Renormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor

We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim $\sigma$-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant $\sigma$-model or possibly by a more general AKSZ model.