On the Scarcity of Weak Coupling in the String Landscape

We study the geometric requirements on a threefold base for the corresponding F-theory compactification to admit a weakly-coupled type IIB limit. We examine both the standard Sen limit and a more restrictive limit, and determine conditions sufficient for their non-existence for both toric bases and more general algebraic bases. In a large ensemble of geometries generated by base changing resolutions we derive an upper bound on the frequency with which a weak-coupling limit may occur, and find that such limits are extremely rare. Our results sharply quantify the widely held notion that the vast number of weakly-coupled IIB vacua is but a tiny fraction of the landscape.


Introduction
The landscape of string theory vacua has provided a rich ensemble of four dimensional effective theories of quantum gravity, and many candidates of an ultraviolet completion of our universe [1][2][3]. A large number of 4d N = 1 theories are realized by weakly-coupled string theory on Calabi-Yau manifolds and orientifolds thereof. Such compactifications have provided a vast set of four-dimensional effective theories enumerated by choices of geometry and flux, and have also yielded many non-trivial computations of operators in the low energy theory, including various candidates for BSM physics, such as interesting cosmologies and particle spectra (see e.g. [4] for recent work). However, it is believed that this weakly-coupled landscape is only a tiny fraction of the landscape of 4d theories coupled to gravity that have UV completions.
In this work we endeavor to determine the prevalence or scarcity of one of the best studied corners of the landscape, weakly coupled type IIB orientifold compactifications, within its natural larger context, F-theory. F-theory [5,6] is an intrinsically strongly-coupled generalization of type IIB string theory on a Calabi-Yau orientifold that incorporates a holomorphically varying axio-dilaton in the structure of an elliptically fibered Calabi-Yau variety X 1 . In some cases a particular limit in the complex structure moduli of X, known as the Sen limit [7,8], recovers weakly coupled type IIB on the base B of X, where B is realized as an orientifold of a Calabi-Yau threefold. Interestingly, unless a strict Sen limit is taken in a way that will be made precise, there are always regions in B where g s is O(1), which motivates us to also study a global weak coupling limit (GWCL) that can avoid the existence of strongly coupled regions. Whether an F-theory geometry X admits a limit in complex structure that gives rise to a Sen limit or GWCL is determined by the geometry of B. We study conditions under which the topology of B forbids such limits, and the frequency with which these conditions hold in large ensembles of geometries.
Our main result is that there are relatively simple conditions on B that determine whether there exists a Sen limit or a GWCL, and in the ensembles of bases B that we study the probability that such a limit exists is extremely small. Specifically, in our ensemble the fraction of bases admitting a GWCL or a Sen limit are bounded by (1. 1) In the discussion at the end we will argue that this probability can only shrink in ensembles of toric bases B larger than ours, and that the low probability of the existence of these limits should also be a feature of other ensembles. These results suggest that the widely studied weakly coupled type IIB landscape is but a small fraction of F-theory, and therefore also of the landscape as whole. While its relative scarcity has been widely anticipated, this attempt to quantify it exacerbates the need to better understand moduli stabilization in strongly coupled limits of string theory, as well as transitions between vacua arising in different low energy effective theories that themselves arise from different strongly coupled compactifications.
At a technical level, our results arise from understanding how seven-brane configurations are modified under topological transitions. For example, specific topological transitions from B to B induce a transition of the associated Calabi-Yau fourfold X to X that gives rise to so-called non-Higgsable clusters (NHC) [6,9,10] with high probability. These are clusters of non-trivial intersecting seven-brane configurations that exist for generic values of the complex structure of X . Moduli stabilization at generic points in moduli space can therefore achieve gauge symmetry [11] and there is increasing evidence [12][13][14] that generic bases give rise to such clusters for generic moduli. For example, in the ensemble of 4 3 × 2.96 × 10 755 geometries from our previous work [14], the fraction of geometries that exhibited non-Higgsable clusters was i.e. essentially every geometry exhibited NHCs. Furthermore, the list of individual non-Higgsable seven-brane (NH7) structures that can exist in a cluster is rather short, though the building blocks can be assembled into a cluster in many ways, and most of the individual seven-brane structures are intrinsically strongly coupled; see [15] for a detailed study. Together, these facts provide strong evidence that generic points in the complex structure moduli space of generic F-theory geometries are strongly coupled. However, this does not preclude the existence of a Sen limit or a GWCL, since if the gauge factors on the seven-brane are sufficiently low rank then there could exist limits in moduli space in which the gauge factor is actually enhanced, and can be described by N D7-branes on top of an O7-plane at weak coupling. These arise from so-called Kodaira I * n singular fibers. This potential loophole is the technical focus of this work, and what we show is that if the low rank strongly coupled NH7 is tuned to I * n , those or other seven-branes in the geometry are forced to enhance beyond I * n , under simple conditions that occur with high probability. These more singular seven-branes carry exceptional gauge groups and forbid either of the weak coupling limits.
The outline of this paper is as follows. In §2 we review the standard Sen limit, and introduce a global weak coupling limit that ensures g s 1 globally on a base B. In §3 we determine the geometric conditions for these weak coupling limits in a large ensemble of toric bases, and find that these conditions are almost never satisfied. In §4 we extend this result to more general bases, constructed via gluing local patches together, where the local patches are crepant resolutions of orbifold singularities. In §5 we conclude.

Non-Trivial Weak Coupling Limits
In this section we review the Sen limit and introduce a stricter limit where no O(1) g s regions occur. The Sen limit is the most well-known method of recovering weakly coupled type IIB from F-theory, by taking the string coupling to be perturbatively small [7,8]. We consider an F-theory compactification on an elliptically fibered Calabi-Yau fourfold π : X → B with section over a smooth algebraic threefold base B. By a result of Nakayama [16] such an elliptic fibration is birational to a Weierstrass model, which is described by the equation ) are generic sections of appropriate powers of the anticanonical bundle on B. The corresponding discriminant locus is given by the vanishing of which specifies the singular fibers and location of 7-branes in B. The complex structure modulus of the elliptic fiber corresponds to the axio-dilaton which varies over the base B. Here the vacuum expectation value of φ determines the string coupling constant as g s = e φ . The J-invariant, defined in terms of the usual j-invariant as J := j/1728, takes the form As we are interested in weak coupling we need to determine the value of the axiodilaton from the J-invariant. Defining q = e 2πiτ , at large |q| the j-invariant admits the expansion In particular, for small |q| we have that J ∼ 1 q where |q| = exp(− 2π gs ), and hence studying the weak coupling limit is equivalent to studying the conditions under which J → ∞. One can solve for g s for each Kodaira singular fiber by inverting the Jfunction to solve for τ ; see e.g. [15]. τ is defined only up to an SL(2, Z) action, and so g s is not uniquely defined. However, each Kodaira singular fiber has a minimal associated g s that cannot be further lowered by an SL(2, Z) transformation. This information is presented in Table 1. In this sense, each 7-brane locus corresponding to a Kodaira singular fiber in Table 1 can be thought of as a boundary condition for the Table 1: Kodaira fiber F i , singularity, and gauge group G i on the seven-brane at x i = 0 for given l i , m i , and n i . In the second last two columns we display the minimal g s with corresponding τ .
axio-dilaton in the internal space, where for a fixed SL(2, Z) frame a 7-brane along a divisor Σ uniquely specifies the axio-dilaton along Σ. Away from these 7-brane loci the axiodilaton depends on complex structure moduli, and can be fixed by flux. From Table 1 we can see that the singularities that allow for weak coupling are the I n series, the I * n series, and I * 0 , which realize SU (n) or Sp( n/2 ), SO(2n − 4) or SO(2n − 5), and SO (8) or SO(7) or G 2 gauge groups, respectively. In fact, it is precisely the SU (n), Sp( n/2 ), SO(2n−4), and SO(8) gauge groups that are realizable in weakly-coupled IIB orientifold theories, and so we should expect the F-theory lift to reflect this somehow.

The Sen limit
Sen's orientifold limit is achieved by taking the following ansatz The J-invariant is then given by The Sen limit is the limit in which is taken to be small, in which case the J-invariant at leading order in is given by It is important to note that we wish to carefully distinguish between the Sen limit, and what we will call the strict Sen limit, in which → 0, strictly. The latter limit takes the string coupling strictly to zero away from any 7-branes, while the former allows for small but non-zero string coupling away from 7-branes. However, even if we take infinitesimally small, but non-zero, we have O(1) g s regions along the loci where f = 0 or g = 0. These sit between components of the residual I 1 locus, part of which collapses to an O7-plane in the limit → 0. Thus, in the Sen limit, we may still have strongly coupled regions along certain loci in the base. In the strict limit the resulting type IIB Calabi-Yau may be identified as a double covering of the base B.
In fact, particular 7-branes known as non-Higgsable 7-branes (NH7) play a major role in obstructing weak coupling limits. NH7s arise along a divisor D when f and g vanish to a minimal non-zero order along D, for all choices of complex structure moduli. This implies we can write f = z nf and g = z mg , where both n and m are greater than zero. A key point is that I 1 and A n fibers cannot arise on NH7s, as the multiplicities of vanishing of f and g are already too high, and so for a weak coupling limit to exist we must tune to I * 0 or D n on any NH7s. Clusters of intersecting NH7s are aptly named non-Higgsable clusters (NHCs), and they have been well-studied in the literature. Some references particularly relevant for this paper were cited in the introduction, but we would like to briefly mention their applications in broader contexts as well. In six dimensions, NHCs are particularly well-understood, beginning with seminal first works [6,9,10] and also later studies [10,13,[17][18][19][20][21] on a variety of topics. Non-Higgsable clusters allow for gauge symmetry to arise in stabilized vacua at generic points in moduli space [11], i.e. without tuning to high codimension in moduli space [22][23][24], and may allow for a simple realization of the Standard Model in F-theory [11]. 4d NHCs [25] are particularly interesting due to the fact that they can realize structure not possible in high dimensions, such as loops [9]. NHCs appear in the geometry with the most number of flux vacua known [26], as well as in large ensembles of geometries [12]. Recently aspects of NHCs have been studied using supervised machine learning [27].
Let us understand this from the perspective of the J-invariant. The minimal non-Higgsable 7-brane corresponds to the type II fiber, so suppose that we start with a divisor with a type II fiber above it. Let the 7-brane locus be given by the vanishing of a local coordinate z. Near z = 0 we can Taylor expand f and g as For a type II fiber we have f 0 = g 0 = 0, and so we can write f = zf and g = zg. The J-invariant thus takes the form, This has strong coupling points along z = 0. In order to get rid of the strong coupling point at z = 0 one can try to tune to remove the factors of z. The only way to do so is to take g = z 2g , which gives a type III fiber. However, this leaves us with so we are again left with strong coupling along z = 0. One can continue on in the same fashion, until we hit the f = z 2f , g = z 3g , which is the case of I * 0 . Here the J-invariant reads which does not necessarily have strong coupling along z = 0. However, if we start with a type IV * fiber it is easy to see in a similar manner that one cannot tune f and g to eliminate all strong coupling points on 7-branes simultaneously. Therefore we confirm that the singular fibers with possible Sen limits, where there are no regions of O(1) g s along 7-branes that carry a gauge group, are I n , I * n , and I * 0 . One feature that distinguishes the Sen limit from the strict Sen limit is the allowed gauge groups, as a monodromy action on the fibers can reduce the rank of the gauge group. In the strict Sen limit the gauge group G 2 is not present, as the monodromy ramification locus necessary to reduce SO(8) to G 2 disappears as → 0 [28]. However, for finite but small this locus can still be present, and G 2 can still be realized.
If one insists on the stronger condition of a global weak-coupling limit, without any regions of O(1) g s , the 7-brane charge needs to be cancelled everywhere locally, which restricts us to a fiber of type I * 0 . In analogy to the Sen limit we now define the global weak coupling limit.

The Global Weak Coupling Limit
We say that a smooth algebraic threefold base B admits a global weak coupling limit (GWCL), if the corresponding Weierstrass model π : X → B can be tuned so that g s 1 globally on the base B. As shown in e.g [15], a necessary condition to achieve weak coupling everywhere in B is to have an I * 0 fiber on all non-Higgsable seven branes. Indeed, in the most generic such case, f and g take the following form can be thought of as complex constants and we have non-Higgsable 7-branes along the vanishing loci of m. Moreover, m should be reduced, possibly with normal crossing singularities. Given such a parametrization, the J-invariant thus takes the form which is indeed constant everywhere on the base. Thus, in order to achieve weak coupling, we must take the expression 4F 3 + 27G 2 sufficiently small. It suffices to parametrize F and G appropriately to take J to be arbitrarily large. Let us take the following ansatz for the constants F and G where˜ is an arbitrary parameter. At small˜ we therefore have We thus obtain the weak coupling limit by taking the parameter˜ → 0, in other words, taking G → −2.
We may thus generalize the Sen parametrization by taking the following ansatz where we simply take the transformation −2 → −2 +˜ in the original Sen parametrization. By taking the limit˜ → 0, we recover Sen's original expressions, while successively taking the limits → 0 and then˜ → 0 recovers our original ansatz and allows us to achieve weak coupling everywhere in the base. For a GWCL, any seven-brane must have an associated singular fiber that is I * 0 , but in general this does not uniquely determine the gauge group, due to monodromy effects. In the case of an I * 0 fiber along a locus Σ the gauge group is determined from a monodromy-coverΣ of Σ, which takes the form ψ 3 + (f /z 2 )| z=0 ψ + (g/z 3 )| z=0 , where ψ is a generic section of a particular line bundle on X [29]. If F, G ∈ Γ(O B ) then the monodromy-cover has no monodromy, and the gauge group is necessarily SO(8), as opposed to SO(7) or G 2 , which are also possible with an I * 0 fiber. This is true even in the presence of multiple I * 0 loci, as one can explicitly check that each Dynkin node is invariant under any possible monodromy action.
We now turn to studying the geometric requirements for both of these weak coupling limits. We first examine this in the toric setting, from which most examples so far have been drawn.

Weak Coupling Limits on Toric Bases
In this section we derive rather constraining requirements for a toric base to admit either of the two weak coupling limits. In [14] an ensemble of 4 3 ×2.96×10 755 toric bases for F-theory was generated by crepant fourfold transitions induced by base changes from certain minimal geometries that we will discuss. We will discuss both general toric bases and those that are part of this ensemble. We will begin by reviewing the construction of the ensemble and then proceed by analyzing the GWCL first as it is more restrictive and therefore simpler than the Sen's limit.
We first review some basic facts about toric varieties. Let B be a smooth toric variety, corresponding to a fan Σ, and denote the rays of Σ as u i ∈ Σ(1). Writing The m's that satisfy these conditions form lattice polytopes, which we denote ∆ g and ∆ f : which correspond to monomials via where each x i is a homogenous coordinate on B corresponding to each v i . The most generic forms for the sections f and g thus take the following form with the a f and a g generic complex coefficients. In the following we will be interested in generating a large ensemble of geometries from crepant base-changing resolutions of some minimal geometries. The minimal geometries we consider are smooth weak Fano toric varieties associated with a fine regular star triangulation (FRST) of a 3d reflexive polytope ∆ • , with corresponding face fan Σ. Of key interest in this work will be the Cox ring, or the homogenous coordinate ring, of the toric variety; in particular, the behavior of sections of line bundles under a crepant resolution. Each of these toric varieties can be viewed as a crepant resolution of a singular toric varietyB, whose coordinate ring is generated by the corners (vertices) of ∆ • . This toric variety may not even be simplicial, though one can make it so by appropriately subdividing the associated fan. ResolvingB by triangulating ∆ • introduces exceptional toric divisors, that correspond to points interior to edges and facets of ∆ • . These divisors are rigid, in the sense that they have no normal bundle deformations. This structure will be important both in determining if there are non-Higgsable clusters, and the presence of either weak coupling limit. On a toric base it will often be the case that a single base-changing resolution introduces at least a type II fiber on one of these crepant exceptional divisors D. Therefore, to realize either of the weak coupling limits we must tune to an I * n fiber for n ≥ 0, as the type II non-Higgsable 7-branes eliminate the possibility of an I n fiber 2 . This requires f and g to factorize as F and G are non-trivial polynomials that parametrize a possible I 1 locus, and c f and c g are constants. However, as a single blowup generically forces type II on many toric divisors, we will be mainly interested in the case that the s i correspond to toric coordinates x i .
In the following we will find that the existence of a generic toric blowup on a toric base B naturally induces type II fibers on nearby crepant exceptional divisors. As singular fibers can only be enhanced, we must engineer transitions to tune to singular fibers supporting a GWCL or Sen limit along the starting type II fibers. It then follows, via simple combinatorics and using necessary conditions on orders of vanishing for transitions in the Weierstrass model, that such bases supporting either limit are scarce, and always admit multiple transitions to bases prohibiting either limit. On the level of the geometry, as we blowup and move away from the Fano range and simultaneously increase the Kodaira dimension, i.e. as the number of sections of the canonical bundle grow, we can image the space of sections of the anticanonical bundle gradually shrinking, and its base locus, i.e. the common locus of all the sections, enlarging, while the multiplicity of the sections increases simultaneously. Such a base locus will typically correspond to rigid crepant exceptional divisors, and therefore the existence of rigid crepant exceptional divisors is crucial to an obstruction of either of the weak coupling limits. Such exceptional divisors are the easiest to move within the base locus and to augment its multiplicity past the orders required for weak coupling limits.

Trees of F-theory Geometries
In this section we give a brief overview of the ensemble of F-theory geometries constructed in [14]. This is the largest ensemble of Calabi-Yau fourfolds known, and the scarcity of weak coupling limits in this ensemble provides strong evidence against the existence of weak coupling limits for a generic Calabi-Yau fourfold. We will henceforth refer to the corresponding bases of the elliptic fibration as toric tree bases. Let B be a smooth weak Fano toric threefold induced by a crepant resolution of a Fano toric threefold corresponding to a 3d reflexive polytope ∆ • via an FRST. Associated to such a B is a fan consisting of 2d and 3d cones corresponding to edges and faces on facets of ∆ • . For a fixed facet, the number of such edges and faces is completely determined by the number of vertices and boundary lattice points interior to each edge which is independent of the choice of triangulation. Beginning from a fixed weak Fano toric threefold, transitions to topologically distinct threefolds will be obtained via smooth toric blowups of curves or points. These birational morphisms can be interpreted purely combinatorially at the level of polytopes. For instance, to blowup a curve, given 2 cones with generators (v 0 , v 1 , v 2 ) and (v 1 , v 2 , v 3 ) we add an additional ray v e := v 1 +v 2 which replaces the original 2 cones with 4 cones given by the . We can repeat this procedure by recursively subdividing, which yields the diagram shown in Fig. 1 in two dimensions, where the edge between v 1 and v 2 corresponds to an edge contained in a facet, and we subdivide this edge by adding additional rays. In particular, we will refer to the cones corresponding to a sequence of blowups as illustrated above colloquially as "trees" which is further emphasized from the green dashed lines corresponding to new edges resulting from these subdivisions. From such a procedure, it is clear that any additional ray v e must take the form v e = i a i v i which is a linear combination of two vertices if it lies over an edge, or three vertices if it lies over a face, where each v i is a lattice point of B i . Such additional rays will also be informally called "leaves" with "height" h defined by h = i a i where the heights of the leaves are labeled in the above diagram. The leaves with h = 1 correspond to lattice points on the facet which we will refer to as "roots". Given a tree obtained from a sequence of subdivisions, we will refer to the height of the tree as the height of a highest leaf.
Instead of considering all possible trees above a fixed facet, which is computationally infeasible, we will consider trees built above individual simplices on the facet. Thus, we will refer to trees above a fixed face or edge as face trees or edge trees, respectively. Given a fixed height, all trees with height at most h can be classified purely via combinatorial techniques. In particular, to visualize such a combinatorial procedure, it is fruitful to picture each tree with its leaves projected to the base edge or face. To illustrate this, given a ground edge where the last subdivided edge is the above tree with all its leaves projected down. In a completely analogous fashion, a subdivision of a face may be pictured as follows where the additional point with height 3 corresponds to adding a ray by summing all three of the roots.
There is a strong constraint on the full moduli space of elliptically fibered Calabi-Yau fourfolds which give consistent F-theory compactifications, namely the exclusion of fourfolds containing (4, 6) divisors. It is easy to see, via a completely combinatorial argument [14], that this (4, 6) condition is satisfied by imposing an upper bound h ≤ 6 on the height of all trees. Thus, for all heights 3 ≤ h ≤ 6, we may enumerate all possible trees which gives the numbers in Table 2. Trees  3  5  2  4  10  17  5  50  4231  6 82 41, 873, 645 Given a 3d reflexive polytope, we may define a corresponding ensemble S ∆ • . Let T (∆ • ) be an FRST of ∆ • . For each face or edge, we add a face or edge tree with maximal height 6. Then, from the above table, one finds that the number of possibilities in the ensemble S ∆ • is and ∆ • 2 , we find that they have the same number of total edges and faces given by #Ẽ = 108 and #F = 72. We find that the cardinality of each ensemble is given by

N # Edge Trees # Face
where the additional factor of 1 3 in |S ∆ • 1 | corresponds to an order 3 symmetry obtained by rotating one of the facets of ∆ • 1 . Adding these numbers gives a lower bound of 4 3 × 2.96 × 10 755 threefold bases supporting topologically distinct F-theory geometries. The ensemble generated by building trees over ∆ • 1 and ∆ • 2 completely dominate the ensemble of tree geometries. These geometries will therefore serve as the "ground" on which we will build our trees, yielding a vast number of topologically distinct bases for elliptically fibered Calabi-Yau fourfolds. In particular, ∆ • 1 and ∆ • 2 each have a facet with 63 edges and 36 faces, which determine much of the structure of the ensemble, including the scarcity of weak coupling limits, as we will find. The corresponding facets are shown in Fig. 2.

The global weak coupling limit
As discussed above we will concentrate on the case where our I * 0 loci are located on the vanishing of single toric coordinates, as opposed to the vanishing of more general polynomials thereof. This will actually provide rather strict requirements on toric bases that admit GWCLs, as blowups generically results in non-Higgsable type II fibers over particular toric divisors, as we will discuss.
Since the existence of GWCL requires that the only singular fibers are I * 0 fibers, it is natural to group the toric divisors into two sets: those that support an I * 0 singularity, and those that do not. Then f and g factorize as Here F, G ∈ Γ(O B ), and we have I * 0 loci on the toric coordinates x 1 , . . . , x p , and smooth elliptic fibers over x p+1 , . . . , x n . Denote the former coordinates x a , and the latter x i . It is simple to see that, for all m corresponding to monomials in g, we have The −6 condition is necessary for a smooth fiber over D i and no I 1 fiber anywhere, where the latter is necessary for a GWCL.
We now consider which toric bases B allow for a GWCL. We will see that under simple geometric conditions on B the GWCL condition 3.8 is violated and F-theory on B does not admit a GWCL. Let us take our base B to be obtained by a sequence of toric blowups from one of the smooth weak Fano toric threefolds given by an FRST of a reflexive 3d polytope ∆ • .
First consider a point p interior to a facet F , corresponding to a toric coordinate x p with divisor D p . Assume that we have a type II fiber on D p , and so to realize a GWCL we need to tune an I * 0 on D p , then we need to include only monomials in g whose corresponding m (see Eq. 3.2) satisfy m·p f = −3. Suppose that B indeed admits a GWCL. Then by linearity across F , these monomials must also satisfy m · v = −3, for all v ∈ F . To see this, express p = are vertices of F , and a 1 , a 2 > 0. By assumption of the existence of the GWCL we have v i · m ∈ {−3, −6} for all m ∈ ∆ g , i = 1, 2, 3, One then sees that the only solution consistent with the GWCL is that v i · m = −3 for all m, i = 1, 2, 3.
This immediately shows that if we start with a type II fiber on D p , we must tune an I * 0 on every point on F to realize a GWCL. A similar result holds for any point p e interior to edges: if we tune an I * 0 on p e , then we must have I * 0 on the entire edge. Let us now start with a toric base geometry corresponding to an FRST of a 3d reflexive polytope, and consider the effect of toric blowups. Here we are simply blowing up in a coordinate patch, so that the blowup divisors are generated by two or three rays in Σ that belong to a common cone. Each blowup ray can then be written as v e = i a i u i . In general, the blowing up of B will reduce the number of sections of O(−4K B ) and O(−6K B ), since adding rays to the fan induces more hyperplane constraints on ∆ f and ∆ g . Without loss of generality we will concentrate on ∆ g , since the result for ∆ f is nearly identical. Assume that we have an I * 0 on a point interior to a facet F , which implies I * 0 fibers on all toric divisors corresponding to points on F . Now consider two intersecting toric divisors D 1 and D 2 , with corresponding rays v 1 and v 2 on F . Let us build a tree above the edge defined by {v 1 , v 2 }. There will then be corresponding divisors of the form D = a 1 D 1 + a 2 D 2 , and we have mult D (g) = m, a 1 v 1 + a 2 v 2 + 6 = −3(a 1 + a 2 ) + 6 = −3h + 6 (3.9) for all m ∈ ∆ g . Thus, for any h ≥ 3 blowup, it must be the case that m / ∈ ∆ g and therefore that the monomials supporting I * 0 fibers above all points in F are absent. That is, the condition (3.8) is violated and the GWCL is spoiled, as the multiplicity of vanishing in f and g is greater than that in I * 0 ; i.e. a IV * , III * , or II * fiber is obtained, all of which correspond to seven-branes with exceptional geometric gauge group, none of which admit a weak coupling limit. An even stricter result holds for face blowups: any face blowup necessarily has h ≥ 3, and so any face blowup necessarily eliminates the monomials allowing for I * 0 fibers, and therefore also spoils the GWCL.
A similar result hold for a point interior to a polytope edge e. Let us assume we have at least a type II fiber on the divisor corresponding to such a point, which implies that there are I * 0 fibers on all toric divisors corresponding to points on e. Now consider two of these toric divisors D 1 and D 2 , with corresponding rays v 1 and v 2 . If we perform a blowup using these rays, such that D = a 1 D 1 + a 2 D 2 , it is simple to see that mult D (g) = −3h + 6, and therefore any h ≥ 3 blowup will eliminate the monomials that give the correct multiplicity of vanishing for the points interior for e to support I * 0 fibers. Then the GWCL is spoiled. Summarizing, we have a strong constraint on the toric tree bases that admit a GWCL: a GWCL can only exist if the base is a weak Fano toric variety, or a toric resolution of a weak Fano toric variety with blowups of height-2.
The physics of the height-2 blowups has simple interpretation as the splitting of branes. Consider a base B with intersecting I * 0 singularities, that admits a GWCL. Such an intersection over a curve C has M OV C (f, g) = (4, 6), and therefore admit a crepant base change, to produce a base B without such intersections. The change of base to B introduces a new divisor in B , which one can shrink to zero size to recover B, and so we should not be surprised that the resolved geometry admits a GWCL as well. However, B also admits another GWCL, distinct from that of B, that results in non-intersecting I * 0 fibers. Clearly, such a phenomena only occurs for height 2 blowups, as B would not admit further base changes obtained by blowing up intersections between the exceptional divisor and the original I * 0 singularities. In particular, such a B must have OOV of (6,9) at triple intersections p of I * 0 singularities which is not sufficient to admit a base change by blowing up p. For that to occur, the OOV must be ≥ (8, 12); see the Appendix of [14] for a detailed discussion.
Let us now turn to bounding the fraction of tree toric bases that admit a GWCL. The ensemble is overwhelmingly generated by trees over ∆ Therefore, the fraction of geometries in the tree ensemble that allow for a GWCL make up an absolutely minuscule fraction of the total geometries.

The Sen limit
Recall from Section 2 that a Sen limit is possible when all the fibers are either smooth, I n , I * 0 , or I * n , which is more general than the GWCL. However, we will find that the bases that allow for a Sen limit are still very constrained. We will proceed in the same manner as before. We consider the effect of a base-changing resolution, which forces certain multiplicities of vanishing of f and g on divisors corresponding to points interior to faces and edges. As before, any blowup along an edge or a face forces at least type II fibers on every point interior to that edge or face. Let the divisors corresponding to such interior points be D i . Assume such a blowup has been done, in which case the fiber type is at least type II for all D i . Then each fiber above the D i must be tuned to I * n to allow for a Sen limit. A necessary condition is then f and g vanish to at least orders 2 and 3, respectively, with M OV (∆) = 6, and so we have a similar set of conditions as in the case of the GWCL, the only difference being that we allow for an I 1 locus in the case of a Sen limit.
First we consider an edge e of the polytope with at least one interior point. Let the points along the edge correspond to coordinates x i , with corresponding divisors D i . Then there must be an m f ∈ ∆ f , corresponding to a monomial in f , such that m f · v n = −2, or there must be an m g ∈ ∆ g , corresponding to a monomial in g, such that m g ·v n = −3. Without loss of generality let us first assume the former is true, since both analyses are nearly identical and yield the same constraints. Linearity constrains the dot products between such an m and the rest of the v i as displayed in Figure 3. First assume that p > 0 for this m. If we perform a height-2 blow up along a one-simplex to the right of v n then this eliminates such an m, as m · (v n + v n+1 ) = (−2) + (−2 − p) < −4 for p > 0. Similarly, if p < 0 then a height-2 blowup to the left of v n will eliminate such an m. Let us perform height-2 blowups on both the left and right of v n , i.e. p = 0. It then must be the case that m·v i = −2 for all v i ∈ e. However, under this hypothesis, a subsequent height ≥ 3 blowup anywhere along e will eliminate any such m. Indeed, any height 3 blowup leads to a ray of the form 2v i + v i+1 and hence we have m · (2v i + v i+1 ) = −6. This violates the condition for m to correspond to a monomial in f : m · v a ≤ −4 ∀ v a ∈ ∆ • , and therefore the monomials protecting higher than I * 0 vanishing along the v i are eliminated by such a blowup. If there are at least three points interior to an edge e one can say more. Let us consider a divisor D corresponding to a point p in the interior of e that is not adjacent to either of the vertices of e. Assume there is a monomial, corresponding to m f ∈ ∆ f , that has MOV of 2 along D in f , which protects against higher than I * 0 M OV in f . Blowing up along e then forces at least type II vanishing along D, v 0 Figure 4: The largest facet F 1 of the 3d reflexive polytope ∆ • 1 , with an arbitrary triangulation. A type II fiber above the divisor corresponding to v 0 places strong constraints on the existence of the Sen limit. and any other divisors corresponding to interior points of e. However, linearity then requires m, ·v i = −2 for all v i interior to e, which brings us to the same situation as as in a GWCL. Therefore a single blowup of height ≥ 3 above e will spoil a Sen limit.
There is a similar condition for 2-simplex blowups, where the 2-simplices are interior to a facet F . A 2-simplex blowup involving only points interior to a face will force > I * n vanishing in both f and g on at least one divisor. An even stronger condition occurs nearly universally in the tree ensemble. First consider trees built over ∆ • 1 . The largest facet F 1 of ∆ • 1 is shown in Figure 4. After a height ≥ 2 blowup anywhere on the face all of the points interior to F 1 will have at least type II fibers above them. Therefore, to admit a Sen limit, we must tune to at least I * 0 on all of these points. The point in the middle v 0 , corresponding to a divisor D 0 , has neighbors in both the vertical and horizontal directions with I * 0 fibers. This implies that there is at least one monomial in f that vanishes to multiplicity 2 along D 0 , or at least one monomial in g that vanishes to multiplicity 3 along D 0 . Without loss of generality let us assume the former is true. We then have m · v 0 = −2 for some m. However, the fact that we have all I * 0 on the points v i interior to F 1 implies m · v i = −2, and therefore m · F 1 = −2. It is clear then any height ≥ 3 blowup on F 1 will eliminate this monomial, and force greater than I * 0 on the divisor D 0 . Therefore a height ≥ 3 blowup on F 1 will obstruct the Sen limit.
A similar analysis can be done for ∆ • 2 . The largest facet F 2 of ∆ • 2 is shown in Figure 5. However, there is no interior point that is bounded both horizontally and vertically by interior points. Instead we consider a point in the second from the top horizontal row of points h 2 , that is bounded horizontally by interior points. An example of such a point is labelled v 0 in Figure 5. Taking the same approach as above, for there to exist a Sen limit there must be a monomial in f , corresponding to a vector m, such that m · v 0 = −2. To have I * 0 on each point interior to F 2 then implies m · h 2 = −2, and furthermore m · h 3 = −3, where h 3 is the row of horizontal points third from the top, and so forth. The corresponding dot products are displayed on the right in Figure 5. This is fairly constraining, but one can make the same argument for a point in h 3 , which switches the ordering of dot products vertically. Then one can immediately conclude that a height ≥ 3 blowup on F 2 will obstruct a Sen limit.
Therefore, we are able to provide a rather strict upper bound on the probability of finding a base that admits a Sen limit in our ensemble, similar to what we found in the GWCL case. Since either a single height-3 blowup internal to one of the large facets F 1 or F 2 eliminates the possibility of a Sen limit, an upper bound on the a fraction of tree geometries admit a weakly coupled Sen limit is calculated to be: We can strengthen this slightly by using the additional constraints that the other facets provide. By requiring that there are height-2 blowups along two 1-simplices on the same edge, coupled with a height-3 blowup along that same edge, on the three edges away from F 1 and F 2 , one picks up an additional factor of 3 × 10 −15 , yielding a fraction of 3×10 −391 . There are other constraints that significantly reduce this number further, but these are triangulation dependent.
Let us summarize the results. The Sen limit is spoiled in general for • A height-2 blowup along two 1-simplices on the same edge, coupled with a height-3 blowup along that same edge.
• A height-3 blowup along a 2-simplex strictly interior to a face.
The conditions in the tree ensemble are even stricter, as we can simply consider the largest facets F 1 and F 2 of the largest 3d reflexive polytopes ∆ • 1 and ∆ • 2 . For the tree ensemble, the Sen limit is spoiled for any height-3 blowup along F 1 or F 2 .

Weak Coupling Limits on More General Algebraic Bases
In the previous section we considered toric threefold bases, as the combinatorial properties of toric varieties allowed us to greatly simplify the computation for determining whether a base admits either of the weak coupling limits. One of the key structures of toric varieties that allowed us to determine this was that the smooth toric varieties could be interpreted as crepant resolutions of a singular, more minimal toric variety. The resolution then introduced rigid exceptional divisors, and further base-changing resolutions forced NH7s on these exceptional divisors. The toric case can be viewed as a two-step process. First, we start with a singular base B , and crepantly resolve it to a smooth base B, without introducing any NH7.
We then build trees of geometries on top of B by performing base-changing resolution. We follow the same procedure in the non-toric case. We will start with a possibly singular B , and resolve it to a new space B. Our assumption, without loss of generality, will be that resolving to B does not produce any non-Higgsable 7-branes, so that the generic Calabi-Yau elliptic fibration over B is smooth. In performing these blowups we naturally generalize the notion of the "height" of a divisor, which we can define inductively. Let a divisor be a blowup along the intersection n divisors. The height of the blowup divisor is given by the sum of the n heights of the divisors whose intersection defines the locus to be blown up. The induction terminates by defining the divisors of a ground geometry to have height-1. Here a geometry is a ground geometry if the generic elliptic fibration over it has no non-Higgsable 7-branes.
The structure of B will be partially determined by the fact that B is a crepant resolution of B , which will allow us to analyze whether blowups of B allow for weak coupling limits. This itself is quite convenient: while a singular variety may admit many crepant resolutions, some of the features of the resolved spaces can be read off from the singular space itself, without needing to explicitly resolve. The toric examples in the previous section are such an example; the sections of O(−nK B ) are calculated only using the data of ∆ • , and the existence of weak-coupling limits did not strongly depend on the resolution, i.e. on the choice of a FRST. We wish to proceed in a similar manner, in the spirit of minimal models, with base geometries that are non-toric.
We will generalize this procedure by constructing global geometries via gluing together local patches. In some cases this will allow us to use the tools of toric geometry locally, without requiring the global, compact variety to be toric. We will consider local patches of the form C 3 /G, where G is a finite subgroup of SL (3, C). Local patches will then be glued together to form a global Kähler manifold X. This type of orbifold is especially convenient to work with because it is known to admit a crepant resolution [30]. In addition, if G is abelian then C 3 /G can be resolved using toric methods [31]. Let X be such an orbifold. The singular points of X can be divided into two categories [32]: 1. Singular points modeled on (C 2 /H) × C, where H is a finite subgroup of SU (2).

Singular points not of the first type.
The singular points of the second type are actually discrete isolated points, which implies that G is abelian [33], and therefore locally admit a toric description. Singularities of the first type correspond to ADE singularities fibered over a curve. Only the A-type singularities are realizable as toric.
Since we do not have a global description of X we do not have a way to determine global sections of the anticanonical bundle and its powers. Instead, we will assume the most general local form and understand the behavior of sections as we resolve. This is sufficient because the general form can only be further restricted by gluing the local patch into a compact variety.

Isolated singularities
We begin by discussing the isolated rational points, which admit a toric description. The space is locally Calabi-Yau, and since we will be considering only crepant resolutions all rays, including those corresponding to blowups, can be taken to lie on a 2d plane, which we will informally refer to as a facet F . The singular point is located at x 1 = x 2 = x 3 = 0, which corresponds to 3d cone C formed by the rays v 1 , v 2 , v 3 .
As an illustrative example we first consider the case where there is a single interior point v z to F , such that v z = (v x 1 + v x 2 + v x 3 )/3. Resolution of C then involves introducing the ray v z , and subdividing F . Let us take a local section of O(−4K) of the form 3 x a 1 x b 2 x c 3 . The toric resolution promotes this to x a 1 x b 2 x c 3 z (a+b+c)/3 . If we tune an I * 0 fiber on D z , such that f vanishes to order 2 along z = 0, then we find a + b + c = 6, which with the assumption I * 0 or smooth fibers along the other divisors, then implies a = b = c = 2. There is an analogous story for the monomials in g. Therefore, we find that introducing an I * 0 fiber on D z forces an I * 0 fiber on D x 1 , D x 2 , and D x 3 . This generalizes readily to more general singularities. Consider a facet F with many interior points. All points in F can be written as where z 1 , . . . , z n are the projective coordinates introduced in the resolution. Let us tune an I * 0 on D z 1 . We then have aα 1 + bβ 1 + c(1 − α 1 − β 1 ) = 2. In order to have a GWCL we need a, b, c ∈ {0, 2}, and we see that aα 1 + bβ 1 + c(1 − α 1 − β 1 ) = 2 then requires a = b = c = 2, which implies the section is of the form Therefore, the existence of an I * 0 on any of the D z i , along with requiring a GWCL, forces an I * 0 on all divisors corresponding to points on F . We now study the existence of a GWCL after building trees above such a patch. First let us blow up a point. Blowing up the intersection of any three of the D z i requires us to tune f to order 8 along the intersection. From Eq. 4.1 we can see by demanding any order of vanishing along an intersection of the D z i this clearly forces at least a type II fiber on all the D z i , as the powers of the z i in Eq. 4.1 all become non-zero. We must therefore tune an I * 0 on all of D z i to get a GWCL. This forces an I * 0 on D x 1 , D x 2 , and D x 3 as well, as requiring the absence of an I 1 locus forces a = b = c = 2. Thus, by contradiction we see that a GWCL is not compatible with such a blowup. However, we immediately see the assumption of only I * 0 on each of these divisors is incompatible with the required OOV of (8,12) to crepantly blowup the point, and so we see that this height-3 blowup spoils the GWCL.
The Sen limit case is most constrained for isolated singularities that require at least three exceptional divisors to crepantly resolve, similar to the toric case. A height ≥ 2 blowup on F will force at least II vanishing on all of the exceptional divisors, and we therefore need to tune to I * 0 vanishing on such divisors. In this case tuning to I * 0 vanishing on all the exceptional divisors forces an I * 0 above D x 1 , D x 2 , and D x 3 . One can see this by noting that tuning an I * 0 on three exceptional divisors is done by solving the equations, for instance, for vanishing to multiplicity two in f : The unique solution is a = b = c = 2, and so any height-3 blowup on F , including above a curve, spoils a Sen limit, as one must then turn off all the monomials with a = b = c = 2 to obtain the necessary multiplicity eight vanishing in f , thereby tuning to > I * 0 vanishing in f . A similar argument applies for monomials in g. This confirms the results for the ∆ • i that we found at the end of Section 3.3, for which the large facets F i are examples of resolutions of isolated singularities, from a more general framework.

A n over a curve
Let us now turn to the singularities of type (1). We begin with A-type singularities, as these admit toric resolutions locally. The toric description of an A n singularity is wellknown; the fan has rays (1, 0) and (1, n + 1), with corresponding coordinates x 1 and x 2 , and the resolution introduces the rays (1, 1), . . . , (1, n), with corresponding exceptional coordinates z i . Let a curve C be the intersection of D z 1 with another divisor, which could be D x 1 , D x 2 , or another exceptional divisor. In order for the base change to be crepant we require that the M OV C (f, g) ≥ (4, 6). Let us focus on monomials in f , which are all of the form for positive α i . First, let us tune a (4, 6) curve C on x 1 = z 1 = 0 by tuning toric monomials in f and g, and then perform a base changing resolution over C by blowing up the locus x 1 = z 1 = 0, which is depicted in Fig 6. Note that 0 < α i < 1, and so we see that the tuning required for a crepant resolution forces at least a type II fiber on D z 1 , because the mentioned tuning forces a = 0 or b = 0. In this case, performing the crepant base change promotes Eq. 4.4 to where the −4 arises from the crepant base changing resolution 4 . In a similar way, we could also tune a (4, 6) fiber above the curve z 1 = z 2 = 0 and perform a base changing resolution, as depicted in Fig. 6. This also results in type II fibers above all of the D z i , and so we see that blowing any curve C that is the intersection of a z i with another divisor forces at least a type II fiber on all of Figure 6: A blowup of the curve x 1 = z 1 = 0, which introduces the exceptional coordinate e. This forces at least type II fibers above all of the D z i . Figure 7: A blowup of the curve z 1 = z 2 = 0, which introduces the exceptional coordinate e. This forces at least type II fibers above all of the D z i . the resolution divisors D z i . In this case, performing the crepant base change promotes Eq. 4.4 to Having performed either resolution, we can now discuss weak coupling limits. We first discuss the GWCL. As the resolution forces a type II fiber over all the D z i we then require an I * 0 fiber above D z 1 , . . . , D zn . Requiring only smooth or I * 0 fibers then forces an I * 0 above D x 1 and D x 2 , and so any monomial in f takes the form Note that by demanding a GWCL the exceptional coordinate e arising from the resolution does not appear in Eq. 4.7 due to the −4, since a = b = 2. If we then wish to perform a height-3 blowup along D e ∩ D i , for some divisor D i , we would need to tune a (4, 6) curve along D e ∩ D i . This would clearly spoil the form of Eq. 4.7, as those monomials would no longer appear in the Weierstrass equation, and therefore a height-3 blowup over any such C spoils the GWCL.
In the case of a Sen limit then we demand an I * 0 or I * n fiber above D z 1 , . . . , D zn . Let us assume that there is a monomial in f preventing M OV za (f ) > 2. The powers of the other variables are constrained by linearity, in the exact same way as the toric edge case, demonstrated in Fig. 3. Such a monomial takes the form . . . z 2−2p a−2 z 2−p a−1 z 2 a z 2+p a+1 z 2+2p a+2 . . . for integer p, which may be positive or negative. Blowing up along two different curves that arise from the resolution of the A n singularity forces p = 0, in order to satisfy the assumption of I * 0 on all the D z i . The argument for this is identical to the one made below Fig. 3. A height ≥ 3 blowup will then eliminate this monomial, eliminating the possibility of the Sen limit. A similar argument holds for a monomial in g preventing MOV of 4 in g.
Remarkably, using only local models we have reproduced the results of Section 3 on weak coupling limits on toric bases, in the case of isolated singularities and A n singularities fibered over curves. This should come as no surprise, as both of these cases admit toric descriptions locally. However, the local model approach has allowed us understand the effects of the base-changing resolutions on the sections of line bundles from the point of view of the Cox ring, or the homogenous coordinate ring, as opposed to the toric-specific structure of fans and polytopes. With this in hand we can now approach the more general case of D n and E n singularities fibered over curves.

D n and E n over a curve
We are finally left with singularities of type (1) for the D and E series. We will find that a single blowup removes the possibility of either weak coupling limit. The D and E series singularities, as well as their resolutions, are most conveniently expressed as complete intersections in affine space. We first consider the case of a D n singularity, that can be realized as a hypersurface in A 3 embedded as X 2 + ZY 2 + Z n−1 = 0 (4.9) Table 3: Generators of the Cox ring for the resolutions of the E n singularities.
coordinate in a given monomial are: In particular, note that mult(z 0 ) − mult(z 3 ) = a + b + d. Therefore, in order to tune an I * 0 fiber above both z 0 and z 3 , there must be a monomial in f that is multiplicity 2 in z 0 , and multiplicity ≥ 2 in z 3 . This monomial must then satisfy a = b = d = 0, and c = 1, as one can see from the form of Z 3 . However, this monomial would result in a type II fiber above z 1 and z 2 , which contradicts the assumptions of I * 0 above all of the exceptional divisors. Therefore crepant resolutions of D 2k singularities do not admit either weak coupling limit. The same result follows in a similar manner for the case of odd n = 2k + 1.
The absence of weak coupling limits for E 6 , E 7 , and E 8 singularities follows in the same way. For each case the generators of the Cox ring are presented in Table 3. Let us consider the E 6 case as an example. From Table 3 we can read off that in order to prevent a greater than I * 0 vanishing along z 0 there must be a monomial in f of the form rZ 1 ⊂ f , where r ∈ C. However, this would imply a type II fiber above z 3 and z 5 , spoiling both weak coupling limits. The same conclusion follows trivially for E 7 and E 8 , from the multiplicity of vanishing of the z 0 coordinate.

Discussion
In this paper we determined conditions on the base space B of an F-theory geometry that are sufficient to preclude the existence of a Sen limit, or of a GWCL. In the toric case, the conditions sufficient for the absence of the limits are • GWCL: any height-3 blowup on a face or an edge with at least one interior point.
• Sen: a height-3 blowup of a point above three exceptional divisors, or a height-2 blowup along two different curves represented by 1-simplices on the same edge, coupled with a height-3 blowup on that edge.
In the tree ensemble the conditions for the Sen limit were also quite restrictive, as a height-3 blowup along either of the large facets is enough to spoil the existence of a Sen limit. Our strong tree ensemble constraints could be further strengthened by a more detailed study of the facets. By understanding the geometry behind these conditions in the toric case, we were also able to move beyond the toric case to more general bases. The bases we considered are generalizations of the toric case, where local patches are constructed via crepant resolutions of orbifold singularities and then glued together. In the cases of a) isolated singularities and b) A n singularities fibered over curves, the conditions sufficient for the absence of the weak coupling limits are the same as the toric case. This was expected, as these types of singularities admit a toric description and resolution. In the case of D n and E n singularities fibered over curves, we found that there ware no weak coupling limits, due to the behavior of the Cox ring under the resolution.
We then performed a geometric analysis of a class of 4 3 ×2.96×10 755 bases that are built from crepant base-changing resolutions of elliptic fibrations over weak Fano toric threefolds; i.e. a set of 10 755 elliptic fourfolds that are related by topological transitions. Using the conditions we derived, we showed that nearly all of these geometries do not admit weak coupling limits. Specifically, the fraction admitting a GWCL or a Sen limit are bounded above by This strengthens the previous weak coupling result of [14] that was based on non-Higgsable clusters. While that work showed that generic points in the complex structure moduli of X over generic bases are strongly coupled, this work showed that essentially none of them admit weak coupling limits; i.e. not only are the generic points strongly coupled, but furthermore there are no subloci in complex structure that become weakly coupled. This was not forbidden a priori, as the bases that give rise to NHC in principle could all be enhanced to I * n fibers and admit a Sen limit or GWCL limit.
It is likely that Sen limits or GWCLs are similarly rare in the complete set of bases for four-dimensional F-theory compactifications, which is currently unknown. The reasoning is that the prevalence of NHC is correlated strongly with moving away from Fano or weak-Fano threefolds via topological transitions. Once a cluster exists, the existence of a Sen limit or GWCL requires that the singular fibers that exist for generic moduli are less singular than Kodaira I * 0 (which is 4 D7s on an O7, from a type IIb perspective) or are I * 0 , and then furthermore that there exists a limit in moduli space in which all of these enhance to I * n . This is a very strong condition to satisfy, and we find it extremely implausible that it is satisfied very often, an expectation that is buttressed by our results. It would be interesting to study this in the future to determine whether there is a no-go for this to occur once an NHC exists.
More broadly, non-Higgsable clusters clearly play a role in obstructing weak coupling limits. However, the precise details of the microphysics that does so is not known, even though the mathematics is clear. It would be interesting to further understand the physics of the obstruction, including the interpretation of the exceptional divisors that arise from crepant resolutions of ADE singularities. Uncovering this mechanism is likely an important step in fully understanding strong coupling in F-theory.