Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions

In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases to be flat over compact Riemann surfaces of genus $g$ in the base. These non-flat fibers contribute Kaehler moduli to the four-fold but also add to the three-form cohomology for $g>0$. In F-/M-theory these sectors are to be interpreted as compactifications of six/five dimensional $\mathcal{N}=(1,0)$ superconformal matter theories. The three-form cohomology leads to additional chiral singlets proportional to the dimension of five dimensional Coulomb branch of those sectors. We construct explicit examples for E-string theories as well as higher rank cases. For the E-string theories we further investigate conifold transitions that remove those non-flat fibers. First, we show how non-flat fibers can be deformed from curves down to isolated points in the base. This removes the chiral singlet of the three-forms and leads to non-perturbative four-point couplings among matter fields which can be understood as remnants of the former E-string. Alternatively, the non-flat fibers can be avoided by performing birational base changes, analogous to 6D tensor branches. For compact bases these transitions alternate all Hodge numbers but leave the Euler number invariant.


Introduction
In recent times, our understanding of strongly coupled super symmetric theories (possibly coupled to gravity) in various dimensions has made great progress by means of geometric methods. Those theories are generically hard to control within the framework of regular quantum field theories. Six dimensional superconformal field theories (SCFT) for example are genuine strongly coupled and were believed to contain massless string excitations in the spectrum [18,[32][33][34][35]. Evidence for their existence could be made in explicit construction made in string theory. The string construction uses of compactification geometries that provide certain divisors that are wrapped by branes and lead to massive BPS strings in the non-compact directions. If those divisors are collapsed to points these strings become tensionless and support the SCFT [56]. For 6D theories with minimal supersymmetry the most flexible tool of choice has been F-theory [49]. F-theory itself geometrizes the type IIB axio-dilation into the complex structure of an auxiliary elliptic fiber which is put on top of the physical compactification space. The total geometry becomes that of an elliptic Calabi-Yau(CY) three-fold which features the consistent description e.g. of branes with large IIB string coupling. Classifying all non-compact elliptic-threefolds that can be shrunk to a (possibly singular) point has lead to an extensive list of 6D SCFTs [36][37][38] (see [39] for a review). This direction is also fruitful to further classify 5D SCFTs using M-theory on the same class of geometries, related to F-theory by a circle compactification [53,54]. To access M-theory though the CY geometry must be fully resolved. This can be done by performing birational base changes until the elliptic fiber admits at most minimal singularities. Such fiber singularities then admit a crepant resolution according to the Tate-algorithm. This resolution strategy naturally reflects the 6D tensor branch followed by the 5D coulomb branch upon the circle reduction. However, there is yet another way to resolve the geometry. This resolution is crepant and hence Calabi-Yau but does not respect the dimension of the elliptic fiber, called non-flat resolution. In such cases a non-minimal fiber singularity in codimension two can be replaced by surfaces E i of complex dimension two. Shrinking those surfaces back to points reaches the respective 5D SCFT point. This point of view has proven to be very efficient to characterize 5D SCFTs and its phase structure [15][16][17]40]. In addition it also allows to study the non-minimal singularities of elliptic three-folds and their F-theory lifts directly, without the need to change the base [3,5,50]. Extending this geometric approach to 4D is of course highly desirable. Related geometric approaches e.g. those pioneered by Seiberg and Witten [41,42] have lead to many new insights in N = 2, 3, 4 SCFTs and their construction [43][44][45][46]. Clearly, one would like to extend this program further to minimal supersymmetric theories in 4D. The reduced amount of supersymmetry though, does not protect the moduli spaces from non-perturbative corrections anymore which can obstruct the SCFT point. This was also observed e.g. in [29] where it was the goal to construct non-trivial SCFTs using F-theory on non-compact elliptic four-folds similar extending the approach of superconformal matter [37]. Using birational base changes loci of non-minimal singularities in codimension three could be removed which introduced new Kähler moduli. However, the existence of this SCFT point at the origin of the Kähler moduli space might be obstructed by the aforementioned quantum corrections. The corrections originate from Euclidean D3 instantons that wrap the collapsing cycle and mix Kähler and complex structure moduli space of the geometry.
This note wants to take a similar approach to [29] and investigate the (classical) deformation space of elliptic four-folds with non-minimal singularities but in codimension two. Opposed to [29] we do not want to focus solely on the resolution phase via a birational base change but also on the non-flat resolution and (partial) Higgs branches. A major difference of elliptic four-folds as opposed to three-folds is that they can have non-trivial three-form cohomology. As we will show in the following, these cohomology classes are naturally related to non-flat fibers in codimension two. The dimension of this moduli space is given by the additional independent Hodge number h 2,1 . Due to their absence in three-folds and the fact that their dimension is self-dual under mirror symmetry [10] makes the three-fold cohomology interesting in their own right. Unfortunately, these contributions have not been given much attention in the literature apart from [8,9]. Moreover, in the application to 4D SCFTs, this contribution yields yet another part in the moduli space of the four-fold which might mix with Kähler and complex structure moduli under quantum corrections.
The non-flat contributions we are considering in this work appear in codimension two which makes them very analogous to the superconformal matter in three-folds. Indeed, one might construct four-folds simply from a three-fold fibered over another P 1 which is a perspective we will large use in this work. Due to this analogy one might wonder weather it is possible to characterize the geometric moduli space of those non-flat fibrations via the 6D/5D SCFT data. Moreover, we want to show that these fibers and hence their non-perturbative contributions can dynamically be created/destroyed by studying conifold transition between four-folds. These transitions generically change the h 2,1 contribution but in fact also all other Hodge numbers. However we will find that transition related to base changes keep the Euler number invariant, which is important in order to connecting 4D F-theory vacua [13,14]. Another class of conifold transitions can be interpreted as a (partial) Higgs branch that keeps the gauge group but removes the non-flat fiber from codimension two. This transition does not fully avoid the non-flat fiber though. Instead they are pushed down to codimension three and hence correspond to points in the base of the four-fold. Scenarios like those were considered in [4] where they were shown to lead to non-perturbative four-point couplings. These superpotential terms are mediated by D1 string instantons between matter curves that meet at the respective points. Those couplings are also present in our cases and involve matter curves that are remnants of the original E-string curve. Moreover, since the same transition admits an origin from a 6D partial Higgs branch the involved matter curve representations of the four-point coupling are enforced geometrically and dictated by 6D anomaly cancellation.
This work is structured as follows: In Section 2 we give an overview of the geometry of elliptic four-folds. We infer the Hodge numbers for non-flat fibrations and link those in Section 2.1 to the 6D/5D SCFT data. Those contribute additional chiral singlets in a four-fold which we review in Section 2.2. In Section 2.3 we also show how (partial) Higgs branches of those SCFTs naturally to the prediction of non-perturbative four-point couplings. Figure 1: The Hodge half-diamond of a CY four-fold and its non-trivial entries. h 2,2 is not independent but related to the other three entries.
In Section 3 we present our two main examples which are families of four-folds with E-string theories that exhibit a (E 6 × U (1))/Z 3 and (SO(10) × U (1) 2 )/Z 4 gauge group. Our main tool to study and construct these compact CY geometries is toric geometry. However we also expect our arguments to be valid beyond those constructions and for non-compact geometries as we. In Section 4 we also give examples with higher rank SCFTs that have an gauge group, to show the validity of our proposal. More details of the toric resolutions are presented in the Appendices A.

Geometric Preliminaries and F-theory
One of the main goals of this note is to give a physical explanation to certain Hodge numbers of elliptic Calabi-Yau four-fold X 4 with non-flat fibers in the context of F/M-theory. For this we start by reviewing the independent Hodge numbers of a Calabi-Yau four-fold. These are depicted in the upper half of the Hodge diamond in Figure 1. A four-fold X 4 admits Kähler and complex structure parameters whose dimensions are counted by h 1,1 (X 4 ) and h 3,1 (X 4 ) analogous to those found in three-folds. Unlike three-folds, four-folds can also exhibit nontrivial three-form cohomology counted by h 2,1 (X 4 ) which is self-dual under mirror symmetry. The dimension of h 2,2 (X 4 ) of a four-fold is not independent but related to the other Hodge numbers as The Euler number can be computed as This quantity will be important later when discussing transitions among smooth four-folds. In addition we are interested in elliptic four-folds with non-flat fibers in codimension two. In order to investigate the contribution of non-flat fibers to the Kähler moduli we first turn to smooth three-folds X 3 . For a smooth elliptic three-fold X 3 we are able to split up the Kähler moduli contributions into pieces that directly connect with the F-theory interpretation. This split is due to the Shioda-Tate-Wazir theorem and given as 3) The first contribution is that of the Mordell-Weil group and the number of n fibral fibral divisors that resolve some ABCDEFG type singularity. Taken together, both parts count the rank of the full F-theory gauge group. Second there is the contribution from base divisors that contribute the universal hypermultiplet and the 6D tensor multiplets. In [5] it was first proposed that non-flat fibrations generally require the addition of new divisor classes in that decomposition. In practice there can be k = 1 . . . n p points in the two-fold base B 2 where the fiber decomposes into E i,k surface components with i = 1 . . . n non-flat,k [3,5,17] as we will further explain in Section 2.1.
Having reviewed the three-fold case, we now turn to four-folds. Inspired from eqn. 2.3 we propose a similar type of Shioda-Tate-Wazir decomposition for four-folds given by The general decomposition is analogous to the three-fold case but we explicitly added contributions from non-flat fibers. In the case of four-folds these are n c codimension two loci in B 3 and as such irreducible Riemann surfaces C α of genus g α with α = 1 . . . n c . Just as for three-folds, these curves admit complex two dimensional surfaces E i,α as their fiber. In analogy to three-folds these non-flat fibers E i,α can be shrunk to singular curves C α in the four-fold geometry controlled by a volume modulus of the surface. Hence these surfaces contribute a additional Kähler parameters to those of the four-fold. Note that there can be additional non-flat degenerations in codimension three on the base B 3 . These types of non-flat fibers are rather generic and do not need to contribute to h 1,1 (X 4 ) in the decomposition of Equation 2.4. The degenerations in codimension three that we are going to study here e.g. in Sections 3 and throughout this paper are relatively mild and indeed do not contribute to h 1,1 (X 4 ).
In the following we want to deduce how non-flat fibers contribute to h 2,1 (X 4 ). Non-flat fibers will in general not be the only source to the three-form cohomology but the one we want to focus on in this work. As we have just extended the non-flat surfaces E i,α to be fibered over a curve their total space is a divisor divisors D i,α with Note that in general, there can be several divisors D i,α labeled with i = 1 . . . n nf,α that restrict to the same α = 1 . . . n c base curves C α . This fibration structure allows to deduce the cohomology of (2.5) from those of fiber and base using the Leray-Hirsch theorem [9,52]. Since the fibers E i,α are compact and connected its Hodge numbers are trivial but h 0,0 (E i,α ) = h 2,2 (E i,α ) = 1. This data is enough to show that the divisors D i,α support a non-trivial one-form cohomology whose dimensions is given by This non-trivial one-form cohomology embeds non-trivially into the four-fold X 4 and is the main source for the three-form cohomology. This fact has been shown in [9,52] via the so called Gysin isomorphism ι. For a smooth toric hypersurface X 4 and under the assumption that D is toric, the Gysin spectral sequence has been evaluated in [52]. The result is that those divisors that admit a non-trivial one-form cohomology essentially inject a non-trivial three-form cohomology in X 4 of that same dimensions Putting the pieces together we deduce the contribution of non-flat fibrations to the threeform cohomology as The Formula (2.8) can be used in general to also construct bases B 3 that exhibit themselves non-trivial three-forms h 2,1 (B 3 ) as has been done in [9]. The trick is simply too look for bases that themselves admit divisors with non-trivial one-form cohomology. This however is not the focus of study for this work. Moreover, although eqn. (2.7) has been proven in the framework of toric hypersurfaces, it is plausible to be of general validity. In the end from an intuitive point we might simply think of those three-forms to come from the sections of the genus g surfaces twisted by the surface component.
As this work mainly focuses on the contribution of non-flat fibers, we will always consider bases with only trivial fibration structure in 3 and 4. Analogous to three-folds and also arguing from a physics perspective in Section 2.1 each of those surface E i,α contributes an additional Kähler parameter 1 . We summarize the contributions of all non-flat surfaces using (2.8) as We can now turn to implications for the physics of F-theory compactifications. The concrete examples presented in the sections below all make use of the toric description in terms of the Batyrev construction. For those exists a nice way to compute the Hodge numbers from the combinatorial data of the polytope ∆ which we review in Appendix B. This allows first to explicitly construct non-flat four-folds via toric geometry and to explicitly show the validity of the expressions above.

(4,6,12) and non-flat resolutions
The structures of non-flat fibers appear naturally when one considers elliptic fibrations that exhibit non-minimal singularities at codimension two (or higher). In the Weierstrass model this amounts to loci where the functions f, g and discriminant ∆ admits a vanishing order in the window 2 (4, 6, 12) ≤ ord van (f, g, ∆) < (8, 12, 24) . (2.10) In order to discuss those cases, we consider three-folds and the occurrence of those singularities in some detail first. In a threefold X 3 these singularities occur over smooth points 1 The exact identification is given again via the Gysin spectral sequence [52] that reads ⊕i,αH 0,0 (Di,α) .. 2 In codimension three [29], this window is bounded by the vanishing orders (12, 24, 36) from above. of the base B 2 and signal the presence of non-perturbative objects that become light in the six dimensional F-theory. The simplest example are those of E-string theories that exist at the lowest end of the window (2.10). These theories are typically engineered by collisions of e 8 and su 1 branes in a point of B 2 . In a threefold X 3 , there are three possibilities to remove those singularities and make the geometry smooth 3 : 1. Performing a complex structure deformation that removes the singularity. This deformation corresponds to the Higgs branch of the 6D strongly coupled theory.
2. Blowing up the intersection point(s) in B 2 until the fiber becomes regular. This resolution corresponds to the tensor branch of the 6D strongly coupled theory.
3. Resolving the fiber in a non-flat way without changing the base. In M-theory this represents a point in the 5D Coulomb branch of the circle reduced 6D SCFT, related via the usual F/M-theory duality.
Note that the two different resolutions (2) and (3) should not be thought of largely different theories as they collapse down to the same singularity. Instead these resolutions should be thought of as different points in the extended Kähler cone of the resolved threefold X 3 [15][16][17] related by flop transitions. Hence their Kähler moduli spaces are the same and thus in 5D the dimension of the Coulomb branches are the same too. Since the 5D Coulomb branch is the sum of the 6D tensor branch plus the rank of the gauge algebra factors we can write For the simplest case of an E-string theory the (4, 6, 12) collision can be avoided by a single blow-up in B 2 which yields dim(Tensor 6D )= 1 and no gauge symmetry. In the same way the non-flat resolution can be performed by a single surface over that point with h 1,1 non-flat = 1 as expected. Other non-trivial higher rank examples are discussed in Section 4.
Having clarified the geometric and physics implications of non-flat fibrations in threefolds we can move to a four-fold. The main point for geometry is that the non-flat resolution over a codimension two locus do not depend on whether it happens over points in a threefold base B 2 or curves in a four-fold base B 3 . The idea is analogous to the resolution of ADE singularities at codimension one in an elliptic K3 and those at codimension one in an elliptic three-fold. The general resolution procedure is exactly the same and they contribute with the same amount of fibral divisors in the Shioda-Tate-Wazir decomposition eqn. 2.3. An important difference in a three-fold though is whether a fiber singularity is (semi-)split or non-split. Such effects are caused by some additional monodromy effects along the codimension one curve in B 2 . Their presence affects the resolution of the fiber singularities by identifying fibral curves and effectively folds the ADE singularity by an outer automorphism to a non-simply laced algebra. Similar to those, one might expect monodromies to also be present in four-folds that act non-trivial on the non-flat fiber. Such cases are not discussed in this work but left for future investigations. In the absence of such monodromies, the number of non-flat resolution surfaces in a four-fold can directly be inferred from those of three-folds X 3 . From a physics perspective this makes sense as the theories might simply be viewed as compactifications of the 6D/(5D) theories on Riemannsurfaces C gα . 4 This physics interpretation allows us to express the contributions of non-flat fibers to h 1,1 (X 4 ) and h 2,1 (X 4 ) in terms of 6D/5D SCFT data from eq. (2.11) and eq. (2.8) as when summing over the n c curves C α in the base. In Section 3 we consider several examples of four-folds with E-string curves as well as higher rank examples in Section 4.

Singlets in M/F-theory duality on elliptic four-folds
In this section we review the contribution of three-form cohomology in the F/M-theory, following [1]. In order to do so we consider M-theory on a four-fold X 4 and consider the contributions that lead to neutral chiral fields in 4D. We first focus on those contributions that solely come from the base B 3 and lift to neutral singlets in the 4D F-theory. There are complex scalars T κ with β = 1 . . . h 1,1 (B 3 ) + 1 that come from the expansion of the Kähler form J and RR four-form C 4 of IIB string theory. In a basis of ω κ ∈ H 1,1 (B 3 ) we expand those as Dualizing the two-forms B κ 2 in 4d gives rise to axions ρ κ that combine with v κ to complexified Kähler moduli T κ . These axions can be gauged and are important for anomaly cancellation. Further chiral singlets originate from complex structure moduli h 3,1 (X 4 ) of the full four-fold. Our main interest though are the chiral singlets that are inherited from h 2,1 (X 4 ) that do not come from the base and in our case are given by These are obtained from the expansion of the M-theory C 3 -form and are denoted by N β with β = 1 . . . h 2,1 (X 4 ) non-flat . These singlets are unfortunately not very well understood. I.e. it is not clear whether the axion parts σ of the N β couple to curvature terms of the form σF ∧ F and σR ∧ R in the 4D effective action. These singlets though appear in an interesting way [47] in the Kähler moduli, given as with w the base intersection form on B 3 and d κ,β,γ a holomorphic function on the complex structure moduli space. The curious observation though is, that the N β enjoys of an additional discrete symmetry N → −N that the other chiral singlet fields do not have. Understanding these fields and their couplings is beyond the scope of this note. Instead this review should serve as a motivation of why these singlets are interesting and that non-flat fibrations naturally produce them.

Higgs branches and non-perturbative couplings
Having discussed the role of the resolutions of non-minimal singularities in three-and fourfolds in Section 2.1 we now want to consider their deformations. These deformations can be of very general type e.g. they can fully remove all singular fibers resulting in a broken gauge group. The specific kind of deformations we want to consider here are those which keep the gauge group G but only create/remove (4, 6, 12) loci at codimension two. For concreteness we fix such a locus to be the vanishing of the ideal I E-string = {z, p}. For this we consider the inverse problem by starting with a three-fold geometry that does not posses any (4,6,12) yet. This geometry should have some non-Abelian gauge algebra G localized over z = 0 and possibly additional Abelian factors. In general these gauge group factors are expected to lead to massless hypers in 6D that carry non-trivial representations R i . These matter representations are found at loci where the vanishing order of the discriminant ∆ in the Weierstrass form enhances. Such loci we denote by the vanishing ideals I matter,i which might be very complicated and not necessarily of complete intersection type. For simplicity we fix one of them to be I matter,0 = {z, p}. At next we perform a complex structure deformation, such that one polynomial factors out one power of z as where b itself is some other polynomial. For some of the matter loci I matter,i we now require that these are themselves non-trivial in {a, p} which we denote as I matter,k (a, z, p). Those ideals we require to become reducible upon the factorization eqn. (2.17) as The deformation eqn. (2.17) forces matter ideals to be moved onto the I matter,0 locus. Since the loci themselves are identified with matter loci, the Weierstrass coefficients (f, g, ∆) admit a non-trivial vanishing order over them before tuning. The factorization in eqn. (2.18) therefore increases the vanishing order of the I matter,0 to the E-string loci I E-string . From the field theory side, such E-string transitions decrease the hypermultiplet sector S and turns them into non-perturbative E-string sector. The way we have set up those transitions above, might seem odd and artificial at first glance. However, those transitions are highly constrained by the 6D anomaly conditions due to the fact that all E-string points should admit a tensor branch where the field theory anomalies are properly canceled. Hence physics tells us that the 6D matter content must change in such a transition and hence the requirement (2.18) is actually very natural to appear. In particular when the gauge group G stays fixed such transitions are essentially unique [3,55]. One obvious constrained is given by the gravitational anomaly which fixes the dimensions of representations in S to k dim(R k ) = 29 as it is the same contribution of a 6D tensor multiplet. Hence the transition can never involve a representation R with dimension larger than 28 5 . Consistency of the physics and hence the geometry allows therefore to deduce the subset of matter ideals I matter,k in I matter,i that admits the correct factorization properties as dictated by the 6D anomalies. These structures generalize in a straight forward manner to four-folds. Indeed, the various polynomials important in the discussion can simply be taken to be sections on the base B 3 . I.e. the same kind of complex structure deformation creates (4, 6, 12) curves specified by the very same codimension two ideals that lead to the factorization of matter curves I matter,k . However we can use this structure in order to make an additional observation in four-folds: The factorization property (2.18) under the deformation (2.17) guarantees the matter loci I matter,0 and I matter,k (a, z, p) to all vanish at the codimension three point(s) z = a = p = 0 before that transition. The prescribed deformation simply ensured that the polynomial a vanishes to first order at z = p = 0 in order to obtain the (4, 6, 12) loci. Hence even when the (4, 6, 12) loci are absent at z = p = 0, it is present at the codimension three locus a = z = p = 0. Indeed this effect is ensured by the reducibility of the matter ideals I matter,k enforced by the 6D E-string transition. As a result we see the (4, 6, 12) curve in the four-fold base B 3 was simply deformed to one codimension lower where all curves of the ideals I matter,k lie on. Enhanced singularities in the elliptic fiber at codimension three are generically interpreted as Yukawa couplings in the 4D superpotential W. In the IIB picture these are induced from intersections of three matter curves in the respective point that is systematically been tracked by the F-theory torus. However when the fiber becomes of (4, 6, 12) type one expects non-perturbative effects to be present similar to the E-string theories in 6D. Indeed codimension three points of such non-minimal type have been studied in [4] where they were shown to lead to gauge invariant four-point couplings in the 4D superpotential. In the IIB picture these couplings are generated by D1 instanton strings that stretch between the involved matter matter curves that meet in the (4, 6, 12) point.
The main point of our geometric construction is, that it allows to interpret those codimension three (4, 6, 12) points naturally in terms of the 6D E-string transition. As the 6D anomalies have dictated the matter in the transition they also fix the very same curves in 4D that meet in the (4, 6, 12) point at codimension three. These points and the matter curves I matter,k that meet them are therefore naturally interpreted as the remnants of the 6D E-string transitions which lead to the non-perturbative four-point coupling of type where representations R k may occur multiple times. E-string transitions that preserve the total gauge group have been classified in [3] including the respective change in the 6D matter spectrum. Hence knowledge can be used to infer the induced 4D non-perturbative couplings.
A simple example is that of an SU (7) × U (1) 6D gauge theory. Anomalies of the 6D tensor branch force the following change in the hypermultiplet sector ∆S = 21 − q 3 ⊕ 7 q ⊕ 1 0 in an E-string transition. By the arguments above, the very same representations must be present in the four-point coupling that resembles the codimension three (4, 6, 12) point in the four-fold. Indeed, the lowest order gauge invariant coupling [57] in the superpotential appears at fourth order as (2.20) In Section 3 geometry and physics of similar examples are discussed in detail.

E-string transitions in three-and four-folds
One way to obtain three-folds with non-flat fibers is to perform conifold transitions [3,5] that originate from those that are flat. We will adopt the very same strategy here for fourfolds. Concretely, we want to perform conifold transitions among three types of (compact) four-folds X 4,a , X 4,b and X 4,c as summarized in Figure 2. To stay close to the analogy of three-folds, we actually construct the four-folds as three-fold fibrations over another P 1 . This allows us to simply extract the three-fold conifold to four-folds. In this way, we are able to use the 6D anomalies of E-string transitions and to extend those to 4D. Note that this is just an auxiliary construction for illustrational purposes i.e. the logic also works for general four-folds. In order to investigate the non-flat fiber structure and its underlying physics, we perform conifold transitions that do not change the codimension one and Mordell-Weil structure i.e the 4D gauge group. We start in X 4,a which admits a non-flat fiber in codimension three which is of course absent in the analogous three-fold X 3,a . We then perform a complex structure deformation to enhance this non-flat fiber to curves in one codimension higher.
Resolving the geometry fully leads to X 4,b . These four-folds will exhibit in general a nontrivial three-form cohomology corresponding to the non-flat surface. In this section we consider E-strings curves only and hence the three-form contribution corresponds directly to the genus of the base curve. The final transition corresponds to a birational base change that removes all non-flat fibers. Keeping track of all Hodge numbers of both three and four-folds allows the computation of the Euler number too. Both in three-and four-folds we will observe how the first transition changes the Euler numbers while the second one does not. Our main example is going to be a (E 6 × U (1))/Z 3 gauge theory. Here we focus on the details of the singular model and present details of the resolution in Appendix A. Section 3.2 concerns similar configurations with gauge group (SO(10) × U (1) × U (1))/Z 4 . Besides being richer in structure which we present more technical details of the resolved geometry directly. All four-folds of consideration are compact and all Hodge numbers are computed via the Batyrev formulas.

Example
The starting theory we want to engineer is that of an (E 6 ×U (1))/Z 3 6D gauge theory whose spectrum admits massless hypermultiplets in the representations Non-flat in codim 2 Non-flat in codim 3 Flat Figure 2: Summary of a chain of conifold transitions in elliptic four-folds. The starting four-fold X 4,a is constructed from a three-fold X 3,a fibered over a P 1 . We then perform conifolds in four-folds inherited from the three-folds. Conifold 1 renders X 4,b non-flat and conifold two removes the non-flat fiber again by a birational base change (partially).
In order to engineer this theory in F-theory we start with the Tate-model , given as The affine coordinates [Y, X, Z] describe the elliptic fiber in an P 2 3,2,1 ambient space. The divisor Z = 0 is the zero-section and a i sections of line bundles of the base, fixed to be powers in the first Chern class of the base [a i ] ∈ c 1 (B) i . The Weierstrass coefficients can easily be computed by completing the square and cube in X and Y which allows a mapping into the simpler Weierstrass form where Weierstrass coefficients and discriminant is related to the Tate-form as We start by engineering an u 1 gauge factor by setting a 6 = 0 globally which leads to an additional holomorphic section s 1 of the torus fiber at and hence a non-trivial Mordell-Weil group. Matter that is charged under this gauge factor is found at the locus a 3 = a 4 = 0 where the fiber becomes of I 2 type. The e 6 factor is engineered over Z : {z = 0} according to the Tate-classification (e.g. see [20]) by employing the factorization of the Tate-coefficients which can be shown to lead to type IV split fibers and hence an e 6 over z = 0. Matter in the 27 1 3 representation 6 is found at the locus z = a 3,2 = 0. Up to now we have not specified the dimension of the base B and hence the CY manifold. We start with a three-fold X 3,a where the matter loci are just points that are counted by intersecting their divisor classes in the base cohomology. For some generic base 7 this leads to the multiplicities where g denotes the genus of the curve Z. The resulting threefold geometry X 3,a admits only minimal singularities. Hence the geometry X 3,a can be fully resolved in a flat manner which we demonstrate in Appendix A. An explicit geometry Y 3,a with F 4 base and e 6 placed over the −4 curve is also given in Appendix A. Using (3.7) one can compute the spectrum which admits two 27 1 3 -plets, 106 charged-and 163 uncharged singlets. The Hodge numbers of this geometry are given as (3.8) The Kähler parameters are exactly as expected from the Shioda-Tate-Wazir theorem and the complex structure parameters precisely contribute the amount of neutral singlets needed for anomaly cancellation.
Next we want to tune E-strings along the e 6 gauge group by further performing the factorization Counting the degrees of freedom we found this change to come at the cost of complex structures in X 3,b . When plugging the factorization into (3.4) one finds that one still obtains an e 6 type of gauge group over {z = 0}. However now the former matter locus z = a 3,2 = 0 got enhanced from a (3,5,9) to a non-minimal (4, 6, 12) singularity. Hence in X 3,b , the 27 1

3
-plets got exchanged to E-string theories. Note that the tuning also involved a change in classes of the 1 1 matter loci. The full charged matter spectrum of the 6D model is given as We summarize the full change in the hypermultiplet sector S in the transition as  locus which merges the local su 2 and e 7 enhancement into a non-flat fiber.
As already implicitly used in the considerations of anomalies before, a well defined SUGRA theory is only obtained when we get rid of those n E-string E-string contributions. This can be done blowing up the base exactly n E-string times, which brings us to geometry X 3,c . Since both theories X 3,a and X 3,c are consistent SUGRA theories one can analyze the possible representations that get lost in such an E-string transitions on very general grounds [3]. From the change of the 6D anomaly lattice one can derive the change of the spectrum in an (E 6 × U (1))Z 3 gauge group which in general must be given by the representations as it is the case in the geometry above. Note that the transition requires a neutral singlet for each E-string it produces. This can be directly checked when considering the smooth non-flat resolution of X 3,b . The exact toric resolution is given in Appendix A as well es for the concrete three-fold Y 3,b in detail. A schematic depiction of the fiber properties is shown in Figure 3. There the non-flat surface component is highlighted as a red box, as well as its reducible components. In the example one observes these reducible components to arrange in a closed loop that is attached to other fibral divisors [15][16][17]. Computing the Hodge numbers for Y 3,b we find (3.14) These Hodge numbers are consistent with the considerations made earlier. We remember that we have created two E-string type non-flat fibers that contribute one Kähler class each. This tuning though came at the cost of one complex structure for each of them. To remove the non-flat fibers/E-string points we can blow-up the −4 curve which results in a −6 curve.
In the toric setup we demonstrate this in Appendix A for geometry Y 3,c and show that it admits the same Hodge numbers (3.17) as three-fold Y 3,b .

The four-fold compactification
Having clarified the conifold transitions in the three-fold geometries and their associated Ftheory physics, we can perform the same strategy in four-folds and consider their associated 4D physics. To keep matters close to 6D we construct the four-fold as a three-fold fibration over another P 1 . In general we have lots of freedom to fiber X 3 over the new P 1 . For simplicity we chose this fibration to be trivial, which extends to the four-fold base as The matter loci of X 4,a are readily obtained from X 3,a . These codimension two loci become Riemann-surfaces whose genus can be computed explicitly as . In comparison to Y 3,a there is exactly one more Kähler class of the additional base P 1 .
Contributions to h 2,1 (Y 4,a ) are trivial, as the fibration is flat and the base with h 2,1 (F 4 ×P 1 ) = 0 does not contribute either. When moving to the matter sector we find the two 27 1 3 plets in 6D to be localized over the genus g = 5 curve in the B 3 base and similarly the singlets over a curve of genus g 1 1 = 1893.
Before performing the first conifold transition, we consider interaction terms between the charged fields present in the 4D superpotential W. A key feature of F-theory which makes it also relevant for phenomenological model building is the presence of Yukawa couplings that are suppressed in perturbative IIB string theory [21][22][23][24][25][26]. An example is that of the top quark in an su 5 theory. This coupling is schematically given by couplings of type 10 · 10 · 5. In F-theory these points lead to type IV * fiber degenerations. Those degenerations highlight local IIB string couplings of order one and hence strongly coupled in nature. Similarly, when one further unhiggses the su 5 to e 6 there are couplings of type 27 · 27 · 27. Over these points, the fiber enhances to a type II * singularity, i.e. a point of e 8 . 8 This fact can be readily checked when adding the term Z 6 z 5 a 6,5 to the Tate-model which breaks the u 1 gauge symmetry. In the following we will call this four-fold with only E 6 gauge group, X 4,0 . For a fully resolved model we have depicted fiber structure of the e 8 point in Figure 4. Note that the exact fiber topology in a smooth model might not be of precise e 8 shape. Instead the fiber structure might only be a bouquet which misses a couple of e 8 nodes [27]. This effect however does not to obstruct the phenomenological implications of the Yukawa couplings in [11,28]. 9 By setting the polynomial a 6,5 in the Tate-model to zero restores the global u 1 gauge factor under which the 27-plets are non-trivially charged. From a field theory perspective one therefore expects the 27 3 Yukawa coupling to be forbidden. Therefore one might come to the conclusion that those intersection points should be absent in a given geometry. However, what we actually observe is, that the e 8 Yukawa points enhance to (4, 6, 12) as discussed in that is N Yukawa (Y 4,a ) = 152 respectively. Finally we comment on the fiber structure of the resolved model X 4,a . Here the non-flat Yukawa point is depicted in Figure 4. As opposed to the codimension one non-flat fibers, these do not contribute additional Kähler parameters as they simply come from the degeneration of one of the exceptional e 6 resolution divisors.

Enhancing to non-flat fibers in codimension two
In the next step we want to perform the matter-transition we did in the three-fold given by (3.9) now in the four-fold geometry X 4,b . From the four-fold perspective, this tuning pushes the codimension three (4,6,12) singularity one codimension higher and enhances the 27 1 In the concrete example of four-fold Y 4,b that is detailed in Appendix A we compute the Hodge numbers as These Hodge numbers are to be understood from the general discussion in Section 2: First there is the E-string theory whose non-flat surface which contributes a single new Kähler class as its 6D rank is one. Note that we have started with two 6D E-string theories, that both contributed an individual Kähler class. Since both contributions are merged along the same genus-five surface in the base, they now contribute a single Kähler class. Second there is the contribution of the five h 2,1 three-forms. These are inherited from the rank one theory multiplied by the genus of its curve. Finally, we can discuss the removal of these non-flat surfaces by changing the B 3 base analogous to the 6D case. We take two equivalent ways by either blowing up the base B 2 to get X 3,c and then compactify this configuration over another P 1 or by blowing up the four-fold base in X 4,b in the B 2 direction to get X 4,c . In both cases we end up with a base B 3 = (BLB 2 ) × P 1 . Notably X 3,c admits no E-string points anymore which might not the case for the four-fold X 4,c . This can be seen from realizing that the E-string curve class in the four-fold is given by (3.22) When we pullback 3c 1 (B 2 ) · Z ∼ 2Z 2 from B 2 this is enough to require absence of E-string points in a three-fold but not sufficient for the four-fold anymore. In order to do so we also need to require the relation H 1 · Z ∼ 0 by e.g. perform additional blow-ups in the base. Hence even when started with a non-higgsable cluster in 6D, i.e. E 6 on a −6 curve that admits neither matter and is fully flat, the new P 1 direction introduce them. The same is true for the explicit non-flat resolution in Y 4,c where we have performed a base change solely in the B 2 direction. Performing this resolution has split the respective genus five in B 3 into six genus-zero curves as depicted in Figure 5. The full four-fold geometry is given in Appendix A and admits the Hodge numbers The Hodge numbers make sense as we have introduced two more resolution divisors to B 2 . This change in the base has split up the single non-flat surface into six disconnected genus zero curves. As all of them host a non-flat fiber, they all contribute additional Kähler parameters to Y 4,c . None of the non-flat surfaces contribute to h 2,1 (Y 4,c ) anymore. In order to obtain the fully flat four-fold, six more base blowups are required to split to remove all genus-zero curves. The exact polytope of Y 4,c is again given in the appendix and it is fully flat. Moreover it admits exactly the same Hodge numbers as Y 4,c given in (3.23). Finally we are in the position to compare the transition between three-folds and those between four-folds. While removing non-flat fibers of Y 3,b via a base change to Y 3,c has not changed the Hodge numbers at all, this was not the case in the analogous transition of fourfolds. Here in fact virtually all Hodge numbers where involved in the transition. However an important observations is that all those transitions left the Euler number invariant. This observation is particularly important when it comes to the inclusion of G 4 flux which we have not addressed in this work. The relevant point is that it makes it easier to satisfy the change in the tadpole cancellation conditions eqn. (5.1) when a consistent G 4 flux configuration has been found in either phase. We will comment on this point again in the conclusion.

Example II:
The second example works similar as the one before but admits a more rich matter structure. Moreover we use the chance to directly give more attention to the explicit toric resolution of the geometry. Again we use E-string transitions in 6D to understand four-point Yukawa couplings and the general change of the matter spectrum in 4D. The model at hand admits two u 1 gauge factors that are engineered with the most general elliptic model with three sections [30]. This model is given by the elliptic curve in an dP 2 which we parametrize by the coordinates {u, v, w} and the two exceptional coordinates e 1 , e 2 . An additional so 10 singularity over Z : x 0 = 0 can be engineered via a toric top [5,6] which we resolve directly via the exceptional coordinates f 2 , f 3 , f 4 , g 1 , g 2 and f 0 being the affine node. The resolved hypersurface 11 is given as The d i are to be interpreted as generalized Tate Table 1: Summary of the 6D matter spectrum. We also give the respective codimension two loci and their multiplicities using eqn. (3.26). The longer (quotient) ideals I 8,9,10 are given in eqn. (3.27).
This model has been analyzed in detail in [5] where more details of the exact spectrum computation are given. We choose a simple two-fold base as B 2 = F 0 and a fibration with bundle choice of the base sections d i : where again Z is the so 10 divisor and H 1 and H 2 the two classes of F 0 . With the help of the Weierstrass model one can find the reducible fiber components which give rise to the 6D matter. In Table 1 we have summarized multiplicity and matter representations. I i singlet loci that are given as: F 0 base y 1 (0,0,0,1) y 0 (0,0,0,-1) x 1 (0,0,-1,0) so 10 top f 0 (0,0,1,0) f 2 (1,0,1,0) f 3 (0,1,1,0) f 4 (1,1,1,0) g 1 (1,1,2,0) g 2 (1,2,2,0) , (3.28) Note that the elliptic fibration structure is inherited from a dP 2 fibration of the ambient spaces. This allows also to directly obtain the projection π from the ambient variety given via a projection onto the two last coordinates in ∆. Using the 42 + 1 neutral singlets also allows to show cancellation of the gravitational anomaly. Next we want to perform an Estring transition which we do by factorizing another power of the so 10 divisor x 0 out of the base section d 1 . This tuning effectively merges two 10 1/2,0 -plets and two 16 −1/4,−1/2 as well as two 1 1,1 and 1 0,1 -plets as one can explicitly see from Table (1) and eqn. (3.27). After this transition we have two non-minimal singularities over d 5 = x 0 = 0. Lets again summarize the total change in the spectrum S This transition is fully consistent with the expectation of a tensor-matter transitions derived in [3] The geometry that admits the two (4, 6, 12) points can be resolved using toric geometry. This is done by simply adding the vertex vertex f 1 = (0, 2, 1, 0) to the polytope (3.28). The addition of the vertex respects the fibration and does not change the base while also leaving the new polytope reflexive. Hence the the anti-canonical hypersurface is still Calabi-Yau, which we denote by Y 3,b with Hodge numbers: In comparison to Y 3,a we find the expected loss of two complex structure parameters that are traded for the two Kähler parameters that come fro the non-flat fibers. In the toric realization, those are given by the ambient divisor f 3 = 0 that intersect the base twice. This can be seen from the new hypersurface which is explicitly given as A comment from the perspective of the toric geometry is in order. Here the generic fiber structure parametrized by u, v, w, e 1 , e 2 has not changed at all but the modification appears only at codimension two over The tuning process and the resulting fiber structures are summarized in Figure 6. From the perspective of the 6D F-theory compactification, the E-string points can be avoided by going to the tensor branch. The blow-up of the ambient variety can be done explicitly by adding the vertices xb 1 : (0, 1, 1, 1), xb 2 : (0, 1, 1, 1) to the polytope which gives a new three-fold Y 3,c with Hodge numbers These Hodge numbers are identical to those, of the non-flat model as expected. The resulting fibration has been made fully flat and anomalies can be shown to cancel as expected.

The four-fold compactification
We follow the same strategy as in Section 3.1 and fiber the three-fold over another P 1 to obtain the four-fold Y 4,a . Torically this is done by simply adding a fifth direction in the polytope (3.28) and by adding the vertices z 0 : (0, 0, 0, 0, 1), z 1 : (0, 0, 0, 0, −1) to fill out the full lattice Z 5 . The fibration structure is still inherited from the ambient variety, where the projection is given onto the last three columns resulting in the base B 3 = (P 1 ) 3 . Again we do not expect to find any three-form cohomology in Y 4,a as the fibration is flat in codimension two and the base B 3 too simple to admit h 2,1 contributions. This is readily checked by computing the Hodge numbers via the Batyrev prescription For convenience we also give the leading order coefficients of the Weierstrass model in x 0 = 0 as with R i being some residual polynomials. The discriminant can be used to find the perturbative Yukawa and non-perturbative four-point couplings. E.g. there are (3,5,9) points when d 7 = 0 and d 9 = 0 which give rise to the expected trilinear Yukawa couplings of the type  Table 1 and (3.27) we find the matter curves to intersect at this point. Similar as in the sections before, these loci are to be interpreted as non-perturbative four-point coupling points [4] generated by D1 instantons. Interestingly, this locus allows for two independent four-point couplings that are possible via the involved representations. The two possible coupling are schematically depicted as W 16 −1/4,−1/2 · 16 −1/4,−1/2 · 10 1/2,0 · 1 0,1 From the perspective of the smooth geometry, given via (3.24), one finds that it is the f 3 = 0 fibral divisor that splits into a non-flat component over d 1 = d 5 = 0. This comes as no surprise as it is going to be the same (ambient divisor) that is pushed to a non-flat surface at codimension two, when performing the conifold transition.

Enhancing to non-flat fibers in codimension two
Now we perform the four-fold analog of the 6D tensor transition which enhances the 16 −1/4,−1/2 curve over x 0 = d 5 = 0 to a (4,6,12) curve via the factorization d 1 → d 1 x 0 , resulting in the four-fold Y 4,b This transition again simply merges parts of the 10-plet and singlet matter curves into that of 16-over which the fiber attains a non-minimal singularity. In the smooth geometry the non-minimal singularity over that curve is resolved by the f 3 non-flat surface, analogous to the three-fold case. In this regard it makes again sense to interpret the non-flat matter curve as the compactification of 6D E-string theories. From the perspective of toric geometry the four-fold Y 4,b is simply obtained by adding the same toric vertex f 1 = (0, 2, 1, 0, 0) to the 5d polytope with CY hypersurface X 4,b which admits the Hodge numbers (h 1,1 , h 2,1 , h 3,1 ) χ (Y 4,b ) = (12, 1, 133) 912 . (3.39) The above data is exactly as expected: First there is the new class that contributes that non-flat surface at f 3 = 0 over the genus-one curve x 0 = d 5 = 0 which also contributes in h 2,1 (Y 4,b ). Finally we want to get rid of this non-flat fiber again be performing a base change of B 3 = (P 1 ) 3 . In the first step we want to deform the curve x 0 = d 5 = 0 by performing a complex structure deformation in the d 5 followed by a blow-up. For this we remember that Hence by a complex structure deformation we enforce first the factorization to the full 5D polytope. Note that the base has become B 3 = (BL 2 F 0 ) × P 1 and admits a triangulation with Stanley-Reisner ideal: The change in the Hodge numbers is explained as before: First there are two more classes from the two blow-ups which has removed all h 2,1 contributions. This genus one curve has split into two P 1 with a non-flat fiber over each of them. Both of these fibers contribute an independent non-flat fiber, which is responsible for the third additional Kähler parameter.
If we want to also get rid of those we need a base that forbids the intersections x 0 = z i = 0 as well which can be done with yet two more blow-ups. The two additional rays that are to be added to the polytope that do the job are given as In the above ideal we have underlined the components that forbid the non-flat curve classes when compared to the blow-down. We conclude by making the same important observation as in the example from Section 3.1: In the transition from Y 4,b to Y 4,c all Hodge numbers and in particular h 2,1 change. However the change is always such that the Euler numbers stay inert. This observation is again important when including G 4 fluxes and to satisfy the condition in eqn. (5.1) which we will comment on in Section 5.
4 High rank cases: In this section we want to consider more complicated examples that admit multiple curves with different superconformal matter curves over each to demonstrate the validity of eqn. (2.12). For this we engineer a generalized Tate model, with an e 8 , an su 2 and an su 3 divisor via Here, the Arten Tate-algorithm can be used to bring the above model into Weierstrass form which makes it easier to read off the singularity structure. The Weierstrass coefficients are given as The zero-section is given by Z = 0 in the generalized Tate-model is not a trivial section of B 3 anymore. This results in the pre-factors b 1 and b 2 to be non-trivial divisors of the base whose zero-locus gives the additional su 3 and su 2 singularities respectively. The base classes of b 1 and b 2 parametrize two new line bundle classes denoted by Z su3 and Z su2 . Engineering the additional e 8 singularity over Z e8 : z = 0 in Tate form is standard and can be taken from the literature (e.g. [20]) by factorizing {a 1 , a 2 , a 3 , a 4 , a 6 } → {za 1,1 , z 2 a 2,2 , z 3 a 3,3 , z 4 a 4,4 , z 5 a 6,5 } . The option to allow for Z su 2 and Z su 3 shifts the classes a i such that one ends up with  as e.g. discussed in [15,16]. In general [e 8 su n ] conformal matter gives rise to a ord van (f, g, ∆) = (4, 6, 12 + n) collisions in the Weierstrass model. The tensor branches of those models have been analyzed in detail in [37]. They are given via a linear chain of P 1 's with self-intersection (−n) given as The superscript denotes additional su k gauge algebra factors hat are enforced in the Weierstrass model upon the base change. Note that bi fundamental matter lies in between these nodes. These codimension two collisions can be resolved by a non-flat fiber. From the ndimensional 6D tensor branch and upon collecting the contributions of the gauge algebra factors the full 5D coulomb branch dimension is given as dim (Coulomb 5D ([e 8 su n ])) = 1 + 1 2 n(n − 1) . (4.7) As argued in Section 2.1 the non-flat fiber resolution must include the same amount of surface contributions with their own Kähler classes. In order to compute the contribution to h 2,1 as well, requires the computation of the genus of those curves in B 3 , given as This can be explicitly checked by constructing a family of four-folds that exhibit a simple toric description. The simplest base to take is B 3 = P 3 . In terms of the base hyperplane class H 1 we can fix all line bundle choices as Here the n sun are positive integers bounded by effectiveness of all b i , a i,k in (4.4). For those constructions, the base is again too simple to contribute h 2,1 . It is also important to note that this particular geometry does not come from a direct compactification of an already present 6D theory. Hodge numbers of the family of four-folds X 4 (n su 2 , n su 3 ) are given as h 1,1 (X 4 (n su 2 , n su 3 )) =10 + (1 − δ (19−2nsu 2 −3nsu 3 ,0) ) + 3(1 − δ (nsu 2 ,0) ) + 6(1 − δ (nsu 3 ,0) ) , h 2,1 (X 4 (n su 2 , n su 3 )) =g E-string (1 − δ (19−2nsu 2 −3nsu 3 ,0) ) +2g e 8 su 2 (1 − δ (nsu 2 ,0) ) + 4g e 8 su 3 (1 − δ (nsu 3 ,0) ) . This is double checked via the Batyrev construction. The 5D polytope ∆(n su 3 , n su 2 ) that realizes the two-parameter family of four-folds X 4 (n su 3 , n su 2 ) is summarized in Table 2.

Conclusion and Outlook
This note has considered a systematic analysis of smooth elliptic four-folds that exhibit non-flat fibers. First we have shown how non-flat fibers in codimension two contribute to the Hodge numbers and in particular to the three-form cohomology. Via the M-theory duality such three-forms lead to additional chiral singlets. In F/M-theory these non-flat configurations are to be interpreted as compactifications of 6D/5D superconformal matter theories on a Riemann-surface. This allowed us to identify their contributions to the Hodge numbers in terms of the 6D/5D tensor(coulomb) branch dimensions and the genus of the Riemann-surface. The validity of this proposal is checked for several examples which include 6D SCFTs of various ranks. Furthermore we have investigated conifold transitions among four-folds that remove those non-flat fibers and hence these specific chiral singlet fields. The first branch of these transitions is analogous to a 6D tensor branch and corresponds to a birational base change. This transition changes all Hodge numbers but most notably does not change the Euler number of the compact model. Note that this work has focused primarily on the geometric aspects and not the inclusion of G 4 flux. The fact that the Euler number does not change in such transitions simplifies the matching of 4D SUSY vacua substantially [13,14] due to the vanishing of the right hand side of Euler number preserving transitions therefore do not require a to change the G 4 flux (norm) or number of D3 branes during the transition. The second type of transitions we have considered is analogous to a (partial) 6D Higgs branch that keeps the total gauge group but moves the non-flat fiber from curves down to points of the base B 3 . These points lead to non-perturbative four-point matter couplings, mediated by D1 instantons and do change the Euler number. The existence of these non-perturbative interaction points is enforced geometrically and can be interpreted as a remnant of the 6D E-string theory. In fact we have argued that the matter representations involved in such couplings can be deduced due to the anomalies of the 6D E-string transition. This work serves as a first step towards the investigation of 4D/3D theories obtained from F/M-theory on non-flat elliptic four-folds. From here on there are several directions to go in the future. First it is important to fully include G 4 fluxes. As argued, the right starting point might be the phase without those fibers compute a consistent configuration and perform the conifold to the non-flat configuration. Moreover in order to fully understand the 4D theory and its possible non-perturbative effects, a better understanding of the 4D effective action is desirable. As analyzed in [29], Euclidean D3 instantons can lead to a mixing of the complex structure and Kähler moduli. This work shows that three-forms are a potential third contribution to those moduli sectors that are characteristic to non-flat resolution. Understanding all those effects is important to clarify how quantum corrections might obstruct 4D SCFT points in the IR. As the four-fold is fully smooth those contributions are best analyzed in the 3D M-theory. There are also further generalizations possible from pure geometric point of view. These include the addition of monodromies that act on the superconformal matter curves, analogous to to split fibers in elliptic three-folds. Such effect would naturally incorporate the folding action familiar from twisted compactifications [53] into the four-dimensional picture. Finally it would also be very interesting to explicitly construct the heterotic duals of those theories and match the contributions of h 2,1 non-flat (X 4 ) to that of the NS5 branes [31].
The intersections allow us to compute the U (1) charges by using the smooth geometry in M-theory and lift it back to F-theory. For this we employ the Shioda map, that is the divisor that gives the U (1) generator in the M-theory expansion of the C 3 form. For simplicity we focus here on the fibral part, which is given in the difference of the two sections. By then demanding orthogonality with respect to all other fibral divisors leaves us with the following linear combination σ(s 1 ) = [e 1 ] − [Z] + 1 3 (4f 1 + 5g 3 + 6h 1 + 4g 2 + 2f 2 + 3g 1 ) . (A. 3) The above concrete form allows us to compute the explicit charges of matter multiplets by intersecting it with the reducible fibral curves at codimension two. E.g. first there are the 1 1 over a 4,3 = a 3,2 = 0. Here one can see, that the coordinate e 1 = 0 becomes reducible. The fiber topology is that of an I 2 curve as given in Figure 4. Since e 1 = 0 is a −1 curve on the fiber ambient space which coincides with the U (1) charge of the singlets using eqn.(A.3). Then there are the 27 1 3 -plets over z = a 3,2 = 0. This can be seen by noting, that the fiber here becomes of e 7 type. From the reducible fiber components one can deduce the e 6 charges using (A.3). For four-folds there is an additional enhancement locus that is of interest. A codimension three by first setting a 3,2 = a 4,3 = 0, where the hypersurface becomes p = e 1 f 1 (g 3 Y 2 + e 1 f 2 2 g 2 X 3 + a 1,1 f 0 f 2 g 1 g 2 g 3 h 1 XY Z + a 2,2 f 2 0 f 2 2 g 2 1 g 2 2 g 3 h 2 1 X 2 Z 2 ) , (A.4) This is also the 1 1 singlet locus, but note that the f 1 e 6 component also factors out. As f 1 = 0 restricts onto the z = 0 locus in the base, setting it to zero is a codimension three locus. Moreover, coordinates {f 2 , g 2 , g 3 , Y, e 1 } are fully unrestricted and they parametrize a two-dimensional surface. In Figure 4 we have computed all intersections of the fibral components over that locus. In the red box we have computed the intersections of the five components that sit inside the non-flat surface f 1 = 0. This is unlike the shape a regular 14 Dynkin diagram shaped fiber should have but it is characteristic for a non-flat fiber. Secondly we the enhanced model with a non-flat fiber at codimension two. This is done, by further factorizing one power of z out of a 4,3 as zb 4,4 . This leads to the codimension two non-flat curve over z = a 3,2 = 0. Note that we use the toric resolution of the fiber as given via the top construction that we are using to construct full three-and four-folds in the next section. We have added the new fibral coordinate f 3 which removes f 1 as one of the e 6 fibral divisors. Notably, it is still present in the ambient space and will be introduced as a non-flat fiber momentarily. The new resolved fibration is given as p = e 1 f 1 f 2 3 g 3 Y 2 + e 2 1 f 1 f 2 2 g 2 X 3 + a 1,1 e 1 f 0 f 1 f 2 f 3 g 1 g 2 g 3 h 1 XY Z + a 2,2 e 1 f 2 0 f 1 f 2 2 g 2 1 g 2 2 g 3 h 2 1 X 2 Z 2 + 3,2 f 2 0 g 1 Y Z 3 + b 4,4 f 4 0 f 1 f 2 2 g 4 1 g 3 2 g 2 3 h 4 1 XZ 4 . All intersections of the e 6 fiber components as well as the 1 1 singlets can be computed as usual and their structure has not changed at all. Notably, over a 3,2 = 0 the fiber becomes P = f 1 (e 2 1 f 2 2 g 2 X 3 + e 1 f 2 3 g 3 Y 2 + a 1,1 e 1 f 0 f 2 f 3 g 1 g 2 g 3 h 1 XY Z + a 2,2 e 1 f 2 0 f 2 2 g 2 1 g 2 2 g 3 h 2 1 X 2 Z 2 + b 4,4 f 4 0 f 2 2 g 4 1 g 3 2 g 2 3 h 4 1 XZ 4 ) . (A.7) We have exactly the same behavior as before, that is f 1 = 0 is a non-flat fiber component but this time it happens over codimension two. Note also that it is exactly the same ambient space coordinate the appears here. Using the SRI above, one can also compute the intersections of the fibral components as given in Figure 5. There we observe again a similar intersection picture as in codimension three, that is a loop of components that sit inside the non-flat surface.

A.2 Explicit Toric three-and four-fold
We give the explicit toric polytope for the three-folds and four-folds that are considered in Section 3. The toric rays of the polytope are given in the tables below. The threefold X 3 is constructed from a regular fine star triangulation of the toric fan associated to the Batyrev polytope ∆. These polytopes are a combination of various rays as given below. The projection onto the bases B is inherited from the toric ambient space and we fix it to be onto the last two coordinates. We then give the various polytope ingredients below. The U(1) restricted Tate-model is a combination of the e 6 top and completed with the choice of a base. The e 6 resolution divisor becomes non-flat upon adding the ray f 1 and the non-flat fibers can be avoided by adding the two −1 curves that correspond to the toric rays b 1 and b 2 to the base. Note that these are next to the −4 curve on which the e 6 is localized and thus reduces its self-intersection by one each time.
From the data above, the polytope ∆ can be composed that allows an easy computation of Hodge numbers via the Batyrev formula as: B Review: Hodge numbers from polytopes The main framework, that we use in order to construct elliptic three-and fourfolds are going to be hypersurfaces in toric varieties. The toric ambient space is encoded by a regular fine start triangulation of the polytope ∆. In particular for fourfolds, the Hodge numbers can be easily computed the polytope [10] with θ being faces of ∆ and δ * of its polar ∆ * . l denotes points and l * interior ones of the respective facets. The sum in the last terms goes over pairs of dual facets θ.