  Motivated by quantum oscillations observed in highly ordered ortho-II
YBa_2Cu_3O_6.5, we study the Fermi surface topology of d-density wave (ddw) or
antiferromagnetic (AF) ordering in the presence of the ortho-II potential. We
find that the electron pocket is unaffected by the presence of the ortho-II
potential. This further strengthens the proposal that quantum oscillations
observed in ortho-II-free YBa_2Cu_4O_8 arise from an electron pocket. On the
other hand, the hole pocket topology is sensitive to the ortho-II potential. We
show there exist generically three distinct quantum oscillations associated
with one electron-like and two hole-like Fermi pockets. We compare our results
to the quantum oscillations observed in experiments. We discuss possible ways
to distinguish between ddw and antiferromagnetic orders in a potential
single-layer ortho-II material.
