| Abstract: |
| With each pair consisting of a marked surface $S$ and a split simple Lie group $G$, Goncharov--Shen have associated a cluster algebra governing the higher Teichmuller space corresponding to $S$ and $G$. In the case where $S$ is a triangle, work by Miantao Liu allows to categorify this cluster algebra using the Higgs category associated to a canonical quiver with potential (constructed uniformly using a relative Calabi--Yau completion). Merlin Christ has conjectured two alternative descriptions of this Higgs category: 1) as a cosingularity category and 2) as the category of triangles in a $1$-cluster category. We will sketch a proof of his conjectures. This is a report on part of Miantao Liu`s ongoing Ph. D. thesis. |
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