Strong Positivity for the Skein Algebras of the 4-Punctured Sphere and of the 1-Punctured Torus

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL 2 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 appli...

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Bibliographic Details
Published in:Communications in mathematical physics 2023-02, Vol.398 (1), p.1-58
Main Author: Bousseau, Pierrick
Format: Article
Language:English
Subjects:
Algebra
Classical and Quantum Gravitation
Complex Systems
Curves
Gauge theory
M theory
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Skeins
Theoretical
Topology
Toruses
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Summary:The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL 2 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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-022-04512-9