ORBIFOLDS WITH DISCRETE TORSION AND MIRROR SYMMETRY

TL;DR

This paper explores how mirror symmetry in certain $N=2$ superconformal field theories can be understood through orbifolds with discrete torsion, extending known constructions and analyzing specific $bZ_2 imes bZ_2$ cases.

Contribution

It extends mirror symmetry constructions to include discrete torsion in orbifolds, providing new insights into the Berglund--Hübsch framework and the special role of $bZ_2$ twists.

Findings

Discrete torsion can produce mirror models in orbifold theories.

The $bZ_2 imes bZ_2$ case exemplifies this mechanism.

The example suggests a coincidence rather than a new mirror symmetry principle.

Abstract

For a large class of $N=2$ SCFTs, which includes minimal models and many $\s$ models on Calabi-Yau manifolds, the mirror theory can be obtained as an orbifold. We show that in such a situation the construction of the mirror can be extended to the presence of discrete torsions. In the case of the $\ZZ_2\ex\ZZ_2$ torus orbifold, discrete torsion between the two generators directly provides the mirror model. Working at the Gepner point it is, however, possible to understand this mirror pair as a special case of the Berglund--H"ubsch construction. This seems to indicate that the $\ZZ_2\ex\ZZ_2$ example is a mere coincidence, due to special properties of $\ZZ_2$ twists, rather than a hint at a new mechanism for mirror symmetry.