Orientifold

In theoretical physics orientifold is a generalization of the notion of orbifold, proposed by Augusto Sagnotti in 1987. The novelty is that in the case of string theory the non-trivial element(s) of the orbifold group includes the reversal of the orientation of the string. Orientifolding therefore produces unoriented strings—strings that carry no "arrow" and whose two opposite orientations are equivalent. Type I string theory is the simplest example of such a theory and can be obtained by orientifolding type IIB string theory.

In mathematical terms, given a smooth manifold ${\displaystyle {\mathcal {M}}}$, two discrete, freely acting, groups ${\displaystyle G_{1}}$ and ${\displaystyle G_{2}}$ and the worldsheet parity operator ${\displaystyle \Omega _{p}}$ (such that ${\displaystyle \Omega _{p}:\sigma \to 2\pi -\sigma }$) an orientifold is expressed as the quotient space ${\displaystyle {\mathcal {M}}/(G_{1}\cup \Omega G_{2})}$. If ${\displaystyle G_{2}}$ is empty, then the quotient space is an orbifold. If ${\displaystyle G_{2}}$ is not empty, then it is an orientifold.