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arXiv · 2503.15037

Skein and cluster algebras of punctured surfaces

Enhan Li
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Abstract

We prove the full Fock--Goncharov conjecture for $\mathcal{A}_{SL_2,Σ_{g,p}}$, the $\mathcal{A}$-cluster variety of the moduli of decorated twisted $SL_2$-local systems on triangulable surfaces $Σ_{g,p}$ with at least 2 punctures. Equivalently, we show that the tagged skein algebra $Sk^{ta}(Σ)$, or the middle cluster algebra $\mathrm{mid}(\mathcal{A})$, coincides with the upper cluster algebra $U(Σ)$. Inspired by the work of Shen--Sun--Weng, we introduce the localized cluster variety $\mathring{\mathcal{A}}$ as the algebraic version of the decorated Teichmüller space $\mathcal{T}^d(Σ)$. We show its global section $Γ(\mathring{\mathcal{A}},\mathcal{O}_{\mathring{\mathcal{A}}})$ equals the classical Roger--Yang skein algebra $Sk^{RY}_{q\to1}(Σ)$, thereby providing a quantization of $\mathcal{T}^d(Σ)$ in terms of the Roger--Yang skein algebra $Sk^{RY}_q(Σ)$. As a consequence of our geometric characterizations, we deduce normality and the Gorenstein property of the tagged skein algebra $Sk^{ta}(Σ)$ and the classical Roger--Yang skein algebra $Sk^{RY}_{q\to1}(Σ)$, as well as finite generation of upper cluster algebra $U(Σ)$.

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BibTeXRIS

Enhan Li. 2025-12-25. Skein and cluster algebras of punctured surfaces. https://arxiv.org/abs/2503.15037

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