Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?

TL;DR

This paper explores new algebraic structures derived from reflexive polyhedra in Calabi-Yau threefolds, revealing generalized graphs that extend Dynkin diagrams and may lead to novel symmetries in physics.

Contribution

It introduces a family of integral matrices and graphs from Calabi-Yau reflexive polyhedra, generalizing Cartan matrices and Dynkin diagrams, suggesting new algebraic frameworks.

Findings

Identified graphs with affine structure from Calabi-Yau polyhedra

Proposed root structures for specific cases

Conjectured these graphs generalize known algebraic systems

Abstract

The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this work we continue to study the structure of graphs obtained from $CY_3$ reflexive polyhedra. We show how some particularly defined integral matrices can be assigned to these diagrams. This family of matrices and its associated graphs may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep however the affine structure, as it was in Kac-Moody Dynkin diagrams. We presented a possible root structure for some simple cases. We conjecture that these generalized graphs and associated link matrices may characterize generalizations of these algebras.