The Witten genus and vertex algebras

Abstract

This article is the first report of an ongoing project aimed at finding a geometric interpretation of the Witten genus and other tmf classes. Section 2 reviews the sheaves of chiral differential operators (CDOs) over a complex manifold, including their construction, obstructions and relation with the Witten genus. In section 3, the structure of each sheaf of CDOs is reorganized in terms of modules over the sheaf of holomorphic functions. This invokes the notion of a differential graded vertex algebroid. The construction of sheaves of CDOs is due to Gorbounov, Malikov and Schechtman, and so is the notion of a vertex algebroid; the differential graded version is first introduced here. Section 4 contains the main result, namely the construction of a sheaf of differential graded conformal vertex algebras that provides a fine resolution of a sheaf of CDOs. This `infinite dimensional Dolbeault complex' plays a role for the Witten genus similar to that of the Dolbeault complex for the Todd genus.