Orientifold
Encyclopedia
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
, two discrete
, freely acting, groups
and 
and the worldsheet
parity
operator
(such that 
) an orientifold is expressed as the quotient space 
. If 
is empty, then the quotient space is an orbifold. If 
is not empty, then it is an orientifold.
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 orbifold group
is restricted to those groups which work crystallographically
on the torus
lattice, i.e. lattice preserving.
is generated by an involution 
, not to be confused with the parameter signifying position along the length of a string. The involution acts on the holomorphic 3-form 
(again, not to be confused with the parity operator above) in different ways depending on the particular string formulation being used.
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 space-time unaffected and so orientifolds can have O-planes of at least dimension 3. In the case of
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.
However, unlike D-branes, O-planes are not dynamical. 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. Since
is an involution, it has eigenvalues 
. The (1,1)-form basis 
, with dimension 
(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
. Since moduli 
are defined by 
and J must transform as listed above under 
, only those moduli paired with 2-form basis elements of the correct parity under 
survive. Therefore 
creates a splitting of the cohomology as 
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. 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 
, in that all of the (1-1)-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.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
orientifold is a generalization of the notion of orbifold
Orbifold
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold is a generalization of a manifold...
, 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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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
Type I string theory
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...
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
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, two discreteDiscrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
, freely acting, groups
and and the worldsheetWorldsheet
In string theory, a worldsheet is a two-dimensional manifold which describes the embedding of a string in spacetime. The term was coined by Leonard Susskind around 1967 as a direct generalization of the world line concept for a point particle in special and general relativity.The type of string,...
parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
operator
(such that ) an orientifold is expressed as the quotient space . If is empty, then the quotient space is an orbifold. If is not empty, then it is an orientifold.Application to String Theory
In string theory 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.Supersymmetry Breaking
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 N=2 supersymmetry and compactifying them directly onto a six dimensional torus increases this to N=8. By using a more general Calabi-Yau 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 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.Effect on Field Content
A simpler alternative to using Calabi-Yaus 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
is restricted to those groups which work crystallographicallyCrystallographic point group
In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...
on the torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
lattice, i.e. lattice preserving.
is generated by an involution , not to be confused with the parameter signifying position along the length of a string. The involution acts on the holomorphic 3-form (again, not to be confused with the parity operator above) in different ways depending on the particular string formulation being used.- Type IIB :
or 
- Type IIA :
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 space-time unaffected and so orientifolds can have O-planes of at least dimension 3. In the case of
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.
However, unlike D-branes, O-planes are not dynamical. 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
- Type IIB :
- Type IIA :
This has the result that the number of moduli
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
parameterising the space is reduced. Since
is an involution, it has eigenvalues . The (1,1)-form basis , with dimension (as defined by the Hodge Diamond of the orientifold's cohomologyCohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
) is written in such a way that each basis form has definite sign under
. Since moduli are defined by and J must transform as listed above under , only those moduli paired with 2-form basis elements of the correct parity under survive. Therefore creates a splitting of the cohomology as 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. 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 , in that all of the (1-1)-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.