A geometric model for the module category of a string algebra

TL;DR

This paper introduces a geometric framework using punctured Riemann surfaces with labels to model string algebras and their modules, providing a new classification method for support tau-tilting modules.

Contribution

It presents a novel geometric construction for string algebras and their modules, extending existing classifications to a broader class using surface dissections.

Findings

Geometric construction of string algebras via punctured Riemann surfaces.

Classification of support tau-tilting modules using arcs in tiled surfaces.

Recovery of known classifications for gentle string algebras.

Abstract

In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a classification of support tau-tilting modules in terms of arcs in such a tiled surface. In the case when the string algebra is gentle, we recover the classification given arXiv:2004.11136.