TITLE: Virasoro action on the elliptic cohomology
QUESTION [17 upvotes]: I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's paper. 
Let $X$ be a Calabi-Yau manifold. (The elliptic genus can be defined for any $X$ with less properties, but the physics is nicest when $X$ is a CY. So let me assume that.) 
In the paper quoted above, a sequence of sheaves $F_k$ on $X$ ($k=0,1,2,$...) were constructed (by taking suitable tensor powers of the tangent bundle), such that $\Phi(q)=\sum_k q^k\chi(F_k)$ gives the elliptic genus of $X$.
Now, the physics construction says that, before taking the Euler characteristic, there is an action of (N=2 superconformal) Virasoro algebra on $\oplus_{i,k} H^i(X,F_k)$; this is the basis of the modularity of the elliptic genus. I presume this action has already been geometrically constructed in the mathematical literature, given the fact that Witten's paper is from 1980s.
So, my question is, where can I find it?

REPLY [2 votes]: I would like to add that (cohomology of) chiral de Rham complex has to be viewed as the large Kahler limit of halftwisted theory. Indeed, it does not use the Kahler data on the CY manifold and does not see the instanton count. So it is not the final answer in any sense, but is a very important step towards rigorous understanding of type II superstring theories.