O(D,D)-covariant two-loop β-functions and Poisson-Lie T-duality

We show that the one- and two-loop $\beta$-functions of closed, bosonic string can be written in a manifestly O($D$,$D$)-covariant form. Based on this result, we prove 1) Poisson-Lie symmetric $\sigma$-models are renormalisable 2) their invariant under T-duality. Moreover, identify distinguished scheme which symmetry is manifest. It simplifies calculation significantly thereby provides powerful new tool to advance into quantum regime integrable generalised T-dualities. As an illustrating example, present $\lambda$- $\eta$-deformation.