Generating functions for ranks of pre-modular categories

论奇数维模张量范畴

On odd-dimensional modular tensor categories

本文研究奇数维及低秩MNSD模张量范畴，利用伴随子范畴秩与可逆对象群阶建立秩下界，并据此分类若干奇秩范畴为点状或完美。

On the classification of non-self-dual modular categories

We develop a symbolic computational approach to classifying low-rank modular categories. We use this technique to classify pseudo-unitary modular categories of rank at most 5 that are non-self-dual, i.e. those for which some object is not isomorphic to its dual object.

利用相对幺半中心构造非半单模范畴

Constructing non-semisimple modular categories with relative monoidal centers

本文研究非半单模范畴的构造问题，通过明确Müger中心化子的模性条件，推导相对幺半中心及两类量子群表示生成非半单模范畴的条件，拓展了该类范畴的构造路径。

Modular categories of types B,C and D

We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières' modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements.

Construction of Modular Functors from Modular Tensor Categories

In this paper we follow the constructions of Turaev's book [Tu] closely, but with small modifications, to construct of a modular functor, in the sense of Kevin Walker, from any modular tensor category. We further show that this modular functor has duality and if the modular tensor category category is Hermitian or unitary, then the resulting modular functor is also Hermitian or unitary respectively.

Modular categories are not determined by their modular data

Arbitrarily many pairwise inequivalent modular categories can share the same modular data. We exhibit a family of examples that are module categories over twisted Drinfeld doubles of finite groups, and thus in particular integral modular categories.

Fermionic Modular Categories and the 16-fold Way

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towar…

Modular Categories of Dimension $p^3m$ with $m$ Square-Free

We give a complete classification of modular categories of dimension $p^3m$ where $p$ is prime and $m$ is a square-free integer. When $p$ is odd, all such categories are pointed. For $p=2$ one encounters modular categories with the same fusion ring as orthogonal quantum groups at certain roots of unity, namely $SO(2m)_2$. As an immediate step we classify a more general class of so-called even metaplectic modular categories with the same fusion rules as $SO(2N)_2$ with $N$ odd.

Congruence Subgroups and Super-Modular Categories

A super-modular category is a unitary pre-modular category with Müger center equivalent to the symmetric unitary category of super-vector spaces. Super-modular categories are important alternatives to modular categories as any unitary pre-modular category is the equivariantization of a either a modular or super-modular category. Physically, super-modular categories describe universal properties of quasiparticles in fermionic topological phases of matter. In general one does not have a representation of the modular group $\mathrm{SL}(2,\mathbb{Z})$ associated to a super-modular category, but it…

Modular categories with transitive Galois actions

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of $SL_2(\mathbb{Z})$ associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category $\mathcal{C}(\mathfrak{sl}_2,p-2)$ for some prime $p > 3$. As a consequence, we complet…

Pre-modular fusion categories of small global dimensions

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 $10$.

Modular categories from finite crossed modules

It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.

The rank of $G$-crossed braided extensions of modular tensor categories

We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.

On Generalized Symmetries and Structure of Modular Categories

Pursuing a generalization of group symmetries of modular categories to category symmetries in topological phases of matter, we study linear Hopf monads. The main goal is a generalization of extension and gauging group symmetries to category symmetries of modular categories, which include also categorical Hopf algebras as special cases. As an application, we propose an analogue of the classification of finite simple groups to modular categories, where we define simple modular categories as the prime ones without any nontrivial normal algebras.

Geometric construction of spinors in orthogonal modular categories

A geometric construction of Z_2-graded orthogonal modular categories is given. Their 0-graded parts coincide with categories previously obtained by Blanchet and the author from the category of tangles modulo the Kauffman skein relations. Quantum dimensions and twist coefficients for 1-graded simple objects (spinors) are calculated. We show that our even orthogonal modular categories admit cohomological refinements and the odd orthogonal ones lead to spin refinements. The relation with the quantum group approach is discussed.

Integral modular categories of Frobenius-Perron dimension $pq^n$

Integral modular categories of Frobenius-Perron dimension $pq^n$, where $p$ and $q$ are primes, are considered. It is already known that such categories are group-theoretical in the cases of $0 \leq n \leq 4$. In the general case we determine that these categories are either group theoretical or contain a Tannakian subcategory of dimension $q^i$ for $i>1$. We then show that all integral modular categories $\mathcal{C}$ with $\mathrm{FPdim}(\mathcal{C})=pq^5$ are group-theoretical, and, if in addition $p<q$, all with $\mathrm{FPdim}(\mathcal{C})=pq^6$ or $pq^7$ are group-theoretical. In the pro…

Pre-modular categories of rank 3

We classify ribbon semisimple monoidal categories with three isomorphism classes of simple objects over the field of complex numbers.

Some remarks on relative modular categories

We study properties of relative modular categories and derive sufficient conditions for their existence. In particular, we derive sufficient conditions for relative pre-modular categories to be non-degenerate and relative modular, and for the quotient of categories taking a particular form by their ideal of negligible modules to be generically semi-simple.

Modular categories, integrality and Egyptian fractions

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. T…

低秩奇数维模范畴的分类

Classification of low-rank odd-dimensional modular categories

本文研究奇数维模范畴的低秩分类问题，证明了秩至多23者均为带点范畴，刻画了秩25的三种可能类型，并给出秩最高为73的部分分类结果。

On invariants of Modular categories beyond modular data

We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.

Classification of Rank 6 Modular Categories with Galois Group $\langle (012)(345)\rangle$

Modular Tensor Categories (MTC's) arise in the study of certain condensed matter systems. There is an ongoing program to classify MTC's of low rank, up to modular data. We present an overview of the methods to classify modular tensor categories of low rank, applied to the specific case of a rank 6 category with Galois group $\langle(012)(345)\rangle$, and show that certain symmetries in this case imply nonunitarizable (hence, nonphysical) MTC's. We show that all the rank 6 MTC's with this Galois group have modular data conjugate to either the product of the semion category with $(A_1, 5)_{\fra…

非半单模范畴的模数据

Modular data of non-semisimple modular categories

本文研究非半单模范畴，通过可分解ribbon Hopf代数的表示扩展半单理论，聚焦Cohen-Westreich模数据及其$S$-矩阵对角化混合融合规则，并详细分析小量子群和Nichols Hopf代数实例。

On the Classification of Weakly Integral Modular Categories

We classify all modular categories of dimension $4m$, where $m$ is an odd square-free integer, and all ranks $6$ and $7$ weakly integral modular categories. This completes the classification of weakly integral modular categories through rank $7$. Our results imply that all integral modular categories of rank at most $7$ are pointed (that is, every simple object has dimension $1$). All strictly weakly integral (weakly integral but non-integral) modular categories of ranks $6$ and $7$ have dimension $4m$, with $m$ an odd square free integer, so their classification is an application of our main …

Integral Metaplectic Modular Categories

A braided fusion category is said to have Property $\textbf{F}$ if the associated braid group representations factor over a finite group. We verify integral metaplectic modular categories have property $\textbf{F}$ by showing these categories are group theoretical. For the special case of integral categories $\mathcal{C}$ with the fusion rules of $SO(8)_2$ we determine the finite group $G$ for which $Rep(D^ωG)$ is braided equivalent to $\mathcal{Z}(\mathcal{C})$. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.

Simple current auto-equivalences of modular tensor categories

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.

关于近群中心与超模范畴

On near-group centers and super-modular categories

本文通过修改与群$\mathbb Z/6$和$\mathbb Z/2\times\mathbb Z/4$相关联的近群融合范畴的Drinfeld中心，实现了Cho等人提出的两个神秘超模数据族，并探讨了近群中心提供超模范畴的条件。

Minimal modular extensions for super-Tannakian categories

In this paper, we continue with the ideas presented in [GVR17]. In this opportunity, we apply the fermionic action concept to classify in cohomology terms the minimal modular extensions of a super-Tannakian category. For this goal, we study some properties of equivariantization and de-equivariantization processes and cohomology data for the fermionic case.

Modular invariants for group-theoretical modular data. I

We classify indecomposable commutative separable (special Frobenius) algebras and their local modules in (untwisted) group-theoretical modular categories. This gives a description of modular invariants for group-theoretical modular data. As a bi-product we provide an answer to the question when (and in how many ways) two group-theoretical modular categories are equivalent as ribbon categories.

A spin decomposition of the Verlinde formula for type A modular categories

A modular category is a braided category with some additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We compute this formulas and discuss reductions and refinements for modular categories related with SU(N).Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer.We introduce the notion of a sp…

关于与仿射李代数相关的带状张量范畴的酉性与模性

On the unitarity and modularity of ribbon tensor categories associated with affine Lie algebras

本文通过数值计算和两个猜想，确定了与仿射李代数相关的带状张量范畴在哪些 $q$ 值下为（伪）酉和模的，并与文献结果完全一致，补充了缺失情况。

具有少量扭的模融合范畴

Modular fusion categories with few twists

本研究针对单对象扭的不同取值少于4个的模融合范畴，证明了对任意正整数$N$，满足扭为$N$次单位根真子集且Frobenius-Schur指数为$N$的此类范畴在辫子等价下仅有有限个，完成了对应分类。

On the classification of almost square-free modular categories

Let $C$ be a modular category of Frobenius-Perron dimension $dq^n$, where $q$ is a prime number and $d$ is a square-free integer. We show that if $q>2$ then $C$ is integral and nilpotent. In particular, $C$ is group-theoretical. In the general case, we describe the structure of $C$ in terms of equivariantizations of group-crossed braided fusion categories.

On Arithmetic Modular Categories

Modular categories are important algebraic structures in a variety of subjects in mathematics and physics. We provide an explicit, motivated and elementary definition of a modular category over a field of characteristic 0 as an equivalence class of solutions to a set of polynomial equations. We conclude that within each class of solutions, there is one which consists entirely of algebraic numbers. These algebraic solutions make it possible to discuss defining algebraic number fields of modular categories and their Galois twists. One motivation for such a definition is an arithmetic theory of m…

来自$\boldsymbol{\frak{osp}(2|2n)}$的相对模范畴

Relative modular categories from $\mathfrak{osp}(2 \vert 2n)$

本文基于$\boldsymbol{\frak{osp}(2|2n)}$的展开量子群加权表示理论，构造了相对模范畴。

小维数极大非自对偶模范畴的分类

Classification of maximally non-self-dual modular categories of small dimension

本文分类了 FP 维数小于 2025 的非点极大非自对偶模范畴，证明其至多有两种类型，并确定了最小 FP 维数的非群理论及非点例子分别为 675 和 243。

Classification of integral modular categories of Frobenius-Perron dimension pq^4 and p^2q^2

We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.

Infinite families of potential modular data related to quadratic categories

We present several infinite families of potential modular data motivated by examples of Drinfeld centers of quadratic categories. In each case, the input is a pair of involutive metric groups with Gauss sums differing by a sign, along with some conditions on the fixed points of the involutions and the relative sizes of the groups. From this input we construct $S$ and $T$ matrices which satisfy the modular relations and whose Verlinde coefficients are non-negative integers. We also check certain restrictions coming from Frobenius-Schur indicators. These families generalize Evans and Gannon's co…

维数为 $p^2q^2$ 的整非群论模范畴

Integral non-group-theoretical modular categories of dimension $p^2q^2$

本文分类构造维数为 $p^2q^2$ 的整非群论模范畴，给出其存在的充要条件及秩公式。

弱群论模融合范畴的广义性质F

Generalized property F for weakly group-theoretical modular fusion categories

本文证明弱群论模融合范畴具有广义性质F，即其Reshetikhin-Turaev映射类群表示对紧致定向曲面具有有限射影像，推广了早期结果并扩展至完整映射类群。

Nondegenerate extensions of near-group braided fusion categories

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…

Prime decomposition of modular tensor categories of local modules of Type D

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $\text{Rep}(H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt}$. For $\mathfrak{g}=\mathfrak{so}_5$ we describe the decomposition of $\mathcal{C}(\mathfrak{g},k)_R^…

利用局部模构造非半单模范畴

Constructing non-semisimple modular categories with local modules

该研究定义非半单模范畴中的刚性Frobenius代数类，证明其局部模范畴仍为模范畴，推广半单情形的已有结论，给出相关示例与关联，并将群代数相关的分类结果推广到任意特征。

用Hopf代数构造模张量范畴

Constructing Modular Tensor Categories with Hopf Algebras

本文通过研究Hopf代数的中心扩张、商代数可因子化条件及ribbon元构造方法，得到两类非半单和一族半单的ribbon可因子化Hopf代数，证明其表示范畴为素模张量范畴且无法通过常规张量积构造，为模张量范畴的构建提供了新途径。

Finitely semisimple spherical categories and modular categories are self-dual

We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.

Module categories, weak Hopf algebras and modular invariants

We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.

From Quantum Groups to Unitary Modular Tensor Categories

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasiz…

Jones-Wassermann subfactors for modular tensor categories

The representation theory of a conformal net is a unitary modular tensor category. It is captured by the bimodule category of the Jones-Wassermann subfactor. In this paper, we construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are self-dual. It generalizes and categorifies the self-duality of finite abelian groups and we call it modular self-duality.

Group-theoretical properties of nilpotent modular categories

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue of the Sylow decomposition. If the simple objects of C have integral Frobenius-Perron dimensions then C is group-theoretical. As a consequence, we obtain that semisimple quasi-Hopf algebras of prime power dimension are group-theoretical. Our arguments are based on a reconstruction of twisted group doubles from Lagrangian subcategories of modular categories (t…

Making nontrivially associated modular categories from finite groups

We show that the non-trivially associated tensor category constructed from left coset representatives of a subgroup of a finite group is a modular category. Also we give a definition of the character of an object in a ribbon category which is the category of representations of a braided Hopf algebra in the category. The definition is shown to be adjoint invariant and multiplicative. A detailed example is given. Finally we show an equivalence of categories between the non-trivially associated double D and the category of representations of the double of the group D(X).

On inner product in modular tensor categories. I

In this paper we study modular tensor categories (braided rigid balanced tensor categories with additional finiteness and non-degeneracy conditions), in particular, representations of quantum groups at roots of unity. We show that the action of modular group on certain spaces of morphisms in MTC is unitary with respect to the natural inner product on these spaces. In a special case of category based on representations of the quantum group U_q sl_n at roots of unity we show that in some of these spaces of morphisms (for U_q sl_2, in all of them) the action of modular group can be written in ter…

Modular categories as representations of the 3-dimensional bordism 2-category

We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.

Modular Categories Associated to Unipotent Groups

Let G be a unipotent algebraic group over an algebraically closed field k of characteristic p > 0 and let l be a prime different from p. Let e be a minimal idempotent in D_G(G), the braided monoidal category of G-equivariant (under conjugation action) \bar{Q_l}-complexes on G. We can associate to G and e a modular category M_{G,e}. In this article, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class C_p^{\pm}.

Lattices, Vertex Algebras and Modular Categories

In this note we give an account of recent progress on the construction of holomorphic vertex algebras as cyclic orbifolds as well as related topics in lattices and modular categories. We present a novel computation of the Schur indicator of a lattice involution orbifold using finite Heisenberg groups and discriminant forms.