A construction of Frobenius manifolds from stability conditions

A finite quiver Q without loops or 2-cycles defines a CY3 triangulated category D(Q) and a finite heart A(Q)subset of D(Q). We show that if Q satisfies some (strong) conditions, then the space of stability conditions Stab(A(Q)) supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in D(Q). In the case of An 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.