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The Global Gauge Group Structure of F-theory Compactification with U(1)s

We show that F-theory compactifications with abelian gauge factors generally exhibit a non-trivial global gauge group structure. The geometric origin of this structure lies with the Shioda map of the Mordell--Weil generators. This results in constraints on the U(1) charges of non-abelian matter consistent with observations made throughout the literature. In particular, we find that F-theory models featuring the Standard Model algebra actually realise the precise gauge group [SU(3)xSU(2)xU(1)]/Z6. Furthermore, we explore the relationship between the gauge group structure and geometric (un-)higgsing. In an explicit class of models, we show that, depending on the global group structure, an SU(2)xU(1) gauge theory can either unhiggs into an SU(2)xSU(2) or an SU(3)xSU(2) theory. We also study implications of the charge constraints as a criterion for the F-theory 'swampland'.

An approach to construct models with both abelian and non-abelian gauge symmetries is to first pick a global fibration known to have sections, and then using the aforementioned techniques to introduce suitable singularities in codimension one. This approach has been used throughout the literature to construct phenomenologically appealing models (in addition to the previous references, see also [28][29][30][31][32][33]). However, as it is so often the case in geometry, the Mordell-Weil group of sections and codimension one singularities are not completely independent of each other. Indeed, it turns out that the existence of torsional sections 1 not only enforces specific codimension one singularities corresponding to a semi-simple Lie algebra g, it also restricts the possible matter representations [34,35] (see also [26,36]). An equivalent formulation is to say that the gauge group is not G-the simply connected Lie group associated to g-but rather G/Z, where Z is a subgroup of the centre of G. It is important to note that in this case, only representations transforming trivially under Z are allowed. Because field theoretically, only non-local operators such as line operators are sensitive to this quotient structure [37], one often refers to G/Z as the structure of the global gauge group, in order to distinguish it from the gauge algebra that is seen by local operators. If Z = {1}, we will refer to the global group structure to be 'non-trivial'.
The analysis of [35] produced only models that have a non-trivial global structure in the non-abelian sector of the gauge group. However, the central subgroup Z can also overlap with a subgroup of the abelian sector. The most prominent example of such a non-trivial gauge group structure is in fact presumed to be the Standard Model of particle physics.
It may therefore seem surprising, that F-theory compactifications realising the Standard Model gauge algebra without an explicit GUT structure [26,32,41,42] actually reproduces exactly the same representations which are invariant under the Z 6 centre of SU (3) × SU (2) × U (1) Y . Furthermore, since there is no evidence for these models to have torsional sections, one might wonder if this agreement is purely coincidental, or if there is some further hidden structure in the geometry giving rise to the non-trivial global gauge group.
In this paper, we will show that the latter is the case. In fact, we will present an argumentvery similar to that for torsional sections in [35]-showing that generically, F-theory compactifications with abelian gauge factors exhibit a non-trivial gauge group structure. We will demonstrate in section 2 that the Shioda map [43][44][45] of sections generating the Mordell-Weil group relates the u(1) charges of matter non-trivially to their representations under the nonabelian part of the gauge algebra. This relationship, which leads to a non-trivial centre of the universal covering of the actual gauge group, can be equivalently understood as a refined charge quantisation condition, which has been previously observed throughout the literature.
Examples hereof will be presented in section 3, including those leading to F-theory 'Standard Models'. In section 4, we address the issue if and how, in F-theory, such non-trivial gauge group structures can arise from the breaking, a.k.a. higgsing, of a larger non-abelian gauge group, similar to breaking SU (5) to the Standard Model. Because of the intricate geometric description of higgsing, we will content ourselves with the discussion of a concrete class of models having su(2) ⊕ u(1) gauge algebra. For these, we demonstrate explicitly, how different gauge group structures arise from different breaking patterns that are captured beautifully in the geometry. An interesting implication of our findings is presented in section 5, where we argue that the geometric properties leading to the non-trivial global gauge group structure can also be interpreted as a criterion for effective field theories to be in the F-theory 'swampland'.
This swampland criterion is formulated in terms of a charge constraint on matter representations of the non-abelian gauge algebra. In section 6, conclusions and outlook for further investigations are presented.

Shioda map and the centre of gauge groups
Because our main argument is based on the Shioda map, we will first present a brief review of its prominent role in F-theory, which will also help to set up the notation. Let π : Y n+1 → B n be a smooth, elliptically fibred Calabi-Yau space of complex dimension n + 1, with (singular) Kodaira fibres over a codimension one locus {θ = 0} ≡ {θ} ⊂ B n , and Mordell-Weil rank m.
In addition to the zero section σ 0 , the Mordell-Weil (MW) group has independent sections σ k , 1 ≤ k ≤ m, which generate the free part (we will call σ k a 'free' generator of the MWgroup). In the following, we will denote the divisor classes of the (zero) sections by (Z) S k . Furthermore, we have the exceptional divisors E i = (P 1 i → {θ}), 1 ≤ i ≤ r, which are P 1fibrations over {θ}. Note that by definition, the zero section does not intersect the exceptional divisors, Z · P 1 i = 0. 3 In this set-up, the Shioda-Tate-Wazir theorem [45] implies that the Néron-Severi (NS) group (i.e., divisors modulo algebraic equivalence) of Y n+1 ≡ Y 4 satisfies (2.1) The subspace T is spanned by the zero section Z, the exceptional divisors E i , and any divisor D B pulled back from the base B with π. Finally, let us introduce the height pairing , : For n ≥ 2, we know [2,3,46] that F-theory compactified on Y n+1 gives rise to a gauge theory in d = 10 − 2n dimensions with gauge algebra u(1) ⊕ r ⊕ g and charged matter arising from singular fibres over codimension two loci of B n . The semi-simple (non-abelian) algebra g is determined by the singularity types over {θ}. In particular, the exceptional divisors E i , i = 1, ..., rank(g) = r, being dual to harmonic (1, 1)-forms ω i , give rise-via the standard Cartan subalgebra h of g. The W-bosons, i.e., states forming the roots of g, originate from M2-branes wrapping the P 1 i fibres of E i . On the other hand, the u(1) gauge fields arise from expanding C 3 along the (1, 1)-forms ω u(1) k which are Poincaré-dual (PD) to divisors ϕ(σ k ) associated with the free generators σ k of the Mordell-Weil group. This so-called Shioda map [43][44][45] is a homomorphism ϕ : MW → NS(Y ) ⊗ Q with ker(ϕ) = MW(Y ) torsion , that satisfies ϕ(σ), D = 0 for any D ∈ T . These conditions can be recast in terms of intersection numbers: The first two conditions ensure that the intersection product of ϕ(σ) with the generic fibre f and any curve C B of the base (lifted by the zero section) vanishes. Physically, this is related to the requirement that the u(1) gauge field lifts properly from d − 1 to d dimensions in the M-/F-theory duality. The last condition is nothing other than the statement that the gauge bosons of g are uncharged under the u(1).
These conditions determine the Shioda map up to an overall scaling: Since ϕ relates a section σ to a divisor class, we expect that ϕ(σ) ∼ S+(correction terms), where S is the class of σ itself. To satisfy the first condition (2.2), the correction terms must contain −Z. 5 The second condition (2.3) introduces a term of the form π −1 (D B ), where the exact divisor D B ∈ NS(B) depends on the concrete model. We will neglect the discussion of this term, since its intersection number with any fibral curve Γ is zero and hence does not contribute to the u(1) charge of localised matter. Finally, the last condition (2.4) gives rise to a correction term of the form i l i E i , where the coefficients l i ∈ Q will be discussed in more detail momentarily.
Accordingly, the u(1) k charge of matter states which arise as M2-branes wrapping fibral curves Γ-given by the intersection number q k (Γ) = ϕ(σ k ) · Γ-are only determined up to an overall scaling, which does not have a direct physical meaning. Therefore, we often find in the literature that the scaling is chosen such that all charges are integral. While there is in principle nothing wrong with such a rescaling, the factor can be misleading when we analyse the global gauge group structure. As we will see, by setting λ = 1, we can read off the global gauge group directly from the coefficients l i . Field theoretically, this points towards a 'preferred' u(1) charge normalisation, in which case the u(1) charge lattice for each representation R of g has lattice spacing 1. In this formulation, we can also interpret the non-trivial global gauge group as a relative shift by a fractional number of the charge lattice for different g-representations.
Of course, these restrictions on the u(1) charges have been previously observed, e.g., they are quantified in the literature [12][13][14]47] for g = su(5), and derived more generally from the consistency of large gauge transformation in the circle reduction of F-theory [48]. The novelty of this paper is the observation that these charge restrictions are explicitly tied to the global gauge group structure for any F-theory compactification with non-trivial Mordell-Weil group.

Fractional U (1) charges in F-theory
In the following, we will focus the discussion on a rank one Mordell-Weil group with a single free generator σ. The generalisation to higher Mordell-Weil rank (and also the inclusion of 5 We could in fact shift by any other section S k instead of Z to satisfy (2.2); however, because ϕ is a homomorphism, we need ϕ(σ0) ! = 0. Therefore, the shift has to be the divisor class Z of the zero section. torsion) is straightforward and will be presented in section 3.2 with explicit examples. For the purpose of these notes, let us fix the factor λ in the Shioda map to 1: Since the following discussion revolves around the fractional coefficients l i , let us recall that they arise from requiring the intersection numbers of ϕ(σ) with the fibre P 1 i s of the exceptional divisors E i to vanish, see (2.4). This imposes Here, C −1 denotes the inverse of C ij = −E i · P 1 j , which is the Cartan matrix of the algebra g. 6 In general, the coefficients l i are fractional numbers that in particular depend on the intersection properties between the divisor S − Z and the fibres P 1 i of the exceptional divisors E i . However, there is always a positive integer κ such that κ l i ∈ Z for all i. For example, we know that the entries of the inverse Cartan matrix of su(n a ) are z/n a with z ∈ Z. Hence, if g = a su(n a ), then the smallest such κ is the least common multiple of all n a . Note that this immediately implies charge quantisation (i.e., we really have a compact abelian gauge factor): Since κ ϕ(σ) is a manifestly integer class, its intersection number with fibral curves is always integral. So the u(1) charges (measured with respect to ϕ(σ)) of all states realised geometrically (i.e., as M2-branes on fibral curves) lie in a lattice of spacing 1/κ.
In fact, the Shioda map (2.6) makes an even more refined statement. Because S and Z are divisor classes of sections, they are manifestly integer, i.e., their intersection product with fibral curves Γ must be integer as well. But then, the charge of the matter state w associated with Γ must satisfy where, in the second line, we have used the standard result that the Dynkin labels of a weight w associated with a fibral curve Γ is given by w i = E i · Γ ∈ Z. Curves Γ w,v localised at the same codim 2 locus, but realising different states w, v of the same g-representation, differ by an integer linear combination µ k P 1 k s, since these P 1 s correspond to the simple roots of the algebra g. 7 For these, we have Thus, we can associate to each g-representation R g a κ-fractional number between 0 and 1, which is independent of the choice of weight w ∈ R g . For a representation (q R , R g ) of u(1)⊕g, this allows to rewrite (2.8) as a condition for the u(1) charge, So for any matter with g-representation R g , the possible u(1) charges arrange in a lattice of integer spacing. However, for different representations, the lattices do not in general align. In fact, from what we have seen above, they can differ by multiples of 1/κ.
The geometric origin of (2.11) lies in the intersection properties of divisors and codimension one singular fibres over {θ}. Indeed, the non-integrality of the coefficients l i (2.7), which leads to the non-trivial integrality condition (2.8), stems from the zero section Z and the generating section S intersecting P 1 fibres of possibly different exceptional divisors E i . This so-called split [13,30] of the fibre structure over {θ} by the section can be easily determined in concrete models, e.g., directly from the polytope in toric constructions [12]. The analysis carried out to obtain (2.11) is essentially equivalent to the study of the fibre splitting patterns in the presence of sections and the allowed u(1) charges, e.g., as in [13,47] for classifying all possible u(1) charges of su(5) matter. Here, we have rephrased it in a way that allows for a more straightforward connection to the global structure of the gauge group. An alternative way of deriving (2.8) is to consider circle compactifications of F-theory and require consistency of the large gauge transformations along the circle [48].
Note that the above discussion, in particular the derivation of (2.11) for matter localised in codimension two, is based purely on codimension one properties. Hence, all arguments and conclusions hold for F-theory compactifications to six, four and two dimensions.

Non-trivial central element from the Shioda map
To see how the above observation relates the Lie algebra u(1) ⊕ g to the global gauge group G glob , first note that G glob has U (1) ×G as a cover, whereG is the simply connected Lie group associated to g. We can now define an element of the centre which has to act trivially on all geometrically realised weights. For that, we first define the element Ξ := q − i l i E i of the Cartan subalgebra u(1)⊕h ⊂ u(1)⊕g, where q is the generator of u(1). Its action on the representation space of an irreducible representation R = (q R , R g ) of u(1) ⊕ g is then simply defined through its action on the weights w ∈ R. Explicitly, denoting by w i the Dynkin labels of w under g, we have where 1 is the identify matrix in the representation R g . By exponentiating this equation, we obtain the action of a group element in U (1) ×G on weights of R, (2.13) Evidently, C-being proportional to the identify element ofG-commutes with every element, i.e., C is in the centre U (1) × Z(G). Let us now restrict the action to representations realised in the F-theory compactification, i.e., weights w that arise from fibral curves Γ. Because the tensor product 8 is bilinear, the expression can be also written as i.e., C acts trivially on weights w arising from fibral curves Γ! But on the other hand, we also know from the previous discussion that there is a positive integer κ such that κ l i ∈ Z for all i, see paragraph after (2.7). 9 Going back to (2.13), for which we introduce the short-hand we see that for weights in a any representation ofG, we have In other words, ξ w × 1 generates a Z κ subgroup of the centre Z(G). But because we have shown that all states in geometrically realised representations must be acted on trivially by Q w ⊗ (ξ w × 1), we conclude that the global gauge group structure should be As mentioned before, the whole discussion applies to F-theory compactified to six, four and two dimensions.
Note that, strictly speaking, the second equality in (2.17) is merely a definition of the notation [U (1) ×G]/Z κ . Indeed, from a purely representation theoretic point of view, we do not know that (Q w ) κ = 1 for every charged state (the charges could be quantised finer than 1/κ). However, we have seen above that the geometry of the F-theory model actually dictates the charges to be quantised in units of 1/κ, i.e., C κ = id. In our discussion, both charge quantisation and the central element C follow from the same observation, namely the integrality condition (2.8). Hence, the notation (U (1) ×G)/Z κ can also be seen as encoding the u(1) charge quanta of an F-theory compactification.
The reader might recognise the above argument, leading up to (2.13), from [35], which related the presence of κ-torsional sections to the Z κ -center of purely non-abelian groups (i.e., no U (1) factor in the cover of G glob ). Indeed, there one arrives by the same logic at (2.15) with Q = 1. In that case the conclusion is simply that ξ w × 1, which generates a subgroup of the centre Z(G), must act trivially.
Finally, we note that even though the above discussion has been limited to a single u(1) factor, the analysis readily extends to multiple sections σ k (free or torsional). Because the Shioda map (2.6) of any Mordell-Weil generator σ (free or torsional) takes the form S − Z + (non-sectional divisors), one quickly realises that each Mordell-Weil generator σ k gives rise to an independent trivially acting central element C k . Thus, the global gauge group structure is a quotient by a product of Z κ k 's. We will come back to explicit examples hereof in section 3.2.

Preferred charge normalisation in F-theory
Let us revisit the possible rescaling (2.5) of the Shioda map and the resulting normalisation of u(1) charges in F-theory. In field theory, the overall scaling of the u(1) charge is unphysical, and can be chosen to our convenience. Likewise, as mentioned before, the Shioda map is only defined up to a constant rescaling. However, in F-theory we have a preferred normalisation provided by the integer divisor classes of sections.
Explicitly, given free generator σ, we know that its divisor class S must be integer and intersecting the generic fibre once. Any rescaling of S cannot preserve these properties. Furthermore, if we rescale ϕ(σ) = S −Z +l i E i by an integer κ, then, depending on the non-abelian gauge algebra and the fibre split structure, κ l i could be integer, which makes the u(1) generator potentially 'blind' to the central element C. Indeed, if we were to repeat the analysis leading to (2.8) with the divisor κ ϕ(σ), then the equivalent expression becomes which does not provide any non-trivial relation between the u(1) charge and the weight vectors w. On the other hand, if we rescale the u(1) charge by a fractional number λ, then it is no longer guaranteed that λ (S − Z) · Γ is always an integer.
Therefore, it is only with the normalisation ϕ(σ) = S − Z + ... of the u(1) generator, that we can make the non-trivial relation (2.8) manifest in any F-theory compactification.
Comparing to the field theory perspective, where any rescaling of u(1) charges has no physical meaning, we conclude that the appropriate field theoretic data associated with this preferred normalisation are the global gauge group structure and the charge quantisation of individual g-representations. Equivalently, by first establishing these data, one is then free to choose any normalisation for the u(1) charge in the field theory.

The global gauge group of F-theory models
In this section, we will apply the above analysis to concrete models with u(1)s that have been constructed over the last few years in the literature.

Models with su(5) ⊕ u(1) singularity
Let us begin with arguably one of the most studied F-theory model, namely the so-called U (1)-restricted Tate model [17] given by the hypersurface mod Z. This is of course consistent with the spectrum, which in terms of the normalised u(1) Note that there are also su(5) ⊕ u(1) models with a Z 5 centre that is embedded differently into the U (1), leading to different charge assignments. One such example can be constructed via 'toric tops' [9,51] in a Bl 1 P 112 -fibration [16]. It is labelled 'top 2' in the appendix of [19], which is equivalent to the model 'Q(4, 2, 1, 1, 0, 0, 2)' in [13]. Without going into the details of this model, we note that the sections Z and S intersect in neighbouring nodes of su(5) fibre (i.e., fibre split type (0|1) in the notation of [13]). The Shioda map is then S − Z + 1 5 (1, 2, 3, 4) i E i , which also leads to a Z 5 centre. However, the u(1) charges are constrained to be 1 5 mod Z for 5-matter and 2 5 mod Z for 10-matter. Correspondingly, the spectrum reads Of course there are also models without a non-trivial global gauge group structure. An example is the model labelled 'top 4' in the appendix of [19], or 'Q(3, 2, 2, 2, 0, 0, 1)' in [13].
Here, both sections Z and S intersect the affine node of su(5) (i.e., fibre split type (01)). So the Shioda map is ϕ(σ) = S − Z, without any shifts by Cartan divisors. The central element (2.13) then just imposes that all charges must be integral. Thus, the global gauge group is , which of course is consistent with the spectrum

Higher Mordell-Weil rank
We have mentioned in the previous section that a higher rank m of the Mordell-Weil group implies that there are possibly m independent non-trivial central elements acting trivially on 10 Recall that the zero section Z intersects the affine node of I5 fibre over {θ}. The affine node is separated from the P 1 3 node by the fibre of E4. In the notation of [13], this is a fibre split type (0||1).
representations. We will illustrate this now with concrete examples, in which the Mordell-Weil rank is 2. The simplest fibration that has two independent free sections arise from a generic cubic in an Bl 2 P 2 = dP 2 fibration [19,20]. 11 We will denote the divisor classes of the two sections σ 1,2 generating the Mordell-Weil group by S 1,2 and stick with the above notation of Z being the zero section.
For simplicity, we focus on models with non-abelian gauge algebra su (2), and label the single Cartan divisor by E 1 . All three such models arising from toric tops have been constructed in [32]. Dubbed tops I, II and III, each of them turn out to have a different global gauge group structure, so it is instructive to analyse each individually. In top I, the Shioda map ϕ takes the sections to Therefore, the u(1) charges (q 1 , q 2 ) of su(2) matter must satisfy q 1 − 1 2 w ∈ Z and q 2 ∈ Z, where w is the Dynkin label of su(2) states. Only the first condition leads to a central element acting non-trivially on su(2) states. Clearly, it is an element of order 2, because 2 q 1 − w ∈ Z. We can also translate the second condition into a central element C 2 = e 2πi q 2 ∈ U (1) 2 . However, this element evidently just imposes charge quantisation q 2 ∈ Z. So the global gauge group structure is The non-abelian part of the spectrum arranges consistently into In the top II model, the Shioda map of the sections are Now both u(1) charges must satisfy q i − 1 2 w ∈ Z. Put differently, there are now two central elements of order 2,

6)
11 A more general model with MW rank 2 has been recently constructed in [27]; as shown there, the Bl2P 2fibration arises as a specialisation of this general construction. of the covering groupG = SU (2) × U (1) 1 × U (1) 2 that have to act trivially on all representations. Each therefore generates a separate Z 2 subgroup ofG, leading to the actual gauge group where it needs to be understood that Z (i) 2 lies in the center of SU (2) × U (1) i . Consequently, the su(2) matter are charged as Finally, there is also the top III with Shioda map which clearly leads to trivial central elements. Hence, the gauge group in this case is just The spectrum in this case is 12 Before we move on, let us briefly comment on a peculiar behaviour of the centre when we rotate the u(1)s. Concretely, it was noted in [32] that, if we redefine the u(1) charges (q a , q b ) = (−q 1 , q 2 − q 1 ) in top II, the spectrum is identical to that of top I. In fact, this is a consequence of a toric symmetry relating tops I and II. How is it compatible with the seemingly different gauge group structures (3.4) and (3.7)? To understand this, let us rewrite the central elements (3.6) in terms of the rotated u(1) charges. Explicitly, we have e 2πi q 1 ⊗ 1 = e −2πi qa ⊗ 1 and 1 ⊗ e 2πi q 2 = 1 ⊗ e 2πi (q b −qa) = e −2πi qa ⊗ e 2πi q b . So the central elements are where here we use • to denote the group multiplication in SU (2) × U (1) 2 . Note that we are dealing with central elements, hence they all commute. In the gauge group (3.7) of top II, 12 Note that we have included a completely uncharged doublet here that was previously missed in [32]. In fact, the codimension two locus of I3 fibres corresponding to this matter was noticed. However, the monodromy around a codimension 3 sublocus interchanging two of the fibre components was misinterpreted as projecting out the matter states. But due to the vanishing charges, this doublet is actually a real representation, i.e., the two fibre components are homologically equivalent. Thus the monodromy in higher codimension exchanging them is not surprising and actually expected geometrically. A similar observation holds for singlets charged under a discrete Z2 symmetry [26,52].
both C II 1,2 must act trivially on all states. The above equation implies that this is equivalent to C II 1 andC II 2 acting trivially. But sinceC II 2 lies in U (1) b , we have (3.10) Now the second quotient structure just imposes that the U (1) b charges are integer for all states, which is also implemented in (3.4) by the second quotient. Therefore, we have shown that by rotating the u(1)s in top II, the global gauge group structure (3.7) (including charge quantisation) turns out to be equivalent to that of top I (3.4).

Inclusion of Mordell-Weil torsion
Let us now look at a model with Mordell-Weil group Z⊕Z 2 . This example-studied extensively in [35] (and also appears in a slightly different fashion in [26])-has, in addition to the zero Because of the 2-torsional property of σ r and ϕ being a homomorphism, we know that ϕ(σ r ) = 0. Analogous to the derivation of (2.8) (with q w = 0), it means that 1 2 (w C − w D ), with w C,D being the weights of SU (2) C × SU (2) D irreps, must be integral. So it defines an- which has to act trivially on representations of the F-theory compactification. Therefore, the global gauge group structure is where, in order for the notation to make sense, we need to clarify that Z

F-theory Standard Models
We now come to a class of somewhat more phenomenologically interesting models, namely elliptic fibrations realising the Standard Model gauge algebra g SM = su(3) ⊕ su(2) ⊕ u(1) in F-theory.
The first model, presented in [26] (and labelled there as X F 11 ) has g SM as the full gauge algebra and the exact Standard Model spectrum (at the level of representations). The inclusion of fluxes in [41] resulted in a first globally consistent three-chiral-family Standard Model construction in F-theory. As mentioned in the introduction, the Standard Model spectrum is consistent with a global gauge group structure (SU (3) × SU (2) × U (1))/Z 6 . With the new insights from section 2, we can now explicitly show that in F-theory, we indeed can construct such a global structure. In the X F 11 model, the Shioda map of the free section is where E h i denotes the Cartan generator(s) of the corresponding subalgebra h. 13 If we denote the weight vectors of su(3) resp. su(2) by (w 1 , w 2 ) resp. ω and the u(1) charge by q, then the integrality condition (2.8) for X F 11 reads Because ω, w i ∈ Z, the smallest positive integer κ such that κ q ∈ Z for all possible charges q is κ = 6. Thus, the central element acting on SU (3) × SU (2) × U (1) representations has order 6, so it defines a Z 6 subgroup of the centre, i.e., the global gauge group is . This exchanges the notion of 3 and 3, making the charges identical to that of the Standard Model.
The condition (3.15) implies that fundamental matter charged only under su(3) must have charges 1 3 mod Z, while pure fundamentals of su(2) have q = 1 2 mod Z. Inspecting the highest weight of the bifundamental, ω = 1, w 1 = 1, w 2 = 0, we see that this representation must have charge 1 6 mod Z. Correspondingly, the geometric spectrum, exists an extensive analysis including G 4 -fluxes [42]. In this case, the Shioda map, which for the first section also yields the hypercharge u(1) generator, reads ) .
Geometrically, the representations of the su(3) ⊕ su(2) ⊕ u(1) Y subalgebra agrees with that of the Standard Model (with additional singlets with no hypercharge). However, the u(1) 2 charge discriminates between states that would be otherwise indistinguishable under the Standard Model algebra. Note that, in order to get in touch with the actual Standard Model, one ultimately needs to lift the second u(1) from the massless spectrum, which would require further investigation, see [42].

Relationship to (un-)higgsing
In this section, we want to explore the origin of the non-trivial global gauge group structure in higgsing processes. In F-theory, u(1)s can often be unhiggsed into the Cartan of non-abelian gauge algebras [16,26,27,53,54]. Given that the Cartan charges of non-abelian matter are naturally integrally quantised, one might wonder if and how this is related to the restrictions on the u(1) charges after breaking the non-abelian symmetry.
In fact, the F-theory Standard Model fibration X F 11 we discussed in section 3.3 is shown in [26] to geometrically unhiggs into a Pati-Salam-like theory with [SU (4) × SU (2) 2 ]/Z 2 gauge group. In this case, the Z 6 centre of the Standard Model is known to arise from the representation theory of [SU (4) × SU (2) 2 ]/Z 2 . 14 One may ask whether it is possible to also unhiggs X F 11 to an SU (5) fibration.
However, the geometric description of unhiggsing is in general quite involved, since one does not a priori know the deformation corresponding to the specific unhiggsing process. In order to gain some further intuition, we therefore restrict our analysis to a specific class of models, for which we have a good handle on the geometry. For these, we show explicitly that the restrictions on the u(1) charges leading to the global gauge group structure arise from a larger, purely non-abelian gauge theory. As we will see, the unhiggsed non-abelian gauge algebra depends on the fibre split structure induced by the section.

Unhiggsing the u(1) in a Bl 1 P 112 fibration
The class of models we analyse have non-abelian gauge algebras engineered in a Bl 1 P 112 fibration, a.k.a. the Morrison-Park model [16]. Such models have a gauge algebra of the form g ⊕ u(1). Note that a broad class of such constructions has been classified through an analogue of Tate's algorithm in [13]. They can be realised as a toric hypersurface defined by the vanishing of the polynomial where [u : v : w] are the projective P 112 coordinates. 15 Furthermore, s is the blow-up coordinate whose vanishing defines the addition rational section generating the Mordell-Weil group.
As already discussed in [16], this fibration has a complex structure deformation, b 2 → 0, which enhances the gauge algebra from u(1) to an su (2)  corresponding to a certain simple gauge algebra g. The above deformation still exists for these gauge enhanced models in the form of b 2,k → 0 (we will abusively write b 2,k ≡ b 2 ). This deformation will then modify the fibres over c 3,k ≡ c 3 to have an su(2) singularity, since a generic choice for {θ} will not affect this codimension one locus. Likewise, the codimension two enhancement leading to the fundamentals of su(2) will still persist in the presence of the divisor {θ}, as it will generically not contain this codimension two locus.

Charge constraints on su(2) matter from higgsing
To keep things simple, we will focus on the easiest example with g = su(2) a from I 2singularities over {θ}, which we construct using 'toric tops'. The subscript a is to distinguish it from the su(2) b gauge algebra, which arises from unhiggsing the u(1). The spectrum of these models consists of singlets with charges 1 and 2, and fundamentals of su(2) a . The charges of these fundamentals depend on the fibre split, which for g = su(2) a can only be of type (01) or (0|1) (see figure 1). With the techniques of [9], one finds two different Bl 1 P 112 fibrations with additional su(2) a singularities, corresponding to the (01) and (0|1) split type, respectively.
In the (01) split model, whose resolved geometry is given by the vanishing of the polynomial (4.5) The su(2) a doublets have charges 3/2 and 1/2, consistent with the integrality condition (2.8) for the Shioda map S − Z + E 1 /2. Therefore, the global gauge group structure is [SU (2) × where Q is a lengthy polynomial in the coefficients b i , c j . However, since b 2 is now an explicit factor of the θ 2 term of the discriminant, tuning b 2 → 0 will clearly enhance the vanishing order of the discriminant in θ. Indeed, we find that after tuning, the discriminant becomes The deformation process of turning on the b 2 term now corresponds to bifundamental higgsing. To see this, let us first look at the group theory. The fundamental and adjoint representations decomposes as (4.8) Hence, the representations of the product algebra decompose according to (1, 2) → 1 (0,−1) ⊕ 1 (0,1) , (4.10) First, note that the charges agrees with the spectrum of the toric (0|1) split su(2) a ⊕ u(1).

Higher rank gauge algebras
We have also repeated the above analysis for (01) and (0|1) split types with higher rank gauge algebras that appear in the classification of 'canonical' Tate-like models in [13]. This contains all A-and D-type algebras up to rank 5 as well as all exceptional algebras and so (7). We find that for any algebra g along {θ} with (01) split, the tuning b 2 → 0 never affects g, i.e., the vanishing order of the discriminant along θ does not enhance further with this tuning. It only leads to an su(2) singularity along {c 3 }, as we have seen before. This is consistent with the tuning corresponding to adjoint (un-)higgsing of u(1) ↔ su (2), as the Shioda map in these cases will always be S − Z, and hence the global gauge group being G × U (1).
For (0|1) split type su(n)⊕u(1) models arising from I n singularities in codimension one, the tuning b 2 → 0 unhiggs the model to su(n + 1) ⊕ su (2). A more general treatment of the group theoretic decomposition (4.9) and (4.10) including 2-index anti-symmetric representations 16 confirms that the charges and global gauge group structure are consistent with bifundamental breaking. For the other singularity types with (0|1) split, we could verify that the tuning always enhances the singularity, i.e., increasing the rank of the gauge algebra along {θ} while still producing an su(2) along {c 3 }. This suggests that the higgsing, that produces the nontrivial global gauge group structure for the (0|1) type g ⊕ u(1) model, is not achieved with adjoints. However, to determine the exact matter content of the (un-)higgsed model requires a more detailed analysis, which we postpone to future works.
We also performed the same analysis for (0||1) split types, i.e., where the section S and zero-section Z intersect next-to-neighbouring nodes of g's affine Dynkin diagram. Naively, one finds an even higher enhancement along {θ} upon setting b 2 → 0 (e.g., for I n singularities we find su(n) → su(n + 2)). However, these tuned geometries always exhibit non-minimal codimension two loci (i.e., where the Weierstrass functions (f, g, ∆) vanish to orders (4, 6, 12)), indicating that there is-at least in F-theory compactifications to 6D-hidden strongly coupled superconformal physics. We hope to return to this issue in the future.

A Criterion for the F-theory swampland
We have seen that the geometry of elliptic fibrations imposes very stringent constraints on the u(1) charges of matter states in F-theory compactifications. A natural question that arises is if these constraints go beyond consistency conditions from a (supersymmetric) effective field theory (EFT) perspective. Put differently, do they give rise to criteria for an EFT to be in the 'swampland' [55,56] of F-theory? Given that in an EFT description, the global gauge group structure is often very obscure (e.g., because the spectrum of line operators is difficult to determine), it would be advantageous to have a criterion based solely on the gauge algebra g ⊕ k u(1) k and the particle spectrum, which usually are directly accessible.
However, from the field theory perspective, there is no physically preferred normalisation for the u(1) charge, whereas the integrality constraints appearing in F-theory models are only manifest in the geometrically preferred normalisation discussed in section 2.3. Thus, to formulate a swampland criterion based on charges, we first need to establish a method to fix the charge normalisation from the field theory perspective. As we would like to argue now, singlet states (i.e., states uncharged under any non-abelian gauge symmetries) should provide such a reference for the normalisation.

Singlet charges as measuring sticks
Recall the geometrically preferred charge normalisation of F-theory discussed in section 2.3.
This normalisation corresponds to having u(1) generators ω k = ϕ(σ k ) that arise from the normalised Shioda map ϕ (2.6) for free Mordell-Weil generators σ i . For simplicity, let us first look at the single u(1) case. In the preferred normalisation, singlets of g have integral charges, because their associated fibral curves satisfy Γ · E i = 0, so q 1 = (S − Z) · Γ ∈ Z. In fact, we observe that in all u(1) models with matter constructed so far in the literature, the charges of singlets computed with ϕ(σ k ) for a free generator σ k of the Mordell-Weil group, are mutually relatively prime (there is at least one pair of coprime numbers). This was expected to hold in [16], and viewed as a geometric incarnation of charge minimality [57][58][59][60], though a precise proof of this statement is to date not available. If this statement holds in general, then singlets uniquely determine the preferred normalisation, since no rescaling of the u(1) generator can preserve the charges to be integral and mutually relatively prime at the same time.
Observe that for a single u (1) Similar to the case of a single u(1), this condition is not proven in general. For the purpose of this discussion, we will assume its validity, noting that it is true in all F-theory models with multiple u(1)s and charged matter constructed so far in the literature [19-24, 27, 33].
One may worry about cases where there are no singlets present at all, e.g., in F-theory models with non-higgsable u(1)s [61][62][63]. However, since we are interested in the interplay of non-abelian matter with the u(1)s, these particular models are not of concern because the tuning required for additional non-abelian algebra is expected to enhance the non-higgsable u(1)s into non-abelian symmetries. 17 Whether this phenomenon persists in all non-higgsable F-theory models, or if there are (higgsable) u(1)s with non-mutually relatively prime singlet charges, requires a more in-depth geometric analysis beyond the scope of this work.
Before we turn to the actual swampland conjecture, we would like to discuss how the normalisation condition carries over in cases where u(1)s are broken by either a higgsing or a fluxed-induced Stückelberg mechanism. In the case of higgsing, i.e., giving a vacuum expectation value to a collection of massless states (in a D-flat manner), the breaking mechanism can be described geometrically by a complex structure deformation of the elliptic fibration. This deformation yields another F-theory compactification, where the massless u(1)s are again realised by a non-trivial Mordell-Weil group. 18 Thus, the above formulation of the normalisation condition carry over directly.
On the other hand, a G 4 -flux-induced breaking mechanism (which in F-theory is only possible in 4D and 2D) is not geometrised. Hence, we have to understand field theoretically how the singlet charges behave in such a situation. In the following, we will restrict our attention to four-dimensional compactifications, noting that the 2D case proceeds analogously [67]. In 4D, the field theoretic description of flux-induced breaking of u(1)s has been worked out in detail in type II (see [68][69][70] for a review) and subsequently in F-theory [71]. In the latter setting, a non-zero flux induces a mass matrix Crucially, one can show that the FI-terms ξ k,α can be taken to be integers due to the quantisation condition of G 4 (see appendix A). Therefore, a non-trivial solution space V of (5.2) can be generated by integer vectors λ s k , s = 1, ...,m = dim V . In other words, the massless u(1)s are generated by integer linear combinationsω s = λ s k ω k = ϕ(λ s k σ k ) of the Mordell-Weil generators. Hence, there must existm free Mordell-Weil generatorsσ s that span V . Since the full singlet charge lattice was by assumption dual to the Mordell-Weil lattice, it must also contain the sublattice dual to Λ = span Z (σ s ). In other words, there is a basisω s = ϕ(σ s ) for the massless u(1) generators, in which the singlet charges are all integer and their integer linear span fills out every lattice site of Zm.
So we have established that in an F-theory compactification with m massless u(1)s, there are always m free Mordell-Weil generators σ k , such that the associated u(1)s are dual to the Shioda-divisors ω k = ϕ(σ k ) given by (2.6). In this geometrically preferred normalisation, the singlet charges are all integer and span the full Z m lattice. From the field theoretic point of view, which only has direct access to the singlet charges, it is crucial to realise that this normalisation is unique up to a unimodular transformation of the u(1) generators, i.e., a change of basis for the Mordell-Weil (sub-)group. To see that, let us denote by q k , k = 1, ..., m the charges of singlet states Q k , which form a basis dual to ω k , i.e., (q k ) i = δ ki . Now suppose that we picked a different basis ω l for the u(1) generators, in which the singlet charges again span Z m , with basis q k dual to the ω k . While the q k and q k correspond to different physical states Q k and Q k , their charge vectors both span Z m , so there must exist a change of basis, i.e., a unimodular matrix U , such that Q k = U kl Q l . For the dual generators ω k and ω k , the corresponding transformation ω k = U −1 kl ω l is then again unimodular. Therefore, the sections σ k = U −1 kl σ l generate the same lattice as σ k , i.e., they are also Mordell-Weil generators. Thus, the u(1) generators ω k = ϕ(σ k ) are also in the geometrically preferred normalisation.

The swampland criterion
Having established that the singlet charges provide a measuring stick for determining the preferred normalisation, we are now in a position to formulate the criterion which needs to be satisfied by EFTs arising from an F-theory compactification. Given a theory with an unbroken u(1) ⊕m ⊕ g gauge symmetry, normalise the u(1)s such that the singlet charges are integer and span Z m . As argued above, we assume that this is always possible in F-theory. In this case, the corresponding u(1) generators ω k are given by the Shioda-map (2.6) of some free Mordell-Weil generators σ k . Then, due to (2.10) and (2.11), the difference q k of the u(1) k charges for any two representations R (i) = (q g . In other words, the condition states that singlets under the non-abelian gauge algebra provide a reference for the spacing of u(1) charges, which has be respected also by all non-abelian matter in a given g-representation, even if these may have fractional charges. Any EFT that does not satisfy this criterion must lie in the F-theory 'swampland', i.e., cannot be an F-theory compactification.
Note that this condition goes beyond anomaly cancellation. An example in 6D is given by a tensorless su(2) ⊕ u(1) theory with 10 uncharged adjoint hypers, 64 fundamental hypers with charge 1/2, 8 fundamental hypers with charge 1, 24 singlet hypers with charge 1, and 79 uncharged hypers. Clearly, the u(1) is properly normalised according to our condition above, since there are only one type of charged singlets with charge 1. However, despite the anomalies [72] been cancelled with Green-Schwarz coefficients a = −3, b u(1) = 4, b su(2) = 6, the presence of both charge 1 and 1/2 su(2) fundamentals does not meet our 'swampland' criterion. In 4D, the constraints are even weaker, since a completely vector-like spectrum is always gauge-anomaly-free, independent of charges or representations.
In summary, our necessary condition for an effective field theory with gauge algebra u(1) ⊕m ⊕ g to be an F-theory compactification requires to first establish a 'preferred' normalisation. This normalisation is determined by having all singlet charges being integer, and their integer span generates Z m . Then, the difference of charges for matter in the same grepresentation must be integer. Any field theory not satisfying this condition must lie in the F-theory swampland. We re-emphasise that this criterion relies on the two key assumptions consistent with the current literature: (a) F-theory models with gauge algebra u(1) ⊕m ⊕ g and charged matter always have singlets, and (b) their charges in the preferred normalisation span the full integer lattice. Were it not for these assumptions, we could always rescale the u(1)s so that all charges are integer, and the above conditions are trivially satisfied. To sharpen our criterion will eventually require a rigorous proof of both assumptions.
Finally, we point out that our arguments are based on intersection properties of the Shioda map divisors with fibral curves, which in the F-theory compactification give rise to massless states in the effective field theory. On the other hand, there are also massive states in string compactifications coming from higher Kaluza-Klein states or KK reductions along non-harmonic forms. In light of the recent development of the weak gravity conjecture (see [56,[73][74][75][76][77][78] for an incomplete list) which puts constraints on u(1) charges of states relative to their masses, an interesting question is whether the global gauge group structure is also respected by massive states in F-theory. If so, it would certainly be interesting to explore if and how our F-theory swampland arguments fit into more general quantum gravity concepts.

Summary and outlook
In this work, we have shown that F-theory compactifications with an abelian gauge factor generically come equipped with a non-trivial gauge group structure [G × U (1)]/Z. The finite subgroup Z = Z κ ⊂ Z(G) × U (1) of the centre is generated by an element C, which we have constructed explicitly from the Shioda map of the Mordell-Weil generator. Geometrically, different centres Z arise from different fibre split types, i.e., relative configurations of the zero and the generating sections on the codimension one singular fibres determining G. At the level of representations, the construction-generalising that for torsional sections [35]imposes specific constraints on the allowed u(1) charges q R of each G-representation R, such that Z acts trivially on (q R , R). These constraints can be equivalently viewed as a refined charge quantisation condition: There is a normalisation (2.6) of the u(1) such that all charges of matter in a given G-representation R span a (one-dimensional) lattice with integer spacing.
A non-trivial gauge group structure is then reflected in a relative shift between charge lattices of different G-representations by multiples of 1/κ.
We have exemplified our findings in several concrete models that have been constructed throughout the literature. Using these examples, we have also demonstrated that the argument straightforwardly generalises to multiple u(1) factors, i.e., higher rank Mordell-Weil groups, and also to cases with both free and torsional sections. Each generator (free or torsional) leads to an independent central element (possibly trivial), such that in general, Z is a product of Z κ k factors. In particular, when applied to the 'F-theory Standard Models' [26,32,41,42], we found that these models realise the physical Standard Model gauge group [SU (3) × SU (2) × U (1) Y ]/Z 6 . Correspondingly, the geometric spectrum completely agrees with the physical representations of the su(3) ⊕ su(2) ⊕ u(1) Y algebra.
We have also explored the connections of u(1) charge restrictions to the process of unhiggsing into a larger non-abelian group. Relying on simple class of geometries with su(2) ⊕ u(1) gauge algebra, we have shown explicitly that the two different global gauge group structures unhiggs, both geometrically and field theoretically, into different non-abelian gauge groups. However, for [G × U (1)]/Z models, where the zero section and the Mordell-Weil generator do not intersect the same or neighbouring fibre components in codimension one, the unhiggsing procedure introduces codimension two non-minimal loci. In compactifications on a threefold, one would usually associate such a non-minimal locus with the existence of tensionless strings and interpret it as a superconformal sector of the 6D field theory. Clearly, it would be exciting to investigate how superconformal physics enters the global gauge group structure, and also gain insight into 4D compactifications, where non-minimal loci are less understood. It is worth pointing out that the centre also plays a crucial role in the higgsing of u(1)s to discrete symmetries [26,79]. With the explicit description of the centre laid out in these notes, we look forward to apply our new insights to phenomenologically more appealing models with discrete symmetries [80].
We have also studied how the non-trivial global gauge group structures give rise to an Ftheory 'swampland' criterion. Formulated in terms of u(1) charges, a field theory with gauge algebra g⊕u(1) that arises from F-theory must satisfy the following condition. By normalising the u(1) such that all singlet charges are integer and span the full integer lattice, charges of matter having the same non-abelian g-representation, which individually can be fractional, must differ from each other by integers. Our analysis also shows that this criterion generalises to the case of multiple u(1)s, which remain massless in the low energy effective theory after a higgsing and/or turning on G 4 -flux. While this condition is stronger than cancellation of field theory anomalies and hence can be used to rule out swampland theories, their validity is based on the assumption that in F-theory compactifications, the singlets serve as a 'measuring stick' for the u(1) charges. Geometrically, it is based on the observation that any F-theory model with u(1)s and charged matter have in particular charged singlets, whose corresponding fibral curve in the elliptic fibration spans a lattice that is dual to the (free) Mordell-Weil lattice under the intersection pairing. Note that this observation extends the conjecture [16] made for a singlet u(1), that singlet charges computed with respect to the normalised Shioda-map (2.6) are integer and mutually relatively prime. To make the swampland criterion precise will therefore require a careful analysis of the intersection structures between sections and codimension two fibres in elliptically fibred Calabi-Yau manifolds. Nevertheless, it would be interesting to study the connection of this swampland criterion to other quantum gravity conditions such as the weak gravity conjecture and extensions thereof.

A Integrality of fluxed-induced Fayet-Iliopoulos terms
In this appendix, we show that the fluxed-induced FI-terms (5.1) can be taken to be integers in (5.2), which determines the massless linear combinations of u(1)s. First, recall the consistency conditions [81] for G 4 -flux in F-theory compactified on an elliptic fourfold π : Y → B: G 4 ∈ H 2,2 (Y ) such that ∀D B ∈ H 1,1 (B) and ∀C B ∈ H 2,2 (B) : where Z is the cohomology form dual to the divisor of the zero-section. These so-called transversality conditions ensure that the flux preserves 4D Lorentz symmetry in the M-/Ftheory duality. Furthermore, in order not to break the non-abelian gauge algebra g with Cartan divisors E i , we impose Because of these conditions, the fluxed-induced FI-terms (5.1) for a u(1) generator ω k = ϕ(σ k ), with σ k a Mordell-Weil generator with divisor class S k , simplifies to The basis elements J α for the Kähler form J B of the base can be taken to be integer cohomology forms dual to divisor. Then we can invoke the quantisation condition [82], where c 2 (Y ) is the second Chern class of Y . Therefore, since the section classes S k are also integer, the FI-terms (A.3) satisfy Because c 2 (Y ) ∈ H 4 (Y, Z), the last term is at most half-integer. In other words, 2ξ k,α ∈ Z for all k and α. Since an overall factor of the FI-terms does not affect the solutions of the masslessness condition (5.2) for u(1)s, we can thus assume from the beginning that ξ k,α ∈ Z.