A Note on Non-Flat Points in the $SU(5)\times U(1)$ PQ F-Theory Model

Non-flat fibrations often appear in F-theory GUT models, and their interpretation is still somewhat mysterious. In this note we explore this issue in a model of particular phenomenological interest, the global $SU(5)\times U(1)$ Peccei-Quinn F-theory model. We present evidence that co-dimension three non-flat fibres give rise to higher order couplings in the effective four-dimensional superpotential. More specifically, in our example we find $\mathbf{10}\, \mathbf{5}\, \mathbf{5}\, \mathbf{5}$ couplings.


Introduction
F-theory [1] models have been extensively studied in the last few years, starting with [2,3,4,5,6], for their promising features for GUT-inspired string theory model building.
A detailed analysis of such models reveals that they sometimes develop "non-flat" points: these are points on the base over which the dimension of the fiber jumps. The goal of this note is to address what happens at these points. We will not give a general solution, but rather analyze in detail a particular example with interesting phenomenological properties. This is the SU (5) × U (1) Peccei-Quinn model analyzed in [7,8,9], and follow-up works. We expect related models to be amenable to an analysis akin to the one we perform here.
However, before we focus on the non-flat points in the SU (5) × U (1) P Q model, we pick up one of the loose ends of [9] and give it a more satisfactory resolution. More concretely, we will look into an issue regarding Q-factorial terminal singularities which was not fully elucidated in [9]. We will analyse them along the lines of [10], but then switch on complex structure deformations such that we do not have to be concerned by these singularities when caring out our study of the co-dimensional three effects we are interested in. The main focus this article lies on the investigation of non-flat torus-fibrations which come about naturally when we relax a constraint on the base of the F-theory fibration which was imposed in [8,9].
We remove this restriction and allow for the simultaneous vanishing of the polynomials, or sections to be more precise, α and c 2 on the GUT-divisor. We find that in the resolved F-Theory four-fold the dimension of the fibre over this point increases, i.e. the fibration becomes non-flat. We study this co-dimension three effect from various angles, and find that the physical interpretation of it is a higher order coupling -in our case a 10 3 5 −1 5 −1 5 −1 coupling.
In the course of this research, we also determine all the fluxes which are induced by the matter curves and the non-flat fibre. We calculate the second Chern class of the fourfold and look at its implications on the flux quantisation. We give the fluxes which must be turned on to satisfy the quantisation condition and show that this flux forbids string states in four dimensions, coming from M5 branes wrapping the non-flat fibre.
We have organized this paper as follows: in section 2 we review the most relevant geometric aspects of the global SU (5) × U (1) P Q as studied in [9]. Then we study the Q-factorial terminal singularities which appear in this setting and discuss how to introduce complex structure deformations so that these singularities do not appear. Afterwards we carefully analyse this fibration over a general base without constraints. In section 3, we list all the fluxes coming from the Mordell-Weil group, the matter surfaces, and the non-flat fibres, respectively, and relate them with the quantisation condition and explain why it forbids strings in four-dimensions. In section 4, we take the weak coupling limit of our setting and study the states and their coupling in the IIB picture. As a check, in section 5 we go to the mirror/IIB side to confirm also from this perspective that the non-flat point gives rise to a higher order coupling. Finally, we present our conclusions in section 6.

The geometric setup
In this section, we review and extend the analysis of the global realisation of the SU (5) × U (1) Peccei-Quinn model in F-theory started in [9]. Hence, let us first recall the geometric setup presented in section 5 of [9]. It was shown there that in order to obtain two 10-curves the SU (5) enhancement has to be imposed in a non-toric way. This in turn does not allow for a resolution of the fibration in a purely torical way. Though we can resolve parts of the hypersurface singularities torically, for the final resolution step we need a complete intersection to represent the smooth Calabi-Yau. The resolved model is then given by the following two hypersurface equations HSE 1 : λ 1 e − λ 2 s P 2 = 0 , (2.1) with the polynomials 3) The two hypersurfaces (2.1) and (2.2) are embedded into the ambient variety with the relations u v w s e 0 e 1 e e 4 λ 1 λ 2 HSE 1 HSE 2  1  1  2  0  0  0 0 0  1  0  1  4  0  1  1  1  0  0 0 0  2  0  2  (2.7) Here [·] means the 'degree' of the respective section or polynomial and c B is the 'degree' of the first Chern class of the base space.
As noted in [9] this complete intersection Calabi-Yau (CICY) still has singularities. A careful analysis of (2.1) and (2.2) yields that there is a remaining singularity at the base loci and fibre coordinates w = v = λ 1 = 0. Indeed, if we assume for the above fibration a twodimensional base then the so-obtained Calabi-Yau threefold will be Q-factorial with terminal singularity points. Such varieties have recently been studied from the F-theory perspective in [10,11]. There it has been pointed out that such singularities can only be resolved in a discrepant way. Furthermore, upon compactification uncharged hypermultiplets localise at these singularities which are needed to cancel the six-dimensional gravitational anomaly. It is not too difficult to show that also the fibration at hand has the right amount of uncharged singlets to be anomaly-free. The reader interested in the explicit calculation is pointed to appendix A.
Although these Q-factorial terminal singularities are present in the original setup as presented in [9], we can smooth them away by switching on complex structure deformations [12]. Since the locus (2.8) lies generically away from the GUT-divisor, these deformation do not interfere with the local geometry at ω = 0 and only alter things away from it. 1 This smooth geometry is the one we will study throughout the rest of the article.
Most of the details along the GUT-divisor of this SU (5)×U (1) P Q fibration have been analysed in [9]. However, due to spectral cover considerations the locus was excluded. But these loci are always presented if we consider the above setting over a generic three-dimensional base. Therefore, we examine these points very carefully in the following after recalling the most important features of the model. We start with the two 10-curves: 10) and the three 5-curves: have been presented in [9]. Besides these couplings, there is the intersection (2.9) between the 10 3 -curve and the 5 −1 -curve for which we cannot write down any gauge invariant three-point interaction. Looking at the second equation in (2.11), we observe that the 5 −1 -curve intersects the 10 3 -curve at the points (2.9) three times, i.e. near α = c 2 = 0 the 5 −1 -curve takes the form with ρ i some constants. This hints already at a four-point coupling 10 3 5 −1 5 −1 5 −1 but to get a better picture of what really happens at these points, we have to look at the full fourfold geometry, especially the fibre structure. As it turns out, these are points where the dimension of the resolved fibre jumps, i.e. the fibration described by (2.1) and (2.2) over a three-dimensional (or higher dimensional) base is non-flat. 2 The dimensionality jump is due to the vanishing of P 2 at α = c 2 = 0. We 'lose' one of the equations which define the fibral curve of E 3 A summary of the curves and the coupling points of this setup is depict in Figure 1 2 This does not imply that the dimension of the fourfold changes nor that it is singular at these points.

(2.17)
To see that the fibre surface FS at the non-flat points is a del Pezzo four surface at a special complex structure sublocus, we give the reduced ambient space: v e 1 w e 4 λ 1 λ 2 HSE red 2 1 1 1 0 2 0 5 3 0 0 1 1 1 0 3 2 0 0 0 0 1 1 2 1 is embedded. The polynomials β, δ, γ, d 2 , and d 3 of before are now effectively coefficients. The toric space (2.18) is a P 1 -fibration over the Hirzebruch surface F 1 ∼ = dP 1 and (2.19) defines a section of this fibration. Since the section degenerates over the points the del Pezzo one surface is blown up at three points. These three points lie along a line. Therefore, the fibre-surface FS is not a generic del Pezzo four surface but a degenerate dP 4 . FS contains several rational curves: the generic fibre and the two special sections of F 1 ; from the blow-ups of the Hirzebruch surface, we have the line going through the blown-up points, the exceptional P 1 's, and the proper transforms of the fibres at these points. The Cartan charges of the rational lines are: (2.21) Regarding P 1 line , we should note that prior to the blow-ups it was equivalent to P 1 sec 1 , i.e. P 1 line is the proper transform of sec 1 going through the three points which are blown-up. Hence, there are two special points for the complex structure deformation of P 1 sec 1 ; one where it splits into bu i and the one, which also exists in F 1 , where it becomes reducible to With these details at hand, we can describe the three-cycle which fuses three5 1 states into a 10 3 state: Figure 2, we sketched FS to better understand the interplay of the rational curves.

Fluxes
Now that we have gained a good understanding of the geometry of our model, we can turn to the F-theory four-form flux of our setup. It has to fulfil the flux quantisation condition [13,14]: with Y 4 the Calabi-Yau four-fold on which we compactify. To see whether (3.1) forces us to switch on half-integer fluxes, we are analysing in the following the Chern class of our four-fold.
The main goal of this study will be to prove that the restriction of G 4 to the non-flat fiber gives rise to a non-trivial homology class. This fact provides a nice simplification of the physics of the system, since it immediately implies that the M5 brane wrapping this divisor is inconsistent [14]. Accordingly, the four dimensional light strings this wrapped M5 would give rise to in four dimensions are absent. 4 Let us also mention that non-trivial flux will potentially induce chirality, and thus anomaly cancellation is a worry. Our goal in this note is to clarify the dynamics arising from the non-flat (codimension-three) point, while anomaly cancellation is a more global phenomenon arising from matter curves, at codimension two. Therefore, we expect our considerations to hold regardless of whether anomalies are ultimately canceled in any specific model, as long as the local behavior is as in our example. Even if not immediately relevant to us, the details of anomaly cancellation could be interesting. For example if there were underlying algebraic relations like the one observed in [16]. We will leave such an analysis for future work.

The second Chern class of the F6-fibration
Let us start by giving the second Chern class of an elliptically fibred fourfoldŶ 4 where the torus fibre is defined by the sixth reflexive polygon (as enumerated in [17]), i.e. Bl 1 P 1,1,2 [4] [18]. For this manifold, where we did not impose an SU (5) singularity yet, the second Chern class reads 5 Hence, depending on the degree of [δ], when considering such an F-theory compactification one might be forced to switch on flux even though no non-abelian gauge groups are present yet. Note that this is different from the U (1) X case [19,20].

U (1)-and matter surface fluxes
As a next step let us write down the different fluxes we can construct from the Mordell-Weil generator and the matter surfaces. From the section S we obtain via the Shioda map [21,22,23] the following expression for the U (1)-flux: In addition, we can use a similar strategy to the one presented in [24] to construct from the matter surfaces the follwing gauge invariant fluxes: With all these expressions at hand, we can now finally give the second Chern class of the fourfold we are considering in this article: where G nf 4 is the flux corresponding to the four cycle FS: The main properties of the G nf 4 flux are that it does not break the SU (5) gauge symmetry and it localises at the non-flat points. To see this we can integrate G nf 4 over all algebraic two-cycles in Y 4 which are accessible to us: where C 10 = E 1 E 4 and C 11 is the four-cycle of the non-flat fibre. In equation (3.8) Γ andΓ are place holders for all possible divisor classes pull back from the base B 3 .

The weak coupling limit and the IIB picture
As we will argue, the F-theory model of interest to us can be taken to weak coupling without breaking any of the GUT symmetries, and without encountering any special behavior along the way. Since we are interested in computing a superpotential coupling, which is a holomorphic quantity, we expect that the result of computing such quantities at weak coupling remains valid all through moduli space.
can be brought via a birational transformation into Tate form with In analogy to [25], we define To take the weak coupling limit, we proceed along the lines of Sen's original work [26] and require b 2 , b 4 , and b 6 to scale (at leading oder) like 0 , 1 , and 2 , respectively, as we take the limit → 0. One way to obtain that behaviour is to take we can write the discriminat in the weak coupling limit as Plugging (4.3) into ∆ w.c. , we obtain the rather lengthy polynomial This is the IIB D-brane locus (without the orientifold plane) for the generic F6-fibration if we take the weak coupling limit as in (4.5). The corresponding Calabi-Yau threefold is given by following double cover of B 3 : where the vanishing set {R = 0} defines the orientifold plane and the orientifold action is naturally induced by ξ ←→ −ξ .  where, for convenience, we switch on only one complex structure deformation compared with [9], cf. footnote 1. Thus, we obtain for the hypersurface of the Calabi-Yau threefold. We do not show the rather lengthy expression for ∆ w.c. because it will turn out that in suitable coordinates the polynomial factorizes and the loci of the brane-image brane pair become evident. We work now close to the singular point 6 where we expect the higher order coupling to arise. In particular, we assume that all d 3 and δ are non-vanishing close to the points of interest. We define now such that the ordinary double point singularity, or conifold, takes the form (4.14) We can represent this confold also in a toric way by introducing the homogeneous coordinate α i , β i with i = 1, 2 and scaling relation: where The affine coordinates from above are expressed in terms of homogeneous ones as (ξ, u, σ, w) = ( 1 2 (α 1 β 2 − α 2 β 1 ), 1 2 (α 1 β 2 + α 2 β 1 ), −α 1 β 1 , α 2 β 2 ). Using these two coordinate changes, we can rewrite the D-brane locus close to the point of interest as follows: This makes it obvious that the flavor brane/image brane pair are respectively located at P 1 = η 0 α 3 1 + η 1 α 2 1 α 2 + η 2 α 1 α 2 2 + η 3 α 3 2 = 0 (4.20) whereas the GUT stack and image-stack are at α 1 = 0 and β 1 = 0, respectively. Locally the η i 's are invertible and we treat them as if they were non-zero complex numbers. Under this assumption, we can further factorise the flavour branes to

Ext groups and Quiver theory
In order to construct the resulting gauge theory, we need to specify all branes participating at the point of interest. Following [15], we employ the method of non-commutative crepant resolutions [27]. This entails describing branes as elements of the derived category of quasi-coherent sheaves on say Y + . Open string states between these are expressed in terms of morphisms between such objects, which in turn are elements of so called Ext groups. (For a review of the relevant background material aimed at physicists, see [28].) We will first briefly review the general form of the construction for the conifold in §4.2.1, and will then apply this construction to our non-flat point in §4.2.2.

Non-commutative crepant resolution of the conifold
Consider again the singular conifold, described by Spec C[ξ, u, w, σ]/ ξ 2 − u 2 − σw . This, as we saw above, is a toric variety The conifold has two small crepant resolutions which correspond in toric language to different subdivisions of its fan. These are also toric varieties with homogeneous coordinates α 1 , ..., β 2 subject to the constraint The two small resolutions are distinguished by the sign of t, and we denote them as Y ± respectively. Applying the orientifold involution (4.18) to (4.26), we see that t ↔ −t, that is to say the two resolutions Y ± are exchanged. This means that the resolution mode corresponding to the P 1 is projected out.
It is, however, possible to describe D-branes on the singular space directly using its noncommutative crepant resolution [27]. By this we mean a non-commutative ring A A = End(M ⊕ R), (4.27) where R = C[ξ, u, w, σ]/ ξ 2 − u 2 − σw and M is Here the map ψ is given by Notice that one could also take Observe that We do not want to delve into the details but simply state that A is derived equivalent to Y ± . More concretely there is a correspondence cf. Theorem 5.1 in [27]. As is well established [28], one can view objects of D b (QCoh(Y ± )) as D-branes in the B-model and morphisms between them correspond to open strings states.
Using the dictionary laid out in [15], we will map certain (complexes of) A-modules to Dbranes of interest. In order describe these effectively note that as R-vector spaces. In particular, e i are idempotents. Any module of A can be encoded as a quiver representation. As laid out in [15], the basic representations from which one builds D7-branes are: These are linear combinations of paths ending at the left and right node of the quiver (4.35), respectively. Clearly morphisms from P 0 to P 1 are generated by α 1,2 and from P 1 to P 0 by β 1,2 .
Together with the assignment where O is the structure sheaf of the resolved conifold, we obtain for instance Here the map α 1 between the sheaves is nothing but the fiberwise multiplication by the homogeneous coordinate. The power of this approach is that computing Ext-groups between complexes of sheaves is easier in the setting of quiver representations. Since all relevant computations were already carried out in [15], we will not demonstrate them but only list the results in the following.
Fractional branes given by D1-branes wrapping the resolution divisor are given by This follows from the fact that the moduli space of representations of dimension (1, 1) is exactly the resolved conifold, see [15] section 3.2.1.
The brane/image brane pairs appearing in this paper are These correspond to D7 branes located at the 5 curve. To see this apply the cokernel to the relevant maps, which is commonly referred to as Tachyon condensation. Moreover there is one pair of objects corresponding to D7 branes located at the 10 curve We also have fractional branes D(-1) instantons described by objects I 0 = S 0 [−1] and I 1 = For more details on this see Appendix A of [15].
We now study the open string states between these branes by computing certain Ext groups between elements of D b (Y + ), where Y + is one of the crepant small resolutions of the conifold. To this end consider the pair The groups Ext i (F 0 , F 1 ) were calculated in [15], but only for the value (A, B) = (0, 1). We claim that these are isomorphic to our Ext groups as the two complexes To see this consider the following automorphism of the conifold f : (α 1 , α 2 , β 1 , β 2 ) → (α 1 , 1 B (α 2 − Aα 1 ), β 1 , β 2 ) ≡ (α 1 ,α 2 , β 1 , β 2 ). (4.58) Observe that Aα 1 + Bα 2 = α 2 . One readily checks that Similarly, we obtain an isomorphism This implies that all Ext groups computed in [15] are isomorphic to the ones we will need, e.g.

The non-flat point at weak coupling
We now describe the relevant branes in our setup. There are the three pairs of objects These correspond to D7 branes located at the 5 curve. Moreover there is one pair of objects corresponding to D7 branes coming from the 10 curve We also have fractional branes D(-1) instantons described by objects I 0 = S 0 [−1] and I 1 = Figure 3: Quiver theory for GUT and flavor branes. Note that one should draw F 1 0 , F 2 0 , F 3 0 separately and connect to the other nodes as indicated. For the sake of clarity only one flavor brane/image brane is shown.
In order to obtain the desired theory after orientifolding one takes the branes G i with multiplicity 5 to generate the GUT stack. A chiral bifundamental string between G 0 and G 1 giving rise to a state in the 10 representation upon orientifolding. This can be derived more rigorously by considering the gauge group on empty nodes. In [15] it was shown that indeed we obtain USp(0).
The flavor branes F i j are each chosen with multiplicity 1. Between G 1 and each F j 0 we have a bifundamental with the same chirality as above giving rise to a 5 state. Instanton effects arise from D1 branes wrapping the nodes I i . We will only consider the case of a single instanton.
Firstly, consider a D1 brane wrapping I 1 . This gives rise to charged zero modes as in Figure  4. Hence, the superpotential reads we obtain the desired coupling. If on the other hand we wrap one D1 brane around the I 0 node, there will be no contribution to the superpotential due to our choice of chirality.

The mirror picture
Finally, it is interesting to see how the superpotential coupling appears from the mirror IIA perspective. This mirror picture gives a useful heuristic understanding of the physics, but the analysis is harder to make fully precise than in the IIB setting, where we have a well defined problem in algebraic geometry. The analysis is very similar to that in [15] (building on previous work in [29,30,31]), so we will be somewhat brief.
For the purposes of computing holomorphic data the topology of the mirror to the conifold can be described by a fibration over C with fiber C * × Σ [32,33], described by uv = W , P (x, y) = W , where W ∈ C parameterizes the base of the fibration, u, v ∈ C parameterize the C * fiber, and x, y ∈ C * describe the (punctured) Riemann surface Σ. For the specific case of the conifold, we can choose a framing [34] such that Here q is a complex structure modulus mirror to the complexified size of the small resolution of the conifold. This equation defines a P 1 punctured at four points. As discussed in detail in [29], for the purposes of computing holomorphic quiver data for our system, it is enough to focus our attention on Σ. In addition to the geometric background itself, we need to describe how the branes wrap the geometry. The case with one U (5) stack and one additional U (1) brane stack was described in detail in [15]. An important difference in our case is that, in addition to the U (5) stack, we have three U (1) flavor branes. We will start by analyzing the case in which all U (1) branes are coincident, leading to a flavor stack with gauge group U (3) × U (5). The restriction of the brane system to Σ can then be determined by identical arguments to those in [15], with the result shown in Figure 5.
There are various features to note in Figure 5. We have the G 0 ∼ G 1 stacks (the identification is due to the orientifold action), associated with the U (5) stack, and the F 0 ∼ F 1 stacks, associated to U (3). We obtain various fields, as these stacks intersect each other 7 , and additional matter fields as the flavor stacks intersect the instanton brane I 1 , with gauge group O(1) = Z 2 . The resulting matter content can be summarized as Note that P is most naturally the (complex conjugate of the) two-index representation of SU (3), which can be identified with the fundamental representation. The worldsheet instantons depic-ted in Figure 5 then generate an effective action for the charged instanton zero modes of the form where raising the index corresponds to going to the complex conjugate representation. The effective non-perturbative superpotential one obtains from integrating out the charged zero modes is then of the form where we have omitted the unknown (but generically nonzero, since the relevant worldsheet instantons have generically finite area) coefficients of the various terms in the superpotential, which depend on various geometric and brane moduli.
It is now a simple job to deform away from the U (3) locus. This can be seen as a Higgsing of the SU (3) flavor symmetry, which will give a mass to at least some of the fields in P , and generically to all of them. 8 We can model this as the deformation of (5.5) given by where, for simplicity, we have set all of the masses equal. Integrating out P then leads to an effective superpotential of the form which in the m → ∞ limit leads to the superpotential that we have argued for in the previous section.

Conclusions
The main focus of this work was to understand the effect on the low-energy dynamics of non-flat fibres in co-dimension three. We found that (at least in a class of interesting models) they are related to higher order couplings.
This information is important for F-theory model building because such points seem to appear rather frequently. There are instances of such points in the literature going back to the start of the intense study of non-abelian gauge groups together with U (1) selection rules [35,36]. Two of the many examples appeared in the context of SU (5)-top constructions over the fibre F6 [17,37,36], i.e. over the Morrison-Park fibre. It was found that both the third and fourth SU (5)-top of the fibre F6 in [36] have a non-flat point. Following our arguments, we see that in the case of third SU (5)-top one obtains the coupling 10 3 5 −1 5 −1 5 −1 from the non-flat points because the setup is almost identical to ours. For the fourth SU (5)-top we expect the coupling 10 −1 5 7 5 −3 5 −3 . A further example shows up in the study of the exceptional gauge groups. When looking at the E 6 -top over the F11 fibre, i.e. over the Grimm-Weigand fibre [19], we find again a non-flat fibre in co-dimension three. Though we do not have a weak coupling limit in this case to carry out the second half of our above analysis, we expect the appearance of a 27 −1 27 −1 27 −1 1 3 coupling.
Here we exploit the fact that over B 2 = P 2 the degree of a homogenous ploynomial is equal to the first Chern class of its asssociated line bundle. The charge 10 states are located at (2.10). It follows from Bezouts theorem that there are deg(ω) · (deg(d 3 ) + deg(c 2 )) = 3 such points on the base. This gives us 30 hyper multiplets.

A.2. Counting uncharged hyper multiplets
The number of uncharged hyper multiplets is computed from the topological Euler characteristic and h 1,1 of our variety. We know that h 1,1 = 6. (A. 13) Strictly speaking this is the Hodge number of a smooth threefold rationally equivalent to our singular variety. The existence of such a deformation is guaranteed by [12]. The Euler characteristic of the singular variety is computed by first computing it for a smooth representative of its rational equivalence class. Then we use the fact that [38] χ(X Sing ) − χ(X smooth ) = P m P , (A.14) where the latter sum runs over the singular points P and m P denotes the Milnor number of such a point. The Euler characteristic χ(X smooth ) is computed using the toric embedding and turns out to be χ(X smooth ) = −132.