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].
Dedicated to Maxim Kontsevich, for his 50th birthday
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Goncharov, A.B. (2017). Ideal Webs, Moduli Spaces of Local Systems, and 3d Calabi–Yau Categories. In: Auroux, D., Katzarkov, L., Pantev, T., Soibelman, Y., Tschinkel, Y. (eds) Algebra, Geometry, and Physics in the 21st Century. Progress in Mathematics, vol 324. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59939-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-59939-7_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-59938-0
Online ISBN: 978-3-319-59939-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)