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

Abstract

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the $S{L}_2$ 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 $\mathcal{N}=2$ $N_f=4$ $SU(2)$ gauge theory.

59 pages, 17 figures. Final version published in Communications in Mathematical Physics

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