Universal Calabi-Yau Algebra: Classification and Enumeration of Fibrations

TL;DR

This paper introduces a universal algebraic framework for classifying and enumerating fibrations of Calabi-Yau spaces across various dimensions, leveraging algebraic and Diophantine methods.

Contribution

It develops a universal Calabi-Yau algebra that extends reflexive weight vectors and provides recurrence formulas for counting fibrations in arbitrary dimensions.

Findings

Recurrence formulas for fibrations in Calabi-Yau spaces

Explicit enumeration for Weierstrass and K3 examples

Extension of algebraic methods to higher-dimensional Calabi-Yau spaces

Abstract

We apply a universal normal Calabi-Yau algebra to the construction and classification of compact complex $n$-dimensional spaces with SU(n) holonomy and their fibrations. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions and a `dual' construction based on the Diophantine decomposition of invariant monomials. The latter provides recurrence formulae for the numbers of fibrations of Calabi-Yau spaces in arbitrary dimensions, which we exhibit explicitly for some Weierstrass and K3 examples.