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Operadic formulation of the notion of vertex operator algebra. (English) Zbl 0848.17030
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 131-148 (1994).
A reformulation of the notion of vertex operator algebra in terms of operads is presented. First, this moduli space naturally induces a certain (partial) operad, which admits “\(\mathbb{C}\)-extensions” constructed from the determinant line bundle. Naturally defined “meromorphic associative algebras” for these extension-operads can then be thought of as “vertex associative algebras.” The main theorem of [Y.-Z. Huang, On the geometric interpretation of vertex operator algebras, Ph. D. thesis, Rutgers University, 1990; Operads and the geometric interpretation of vertex operator algebras. I (to appear)], and the announcement [Y.-Z. Huang, Geometric interpretation of vertex operator algebras, Proc. Natl. Acad. Sci. USA 88, 9964-9968 (1991; Zbl 0810.17019)] then in fact says that the category of vertex operator algebras with central charge say \(c\) is isomorphic to the category of vertex associative algebras with central charge \(c\). 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. The viewpoint and results in this paper are briefly summarized in [Y.-Z. Huang and J. Lepowsky, Vertex operator algebras and operads, The Gelfand Seminars, 1990-1992, 145-161 (1993; Zbl 0807.17024)].
For the entire collection see [Zbl 0801.00049].

MSC:
17B69 Vertex operators; vertex operator algebras and related structures
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
55N99 Homology and cohomology theories in algebraic topology
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