Phases of 5d SCFTs from M-/F-theory on Non-Flat Fibrations

We initiate the systematic investigation of non-flat resolutions of non-minimal singularities in elliptically fibered Calabi--Yau threefolds. Compactification of M-theory on these geometries provides an alternative approach to studying phases of five-dimensional superconformal field theories (5d SCFTs). We argue that such resolutions capture non-trivial holonomies in the circle reduction of the 6d conformal matter theory that is the F-theory interpretation of the singular fibration. As these holonomies become mass deformations in the 5d theory, non-flat resolutions furnish a novel method in the attempt to classify 5d SCFTs through 6d SCFTs on a circle. A particularly pleasant aspect of this proposal is the explicit embedding of the 5d SCFT's enhanced flavor group inside that of the parent 6d SCFT, which can be read off from the geometry. We demonstrate these features in toric examples which realize 5d theories up to rank four.


Introduction
One of the remarkable achievements of string theory is to provide evidence for the existence of higher dimensional super(symmetric-)conformal field theories (SCFTs). Such theories are always strongly coupled and lack a canonical Lagrangian description, thus making them difficult to approach using traditional quantum field theory techniques. Compared to these methods, the crucial advantage of string theory comes from the geometrization of supersymmetric gauge dynamics-including non-perturbative effects-which via dualities can be described by various brane constructions. Perhaps one of the most recent success stories in this context is the classification of 6d N = (1, 0) SCFTs via compactifications of string and M-/F-theory .
Renormalization group (RG) flows triggered by mass deformations connect different 5d N = 1 supersymmetric gauge theories. In the moduli space of each of these different gauge theories a 5d UV fixed point can be present. On the Coulomb branch of such gauge theories, the dynamics is described by a supersymmetric quantity called prepotential, F, which is a realvalued function in the Coulomb branch parameters. Its derivatives encode the metric on the Coulomb branch, the related kinetic terms, as well as the tension of non-perturbative objects such as monopole strings. Traditionally, one can infer the existence of a 5d conformal fixed point as well as the transitions between different phases of the theory by positivity conditions of derivatives of F [22]. Using geometric engineering, these conditions have been translated into properties of Calabi-Yau threefolds [60,61,63,64] and subsequently refined in [65,66], setting up the stage for a classification program which was recently successfully applied to 5d honest 5d fixed point, because morally speaking, the degrees of freedom in the UV reassemble into those of a 6d SCFT (hence the name). However, by mass-deforming a 5d KK-theory, which from the circle reduction perspective amounts to turning on non-trivial holonomies, one can now flow to a different 5d theory that has a genuine 5d UV fixed point.
Given that all 6d SCFTs are classified by F-theory, one way to test the above conjecture would be, at least in principle, to dimensionally reduce all F-theory constructions and study the possible 5d theories obtainable by mass deformation and RG flow. From a field theoretic perspective, this is anything but an easy task. The difficulty comes from our incomplete understanding of strongly coupled dynamics, in particular what the possible mass deformations of a given 6d SCFT on an S 1 are.
From a stringy perspective, however, the geometric construction allows us to relate 6d and 5d theories via the duality between F-and M-theory [7,69,70]: M-theory on elliptically fibered Calabi-Yau threefold Y 3 ∼ = F-theory on Y 3 × S 1 . (1.1) For weakly coupled 6d theories, the corresponding threefold has minimal singularities according to the Kodaira classification. Under the duality, (partial) blow-up resolutions of these singularities correspond to turning on holonomies of 6d gauge fields along the S 1 when descending to 5d, which pushes the 5d theory onto its Coulomb branch.
On the other hand, 6d SCFTs are known to arise from non-minimal singularities of the elliptic fibration [9,17] over codimension 1 two loci, i.e., points p in the base. In terms of a Weierstrass model of π : Y 3 → B 2 , with discriminant δ = 27g 2 + 4f 3 , such singularities are characterized by the vanishing orders ord(f | p , g| p , δ| p ) ≥ (4, 6, 12) (1. 3) of the Weierstrass functions at p. To make sense of such singularities, one typically blows up such points p in the base into a collection of rational curves Σ i , until the resulting total space has only minimal singularities. Physically, this corresponds to pushing the 6d SCFT onto its weakly coupled tensor branch. Indeed, the restriction coming from the geometry on how such base blow-ups are compatible with the Calabi-Yau condition on Y 3 is the basis for the 6d classification [19,20].
Given the better handle-both physically and geometrically-of the 6d theory on the weakly coupled tensor branch, one approach to study the relationship between 6d and 5d 1 Here and in what follows codimension refers to the codimension of the base of the elliptic fibration.
SCFTs is to first reduce the 6d tensor branch theory on an S 1 . This yields a weakly coupled 5d gauge theory whose UV limit is an aforementioned 5d KK-theory. Then one can mass deform this gauge theory, leading to phases which can have 5d UV fixed points. The geometric counterpart of this process is to consider M-theory on the base-blown-up threefold-yielding the weakly coupled phase of the KK-theory-and then consider geometric transitions to obtain suitable geometries supporting 5d SCFTs. Indeed, there has been a lot of recent activity along these lines [63,65,66,68].
The classification of 6d SCFTs revealed the existence of so-called conformal matter (which are 6d SCFTs by themselves) as building blocks for the generalized quiver structure of 6d SCFTs [18]. Such theories are constructed in F-theory via a collision of two non-compact divisors W 1,2 carrying ADE groups G 1,2 at a smooth point p in the F-theory base B 2 . If the fiber singularity at the collision point p was of minimal type, this would just correspond to ordinary charged matter. In that analogy, one can think of the strongly coupled sector at the non-minimal singularity over p as a type of generalized matter charged under the flavor symmetry G 1,2 .
For the circle reduction of conformal matter theories, we propose an alternative geometric procedure to analyze the resulting 5d theories: Instead of blowing up the base to reach a fibration with only Kodaira fibers, as was done in [18], one can also resolve the non-minimal singularities of the total space Y 3 via fiber blow-ups which do not change the base B 2 . Because of the severity of the singularity, such a resolution introduces, in addition to curves, also surface components into the fiber. The resulting fibration is thus non-flat, that is, it does not have equi-dimensional fibers.
Being divisors in a Calabi-Yau threefold, these compact surface components immediately give rise to gauged u(1)s in the M-theory compactification. Since they are typically ruled surfaces, they can be blown-down to a curve, which enhances the u(1)s to a non-abelian algebra [60,61]. The basic field content of these gauge theories come from the spectrum of M2-branes wrapping fibral curves of the ruling in these surfaces. The careful analysis of the Kähler cone of the resolved Calabi-Yau matches the 5d prepotential analysis which describes the Coulomb branches of the 5d gauge theory phases. Furthermore, since by construction the surfaces can be blown down to a point (namely the fiber singularity), the corresponding gauge theory also has a strong coupling limit.
In fact, different blow-up resolutions of the same non-minimal singularity will in general lead to different geometries for the surface components, thus yielding different weakly coupled gauge theories and SCFT limits. One key point of the fiber resolution picture, however, is that we can easily identify the "parent" 6d conformal matter theory from which these different 5d phases originate: It is the F-theory compactification defined on the singular elliptic threefold.
Compared to the local analysis, where the focus is just on the geometry of the compact surfaces, our approach provides another advantage, namely a geometric way to identify the flavor symmetry in the strong coupling limit. Flavor symmetry enhancements in 5d have already been successfully studied with other methods, such as analyzing the spectrum of operators charged under the instantonic U (1) symmetry and the superconformal index [71][72][73][74][75][76][77][78], or exploring the Higgs branch at infinite coupling by compactifying on a T 2 to a 3d N = 4 theory and constructing the Coulomb branch of the mirror dual Lagrangian theory [79,80].
A direct method for identifying flavor symmetries from the geometry has been previously presented in [64]. There, the authors conjectured that any canonical singularity in a noncompact Calabi-Yau threefold defines a 5d N = 1 SCFT. Using toric examples, it was shown that resolutions of these singularities encode global symmetries, both at weak and strong coupling, in terms of collapsing non-compact divisors. Our approach is similar in practice, but conceptually it identifies these collapsing divisors as the exceptional divisors that resolve the minimal singularities over the codimension one loci W i , thus revealing the higher dimensional origin of the 5d global symmetry.
This identification is possible because we not only perform the fiber resolution at the nonminimal point, but also over W i . Concretely, it allows us to track how the generic Kodaira fibers ofĜ i (the affinized version of G i ) split and become curves in the non-flat surfaces. Since the generic fiber is homologous to the sum of its split products at a special fiber, it is not hard to see that, by blowing down the non-flat surfaces (either to a curve or a point), some of the curve components over W i may be forced to shrink too, thus enhancing the singularity over W i to H i ⊂Ĝ i . In this way, the global symmetry H i of the 5d theory appears as a subgroup of the flavor symmetry of the 6d parent theory (affinized by the KK-U (1) upon circle reduction).
In this paper, we will exemplify our proposal by considering non-flat resolutions of singularities which in 6d give rise to conformal matter with (E 8 , G) flavor symmetry, where G ∈ {∅, SU (2), SU (3)}. Their circle reductions yield 5d theories of rank 1, 2 and 4, respectively. For simplicity, we will restrict ourselves to constructions based on so-called "tops" [81,82], where the elliptic fibration Y 3 is resolved through a hypersurface embedding into a (semi-)toric ambient space X 4 . As we will demonstrate, our method to identify the global symmetries agrees with known results.
The paper is organized as follows. In section 2, we briefly review the gauge dynamics of 5d SCFTs and their construction via geometric engineering in M-theory. Section 3 then makes connection to 6d theories via M-/F-theory duality, and discusses the physical difference of base and fiber blow-ups. This motivates the study of non-flat resolutions of non-minimal elliptic singularities, which is detailed in section 4. Here, we also explain the relationship between flavor symmetries in 6d and 5d, and how the non-flat geometry makes them manifest.
To demonstrate these ideas, we then turn to an explicit construction of rank one theories in section 5, where we analyze in detail all phases that are torically available. Because the higher rank theories have a more complex RG-flow network, we will leave a full classification of their non-flat resolutions for future works. Instead, section 6 will contain isolated cases of rank two and four theories, where we focus on the appearance of 5d quiver theories and dualities between them. Finally, after a summary, section 7 will touch upon some aspects and open questions which we believe are worthwhile pursuing in the future.
Note Added: In the final stages of preparing this manuscript, the work [83] appeared which extends the results of [68]. While the motivation there is similar to ours, namely to study the Coulomb branch of 5d SCFTs from a parent 6d SCFT via M-/F-theory duality, the practical approaches are different. The starting point of [83] is the 6d tensor branch, that is, the elliptic threefold after blow-ups in the base. The authors argue that they can reach the geometries (which cannot have a flat fibration) that describe the 5d gauge theory phases with UV fixed point in their Coulomb branch. This is done by a complicated series of flop transitions. In contrast to this, for a large class of examples our approach gives directly these geometries, which manifestly describe the 5d gauge theories. Our methods also allow us to study the existence of a 5d fixed point in the moduli space of these theories, as well as the enhanced flavor symmetries.

5d Supersymmetric Gauge Theories
In this section, we collect some basic facts about five dimensional supersymmetric gauge theories. For a recent overview see, e.g., [65,66] and references therein. Throughout this paper, unless otherwise stated, we will adopt the notation of using letters in lower case Fraktur (g, The field content of a 5d gauge theory is given by vector multiplets A and hypermultiplets h. A vector multiplet consists of a 5d gauge potential A µ transforming in the adjoint representation of g, a collection of real scalars φ i , i = 1, . . . , rank(g), and an adjoint fermion λ.
Hypermultiplets have two complex scalars (and a fermion ψ) transforming in an irreducible representation and its conjugate, R g ⊕ R c g , respectively. If R g is real or pseudo-real (i.e., the conjugate representation is the same), the matter comes in half-hypermultiplets transforming in R g . To summarize we have the following multiplets: A = (A µ , φ i , λ), A µ , λ ∈ Adj g , i = 1, . . . , rank(g) , Gauge theories in 5d can have a discrete theta angle which signals the sign flip of the partition function when integrated over spaces of different topology. It is defined as an element of π 4 (G), where G is the simply connected Lie group associated to g. For instance, we will see shortly that some of the rank one theories with g = su (2) can have a non-trivial discrete theta angle, as π 4 (SU (2)) ∼ = Z 2 .
There exists a non-renormalizable effective Lagrangian on the Coulomb branch, which can be written as 3) The couplings that appear, namely the metric G ij of the Coulomb moduli space and the Chern-Simons couplings c ijk can be derived from a quantity called the prepotential, F: Moreover, it is important to notice that ∂ i ∂ j ∂ k F = c ijk are constant integers, c ijk ∈ Z. This is because, even though the Chern-Simons term is not gauge invariant, its integral should be well defined modulo 2πi Z. The prepotential receives in general a classical contribution which where h ij is the Cartan matrix of the Lie algebra g and d ijk = 1 2 tr fund (T a (T b T c + T c T b )). The quantum contribution can be computed by evaluating the one-loop AAA amplitude with fermions charged under the gauge symmetry that run in the loop [84]: Here, α are the roots in the root lattice L, φ ≡ φ is the vev that parametrize the Coulomb branch, f is the number of hypermultiplets in a certain representation R f (and its conjugate) of g, W f are the weights of R f , and · is the bilinear product on the root lattice constructed via h ij . Gauge invariance and quantization of the integral of the Chern-Simons term implies that the quantum contribution is exact at one-loop (by dimensional analysis higher loops will violate the condition c ijk ∈ Z). Furthermore, it is also important to highlight the fact that the fermions in the matter hypermultiplets contribute opposite sign with respect to the fermions in the vector multiplet. The total prepotential is given by The Coulomb branch moduli space is known to have a wedge structure, and the Weyl chambers are defined by α · φ being positive (or negative) everywhere in a connected region; α · φ = 0 are the boundaries of the chambers. On the other hand, it is still possible for w · φ + m f to change sign in a single Weyl chamber, which means that, because of the absolute values in (2.6), there can be different prepotentials within a single Weyl chamber.
The first derivative of the prepotential describes the tension of non-perturbative objects of the theory. The BPS object in 5d gauge theories are monopole strings and electric particles (under compactification on a circle to four dimensions, these become magnetic monopoles and electric particles), whose central charges are given by where n i e , n i m are integral electric and magnetic charges. Moreover, the instanton charge is defined as Tr(F ∧ F ), (2.9) where Tr ≡ tr ρ /Index(ρ) is the normalized trace with respect any representation ρ of g.
Finally, T (φ) i is the tension of the monopole strings given by These tensions can be thought of as a dual coordinate base on the Coulomb branch with respect to φ i .
Lastly, the first and second derivatives of the prepotential must be continuous all over the Coulomb branch, whereas c ijk can jump in integer values. In particular, such jumps at the boundaries of the sub-wedges signals the presence of charged matter, which become massless at these boundaries.

5d fixed points
For unitarity and consistency of the theory, the gauge kinetic term must be positive which implies that is positive semi-definite. In general, the metric takes the following form, and when the quantum contribution c quantum, ijk is positive in the Weyl chamber, it is possible to have a scale invariant fixed point where the classical mass and inverse coupling are set to zero, m 0 = g −2 classical = 0. This is not possible when the quantum contribution is negative due to the positivity constraint (2.11). So, the existence of a fixed point depends on the positivity of the second derivative of the quantum prepotential δF quantum , or, in other words, whether the prepotential is a convex function throughout the Weyl chamber. Moreover, from (2.11), we can see that charged matter hypermultiplets give concave contribution to F, which means that there is an upper bound on the matter content. However, the positivity of the second derivative is not the only constraint. For instance one should also verify that the tensions of the monopole strings stay positive throughout the chamber, T (φ) i > 0, ∀i = 1, . . . , rank(g). (2.13) Sometimes, it can happen that the metric is not positive in some region of the Coulomb branch.
However, before the metric becomes negative definite, the tension of some BPS object will change sign. This signals a break-down of the effective description, since the vanishing tension of a BPS object indicates non-perturbative physics at that point. In fact, in these cases there is usually an alternative effective description which is perturbatively valid at that point [65], which fully describes the dynamic of the Coulomb branch. These more refined analysis allows also for quiver gauge theories, [24], and it will become more clear in the geometric description.
Finally, we note that many different gauge/quiver theories can have the same fixed point in the UV. A 5d SCFT, if it exist in the moduli space of a gauge/quiver theory, is then believed to be characterized by their rank, which is the same as the Coulomb branch dimension, and by the enhanced flavor symmetry group.

su(2) gauge theory with matter
Let us discuss the simple example of rank one theories, i.e., g = sp(1) ∼ = su (2) (2.14) It is clear from this that, in order to have a non-trivial fixed point, we need N f ≤ 7, whereas for N f = 8 there is no singularity in the moduli space, since the metric (2.12) in the limit m 0 → 0 is trivial. For this reason, the theory with N f = 8 has no 5d fixed point, but it has a 6d UV completion, i.e., the E-string theory.
For N f ≤ 7, the strong coupling limit g −2 classical → 0 enhances the flavor symmetry to E n f +1 [22], where (10) , (2.15) and E 6 , E 7 , E 8 are the exceptional Lie groups. Moreover there are two outliers where the flavor symmetry at infinite coupling (conformal point) is E 0 = ∅ and E 1 = U (1). The E 1 and E 1 gauge theories are distinguished by a different discrete theta angle for the gauge group, which is SU (2) 0 for E 1 and SU (2) π for E 1 .
As we will see later, we can reproduce this result also geometrically, including the two outliers which are difficult to track field theoretically. Because we limit ourselves to toric constructions, our geometric examples do not cover cases N f ≥ 3. However, as we will emphasize in section 4, we expect this limitation to be lifted for general, non-toric models.

5d gauge theories and fixed points from geometry: brief summary
Here we briefly list the fundamental ingredients to understand 5d gauge theories and their Coulomb branch phases from M-theory on singular Calabi-Yau threefold and their resolutions, following the work [61].
We study M-theory on singular Calabi-Yau threefolds Y , whose blow-up resolutionŶ contains a bouquet of complex surfaces S i . Though the precise geometry of the S i depend on the resolution procedure, these surfaces are generically ruled, i.e., carry a fibration structure S i → β i with the generic fiber i being a P 1 . The collection of surfaces is usually arranged as the Dynkin diagrams of the associated gauge theory. When the surfaces intersect along (multi-)sections γ ij of the rulings on both S i and S j , one can shrink the surfaces to a curve by blowing down the fibers i of the rulings. 2 This produces a curve-worth of singularities insidê Y , which realizes a gauge theory as follows: • Vector multiplets: -The simple roots of the adjoint weights are given by the M2-brane states wrapping the fiber curves i .
• Fundamental hypermultiplets: These are given by M2-branes wrapping special fibers ±[σ i ] of the ruled surfaces S i with intersection numbers ∓1, i = j + 1, 0, i = j, j + 1. (2.16) • Adjoint hypermultiplets: These arise from the moduli of M2-branes wrapping i . Since i is a fiber, its moduli space is precisely the base β i with genus g, and hence there are g adjoint hypermultiplets [84].
• Hypermultiplets with more exotic representations: If the base β i of intersecting surfaces are of different genera, M2-branes on special fiber components σ can carry weights of (anti-)symmetric representations. 2 For a visualization of these configurations, we again refer the reader to [61].
Because we are shrinking the fibers, all the M2-brane states listed above will become massless, and hence give rise to well-defined gauge dynamics. The precise gauge algebra depends on the topology of the bases β i and the gluing curves γ ij . For example, to realize an su(n) gauge symmetry, n − 1 surfaces, all ruled over a genus g curve, must intersect along a chain such that γ i,i+1 = S i ∩ S i+1 is a section of the rulings on S i and S i+1 .
To go to strong coupling, all surfaces must be further blown down to a point, At this stage, also M2-branes states wrapping the bases β i will become massless, signaling additional light degrees of freedom in the spectrum. In fact, one can dually see massless excitations of monopole strings, which come from M5-branes wrapping the surfaces that are tensionless in the singular limit. These light states indicate a break down of perturbative physics.
Let us now relate the Coulomb branch scalars to geometric quantities. The Coulomb branch is identified with the negative Kähler cone ofŶ , with an the extra condition imposed by the shrinkability of curves in the singular limitŶ → Y : The prepotential is computed geometrically by Lastly, a 5d gauge theory has one further discrete label. E.g., su(n) theories with n ≥ 3 are labelled by Chern-Simons level κ, which one can in principle determine this geometrically by comparing the prepotential (2.18) with the field theory data (2.5) -(2.7). For su (2) and sp(n) theories, there is a discrete theta angle θ = 0 or π.

SCFTs in M-/F-theory Duality
In this section, we will briefly review the F-theoretic description of 6d conformal matter theories. We then discuss the state of the art methods to relate these to 5d SCFTs via the duality between M-and F-theory and motivate the study of non-flat resolutions.

F-theory on elliptic threefolds with non-minimal singularities
While the classification of 5d SCFTs is an open problem, the situation in 6d is much more pleasant: there, all N = (1, 0) SCFTs 3 have been classified using the language of F-theory [19,20] (for a recent review, see [85]). That is, any 6d SCFT can be described geometrically by a singular elliptically fibered Calabi-Yau threefold π : Y 3 → B 2 .
The physics of an F-theory compactification is determined by a Weierstrass model for Y 3 , where K B 2 is the canonical bundle of the base B 2 . The degeneration locus of the elliptic fiber, given by the vanishing of the discriminant, corresponds to the location of 7-branes in the non-perturbative type IIB interpretation of F-theory [69]. The worldvolume dynamics of these branes gives rise to 6d N = (1, 0) Yang-Mills theories, whose gauge algebras g i are of the same ADE-type as the Kodaira singularities determined by the vanishing orders of (f, g, δ) as well as monodromy effects along irreducible components W i of δ [7,70,[86][87][88] (see also [89,90]).
Furthermore, at intersection points p = W i ∩ W j , the fiber singularity in general enhances, indicated by a higher vanishing order of (f, g, δ). For minimal singularities, i.e., ord(f | p , g| p , δ| p ) < (4, 6,12), the corresponding ADE algebra h p ⊃ g i ⊕ g j indicates the presence of matter states in representations R k according to the "Katz-Vafa" rule [86,91]: However, if the singularity types on Σ i,j are too severe, their collision will lead to a so-called non-minimal singularity with ord(f | p , g| p , δ| p ) ≥ (4, 6,12). For such singularities, there is no associated ADE-algebra, and hence no conventional matter states.
To make sense of them in F-theory compactifications to 6d, one can blow-up the base B 2 at the point p. This procedure introduces a collection of rational curves Σ i ⊂B 2 in the blown-up base, over which the (pulled-back) elliptic fibrationπ :Ỹ 3 →B 2 only has minimal singularities, i.e., ordinary gauge algebras g i and matter representations.
Physically, the blow-up curves Σ i support 6d tensor multiplets dual to BPS strings, the latter arising from D3-branes wrapping Σ i in the IIB picture. The volume of Σ i correspond to the vacuum expectation value of the scalar in the tensor multiplet, and parametrize the socalled tensor branch of the 6d theory. Furthermore, if Σ i carries singular fibers, i.e., supports a gauge theory, its squared gauge coupling is proportional to vol(Σ i ) −1 .
Blowing down these curves, thus reproducing the non-minimal singularity, we immediately see that the gauge coupling becomes formally infinite. In addition, even if Σ i carries no gauge symmetry, the strings from wrapped D3-branes still become tensionless, such that their excitations give rise to infinitely many light degrees of freedom. Both observations indicate a strongly coupled sector without a Lagrangian description. To obtain an honest SCFT, the F-theory base B 2 has to decompactify in order to decouple gravity. In this limit, all the irreducible components W i of the discriminant decompatify as well. Since the gauge coupling is inversely proportional to their volume, we see that the gauge symmetries on W i become non-dynamical, i.e., global, or flavor symmetries of the SCFT.
Some 6d SCFTs can also arise solely from a singular point p of the base, without colliding codimension one singularities [18]. These SCFTs typically have no global symmetries. In addition, it is not entirely clear how the concept of fiber resolutions works in this context (see, however, [92] for recent studies of such examples). For the purpose of this paper, we will therefore assume that p ∈ B 2 is smooth, in which case the non-minimal singularity in the fiber over p has to come from the collision of two non-compact divisors W 1,2 with ADE fibers. The associated Lie groups G 1,2 constitute then (a subgroup of) the global symmetry of the SCFT at p. These theories are called 6d conformal matter theories and are important building blocks of 6d SCFTs.

Circle reduction of conformal matter theories and fiber resolutions
With the classification of 6d SCFTs on one hand, and the motivation to better understand 5d SCFTs on the other, a natural object to study is the circle reduction from 6d to 5d. This is particularly appealing in the spirit of the M-/F-theory duality, as it connects the 6d classification directly to geometric engineering of 5d theories via M-theory.
Due the complications arising from the strong coupling nature of SCFTs, the practical way found in the literature is to first push the 6d F-theory model onto the tensor branch by blowing up the base intoB 2 , and to then study the 5d theory obtained by compactifying M-theory onỸ 3 →B 2 [63,68] (see also [65,66]). This applies in particular to 6d conformal matter theories, which will be the only type of 6d SCFTs we consider for the rest of the paper.
After moving onto the tensor branch of the 6d theory, i.e., when we consider the three-foldỸ 3 , the M-theory compactification sees a collection of compact surfaces inỸ 3 , which in general are blown-down to curves. Indeed, the elliptic fibrationỸ 3 will generically have ADE-singularities of type g i over the blow-up curves Σ i . Blowing up these singularities will yield rank(g i ) compact divisors S i;a , a = 1, ..., rank(g i ) that are ruled surfaces over Σ i , whose P 1 fibers intersect in the Dynkin diagram of g i . Following the discussion in section 2.3, we conclude the presence of a 5d gauge theory with gauge algebra g i , when we shrink the fibers of all S i;a .
OnỸ 3 , there is another divisor S i;0 ruled over Σ i , whose fiber is the affine component of the Kodaira fiber over Σ i . In 5d, one is allowed to shrink also the affine node, as long as at least one of the other fiber components remains at finite size. 4 The enhanced 5d gauge symmetry is then a sub-algebra h i of the "affinized" versionĝ i of g i . In a certain sense, one can viewĝ as augmenting the gauge symmetry g by the KK-U (1) from the circle reduction.
However, we cannot enhance this full symmetry without simultaneously unwinding the S 1 and ending up in 6d.
Nevertheless, given that a further blow-down of Σ i in the base reduces the surfaces with blown-down fibers to a point, one might be tempted to say that this is the 5d SCFT limit of the h i gauge theory. However, as pointed out in [63,66], this limit is a so-called 5d KK-theory, as the light degrees of freedom are all accompanied by a tower of massive KK-states. To obtain theories with non-trivial 5d UV fix points, one has to mass deform the KK-theory.
Within the duality (1.1) between M-and F-theory, such mass deformations correspond to turning on non-trivial holonomies of the flavor symmetry along the S 1 on which we reduce the 6d F-theory. Locally, these parameters change the volumes of special fibers inside the ruled surfaces S i;a supporting the gauge symmetry. These special fibers give rise to charged matter states via wrapped M2-branes. By tuning such volume parameters to formally change sign, the geometry undergoes a flop transition, whereby a holomorphic curve shrinks to zero size, and another one gets blown-up. In extreme cases, when the flopped curves are generic fibers of a fibered surface, the surface will have to shrink to a curve of singularities before blown up again into a different surface. Strictly speaking, such a process is a geometric transition rather than a flop.
The resulting surfaces S k will in general no longer be compatible with an elliptic fibration.
For example, in the cases of 5d rank one theories, the KK theory is obtained via compactification of the tensor branch of the 6d E-string theory. Geometrically, we have a single compact blow-up curve Σ i ∼ = P 1 ⊂B 2 over which the elliptic fibrationỸ 3 is generically smooth. The resulting elliptic surface S i;1 =π −1 (Σ i ) ∼ = dP 9 has many curves (in fact, infinitely many) in form of sections, which upon (successive) flopping produces the other del Pezzo surfaces S ∼ = dP n≤8 . Since these surfaces have no elliptic fibration structure, the full threefold cannot possibly be elliptically fibered-at least not in the ordinary way. Thus, it seems that the geometry after such a transition is only distantly related to the original F-theory geometry one started with.
However, from the perspective of M-/F-theory duality, non-trivial S 1 holonomies of a gauge field supported on a base divisor W ⊂ {δ = 0} precisely correspond to a blow-up resolution of the fiber singularities over W . For F-theory compactifications with only minimal singularities, i.e., only ordinary matter representations, such holonomies will generically make all matter states charged under the gauge algebra over W massive. Geometrically, this means that all singularities in codimension two in the base are also resolved through the appearance of additional curves in the fiber. 5 Furthermore, it is well-known that there can different phases of the fiber resolution related by flop transitions, which differ by the fiber structures in codimension two in the base.
With this in mind, we therefore ask if a fiber-resolution of non-minimal singularities can teach us something non-trivial about the associated 5d SCFTs. 6 Following the above intuition, we further expect the resolution of codimension one fibers over W 1,2 ⊂ B 2 to give rise to a non-trivial mass deformation of the 5d KK-theory, which is obtained via S 1 -reduction of the 6d SCFT living at the intersection p = W 1 ∩ W 2 .
Given that an M-theory realization of 5d gauge theory requires (possibly collapsed) compact surfaces in the geometry, it might seem puzzling that a fiber resolution of an elliptic fibration alone could produce such a setting. However, because of the non-minimal singularity type, the fiber resolution actually introduces surface components. As such, the resulting total space is what is called a non-flat fibration.

Non-flat Resolutions of Non-Minimal Singularities
Resolutions of minimal singularities of elliptic fibrations always introduces rational curves into the fiber. As such, the fiber of the resolved fibrationπ :Ŷ 3 → B 2 always has (complex) dimension one. In algebraic geometry, such a fibration is called flat. Conversely, given a fibration where the dimension of the fiber jumps from a generic to a special fiber, one typically refers to it as being non-flat. For an elliptic fibration π : Y 3 → B 2 with a non-minimal singularity over a smooth point p ∈ B 2 , we conjecture that a full resolutionŶ 3 of Y 3 without changing the base B 2 will generically introduce surface components intoπ −1 (p), i.e.,π :Ŷ 3 → 5 For our purposes, we assume that there are no terminal singularities in the vicinity of the non-minimal singularities. In general, their physical significance has been discussed in [93,94]. 6 See [95][96][97] for earlier works on circle compactifications of the E-string in this spirit.
Although we do not know of a strict mathematical proof of this statement, we point out that this phenomenon has been observed frequently in the F-theory literature [98][99][100][101][102][103][104][105][106][107][108][109][110][111][112]. 7 More importantly, however, we expect this phenomenon from the M-/F-theory duality! As discussed in the previous section, a fiber resolution corresponds to turning on holonomies along the S 1 , which in turn generates a mass deformation of the 5d KK-theory. For generic deformations, this will lead to a weakly coupled 5d gauge theory, which in M-theory has to be realized on compact surfaces. Without changing the base or the codimension one structure of the fibration, the only "place" these surfaces S k can appear is therefore in the fiber over p.
As anticipated in subsection 2.3, these surface components are generically ruled, and have reducible special fibers. Moreover, the surfaces will generically intersect each other along curves. In addition to the surface components S k , the fiberπ −1 (p) also contains other curves which lie completely outside the surfaces, with some intersecting the surfaces only in points.
Focusing just on the local geometry of the intersecting surfaces, one can apply the criteria from [60,61,65,66] to analyze the resulting M-theory gauge theory (see also section 2). In addition, one will find curves Γ inside the surfaces which have zero intersection number with the canonical divisors of the surfaces: Because of this, the volume of these curves are not controlled by the Kähler parameters dual to the compact surfaces. From the gauge theory perspective, it means that the masses of M2-brane states on these curves are not set by the Coulomb branch parameters, and hence correspond to external parameters of the gauge theory.
However, recall that the non-minimal singularity was a result of the collision of two ADE singularities G 1,2 along W 1,2 . Therefore, the fiber components over W i , which at a generic point q ∈ W i form nodes of the affine Dynkin diagram of G i , must also appear within the non-flat fiberπ −1 (p). Let us denote the (non-compact) exceptional resolution divisors over W i by A i;a , a = 1, ..., rank(G i ), and their fibers by P 1 i;a . If the singularity type at p was minimal, then it is a well-known story that some of the P 1 i;a would split over p, giving rise to matter charged under G i . 8 The same situation now also occurs within the non-flat fiber, except for 7 Non-flat fibers occur frequently in elliptic fourfolds, which were mostly studied for phenomenological purposes. However, there, surface components in non-flat fibers in codimension three do not define divisors in a fourfold, and thus appear on different footings as in threefolds. Consequently, we lack a good understanding of the resulting 4d/3d physics of F-/M-theory. 8 For example, in a collision of G1 = SU (N ) with G2 = "SU (1)", the latter corresponding to an I1 fiber over W2, one of the N fibers over W1 that corresponded to a simple root of SU (N ) would split into two curves carrying the weights of an (anti-)fundamental state. the appearance of the surfaces S k ⊂π −1 (p) [102].
In particular, the curves Γ contained inside some S k with Γ·S k = 0, i.e., those whose volume correspond to the mass parameters of the gauge theory defined on S k , can be identified with (split) components of the codimension one fibers. We will argue in the following that this fact allows us to read off the global symmetries of the 5d gauge theory and its SCFT limit.

Global symmetry from the global resolution
It was anticipated in [64] via toric models that one can determine the global symmetries of a 5d SCFT explicitly from the degeneration of non-compact divisors of the Calabi-Yau threefold Y . As we will describe now, these divisors are naturally identified with the exceptional divisors which resolve the ADE singularities of the elliptic fibration in codimension one in the base.
Within the M-/F-theory duality, we thus identify the global symmetry as a subgroup of the global symmetries that arise from circle reducing the 6d theory.
Let us spell out these ideas in more detail. First, we can identify the rank of the global symmetry group G f with the rank of the matrix D l · Γ m , where Γ m are all independent classes of curves inside the compact surfaces S k . D l on the other hand are all independent classes of non-compact divisors ofŶ , which include the exceptional divisors A i;a and (multi-)sections.
These divisors are dual to Cartan U (1)s of the flavor symmetry in 6d and the KK-U (1), from which we expect 5d global symmetries to originate.
In order to determine the non-abelian part of G f , we use the basic Kähler geometry fact that algebraic curves homologous to each other insideŶ have the same volume. Therefore, if one curve within a class shrinks to zero volume, then so do all the others. Now, as we have pointed out above, some of the resolution curves over the codimension one loci W 1,2 may split into several curves at p = W 1 ∩ W 2 , of which one or some could be contained in a surface S k .
In particular, it can also happen that a codimension one fibral curve remains unchanged over p, and simply sit as a whole inside one of the surfaces. See figure 1 for illustrations of these cases.
When the elliptic fibration is fully resolved, all curves and surfaces are at finite size.
Then, the M-theory compactification is at a generic point on its Coulomb branch, where all charged states are massive. The gauge symmetry here is just u(1) n , where n is the number of compact surfaces S k . If the surfaces S k are ruled, then by blowing down their generic fiber, the surfaces shrink to curves (which form singular loci of the total space), thus enhancing the gauge symmetry to a rank n non-abelian algebra g. Because we are blowing down the generic fiber of S k , any individual component of special fibers inside S k also shrinks.
In the fully blown-up case, i.e., where the singularities over W i are resolved, one can easily track how the generic fibers P 1 i;a splits into curves Γ l insideπ −1 (p) (including possible multiplicities). But because fibral curves of A i;a are homologous to each other, we have, in terms of homology classes, The relative locations of the split components Γ l and the non-flat surfaces now allow for several possibilities.
1. One of the Γ l is not contained in any non-flat surface S k . In this case, P 1 i;a does no shrink regardless of the fate of S k .
2. All Γ l are inside the non-flat surfaces, and become fibers of the ruling on S k . In this case, once we enhance the gauge symmetry to g by shrinking along the ruling, P 1 i;a shrinks everywhere over W i .
3. All Γ l are inside the non-flat surfaces, but some form a (multi-)section of the ruling on S k . In this case, P 1 i;a will only shrink when we blow down S k to a point, i.e., go to the SCFT limit of the gauge theory.
In general, the shrunken codimension one fibers form a subset of nodes of the affine Dynkin diagram of G i . In other words, blowing down S k (to either a curve or a point) will in general lead to a fibration of canonical singularities over W i , whose ADE type H i correspond to the global symmetry G f of the 5d theory [63,64]. If the fiber splittings are only of case 1., then the 5d theory only has abelian flavor symmetries. In case 2., the weakly coupled gauge theory exhibits a non-abelian flavor symmetry. As case 3. illustrates, we generically expect a further non-abelian enhancement of G f from additional singularities over W i in the SCFT limit of the gauge theory. We have summarized these possibilities in figure 1.
Differently from the work of [64], our approach provides an explicit geometric link between 6d and 5d SCFTs and their flavor symmetries. In particular, it is known that the only noncompact divisors in crepant resolutions of canonical Calabi-Yau threefold singularities are fibrations of resolved surface ADE singularities (see, e.g., footnote 5 in [66]). Therefore it seems natural to identify these singularities as the fiber singularities of an non-flat elliptically fibered threefolds, which, as we argued, always has a 6d SCFT with prescribed flavor symmetries associated with it. In what follows we work under the assumption that the starting point of our analysis is the singular non-minimal elliptic fibration, which manifestly has the largest possible enhanced 6d flavor symmetry. We believe that this is sufficient in order to see all the B < l a t e x i t s h a 1 _ b a s e 6 4 = " o m i o 5 9 k v E V Q 4 5 s Y I D z D K 7 w 5 j 8 6 L 8 + 5 8 L F s L T j 5 z C n / g f P 4 A l G G M x g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o m i o 5 9 k v E V Q 4 5 s Y I D z D K 7 w 5 j 8 6 L 8 + 5 8 L F s L T j 5 z C n / g f P 4 A l G G M x g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o m i o 5 9 k v E V Q 4 5 s Y I D z D K 7 w 5 j 8 6 L 8 + 5 8 L F s L T j 5 z C n / g f P 4 A l G G M x g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o m i o 5 9 k v E V Q 4 5 Z d X S f u i 7 r l 1 7 / 6 y 1 r g r 4 i j D C Z z C O X h w B Q 2 4 h S a 0 g M E E n u E V 3 p z E e X H e n Y 9 F a 8 k p Z o 7 h D 5 z P H 4 M 8 j 7 E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 Z d X S f u i 7 r l 1 7 / 6 y 1 r g r 4 i j D C Z z C O X h w B Q 2 4 h S a 0 g M E E n u E V 3 p z E e X H e n Y 9 F a 8 k p Z o 7 h D 5 z P H 4 M 8 j 7 E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 Z d X S f u i 7 r l 1 7 / 6 y 1 r g r 4 i j D C Z z C O X h w B Q 2 4 h S a 0 g M E E n u E V 3 p z E e X H e n Y 9 F a 8 k p Z o 7 h D 5 z P H 4 M 8 j 7 E = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 Z d X S f u i 7 r l 1 7 / 6 y 1 r g r 4 i j D C Z z C O X h w B Q 2 4 h S a 0 g M E E n u E V 3 p z E e X H e n Y 9 F a 8 k p Z o 7 h D 5 z P H 4 M 8 j 7 E = < / l a t e x i t > Figure 1: A cartoon illustrating the splitting of fibers after blowing up the non-minimal singularity over a point of the base. The gray squares represent the surface components of the non-flat fiber. In general, codimension one fibers can split into several curves which all (green) or only partly (red) lie inside the surfaces. They can also remain irreducible, and sit inside (blue) or outside (black) of any surface. By blowing down the surfaces to a point, the blue and green fibral curves are forced to shrink over the codimension one loci of the base. In the case depicted here, this would lead to an enhancement of an SU (3) subgroup of the full E 6 flavor symmetry. possible enhancements in 5d. For example, we will consider the 6d E-string theory described by F-theory with an E 8 locus colliding an I 1 , instead of two colliding SO(8)s.
Moreover, by combining the two conjectures that a), all 5d SCFTs arise from circle reductions of 6d SCFTs, and b), all canonical threefold singularities define a 5d SCFT, it seems natural to attempt a classification of threefold singularities via elliptically fibered Calabi-Yaus and F-theory. Following our arguments, we believe that non-flat fibers would play a central role in this endeavor. We leave a more thorough investigation of these speculations for future works.
For the remainder of this paper, we will instead provide examples of the interplay between non-flat fibers and 6d/5d SCFTs. To have an easy method for efficiently producing different resolutions of non-minimal singularities, we will use toric methods, where some geometric aspects can be described using combinatorics. Thus, before presenting the models, we will briefly review the necessary toric geometry tools.

Non-flat resolutions from toric constructions
The canonical singularities considered in [64] allowed for a toric resolution. That is, the compact divisors which are used to blow up the singularities were toric, and could be described via a 2d lattice polygon. There, the authors make no reference to an elliptic fibration in which these surfaces are embedded in.
However, as we will describe now, such singularities and the corresponding resolutions appear naturally in toric constructions of elliptically fibered Calabi-Yau hypersurfaces. In particular, a convenient way to resolve elliptic singularities torically is via so-called tops [81,82].
In the following we will give a qualitative description about these methods, focusing on the appearance of non-flat fibers and the combinatorics associated with different resolutions and flop transitions.
The construction is based on Batyrev's insight [113] that a pair of reflexive polytopes, (♦, ♦ * ), define a Calabi-Yau hypersurfaceŶ = {P = 0} inside the toric variety X whose fan is given by a triangulation T of ♦. The precise form of P is determined by the dual ♦ * . A suitable ♦ will give rise to an elliptically fibered hypersurface. 9 Batyrev's construction can be further specialized in order to engineer elliptic fibrations with a desired codimension one Kodaira singularity. The basic idea proposed in [81] is to "chop up" ♦ into two pieces, a "top" ∆ and a "bottom" ∇. Then, following the classification in [82], constructing the top specifies an elliptic fibration with a prescribed fiber degeneration over codimension non-compact one loci W i . Including the bottom can be then thought of as compactifying the base, i.e., embedding the local structures coming from the top into a global, compact elliptic Calabi-Yau. Since we are interested in SCFTs, a local description of the singularity suffices, so the information contained in the top ∆ and its dual ∆ * is all that we need.
Given a top ∆, the polynomial cutting out the hypersurfaceŶ takes the form Note that X ϕ → B is itself fibered over B, and the coefficients b k are sections of suitable line bundles on B. The fibers of ϕ are toric spaces with homogeneous coordinates q l corresponding to lattice vectors in ∆, but B can be generically non-toric. The elliptic fibration can be viewed as elliptic curves defined by {P = 0} ≡ {P } inside the toric space ϕ −1 (p), fibered over points p ∈ B of the base. The precise form of P can be derived from the dual top ∆ * , see [82].
Irrespective of the expression of P , the geometry ofŶ has some generic features which we discuss now. For that, let us denote the coordinates of the lattice vectors by v = (x 1 , x 2 , x 3 ).
One 2d facet of ∆ is one of the 16 reflexive 2d polygon F m , which we can assume to sit at x 3 = 0. Vectors at vertices of F m define toric divisors of X which intersect the generic smooth elliptic fiber ofπ :Ŷ → B and hence correspond to (multi-)sections ofŶ [103,115,116].
Divisors with vectors interior to an edge E of F m , on the other hand, do not intersect the generic fiber, but instead restrict to exceptional divisors which resolve an ADE singularity of type G E , fibered over a non-compact divisor W E ⊂ B. The codimension one structures and the resulting F-theory physics of the elliptic fibrations associated with a choice of F m have been classified in [117]. For an example, which we will analyze in detail later on, see figure 3.
In addition to these structures coming from F m , one can further engineer Kodaira singularities of ADE type G over a codimension one locus W in the base by including vectors v with x 3 > 0, which fill out the top ∆. A full classification of possible tops for a given polygon F m at x 3 = 0 has been worked out in [82]. The toric divisors corresponding to vectors v with x 3 > 0, which sit on edges (i.e., 1d facets) of ∆, give rise to the exceptional divisors resolving the singularities over W .
However, the top can also have (2d) facets which contain interior vectors with x 3 > 0.
In the ambient space X , their corresponding divisors D int are also fibered over W ⊂ B, i.e., ϕ(D int ) = W . However, they intersect the elliptically fibered hypersurface {P = 0} ≡ {P } over a codimension two locus of the base [100,103,118]. For a fibration over a two dimensional base, this means that the four-cycle D int ∩ {P } must be completely contained inside the fiber at that locus! These four-cycles therefore give rise to non-flat fiber components ofŶ .
In the top ∆, each edge E of F m bounds a facet F E . Any internal vector of F E will then correspond to a non-flat fiber component over the point W ∩ W E . In M-theory, these compact surfaces support the 5d gauge sector, which arise from a circle reduction with non-trivial holonomies of a 6d (G, G E ) conformal matter theory.
As an illustration, we have displayed in figure 2 a graphical representation of the top that specifies the geometries we will consider in the next section.

Resolution phases and flop transitions from triangulations
To fully specify the toric geometry, one needs to define fan among the vectors of the toric divisors. In the case of tops, such a fan is provided by a triangulation T of ∆. 10 For a three dimensional top, any triangulation T induces a triangulation T (Σ) of its surface. Geometrically,  Instead, there will be a different line connecting v 1 with, say, v 3 , giving rise to a curve C 13 that was absent in T (Σ) . Thus, changing the triangulation precisely corresponds to a transition that flops C 12 into C 13 .
Since we are only interested in the local geometry of the non-flat fiber components, we actually do not need a full triangulation of the top, but only the triangulation T F it induces on a facet F. In particular, each non-flat fiber component S k corresponding to a toric divisor D v k is a toric surface described itself by a 2d polygon P k with a single interior vector. This polygon is simply a sub-polygon of the facet F with v k being the interior vector and the boundary vectors given by all vectors which are connected to v k as dictated by the triangulation T F . Note that the resulting 2d polygon also appear in [64] for toric resolutions of canonical threefold singularities. Here, we see that these models can in principle arise as non-minimal elliptic singularities, which can be resolved in a non-flat manner via tops.
Before presenting the top we will be focusing on in the rest of this paper, we have to point out two main caveats of restricting to toric constructions. First, even though there are many tops that give rise to a non-flat fibration (see, e.g., [110]), we do not have a general argument that we can have a suitable toric construction that resolves any given 6d conformal matter model. Second, not all resolution phases of a given elliptic fibration can be realized torically via a top (we will see this drawback explicitly later on). We suspect that both of these restrictions can be overcome with non-toric resolution methods, such as in [102,[119][120][121]. However, we will defer an explicit analysis of this sort for future work [122], and use toric constructions here for convenience and flexibility.

Explicit example: E 8 -top over F 10
To demonstrate our proposal of studying 5d SCFTs on non-flat fibrations, we will consider three explicit types of conformal matter theories in 6d, namely collisions of an E 8 singularity with 1. an I 1 locus, 2. an SU (2) locus, 3. an SU (3) locus.
In 6d, these theories are also known as the E-string theories of rank 1, 2 and the (E 8 , SU (3)) conformal matter theory, whose 6d tensor branch in the notation of [19] reads (4.4) The numbers describe the negative self-intersection of the compact curves Σ i used to blow up the base. The resulting threefold has only minimal singularities, with possible Kodaira fibers over the compact curves supporting the indicated gauge algebra.
These three different SCFT sectors can all be realized at different non-minimal singularities within the same elliptic fibration described by a single top, namely the only possible E 8 top over the polygon F 10 [82]. This polygon has one edge of length 1, one of length 2 and one of length 3 (see figure 3). Figure 3: Toric diagram of F 10 with blow-ups on the edges. The vertices ( x, y, z) give rise to bi-, tri-, and rational sections, respectively, of the elliptic fibration. The SU (2) singularity over {s 4 = 0} is resolved by j, and the SU (3) over {s 8 = 0} is resolved by f 1 and f 2 . The edges of length l are labelled by E l .
The elliptic Calabi-Yau threefoldŶ 3 is the vanishing of the polynomial [117] where the coefficients b i and s j are holomorphic sections of suitable line bundles over B 2 . Their vanishing loci define vertical divisors, i.e., pull-back of divisors on B 2 . All other variables that appear in (4.5) define toric divisors.
As shown in [117], the resulting elliptic fibration has an SU (2) and SU (3) locus, located over {s 4 = 0} ≡ {s 4 } and {s 8 = 0} ≡ {s 8 }, respectively (we will from now on use the notation {F } to denote the vanishing locus of F ). To see this, we can set s 4 respectively s 8 to 0 in (4.5), in which case the hypersurface polynomial P factorizes: From this, one can see that over {s 4 }, the fiber splits into two components, and over {s 8 } into three. A more close analysis (see appendix A) reveals that the Kodaira types are I 2 and I 3 , respectively, hence give rise to the promised SU (2) and SU (3) gauge groups in F-theory.
Finally, there is, as usual, also a divisor (the residual discriminant) in B 2 carrying I 1 fibers, see appendix A. Now, to incorporate an E 8 singularity along a divisor W ⊂ B 2 , we construct the associated top over F 10 . The ruled divisors A i , i = 0, ..., 8, whose fibers give rise to the affine E 8 diagram in codimension one have associated polytope vectors α i given by (4.8) The convex hull of these lattice points and those in F 10 form a top ∆ with three facets (not counting the one at height 0, which is just F 10 itself), each bounded by one of the edges E l of F 10 . It can be easily checked that these facets contain the following interior vectors (cf. fig. 2): Face over E 1 : r = (0, 0, 1) , All vectors of the top above height x 3 = 0 introduce additional variables, but no extra terms, into the hypersurface polynomial (4.5). The modified polynomial now readŝ (4.10) The Likewise, setting s 4 to 0 leads to a factorization of the form P | s 4 =0 = j t 1 t 2 (...) . is part of the resolved I 2 fiber ofŶ 3 over that locus [117].
Finally, setting s 8 to 0 in (4.10) yields a factorization

5d Rank 1 Theories from 6d E-string
In this section, we turn to the familiar example of rank one theories in 5d, which are known to arise from circle reductions of the 6d E-string. Our focus will be on demonstrating how the non-flat fiber captures the 5d physics, and how we can read off the global symmetries from the singularity enhancements over the codimension one fibers, which perfectly matches the already know flavor symmetry enhancements argued by other methods.
First, for the purpose of studying the fiber geometry around {b 6 } ∩ W , we can simplify the hypersurface polynomial (4.10) by setting the coordinates j, f l , t m , u n to one, since they do not vanish over this point in the base. Put differently, their associated divisors do not intersect the fiber over {b 6 } ∩ W . The resulting polynomial defining the (local patch of the) Calabi-Yau threefold is then (5.1) Over the generic point of the codimension one locus W ⊂ B 2 , the elliptic fiber degenerates into the affine Dynkin diagram of E 8 , whose fiber components are described by the equations We focus on the non-flat fiber over {b 6 } ∩ W ∈ B 2 . The only compact surface S in this fiber, given by {b 6 } ∩ {r}, supports a 5d rank one theory when we compactify M-theory on it.
By construction, this fiber component can be shrunk to a point (re-creating the singularity of the elliptic fibration). In general, this can happen without shrinking all codimension one fiber components over W . As we recall from section 3.2, this implies a non-trivial S 1 holonomy of the E 8 in the reduction of F-to M-theory. Thus, the 5d theory is expected to be a mass deformation of the 5d KK-theory, and hence should have a non-trivial SCFT limit.
When the non-flat fiber is at finite size, it defines a weakly coupled phase of the SCFT. For For each such surface S, we can compute the associated number of flavors N f via [61] 8 where the second equality follows from adjunction formula on an Calabi-Yau threefoldŶ 3 .
Obviously, this formula does not apply for the P 2 case, cf. figure 4(f), because this phase has no gauge symmetry at weak coupling [60], and hence also no notion of flavors.

Gauge enhancement at weak coupling
Having a toric description of the surface S immensely simplifies the analysis of the weakly coupled phase (see also [64]). First, to establish the notation, we will denote the toric curves {q} ∩ {r}, for q a lattice point on ∆, simply by {q} whenever we are only focusing on the local geometry of S. Furthermore, we will denote the divisor class of the curve {q} in S by [q]. Finally, as usual, K S is the canonical divisor of S. With that, one can immediately show with basic toric geometry that all toric curves appearing here are P 1 s Now, to have a non-trivial gauge theory at weak coupling, the surface S has to be ruled.
In the toric case, such rulings can be easily determined from the diagram F S . Concretely, any lattice projection ψ S : F S → L mapping the interior vector of F S onto the origin of a one-dimensional sub-lattice L, which collapses cones of F S onto cones of L, indicates a ruling r y x α 7 α 6 α 5 of S. 11 Note that cones of L are simply the "left" or "right" from the origin. For the toric diagrams in 4, one can easily spot such a projection ψ for all phases except the P 2 : It is giving by projecting along the vertical axis (i.e., it projects the red lines to the origin). The sub-lattice L in this case is the "orthogonal" sub-lattice spanned by ±( α 8 − r). In all cases, the vectors y and α 7 project onto the origin. This signals that in S, {y} and {α 7 } define sections of the ruling. The other curves correspond to fibers of the ruling.
Furthermore, a simple toric computation reveals that K S · [x] = −2 for all relevant phases, which means that {x} is generic fiber of ψ S . 12 This also immediately implies that by collapsing the generic fiber of S (except when S ∼ = P 2 ) which shrinks S to a curve, we obtain an su (2) gauge symmetry, whose W-bosons come from M2-branes wrapping curves in [x].
Lastly, S also contains fibral curves with intersection numbers −1 and 0 with K S . For Further mass deforming the N f = 1 phase yields a pure su(2) gauge theory with no flavors. Physically, however, there are two distinct N f = 0 theories with different θ-angle. 11 Such a lattice projection gives rise to a so-called toric morphism, i.e., a map between two toric spaces. Here, this map is simply the projection map of the ruling on S. 12 Because {x} is toric, it is a P 1 , so we know that phase, corresponding to the θ = 0 theory, while the second flop produces the θ = π theory on S ∼ = dP 1 . Lastly, the θ = π theory allows for another transition, which geometrically is described by the flop transition from dP 1 to P 2 . Note that the two N f = 0 phases are not related by a simple flop transition, but rather by an extremal transition, in which the volume of the surface has to pass through zero [60]. In the toric diagrams, we can see that, by starting from F 2 in figure 4(d), one would have to first blow down the curve {α 6 }, which corresponds to a generic fiber of F 2 . Hence, this blow-down would contract F 2 to a curve.

Global symmetries at weak and strong coupling
So far, we have simply reproduced the known results on the gauge dynamics by considering M-theory on the local surface geometry, similarly to [60,61,65,66]. In the following, we will discuss the flavor symmetries G f of these theories, both at weak and strong coupling. The idea is analogous to that presented in [64], namely that the toric diagram of the compact surface S indicates the existence of ADE singularities along a non-compact curve in the Calabi-Yau threefoldŶ , once S is partially or complete blown down. The novelty here will be to identify these singularities explicitly as codimension one singularities ofπ :Ŷ → B which corresponds to the remnant flavor symmetries of the 6d conformal matter theory from F-theory onŶ . We will match these exactly with the known field theoretic results presented in section 2.2, which we summarize here again for convenience: To begin, let us first observe that the toric phases of the top (see figure 4) only allow nontrivial intersections between the non-flat surface S and the fibers of the exceptional divisors The global symmetry G f of this theory is expected to be a subgroup of the global E 8 symmetry of the 6d SCFT affinized by the Kaluza-Klein U (1) from the circle reduction. As explained in [64], the rank of the flavor symmetry G f is #(F S )−3, where #(F S ) is the number of lattice points on the boundary of the toric diagram of S.
We remark that this number agrees in all cases with the rank of the intersection matrix of all toric curves in S with all non-compact divisors ofŶ , which is spanned by the exceptional divisors A i and a (multi-)section of the fibration. Physically, this is expected if the 5d gauge sector indeed arises from an F-theory conformal matter theory onŶ , because the full 5d flavor symmetry must be generated by the 6d flavor group realized via A i and the KK-U (1) dual to the (multi-)section. 13 In the N f = 2 phase, we see that the toric diagram of S ∼ = P 231 has 6 boundary vectors (cf. figure 4(b)), hence rank(G f ) = 3. To go beyond rank counting, we will identify how the codimension one fibers ofŶ split at the non-flat fiber. This requires us to re-evaluate the 13 Here, we are ignoring possible abelian factors of the 6d flavor symmetry, which could arise from a non-trivial Mordell-Weil group ofŶ , see [123]. In principle, these can also be captured in toric models via the "base" polygon Fm of the top, though they are trivial for F10. We leave a more thorough analysis of other examples for the future.
. From this analysis, we see that the fibers of the exceptional divisors A 6 , A 7 and A 8 do not split, and are in fact entirely contained in the surface S. Furthermore, recall from above that the corresponding fibers of {α 6 } and {α 8 } are also fibral curves with respect to the ruling on S. Following our general discussion in 4.1, we therefore conclude that, when the fiber of S is blown down, the codimension one fibers of A 6 and A 8 are also shrunk. Being two disconnected nodes of the E 8 's Dynkin diagram, the threefoldŶ 3 therefore develops two SU (2) singularities in the generic fiber over W .
Physically, one thus sees an enhancement of an SU (2) 2 ∼ = SO(4) = SO(2N f ) flavor subgroup of the E 8 , when the surface S shrinks to a curve, i.e., when the gauge sector is a weakly coupled su (2) theory with N f flavors. Since this non-abelian flavor symmetry has rank 2, there must be a remaining U (1) part. Indeed, this U (1) must be the topological U (1) T inherent to any 5d gauge theory! There is some redundancies in identifying the U (1)s in terms of the non-compact divisors. One natural choice for the Cartans of the SO(4) flavor group is simply A 6 and A 8 . The topological U (1) T is determined by requiring all massless states of the weakly coupled gauge theory to be uncharged under it. One such divisor is D T = A 5 + A 6 + A 7 − Y , which makes apparent the involvements of the KK-U (1) in form of Y and the Cartan generator Finally, we can also see how the flavor symmetry enhances when we pass to strong coupling.
To do that, we have to shrink the surface S ∼ = P 231 completely to a point. This amounts for not only to blow down the fiber, but also the base of the ruling on S. Recall from earlier that {α 7 } is a section, i.e., a copy of the base of S. But in this resolution phase, it is also homologically equivalent to the generic fiber of the exceptional divisor A 7 . Thus, blowing down S to a point will inevitable force the fibers of A 6 , A 7 and A 8 to shrink everywhere over W .
We note that this is in accord with the statement in [64] that the intersection pattern of vectors on the interior of edges of F S corresponds to the Dynkin diagram of the non-abelian part of G f . In this case, we see from figure 4(b) that there are two edges with interior vectors, one having only a single vector ( α 8 ) and one with two connected vectors ( α 6 and α 7 ). The first corresponds to the Dynkin diagram of SU (2), and the second is that of SU (3), compatible with the embedding of G f into the E 8 that we saw above.

N f = 1 phase
We can repeat the analysis in analogous fashion for the N f = 1 phase, where the surface geometry is S ∼ = Bl 1 F 2 . Since the corresponding toric diagram F S has 5 boundary vectors, the rank of the global flavor symmetry is 2.
Utilizing the facet triangulation depicted in figure 4(c), we see that the relevant splittings of the codimension one fibers are only for A 6,7,8 . In particular, we find that the fiber of A 6 in To go to strong coupling, we now have to also shrink the base of the ruling on S, which again is given by {α 7 }. Like before, this curve is homologous to the generic fiber of A 7 , because it did not split at the non-flat point. This means that at strong coupling, we find a single node of the E 8 Dynkin diagram that is blown down. Therefore, we confirm that in the N f = 1 phase, the global symmetry enhances from SO(2) × U (1) at weak coupling to SU (2) × U (1) ∼ = E 2 at strong coupling. This is also compatible with [64], since the polygon F S in this case has only one vector ( α 7 ) interior to edges. Furthermore, we can verify that in the S ∼ = F 2 phase, the fiber of S contains only one of the two components, into which the fiber of A 6 splits, see figure 4(d) and equation (5.5). Similarly, the fiber of S ∼ = dP 1 only contains a split component of A 8 , cf. figure 4(e) and equation (5.7).
Thus blowing down S to a curve along its ruling does not produce any singularities over W , thus the global symmetry is just the topological U (1) in both cases.
However, in the S ∼ = F 2 phase, the surface contains the full fiber of A 7 , which does not split.
Thus, when we shrink S to a point, we do observe an enhancement of the codimension one singularity. This indicates the enhancement of the global symmetry, Both cases are of course in agreement with [64].
On the other hand, when S ∼ = dP 1 , the fiber of A 7 also splits, see figure 4(e) and equation (5.7), such that the base of S is identified with only one of the two components. Thus, even when the surface is blown down to a point, the generic fiber of A 7 is still at finite size. Hence, the symmetry at strong coupling remains a U (1), which has been labelledẼ 1 in [60]. Before we proceed to the higher rank theories, let us briefly mention that the triangulations of the facet in figure (4) allow us to determine the full structure of the non-flat fibers in each case, which we have collected in appendix C. From the hopefully intuitive notation there, one can very quickly read off the non-abelian flavor symmetry enhancements of the SCFT.

Higher Rank Theories from Non-flat Fibers
In this section, we provide further examples of non-flat fibers which realize higher rank 5d SCFTs. These theories come from the other two facets of the E 8 top and can be analyzed similarly as the rank one facet.
6.1 5d Rank 2 Theories from 6d (E 8 , SU (2)) Conformal Matter Let us turn to examples of 5d rank two theories obtained from circle reductions of the 6d (E 8 , SU (2)) conformal matter theory, also known as the 6d rank 2 E-string. In the singular elliptic fibration Y 3 , this theory is supported at the non-minimal singularity over W ∩ {s 4 }, where the E 8 and SU (2) loci collide. It is known that the circle reduction to 5d (without automorphism twists) will yield a rank two theory. And indeed, this is realized by the top.  As previously, the physics of the 5d theory will depend on the resolution phase of the geometry. A subset of these phases is encoded via the triangulation of the top, which induces a triangulation of the facet in figure 5. While for the rank one case, the combinatorics only allows for five different triangulations (see figure 4(a)), the number for the rank two facet is 156.
Of these, 131 lead to geometries where the two non-flat fibers intersect; the other 25 will give two independent su(2) theories. Those configuration where the surfaces intersect can realize su(3) or su(2) × su(2) gauge theories, but never sp (2). To realize the latter, the two surfaces S 1 ≡ {t 1 } and S 2 ≡ {t 2 } would have to intersect along a curve which is a bisection on one of the two [61]. However, in the present toric realization, the intersection curve is always a toric curve, and thus cannot be a bisection of either. 14 Still, it would be interesting to identify the torically realized theories here with those appearing in the classification [66].
We will leave this for future works.

Non-flat fiber with su(3) + 6F theory
The first example we examine comes a triangulation which involves the all vectors of the facet, i.e., all vectors in figure 5 give rise to a curve on one of the two surfaces. The explicit triangulation is depicted in figure 6. The topology of the surface {t 1 } is that of a dP 2 , blown Right away, we can see that both surfaces are ruled, and glued along a curve which is a section of both rulings. Indeed, the diagram admits a projection along the horizontal axis which maps 2d-cones (triangles in figure 6) onto 1d-cones (vertical lines starting at origin, which is the projection of t i ). Therefore, the weakly coupled gauge theory is has an su (3) algebra [61]. One can easily verify that the curves {y}∩{t 1 } and {y}∩{t 2 } have the appropriate intersection numbers to support the simple roots of su(3). This is of course no surprise, since they are also the generic fibers of the rulings on either surface.   Based on some combinatorial methods [124], from this collection of curves one can form connected linear combinations with genus 0, on which the M2-brane states would fill out N f = 6 fundamental hypermultiplets of su (3).
Using the intuition about the global symmetries, we can also see this arising as follows: With the triangulation structure on the facet as in figure 6, we can straightforwardly check that out of the E 8 exceptional divisors A i , i = 0, ..., 8, only A 1 and A 5 factorize into two.
While the fiber splitting of A 5 yields a curve ({α 5 } ∩ {α 8 }) not inside the non-flat surfaces, the factors of A 1 's fiber are each contained inside {t 1 } and {t 2 }, respectively: Furthermore, because the toric diagram has 10 vectors on the boundary, the rank of the flavor group must be 7 [64]. Thus, the full flavor symmetry at weak coupling must be Lastly, we observe that in order to reach strong coupling, we have to also blow down the base of the fibrations, which are the curves {t 1 } ∩ {j} and {t 2 } ∩ {α 8 }. Again, it is easily verified that, the triangulation (figure 6) for this resolution phase will not split the fibers of the corresponding exceptional divisors J and A 8 at the non-minimal point. Hence, in the SCFT limit, we further enhance the singularities ofŶ 3 in codimension one by shrinking two disjoint P 1 fibers (one over {s 4 } and one over W ), which furthermore are also not connected to the SU (6) singularity. In total, we therefore find an SU (6) × SU (2) × SU (2) flavor symmetry at the SCFT point, consistent with the structure of the vectors on edge interiors.
This flavor symmetry enhancement is only compatible with the su(3) + 6F theory if the Chern-Simons level is κ = 0 (see, e.g., [75]). In fact, the prepotential of an SU (3) gauge theory with N f hypermultiplets reads where the product between matrices is the standard Cartesian one, and we have The contributions from the hypermultiplets splits accordingly to the sign of the middle terms in the third and fourth row.
. For N f = 6 the match between (6.5) and the geometric prepotential computed with the intersection numbers (6.2) fixes a = 2 and Chern-Simons level κ = 0. flop a curve "out" from one surface "into" the other. 15 However, these flops clearly does not change the "combined" polygon, which is simply the full facet. Thus, according to [64], the singular limit where both surfaces shrink to a point is the same. Hence the 5d SCFT should also be the same.
However, depending on the triangulation, the geometry can also realize an su(2) × su (2) quiver theory. Indeed, the phases with corresponding diagrams 7(c) and 7(d) admit a separate ruling for each surface {t i }, indicated by the blue lines. Note that diagram 7(c) is precisely the phase we considered previously. One can quickly convince oneself that the corresponding gauge theory includes, in addition to the bifundamental hypermultiplet, two flavors for each The apparent paradox with the quiver having an SCFT limit [61] is resolved by the obvious duality to the su(3) phase, which is provided by the alternative ruling of both surfaces (red lines in figure 7). Meanwhile, in phases corresponding to triangulations 7(a) and 7(b), only the surface {t 2 } admits a separate ruling. Blowing down {t 2 } along this ruling would degenerate {t 1 } to a (singular) curve, thus introducing additional massless instantonic states which indicate the break-down of a potential su(2) gauge theory interpretation. This is also confirmed by the prepotential analysis, and for this the hypermultiplets masses can be ignored. In fact, similarly to [65], in the wedge φ 2 > φ 1 > 0 (where for this analysis we set the hypermultiplets masses to zero) the tension T 1 = ∂F/∂φ 1 of monopole strings changes sign at φ 1 = 2φ 2 . This signals the appearance of extra light degrees of freedom, and the breakdown of the quiver gauge theory, which is valid only in the subwedge φ 2 > φ 1 > φ 2 /2 > 0. Hence, the gauge theory which is allowed by positivity of the metric of the Coulomb branch and string tensions in the entire chamber is su(3) + 6F. This gauge theory interpretation given by the ruling of these surfaces is always present in these triangulations. In particular, the triple intersection numbers of phase 7(a), match perfectly with the prepotential, (6.5), of su(3) + 6F with a = κ = 0.
Since all these geometries are related by flops, the corresponding 5d UV fixed points must be equivalent. Concretely, we believe that the four triangulations of figure 6 are physically equivalent to the four geometries appearing in the classification [66] which realize su(3) with six flavors at CS-level 0. 16 More specifically, triangulation 1 in figure 7(a) should be physically equivalent to the geometry Bl 6 F 4 F +E ∪ F 0 . This is supported by the fact that the blue polygon indeed corresponds to a six-fold blow-up (at non-generic points) of an F 4 (spanned by the vectors ( y, t 1 , z, α 8 ), cf. figure 7(a)), and F 0 is believed to be physically equivalent to F 2 which is the green polygon. Continuing along the chains of flops, we can then identify the other three geometries with the triangulations presented here, with agreement on the existence of the su(2) quiver phase. 16 Such a theory has seven mass deformations, see figure 16 of [65].
Interestingly, this example seem to suggest that, in order to embed the rank two geometries presented in the classification [66] into a non-flat elliptic fibration, one would in general need to find physically equivalent surfaces that are not generic blow-ups of Hirzebruch or del Pezzo surfaces. A better understanding of the possibilities here, and also for higher rank, would be essential for an attempt to classify all possible circle reductions of 6d SCFTs via M-/F-theory duality.

5d Rank 4 Theories from 6d (E 8 , SU (3)) Conformal Matter
In this part, we will take a look at the rank four facet of the E 8 top. This facet, sitting over the edge E 3 of the base polygon, is depicted in figure 8. This facet describes the non-flat fiber over the collision locus W ∩ {s 8 } of the E 8 and SU (3) divisors. In F-theory, such collision realizes an (E 8 , SU (3)) conformal matter system.
x The fiber resolution of the singularities is described by the vanishing locus of the following polynomial:P We observe the appearance of the SU (3) divisors {f i } over {s 8 }, as well as the four non-flat The precise fiber structure depends on the triangulation of the facet. Again, there is an enormous number of possible triangulations (∼ 14000), which we do not attempt to classify here. However, there are some universal properties which can be inferred from the toric realization right away.
First, as we already noted in the rank two case, the toric setting does not admit gauge groups of non-simply laced ADE type, because the gluing curve dictated by triangulations is always a P 1 in every surface. Second, for the facet in figure 8, we can only have quiver gauge theories at weak coupling, but no single gauge factor theories, i.e., no su(5) or so (8). Again, we expect these restrictions to be lifted in non-toric resolutions of the non-flat fiber.
Some of these phases should also be only one flop away from a toric resolution. For now, we will content ourselves with the torically available phases and leave detailed studies of such flops for the future [122].

Rank 4 quiver theories
The resolution phase we will consider in the following is given by the triangulation depicted in figure 9(a). This triangulation allows for two dual descriptions at weak coupling, which can be reached by blowing down the surfaces along different rulings.
The first choice, corresponding to the rulings marked by red lines in figure (4). With some combinatorics, we find six fundamental hypermultiplets. Thus, the weakly coupled theory obtained via this ruling is ]. x (a) Triangulation of the rank four facet.
x f 2 (b) One choice of ruling which gives rise to an su(2) × su (4) quiver.
x f 2 (c) A different ruling giving rise to an su(2) × su(3) × su(2) quiver. Figure 9: In this triangulation, the surface {u 1 } has polygon bounded by Depending on the rulings, along which the fibers are shrunk, one obtains a different gauge theory at weak coupling.
By computing the rank of the intersection matrix between all non-compact divisors with vectors on the boundary and curves inside the surfaces, 17 we find a rank 9 flavor symmetry group G f . Using the same method as in the lower rank examples, we can easily verify that by blowing down along the ruling indicated by the red lines in figure 9(b), the codimension where U (1) B 1 is the baryonic symmetry of the su(4). The remaining three abelian factors are By construction, these two seemingly different gauge theories must have the same SCFT limit, which is reached by blowing down the surfaces to a point. We can read off the flavor symmetry enhancement at this limit to be SU ( Finally, we note that the enhanced flavor symmetry matches the field theory analysis in [72,75,76], so based on this we claim that the Chern-Simons levels and su(2) theta angle can also confirm these Chern-Simons levels geometrically from the intersection numbers. The explicit computation is detailed in appendix B.
In this paper we have motivated a systematic study of elliptic Calabi-Yau threefoldsŶ with non-flat fibers. These geometries connect 6d conformal matter theories defined via F-theory on the singular limit Y with their circle reduced 5d SCFTs from M-theory on degenerations ofŶ . Compared to previous approaches [63,65,66,68], the non-flat geometries have both the 6d and the 5d SCFT manifest.
To see the difference, recall that in those references, the non-minimal singularities of Y are resolved by first blowing up the base, which pushes the 6d theory onto the tensor branch. A subsequent circle reduction is a 5d KK-theory, which does not have an honest strong coupling limit in 5d. To obtain 5d SCFTs, one has to further mass deform, which is equivalent to turning on holonomies in the circle reduction. Our approach is based on the intuition that in the M-/F-theory duality, resolving the fiber corresponds precisely to such non-trivial holonomies.
The proposal is confirmed by the observation that any crepant fiber resolution of nonminimal elliptic singularities introduces surface components in the fiber, hence being non-flat.
Physically, these surfaces are precisely the compact divisors of a (non-compact) Calabi-Yau threefold which support 5d gauge theories [61]. By construction, they can be collapsed to a point without shrinking the generic elliptic fiber, hence they have a well-defined SCFT limit in 5d.
The non-flat fibers have another feature, namely they allow us to identify the global flavor symmetry of the 5d system, both at weak and at strong coupling, as a subgroup of the flavor symmetry of the 6d conformal matter theory times the KK-U (1). Physically, this is clearly expected if the 5d theory ought to arise from a reduction of a 6d SCFT. What makes this identification possible is a clean way to track how the codimension one fibers of the resolved elliptic fibrationŶ split and become parts of the non-flat fiber components. Blowing down these components then leads to curves of ADE singularities inŶ , which are the Kodaira singularities ofŶ over codimension one loci. While it has been known that such singularities are the source of 5d flavor symmetries [63,64], the non-flat fibration provides an explicit embedding of these into the 6d theory. 19 We have tested our proposal using 5d theories of rank 1, 2 and 4, which arise from the 6d E-string theories of rank 1, 2 and the (E 8 , SU (3)) conformal matter. For simplicity, we have restricted ourselves to elliptic fibrations constructed via so-called tops which provide 19 More precisely, this holds for 5d theories arising from 6d conformal matter theories.
resolutions in terms of toric geometry. The advantage of these constructions is that they incorporate many different resolution phases, and thus different 5d theories, in terms of combinatorial data. For these, we showed that the "usual" fiber analysis in spirit of [102,[119][120][121] matches previous toric constructions of 5d SCFTs [64].
However, we also saw clear restrictions of such toric models. For example, it is impossible to obtain non-simply laced gauge algebras, or non-SU (N ) types of flavor symmetries. On the other hand, we also know [63,65,66] that such geometric phases must exist and be related to toric resolutions via flops. We believe that an explicit construction and analysis of these phases will provide further evidence for the efficacy of our proposal.
Of course, the ideal scenario would be to have a classification of all non-flat fiber resolutions of non-minimal singularities, similar to the Kodaira classification of minimal ones. Such a classification could provide a systematic approach, orthogonal to that of [68], to find all circle reductions of 6d SCFTs, which hopefully brings us closer to a classification of all 5d SCFTs.
Note that there is also some evidence that fiber resolutions sometimes goes hand in hand with twisting by discrete automorphisms (see [125] for the classification of 6D automorphisms compatible with the string lattice), which would reduce the rank of 5d theory [126,127]. A precise understanding of such phenomena [128] could vastly increase the predictive power of our proposal.
Mathematically, our proposal might also open up a new perspective on the classification of canonical Calabi-Yau threefold singularities. Indeed, combining the two conjectures-1), all 5d SCFTs arise from circle reductions [66], and 2), any canonical singularity gives rise to a 5d SCFT [64]-seems to suggest the classification [19,20] of 6d SCFTs via elliptic fibrations also encodes one for canonical threefold singularities. If true, then our analysis suggests that non-flat fibers will play a central role to decipher this code.
Additionally, it is worth pointing out that the phenomenon of non-flat fibers has haunted the F-theory model building community for some time. However, in these constructions one encounters them as surface components in codimension three fibers of elliptic Calabi-Yau fourfolds [98][99][100][101][102][103][104][105][106]. 20 Hence, though geometrically they are similar to what we have studied in this work, the physics is qualitatively different as surfaces on a fourfold do not give rise to any gauge sectors in M-theory. Though there had been some recent progress in this direction [111], a more thorough investigation is needed to fully understand the physical ramifications in 4d/3d.
As a last comment, we highlight that our construction can be easily embedded inside a 20 Similar degenerations also occur in codimension four on Calabi-Yau fivefolds, which define 2d (0, 2) theories [129,130].
compact Calabi-Yau threefold. In the top construction, this simply amounts to include a "bottom", see section 4.2. In some limits of the moduli space of M-theory on this compact Calabi-Yau, a complicated quiver description of the effective theory is likely to exist and probably easy to determine from our constructions. In the putative quiver all flavor symmetries determined in the non-compact setting will be gauged. Computing BPS invariants on compact Calabi-Yau threefolds is in general a difficult task, and the computation of these invariants involves evaluating at all loops the topological string partition function [131,132].
A description of the compact threefold in terms of a quiver with 5d gauge theories/SCFTs building blocks [133] might, in combination with the correspondences between the partition function (or index) of gauge/superconformal theories and topological string partition function [134][135][136][137], be important for the computation of these invariants.

A Weierstrass Model of the E 8 top on F 10
For completeness, we present the Weierstrass model of the elliptic fibrationŶ which we constructed through the E 8 top on the polygon F 10 . This is obtained first by mapping the generic fibration in F 10 , without the specialization from the top, into Weierstrass form. Recall that the hypersurface polynomial in this case is where we have omitted the blow-up coordinates j and f i compared to (4.5). The careful reader might notice the resemblance with the ordinary Tate All in all, the Weierstrass functions of the elliptic fibration (4.10) are With these, one can easily verify the codimension one singularities to be I 2 over {s 4 }, I 3 over

B Geometric Computation of Chern-Simons Levels for Rank Four Example
In this appendix, we present the details of the prepotential computation for the rank four example discussed in section 6.2.1. The intersection numbers for this geometry can be read off from the toric diagram 9(a): where {u i } are the four non-flat surfaces. By expanding the Kähler form as recall from section 2 that these intersection numbers determine the prepotential via We wish to match this with the field theoretic expression of the prepotential, which for where we have already fixed the signs of the absolute values in (2.6). We can now see that the geometric prepotential computed with the intersection numbers (B.1) matches the field theory one (B.3) with N f 1 = N f 2 = 2, and κ = −1.
Likewise, the field theory prepotential for su(2) × [su (4) < l a t e x i t s h a 1 _ b a s e 6 4 = " z 9 d K y 1 t p n I q M x 9 4 + + H e l u G h S 8 D s = " < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 b 6 1 7 k 7 z t t T P 3 Z K e B j s 2 f r n H g U s = "

k 7 z t t T P 3 Z K e B j s 2 f r n H g U s = " > A A A B 6 X i c b Z B N S 8 N A E I Y n 9 a v W r 6 p H L 4 t F 8 C A l k U I 9 F r x 4 r G I / o A 1 l s 9 2 0 S z e b s D s R S u g / 8 O J B E a / + I 2 / + G 7 d t D t r 6 w s L D O z P s z B s k U h h 0 3 W + n s L G 5 t b 1 T 3 C 3 t 7 R 8 c H p W P T 9 o m T j X j L R b L W H c D a r g U i r d Q o O T d R H M a B Z J 3 g s n t v N 5 5 4 t q I W D 3 i N O F + R E d K h I J R t N Z D X Q / K F b f q L k T W w c u h A r m a g / J X f x i z N O I K m a T G 9 D w 3 Q T + j G g W T f F b q p 4 Y n l E 3 o i P c s K h p x 4 2 e L T W f k w j p D E s b a P o V k 4 f 6 e y G h k z D
h z Z k 4 L 8 6 7 8 7 F s L T j 5 z C n 8 k f P 5 A 0 8 L j S E = < / l a t e x i t >  7 v Z S Y s q D K c C Z y 5 / V x j R t m E j r B n U d I E d V g s F p 2 R c + s M S Z w q + 6 Q h C / f 3 R E E T r a d J Z D s T a s Z 6 t T Y 3 / 6 v 1 c h P f h g W X W W 5 Q s u V H c S 6 I S c n 8 a j L k C p k R U w u U K W 5 3 J W x M F W X G Z u P a E P z V k 9 e h f d X w L d e b l 2 U Y V T i F M 7 g A H 2 6 g C f f Q g g A Y I L z A G 7 w 7 T 8 6 r 8 7 F s r D j l x A n 8 k f P 5 A w 5 j i 3 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d h B l t y E h / O j + x a 7 b T Y L 9 p F f 9 9 b 0 = " > A A A B 5 H i c b Z B N S 8 N A E I Y n 9 a v G r + r V y 2 I R P E h J R N B j w Y v H C q Y t t K F s t p N 2 7 W Y T d j d C C f 0 F X j w o X v 1 N 3 v w 3 b t s c t P W F h Y d 3 Z t i Z N 8 o E 1 8 b z v p 3 K x u b W 9 k 5 1 1 9 3 b P z g 8 q r n H b Z 3 m i m H A U p G q b k Q 1 C i 4 x M N w I 7 G Y K a R I J 7 E S T u 3 m 9 8 4 x K 8 1 Q + m m m G Y U J H k s e c U W O t h + t B r e 4 1 v I X I O v g l 1 K F U a 1 D 7 6 g 9 T l i c o D R N U 6 5 7 v Z S Y s q D K c C Z y 5 / V x j R t m E j r B n U d I E d V g s F p 2 R c + s M S Z w q + 6 Q h C / f 3 R E E T r a d J Z D s T a s Z 6 t T Y 3 / 6 v 1 c h P f h g W X W W 5 Q s u V H c S 6 I S c n 8 a j L k C p k R U w u U K W 5 3 J W x M F W X G Z u P a E P z V k 9 e h f d X w L d e b l 2 U Y V T i F M 7 g A H 2 6 g C f f Q g g A Y I L z A G 7 w 7 T 8 6 r 8 7 F s r D j l x A n 8 k f P 5 A w 5 j i 3 g = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " d h B l t y E h / O j + x a 7 b T Y L 9 p F f 9 9 b 0 = " > A A A B 5 H i c b Z B N S 8 N A E I Y n 9 a v G r + r V y 2 I R P E h J R N B j w Y v H C q Y t t K F s t p N 2 7 W Y T d j d C C f 0 F X j w o X v 1 N 3 v w 3 b t s c t P W F h Y d 3 Z t i Z N 8 o E 1 8 b z v p 3 K x u b W 9 k 5 1 1 9 3 b P z g 8 q r n H b Z 3 m i m H A U p G q b k Q 1 C i 4 x M N w I 7 G Y K a R I J 7 E S T u 3 m 9 8 4 x K 8 1 Q + m m m G Y U J H k s e c U W O t h + t B r e 4 1 v I X I O v g l 1 K F U a 1 D 7 6 g 9 T l i c o D R N U 6 5 7 v Z S Y s q D K c C Z y 5 / V x j R t m E j r B n U d I E d V g s F p 2 R c + s M S Z w q + 6 Q h C / f 3 R E E T r a d J Z D s T a s Z 6 t T Y 3 / 6 v 1 c h P f h g W X W W 5 Q s u V H c S 6 I S c n 8 a j L k C p k R U w u U K W 5 3 J W x M F W X G Z u P a E P z V k 9 e h f d X w L d e b l 2 U Y V T i F M 7 g A H 2 6 g C f f Q g g A Y I L z A G 7 w 7 T 8 6 r 8 7 F s r D j l x A n 8 k f P 5 A w 5 j i 3 g = < / l a t e x i t >    The only exception is the curve 5A, which is the vanishing of α 5 and a two-term polynomial (see equation (5.4)). Lines indicate pairwise intersections; ⊥ signals that three P 1 s intersect at one point. The gray box with enclosed circles represents the non-flat surface with curves contained inside of it.