Meromorphic open-string vertex algebras and Riemannian manifolds

Abstract

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space 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. We construct a sheaf $\mathcal{V}$ 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 $\mathcal{W}$ of left $\mathcal{V}$-modules from the sheaf of smooth functions. In particular, we obtain a meromorphic open-string vertex algebra $V_{M}$ of the global sections on $M$ of the sheaf $\mathcal{V}$ and a left $V_{M}$-module $W_{M}$ of the global sections on $M$ of the sheaf $\mathcal{W}$. 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 $M$ is in fact a component of a vertex operator for the left $V_{M}$-module $W_{M}$ restricted to the space of smooth functions.