Decorated marked surfaces: spherical twists versus braid twists

TL;DR

This paper explores the relationship between spherical twist groups in 3-Calabi-Yau categories from marked surfaces and braid twists in the surface's mapping class group, revealing topological properties of stability spaces.

Contribution

It establishes an isomorphism between the spherical twist group and a subgroup of the mapping class group for decorated surfaces, linking algebraic and geometric structures.

Findings

Spherical twist group is isomorphic to a subgroup of the mapping class group.

The space of stability conditions on certain categories is contractible for specific surfaces.

Provides a geometric interpretation of algebraic autoequivalences in derived categories.

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 $\mathbf{S}_{\bigtriangleup}$. Here $\mathbf{S}_{\bigtriangleup}$ 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.