TITLE: Equivariant cohomology of $\text{Diff}S^1/ S^1$ and Virasoro
QUESTION [8 upvotes]: Consider 
$$\mathcal{M}\ =\ \text{Diff}S^1/S^1$$
which is a contractible complex manifold with an action of $\text{Diff}S^1$ by translations. It is claimed in page 358 of [1] that $\mathcal{M}$ has single-dimensional equivariant cohomology $H^2_{\text{Diff}S^1}(\mathcal{M})$, which is related to the central charge in the Virasoro algebra.

Question: is $H^*_{\text{Diff}S^1}(\mathcal{M})$ known in general, and what relation does it have to the Virasoro algebra?

I think the first part might be easy because (I think) the stack $\mathcal{M}/\text{Diff}S^1$ is homotopic to $B\text{Diff}S^1$, which in turn is homotopic to $B \text{U}(1)$ which has cohomology $\mathbf{Z}[x]$.
[1] M. Bowick, S. Rajeev: The holomorphic geometry of closed bosonic string theory and $\text{Diff}S^1/S^1$.

Edit: I should maybe explain what [1] says.  It says that the two-form $\text{Ric}$ (Ricci curvature) satisfies
$$\text{Ric}(L_m,L_n)\ =\ \left(\frac{-26}{12}m^3+\frac{1}{6}m\right) \delta_{-m,n}$$
where $L_n$ are the  vector fields on $\mathcal{M}$ generating the action of $\text{Diff}S^1$. It then goes on to make the relationship between the generator of $H^2$ and the central charge more precise (I've not read the paper closely but I don't think it contains an answer to my question).

REPLY [5 votes]: For any group $G$ and subgroup $K$, there is an isomorphism
$$
H^*_G(G/K)= H^*_K(pt)
$$
So the cohomology in question is $H^*_{S^1}(pt)=\mathbb Z[x]$, and it does not seem to be very related to the Virasoro algebra.
(Here, the generator $x\in H^2_{S^1}(pt)$ is in degree two.)

If I remember correctly, the Lie algebra cohomology of the Witt algebra (with coefficients in the trivial module $\mathbb C$) is a polynomial algebra generated by a class in degree $2$ tensor an exterior algebra generated by a class in degree $3$. From this, I think that one can deduce that the Lie algebra cohomology of the Virasoro algebra (= the universal central extension of the Witt algebra) is an exterior algebra generated by a class in degree $3$.