The cluster category of a surface with punctures via group actions

TL;DR

This paper constructs a surface covering for a punctured surface, analyzes the group actions on associated quivers and Ginzburg algebras, and describes indecomposable objects in cluster categories via surface curves.

Contribution

It introduces a method to relate cluster categories of punctured surfaces to those of unpunctured surfaces through group actions and coverings, providing a complete classification of indecomposables.

Findings

Constructed a punctured surface cover with a group action.

Proved the Ginzburg dg algebras are skew group algebras up to Morita equivalence.

Described indecomposable objects in cluster categories via surface curves.

Abstract

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient by this action recovers the original surface. We show that the group acts on the quivers with potentials associated to the surfaces, and that their Ginzburg dg algebras are skew group algebras of each other, up to Morita equivalence. We then use these results to construct functors between the generalized cluster categories associated to the triangulations. This allows us to give a complete description of the indecomposable objects of these categories in terms of curves on the surface, when the surface has punctures and non-empty boundary.