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dc.contributor.advisorRowell, Eric
dc.creatorCreamer, Daniel Edward
dc.date.accessioned2019-01-18T15:13:25Z
dc.date.available2020-08-01T06:39:30Z
dc.date.created2018-08
dc.date.issued2018-08-07
dc.date.submittedAugust 2018
dc.identifier.urihttp://hdl.handle.net/1969.1/174018
dc.description.abstractThe dissertation introduces a computational approach to classifying low rank modular categories up to their modular data. The modular data of a modular category is a pair of matrices, (S; T). Virtually all the numerical information of the category is contained within or derived from the modular data. The modular data satisfy a variety of criteria that Bruillard, Ng, Rowell, and Wang call the admissibility criteria. Of note is the Galois group of the S matrix is an abelian group that acts faithfully on the columns of the eigenvalue matrix, s = ( SvijS/v0j). This gives an injection from Gal(Q(S);Q) !Symr, where r is the rank of the category. Our approach begins by listing all the possible abelian subgroups of Symv6 and building all the possible modular data for each group. We run each set of modular data through a series of Gröbner basis calculations until we either find a contradiction or solve for the modular data. The effectiveness of this approach is shown by the two main results. The first is a complete classification of rank 6 non-self-dual and non-integral modular tensor categories, specifically any rank 6 non-integral non-self-dual modular category is isomorphic to a tensor product The second is a partial classification of the subgroups of Sym6 that give rise to self-dual non-integral modular tensor categories. Specifically, we show that the following groups have no associated modular category, ⟨(01234)⟩, ⟨(0123)⟩, ⟨(01)(23); (02)(13)⟩, ⟨(0123)(45)⟩, ⟨(012); (345)⟩, ⟨(01); (2345)⟩, ⟨(01)(2345)⟩, ⟨(01); (23)(45), (24)(35)⟩, ⟨(01)(23)(45); (24)(35)⟩, ⟨(01)(23)(45)⟩, or ⟨(01)(23), (23)(45)⟩. It is known that following groups do have categories associated with them, ⟨(012)⟩, ⟨(01)(23)⟩, ⟨(012)(345)⟩, ⟨(01)(23)(45); (02)(13)⟩, and ⟨(012345)⟩. It is unknown but conjectured that the following groups do not have a modular category associated to them, ⟨(01)⟩, ⟨(01); (234)⟩, and ⟨(01); (23)(45)⟩.en
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.subjectModular tensor categoriesen
dc.titleA Computational Approach to Classifying Low Rank Modular Tensor Categoriesen
dc.typeThesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorTexas A & M Universityen
thesis.degree.nameDoctor of Philosophyen
thesis.degree.levelDoctoralen
dc.contributor.committeeMemberKlappenecker, Andreas
dc.contributor.committeeMemberPapanikolas, Matthew
dc.contributor.committeeMemberWitherspoon, Sarah
dc.type.materialtexten
dc.date.updated2019-01-18T15:13:26Z
local.embargo.terms2020-08-01
local.etdauthor.orcid0000-0002-9853-972X


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