  Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$
the group of symplectomorphisms preserving the origin and the boundary
$\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism
on $\rm{Symp}(D,0)_{\rm{rel}}$ 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 $D$.
