  We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections
in superstring theory can be conveniently understood by the inclusion of
fermionic invertible phases, or equivalently topological superconductors, on
the worldsheet. This allows us to find that the unoriented Type $\rm 0$ string
theory with $\Omega^2=(-1)^{\sf f}$ admits different GSO projections
parameterized by $n$ mod 8, depending on the number of Kitaev chains on the
worldsheet. The presence of $n$ boundary Majorana fermions then leads to the
classification of D-branes by $KO^n(X)\oplus KO^{-n}(X)$ in these theories,
which we also confirm by the study of the D-brane boundary states. Finally, we
show that there is no essentially new GSO projection for the Type $\rm I$
worldsheet theory by studying the relevant bordism group, which classifies
corresponding invertible phases.
  This paper provides the details for the results announced in the letter
\cite{Kaidi:2019pzj}.
