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The $G_2$-MSSM is an MSSM model (in particle physics) motivated from compactification of M-theory on a 7-dimensional singular G2-manifold, hence a model in string phenomenology: M-theory on G2-manifolds.
The G2-MSSM is derived in the context of effective N=1 D=4 supergravity. The model is not based on a specific example of a G2 manifold but rather, it relies on the most general known properties of G2 compactifications of M-theory. For example, the Kähler potential is taken to be of the most general form consistent with G2 holonomy. Unlike Calabi-Yau manifold compactifications of string theory where one must stabilize different classes of moduli, which often require a different stabilization mechanism for each class, in $G_2$ compactifications of M-theory all the geometric moduli are on an equal footing. In other words, there is only one class of moduli - the periods of the associative 3-form over the basis 3-cycles. Furthermore, each geometric modulus is paired up with an axionic partner to form complex scalar fields, which possess a PQ-type shift symmetry. This symmetry originates from the gauge symmetry of the supergravity C-field in 11-dimensional supergravity. The shift symmetry is exact to all orders in perturbation theory and can only be broken by non-perturbative effects. Therefore, in the absence of fluxes the superpotential? in the corresponding N=1 d=4 supergravity is purely non-perturbative. This unique feature of the superpotential in G2 compactifications of M-theory provides a natural mechanism to generate the hierarchy between the Planck scale and the scale of supersymmetry breaking, and ultimately the electroweak breaking scale in the context of the G2-MSSM.
Another crucial feature that distinguishes the G2-MSSM from other known scenarios is the exponentially large split between masses of the geometric moduli and their axionic partners once the moduli are stabilized. This happens because the two dominant terms in the superpotential depend only on a single linear combination of all the moduli. This linear combination represents the volume of an associative three-cycle that supports the hidden sector non-Abelian gauge theory, which is taken to be Poincare dual (up to a positive real number) to the co-associative four-form of the $G_2$-manifold. Therefore, the model not only provides a natural QCD axion candidate, but also explicitly realizes the String Axiverse scenario? proposed by Arvanitaki et.al. Incidentally, this very feature of the superpotential ultimately explains the absence of large CP-violating phases in the soft supersymmetry breaking terms.
Another distinctive feature of the G2-MSSM is a slightly split spectrum of superpartners. The gaugino masses are suppressed relative to the gravitino mass by a factor of $\mathcal{O}(60)$ due to the dynamics of the moduli stabilization. On the other hand, the scalar masses and the trilinear couplings are as heavy as the gravitino mass. The higgsinos may or may not be suppressed relative to the gravitino mass depending on the details of the solution to the doublet-triple splitting problem.
Apart from the very specific form of the soft terms, the difference between the G2-MSSM and a generic bottom-up Split Supersymmetry scenario is that in Split SUSY both gaugino masses and trilinear couplings are protected by an R-symmetry and therefore remain light, while the scalar masses can be as heavy as the GUT scale.
The slightly split G2-MSSM spectrum naturally accommodates the Higgs particle at the observed energy of around 125 GeV together with reasonable dark matter candidates in the form of axions and/or W-ino WIMPs, which are generated non-thermally via late-time moduli decays.
A comprehensive account is in
Original articles include
Bobby Acharya, Konstantin Bobkov, Gordon Kane, Piyush Kumar, Jing Shao, Explaining the Electroweak Scale and Stabilizing Moduli in M Theory (arXiv:hep-th/0701034)
Bobby Acharya, Konstantin Bobkov, Gordon Kane, Piyush Kumar, Diana Vaman, An M theory Solution to the Hierarchy Problem (arXiv:hep-th/0606262)
Bobby Acharya, Konstantin Bobkov, Kähler Independence of the $G_2$-MSSM, JHEP (arXiv:0810.3285)
A review of the specific phenomenological properties is in