$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are isomorphism classes of annulus-shaped Riemann surfaces, with parametrized boundary.

Both $\mathit{Diff}(S^1)$ and $\mathcal A$ have central extensions by $\mathbb R$, and my question is about their relationship.

♦ The group $\mathit{Diff}(S^1)$ carries the so-called *Bott-Virasoro cocycle*, which is given by
$$
B(f,g) = \int_{S^1}\ln(f'\circ g)\;\; d\;\ln(g').
$$
The corresponding centrally extended group is $\widetilde{\mathit{Diff}(S^1)}:=\mathit{Diff}(S^1)\times \mathbb R$, with product given by $(f,a)\cdot(g,b):=(f\circ g,a+b+B(f,g))$.

♦ The elements of the central extension $\widetilde{\mathcal A}$ of $\mathcal A$ have a very different description.

An element $\widetilde{\Sigma}\in\widetilde{\mathcal A}$ sitting above $\Sigma\in\mathcal A$
is an equivalence class of pairs $(g,a)$, where $g$ is a Riemannian metric on $\Sigma$ compatible with the complex structure, and $a\in\mathbb R$.
There's the extra requirement that the boundary circles of $\Sigma$ be constant speed geodesics for $g$.

The equivalence relation involves the *Liouville functional*:
one declares $(g_1,a_1)\sim (g_2,a_2)$ if $g_2=e^{2\varphi}g_1$, and
$$
a_2-a_1=\int_\Sigma {\textstyle\frac 1 2}(d\varphi\wedge \ast d\varphi+4\varphi R),
$$
where $R$ is the curvature 2-form of the metric $g_1$.

It is reasonable to believe that the restriction of the central extension $\widetilde{\mathcal A}$ to the "subgroup" $\mathit{Diff}(S^1)\subset \mathcal A$ is $\widetilde{\mathit{Diff}(S^1)}$. But I really don't see why that's should be the case.

Any insight? How does one relate the Bott-Virasoro cocycle to the Liouville functional??