T-dualization of G\"{o}del string cosmologies via Poisson-Lie T-duality approach

Using the homogeneous G"{o}del spacetimes we find some new solutions for the field equations of bosonic string effective action up to first order in ${\alpha }^{\mathrm{\prime }}$ including both dilaton and axion fields. We then discuss in detail the (non-)Abelian T-dualization of G"{o}del string cosmologies via the Poisson-Lie (PL) T-duality approach. In studying Abelian T-duality of the models we get seven dual models in such a way that they are constructed by one-, two- and three-dimensional Abelian Lie groups acting freely on the target space manifold. The results of our study show that the Abelian T-dual models are, under some of the special conditions, self-dual; moreover, by applying the usual rules of Abelian T-duality without further corrections, we are still able to obtain two-loop solutions. We also study the Abelian T-duality of G"{o}del string cosmologies up to ${\alpha }^{\mathrm{\prime }}$-corrections by using the T-duality rules at two-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian duals of the G"{o}del spacetimes are constructed by two- and three-dimensional non-Abelian Lie groups such as $A_2$, $A_2\oplus A_1$ and $SL(2,\mathbb{R})$. In this way, the PL self-duality of $Ad{S}_3\times \mathbb{R}$ space is discussed.