Work

Rank-finiteness for modular categories

Year: 2015

Type: article

Abstract: We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category C \mathcal {C} with N = ord ( T ) N= \textrm {ord}(T) , the order of the modular T T -matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension D 2 D^2 in the Dedekind domain Z [ e 2 π i N ] \mathbb {Z}[e^{\frac {2\pi i}{N}}] is identical to that of N N .

Source: Journal of the American Mathematical Society

DOI: 10.1090/jams/842

Authors: Paul Bruillard, Siu-Hung Ng, Eric Rowell, Zhenghan Wang

Institutions: Pacific Northwest National Laboratory, Texas A&M University, Louisiana State University, University of California, Santa Barbara

Language: English

Topic: Algebraic structures and combinatorial models

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: Hopf algebras, modular categories and their invariants, Collaborative Research: Topological Phases of Matter and their Application to Quantum Computing, Hopf algebras, modular categories and their invariants