We investigate trace functions of modules for vertex operator algebras satisfying C 2 -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Zh]: A(V ) is semisimple and C 2 -cofiniteness. We show that C 2 -cofiniteness is enough to prove a modular invariance property. For example, if a VOA V = ⊕ ∞ m=0 V m is C 2 -cofinite, then the space spanned by generalized characters (pseudo-trace functions of the vacuum element) of V -modules is a finite dimensional SL 2 (Z)-invariant space and the central charge and conformal weights are all rational numbers. Namely we show that C 2 -cofiniteness implies "rational conformal field theory" in a sense as expected in [GN]. Viewing a trace map as one of symmetric linear maps and using a result of symmetric algebras, we introduce "pseudo-traces" and pseudo-trace functions and then show that the space spanned by such pseudo-trace functions has a modular invariance property. We also show that C 2 -cofiniteness is equivalent to the condition that every weak module is an N-graded weak module which is a direct sum of generalized eigenspaces of L(0).

We define automorphisms of vertex operator algebra using the representations of the Virasoro algebra. In particular, we show that the existence of a special 1 element, which we will call a ''rational conformal vector with central charge ,'' 2 implies the existence of an automorphism of a vertex operator algebra. This result offers a simple construction of triality involutions of the Moonshine module V h .We also study the structures of Griess algebras and prove a conjecture given by Meyer Neutsch that the maximal dimension of associative subalgebras of the Griess Monster algebra is 48. ᮊ

We give a new construction of the moonshine module vertex operator algebra V , which was originally constructed in [FLM2]. We construct it as a framed VOA over the real number field R. We also offer ways to transform a structure of framed VOA into another framed VOA. As applications, we study the five framed VOA structures on V E8 and construct many framed VOAs including V from a small VOA. One of the advantages of our construction is that we are able to construct V as a framed VOA with a positive definite invariant bilinear form and we can easily prove that Aut(V ) is the Monster simple group. By similar ways, we also construct an infinite series of holomorphic framed VOAs with finite full automorphism groups. At the end of the paper, we calculate the character of a 3C element of the Monster simple group.

We prove an orbifold conjecture for a solvable automorphism group. Namely, we show that if V is a C 2 -cofinite simple vertex operator algebra and G is a finite solvable automorphism group of V , then the fixed point vertex operator subalgebra V G is also C 2 -cofinite. This offers a mathematically rigorous background to orbifold theories with solvable automorphism groups.

We study a vertex operator algebra whose Virasoro element is a sum of pairwise 1 orthogonal rational conformal vectors with central charge . The most important 2 example is the moonshine module V h . In particular, we construct a series of vertex operator algebras whose full automorphism groups are finite. Namely, we construct ϱ Ž .

1. Introduction. Throughout this paper, V denotes a vertex operator algebra, or VOA, (⊕ ∞ n=0 V n , Y, 1, ω) with central charge c and Y (v,z) = v(n)z −n−1 denotes a vertex operator of v. (Abusing the notation, we also use it for vertex operators of v for V -modules.) o(v) denotes the grade-keeping operator of v, which is given by v(m − 1) for v ∈ V m and defined by extending it for all elements of V linearly. InWe call V a rational vertex operator algebra in the case when each V -module is a direct sum of simple modules. Define C 2 (V ) to be the subspace of V spanned by elements u(−2)v for u, v ∈ V . We say that V satisfies condition C 2 if C 2 (V ) has finite codimension in V . For a V -module M with grading M = ⊕M m , we define the formal character asIn this paper, we consider these functions less formally by taking q to be the usual local parameter q = q τ = e 2πιτ at infinity in the upper half-planeAlthough it is often said that a VOA is a conformal field theory with mathematically rigorous axioms, the axioms of VOA do not assume the modular invariance. However, Zhu [Z] showed the modular (SL 2 (Z)) invariance of the space q |a 1 | 1 · · · q |a n | n tr W Y (a 1 , q 1 ) · · · Y (a n , q n )q L(0)−c/24 : W irreducible V -modules (2) for a rational VOA V with central charge c and a i ∈ V |a i | under condition C 2 , which are satisfied by many known examples, where q j = q z j = e 2πιz j and |a i | denotes the

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