Non-Hamiltonian Actions and Lie-Algebra Cohomology of Vector Fields

TL;DR

This paper investigates the cohomological obstructions to the existence of equivariant moment maps for certain invariant two-forms on the circle, linking them to nontrivial Lie algebra cohomology classes.

Contribution

It provides explicit examples of invariant forms that lack equivariant moment maps and computes their associated cohomological obstructions.

Findings

Identifies invariant two-forms without equivariant moment maps.

Computes the cohomological obstruction as a nontrivial Lie algebra cohomology class.

Establishes a connection between geometric forms and Lie algebra cohomology.

Abstract

Two examples of $\mathrm{Diff}^+S^1$-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on $H^2(\mathfrak{X}(S^1))$.