Witten-Nester Energy in Topologically Massive Gravity

TL;DR

This work develops a Witten-Nester energy framework for topologically massive gravity with a cosmological constant by embedding CTMG in a first-order, N=(1,0) supergravity setting. It derives the Noether supercurrent and superpotential, and shows that the resulting Witten-Nester charges at the boundary reproduce Abbott-Deser-Tekin energies for asymptotically AdS spacetimes, with explicit matches in the left and right sectors. A generalized Witten equation for spinors is analyzed, yielding a bulk energy bound that depends on the Cotton tensor and implies positivity for globally well-defined spinor solutions, notably in Petrov type N cases under standard Brown-Henneaux boundary conditions. The paper computes ADT charges under HMT boundary conditions, explores exact vanishing-Cotton solutions, and examines chiral pp-waves, highlighting how boundary conditions and global spinor existence govern energy positivity in CTMG at and away from the chiral point. These results illuminate the interplay between boundary conditions, spinorial methods, and nonperturbative energy positivity in 3D gravity with Chern-Simons terms.

Abstract

We formulate topologically massive supergravity with cosmological constant in the first order formalism, and construct the Noether supercurrent and superpotential associated with its local supersymmetry. Using these results, we construct in ordinary topologically massive gravity the Witten-Nester integral for conserved charges containing spinors which satisfy a generalized version of Witten equation on the initial value surface. We show that the Witten-Nester charge, represented as an integral over the boundary of the initial value surface produces the Abbott-Deser-Tekin energy for asymptotically anti de Sitter spacetimes. We consider all values of the Chern-Simons coupling constant, including the critical value known as the chiral point, and study the cases of standard Brown-Henneaux boundary conditions, as well as their weaker version that allow a slower fall-off. Studying the Witten-Nester energy as a bulk integral over the initial value surface instead, we find a bound on the energy, and through it the sufficient condition for the positivity of the energy. In particular, we find that spacetimes of Petrov type N that admit globally well defined solutions of the generalized Witten equation have positive energy.