The flux homomorphism and central extensions of diffeomorphism groups

Let $D$ be a 2-dimensional closed unit disk and $\mathrm{S}\mathrm{y}\mathrm{m}\mathrm{p}(D,0{)}_{\mathrm{r}\mathrm{e}\mathrm{l}}$ the group of symplectomorphisms preserving the origin and the boundary $\mathrm{\partial }D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\mathrm{S}\mathrm{y}\mathrm{m}\mathrm{p}(D,0{)}_{\mathrm{r}\mathrm{e}\mathrm{l}}$ and define the central $\mathbb{R}$-extension called the $\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.