Minuscule Schubert Varieties and Mirror Symmetry

TL;DR

This paper explores mirror symmetry for Calabi-Yau 3-folds within minuscule Schubert varieties, identifying new examples and calculating their invariants through degenerations to Hibi toric varieties.

Contribution

It introduces a new smooth Calabi-Yau 3-fold in a Schubert variety of the Cayley plane and analyzes its topological invariants and mirror symmetry properties.

Findings

List of all possible Calabi-Yau 3-folds in this setting.

Discovery of a new Calabi-Yau 3-fold with Picard number one.

Calculation of topological invariants and BPS numbers.

Abstract

We consider smooth complete intersection Calabi-Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi-Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi-Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fourier-Mukai partner.