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.

中文翻译：

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Orientifolds 的符号选择

我们分析了在 II 型弦理论的定向折叠中为 O 平面分配符号选择的问题。我们表明存在一个具有 $p\geq-1$ 的不变 $p$-gerbes 序列，它们引起符号选择并通过共边界图相关联。我们证明了符号选择同态随着三边形的维数而稳定，并且我们推导出对可能的符号配置的拓扑约束。对球形和环形立体折线进行了具体计算，特别是我们展示了一个四维立体折线，其中并非每个符号选择都是几何上可实现的。我们阐明了与 $p=-1,0,1$ 的不变 $p$-gerbes 相关的 $K$-理论群如何与共边界图相互作用。这使我们能够将由于 Gao 和 Hori 的 $K$-theory 的概念解释为扭曲的 $KR$-theory 的特例，