A ug 2 01 8 GLOBAL DIMENSION FUNCTION , GEPNER EQUATIONS AND X-STABILITY CONDITIONS

We study the global dimension function gl. dim: C\StabD /Aut → R≥0 on the quotient space of Bridgeland’s stability conditions on a triangulated category D and Toda’s Gepner equation Φ(σ) = s ·σ for some σ ∈ StabD and (Φ, s) ∈ AutD×C. We show that Kajiura-Saito-Takahashi’s solution σG of τ (σ) = (−2/h) · σ, for the category of graded matrix factorization D∞(Q) = D (kQ) for a Dynkin quiver Q, gives the unique minimal value 1 − 2/h of gl. dim. Here τ is the Auslander-Reiten functor and h is the Coxeter number. Moreover, we show that, for each Lagrangian immersion L : D∞(Q) → DX(Q), the Gepner point σG in StabD∞(Q) induces a solution σ L G,s of τ L X (σ) = (−2/h)·σ in IkedaQiu’s s-fiber XStabs DX(Q) of X-stability condition on the Calabi-Yau-X category DX(Q), provided Re(s) ≥ 2− 2/h. Here XStabs DX(Q) is defined by another Gepner equation X(σ) = s ·σ and τ X = [X− 2] ◦ ζ L Q, where ζ L Q is a root of the generator of the center of the spherical twist group STX(Q).