No Mirror Symmetry in Landau-Ginzburg Spectra!

TL;DR

The paper tackles whether mirror symmetry holds for Landau-Ginzburg descriptions of $N=2$ SCFTs with $c=9$ by classifying all non-degenerate quasihomogeneous polynomials with $D=3$, computing Hodge numbers from the chiral ring via generalized Poincare polynomials, and analyzing mirror partners. It finds that mirror symmetry is strong for invertible LG models—consistent with the Berglund-Hübsch construction—but collapses for a large fraction of non-invertible models that require links, many of which have no mirror partners. The methodology combines a graph-based non-degeneracy criterion, finite enumeration of skeleton graphs, and efficient formulas for Hodge data, yielding extensive catalogs of spectra and Euler numbers. The results suggest that MS in this setting may be restricted to certain LG subclasses and motivate exploring abelian orbifolds and beyond-LG interpretations to realize more realistic, potentially three-generation vacua.

Abstract

We use a recent classification of non-degenerate quasihomogeneous polynomials to construct all Landau-Ginzburg (LG) potentials for N=2 superconformal field theories with c=9 and calculate the corresponding Hodge numbers. Surprisingly, the resulting spectra are less symmetric than the existing incomplete results. It turns out that models belonging to the large class for which an explicit construction of a mirror model as an orbifold is known show remarkable mirror symmetry. On the other hand, half of the remaining 15\% of all models have no mirror partners. This lack of mirror symmetry may point beyond the class of LG-orbifolds.