The Witten genus and vertex algebras

TL;DR

This paper explores a geometric interpretation of the Witten genus through the construction of sheaves of chiral differential operators and differential graded vertex algebroids, linking complex geometry with topological invariants.

Contribution

It introduces the notion of differential graded vertex algebroids and constructs a sheaf of differential graded conformal vertex algebras that resolves sheaves of CDOs, advancing the understanding of the Witten genus.

Findings

Constructed a sheaf of differential graded conformal vertex algebras.

Reorganized sheaves of CDOs using modules over holomorphic functions.

Established an analogy between the infinite dimensional Dolbeault complex and the Witten genus.

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.