Existence of 'Darboux chart' on loop space

Abstract

For a finite dimensional symplectic manifold $(M,\omega )$ with a symplectic form $\omega$, corresponding loop space ($LM=C^{\infty }(S^1,M)$) admits a weak symplectic form ${\Omega }^{\omega }$. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure ${\Omega }^{\omega }$. 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.

9 pages[Updated]

References in corpus (2)

• The differential topology of loop spaces
• Darboux chart on Projective limit of weak symplectic Banach manifold

Cited by in corpus (1)

• Geodesics on path spaces and double category