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Decorated marked surfaces: spherical twists versus braid twists

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arxiv 1407.0806 v4 pith:S2VHUYWR submitted 2014-07-03 math.RT math.AGmath.CTmath.GT

Yu Qiu This is my paper · ORCID
classification math.RTmath.AGmath.CTmath.GT
keywords mathbfdecoratedmarkedmathcalsurfacetwistsbigtriangleupbraid
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

    math.RT 2025-01 conditional novelty 7.0 of 10

    Introduces a piecewise-linear flow on cluster complexes whose leaves generalize green mutation, proving these complexes are spheres for Dynkin quivers and contractible for Euclidean quivers.

  2. Topological model for derived category associated to sphere with four binaries

    math.RT 2026-07 conditional novelty 6.0 of 10

    Indecomposable rigid objects in the derived category of the (2,2,2,2)-weighted projective line are shown to correspond to graded simple arcs on a sphere with four binaries, with Hom-dimensions given by oriented inters...

  3. Contractibility and total semi-stability conditions of Euclidean quivers

    math.RT 2025-01 conditional novelty 6.0 of 10

    The moduli space of total semi-stability conditions on Euclidean quivers is described by partition data and contracts linearly to any non-concentrated stability condition.

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