2000/05/19 by Antonella Grassi, Grassi, Antonella, David R. Morrison +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.math/0005196
openalex publication_date 2000/05/19 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
To every elliptic Calabi-Yau threefold with a section X there can be\nassociated a Lie group G and a representation \ρ of that group. The group\nis determined from the Weierstrass model, which has singularities that are\ngenerically rational double points; these double points lead to local factors\nof G which are either the corresponding A-D-E groups or some associated\nnon-simply laced groups.\n The representation \ρ is a sum of representations coming from the local\nfactors of G, and of other representations which can be associated to the\npoints at which the singularities are worse than generic.\n This construction first arose in physics, and the requirement of anomaly\ncancellation in the associated physical theory makes some surprising\npredictions about the connection between X and \ρ. In particular, an\nexplicit formula (in terms of \ρ) for the Euler characteristic of X is\npredicted. We give a purely mathematical proof of that formula in this paper,\nintroducing along the way a new invariant of elliptic Calabi-Yau threefolds. We\nalso verify the other geometric predictions which are consequences of anomaly\ncancellation, under some (mild) hypotheses about the types of singularities\nwhich occur.\n As a byproduct we also discover a novel relation between the Coxeter number\nand the rank in the case of the simply laced groups in the ``exceptional\nseries'' studied by Deligne.\n