  The Fateev-Zamolodchikov-Zamolodchikov (FZZ) duality relates Witten's cigar
model to sine-Liouville theory. This duality was proven in the path integral
formulation and extended to the case of higher genus closed Riemann surfaces by
Schomerus and one of the authors. In this note we further extend the duality to
the case with boundary. Specifically, we relate D1-branes in the cigar model to
D2-branes in the sine-Liouville theory. In particular, the boundary action for
D2-branes in the sine-Liouville theory is constructed. We also consider the
fermionic version of the FZZ duality. This duality was proven as a mirror
symmetry by Hori and Kapustin, but we give an alternative proof in the path
integral formulation which directly relates correlation functions. Also here
the case with boundary is investigated and the results are consistent with
those for branes in N=2 super Liouville field theory obtained by Hosomichi.
