Fusion rules for the charge conjugation orbifold

Cyclic orbifolds of lattice vertex operator algebras having group like fusions

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (or equivalently $(1-g)L^*<L$). We also determine their fusion rings and the corresponding quadratic space structures when $g$ is fixed point free on $L$. By applying our method to some coinvariant sublattices of the Leech lattice $Λ$, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic…

置换轨形中扭曲模的融合积：II

Fusion Products of Twisted Modules in Permutation Orbifolds: II

本文针对 CFT 型简单有理 $C_2$-余有限顶点算子代数的张量积，确定了任意置换生成循环群作用下扭曲模的融合积。

Conjugation Properties of Tensor Product and Fusion Coefficients

We review some recent results on properties of tensor product and fusion coefficients under complex conjugation of one of the factors. Some of these results have been proven, some others are conjectures awaiting a proof, one of them involving hitherto unnoticed observations on ordinary representation theory of finite simple groups of Lie type

Quantum dimensions and fusion rules of the VOA $ V^τ_{L_{C \times D}}$

In this article, we determine quantum dimensions and fusion rules for the orbifold code VOA $ V^τ_{L_{C \times D}}$. As an application, we also construct certain $3$-local subgroups inside the automorphism group of the VOA $V^\sharp$, where $V^\sharp$ is a holomorphic VOA obtained by the $\mathbb{Z}_3$-orbifold construction on the Leech lattice VOA.

Representations and fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{\mathbb{Z}_3}$

For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study the representations of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreducible modules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classified and constructed explicitly. Quantum dimensions and fusion rules for the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.

The 3-permutation orbifold of a lattice vertex operator algebra

Irreducible modules of the 3-permutation orbifold of a rank one lattice vertex operator algebra are listed explicitly. Fusion rules are determined by using the quantum dimensions. The $S$-matrix is also given.

Fusion rules for $\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra $K(sl_2,k)$ associated to the integrable highest weight modules for the affine Kac-Moody algebra $A_1^{(1)}$ is the building block of the general parafermion vertex operator $K(\mathfrak{g},k)$ for any finite dimensional simple Lie algebra $\mathfrak{g}$ and any positive integer $k$. We first classify the irreducible modules of $\mathbb{Z}_{2}$-orbifold of the simple affine vertex operator algebra of type $A_1^{(1)}$ and determine their fusion rul…

Fibonacci-type orbifold data in Ising modular categories

An orbifold datum is a collection $\mathbb{A}$ of algebraic data in a modular fusion category $\mathcal{C}$. It allows one to define a new modular fusion category $\mathcal{C}_{\mathbb{A}}$ in a construction that is a generalisation of taking the Drinfeld centre of a fusion category. Under certain simplifying assumptions we characterise orbifold data $\mathbb{A}$ in terms of scalars satisfying polynomial equations and give an explicit expression which computes the number of isomorphism classes of simple objects in $\mathcal{C}_{\mathbb{A}}$. In Ising-type modular categories we find new example…

A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications

In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra $V^G$ and show that $V^G$ has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the $V_1$-structures of suitably regular, holomorphic vertex oper…

Constructing modular categories from orbifold data

In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simp…

仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 的轨形理论

Orbifold theory of the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$

本文开展仿射顶点算子超代数的轨形理论研究，完成单仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 轨形的不可约模分类并确定其融合规则。

An algorithm for twisted fusion rules

We present an algorithm for an efficient calculation of the fusion rules of twisted representations of untwisted affine Lie algebras. These fusion rules appear in WZW orbifold theories and as annulus coefficients in boundary WZW theories; they provide NIM-reps of the WZW fusion rules.

Conjugation properties of tensor product multiplicities

It was recently proven that the total multiplicity in the decomposition into irreducibles of the tensor product lambda x mu of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them; at a given level, this also applies to the fusion multiplicities of affine algebras. Here, we show that, in the case of SU(3), the lists of multiplicities, in the tensor products lambda x mu and lambda x bar{mu}, are identical up to permutations. This latter property does not hold in general for other Lie algebras. We conjecture that the same property should hold for …

通过有限子群扩展融合规则：Z_N扩展共形场论的一般构造及其 orbifoldings

Extending fusion rules with finite subgroups: A general construction of $Z_{N}$ extended conformal field theories and their orbifoldings

本文提出了一种构造Z_N扩展共形场论及其orbifoldings的方法，通过非异常子群Z_n(⊂Z_N)构建了 bulk 和 chiral 理论的融合环及模分拆函数，并探讨了其在拓扑场论和边界共形场论中的应用。

Cyclic permutations of lattice vertex operator algebras

The irreducible modules of the 2-cycle permutation orbifold models of lattice vertex operator algebras of rank 1 are classified, the quantum dimensions of irreducible modules and the fusion rules are determined.

Generalized Orbifold Construction for Conformal Nets

Let $\mathcal{B}$ be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet $\mathcal{B}^K$ of $\mathcal{B}$, which generalizes the $G$-orbifold. Conversely, we show that if $\mathcal{A}\subset \mathcal{B}$ is a finite inclusion of conformal nets, then $\mathcal{A}$ is a generalized orbifold $\mathcal{A}=\mathcal{B}^K$ of the conformal net $\mathcal{B}$ by a unique f…

Some applications and constructions of intertwining operators in LCFT

We discuss some applications of fusion rules and intertwining operators in the representation theory of cyclic orbifolds of the triplet vertex operator algebra. We prove that the classification of irreducible modules for the orbifold vertex algebra W(p)^{A_m} follows from a conjectural fusion rules formula for the singlet vertex algebra modules. In the p=2 case, we computed fusion rules for the irreducible singlet vertex algebra modules by using intertwining operators. This result implies the classification of irreducible modules for W(2)^{A_m}, conjectured previously in [4]. The main technica…

顶点算子代数的扭曲 Verlinde 公式

Twisted Verlinde formula for vertex operator algebras

对于具有素数阶自同构群的有理且 $C_2$-余有限顶点算子代数，本文建立了扭曲 Verlinde 公式将扭曲模融合规则与 orbifold 理论中的 $S$ 矩阵相联系，并以此证明了 Kac-Walton 公式的扭曲类比。

对非可逆对称性的规范化：拓扑界面与广义 orbifold 群oids 在 2d 量子场论中

Gauging Non-Invertible Symmetries: Topological Interfaces and Generalized Orbifold Groupoid in 2d QFT

本文研究了二维量子场论中非可逆对称性的规范化，通过拓扑界面和广义 orbifold 群oids 探讨其结构和性质，揭示了规范化的数学框架和物理意义。

Quantum dimensions and fusion products for irreducible V_Q^s-modules, where s is an isometry of Q with s^2=1

Every isometry s of a positive-definite even lattice Q can be lifted to an automorphism of the lattice vertex algebra V_Q. An important problem in vertex algebra theory and conformal field theory is to classify the representations of the s-invariant subalgebra V_Q^s of V_Q, known as an orbifold. In the case when s is an isometry of Q of order two, we have classified the irreducible modules of the orbifold vertex algebra V_Q^s and identified them as submodules of twisted or untwisted V_Q-modules in [Bavalov-Elsinger]. Here we calculate their quantum dimensions and fusion products. The examples …

2-permutations of lattice vertex operator algebras: Higher rank

The fusion rules of the 2-permutation orbifold of an arbitrary lattice vertex operator algebra are determined by using the theory of quantum dimension.

Twisted boundary states and representation of generalized fusion algebra

The mutual consistency of boundary conditions twisted by an automorphism group G of the chiral algebra is studied for general modular invariants of rational conformal field theories. We show that a consistent set of twisted boundary states associated with any modular invariant realizes a non-negative integer matrix representation (NIM-rep) of the generalized fusion algebra, an extension of the fusion algebra by representations of the twisted chiral algebra associated with the automorphism group G. We check this result for several concrete cases. In particular, we find that two NIM-reps of the …

带点顶点算子代数的轨形 I

Orbifolds of Pointed Vertex Operator Algebras I

本文系统研究带点顶点算子代数的轨形与张量范畴论的相互作用，证明了 Dijkgraaf-Witten 猜想，实现了任意幂零群作为格 VOA 自同构，推广了 Tambara-Yamagami 范畴，并构造了广泛的辫化交叉融合范畴和模张量范畴。

Heisenberg顶点算子代数的Z₂-orbifold的张量范畴及其应用

Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications

本文证明Heisenberg顶点算子代数的Z₂-orbifold M(1)^+ 的有限长度模的张量范畴是顶点和braided张量范畴，并探讨其在简单affine顶点代数的泛化模中的应用。

Orbifolds and minimal modular extensions

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the category ${\cal E}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category ${\cal E}={\rm Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\cal C}_{V^G}$ is a minimal modular extension of ${\cal E},$ and is equivalent to the Drinfeld center ${\cal Z}({\rm Vec}_G^α)$ as modular te…

On sums of tensor and fusion multiplicities

The total multiplicity in the decomposition into irreducibles of the tensor product i x j of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them sum_k N_{i j}^{k}= sum_k N_{ibar j}^{k}. This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S matrix, Sigma(k)= sum_j S_{j k}=0 if k is a complex representation. Curiously, this vanishing of Sigma(k) also holds when k is a quaternionic representation. We provide proofs of all these statem…

任意子与固有复杂的F符号

Anyons and Inherently Complex F-symbols

研究二维加一维空间中任意子的F符号，通过展示特定编织融合范畴的F符号不能取实数值，表明其“固有复杂”，分析了最小秩的此类范畴及对应任意子模型，还将例子与任意子分类的最新结果相联系。

On the tensor structure of modules for compact orbifold vertex operator algebras

Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-m…

对任意子融合规则进行分类：一种通用的代数公式用于域壁问题和量子相变

Classifying fusion rules of anyons or SymTFTs: A general algebraic formula for domain wall problems and quantum phase transitions

本文提出一种公式，用于描述在拓扑有序相或拓扑量子场论中任意子的变换规律，通过有隙或对称保持的域壁。基于复数线性交换融合环的环同态，结合Verlinde公式，提供任意子分类及CFT与TQFT之间的无质量重整化群流和测量诱导量子相变的描述。

酉顶点算子代数模的融合积的几何正性

Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules

本文通过几何刻画融合积上 Hermite 型的正性，证明不可约可酉化融合积必正定，并将其用于循环轨形与 simple current 扩张的完全酉性。

A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$ via a logarithmic Kazhdan-Lusztig correspondence. We first investigate the representation theory of $\overline{U}^H_i(\mathfrak{sl}_3)$, the unrolled restricted quantum group of $\mathfrak{sl}_3$ at fourth root of unity. In particular, we analyse its finite-dimensional weigh…

有理c=1顶点算子代数与顶点算子超代数的分类

Classification of Rational $c=1$ Vertex Operator Algebras and Vertex Operator Superalgebras

该文严格分类了有理c=1顶点算子代数与顶点算子超代数，证实其为秩1整格格VO(S)A或其有限群自同构的orbifold，为后续完整CFT分类奠定基础。

A型中某些W-(超)代数的模范畴上的刚性张量结构

Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$

本文为A型次正则W-代数及其Feigin-Semikhatov对偶W-超代数的有限生成权模建立刚性张量范畴结构，通过单流扩张方法给出不可分解投射模和融合规则，并得到仿射sl2在非整可容许水平的首批ribbon范畴例子。