Vertex operator algebras and operads

Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, $n$-ary operations for all $n$ greater than or equal to $0$, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (two-dimensional) ``complex'' geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) ``real'' geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level.Comment: 16 pages. Only the definitions of "partial operad" and of "rescaling group" have been improve

Virasoro vertex operator algebras, the (nonmeromorphic) operator product expansion and the tensor product theory

In the references [HL1]--[HL5] and [H1], a theory of tensor products of modules for a vertex operator algebra is being developed. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra satisfy the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic) associativity (operator product expansion). Combined with a result announced in [HL4], the results of the present paper also show that the category of modules for such a vertex operator algebra has a natural structure of a braided tensor category. In particular, for any pair $p, q$ of relatively prime positive integers larger than $1$, the category of minimal modules of central charge $1-6\frac{(p-q)^{2}}{pq}$ for the Virasoro algebra has a natural structure of a braided tensor category.Comment: LaTeX file. 37 page

Hilbert series for twisted commutative algebras

Suppose that for each n >= 0 we have a representation $M_n$ of the symmetric group S_n. Such sequences arise in a wide variety of contexts, and often exhibit uniformity in some way. We prove a number of general results along these lines in this paper: our prototypical theorem states that if $M_n$ can be given a suitable module structure over a twisted commutative algebra then the sequence $M_n$ follows a predictable pattern. We phrase these results precisely in the language of Hilbert series (or Poincar\'e series, or formal characters) of modules over tca's.Comment: 28 page