Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates
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| Format: | Preprint |
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2022
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| _version_ | 1866916818425217024 |
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| author | Ishibashi, Tsukasa Kano, Shunsuke |
| author_facet | Ishibashi, Tsukasa Kano, Shunsuke |
| contents | Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,Σ}$, which we call unbounded $\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs. We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $\mathcal{P}_{PGL_3,Σ}$ of Goncharov--Shen, we describe a geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with pinnings via ``shearings''. We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads to a quantum duality map. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_08947 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates Ishibashi, Tsukasa Kano, Shunsuke Geometric Topology Quantum Algebra 13F60, 57K20, 57K31 Generalizing the work of Fock--Goncharov on rational unbounded laminations, we give a geometric model of the tropical points of the cluster variety $\mathcal{X}_{\mathfrak{sl}_3,Σ}$, which we call unbounded $\mathfrak{sl}_3$-laminations, based on the Kuperberg's $\mathfrak{sl}_3$-webs. We introduce their tropical cluster coordinates as an $\mathfrak{sl}_3$-analogue of the Thurston's shear coordinates associated with any ideal triangulation. As a tropical analogue of gluing morphisms among the moduli spaces $\mathcal{P}_{PGL_3,Σ}$ of Goncharov--Shen, we describe a geometric gluing procedure of unbounded $\mathfrak{sl}_3$-laminations with pinnings via ``shearings''. We also investigate a relation to the graphical basis of the $\mathfrak{sl}_3$-skein algebra [IY23], which conjecturally leads to a quantum duality map. |
| title | Unbounded $\mathfrak{sl}_3$-laminations and their shear coordinates |
| topic | Geometric Topology Quantum Algebra 13F60, 57K20, 57K31 |
| url | https://arxiv.org/abs/2204.08947 |