Chern-Simons-like Gravity Theories

TL;DR

We develop a Hamiltonian analysis for CS-like gravity theories in 3D, starting from a metric-independent Lagrangian 3-form $L=rac{1}{2} g_{rs} a^r\cdot da^s + \frac{1}{6} f_{rst} a^r\cdot(a^s\times a^t)$. Applying the framework to Einstein-Cartan GR, GMG, ZDG, and GZDG, we count local degrees of freedom by examining primary and secondary constraints and their Poisson brackets, showing that GMG and ghost-free ZDG/GZDG propagate two massive spin-2 modes (i.e., two local DoF) while Einstein-Cartan GR has none. Ghost modes (the Boulware-Deser ghost) appear in generic ZDG parameter regions and are removed by assuming invertibility of a linear combination of dreibeine; the two secondary constraints in the invertible cases restore a consistent constraint count. The results illuminate how parity-violating extensions (GZDG) maintain the same DoF as GMG, and demonstrate a robust, linearization-free method for DoF counting with potential implications for AdS/CFT via central charges.

Abstract

A wide class of three-dimensional gravity models can be put into "Chern-Simons-like" form. We perform a Hamiltonian analysis of the general model and then specialise to Einstein-Cartan Gravity, General Massive Gravity, the recently proposed Zwei-Dreibein Gravity and a further parity violating generalisation combining the latter two.