Global dimension function on stability conditions and Gepner equations
Abstract
We study the global dimension function $\operatorname{gldim}\colon\operatorname{Aut}\backslash\operatorname{Stab}\mathcal{D}/\mathbb{C}\to\mathbb{R}_{\ge0}$ on a 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 $\sigma\in\operatorname{Stab}\mathcal{D}$ and $(\Phi,s)\in\operatorname{Aut}\mathcal{D}\times\mathbb{C}$. We prove the uniqueness (up to the $\mathbb{C}$action) of the solution of the Gepner equation $\tau(\sigma)=(2/h)\cdot\sigma$ for the bounded derived category $\mathcal{D}^b(\mathbf{k} Q)$ of a Dynkin quiver $Q$. Here $\tau$ is the AuslanderReiten functor and $h$ is the Coxeter number. This solution $\sigma_G$ was constructed by KajiuraSaitoTakahashi. Moreover, we show that $\operatorname{gldim}$ has minimal value $12/h$, which is only attained by this solution. We also show that for an acyclic nonDynkin quiver $Q$, the minimal value of $\operatorname{gldim}$ is $1$. Our philosophy is that the infimum of $\operatorname{gldim}$ on $\operatorname{Stab}\mathcal{D}$ is the global dimension for the triangulated category $\mathcal{D}$. We explain how this notion could shed light on the problem of the contractibility of the space of stability conditions.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1807.00010
 Bibcode:
 2018arXiv180700010Q
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry
 EPrint:
 Fix a proof and many typos and change the structure