Sign choices for orientifolds

We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant $p$-gerbes with $p\geq-1$, which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the $K$-theory groups associated with invariant $p$-gerbes for $p=-1,0,1$ interact with the coboundary maps. This allows us to interpret a notion of $K$-theory due to Gao and Hori as a special case of twisted $KR$-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction.