Global dimension function on stability conditions and Gepner equations

Abstract

We study the global dimension function \({\text {gldim}}:{\text {Aut}}\backslash {\text {Stab}}{\mathcal {D}}/\mathbb {C}\rightarrow \mathbb {R}_{\ge 0}\) on the quotient of the space of Bridgeland stability conditions on a triangulated category \({\mathcal {D}}\) as well as Toda’s Gepner equation \(\Phi (\sigma )=s\cdot \sigma \) for some \(\sigma \in {\text {Stab}}{\mathcal {D}}\) and \((\Phi ,s)\in {\text {Aut}}{\mathcal {D}}\times \mathbb {C}\). For the bounded derived category \({\mathcal {D}}^b(\textbf{k}Q)\) of a Dynkin quiver Q, we show that there is a unique minimal point \(\sigma _G\) of \({\text {gldim}}\) (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 \({\text {gldim}}\) is 1. Our philosophy is that the infimum of \({\text {gldim}}\) on \({\text {Stab}}{\mathcal {D}}\) 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.