2004/08/03 by John Ellis, J. Ellis, E. Torrente-Lujan +6
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/0408027
arxiv created 2004/08/03 · openalex publication_date 2004/08/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 CY3 reflexive polyhedra. The objective is to describe the ``simply laced'' cases, those graphs obtained from three dimensional spaces with K3 fibers which lead to symmetric matrices. We study both the affine and, derived from them, non-affine cases. We present root and weight structurea for them. We study in particular those graphs leading to generalizations of the exceptional simply laced cases E6,7,8 and E6,7,8(1). We show how these integral matrices can be assigned: they 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 present in Kac-Moody Dynkin diagrams. We conjecture that these generalized simply laced graphs and associated link matrices may characterize generalizations of Cartan-Lie and affine Kac-Moody algebras.