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

TL;DR

This paper introduces ideal bipartite graphs on decorated surfaces to construct cluster coordinate systems, relate them to 3d Calabi-Yau categories, and explore their connections to Hitchin systems and Fukaya categories.

Contribution

It develops a new framework linking ideal bipartite graphs to cluster coordinates, 3d Calabi-Yau categories, and the geometry of Hitchin systems, extending prior triangulation-based approaches.

Findings

Construction of cluster coordinate systems from bipartite graphs.

Association of bipartite graphs with quivers and 3d CY categories.

Conjectured equivalence with Fukaya categories of certain threefolds.

Abstract

A decorated surface S is an oriented surface with punctures and a finite set of marked points on the boundary, such 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 A, and gives rise to cluster coordinate systems on certain spaces of G-local systems on S. These coordinate systems generalize the ones assigned to ideal triangulations of S. A bipartite graph on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d CY category with a cluster collection of spherical objects. Given an ideal bipartite graph on S, we define an extension of the mapping class group of S which acts by symmetries of the category. There is a family of open CY 3-folds over the universal Hitchin base, whose intermediate Jacobians describe the Hitchin system. We conjecture that the 3d CY category with cluster collection 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 SL(2) a substantial part of the story is already known thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We hope that ideal bipartite graphs provide special examples of the Gaiotto-Moore-Neitzke spectral networks.