# Global dimension function, Gepner equations and $q$-stability conditions

29 Jun 2018Yu Qiu

We study the global dimension function $\operatorname{gl.dim}\colon\mathbb{C}\backslash\operatorname{Stab}\mathcal{D}/\operatorname{Aut}\to\mathbb{R}_{\ge0}$ on the quotient space of Bridgeland's stability conditions on a triangulated category $\mathcal{D}$ and Toda's Gepner equation $\Phi(\sigma)=s\cdot\sigma$ for some $\sigma\in\operatorname{Stab}\mathcal{D}$ and $(\Phi,s)\in\operatorname{Aut}\mathcal{D}\times\mathbb{C}$. We show that Kajiura-Saito-Takahashi's solution $\sigma_G$ of $\tau(\sigma)=(-2/h)\cdot\sigma$, for the category of graded matrix factorization $\mathcal{D}_\infty(Q)=\mathcal{D}^b(\mathbf{k}Q)$ for a Dynkin quiver $Q$, gives the unique minimal value $1-2/h$ of $\operatorname{gl.dim}$... (read more)

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