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

TL;DR

This paper computes specific differentiable cohomology groups of the diffeomorphism group of the circle and introduces three new invariant cocycles that generalize the Schwarzian derivative, expanding understanding of geometric structures on $S^1$.

Contribution

It calculates new cohomology groups of $ ext{Diff}(S^1)$ and introduces three novel $PSL(2,R)$-invariant cocycles generalizing the Schwarzian derivative.

Findings

Computed differentiable cohomology groups vanishing on $PSL(2,R)$

Introduced three new $PSL(2,R)$-invariant cocycles

Generalized the Schwarzian derivative to new cocycles

Abstract

The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the M\"obius 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.