Elementary Moves on Ideal Webs, Local Monodromies on Hitchin Fibrations, and Tropical Mutations on $\mathrm{K_0}$ Groups

arXiv:2610.35898v1 Announce Type: new 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 $K_0(\mathcal{D}_{\Gamma})$ associated with the quivers with potential $(Q_{\Gamma},W_{\Gamma})$ assigned to corresponding bipartite $A_{m-1}$-graphs. We show that the automorphisms on the non-frozen quiver lattices $\Lambda_{\Gamma}^{\circ}$ 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 $H_1(\operatorname{Prym}(\Sigma);\mathbb{Z})$ locally around discriminant loci. And the Gauss--Manin monodromy actions agree with the automorphisms on the Grothendieck groups $K_0(\mathcal{D}_{\Gamma}^{\circ})$ induced by compositions of mutations on the quivers with potential $(Q_{\Gamma},W_{\Gamma})$.

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