GLOBAL DIMENSION FUNCTION ON STABILITY CONDITIONS AND GEPNER EQUATIONS

We study the global dimension function gldim: Aut \StabD /C → R≥0 on a quotient of the space of Bridgeland stability conditions on a triangulated category D as well as Toda’s Gepner equation Φ(σ) = s · σ for some σ ∈ StabD and (Φ, s) ∈ AutD×C. We prove the uniqueness (up to the C-action) of the solution of the Gepner equation τ (σ) = (−2/h) · σ for the bounded derived category D(kQ) of a Dynkin quiver Q. Here τ is the Auslander-Reiten functor and h is the Coxeter number. This solution σG was constructed by Kajiura-Saito-Takahashi. Moreover, we show that gldim has minimal value 1− 2/h, which is only attained by this solution. We also show that for an acyclic non-Dynkin quiver Q, the minimal value of gldim is 1. Our philosophy is that the infimum of gldim on StabD is the global dimension for the triangulated category D. We explain how this notion could shed light on the problem of the contractibility of the space of stability conditions.