v0 v1 v2 v3 v4 v5 v6 v7
1 0 0
0 1 0
1
−1 0
−1 0 0
0 1 2
−1 0
−2
−1
−2
−2
2 1 2
Table 5.13: Vertices of the three-dimensional reflexive polytope with PALP id 3415.
v8 v9 v10 v11 v12 v13
0
−1
−1
1 0 1
−1
−1
−1
0 0 1
−1
−1
−2
0 0 0
v14 v15 v16 v17 v18
1 1 2
0 0
−1
1 1 1
−1 0
−1
0 1 1
Table 5.14: Integral points of the reflexive polytope with PALP id 3415 that are neither ver-tices nor the origin. In order to fully resolve every fibration of the nef partition (5.3.55) one must use all of these points as rays of the toric fan.
the universal covering group isSU(n), which, without Mordell-Weil torsion, constitutes the gauge group of the F-theory model. In the presence of a non-trivial Mordell-Weil torsion group Zk this changes: The non-Abelian gauge group becomes SU(n)/Zk. By construction the universal covering group has a trivial first fundamental group, and therefore the effect of non-trivial Mordell-Weil torsion is that the non-Abelian gauge group of the low-energy effective theory is no longer simply connected:
π1(SU(n)/Zk) =Zk. (5.3.54)
In the examples studied in [160] Mordell-Weil torsion groupsZ2 andZ3 always came accom-panied by gauge groups of type SU(2n) and SU(3n), respectively. Since SU(n) has a Zn
center generated by the identity matrix times e2πin , one can mod out Zk by eliminating the center (or a subgroup thereof) ofSU(k·n).
The corresponding reflexive polytope has PALP id 3415 and we list its defining data in table 5.13. It has a single nef partition, namely
∇1 =hv0, v3, v5, v6iconv, ∇2 =hv1, v2, v4, v7iconv. (5.3.55) In order to write down the most general complete intersection corresponding to this nef partition, we must use every integral point of the polytope defined in table 5.13 apart from the origin. The additional eleven points are listed intable 5.14.
130 CHAPTER 5. FIBERED CALABI-YAU MANIFOLDS After resolution, the complete intersection defined by (5.3.55) is defined by the following two polynomials:
p1 =a0z0z3z5z6z8z10z12z15z17+a1z20z72z8z9z14z15z16+a2z32z42z10z11z14z17z18
p2 =b0z21z52z12z15z16z17z18+b1z22z62z8z9z10z11z12+b2z1z2z4z7z9z11z14z16z18. (5.3.56) This time we are not interested in engineering additional singularities, but rather in confirming that models with this fiber contain theSU(4) gauge factors that we expect to exist. To this end we compute the discriminant of the elliptic curve and find
f =−1
48 · 16a21a22b20b21−16a20a1a2b0b1b22+a40b42
(5.3.57) g= 1
864· 8a1a2b0b1−a20b22
· 8a21a22b20b21+ 16a20a1a2b0b1b22−a40b42
(5.3.58)
∆ =−1
16 ·a20·b22·a41·a42·b40·b41· −16a1a2b0b1+a20b22
. (5.3.59)
From the vanishing orders we see that there are twoI2 and four I4 singularities. Since 9g
2f a
1=0= 9g 2f a
2=0 = 9g 2f b
1=0= 9g 2f b
2=0 =−1
4a20b23 (5.3.60) theI4 singularities are of split type (see [134] orAppendix B) and we therefore see that there is indeed a non-toricSU(2)2×SU(4)4/Z4 gauge group. One can mod out the Z4 torsion by identifying it with the diagonal subgroup of the centerZ⊕44 of the SU(4) gauge group part.
It is interesting to see that up to a lattice isomorphism the reflexive polytope∇◦ associ-ated to the nef partition (5.3.55) is precisely the polytope with PALP id 0. Under the same lattice isomorphism, the ∆i of (5.3.55) are mapped to the ∇i of (5.3.37) and we therefore see that the fiber considered in this subsection is mirror-dual to the fiber ofsubsection 5.3.3.
In particular, it appears that under this duality the discrete gauge group part is mapped to the torsion part of the Mordell-Weil group and vice versa. The same behavior was observed in [144] for hypersurface fibers and, as noted in subsection 3.9.4, it is intriguing to speculate about a possible physical reason underlying this observation.
Finally, let us note that it would be interesting to study explicit realizations of such fibrations. While this is possible in principle, the large number of involved points might make it technically challenging to find a triangulation that gives rise to an appropriate toric fan of the ambient variety. In the recent work [179] it was used that the relevant triangulations are star triangulations with respect to the origin in order to speed up the calculation. It would be exciting to incorporate such an algorithm in the Sage software package and apply it to these spaces.
Part III
Effective Actions
131
133 One of the central objects of field theory, both classical and quantum, is the Lagrangian action. While there are field theories for which no such Lagrangian can be defined [180–184], its study is of crucial importance whenever it does exist. In our specific context, we are not interested in the action at energies near the string scale, but at energies small compared to both the string scale and the scale of the compactification manifold. Such a low-energy effective action will usually contain only finitely many fields (as opposed to the infinitely many massive string excitations) and its computation provides the link between string theory and the quantum field theories we use to describe our observed universe.
In the third and final part of this dissertation, we therefore study the field theories that arise as the low-energy limits of F-theory compactifications on the geometries introduced in the previous chapters. Ideally, one would wish to be able to do two things:
• Given any genus-one fibered Calabi-Yau manifold, one would want to compute the quantum field theory it gives rise to as precisely as possible.
• Given a set of physical observables, one would like to determine as many geometrical properties as possible that the compactification manifold must have.
Stated in such generality, these are clearly two very difficult problems and solving them is currently (and will possibly always be) simply too hard. To nevertheless make progress in this direction, it has proven very fruitful to isolate particular physical quantities and attempt to study them on their own. One such example is the local study of GUTs in F-theory [55–
57,185], in the course of which it was realized that much of the essential information governing the non-Abelian gauge dynamics is captured already by the geometry of the neighborhood of the branes the gauge theories are located at.
To study gauge theories with Abelian gauge groups we take a different approach. Since Abelian gauge symmetries in F-theory are inextricably linked to global properties of the com-pactification manifold, it does not seem justifiable to take a local limit. However, matter charged under Abelian gauge groups in F-theory is essentially asix-dimensional quantity in F-theory, as it is localized along loci of complex codimension two in the base of the com-pactification manifold. While F-theory comcom-pactifications to four dimensions are considerably richer due to the additional presence of G-flux and Yukawa couplings, it suffices to study their six-dimensional siblings to understand most of their features. In fact, as we have seen inPart II, much of the information specifying the gauge theory is contained already in the fiber geometry (i.e. the top) and does not depend on whether one completes the top to a Calabi-Yau threefold or a Calabi-Yau fourfold.
Consequently, most of our effort is concentrated on studying F-theory in six dimensions.
As outlined in the introduction of this thesis, we use M-/F-theory duality in order to obtain the F-theory effective action. We begin inchapter 6by recalling on the one side the effective actions of M-theory compactified on a Calabi-Yau threefold with section and reduce on the other side the general six-dimensional supergravity action with N = (1,0) supersymmetry
134
on a circle. After matching both sides in chapter 7, we obtain the effective F-theory action in six dimensions for elliptically fibered Calabi-Yau manifolds. This analysis is extended to compactifications without section in chapter 8, where we encounter massive Abelian gauge fields. Inchapter 9we illustrate the general concepts obtained thus far by computing the low-energy effective matter spectrum of various genus-one fibered Calabi-Yau threefolds, including manifolds with multiple sections and those without section. Finally, we extend our study of F-theory on genus-one fibrations without section to four-dimensional models by examining the impact of discrete symmetries on the Yukawa couplings of the effective theory inchapter 10.
Chapter 6
Five-Dimensional Supergravity Reductions
As discussed in Part I, effective actions of F-theory compactifications can be obtained by using the chain of S- and T-dualities that connect M-theory on a Calabi-Yau manifold with torus fibration to Type IIB superstring theory on the base of the fibration times a circle. To employ the duality to obtain the six-dimensional F-theory effective action for theories with Abelian gauge symmetries, it is necessary to compute the effective action of M-theory on a Calabi-Yau threefold as well as to reduce a generalN = (1,0) supergravity theory on a circle.
We illustrated this procedure infigure 2.3.
Since the five- and six-dimensional supergravity theories discussed in this chapter may not be overly familiar, we summarize the relevant matter multiplets in tables 6.1 and 6.2 before proceeding with the reductions. Note that in six dimensions there is an additional
Multiplet Field Content
Gravity 1 graviton, 1 self-dual two-form, 1 left-handed Weyl gravitino Vector 1 vector, 1 left-handed Weyl gaugino
Tensor 1 anti-self-dual two-form, 1 real scalar, 1 right-handed Weyl ten-sorino
Hyper 4 real scalars , 1 right-handed Weyl hyperino
Table 6.1: The massless spectrum of six-dimensionalN = (1,0) supergravity. Note that one can substitute each Weyl spinor by two symplectic Majorana-Weyl spinors. The gravity multiplet has 24 real degrees of freedom, while the other three multiplets all have eight degrees of freedom.
135
136 CHAPTER 6. FIVE-DIMENSIONAL SUPERGRAVITY REDUCTIONS
Multiplet Field Content
Gravity 1 graviton, 1 vector, 1 Dirac gravitino Vector 1 vector, 1 real scalar, 1 Dirac gaugino Hyper 4 real scalars, 1 Dirac hyperino
Table 6.2: The massless spectrum of five-dimensional N = 2 supergravity. The gravity multiplet has 16 real degrees of freedom, while the other two multiplets both have eight degrees of freedom.
massless multiplet, the tensor multiplet, that contains as part of its bosonic field content an anti-self-dual two-form. The existence of such a two-form can be understood via group theory.
The massless fields in six dimensions are classified via the representations of SO(4) and, in particular, there exists a completely antisymmetric tensorijkl invariant underSO(4) which can be used to impose an (anti-)self-duality condition on the antisymmetric representation with two indices. Notably, this does not work anymore for massless five-dimensional fields, since their representations are those of SO(3). Here the invariant tensor is ijk and it can be used to dualize the antisymmetric representation into a vector. Upon reduction to five dimensions, a massless two-form field can therefore be dualized into a massless vector field and therefore the tensor multiplet reduces to a vector multiplet in five dimensions. We emphasize, however, that this is true only for massless fields. The representations of massive fields in six and five dimensions are those ofSO(5) and SO(4), respectively and thus there do exist massive tensor multiplets in five dimensions. For a more detailed discussion of such massive tensor fields and the reduction of tensor multiplets we refer to [186,187].
Before proceeding with the reductions of M-theory on a Calabi-Yau threefold insection 6.2 and N = (1,0)-supergravity on a circle in section 6.3, we first introduce in section 6.1 the basis of divisors and their respective dual (1,1)-forms that we will perform the reduction on.
Note that in the following most of our fields will be five-dimensional. To emphasize when that is not the case, we use hatted fields for fields living in eleven or six dimensions. For the remainder of this chapter, we assume that our compactification manifold is elliptically fibered, that is we assume the existence of a (not necessarily holomorphic) section of the torus fibration.
6.1 A Basis of Divisors for an Elliptically Fibered Calabi-Yau
Let us now fix our notation and choose a convenient divisor basis of the elliptically fibered Calabi-Yau manifold. Our conventions are essentially the same as in [54, 95, 113, 157]. As before, we assumeY →Ysingto be the smooth blow-up ofYsingalong all singular loci. We then choose the following basis of divisorsDΛand their respective dual two-formsωΛ∈H1,1(Y,Z):