Universal Calabi-Yau Algebra: Towards an Unification of Complex Geometry

TL;DR

This paper introduces a universal algebraic framework for constructing and classifying Calabi-Yau spaces across all dimensions, unifying complex geometry concepts and providing explicit recurrence relations and visualizations.

Contribution

It develops a universal normal algebra that extends reflexive weight vectors, incorporates dual Diophantine decompositions, and relates Calabi-Yau spaces across dimensions, advancing the classification and understanding of these geometries.

Findings

Provides explicit recurrence formulas for Calabi-Yau counts in various dimensions.

Establishes algebraic relations between Calabi-Yau chains in different dimensions.

Offers visualizations of singularities linked to Lie algebras.

Abstract

We present a universal normal algebra suitable for constructing and classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a `dual' construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi-Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi-Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan-Lie algebras. This Universal Calabi-Yau Algebra is a powerful tool for decyphering the Calabi-Yau genome in all dimensions.