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