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Mirror partner for a Klein quartic polynomial
Journal of Geometry and Physics. 2025. Vol. 215. Article 105538.
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 $\GL(3,\CC)$, 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 $\ZZ/2\ZZ$--extension of a Klein four--group.
Keywords: mirror symmetry
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 8 С. 110–123
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