Chiral differential operators: formal loop group actions and associated modules

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.