Vertex Operator Solutions of¶2d Dimensionally Reduced Gravity

We apply algebraic and vertex operator techniques to solve two dimensional reduced vacuum Einstein's equations. This leads to explicit expressions for the coefficients of metrics solutions of the vacuum equations as ratios of determinants. No quadratures are left. These formulas rely on the identification of dual pairs of vertex operators corresponding to dual metrics related by the Kramer-Neugebauer symmetry.     

The Intermediate Vertex Subalgebras of the Lattice Vertex Operator Algebras

A notion of intermediate vertex subalgebras of lattice vertex operator algebras is introduced, as a generalization of the notion of principal subspaces. Bases and the graded dimensions of such subalgebras are given. As an application, it is shown that the characters of some modules of an intermediate vertex subalgebra between E ₇ and E ₈ lattice vertex operator algebras satisfy some modular differential equations. This result is an analogue of the result concerning the “hole” of the Deligne dimension formulas and the intermediate Lie algebra between the simple Lie algebras E ₇ and E ₈.     

Vertex Operator Algebras Associated to Admissible Representations of

The Kac-Wakimoto admissible modules for are studied from the point of view of vertex operator algebras. It is shown that the vertex operator algebra L(l,0) associated to irreducible highest weight modules at admissible level is not rational if l is not a positive integer. However, a suitable change of the Virasoro algebra makes L(l,0) a rational vertex operator algebra whose irreducible modules are exactly these admissible modules for and for which the fusion rules are calculated. It is also shown that the q-dimensions with respect to the new Virasoro algebra are modular functions. Received: 4 April 1996/Accepted: 1 August 1996     

Mirror Extensions of Vertex Operator Algebras

The mirror extensions for vertex operator algebras are studied. Two explicit examples of extensions of affine vertex operator algebras of type A are given which are not simple current extensions.     

19顶角模型反射方程的常数解

通过直接解反射方程,给出了19顶角模型A2(2)模型反射方程的所有矩阵元非零形式以及其它几种非对角形式的常数解.     

15顶角模型反射方程的常数解

得到了15顶角模型A2(1)模型和超对称t–J模型反射方程的非对角解,结果发现,A2(1)模型具有三种形式的非对角解,超对称t–J模型具有一种形式的非对角解,每种形式的非对角解均含有两个解,每个非对角解中均含有三个任意参数.关于对角解也得到了一些新的形式的解.     

Modularity in Orbifold Theory for Vertex Operator Superalgebras

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 C₂-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C₂-cofinite and g-rational for any g ∈ G.Supported by NSF grants, China NSF grant 10328102 and a Faculty research grant from the University of California at Santa Cruz     

The q-Analogous Vertex Operator and Quantum Affine Algebras

The extended quantum affine algebras of q-analogous vertex operator have been constructed in this Jet ter, and the similar algebra structure of q-analogous fermionic vertex operator is also analysed.     

体积算符对顶角作用的重耦矩阵

用重耦理论的图形计算法,系统推出了非微扰量子引力自旋结网圈表象体积算符对顶角作用的重耦矩阵的图形表式和记号表式     

Framed Vertex Operator Algebras, Codes and the Moonshine Module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ?, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ? are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras. Received: 14 July 1997 / Accepted: 8 September 1997     

Generalized Twisted Modules Associated to General Automorphisms of a Vertex Operator Algebra

We introduce a notion of a strongly ${\mathbb{C}^{\times}}We introduce a notion of a strongly \mathbbC^×{\mathbb{C}^{\times}}-graded, or equivalently, \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{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 a strongly \mathbbC{\mathbb{C}}-graded generalized g-twisted V-module if V admits an additional \mathbbC{\mathbb{C}}-grading compatible with g. Let V=\coprod_{n ? \mathbbZ}V_(n){V=\coprod_{n\in \mathbb{Z}}V_{(n)}} be a vertex operator algebra such that V₍₀₎=\mathbbC1{V_{(0)}=\mathbb{C}\mathbf{1}} 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 2p?{-1}  Res_x Y(u, x){2\pi \sqrt{-1}\; {\rm Res}_{x} Y(u, x)} is an automorphism g _u of V. In this case, a strongly \mathbbC{\mathbb{C}}-graded generalized g _u -twisted V-module is constructed from a strongly \mathbbC{\mathbb{C}}-graded generalized V-module with a compatible action of g _u by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group \mathbbC/\mathbbZ{\mathbb{C}/\mathbb{Z}} or \mathbbC{\mathbb{C}} and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.     

Operator Systems from Discrete Groups

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group ${\mathbb{F}_n}$ on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group ${\mathbb{Z}_2}$ . We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.     

Twisted Sectors for Tensor Product Vertex Operator Algebras Associated to Permutation Groups

Let V be a vertex operator algebra, and for k a positive integer, let g be a k-cycle permutation of the vertex operator algebra V ^{⊗ k} . We prove that the categories of weak, weak admissible and ordinary g-twisted modules for the tensor product vertex operator algebra V ^{⊗ k} are isomorphic to the categories of weak, weak admissible and ordinary V-modules, respectively. The main result is an explicit construction of the weak g-twisted V ^{⊗ k} -modules from weak V-modules. For an arbitrary permutation automorphism g of V ^{⊗ k} the category of weak admissible g-twisted modules for V ^{⊗ k} is semisimple and the simple objects are determined if V is rational. In addition, we extend these results to the more general setting of γg-twisted V ^{⊗ k} -modules for γ a general automorphism of V acting diagonally on V ^{⊗ k} and g a permutation automorphism of V ^{⊗ k} . Received: 20 April 2000 / Accepted: 20 January 2002     

Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group \({\exp(-\mathrm{i} t H_0)}\) of the discrete Laguerre operator H₀ (i.e., the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression, we show that the norm of the evolution group acting from \({\ell^1}\) to \({\ell^\infty}\) is given by \({(1+t^2)^{-1/2}}\).     

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of V and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of V-modules are equivalent.     

Affine Vertex Operator Algebras and Modular Linear Differential Equations

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.     

The Structure of Parafermion Vertex Operator Algebras: General Case

The structure of the parafermion vertex operator algebra associated to an integrable highest weight module for any affine Kac-Moody algebra is studied. In particular, a set of generators for this algebra has been determined.     

Modular Invariance for Twisted Modules over a Vertex Operator Superalgebra

The purpose of this paper is to generalize Zhu’s theorem about characters of modules over a vertex operator algebra graded by integer conformal weights, to the setting of a vertex operator superalgebra graded by rational conformal weights. To recover ${SL_2(\mathbb{Z})}$ S L 2 ( Z ) -invariance of the characters it turns out to be necessary to consider twisted modules alongside ordinary ones. It also turns out to be necessary, in describing the space of conformal blocks in the supersymmetric case, to include certain ‘odd traces’ on modules alongside traces and supertraces. We prove that the set of supertrace functions, thus supplemented, spans a finite dimensional ${SL_2(\mathbb{Z})}$ S L 2 ( Z ) -invariant space. We close the paper with several examples.     

On the Constructions of Holomorphic Vertex Operator Algebras of Central Charge 24

In this article, we construct explicitly several holomorphic vertex operator algebras of central charge 24 using Virasoro frames. The Lie algebras associated to their weight one subspaces are of the types A_1,2 A_3,4⁴, A_1,2D_5,8, A_1,1³A_7,4{A_{1,2} {A_{3,4}}^4, A_{1,2}D_{5,8}, {A_{1,1}}^3A_{7,4}} , A_1,1² C_3,2 D_5,4, A_2,1² A_5,2² C_2,1, A_3,1 A_7,2 C_3,1², A_3,1C_7,2{{A_{1,1}}^2 C_{3,2} D_{5,4}, {A_{2,1}}^2 {A_{5,2}}^2 C_{2,1}, A_{3,1} A_{7,2} {C_{3,1}}^2, A_{3,1}C_{7,2}} , A_4,1 A_9,2B_3,1, B_4,1 C_6,1²{A_{4,1} A_{9,2}B_{3,1}, B_{4,1} {C_{6,1}}^2} and B _6,1 C _10,1. These vertex operator algebras correspond to number 7, 10, 18, 19, 26, 33, 35, 40, 48 and 56 in Schellekens’ list Schellekens (Commun Math Phys 153:159–185, 1993).     

Modular Forms and Second Order Ordinary Differential Equations: Applications to Vertex Operator Algebras

We study the relation between the Kaneko–Zagier equation and the Mathur–Mukhi–Sen classification, and extend it to the case of solutions with logarithmic terms, which correspond to pseudo-characters of non-rational vertex operator algebras. As an application, we prove a non-existence theorem of rational vertex operator algebras.