A Correspondence between Rigid Modules Over Path Algebras and Simple Curves on Riemann Surfaces

Abstract

We propose a conjectural correspondence between the set of rigid indecomposable modules over the path algebras of acyclic quivers and the set of certain non-self-intersecting curves on Riemann surfaces, and prove the correspondence for the two-complete rank 3 quivers.

Keywords:

• quiver representation
• real Schur root
• Riemann surface

Acknowledgments

We thank Cheol-Hyun Cho, Christof Geiss, Ralf Schiffler, Hugh Thomas, Pavel Tumarkin, Jerzy Weyman, and Nathan Williams for helpful discussions. We also thank an anonymous referee for letting us know of [Crawley-Boevey 92]. K.-H. L. gratefully acknowledges support from the Simons Center for Geometry and Physics at which some of the research for this article was performed.

Notes

1 After the first version of this paper was posted on the arXiv, Felikson and Tumarkin [Felikson and Tumarkin 17] proved Conjecture 1.1 for all 2-complete quivers. Moreover they characterized c-vectors in the same seed, using a collection of pairwise non-crossing admissible curves satisfying a certain word property.

2 The punctured discs appeared in Bessis’ work [Bessis 06]. For better visualization, here we prefer to use an alternative description using compact Riemann surfaces with one or two marked points.

Additional information

Funding

K.-H. L. work was partially supported by a grant from the Simons Foundation (#318706). K. L. work was partially supported by the University of Nebraska–Lincoln, Korea Institute for Advanced Study, AMS Centennial Fellowship, and NSA grant H98230-16-1-0059.