Decorated marked surfaces III: The derived category of a decorated marked surface

We study the Ginzburg dg algebra ${\mathrm{\Gamma }}_{\mathbf{T}}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface ${\mathbf{S}}_{△}$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories ${\mathcal{D}}_{fd}({\mathrm{\Gamma }}_{\mathbf{T}})$, denoted by ${\mathcal{D}}_{fd}({\mathbf{S}}_{△})$. As an application, we show that the spherical twist group $\mathrm{ST}({\mathbf{S}}_{△})$ associated to ${\mathcal{D}}_{fd}({\mathbf{S}}_{△})$ acts faithfully on its space of stability conditions.