In all string compactifications studied in previous chapters, the massless spectrum contains a large number of moduli, including the dilaton and the CY geometric moduli. These are problematic, as they couple to matter particles and easily lead to deviations from universality of gravitational interactions (fifth forces), which have not been observed experimentally. Therefore, any serious attempt to reproduce realistic string models of particle physics and/or cosmology, must address the issue of moduli stabilization; namely, the generation of a scalar potential fixing the vevs of the moduli fields and giving them large enough masses to overcome such phenomenological problems. In this chapter we review a very general and systematic mechanism to fix large numbers of (or even all) moduli. It is based on a generalization of the simplest compactification ansatz, allowing for non-trivial backgrounds for additional 10d fields. Most prominently, the compactifications include non-trivial fluxes for the field strength tensor of the diverse p-form fields in the 10d theory. Application of string dualities to these motivates additional possible backgrounds, termed geometric and non-geometric fluxes. Compactifications including field strength fluxes, or their generalizations, are thus known as “flux compactifications.” This chapter focuses on their definition and impact on moduli stabilization, while their role in SUSY breaking is further explored in Chapter 15.

Type IIB with 3-form fluxes

A prototypical class of flux compactifications is obtained from type IIB (orientifolds) on CY geometries with non-trivial fluxes for the NSNS and RR 3-form field strengths (and their F-theory generalization, which we briefly touch upon).

In Chapter 7 we have studied the construction of 4d N =1 heterotic string vacua by compactification on smooth CY manifolds. The analysis of such compactifications is however limited because their worldsheet theory is not exactly solvable. Therefore, we must rely on the Kaluza–Klein reduction of the 10d supergravity action, which provides a good approximation to the 4d physics only in the large volume regime. In this chapter we study 4d N =1 heterotic string vacua obtained from toroidal orbifold compactifications and other exact CFT constructions, which overcome this limitation. These are simple α′-exact compactifications, which share many properties with more general CY compactifications (and in fact are often closely related to them), and which allow a very explicit construction of phenomenologically interesting particle physics models. We describe toroidal orbifolds, which provide a free CFT description of geometries which can be regarded as singular limits of CY spaces. We also introduce asymmetric orbifolds and free fermionic models, which are also described by free worldsheet CFTs but in general do not admit a geometric interpretation. Finally we describe Gepner models (and orbifolds thereof), defined by interacting but solvable CFTs and which can often be regarded as compactifications on CY spaces of stringy size. We focus on the E8 × E8 heterotic string theory, although the basic rules apply in complete analogy to the SO(32) theory.

Toroidal orbifolds

Let us start by describing the geometry of toroidal orbifolds. The main ingredients are useful for heterotic string compactifications, as well as for other string theories, like type II orientifolds.

In Chapter 5 we showed that toroidal compactifications of superstrings can lead at low energies to 4d theories, albeit with too much supersymmetry to allow for chirality, an essential ingredient of particle physics. We will thus be interested in constructing more general classes of compactifications leading to chiral theories in 4d, an enterprise starting in this chapter and reaching until Chapter 12, and for which it is convenient to give an aerial view.

A road map for string compactifications

There are different options to achieve the construction of chiral 4d compactifications in string theory, summarized in Figure 7.1, and roughly classified according to the underlying 10d or 11d theory taken as starting point. In this book we will study in turn heterotic compactifications, type IIA orientifolds and type IIB orientifolds. In this chapter and Chapter 8 we describe different varieties of heterotic string compactifications, whose 4d effective action is studied in Chapter 9. We consider models in the geometric realm, using Calabi–Yau manifolds and toroidal orbifolds, and in the more abstract non-geometric conformal field theory (CFT) framework, using asymmetric orbifolds, free fermions, and Gepner constructions. In Chapter 10 we study type IIA orientifold constructions with gauge fields localized on D6-branes, and matter fields at their intersections. We consider explicit examples in toroidal orbifold compactifications, as well as Gepner constructions. Supersymmetric type IIA models with intersecting D6-branes are related to purely geometric compactifications of M-theory on manifolds with G2 holonomy, as we will also review.

In this chapter we consider different aspects of the low-energy 4d effective action of heterotic compactifications. After discussing the structure of the effective action for the general CY compactifications of Chapter 7, we concentrate on the case of the abelian orbifold compactifications of Chapter 8, in which several additional aspects can be described in an explicit quantitative fashion. We discuss the form of the Kähler potential and the superpotential, as well as the gauge kinetic function and their one-loop corrections. In toroidal orbifolds the low-energy effective actionmust bemodular invariant under T-duality transformations, and this leads to strong constraints on its structure. A general feature of heterotic compactifications is that they often have a non-trivial cancellation of mixed U(1) anomalies, involving a generalized Green×Schwarz mechanism. This renders some U(1) gauge bosons massive, and introduces a one-loop re-stabilization of the vacuum, triggering the breaking of extra gauge symmetries and the disappearance of charged matter multiplets. This phenomenon can be efficiently exploited in the construction of improved models of particle physics, as we describe using explicit examples.

A first look at the heterotic 4d N =1 effective action

We start by motivating the general structure of the effective action of heterotic string geometric (CY or orbifold) compactifications by using an intuitive description of the truncation to the zero mode sector. The resulting structure is quite general, as can be shown to largely follow from symmetries of the system. We finally describe more explicit results for CY compactifications with standard embedding.

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.

In the preceding chapters we have displayed the wealth of possibilities to build specific 4d string vacua. In this chapter we take a more general viewpoint and present an aerial view of the space of string vacua (at our present level of knowledge), in particular of those corners with low-energy physics resembling closely the SM of particle physics. In spite of the apparent enormous number of consistent vacua, the space of field theories obtainable as low-energy effective descriptions of string theories can be argued to be strongly constrained, forming a minute subset in the space of all possible field theories. We present some of these general properties of string vacua, and then turn to overviewing the status of realistic vacua searches in different string compactification setups. These explicit models are presumably only the tip of the iceberg in the set of semi-realistic models in string theory, which we term the flavour landscape. In addition, the latter may be complemented by additional ingredients like flux degrees of freedom, yielding a large multiplicity of vacua, known as the flux landscape. Our outlook summarizes some general reflections on string theory and particle physics, inspired by this general perspective.

General properties of the massless spectrum in string compactifications

The spectrum and symmetries of light particles in 4d string compactifications is very model dependent. There are, however, several important general properties valid in any compactification, and which can provide non-trivial constraints in model building.