Representations on the cohomology of hypersurfaces and mirror symmetry

TL;DR

This paper investigates the group actions on the cohomology of hypersurfaces in toric varieties, proposes a new mirror symmetry conjecture in representation-theoretic terms, and proves it in specific cases, revealing new mirror pairs of Calabi-Yau orbifolds.

Contribution

It provides an explicit description of group representations on hypersurface cohomology and introduces a conjectural, representation-theoretic form of mirror symmetry, proven for certain smooth or low-dimensional cases.

Findings

Explicit description of group representations on hypersurface cohomology.

Formulation of a conjectural mirror symmetry in representation theory.

Existence of mirror pairs of Calabi-Yau orbifolds with mirror Hodge diamonds.

Abstract

We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst quotient singularities. As an application, we conjecture a representation-theoretic version of Batyrev and Borisov's mirror symmetry between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces are both smooth or have dimension at most 3. An interesting consequence is the existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror, with respect to the usual Hodge structure on singular cohomology.