$q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories

Abstract

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.

Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras

“…One interesting question is about the relation between these two works. In [6,7], we introduce the q-deformation of stability conditions and quadratic differentials to give an answer. On the categorical level, HKK's topological Fukaya category D ∞ (S) can be embedded into a Calabi-Yau category D X (S) with distinguish automorphism [X] as another (grading) shift functor.…”

Stability conditions and braid group actions on affine $A_n$ quivers

“…By applying the framework of Bridgeland and Smith [4], we prove that there is a C * -equivaraint isomorphism Quad(N, n) * Stab • (D( Ãn,N ))/Sph( Ãn,N ). This is also proved in [10] by a different method.…”

Geometric classification of totally stable stability spaces

“…For all Dynkin case, there is also a conjectural description on the (almost) Frobenius structure on Stab D ? (Q), which has been proved in [IQ2] for type A. More precisely, [IQ1] introduces the Calabi-Yau-X category to link the Calabi-Yau-∞ category and the Calabi-Yau-2 one.…”

“…Motivated by q-deformation of stability conditions in [IQ1,IQ2], Ikeda-Qiu introduce the notion of global dimension (R ≥0 -valued) function gldim of a stability condition σ to measure how stable a stability condition is. Qiu [Q2] shows that σ is totally stable if and only if gldim σ < 1, which is very rare.…”