Cluster exchange groupoids and framed quadratic differentials

Alastair King Uviersity of Bath Yu Qiu Tsinghua University

Algebraic Geometry arXiv subject: Geometric Topology (math.GT) arXiv subject: Representation Theory (math.RT) arXiv subject: Dynamical Systems (math.DS) mathscidoc:2012.45001

Inventiones mathematicae volume, 220, 479-523, 2020.3
We introduce the cluster exchange groupoid associated to a non-degenerate quiver with potential, as an enhancement of the cluster exchange graph. In the case that arises from an (unpunctured) marked surface, where the exchange graph is modelled on the graph of triangulations of the marked surface, we show that the universal cover of this groupoid can be constructed using the covering graph of triangulations of the surface with extra decorations. This covering graph is a skeleton for a space of suitably framed quadratic differentials on the surface, which in turn models the space of Bridgeland stability conditions for the 3-Calabi–Yau category associated to the marked surface. By showing that the relations in the covering groupoid are homotopically trivial when interpreted as loops in the space of stability conditions, we show that this space is simply connected.
Calabi-Yau categories, cluster theory, quadratic differentials, stability conditions, braid groups
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  title={Cluster exchange groupoids and framed quadratic differentials},
  author={Alastair King, and Yu Qiu},
  booktitle={Inventiones mathematicae volume},
Alastair King, and Yu Qiu. Cluster exchange groupoids and framed quadratic differentials. 2020. Vol. 220. In Inventiones mathematicae volume. pp.479-523.
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