On Calabi--Yau fractional complete intersections

TL;DR

This paper explores mirror symmetry for singular Calabi--Yau manifolds that are double covers of toric varieties, introducing fractional analogues of period integrals to solve associated Riemann--Hilbert problems and advance understanding of their mirror symmetry.

Contribution

It introduces fractional period integrals for singular Calabi--Yau double covers and applies them to resolve Riemann--Hilbert problems, deepening the understanding of mirror symmetry in this context.

Findings

Fractional period integrals effectively describe mirror symmetry for these manifolds.

Riemann--Hilbert problems can be solved using the new fractional structures.

Results provide definitive answers about mirror symmetry for this class of Calabi--Yau manifolds.

Abstract

In this article, we study mirror symmetry for pairs of singular Calabi--Yau manifolds which are double covers of toric manifolds. Their period integrals can be seen as certain `fractional' analogues of those of ordinary complete intersections. This new structure can then be used to solve their Riemann--Hilbert problems. The latter can then be used to answer definitively questions about mirror symmetry for this class of Calabi--Yau manifolds.