In this chapter we construct orientifolds of type IIB 4d compactifications. There are several constructions in this class, including toroidal orientifolds, systems of D3-branes at singularities, and magnetized D-branes, providing the mirrors of type IIA models in Chapter 10. We also introduce F-theory and its compactifications. These setups provide a rich arena for particle physics model building in string theory.

Generalities of type IIB orientifold actions

Type IIB orientifolds are obtained by considering IIB theory on a CY X6 and modding out by ΩR, where R is a geometric symmetry acting holomorphically on the complex coordinates on X6. For instance, the simplest orientifold action is just Ω, with trivial R, thus acting holomorphically zi → zi in a trivial way. These models involve 10d spacetime filling orientifold planes (O9-planes) and open string sectors (D9-branes), and so correspond to type I compactifications on X6, which are thus included as particular type IIB orientifold compactifications. Additional constructions are obtained for non-trivial actions R, as follows:

• O7/D7 models: Consider the orientifold action ΩRi (–1)FL, where Ri acts as zi →–zi leaving other complex coordinates invariant; the factor (–1)FL, with FL being the leftmoving fermion number, is necessary for the orientifold action to square to the identity operator. The quotient introduces O7-planes at the fixed points of Ri, namely transverse to the coordinate zi, and wrapped on a 4-cycle parametrized by the remaining two complex coordinates.