Elliptic genera of toric varieties and applications to mirror symmetry

Abstract

The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities.

Vertex Algebras and Algebraic Curves

“…Recall from §6.1 that any smooth scheme S of dimension N carries a canonical (W N , Aut O N )-structure S. Applying the construction of §6.1, we attach to Ω • N a D-module on S. The sheaf of horizontal sections of this D-module is a sheaf of vertex superalgebras (with a structure of complex) on S. This is the chiral deRham complex of S, introduced in [MSV]. For its applications to mirror symmetry and elliptic cohomology, see [BoL,Bo1]. Now consider a purely bosonic analogue of this construction (i.e., replace Ω N with H N ).…”

Higher Todd Classes and Holomorphic Group Actions

“…Presumably there is some simultaneous generalization that applies to higher elliptic genera. Some useful references that discuss elliptic genera in the context of birational geometry are [47], [9], [10], [11], [13], [45]. (2) Are there examples that show that our vanishing theorem for higher Todd genera are false for finite π, if one does not rationalize?…”

Chiral Algebras of (0, 2) Models: Beyond Perturbation Theory

“…The theory of CDOs had first been introduced and developed in a series of seminal papers by Malikov et al [2,3]. It has since found interesting applications in geometry and representation theory, such as mirror symmetry [4] and the study of elliptic genera [5,6,7], just to name a few. The mathematical relevance of twisted (0, 2) sigma models is thus clear in this respect.…”