## Abstract

The notion of vertex algebra, introduced by Borcherds [**Bo**] in a mathematical setting, and by Prenkel, Lepowsky and Meurman [**FLM2**] in the modified form, “vertex operator algebra,” that we will need to use in the present work, arose naturally in Frenkel, Lepowsky and Meurman’s vertex operator construction [**FLM1**] [**FLM2**] of the Monster sporadic finite simple group [**Gr**]. In fact, they realized the Monster as the symmetry group of a special vertex operator algebra, the “moonshine module.” (See the introduction of [**FLM2**] for a historical discussion, including the important contribution of Borcherds’ announcement [**Bo**].) Meanwhile, conformai field theory, a physical theory whose algebraic structure was developed systematically by Belavin, Polyakov and Zamolodchikov [**BPZ**] in a physical context, has been becoming more and more attractive to mathematicians. In the study of conformal field theories, physicists also arrived at, though perhaps without a completely rigorous definition, the notion of “chiral algebra,” a notion which essentially coincides with the notion of vertex operator algebra (see e.g. [**MS**]).

### Keywords

Manifold Lution Dinates Mirror Symmetry## Preview

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