On the Classification of Low-Rank Braided Fusion Categories
MetadataShow full item record
A physical system is said to be in topological phase if at low energies and long wavelengths the observable quantities are invariant under diffeomorphisms. Such physical systems are of great interest in condensed matter physics and computer science where they can be applied to form topological insulators and fault–tolerant quantum computers. Physical systems in topological phase may be rigorously studied through their algebraic manifestations, (pre)modular categories. A complete classification of these categories would lead to a taxonomy of the topological phases of matter. Beyond their ties to physical systems, premodular categories are of general mathematical interest as they govern the representation theories of quasi–Hopf algebras, lead to manifold and link invariants, and provide insights into the braid group. In the course of this work, we study the classification problem for (pre)modular categories with particular attention paid to their arithmetic properties. Central to our analysis is the question of rank finiteness for modular categories, also known as Wang’s Conjecture. In this work, we lay this problem to rest by exploiting certain arithmetic properties of modular categories. While the rank finiteness problem for premodular categories is still open, we provide new methods for approaching this problem. The arithmetic techniques suggested by the rank finiteness analysis are particularly pronounced in the (weakly) integral setting. There, we use Diophantine techniques to classify all weakly integral modular categories through rank 6 up to Grothendieck equivalence. In the case that the category is not only weakly integral, but actually integral, the analysis is further extended to produce a classification of integral modular categories up to Grothendieck equivalence through rank 7. It is observed that such classification can be extended provided some mild assumptions are made. For instance, if we further assume that the category is also odd–dimensional, then the classification up to Grothendieck equivalence is completed through rank 11. Moving beyond modular categories has historically been difficult. We suggest new methods for doing this inspired by our work on (weakly) integral modular categories and related problems in algebraic number theory. The allows us to produce a Grothendieck classification of rank 4 premodular categories thereby extending the previously known rank 3 classification.
braided fusion categories
Bruillard, Paul Joseph (2013). On the Classification of Low-Rank Braided Fusion Categories. Doctoral dissertation, Texas A & M University. Available electronically from
Showing items related by title, author, creator and subject.
Parallel distributed processing models using the back-propagation rule for studying analytic and holistic modes of processing in category learning Bauer, Niels Konrad (Texas A&M University, 1988)Not available
Belcher, Brian Neil (Texas A&M University, 1998)There are a variety of studies dealing with the persistence of many meteorological variables, including temperature. The resulting sequences of days with similar conditions have also been studied along with these persistence ...
Chavez Dominguez, Javier (2012-10-19)We study analogues, in the Lipschitz and Operator Spaces categories, of several classical ideals of operators between Banach spaces. We introduce the concept of a Banach-space-valued molecule, which is used to develop a ...