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Skein and cluster algebras of punctured surfaces

Enhan Li

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 $U(Σ)$.

Skein and cluster algebras of punctured surfaces

Abstract

We prove the full Fock--Goncharov conjecture for , the -cluster variety of the moduli of decorated twisted -local systems on triangulable surfaces with at least 2 punctures. Equivalently, we show that the tagged skein algebra , or the middle cluster algebra , coincides with the upper cluster algebra . Inspired by the work of Shen--Sun--Weng, we introduce the localized cluster variety as the algebraic version of the decorated Teichmüller space . We show its global section equals the classical Roger--Yang skein algebra , thereby providing a quantization of in terms of the Roger--Yang skein algebra . As a consequence of our geometric characterizations, we deduce normality and the Gorenstein property of the tagged skein algebra and the classical Roger--Yang skein algebra , as well as finite generation of upper cluster algebra .

Paper Structure

This paper contains 23 sections, 38 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Fock--Goncharov duality conjecture
  3. Decorated Teichmüller space and Roger--Yang's program
  4. (Lack of) local acyclicity and estimation of Krull dimension
  5. Preliminaries
  6. Roger--Yang and tagged skein algebras
  7. Cluster algebras and cluster varieties associated to surfaces
  8. Freezing, cutting and localization
  9. Fock--Goncharov duality conjecture and compatibility of skein and cluster algebras
  10. Canonical bases of cluster algebras
  11. Birational geometry of skein and cluster algebras
  12. Geometry of $\mathrm{Spec}(Sk^{ta}(\Sigma))$ and $\mathrm{Spec}(Sk^{RY}(\Sigma))$
  13. Cluster variety and the spectrum of skein algebras
  14. Estimation of Krull dimension
  15. Geometric intersection number and skein product
  16. ...and 8 more sections

Key Result

Theorem 1.1

(1) For a triangulable surface $\Sigma=\Sigma_{g,p}$ with $p\geq 2$, there is an open immersion $\mathcal{A}\to\mathrm{Spec}(Sk^{ta}(\Sigma))$ from the cluster variety to the spectrum of tagged skein algebra, which is a morphism between integral Noetherian schemes. The complement of the image of $\m

Theorems & Definitions (83)

  • Theorem 1.1: Theorem \ref{['mainthmproof']}, Theorem \ref{['oncepuncturedconj']}
  • Corollary 1.2: Theorem \ref{['mainthmproof']}
  • Theorem 1.3: Corollary \ref{['localized']}
  • Proposition 1.4: Proposition \ref{['RYtaggedbirational']}
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.6
  • Definition 2.7: Tagged arcs, compatibility and tagged curve systems
  • Remark 2.8
  • ...and 73 more