Abstract

Spectral networks and non-abelianization were introduced by Gaiotto-Moore-Neitzke and theyhave many applications in mathematics and physics. in a recent work by Nho, he proved that thenon-abelianization of an almost flat local system over the spectral curve of a meromorphicguadratic differential is actually the same as the family Floer construction. Based on the mirrorsymmetry philosophy, it is then natural to ask how holomorphic vector bundles arise from spectralnetworks and non-abelianization. In this paper, we construct toric vector bundles on complete toricsurfaces via spectral networks and non-abelianization arising from Lagrangian multi-sections.