Geometric Stability Framework

• The Geometric Stability Framework is defined as a rigorous method linking total stability in derived categories with convex polygon moduli based on Dynkin diagrams.
• It employs explicit geometric constructions of decorated h-gons in the complex plane using central charges and combinatorial constraints to mirror stability conditions.
• The framework unifies categorical and geometric stability perspectives, enabling clear analyses of wall-crossing, tiling theory, and applications in mirror symmetry and cluster theory.

The geometric stability framework encompasses a class of rigorous, structurally grounded approaches to stability conditions, typically in algebraic geometry and representation theory, that identify explicit global moduli spaces of stability data with moduli of polygons or, more generally, decorated geometric objects associated to the combinatorics of Dynkin diagrams. In the context of derived categories associated to Dynkin quivers, this framework establishes an explicit bijective correspondence between total stability conditions (i.e., stability conditions in which every indecomposable is stable) and moduli of convex, decorated polygons in the complex plane, with geometric constraints determined by the representation type. This correspondence unifies categorical and geometric perspectives on stability, leading to a transparent, calculable description of stability spaces for an important class of triangulated categories.

1. Total Stability and Categorical Foundations

Let $\mathcal{D}$ be a triangulated category, such as the bounded derived category $D^b(Q)$ of a Dynkin quiver $Q$. A stability condition on $\mathcal{D}$ consists of a central charge $Z:K_0(\mathcal D)\to\mathbb{C}$ and a slicing $\mathcal{P}$—that is, a collection of semistable subcategories $\mathcal P(\phi)$ indexed by phase $\phi\in\mathbb{R}$—subject to Bridgeland’s axioms (support property, Harder–Narasimhan filtration, Hom-vanishing off by phase).

A stability condition $\sigma=(Z, \mathcal P)$ is called totally (semi)stable if every indecomposable object is (semi)stable. The categorical characterization is via the global dimension invariant: $$\sigma\text{ is totally stable}\ \Longleftrightarrow\ \mathrm{gldim}(\sigma)<1,$$ where

$$\mathrm{gldim}(\sigma) := \sup\{\phi_2-\phi_1\mid \operatorname{Hom}(\mathcal P(\phi_1), \mathcal P(\phi_2))\neq 0\}.$$

Total stability conditions exist if and only if $\mathcal{D}$ is derived-equivalent to $D^b(Q)$ for $Q$ Dynkin, i.e., when the representation category is of finite type. The set of total stability conditions is denoted $ToSt(Q)$. For non-Dynkin types, no such total stability conditions exist.

2. The Polygonal $h_Q$-gon Model and Associated Constraints

For each Dynkin type $Q$ with Coxeter number $h_Q$, the geometric stability framework describes $ToSt(Q)$ via a moduli space of decorated $h_Q$-gons in $\mathbb{C}$. The construction is as follows:

• Select a distinguished “far-end” $\tau$-orbit in $\mathcal{R}_Q:=D^b(Q)/[2]$, giving indecomposables $M$, $\tau M$, …, $\tau^{h_Q-1} M$.
• Let vertices $V_0,\dots,V_{h_Q-1}$ in $\mathbb{C}$ correspond to these objects, with edges $z_j = V_{j-1} V_j = Z(\tau^{j-1} M)$.
• The collection $(V_j)$ forms a closed polygon, as $\sum_j z_j=0$.
• The phases of $Z(\tau^j M)$ must increase strictly in $[0,2\pi)$, yielding a positively convex orientation.

Further constraints depend on the Dynkin type:

• Type $A_n$:

Any positively convex $(n+1)$-gon suffices. No further relations are imposed among the edge vectors $z_j$.

• Type $D_n$ ($h=2(n-1)$):

The polygon is centrally symmetric with two punctures $B_+$, $B_-$ at the center, with $V_{j+(n-1)}=-V_j$ and $B_+=-B_-$. The punctures must lie strictly inside the level-$(n-2)$ diagonal polygon.

• Type $E_6$ ($h=12$):

The $12$-gon must satisfy specific four triangle-relations ($z_j + z_{j+4} + z_{j+8}=0$) and three square-relations ($z_j - z_{j-3} + z_{j-6} - z_{j-9}=0$). Interior “ice-core” and “fire-core” hexagons generated from these relations must lie inside the level-3 diagonal polygon.

• Type $E_7$ ($h=18$):

Central symmetry and three hexagon-relations are imposed. Both the 9-gon fire-core and the ice-core must be strictly internal to the level-4 diagonal polygon.

• Type $E_8$ ($h=30$):

Central symmetry plus five triangle relations and three pentagonal relations structure the polygon. The corresponding 15-gon cores must lie strictly inside the level-5 diagonal polygon.

• Folding to Non-Simply-Laced Types:

For $B_n$, $C_n$, $F_4$, $G_2$, analogous polygonal models are defined by folding symmetric configurations in $D_{n+1}, D_{n+1}, E_6, D_4$ respectively.

The definition of “stable $h_Q$-gon of type $Q$” encodes all such combinatorial and convexity/puncture/core constraints.

3. Moduli Space Identification and Isomorphism Theorem

Let $\mathrm{Stgon}(Q)$ denote the space of stable $h_Q$-gons (up to translation in $\mathbb{C}$) as above. A central result is:

Main Theorem:

For each Dynkin $Q$ of rank $n$, the natural map

$Z_h: ToSt(Q)/[2] \to \mathrm{Stgon}(Q)$

from total stability conditions (modulo [2]-shifts) to stable polygons is a bijective local isomorphism of complex manifolds, and thus a global isomorphism,

$ToSt(Q)/[2] \cong \mathrm{Stgon}(Q).$

The $C^*$-action on $ToSt(Q)$ by phase-shift corresponds to rotation and rescaling of the polygon, so

$$ToSt(Q)/\mathbb{C}^* \cong \mathrm{Stgon}(Q)/\mathbb{C}^*.$$

For each polygon, a slicing $\mathcal{P}$ is explicitly reconstructed: assign the phase $\phi(X) = \frac{1}{\pi}\arg(Z(X))$ (modulo shifts), and use the convexity/core constraints to guarantee all $\operatorname{Hom}$-vanishing and stability of indecomposables.

4. Explicit Examples

• $A_2$ (triangle, $h=3$):

Any positively oriented triangle in $\mathbb{C}$ corresponds to a total stability; for example, an equilateral triangle gives the “Gepner point” satisfying $\tau \sigma = e^{-2 \pi i/3} \sigma$.

• $D_4$ (hexagon, $h=6$):

The model is a centrally-symmetric convex hexagon with two punctures in the center, corresponding to distinct boundary $\tau$-orbits.

• $E_6$ (dodecagon, $h=12$):

A convex dodecagon admits explicit triangle and square relations. Two hexagonal cores (ice and fire) are formed within, governing the stability region.

• $E_7$ and $E_8$:

Analogous constructions yield 18- and 30-gons with cores and central symmetry, following linear relation constraints appropriate to the underlying root system.

Each structure can be represented diagrammatically by constructing the relevant convex polygon, placing punctures and cores as prescribed by type, and verifying the interiority of the stability-determining features.

5. Uniform Classification and Significance

For every simply-laced Dynkin diagram $Q$ of rank $n$ and Coxeter number $h_Q$, the geometric stability framework gives an explicit moduli-theoretic classification: $ToSt(Q)/[2] \longrightarrow \{\text{positively convex stable %%%%81%%%%-gons in } \mathbb{C}\}/\{\text{translation}\}$ with explicit combinatorial constraints matching the structural features of the derived category. The entire space of total stability conditions is thus identified with a finite-dimensional moduli space of decorated convex polygons.

This unifies categorical and geometric descriptions, and allows direct calculation of stability spaces, wall-crossing phenomena, and automorphism group actions entirely in terms of complex polygon geometry.

6. Broader Implications and Extensions

The geometric stability framework provides a complete, uniform description of total stability spaces for all Dynkin quivers, bridging triangulated category theory and planar geometry. The explicit moduli isomorphism offers a calculable and visualizable tool for representation theorists and algebraic geometers, enabling deeper analysis of wall-crossing, autoequivalences, and connections to cluster theory and mirror symmetry. Extensions to non-simply-laced and folding types are achieved via combinatorial symmetrization of the core polygon model. This approach stands as a canonical model for understanding total stability as a geometric problem, with direct links to categorical dynamics, tiling theory, and related domains in mathematical physics and geometry.