Multi-dimensional Schwarzian derivatives and differential operators

Abstract

Lie algebras of vector fields on a smooth manifold M became popular in mathematics and physics after the discovery of the Virasoro algebra by Gelfand and Fuchs in 1967. Gelfand and Fuchs, Bott, Segal, Haefliger and many others studied cohomology of Lie algebras of vector fields and diffeomorphism groups with coefficients in spaces of tensor fields. This theory attracted much attention in the last three decades, many important problems were solved and many beautiful applications, such as characteristic classes of foliations, were found. In this chapter we consider cohomology of Lie algebras of vector fields and of diffeomorphism groups with coefficients in various spaces of differential operators; this is a generalization of Gelfand–Fuchs cohomology. Only a few results are available so far, mostly for the first cohomology spaces. The main motivation is to study the space of differential operators D λ,μ ( M ), viewed as a module over the group of diffeomorphisms. This cohomology is closely related to projective differential geometry and, in particular, to the Schwarzian derivative. The classic Schwarzian derivative is a 1-cocycle on the group Diff( S 1 ), related to the module of Sturm–Liouville operators. Multi-dimensional analogs of the Schwarzian derivative are defined as projectively invariant 1-cocycles on diffeomorphism groups with values in spaces of differential operators.