Global dimension function on stability conditions and Gepner equations

Abstract

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

arXiv commentFinal version, Math. Zeit. to appear

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Cited by in corpus (5)

• Cambrian combinatorics on quiver representations (type A)
• q-Stability conditions on Calabi-Yau-X categories
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Contractibility of space of stability conditions on the projective plane via global dimension function
• Contractible flow of stability conditions via global dimension function