Common use of Summary and Discussion Clause in Contracts

Summary and Discussion. N ≥ 1. In total we find 60 Q-classes (point groups) that lead to N ≥ 1 SUSY. 2. These Q-classes decompose in • 22 with an Abelian point group with one or two generators, i.e. ZN or ZN × ZM , out of which 17 lead to exactly N = 1 SUSY, and • 38 with a non-Abelian point group with two or three generators, such as S3 or ∆(216), out of which 35 lead to exactly N = 1 SUSY. That is, there are 52 Q-classes that can lead to models yielding the supersymmet- ric standard model. As we have explained in detail, Q-classes (or point groups) can come with in- equivalent lattices, classified by the so-called Z-classes. In the traditional orbifold literature, Z-classes are given by Lie lattices and a given choice fixes an orbifold geometry. However, as we have pointed out, not all lattices can be described by Lie lattices. Our results on Q-classes potentially relevant for supersymmetric model build- ing are as follows. 3. We find that there are 186 Z-classes, or, in other words, orbifold geometries that lead to N ≥ 1 SUSY. 4. These Z-classes decompose in 71 with an Abelian point group, out of which 60 lead to exactly = 1 ▇▇▇▇, and • 115 with a non-Abelian point group, out of which 108 lead to exactly N = 1 SUSY. Furthermore, space groups can be extended by so-called roto-translations, a com- bination of a twist and a (non-lattice) translation. We provide a full classification of all roto-translations in terms of affine classes, which are, as we discuss, the most suitable objects to classify inequivalent space groups. 5. We find 520 affine classes that lead to N ≥ 1 SUSY. 6. These affine classes decompose in • 162 with an Abelian point group, out of which 138 lead to exactly N = 1 SUSY, and • 358 with a non-Abelian point group, out of which 331 lead to exactly N = 1 SUSY. An important aspect of our classification is that we provide the data for all 138 space groups with Abelian point group and = 1 SUSY required to construct the corresponding models with the C++ orbifolder [46]. Among other things, this allows one to obtain a statistical survey of the properties of the models, which has so far only been performed for the Z6-II orbifold [42]. Our classification also has conceivable importance for phenomenology. For instance, one of the questions is how the ten-dimensional gauge group (i.e. E8 E8 or SO(32)) of the heterotic string gets broken by orbifolding. In most of the models discussed so far, the larger symmetry gets broken locally at some fixed point. Yet it has been argued that ‘non-local’ GUT symmetry breaking, as utilized in the context of smooth compactifications of the heterotic string [6, 9, 8, 2], has certain phenomenological advantages [33, 1]. Explicit MSSM candidate models, based on the DW classification, featuring non-local GUT breaking have been constructed recently [4, 38]. As we have seen, there are 31 affine classes of space groups, based on the Q-classes Z2 Z2, Z2 Z4 and Z3 Z3, that lead to an orbifold with a non-trivial fundamental group, thus allowing us to introduce a ▇▇▇▇▇▇ line that breaks the GUT symmetry. In other words, we have identified a large set of geometries that can give rise to non-local GUT breaking. This might also allow for a dynamical stabilization of some of the moduli in the early universe, similar as in toroidal compactifications [7]. In this study, we have focused on symmetric toroidal orbifolds, which have a rather clear geometric interpretation, such that crystallographic methods can be applied in a straightforward way. We have focused on the geometrical aspects. On the other hand, it is known that background fields, i.e. the so-called discrete ▇▇▇▇▇▇ lines [36] and discrete torsion [53, 54, 52, 29, 50], play a crucial role in model building. It will be interesting to work out the conditions on such background fields in the geometries of our classification. Further, it is, of course, clear that there are other orbifolds, such as T-folds [34, 16], asymmetric and/or non-toroidal orbifolds, whose classification is beyond the scope of this study. Let us also mention, we implicitly assumed that the radii are away from the self- dual point. As we are using crystallographic methods our classification strategy is independent of this assumption. Still, it might be interesting to study what happens if one sends one or more T -moduli to the self-dual values. In this case one may make contact with the free fermionic formulation, where also interesting models have been constructed [15]. In addition, our results may also be applied to compactifications of type II string theory on orientifolds (see e.g. [30, 22, 31] for some interesting models and [5] for a review).

Summary and Discussion. N ≥ 1. In total we find 60 Q-classes (point groups) that lead to N ≥ 1 SUSY. 2. These Q-classes decompose in • 22 with an Abelian point group with one or two generators, i.e. ZN or ZN × ZM , out of which 17 lead to exactly N = 1 SUSY, and • 38 with a non-Abelian point group with two or three generators, such as S3 or ∆(216), out of which 35 lead to exactly N = 1 SUSY. That is, there are 52 Q-classes that can lead to models yielding the supersymmet- ric standard model. As we have explained in detail, Q-classes (or point groups) can come with in- equivalent lattices, classified by the so-called Z-classes. In the traditional orbifold literature, Z-classes are given by Lie lattices and a given choice fixes an orbifold geometry. However, as we have pointed out, not all lattices can be described by Lie lattices. Our results on Q-classes potentially relevant for supersymmetric model build- ing are as follows. 3. We find that there are 186 Z-classes, or, in other words, orbifold geometries that lead to N ≥ 1 SUSY. 4. These Z-classes decompose in 71 with an Abelian point group, out of which 60 lead to exactly = 1 ▇▇▇▇SUSY, and • 115 with a non-Abelian point group, out of which 108 lead to exactly N = 1 SUSY. Furthermore, space groups can be extended by so-called roto-translations, a com- bination of a twist and a (non-lattice) translation. We provide a full classification of all roto-translations in terms of affine classes, which are, as we discuss, the most suitable objects to classify inequivalent space groups. 5. We find 520 affine classes that lead to N ≥ 1 SUSY. 6. These affine classes decompose in • 162 with an Abelian point group, out of which 138 lead to exactly N = 1 SUSY, and • 358 with a non-Abelian point group, out of which 331 lead to exactly N = 1 SUSY. An important aspect of our classification is that we provide the data for all 138 space groups with Abelian point group and = 1 SUSY required to construct the corresponding models with the C++ orbifolder [46]. Among other things, this allows one to obtain a statistical survey of the properties of the models, which has so far only been performed for the Z6-II orbifold [42]. Our classification also has conceivable importance for phenomenology. For instance, one of the questions is how the ten-dimensional gauge group (i.e. E8 E8 or SO(32)) of the heterotic string gets broken by orbifolding. In most of the models discussed so far, the larger symmetry gets broken locally at some fixed point. Yet it has been argued that ‘non-local’ GUT symmetry breaking, as utilized in the context of smooth compactifications of the heterotic string [6, 9, 8, 2], has certain phenomenological advantages [33, 1]. Explicit MSSM candidate models, based on the DW classification, featuring non-local GUT breaking have been constructed recently [4, 38]. As we have seen, there are 31 affine classes of space groups, based on the Q-classes Z2 Z2, Z2 Z4 and Z3 Z3, that lead to an orbifold with a non-trivial fundamental group, thus allowing us to introduce a ▇▇▇▇▇▇ line that breaks the GUT symmetry. In other words, we have identified a large set of geometries that can give rise to non-local GUT breaking. This might also allow for a dynamical stabilization of some of the moduli in the early universe, similar as in toroidal compactifications [7]. In this study, we have focused on symmetric toroidal orbifolds, which have a rather clear geometric interpretation, such that crystallographic methods can be applied in a straightforward way. We have focused on the geometrical aspects. On the other hand, it is known that background fields, i.e. the so-called discrete ▇▇▇▇▇▇ lines [36] and discrete torsion [53, 54, 52, 29, 50], play a crucial role in model building. It will be interesting to work out the conditions on such background fields in the geometries of our classification. Further, it is, of course, clear that there are other orbifolds, such as T-folds [34, 16], asymmetric and/or non-toroidal orbifolds, whose classification is beyond the scope of this study. Let us also mention, we implicitly assumed that the radii are away from the self- dual point. As we are using crystallographic methods our classification strategy is independent of this assumption. Still, it might be interesting to study what happens if one sends one or more T -moduli to the self-dual values. In this case one may make contact with the free fermionic formulation, where also interesting models have been constructed [15]. In addition, our results may also be applied to compactifications of type II string theory on orientifolds (see e.g. [30, 22, 31] for some interesting models and [5] for a review).