Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum duality maps, skein algebras and their ensemble compatibility

Tsukasa Ishibashi, Hiroaki Karuo

TL;DR

The paper develops a skein-theoretic realization of quantum duality maps for general marked surfaces by leveraging reduced stated skein algebras and skein lifting. It establishes an ensemble-compatible framework linking the A- and X- sides of quantum cluster varieties via trace and cutting maps, and proves strong positivity of the bracelets basis on key small surfaces (marked disks and annuli). The construction is anchored in Lê’s and Muller’s skein theories and connects to the Mandel–Qin quantum theta basis, providing a robust bridge between skein algebras, quantum Teichmüller theory, and cluster algebras. These results deepen the understanding of quantum dualities, positivity phenomena, and the algebraic structure of moduli spaces of framed local systems, with potential implications for quantization and representation theory in low-rank settings.

Abstract

We generalize the quantum duality map $\mathbb{I}_{\mathcal{A}}$ of Allegretti--Kim [AK17] for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $\mathbb{I}_{\mathcal{X}}$ on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral $\mathcal{A}$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.

Quantum duality maps, skein algebras and their ensemble compatibility

TL;DR

The paper develops a skein-theoretic realization of quantum duality maps for general marked surfaces by leveraging reduced stated skein algebras and skein lifting. It establishes an ensemble-compatible framework linking the A- and X- sides of quantum cluster varieties via trace and cutting maps, and proves strong positivity of the bracelets basis on key small surfaces (marked disks and annuli). The construction is anchored in Lê’s and Muller’s skein theories and connects to the Mandel–Qin quantum theta basis, providing a robust bridge between skein algebras, quantum Teichmüller theory, and cluster algebras. These results deepen the understanding of quantum dualities, positivity phenomena, and the algebraic structure of moduli spaces of framed local systems, with potential implications for quantization and representation theory in low-rank settings.

Abstract

We generalize the quantum duality map of Allegretti--Kim [AK17] for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral -laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus.
Paper Structure (40 sections, 42 theorems, 115 equations, 15 figures)

This paper contains 40 sections, 42 theorems, 115 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Background: quantum Teichmüller theory and the skein algebra
  3. A construction of quantum duality maps via stated skein algebras
  4. Skein proof of structure constant positivity for small surfaces
  5. Marked surfaces and associated quantum tori
  6. Marked surfaces and their triangulations
  7. Quantum torus
  8. Matrices and quantum tori associated with triangulations
  9. The Chekhov--Fock square-root algebra
  10. Quantum cluster Poisson torus
  11. Quantum cluster $K_2$-torus
  12. Skein algebras
  13. $\mathbb{M}_\partial$-tangles and $\mathbb B$-tangles
  14. Picture conventions.
  15. Muller skein algebras
  16. ...and 25 more sections

Key Result

Theorem 1

When $\Sigma$ has no interior marked points, the map eq:our_duality fits into the following commutative diagram (in the category of sets): \begin{tikzcd} \A_\Sigma(\bZ^\sfT) \ar[r,"\bI_\A"] \ar[d,"\check{p}_\Sigma^\sfT"'] & \cO_v(\X_\Sigma) \ar[d,"p_\Sigma^\ast"] \\ \X_\Sigma(\bZ^\sfT) \ar[r

Figures (15)

  • Figure 1: Left: Reidemeister move II, Middle: Reidemeister move II', Right: Reidemeister move III
  • Figure 2: A bad arc
  • Figure 3: The generators of the Chekhov--Fock algebras
  • Figure 4: The splitting along an ideal arc $\alpha$.
  • Figure 5: The operation from left to right is "moving right"
  • ...and 10 more figures

Theorems & Definitions (118)

  • Theorem 1: Ensemble compatibility: \ref{['thm:compatibility']}
  • Theorem 2: \ref{['thm:skein_q-Poisson_isom']}
  • Theorem 3: \ref{['thm_skein', 'thm_S_A']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 108 more