# Discrete Wilson Lines in F-Theory

Authors: Volker Braun
Journal: Advances in High Energy Physics
 (2011)

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## Abstract

F-theory models are constructed where the7-brane has a nontrivial fundamental group. The base manifolds used are a toric Fano variety and a smooth toric threefold coming from a reflexive polyhedron. The discriminant locus of the elliptically fibered Calabi-Yau fourfold can be chosen such that one irreducible component is not simply connected (namely, an Enriques surface) and supports a non-Abelian gauge theory.


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## Body

## 1. Introduction

F-theory [1] is a way to use geometry as a tool to understand certain compactifications of string theory that are otherwise not entirely geometric [2]. It uses an auxiliary elliptically fibered Calabi-Yau fourfold, not to be confused with the space-time manifold to study string theory in a regime away from any known weakly coupled perturbative description. Recently [3, 4] a particular model building Ansatz has been suggested where the GUT gauge group arises from a 7-brane wrapped on a contractible del Pezzo surface. Various models [5–10] and more have been constructed along these lines.One key feature of this Ansatz is that the scales of gravity and gauge physics can be decoupled as one can decompactify the Calabi-Yau manifold without changing the del Pezzo surface. However, the price one has to pay for this is that the usual way of GUT symmetry breaking in string theory, namely, the Hosotani mechanism [11, 12] using discrete Wilson lines, no longer works: all del Pezzo surfaces are simply connected. Alternatives have been developed [3, 4, 13], but require one to give a vacuum expectation value to fields locally and not just make global nontrivial identifications. Turning on fields locally then affects the running of the coupling constants and, potentially, defocuses the gauge coupling unification [14, 15].In this paper I will advocate for a different Ansatz for GUT model building and symmetry breaking in F-theory, namely, by wrapping the GUT 7-brane on a nonsimply connected divisor in the base of the elliptic fibration. This allows one to choose a globally nontrivial identification of the gauge bundle while keeping it locally trivial, breaking the GUT gauge group by the usual Hosotani mechanism. For what it is worth, this setup also implies that there is no gravity/gauge theory decoupling limit.Of course this raises the question of whether there are any such divisors in threefolds that are suitable as bases for elliptically fibered Calabi-Yau manifolds. In this paper I will answer this question and work out a rather simple example of an Enriques surface embedded into a toric threefold associated to a reflexive 3-dimensional polytope. There is nothing particularly unique about this example. It just combines the most simple surface withℤ2 fundamental group and the class of threefolds we are most used to work with. All toric geometry computations used in this paper were done using [16–18].


## 2. Base Threefold

### 2.1. Foreword

An Enriques surface is a free quotient of aK3 surface by a freely acting holomorphic involution and is probably the best-known example of a complex surface S with fundamental group π1(S)=ℤ2. Its first Chern class c1(S) is the torsion element in H2(S,ℤ)≃ℤ10⊕ℤ2, so it admits a Ricci-flat metric but has no covariantly constant spinor (equivalently, no covariantly constant (2,0)-form). Some, but not all, K3 surfaces can be realized [19] as quartics in ℙ3. Somewhat unfortunately, the locus of quartic K3s and the locus of K3 surfaces with an Enriques involution do not intersect in the moduli space of smooth K3 surfaces. In other words, no smooth quartic in ℙ3 carries an Enriques involution. Therefore, out of necessity one is forced to look at singular (birational) models and then resolve these singularities. This will be the central theme in the following.To explicitly construct and resolve the singularities, I will make extensive use of toric geometry. However, before delving into these technical details let me first give an overview. The basic idea is to look at the followingℤ4 action on ℙ3:
(2.1)g:P3→P3,[x0:x1:x2:x3][x0:ix1:i2x2:i3x3].
The fixed point set of g2 are the two disjoint rational curves(2.2)P1∪P1={x0=x2=0}∪{x1=x3=0},
and the fixed points of g are the north and south poles on these (4 points altogether). A sufficiently generic ℤ4 invariant quartic q(x0,x1,x2,x3) is then a (singular) Enriques surface on the quotient (this construction is rather similar to the way to construct nonsimply connected Calabi-Yau threefolds [20–24], except that I will not be looking at sections of the anticanonical bundle (which would be Calabi-Yau)). The fastest way to see this is to note that the would-be (2,0)-form
(2.3)Ω(2,0)=∮εijklxidxj∧dxk∧dxlq(x0,x1,x2,x3)
is projected out by g.Here is where this paper essentially begins, because so far we only have a singular Enriques surface in an even more singular ambient space. Clearly, one wants to resolve the singularities. The first step is to resolve the curves ofℤ2 singularities, for which there is a unique crepant resolution. Then one has to deal with the remaining 4 ℤ4 singularities. By a happy coincidence, the above ℤ4 quotient of ℙ3 is itself a toric variety. Hence, the methods of toric geometry can be applied and allow us to construct partial and complete resolutions explicitly as toric varieties.


### 2.2. Toric Geometry

As a warm-up, I will first review some basic notions of toric geometry. The defining data is a rational polyhedral fan in a latticeN≃ℤd, where d is the complex dimension of the variety. A fan Σ is a finite set of cones σ∈Σ, closed under taking faces. Often, the fan will be the cones over the faces of a polytope. This is called the face fan of the polytope.Amongst the different, but equivalent, ways to define the corresponding complex algebraic variety from the fan data, I will use the Cox homogeneous coordinate [25] description in the following. The basic idea is to associate one complex-valued homogeneous coordinate to each ray (one-dimensional cone) of the fan. Then one has to remove a codimension-2 or higher algebraic subset and mod out generalized homogeneous rescalings. This construction will be reviewed and applied in much more detail in Section 2.4. For now, let us just consider ℙ3 as an example. Its fan consists of the cones (2.4)ΣP3={〈0〉,〈e1〉,〈e2〉,〈e3〉,〈-∑ei〉,〈e1,e2〉,〈e1,e3〉,〈e1,-∑ei〉,〈e2,e3〉,〈e2,-∑ei〉,〈e3,-∑ei〉,〈e1,e2,e3〉,〈e1,e2,-∑ei〉,〈e1,e3,-∑ei〉,〈e2,e3,-∑ei〉,},
where e1, e2, e3 are a basis for N≃ℤ3. There are 4 one-dimensional rays satisfying a unique linear relation, which translates into 4 homogeneous coordinates with the usual identification (2.5)[x0:x1:x2:x3]=[λx0:λx1:λx2:λx3],λ∈C×.
A map of fans is a map of ambient lattices such that every cone of the domain maps into a cone of the range fan. Any such fan morphism defines a morphism of toric varieties in a covariantly functorial way. For the purposes of this paper (except for A), we will only consider the case where the lattice map is the identity. In this case the domain fan is simply a subdivision of the range fan. The toric map corresponding to a subdivision of a cone σ is the blow-up along a toric subvariety of dimension equal to codim(σ).A toric divisor (all divisors in this paper will be Cartier divisors, even though we will be working with auxiliary singular varieties where not all divisors are Cartier) is a formal linear combinationD=∑aiV(xi) of the codimension-one subvarieties (2.6)V(xi)={xi=0}
corresponding to the one-dimensional cones of the fan. There are two basic constructions associated to such a toric divisor that will be important in the following.  (i)
Every coefficientai can be thought of as the value of a function f:N→ℤ on the generating lattice point ρi of the i-th one-cone. If every cone is simplicial, then there is a uniquely defined continuous function on the fan with the above property. The pull-back of the function on the fan corresponds to the pull-back of the divisor by the toric map.(ii)
The divisor also defines a polytope(2.7)PD={m∈MR|〈m,ρi〉≥-ai},
where M=N∨ is the dual lattice. The global sections Γ𝒪(D) are in one-to-one correspondence with the integral lattice points M∩PD and can easily be counted for any given divisor.A particularly relevant divisor is the anticanonical divisor-K=∑V(xi). Given a lattice polytope ∇∈N, we can construct its face fan and the polytope Δ=P-K⊂M. If Δ is again a lattice polytope, then ∇ is called reflexive.Finally, note thatH2(ℙ3)=ℤ. Hence, the line bundles on ℙ3 are classified by a single integer, their first Chern class. The toric divisors, on the other hand, are defined by 4 integers. Clearly, there is no one-to-one correspondence between divisors D and the isomorphism class of the associated line bundle 𝒪(D). To make this into a bijection, one must mod out linear equivalence of divisors. That is, one has to identify the piecewise linear functions modulo linear functions. In particular, one can easily see that (2.8)D=∑i=03aiV(xi)∼(a0+a1+a2+a3)V(x0)
on ℙ3.


### 2.3. Three Birational Models

We now begin with the core of this paper and define the base threefold of the elliptically fibered Calabi-Yau fourfold. In fact, I will choose a smooth toric varietyB̂ as the base manifold, containing a nonsimply connected divisor D̂. However, directly analyzing B̂ will be overly complicated. In particular, B̂ contains exceptional divisors that do not intersect the divisor D̂ we are interested in. Therefore, to better understand D̂⊂B̂, I will blow down these additional exceptional divisors. This will produce a singular variety B containing the same divisor D=D̂. Finally, I will blow down two more curves in B to obtain a (even more singular) three-dimensional variety B̲. The blown-down divisor D̲⊂B̲ is the most suitable one to compute the fundamental groups. To summarize, I am going to define successive blow-ups (2.9)B̂⟶π̂B⟶π̲B̲
of three-dimensional toric varieties. Both of the maps π̂,π̲ are toric morphisms defined in the obvious way by combining cones of the fan into bigger cones, discarding all rays that are no longer part of the more coarse (blown-down) fan.I now define the fansΣ corresponding to the toric varieties. Let me start by the rays Σ(1). The most singular variety has (2.10)∑B̲(1)={(-3,-2,4),(0,1,0),(1,0,0),(2,1,-4)},
see Figure 1. The convex hull of these four points is a tetrahedron, but not a minimal lattice simplex. In addition to the origin (which is an interior point), it contains the two points (-1,-1,2) and (1,1,-2) along two different edges. The variety B will be the maximal crepant partial resolution of B̲, that is, the (in this case unique) maximal triangulation of the convex hull conv(ΣB̲(1)). Hence, one must add the additional integral points to the ray generators, (2.11)∑B(1)=ΣB̲(1)∪{(-1,-1,2),(1,1,-2)}.
Neither the variety B̲ nor its maximal crepant partial resolution B is smooth, related to the fact that the polytope conv(ΣB̲(1))=conv(ΣB(1)) is not reflexive. One again needs to add rays to resolve all singularities; however these time the generators are necessarily outside of conv(ΣB̲(1)). One particular choice I am going to make is the rays generated by the 18 points listed in Table 1.Table 1
Ray generators and the associated Cox homogeneous variables.
i01234567891011121314151617i-th ray−3012−11−2−2−1−1−10011122−2101−11−1−1−1000000111400−42−223112−11−2−1−1−3−2B̲z̲0z̲1z̲2z̲3Bz0z1z2z3z4z5B̂ẑ0ẑ1ẑ2ẑ3ẑ4ẑ5ẑ6ẑ7ẑ8ẑ9ẑ10ẑ11ẑ12ẑ13ẑ14ẑ15ẑ16ẑ17Figure 1
The raysΣB̲(1) (red dots) and ΣB(1) (red and blue dots).The 8 facets of the convex hullconv(ΣB̂(1)) are given by the inequalities (2.12)(nx,ny,nz)⋅(-1-1-1-10112-1012-1-11-1-1-1000011)≥(1,1,1,1,1,1,1,1),
which is, therefore, a reflexive polytope.Finally, to completely specify the toric varietiesB̲, B, and B̂, let me define the generating cones of the respective fans:  (i)
ΣB̲ is the face fan of the polytope conv(ΣB̲(1)), see Figure 1;(ii)
ΣB is the unique maximal subdivision of ΣB̲;(iii)
ΣB̂ is a maximal subdivision of the face fan of the polytope conv(ΣB̂(1)), see Figure 2.Figure 2
The raysΣB̂(1).As there are many different maximal subdivisions of the face fan, this alone does not uniquely specify the fanΣB̂. For concreteness, I will fix the one listed in Table 3. Note that not all combinatorial symmetries of the graph in 3 are actually symmetries of the fan.


### 2.4. Homogeneous Coordinates

For future reference, let me list the toric Chow groups [26]:(2.13)Ak(B̂)={1Z15Z151,Ak(B)={1Z3×Z2Z3×Z241,Ak(B̲)={1k=3Z×Z4k=2Z×Z22k=11k=0.
Since all three toric varieties have at most orbifold singularities, the Hodge numbers are hp,p=rankAp and hp,q=0 if p≠q.The appearance of torsion in the Chow group slightly complicates the Cox homogeneous coordinate [25] construction of the toric varieties, so let me spell out the details. In general, a simplicial (that is, with at most orbifold singularities) d-dimensional toric variety X can be written as a geometric quotient (2.14)X=CΣX(1)-ZHom(Ad-1(X),C×)≃Cr-Z(C*)d-r×Ad-1(X)tors,
where r is the number of rays in the fan ΣX. The exceptional set Z is the variety defined by the irrelevant ideal. A more catchy way of remembering Z is that it forbids homogeneous coordinates from vanishing simultaneously if and only if their product is a monomial in the Stanley-Reisner ideal. The latter is (2.15)SR(ΣB̲)=〈z̲0z̲1z̲2z̲3〉,SR(ΣB)=〈z0z2,z1z3,z4z5〉,SR(ΣB̂)=〈⋯105quadricmonomials⋯〉.
It remains to describe the groups in the denominator of (2.14). For the most singular variety B̲, one finds (2.16)[z̲0:z̲1:z̲2:z̲3]=[λz̲0:λz̲1:λz̲2:λz̲3]∀λ∈C×,[z̲0:z̲1:z̲2:z̲3]=[z̲0:μz̲1:μ2z̲2:μ3z̲3]∀μ∈{1,i,i2,i3}≃Z4
and for the intermediate blow-up B(2.17)[z0:z1:z2:z3:z4:z5]=[λz0:λz1:λz2:λz3:z4:z5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[λz0:z1:λz2:z3:λ-2z4:z5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[z0:z1:z2:z3:λz4:λz5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[μz0:μz1:z2:z3:μz4:z5]∀μ∈{1,-1}≃Z2.


### 2.5. A Nonsimply Connected Divisor

I am now going to define a divisorD̲⊂B̲ in the same linear system as the toric divisor (V(z̲i) denotes the toric divisor {z̲i=0} associated to the i-th ray) (2.18)D̲~4V(z̲0),
that is, as the zero set of a sufficiently generic section of the line bundle 𝒪(D̲)=𝒪(V(z̲0))4. (By ~, we will always denote rational equivalence of divisors. That is, D1~D2 means that there is a one-parameter family of divisors interpolating between D1 and D2. Equivalently, the Chow cycle defined by D1 and D2 is the same.) A basis for the sections is (2.19)H0(B̲,O(D̲))=〈z̲04,z̲24,z̲14,z̲34,z̲1z̲22z̲3,z̲12z̲32,z̲0z̲2z̲32,z̲0z̲12z̲2,z̲02z̲22,z̲02z̲1z̲3〉,
corresponding to the points of the polytope (2.20)PD̲=conv{(0,0,0),(4,0,2),(0,4,1),(0,0,-1)}⊂M.
Note that the fan ΣB̲ is precisely the normal fan of the Newton polytope PD̲. In this sense, B̲ is the “natural” ambient toric variety for the surface D̲.For explicitness, let me fix once and for all a linear combination of the monomials as the defining equation of the divisorD̲. I will select the vertices of the Newton polytope and define (2.21)D̲={z̲04+z̲14+z̲24+z̲34=0}⊂B̲.
This surface is known to be an Enriques surface since it projects out the potential (2,0)-form as mentioned in Section 2.1. In fact, this example has been known for some time, see Remark 3.6 in [27].


### 2.6. Kähler Cone and Canonical Divisors

The content of this subsection is not necessary for the understanding of the paper, but I would like to pause for a moment and mention how the “Fermat quartic” in (2.21) fails to define a K3 surface. In other words, how does the divisor D̲=4V(z̲0) differ from the anticanonical divisor (2.22)-KB̲=V(z̲0)+V(z̲1)+V(z̲2)+V(z̲3)
of B̲? Comparing with ℙ3, see (2.8), one might have thought that they were linearly equivalent.Similarly to the quarticK3⊂ℙ3, one can also define a Calabi-Yau variety in B̲ as the zero locus of a section of the anticanonical bundle (after resolution of singularities, the anticanonical divisor will be a smooth 2-dimensional Calabi-Yau manifold, that is, again a K3 surface). The available sections are (2.23)H0(B̲,O(-KB̲))=〈z̲02z̲32,z̲03z̲2,z̲22z̲32,z̲0z̲23,z̲1z̲33,z̲0z̲1z̲2z̲3,z̲02z̲12,z̲12z̲22,z̲13z̲3〉.
Note that this differs from the sections of D̲, see (2.19). Therefore, the two divisors are not linearly equivalent. Nevertheless, -KB̲ and D̲ are very close to being linearly equivalent. In fact, it is easy to see that they are in the same rational divisor class since the rational divisor class group is one-dimensional, dim(A2(B̲)⊗ℤℝ)=1. (The toric divisor class group of a variety X is often written as Cl(X). I will not use this notation in the following, but opt for Adim(X)-1(X) instead.) However, their difference is a 2-torsion element in the (integral) divisor class (2.24)KB̲+D̲≠0,2(KB̲+D̲)=0∈A2(B̲)≃Z⊕Z4.
The same is true on the crepant partial resolution, where KB+D is again a 2-torsion element in A2(B).On the final smooth resolutionB̂ the divisor class group A2(B̂)=ℤ15 is torsion-free. However, the last blow-up π̂:B̂→B is not crepant, so (2.25)π̂*(KB)≠KB̂.
Therefore, the divisors -KB̂ and D̂=π̂*(D) are no longer in the same rational equivalence class.Finally, let me describe the Kähler cones of these varieties. First, let me remind the reader that the Kähler cone of a toric variety is an open rational polyhedral cone in the rational divisor class group corresponding to the cone of convex piecewise linear support functions on the fan. For the two singular varieties, one obtains(2.26)K(B̲)=〈V(z̲0)〉⊂A2(B̲)⊗ZR≃RK(B)=〈V(z0),V(z1),2V(z0)+V(z4)〉⊂A2(B)⊗ZR≃R3.
As the anticanonical class -KB̲ is rationally equivalent to 4V(z̲0), we see the following.  (i)
B̲ is a (singular) Fano variety.(ii)
B is not Fano, but the anticanonical class is on the boundary of the Kähler cone. In other words, -KB is nef but not ample.On the smooth blow-upB̂, the Kähler cone (2.27)K(B̂)⊂A2(B̂)⊗ZR≃R15
is rather complicated, and we will refrain from listing it explicitly. It is spanned by the origin and 169 rays and has 20 facets (that is, 14-dimensional faces). The anticanonical divisor -KB̂ as well as D̂ sits on the boundary of the Kähler cone, that is, is nef but not ample. However, each satisfies a different subset of 16 out of the 20 facet equations, so they lie on different faces of the Kähler cone.


### 2.7. Pull-Back Divisors

By the usual dictionary of toric geometry, the toric divisor4V(z̲0)~D corresponds to a continuous piecewise linear function on Nℝ≃ℝ3. Explicitly, the function is (2.28)f(nx,ny,nz)={nzifn⃗∈〈0,1,2〉,-4nx-2nzifn⃗∈〈0,1,3〉,-4ny-nzifn⃗∈〈0,2,3〉,0ifn⃗∈〈1,2,3〉.
The pull-back of this toric divisor by the toric morphisms π̲ and π̲∘π̂ is simply given by the pull-back of the piecewise linear function. Therefore, (2.29)D̲∼4V(z̲0),D∼4V(z0)+2V(z4),D̂∼4V(ẑ0)+2V(ẑ4)+4V(ẑ6)+3V(ẑ7)+3V(ẑ8)+2V(ẑ9)+2V(ẑ10)+2V(ẑ11)+V(ẑ12)+2V(ẑ13)+V(ẑ14).
What is the exceptional set of the first blow-up π̲? Recall that it corresponds to the subdivisions along the 2-cones (2.30)〈0,4〉∪〈2,4〉⟶〈0,2〉,〈1,5〉∪〈3,5〉⟶〈1,3〉,
see Figure 1. Therefore, π̲ is the blow-up along two disjoint rational curves of ℤ2-singularities in B̲. A standard intersection computation in the Chow group [26] yields that each curve intersects the divisor D̲ in two points. Therefore, the proper transform of D⊂B is D̲ blown up in four points. The final blow-up π̂:B̂→B does not further subdivide the 2-skeleton ΣB(2) and, therefore, corresponds to the blow-up of points in B. Any sufficiently generic divisor D misses these blow-up points, and, therefore, the surfaces D and D̂ are isomorphic. To summarize,  (i)
D̲ is a singular Enriques surface with four ℤ2-orbifold points;(ii)
D̂ and D are the same smooth Enriques surface after blowing up the orbifold points;(iii)
Since the blow-up at a point does not change the fundamental group, we find that(2.31)π1(D̲)=π1(D)=π1(D̂)=Z2.It is important to remember that the actual divisor is a fixed subvariety defined as the zero locus of an equation. To relate this defining equation before and after the blow-up, it is instructional to write the first blow-up mapπ̲:B→B̲ explicitly in terms of its action on homogeneous coordinates. One finds (2.32)π̲([z0:z1:z2:z3:z4:z5])=[z0z4:z1z5:z2z4:z3z5].
Note that this map is well defined on the equivalence classes (2.17) thanks to the identifications (2.16). One sees that, for example, the section z̲04 corresponds to the section z04z42 under the π̲* pull-back. I leave the analogous expression for π̂ as an exercise to the reader.To summarize, the equation for the divisorD̲ determines the equation satisfied by the proper transforms D and D̂ on the blow-ups. They are (2.33)D̲={z̲04+z̲14+z̲24+z̲34=0},D={z04z42+z14z52+z24z42+z34z52=0},D̂={ẑ04ẑ42ẑ64ẑ73ẑ83ẑ92ẑ102ẑ112ẑ12ẑ132ẑ14+ẑ14ẑ52ẑ62ẑ72ẑ93ẑ104ẑ11ẑ122ẑ153ẑ16ẑ172+ẑ24ẑ42ẑ7ẑ8ẑ102ẑ123ẑ132ẑ143ẑ152ẑ162ẑ174+ẑ34ẑ52ẑ62ẑ82ẑ9ẑ113ẑ134ẑ142ẑ15ẑ163ẑ172=0}.


## 2.1. Foreword

An Enriques surface is a free quotient of aK3 surface by a freely acting holomorphic involution and is probably the best-known example of a complex surface S with fundamental group π1(S)=ℤ2. Its first Chern class c1(S) is the torsion element in H2(S,ℤ)≃ℤ10⊕ℤ2, so it admits a Ricci-flat metric but has no covariantly constant spinor (equivalently, no covariantly constant (2,0)-form). Some, but not all, K3 surfaces can be realized [19] as quartics in ℙ3. Somewhat unfortunately, the locus of quartic K3s and the locus of K3 surfaces with an Enriques involution do not intersect in the moduli space of smooth K3 surfaces. In other words, no smooth quartic in ℙ3 carries an Enriques involution. Therefore, out of necessity one is forced to look at singular (birational) models and then resolve these singularities. This will be the central theme in the following.To explicitly construct and resolve the singularities, I will make extensive use of toric geometry. However, before delving into these technical details let me first give an overview. The basic idea is to look at the followingℤ4 action on ℙ3:
(2.1)g:P3→P3,[x0:x1:x2:x3][x0:ix1:i2x2:i3x3].
The fixed point set of g2 are the two disjoint rational curves(2.2)P1∪P1={x0=x2=0}∪{x1=x3=0},
and the fixed points of g are the north and south poles on these (4 points altogether). A sufficiently generic ℤ4 invariant quartic q(x0,x1,x2,x3) is then a (singular) Enriques surface on the quotient (this construction is rather similar to the way to construct nonsimply connected Calabi-Yau threefolds [20–24], except that I will not be looking at sections of the anticanonical bundle (which would be Calabi-Yau)). The fastest way to see this is to note that the would-be (2,0)-form
(2.3)Ω(2,0)=∮εijklxidxj∧dxk∧dxlq(x0,x1,x2,x3)
is projected out by g.Here is where this paper essentially begins, because so far we only have a singular Enriques surface in an even more singular ambient space. Clearly, one wants to resolve the singularities. The first step is to resolve the curves ofℤ2 singularities, for which there is a unique crepant resolution. Then one has to deal with the remaining 4 ℤ4 singularities. By a happy coincidence, the above ℤ4 quotient of ℙ3 is itself a toric variety. Hence, the methods of toric geometry can be applied and allow us to construct partial and complete resolutions explicitly as toric varieties.


## 2.2. Toric Geometry

As a warm-up, I will first review some basic notions of toric geometry. The defining data is a rational polyhedral fan in a latticeN≃ℤd, where d is the complex dimension of the variety. A fan Σ is a finite set of cones σ∈Σ, closed under taking faces. Often, the fan will be the cones over the faces of a polytope. This is called the face fan of the polytope.Amongst the different, but equivalent, ways to define the corresponding complex algebraic variety from the fan data, I will use the Cox homogeneous coordinate [25] description in the following. The basic idea is to associate one complex-valued homogeneous coordinate to each ray (one-dimensional cone) of the fan. Then one has to remove a codimension-2 or higher algebraic subset and mod out generalized homogeneous rescalings. This construction will be reviewed and applied in much more detail in Section 2.4. For now, let us just consider ℙ3 as an example. Its fan consists of the cones (2.4)ΣP3={〈0〉,〈e1〉,〈e2〉,〈e3〉,〈-∑ei〉,〈e1,e2〉,〈e1,e3〉,〈e1,-∑ei〉,〈e2,e3〉,〈e2,-∑ei〉,〈e3,-∑ei〉,〈e1,e2,e3〉,〈e1,e2,-∑ei〉,〈e1,e3,-∑ei〉,〈e2,e3,-∑ei〉,},
where e1, e2, e3 are a basis for N≃ℤ3. There are 4 one-dimensional rays satisfying a unique linear relation, which translates into 4 homogeneous coordinates with the usual identification (2.5)[x0:x1:x2:x3]=[λx0:λx1:λx2:λx3],λ∈C×.
A map of fans is a map of ambient lattices such that every cone of the domain maps into a cone of the range fan. Any such fan morphism defines a morphism of toric varieties in a covariantly functorial way. For the purposes of this paper (except for A), we will only consider the case where the lattice map is the identity. In this case the domain fan is simply a subdivision of the range fan. The toric map corresponding to a subdivision of a cone σ is the blow-up along a toric subvariety of dimension equal to codim(σ).A toric divisor (all divisors in this paper will be Cartier divisors, even though we will be working with auxiliary singular varieties where not all divisors are Cartier) is a formal linear combinationD=∑aiV(xi) of the codimension-one subvarieties (2.6)V(xi)={xi=0}
corresponding to the one-dimensional cones of the fan. There are two basic constructions associated to such a toric divisor that will be important in the following.  (i)
Every coefficientai can be thought of as the value of a function f:N→ℤ on the generating lattice point ρi of the i-th one-cone. If every cone is simplicial, then there is a uniquely defined continuous function on the fan with the above property. The pull-back of the function on the fan corresponds to the pull-back of the divisor by the toric map.(ii)
The divisor also defines a polytope(2.7)PD={m∈MR|〈m,ρi〉≥-ai},
where M=N∨ is the dual lattice. The global sections Γ𝒪(D) are in one-to-one correspondence with the integral lattice points M∩PD and can easily be counted for any given divisor.A particularly relevant divisor is the anticanonical divisor-K=∑V(xi). Given a lattice polytope ∇∈N, we can construct its face fan and the polytope Δ=P-K⊂M. If Δ is again a lattice polytope, then ∇ is called reflexive.Finally, note thatH2(ℙ3)=ℤ. Hence, the line bundles on ℙ3 are classified by a single integer, their first Chern class. The toric divisors, on the other hand, are defined by 4 integers. Clearly, there is no one-to-one correspondence between divisors D and the isomorphism class of the associated line bundle 𝒪(D). To make this into a bijection, one must mod out linear equivalence of divisors. That is, one has to identify the piecewise linear functions modulo linear functions. In particular, one can easily see that (2.8)D=∑i=03aiV(xi)∼(a0+a1+a2+a3)V(x0)
on ℙ3.


## 2.3. Three Birational Models

We now begin with the core of this paper and define the base threefold of the elliptically fibered Calabi-Yau fourfold. In fact, I will choose a smooth toric varietyB̂ as the base manifold, containing a nonsimply connected divisor D̂. However, directly analyzing B̂ will be overly complicated. In particular, B̂ contains exceptional divisors that do not intersect the divisor D̂ we are interested in. Therefore, to better understand D̂⊂B̂, I will blow down these additional exceptional divisors. This will produce a singular variety B containing the same divisor D=D̂. Finally, I will blow down two more curves in B to obtain a (even more singular) three-dimensional variety B̲. The blown-down divisor D̲⊂B̲ is the most suitable one to compute the fundamental groups. To summarize, I am going to define successive blow-ups (2.9)B̂⟶π̂B⟶π̲B̲
of three-dimensional toric varieties. Both of the maps π̂,π̲ are toric morphisms defined in the obvious way by combining cones of the fan into bigger cones, discarding all rays that are no longer part of the more coarse (blown-down) fan.I now define the fansΣ corresponding to the toric varieties. Let me start by the rays Σ(1). The most singular variety has (2.10)∑B̲(1)={(-3,-2,4),(0,1,0),(1,0,0),(2,1,-4)},
see Figure 1. The convex hull of these four points is a tetrahedron, but not a minimal lattice simplex. In addition to the origin (which is an interior point), it contains the two points (-1,-1,2) and (1,1,-2) along two different edges. The variety B will be the maximal crepant partial resolution of B̲, that is, the (in this case unique) maximal triangulation of the convex hull conv(ΣB̲(1)). Hence, one must add the additional integral points to the ray generators, (2.11)∑B(1)=ΣB̲(1)∪{(-1,-1,2),(1,1,-2)}.
Neither the variety B̲ nor its maximal crepant partial resolution B is smooth, related to the fact that the polytope conv(ΣB̲(1))=conv(ΣB(1)) is not reflexive. One again needs to add rays to resolve all singularities; however these time the generators are necessarily outside of conv(ΣB̲(1)). One particular choice I am going to make is the rays generated by the 18 points listed in Table 1.Table 1
Ray generators and the associated Cox homogeneous variables.
i01234567891011121314151617i-th ray−3012−11−2−2−1−1−10011122−2101−11−1−1−1000000111400−42−223112−11−2−1−1−3−2B̲z̲0z̲1z̲2z̲3Bz0z1z2z3z4z5B̂ẑ0ẑ1ẑ2ẑ3ẑ4ẑ5ẑ6ẑ7ẑ8ẑ9ẑ10ẑ11ẑ12ẑ13ẑ14ẑ15ẑ16ẑ17Figure 1
The raysΣB̲(1) (red dots) and ΣB(1) (red and blue dots).The 8 facets of the convex hullconv(ΣB̂(1)) are given by the inequalities (2.12)(nx,ny,nz)⋅(-1-1-1-10112-1012-1-11-1-1-1000011)≥(1,1,1,1,1,1,1,1),
which is, therefore, a reflexive polytope.Finally, to completely specify the toric varietiesB̲, B, and B̂, let me define the generating cones of the respective fans:  (i)
ΣB̲ is the face fan of the polytope conv(ΣB̲(1)), see Figure 1;(ii)
ΣB is the unique maximal subdivision of ΣB̲;(iii)
ΣB̂ is a maximal subdivision of the face fan of the polytope conv(ΣB̂(1)), see Figure 2.Figure 2
The raysΣB̂(1).As there are many different maximal subdivisions of the face fan, this alone does not uniquely specify the fanΣB̂. For concreteness, I will fix the one listed in Table 3. Note that not all combinatorial symmetries of the graph in 3 are actually symmetries of the fan.


## 2.4. Homogeneous Coordinates

For future reference, let me list the toric Chow groups [26]:(2.13)Ak(B̂)={1Z15Z151,Ak(B)={1Z3×Z2Z3×Z241,Ak(B̲)={1k=3Z×Z4k=2Z×Z22k=11k=0.
Since all three toric varieties have at most orbifold singularities, the Hodge numbers are hp,p=rankAp and hp,q=0 if p≠q.The appearance of torsion in the Chow group slightly complicates the Cox homogeneous coordinate [25] construction of the toric varieties, so let me spell out the details. In general, a simplicial (that is, with at most orbifold singularities) d-dimensional toric variety X can be written as a geometric quotient (2.14)X=CΣX(1)-ZHom(Ad-1(X),C×)≃Cr-Z(C*)d-r×Ad-1(X)tors,
where r is the number of rays in the fan ΣX. The exceptional set Z is the variety defined by the irrelevant ideal. A more catchy way of remembering Z is that it forbids homogeneous coordinates from vanishing simultaneously if and only if their product is a monomial in the Stanley-Reisner ideal. The latter is (2.15)SR(ΣB̲)=〈z̲0z̲1z̲2z̲3〉,SR(ΣB)=〈z0z2,z1z3,z4z5〉,SR(ΣB̂)=〈⋯105quadricmonomials⋯〉.
It remains to describe the groups in the denominator of (2.14). For the most singular variety B̲, one finds (2.16)[z̲0:z̲1:z̲2:z̲3]=[λz̲0:λz̲1:λz̲2:λz̲3]∀λ∈C×,[z̲0:z̲1:z̲2:z̲3]=[z̲0:μz̲1:μ2z̲2:μ3z̲3]∀μ∈{1,i,i2,i3}≃Z4
and for the intermediate blow-up B(2.17)[z0:z1:z2:z3:z4:z5]=[λz0:λz1:λz2:λz3:z4:z5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[λz0:z1:λz2:z3:λ-2z4:z5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[z0:z1:z2:z3:λz4:λz5]∀λ∈C×,[z0:z1:z2:z3:z4:z5]=[μz0:μz1:z2:z3:μz4:z5]∀μ∈{1,-1}≃Z2.


## 2.5. A Nonsimply Connected Divisor

I am now going to define a divisorD̲⊂B̲ in the same linear system as the toric divisor (V(z̲i) denotes the toric divisor {z̲i=0} associated to the i-th ray) (2.18)D̲~4V(z̲0),
that is, as the zero set of a sufficiently generic section of the line bundle 𝒪(D̲)=𝒪(V(z̲0))4. (By ~, we will always denote rational equivalence of divisors. That is, D1~D2 means that there is a one-parameter family of divisors interpolating between D1 and D2. Equivalently, the Chow cycle defined by D1 and D2 is the same.) A basis for the sections is (2.19)H0(B̲,O(D̲))=〈z̲04,z̲24,z̲14,z̲34,z̲1z̲22z̲3,z̲12z̲32,z̲0z̲2z̲32,z̲0z̲12z̲2,z̲02z̲22,z̲02z̲1z̲3〉,
corresponding to the points of the polytope (2.20)PD̲=conv{(0,0,0),(4,0,2),(0,4,1),(0,0,-1)}⊂M.
Note that the fan ΣB̲ is precisely the normal fan of the Newton polytope PD̲. In this sense, B̲ is the “natural” ambient toric variety for the surface D̲.For explicitness, let me fix once and for all a linear combination of the monomials as the defining equation of the divisorD̲. I will select the vertices of the Newton polytope and define (2.21)D̲={z̲04+z̲14+z̲24+z̲34=0}⊂B̲.
This surface is known to be an Enriques surface since it projects out the potential (2,0)-form as mentioned in Section 2.1. In fact, this example has been known for some time, see Remark 3.6 in [27].


## 2.6. Kähler Cone and Canonical Divisors

The content of this subsection is not necessary for the understanding of the paper, but I would like to pause for a moment and mention how the “Fermat quartic” in (2.21) fails to define a K3 surface. In other words, how does the divisor D̲=4V(z̲0) differ from the anticanonical divisor (2.22)-KB̲=V(z̲0)+V(z̲1)+V(z̲2)+V(z̲3)
of B̲? Comparing with ℙ3, see (2.8), one might have thought that they were linearly equivalent.Similarly to the quarticK3⊂ℙ3, one can also define a Calabi-Yau variety in B̲ as the zero locus of a section of the anticanonical bundle (after resolution of singularities, the anticanonical divisor will be a smooth 2-dimensional Calabi-Yau manifold, that is, again a K3 surface). The available sections are (2.23)H0(B̲,O(-KB̲))=〈z̲02z̲32,z̲03z̲2,z̲22z̲32,z̲0z̲23,z̲1z̲33,z̲0z̲1z̲2z̲3,z̲02z̲12,z̲12z̲22,z̲13z̲3〉.
Note that this differs from the sections of D̲, see (2.19). Therefore, the two divisors are not linearly equivalent. Nevertheless, -KB̲ and D̲ are very close to being linearly equivalent. In fact, it is easy to see that they are in the same rational divisor class since the rational divisor class group is one-dimensional, dim(A2(B̲)⊗ℤℝ)=1. (The toric divisor class group of a variety X is often written as Cl(X). I will not use this notation in the following, but opt for Adim(X)-1(X) instead.) However, their difference is a 2-torsion element in the (integral) divisor class (2.24)KB̲+D̲≠0,2(KB̲+D̲)=0∈A2(B̲)≃Z⊕Z4.
The same is true on the crepant partial resolution, where KB+D is again a 2-torsion element in A2(B).On the final smooth resolutionB̂ the divisor class group A2(B̂)=ℤ15 is torsion-free. However, the last blow-up π̂:B̂→B is not crepant, so (2.25)π̂*(KB)≠KB̂.
Therefore, the divisors -KB̂ and D̂=π̂*(D) are no longer in the same rational equivalence class.Finally, let me describe the Kähler cones of these varieties. First, let me remind the reader that the Kähler cone of a toric variety is an open rational polyhedral cone in the rational divisor class group corresponding to the cone of convex piecewise linear support functions on the fan. For the two singular varieties, one obtains(2.26)K(B̲)=〈V(z̲0)〉⊂A2(B̲)⊗ZR≃RK(B)=〈V(z0),V(z1),2V(z0)+V(z4)〉⊂A2(B)⊗ZR≃R3.
As the anticanonical class -KB̲ is rationally equivalent to 4V(z̲0), we see the following.  (i)
B̲ is a (singular) Fano variety.(ii)
B is not Fano, but the anticanonical class is on the boundary of the Kähler cone. In other words, -KB is nef but not ample.On the smooth blow-upB̂, the Kähler cone (2.27)K(B̂)⊂A2(B̂)⊗ZR≃R15
is rather complicated, and we will refrain from listing it explicitly. It is spanned by the origin and 169 rays and has 20 facets (that is, 14-dimensional faces). The anticanonical divisor -KB̂ as well as D̂ sits on the boundary of the Kähler cone, that is, is nef but not ample. However, each satisfies a different subset of 16 out of the 20 facet equations, so they lie on different faces of the Kähler cone.


## 2.7. Pull-Back Divisors

By the usual dictionary of toric geometry, the toric divisor4V(z̲0)~D corresponds to a continuous piecewise linear function on Nℝ≃ℝ3. Explicitly, the function is (2.28)f(nx,ny,nz)={nzifn⃗∈〈0,1,2〉,-4nx-2nzifn⃗∈〈0,1,3〉,-4ny-nzifn⃗∈〈0,2,3〉,0ifn⃗∈〈1,2,3〉.
The pull-back of this toric divisor by the toric morphisms π̲ and π̲∘π̂ is simply given by the pull-back of the piecewise linear function. Therefore, (2.29)D̲∼4V(z̲0),D∼4V(z0)+2V(z4),D̂∼4V(ẑ0)+2V(ẑ4)+4V(ẑ6)+3V(ẑ7)+3V(ẑ8)+2V(ẑ9)+2V(ẑ10)+2V(ẑ11)+V(ẑ12)+2V(ẑ13)+V(ẑ14).
What is the exceptional set of the first blow-up π̲? Recall that it corresponds to the subdivisions along the 2-cones (2.30)〈0,4〉∪〈2,4〉⟶〈0,2〉,〈1,5〉∪〈3,5〉⟶〈1,3〉,
see Figure 1. Therefore, π̲ is the blow-up along two disjoint rational curves of ℤ2-singularities in B̲. A standard intersection computation in the Chow group [26] yields that each curve intersects the divisor D̲ in two points. Therefore, the proper transform of D⊂B is D̲ blown up in four points. The final blow-up π̂:B̂→B does not further subdivide the 2-skeleton ΣB(2) and, therefore, corresponds to the blow-up of points in B. Any sufficiently generic divisor D misses these blow-up points, and, therefore, the surfaces D and D̂ are isomorphic. To summarize,  (i)
D̲ is a singular Enriques surface with four ℤ2-orbifold points;(ii)
D̂ and D are the same smooth Enriques surface after blowing up the orbifold points;(iii)
Since the blow-up at a point does not change the fundamental group, we find that(2.31)π1(D̲)=π1(D)=π1(D̂)=Z2.It is important to remember that the actual divisor is a fixed subvariety defined as the zero locus of an equation. To relate this defining equation before and after the blow-up, it is instructional to write the first blow-up mapπ̲:B→B̲ explicitly in terms of its action on homogeneous coordinates. One finds (2.32)π̲([z0:z1:z2:z3:z4:z5])=[z0z4:z1z5:z2z4:z3z5].
Note that this map is well defined on the equivalence classes (2.17) thanks to the identifications (2.16). One sees that, for example, the section z̲04 corresponds to the section z04z42 under the π̲* pull-back. I leave the analogous expression for π̂ as an exercise to the reader.To summarize, the equation for the divisorD̲ determines the equation satisfied by the proper transforms D and D̂ on the blow-ups. They are (2.33)D̲={z̲04+z̲14+z̲24+z̲34=0},D={z04z42+z14z52+z24z42+z34z52=0},D̂={ẑ04ẑ42ẑ64ẑ73ẑ83ẑ92ẑ102ẑ112ẑ12ẑ132ẑ14+ẑ14ẑ52ẑ62ẑ72ẑ93ẑ104ẑ11ẑ122ẑ153ẑ16ẑ172+ẑ24ẑ42ẑ7ẑ8ẑ102ẑ123ẑ132ẑ143ẑ152ẑ162ẑ174+ẑ34ẑ52ẑ62ẑ82ẑ9ẑ113ẑ134ẑ142ẑ15ẑ163ẑ172=0}.


## 3. Elliptic Fibration

So far, I have constructed(i)
a three-dimensional (singular) Fano varietyB̲,(ii)
a quasismooth divisorD̲ in B̲ with π1(D̲)=ℤ2,(iii)
a smooth three-dimensional toric varietyB̂, corresponding to a maximal subdivision of a reflexive polytope,(iv)
a smooth divisorD̂ in B̂ with π1(D̂)=ℤ2. This divisor is a smooth Enriques surface.I will now proceed and construct four-dimensional elliptically fibered Calabi-Yau varietiesY̲, Ŷ over B̲ and B̂ whose discriminant contains D̲ and D̂, respectively.


### 3.1. Weierstrass Models

Ideally, one would like to classify all elliptic fibrations over the base manifold. Unfortunately it is not known how to do so in this generality. It is known, however, that there exists a Weierstrass model (not necessarily over the same base) which is a (in general) different elliptic fibration [28, 29], at least assuming that the base is smooth and the discriminant is a normal crossing divisor. The Weierstrass model and the original elliptic fibration are birational to each other, but apart from that their relationship is arduous at best.Having said this, let us define the elliptically fibered varietyY in the most unimaginative way possible as a (global) Calabi-Yau Weierstrass model (3.1)Y={y2z=x3+f(ζ⃗)xz2+g(ζ⃗)z3}⊂P(O⊕O(-2KZ)⊕O(-3KZ)),
on a base variety Z with coordinates ζ. The remaining coordinates, z, x, and y, are sections (3.2)z∈ΓO,x∈ΓO(-2KZ),y∈ΓO(-3KZ).
The defining data of the Weierstrass model is the choice of coefficients in the Weierstrass equation, that is, the choice of sections (3.3)f∈ΓO(-4KZ),g∈ΓO(-6KZ).To engineer gauge theories on 7-branes wrapped on a divisor{ζ=0}⊂Z, one needs suitable singularities. In addition, the singularity must be of the correct split or nonsplit type as in Tate’s algorithm [30]. For this purpose it is convenient to parametrize the Weierstrass (technically, the singularity appears after blowing down all fiber components of the Weierstrass model not intersecting the zero section, but we will not dwell on this) model by polynomials (that is, sections of suitable line bundles) a1, a2, a3, a4, a6 as (3.4)f=-148a14-16a12a2+12a1a3-13a22+a4g=1864a16+172a14a2-124a13a3+118a12a22-112a12a4-16a1a2a3+227a23-13a2a4+14a32+a6.
The degree of vanishing of the aℓ(ζ) then determines (except for a few special cases that will be of no relevance for us) the low-energy effective gauge theory, see [31, 32]. For everything to be globally defined, the aℓ need to be sections of (3.5)al∈ΓO(-lKZ).


### 3.2. Weierstrass Model on the Singular Base

To engineer aSU(5) gauge theory coming from a 7-brane wrapped on the divisor D̲, one needs a split A4 singularity [31]. This translates into aℓ vanishing to degree ℓ-1 on D̲. In other words, aℓ must be divisible by d̲ℓ-1=0, where d̲ is the defining equation for the divisor D̲ as given in (2.33). Put yet differently, (3.6)ald̲l-1∈ΓO(-lKB̲-(l-1)D̲).
The number of sections is tabulated in Table 2. Note how the rows repeat with periodicity 2. This again follows from the fact that KB̲ and D̲ differ by 2-torsion in the divisor class group, see (2.24). Hence, there are plenty sections available for a1,…,a6, and one can easily find an elliptic fibration with a split A4 over D̲.Table 2
Number of sections of𝒪(-κKB̲-δD̲).
dimΓ𝒪(-κKB̲-δD̲)κ=1κ=2κ=3κ=4κ=5κ=6δ=0943115245445735δ=101042116244446δ=201943115245δ=30001042116δ=40001943δ=50000010δ=6000001δ≥7000000Table 3
Number of sections of𝒪(-κKB̂-δD̂), see also Figure 2.
dimΓ𝒪(-κKB̂-δD̂)κ=1κ=2κ=3κ=4κ=5κ=6δ=093591189341559δ=1021860140270δ=200032785δ=3000004δ≥4000000


### 3.3. Weierstrass Model on the Smooth Base

Let me now turn to the smooth threefoldB̂ and construct a suitable singularity over the smooth divisor D̂. The main difference is that now, after resolving the singularity, the anticanonical divisor is “smaller” than D̂, by which I mean that there are strictly less sections available for the Weierstrass model. See Figure 3 for details. Note that, if one always imposes the maximal degree of vanishing such that there are still nonzero sections, one can at most implement a split A2 singularity leading to a low-energy SU(3) gauge theory.Figure 3
The generating cones of the fanΣB̂ (yellow, outer circle), ΣB (blue), and ΣB̲ (red, inner circle). By 〈i,j,k〉 we denote the cone spanned by the rays number i, j, and k in 1, and arrows mean “contained in.”Having being dealt this lemon, let me try to make some lemonade. As in the previous subsection, I will writeD̂={d̂=0} for the defining equation, see (2.33). The split A2 singularity corresponds to a factorized form (3.7)a1=α1α1∈ΓO(-KB̂),a2=d̂α2α2∈ΓO(-2KB̂-D̂),a3=d̂α3α3∈ΓO(-3KB̂-D̂),a4=d̂2α4α4∈ΓO(-4KB̂-2D̂),a6=d̂3α6α6∈ΓO(-6KB̂-3D̂).
A basis for all sections can, of course, be written as in terms of homogeneous monomials in the 18 homogeneous coordinates ẑ0,…,ẑ17. To save a tree I will now switch to inhomogeneous coordinates (ξ0,ξ1,ξ2)∈ℂ3 for the coordinate patch, say, corresponding to the cone 〈0,4,7〉. This amounts to replacing the homogeneous coordinates with (3.8)[ẑ0:ẑ1:⋯:ẑ17]=[ξ0:1:1:1:ξ1:1:1:ξ2:1:⋯:1].
In this patch, (3.9)d̂=ξ04ξ12ξ23+ξ12ξ2+ξ22+1,
and the sections of the relevant line bundles are (3.10)ΓO(-KB̂)=〈1,ξ1,ξ2,ξ0ξ12ξ2,ξ1ξ2,ξ02ξ1ξ2,ξ02ξ1ξ22,ξ03ξ12ξ22,ξ0ξ1ξ2〉,ΓO(-2KB̂-D̂)=〈1,ξ02ξ12ξ2〉,ΓO(-3KB̂-D̂)=〈1,ξ1,ξ2,ξ1ξ2,ξ02ξ13ξ2,ξ03ξ14ξ22,ξ02ξ13ξ22,ξ02ξ1ξ2,ξ02ξ1ξ22,ξ04ξ13ξ22,ξ04ξ13ξ23,ξ05ξ14ξ23,ξ0ξ12ξ2,ξ02ξ12ξ22,ξ03ξ13ξ22,ξ0ξ1ξ2,ξ02ξ12ξ2,ξ03ξ12ξ22〉,ΓO(-4KB̂-2D̂)=〈1,ξ04ξ14ξ22,ξ02ξ12ξ2〉,ΓO(-6KB̂-3D̂)=〈1,ξ06ξ16ξ23,ξ02ξ12ξ2,ξ04ξ14ξ22〉.
For simplicity I will choose αℓ to be the sum of the monomials corresponding to the vertices of the Newton polyhedron, that is, (3.11)α1=ξ03ξ12ξ22+ξ02ξ1ξ22+ξ02ξ1ξ2+ξ0ξ12ξ2+ξ1ξ2+ξ1+ξ2+1,α2=ξ02ξ12ξ2+1,α3=ξ05ξ14ξ23+ξ04ξ13ξ23+ξ04ξ13ξ22+ξ03ξ14ξ22+ξ02ξ13ξ22+ξ02ξ13ξ2+ξ02ξ1ξ22+ξ02ξ1ξ2+ξ1ξ2+ξ1+ξ2+1,α4=ξ04ξ14ξ22+1,α6=ξ06ξ16ξ23+1.
For all purposes in the following, this choice is generic. By construction, the discriminant then factorizes as (3.12)Δ=4f3+27g2=d̂3r̂,
where the remainder (in fact, r̂ is a polynomial consisting of 1083 monomials in ξ0, ξ1, and ξ2) defines a new divisor R̂=def{r̂=0}~-12KB̂-3D̂. In particular, the homology class splits as (3.13)[Δ]=3[D̂]+[R̂].
Using the explicit equations, one can check [18] that  (i)
R̂ is an irreducible divisor;(ii)
Neitherf nor g vanishes at a generic point of R̂. Hence it supports I0 Kodaira fibers in the Weierstrass model;(iii)
D̂ is smooth;(iv)
R̂ is not smooth, for example, (ξ0,ξ1,ξ2)=(1,1,-1) is a singular point;(v)
The curveD̂∩R̂ is not a complete intersection.Let me further investigate the intersection curveD̂∩R̂. One component (in the 〈0,4,7〉 patch) is given by the surprisingly simple expression (3.14)c:C⟶D̂∩R̂,t⟼(t,0,i).
Therefore, (ξ1,ξ2) are good normal coordinates. Taylor expanding along the normal directions for a generic point c(t), we see that D̂ and R̂ share the same tangent plane but do not osculate to any higher degree. Therefore, the degree of vanishing of the discriminant jumps from 5 to 7 along the intersection locus D̂∩R̂, corresponding to worsening of the A2 singularity to an A4 singularity.


## 3.1. Weierstrass Models

Ideally, one would like to classify all elliptic fibrations over the base manifold. Unfortunately it is not known how to do so in this generality. It is known, however, that there exists a Weierstrass model (not necessarily over the same base) which is a (in general) different elliptic fibration [28, 29], at least assuming that the base is smooth and the discriminant is a normal crossing divisor. The Weierstrass model and the original elliptic fibration are birational to each other, but apart from that their relationship is arduous at best.Having said this, let us define the elliptically fibered varietyY in the most unimaginative way possible as a (global) Calabi-Yau Weierstrass model (3.1)Y={y2z=x3+f(ζ⃗)xz2+g(ζ⃗)z3}⊂P(O⊕O(-2KZ)⊕O(-3KZ)),
on a base variety Z with coordinates ζ. The remaining coordinates, z, x, and y, are sections (3.2)z∈ΓO,x∈ΓO(-2KZ),y∈ΓO(-3KZ).
The defining data of the Weierstrass model is the choice of coefficients in the Weierstrass equation, that is, the choice of sections (3.3)f∈ΓO(-4KZ),g∈ΓO(-6KZ).To engineer gauge theories on 7-branes wrapped on a divisor{ζ=0}⊂Z, one needs suitable singularities. In addition, the singularity must be of the correct split or nonsplit type as in Tate’s algorithm [30]. For this purpose it is convenient to parametrize the Weierstrass (technically, the singularity appears after blowing down all fiber components of the Weierstrass model not intersecting the zero section, but we will not dwell on this) model by polynomials (that is, sections of suitable line bundles) a1, a2, a3, a4, a6 as (3.4)f=-148a14-16a12a2+12a1a3-13a22+a4g=1864a16+172a14a2-124a13a3+118a12a22-112a12a4-16a1a2a3+227a23-13a2a4+14a32+a6.
The degree of vanishing of the aℓ(ζ) then determines (except for a few special cases that will be of no relevance for us) the low-energy effective gauge theory, see [31, 32]. For everything to be globally defined, the aℓ need to be sections of (3.5)al∈ΓO(-lKZ).


## 3.2. Weierstrass Model on the Singular Base

To engineer aSU(5) gauge theory coming from a 7-brane wrapped on the divisor D̲, one needs a split A4 singularity [31]. This translates into aℓ vanishing to degree ℓ-1 on D̲. In other words, aℓ must be divisible by d̲ℓ-1=0, where d̲ is the defining equation for the divisor D̲ as given in (2.33). Put yet differently, (3.6)ald̲l-1∈ΓO(-lKB̲-(l-1)D̲).
The number of sections is tabulated in Table 2. Note how the rows repeat with periodicity 2. This again follows from the fact that KB̲ and D̲ differ by 2-torsion in the divisor class group, see (2.24). Hence, there are plenty sections available for a1,…,a6, and one can easily find an elliptic fibration with a split A4 over D̲.Table 2
Number of sections of𝒪(-κKB̲-δD̲).
dimΓ𝒪(-κKB̲-δD̲)κ=1κ=2κ=3κ=4κ=5κ=6δ=0943115245445735δ=101042116244446δ=201943115245δ=30001042116δ=40001943δ=50000010δ=6000001δ≥7000000Table 3
Number of sections of𝒪(-κKB̂-δD̂), see also Figure 2.
dimΓ𝒪(-κKB̂-δD̂)κ=1κ=2κ=3κ=4κ=5κ=6δ=093591189341559δ=1021860140270δ=200032785δ=3000004δ≥4000000


## 3.3. Weierstrass Model on the Smooth Base

Let me now turn to the smooth threefoldB̂ and construct a suitable singularity over the smooth divisor D̂. The main difference is that now, after resolving the singularity, the anticanonical divisor is “smaller” than D̂, by which I mean that there are strictly less sections available for the Weierstrass model. See Figure 3 for details. Note that, if one always imposes the maximal degree of vanishing such that there are still nonzero sections, one can at most implement a split A2 singularity leading to a low-energy SU(3) gauge theory.Figure 3
The generating cones of the fanΣB̂ (yellow, outer circle), ΣB (blue), and ΣB̲ (red, inner circle). By 〈i,j,k〉 we denote the cone spanned by the rays number i, j, and k in 1, and arrows mean “contained in.”Having being dealt this lemon, let me try to make some lemonade. As in the previous subsection, I will writeD̂={d̂=0} for the defining equation, see (2.33). The split A2 singularity corresponds to a factorized form (3.7)a1=α1α1∈ΓO(-KB̂),a2=d̂α2α2∈ΓO(-2KB̂-D̂),a3=d̂α3α3∈ΓO(-3KB̂-D̂),a4=d̂2α4α4∈ΓO(-4KB̂-2D̂),a6=d̂3α6α6∈ΓO(-6KB̂-3D̂).
A basis for all sections can, of course, be written as in terms of homogeneous monomials in the 18 homogeneous coordinates ẑ0,…,ẑ17. To save a tree I will now switch to inhomogeneous coordinates (ξ0,ξ1,ξ2)∈ℂ3 for the coordinate patch, say, corresponding to the cone 〈0,4,7〉. This amounts to replacing the homogeneous coordinates with (3.8)[ẑ0:ẑ1:⋯:ẑ17]=[ξ0:1:1:1:ξ1:1:1:ξ2:1:⋯:1].
In this patch, (3.9)d̂=ξ04ξ12ξ23+ξ12ξ2+ξ22+1,
and the sections of the relevant line bundles are (3.10)ΓO(-KB̂)=〈1,ξ1,ξ2,ξ0ξ12ξ2,ξ1ξ2,ξ02ξ1ξ2,ξ02ξ1ξ22,ξ03ξ12ξ22,ξ0ξ1ξ2〉,ΓO(-2KB̂-D̂)=〈1,ξ02ξ12ξ2〉,ΓO(-3KB̂-D̂)=〈1,ξ1,ξ2,ξ1ξ2,ξ02ξ13ξ2,ξ03ξ14ξ22,ξ02ξ13ξ22,ξ02ξ1ξ2,ξ02ξ1ξ22,ξ04ξ13ξ22,ξ04ξ13ξ23,ξ05ξ14ξ23,ξ0ξ12ξ2,ξ02ξ12ξ22,ξ03ξ13ξ22,ξ0ξ1ξ2,ξ02ξ12ξ2,ξ03ξ12ξ22〉,ΓO(-4KB̂-2D̂)=〈1,ξ04ξ14ξ22,ξ02ξ12ξ2〉,ΓO(-6KB̂-3D̂)=〈1,ξ06ξ16ξ23,ξ02ξ12ξ2,ξ04ξ14ξ22〉.
For simplicity I will choose αℓ to be the sum of the monomials corresponding to the vertices of the Newton polyhedron, that is, (3.11)α1=ξ03ξ12ξ22+ξ02ξ1ξ22+ξ02ξ1ξ2+ξ0ξ12ξ2+ξ1ξ2+ξ1+ξ2+1,α2=ξ02ξ12ξ2+1,α3=ξ05ξ14ξ23+ξ04ξ13ξ23+ξ04ξ13ξ22+ξ03ξ14ξ22+ξ02ξ13ξ22+ξ02ξ13ξ2+ξ02ξ1ξ22+ξ02ξ1ξ2+ξ1ξ2+ξ1+ξ2+1,α4=ξ04ξ14ξ22+1,α6=ξ06ξ16ξ23+1.
For all purposes in the following, this choice is generic. By construction, the discriminant then factorizes as (3.12)Δ=4f3+27g2=d̂3r̂,
where the remainder (in fact, r̂ is a polynomial consisting of 1083 monomials in ξ0, ξ1, and ξ2) defines a new divisor R̂=def{r̂=0}~-12KB̂-3D̂. In particular, the homology class splits as (3.13)[Δ]=3[D̂]+[R̂].
Using the explicit equations, one can check [18] that  (i)
R̂ is an irreducible divisor;(ii)
Neitherf nor g vanishes at a generic point of R̂. Hence it supports I0 Kodaira fibers in the Weierstrass model;(iii)
D̂ is smooth;(iv)
R̂ is not smooth, for example, (ξ0,ξ1,ξ2)=(1,1,-1) is a singular point;(v)
The curveD̂∩R̂ is not a complete intersection.Let me further investigate the intersection curveD̂∩R̂. One component (in the 〈0,4,7〉 patch) is given by the surprisingly simple expression (3.14)c:C⟶D̂∩R̂,t⟼(t,0,i).
Therefore, (ξ1,ξ2) are good normal coordinates. Taylor expanding along the normal directions for a generic point c(t), we see that D̂ and R̂ share the same tangent plane but do not osculate to any higher degree. Therefore, the degree of vanishing of the discriminant jumps from 5 to 7 along the intersection locus D̂∩R̂, corresponding to worsening of the A2 singularity to an A4 singularity.


## 4. Conclusions

In this paper I have constructed F-theory models with, a priori,SU(5) gauge theory on a singular Fano threefold and a SU(3) gauge theory on the blown-up smooth threefold. In both cases the non-Abelian gauge theory comes from a 7-brane wrapped on an Enriques surface, which has fundamental group ℤ2. Therefore, in both cases one can switch on a discrete Wilson line and break the gauge group below the compactification scale in the usual manner.The fact that the only partially resolved baseB allows for a higher rank gauge group on the 7-brane than its smooth blow-up is curious: one might be tempted to interpret the Kähler deformation as the usual Higgs mechanism; however the singular points are disjoint from the 7-brane. In any case, there must be further physical degrees of freedom associated to the singularities in the base, and it would be nice to have a more concise F-theory dictionary for them.



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*Source: 404691-2011-09-19.xml*