TITLE: central extensions of Diff(S^1) and of the semigroup of annuli
QUESTION [20 upvotes]: $\mathit{Diff}(S^1)$ 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??

REPLY [4 votes]: Let me give a method for answering the problem.  I haven't yet done the relevant integral to get an actual answer.
In the setting you originally laid out for $\mathcal{A}$, actual diffeomorphisms are not included; however, you can find an annulus close to a diffeomorphism $f$ by taking an annulus of the form $[0,\epsilon] \times S^1$ with $\epsilon$ very small, and parametrizing the two sides differently: parametrize the left boundary by $x \mapsto (0,f(x))$ and parametrize the right boundary by $x \mapsto (1,x)$.  Call this parametrized annulus $A_0(f)$.
$A_0(f)$ is not of the type that you apply the Liouville functional to, since the left boundary is not constant-speed geodesic.  So let's construct another metrized torus where both boundaries are constant-speed.  Let $B_\epsilon:[0,\epsilon] \to [0,1]$ be a bump function which is $1$ in a neighborhood of $0$ and $0$ in a neighborhood of $\epsilon$.  Let $A_1(f)$ be the annulus $A_0(f)$ with the metric rescaled as
$$
g_1(s,t) = \exp\Bigl(-2B_\epsilon(s)\ln \bigl(f'(t)\bigr)\Bigr)g_0(s,t) =
\exp\Bigl(-2B_\epsilon(s)\ln \bigl(f'(t)\bigr)\Bigr)(ds^2 + dt^2).
$$
Then $A_1(f)$, with the same boundary parametrization as in $A_0(f)$, is now parametrized by constant-speed geodesics.  Call $(A_1(f), 0) \in \widetilde{\mathcal{A}}$ the canonical annulus of the diffeomorphism $f$.
Now suppose we have two diffeomorphisms $f,g$.  We can glue the associated cannonical annuli $A_1(f), A_1(g)$.  The result will represent the diffeomorphism $g \circ f$; but the metric will not be the canonical metric on $A_1(g \circ f)$.  We have to rescale the metric by an appropriate scaling factor $\phi$ to get to the canonical metric.  (Actually, the conformal width of the annulus is also now $2\epsilon$ rather than $\epsilon$, but that shouldn't matter.) The integral in the problem statement for this rescaling presumably reduces to the Bott-Virasoro integral as $\epsilon$ gets sufficiently small.
I'd still like to understand this geometrically well enough to not have to do the integral.