Yi-Zhi Huang
We discuss a recent proof by the author of a general version of the Verlinde conjecture in the framework of vertex operator algebras and the application of this result to the construction of modular tensor tensor category structure on the category of modules for a vertex operator algebra.
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Vertex operator algebras, partition functions and Teichmüller modular forms
本文通过将每个全纯顶点算子代数与任意亏格曲线模空间上的Hodge线束幂次的截面（即任意亏格的Teichmüller模形式）显式关联，在二维共形场论的几何框架下，连接了曲线模空间几何与全纯顶点代数分类，并探讨了如何利用全纯顶点代数理论获得曲线模空间的新结果，最后研究了如何利用这些形式重建顶点代数。
Based on the vertex-face correspondence, we give an algebraic analysis formulation of correlation functions of the $k\times k$ fusion eight-vertex model in terms of the corresponding fusion SOS model. Here $k\in Z_{>0}$. A general formula for correlation functions is derived as a trace over the space of states of lattice operators such as the corner transfer matrices, the half transfer matrices (vertex operators) and the tail operator. We give a realization of these lattice operators as well as the space of states as objects in the level $k$ representation theory of the elliptic algebra $U_{q,…
This is the first part in a two-part series of papers constructing a unitary structure for the modular tensor category (MTC) associated to a unitary rational vertex operator algebra (VOA).
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.
This is an expository article invited for the ``Commentary'' section of PNAS in connection with Y.-Z. Huang's article, ``Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,'' appearing in the same issue of PNAS. Huang's solution of the mathematical problem of constructing modular tensor categories from the representation theory of vertex operator algebras is very briefly discussed, along with background material. The hypotheses of the theorems entering into the solution are very general, natural and purely algebraic, and have been verified in a wide range of famil…
Classification of connected étale algebras in multiplicity-free modular fusion categories up to rank nine
本文分类了秩不超过9的重数无关模融合范畴中的连通étale代数及其右模范畴，并应用于约束重正化群流，通过非幺正极小模型预测基态简并度与证明自发对称性破缺。
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 introduce a notion of quasi-lisse vertex algebras, which generalizes admissible affine vertex algebras. We show that the normalized character of an ordinary module over a quasi-lisse vertex operator algebra has a modular invariance property, in the sense that it satisfies a modular linear differential equation. As an application we obtain the explicit character formulas of simple affine vertex algebras associated with the Deligne exceptional series at level $-h^{\vee}/6-1$, which express the homogeneous Schur indices of 4d SCFTs studied by Beem, Lemos, Liendo, Peelaers, Rastelli and van Ree…
This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl(2) at an arbitrary admissible level k. Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce non-negative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck…
We give a construction and algorithmic description of the fusion ring of permutation extensions of an arbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons and symmetry defects in bosonic topological phases of matter. The definition is illustrated with examples, namely bilayer symmetry defects and $S_3$-extensions of small modular tensor categories like the Ising and Fibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematica package. We introduce the notions of confinement and deconfinement of anyons and d…
The matrix elements of the $2\times 2$ fusion of Baxter's elliptic $R$-matrix, $R^{(2,2)}(u)$, are given explicitly. Based on a note by Jimbo, we give a formula which show that $R^{(2,2)}(u)$ is gauge equivalent to Fateev's $R$-matrix for the 21-vertex model. Then the crossing symmetry formula for $R^{(2,2)}(u)$ is derived. We also consider the fusion of the vertex-face correspondence relation and derive a crossing symmetry relation between the fusion of the intertwining vectors and their dual vectors.
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 $\widehat{gl(1 \vert 1)}$.
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 $10$.
This is the second half of a two-part series studying tensor categories of unitary vertex operator algebras from a unitary point of view.
Classification of connected étale algebras in pre-modular fusion categories up to rank three
本文分类了秩不超过三的预模融合范畴中的连通étale代数，涵盖退化与非（伪）酉情形，并探讨了其在基态简并和自发对称性破缺中的物理应用。
We introduce theta-functions of VOA-modules and show that the space spanned by them has a modular invariance property.
In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $\mathcal{H}$ and we then introduce a notion of $\mathcal{H}$-module vertex algebra and a notion of $(V,\mathcal{H})$-module for an $\mathcal{H}$-module vertex algebra $V$. Then we give a modular version of Frenkel-Huang-Lepowsky's theory and study invariant bilinear forms on an $\mathcal{H}$-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general $\mathcal{H}$-mo…
Vertex operator algebra bundles on modular curves and their associated modular forms
本文研究了椭圆模曲线上的顶点算子代数束及其相关的模形式，揭示了其与准模形式的联系，并建立了模形式与顶点算子代数的同构关系。
Modular invariance of characters of quasi-lisse vertex algebras
研究椭圆曲线上线丛的共形块空间，证明在满足有限性和半单性条件的顶点代数（包括所有可容许仿射顶点代数和标准Levi型幂零元对应的可容许W-代数）中，共形块层在丛模空间上的全纯性，并证明相关Jacobi不变联络的平坦截面由模上的迹函数张成，从而将Yongchang Zhu的著名定理推广到拟lisse顶点代数。
Fusion rules among irreducible modules of the free bosonic orbifold vertex operator algebra are completely determined.
We consider the quantum-mechanical algebra of observables generated by canonical quantization of $SL(2,R)$ Chern-Simons theory with rational charge on a space manifold with torus topology. We produce modular representations generalizing the representations associated to the $SU(2)$ WZW models and we exhibit the explicit polynomial representations of the corresponding fusion algebras. The relation to Kac-Wakimoto characters of highest weight $\widehat{sl}(2)$ representations with rational level is illustrated. Talk given at the Como Conference on ``Integrable Quantum Field Theories'', September…
In this article, using an idea of the physics superselection principal, we study a modularity on vertex operator algebras arising from semisimple primary vectors. We generalizes the theta functions on vertex operator algebras and prove that the internal automorphisms do not change the genus one twisted conformal blocks.
This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-…
We develop some basic properties such as $p$-centers of affine vertex algebras and free field vertex algebras in prime characteristic. We show that the Wakimoto-Feigin-Frenkel homomorphism preserves the $p$-centers by providing explicit formulas. This allows us to formulate the notion of baby Wakimoto modules, which in particular provides an interpretation in the context of modular vertex algebras for Mathieu's irreducible character formula of modular affine Lie algebras at the critical level.
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.
The fusion rules for vertex operator algebra $V_{L_{2}}^{A_{4}}$ are determined.
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。
Modularity of Vertex Operator Algebra Correlators with Zero Modes
本研究针对带零模的顶点算子代数关联函数的模变换问题，通过推导递推关系确立其模性，发现纯零模关联函数为拟模形式、混合关联函数为拟雅可比形式，还引入高权场代数并简化了权1变换性质的原有证明。
Modular functors from conformal blocks of rational vertex operator algebras
本文证明强有理顶点算子代数的共形块空间构成模函子，进而导出模融合范畴结构，并延拓为三维拓扑场论。
In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $\mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $\mathbb{N}$-graded modules and they are $C_2$-cofinite.
Using techniques from the theory of mock modular forms and harmonic Maass forms, especially Weierstrass mock modular forms, we establish several dimension formulas for certain strongly rational, holomorphic vertex operator algebras, complementing previous work by van Ekeren, Möller, and Scheithauer.
Graphical calculus for quantum vertex operators, I: The dynamical fusion operator
本文作为量子顶点算子图形演算系列首篇，建立了带扭转的带状范畴与辫子幺半范畴的图形演算基础，并将其应用于量子群模范畴，推广了动力学融合算子的线性算子方程为q-KZ型方程组。
We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/η(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.
The rank 1 bosonic ghost vertex algebra, also known as the $βγ$ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor $C_2$-cofinite. We identify a module category, denoted category $\mathscr{F}$, which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category $\ma…
The parafermionic cosets $C_k = \mathrm{Com} (H, L_k(\mathfrak{sl}_2) )$ are studied for negative admissible levels $k$, as are certain infinite-order simple current extensions $B_k$ of $C_k$. Under the assumption that the tensor theory considerations of Huang, Lepowsky and Zhang apply to $C_k$, all irreducible $C_k$- and $B_k$-modules are obtained from those of $L_k(\mathfrak{sl}_2)$, as are the Grothendieck fusion rules of these irreducible modules. Notably, there are only finitely many irreducible $B_k$-modules. The irreducible $C_k$- and $B_k$-characters are computed and the latter are sho…
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 $p^2$.
Classification of connected étale algebras in modular fusion categories up to rank five
本文分类了秩不超过五的模融合范畴中的连通étale代数，并利用该分类证明了自发对称性破缺，预言了非酉最小模型大质量重整化群流中的基态简并度。
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.
This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.
The modular data of a modular category $\mathcal{C}$, consisting of the $S$-matrix and the $T$-matrix, is known to be an incomplete invariant of $\mathcal{C}$. More generally, the invariants of framed links and knots defined by a modular category as part of a topological quantum field theory can be viewed as numerical invariants of the category. Among these invariants, we study the invariant defined by the Borromean link colored by three objects. Thus we obtain a tensor that we call $B$. We derive a formula for the Borromean tensor for the twisted Drinfeld doubles of finite groups. Along with …
We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.
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…
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.
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。
We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We…
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 study modularity of the characters of a vertex (super)algebra equipped with a family of conformal structures. Along the way we introduce the notions of rationality and cofiniteness relative to such a family. We apply the results to determine modular transformations of trace functions on admissible modules over affine Kac-Moody algebras and, via BRST reduction, trace functions on minimal series representations of principal affine W-algebras.
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。
We study the trace functions in orbiford theory for Z-graded vertex operator superalgebras and obtain a modular invariance result. More precisely, let V be a C_2-cofinite Z-graded vertex operator superalgebra and G a finite automorphism group of V. Then for any commuting pairs (g,h) in G, the hσ-trace functions associated to the simple g-twisted V-modules are holomorphic in the upper half plane where σis the canonical involution on V coming from the superspace structure of V. If V is further g-rational for every g n G, the trace unctions afford a representation for the full modular group SL(2,…
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 $L_{\widehat{\frak{sl}_2}}(k,0)^{K}$.
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 vertex operator algebra of lattice type ADE has a standard integral form which extends a Chevalley basis for its degree 1 Lie algebra. This integral form may be used to define a vertex algebra over a commutative ring $R$ and to get a Chevalley group over $R$ of the same type, acting as automorphisms of this vertex algebra. We define vertex algebras of types BCFG over a commutative ring and certain reduced VAs, then get analogous results about automorphism groups. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for …
Zesting produces modular isotopes and explains their topological invariants
本文证明ribbon zesting构造能产生具有相同模数据的模同位素，并基于Reshetikhin-Turaev不变量的分解性质，为拓扑序分类提供了新视角。
Conformal blocks for modular vertex algebras
本文研究了模顶点代数的共形块理论，将其从复数域推广到任意代数闭域，并移除了 $C_1$-cofinite 和 CFT 型等标准假设，从而扩展了该理论的适用范围。
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…
To an RCFT corresponds two combinatorial structures: the amplitude of a torus (the 1-loop partition function of a closed string, sometimes called a modular invariant), and a representation of the fusion ring (called a NIM-rep or equivalently a fusion graph, and closely related to the 1-loop partition function of an open string). In this paper we develop some basic theory of NIM-reps, obtain several new NIM-rep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely …
Modular invariants and NIM-reps
本文引入了环绕模的概念，并证明其与NIM-表示在融合代数上同构，从而推广了Böckenhauer等人关于模不变量的结果，同时展示了TM在模不变性下的自动满足条件。
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 $V_L^{T_{χ_1}} \boxtimes_{V_L} V_L^{T_{χ_2}}$.
The SL(2,Z) representation $π$ on the center of the restricted quantum group U_{q}sl(2) at the primitive 2p-th root of unity is shown to be equivalent to the SL(2,Z) representation on the extended characters of the logarithmic (1,p) conformal field theory model. The multiplicative Jordan decomposition of the U_{q}sl(2) ribbon element determines the decomposition of $π$ into a ``pointwise'' product of two commuting SL(2,Z) representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2,Z) representation on the (1,p)-characters, related to the fusion …
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