Anti-Quill Part 2: Orientifolds

 Like I said in Anti-Quill Part 1, when doing compactifications we also have to consider D-branes and O-planes. These orientifolds are quite mysterious; in a mathematical context, these arise from RR tadpole cancellation in type IIA/B superstring theories and are related to orbifolds. For now, however, we will approach orientifolds from the second and the more characteristic nature of orientation reversal. To see this, recall that strings on the worldsheet in closed string theories decompose into left and right moving modes $X^{\mu }_{L}(t+\sigma )$ and $X^{\mu }_{R}(t-\sigma )$. With periodicity $\pi $, we can introduce a parity conjugating operator $\mathcal{P}$ so that $\mathcal{P} : X^{\mu }_{L}(t+\sigma )\to X^{\mu }_{R}(t-\sigma )$. In the grand scheme of M-theory, you have the duality between type IIA and type IIB superstring theories via the T-duality, and you can relate type I superstring theory to type IIB and conversely type IIA by using a combination of T-duality and the action of $\mathcal{P}$. In this sense, taking an un-oriented version of type I string theory on $S^{1}$ and acting on it with T-duality gives you a type IIA string theory. In this language of O-planes, a type I theory is a type IIB theory with an O9-plane, whereas with type IIA theory it would be an O8-plane. Going from type IIB to type I theory, the O-planes break half the supersymmetry of the full theory.