Geometric quantization via SYZ transforms

The so-called quantization problem in geometric quantization is asking whether the space of wave functions is independent of the choice of polarization. In this paper, we apply SYZ transforms to solve the quantization problem in two cases: 1) semi-flat Lagrangian torus fibrations over complete compact integral affine manifolds, and 2) projective toric manifolds. More precisely, we prove that the space of wave functions associated to the real polarization is canonically isomorphic to that associated to a complex polarization via SYZ transforms in both cases.