All Abelian Symmetries of Landau-Ginzburg Potentials

TL;DR

This paper develops an algorithm to find all abelian symmetries of Landau-Ginzburg potentials, expanding known spectra of superconformal theories and exploring mirror symmetry and orbifolds with torsion.

Contribution

It introduces a new algorithm for classifying abelian symmetries of Landau-Ginzburg potentials and applies it to expand the set of known superconformal spectra.

Findings

Increases known spectra by about one third.

Mirror symmetry remains limited despite new potentials.

Initial exploration of orbifolds with a52 torsion.

Abstract

We present an algorithm for determining all inequivalent abelian symmetries of non-degenerate quasi-homogeneous polynomials and apply it to the recently constructed complete set of Landau--Ginzburg potentials for $N=2$ superconformal field theories with $c=9$. A complete calculation of the resulting orbifolds without torsion increases the number of known spectra by about one third. The mirror symmetry of these spectra, however, remains at the same low level as for untwisted Landau--Ginzburg models. This happens in spite of the fact that the subclass of potentials for which the Berglund--H\"ubsch construction works features perfect mirror symmetry. We also make first steps into the space of orbifolds with $\ZZ_2$ torsions by including extra trivial fields.