Elementary Moves on Ideal Webs, Local Monodromies on Hitchin Fibrations, and Tropical Mutations on $\mathrm{K_0}$ Groups
Year: 2026
Type: preprint
Abstract: This paper builds on the work of Goncharov by studying the correspondence between topological spectral covers constructed from generic $A_{m-1}$-webs with rank $m\ge2$, Hitchin spectral curves over good Hitchin loci, and the Grothendieck groups $K_0(\mathcal{D}_Γ)$ associated with the quivers with potential $(Q_Γ,W_Γ)$ assigned to corresponding bipartite $A_{m-1}$-graphs. We show that the automorphisms on the non-frozen quiver lattices $Λ_Γ^{\circ}$ induced by compositions of elementary moves on ideal webs/bipartite graphs are the same as the Gauss--Manin monodromy actions on the $SL_m$ Hitchin fiber lattices $H_1(\operatorname{Prym}(Σ);\mathbb{Z})$ locally around discriminant loci. And the Gauss--Manin monodromy actions agree with the automorphisms on the Grothendieck groups $K_0(\mathcal{D}_Γ^{\circ})$ induced by compositions of mutations on the quivers with potential $(Q_Γ,W_Γ)$.
Source: arXiv (Cornell University)
Author: Liao, Yuhan
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: University of Melbourne
OpenAlex ID: W7224097490