Existence of 'Darboux chart' on loop space

Abstract

For a finite dimensional symplectic manifold $(M,ω)$ with a symplectic form $ω$, corresponding loop space ($LM=C^\infty(S^1,M)$) admits a weak symplectic form $Ω^ω$. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure $Ω^ω$. Further, we show that inclusion map from the symplectic cohomology (as defined by Kriegl and Michor \cite{KM}) of the loop space over $\mathbb R^n$ to the De Rham cohomology of the loop space is an isomorphism.