T. Quella, I. Runkel, C. Schweigert
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.
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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 …
The comultiplication formula for fusion products of untwisted representations of the chiral algebra is generalised to include arbitrary twisted representations. We show that the formulae define a tensor product with suitable properties, and determine the analogue of Zhu's algebra for arbitrary twisted representations. As an example we study the fusion of representations of the Ramond sector of the N=1 and N=2 superconformal algebra. In the latter case, certain subtleties arise which we describe in detail.
Let $V$ be a vertex operator algebra, $k$ a positive integer and $σ$ a permutation automorphism of the vertex operator algebra $V^{\otimes k}$. In this paper, we determine the fusion product of any $V^{\otimes k}$-module with any $σ$-twisted $V^{\otimes k}$-module.
We describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion …
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。
With any skew Young diagram one can associate a one parameter family of "elementary" modules over the Yangian $\Yg(\gł_N)$. Consider the twisted Yangian $\Yg(\g_N)\subset \Yg(\gł_N)$ associated with a classical matrix Lie algebra $\g_N\subset\gł_N$. Regard the tensor product of elementary Yangian modules as a module over $\Yg(\g_N)$ by restriction. We prove its irreducibility for generic values of the parameters.
Twisted Fusion Products and Quantum Twisted $Q$-Systems
本文研究特殊扭曲流代数上 Kirillov-Reshetikhin 模的融合积，刻画对偶矩阵元空间，并证明分次重数同时由 $q$-分次费米子和与量子扭曲 $Q$-系统解的常数项表示。
Twisted restricted conformal blocks of vertex operator algebras I: $g$-twisted correlation functions and fusion rules
本文引入了与未扭曲模块M¹及两个g-扭曲模块M²、M³的底层相关的三点扭曲射影线上的g-扭曲限制共形块概念，并展示了该空间与g-扭曲关联函数及这些扭曲模块间的 intertwining 运算符空间同构，推导了扭曲版本的融合规则定理。
In this note, we describe two analogues of the Verlinde formula for modular categories in a twisted setting. The classical Verlinde formula for a modular category $\mathscr{C}$ describes the fusion coefficients of $\mathscr{C}$ in terms of the corresponding S-matrix $S(\mathscr{C})$. Now let us suppose that we also have an invertible $\mathscr{C}$-module category $\mathscr{M}$ equipped with a $\mathscr{C}$-module trace. This gives rise to a modular autoequivalence $F:\mathscr{C}\xrightarrow{\cong}\mathscr{C}$. In this setting, we can define a crossed S-matrix $S(\mathscr{C},\mathscr{M})$. As o…
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。
Cofiniteness for Twisted Fusion Products in Vertex Operator Algebra Theory
本文研究了在顶点算子代数中两个交换有限阶自同构下的扭曲模的有限性问题，证明了在特定条件下扭曲融合积也是C1-有限的，并应用此结果建立了有限融合规则。
The fusion products of admissible representations of the su(2) WZW model at the fractional level k=-4/3 are analysed. It is found that some fusion products define representations for which the spectrum of L_0 is not bounded from below. Furthermore, the fusion products generate representations that are not completely reducible and for which the action of L_0 is not diagonalisable. The complete set of representations that is closed under fusion is identified, and the corresponding fusion rules are derived.
We present a general scheme for describing su(N)_k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way a complete solution of su(4)_k fusion rules is obtained.
In this paper we determine the fusion rules of the logarithmic ${\calW}_{p,q}$ triplet theory and construct the Grothendieck group with subgroups for which consistent product structures can be defined. The fusion rules are then used to determine projective covers. This allows us also to write down a candidate for a modular invariant partition function. Our results demonstrate that recent work on the ${\cal W}_{2,3}$ model generalises naturally to arbitrary (p,q).
We show that Galois theory of cyclotomic number fields provides a powerful tool to construct systematically integer-valued matrices commuting with the modular matrix S, as well as automorphisms of the fusion rules. Both of these prescriptions allow the construction of modular invariants and offer new insight in the structure of known exceptional invariants.
We recall the classification of the irreducible representations of $SL(2)_q$, and then give fusion rules for these representations. We also consider the problem of $\cR$-matrices, intertwiners of the differently ordered tensor products of these representations, and satisfying altogether Yang--Baxter equations.
Algebraic structures in group-theoretical fusion categories
本文推广Ostrik和Natale的结果，在群论融合范畴中构造不可分解可分代数的显式Morita等价类代表，并证明其具有Frobenius代数结构及良好性质。
Bimodules over twisted Zhu algebras and twisted fusion rules theorem for vertex operator algebras
本文研究了扭曲Zhu代数上的双模构造，并基于此推导出顶点算子代数的扭曲融合规则定理，通过构造特定模并证明其子模的universal性质。
We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories.
We find the solution of the $\hat{sl}(3)_k$ singular vector decoupling equations on 3-point functions for the particular case when one of the fields is of weight $w_0\cdot kΛ_0$. The result is a function with non-trivial singularities in the flag variables, namely a linear combination of 2F1 hypergeometric functions. This calculation fills in a gap in [1] and confirms the $\hat{sl}(3)_k$ fusion rules determined there both for generic $κ\not \in \IQ$ and fractional levels. We have also analysed the fusion in $\hat{sl}(3)_k$ using algebraic methods generalising those of Feigin and Fuchs and agai…
The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmann's derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result, however. Only a modified version of the Littlewood-Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of affine $A_r$ algebras. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl gro…
We derive fusion rules for the composition of $q$-deformed classical representations (arising in tensor products of the fundamental representation) with semi-periodic representations of $SL(N)_q$ at roots of unity. We obtain full reducibility into semi-periodic representations. On the other hand, heterogeneous $\cR$-matrices which intertwine tensor products of periodic or semi-periodic representations with $q$-deformed classical representations are given. These $\cR$-matrices satisfy all the possible Yang Baxter equations with one another and, when they exist, with the $\cR$-matrices intertwin…
We completely determine the fusion rules for the vertex operator algebra $V_L^+$ for a rank one even lattice $L$.
Fusion rules among irreducible modules of the free bosonic orbifold vertex operator algebra are completely determined.
The fusion rules for vertex operator algebra $V_{L_{2}}^{A_{4}}$ are determined.
We determine fusion rules (dimensions of the space of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.
We address the question whether the condition on a fusion category being solvable or not is determined by its fusion rules. We prove that the answer is affirmative for some families of non-solvable examples arising from representations of semisimple Hopf algebras associated to exact factorizations of the symmetric and alternating groups. In the context of spherical fusion categories, we also consider the invariant provided by the $S$-matrix of the Drinfeld center and show that this invariant does determine the solvability of a fusion category provided it is group-theoretical.
Bimodues associated to twisted modules of vertex operator algebras and fusion rules
本文研究了顶点算子代数扭模块的双模结构及其与融合规则的关系，通过构造特定双模来探索模块间的作用和联系。
The construction of the Q-operator for twisted affine superalgebra $C^{(2)}_q(2)$ is given. It is shown that the corresponding prefundamental representations give rise to evaluation modules some of which do not have a classical limit, which nevertheless appear to be a necessary part of fusion relations.
This is an expository introduction to fusion rules for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Many explicit examples are included with figures illustrating the rank 2 cases. New results relating fusion coefficients to tensor product coefficients are proved, and a conjecture is given which shows that the Frenkel-Zhu affine fusion rule theorem can be seen as a beautiful generalization of the Parasarathy-Ranga Rao-Varadarajan tensor product theorem. Previous work of the author and collabo…
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions
We determine the fusion rules of the equivariantization of a fusion category $\mathcal{C}$ under the action of a finite group $G$ in terms of the fusion rules of $\mathcal{C}$ and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category.
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.
Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
We show that certain tensor product multiplicities in semisimple braided sovereign tensor categories must be even. The quantity governing this behavior is the Frobenius-Schur indicator. The result applies in particular to the representation categories of large classes of groups, Lie algebras, Hopf algebras and vertex algebras.
For the Klein group $K$ and a positive integr $k$, irreducible modules of the orbifold vertex operator algebra $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{K}$ have been classified and constructed in \cite{JWa}. In this paper, we determine completely the fusion rules of $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$.
In this paper we prove the fusion rules of $L(c_{1,q},0)$ for all $q\geq1$. Roughly speaking, we consider $L(c_{1,q},0)$ as the limitation of $L(c_{n,nq-1},0)$, where $n\rightarrow\infty$, and the fusion rules of $L(c_{1,q},0)$ follow as the limitation of the fusion rules of $L(c_{n,nq-1},0)$.
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.
In this note, we examine the gauging of the $\mathbb{Z}/2\mathbb{Z}$ permutation action on the tensor square of a modular tensor category. When $\mathcal{C}$ has no nontrivial invertible objects, we provide formulas for the fusion rules of both the extensions, expressed in terms of the fusion rules of $\mathcal{C}$, and the subsequent equivariantizations, which additionally requires the modular data of $\mathcal{C}$. We discuss several examples related to quantum groups at roots of unity.
The fusion rules for the vertex operator algebras M(1)^+ (of any rank) and V_L^+ (for any positive definite even lattice L) are determined completely.
The quantum dimensions and the fusion rules for the parafermion vertex operator algebra associated to the irreducible highest weight module for the affine Kac-Moody algebra A_1^{(1)} of level k are determined.
Fusion rules for V_{L_2}^{S_4}
本文确定了顶点算子代数V_{L_2}^{S_4}的融合规则。
The irreducible modules for the parafermion vertex operator algebra associated to any finite dimensional Lie algebra and any positive integer are identified, the quantum dimensions are computed and the fusion rules are determined.
For a positive-definite, even, integral lattice $L$, the lattice vertex operator algebra $V_L$ is known to be rational and $C_2$-cofinite, and thus the fusion products of its modules always exist. The fusion product of two untwisted irreducible $V_L$-modules is well-known, namely $V_{L+λ} \boxtimes_{V_L} V_{L+μ} = V_{L + λ+ μ}$. In this paper, we determine the other two fusion products: $V_{L+λ} \boxtimes_{V_L} V_L^{T_χ}$ and $V_L^{T_{χ_1}} \boxtimes_{V_L} V_L^{T_{χ_2}}$.
In this paper we prove a formula for fusion coefficients of affine Kac-Moody algebras first conjectured by Walton [Wal2], and rediscovered in [Fe]. It is a reformulation of the Frenkel-Zhu affine fusion rule theorem [FZ], written so that it can be seen as a beautiful generalization of the classical Parasarathy-Ranga Rao-Varadarajan tensor product theorem [PRV].
Gauging a finite group 0-form symmetry $G$ of a quantum field theory (QFT) results in a QFT with a Rep$(G)$ symmetry implemented by Wilson lines. The group $G$ determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine $G$. In this paper, we study the properties of $G$ that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine th…
Axial algebras are commutative algebras generated by idempotents; they generalise associative algebras by allowing the idempotents to have additional eigenvectors, controlled by fusion rules. If the fusion rules are $\mathbb{Z}/2$-graded, axial algebras afford representations of transposition groups. We consider axial representations of Weyl groups of simply-laced root systems, which are examples of regular $3$-transposition groups. We introduce coset axes, a special class of idempotents based on embeddings of transposition groups, and use them to study the propagation of fusion rules in axial…
Characters and fusion rules of boundary W-algebras
本文研究例外W-代数的q-特征标与模数据，建立边界W-代数间的相等性，从而确定A型例外W-代数的融合规则。
We collate information about the fusion categories with $A_n$ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres. The first section describes the classification of these categories (as monoidal, dagger, pivotal, and braided categories). The second section describes the properties of these categories.
In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.
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…
Let $\mathcal{F}_{25}$ be the family of irreducible highest weight modules for the Virasoro algebra of central charge $25$ which are not isomorphic to Verma modules. Let $L(25,0)$ be the Virasoro vertex operator algebra of central charge 25. We prove that the fusion rules for the $L(25,0)$-modules in $\mathcal{F}_{25}$ are in correspondence with the tensor rules for the irreducible finite dimensional representations of $sl(2, \mathbb{C})$, extending the known correspondence between modules for the Virasoro algebras of dual central charges 1 and 25.
Fusion Rings over Drinfeld Doubles
本文研究了Drinfeld双代数上的融合环，证明了二面体群和四面体群的乘积自由性，并揭示了其与共形场论的关联。
As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin diagram automorphisms. Combinatorial properties of twisted AR-quivers are useful to understand not only Dorey's rule but also other notions in the representation theory of the quantum affine algebra $U_q'(C_n^{(1)})$ such as denominator formulas. In addition, unlike twisted adapted classes of type $A_{2n-1}$ in the previous paper, we show twisted AR-quivers…
We prove that the space of intertwining operators associated with certain admissible modules over vertex operator algebras is isomorphic to a quotient of the vector space of conformal blocks on a three-pointed rational curve defined by the same data. This provides a new proof and alternative version of Frenkel and Zhu's fusion rules theorem in terms of the dimension of certain bimodules over Zhu's algebra, without the assumption of rationality.
We show that all orthogonal free quantum groups are isomorphic to variants of the free orthogonal Wang algebra, the hyperoctahedral quantum group or the quantum permutation group. We also obtain a description of their free complexification. In particular we complete the calculation of fusion rules of all orthogonal free quantum groups and their free complexifications.
We show how the fusion rules for an affine Kac-Moody Lie algebra g of type A_{n-1}, n = 2 or 3, for all positive integral level k, can be obtained from elementary group theory. The orbits of the kth symmetric group, S_k, acting on k-tuples of integers modulo n, Z_n^k, are in one-to-one correspondence with a basis of the level k fusion algebra for g. If [a],[b],[c] are any three orbits, then S_k acts on T([a],[b],[c]) = {(x,y,z)\in [a]x[b]x[c] such that x+y+z=0}, which decomposes into a finite number, M([a],[b],[c]), of orbits under that action. Let N = N([a],[b],[c]) denote the fusion coeffici…
In this paper we find the fusion rules of the free wreath products $\widehatΓ\wr_*S_N^+$ for any (discrete) group $Γ$. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced $C^*$-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type $II_1$-factor and that the dual of $\widehatΓ\wr_*S_N^+$ has the Ha…
In this paper we classify all semisimple tensor categories with the same fusion rules as $\operatorname{Rep}(SO(4))$, or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex numbers. Furthermore we show these tensor categories are always braided, and there exist exactly 8 braidings.
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.
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