We summarize the foliation approach to ${cal N}=1$ compactifications of eleven-dimensional supergravity on eight-manifolds $M$ down to $mathrm{AdS}_3$ spaces for the case when the internal part $xi$ of the supersymmetry generator is chiral on some pr
oper subset ${cal W}$ of $M$. In this case, a topological no-go theorem implies that the complement $Msetminus {cal W}$ must be a dense open subset, while $M$ admits a singular foliation ${bar {cal F}}$ (in the sense of Haefliger) which is defined by a closed one-form $boldsymbol{omega}$ and is endowed with a longitudinal $G_2$ structure. The geometry of this foliation is determined by the supersymmetry conditions. We also describe the topology of ${bar {cal F}}$ in the case when $boldsymbol{omega}$ is a Morse form.
In MSSM models with various boundary conditions for the soft breaking terms (m_{soft}) and for a higgs mass of 126 GeV, there is a (minimal) electroweak fine-tuning Deltaapprox 800 to 1000 for the constrained MSSM and Deltaapprox 500 for non-universa
l gaugino masses. These values, often regarded as unacceptably large, may indicate a problem of supersymmetry (SUSY) breaking, rather than of SUSY itself. A minimal modification of these models is to lower the SUSY breaking scale in the hidden sector (sqrt f) to few TeV, which we show to restore naturalness to more acceptable levels Deltaapprox 80 for the most conservative case of low tan_beta and ultraviolet boundary conditions as in the constrained MSSM. This is done without introducing additional fields in the visible sector, unlike other models that attempt to reduce Delta. In the present case Delta is reduced due to additional (effective) quartic higgs couplings proportional to the ratio m_{soft}/(sqrt f) of the visible to the hidden sector SUSY breaking scales. These couplings are generated by the auxiliary component of the goldstino superfield. The model is discussed in the limit its sgoldstino component is integrated out so this superfield is realized non-linearly (hence the name of the model) while the other MSSM superfields are in their linear realization. By increasing the hidden sector scale sqrt f one obtains a continuous transition for fine-tuning values, from this model to the usual (gravity mediated) MSSM-like models.
We use geometric algebra techniques to give a synthetic and computationally efficient approach to Fierz identities in arbitrary dimensions and signatures, thus generalizing previous work. Our approach leads to a formulation which displays the underly
ing real, complex or quaternionic structure in an explicit and conceptually clear manner and is amenable to implementation in various symbolic computation systems. We illustrate our methods and results with a few examples which display the basic features of the three classes of pin representations governing the structure of such identities in various dimensions and signatures.
Under the hypotheses of analyticity, locality, Lorentz covariance, and Poincare invariance of the deformations, combined with the requirement that the interaction vertices contain at most two spatiotemporal derivatives of the fields, we investigate t
he consistent selfinteractions that can be added to a collection of massless tensor fields with the mixed symmetry (3,1) and respectively (2,2). The computations are done with the help of the deformation theory based on a cohomological approach, in the context of the antifield-BRST formalism. Our result is that no selfinteractions that deform the original gauge transformations emerge. In the case of the collection of (2,2) tensor fields it is possible to add a sum of cosmological terms to the free Lagrangian.
