Elliptic Calabi-Yau fivefolds and 2d (0,2) F-theory landscape

In this paper, we initiate the study of the 2d F-theory landscape based on compact elliptic Calabi-Yau fivefolds. In particular, we determine the boundary models of the landscape using Calabi-Yau fivefolds with the largest Hodge numbers $h^{1,1}$ and $h^{4,1}$. The former gives rise to the largest geometric gauge group in the currently known 2d (0,2) supergravity landscape, which is $E_8^{482\,632\,421}\times F_4^{3\,224\,195\,728}\times G_2^{11\,927\,989\,964}\times SU(2)^{25\,625\,222\,180}$. Besides that, we systematically study the hypersurfaces in weighted projective spaces with small degrees, and check the gravitational anomaly cancellation. Moreover, we also initiate the study of singular bases in 2d F-theory. We find that orbifold singularities on the base fourfold have non-zero contributions to the gravitational anomaly.


Introduction
In the pursuit of the gobal set of consistent quantum gravity theories, it is very important to identify the boundaries of the string theory landscape, in order to compare them with the swampland bounds [1]. For example, one can ask the following question: In a given space-time dimension and amount of supersymmetry, what is the maximal number of fields of a given type in a string compactification model?
For non-chiral theories with 16 supercharges in d > 3 space-time dimensions, the maximal rank of gauge group is given by r G = 26 − d, and it was matched with the swampland bounds [2].
For theories with four supercharges, such as 4d N = 1 supergravity, the maximal rank of gauge group r G = 121 328 is given by F-theory on the elliptic Calabi-Yau fourfold X 4 with maximal h 1,1 [3,11]: (h 1,1 , h 2,1 , h 3,1 ) = (303 148, 0, 252) . (1. 2) The same model also leads to the largest number of axions N (axion) = 181 820 . On the other hand, F-theory on the mirror Calabi-Yau fourfold with the largest h 3,1 would lead to the largest number of complex structure moduli and number of flux vacua on a single geometry [12].
As a general pattern, the F-theory landscape seems to always provide the answer to the above question in even space-time dimensions. In particular, the point of interest is always the elliptic Calabi-Yau manifold with the largest Hodge numbers.
In this paper, we will extend this logic to the case of 2d (0,2) supergravity with two supercharges, which comes from F-theory on a compact elliptic Calabi-Yau fivefold [13,14]. In particular, we will study the details of the elliptic Calabi-Yau fivefolds with maximal h 1,1 or h 4,1 . For the case of maximal h 1,1 : The total rank of gauge group is r G = 66 239 044 388 , (1.6) which is conjectured to be the largest in the whole 2d (0,2) landscape. The construction of the corresponding fourfold base with h 1,1 (B 4 ) = 181 299 558 192 is similar to the 4d case [11]. We tune E 8 gauge groups on the toric divisors of a starting point toric fourfold, and then blow up all the non-minimal loci in codimension-two, three and four.
-2 -Besides this particular geometric model, we also present the first attempt of studying the set of elliptic Calabi-Yau fivefolds and the 2d F-theory geometric landscape. The constructions of Calabi-Yau fivefolds were explored in [15][16][17], but the elliptic fibration structures have not been discussed in the literature. Namely, we study the Calabi-Yau hypersurfaces of reflexive weighted projective spaces up to degree d ≤ 150 that have an elliptic fibration structure. For example, the generic fibration over a "generalized Hirzebruch fourfold" is given by a Calabi-Yau hypersurface inside P 1,1,1,1,n,2n+8,3n+12 . We also find Calabi-Yau fivefolds with non-zero Hodge numbers h 2,1 and h 3,1 . The ones with nonzero h 3,1 describes 2d (0,2) supergravity coupled to 2d Fermi multiplets. The full table of these geometries is listed in Appendix B.
Finally, we checked the 2d gravitational anomaly cancellation conditions [14,18] in several cases with or without non-Abelian gauge groups. More interestingly, we also analyzed cases with a singular base, and we found that these orbifold singularities also have a non-zero contribution to the gravitational anomaly.
The structure of this paper is as follows: in section 2, we briefly recap the formulation of 2d F-theory and the gravitational anomaly computation. In section 3, we present the detailed construction of the elliptic Calabi-Yau fivefolds with either largest h 1,1 or h 4,1 .
In section 4, we study the geometric structure of a number of other elliptic Calabi-Yau fivefolds. In section 5, we check gravitational anomaly cancellation, including the models with a singular base.

Mathematics and physics of 2d F-theory compactifications
In this section, we introduce the basics of globally consistent compactification of F-theory to 1+1 dimensions on compact elliptic Calabi-Yau fivefolds, including the geometric tools and the gravitational anomaly computation of the low energy effective theory. In section 2.1 we introduce compactification of F-theory on elliptic Calabi-Yau fivefolds with an emphasis on the computation of the massless spectrum of the low energy effective theory. In section 2.2 we discuss the derivation of gravitational anomaly of the 2d effective theory. The materials in section 2.1 and 2.2 are not new and are all covered in [13,14,18]. In section 2.3 we review the basic toric geometry tools that we will make use of to construct examples of elliptic Calabi-Yau fivefolds.

Basic setup of 2d F-theory
We consider compactification of F-theory on an elliptic Calabi-Yau fivefold X 5 whose low energy effective theory is a 2d N = (0, 2) supersymmetric field theory coupled to gravity. In general, an elliptic Calabi-Yau (n + 1)-fold has the following form: -3 -and we will mainly focus on the n = 4 cases. We further assume that the fibration has a zero section therefore it can be described by a Weierstrass model: Here K B is the canonical bundle of the base fourfold B 4 . We will mainly working in the local chart where we can set z = 1. Singularities of the elliptic fibration at different codimensions of the base B 4 correspond to different physical contents and we list such correspondences in Table 1.

Codimension
Physical data 1 Gauge groups 2 Matters in R ⊕ R Bulk-surface matter couplings 3 Holomorphic matter couplings 4 Table 1: Singularities and the corresponding physical data of the low energy 2d N = (0, 2) field theory.
For our purpose it is sufficient to discuss the codimension-1 and 2 singularities on B 4 as we will focus only on the gauge groups and matters in this paper. Codimension-1 singularities are characterized by the vanishing of the discriminant locus: In the IIB physics, the locus ∆ = 0 is wrapped by 7-branes, and the gauge group G S along the codimension-1 locus S is determined by the order of vanishing of (f, g, ∆) along S. The matters are localized at codimension-2 locus of B 4 where the order of vanishing of (f, g, ∆) along S enhances. The matter representations can be determined following Katz-Vafa [19]. There is also bulk matter that is not localized as we will discuss later. For us it is important to know that with gauge invariant G 4 flux, the bulk matter transforms in the adjoint representation of the gauge group G S and it will contribute to the anomaly.
Besides the 7-branes wrapping codimension-1 loci of B 4 , there will also be D3-branes wrapping codimension-2 loci of B 4 due to tadpole cancellation. The interplay between D3-brane sector and 7-brane sector will also contribute to gravitational anomaly in 2d.
Another indispensable ingredient in the F-theory compactification is the G 4 flux which must satisfy the following condition: in order for the M-theory compactification on Y 5 to preserve two supercharges [17]. We will see that G 4 flux contributes to the gravitational anomaly from the 3-7 sector. We will summarize some properties of the the supermultiplets in the 2d N = (0, 2) field theory. They include vector multiplets with one negative chirality complex fermion, -4 -chiral multiplets with one positive chirality Weyl fermion, Fermi multiplets with one negative chirality complex fermion and a single gravity multiplet with one positive chirality complex dilatino and one negative chirality gravitino. In 2d there are also tensor multiplets containing real axionic scalar fields arising from KK reduction of the F-theory 4-form field C 4 . The tensor multiplets will play an important role in the Green-Schwarz mechanism of anomaly cancellation as will be discussed in the next section.

Gravitational anomaly cancellation
In 2d the gravitational and gauge anomaly can be described by a gauge invariant polynomial of degree 2 in gauge field strength F and the curvature 2-form R: where I s (R) is the anomaly polynomial of a single spin s matter field in representation R and n s (R) is the multiplicity of that matter field. In general I 4 does not have to vanish in a consistent quantum field theory. A gauge variant Green-Schwarz counter-term at tree level can cancel I 4 if I 4 factorizes suitably. This is possible in 2d because of the existence of an axionic scalar field c α that gives rise to a self-dual one-form H α = dc α + Θ α i A i , such that: The gauge variant pseudo-action that contains c α and H α is: where dH α = X α and X α = Θ α i F i , F i is the field strength of the abelian gauge group factor U (1) i . The axionic symmetry of c α is gauged by A i with the following transformation rule: It is then easy to obtain the gauge variation of S GS is: Using the descent equations: we have: We require: It is easy to see that since I 4,GS contains only the field strengths of abelian gauge groups, the cancellation is possible only if the gravitational and non-abelian gauge anomalies vanish by themselves and the abelian gauge anomalies factorize suitably. In this paper, we will denote by I 4 the gravitational anomaly of the low energy effective theory from a 2d F-theory construction, and we will check if I 4 = 0 for a series of examples. For simplicity we first consider the gravitational sector of F-theory compactification on a smooth Calabi-Yau fivefold X 5 . Using the duality between F-theory and IIB orientifold we have the following spectrum in the moduli and gravitational sector [14] in table 2.
2d multiplet Multiplicity Chiral Here the signature τ (B 4 ) is given by Summing up the contributions of chiral, Fermi and tensor multiplets (+1 for chiral multiplets and (−1) for Fermi and tensor multiplets) to the 2d anomaly polynomial we have: (2.9) where we have used the relation h 1,1 (X 5 ) = 1 + h 1,1 (B 4 ) and the definition of arithmetic genus: The gravitational anomaly from the gravity multiplet is: (2.11) We then consider the spectrum of 3-7 sector when a D3 brane wraps genus g curve C in B 4 . The spectrum is summarized in the table 3. Summing up the contributions from chiral and Fermi multiplets (note again they have opposite contributions), we have: (2.12) The various arithmetic genus above can be computed via index theorem and we have: Here c i is the i th Chern class of the base B 4 . For a smooth Calabi-Yau fivefold we have: Here π : X 5 → B 4 is the fibration map, and π * is the push forward map from X 5 to B 4 . Summing up all the contributions we have: where: Recall that for a base B 4 to support a smooth elliptic fibration for a Calabi-Yau fivefold, we have h 0,0 (B 4 ) = 1 and h k,0 (B 4 ) = 0 for k = 0. Therefore χ 0 (B 4 ) = 1 and the gravitational anomaly is cancelled for smooth elliptic Calabi-Yau fivefolds. We now assume that the fibration contains non-abelian gauge groups from 7-branes and charged 7-7 matters. In addition we turn on G 4 flux. In this case the terms above needs slight modification ad there will be a new term I 4,7−7 contributing to the gravitational anomaly from the 7-brane sector.
Suppose that the divisor S ⊂ B 4 is wrapped by 7-branes. The Kodaira fiber is singular over S and the Calabi-Yau fivefold X 5 is singular. We assume that the singular X 5 admits -7 -a crepant resolutionf :X 5 → X 5 and G 4 ∈ H 2,2 vert (X 5 ). In this situation the χ(X 5 ) term in I 4,moduli (2.9) is replaced by χ(X 5 ). The D3-brane class [C] is corrected to: The anomaly polynomial from the non-trivial 7-brane sector is: (2.20) In section 5, we investigate cases with only non-Higgsable gauge groups and χ(adj) is purely geometric. To cancel the gravitational anomaly the following relation must hold: The above equation puts a set of topological constraints that every crepant resolutioñ X 5 → X 5 with consistent background G 4 flux onX 5 must satisfy. It will be verified on a set of Calabi-Yau fivefoldsX 5 in section 5.

Construction of Calabi-Yau fivefold hypersurfaces
In this section, we will review some basics tools of toric geometry that we will use to construct Calabi-Yau fivefolds as hypersurfaces in toric sixfolds. The techniques are standard and can be found in [20]. We will use Batyrev's construction [21] to construct Calabi-Yau hypersurfaces in a reflexive polytope. We will explain the details in a moment. We will start with an (n + 1)-d reflexive polytope ∆ in an (n + 1)-d lattice in M R . That is, ∆ ⊂ M R contains 0 and both ∆ and ∆ * are lattice polytopes where ∆ * ⊂ N R is defined as: where N R is the dual lattice of M R .
The polytope ∆ * defines a toric fan Σ and to each point v i on the boundary of ∆ * one can associate a homogeneous coordinate z i . We denote by Y n+1 the (n + 1)-d toric variety defined by Σ. To each point u i ∈ ∆ one can associate a monomial m i = j z Note that there is no guarantee that X n is smooth when n > 3.
For the Calabi-Yau n-fold defined from the reflexive pair (∆ * , ∆), then the (stringy) Hodge numbers of the Calabi-Yau hypersurface X d can be computed with Batyrev formula [21,22]: Here Θ * and Θ means the faces on ∆ * and ∆ respectively. l(.) means the number of integral points in a polytope, and l (.) means the number of interior points on a face. For the cases we will discuss in this paper, they are all n-d hypersurfaces defined in some (n + 1)-d ambient toric varieties that are also elliptically fibered over some (n − 1)d bases. Such a fibration structure can be easily read off by studying the toric fans of their ambient toric varieties. For all the examples in this paper, after a suitable SL(6, Z) transformation, the vertices of ∆ * can be put into the following form: This is of the form introduced in [23] and is known to be a P 2,3,1 fibration over a base toric variety B 4 . The fan of B 4 has toric rays v i , and we denote the convex hull of it by the polytope ∆ B 4 .
The Calabi-Yau hypersurface defined by the pair (∆ * , ∆) is thus an elliptic fibration over B 4 . Note that to fully specify the toric variety corresponding to ∆ * , a triangulation is also required. We require the triangulation to be fine (uses all the points in ∆ * ), regular (resulting variety is projective and Kähler) and star (the simplices define the cones of a toric fan). Though a triangulation of ∆ B 4 is needed to compute some detailed geometrical data such as intersection numbers on B 4 , the computation of the Hodge numbers and the characteristic classes of B 4 depends only on the rays in the fan Σ B 4 associated with ∆ B 4 . Therefore in later sections where we compute Hodge numbers and characteristic classes of B 4 and Y 5 , we will choose a convenient triangulation to facilitate our computations and the results are indeed independent from our choices.
The base varieties of the examples in Section 5 are particularly easy in this sense since their triangulations are unique. In contrast, the triangulations of the bases of the examples in Section 3 are far from being unique, but one does not need to worry about any specific choice of triangulation since we will be computing Hodge numbers only and the key data involed this computation are the numbers of cones in various codimensions which are constants across all fine-star-regular triangulations (FRST).
For most examples in our paper with E 8 geometric gauge groups, we will try to construct a smooth base B 4 that supports a flat fibration. To do that, we will first pick all the -9 -primitive rays ρ inside ∆ B 4 and this we will denote by S this set of primitive rays. We will denote by B toric the toric variety given by S (and a suitable triangulation of it). We then pick the subset S E 8 ⊂ S whose elements are the rays that correspond to divisor supporting Kodaira II * fiber, that is, carrying an E 8 gauge group. To find these rays we consider the following two polytopes: The points in ∆ F correspond to monomials in the class −4K B toric and the points in ∆ G correspond to monomials in the class −6K B toric . The orders of vanishing of the polynomials f ∈ O(−4K B toric ) and g ∈ O(−6K B toric ) in the Weierstrass model along a divisor D i corresponding to the primitive ray u i ∈ S are: Usually the set S E 8 does not give rise to a compact base and we need to add several rays manually. After adding these rays by hand we arrive at a base we call B seed . This base needs to be blown-up to be free from codimension-two (4, 6) locus, codimension-three (8,12) and codimension-four (12, 18) non-minimal loci. Focusing on S E 8 , we can compute the number of 4d cones in S E 8 , n 4D . By assigning a convenient triangulation to S E 8 we can then compute the number of 3d and 2d cones in S E 8 , n 3D and n 2D respectively and n 1D is simply the number of rays in S E 8 . Note that the n 4D , n 3D , n 2D and n 1D are all indeed independent of triangulation and our choice is simply to make the computation easier. There is the following correspondence between those numbers and the gauge web structure over S E 8 : For each of the above intersecting E 8 structure there is a sequence of blow-ups one needs to perform over B seed to finally arrive at a smooth base B 4 . We will present the process in the Appendix A.

Calabi-Yau d-fold with extremal Hodge numbers
We first compute the ambient reflexive polytope for Calabi-Yau d-fold with extremal Hodge numbers, which is a generalization of the sequence (3.3) in [24]. We first define a sequence -10 -of integers m k , with The first a few m i are Then the ambient reflexive polytope is a (d + 1)-dimensional weighted projective space The weights are computed as: For the elliptic CY3 X 3 with (h 1,1 , h 2,1 ) = (11, 491), the ambient weighted projective space is P 1,1,12,28,42 .
Using the terminologies in section 2.3, a weighted projective space P 1,w 1 ,...,w d+1 corresponds to an ambient polytope ∆ * with vertices: As one can check, the pairs (∆ * , ∆) above are all reflexive.
The other Hodge numbers can be computed by Landau-Ginzburg methods [25]: They satisfy the relation [17]: We perform an SL(6, Z) rotation on v i : The resulting vertices arẽ Hence it is in form of P 1,2,3 bundle over a 4d base B 4 , whose 4d polytope ∆ B 4 has the following vertices: (3.11) The base B 4 of X 5 is a B 3 fibration over P 1 . B 3 is exactly the threefold base for the elliptic CY4 X 4 with h 1,1 = h 3,1 = 151 700, as similar phenomenon is observed in the lower dimensional case [12]. Note that X 4 has an elliptic fibration with geometric gauge groups [3] G 4d = E 1 285 To construct the rays and cones on B 4 and B 3 . We first compute the set of lattice points {x, y, z, w} in the polytope (3.11), with the following condition: Among these points, we select the ones that correspond to divisors with E 8 gauge group, which form the set S E 8 . Such a point v satisfy the following condition: where the ∆ G polytope is the set of lattice points u = (u x , u y , u z , u w ) satisfying It turns out that there are 1 285 points satisfying the conditions, and they are all in the form of (0, y, z, w). Then we can construct a non-compact toric threefold B (3) E 8 with the 3d rays (y, z, w). After a triangulation, we find that there are 2 508 (E 8 , E 8 , E 8 ) 3d cones and 3 792 (E 8 , E 8 ) 2d cones on B (3) E 8 . Then we add three additional rays (1, 0, 0), (0, 1, 0), (0, 0, 1) and a number of additional 3d cones into B E 8 , such that the resulting base is a compact one B seed . Finally, after we blow up the (E 8 , E 8 , E 8 ) 3d cones and (E 8 , E 8 ) 2d cones according to [11] (also see appendix A), we get a base B seed with 1 288 rays. Then for each of the 2 508 (E 8 , E 8 , E 8 ) 3d cones, we need to add 19 additional rays in the interior. For each of the 3 792 (E 8 , E 8 ) 2d cones, we need to add 11 additional rays on it. Thus these numbers add up to 90 652. Finally, to get the base B 3 , we checked that there are 310 E 8 divisors p on B toric with non-toric (4, 6)-curves. This can be checked by the following criterion: It turns out that all of these non-toric (4, 6)-curves are irreducible. After these curves are blown up, we get the non-toric base B 3 with h 1,1 (B 3 ) = 90 959. After adding up the rank of geometric gauge group, we get exactly the following Shioda-Tate-Wazir formula in CY4 case:

Maximal h 1,1
In this section, we construct Calabi-Yau fivefold X 5 with the largest h 1,1 , along with its elliptic fibration structure. We take (3.6), and perform an SL(6, Z) rotation:  Now we write down the set of rays S E 8 whose corresponding toric divisor supports E 8 gauge algebra: There are in total n 1D = 482 632 421 (3.24) integral points in this set. Now we are going to construct the non-compact toric fourfold B E 8 with rays in the set S E 8 . We denote by ∆ E 8 the convex hull polytope of S E 8 . ∆ E 8 has a shape of hyper truncated pyramid, with the following 16 vertices, see figure 1: It is a fact that the number of simplicial 4d cones is independent of the choice of triangulation of the 4d fan given by the primitive rays in S E 8 . To compute the number of To compute the total number of 3d cones on B E 8 , one can use the following trick. On a compact toric fourfold, each 4d cone contains four 3d cones, while each 3d cone is shared by two 4d cones. Hence the number of 3d cones on a compact toric fourfold should be the twice of the number of 4d cones. However, the base B E 8 is non-compact, with the following boundary 2d faces: (3.30) In the above list, we take the 2d faces inside a single 3d face with non-zero contribution to the 4d volume of ∆ E 8 . Now one takes two times the number of 4d cones (3.29), plus additional 3d cones from the boundary set (3.30) divided by two. We get n 3D = 2n 4D + 1 2 × 7 056 216 = 5 479 598 508 . To compute the h 1,1 (X 5 ) of this elliptic Calabi-Yau fivefold. We add the rank of non-Higgsable gauge groups: for each 4d cone, there is a single SU (2); for each 3d cone, the additional gauge group is G 2 × SU (2) 3 ; for each 2d cone, the additional gauge group is F 4 × G 2 2 × SU (2) 2 ; for each E 8 ray, the gauge rank is 8. Thus we have (2.25) h 1,1 (X 5 ) = h 1,1 (B 4 ) + 8n 1D + n 4D + 5n 3D + 10n 2D + 1 = 247 538 602 581 .  In this section, we explicitly study a number of elliptic Calabi-Yau fivefolds as hypersurfaces of P 1,w 1 ,w 2 ,w 3 ,w 4 ,w 5 ,w 6 , which constructed from a reflexive polytope. While the full list for 6 i=1 w i < 150 is presented in Appendix B, we will discuss a few examples in full detail and explain the origin of the non-vanishing Hodge numbers h 2,1 (X 5 ) and h 3,1 (X 5 ).
The non-zero h 3,1 (X 5 ) is explained in another way. Denote the base coordinates of B 4 by z 1 , z 2 , . . . , z 9 . The local Tate model near the divisor D 6 with SU (2) is [28] Here F i is generic homogeneous polynomial of degree i. Note that the coefficients b 3 and b 6 of Tate model do not depend on z 4 and z 9 , although D 4 and D 9 intersect D 6 . Thus b 3 and b 6 can be thought as sections of line bundles on P 2 × P 1 . The coordinates of P 2 are z 1 , z 2 , z 3 and the coordinates of P 1 are z 4 and z 9 .

Smooth base with non-Higgsable gauge groups
As introduced in section 2.2, to cancel the gravitational anomaly, the following expression needs to vanish:
in [13,18], and these formula can also be found in an analogous computation for the elliptic Calabi-Yau fourfolds in [31,32]. We test the gravitational anomaly cancellation using the series of CY fivefolds with non-abelian gauge groups we constructed in Section 4.1. The base manifolds are generalized Hirzebruch fourfolds B n,4 . The CY5s in this series all have non-Higgsable non-abelian gauge groups and have matter only in the adjoint representation of the gauge group. The anomaly can be cancelled when c 1 (B 4 ) · π * (G 4 · G 4 ) vanishes. For all the bases B n,4 , the divisor carrying non-Abelian gauge group is S = P 3 , and we have the purely geometric χ(adj): Since the gravitational anomaly has already been cancelled, according to (5.5) any consistent G 4 -flux on these bases must satisfy the condition For example, we considerX 5 π − → B 6,4 for which h 1,1 (X 5 ) = 5. The five generators of H 1,1 (X 5 ) are two vertical divisors D 1 and D 2 , two exceptional divisors E 1 and E 2 and the zero section σ of the elliptic fibration. For simplicity we set:  Table 7: Gravitational anomaly coefficients for the generic elliptic fivefold over the generalized Hirzebruch fourfolds B n,4 .
We consider the G 4 -fluxes in the vertical cohomology group H 2,2 V (X 5 ) therefore we have: In order for G 4 to uplift to fluxes in F-theory, it must satisfy the transversality conditions: We also require that G 4 does not break non-abelian gauge groups, therefore it satisfies: (5.14) It is then easy to show that any G 4 = i,j n ij H i · H j such that the n ij 's satisfy the above restrictions also satisfies c 1 (B 6,4 ) · π * (G 4 · G 4 ) = 0.
Here we also prove that for an X 5 over a smooth B 4 with no gauge group (codimensionone singular fiber), we always have We prove the uplift of such equality in X 5 : Denote the zero section by S 0 and vertical divisors in X 5 by D i , the general form of vertical G 4 -flux is We write the anticanonical divisor of B 4 as where m i ∈ Z are coefficients associated to B 4 and the choice of basis D i . Then we have the following intersection number relations in X 5 : (5.20) Besides these relations, we have obviously D i · D j · D k · D l · D m = 0. The transversality conditions (5.12) on G 4 become: for any i, j, k. Plug in (5.17) and using (5.19, 5.20, 5.21), they are further reduced to Note that these equations are equivalent to the following equations: Now we can rewrite (5.16): For example, we can simply check that the generic fibration X 5 over P 4 , with Hodge numbers (h 1,1 , h 2,1 , h 3,1 , h 4,1 ) = (2, 0, 0, 56 977) satisfies the gravitional anomaly cancellation with G 4 flux. This also holds for B 4,4 .

Orbifold singularity and anomaly
In this section, we consider a number of bases with orbifold singularity, and check the gravitational anomaly cancellation in these cases. As a result, we found that there need to be finite contributions from the orbifold singularities to cancel the anomaly.
The bases we considered are the weighted projective space P 1,1,1,1,n , where n takes the values in table 4. The rays of the base are As the volume of the 4d cone Vol(v 1 v 2 v 3 v 5 ) = n, there is a C 4 /Z n orbifold singularity at z 1 = z 2 = z 3 = z 5 = 0. Unlike B n , there is no toric divisor carrying non-Higgsable gauge group on P 1,1,1,1,n . Now we compute the topological quantities involved in the gravitational anomaly cancellation (5.5). For the divisors D i corresponds to the ray v i , we have the linear equivalence relation: However, in this case the formula (5.6) and (5.7) will no longer hold, as they give rise to fractional numbers for a general n. For a singular toric variety, the topological numbers -29 - Table 8: The additional gravitational anomaly contribution from C 4 /Z n orbifold singularity on a compact singular base.
τ and χ 1 are computed in a combinatoric way instead [33]. In particular, these numbers for P 1,1,1,1,n are exactly the same as the ones of P 4 : τ (P 1,1,1,1,n ) = 1 Adding up the contributions in (5.5), we get the total gravitational anomaly: On the other hand, the Hodge numbers of the smooth X 5 over P 1,1,1,1,n is the same as the generic elliptic CY5 over B n , given in table 4. The reason is that the 6d reflexive polytopes for X 5 are exactly the same in the two cases, and the Batyrev formula (2.22, 2.23, 2.24) hold. Plug in the χ(X 5 ) from table 4 into (5.34), we found that A grav is always non-zero. To compensate this, we propose a new 2d sector from the orbifold singularity, which has the contribution A orbifold to A grav in table 8. Notably, the case of Z 2 and Z 3 has a contribution (−1) and (+1), respectively. Hence a C 4 /Z 2 singularity would effectively act as a Fermi or tensor multiplet, while a C 4 /Z 3 singularity effectively acts as a chiral multiplet.
Finally we make more comments on the physics of singular bases in F-theory. In the case of singular base surface in 6d F-theory, such as the Z 3 orbifold in [34], there is a localized SCFT sector coupled to gravity. The gravitational anomaly will cancel after the contribution of the SCFT is included. In the case of 2d F-theory here, we expect a similar story. Nonetheless, for the case of Z 2 and Z 3 , one cannot blow up the singular loci and still get a X 5 with the same Hodge numbers. One can also check this from the Hodge numbers of X 5 in table 4, where the h 1,1 (X 5 ) over B 2 and B 3 are the same as the ones over P 4 . The SCFT sector in these cases will not have a Coulomb branch after the dimensional reduction to 1d.
- 30 -In this paper, we constructed the elliptic fibration structure for a variety of elliptic Calabi-Yau fivefolds. Especially, we studied the elliptic Calabi-Yau fivefolds with the largest h 1,1 or h 4,1 , as well as hypersurfaces of weighted projective spaces with small degrees. The non-vanishing h 2,1 and h 3,1 in some examples are explained as well. Nonetheless, we have not studied the detailed condition on G 4 flux in many of these geometries. For example, for the Calabi-Yau fivefold with the largest h 1,1 , one needs to know whether a non-vanishing G 4 is required, and if a generic G 4 flux choice would break any gauge symmetry. This would be a question for the future work.
Moreover, one can also study the set of smooth compact fourfold bases in a more systematic way, applying the methods in 4d F-theory, such as P 1 fibrations [35], Monte Carlo and random walk methods [27,36], and systematic blow ups from weak-Fano bases [37].
Besides the cases with a smooth fourfold base, we have also initiated the study of singular base fourfold in 2d F-theory. This question is especially interesting in 2d, because of the presence of pure gravitational anomaly and one can study the correction term from base singularities. In this paper, we have studied the contribution of an orbifold singularity C 4 /Z n with n = 2, 3, 4, 6, 8, 12, 24 in table 8. For more general types of base singularities, we will study them in the future. Of course, it is also crucial to explain the physical origin of these effects.
Finally, one can ask what are the details of the 2d (0,2) SCFT constructed from either a base singularity or a non-minimal loci. It would be curious to relate them with the existing 2d (0,2) literature [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56].  Then there are six new 3d cones with (E 8 , E 8 , SU (2)) Collisions: Then we can blow up these codimension-three loci, according to figure 3. Note that the order of vanishing of g on the new exceptional divisor is zero. The whole tetrahedron will look like figure 4 after this step (we did not draw out all the subdivision cones).  After these blow ups, there are four (E 8 , SU (2)) collisions in the middle of the tetrahedron, which need to be blown up twice for each. Finally, we can just blow up the four (E 8 , E 8 , E 8 ) collisions on the faces, according to [11]. For convenience purpose, we plot the blow up of (E 8 , E 8 , E 8 ) collisions in figure 5. We also show the fully subdivided (E 8 , E 8 , SU (2)) collision in figure 6. The final geometric configuration will be absent of non-minimal loci, and the elliptic fibration is flat.

B A list of elliptic Calabi-Yau fivefolds
In this appendix, we list the elliptic CY5 as hypersurfaces of weighted projective spaces P 1,w 1 ,w 2 ,w 3 ,w 4 ,w 5 ,w 6 . We impose the following conditions: 1. The lattice polytope associated to P 1,w 1 ,w 2 ,w 3 ,w 4 ,w 5 ,w 6 is reflexive.
To get a finite list, we require that the degree 3 d ≡ 1 + w 1 + w 2 + w 3 + w 4 + w 5 + w 6 ≤ 150 . (B.1) We list these models along with the Hodge numbers of CY5 and the 2d F-theory geometric gauge group in table 9-12. Note that for many cases, the base fourfold has to be singular. We do not list all the possible base topologies in detail. 3 The complete list of weights giving rise to reflexive polytopes in this category was already worked out in [15]. (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 Figure 6: The fully blown up (E 8 , E 8 , SU (2)) collision.