All timelike supersymmetric solutions of three-dimensional half-maximal supergravity

TL;DR

This paper classifies all timelike supersymmetric solutions in three-dimensional half-maximal supergravity, showing they are composed of solutions to integrable Liouville and SU(3) Toda systems, completing the solution space characterization.

Contribution

It provides a complete classification of all timelike supersymmetric solutions in the theory, including their explicit construction from integrable systems.

Findings

Solutions preserve 8, 16, or 24 supersymmetries.

All solutions can be expressed via Liouville and SU(3) Toda equations.

The null case solutions are already known, completing the classification.

Abstract

We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector coming from the Killing spinor bilinear by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve $16/2^n, 1 \leq n \leq 3$ real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that all such solutions can be expressed as a direct sum of solutions of the integrable Liouville and SU(3) Toda systems. This completes the construction of all supersymmetric solutions of the model since the null case has already been solved.