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Monoidal categorification of genus zero skein algebras

Dylan G. L. Allegretti, Hyun Kyu Kim, Peng Shan

TL;DR

The article proves that the Kauffman bracket skein algebra of a genus-zero surface with boundary is isomorphic to a quantized K-theoretic Coulomb branch, thereby providing a monoidal categorification via the derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples. It constructs a polynomial representation of the skein algebra and compares it with a monopole-operator-based representation of the Coulomb branch, establishing a precise algebra isomorphism at the level of localized algebras and integral forms. The work also frames the skein algebra as the Grothendieck ring of a monoidal category, connecting to canonical bases and positivity, and outlines future directions for higher-genus surfaces and cluster-algebraic perspectives. Overall, the paper bridges quantum topology and geometric representation theory, offering tools to study positivity phenomena and canonical bases in skein algebras through categorification and Coulomb-branch techniques.

Abstract

We prove a conjecture of the first and third named authors relating the Kauffman bracket skein algebra of a genus zero surface with boundary to a quantized $K$-theoretic Coulomb branch. As a consequence, we see that our skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples with monoidal structure defined by the convolution product. We thus give a monoidal categorification of the skein algebra, partially answering a question posed by D. Thurston.

Monoidal categorification of genus zero skein algebras

TL;DR

The article proves that the Kauffman bracket skein algebra of a genus-zero surface with boundary is isomorphic to a quantized K-theoretic Coulomb branch, thereby providing a monoidal categorification via the derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples. It constructs a polynomial representation of the skein algebra and compares it with a monopole-operator-based representation of the Coulomb branch, establishing a precise algebra isomorphism at the level of localized algebras and integral forms. The work also frames the skein algebra as the Grothendieck ring of a monoidal category, connecting to canonical bases and positivity, and outlines future directions for higher-genus surfaces and cluster-algebraic perspectives. Overall, the paper bridges quantum topology and geometric representation theory, offering tools to study positivity phenomena and canonical bases in skein algebras through categorification and Coulomb-branch techniques.

Abstract

We prove a conjecture of the first and third named authors relating the Kauffman bracket skein algebra of a genus zero surface with boundary to a quantized -theoretic Coulomb branch. As a consequence, we see that our skein algebra arises as the Grothendieck ring of the bounded derived category of equivariant coherent sheaves on the Braverman-Finkelberg-Nakajima variety of triples with monoidal structure defined by the convolution product. We thus give a monoidal categorification of the skein algebra, partially answering a question posed by D. Thurston.
Paper Structure (38 sections, 36 theorems, 134 equations, 15 figures)

This paper contains 38 sections, 36 theorems, 134 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Skein algebras and quantized Coulomb branches
  3. Categorification of skein algebras
  4. Further directions
  5. Higher genus surfaces
  6. Canonical bases and positivity
  7. Relation to other categorifications
  8. Genus zero skein algebras and their generators
  9. The skein module
  10. The skein algebra
  11. Generators for skein algebras
  12. The associated graded algebra
  13. The quantum trace map
  14. Skeins from ribbon graphs
  15. Fusion rules
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\widetilde{G}=G\times F$ and $N$ be the group and representation associated to a surface $S$ of genus zero, and identify $\Bbbk\coloneqq\mathbb{C}[A^{\pm1},\lambda_i^{\pm1}]\cong\mathbb{C}[q^{\pm\frac{1}{2}},t_i^{\pm1}]$ by mapping $A\mapsto q^{-\frac{1}{2}}$ and $\lambda_i\mapsto t_i$ for all from the skein algebra to the quantized Coulomb branch.

Figures (15)

  • Figure 1: The Kauffman bracket skein relations.
  • Figure 2: Additional relation associated to a boundary component.
  • Figure 3: A genus zero surface.
  • Figure 4: Curves on a genus zero surface, $1\leq i, j\leq n$.
  • Figure 5: Curves in the proof of Proposition \ref{['prop:generators']}.
  • ...and 10 more figures

Theorems & Definitions (72)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: C22, Theorem 5.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • ...and 62 more