Pre-modular categories of rank 3

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…

略退化范畴与$\u2124$-模数据

Slightly degenerate categories and $\mathbb{Z}$-modular data

本研究针对略退化融合范畴，突破球面范畴限制采用枢轴范畴，证明其可自然生成Z-模数据，并在此框架下解释了循环复反射群非平凡族对应Z-模数据的Bonnafé-Rouquier范畴化。

Integral almost square-free modular categories

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius-Perron dimension $p^nm$, where $p$ is a prime number, $m$ is a square-free natural number and ${\rm gcd}(p,m)=1$. We prove that if $n\leq 5$ or $m$ is prime with $m<p$ then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and H. Tucker. As an application, we prove that an integral modular category whose Frobenius-Perron dimensions is odd and less than $1125$ is group-theoretical.

Pre-modular fusion categories of global dimensions $p^2$

Let $p\geq5$ be a prime, we show that a non-pointed modular fusion category $\mathcal{C}$ is Grothendieck equivalent to $\mathcal{C}(\mathfrak{sl}_2,2(p-1))_A^0$ if and only if $\dim(\mathcal{C})=p\cdot u$, where $u$ is a certain totally positive algebraic unit and $A$ is the regular algebra of the Tannakian subcategory $\text{Rep}(\mathbb{Z}_2)\subseteq\mathcal{C}(\mathfrak{sl}_2,2(p-1))$. As a direct corollary, we classify non-simple modular fusion categories of global dimensions $p^2$.

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 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.

A Geometric Construction for Permutation Equivariant Categories from Modular Functors

Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper arXiv:0812.0986 [math.CT].

A Computational Approach to Classifying Low Rank Modular Categories

This paper introduces a computational approach to classifying low rank modular categories up to their modular data. The modular data of a modular category is a pair of matrices, $(S,T)$. Virtually all the numerical information of the category is contained within or derived from the modular data. The modular data satisfy a variety of criteria that Bruillard, Ng, Rowell, and Wang call the admissibility criteria. Of note is the Galois group of the $S$ matrix is an abelian group that acts faithfully on the columns of the eigenvalue matrix, $s = (\frac{S_{ij}}{S_{0j}})$. This gives an injection fro…

Equivariant Modular Categories via Dijkgraaf-Witten Theory

Based on a weak action of a finite group J on a finite group G, we present a geometric construction of J-equivariant Dijkgraaf-Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group J on a group G, the category is described as the representation category of a J-ribbon algebra that generalizes the Drinfel'd double of the finite group G.

Verlinde模范畴的辫状Zesting及其模数据

Braided Zestings of Verlinde Modular Categories and Their Modular Data

本文针对单位根处量子群相关的模范畴，分类并构造了所有辫状zesting数据，还基于对应李代数的根系推导了新模范畴模数据的闭式公式。

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…

Classification of super-modular categories by rank

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank=$6$, and spin modular categories up to rank=$11$. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank $2,4$ and $6$, namely $PSU(2)_{4k+2}$ for $k=0,1$ and $2$. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.

Modular categories and orbifold models II

This is a continuation of the paper "Modular tensor categories and orbifold theories", arXiv:math.QA/0104242. It discusses orbifold models of conformal filed theory, or, in mathematical language, question of constructing the category of representations of the fixed point algebra $V^G$ for a given vertex operator algebra $V$ with an action of a finite group $G$. The previous paper gave a proof of well-known conjecture of Dijkgraaf-Vafa-Verlinde-Verlinde giving a complete answer to this question in the holomorphic case (when $V$ has a unique simple module, $V$ itself) under the assumption that c…

Modular tensor categories, subcategories, and Galois orbits

We establish a set of general results to study how the Galois action on modular tensor categories interacts with fusion subcategories. This includes a characterization of fusion subcategories of modular tensor categories which are closed under the Galois action, and a classifcation of modular tensor categories which factor as a product of pointed and transitive categories in terms of pseudoinvertible objects. As an application, we classify modular tensor categories with two Galois orbits of simple objects and a nontrivial grading group.

Classification of Metaplectic Modular Categories

We obtain a classification of metaplectic modular categories: every metaplectic modular category is a gauging of the particle-hole symmetry of a cyclic modular category. Our classification suggests a conjecture that every weakly-integral modular category can be obtained by gauging a symmetry of a pointed modular category.

On classification of modular categories by rank

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=$5$ modular categories and describe the corresponding monoidal equivalence classes.

Computing Modular Data for Pointed Fusion Categories

A formula for the modular data of $\mathcal{Z}(Vec^ωG)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^ωG$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimensi…

Classification of super-modular categories

We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank $8$. In particular we find three distinct families of prime categories in rank $8$ in contrast to the lower rank cases for which there is only one such family.

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.

共轭类的Frobenius-Perron维数与模融合范畴中的Ito-Michler型结果

Frobenius-Perron Dimensions of Conjugacy Classes and an Ito-Michler-Type Result in Modular Fusion Categories

本文研究有限群共轭类大小的算术条件在融合范畴中的类比，并为模融合范畴建立了Ito-Michler型结果。

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 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.

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.

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…

Rank-finiteness for modular categories

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 $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2πi}{N}}]$ is identical to that of $N$.

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。

Generating functions for ranks of pre-modular categories

We derive generating functions for the ranks of pre-modular categories associated to quantum groups at roots of unity.

On $G$--equivariant modular categories

In this paper, we study $G$-equivariant tensor categories for a finite group $G$. These categories were introduced by Turaev under the name of $G$-crossed categories; the motivating example of such a category is the category of twisted modules over a vertex operator algebra $V$ with a finite group of automorphisms $G$. We discuss the notion of "orbifold quotient" of such a category (in the example above, this quotient is the category of modules over the subalgebra of invariants $V^G$). We introduce an extended Verlinde algebra for a $G$-equivariant tensor category and give a simple description…

A modular functor from state sums for finite tensor categories and their bimodules

We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule …

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…

Conformal Net Realizability of Tambara-Yamagami Categories and Generalized Metaplectic Modular Categories

We show that all isomorphism classes of even rank Tambara-Yamagami categories arise as $\mathbb{Z}_2$-twisted representations of conformal nets. As a consequence, we show that their Drinfel'd centers are realized by (generalized) orbifolds of conformal nets associated with (self-dual) lattices. The quantum double subfactors of even rank Tambara-Yamagami categories are Bisch-Haagerup subfactors and we describe their (dual) principal graphs. For every abelian group of odd order the Drinfel'd centers of the associated Tambara-Yamagami categories give a fusion ring generalizing the Verlinde ring $…

Higher Gauss sums of modular categories

The definitions of the $n^{th}$ Gauss sum and the associated $n^{th}$ central charge are introduced for premodular categories $\mathcal{C}$ and $n\in\mathbb{Z}$. We first derive an expression of the $n^{th}$ Gauss sum of a modular category $\mathcal{C}$, for any integer $n$ coprime to the order of the T-matrix of $\mathcal{C}$, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these $n$, the higher Gauss sums are $d$-numbers and the associated central charges are roots of unity. In particular, if $\mathcal{C}$ is the Dri…

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。

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.

Module categories for permutation modular invariants

We show that a braided monoidal category C can be endowed with the structure of a right (and left) module category over C \times C. In fact, there is a family of such module category structures, and they are mutually isomorphic if and only if C allows for a twist. For the case that C is premodular we compute the internal End of the tensor unit of C, and we show that it is an Azumaya algebra if C is modular. As an application to two-dimensional rational conformal field theory, we show that the module categories describe the permutation modular invariant for models based on the product of two id…

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.

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.

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。