Reflexive Numbers and Berger Graphs from Calabi-Yau Spaces

TL;DR

This paper explores the mathematical structures of Calabi-Yau spaces using reflexive weight vectors, introducing recursive construction methods, and linking these to Berger graphs and potential new algebraic frameworks.

Contribution

It presents a recursive approach to construct reflexive numbers, relates them to Berger graphs, and discusses their classification within Calabi-Yau spaces using quantum field theory concepts.

Findings

Recursive construction of reflexive numbers from weight vectors

Connection between reflexive numbers and Berger graphs

Potential new algebraic structures from Berger matrices

Abstract

We review the Batyrev approach to Calabi-Yau spaces based on reflexive weight vectors. The Universal CY algebra gives a possibility to construct the corresponding reflexive numbers in a recursive way. A physical interpretation of the Batyrev expression for the Calabi-Yau manifolds is presented. Important classes of these manifolds are related to the simple-laced and quasi-simple-laced numbers. We discuss the classification and recurrence relations for them in the framework of quantum field theory methods. A relation between the reflexive numbers and the so-called Berger graphs is studied. In this correspondence the role played by the generalized Coxeter labels is highlighted. Sets of positive roots are investigated in order to connect them to possible new algebraic structures stemming from the Berger matrices.