  We construct "Flying Saucer" solitons in supersymmetric N=2 gauge theory
which is known to support BPS domain walls with a U(1) gauge field localized on
its worldvolume. We demonstrate that this model supports exotic particle-like
solitons whose topology is largely arbitrary: closed orientable surfaces in
three-dimensional space which can have arbitrary genus. In particular, we focus
on Q tori. Q tori and similar solitons of higher genera are obtained by folding
the domain wall into an appropriate surface. Nontrivial cycles on the domain
wall worldvolume (handles) are stabilized by crossed electric and magnetic
fields inside the folded domain wall. Three distinct frameworks are used to
prove the existence of these Flying Saucer solitons and study their properties:
the worldvolume description (including the Dirac-Born-Infeld action), the bulk
theory description in the sigma-model limit, and the bulk theory description in
the thin-edge approximation. In the sigma-model framework the Q torus is shown
to be related to the Hopf Skyrmion studied previously.
