Mirror partner for a Klein quartic polynomial
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918267484897280 |
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| author | Basalaev, Alexey |
| author_facet | Basalaev, Alexey |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_12490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mirror partner for a Klein quartic polynomial Basalaev, Alexey Algebraic Geometry 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. |
| title | Mirror partner for a Klein quartic polynomial |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2406.12490 |