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

Abstract

We classify 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, by the work of Amiot (see also Burban and Buan), two main classes: one denoted by An,t called of type A, and the other denoted by Dn,t called of type D. Using the geometric model of torsion pairs in cluster categories of type A, or type D in Holm, we give a geometric description of torsion pairs in An,t or Dn,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.