D-branes in $\lambda$-deformations

We show that the geometric interpretation of D-branes in WZW models as twisted conjugacy classes persists in the $\lambda$–deformed theory. We obtain such configurations by demanding that a monodromy matrix constructed from the Lax connection of the $\lambda$–deformed theory continues to produce conserved charges in the presence of boundaries. In this way the D-brane configurations obtained correspond to `integrable’ boundary configurations. We illustrate this with examples based on $SU(2)$ and $SL(2,\mathbb{R})$, and comment on the relation of these D-branes to both non-Abelian T-duality and Poisson-Lie T-duality. We show that the D2 supported by D0 charge in the $\lambda$–deformed theory map, under analytic continuation together with Poisson-Lie T-duality, to D3 branes in the $\eta$-deformation of the principal chiral model.