  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.
