We discuss the structure of charged matter couplings in 4-dimensional Ftheory compactifications. Charged matter is known to arise from M2-branes wrapping fibral curves on an elliptic or genus-one fibration Y . If a set of fibral curves satisfies a homological relation in the fibre homology, a coupling involving the states can arise without exponential volume suppression due to a splitting and joining of the M2-branes. If the fibral curves only sum to zero in the integral homology of the full

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... ration, no such coupling is possible. In this case an M2-instanton wrapping a 3-chain bounded by the fibral matter curves can induce a D-term which is volume suppressed. We elucidate the consequences of this pattern for the appearance of massive U(1) symmetries in F-theory and analyse the structure of discrete selection rules in the coupling sector. The weakly coupled analogue of said M2-instantons is worked out to be given by D1-F1 instantons. The generation of an exponentially suppressed F-term requires the formation of half-BPS bound states of M2 and M5-instantons. This effect and its description in terms of fluxed M5-instantons is discussed in a companion paper. expected from the perspective of M-theory on G 2 -manifolds [10, 11, 14, 24] . However, since in F-theory an M2-instanton along a 3-chain on a Calabi-Yau 4-fold is non-BPSunlike the situation for G 2 -manifolds -M2-instanton induced couplings can arise at the level of D-terms in the 4-dimensional F-theory effective action, but not as F-term couplings. Consequently, the superpotential and other F-terms are in fact protected against such non-perturbative corrections generated by M2-instantons. On the other hand, the instantonic M2-branes along the 3-chains Γ can form supersymmetric bound states with M5-instantons, described by an M5-instanton supporting a non-trivial world-volume flux. Such fluxed M5-instanton can contribute to the F-term sector of the effective action, producing couplings exponentially suppressed by e −2πvol 6 (M5) . We will discuss such corrections, the associated selection rules as well as their weakly coupled description in terms of fluxed D3-brane instantons in a companion paper [25] . The viewpoint developed in the present work allows us, in particular, to investigate the selection rules associated with massive U(1)s directly in F/M-theory. As in perturbative compactifications, one obvious way to obtain an effectively massive U(1) symmetry is by means of a flux-induced Stückelberg mechanism. More challenging from a conceptual point of view is the realisation of a massive U(1) symmetry even in absence of gauge fluxes. The perturbative analogue of this mechanism is sometimes called 'geometric Stückelberg mechanism' and involves axionic fields from reduction of the orientifold odd 2-forms C 2 . This can be mimicked in F/M-theory by means of a set of non-harmonic 2-and 3-forms in terms of which the M-theory 3-form C 3 is expanded [26, 27] . More precisely a ktorsional [11, 13] 3-form α 3 with k α 3 = dw 2 gives rise to a vector field A and a Stückelberg axion c by expanding C 3 = A ∧ w 2 + c α 3 . The relation between α 3 and w 2 ensures that A is massive by 'eating' the axionic field. This structure has been explicitly confirmed in [28] for the simplest case of a Z 2 symmetry in a six-dimensional F-theory compactification. 1 Another, in fact earlier, example in this spirit is the realisation of a 'Z 1 ' symmetry presented in [32] . It is worth noting that in all these setups the massive U(1) symmetry can be interpreted as a Kaluza-Klein mode of the 11-dimensional field C 3 . Indeed the geometric mass scale for the modes A and c is generically the compactification scale. In this sense the quantum gravity folk theorem about the inevitable gauging of a global symmetry in the ultra-violet is explicitly respected. In the explicit 4-dimensional examples with gauge group G×Z k for various non-abelian gauge groups G in [31, 33, 34] all Z k compatible couplings arise already at the perturbative level. More generally, whenever the Z k symmetry can be unhiggsed to a massless U(1) via a localised Higgs field of charge k as in [30] , all Z k compatible couplings are realised perturbatively provided in the unhiggsed model all Yukawa couplings involving the Higgs field allowed by gauge invariance appear as perturbative couplings. Indeed, for generic choice of base spaces the fibrations [31, 33, 34] are of this type [28, 30, 31, [33] [34] [35] [36] . Nonetheless for specific patterns of intersection numbers in the base, it can happen that the full set of operators allowed by the Z k symmetry arises only once non-perturbative coup-1 The geometry associated with a Z k symmetry is much richer, though, and involves a set of k − 1 additional genus-one fibrations classified by the Tate-Shafarevich group [29] of the original elliptic fibration, whose associated effective action becomes equivalent only in the F-theory limit [28, 30, 31] .