Global dimension function on stability conditions and Gepner equations

We study the global dimension function $${\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}$$ on quotient of space Bridgeland stability conditions a triangulated category $${\mathcal {D}}$$ as well Toda’s Gepner equation $$\Phi (\sigma )=s\cdot \sigma $$ for some $$\sigma \in and $$(\Phi ,s)\in {Aut}}{\mathcal {D}}\times {C}$$ . For bounded derived {D}}^b(\textbf{k}Q)$$ Dynkin quiver Q, we show that there is unique minimal point _G$$ {gldim}}$$ (up to $$\mathbb -action), with value $$1-2/h$$ which solution $$\tau )=(-2/h)\cdot Here Auslander–Reiten functor h Coxeter number. This was constructed by Kajiura–Saito–Takahashi. also an acyclic non-Dynkin 1. Our philosophy infimum explain how this notion could shed light contractibility conjecture conditions.