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Skein algebras and quantized Coulomb branches

Dylan G. L. Allegretti, Peng Shan

Abstract

To a compact oriented surface of genus at most one with boundary, we associate a quantized $K$-theoretic Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima. In the case where the surface is a three- or four-holed sphere or a one-holed torus, we describe a relationship between this quantized Coulomb branch and the Kauffman bracket skein algebra of the surface. We formulate a general conjecture relating these algebras.

Skein algebras and quantized Coulomb branches

Abstract

To a compact oriented surface of genus at most one with boundary, we associate a quantized -theoretic Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima. In the case where the surface is a three- or four-holed sphere or a one-holed torus, we describe a relationship between this quantized Coulomb branch and the Kauffman bracket skein algebra of the surface. We formulate a general conjecture relating these algebras.
Paper Structure (25 sections, 26 theorems, 106 equations, 8 figures)

This paper contains 25 sections, 26 theorems, 106 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Two quantized algebras
  3. Summary of the main results
  4. Conjecture and generalizations
  5. The Kauffman bracket skein algebra
  6. The $\mathrm{SL}_2$-character variety
  7. The skein algebra
  8. Presentations of skein algebras
  9. Spherical DAHA of type $(C_1^\vee,C_1)$
  10. A representation of $\mathop{\mathrm{\operatorname{Sk}}}\nolimits_{A,\boldsymbol{\lambda}}(S_{0,4})$
  11. Spherical DAHA of type $A_1$
  12. A representation of $\mathop{\mathrm{\operatorname{Sk}}}\nolimits_{A,\lambda}(S_{1,1})$
  13. Quantized $K$-theoretic Coulomb branches
  14. The variety of triples
  15. Equivariant $K$-theory
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $(\widetilde{G},N)$ be the group and representation associated to the three-holed sphere $S_{0,3}$. Then there is a $\mathbb{C}$-algebra isomorphism

Figures (8)

  • Figure 1: The Kauffman bracket skein relations.
  • Figure 2: Additional relation associated to a boundary component.
  • Figure 3: Generators of $\mathop{\mathrm{\operatorname{Sk}}}\nolimits_A(S_{0,4})$.
  • Figure 4: Generators of $\mathop{\mathrm{\operatorname{Sk}}}\nolimits_A(S_{1,1})$.
  • Figure 5: A family of curves on $S_{0,4}$.
  • ...and 3 more figures

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Definition 2.1
  • Proposition 2.2: BP00, Theorem 3.1
  • Proposition 2.3: BP00, Theorem 2.1
  • Proposition 2.4: BS18, Theorem 2.7
  • Proposition 2.5
  • proof
  • ...and 40 more