Spherical objects and stability conditions on 2-Calabi–Yau quiver categories

Bapat, Asilata; Deopurkar, Anand; Licata, Anthony

doi:10.1007/s00209-022-03172-8

Cited by 1 publication

(22 citation statements)

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“…There is a homeomorphism Q 𝑞 I 𝑞 → R ∪ {∞}, which restricts to the identity on the left and right q-deformed rationals and the q-deformed irrationals. So, by Part (2),…”

Section: The Boundarymentioning

confidence: 93%

“…The next lemma describes when these inequalities degenerate into equalities; it is the q-analogue of the phenomenon noted at [2,Remark 5.15]. Lemma 5.11 (q-Degeneracy).…”

Section: Embedding the Boundary Of 𝑀mentioning

confidence: 99%

“…Recall the definition of standard stability conditions from Definition 4.1. The subset of the space of stability conditions that corresponds to the first two Farey triangles is given by the following definition: Definition 5.7 [2,Section 5.3]. A standard stability condition 𝜏 is type-A if 𝑃 21 is semistable with respect to 𝜏, and type-B if 𝑃 12 is semistable with respect to 𝜏.…”

Section: Analogy With the Q-farey Tessellationmentioning

confidence: 99%

“…To explain this, in Section 5, we associate to each positive real q a compactification of the space of Bridgeland stability conditions on , the 2-Calabi–Yau (2-CY) triangulated category for the quiver. This compactification is a generalisation of a construction of [2], which corresponds to the case , and proceeds by embedding the space of stability conditions into an infinite dimensional projective space, and taking the closure there. We set to be the image of the space of stability conditions under this embedding, to be its closure and to be .…”

Section: Introductionmentioning

confidence: 99%

“…It is a theorem of Rouquier–Zimmermann [17] that this bijection intertwines the braid group action on the spherical objects with the action on . Details about this construction may be found in [2] and references therein.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

q-deformed rational numbers and the 2-Calabi–Yau category of type

Bapat

Becker

Licata

2023

Forum of Mathematics, Sigma

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We describe a family of compactifications of the space of Bridgeland stability conditions of a triangulated category, following earlier work by Bapat, Deopurkar and Licata. We particularly consider the case of the 2-Calabi–Yau category of the $A_2$ quiver. The compactification is the closure of an embedding (depending on q) of the stability space into an infinite-dimensional projective space. In the $A_2$ case, the three-strand braid group $B_3$ acts on this closure. We describe two distinguished braid group orbits in the boundary, points of which can be identified with certain rational functions in q. Points in one of the orbits are exactly the q-deformed rational numbers recently introduced by Morier-Genoud and Ovsienko, while the other orbit gives a new q-deformation of the rational numbers. Specialising q to a positive real number, we obtain a complete description of the boundary of the compactification.

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“…There is a homeomorphism Q 𝑞 I 𝑞 → R ∪ {∞}, which restricts to the identity on the left and right q-deformed rationals and the q-deformed irrationals. So, by Part (2),…”

Section: The Boundarymentioning

confidence: 93%

“…The next lemma describes when these inequalities degenerate into equalities; it is the q-analogue of the phenomenon noted at [2,Remark 5.15]. Lemma 5.11 (q-Degeneracy).…”

Section: Embedding the Boundary Of 𝑀mentioning

confidence: 99%

Section: Analogy With the Q-farey Tessellationmentioning

confidence: 99%