Tracing symmetries and their breakdown through phases of heterotic (2,2) compactifications

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.