Virasoro action on the elliptic cohomology

Question

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?

Answer 1

There is an extensive math literature on related constructions. The key word is "chiral de Rham complex", introduced by Malikov, Schechtman and Vaintrob here and further developed in many many papers, starting with work of Gorbounov-Malikov-Schechtman. The connections to elliptic genera are perhaps most extensively developed in papers of Borisov-Libgober, the first of a series of which is here. The chiral de Rham complex in the case of a Calabi-Yau is a (sheaf of) N=2 superconformal vertex algebras whose genus one partition function calculates the (two-variable) elliptic genus. Its physics interpretation is not precisely the sigma model in Witten's original paper, rather it's given by a "half-twisted" model as explained physically by Kapustin and Witten.

However perhaps the mathematical construction most explicitly related to the physics of Witten's original paper is described in Costello's ICM address here.

You might also be interested in the general program that began from Segal's landmark

"Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)", Séminaire Bourbaki, Vol. 1987/88. Astérisque No. 161-162 (1988)

on "elliptic objects", the attempt to DEFINE elliptic cohomology in terms of conformal field theory. This program is not yet complete but has been advanced significantly in work of Stolz and Teichner, see in particular "What is an elliptic object?" Topology, geometry and quantum field theory, 247–343, London Math. Soc. Lecture Note Ser., 308.

Answer 2

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.