G -flux, M5 instantons, and U ( 1 ) symmetries in F -theory

Local aspects of singular $F$-theory compactifications for supersymmetric grand unified theory model building are fairly well understood in terms of Higgs bundles and their spectral data. Several global issues remain, however, including a description of $G$-fluxes, which are key to constructing chiral matter and stabilizing moduli, and the global realization of $U(1)$ symmetries that can forbid phenomenologically unfavorable couplings. In this paper, we sharpen our earlier proposal for describing $G$-fluxes and $U(1)$s through a distinguished ``Tate'' divisor. As an application, we give a general discussion of M5-instanton contributions in the presence of $G$-flux and exemplify this in a concrete example, where we comment on the ability of instanton-induced superpotential couplings to stabilize K\"ahler moduli.