Discrete Wilson Lines in F-Theory

F-theory models are constructed where the 7-brane has a non-trivial 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 it is not simply connected (namely, an Enriques surface) and supports a non-Abelian gauge theory.

The generating cones of the fans ΣB, Σ B , and Σ B . . . . . . . . . . . . . 8 4 The fans of P 1 × P 1 /Z 2 and its blow-up. . . . . . . . . . . . . . . . . 18 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 [5,6,7,8,9,10] 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 non-trivial identifications. Turning on fields locally then affects the running of the coupling constants and, potentially, defocus 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 non-trivial identification of the gauge bundle while keeping it locally trivial, breaking the GUT gauge group by the usual Hosotani mechanism. For what its 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 with Z 2 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 [16,17,18].

Foreword
An Enriques surface is a free quotient of a K3 surface by a freely-acting holomorphic involution and is probably the best-known example of a complex surface S with fundamental group π 1 (S) = Z 2 . Its first Chern class c 1 (S) is the torsion element in H 2 (Z, Z) Z 10 ⊕ Z 2 , so it admits a Ricci-flat metric but has no covariantly constant spinor. 1 Some, but not all, K3 surfaces can be realized [19] as quartics in P 3 . 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 P 3 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 following Z 4 action on P 3 , The fixed point set of g 2 are the two disjoint rational curves and the fixed points of g are the north and south poles on these (4 points altogether). A sufficiently generic Z 4 invariant quartic q(x 0 , x 1 , x 2 , x 3 ) is then a (singular) Enriques surface on the quotient. 2 The fastest way to see this is to note that the would-be (2, 0)-form 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 of Z 2 singularities, for which there is a unique crepant resolution. Then one has to deal with the remaining 4 Z 4 singularities. By a happy coincidence, the above Z 4 quotient of P 3 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.

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 lattice N Z d , 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 Subsection 2.4. For now, let us just consider P 3 as an example. Its fan consists of the cones where e 1 , e 2 , e 3 are a basis for N Z 3 . There are 4 one-dimensional rays satisfying a unique linear relation, which translates into 4 homogeneous coordinates with the usual identification 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, 3 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 4 is a formal linear combination D = a i V (x i ) of the codimensionone subvarieties 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: • Every coefficient a i can be thought of as the value of a function f : N → Z on the generating lattice point ρ i 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.
• The divisor also defines a polytope where M = N ∨ is the dual lattice. The global sections ΓO(D) are in one-toone correspondence with the integral lattice points M ∩ P D and can easily be counted for any given divisor.
A particularly relevant divisor is the anti-canonical divisor −K = V (x i ). Given a polytope ∇ ∈ N , we can construct its face fan and the polytope ∆ = P −K ⊂ M . If ∆ is again a lattice polytope, then ∇ is called reflexive.
Finally, note that H 2 (P 3 ) = Z. Hence, the line bundles on P 3 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 on P 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 varietyB as the base manifold, containing a non-simply connected divisorD. However, directly analyzingB will be overly complicated. In particular,B contains exceptional divisors that do not intersect the divisorD we are interested in. Therefore, to better understandD ⊂B, I will blow-down these additional exceptional divisors. This will produce a singular variety B containing the same divisor D =D. Finally, I will blow-down two more curves in B to obtain an (even more singular) three-dimensional variety B. The blown-down divisor D ⊂ B is the most suitable one to compute the fundamental groups. To summarize, I am going to define successive blow-upŝ of three-dimensional toric varieties. Both of the mapsπ, π 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 fans Σ corresponding to the toric varieties. Let me start by the rays Σ (1) . The most singular variety has Σ (1) 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 B, that is, the (in this case unique) maximal triangulation of the convex hull conv(Σ (1) B ). Hence, one must add the additional integral points to the ray generators, Neither the variety B nor its maximal crepant partial resolution B are smooth, related to the fact that the polytope conv(Σ is not reflexive. One again needs to add rays to resolve all singularities, however this time the generators are necessarily outside of conv(Σ (1) B ). One particular choice I am going to make are the rays generated by the 18 points listed in Table 1.
Finally, to completely specify the toric varieties B, B, andB, let me define the generating cones of the respective fans: • Σ B is the face fan of the polytope conv(Σ As there are many different maximal subdivisions of the face fan, this alone does not uniquely specify the fan ΣB. For concreteness, I will fix the one listed in Figure 3. Note that not all combinatorial symmetries of the graph in Figure 3 are actually symmetries of the fan.

Homogeneous Coordinates
For future reference, let me list the toric Chow groups [26]: Since all three toric varieties have at most orbifold singularities, the Hodge numbers are h p,p = rank A p and h p,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 5 d-dimensional toric variety X can be written as a geometric quotient where r is the number of rays in the fan Σ X . 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 SR(ΣB) = · · · 105 quadric monomials · · · .
5 That is, with at most orbifold singularities.

A Non-Simply Connnected Divisor
I am now going to define a divisor D ⊂ B in the same linear system as the toric divisor 67 Note that the fan Σ B is precisely the normal fan of the Newton polytope P D . In this sense, B is the "natural" ambient toric variety for the surface D.
For explicitness, let me fix once and for all a linear combination of the monomials as the defining equation of the divisor D. I will select the vertices of the Newton polytope and define This surface is known to be an Enriques surface since it projects out the potential (2, 0)-form as mentioned in Subsection 2.1. In fact, this example has been known for some time, see Remark 3.6 in [27].

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 eq. (21) fails to define a K3 surface. In other words, how does the divisor D = 4V (z 0 ) differ from the anticanonical divisor of B? Comparing with P 3 , see eq. (8), one might have thought that they were linearly equivalent.
Similarly to the quartic K3 ⊂ P 3 , one can also define a Calabi-Yau variety in B as the zero locus of a section of the anticanonical bundle. 8 The available sections are Note that this differs from the sections of D, see eq. (23). Therefore, the two divisor are not linearly equivalent. Nevertheless, −K B and D are very close to being linearly equivalent. In fact, its easy to see that they are in the same rational divisor class 9 since the rational divisor class group is one-dimensional, dim A 2 (B) ⊗ Z R = 1. However, their difference is a 2-torsion element in the (integral) divisor class The same is true on the crepant partial resolution, where K B + D is again a 2-torsion element in A 2 (B). On the final smooth resolutionB the divisor class group A 2 (B) = Z 15 is torsion free. However, the last blow-upπ :B → B is not crepant, sô Therefore, the divisors −KB andD =π * (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 As the anticanonical class −K B is rationally equivalent to 4V (z 0 ), we see that • B is a (singular) Fano variety.
• B is not Fano, but the anticanonical class is on the boundary of the Kähler cone. In other words, −K B is nef but not ample.
On the smooth blow-upB, the Kähler cone is rather complicated and we will refrain from listing it explicitly. It is spanned by the origin and 169 rays and has 20 facets. 10 The anticanonical divisor −KB as well asD sit on the boundary of the Kähler cone, that is, are 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.

Pull-Back Divisors
By the usual dictionary of toric geometry, the toric divisor 4V (z 0 ) ∼ D corresponds to a continuous piecewise linear function on N R R 3 . Explicitly, the function is The pull-back of this toric divisor by the toric morphisms π and π •π is simply given by the pull-back of the piecewise linear function. Therefore, What is the exceptional set of the first blow-up π? Recall that it corresponds to the subdivisions along the 2-cones see Figure 1. Therefore, π is the blow-up along two disjoint rational curves of Z 2singularities in B. A standard intersection computation in the Chow group [26] yields that each curve intersects the divisor D in two points. Therefore, the proper transform of D ⊂ B is D blown up in four points. The final blow-upπ :B → B does not further subdivide the 2-skeleton Σ (2) B 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 andD are isomorphic. To summarize, • D is a singular Enriques surface with four Z 2 -orbifold points.
•D and D are the same smooth Enriques surface after blowing up the orbifold points.
• Since the blow-up at a point does change the fundamental group, we find that 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 map π : B → B explicitly in terms of its action on homogeneous coordinates. One finds π z 0 : z 1 : z 2 : z 3 : z 4 : Note that this map is well-defined on the equivalence classes eq. (17) thanks to the identifications eq. (16). One sees that, for example, the section z 4 0 corresponds to the section z 4 0 z 2 4 under the π * pull-back. I leave the analogous expression forπ as an exercise to the reader.
To summarize, the equation for the divisor D determines the equation satisfied by the proper transforms D andD on the blow-ups. They are

Elliptic Fibration
So far, I have constructed • a three-dimensional (singular) Fano variety B, • a quasi-smooth divisor D in B with π 1 (D) = Z 2 , • a smooth three-dimensional toric varietyB, corresponding to a maximal subdivision of a reflexive polytope, and • a smooth divisorD inB with π 1 (D) = Z 2 . This divisor is a smooth Enriques surface.
I will now proceed and construct four-dimensional elliptically fibered Calabi-Yau varieties Y ,Ŷ over B andB whose discriminant contains D andD, respectively.

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 variety Y in the most unimaginative way possible as a (global) Calabi-Yau Weierstrass model on a base variety Z with coordinates ζ. The remaining coordinates, z, x, and y are sections The defining data of the Weierstrass model is the choice of coefficients in the Weierstrass equation, that is, the choice of sections 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 11 model by polynomials (that is, sections of suitable line bundles) a 1 , a 2 , a 3 , a 4 , a 6 as The degree of vanishing of the a (ζ) then determines 12 the low-energy effective gauge theory, see [31,32]. For everything to be globally defined, the a need to be sections of a ∈ ΓO − KB .

Weierstrass Model on the Singular Base
To engineer a SU (5) gauge theory coming from a 7-brane wrapped on the divisor D, one needs a split A 4 singularity [31]. This translates into a vanishing to degree − 1 on D. In other words, a must be divisible by d −1 = 0 where d is the defining equation for the divisor D as given in eq. (33). Put yet differently, The number of sections is tabulated in Table 2; Note how the rows repeat with periodicity 2. This again follows from the fact that K B and D differ by 2-torsion in the divisor class group, see eq. (24). Hence, there are plenty sections available for a 1 , . . . , a 6 and one can easily find an elliptic fibration with a split A 4 over D.

Weierstrass Model on the Smooth Base
Let me now turn to the smooth threefoldB and construct a suitable singularity over the smooth divisorD. The main difference is that now, after resolving the singularity, the anticanonical divisor is "smaller" thanD, by which I mean that there are strictly less sections available for the Weierstrass model. See Table 3 for details. Note that, if one always imposes the maximal degree of vanishing such that there are still non-zero sections, one can at most implement a split A 2 singularity leading to a low-energy SU (3) gauge theory.
• Neither f nor g vanish at a generic point ofR. Hence it supports I 0 Kodaira fibers in the Weierstrass model.
•D is smooth.
• The curveD ∩R is not a complete intersection.
Let me further investigate the intersection curveD ∩R. One component (in the 0, 4, 7 patch) is given by the surprisingly simple expression c : C →D ∩R, t → (t, 0, i).
(47) Therefore, (ξ 1 , ξ 2 ) are good normal coordinates. Taylor expanding along the normal directions for a generic point c(t), we see thatD andR 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 locusD ∩R, corresponding to worsening of the A 2 singularity to an A 4 singularity.

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 Z 2 . 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 base B allows for a higher rank gauge group on the 7-brane than its smooth blow-up is curious: One might be tempted to 13 In fact,r is a polynomial consisting of 1083 monomials in ξ 0 , ξ 1 , and ξ 2 . 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.

A Fibrations of the Base
The base manifolds B, B, andB are fibered in an interesting manner which I will describe in this appendix. The map to the 2-dimensional base is given by the N -lattice  However, consistently mapping the rays of the fans is not enough to define a toric morphism. Checking all higher-dimensional cones with respect to the lattice homomorphism φ, one finds that • The variety B is not fibered.
• The variety B is a P 1 -fibrations over S.
• The smooth threefoldB is a P 1 -fibration over S, but not over the crepant reso-lutionŜ.
It is, perhaps, vexing that the resolved threefoldB is not a fibration over the resolved baseŜ. However, a closer investigation reveals that one can flop 4 offending curves, corresponding to the 4 bistellar flips of the fan ΣB. The flopped threefold is then a P 1 fibration over the resolved basê S. Of course the flopped threefold is then only birational to B, B and no longer a direct blow-up. However, it supports essentially the same elliptic fibration asB as constructed in Section 3.