In this paper, we study the space of stability conditions on a certain N -Calabi-Yau (CY N ) category associated to an An-quiver. Recently, Bridgeland and Smith constructed stability conditions on some CY 3 categories from meromorphic quadratic differentials with simple zeros. Generalizing their results to higher dimensional Calabi-Yau categories, we describe the space of stability conditions as the universal cover of the space of polynomials of degree n+1 with simple zeros. In particular, central charges of stability conditions on CY N categories are constructed as the periods of quadratic differentials with zeros of order N − 2 which are associated to polynomials.
We introduce q-stability conditions (σ, s) on Calabi-Yau-X categories D X , where σ is a stability condition on D X and s a complex number. Sufficient and necessary conditions are given, for a stability condition on an X-baric heart D∞ of D X to induce q-stability conditions on D X . As a consequence, we show that the space QStab ⊕ D X of (induced) open q-stability conditions is a complex manifold, whose fibers (fixing s) give usual type of spaces of stability conditions. Our motivating examples for D X are coming from Calabi-Yau-X completions of dg algebras.A geometric application is that, for type A quiver Q, the corresponding space QStab • s D X (Q) of q-stability conditions admits almost Frobenius structure while the central charge Zs corresponds to the twisted period Pν, for ν = (s − 2)/2, where s ∈ C with Re(s) ≥ 2. A categorical application is that we realize perfect derived categories as cluster(-X) categories for acyclic quiver Q.In the sequel, we construct quivers with superpotential from flat surfaces with the corresponding Calabi-Yau-X categories and realize open/closed q-stability conditions as q-quadratic differentials.
We construct a quiver with superpotential (QT, WT) from a marked surface S with full formal arc system T. Categorically, we show that the associated cluster-X category is Haiden-Katzarkov-Kontsevich's topological Fukaya category D∞(T) of S, which is also an X-baric heart of the Calabi-Yau-X category D X (T) of (QT, WT). Thus stability conditions on D∞(T) induces q-stability conditions on D X (T).Geometrically, we identify the space of q-quadratic differentials on the logarithm surface log S∆, with the space of induced q-stability conditions on D X (T), with parameter s satisfying Re(s) ≫ 1. When s = N is an integer, the result gives an N -analogue of Bridgeland-Smith's result for realizing stability conditions on the orbit Calabi-Yau-N category D X (T)/ /[X−N ] via quadratic differentials with zeroes of order N − 2. When the genus of S is zero, the space of q-quadratic differentials can be also identified with framed Hurwitz spaces.
The aim of this paper is to study the space of stability conditions on the bounded derived category of nilpotent modules over the preprojective algebra associated with a quiver without loops. We describe this space as a covering space of some open set determined by the root system of the Kac-Moody Lie algebra associated with the quiver.
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