论文标题
$ q $ - 已定性的有理数和2-calabi-YAU类型$ a_2 $类别
$q$-deformed rational numbers and the 2-Calabi--Yau category of type $A_2$
论文作者
论文摘要
我们描述了Bapat，Deopurkar和Licata早期工作的任何三角策略类别的Bridgeland稳定性条件空间的紧凑型家族。我们特别考虑$ a_2 $ Quiver的2-卡拉比 - YAU类别的情况。紧凑型是将稳定空间的嵌入（取决于$ Q $）封闭到无限维投影空间中。 在$ a_2 $的情况下，三链辫子组$ b_3 $在此关闭上行动。我们描述了边界中的两个杰出的编织组轨道，可以用$ q $的某些理性功能识别其点。其中一个轨道中的点正是Morier-Genoud和Ovsienko最近引入的$ Q $成型的合理数字，而另一个轨道给出了新的$ Q $ $ Q $ - Q $ of-q $ quartional的数字。将$ Q $专门为正实数，我们获得了紧凑型边界的完整描述。
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.
