Calabi-Yau Three-folds: Poincare Polynomials and Fractals

TL;DR

This paper investigates the Poincare polynomials of Calabi-Yau three-folds, revealing fractal patterns in their roots related to symmetries and mirror phenomena, thus extending understanding of their geometric properties.

Contribution

It introduces a novel analysis of Poincare polynomials as Littlewood-type constrained polynomials, uncovering fractal behaviors linked to Calabi-Yau symmetries and mirror symmetry.

Findings

Fractal patterns observed in polynomial roots

Connections between roots and geometric symmetries

Insights into distribution of Euler characteristics and Hodge numbers

Abstract

We study the Poincare polynomials of all known Calabi-Yau three-folds as constrained polynomials of Littlewood type, thus generalising the well-known investigation into the distribution of the Euler characteristic and Hodge numbers. We find interesting fractal behaviour in the roots of these polynomials in relation to the existence of isometries, distribution versus typicality, and mirror symmetry.