Biholomorphism between Frobenius manifold orbit spaces and stability (q-stability) condition quotients

Establish the following correspondences for a tuple A = (a1,a2,a3) with χ_A > 0: (1) Prove a biholomorphism XA/W_A ≅ Stab(DA)/ST^1(DA), where DA = D^b(mod(CT_A)) and ST^1(DA) is the subgroup generated by 1-spherical twists via the Lagrangian immersion; (2) For any complex parameter s with Re(s) ≥ 2, prove a biholomorphism X_reg_A/W_A ≅ QStab^◦(DX_A)_s/ST(DX_A), where QStab^◦(DX_A)_s denotes the principal component of the space of q-stability conditions on the X-Calabi–Yau triangulated category DX_A.

Background

Earlier sections identify a Frobenius manifold structure on the orbit space XA/W_A associated to a generalized root system of affine ADE type and relate its monodromy group to W_A (Corollary 3.14 and Corollary 3.16). Motivated by mirror symmetry and Conjecture 5.16, the authors propose precise correspondences between these orbit spaces and quotient spaces of stability and q-stability conditions on DA and DX_A.

Proposition 5.18 confirms Conjecture 5.17 in the special case A = (1,p,q) and s ∈ Z≥3, supporting the broader conjectural picture.