TY - JOUR
T1 - Tropical Lagrangians in toric del-Pezzo surfaces
AU - Hicks, Jeffrey
N1 - This work was partially supported by NSF Grants DMS-1406274 and DMS-1344991 and by a Simons Foundation Grant (# 385573, Simons Collaboration on Homological Mirror Symmetry).
PY - 2021/1/6
Y1 - 2021/1/6
N2 - We look at how one can construct from the data of a dimer model a Lagrangian submanifold in (C∗)n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori LT2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair (CP2\ E, W) , Xˇ 9111. We find a symplectomorphism of CP2\ E interchanging LT2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on Xˇ 9111.
AB - We look at how one can construct from the data of a dimer model a Lagrangian submanifold in (C∗)n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori LT2 in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair (CP2\ E, W) , Xˇ 9111. We find a symplectomorphism of CP2\ E interchanging LT2 and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on Xˇ 9111.
U2 - 10.1007/s00029-020-00614-1
DO - 10.1007/s00029-020-00614-1
M3 - Article
AN - SCOPUS:85098847775
SN - 1022-1824
VL - 27
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 1
M1 - 3
ER -