Ideal webs, moduli spaces of local systems, and 3d Calabi–Yau categories

Abstract

A decorated surface S is an oriented surface with punctures, and a finite set of marked points on the boundary, considered modulo isotopy. We assume that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type Am or GL m, and gives rise to cluster coordinate systems on certain moduli spaces of G-local systems on S. These coordinate systems generalize the ones assigned in [FG1] to ideal triangulations of S. A bipartite graph W on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d Calabi–Yau A ∞-category C W with a cluster collection S W – a generating collection of spherical objects of special kind [KS1]. Let W be an ideal bipartite graph on S of type G.We define an extension ГG,S of the mapping class group of S, and prove that it acts by symmetries of the category CW. There is a family of open CY threefolds over the universal Hitchin base BG,S, whose intermediate Jacobians describe Hitchin’s integrable system [DDDHP], [DDP], [G], [KS3], [Sm]. We conjecture that the 3d CY category with cluster collection (C W, S W) is equivalent to a full subcategory of the Fukaya category of a generic threefold of the family, equipped with a cluster collection of special Lagrangian spheres. For G = SL 2 a substantial part of the story is already known thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others, see [BrS], [Sm]. We hope that ideal bipartite graphs provide special examples of the Gaiotto–Moore–Neitzke spectral networks [GMN4].