6533b853fe1ef96bd12ac2b7
RESEARCH PRODUCT
A construction of Frobenius manifolds from stability conditions
Tom SutherlandAnna BarbieriJacopo Stoppasubject
High Energy Physics - TheoryMathematics - Differential GeometryFrobenius manifoldPure mathematics010308 nuclear & particles physicsTriangulated categoryGeneral MathematicsAnalytic continuation010102 general mathematicsQuiverStructure (category theory)FOS: Physical sciencesSpace (mathematics)01 natural sciencesMathematics - Algebraic GeometrySingularityHigh Energy Physics - Theory (hep-th)Differential Geometry (math.DG)0103 physical sciencesMutation (knot theory)FOS: MathematicsSettore MAT/03 - Geometria0101 mathematicsAlgebraic Geometry (math.AG)Mathematicsdescription
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 $A_n$, $n \leq 5$.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-11-23 |