  Spectral networks and non-abelianization were introduced by
Gaiotto-Moore-Neitzke and they have many applications in mathematics and
physics. In a recent work by Nho, he proved that the non-abelianization of an
almost flat local system over the spectral curve of a meromorphic quadratic
differential is actually the same as the family Floer construction. Based on
the mirror symmetry philosophy, it is then natural to ask how holomorphic
vector bundles arise from spectral networks and non-abelianization. In this
paper, we construct toric vector bundles on complete toric surfaces via
spectral networks and non-abelianization.
