Brownian motion and Ricci curvature on an infinite dimensional symplectic group related to the diffeomorphism group of the circle

TL;DR

This paper explores the embedding of the diffeomorphism group of the circle into an infinite-dimensional symplectic group, constructs a Brownian motion on this group, and analyzes its Ricci curvature, revealing predominantly negative infinity values.

Contribution

It demonstrates that the embedding of $ ext{Diff}(S^1)$ into $ ext{Sp}( ext{infinity})$ is not surjective, constructs a Brownian motion on $ ext{Sp}( ext{infinity})$, and computes its Ricci curvature.

Findings

Embedding is not surjective.

Brownian motion on $ ext{Sp}( ext{infinity})$ is constructed.

Ricci curvature is negative infinity in almost all directions.

Abstract

An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin. The Ricci curvature of the infinite-dimensional symplectic group is computed. The result shows that in almost all directions, the Ricci curvature is negative infinity.