Chiral differential operators on supermanifolds

TL;DR

This paper introduces a geometric formulation of chiral differential operators on supermanifolds, constructs chiral Dolbeault complexes, and relates their cohomology to important genera like the Witten genus.

Contribution

It provides a new global geometric approach to defining CDOs, constructs chiral Dolbeault complexes, and connects these structures to topological invariants such as the Witten genus.

Findings

A recipe to define sheaves of CDOs using an affine connection and a 3-form.

Construction of chiral Dolbeault complexes and analysis of their conformal structures.

Cohomology of these complexes computes the Witten genus and elliptic genera.

Abstract

The first part of this paper provides a new formulation of chiral differential operators (CDOs) in terms of global geometric quantities. The main result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its ingredients consist of an affine connection and an even 3-form that trivializes the first Pontrjagin form. With the connection fixed, two suitable 3-forms define isomorphic sheaves of CDOs if and only if their difference is exact. Moreover, conformal structures are in one-to-one correspondence with even 1-forms that trivialize the first Chern form. Applying our work in the first part, we construct what may be called "chiral Dolbeault complexes" of a complex manifold M, and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings (fermion numbers). When M is compact, their cohomology computes (in various cases) the Witten genus, the two-variable elliptic genus and a spin-c version of the Witten genus. This part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic CDOs and sigma models.