Meromorphic open-string vertex algebras and Riemannian manifolds
arXiv:1205.2977
Abstract
Let be a Riemannian manifold. For , the tensor algebra of the negative part of the (complex) affinization of the tangent space of at has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over with a connection. We construct a sheaf of meromorphic open-string vertex algebras on the sheaf of parallel sections of this vector bundle. Using covariant derivatives, we construct representations on the spaces of complex smooth functions of the algebras of parallel tensor fields. These representations are used to construct a sheaf of left -modules from the sheaf of smooth functions. In particular, we obtain a meromorphic open-string vertex algebra of the global sections on of the sheaf and a left -module of the global sections on of the sheaf . By the definitions of meromorphic open-string vertex algebra and left module, we obtain, among many other properties, operator product expansion for vertex operators. We also show that the Laplacian on is in fact a component of a vertex operator for the left -module restricted to the space of smooth functions.
25 pages. The published version (in a special issue of Chinese Annals of Mathematics, series B in memory of Hu Hesheng) was mistakenly revised based on an old version of the paper. Here is the correct final version of the paper