From Local to Global in F-Theory Model Building

TL;DR

This work argues that local F-theory model-building is globally constrained: the D7-brane configuration cannot be treated independently from the rest of the compactification, as global tadpole and cohomology conditions bind branes together and introduce unavoidable matter curves and intrinsic singularities. By examining the discriminant locus and its codimension-1 to -3 structures, the authors illustrate how global geometry forces additional intersections and singularities (such as cusp curves and higher double-point curves) that shape the low-energy spectrum and Yukawa couplings. The SU(5) GUT example demonstrates how matter curves (h and P) arise from global discriminant data, how separation cases may or may not remove certain curves, and how Euler characteristics and tadpole counts relate to heterotic duals. Collectively, the paper emphasizes the necessity of global consistency in predicting phenomenology and in computing topological invariants like e(X) for resolved models.

Abstract

When locally engineering F-theory models some D7-branes for the gauge group factors are specified and matter is localized on the intersection curves of the compact parts of the world-volumes. In this note we discuss to what extent one can draw conclusions about F-theory models by just restricting the attention locally to a particular seven-brane. Globally the possible D7-branes are not independent from each other and the (compact part of the) D7-brane can have unavoidable intrinsic singularities. Many special intersecting loci which were not chosen by hand occur inevitably, notably codimension three loci which are not intersections of matter curves. We describe these complications specifically in a global SU(5) model and also their impact on the tadpole cancellation condition.