# Is there a canonical map between the cohomology of orbifold Chiral de Rham on an orbifold and the cohomology of Chiral de Rham on a crepant resolution?

The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold is Calabi-Yau. It is defined by constructing a formal power series in K-theory out of exterior and symmetric powers of tangent and cotangent bundles, and taking its Euler characteristic. It seems to capture some stringy geometry of a manifold $X$, in the sense that it is (under the assumption that such objects can be precisely defined) the partition function of the half-twisted $\mathcal{N}=(2,2)$ superconformal $\sigma$-model with target $X$. The modularity properties seem to reflect some correspondence between traces of certain operators and small deformations of a nodal genus 1 curve.

Work by Borisov and Libgober in the turn of the century yielded two generalizations of elliptic genus:

1. a notion of orbifold elliptic genus, where one adds contributions from fixed loci of conjugacy classes. This seems to be related to the fact that the space of small loops on a stack is more or less its inertia stack.
2. a notion of singular elliptic genus, defined on certain singular varieties by taking a resolution. It is independent of the resolution.

The authors showed that when the coarse moduli space of an orbifold has a crepant resolution, the elliptic genus of that resolution is equal (up to a factor involving a theta function and its derivative) to the orbifold elliptic genus. For example, one could calculate the elliptic genus of a K3 surface by computing the orbifold elliptic genus of the $[\pm 1]$-quotient of an abelian surface, since any K3 is diffeomorphic to the minimal resolution of a Kummer surface.

There is a way to interpret the elliptic genus mathematically in a way that is closer to the physicists' method, by the Chiral de Rham complex (see also Yuji's question). It is defined as a sheaf of vertex superalgebras on the manifold $X$, and its global cohomology yields the elliptic genus as graded characters of certain operators. Physically, according to Kapustin, the Chiral de Rham complex is the perturbative part of the half-twisted SCFT. Naturally, one can construct this complex on a crepant resolution of an orbifold, and Frenkel and Szczesny constructed a version of Chiral de Rham on orbifolds, and showed that its cohomology yields the orbifold elliptic genus.

Question: Is there a vertex superalgebra isomorphism between the cohomology of the orbifold Chiral de Rham (possibly tensored with some superalgebra whose character involves a theta function and its derivative) and the cohomology of the Chiral de Rham of a crepant resolution?

The character equality makes it clear that some map should exist on the level of vector spaces, but it would be nice if there were a more categorified correspondence. Vague physical explanations for such a map and references to partial answers would also be greatly appreciated.

-