We perform a test of the Gamma Conjecture in the setting of local mirror
symmetry. Given a coherent sheaf on the canonical bundle of a smooth toric
surface defined by an ample curve, we identify a 3-cycle in the mirror by
lifting a tropical 1-cycle in the base space and compute its period. We show
that this period agrees with the central charge of the given coherent sheaf, as
conjectured by Hosono.Comment: 31 pages, 12 figure

Wang, Junxiao

English

arXiv

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.Comment: 46 pages, 13 figures; generalization to the case of tropical curves
  in higher dimensional variety; computation of the subleading terms; details
  about the construction of Lagrangian submanifolds near the end

The Gamma Conjecture for Tropical Curves in Local Mirror Symmetry

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