q-deformed rational numbers and the 2-Calabi–Yau category of type $A_{2}$

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.