Cluster exchange groupoids and framed quadratic differentials
arXiv:1805.00030 · doi:10.1007/s00222-019-00932-y
Abstract
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.
arXiv commentFinal version, 37 pages, many figures, to appear in Invent. Math
References in corpus (10)
- Connected components of the moduli spaces of Abelian differentials with prescribed singularities
- Mutation in triangulated categories and rigid Cohen-Macaulay modules
- Flat surfaces and stability structures
- Quivers with potentials associated to triangulated surfaces
- Cluster algebras via cluster categories with infinite-dimensional morphism spaces
- Exchange graphs and Ext quivers
- Stability conditions and quantum dilogarithm identities for Dynkin quivers
- On the Cluster Category of a Marked Surface
- Decorated marked surfaces: spherical twists versus braid twists
- Contractible stability spaces and faithful braid group actions