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We introduce a sheaf-theoretic stability condition for finite acyclicquivers. Our main result establishes that for representations of affine type$\widetilde{\mathbb{A}}$ quivers, there is a precise relationship between theassociated Harder-Narasimhan filtration and the barcode of the periodic zigzagpersistence module obtained by unwinding the underlying quiver.

Harder-Narasimhan Filtrations and Zigzag Persistence