A construction of Frobenius manifolds from stability conditions

TL;DR

This paper constructs semisimple Frobenius manifold structures on the space of stability conditions for certain quivers, linking them to singularity theory and demonstrating how mutations produce different branches of these structures.

Contribution

It introduces a method to build Frobenius manifolds from stability conditions on 3CY categories associated with quivers, revealing new connections to singularity theory.

Findings

Constructs Frobenius structures from stability conditions on quivers.

Recovers known Frobenius structures for $A_n$ singularities.

Shows mutations lead to different branches of Frobenius manifolds.

Abstract

A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. 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 $A_n$, $n \leq 5$.