$\mathit{Diff}(S^1)$ 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??
Best Answer
There's a very geometric interpretation of the Virasoro-Bott cocycle in terms of projective structures on Riemann surfaces, which I hope should relate directly to the Liouville functional.
Namely to describe a 1-dimensional central extension of a Lie algebra it's equivalent to give an affine space over the dual of the Lie algebra with a compatible "coadjoint" action of the Lie algebra. (This models the hyperplane in the dual to the central extension, consisting of functionals taking value one on the central element.)
In the case of vector fields on the circle (I'm thinking of the Laurent series model, but shouldn't be hard to translate to smooth functions) the dual to the complexified Lie algebra is canonically the space of quadratic differentials (with Laurent coefficients) with its canonical Diff S^1 action by rotation. There's a canonical affine space over quadratic differentials, namely the space of projective structures (ie atlases into projective space with Mobius transitions). The cocycle describing this affine space (ie the transformation property of projective structures under general, not just Mobius, changes of coordinates) is the Schwarzian derivative, which translates directly into the Virasoro-Bott cocycle when you write the corresponding central extension (this is all explained in detail eg in my book with Frenkel, Vertex Algebras on Algebraic Curves).
In any case the Liouville functional relates to the conformal factor you need to change your given metric to one with constant curvature, so should have a natural geometric relation with projective structures, but I don't see it right now..