fiber polygon corresponding to a toric sections and denote the lattice points corresponding to thei-th fiber component by wi, then sintersects thei-th fiber component if and only ifv andwi share an edge. Using this prescription, we have determined the intersection numbers for allSU(5) tops infigure 4.3by listing the sections intersecting a certain exceptional divisor next to the corresponding lattice point of thez= 1 facet of the top.
Figure 4.5: Two different three-dimensional visualizations of the entire topτ5,3. Fiber ver-tices corresponding to sections and lattice points associated to exceptional di-visors intersecting them are colored red, as are the edges connecting them.
To give an example, consider the top τ5,3. Fromtable 3.9, we see that the toric sections generate a subgroup MWT ∼=Z⊕Zof the entire Mordell-Weil group. Now, picks0as the zero section and assign simple rootsαi in clockwise order to the boundary points of τ5,3. Taking σ0 =s1−s0 and σ1 =s2−s0 as generators ofU(1)1 and U(1)2, we find that the charges of the5 representations must satisfy
QU(1)1(5)≡1 mod 5 and QU(1)2(5)≡3 mod 5. (4.4.4) Infigure 4.5we present a three-dimensional visualization of the intersection structure forτ5,3. Finally, let us point out that the above notion of splits agrees with the cases that have been analyzed with the split spectral cover constructions [60, 78–80] only in the case of a singleU(1). As soon as there are multiple Abelian gauge symmetries, our notation describes the “split” between the section generating the particularU(1) symmetry and the zero section, whereas the split spectral cover constructions denote by split the factorization pattern of the
102 CHAPTER 4. NON-ABELIAN SINGULARITIES FROM TOPS spectral cover. Hence, when there are multiple U(1)s we determine a split with respect to each one of them.
4.5 A No-Go-Theorem for Antisymmetric Representations
We now turn to the 10 matter fields of the SU(5) gauge theory. Somewhat surprisingly, their geometric origin is different from the5matter fields. The5 matter fields come from an individualP1 in theI5 Kodaira fiber degenerating into two irreducible components, but this kind of degeneration will never yield a codimension-twoI1∗ Kodaira fiber where the10matter field is localized: Splitting nodes of theI5 Kodaira fiber will never eliminate the fundamental group π1(I5) = Z of the Kodaira fiber, but π1(I1∗) = 0. The only way to obtain a simply connected fiber is to have the hypersurface equation vanish identically on a toric curve of the top. That is, along the intersection of the irreducible components of the toric surfaces in the fiber of the ambient toric variety. Note that the irreducible components of the two-dimensional ambient space fiber correspond to the vertices of the top that are not interior to a facet and not part of the fiber polygon. They intersect in a toric curve'P1 whenever the triangulation induced by the fan joins two vertices.
As we will see in chapter 5, if anSU(5)-top contains a point interior to a facet then the fibration is not flat, i.e. there are base loci over which the fiber dimension jumps. Non-flat fibrations leads to low-energy theories that are not ordinary gauge theories and therefore we only have to focus on tops without facet interior points. For an SU(5) top this means that the facet at height z = 1 is a degenerate lattice pentagon with one of the lattice points at a midpoint of an edge. Up to isomorphism, there is only a single such lattice pentagon, see figure 4.3. There are two fine triangulations T1 and T2 of this boundary facet and they are shown on the left hand side offigure 4.6. Regardless of the triangulation, the degenerate toric ambient space fiber consists of five irreducible surfacesV(e0),. . .,V(e4). These always inter-sect cyclically in toric curves, that is,V(ei)∩V(ei+1)'P1. Depending on the triangulation, they additionally intersect as the internal one-simplices in the triangulation, that is,
• Triangulation T1: V(e0)∩V(e3)'P1 and V(e0)∩V(e2)'P1,
• Triangulation T2: V(e0)∩V(e3)'P1 and V(e1)∩V(e3)'P1.
The Calabi-Yau hypersurface generically intersects the toricP1s corresponding to the bound-ary one-simplices in a point, and is a non-zero constant on the toricP1 corresponding to the internal one-simplices. As argued in the beginning of this section, the 10matter is localized when the whole toric P1 is contained in the hypersurface, that is, where the above constant happens to be zero.5 Since the internal one-simplices are internal to the same facet of the top, the hypersurface always vanishes simultaneously on both toric curves. These two toric
5This is at a codimension-one curve of the discriminant, that is, it is of codimension two in the base.
4.5. A NO-GO-THEOREM FOR ANTISYMMETRIC REPRESENTATIONS 103 P1s intersect in a toric point, the containing two-simplex. Hence they form two nodes joined by an edge in the dual fiber diagram, which will turn out to be the middle two nodes of the D˜5 extended Dynkin diagram.
Intersecting the hypersurfaceY with the ambient space irreducible surface components of a fiber yields additional curve components for the degenerate elliptic fiber. These necessarily contain the toric curves of the adjacent internal one-simplices as irreducible components.
For example, in triangulation T1 the intersection Y ∩ V(e0) contains both toric surfaces V(e0)∩V(e3) and V(e0)∩V(e2) as irreducible components. Likewise, Y ∩V(e4) contains none of the toricP1 since the vertex e4 is not adjacent to an interior one-simplex. This fixes the degeneration of the I5 Kodaira fiber, that is the five curves Y ∩V(ei) away from the matter curve, to be the one shown on the right hand side offigure 4.6.
This is the key observation: The triangulation of the top fixes the degeneration of the codimension-one Kodaira fiber at the codimension-two 10 matter curves of a toric hyper-surface. Since the triangulation is fixed for a given manifold, the degeneration is the same for all 10. Importantly, this behavior is different from that of the 5 matter curves, where different degenerations occur over different codimension-two fibers. As a corollary, the U(1) charges of all10matter representations are equal. In other words, if one wants to construct F-theory GUTs such that the10fields carry differentU(1) charges then one needs to consider complete intersections such that the fiber is at least codimension-two in the ambient space fiber [89,158,159].
e0 e1
e2
e4 e3
e3
e1 e4
e2 e0
e0 e1
e2
e4 e3
e0 e1
e2 e4
e3 T1:
T2:
⇒
⇒
Figure 4.6: Left: The two possible fine triangulations of the lattice polygon at heightz= 1 in the SU(5)-top. Right: The corresponding degeneration of the I5 → I1∗ Kodaira fiber.
104 CHAPTER 4. NON-ABELIAN SINGULARITIES FROM TOPS 4.6 Tops for Complete Intersections
As a direct conclusion from the preceding section we are led to complete intersection elliptic curves. In fact, constructing F-theory models with multiple antisymmetric SU(5) represen-tations was part of the motivation to provide the framework ofsection 3.4and to classify the toric Mordell-Weil groups for three-dimensional Gorenstein Fano varieties.
In principle the toric machinery applies equally to Calabi-Yau manifolds with fibers em-bedded in higher-dimensional ambient spaces and one can extend the definition of a top to higher dimensions: A genus-one fibration with codimension-dfibers has (d+ 1)-dimensional tops, which can then be combined to form a reflexive polytope. Unfortunately, however, there does not yet exist an analogous classification to that of [161] and therefore the exhaustive list of SU(5) tops is not yet known for higher-dimensional tops. Nevertheless, it is possible to constructsome SU(5) tops simply by making an ansatz and confirming that it leads to an SU(5) singularity in the blow-down limit, as we will show in subsection 5.3.2.
Chapter 5
Fibered Calabi-Yau Manifolds
With the building blocks studied in the previous chapter at hand, the last remaining step and the goal of this chapter is to combine them with a base manifold into a full-fledged Calabi-Yau manifold. Once that is achieved, one can then attempt to answer so far unresolved questions that depend not only on details of the fiber geometry, but also on properties of the full fibration.
As elaborated on in the introduction to Part II, the ultimate goal in studying string compactifications is not only to construct a single manifold satisfying a set of criteria, but rather to identify all such spaces. Achieving the latter objective remains far out of reach, but at least some progress can be noted: Given a top and a toric base, we explain insection 5.1 how to obtain all varieties corresponding to the reflexive polytopes made up of these building blocks. In possession of an algorithm to construct explicit fibrations, we proceed with the study of global properties of the compactification. As we will show, Calabi-Yau manifolds constructed inside toric ambient spaces may have fibrations that arenon-flat, i.e. their fiber dimension increases over certain base loci. Crucially, this happens generically already for Calabi-Yau fourfolds with a resolved SU(5)-singularity. Since flat fibrations appear to be essential for phenomenologically viable F-theory models,section 5.2is dedicated to studying the conditions under which fibrations are flat. Using different examples, we show that for cer-tain combinations of top and base one cannot construct flat fibrations. Finally, we construct a range of different example manifolds insection 5.3to illustrate as concretely as possible how to handle non-toric U(1)s and manipulate complete intersection fibers giving rise to SU(5) models with multiple antisymmetric representations and additional discrete symmetries.
5.1 The Auxiliary Polytope of All Fibrations
By definition, the top describes the degeneration of the ambient space fibration and thus that of the genus-one fibration over a toric divisor in the base. This base divisor is defined by one of the rays in the base fan. The obvious question is how this data can be completed into
105
106 CHAPTER 5. FIBERED CALABI-YAU MANIFOLDS that of a compact Calabi-Yau manifold, that is, how to combine the top and the choice of base fan to a reflexive polytope. In fact, this has a nice answer: The remaining choices for a lattice polytope after fixing the tops and the base again are parametrized by the integral points of a further auxiliary polytope.1 This just follows from convexity, and one needs to verify reflexivity and flatness of the fibration by hand.
In particular, we will be interested in the case of a single top together with trivial tops over the remaining rays of the base fan. For the purposes of this section, we will only consider the case where the base fan equalsPn, whose rays are generated by the unit vectors e1,. . ., en together with −P
ei. Then
• The fixed top can be chosen to project to [0, e1].
• The single point generating the trivial top over each ofe2,. . .,encan be chosen to have fiber coordinates (0,0) by aGL(n,Z) rotation fixing all previous tops.
• The final point, generating the trivial top over −P
ei, has coordinates (p1, p2) ∈ Z2 with no remaining freedom of coordinate redefinition.
This parametrizes the choices of completion to a polytope by a pair of integers (p1, p2). These are constrained by convexity: Having fixed the height-one points of the other tops, there is only a finite range of (p1, p2) such that the fiber (preimage of the origin in the base) of the convex hull does not exceed the chosen fiber polygon. These are linear constraints, turning the allowed region for (p1, p2) into a polygon (with not necessarily integral vertices). Note that the pk correspond to choosing the line bundles that the homogeneous coordinates (and therefore their coefficients) are sections of. In fact, one can derive the same linear constraints by demanding that all the line bundle that the complex structure coefficients are sections of do indeed admit a section [167].
It turns out that all lattice polytopes for a single SU(5) top overPnthat one constructs just by demanding convexity, as above, are automatically reflexive. Their total number for small values of n is listed in table 5.1. We included also the cases P4 and P5 that, when used as base of an F-theory compactification, would not lead to a gauge theory in four or six dimensions. However, the construction can be supplemented by additional polynomials specifying the actual base as hypersurface inP4 or complete intersection inP5. For example, the Fano threefold obtained by a quartic constraint inP4 is a viable choice for the base. Note that realizing the base itself as hypersurface or complete intersection can be also phenomeno-logically motivated. Such realizations allow for more exhaustive choices of fluxes on the GUT brane as demonstrated in the models of [168,169]. This applies in particular to hypercharge flux [57,64,170] that is non-trivial on the GUT brane but trivial on the entire base manifold.
Our construction thus extends straightforwardly to these more involved Calabi-Yau fourfold examples.
1This polytope is not necessarily integral, that is, its vertices are in general rational.