On decorated representation spaces associated to spherical surfaces

We analyse local features of the spaces of representations of the fundamental group of a punctured surface in $\mathrm{SU}_2$ equipped with a decoration, namely a choice of a logarithm of the representation at peripheral loops. Such decorated representations naturally arise as monodromies of spherical surfaces with conical points. Among other things, in this paper we determine the smooth locus of such absolute and relative decorated representation spaces: in particular, in the relative case (with few special exceptions) such smooth locus is dense, connected, and exactly consists of non-coaxial representations. The present study sheds some light on the local structure of the moduli space of spherical surfaces with conical points, which is locally modelled on the above-mentioned decorated representation spaces.