2017/04/26 by Mboyo Esole, Esole, Mboyo, Patrick Jefferson +3 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1704.08251
We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate's algorithm. We call such geometries F4-models, as the dual graph of their special fiber is the twisted affine Dynkin diagram \widetildeF4t. These geometries are used in string theory to model gauge theories with the exceptional Lie group F4 on a smooth divisor S of the base. Starting with a singular Weierstrass model of an F4-model, we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over S. These are ℙ1-bundles over S or double covers of ℙ1-bundles over S. We compute basic topological invariants such as the double and triple intersection numbers of the fibral divisors and the Euler characteristic of the F4-model. In the case of Calabi-Yau threefolds, we compute the linear form induced by the second Chern class and the Hodge numbers. We also explore the meaning of these geometries for the physics of gauge theories in five and six-dimensional minimal supergravity theories with eight supercharges. We also introduce the notion of "frozen representations" and explore the role of the Stein factorization in the study of fibral divisors of elliptic fibrations.