiklar som kallas icke-abelska anyoner. Partikelstatistik i två dimensioner är relaterat till flätgruppen via fundamental gruppen av konfigurationsrummet av icke-särskiljbara partiklar i två dimensioner, och icke-abelska anyoner svarar mot icke-kommutativa unitära representationerav flätgruppen. I syfte att förstå dessa kopplingar och studera de matematiska möjligheterna för icke-abelska anyoner presenteras och undersöks algebraiska och topologiska definitioner såväl som resultat för flätor och deras representationer. Fokus är på representationer med låg dimension, och en karaktärisering av lågdimensionella irreducibla komplexa representationer analyseras. Vidarebetraktas unitariserbarheten av sådana representationer och konsekvenserna för icke-abelska anyoner.

Place, publisher, year, edition, pages
2015. , 44 p.
National Category
URN: urn:nbn:se:kth:diva-167993OAI: diva2:813777
Available from: 2015-05-25 Created: 2015-05-25 Last updated: 2016-06-22Bibliographically approved

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