Skein-valued mirror curves for toric CY3 strips

TL;DR

This paper demonstrates that the enumeration of all-genus holomorphic curves with boundary on a specific brane in certain toric Calabi-Yau 3-folds can be obtained through a skein-valued quantization of the mirror curve, providing explicit formulas.

Contribution

It introduces a skein-valued quantization framework for mirror curves that determines all-genus holomorphic curve counts in toric CY3 strips, with explicit equations and solutions.

Findings

Holomorphic curve counts are governed by skein-valued quantization.

Explicit mirror curve equations and solutions are provided.

The approach links curve enumeration with skein theory in mirror symmetry.

Abstract

For a smooth semi-projective toric Calabi-Yau 3-fold containing no compact surface, we show the count of all-genus holomorphic curves with boundary on a single Aganagic-Vafa brane is annihilated by a skein-valued quantization of the mirror curve, and that this determines the count. We give explicit expressions for the equation and its solution.