Mirror partner for a Klein quartic polynomial
Alexey Basalaev
TL;DR
We address LG-CY mirror symmetry for the Klein quartic by computing Hochschild cohomology ${\mathsf{HH}}^*(f,G)$ for all symmetry subgroups $G\subseteq {\mathrm{GL}}_f$ preserving the Klein quartic polynomial $f$ and its pairing, including non-abelian actions. Using Berglund–Hübsch–Henningson duality and a central $\mathbb{Z}/2\mathbb{Z}$ extension to form $\widehat{G}$, we compare ${\mathsf{HH}}^*(f,G)$ with the orbifold cohomology $H^*(S_3)$ of the genus-3 curve $S_3$. A key result is that $\dim {\mathsf{HH}}^*(f,G) > \dim H^*(S_3)$ for $G\subseteq {\mathrm{SL}}_f$, while $\dim {\mathsf{HH}}^*(f,\widehat{G}) = \dim H^*(S_3)$ iff $G\cong V_4$, identifying the mirror partner as $(f,\widehat{V}_4)$. The work extends LG mirror symmetry beyond diagonal groups, providing an explicit classification of $G$ and detailed HH-structure, including several non-diagonal and non-abelian cases, and highlights the role of $\mathbb{Z}/2\mathbb{Z}$-extensions in achieving the CY-dual correspondence for curves.
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.