Surveying the Three-Dimensional Fixed Points of T-Duality

We explore the family of fixed points of T-Duality transformations in three dimensions. For the simplest nontrivial self-duality conditions it is possible to show that, additionally to the spacelike isometry in which the T-Duality transformation is performed, these backgrounds must be necessarily stationary. This allows to prove that for nontrivial string coupling, the low energy bosonic string backgrounds which are additionally self-T-dual along an isometry direction generated by a constant norm Killing vector are uniquely described by a two-parametric class, including only three nonsingular cases: the charged black string, the exact gravitational wave propagating along the extremal black string, and the flat space with a linear dilaton. Besides, for constant string coupling, the only self-T-dual lower energy string background under the same assumptions corresponds to the Coussaert-Henneaux spacetime. Thus, we identify minimum criteria that yield a classification of these quoted examples and only these. All these T-dual fixed points describe exact backgrounds of string theory.