Toric vector bundles, non-abelianization, and spectral networks

Abstract

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 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 arising from Lagrangian multi-sections. As an application, we deduce that the moduli space of rank 2 toric vector bundles over toric surfaces admit an $A$-type $\mathcal{X}$-cluster structure.

arXiv comment21 pages, 11 figures. The introduction and the body of the paper are reorganized based on the useful comments of two anonymous referees. Some typos are fixed, and a new section is added to prove that the moduli space of rank 2 toric vector bundles over toric surfaces admits a cluster structure. Comments are welcome!

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