Toric Vector Bundles, Non-abelianization, and Spectral Networks

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...

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Bibliographic Details
Published in:International mathematics research notices 2024-12, Vol.2024 (24), p.14576-14599
Main Author: Suen, Yat-Hin
Format: Article
Language:English
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Summary: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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnae250