Skip to main content

Ideal Webs, Moduli Spaces of Local Systems, and 3d Calabi–Yau Categories

Part of the Progress in Mathematics book series (PM,volume 324)

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].

Mathematics Subject Classification (2010).

  • 14M99

Keywords

  • Calabi–Yau categories
  • cluster coordinates
  • bipartite graphs
  • Fukaya categories

Dedicated to Maxim Kontsevich, for his 50th birthday

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-59939-7_2
  • Chapter length: 67 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-59939-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   169.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. B. Goncharov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints 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