A construction of Frobenius manifolds from stability conditions. Issue 6 (23rd November 2018) - Barbieri, Anna; Stoppa, Jacopo; Sutherland, Tom
Abstract: A finite quiver Q without loops or 2‐cycles defines a CY 3 triangulated category D ( Q ) and a finite heart A ( Q ) ⊂ 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 A n evaluating the family at a special point, we recover a branch of the Saito Frobenius structure of the A n singularity y 2 = x n + 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 A n, n ⩽ 5 .