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

TL;DR

This work shows that a broad class of three-dimensional gravity theories with torsion and nonmetricity can be formulated as Chern-Simons gauge theories with appropriate groups, yielding topological dynamics and no propagating gravitons in the MB case. It unifies MB gravity (with torsion) as an $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ CS theory depending on $\Lambda_{\text{eff}}$, and extends to Weyl geometries via an $SO(3,2)$ CS description where a dilatation field generates trace nonmetricity. The paper further introduces a novel SL$(4,\mathbb{R})$ CS construction that geometrizes a general metric-affine gravity in 3D, with explicit decompositions into $e^a$, $\omega^{ab}$, and nonmetricity data, and shows how partial gauge fixing produces a residual theory with diffeomorphism and GL$(3,\mathbb{R})$ invariance. Overall, the results offer a field-theoretical, CS-based route to analyze 3D gravity with torsion and nonmetricity and lay groundwork for exploring asymptotics, dualities, and matter couplings in lower-dimensional gravity.

Abstract

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) or SL(2,R) x SL(2,R), and the fact that they admit two independent coupling constants, we obtain the Mielke-Baekler model for zero, positive or negative effective cosmological constant respectively. Choosing SO(3,2) as 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.