Let M be a Riemannian manifold. For p ∈ M, the tensor algebra \(T({\widehat {T_{p}M}}\_)\) of the negative part of the affinization \({\widehat {T_{p}M}}\) of the tangent space TpM of M at p has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over M with a connection. The author constructs a sheaf \({\cal V}\) of meromorphic open-string vertex algebras on the sheaf of parallel sections of this vector bundle. Using covariant derivatives, he constructs a representation on the space of smooth functions of the algebra of parallel tensor fields. These representations are used to construct a sheaf \({\cal W}\) of left \({\cal V}\) -modules generated by the sheaf of smooth functions. In particular, the author obtains a meromorphic open-string vertex algebra VM as the global sections on M of the sheaf \({\cal V}\) and a left VM-module WM as the global sections on M of the sheaf \({\cal W}\) . He shows that the Laplacian on M is in fact a component of a vertex operator for the left VM-module WM restricted to the space of smooth functions.
