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Supersymmetric Grand Unified Theories pp 267–308Cite as

Embedding Orbifold GUTs in the Heterotic String

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Part of the book series: Lecture Notes in Physics ((LNP,volume 939))

Abstract

We have seen that supersymmetric orbifold GUTs retain many of the nice features of 4D GUTs, such as gauge coupling unification and GUT relations for Yukawa couplings. In addition, they replace the complicated GUT breaking and doublet-triplet splitting sectors of the 4D theory by the very elegant mathematics of orbifolding with discrete symmetries. They also open up new mechanisms for supersymmetry breaking. However, on the down side, these orbifold GUT theories are non-renormalizable and therefore an explicit cut-off must be introduced. In this chapter we begin the discussion of embedding supersymmetric orbifold GUTs into string theory. In a string theory, the arbitrary cut-off is replaced by the physical string scale. Moreover, string theory has the benefit of also including quantum gravity.

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Notes

  1. 1.

    States which are chiral under the Standard Model gauge group can only obtain mass by coupling to the Higgs boson. Thus their mass is necessarily of order the weak scale. Hence there are stringent experimental bounds on such states.

  2. 2.

    By definition, a vector-like exotic can obtain mass without breaking any Standard Model gauge symmetry.

  3. 3.

    We could have applied the same analysis to Eq. (20.25).

  4. 4.

    A GSO projection is needed to guarantee space-time SUSY.

  5. 5.

    A discussion of the property of bosonization of fermions in 1 + 1 dimensions is discussed in more detail in Sect. 20.0.1.

  6. 6.

    Note, if the coordinates are compactified on a particular lattice, then the momenta are compactified on the dual lattice. However, the lattice Γ 8 is self-dual. Therefore the momenta also reside in the weight space of E 8 × E 8.

  7. 7.

    Note if the backgrounds, G,  B depend on X, then the model is equivalent to the non-linear sigma model. If one requires that the theory remain conformally invariant, i.e. the relevant beta functions vanish, one obtains at one loop the equations of motion for the fields G,  B which are the Einstein equations of gravity coupled to the anti-symmetric tensor field, see Chap. 3.4 of [415].

  8. 8.

    Rotations by 4π on fermions is equivalent to the identity and one requires that the rotation generators in SU(3) are orthogonal to the U(1) in the decomposition of SO(6) → SU(3) × U(1). The U(1) generator is given by J 12 + J 34 + J 56 which gives ∑ i = 1 3 v i  = 0 (see Sect. 5.7 for the definition of vector and spinor representations in SO(10), since the mathematics is identical for any SO(2N) group).

  9. 9.

    In the heterotic string, this statement is a consequence of modular invariance, which is a subject which goes way beyond the scope of these lectures. Modular invariance is the reason why string theory is finite, i.e. no loop divergences. See vol. 2 of [415] or Chap. 6 of [416].

  10. 10.

    In general, the normal ordering coefficient for real bosons is given by

    $$\displaystyle{ a_{B}(\eta ) \equiv -\frac{1} {2}\ \sum _{n=0}^{\infty }\ (n+\eta ) = \frac{1} {24} -\frac{1} {4}\ \eta (1-\eta ) \equiv - \frac{1} {48} + \frac{1} {4}(\eta -\frac{1} {2})^{2}. }$$
    (20.100)

    For real fermions we have

    $$\displaystyle{ a_{F}(\eta ) = -a_{B}(\eta ). }$$
    (20.101)
  11. 11.

    See, for example, Chap. 3.2.4, [415].

  12. 12.

    Underline means include all permutations.

  13. 13.

    Note, the vectors in Eq. (20.125) satisfy

    $$\displaystyle{ \mathbf{P} + \mathbf{V} = (\frac{2} {3}, \frac{2} {3}, \frac{2} {3},\ 0^{5}),\;\; (-\frac{1} {3},-\frac{1} {3},-\frac{1} {3},\ \underline{\pm 1, 0^{4}}),\;\; \frac{1} {2}(\frac{1} {3}, \frac{1} {3}, \frac{1} {3},\ [+1, +1, +1, +1,-1]). }$$
    (20.127)

    Thus the vectors sit at the origin in the shifted SU(3) lattice. Note, the simple roots of SU(3) are given by the unit vector α 1 = e 1 −e 2 and α 2 = e 2 −e 3 with e i ⋅ e j = δ ij .

  14. 14.

    Note, the vectors in Eq. (20.130) satisfy

    $$\displaystyle{ \mathbf{P} + \mathbf{V} = (\frac{2} {3},-\frac{1} {3},-\frac{1} {3},\ 0^{5}),\;\; (-\frac{1} {3},\underline{ \frac{2} {3},-\frac{1} {3}},\ 0^{5}). }$$
    (20.128)

    Thus the vectors are in the \(\mathbf{\bar{3}}\) of the shifted SU(3) lattice.

  15. 15.

    Prime-order orbifold models (such as the \(\mathbb{Z}_{3}\) orbifold models) with Wilson lines [401, 420–424] and non-prime-order orbifold models without Wilson lines [425, 426] have been extensively studied in the literature. Non-prime-order orbifold models with Wilson lines, on the other hand, possess a number of complications, and to our knowledge they have not been studied to the same extent. Our work can be regarded as the first serious attempt at constructing three-family models from non-prime-order orbifolds.

  16. 16.

    By N = 2 supersymmetry in 5 or 6D, we mean the minimal number of supersymmetries in these dimensions, (i.e. the fermions satisfy the pseudo-reality condition). It reduces to N = 2 in 4D by dimensional reduction and is sometimes called N = 1 supersymmetry in the literature.

  17. 17.

    Together with r 4 = (0, 0, 0, 1), they form the set of positive weights of the 8 v representation of the SO(8), the little group in 10d. ±r 4 represent the two uncompactified dimensions in the light-cone gauge. Their space-time fermionic partners have weights \(\mathbf{r} = (\pm \frac{1} {2},\pm \frac{1} {2},\pm \frac{1} {2},\pm \frac{1} {2})\) with even numbers of positive signs; they are in the 8 s representation of SO(8). In this notation, the fourth component of v 6 is zero.

  18. 18.

    The E 8 root lattice is given by the set of states \(\mathbf{P} =\{ n_{1},n_{2},\cdots \,,n_{8}\},\ \{n_{1} + \frac{1} {2},n_{2} + \frac{1} {2},\cdots \,,n_{8} + \frac{1} {2}\}\) satisfying \(n_{i} \in \mathbb{Z},\ \sum _{i=1}^{8}n_{i} = 2\mathbb{Z}\).

  19. 19.

    It should be obvious that our construction can be generalized to 6D models, simply by taking both R and R ′ large compared to the string length scale. These models are related to 6D orbifold GUTs compactified on T\(^{2}/\mathbb{Z}_{2}\).

  20. 20.

    In terms of 4D N = 1 chiral superfields.

  21. 21.

    Together with r 4 = (0, 0, 0, 1), they form the set of positive weights of the 8 v representation of the SO(8), the little group in 10d. ±r 4 represent the two uncompactified dimensions in the light-cone gauge. Their space-time fermionic partners have weights \(\mathbf{r} = (\pm \frac{1} {2},\pm \frac{1} {2},\pm \frac{1} {2},\pm \frac{1} {2})\) with even numbers of positive signs; they are in the 8 s representation of SO(8). In this notation, the fourth component of v 6 is zero.

  22. 22.

    Wilson lines can be used to reduce the number of chiral families. In all our models, we find it is sufficient to get three-generation models with two Wilson lines, one of degree 2 and one of degree 3. Note, however, that with two Wilson lines in the SO(4) torus we can break SO(10) directly to SU(3) × SU(2) × U(1) Y × U(1) X (see for example, [319, 435]).

  23. 23.

    For gauge and untwisted-sector states, p are trivial. For non-oscillator states in the T 2, 4 twisted sectors, p = γ are the eigenvalues of the G 2-plane fixed points under the \(\mathbb{Z}_{2}\) twist. Note that p = + and − states have multiplicities 2 and 1 respectively since the corresponding numbers of fixed points in the G 2 plane are 2 and 1.

  24. 24.

    D 4 is a non-abelian subgroup of SU(2). It is also equivalent to a subgroup of O(2).

  25. 25.

    See problem 13.

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  1. Department of Physics, The Ohio State University, Columbus, Ohio, USA

    Stuart Raby

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Raby, S. (2017). Embedding Orbifold GUTs in the Heterotic String. In: Supersymmetric Grand Unified Theories. Lecture Notes in Physics, vol 939. Springer, Cham. https://doi.org/10.1007/978-3-319-55255-2_20

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