Decorated marked surfaces: spherical twists versus braid twists

Abstract

We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group of the decorated marked surface ${\mathbf{S}}_{△}$. Here ${\mathbf{S}}_{△}$ is the surface obtained from $\mathbf{S}$ by decorating with a set of decorated points, where the number of points equals the number of triangles in any triangulations of $\mathbf{S}$. For instance, when $\mathbf{S}$ is an annulus, the result implies the corresponding spaces of stability conditions on $\mathcal{D}$ is contractible.

Updated version

References in corpus (4)

• The representation type of Jacobian algebras
• Cluster categories for marked surfaces: punctured case
• Deformed Calabi-Yau Completions
• Decorated marked surfaces II: Intersection numbers and dimensions of Homs

Cited by in corpus (18)

• Cluster categories for marked surfaces: punctured case
• Exact WKB analysis and cluster algebras II: Simple poles, orbifold points, and generalized cluster algebras
• Contractible stability spaces and faithful braid group actions
• Braid groups and quiver mutation
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• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Cluster exchange groupoids and framed quadratic differentials
• Decorated marked surfaces II: Intersection numbers and dimensions of Homs
• Contractibility of space of stability conditions on the projective plane via global dimension function
• Topological structure of spaces of stability conditions and topological Fukaya type categories
• Finite presentations for spherical/braid twist groups from decorated marked surfaces
• Stability conditions and braid group actions on affine $A_n$ quivers
• A geometric model for the module category of a skew-gentle algebra
• Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras
• The braid group for a quiver with superpotential
• Decorated Marked Surfaces: Calabi-Yau categories and related topics
• A geometric realization of silting theory for gentle algebras
• Quadratic differentials as stability conditions: collapsing subsurfaces