T-structures and torsion pairs in a 2-Calabi-Yau triangulated category

For a Calabi-Yau triangulated category C of Calabi-Yau dimension d with a d−cluster tilting subcategory T , the decomposition of C is determined by the decomposition of T satisfying ”vanishing condition” of negative extension groups, namely, C = ⊕i∈ICi, where Ci, i ∈ I are triangulated subcategories, if and only if T = ⊕i∈ITi, where Ti, i ∈ I are subcategories with HomC(Ti[t],T j) = 0,∀1 ≤ t ≤ d − 2 and i , j. This induces that for any two cluster tilting objects T,T ′ in a 2−Calabi-Yau triangulated category C, the Gabriel quivers of endomorphism algebra EndCT of T is connected if and only if so is EndCT ′. As an application, we prove that indecomposable 2−Calabi-Yau triangulated categories with cluster tilting objects have no non-trivial t-structures and no non-trivial co-t-structures. This allows us to give a classification of torsion pairs in those triangulated categories, and to determine further the hearts of torsion pairs in the sense of Nakaoka, which are equivalent to the module categories over the endomorphism algebras of the cores of the torsion pairs.