Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects

TL;DR

This paper classifies (co)torsion pairs in certain finite 2-Calabi-Yau triangulated categories with maximal rigid objects, using geometric models and Ptolemy diagrams, and determines their hearts.

Contribution

It provides a complete classification of (co)torsion pairs in categories of type A and D, introducing geometric descriptions and counting methods.

Findings

Classified (co)torsion pairs in categories of type A and D.

Developed geometric models using Ptolemy diagrams.

Determined hearts of (co)torsion pairs via quivers and relations.

Abstract

We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\mathcal{A}_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\mathcal{A}_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to count the number of (co)torsion pairs in these categories. Finally, we determine the hearts of (co)torsion pairs in all finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting via quivers and relations.