AbstractUsing the homogeneous Gödel spacetimes we find some new solutions for the field equations of bosonic string effective action up to first order in $$\alpha '$$
α
′
including both dilaton and axion fields. We then discuss in detail the (non-)Abelian T-dualization of Gödel 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 Gödel string cosmologies up to $$\alpha '$$
α
′
-corrections by using the T-duality rules at two-loop order derived by Kaloper and Meissner. Afterwards, non-Abelian duals of the Gödel spacetimes are constructed by two- and three-dimensional non-Abelian Lie groups such as $$A_2$$
A
2
, $$A_2 \oplus A_1$$
A
2
⊕
A
1
and $$SL(2, \mathbb {R})$$
S
L
(
2
,
R
)
. In this way, the PL self-duality of $$AdS_3 \times \mathbb {R}$$
A
d
S
3
×
R
space is discussed.
Erratum to: T duality and Wald entropy formula in the Heterotic Superstring effective action at first-order in α′
