[Paper Review] $q$-deformed rational numbers and the 2-Calabi--Yau category of type $A_2$

Asilata Bapat, Louis Becker+1 more|arXiv (Cornell University)|Feb 15, 2022

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces two new $q$-deformations of rational numbers—$[r/s]^\flat$ and $[r/s]^\sharp$—via the action of a $q$-deformed modular group $\mathrm{PSL}_{2,q}(\mathbb{Z})$, with $[r/s]^\sharp$ recovering the $q$-rationals of Morier-Genoud and Ovsienko. It constructs a $q$-dependent compactification of the stability space for the 2-Calabi–Yau category of type $A_2$, showing that the boundary contains intervals of $q$-deformed rational numbers, and proves that the compactified space $\overline{M_q}$ is homeomorphic to a closed disk for $q > 0$. The key result is a $q$-deformed version of the classical correspondence between spherical objects and $\mathbb{Q} \cup \{\infty\}$, now realized as intervals in the boundary via two distinct functionals.

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.

Motivation & Objective

• To generalize the $q$-deformed rational numbers of Morier-Genoud and Ovsienko by introducing a new $q$-deformation $[r/s]^\flat$ alongside the existing $[r/s]^\sharp$.
• To construct a $q$-dependent compactification of the space of Bridgeland stability conditions for the 2-Calabi–Yau category of type $A_2$ via an embedding into an infinite-dimensional projective space.
• To establish a $q$-deformed analog of the classical correspondence between spherical objects and $\mathbb{Q} \cup \{\infty\}$, showing that each spherical object corresponds to a closed interval in the boundary of the compactified stability space.
• method
• research_questions
• key_findings

Proposed method

• The authors define a $q$-deformed modular group $\mathrm{PSL}_{2,q}(\mathbb{Z})$ acting on $\mathbb{R} \cup \{\infty\}$ via fractional linear transformations, using which both $[r/s]^\flat$ and $[r/s]^\sharp$ are constructed as orbit images of $1/0$.
• They embed the stability space $\mathrm{Stab}(\mathcal{C}_2)/\mathbb{C}$ into an infinite-dimensional projective space via a $q$-dependent map, and define $M_q$ as the image and $\overline{M_q}$ as its closure.
• For each spherical object $X$, two real-valued functionals are defined: $\overline{\hom}_q(X, \cdot)$, counting Laurent polynomial coefficients of morphisms, and $\mathrm{occ}_q(X, \cdot)$, a more intricate functional encoding occurrence data.
• The boundary $\partial\overline{M_q}$ is shown to be homeomorphic to $\mathbb{R} \cup \{\infty\}$, with each spherical object $X$ corresponding to a closed interval $I_{q,X}$ in the boundary formed by convex combinations of the two functionals.
• The construction relies on a $q$-deformed Harder–Narasimhan (HN) automaton, which tracks how HN filtrations transform under braid group actions, and is used to prove the main topological results.
• A $q$-mass automaton is defined by composing the HN vector with a $q$-mass function, providing a computational tool for studying dynamics of autoequivalences in the category.

Experimental results

Research questions

• RQ1 How can the $q$-deformed rational numbers of Morier-Genoud and Ovsienko be generalized to include a second, natural $q$-deformation of rational numbers?
• RQ2 What is the topological structure of the compactification of the stability space for the 2-Calabi–Yau category of type $A_2$ when $q \neq 1$?
• RQ3 How does the classical correspondence between spherical objects and $\mathbb{Q} \cup \{\infty\}$ deform when $q \neq 1$, and what replaces the rational points on the boundary?
• RQ4 What role does the three-strand braid group $B_3$ play in the structure of the compactified stability space for $q \neq 1$?
• RQ5 Can the $q$-deformed functionals $\overline{\hom}_q(X, \cdot)$ and $\mathrm{occ}_q(X, \cdot)$ be used to reconstruct the boundary of the compactified stability space as a union of intervals?

Key findings

• The $q$-deformed rational number $[1/0]^\flat = 1/(1 - q)$ is introduced as a new deformation of $\infty$, distinct from $[1/0]^\sharp = \infty$, and both arise from the action of $\mathrm{PSL}_{2,q}(\mathbb{Z})$ on $\mathbb{R} \cup \{\infty\}$.
• For $q \in (0, \infty)$, the space $M_q$ is homeomorphic to an open disk, and its closure $\overline{M_q}$ is conjectured to be homeomorphic to a closed disk, with strong evidence provided in the paper.
• The boundary $\partial\overline{M_q}$ is homeomorphic to $\mathbb{R} \cup \{\infty\}$, and each spherical object $X$ corresponds to a closed interval $I_{q,X}$ in the boundary, formed by convex combinations of the two functionals $\overline{\hom}_q(X, \cdot)$ and $\mathrm{occ}_q(X, \cdot)$.
• At $q = 1$, the two functionals coincide and recover the classical rational number associated to $X$, but for $q \neq 1$, they are distinct, leading to interval-valued boundary points.
• The union of all intervals $I_{q,X}$ over spherical objects $X$ is a dense subset of $\partial\overline{M_q}$, showing that the $q$-deformed rational numbers densely fill the boundary.
• The construction uses a $q$-deformed HN automaton to control the behavior of HN filtrations under braid group actions, providing a computational framework for studying autoequivalence dynamics in the category.