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  4. "Three Fermion Generations from Octonions"
  6. [**Carlos De Aldazabal$^{(a)}$, Andrea Sagnotti$^{(b)}$**]{}
  8. [*(a) Dipartimento di Fisica Teorica, Università di Torino, and I.N.F.N., Sezione di Torino*]{}\
  9. [*via P. Giuria 1, I-10125 Torino, Italy*]{}\
  10. [**]{}
  12. [*(b) INFN, Laboratori Nazionali di Frascati, P.O. Box 13, I-00044 Frascati, Italy*]{}\
  13. [**]{}
  15. **Abstract**
  17. We show that the simplest consistent theory of three generations of fermions containing at least one six-dimensional orbifold-projected field can be naturally derived from six-dimensional supersymmetric theories, in a way which requires no large hierarchies. It predicts three copies of the SM gauge group and a family of new vector-like particles, some of which are completely stabilized by orbifold symmetries. It also introduces an important ingredient, crucial for the stability of the compactification, namely a non-polynomial superpotential with a new “hair” for the complex structure of the Calabi-Yau manifold.
  19. Introduction
  20. ============
  22. The status of Grand Unified Theories (GUTs) in six dimensions is by now well established. It has been shown that, given the crucial role played by an $SO(10)$ GUT group in string theory, any string compactification containing six-dimensional supergravity will necessarily lead to an $SO(10)$ GUT theory with 6+1D supersymmetry.
  24. String models have also proven to be quite flexible and powerful tools to achieve this goal, by giving a description of the quantum numbers of the fundamental field multiplets in the GUT theory. It is usually quite easy to identify a minimal set of gauge groups and matter states, which can be identified with the ones which occur in the Standard Model (SM) of particle physics.
  26. However, until a few years ago, it was not known how to get beyond the minimal set, i.e. how to build up the GUT theory starting from a minimal set of gauge factors. Indeed, starting from a 6+1D theory, the natural question was whether it was possible to embed it into a larger 6+1D theory with still smaller number of gauge groups. If the starting theory has too many gauge groups, this would lead to a long chain of GUTs with more and more gauge factors, with less and less unification of couplings, ultimately in disagreement with experimental data. The search for minimal GUT theories has therefore been the subject of a huge number of papers, as reviewed e.g. in [@Ferrara:2010in].
  28. A common feature of the minimal GUTs is the presence of one set of “hidden” particles, whose number is equal to the number of GUT multiplets that are “visible” in the 4+1D theory. The fact that these particles are hidden has been interpreted as an evidence that these are either strings or string-like objects. If this is the case, then the “hidden” particles cannot be counted as fundamental states of the theory, but rather are objects that are only “emergent” from the compactification of a higher-dimensional theory. String-like objects, on the other hand, can be counted as fundamental states of the compactification. It is therefore quite natural to interpret this fact as an evidence of the non-renormalizability of the theory, in analogy with what happens in 4+1D theories.
  30. However, the standard set of assumptions and arguments used to explain the need for extra hidden states have not been applied to GUTs in 6+1D. The aim of the present paper is to try to fill this gap. In particular, we will give some arguments which show that the presence of one set of “hidden” particles is unavoidable, and that they cannot be elementary objects.
  32. The use of orbifolds as a tool to derive new GUT models has been developed in several contexts. It was applied in the first attempts to derive a complete string-GUT model [@Ibanez:1986tp], in the context of $E_8 \times E_8$ heterotic string theory [@Antoniadis:1993qg; @Dixon:1986qv], and in the context of $SO(32)$ heterotic string theory [@Dixon:1986qv; @Ibanez:1986tp]. It was applied, in an apparently different context, to the case of $SO(10)$ heterotic string theory [@Dixon:1986qv] (see also [@Ibanez:1987pj] and [@Ibanez:1987pk]), as well as to the case of heterotic $SO(16)$ models [@Kallosh:1998nx] (see also [@Kallosh:1998ju]). It was further used to derive $SO(10)$ GUT models starting from string theories based on different kinds of Calabi-Yau manifolds [@Dixon:1990pc; @Bailin:1997ns; @Bailin:1998xg; @Leontaris:2001yb; @Leontaris:2009wi]. It was finally applied in an attempt to study the possibility of having more than one set of hidden states [@Lebedev:2010yq], and to derive a phenomenologically viable model of three generations [@Cvetic:2012gd].[^1]
  34. In the present paper, we will further pursue this line of research and consider the possibility of having three generations of quarks and leptons in the context of string theory. For this purpose, we will apply the methods of [@Dixon:1990pc; @Bailin:1997ns; @Bailin:1998xg; @Leontaris:2001yb; @Leontaris:2009wi] to the case of orbifolds of six-dimensional supersymmetric theories with vector-like states in the bulk.
  36. It is well known that compactification on an orbifold is an interesting method to reduce the size of the extra dimensions, since it is related to the breaking of a larger space into smaller sub-spaces, by requiring some states to have certain fixed values of their quantum numbers. In the cases where the six extra dimensions are reduced to just a single point, the orbifold becomes a torus. In this paper we will consider only the simplest case of the so-called [*generalized*]{} orbifolds, which are characterized by a two-torus with a non-trivial holonomy group for its first homology group [@Dixon:1990pc].
  38. The main results of the paper are threefold. First, we will show that, for all orbifolds of the kind considered here, the low-energy theory on the 4+1D brane will always contain one set of hidden states, just as in the case of the simplest orbifolds of 6+1D supersymmetric theories. This fact does not seem to be in contradiction with the fact that these orbifolds are obtained by the breaking of a higher dimensional supersymmetric theory, since the low-energy theory on the 4+1D brane is an effective theory and can be seen as a result of integrating out the extra states in the 6+1D theory. This is exactly what happens in the original construction of $E_8 \times E_8$ heterotic string theory [@Ibanez:1986tp], and it is also true in all the models derived from Calabi-Yau manifolds [@Dixon:1990pc; @Bailin:1997ns; @Bailin:1998xg; @Leontaris:2001yb; @Leontaris:2009wi] (see [@Lebedev:2010yq] for an attempt to have a supersymmetric GUT without extra states).
  40. The second point is that, given a $U(N)$ gauge theory with a non-polynomial superpotential, one can derive a set of supersymmetric theories with the following properties:
  42. - [*two*]{} sets of $N$ fields in the fundamental representation,
  44. - a non-polynomial superpotential of the type discussed in [@Lebedev:2010yq],
  46. - two U(1)s, related to each other by a Higgsing.
  48. The third point is that the conditions derived in [@Lebedev:2010yq] have also the consequence of having a set of states whose mass is bounded from below. This is the case of hidden states
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