Monoidal categorification of genus zero skein algebras

TL;DR

This paper establishes a monoidal categorification of the genus zero skein algebra by linking it to a quantized K-theoretic Coulomb branch and the derived category of equivariant coherent sheaves on a specific variety.

Contribution

It proves a conjecture connecting skein algebras with Coulomb branches and provides a monoidal categorification, addressing a question by Thurston.

Findings

Skein algebra is isomorphic to the Grothendieck ring of a derived category.

The categorification uses convolution product on equivariant coherent sheaves.

Connects topological invariants with algebraic and geometric structures.

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.