The Classical Solutions of the Dimensionally Reduced Gravitational Chern-Simons Theory

TL;DR

This work analyzes the dimensionally reduced gravitational Chern-Simons term, recast as a 2D gravity–gauge theory equivalent to a Poisson-Sigma model with a 4D target and a degenerate Poisson tensor. The classical solution space is fully classified by two Casimir invariants, the charge $c$ and the geometric energy $\mathcal{C}^{(g)}$, and all solutions are constructed patchwise using first-order gravity methods, enabling a complete global Carter-Penrose diagrammatic description. The authors derive explicit forms for constant-dilaton vacua and generic EF-gauge solutions, determine the horizon structure via the zeros of the Killing norm, and present a global patching scheme with detailed CP diagrams, including special tunings that reproduce prior results (hep-th/0305117). They also discuss the potential generalizations and highlight how the horizon structure transitions between 0 and 4 horizons depending on the constants, illuminating the global geometry of this dimensionally reduced theory.

Abstract

The Kaluza-Klein reduction of the 3d gravitational Chern-Simons term to a 2d theory is equivalent to a Poisson-sigma model with fourdimensional target space and degenerate Poisson tensor of rank 2. Thus two constants of motion (Casimir functions) exist, namely charge and energy. The application of well-known methods developed in the framework of first order gravity allows to construct all classical solutions straightforwardly and to discuss their global structure. For a certain fine tuning of the values of the constants of motion the solutions of hep-th/0305117 are reproduced. Possible generalizations are pointed out.