Self-dual solutions of 2+1 Einstein gravity with a negative cosmological constant

TL;DR

The authors classify and construct self-dual solutions of 2+1 Einstein gravity with a negative cosmological constant by quotienting AdS$_{2+1}$ along orbits of self-dual Killing vectors in so$(2,2)$, yielding explicit globally defined metrics. They organize the quotients into three types (A,B,C) and compute their holonomies, Killing vectors, and Killing spinors, showing that the Type A family is causally regular and geodesically complete, while Types B and C exhibit closed causal curves. The results are presented in a group-theoretic, AdS$_{2+1}$ group-manifold framework, with a detailed Chern–Simons holonomy description and analysis of supersymmetry, including a demonstration of invariance under string duality along the angular direction. These self-dual quotients reveal a rich structure of globally well-behaved spacetimes in 2+1 gravity and connect to holographic and string-duality perspectives within AdS$_3$ contexts.

Abstract

All the causally regular geometries obtained from (2+1)-anti-de Sitter space by identifications by isometries of the form $P \rightarrow (\exp πξ) P$, where $ξ$ is a self-dual Killing vector of $so(2,2)$, are explicitely constructed. Their remarkable properties (Killing vectors, Killing spinors) are listed. These solutions of Einstein gravity with negative cosmological constant are also invariant under the string duality transformation applied to the angular translational symmetry $φ\rightarrow φ+a$ The analysis is made particularly convenient through the construction of {\em global} coordinates adapted to the identifications.}