A tensor category $\mathcal{C}$ over a field $\mathbb{K}$ is said to be invertible if there's a tensor category $\mathcal{D}$ such that $\mathcal{C}\boxtimes\mathcal{D}$ is Morita equivalent to $\mathrm{Vec}_{\mathbb{K}}$. When $\mathbb{K}$ is algebraically closed, it is well-known that the only invertible fusion category is $\mathrm{Vec}_{\mathbb{K}}$, and any invertible multi-fusion category is Morita equivalent to $\mathrm{Vec}_{\mathbb{K}}$. By contrast, we show that for general $\mathbb{K}$ the invertible multi-fusion categories over a field $\mathbb{K}$ are classified (up to Morita equivalence) by $H^3(\mathbb{K};\mathbb{G}_m)$, the third Galois cohomology of the absolute Galois group of $\mathbb{K}$. We explicitly construct a representative of each class that is fusion (but not split fusion) in the sense that the unit object is simple (but not split simple). One consequence of our results is that fusion categories with braided equivalent Drinfeld centers need not be Morita equivalent when this cohomology group is nontrivial.
We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the s…
In this short note we investigate the process of constructing auto-equivalences of modular tensor categories using invertible objects. We derive conditions on the invertible object for the resulting auto-equivalence to be either monoidal, braided, or pivotal. We also discuss the composition of these auto-equivalences constructed from invertible objects. To demonstrate the practicality of this construction, we construct auto-equivalences of several real-world examples of modular tensor categories.
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.
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 generali…
We describe the structure of a generalized near-group fusion category and present an example of this class of fusion categories which arises from the extension of a Fibonacci category. We then classify slightly degenerate generalized near-group fusion categories. We also prove a structure result for braided generalized Tambara-Yamagami fusion categories.
In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions $2p^nd$ and $4p^nd$ are nilpotent, where $p$ is an odd prime and $d$ is an odd square-free integer. Then we classify slightly degenerate generalized Tambara-Yamagami fusion categories and weakly integral slightly degenerate fusion categories of particular dimensions.
We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all formal codegrees of spherical fusion categories with square-free norm are rational integers. This implies, with three exceptions, that every spherical braided fusion category whose global dimension has prime norm is pointed. The reason exceptions occur is related to the classical Schur-Siegel-Smyth p…
We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, ζ}$, where $N \geq 1$ is a natural number and $ζ$ is a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order $\mathbb Z_{2^N}$. We show that every braided $N$-Ising fusion category is prime and also that there exists a slightly degenerate $N$-Ising braided fusion category for all $N > 2$. We also prove a structure result for braided extension…
In this paper, we show that non-degenerate fusion categories of FP-dimensions $p^2q^3d$ and $p^3q^3d$ are group-theoretical, where $p, q$ are odd primes, $d$ is a square-free integer such that $(pq,d) = 1$.
The Grothendieck Ring of Strictly Weakly Integral Fusion Categories of Rank 6
College of Mathematical Science, Yangzhou University（扬州大学数学科学学院）；College of Mathematics and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）
In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius-Schur exponents. We classify spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius-Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius-Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.
In this paper, we study a family of fusion and modular systems realizing fusion categories Grothendieck equivalent to the representation category for $so(2p+1)_2$. These categories describe non-abelian anyons dubbed `metaplectic anyons'. We obtain explicit expressions for all the $F$- and $R$-symbols. Based on these, we conjecture a classification for their monoidal equivalence classes from an analysis of their gauge invariants and define a function which gives us the number of classes.
We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.
We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to $10$.
A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action of $G$ on the center $Z(\mathcal C)$. If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action. We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.
