Mirror partner for a Klein quartic polynomial

TL;DR

This paper explores the mirror symmetry of a Klein quartic polynomial within Landau-Ginzburg models, analyzing various symmetry groups beyond the diagonal case, and identifies the structure of its mirror orbifold.

Contribution

It extends mirror symmetry analysis to non-abelian symmetry groups of the Klein quartic polynomial, revealing new mirror orbifold structures.

Findings

Mirror Landau-Ginzburg orbifold corresponds to a specific non-abelian symmetry group.

The symmetry group G is a Z/2Z-extension of a Klein four-group.

The Klein quartic's mirror involves non-diagonal, non-abelian symmetries.

Abstract

The results of A.Chiodo, Y.Ruan and M.Krawitz associate the mirror partner Calabi-Yau variety $X$ to a Landau--Ginzburg orbifold $(f,G)$ if $f$ is an invertible polynomial satisfying Calabi-Yau condition and the group $G$ is a diagonal symmetry group of $f$. In this paper we investigate the Landau-Ginzburg orbifolds with a Klein quartic polynomial $f = x_1^3x_2 + x_2^3x_3+x_3^3x_1$ and $G$ being all possible subgroups of $\mathrm{GL}(3,\mathbb{C})$, preserving the polynomial $f$ and also the pairing in its Jacobian algebra. In particular, $G$ is not necessarily abelian or diagonal. The zero-set of polynomial $f$, called Klein quartic, is a genus $3$ smooth compact Riemann surface. We show that its mirror Landau-Ginzburg orbifold is $(f,G)$ with $G$ being a $\mathbb{Z}/2\mathbb{Z}$-extension of a Klein four-group.