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.

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.

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.

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.

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.

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.

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.

Level-1/2 Realization of Quantum N-Toroidal Algebra in Type C_(n)

We construct a level-1/2 vertex representation of the quantum N-toroidal algebra of type C_(n),which is a natural generalization of the usual quantum toroidal algebra.The construction also provides a vertex representation of the quantum toroidal algebra for type C_(n) as a by-product.

Permutation orbifolds and associative algebras

Let V be a vertex operator algebra and g =(1 2 ··· k) be a k-cycle which is viewed as an automorphism of the vertex operator algebra V?k. It is proved that Dong-Li-Mason’s associated associative algebra Ag(V?k) is isomorphic to Zhu’s algebra A(V) explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible g-twisted V?k-modules and irreducible V-modules.

The Classification of Extensions of Lst3 （k, O）

In this paper, rational extensions of affine vertex operator algebras Lsl3 （k, O） with k Institute of Mathematics, University of Tsukuba, Tsukuba, Japan Z＋ are classified by modular invariants.

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.

Application of SNi PER framework to BESⅢ physics analysis

A fast physics analysis framework has been developed based on SNi PER to process the increasingly large data sample collected by BESⅢ.In this framework,a reconstructed event data model with Smart Ref is designed to improve the speed of Input/Output operations,and necessary physics analysis tools are migrated from BOSS to SNi PER.A real physics analysis e~＋e^-→ π~＋π^-J/ψ is used to test the new framework,and achieves a factor of10.3 improvement in Input/Output speed compared to BOSS.Further tests show that the improvement is mainly attributed to the new reconstructed event data model and the lazy-loading functionality provided by Smart Ref.