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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.

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The short answer is no, one does not expect such isomorphism. What one does expect is some kind of "flat" family of super-vertex algebras that interpolates from one to the other. The base of the family should be the Kahler parameters of the model, i.e. complex parameter of the mirror, if such exists. In the case of CY hypersurfaces in toric varieties, this seems plausible, up to some technicalities.

If one passes to the chiral rings, then this pretty much becomes the Ruan's conjecture about small quantum cohomology of different crepant resolutions.

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Thank you. This is a great answer. –  S. Carnahan Aug 25 '13 at 2:37
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