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

Journal article

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 = 4SU(2) gauge theory.

Authors: Bousseau, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Communications in Mathematical Physics
• Volume: 398
• Issue: 1
• Pages: 1-58
• Publication date: 2022-11-19
• Acceptance date: 2022-08-25
• DOI: 10.1007/s00220-022-04512-9
• EISSN: 1432-0916
• ISSN: 0010-3616
• Language: English
• Keywords: skein; bracket polynomial; connection (principal bundle); skein relation; torus; algebra over a field; structure constants; linear algebra basis; quantum algebra; signal processing quantization