On the Elliptic Calabi-Yau Fourfold with Maximal $h^{1,1}$

In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known $h^{1,1}=303\,148$. It is generated by blowing up a smooth toric"seed"base threefold with $(E_8,E_8,E_8)$ collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, $E_8^{2\,561}\times F_4^{7\,576}\times G_2^{20\,168}\times SU(2)^{30\,200}$, and the largest number of axions, $181\,820$, in the known 4d $\mathcal{N}=1$ supergravity landscape. We also prove that there are at least $1100^{15\,048}\approx 7.5\times 10^{45\,766}$ different flip and flop phases of this base threefold. Moreover, we find that many other base threefolds with large $h^{1,1}$ in the 4d F-theory landscape can be constructed in a similar way as well.


Introduction
In the recent years, there have been a lot of activities to determine the boundary of the string landscape, see [1] for a brief overview. In even space-time dimensions, F-theory compactification models [2][3][4][5][6] give rise to a large class of supersymmetric vacuum solutions. In fact, the F-theory geometric landscape has the largest known number of vacuum solutions, along with the largest gauge groups and the number of moduli fields on a particular geometry.
In particular, the elliptic Calabi-Yau threefold with (h 1,1 , h 2,1 ) = (491, 11) has the largest known number of Kähler moduli, along with the largest geometric gauge group: 17 8 × F 16 4 × G 32 2 × SU (2) 32 (1.1) in the known 6d (1,0) supergravity landscape. It also has the largest h 1,1 in the known set of compact Calabi-Yau threefolds [20]. For the 4d N = 1 supergravity from F-theory on an elliptic Calabi-Yau fourfold X 4 , even the set of toric base threefolds has not been fully classified. If we allow bases that only support non-flat fibration over complex curves, then the lower bound of such toric threefolds was proved to be 4 3 × 2.96 × 10 755 in [21], and further estimated to be ∼ 10 3,000 in [22]. In [22], the notion "good base" was introduced to describe the subset of bases that support a flat fibration over complex curves. This fibration cannot have a complex three-dimensional fiber over a point either. In the language of Weierstrass polynomials, it is required that (f, g) do not vanish to order (4,6) or higher on any complex curve, and (f, g) do not vanish to order (8,12) or higher over any point. Such bases are convenient for the computation of h 1,1 (X 4 ), because the Tate-Shioda-Wazir formula h 1,1 (X 4 ) = h 1,1 (B 3 ) + rank(G) + 1 (1.2) can be applied. The lower bound of the total number of good bases is estimated to be ∼ 10 48 for the bases with h 1,1 < 130 [23] and ∼ 10 250 for the bases with h 1,1 > 1000 [22]. It was also found that the h 1,1 of such good bases are concentrated at certain discrete values.
Hence it was postulated that non-trivial structures exist on this subset. These large exponential numbers arise from the large number of flip and flop operations on the base threefold, which potentially leads to different matter curves and 4d low energy physics [23]. The number of possible flips and flops grows as h 1,1 of the base grows. Hence to get the largest number of bases, the natural point of interest is the base with the largest h 1,1 , which supports X 4 with the largest h 1,1 as well. Among the known set of Calabi-Yau fourfolds [24][25][26][27][28][29], such an X 4 has Hodge numbers (h 1,1 , h 2,1 , h 3,1 ) = (303 148, 0, 252) . (1. 3) It was originally constructed as the Calabi-Yau hypersurface in the dual polytope of weighted projective space P 1,1,84,516,1204,1816 [30], and the 4d F-theory gauge group on X 4 was read off by toric top methods: However, the smooth base threefold B 3 for X 4 with h 1,1 (B 3 ) = 181 819 has not been constructed yet. This will be the main focus of this paper, which is dicussed in section 2 with the following steps: 1. We start with a smooth non-compact toric threefold B E 8 with 2561 rays and 5016 3d cones. On each of these 2561 toric divisors, we tune a Kodaira type II * singular fiber, which corresponds to E 8 gauge group in 4d F-theory. Then each of the 5016 3d cones give rise to an "E 8 − E 8 − E 8 Yukawa point" [31]. There are also 7576 2d cones with 4d (E 8 , E 8 ) conformal matter.
2. Then we add two more rays into B E 8 to make a smooth compact toric threefold B seed , which is called the "seed" of B 3 .
3. We blow up each of the 5016 3d cones, such that the resulting base does not have any toric curves where the (f, g) vanish to order to (4,6) or higher, or any toric points where (f, g) vanish to order (8,12) or higher. We arrive at a toric base B toric with h 1,1 (B toric ) = 181 200 after this step. 4. Finally, it can be checked that for a generic fibration over B toric , there are 619 toric divisors with a non-Higgsable E 8 gauge group, such that (f, g) vanish to order (4, 6) over a non-toric curve on each of these divisors. We then blow up these 619 non-toric curves and get the final B 3 with h 1,1 (B 3 ) = 181 819.
In section 3, we discuss some physical aspects of the 4d F-theory on X 4 , including an argument of the "saturation" of geometric gauge groups on the base B 3 . We also estimated the number of self-dual G 4 flux choices: 10 194 000 on X 4 . This number is much smaller than the number 10 224 000 on the elliptic Calabi-Yau fourfold with largest h 3,1 [32].
In section 4, we give a lower bound on the number of different smooth base threefolds that are related by a number of flips and flops. This bound is proven to be 1100 15 048 ≈ 7.5 × 10 45 766 , which is much larger than any previous estimations in the literature. In particular, there exists a base that support a truly flat and smooth fibration X 4 , see the local structure in figure 5.
In section 5, we revisit the "end point" bases with 1000 < h 1,1 < 13 000 studied in [22]. It is found that a number of these end point bases can be constructed from a seed base with E 8 gauge groups in a similar way. We also explain the approximate ratio between the number of each gauge group and h 1,1 (B 3 ), which is observed in [22], as well as an approximate formula for the elliptic Calabi-Yau fourfolds with large h 1,1 . The number of base flips and flops is lower bounded by the approximate formula Finally, in section 6, we discuss the interpretation of large rank conformal matter coupled to gravity, as well as standard model building aspects.
In [30], the elliptic Calabi-Yau fourfold X 4 is constructed as the anticanonical hypersurface of the dual polytope of P 1,1,84,516,1204,1806 . Here we first perform an SL(5, Z) rotation on P 1,1,84,516,1204,1806 , and get a polytope ∆ 5 with the following vertices: Its dual polytope ∆ • 5 has the following vertices: which has a structure of P 2,3,1 bundle fibered over a 3d polytope ∆ • 3 with vertices: To construct a smooth toric base threefold, we only select the subset S ⊂ ∆ • 3 of lattice In order to construct the list of 3d cones Σ 3 (B toric ) of B toric , we first pick the subset S E 8 ⊂ S that corresponds to the divisors supporting Kodaira type II * singular fiber in X 4 (which carry E 8 geometric gauge groups in the 4d F-theory picture). To determine these rays, we consider the F and G polytope of B toric , defined as: Especially, the polytope G has vertices: The order of vanishing of the Weierstrass polynomials f ∈ O(−4K B toric ) and g ∈ O(−6K B toric ) on v i are given by There are in total 2561 v i ∈ S with ord v i (f ) = 4, ord v i (g) = 5, which carries type II * singular fiber and non-Higgsable E 8 gauge group [7]. We define the set of these v i to be S E 8 . More explicitly, this set includes the following lattice points: (2.12) Note that there does not exist a point in S E 8 on the z > 0 half-plane, hence the rays of S E 8 only form a non-compact toric threefold B E 8 . Nonetheless, there exists a triangulation of the convex hull of S E 8 , which gives rise to the set Σ 3 (B E 8 ) of 5016 3d cones with unit volume (not including the z = 0 plane). The intersection of 3d cones in Σ 3 (B E 8 ) gives the set Σ 2 (B E 8 ) of 7576 2d cones. We present the detailed list of the lattice points and 3d cones in the supplementary Mathematica file [33].
As B E 8 is still non-compact, we add two more rays (2.14) With these 2563 rays and the 3d cones, we define a compact toric threefold denoted as the "seed" threefold B seed . Note that B seed does not correspond to a reflexive polytope, and it is not weak Fano. Hence it can have a much larger h 1,1 than the ones discussed in [21,34]. From the 4d F-theory perspective, we tune 2561 type II * singular fiber (E 8 gauge groups) on the rays in the set S E 8 . Then on the base B seed , there are 5016 toric points where (f, g) vanishes to order equal or higher than (8, 12) ((8, 12)-points) and 7576 toric curves where (f, g) vanishes to order equal or higher than (4, 6) ((4, 6)-curves). Thus the elliptic fibration is not flat, and we need to perform a sequence of base blow ups. Note that the blow up of (4, 6)-curve or (8, 12)-point does not change the number of complex structure moduli of the elliptic Calabi-Yau fourfold over it, because the set of Weierstrass monomials is unchanged after the blow up. In [31], this type of base structure was called "E 8 − E 8 − E 8 Yukawa point" along with 4d (E 8 , E 8 ) conformal matter, and the base blow up sequence was already constructed. We will use such a blow up procedure in this section.
For each of the 5016 3d cones we denote the linear combination av i 1 + bv i 2 + cv i 3 by abc. The blow up of a toric point v i 1 v i 2 v i 3 is then denoted by (100, 010, 001; 111), and the blow up of a toric curve v i 1 v i 2 is denoted by (100, 010; 110). We perform the following sequences of blow ups: (2.17) The final cones from blowing up σ 3 is shown in figure 1, which is the dual graph of figure. 8 in [31]. There is no toric (8, 12) point or toric (4, 6) curve after the blow up sequence. The Kodaira singular fiber type and geometric non-Higgsable gauge groups on each toric divisor are given by (we also labeled the type II singular fiber, which does not have a non-Higgsable gauge group): Note that the sequence of non-Higgsable gauge groups on each edge v 1 v 2 , v 1 v 3 , v 2 v 3 are exactly the same as the tensor branch of 6d minimal (E 8 , E 8 ) conformal matter [35]: (2.20) Finally, we get the smooth toric threefold B toric after all the 5016 3d cones are blown up in this way. More precisely, if all the 5016 3d cones are blown up into figure 1, one needs to first perform the blow ups (2.15) for all of these 3d cones. Then one perform the blow ups (2.16) for all the 7576 2d cones. Finally, one perform the blow ups (2.17) for all the 5016 3d cones again. Note that the cones in (2.14) are also subdivided in the process. We provide the full list of 181 203 1d rays and 362 402 3d cones of B toric in the supplementary Mathematica file [33].
The total numbers of non-Higgsable gauge groups are computed as follows. There is a single F 4 on each of the 7576 edges on B E 8 , so the total number of F 4 is 7,576. For G 2 , SU (2) SU (2) SU (2) SU (2) SU (2) SU (2) SU (2) SU (2) II II   II   II   II   II   II II   II   II   II   II   II   II   II   II   II II II Figure 1: The final 3d cones after blowing up the 3d cone v 1 v 2 v 3 , where there are three E 8 geometric gauge groups on v i . Each vertex abc denotes an 1d ray av 1 + bv 2 + cv 3 . It can be checked that all the 3d cones have unit volume if the original cone v 1 v 2 v 3 has unit volume. The geometric non-Higgsable gauge groups are also labeled on each vertex. We also label the Kodaira type II singular fiber on the divisors.
there are two of them on each of the 7576 edges, and one in each of the 5016 3d cones of B E 8 , hence its total number is 20 168. For SU (2), there are two of them on each of the 7576 edges, and three of them in each of the 5016 3d cones of B E 8 , hence its total number is 30 200. On B toric , there are still a number of non-toric (4, 6)-curves, which can be identified as follows. Given a ray v i with a non-Higgsable E 8 gauge group, we construct the set which corresponds to the monomials in the polynomial g 5 (v i ) of the expansion of Weier-strass polynomial g: Here z i = 0 is the local hypersurface equation of the divisor corresponding to v i . If there are more than one monomial in g 5 (v i ), then the equation defines a (4, 6)-curve on the base B toric , which needs to be blown up to get a fully flat elliptic fibration. On the other hand, if g 5 (v i ) only has a single monomial, then it has to be a constant complex number, as all the toric (4, 6)-curves are already blown up. In total, there are 619 different v i s with |G 5 (v i )| > 1, and we blow up the corresponding 619 (4, 6)-curves (which are all irreducible). Note that the location of these non-toric curves depend on the coefficients of f and g, which correspond to the complex structure moduli of the elliptic Calabi-Yau fourfold. After these non-toric blown ups, we get the non-toric base threefold B 3 with The generic elliptic fibration over B 3 has no codimension-two non-flat fiber. However, there are still codimension-three non-flat fiber over the points where (f, g) vanish to order (4, 6) or higher. In figure 1, such points locate at the intersection of three divisors with F 4 (IV * ns ) − II − II and G 2 (I * 0,ns ) − SU (2) − II singular fiber. Similar to the codimensionthree non-flat fibers studied in the literature [36][37][38][39][40][41], they will potentially lead to a tower of massless states and new Yukawa coupling terms. It is also notable that because of the II − II collisions, the elliptic Calabi-Yau fourfold will have terminal singularities after the crepant resolution.
Nonetheless, in section 4, we will show that there exists another configuration of 3d cones, such that the aforementioned loci are all absent. Thus it is possible to construct a smooth threefold base that supports a flat fibration X 4 .

X 4 and the physical fields
The 4d axions in the supergravity theories are given by the imaginary part of the Kähler moduli for the divisor classes on B 3 , along with the reduction of 10d axiodilaton. The total number of axion fields is given by [42]: As X 4 is a generic elliptic fibration over B 3 , the geometric non-Higgsable gauge groups are already given in section 2: The Hodge number h 1,1 (X 4 ) can be verified as: Finally, there are also a number of D3-branes in the 4d F-theory picture that can carry Abelian and non-Abelian gauge groups. The total number N D 3 is bounded by the tadpole cancellation equation in the M-theory dual picture 2 : The self-dual G 4 flux satisfies

Saturation of gauge groups
On B 3 , one cannot tune any larger non-Abelian gauge group on any toric divisor, because of the "conformal matter" structure in figure 1. Namely, the line (100,010), (100,001) and (010,001) has a sequence of gauge groups (2.20) that is the same as the tensor branch of 6d minimal (E 8 , E 8 ) conformal matter. The line (100,011), (010,101) and (001,110) also has the structure of 6d minimal (E 8 , F 4 ) conformal matter. Then any further tuning of gauge group on a toric divisor will lead to additional toric (4, 6)-curves on the base, which need to be blown up. For example, if we tune any non-Abelian gauge group on the divisor labeled by 131, then the curve (010,141) would be a (4, 6)-curve. For the points v 2562 = (−6, −6, 1) and v 2563 = (−6, 37, −6) that are not in B E 8 , they also intersect divisors D i with E 8 . Hence the presence of non-Abelian gauge group on these divisors would lead to toric (4, 6)-curves D i · D 2562 and D i · D 2563 as well. However, one can check that the base B 3 (and B toric ) has an "end point" property (see [22]). Namely, any further toric blow up leads to an invalid base with codimension-one locus with ord(f, g) ≥ (4, 6), for any fibration on the base. Hence we conclude that it is impossible to tune any larger non-Abelian gauge groups on the toric divisors.
For the non-toric divisors on B 3 , they either lie in the interior of the effective cone of B 3 or intersects a divisor with non-Higgsable E 8 gauge group. Thus it is expected that the tuning of any non-Abelian gauge group on these non-toric divisors would remove monomials in f and g and lead to codimension-one (4, 6) locus as well. Similarly, if one tune Abelian gauge groups generated by a non-trivial rational sections, it is also necessary to remove a number of monomials in f and g [12,44]. Hence we conjecture that any tuning of Abelian or non-Abelian gauge groups would lead to an invalid base with codimension-one (4,6) loci.
Moreover, if one conjectures that the h 1,1 of elliptic Calabi-Yau fourfold (or even Calabi-Yau fourfold) is bounded by 303 148, then any tuning of gauge groups on B 3 will exceed this bound. It is very interesting but difficult to prove this bound mathematically.

Number of flux choices
In [32], the total number of G 4 flux choices on the Calabi-Yau fourfold M max with largest h 3,1 has been estimated along the logics of [45][46][47][48]. Namely, the different self-dual G 4 flux choices on a Calabi-Yau fourfold X 4 can be thought as lattice points in a sphere with radious √ it has the same Euler characteristic with M max , but a smaller b 4 = 1 214 150. Thus we expect that the total number of G 4 flux choices on X 4 is much smaller than that on M max . More explicitly, the counting of lattice points can be computed by [49]: where the integration goes from i∞ to −i∞, and ϑ 3 is the Jacobi Theta function. We use a saddle point approximation In our case, b 4 ≈ 16Q, hence the critical point is and we can estimate This number is much smaller than the estimated self-dual flux choices on M max , which is 10 224 000 [32]. Note that we have not taken into account the non-trivial metric on the space of selfdual G 4 flux 3 , and we have not computed the number of flux vacua associated to each flux choices either. Again, if the higher derivative terms with G 4 in the M-theory effective action are included, the number is going to be corrected.

Flip and flop phases
In this section, we give a lower bound on the total number of topologically different smooth bases that are related to B 3 by a sequence of toric flips and flops. Such a local operation leads to a different set of 3d cones, while the set of 1d rays remains the same, see figure 2. In general, it is required that the four 1d rays involved satisfy av 1 + bv 3 = cv 2 + dv 4 , (a, b, c, d ∈ Z). In particular, this operation is a flop if and only if a = b = c = d = 1. In this section, we simply consider the flips and flops of the toric base B toric , and then perform the 619 blow ups along the non-toric curves.
First, note that in figure 1, the divisors on the three (E 8 , E 8 ) edges are completely fixed. This means that we can consider the flips and flops inside such an (E 8 , E 8 , E 8 ) triangle, and any set of 3d cones will be compatible with the other adjacent (E 8 , E 8 , E 8 ) triangles. Moveover, we subdivide the triangle (100, 010, 001) into six subsets: the three triangles (100, 010, 221), (010, 001, 122), (100, 001, 212) and the three polygons (100, 221, 010, 111), (100, 111, 001, 212), (010, 111, 001, 122). We can individually evaluate the number of different sets of 3d cones in each of these subsets, and multiply these numbers together to get a lower bound on the total number of different base configurations within a single ( For the smaller triangle (100, 010, 221), it can be further subdivided into two identical triangles (010, 110, 221) and (110, 100, 221). We plot all the five possible configurations of 3d cones of the triangle (010, 110, 221) in figure 3. Note that the map from the bottom left to the bottom right configuration is a combination of two toric flips, instead of a simple toric flop. One can check that the volume of each 3d cone always equal to one, thus the base threefold is always smooth. Since there are in total six identical triangles of this shape in   It is amusing that the after we multiply this number by the estimation of self-dual flux choices in (3.15), we get a number ≈ 10 240 000 . It is bigger than the estimated number of self-dual flux choices on M max , which is 10 224 000 [32].
It is also notable that if one choose the bottom right configuration of 3d cones in the 5016 3d cones again. Finally, one perform (4.6) for all the 7576 2d cones again.

End point bases and mirror pairs
In [22], the set of toric base threefolds is probed by random blow up sequences from a starting point base, such as P 3 . The random blow up sequences terminate at an "end point" base, where any further toric blow up would lead to an invalid base with codimensionone (4,6) loci in the generic fibration. It is found that the h 1,1 of end point bases are  (2) SU (2) SU (2) SU (2) SU (2) SU (2) SU (2) SU (2) II II   II   II   II   II   II   II   II   II   II   II   II   II   II   II II II Figure 5: The configuration of 3d cones in an (E 8 , E 8 , E 8 ) triangle, such that all the codimension-three non-minimal loci and II − II collision are absent. Each vertex abc denotes an 1d ray av 1 + bv 2 + cv 3 . It can be checked that all the 3d cones have unit volume if the original cone v 1 v 2 v 3 has unit volume. The geometric non-Higgsable gauge groups and the Kodaira type II singular fiber are labeled.
concentrated at certain numbers. For example, about 10% of the random blow up sequences from P 3 end up with toric bases with h 1,1 (B toric ) = 1943.
In this paper, we show that a number of end point bases in [22] can be constructed by blowing up compact toric "seed bases" B seed with tuned E 8 on the toric divisors. For any 3d cones with E 8 on all the three 1d rays, the structures of rays and cones can be chosen as figure 1. Then one can perform flips and flops to get more topologically distinct bases, as in section 4.

(5.8)
After we tune E 8 on each of the 29 rays and perform the toric blow ups, the resulting base is also an end point base with h 1,1 (B toric ) = 1943. Nonetheless, the convex hull of (5.7) is different from (5.1). In fact, the dual polytope of (5.7) has vertices which is the toric rays of a twisted P 2 fibered over P 1 . For the end points with h 1,1 (B toric ) = 1727, 2015, 2303 and 2591, we have checked that they can be generated by a simple seed base as well. Note that for h 1,1 (B toric ) = 2591, the seed base is exactly given by the maximal reflexive polytope in [21,34]. The details of the seed bases will be presented in appendix A, and they form mirror pairs in a similar way.

Seed bases with F 4
For other end point bases in [22], they may be generated in a similar way with different seed bases. In general, one can pick an end point base with large h 1,1 , and select the 1d rays that carry non-Higgsable E 8 . Such rays will naturally form the rays of the toric seed base, and the list of 3d cones can be generated by the triangulation of the convex hull of these rays. However, one need to check if the computed N 1 (B toric ) after the blow ups matches the correct number of 1d ray. If they do not match, then the seed base needs to include more rays with other non-Higgsable gauge groups as well.

(5.11)
Now we tune 33 E 8 gauge groups on the rays v 1 , . . . , v 33 and an F 4 gauge group on v 34 . Then the first 61 3d cones of (5.11) can be blown up according to figure 1. But the last three of the 3d cones correspond to (E 8 , E 8 , F 4 ) collision, which can be blown up to figure 6. We denote the two rays with E 8 by v 1 , v 2 and the ray with F 4 by v 3 . We use abc to denote a ray av 1 + bv 2 + cv 3 . Then the blow up sequence is:  The non-Higgsable gauge groups on B toric can be counted: After the toric blow ups, there are still 21 E 8 divisors with non-toric (4, 6)-curve. From B toric to B 3 , one needs to blow up these non-toric curves. Hence the total h 1,1 (X 4 ) is (5.14)

Estimated number of gauge factors
Here we provide an explanation of the gauge group counting formula (4.1) in [22]. In a seed base with a large number N of 1d rays, the number of 2d and 3d cones are generally scaled as: Assuming that each 1d ray has a geometric E 8 gauge group, then after the seed base is fully blown up to the base B toric with no toric codimension-two (4,6) locus, the total number of 1d rays is given by The number N 1 (B toric ) ≈ h 1,1 (B toric ) if the number is large. The total numbers of geometric non-Higgsable gauge groups of each type are given by: Thus we have the approximate formula for the number of each type of gauge groups N (SU (2)) ≈ h 1,1 (B toric ) 6 .

(5.18)
We can also estimate the ratio between h 1,1 (B 3 ) and h 1,1 (X 4 ) based on (5.18) and We use an approximation h 1,1 (B 3 ) ≈ h 1,1 (B toric ), although in principle the B 3 base is generated by blowing up B toric along non-toric curves. Then we arrive at the following approximate formula for the end point bases with large h 1,1 (B 3 ): For the number of flip and flop phases on B toric , the lower bound can also be estimated with the methods in section 4: Note that if there exists toric automorphism on the seed base B seed , then this number will be reduced by O(10 1 ), which has no significant change on the exponential. This number is much bigger than the estimated number of "good bases" (the bases without codimension-two (4,6) locus) with the same h 1,1 in [22], which is around 10 200 . This suggests that the statistical methods in [22] lead to a systematic underestimation.

Supergravity coupled to conformal matter
From the construction of base threefolds with large h 1,1 , we see that the (E 8 , E 8 , E 8 ) collision is prevalent. In fact, the Calabi-Yau with largest known h 1,1 provides the known example of the highest rank conformal matter coupled to supergravity. For 6d (1,0) theories, the elliptic Calabi-Yau threefold X 3 with the largest h 1,1 (X 3 ) = 491 has the following toric base geometry [9]: To get a base without (4, 6)-points, the two (−11)-curves need to be blown up at a non-toric point as well. One can see that the non-minimal (E 8 , E 8 ) conformal matter [35,53] with order N = 16 can be embedded into this base. Such a 6d (1,0) SCFT has the following tensor branch in the standard notation: where there are 15 (−12)-curves with e 8 gauge group in the middle. For 5d N = 1 theories, similarly one can consider M-theory on the resolved compact Calabi-Yau threefold X 3 . Then we can couple 5d supergravity with the KK reduction of the non-minimal (E 8 , E 8 ) conformal matter with N = 16, which has the following 5d IR quiver gauge theory description with rank r = 471 [53]: However, to get a genuine 5d SCFT fixed point, one needs to douple an SU (16) vector multiplet from the theory, which geometrically corresponds to decompactifying 15 divisors in X 3 [54,55]. After the decompactification, the gravity sector will be decoupled again. Nonetheless, the 5d quiver (6.4) is still the quiver gauge theory with the known largest rank that can be coupled to 5d N = 1 supergravity.
Finally, for the 4d N = 1 theories, it is unclear whether the E 8 − E 8 − E 8 Yukawa point actually corresponds to an SCFT fixed point or not [31]. Nonetheless, the X 4 with largest h 1,1 (X 4 ) = 303 148 in this paper provides the known example of 4d supergravity coupled to an E 8 quiver network with the largest number (2561) of E 8 gauge groups.
It would be fascinating to have a swampland bound argument along the philosophy of [56][57][58][59], for the various cases discussed here: 6d (1,0), 5d N = 1 and 4d N = 1. Alternatively, one can also attempt to challenge these bounds in other parts of string landscape with a supergravity sector.

Standard model building
Another interesting question is whether the 4d F-theory model on X 4 with the largest h 1,1 (X 4 ) has any model building implications. Since this model has the largest number of axions N (axions) = 181 820 (6.5) in the known 4d superstring landscape, it is potentially useful for the inflation models with large number of axions, e. g. [60][61][62][63]. Nonetheless, it is hard to realize the standard model gauge group on X 4 geometrically, because of the similar problem discussed in [32,64]. If we embed G sm = SU (3) × SU (2) × U (1) into a single E 8 , then it is implausible to get chiral families [65]. On the other hand, if we embed G sm = SU (3) × SU (2) × U (1) into a single F 4 , the branching rule does not give the correct standard model hypercharges. Moreover, it is impossible to tune any larger gauge groups on the base, if we accept that h 1,1 (X 4 ) is already maximal (also see the discussions in section 3.2).
Thus it is more plausible to realize a part or all of the G sm as the gauge bosons from D3 branes. Then the geometric gauge groups will be treated as dark matter sector, if they do not intersect these D3 branes.