Chiral differential operators: formal loop group actions and associated modules

TL;DR

This paper explores how formal loop group actions can be lifted to chiral differential operators on manifolds, linking geometric structures with vertex algebras and modules, with implications for string geometry and the Witten genus.

Contribution

It introduces conditions for lifting Lie group actions to CDOs via formal loop groups and constructs modules over CDOs using semi-infinite cohomology, advancing the understanding of their geometric and algebraic structures.

Findings

Lifted group actions depend on the equivariant Pontrjagin class.

Constructed modules have geometric interpretations in formal loop spaces.

Provided a new construction of CDO algebras and modules.

Abstract

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of two-dimensional sigma models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a `formal loop group action' on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said `formal loop group actions' and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of `formal loop spaces'. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.