Intersections of Dual \(SL_3\)-Webs

We introduce a topological intersection number for an ordered pair of \(\operatorname{SL}_3\)-webs on a decorated surface. Using this intersection pairing between reduced \((\operatorname{SL}_3,\mathcal{A})\)-webs and a collection of \((\operatorname{SL}_3,\mathcal{X})\)-webs associated with the Foc...

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Bibliographic Details
Published in:arXiv.org 2023-11
Main Authors: Shen, Linhui, Sun, Zhe, Weng, Daping
Format: Article
Language:English
Subjects:
Combinatorial analysis
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Summary:We introduce a topological intersection number for an ordered pair of \(\operatorname{SL}_3\)-webs on a decorated surface. Using this intersection pairing between reduced \((\operatorname{SL}_3,\mathcal{A})\)-webs and a collection of \((\operatorname{SL}_3,\mathcal{X})\)-webs associated with the Fock--Goncharov cluster coordinates, we provide a natural combinatorial interpretation of the bijection from the set of reduced \((\operatorname{SL}_3,\mathcal{A})\)-webs to the tropical set \(\mathcal{A}^+_{\operatorname{PGL}_3,\hat{S}}(\mathbb{Z}^t)\), as established by Douglas and Sun in \cite{DS20a, DS20b}. We provide a new proof of the flip equivariance of the above bijection, which is crucial for proving the Fock--Goncharov duality conjecture of higher Teichm\"uller spaces for \(\operatorname{SL}_3\).
ISSN:2331-8422