Introduction to vertex operator algebras I

This is the first 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 review the definitions of vertex operator algebras and twisted modules, and discuss examples.Comment: LaTeX file, 26 page

Representations of vertex operator algebras and bimodules

For a vertex operator algebra V, a V-module M and a nonnegative integer n, an A_n(V)-bimodule A_n(M) is constructed and studied. The connection between A_n(M) and intertwining operators are discussed. In the case that V is rational, A_n(M) for irreducible V-module M is given explicitly.Comment: 17 page

Unitary vertex operator algebras

Unitary vertex operator algebras are introduced and studied. It is proved that most well-known rational vertex operator algebras are unitary. The classification of unitary vertex operator algebras with central charge c less than or equal to 1 is also discussed.Comment: 31 page

Integrability of C_2-cofinite vertex operator algebras

The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions(C_2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra {\frak g} of the weight one subspace V_1 is isomorphic to the irreducible highest weight \hat{\frak g}-module L(k, 0) for a positive integer k, and V is an integrable \hat{\frak g}-module. The case in which {\frak g} is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed.Comment: 13 page

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 C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.Comment: 31 page

Bimodules associated to vertex operator algebras

Let V be a vertex operator algebra and m,n be nonnegative integers. We construct an A_n(V)-A_m(V)-bimodule A_{n,m}(V) which determines the action of V from the level m subspace to level n subspace of an admissible V-module. We show how to use A_{n,m}(V) to construct naturally admissible V-modules from A_m(V)-modules. We also determine the structure of A_{n,m}(V) when V is rational.Comment: a minor chang

Twisted representations of vertex operator superalgebras

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.Comment: 23 page

Some finite properties for vertex operator superalgebras

Vertex operator superalgebras are studied and various results on rational Vertex operator superalgebras are obtained. In particular, the vertex operator super subalgebras generated by the weight 1/2 and weight 1 subspaces are determined. It is also established that if the even part $V_{\bar 0}$ of a vertex operator superalgebra $V$ is rational, so is $V.$Comment: 18 page

Classification of irreducible modules for the vertex operator algebra M(1)^+, II: higher rank

The vertex operator algebra M(1)^+ is the fixed point set of free bosonic vertex operator algebra M(1) under the -1 automorphism. All irreducible modules for M(1)^+ are classified in this paper for all ranks.Comment: latex, 40 page

Representations of vertex operator algebras

This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras A_n(V) and their bimodules. A new result on the rationality is given. That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is a semisimple associative algebra and each irreducible admissible $V$-module is ordinary.Comment: 13 pages, final version for publicatio