Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity

Recently, a new choice of variables was identified to understand how the quantum group structure appeared in three-dimensional gravity [M. Dupuis, L. Freidel, E Girelli, A. Osumanu, and J. Rennert, On the origin of the quantum group symmetry in 3d quantum gravity, arXiv:2006.10105]. These variables arc introduced via a canonical transformation generated by a boundary term. We show that this boundary term can actually be taken to be the volume of the boundary and that the new variables can be defined in any dimension greater than three. In addition, we study the associated metric and teleparallel formalisms. The former is a variant of the Henneaux-Teitelboim model for unimodular gravity. The latter provides a non-Abelian generalization of the usual Abelian teleparallel formulation.