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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
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