The Gamma Conjecture in mirror symmetry relates the central charges of dual
objects. Mathematically, periods of a Lagrangian submanifold are related to
characteristic classes of the mirror coherent sheaf. In this talk, I will test the
Gamma Conjecture in the setting of local mirror symmetry. For a given coherent
sheaf on the canonical bundle of a smooth toric surface, I will identify a 3-cycle
in the mirror using tropical geometry by comparing its period with the central charge
of the coherent sheaf through the Gamma Conjecture. If time permits, I will also discuss
about some higher dimensional cases. This work is based on Ruddat and Siebert's work on the
period computation and is inspired by Abouzaid, Ganatra, Iritani and Sheridan's work on
the Gamma Conjecture.