Heterotic Orbifold Models

First and foremost, we wish to obtain a quantum field theory (QFT) that is consistent with the SM (see e.g. [10] for a detailed description). The latter is based on the continuous gauge symmetry¹¹1Strictly speaking we do not really know the gauge group of the SM but only its Lie algebra, a subtlety which we will, like most of the literature, not discuss in detail.

$$G_{\text{SM}}=\text{SU}(3)_{\text{C}}\times\text{SU}(2)_{\text{L}}\times\text{U}(1)_{\text{Y}}\;.$$

(1)

The matter content consists of three generations of quarks and leptons, left-chiral Weyl fermions which transform as

	quarks	$\displaystyle:~{}q_{f}=(\boldsymbol{3},\boldsymbol{2})_{\nicefrac{{1}}{{6}}}\;,~{}\bar{u}_{f}=(\overline{\boldsymbol{3}},\boldsymbol{1})_{-\nicefrac{{2}}{{3}}}\;,~{}\bar{d}_{f}=(\overline{\boldsymbol{3}},\boldsymbol{1})_{\nicefrac{{1}}{{3}}}\;,$		(2a)
	leptons	$\displaystyle:~{}\ell_{f}=(\boldsymbol{1},\boldsymbol{2})_{-\nicefrac{{1}}{{2}}}\;,~{}\bar{e}_{f}=(\boldsymbol{1},\boldsymbol{1})_{1}$		(2b)

under $G_{\text{SM}}$. Here, $f\in\{1,2,3\}$ labels the generations. In addition, there is the so-called Higgs field, a complex scalar carrying the quantum numbers $h=(\boldsymbol{1},\boldsymbol{2})_{\nicefrac{{1}}{{2}}}$. The Higgs acquires a vacuum expectation value (VEV), $\langle h\rangle\sim 100\,\text{GeV}$, which breaks $G_{\text{SM}}$ down to $\text{SU}(3)_{\text{C}}\times\text{U}(1)_{\text{em}}$, under which (2) are vector-like and acquire masses which are given by the product of so-called Yukawa couplings $Y_{u,d,e}$ and $\langle h\rangle$. In the SM, the Yukawa couplings are input parameters which are adjusted to fit data. A curious fact about the SM is that the combination of charge conjugation and parity, $\mathcal{CP}$, is broken in the flavor sector, i.e. by the Yukawa couplings, but seemingly not in the strong interactions. This mismatch gets referred to as the strong $\mathcal{CP}$ problem. The neutrinos, which are part of the $\ell_{f}$, are also massive, yet it is currently not known which operator describes their mass. The most plausible options are the Weinberg operator, $\kappa_{gf}(\ell^{g}h)(\ell^{f}h)$, or Dirac neutrino masses, in which case one has to amend (2) by right-handed neutrinos $\nu^{f}$. The neutrino masses are much smaller than the masses of the charged fermions.

It is instructive to survey the continuous parameters of the SM. They comprise (i) $3$gauge couplings; (ii) $\theta_{\mathrm{QCD}}$; (iii) $2$Higgs parameters; (iv) $12$masses; (v) $8$or $10$ mixing parameters, depending on whether neutrinos are Dirac or Majorana particles. In a stringy completion, these parameters should be predicted rather than adjusted, and, as we shall discuss in Section 6, the requirement to reproduce these observables remains one of the greatest challenges in string model building. Note also that currently the only other parameters that we need to describe observation are the Planck mass $M_{\text{P}}$ (or, equivalently, $G_{\text{Newton}}$), the vacuum energy $\rho_{\text{vacuum}}$ and the density to dark matter, $\rho_{\text{DM}}$. That is, currently the bulk of the (ununderstood) parameters of Nature resides in the flavor sector of the SM.

An important fact about the SM is that it does not only provide us with couplings and interactions that have been confirmed in experiments, but it also comes with tight constraints on additional particles and interactions. In particular, it is extremely hard to make extra states which are chiral w.r.t. $G_{\text{SM}}$ consistent with observation. Also, while the Weinberg operator is a nonrenormalizable operator that is “good” in the sense that it can describe neutrino masses, other higher-dimensional operators are highly constrained. For instance, the suppression scale of the dimension-6 operators leading to proton decay has to exceed $10^{15}\,\text{GeV}$.

The early universe provides us with important insights into high energy physics (see e.g. [11]). For instance, big bang nucleosynthesis (BBN) works very well within the SM, and extra particles may be inconsistent with the primordial formation of the elements if they decay late or increase the Hubble expansion rate too much. In addition, fractionally charged particles are often stable since they cannot decay into SM states, and there are stringent constraints on their relic abundance (cf. e.g. [12, 13]).

However, the early universe also requires physics beyond the SM. Most notably we need a field or sector that drive inflation, or another ingredient which provides us with solutions to the so-called horizon and flatness problems. In addition, there is very compelling evidence for dark matter which cannot be made of SM particles. Furthermore, the baryon asymmetry of the universe requires physics beyond the SM, too.

Having seen that physics beyond the SM is required to accommodate astrophysics and cosmology, let us spend some words on beyond the standard model (BSM) scenarios.

The supersymmetric variants of the SM (see e.g. [14] for an introduction), most notably the minimal supersymmetric standard model (MSSM), have received substantial attention in the past decades. This is because, assuming low-energy supersymmetry (SUSY), the electroweak scale gets stabilized against quantum corrections. Of course, given the absence of clear signals for SUSY at the Large Hadron Collider (LHC), this scheme has lost some of its popularity in the recent years, yet the MSSM is arguably still one of the best motivated and well-defined BSM scenarios. The MSSM has a number of shortcomings which one may hope to solve in ultraviolet (UV) completions, and the purpose of this review is to discuss these solutions in Section 7. In order to understand some of these shortcomings, let us look at the $G_{\text{SM}}$ invariant superpotential terms up to order 4,

	$\displaystyle\mathscr{W}_{\text{gauge~{}invariant}}=\mu\,{h}_{d}{h}_{u}+\kappa_{i}\,{\ell}_{i}{h}_{u}$
	$\displaystyle\quad{}+Y_{e}^{gf}\,{\ell}_{g}{h}_{d}\overline{{e}}_{f}+Y_{d}^{gf}\,{q}_{g}{h}_{d}\overline{{d}}_{f}+Y_{u}^{gf}\,{q}_{g}{h}_{u}\overline{{u}}_{f}$
	$\displaystyle\quad{}+\lambda_{gfk}\,{\ell}_{g}{\ell}_{f}\overline{{e}}_{k}+\lambda^{\prime}_{gfk}\,{\ell}_{g}{q}_{f}\overline{{d}}_{k}+\lambda^{\prime\prime}_{gfk}\,\overline{{u}}_{g}\overline{{d}}_{f}\overline{{d}}_{k}$
	$\displaystyle\quad{}+\kappa_{gf}\,{h}_{u}{\ell}_{g}\,{h}_{u}{\ell}_{f}+\kappa^{(1)}_{gfk\ell}\,{q}_{g}{q}_{f}{q}_{k}{\ell}_{\ell}+\kappa^{(2)}_{gfk\ell}\,\overline{{u}}_{g}\overline{{u}}_{f}\overline{{d}}_{k}\overline{{e}}_{\ell}\;,$		(3)

where we, in a slight abuse of notation, denoted the superfields by the same symbols as the SM fields in Equation 2. Note also that the MSSM has two Higgs doublets, $h_{u}$ and $h_{d}$. The couplings $Y_{u}$, $Y_{d}$ and $Y_{e}$ are the Yukawa couplings which yield the masses of quarks and charged leptons. The $R$-parity violating terms $\kappa_{i}$, $\lambda_{gfk}$, $\lambda_{gfk}^{\prime}$ and $\lambda_{gfk}^{\prime\prime}$ have to be highly suppressed, and get often forbidden by $R$ (or matter) parity, the origin of which is to be clarified in a UV completion of the model. The $\mu$ term in the first line of Equation 3 can a priori have any size, but in order to have proper electroweak symmetry breaking and sufficiently heavy Higgsinos it should be of the order TeV or so, which is a common choice for the soft SUSY breaking masses. Explanations of this fact comprise the Kim–Nilles [15] and Giudice–Masiero [16] mechanisms, and we will see later in Section 7 both are realized in explicit stringy completions of the MSSM. Further, while the $\kappa$ term in the last line of Equation 3 can describe neutrino masses, the $\kappa^{(i)}$ terms have to be very small, e.g. the coefficients $\kappa^{(1)}_{1121}$ and $\kappa^{(1)}_{1122}$ have to be suppressed by more than $10^{8}\cdot M_{\text{P}}$, where $\sqrt{8\pi}\,M_{\text{P}}=M_{\text{Planck}}\simeq 1.2\cdot 10^{19}\,\text{GeV}$. A proper understanding of this suppression arguably requires a solution within a consistent theory of quantum gravity, such as the explicit string models we consider here.

Another appealing feature of the MSSM with low-energy SUSY is that gauge couplings unify remarkably well at a scale $M_{\text{GUT}}\simeq\text{few}\times 10^{16}\,\text{GeV}$ [17]. This has led to the scheme of SUSY Grand Unified Theorys (see e.g. [18] for an extended discussion), in which a unified symmetry like $\text{SU}(5)$ or $\text{SO}(10)$ gets broken at $M_{\text{GUT}}$ down to $G_{\text{SM}}$. An arguably even stronger motivation for GUTs is the structure of matter since one generation of the SM (cf. Equation 2) including the right-handed neutrino its into a $\boldsymbol{16}$-plet of $\text{SO}(10)$,

$$\boldsymbol{16}=\underbrace{(\boldsymbol{3},\boldsymbol{2})_{\nicefrac{{1}}{{6}}}+(\boldsymbol{\overline{3}},\boldsymbol{1})_{-\nicefrac{{2}}{{3}}}+(\boldsymbol{1},\boldsymbol{1})_{1}}_{{}={\,}\boldsymbol{10}}+\underbrace{(\boldsymbol{\overline{3}},\boldsymbol{1})_{\nicefrac{{1}}{{3}}}+(\boldsymbol{1},\boldsymbol{2})_{-\nicefrac{{1}}{{2}}}}_{{}={\,}\boldsymbol{\overline{5}}}+(\boldsymbol{1},\boldsymbol{1})_{0}\;,$$

(4)

where we also indicated the $\text{SU}(5)$ representations in underbraces. While the GUT symmetries work very well for the matter, they fail for the SM Higgs. The smallest $\text{SU}(5)$ representations that contain the MSSM Higgs doublets are $\boldsymbol{5}+\overline{\boldsymbol{5}}$, which combine to a $\boldsymbol{10}$-plet of $\text{SO}(10)$. The additional $\text{SU}(3)_{\text{C}}$ $\boldsymbol{3}$-plets contained in these representations pose major threats to the model as they typically mediate unacceptably large proton decay unless their mass exceeds the Planck scale. This is one facet of the doublet-triplet splitting problems which haunt GUTs in 4D. On the other hand, as has been pointed out early on in the context of string model building, in higher-dimensional, in particular stringy, models the same mechanism that breaks the GUT symmetry can also split the doublets from the triplets [5, 19].

Of course, it will be reassuring to find a compactification which reproduces the SM in great detail, regardless of whether or not the underlying construction is unique. What is more, given such a model, we will be in a unique position to answer some of the most popular questions of our time:

1. 1.

   What is the origin of flavor and $\mathcal{CP}$ violation?

2. 2.

   What is the nature of dark matter and what are the properties of the dark (aka hidden) sector?

3. 3.

   What drives cosmic inflation?

While these questions might be answered separately, the power of addressing them in explicit string models is that the answers are much more specific and related in intriguing ways.

In this section, we collect some basic facts on the heterotic string. For further details and a broader overview see [20]. The term ‘heterotic’ derives from the Greek word ‘hetero’, which translates as ‘other’, and in biology is related to ‘vigorous hybrid’, which arguably reflects the nature of the heterotic string. The heterotic string theory [21] is the result of combining a 10D superstring and a 26D bosonic string. The former can equip the theory with $\mathcal{N}=1$ supersymmetry in ten dimensions whereas the bosonic string provides us with a non-Abelian gauge group of rank 16,²²2The nonsupersymmetric heterotic string can be obtained from this version, as we briefly describe in Section 7.2.

$$G_{\text{het}}=\text{E}_{8}\times\text{E}_{8}\quad\text{or}\quad\text{SO}(32)\;.$$

(5)

Note that the most general compactification can have continuous enhancements of these gauge symmetries, yet we will mainly focus on (5). The heterotic theories contain only oriented closed strings propagating in ten dimensions.

In lightcone gauge, there are 8 right-moving bosonic string coordinates $X_{\text{R}}^{i}(t-\sigma)$ and 8 right-moving fermions $\psi_{\text{R}}^{i}(t-\sigma)$, where $2\leq i\leq 9$. $t$ and $\sigma$ denote the worldsheet coordinates. There are in total 24 left-moving coordinates $X_{\text{L}}^{M}(t+\sigma)$. In symmetric compactifications they get decomposed into $X_{\text{L}}^{i}(t+\sigma)$ with $2\leq i\leq 9$ as in the right-handed sector, and $X_{\text{L}}^{I}(t+\sigma)$ with $1\leq I\leq 16$. This decomposition gives rise 8 combinations of ordinary physical coordinates, $X^{i}(\tau,\sigma)=X_{\text{L}}^{i}(t+\sigma)+X_{\text{R}}^{i}(t-\sigma)$. The additional left-moving coordinates $X_{\text{L}}^{I}(t+\sigma)$ are responsible for the gauge symmetries $G_{\text{het}}$, cf. Equation 5.

For the sake of keeping this review short, we specialize on symmetric compactifications, cf. Figure 1. Interestingly, the so-called Free Fermionic Formulation (FFF) and $\mathds{Z}_{2}\times\mathds{Z}_{2}$ geometric orbifolds are related by a dictionary [22, 23]. There are possibilities to go more general, and consider e.g. asymmetric orbifolds [24, 25] or Gepner models [26, 27], which is beyond the scope of this review.

We will start our discussion with heterotic orbifolds [6, 7], which allow one to explicitly “see” the strings. For simplicity, we focus on symmetric toroidal orbifolds, which emerge by dividing tori by some of their symmetries. The tori are given by $\mathds{R}^{n}/\Lambda$ or $\mathds{C}^{n/2}/\Lambda$, where $\Lambda$ denotes a lattice, or, more precisely, the group of lattice translations. We will be interested in $n=6$. Therefore, $\mathds{O}=\mathds{R}^{n}/\mathds{S}$ with $\mathds{S}$ denoting the so-called space group, which is comprised of lattice translations and additional operations such as rotations and so-called roto-translations, and forms a discrete subgroup of the $n$-dimensional Euclidean group. Crucially, these operations are also embedded in the gauge sector, which breaks $G_{\text{het}}$ down to a subgroup. Moreover, they also break SUSY, which facilitates the construction of chiral 4D models with $\mathcal{N}=1$ or no SUSY.

In more detail, in the geometric formulation elements of space group $\mathds{S}$ are conveniently denoted by $g=(\vartheta^{r},m_{\alpha}e_{\alpha})$, where $r,m_{\alpha}\in\mathds{N}_{0}$. The $e_{\alpha}$ ($1\leq\alpha\leq 6$) are the basis vectors of the underlying torus. The set of $\vartheta\in\text{O}(n)$ form a finite group, called the point group $\mathds{P}$, which determines the holonomy group, and thus the amount of SUSY that survives the compactification. In fact, in order to classify the physically inequivalent orbifolds, one only needs to find the different affine classes [28], but we refrain from spelling this discussion out in detail. If $\mathds{P}$ is Abelian, it is either $\mathds{Z}_{N}$ or $\mathds{Z}_{N}\times\mathds{Z}_{M}$. If in addition $\mathcal{N}\geq 1$ SUSY is preserved, then a given $\mathds{Z}_{N}$ transformation can be encoded in a so-called 3-component twist vector $v$, which describes the rotations of three complex coordinates. In general, $g$ acts on string coordinates $X$ as

$$X\xmapsto{~{}(\vartheta^{r},m_{\alpha}e_{\alpha})~{}}\vartheta^{r}\,X+m_{\alpha}e_{\alpha}\;.$$

(6)

The space group is to be embedded in the gauge degrees of freedom. Loosely speaking, the point group elements get mapped to so-called shift vectors $V$. This embedding has to preserve the order, i.e. if $\vartheta\in P$ satisfies $\vartheta^{N}=\mathds{1}$ then $N\,V\in\Lambda_{\mathfrak{g}_{\text{het}}}$, where $\Lambda_{\mathfrak{g}_{\text{het}}}$ denotes the root lattice of $G_{\text{het}}$, cf. Equation 5. In $\mathds{Z}_{N}$ orbifolds, if, say, the shifts of two models differ by lattice vectors $\lambda\in\Lambda_{\mathfrak{g}_{\text{het}}}$, the resulting models are identical (cf. [29]). This is no longer true in $\mathds{Z}_{N}\times\mathds{Z}_{M}$ orbifolds, where gauge embeddings differing by lattice vectors may be inequivalent [30], and be related via what is known as discrete torsion [31]. Since $\Lambda_{\mathfrak{g}_{\text{het}}}$ is even and self-dual, one can find an Euclidean basis in which the lattice vectors are given by

$$p=(n_{1},\dots,n_{d})\quad\text{or}\quad p=(n_{1}+\nicefrac{{1}}{{2}},\dots,n_{d}+\nicefrac{{1}}{{2}})\;,$$

(7)

where $n_{i}\in\mathds{Z}$, $\sum_{i}^{d}n_{i}\in 2\mathds{Z}$ and $d\in\{8,16\}$ denotes the dimensions of the Lie algebras $\text{E}_{8}$ or $\mathfrak{so}(32)$, respectively. The gauge embedding of each translation $e_{\alpha}$ is a so-called discrete Wilson line $W_{\alpha}$. The Wilson lines are constrained by geometry. In more detail, since lattice vectors get mapped onto each other by the rotations, the analogous relations have to hold for the Wilson lines,

$$g\,e_{\alpha}=\sum_{\beta=1}^{6}a_{\alpha}{}^{\beta}e_{\beta}\qquad\Longrightarrow\qquad W_{\alpha}=\sum_{\beta=1}^{6}a_{\alpha}{}^{\beta}W_{\beta}+\lambda\quad\text{with}\quad\lambda\in\Lambda_{\mathfrak{g}_{\text{het}}}\;.$$

(8)

For instance, in a $\mathds{Z}_{3}$ orbifold plane one has $\vartheta e_{1}=e_{2}$ and $\vartheta e_{2}=-e_{1}-e_{2}$. Therefore, $W_{1}\equiv W_{2}$ and $W_{2}\equiv-W_{1}-W_{2}$, where “$\equiv$” means “equal up to $\lambda\in\Lambda_{\mathfrak{g}_{\text{het}}}$”. Thus $3W_{1}\in\Lambda_{\mathfrak{g}_{\text{het}}}$. This generalizes to other geometries, i.e. for a given Wilson line $W_{\alpha}$ there is an integer $M_{\alpha}$ such that $M_{\alpha}\,W_{\alpha}\in\Lambda_{\mathfrak{g}_{\text{het}}}$, with no summation over $\alpha$. As a consequence, the coefficients $a_{\alpha}{}^{\beta}$ in (8) are integer.

In addition, the orbifold parameters and their gauge embeddings must satisfy a series of constraints in order to ensure world-sheet modular invariance, which guarantees the UV consistency of the model [7, 31]. For $\mathds{Z}_{N}$ orbifolds, these conditions take the form [30]

	$\displaystyle N(V^{2}-v^{2})$	$\displaystyle=0\mod 2\;,$	$\displaystyle M_{\alpha}V\cdot W_{\alpha}$	$\displaystyle=0\mod 2\;,$
	$\displaystyle M_{\alpha}W_{\alpha}^{2}$	$\displaystyle=0\mod 2\;,$	$\displaystyle\operatorname{gcd}(M_{\alpha},M_{\beta})W_{\alpha}\cdot W_{\beta}$	$\displaystyle=0\mod 2\;,$		(9)

where no summation over $\alpha$ nor $\beta$ is implied.

While early classifications of viable toroidal orbifolds focused on special kinds of lattices [32], more recently a richer set of possibilities has been uncovered in the $\mathds{Z}_{2}\times\mathds{Z}_{2}$ orbifold [22] and generalized to other point groups [28, 33]. Loosely speaking, the new ingredient of the additional possibilities are space groups which contain roto-translations $(\vartheta^{r},m_{\alpha}e_{\alpha})\in\mathds{S}$ but $(\vartheta^{r},0)\not\in\mathds{S}$ (cf. [34]). As a consequence, the fundamental group of the orbifolds (and not just the underlying tori) can be nontrivial. Among other things, this allows for non-local, or Wilson line, breaking of the gauge symmetry, which is also being utilized in the context of smooth compactifications [35]. Another innovation is the consistent construction of non-Abelian orbifolds [36, 33]. In particular, there are 138 Abelian and 331 non-Abelian space groups [28] of toroidal symmetric orbifolds preserving $\mathcal{N}=1$ SUSY in 4D. These geometries have been shown to host many models with gauge symmetry and chiral spectrum of the MSSM [37, 38, 39], yet their detailed phenomenological properties have not been worked out so far.

Because of the gauge symmetries of the heterotic string, heterotic models comply well with the idea of grand unification. Breaking the GUT symmetry via compactification allows one to elegantly avoid the major shortcomings of 4D GUTs, most notably the doublet-triplet splitting challenge and its associated proton decay problems. However, there is a tension between the scale of gauge coupling unification in the MSSM, $M_{\text{GUT}}\simeq\text{few}\cdot 10^{16}\text{GeV}$, and typical compactification radii. This is because string theory also describes gravity, and the effective 4D Planck mass is sensitive to the volume of compact space. In some more detail, Newton’s constant $G_{\text{N}}$ is related to the fine structure ‘constant’ at the GUT/compactification scale, $\alpha_{\text{\acs{GUT}}}$, and the string tension, $\alpha^{\prime}$, via [40]

$$G_{\text{N}}=\frac{\alpha_{\text{\acs{GUT}}}\,\alpha^{\prime}}{4}\quad\text{impying that }G_{\text{N}}\gtrsim\frac{\alpha_{\text{\acs{GUT}}}^{4/3}}{M_{\text{\acs{GUT}}}^{2}}$$

(10)

for a weakly coupled theory. This value of $G_{\text{N}}$ is too large for typical values of $M_{\text{\acs{GUT}}}$ and $\alpha_{\text{\acs{GUT}}}$ (cf. our ealier discussion around Equation 4). There are various proposals to fix this issue (see e.g. [41]). The perhaps most ingenious way to address this problem is M-theory [40]. However, the problem can also be ameliorated in anisotropic compactifications [40, Footnote 3]. A detailed analysis [34] suggests that this solution barely fails, but by the own admission of the authors the presented bound is too conservative. In fact, if one uses the appropriate volume of the orbifold for the analysis rather than the underlying torus, one finds that anisotropic compactifications can work, even though the parameter space of solutions is not too generous. This implies that there is an intermediate orbifold GUT symmetry (see e.g. [42] for a review). However, this also means that the smaller radii are of the order of the string scale, and as stressed in [40], one must use conformal field theory (CFT) (rather than classical geometry) to analyze the model. This is one of the reasons why this review focuses on orbifold constructions.

In general, only a subset of the $G_{\text{het}}$ gauge fields survive the orbifold projections. They can be determined by finding the roots $p\in\Lambda_{\mathfrak{g}_{\text{het}}}$, i.e. $p\cdot p=2$, which satisfy the projection conditions

$$p\cdot V=p\cdot W=0\pmod{1}$$

(11)

with the $p$ from Equation 7 for all shift $V$ and Wilson line $W$ vectors (cf. Section 3.2).

The zero modes are solutions of the mass equation to vanishing mass. In all known examples they are chiral w.r.t. some, possibly discrete, symmetry. The computation of the massless spectrum, i.e. gauge and chiral zero modes, is straightforward though tedious if done by hand, and can conveniently be performed with dedicated tools such as the Orbifolder [43].

Explicit string models exhibit states beyond the SM or MSSM at some level. The additional states include the moduli as well as the winding and Kaluza–Klein (KK) modes, which we will review in Sections 4.3 and 4.4, respectively. In addition, there are often vector-like states w.r.t. the SM gauge group which are neither KK nor winding modes. Whether or not these vector-like states are massless often depends on the point of moduli space under consideration. For instance, vector-like states may attain masses when giving VEVs to blow-up modes that smoothen out orbifold singularities, and break symmetries w.r.t. which these states are chiral. However, it would be arguably wrong to refer to the smoothened out version as “cleaner” since it is really the same model. In fact, often important properties of a given construction are much more directly accessible by studying the symmetry-enhanced point in field space, which is given by the orbifold point in this example, even though the vacuum is away from this point.

Virtually every supersymmetric string compactification contains fields which are classically flat directions, and this is in particular true for models that come close to the real world. Some of these so-called moduli do not have any charge under $G_{\text{het}}$, and comprise the Kähler moduli $\mathcal{T}_{i}$, the complex structure moduli $\mathcal{U}_{i}$ and the dilaton $\mathcal{S}$. Yet also some of the charged fields can attain VEVs because they are along $D$-flat directions. The VEVs of these fields determine, among other things, geometric properties of compact space. Classical flat directions can attain a nontrivial potential at the quantum level, in particular through nonperturbative effects. It is generally challenging to compute these potentials in full detail and thus determine the VEVs at the minima, see [44] for more details of the analogous discussion in other versions of string theory.

The existence of winding and KK modes is one of the most important features of string compactifications, and required to make the theory UV complete. The properties of these modes are particularly accessible in torus-based compactifications such as toroidal orbifolds.

$\mathcal{N}=1$ SUSY (cf. Section 2.3) has long been a standard ingredient of string models. Whether or not $\mathcal{N}=1$ SUSY is preserved by the compactifaction depends on the holonomy group of the compact space [5]. In the case of a smooth compactification, the requirement that the compactification preserves $\mathcal{N}=1$ SUSY dictates that the manifold has to be of the Calabi–Yau (CY) type, and in orbifolds it requires the twist to fit into $\text{SU}(3)$, the holonomy group of CY manifolds.

After compactification the residual continuous gauge symmetry, $G_{\text{gauge}}$, of a realistic model has to contain the gauge symmetry of the SM (1). The gauge symmetry follows already from our discussion in Section 4.1. Apart from the obvious option that $G_{\text{gauge}}=G_{\text{SM}}\times G_{\text{beyond}}$ promising models may also replace $G_{\text{SM}}$ by the Pati–Salam [1] (PS) group $G_{\text{PS}}=\mathrm{SU}(4)_{\text{C}}\times\mathrm{SU}(2)_{\text{L}}\times\mathrm{SU}(2)_{\text{R}}$, the so-called left-right symmetry [51] $G_{\text{LR}}=\mathrm{SU}(3)_{\text{C}}\times\mathrm{SU}(2)_{\text{L}}\times\mathrm{SU}(2)_{\text{R}}\times\mathrm{U}(1)_{{B}-{L}}$, or the flipped $\mathrm{SU}(5)$ symmetry [52]. Other grand unified symmetries are in principle possible but may be challenged by doublet-triplet splitting problems and the lack of appropriate Higgs fields that break the larger symmetry to $G_{\text{SM}}$.

Very often in geometric orbifolds with $\mathcal{N}=1$ SUSY one $\text{U}(1)$ factor appears anomalous, with the anomaly being cancelled by the Green–Schwarz [3] (GS) mechanism. As a consequence, the $D$-term potential contains an Fayet–Iliopoulos [2] (FI) term, $\mathscr{V}_{D}\supset g^{2}D_{\text{anom}}^{2}$, where

$\displaystyle D_{\text{anom}}=\sum_{i}q_{\text{anom}}^{i}\left|\varphi_{i}\right|^{2}+\xi=0\quad\text{with }\xi=\frac{g^{2}\,\operatorname{Tr}\mathsf{t}_{\text{anom}}}{192\pi^{2}}M_{\text{P}}^{2}\;.$

(12)

$g$ denotes the gauge coupling, and $\mathsf{t}_{\text{anom}}$ the generator of $\text{U}(1)_{\text{anom}}$. The requirement of a vanishing of the $D$-term potential induces VEVs of $\text{U}(1)_{\text{anom}}$ charged fields $\varphi_{i}$ that breaks $\text{U}(1)_{\text{anom}}$ and in the overwhelming majority of cases further symmetries [53]. Clearly, in realistic models the fields $\varphi_{i}$ acquiring large VEVs must be SM singlets. Configurations with vanishing $D$-terms can be identified by constructing holomorphic monomials which are invariant under all gauge symmetries but carry nontrivial charge under $\text{U}(1)_{\text{anom}}$ [54, 55, 56]. A complete basis of such monomials can be obtained via the Hilbert basis [57]. In the vast majority of explicit models, $\operatorname{Tr}\mathsf{t}_{\text{anom}}\sim\mathcal{O}(100)$, so that $\sqrt{\xi}/M_{\text{P}}$ is of the order of the Cabibbo angle, and, therefore, the VEVs induced by (12) may conceivably play a role in explaining flavor hierarchies [58].

The extra symmetry $G_{\text{beyond}}$ is usually partly broken by the VEVs that cancel the FI term. The residual continuous part can conceivably provide us with a hidden sector leading to dynamical SUSY breakdown [59, 60]. There are also usually discrete symmetries, which can be determined systematically with the Smith normal form [61].

Starting at a symmetry-enhanced point has various benefits compared to analyzing generic points in moduli space. The models reviewed in this review give rise to a variety of mildly broken, and thus approximate, symmetries. As already mentioned above, the latter may conceivably explain the observed hierarchies in the flavor sector [58]. They may also provide us with solutions to the $\mu$ and/or strong $\mathcal{CP}$ problems (cf. [74, 75]). They may explain the scales in models of dynamical supersymmetry breaking (such as [60], which requires an explicit mass for some pairs of vector-like states) or the messengers of gauge mediated SUSY breaking [76].

Rather than reviewing extensive model scans, let us focus on a particular example, the model of [80]. The orbifold has noncontractible cycles (cf. Section 3.3) which allow one to break an $\text{SU}(5)$ grand unified symmetry nonlocally down to $G_{\text{SM}}$. This type of GUT breaking avoids fractionally charged exotics. The spectrum consists of 3 chiral generations of quark and leptons plus additional states which are vectorlike w.r.t. $G_{\text{SM}}$ but massless at the orbifold point, see Table 1. Like the majority of models of this type, there is an FI term which has to be cancelled consistently with vanishing of the $F$- and (other) $D$-terms. The corresponding VEVs break the gauge symmetry at the orbifold point down to

$$G_{\text{residual}}=G_{\text{SM}}\times\mathds{Z}_{4}^{R}\times\text{SU}(2)_{\text{hid}}\;,$$

(24)

where $G_{\text{SM}}$ and $\text{SU}(2)_{\text{hid}}$ stem from two different $\text{E}_{8}$ factors, and none of the SM matter is charged under $\text{SU}(2)_{\text{hid}}$.

These VEVs also provide mass terms for all SM charged exotics, yet the $\mathds{Z}_{4}^{R}$ symmetry forbids the mass of one linear combination of Higgs fields, which gets identified with the MSSM Higgs pair. This pair acquires a mass after $\mathds{Z}_{4}^{R}$ breaking. The order parameter of $R$ symmetry breaking is the gravitino mass, i.e. of the order of the soft terms, which are assumed to be not too far above the electroweak scale. That is, the $\mathds{Z}_{4}^{R}$, which is a discrete remnant of the Lorentz symmetry of compact space, can provide us with a solution to the $\mu$ problem along the lines of [81]. In addition, this $\mathds{Z}_{4}^{R}$ suppresses dimension-5 proton decay operators enough to be consistent with observation.

It has been checked that qualitatively realistic fermion masses arise, i.e. the Yukawa couplings have full rank, exhibit hierarchies and lead to nontrivial flavor mixing, and neutrino masses are see-saw suppressed. However, this is not to say that they are fully realistic, cf. our discussion of C5.

Altogether this example shows that explicit string models can successfully address some of the most pressing questions of (traditional) unified model building, including the $\mu$ and proton decay problems. However, it also illustrates that there is still a long way to go before we can claim to have found “the” stringy SM. Apart from the question whether or not low-energy SUSY is realized in Nature, one has to successfully fix the moduli. While this is a topic on its own, which is covered in [44], the $\mathds{Z}_{4}^{R}$ symmetry and charges can be used to show that generically there are no flat directions. States with odd $\mathds{Z}_{4}^{R}$ acquire masses because the mass terms carry $\mathds{Z}_{4}^{R}$ charge $2\pmod{4}$, and the fields of $\mathds{Z}_{4}^{R}$ charge 2 pair up with linear combinations of $\mathds{Z}_{4}^{R}$ charge 0 fields. Of course, generic statements do not always lead to the correct conclusions, and one has to verify explicitly that there are no flat directions, what the possible VEVs of the $\mathds{Z}_{4}^{R}$ charge 0 fields are, and whether they leads to phenomenologically viable couplings in the SM sector.

There is a consistent nonsupersymmetric heterotic string [82, 7, 83], which can be understood as a freely acting $\mathds{Z}_{2}$ orbifold of a $\mathcal{N}=1$ heterotic string [7, 83]. Given the absence of evidence of supersymmetry at colliders, this version of the heterotic string deserves increasing attention, even though it does not exhibit the protection that SUSY offers against the appearance of tachyons, quadratic divergences and a large cosmological constant [84].

The massless spectrum of this theory comprises three components: the gravitational part includes the graviton, the antisymmetric 2-form $B_{MN}$ and the dilaton; the gauge bosons of $\mathrm{SO}(16)\times\mathrm{SO}(16)$ arise in the gauge sector; and the charged matter states build the representations $(\boldsymbol{128},\boldsymbol{1})+(\boldsymbol{1},\boldsymbol{128})+(\boldsymbol{16},\boldsymbol{16})$.

Applying similar compactification techniques such as orbifolds as in the supersymmetric case, there has been some effort to study the phenomenology of compactifications of this string theory in 4D, including models with a tachyon-free GUTs or SM massless spectrum [85, 86, 87, 88]. Although the progress does not yet compare to the supersymmetric case, some general features in SM-like models are known. In particular, the following properties of the massless spectrum are found: (i) at perturbative level, tachyons can be avoided; (ii) models with only one SM Higgs exist, but most of them exhibit a larger number of Higgses; (iii) there appear many fermion and scalar exotic states although there are models with a very small exotic spectrum; (iv) among the exotics, there are $\mathcal{O}(100)$ right-handed neutrinos; (v) leptoquark scalars are present in different amounts; and (vi) the number of fermions and bosons can coincide, yielding the possibility of an exponentially suppressed one-loop cosmological constant.

As an example, let us focus on the model 2 of [88], based on an Abelian orbifold compactification of the $\mathcal{N}=0$ string (in the bosonic formulation). It includes the SM gauge group and additional $\text{SU}(2)$ and $\text{U}(1)$ factors. In the fermionic sector, there are only three SM generations arising from twisted sectors and 119 right-handed neutrinos. In the scalar sector, besides 9 Higgs doublets and 9 scalar leptoquarks, there are 30 SM singlets, which may be considered flavons of a traditional $\Delta(54)$ flavor symmetry. The modular flavor features of this kind of models are not known.