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?