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$q$-deformed rational numbers and the 2-Calabi--Yau category of type $A_2$
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abstract
We describe a family of compactifications of the space of Bridgeland stability conditions of any 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.
Forward citations
Cited by 2 Pith papers
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Nuancing the unicity of $q$-rationals
Exactly two modular-group-equivariant deformations of rationals reproduce the standard q-integers, and the newly identified one has positive coefficients and directly yields Jones polynomials of rational knots.
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The Thurston compactification of the stability manifold of a generic analytic K3 surface
For an analytic K3 surface with Picard group zero, the mass map from the projective stability manifold to the space of masses of semi-rigid objects is a homeomorphism onto an open disk whose closure is a closed disk.
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