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 nontrivial 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 \mathcal{M}, two discrete, freely acting, groups G_{1} and G_{2} and the worldsheet parity operator \Omega_{p} (such that \Omega_{p} : \sigma \to 2\pi  \sigma) an orientifold is expressed as the quotient space \mathcal{M}/(G_{1} \cup \Omega G_{2}). If G_{2} is empty, then the quotient space is an orbifold. If G_{2} is not empty, then it is an orientifold.
Contents

Application to string theory 1
 Supersymmetry breaking 1.1
 Effect on field content 1.2
 Notes 2
 References 3
Application to string theory
In string theory \mathcal{M} is the compact space formed by rolling up the theory's extra dimensions, specifically a sixdimensional CalabiYau space. The simplest viable compact spaces are those formed by modifying a torus.
Supersymmetry breaking
The six dimensions take the form of a CalabiYau for reasons of partially breaking the supersymmetry of the string theory to make it more phenomenologically viable. The Type II string theories have N=2 supersymmetry and compactifying them directly onto a sixdimensional torus increases this to N=8. By using a more general CalabiYau instead of a torus 3/4 of the supersymmetry is removed to give an N=2 theory again, but now with only 3 large spatial dimensions. To break this further to the only nontrivial phenomenologically viable supersymmetry, N=1, half of the supersymmetry generators must be projected out and this is achieved by applying the orientifold projection.
Effect on field content
A simpler alternative to using CalabiYaus to break to N=2 is to use an orbifold originally formed from a torus. In such cases it is simpler to examine the symmetry group associated to the space as the group is given in the definition of the space.
The orbifold group G_{1} is restricted to those groups which work crystallographically on the torus lattice,^{[1]} i.e. lattice preserving. G_{2} is generated by an involution \sigma, not to be confused with the parameter signifying position along the length of a string. The involution acts on the holomorphic 3form \Omega (again, not to be confused with the parity operator above) in different ways depending on the particular string formulation being used.^{[2]}
 Type IIB : \sigma (\Omega) = \Omega or \sigma (\Omega) = \Omega
 Type IIA : \sigma (\Omega) = \bar{\Omega}
The locus where the orientifold action reduces to the change of the string orientation is called the orientifold plane. The involution leaves the large dimensions of spacetime unaffected and so orientifolds can have Oplanes of at least dimension 3. In the case of \sigma (\Omega) = \Omega it is possible that all spatial dimensions are left unchanged and O9 planes can exist. The orientifold plane in type I string theory is the spacetimefilling O9plane.
More generally, one can consider orientifold Opplanes where the dimension p is counted in analogy with Dpbranes. Oplanes and Dbranes can be used within the same construction and generally carry opposite tension to one another.
However, unlike Dbranes, Oplanes are not dynamical. They are defined entirely by the action of the involution, not by string boundary conditions as Dbranes are. Both Oplanes and Dbranes must be taken into account when computing tadpole constraints.
The involution also acts on the complex structure (1,1)form J
 Type IIB : \sigma (J) = J
 Type IIA : \sigma (J) = J
This has the result that the number of moduli parameterising the space is reduced. Since \sigma is an involution, it has eigenvalues \pm 1. The (1,1)form basis \omega_{i}, with dimension h^{1,1} (as defined by the Hodge Diamond of the orientifold's cohomology) is written in such a way that each basis form has definite sign under \sigma. Since moduli A_{i} are defined by J = A_{i}\omega_{i} and J must transform as listed above under \sigma, only those moduli paired with 2form basis elements of the correct parity under \sigma survive. Therefore \sigma creates a splitting of the cohomology as h^{1,1} = h^{1,1}_{+} + h^{1,1}_{} and the number of moduli used to describe the orientifold is, in general, less than the number of moduli used to describe the orbifold used to construct the orientifold.^{[3]} It is important to note that although the orientifold projects out half of the supersymmetry generators the number of moduli it projects out can vary from space to space. In some cases h^{1,1} = h^{1,1}_{\pm}, in that all of the (11)forms have the same parity under the orientifold projection. In such cases the way in which the different supersymmetry content enters into the moduli behaviour is through the flux dependent scalar potential the moduli experience,the N=1 case is different from the N=2 case.
Notes
 ^ Lust; Reffert; Schulgin; Stieberger (2006). "Moduli Stabilization in Type IIB Orientifolds, Lust et al.".
 ^ Aldazabal; Camara; Font; Ibanez (2006). "More Dual Fluxes and Moduli Fixing, Font et al.".
 ^ Matthias Ihl; Daniel Robbins; Timm Wrase (2007). "Toroidal Orientifolds in IIA with General NSNS Fluxes,".
References
 A. Dabholkar (1998). "Lectures on orientifolds and duality".
 C. Angelantonj & A. Sagnotti (2002). "Open strings".