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Register a new userA finiteness property for braided fusion categories
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We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.
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We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.
For a braided fusion category $mathcal{V}$, a $mathcal{V}$-fusion category is a fusion category $mathcal{C}$ equipped with a braided monoidal functor $mathcal{F}:mathcal{V} to Z(mathcal{C})$. Given a fixed $mathcal{V}$-fusion category $(mathcal{C}, mathcal{F})$ and a fixed $G$-graded extension $mathcal{C}subseteq mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $widetilde{mathcal{F}}:mathcal{V} to Z(mathcal{D})$ of $mathcal{D}$ which are compatible with the enrichment of $mathcal{C}$. We show that G-crossed extensions of a braided fusion category $mathcal{C}$ are G-extensions of the canonical enrichment of $mathcal{C}$ over itself. As an application, we parameterize the set of $G$-crossed braidings on a fixed $G$-graded fusion category in terms of certain subcategories of its center, extending Nikshychs classification of the braidings on a fusion category.
The Witt group of nondegenerate braided fusion categories $mathcal{W}$ contains a subgroup $mathcal{W}_text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $mathcal{C}(mathfrak{g},k)$ consisting of highest weight integerable $hat{g}$-modules of level $k$ where $hat{mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[mathcal{C}(mathfrak{sl}_2,k)]$, $kgeq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[mathcal{C}(mathfrak{sl}_3,k)]$, $kgeq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $mathfrak{g}$ and positive integer levels $k$.
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.
We show that braidings on a fusion category $mathcal{C}$ correspond to certain fusion subcategories of the center of $mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.
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