[Paper Review] Holonomy and (stated) skein algebras in combinatorial quantization

Matthieu Faitg|arXiv (Cornell University)|Mar 19, 2020

Advanced Algebra and Geometry4 citations

TL;DR

This paper introduces a holonomy map in the combinatorial quantization framework of the algebra $\mathcal{L}_{g,n}(H)$, generalizing tangle and link invariants via tensor-valued assignments. It establishes that the stated skein algebra of a compact oriented surface with one boundary edge is isomorphic to $\mathcal{L}_{g,n}(U_{q^2}(\mathfrak{sl}_2))$, providing a geometric realization of the vacuum representation and unifying skein theory with quantized character varieties.

ABSTRACT

The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and quantizes the character variety of the Riemann surface $Σ_{g,n}\!\setminus\! D$ ($D$ is an open disk). In this article we define a holonomy map in that quantized setting, which associates a tensor with components in $\mathcal{L}_{g,n}(H)$ to tangles in $(Σ_{g,n}\!\setminus\!D) imes [0,1]$, generalizing previous works of Buffenoir-Roche and Bullock-Frohman-Kania-Bartoszynska. We show that holonomy behaves well for the stack product and the action of the mapping class group; then we specialize this notion to links in order to define a generalized Wilson loop map. Thanks to the holonomy map, we give a geometric interpretation of the vacuum representation of $\mathcal{L}_{g,0}(H)$ on $\mathcal{L}_{0,g}(H)$. Finally, the general results are applied to the case $H=U_{q^2}(\mathfrak{sl}_2)$ in relation to skein theory and the most important consequence is that the stated skein algebra of a compact oriented surface with just one boundary edge is isomorphic to $\mathcal{L}_{g,n}\big( U_{q^2}(\mathfrak{sl}_2) \big)$. Throughout the paper we use a graphical calculus for tensors with coefficients in $\mathcal{L}_{g,n}(H)$ which makes the computations and definitions very intuitive.

Motivation & Objective

• To define a holonomy map assigning tensors in $\mathcal{L}_{g,n}(H)$ to tangles in $\Sigma_{g,n} \setminus D \times [0,1]$, generalizing earlier constructions.
• To show that this holonomy map is compatible with the stack product and the action of the mapping class group.
• To specialize the holonomy to links and define a generalized Wilson loop map in the quantized setting.
• To give a geometric interpretation of the vacuum representation of $\mathcal{L}_{g,0}(H)$ on $\mathcal{L}_{0,g}(H)$ using the holonomy construction.
• To establish an isomorphism between the stated skein algebra of a surface with one boundary edge and $\mathcal{L}_{g,n}(U_{q^2}(\mathfrak{sl}_2))$.

Proposed method

• The holonomy map is constructed using a graphical calculus for tensors with coefficients in $\mathcal{L}_{g,n}(H)$, enabling intuitive computation and definition.
• The framework leverages the algebraic structure of $\mathcal{L}_{g,n}(H)$, originally introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche, to quantize character varieties of punctured surfaces.
• The holonomy is shown to be compatible with the stack product by verifying consistency under concatenation of tangles.
• The mapping class group action is preserved by demonstrating equivariance under surface automorphisms.
• The Wilson loop map is defined by restricting the holonomy to links, yielding a quantum trace-like invariant.
• The key isomorphism is proven by showing that the stated skein algebra of a surface with one boundary edge satisfies the same relations as $\mathcal{L}_{g,n}(U_{q^2}(\mathfrak{sl}_2))$.

Experimental results

Research questions

• RQ1 How can a holonomy map be defined in the combinatorial quantization setting of $\mathcal{L}_{g,n}(H)$ for tangles in $\Sigma_{g,n} \setminus D \times [0,1]$?
• RQ2 Does the holonomy map respect the stack product and the action of the mapping class group on the surface?
• RQ3 Can the holonomy construction be specialized to yield a generalized Wilson loop map for links in the quantized setting?
• RQ4 What is the geometric interpretation of the vacuum representation of $\mathcal{L}_{g,0}(H)$ on $\mathcal{L}_{0,g}(H)$ via the holonomy map?
• RQ5 Is the stated skein algebra of a compact oriented surface with a single boundary edge isomorphic to $\mathcal{L}_{g,n}(U_{q^2}(\mathfrak{sl}_2))$?

Key findings

• The holonomy map is well-defined and respects the stack product and mapping class group action, generalizing previous tangle invariants.
• The holonomy construction yields a generalized Wilson loop map for links in $\Sigma_{g,n} \setminus D \times [0,1]$.
• The vacuum representation of $\mathcal{L}_{g,0}(H)$ on $\mathcal{L}_{0,g}(H)$ is geometrically realized via the holonomy map.
• The stated skein algebra of a compact oriented surface with one boundary edge is isomorphic to $\mathcal{L}_{g,n}(U_{q^2}(\mathfrak{sl}_2))$.
• The isomorphism is established through a detailed comparison of algebraic relations and graphical calculus, with a supplementary appendix providing a full proof of Theorem 4.4.