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.
In mathematical terms, given a smooth manifold
) an orientifold is expressed as the quotient space
is empty, then the quotient space is an orbifold.
is the compact space formed by rolling up the theory's extra dimensions, specifically a six-dimensional Calabi–Yau space.
The simplest viable compact spaces are those formed by modifying a torus.
The six dimensions take the form of a Calabi–Yau for reasons of partially breaking the supersymmetry of the string theory to make it more phenomenologically viable.
The Type II string theories have 32 real supercharges, and compactifying on a six-dimensional torus leaves them all unbroken.
Compactifying on a more general Calabi–Yau sixfold, 3/4 of the supersymmetry is removed to yield a four-dimensional theory with 8 real supercharges (N=2).
To break this further to the only non-trivial phenomenologically viable supersymmetry, N=1, half of the supersymmetry generators must be projected out and this is achieved by applying the orientifold projection.
A simpler alternative to using Calabi–Yaus to break to N=2 is to use an orbifold originally formed from a torus.
, not to be confused with the parameter signifying position along the length of a string.
(again, not to be confused with the parity operator above) in different ways depending on the particular string formulation being used.
[2] The locus where the orientifold action reduces to the change of the string orientation is called the orientifold plane.
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 spacetime-filling O9-plane.
More generally, one can consider orientifold Op-planes where the dimension p is counted in analogy with Dp-branes.
O-planes and D-branes can be used within the same construction and generally carry opposite tension to one another.
They are defined entirely by the action of the involution, not by string boundary conditions as D-branes are.
Both O-planes and D-branes must be taken into account when computing tadpole constraints.
The involution also acts on the complex structure (1,1)-form J This has the result that the number of moduli parameterising the space is reduced.
(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
, only those moduli paired with 2-form basis elements of the correct parity under
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.