A construction of Frobenius manifolds from stability conditions

A finite quiver Q) without loops or 2-cycles defines a CY2 triangulated category D(Q) and a finite heart (A(Q) ⸦ D(Q). We show that if if Q satisfies some (strong) conditions, then the space of stability conditions (A(Q)) supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in (Formula presented.). In the case of D(Q) evaluating the family at a special point, we recover a branch of the Saito Frobenius structure of the An singularity (y2 = xn+1. We give examples where applying the construction to each mutation of q and evaluating the families at a special point yields a different branch of the maximal analytic continuation of the same semisimple Frobenius manifold. In particular, we check that this holds in the case of An n ≤ 5.