Formula omitted]-flux, chiral matter and singularity resolution in F-theory compactifications

Abstract

We construct a set of chirality inducing G 4 -fluxes in global F-theory compactifications on Calabi–Yau four-folds. Special emphasis is put on models with gauge group SU ( 5 ) × U ( 1 ) X relevant in the context of F-theory GUT model building, which are described in terms of a U ( 1 ) -restricted Tate model. A G 4 -flux arises in a manner completely analogous to the U ( 1 ) X gauge potential. We describe in detail the resolution by blow-up of the various singularities responsible for the U ( 1 ) X factor and the standard SU ( 5 ) gauge group and match the result with techniques applied in the context of toric geometry. This provides an explicit identification of the structure of the resolved fibre over the matter curves and over the enhancement points relevant for Yukawa couplings. We compute the flux-induced chiral index both of SU ( 5 ) charged matter and of SU ( 5 ) singlets charged only under U ( 1 ) X localised on curves which are not contained in the SU ( 5 ) locus. We furthermore discuss global consistency conditions such as D3-tadpole cancellation, D-term supersymmetry and Freed–Witten quantisation. The U ( 1 ) X gauge flux is a global extension of a class of split spectral cover bundles. It constitutes an essential ingredient in the construction of globally defined F-theory compactifications with chiral matter. We exemplify this in a three-generation SU ( 5 ) × U ( 1 ) X model whose flux satisfies all of the above global consistency conditions. We also extend our results to chiral fluxes in models without U ( 1 ) restriction.