Chern-Simons formulation of three-dimensional gravity with torsion and nonmetricity

We consider various models of three-dimensional gravity with torsion or nonmetricity (metric affine gravity), and show that they can be written as Chern-Simons theories with suitable gauge groups. Using the groups ISO(2,1), SL(2,C) and SL(2,R)xSL(2,R), and the fact that they admit two independent coupling constants, we obtain the Mielke-Baekler model for zero, positive and negative effective cosmological constant respectively. Choosing SO(3,2) as the gauge group, one gets a generalization of conformal gravity that has zero torsion and only the trace part of the nonmetricity. This characterizes a Weyl structure. Finally, we present a new topological model of metric affine gravity in three dimensions arising from an SL(4,R) Chern-Simons theory.