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... standard heart HΓ generated by simple Γ-modules Se, for e ∈ Q0, each of which is a 3-spherical object. Recall that an object S is N -spherical when Hom•(S, S) = k ⊕ k[−N ]. Moreover, we recall (e.g. =-=[27]-=-) a distinguished family of auto-equivalences of Dfd(Γ). 16 THOMAS BRÜSTLE AND YU QIU Definition 4.3. The twist functor φ of a spherical object S is defined by φS(X) = Cone ( X → S ⊗Hom•(X,S)∨ ) [−1]...

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...ide S. We do not consider closed surfaces in this paper, that is we assume S to have a non-empty boundary. Studying the relation between Teichmüller theory and cluster algebras, it has been shown in =-=[15, 12, 13]-=- that the flip of an arc in a (ideal) triangulation corresponds to a mutation of a cluster variable. However, self-folded triangles do not permit to flip the internal arc, whereas the mutation of a cl...

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... C(S) (1.1) The category C(S) admits a distinguished automorphism, the Auslander Reiten translation, which in this context coincides with the suspension functor of the triangulated category C(S), see =-=[18]-=-. The first aim of this article is to show that the Auslander-Reiten translation can be realized by an element in the tagged mapping class group MCG×(S). For each boundary component Y with m marked po...

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...th properties of the categorification C(S) of the cluster algebra A(S): LabardiniFragoso associated in [21] a quiver with potential to the marked surface S which allows to use Amiot’s construction in =-=[1]-=- to define a 2-Calabi-Yau category C(S). It is shown in [9] that the cluster-tilting objects in C(S) correspond bijectively to the clusters of the cluster algebra A(S). Moreover, the set A×(S) of tagg...

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...ide S. We do not consider closed surfaces in this paper, that is we assume S to have a non-empty boundary. Studying the relation between Teichmüller theory and cluster algebras, it has been shown in =-=[15, 12, 13]-=- that the flip of an arc in a (ideal) triangulation corresponds to a mutation of a cluster variable. However, self-folded triangles do not permit to flip the internal arc, whereas the mutation of a cl...

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...r instance, the quiver for a triangle is shown in Figure 2. • The potential W =W (T×) is defined as a sum of certain cycles in the complete path algebra k̂Q, where k is an algebraic closed field, see =-=[8]-=-. If ∂S 6= ∅, this potential is rigid (and thus non-degenerate) by [21]. 2.2. Jacobian algebra and Ginzburg algebra. Let Q be a finite quiver and W a potential on Q. The Jacobian algebra of the quiver...

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... puncture (i.e. of type A or D). Key words: Mapping class group, Auslander-Reiten translation, triangulated surface, cluster theory, braid group 1. Introduction Fomin, Shapiro and Thurston studied in =-=[14]-=- the cluster combinatorics of a marked surface, that is, an oriented surface S with a finite set of marked points M on the boundary and a finite set of punctures P inside S. We do not consider closed ...

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...hange graph of C(S), which can be obtained from CEG∗(S) by replacing each unoriented edge with a 2-cycle. By [1] (cf. [20, Theorem 8.6]), there is a map υ : EG◦(D(S))→ CEG(S) (4.3) which induces (cf. =-=[16]-=- for the general case and [20] for the acyclic case) an isomorphism CEG(S) ∼= EG◦(D(S))/BrS. More precisely, the map υ is defined as follows. Let H be a heart in EG◦(D(S)) which corresponds to a t-str...

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...imultaneous change of tagging. As one of the main results of this paper we show in Theorem 3.8 that the tagged rotation ̺ ∈ MCG×(S) on A ×(S) becomes the shift [1] on C×(S). This result is known from =-=[26]-=- for the case of a punctured disc and from [5] for all unpunctured surfaces. In fact, the cluster category of an unpunctured surface is explicitly given by a combinatorial description of string and ba...

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... be obtained from CEG∗(S) by replacing each unoriented edge with a 2-cycle. By [1] (cf. [20, Theorem 8.6]), there is a map υ : EG◦(D(S))→ CEG(S) (4.3) which induces (cf. [16] for the general case and =-=[20]-=- for the acyclic case) an isomorphism CEG(S) ∼= EG◦(D(S))/BrS. More precisely, the map υ is defined as follows. Let H be a heart in EG◦(D(S)) which corresponds to a t-structure P in Dfd(Γ). Lift P to ...

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...lgebra A(S): LabardiniFragoso associated in [21] a quiver with potential to the marked surface S which allows to use Amiot’s construction in [1] to define a 2-Calabi-Yau category C(S). It is shown in =-=[9]-=- that the cluster-tilting objects in C(S) correspond bijectively to the clusters of the cluster algebra A(S). Moreover, the set A×(S) of tagged arcs in S corresponds bijectively to the set C×(S) of re...

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...in results of this paper we show in Theorem 3.8 that the tagged rotation ̺ ∈ MCG×(S) on A ×(S) becomes the shift [1] on C×(S). This result is known from [26] for the case of a punctured disc and from =-=[5]-=- for all unpunctured surfaces. In fact, the cluster category of an unpunctured surface is explicitly given by a combinatorial description of string and band objects, and the shift functor [1] is well-...

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...on-empty boundary are connected by a sequence of flips. In [22] it is further shown that the corresponding quivers with potential are also related by a sequence of mutations, and it follows then from =-=[17]-=- that the corresponding categories Dfd(Γ) are equivalent as triangulated categories, likewise for C(Γ). We therefore write simply D(S) = Dfd(Γ) and C(S) = C(Γ) in this case. Moreover, we denote by CEG...

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... exchange graph of reachable cluster tilting sets of C(S), that is, the connected unoriented graph whose vertices are the reachable cluster tilting sets and whose edges are the mutations. We refer to =-=[4]-=- for more details on and equivalent definitions of the graph CEG∗(S). Let C ×(S) be the set consisting of objects that appear in some cluster tilting set P in CEG∗(S). 2.3. The canonical bijection. By...

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...C(S) from Lemma 3.1 is an isomorphism. The result can be obtained by cutting the given marked surface along arcs and using induction similar to the methods in section 3. The details are worked out in =-=[3]-=-, from where we derive the main result of this subsection: Theorem 4.7. The canonical injection ιS : MCG×(S) →֒ Aut0 C(S) is an isomorphism except when S is a once-punctured disc with 2 or 4 marked po...

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...inzburg dg algebra associated to the quiver Q with zero potential W = 0, then there is a quotient map π : BrQ → BrΓ since the spherical twists satisfy the braid relation cf. [20, (7.4)]. Example 4.5. =-=[23]-=- Let S be an (n + 3)-gon as shown in the left picture of Figure 12. Then the tagged rotation has order n+ 3. Further, for the shift [1] in D(S), we have π(z̃2) = [n+ 3]. where Q is a quiver of type An...

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...RANSLATION 9 C + DC Figure 4. The Dehn twist M in Y to the next marked point in Y . The mY−th power ρ mY Y of the rotation ρY coincides with the (positive) Dehn twist DY along Y (cf. Figure 4 and see =-=[11]-=- for a definition of Dehn twists). As an element in the mapping class group MCG•(S), ρY acts on the set of arcs of S and thus induces a permutation on A×(S) where the tagging of ρY (α) at any puncture...

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...: AUSLANDER-REITEN TRANSLATION 3 allows to determine the tagged triangulation for the endpoint of any maximal green mutation sequence from the tagged triangulation of the starting point, see [6] (cf. =-=[24]-=-). Note that we do not prove the existence of such a (finite) maximal green sequence, we can only provide a guidance in the search for maximal green sequences: if one exists, it needs to end in the tr...

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...isms of C(S), since these are encoded in the Auslander-Reiten triangles of the triangulated category C(S). A further study aiming to generalize results in [5] to the unpunctured case is undertaken in =-=[25]-=-. One motivation to study such a geometric realization of the shift functor comes from physics: to compute the complete spectrum of a BPS particle, one studies maximal green sequences which go from a ...