Naihuan Jing, Weiqiang Wang
We establish a new group-theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras of ADE types in the same spirit of our new approach to the McKay correspondence. Our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group.
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In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
This paper gives an analogue of A_g(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and A_g(V)-modules is established. It is proved that if V is g-rational, then A_g(V) is finite dimensional semisimple associative algebra and there are only finitely many irreducible g-twisted V-modules.
In this paper we develop a formalism for working with twisted realizations of vertex and conformal algebras. As an example, we study realizations of conformal algebras by twisted formal power series. The main application of our technique is the construction of a very large family of representations for the vertex superalgebra $\goth V_Λ$ corresponding to an integer lattice $Λ$. For an automorphism $\^σ:\goth V_Λ\to\goth V_Λ$ coming from a finite order automorphism $σ:Λ\to Λ$ we define a category $\cal O_{\^σ}$ of twisted representations of $\goth V_Λ$ and show that this category is semisimple …
Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(kΛ_0)$ called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor product of the twisted module $V_L^T$ for the lattice vertex operator algebra $V_L$. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in arXiv:math/0102113.
We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$ algebra with a slope $n'/n$ using vertex operators of quantum affine $\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of an integrable level-one $U_q(\widehat{\mathfrak{gl}}_n)$-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Neguţ conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of $(\mathfrak{gl}_1,\mathfrak{gl}_n)$-duality.
We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.
This paper studies the twisted representations of vertex operator algebras. Let V be a vertex operator algebra and g an automorphism of V of finite order T. For any m,n in (1/T)Z_+, an A_{g,n}(V)-A_{g,m}(V)-bimodule A_{g,n,m}(V) is constructed. The collection of these bimodules determines any admissible g-twisted V-module completely. A Verma type admissible g-twisted V-module is constructed naturally from any A_{g,m}(V)-module. Furthermore, it is shown with the help of bimodule theory that a simple vertex operator algebra V is g-rational if and only if its twisted associative algebra A_g(V) is…
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.
$C_n$-cofinite twisted modules for $C_2$-cofinite vertex operator algebras
该研究针对带一般自同构的顶点算子代数，引入了弱扭曲模的$C_n$-余有限性概念，并证明$C_2$-余有限的CFT型顶点算子代数的所有有限生成弱扭曲模均满足$C_n$-余有限性。
Twisted vertex representations via spin groups and the McKay correspondence
本文利用有限群虚特征标与自旋圈积群的超自旋特征标构造扭顶点算子，实现扭仿射和扭环形代数基本表示，并导出自旋特征标表。
We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove duality, weak associativity, a Jacobi identity, a generalized commutator formula, generalized weak commutativity, and convergence and commutativity for products of more than two operators involving twist vertex operators. These properties of twist vertex operators play an important role in the author's recent general, direct and explicit construction of (lower-bounded generalized) twisted modules.
We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.
We continue with [LY] to construct and classify graded simple twisted modules for the $\N$-graded vertex algebras constructed by Gorbounov, Malikov and Schechtman from vertex algebroids. Meanwhile we determine the full automorphism groups of those $\N$-graded vertex algebras in terms of the automorphism groups of the corresponding vertex algebroids.
We classify irreducible $σ$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $σ$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $σ$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of g…
Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.
This is a paper in a series systematically to study toroidal vertex algebras. Previously, a theory of toroidal vertex algebras and modules was developed and toroidal vertex algebras were explicitly associated to toroidal Lie algebras. In this paper, we study twisted modules for toroidal vertex algebras. More specifically, we introduce a notion of twisted module for a general toroidal vertex algebra with a finite order automorphism and we give a general construction of toroidal vertex algebras and twisted modules. We then use this construction to establish a natural association of toroidal vert…
Given a weight-one element $u$ of a vertex operator algebra $V$, we construct an automorphism of the category of generalized $g$-twisted modules for automorphisms $g$ of $V$ fixing $u$. We apply this construction to the case that $V$ is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules.
In this paper, we study a new kind of vertex operator algebra related to the twisted Heisenberg-Virasoro algebra, which we call the twisted Heisenberg-Virasoro vertex operator algebra, and its modules. Specifically, we present some results concerning the relationship between the restricted module categories of twisted Heisenberg-Virasoro algebras of rank one and rank two and several different kinds of module categories of their corresponding vertex algebras. We also study fully the structures of the twisted Heisenberg-Virasoro vertex operator algebra, give a characterization of it as a tensor …
We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding irreducible twisted modules through the twisted Zhu algebra.
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
In this paper we introduce and study a twisted tensor product construction of nonlocal vertex algebras. Among the main results, we establish a universal property and give a characterization of a twisted tensor product. Furthermore, we give a construction of modules for a twisted tensor product. We also show that smash products studied by one of us before can be realized as twisted tensor products.
This paper studies restricted modules of gap-$p$ Virasoro algebra $Ł$ and their intrinsic connection to twisted modules of certain vertex algebras. We first establish an equivalence between the category of restricted $Ł$-modules of level $\el$ and the category of twisted modules of vertex algebra $V_{\mathcal{N}_{p}}(\el,0)$, where $\mathcal{N}_{p}$ is a new Lie algebra, $\el:=(\ell_{0},0,\cdots,0)\in\C^{\halfp+1}$, $\ell_{0}\in\C$ is the action of the Virasoro center. Then we focus on the construction and classification of simple restricted $Ł$-modules of level $\el$. More explicitly, we give…
We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $σ$ of order $T$ with $M$ as a $σ$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $σ$ of order $T$, giving a $σ$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from…
We continue a previous study on $Γ$-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any $\Z$-graded vertex algebra $V$ and a positive integer $N$, the category of $V$-modules is naturally isomorphic to the category of quasimodules of a certain type for $V^{\otimes N}$. We also study certain generalizations of twisted affine Lie algebras and we relate such Lie algebras to vertex algebras and their quasimodules, in a way similar to that twisted affine Lie algebras are related to vertex algebras and their twisted modules.
Motivated by logarithmic conformal field theory and Gromov-Witten theory, we introduce a notion of a twisted module of a vertex algebra under an arbitrary (not necessarily semisimple) automorphism. Its main feature is that the twisted fields involve the logarithm of the formal variable. We develop the theory of such twisted modules and, in particular, derive a Borcherds identity and commutator formula for them. We investigate in detail the examples of affine and Heisenberg vertex algebras.
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $\mathcal{L}_{t}$, and show that $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules are in one-to-one correspondence with restricted $\mathcal{L}_{t}$-modules of level $\ell_{13}$, where $σ_{t}$ is an order $t$ automorphism of $V_{\mathcal{L}}(\ell_{123},0)$. At the end, we give a complete list of irreducible $σ_{t}$-twisted $V_{\mathcal{L}}(\ell_{123},0)$($\ell_{2}=0$)-modules.
We use a result of Barron, Dong and Mason to give a natural isomorphism between the category of twisted modules and the category of quasi-modules of a certain type for a general vertex operator algebra.
Twisted regular representations and bimodules in vertex operator algebra theory
本文利用顶点算子代数的扭状正则表示理论，构建扭状Zhu代数上的双模，扩展了Li在非扭情况下的工作，并证实了Dong和Jiang关于双模理论的猜想。
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。
The boson-fermion correspondence of type A is an isomorphism between two super vertex algebras (and so has singularities in the operator product expansions only at $z = w$). The boson-fermion correspondence of type B plays similarly important role in many areas, including representation theory, integrable systems, random matrix theory and random processes. But the vertex operators describing it have singularities in their operator product expansions at both $z = w$ and $z = -w$, and thus need a more general notion than that of a super vertex algebra. In this paper we present such a notion: the…
We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$ of $V$. In particular, when $g$ is the identity, we obtain lower-bounded generalized $V$-modules satisfying a universal property. Let $W$ be a lower-bounded graded vector space equipped with a set of "generating twisted fields" and a set of "generator twist fields" satisfying a weak commutativity for generating twisted fields, a generalized weak commutativity for one generating twist…
Using general principles of the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. This is explained through results in the general theory of vertex operator algebras, including a new identity, which we call ``modified weak associativity.'' This paper is an announcement. The detailed proofs will appear elsewhere.
We introduce a notion of strongly C^{\times}-graded, or equivalently, C/Z-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of strongly C-graded generalized g-twisted V-module if V admits an additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let u be an element of V of weight 1 such that L(1)u=0. Then the exponential of 2π\sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this …
We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called $g$-twisted zero-mode algebra, is a subquotient of what we call $g$-twisted universal enveloping algebra of $V$. These algebras are generalizations of the corresponding algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the (untwisted) case that $g$ is the identity. The other is a generalization of the $g$-twisted version of Zhu's algebra for suitable $g$-twisted modules constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly…
Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, ``relativized'' twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lat…
Twisted modules for N=2 supersymmetric vertex operator superalgebras are classified for the vertex operator superalgebra automorphisms which are lifts of a finite automorphism of the N=2 Neveu-Schwarz Lie superalgebra representation. These include the Ramond-twisted sectors and mirror-twisted sectors for N=2 vertex operator superalgebras, as well as twisted modules related to more general "spectral flow" representations of the N=2 Neveu-Schwarz algebra. We present the construct of twisted modules for free N=2 vertex operator superalgebras for all of the N=2 Neveu-Schwarz Lie superalgebra autom…
We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$, and $C$ an algebraic curve such that $H \subset \on{Aut}(C)$. We show that a suitable collection of twisted $V$--modules gives rise to a section of a certain sheaf on the quotient $X=C/H$. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac-Moody vertex algebras. We also give a chiral …
This is the third in a series of papers studying the vertex-algebraic structure of principal subspaces of twisted modules for lattice vertex operator algebras. We focus primarily on lattices $L$ whose Gram matrix contains only non-negative entries. We develop further ideas originally presented by Calinescu, Lepowsky, and Milas to find presentations (generators and relations) of the principal subspace of a certain natural twisted module for the vertex operator algebra $V_L$. We then use these presentations to construct exact sequences involving this principal subspace, which give a set of recur…
We investigate a general theory of the Z_2-twisted representations of vertex operator superalgebras. Certain one-to-one correspondence theorems are established. We also give an explicit realization of the Ising model SVOA and its Z_2-twisted modules. As an application, we obtain the Gerald Hoehn's Babymonster SVOA VB and its Z_2-twisted module VB_{tw} from the moonshine VOA V^\nat by cutting off the Ising models. It is also shown in this paper that Aut (VB) is finite.
Twisted modules of $\frac{1}{2}\mathbb{Z}$-graded modular vertex superalgebras
本文研究了模1/2Z-graded顶点超代数的g-扭模理论，构建了Zhu代数的扭版本，并建立了简单模之间的对应关系，同时研究了affine Lie超代数和模Neveu-Schwarz顶点超代数的扭模结构。
For any integral lattice $Q$, one can construct a vertex algebra $V_Q$ called a lattice vertex algebra. If $σ$ is an automorphism of $Q$ of finite order, it can be lifted to an automorphism of $V_Q$. In this paper we classify the irreducible $σ$-twisted $V_Q$-modules. We show that the category of $σ$-twisted $V_Q$-modules is a semisimple abelian category with finitely many isomorphism classes of simple objects.
This contribution is mainly based on joint papers with Lepowsky and Milas, and some parts of these papers are reproduced here. These papers further extended works by Lepowsky and by Milas. Following our joint papers, I explain the general principles of twisted modules for vertex operator algebras in their powerful formulation using formal series, and derive general relations satisfied by twisted and untwisted vertex operators. Using these, I prove new "equivalence" and "construction" theorems, identifying a set of sufficient conditions in order to have a twisted module for a vertex operator al…
Twisted associative algebras associated to vertex algebras
本文构造了与顶点代数相关且不依赖共形结构的扭曲结合代数序列，并证明顶点算子代数的 $g$-有理性、$g$-正则性与扭曲融合规则独立于共形向量的选取。
In this paper we study fields satisfying $N$-point locality and their properties. We obtain residue formulae for $N$-point local fields in terms of derivatives of delta functions and Bell polynomials. We introduce the notion of the space of descendants of $N$-point local fields which includes normal ordered products and coefficients of operator product expansions. We show that examples of $N$-point local fields include the vertex operators generating the boson-fermion correspondences of type B, C and D-A. We apply the normal ordered products of these vertex operators to the setting of the repr…
For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module $\widehat{M}^{[g]}_{B}$ constructed by the author in \cite{H-const-twisted-mod}. We prove that there exist irreducible lower-bounded generalized $g$-twisted $V$-modules by showing that there exists a maximal proper submodule of $\widehat{M}^{[g]}_{B}$ for a one-dimensional space $M$. We then give several spanning sets of $\widehat{M}^{[g]}_{B}$ and discuss the relations among elements of the spanning …
Twisted logarithmic modules of lattice vertex algebras
本文研究了格顶点代数的扭曲对数模，将其分类问题归结为对特定群上模的分类，该群由离散海森堡群与格加法群中心扩张的半直积构成。
We give a new construction of functors from the category of modules for the associative algebras $A_n(V)$ and $A_g(V)$ associated with a vertex operator algebra $V$, defined by Dong, Li and Mason, to the category of admissible $V$-modules and admissible twisted $V$-modules, respectively, using the method developed in the joint work \cite{HY1} with Y.-Z. Huang. The functors were first constructed by Dong, Li and Mason, but the importance of the new method, as in \cite{HY1}, is that we can apply the method to study objects without the commutator formula in the representation theory of vertex ope…
Let $V$ be a vertex operator algebra and $g$ an automorphism of finite order. We construct an associative algebra $A_g(V)$ and a pair of functors between the category of $A_g(V)$-modules and a certain category of admissible $g$-twisted $V$-modules. In particular, these functors exhibit a bijection between the simple modules in each category. We give various applications, including the fact that the complete reducibility of admissible $g$-twisted modules implies both the finite-dimensionality of homogeneous spaces and the finiteness of the number of simple $g$-twisted modules.
Using general principles in the theory of vertex operator algebras and their twisted modules, we obtain a bosonic, twisted construction of a certain central extension of a Lie algebra of differential operators on the circle, for an arbitrary twisting automorphism. The construction involves the Bernoulli polynomials in a fundamental way. We develop new identities and principles in the theory of vertex operator algebras and their twisted modules, and explain the construction by applying general results, including an identity that we call "modified weak associativity", to the Heisenberg vertex op…
Orbifold Vertex Operator Algebras and the Positivity Condition
该研究通过分析Dong、Li和Mason的扭模不变性，证明全纯$C_2$-余有限顶点算子代数的不可约扭模$L_0$-权下界，推出CFT型代数扭模几乎严格正分次，且有限可解自同构群对应的不动点子代数几乎满足正性条件。
We introduce a twisted quantum affine algebra associated to each simply laced finite dimensional simple Lie algebra. This new algebra is a Hopf algebra with a Drinfeld-type comultiplication. We obtain this algebra by considering its vertex representation. The vertex representation quantizes the twisted vertex operators of Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We also introduce a twisted quantum loop algebra for the Kac-Moody case and give its level one representation.
This paper is to study what we call twisted regular representations for vertex operator algebras. Let $V$ be a vertex operator algebra, let $σ_1,σ_2$ be commuting finite-order automorphisms of $V$ and let $σ=(σ_1σ_2)^{-1}$. Among the main results, for any $σ$-twisted $V$-module $W$ and any nonzero complex number $z$, we construct a weak $σ_1\otimes σ_2$-twisted $V\otimes V$-module $\mathfrak{D}_{σ_1,σ_2}^{(z)}(W)$ inside $W^{*}$. Let $W_1,W_2$ be $σ_1$-twisted, $σ_2$-twisted $V$-modules, respectively. We show that $P(z)$-intertwining maps from $W_1\otimes W_2$ to $W^{*}$ are the same as homomo…
We construct and classify $(1 \; 2\; \cdots \; k)$-twisted $V^{\otimes k}$-modules for $k$ odd and for $V$ a vertex operator superalgebra. This extends previous results of the author, along with Dong and Mason, classifying all permutation-twisted modules for tensor product vertex operator algebras, to the setting of vertex operator superalgebras for odd order permutations. We show why this construction does not extend to the case of permutations of even order in the superalgebra case and how the construction and classification in the even order case is fundamentally different than that for the…
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence, of type D-A. Further, we define a new concept of twisted vertex algebra of order $N$, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing …
Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of ${\rm Aut} V$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and ${\mathscr S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur--Weyl type duality for the actions of ${\mathscr A}_α(G,{\mathscr S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in ${\mathscr S}$ where ${\mathscr A}_α(G,{\mathscr S})$ is a finite dimensional semisimple associative algebra associated with $G,{\mat…
We construct explicitly the $q$-vertex operators (intertwining operators) for the level one modules $V(Λ_i)$ of the classical quantum affine algebras of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$, $i=0$ for $D_4^{(3)}$, $i=0, n$ for $D_{n+1}^{(2)}$, and $i=n$ for $A_{2n}^{(2)}$. A perfect crystal graph for $D_4^{(3)}$ is constructed as a by-product.
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。
We introduce a notion of a (V,T)-module over a vertex algebra V for an arbitrary positive integer T, which is a generalization of a twisted V-module. Under some conditions on V, we construct an associative algebra A^{T}_{m}(V) for m\in(1/T)\N and an A^{T}_{m}(V)-A^{T}_{n}(V)-bimodule A^{T}_{n,m}(V) for n,m\in(1/T)\N and we establish a one-to-one correspondence between the set of isomorphism classes of simple left A^{T}_{0}(V)-modules and that of simple (1/T)\N-graded (V,T)-modules.
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