Abstract
We are considering the class of heterotic $\mathcal{N}=(2,2)$ Landau-Ginzburg orbifolds with 9 fields corresponding to $A_1^9$ Gepner models. We classify all of its Abelian discrete quotients and obtain 152 inequivalent models closed under mirror symmetry with $\mathcal{N}=1,2$ and $4$ supersymmetry in 4D. We compute the full massless matter spectrum at the Fermat locus and find a universal relation satisfied by all models. In addition we give prescriptions of how to compute all quantum numbers of the 4D states including their discrete R-symmetries. Using mirror symmetry of rigid geometries we describe orbifold and smooth Calabi-Yau phases as deformations away from the Landau-Ginzburg Fermat locus in two explicit examples. We match the non-Fermat deformations to the 4D Higgs mechanism and study the conservation of R-symmetries. The first example is a $\mathbb{Z}_3$ orbifold on an E$_6$ lattice where the R-symmetry is preserved. Due to a permutation symmetry of blow-up and torus K\"{a}hler parameters the R-symmetry stays conserved also smooth Calabi-Yau phase. In the second example the R-symmetry gets broken once we deform to the geometric $\mathbb{Z}_3 \times \mathbb{Z}_{3,\text{free}}$ orbifold regime.
Highlights
With the virtue of having a UV complete theory at hand it is a great desire to classify all the possible compactification spaces and obtain their properties and possible connections
We classify all of its Abelian discrete quotients and obtain 152 inequivalent models closed under mirror symmetry with N = 1, 2 and 4 supersymmetry in 4D
Those symmetries have a very nice geometrical interpretation: non-Abelian Flavor symmetries arise from permutation symmetries of orbifold singularities [22, 23] and discrete R-symmetries are remnants of the 10D Lorentz symmetry preserved by the orbifold action
Summary
We review the tools necessary to analyze the massless spectrum of LandauGinzburg orbifold models. We stick to a very specify class of models this review is general and applicable to other cases as well. At first we review the methods that are needed to calculate the massless spectrum. In the second part we give the methods needed to calculate all charges of the spectrum under all discrete and continuous symmetries
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