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

The first group of differentiable cohomology of $\text{Diff}(S^1)$, vanishing on the M"obius subgroup $PSL(2,R)\subset \text{Diff}(S^1)$, 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 $\text{Diff}(S^1)$ generalizing the Schwarzian derivative.