Flux homomorphism - Knowlpedia

Definition

Let $(M,\omega)$ be a closed connected symplectic manifold. For a symplectic isotopy $\phi_t$ with generating vector field $X_t$ , its flux is

\operatorname{Flux}(\{\phi_t\}) =\left[\int_0^1\iota_{X_t}\omega\,dt\right] \in H^1_{\mathrm{dR}}(M;\mathbb R).

The contraction one-forms are closed, so the integral defines a class in the first de Rham cohomology group. Flux depends on the homotopy class of the path with endpoints fixed, rather than only on the endpoint.

Consequently it defines a homomorphism

\operatorname{Flux}: \widetilde{\operatorname{Symp}}_0(M,\omega) \longrightarrow H^1_{\mathrm{dR}}(M;\mathbb R),

where the domain is the universal covering group of the identity component of the symplectomorphism group. The action of a map isotopic to the identity on cohomology is trivial, which makes the concatenation formula additive.

Hamiltonian kernel

Every Hamiltonian isotopy has zero flux because $\iota_{X_t}\omega=dH_t$ . The converse at the level of path classes is the flux theorem: a symplectic path class has zero flux exactly when it has a Hamiltonian representative with the same endpoints. Thus zero flux is a statement about a relative-endpoint homotopy class, not necessarily about the original parameterized path being pointwise Hamiltonian.

Descending to endpoints

Loops at the identity may have nonzero flux. Their image is the flux group $\Gamma_\omega$ . Quotienting by this ambiguity gives an endpoint homomorphism

\overline{\operatorname{Flux}}: \operatorname{Symp}_0(M,\omega) \longrightarrow H^1_{\mathrm{dR}}(M;\mathbb R)/\Gamma_\omega

whose kernel is $\operatorname{Ham}(M,\omega)$ .

Noncompact variants

The displayed definition uses the closed-manifold convention. For compactly supported isotopies on a noncompact manifold, flux naturally takes values in compactly supported de Rham cohomology, and the kernel and endpoint statements require the corresponding support hypotheses. One should not reuse the closed formula without specifying that variant.

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: §10.2, definition and exact sequence for flux.
Augustin Banyaga, The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, 1997. DOI record. Relevant: Chapter 7.