The Gamma Conjecture for Tropical Curves in Local Mirror Symmetry
Junxiao Wang
TL;DR
This paper proves the Gamma conjecture for local mirror symmetry by linking coherent sheaves and Lagrangian submanifolds via tropical curves, confirming their role in the Gross-Siebert model.
Contribution
It establishes a novel connection between tropical geometry and mirror symmetry, providing a proof of the Gamma conjecture in this context.
Findings
Proves Gamma conjecture for local mirror symmetry.
Relates tropical curves to central charges in mirror symmetry.
Confirms parameters in Gross-Siebert model as canonical coordinates.
Abstract
We relate a coherent sheaf supported on a holomorphic curve with its mirror Langrangian submanifold in local mirror symmetry through a tropical curve by interpreting their central charges using the combinatorial information of the tropical curve, which proves the Gamma conjecture for local mirror symmetry in this specific case. Furthermore, we put this description in the Gross-Siebert model of local mirror symmetry and confirm that the parameters in the Gross-Siebert model are the canonical coordinates in mirror symmetry.
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Taxonomy
TopicsNonlinear Waves and Solitons · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models