3D gravity with torsion as a Chern-Simons gauge theory

TL;DR

The paper shows that topological 3D gravity with torsion in the Mielke–Baekler model, in the AdS sector with $\Lambda_{\rm eff}<0$, can be reformulated as a pair of Chern–Simons theories for $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. It derives the CS representation, identifies the two independent gauge sectors, and demonstrates that the boundary dynamics are governed by a conformal field theory with two distinct central charges, $c_1$ and $c_2$. The work also connects the bulk MB parameters to CS couplings and clarifies how the asymptotic AdS boundary conditions lead to two chiral boundary modes, in contrast to GR$_{\Lambda}$ and teleparallel gravity where the central charges coincide. It discusses the role of Liouville-type actions and points to future work in formulating an effective 2D action that captures both central charges.

Abstract

We show that topological 3D gravity with torsion can be formulated as a Chern-Simons gauge theory, provided a specific parameter, known as the effective cosmological constant, is negative. In that case, the boundary dynamics of the theory corresponding to anti-de Sitter boundary conditions is described by a conformal field theory with two different central charges.