The General Supersymmetric Solution of Topologically Massive Supergravity

TL;DR

This work derives the complete set of fully nonlinear supersymmetric solutions of N=1 topologically massive supergravity admitting a Killing spinor, showing they are plane-wave Kerr–Schild spacetimes on AdS$_3$ with a null Killing vector. The general solution is encoded by a single function $f(u)$ and, for generic $\\mu$, yields asymptotically regular but non-compactifiable geometries; at the critical values $\\mu=\\pm1$ logarithmic boundary terms appear, obstructing smooth conformal compactifications. A Nester–Witten energy identity is established, but the Cotton tensor term prevents a universal positivity proof, though the BTZ black hole case provides a well-defined charge $M+J$ for a specific Killing vector. The results indicate identical local degrees of freedom across $\\mu$, with the introduced solutions remaining robust under quantum corrections (cogihepo) and clarifying the relation between supersymmetry, global structure, and boundary conditions in topologically massive supergravity.

Abstract

We find the general fully non-linear solution of topologically massive supergravity admitting a Killing spinor. It is of plane-wave type, with a null Killing vector field. Conversely, we show that all solutions with a null Killing vector are supersymmetric for one or the other choice of sign for the Chern-Simons coupling constant μ. If μdoes not take the critical value μ=\pm 1, these solutions are asymptotically regular on a Poincaré patch, but do not admit a smooth global compactification with boundary S^1\times\R. In the critical case, the solutions have a logarithmic singularity on the boundary of the Poincaré patch. We derive a Nester-Witten identity, which allows us to identify the associated charges, but we conclude that the presence of the Chern-Simons term prevents us from making a statement about their positivity. The Nester-Witten procedure is applied to the BTZ black hole.