Landau-Ginzburg Orbifolds, Mirror Symmetry and the Elliptic Genus

TL;DR

This paper computes the elliptic genus for 2D N=2 Landau-Ginzburg orbifolds to identify mirror pairs, revealing conjugate models with matching elliptic genera and interchanged chiral rings, and proposing new mirror candidates.

Contribution

It introduces a method to find mirror pairs of Landau-Ginzburg orbifolds via elliptic genus calculations and establishes conditions for conjugacy between models.

Findings

Elliptic genus computed for arbitrary Landau-Ginzburg orbifolds.

Mirror pairs characterized by conjugacy and elliptic genus equivalence.

New mirror pairs generated through product constructions.

Abstract

We compute the elliptic genus for arbitrary two dimensional $N=2$ Landau-Ginzburg orbifolds. This is used to search for possible mirror pairs of such models. We show that if two Landau-Ginzburg models are conjugate to each other in a certain sense, then to every orbifold of the first theory corresponds an orbifold of the second theory with the same elliptic genus (up to a sign) and with the roles of the chiral and anti-chiral rings interchanged. These orbifolds thus constitute a possible mirror pair. Furthermore, new pairs of conjugate models may be obtained by taking the product of old ones. We also give a sufficient (and possibly necessary) condition for two models to be conjugate, and show that it is satisfied by the mirror pairs proposed by one of the authors and~H\"ubsch.