Vertex Operators, Littlewood-Richardson Rule for Generalized Symplectic Schur Functions

Littlewood-Richardson rule gives the expansion formula for decomposing a product of two Schur functions as a linear sum of Schur functions,while the decomposition formula for the multiplication of two symplectic Schur function is also given by the combinatorial method.In this paper,we will construct the algebraic forms of the decomposition formula for the product of two symplectic Schur functions by using the generating functions and vertex operator realizations,and then extend these results to generalized symplectic Schur functions.

Realization of Vertex Operators of 7-Twisted Affine Lie Algebra ■[θ]

In this paper, we construct a new algebra structure 7-twisted atone Lie algebra sl（3,C）[θ] and study its vertex operator representations.

Higher Genus Characters for Vertex Operator Superalgebras on Sewn Riemann Surfaces

We review our recent results on computation of the higher genus characters for vertex operator superalgebras modules. The vertex operator formal parameters are associated to local parameters on Riemann surfaces formed in one of two schemes of (self- or tori- ) sewing of lower genus Riemann surfaces. For the free fermion vertex operator superalgebra we present a closed formula for the genus two continuous orbifold partition functions (in either sewings) in terms of an infinite dimensional determinant with entries arising from the original torus Szeg? kernel. This partition function is holomorphic in the sewing parameters on a given suitable domain and possesses natural modular properties. Several higher genus generalizations of classical (including Fay’s and Jacobi triple product) identities show up in a natural way in the vertex operator algebra approach.

A general mirror equivalence theorem for coset vertex operator algebras

We prove a general mirror duality theorem for a subalgebra U of a simple conformal vertex algebra A and its commutant V=ComA(U).Specifically,we assume that A≌■_(i∈I)U_(i)■V_(i) as a U■V-module,where the U-modules Uiare simple and distinct and are objects of a semisimple braided ribbon category of Umodules,and the V-modules Viare semisimple and contained in a(not necessarily rigid) braided tensor category of V-modules.We also assume U=ComA(V).Under these conditions,we construct a braid-reversed tensor equivalence τ:u_(A)→v_(A),where u_(A)is the semisimple category of U-modules with simple objects Ui,i∈I,and v_(A)is the category of V-modules whose objects are finite direct sums of Vi.In particular,the V-modules Viare simple and distinct,and v_(A)is a rigid tensor category.As an application,we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge 13+6p+6p^(-1),p∈Z_(≥2), which is braided tensor equivalent to an abelian 3-cocycle twist of the category of finite-dimensional sl2-modules.Consequently,the Virasoro vertex operator algebra at central charge 13+6p+6p^(-1)is the PSL_(2)(C)-fixed-point subalgebra of a simple conformal vertex algebra w(-p),analogous to the realization of the Virasoro vertex operator algebra at central charge 13-6p-6p^(-1)as the PSL_(2)(C)-fixed-point subalgebra of the triplet algebra W(p).

Resistance Distance and Kirchhoff Index of a Class of Join Graphs

For some complicated graphs obtained by graph operations,it is very difficult to compute resistance distance and Kirchhoff index.Define a new graph operation,and obtain a class of new join graphs:the subdivision-vertex-vertex join G_1* G_2.Then,describe the Laplacian matrix of the graph G_1 * G_2 and use generalized inverse of the Laplacian matrix to get formulas for resistance distance and Kirchhoff index.Through the obtained formulas,the resistance distance of any pairs of vertices and Kirchhoff index of the join graph can be computed.

Vertex Representations of Toroidal Special Linear Lie Superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.

The varieties of Heisenberg vertex operator algebras

For a vertex operator algebra V with conformal vector w, we consider a class of vertex operator subalgebras and their conformal vectors. They are called semi-conformal vertex operator subalgebras and semi- conformal vectors of （V, w）, respectively, and were used to study duality theory of vertex operator algebras via coset constructions. Using these objects attached to （V,w）, we shall understand the structure of the vertex operator algebra （V,w）. At first, we define the set Sc（V,w） of semi-conformal vectors of V, then we prove that Sc（V,w） is an aiYine algebraic variety with a partial ordering and an involution map. Corresponding to each semi-conformal vector, there is a unique maximal semi-conformal vertex operator subalgebra containing it. The properties of these subalgebras are invariants of vertex operator algebras. As an example, we describe the corresponding varieties of semi-conformal vectors for Heisenberg vertex operator algebras. As an application, we give two characterizations of Heisenberg vertex operator algebras using the properties of these varieties.

Extension of Vertex Operator Algebra VH4（e, 0）

We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra H4 with some natural conditions. It turns out that the representation theory of H4 is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra VH4 （e, 0） by the even lattice L. We give the structure of the extension VH4 （e, 0） [L] and its irreducible modules via irreducible representations of VH4（e, 0） viewed as a vertex algebra.

Bimodules associated to vertex operator superalgebras

Let V be a vertex operator superalgebra and m, n ∈ 1/2 ?+. We construct an A n (V)-A m (V)-bimodule A n,m (V) which characterizes the action of V from the level m subspace to level n subspace of an admissible V-module. We also construct the Verma type admissible V-module from an A m (V)-module by using bimodules

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.

Coset vertex operator algebras and W-algebras of A-type

We give an explicit description for a weight three generator of the coset vertex operator algebra C_L_(sln)(l,0)L_(sln)(1,0)(L_(sln)(l+1,0),for n≥2, l≥1. Furthermore, we prove that the nommutant C_L_(sl3)(l,0)L_(sl3)(1,0)(L_(sl3)(l+1,0)) is isomorphic to the W-algebra W_(-3+(l+3)/(l+4))(sl_3), which confirms the conjecture for the sl_3 case that C_L_g(l,0)L_g(1,0)(L_g(l + 1,0)) is isomorphic to W_(-h+(l+h)/(l+h+1))(g) for simaly-laced Lie algebras g with its Coxeter number h for a positive integer l.

Representations of Code Vertex Operator Superalgebras

We study the representations of code vertex operator superalgebras resulting from a binary linear code which contains codewords of odd weight. We also show that there exists only one set of seven mutually orthogonal conformal vectors with central charge 1/2 in the Hamming code vertex operator superalgebra MH7. Phrthermore, we classify all the irreducible weak MH7-modules.

Bimodule and twisted representation of vertex operator algebras

In this paper, for a vertex operator algebra V with an automorphism g of order T, an admissible V-module M and a fixed nonnegative rational number n ∈1/T Z_+, we construct an A_(g,n)(V)-bimodule Ag,n(M) and study its properties, discuss the connections between bimodule A_(g,n)(M) and intertwining operators. Especially, bimodule A _(g,n)-1T(M) is a natural quotient of A_(g,n)(M) and there is a linear isomorphism between the space IM^k M Mjof intertwining operators and the space of homomorphisms HomA_(g,n)(V)(A_(g,n)(M) A_(g,n)(V)M^j(s), M^k(t)) for s, t ≤ n, M^j, M^k are g-twisted V modules, if V is g-rational.

Application of a vertex chain operation algorithm on topological analysis of three-dimensional fractured rock masses

Identifying the morphology of rock blocks is vital to accurate modelling of rock mass structures. This paper applies the concepts of directed edges and vertex chain operations which are typical for block tracing approach to block assembling approach to construct the structure of three-dimensional fractured rock masses. Polygon subtraction and union algorithms that rely merely on vertex chain operation are proposed, which allow a fast and convenient construction of complex faces/loops. Apart from its robustness in dealing with finite discontinuities and complex geometries, the advantages of the current methodology in tackling some challenging issues associated with the morphological analysis of rock blocks are addressed. In particular, the identification of complex blocks with interior voids such as cavity, pit and toms can be readily achieved based on the number and the type of loops. The improved morphology visualization approach can benefit the pre-processing stage when analyzing the stability of rock masses subject to various engineering impacts using the block theory and the discrete element method.

C2-Cofiniteness of Cyclic-Orbifold Vertex Operator Superalgebras

In this paper, it is proved that for any C2-cofinite, simple, CFT-type vertex operator superalgebra V and a finite solvable group G consisting of automorphisms of V, the fixed point subalgebra Vc is a C2-cofinite vertex operator superalgebra.

Representations and Fusion Rules for the Orbifold Vertex Operator Algebras L_(sl_(2))(k,0)^(Z_(3))

For the cyclic group Z_(3)and a positive integer k,we study the representations of the orbifold vertex operator algebra L_(sl_(2))(k,0)^(Z_(3)).All the irreducible modules for L_(sl_(2))(k,0)^(Z_(3))are classified and constructed explicitly.Quantum dimensions and fusion rules for L_(sl_(2))(k,0)^(Z_(3))are completely determined.

Constructing tensor products of modules for C2-cofinite vertex operator superalgebras

For any C2-cofinite vertex product and the P（z）-tensor product finite length are proved to exist, which operator superalgebra V, the tensor of any two admissible V-modules of are shown to be isomorphic, and their constructions are given explicitly in this paper.

IRREDUCIBLE REPRESENTATIONS FOR THE AFFINE-VIRASORO LIE ALGEBRA OF TYPE B_l

An explicit construction of irreducible representations for the affine-Virasoro Lie algebra of type Bl, through the use of vertex operators and certain oscillator representations of the Virasoro algebra, is given.

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory.The author reviews some main convergence results,conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras.He also reviews the related analytic extension results,conjectures and problems.He discusses the convergence and analytic extensions of products of intertwining operators(chiral conformal fields)and of q-traces and pseudo-q-traces of products of intertwining operators.He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result.He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Tau functions of the charged free bosons

We study bosonic tau functions in relation with the charged free bosonic fields.It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector,and some tau functions are given in the completion M by using Schur functions.We also give a new proof of Borchardt’s identity and obtain several q-series identities by using the boson-boson correspondence.