Strong positivity for the skein algebras of the 4-punctured sphere and of the 1-punctured torus

2020/09/04 by Pierrick Bousseau, Bousseau, Pierrick · 2 citations
Mathematics · #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2009.02266

openalex publication_date 2020/09/04 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL₂ character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov-Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the N=2 N_f=4 SU(2) gauge theory.

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