Berglund-H\"ubsch Transpose Rule and Sasakian Geometry

TL;DR

This paper connects BHK mirror symmetry with Sasakian geometry, showing how invertible polynomials define Sasaki manifolds with positive Ricci curvature, including Sasaki-Einstein examples, from Calabi-Yau orbifolds.

Contribution

It introduces a novel application of the Berglund-H"ubsch transpose rule to construct Sasaki manifolds with positive Ricci curvature from Calabi-Yau orbifolds.

Findings

Constructs four Sasaki manifolds from each Calabi-Yau orbifold.

Identifies two Sasaki-Einstein manifolds among the constructed examples.

Demonstrates existence of seven-dimensional Sasakian manifolds with positive Ricci curvature from K3 orbifolds.

Abstract

We apply the Berglund-H\"ubsch transpose rule from BHK mirror symmetry to show that to an $n-1$-dimensional Calabi-Yau orbifold in weighted projective space defined by an invertible polynomial, we can associate four (possibly) distinct Sasaki manifolds of dimension $2n+1$ which are $n-1$-connected and admit a metric of positive Ricci curvature. We apply this theorem to show that for a given K3 orbifold, there exists four seven dimensional Sasakian manifolds of positive Ricci curvature, two of which are actually Sasaki-Einstein.