# A correspondence between rigid modules over path algebras and simple curves on Riemann surfaces

## TL;DR

This paper explores a conjectural link between algebraic structures called rigid modules over path algebras and geometric objects like simple curves on Riemann surfaces, establishing the connection for specific quivers.

## Contribution

It introduces a conjectural correspondence between rigid modules and simple curves, proving it for 2-complete rank 3 quivers, bridging algebra and geometry.

## Key findings

- Established the correspondence for 2-complete rank 3 quivers
- Proposed a conjectural link between modules and curves
- Bridged algebraic and geometric perspectives

## 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 2-complete rank 3 quivers.