Two constructions of grading-restricted vertex (super)algebras

We give two constructions of grading-restricted vertex (super)algebras. We first give a new construction of a class of grading-restricted vertex (super)algebras originally obtained by Meurman and Primc using a different method. This construction is based on a new definition of vertex operators and a new method. Our second construction is a generalization of the author's construction of the moonshine module vertex operator algebra and a related vertex operator superalgebra. This construction needs properties of intertwining operators formulated and proved by the author.Comment: 26 pages. Misprints are corrected. To appear in Journal of Pure and Applied Algebr

A functional-analytic theory of vertex (operator) algebras, I

This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V, Y, 1) satisfying the standard grading-restriction axioms, a locally convex topological completion H of V is constructed. By the geometric interpretation of vertex (operator) algebras, there is a canonical linear map from the tensor product of V and V to the algebraic completion of V realizing linearly the conformal equivalence class of a genus-zero Riemann surface with analytically parametrized boundary obtained by deleting two ordered disjoint disks from the unit disk and by giving the obvious parametrizations to the boundary components. We extend such a linear map to a linear map from the completed tensor product of H and H to H, and prove the continuity of the extension. For any finitely-generated C-graded V-module (W, Y_W) satisfying the standard grading-restriction axioms, the same method also gives a topological completion H^W of W and gives the continuous extensions from the completed tensor product of H and H^W to H^W of the linear maps from the tensor product of V and W to the algenbraic completion of W realizing linearly the above conformal equivalence classes of the genus-zero Riemann surfaces with analytically parametrized boundaries.Comment: LaTeX file. 31 pages, 1 figur

Vertex operator algebras, fusion rules and modular transformations

We discuss a recent proof by the author of a general version of the Verlinde conjecture in the framework of vertex operator algebras and the application of this result to the construction of modular tensor tensor category structure on the category of modules for a vertex operator algebra.Comment: 21 pages. To appear in Proceedings of the conference "Non-commutative Geometry and Representation Theory in Mathematical Physics," July 5-10, 2004, Karlstad University, Swede