arXiv subjects
Publications and source records attributed to Haisheng Li.
We give a new, construction-free proof of the associativity of tensor product for modules for rational vertex operator algebras under certain convergence conditions.
The admissible modules for $\hat{sl}_2$ are studied from the point of view of vertex operator algebra. If $l$ is rational such that $l+2={p\over q}$ for some coprime positive integers $p\ge 2$ and $q$, Kac and Wakimoto found finitely many distinguished irreducible representations for $\hat{sl}_2$, called admissible representations. In this paper we prove that the vertex operator algebra $L(l,0)$ associated to irreducible highest weight representation of $l$ is not rational if $l$ is not a positive integer. However if we change the Virasoro algebra in certain way, $L(l,0)$ becomes a rational vertex operator algebra whose irreducible representations are exactly those admissible representations. We show that the $q$-dimensions with respect to the new Virasoro algebra are modular functions. We aslo calculate the fusions rules.
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.
A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{\natural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro algebra and the vertex operator algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.
We formulate an interpretation of the theory of physical superselection sectors in terms of vertex operator algebra language. Using this formulation we give a construction of simple current from a primary semisimple element of weight one. We then prove that if a rational vertex operator algebra $V$ has a simple current $M$ satisfying certain conditions, then $V\oplus M$ has a natural rational vertex operator (super)algebra structure. Applying our results to a vertex operator algebra associated to an affine Lie algebra, we construct its simple currents and the extension by a simple current. We also present two essentially equivalent constructions for twisted modules for an inner automorphism from the adjoint module or any untwisted module.
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 $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.
This is the second part of the revised versions of the notes of three consecutive expository lectures given by Chongying Dong, Haisheng Li and Yi-Zhi Huang in the conference on Monster and vertex operator algebras at the Research Institute of Mathematical Sciences, Kyoto, September 4-9, 1994. In this part, we discuss the commutativity, the associativity, the Jacobi identity and the relations among them. We also discuss an analogue of endomorphism ring of a vector space in vertex operator algebra theory.
For the moonshine module $V^{\natural},$ whose automorphism is the Monster ${\Bbb M},$ We show how to give a uniform existence proof for irreducible $g$-twisted modules for elements of type $2A,$ $2B$ and $4A$ in ${\Bbb M}.$ The most interesting of these is the twisted sector $V^{\natural}(2A),$ whose automorphism group is essentially the centralizer of $2A$ in ${\Bbb M}.$ This is a 2-fold central extension of the Baby Monster, the second largest sporadic simple group. We also establish uniqueness of the twisted sectors and the hauptmodul property for the graded traces of automorphisms of odd order, as predicted by conformal field theory.
We consider how a vertex operator algebra can be extended to an abelian intertwining algebra by a family of weak twisted modules which are {\em simple currents} associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except for $E_8$) and to get various extensions of the vertex operator algebras associated with integrable representations.