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

Abstract

Categorically, we introduce the Calabi-Yau-$\mathbb{X}$ categories ${\mathcal{D}}_{\mathbb{X}}$ of a graded marked surface ${\mathbf{S}}^{\lambda }$, as a $q$-deformation of the topological Fukaya category ${\mathcal{D}}_{\infty }$ of ${\mathbf{S}}^{\lambda }$. We show that ${\mathcal{D}}_{\infty }$ can be identified with the cluster-$\mathbb{X}$ category associated to ${\mathcal{D}}_{\mathbb{X}}$. Geometrically, we construct and identify the space of $q$-quadratic differentials on the logarithm surface ${\log }_{\mathbf{c}}{\mathbf{S}}_{△}^{\lambda }$ with the space of induced $q$-stability conditions on ${\mathcal{D}}_{\mathbb{X}}$, for a complex parameter $s$ satisfying $\mathrm{Re}(s)\gg 1$. 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 ${\mathcal{D}}_{\mathbb{X}}//[\mathbb{X}-N]$ via CY-$N$ type quadratic differentials. When the genus of $\mathbf{S}$ is zero, the space of $q$-quadratic differentials can be also identified with framed Hurwitz spaces. As a byproduct, the result confirms the conjectural almost Frobenius structure on spaces of $q$-stability conditions for type $A$.

arXiv commentMemoirs of Amer. Math. Soc. to appear

References in corpus (27)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• On triangulated orbit categories
• Flat surfaces and stability structures
• Stability conditions on a non-compact Calabi-Yau threefold
• Exchange graphs and Ext quivers
• Spaces of stability conditions
• Stability conditions and quantum dilogarithm identities for Dynkin quivers
• Cluster categories for marked surfaces: punctured case
• Deformed Calabi-Yau Completions
• Tilting theory and cluster combinatorics
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• Decorated marked surfaces: spherical twists versus braid twists
• Stability conditions and the $A_2$ quiver
• Auslander orders over nodal stacky curves and partially wrapped Fukaya categories
• Derived categories of graded gentle one-cycle algebras
• Picard-Lefschetz theory and dilating C^*-actions
• Stability conditions on $A_n$-singularities
• Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot--Guo--Keller
• q-Stability conditions on Calabi-Yau-X categories
• Erratum to "Deformed Calabi-Yau completions"
• Relative Calabi-Yau completions
• On almost duality for Frobenius manifolds
• Global dimension function on stability conditions and Gepner equations
• Decorated marked surfaces III: The derived category of a decorated marked surface
• Stability conditions on ${\text{CY}}_N$ categories associated to $A_n$-quivers and period maps
• Topological structure of spaces of stability conditions and topological Fukaya type categories
• The (he)art of gluing

Cited by in corpus (8)

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• q-Stability conditions on Calabi-Yau-X categories
• Global dimension function on stability conditions and Gepner equations
• Contractibility of space of stability conditions on the projective plane via global dimension function
• Stability conditions and braid group actions on affine $A_n$ quivers
• Contractible flow of stability conditions via global dimension function
• Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras
• Categorical dynamical systems arising from sign-stable mutation loops