Three Cocycles on $\Diff(S^1)$ Generalizing the Schwarzian Derivative

Abstract

The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the Möbius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian derivative.