2017/10/27 by Paul Bruillard, Bruillard, Paul, Paul Gustafson +5
Mathematics · #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.1710.10284
openalex publication_date 2017/10/27 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We discuss several useful interpretations of the categorical dimension of\nobjects in a braided fusion category, as well as some conjectures demonstrating\nthe value of quantum dimension as a quantum statistic for detecting certain\nbehaviors of anyons in topological phases of matter. From this discussion we\nfind that objects in braided fusion categories with integral squared dimension\nhave distinctive properties. A large and interesting class of non-integral\nmodular categories such that every simple object has integral\nsquared-dimensions are the metaplectic categories that have the same fusion\nrules as SO(N)2 for some N. We describe and complete their classification\nand enumeration, by recognizing them as \ℤ2-gaugings of cyclic\nmodular categories (i.e. metric groups). We prove that any modular category of\ndimension 2km with m square-free and k\≤ 4, satisfying some additional\nassumptions, is a metaplectic category. This illustrates anew that dimension\ncan, in some circumstances, determine a surprising amount of the category's\nstructure.\n