A Computational Approach to Classifying Low Rank Modular Tensor Categories
Abstract
The 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)⟩.
Subject
Modular tensor categoriesCitation
Creamer, Daniel Edward (2018). A Computational Approach to Classifying Low Rank Modular Tensor Categories. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /174018.