In a recent paper, we introduced a new discretization scheme for gravity in 2+1 dimensions. Starting from the continuum theory, this new scheme allowed us to rigorously obtain the discrete phase space of loop gravity, coupled to particle-like edge mode degrees of freedom. In this work, we expand on that result by considering the most general choice of integration during the discretization process. We obtain a family of polarizations of the discrete phase space. In particular, one member of this family corresponds to the usual loop gravity phase space, while another corresponds to a new polarization, dual to the usual one in several ways. We study its properties, including the relevant constraints and the symmetries they generate. Furthermore, we motivate a relation between the dual polarization and teleparallel gravity.
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