Topological defects in string theory orbifolds with target spaces $\mathbb{C}/\mathbb{Z}_N$ and $S^1/\mathbb{Z}_2$

We study conformal defects in two important examples of string theory orbifolds. First, we show that topological defects in the language of Landau-Ginzburg models carry information about the RG flow between the non-compact orbifolds $\mathbb{C}/\mathbb{Z}_d$. Such defects are shown to correctly implement the bulk-induced RG flow on the boundary. Secondly, we study what the possible conformal defects are between the $c=1$ bosonic 2D conformal field theories with target space $S^1/\mathbb{Z}_2$. The defects cataloged here are obtained from boundary states corresponding to D-branes in the $c=2$ free theory with target space $S^1/\mathbb{Z}_2 \times S^1/\mathbb{Z}_2$. Via the unfolding procedure, such boundary states are later mapped to defects between the circle orbifolds. Furthermore, we compute the algebra of the topological class of defects at different radii.