Combinatorics in N = 1 Heterotic Vacua

We briefly review an algorithmic strategy to explore the landscape of heterotic E8 \times E8 vacua, in the context of compactifying smooth Calabi-Yau three-folds with vector bundles. The Calabi-Yau three-folds are algebraically realised as hypersurfaces in toric varieties and a large class of vector bundles are constructed thereon as monads. In the spirit of searching for Standard-like heterotic vacua, emphasis is placed on the integer combinatorics of the model-building programme.


Introduction
Compactifications of E 8 × E 8 heterotic theory [1,2] and heterotic M-theory [3][4][5][6][7] on smooth Calabi-Yau three-folds is a simple and compelling way to reach N = 1 supersymmetry at four dimensions. A Calabi-Yau three-fold necessarily admits a Ricci-flat metric g αβ , where α,β = 1, 2, 3 are respectively, the holomorphic and anti-holomorphic indices. One also turns on an internal gauge field, in a subalgebra G of the full E 8 , resulting in the reduction of the four-dimensional gauge group down to the commutant H of G ⊂ E 8 . To preserve supersymmetry, the gauge field should satisfy the Hermitian Yang-Mills equations SU (5) SU (5) (1, 24) ⊕ (5, 10) ⊕ (5, 10) ⊕ (10, 5) ⊕ (10, 5) ⊕ (24, 1) Table 1: A vector bundle V with structure group G = SU (4) or SU (5), respectively, breaks the E 8 group of the heterotic string into the Grand Unifying group H = SO (10) or SU (5). The four-dimensional low-energy representation contents of H arise from the branching of the 248 adjoint of E 8 under G × H ⊂ E 8 . The particle spectrum is obtained as various bundle-valued cohomology groups.
In this paper, we shall make it clear how the construction of Standard-like heterotic vacua turns into the integer combinatorics for a discrete system. Specifically, the Calabi-Yau three-folds will be torically constructed and described by the combinatorics of reflexive lattice polytopes [10]. 2 Next, monad vector bundles [11] will be constructed thereon, equivalent of turning on internal gauge fluxes over the Calabi-Yau three-folds.
The remainder of this paper is structured as follows. In the ensuing two sections, we lay down the foundation by explaining the basic mathematical toolkit for describing N = 1 heterotic vacua.
Next, in section 4, further constraints will be imposed on the internal geometry so that the resulting 1 The choice G = SU (3) also gives rise to E 6 GUT models. However, they have an inherent trouble in doublet-triplet splitting of Higgs multiplet (see, for a recent example, Ref. [49]) and hence, we will not address the models of this type here. 2 In algebraic geometry, Calabi-Yau three-folds are, in general, realised as complete intersections of hypersurfaces in toric varieties of dimension greater than or equal to four, but this paper will only be dealing with single-hypersurface cases. N = 1 four-dimensional effective theory may mimic the Standard Model. We shall conclude in section 5 with a summary and outlook.
2 Toric Construction of Calabi-Yau three-folds Soon after the famous 7890 Calabi-Yau three-folds were realised as complete intersections of hypersurfaces in multi-projective spaces [18][19][20][21][22], Kreuzer and Skarke have classified the Calabi-Yau three-folds that arise as codimension-one hypersurfaces in toric four-folds, comprising a much bigger dataset [23][24][25]. This construction, first proposed by Batyrev [10], involves an extensive usage of toric geometry. Here we do not intend by any means to give a pedagogical introduction to toric geometry. The readers interested in the details of this subject are referred either to the maths texts [26][27][28][29] or to the excellent, introductory reviews for physicists [30,31].

Ambient Toric four-folds
The Calabi-Yau three-folds X are embedded in toric four-folds A as hypersurfaces and therefore, we shall start with the description of these ambient toric varieties. A toric four-fold is described by the combinatorial data called a fan in R 4 , which is a collection of convex cones in R 4 with their common apex at the origin O = (0, 0, 0, 0). For the sake of Calabi-Yau sub-varieties, however, every fan is not appropriate. We first define a certain class of convex polytopes in R 4 , of which fans of a special kind are made.
The polytopes considered here must contain the origin O as the unique interior lattice point and all the vertices must lie in the lattice Z 4 ⊂ R 4 . Such polytopes are called reflexive. It can be shown that for a given reflexive polytope ∆ in R 4 , the dual polytope ∆ • defined by also has all its vertices on the lattice Z 4 , like the original polytope ∆ does. To this dual polytope ∆ • we can associate a collection of the convex cones over all its faces, forming the fan for our toric four-fold A. Now, as for the construction of toric four-fold from a given fan in R 4 , several equivalent methods are known. What best suits our purpose amongst them is Cox's homogenous-coordinate approach [32], where a complex homogeneous coordinate x ρ is associated to each one-dimensional cone ρ in the fan. Thus, if the fan has k edges, there are k homogeneous coordinates (x 1 , · · · , x k ) for C k .
The next task is to identify a certain measure zero subset Z of C k which should be removed. Let S be a set of edges that do not span any cone in the fan and let Z(S) ⊂ C k be the linear subspace defined by setting x ρ = 0 , ∀ ρ ∈ S. Now let Z ⊂ C k be the union of the subspaces Z(S) for all such S. Then the toric four-fold is constructed as a quotient of C k − Z by the following (C * ) k−4 action: where the coefficients β r ρ are defined by the linear relations k ρ=1 β r ρ v ρ = 0 amongst the edges. Hence, β r ρ form a (k − 4) × k matrix which is often referred to as a charge matrix [33]. The identification rule in Eq. (2.2) can be schematically written as Note that the construction of toric four-folds in Eq. (2.3) naturally generalises that of projective space P 4 , the simplest toric four-fold, in which case Z = {O} and k = 5, that is,

Calabi-Yau three-folds
A Calabi-Yau hypersurface X to the toric four-fold A is constructed in a straight-forward manner without requiring any further data: as long as the polytope ∆ is reflexive it also defines X ⊂ A.
Note that in this case, ∆ • is also a reflexive polytope since (∆ • ) • = ∆. To a reflexive polytope ∆ in R 4 , we can associate a family of Calabi-Yau three-folds X defined as the vanishing loci of the polynomials of the form where x ρ=1,...,k are the k homogeneous coordinates of A associated to the lattice vertices v ρ=1,...,k of ∆ • , and C m are numerical coefficients parameterising the complex structure of X.
Heterotic compactifications ask for compact Calabi-Yau three-folds that are smooth. However, a toric four-fold A constructed by Eq. (2.3) usually bears singularities and they in general descend to the hypersurfaces X too. In order to make X nonsingular, we partially de-singularise A so that the hypersurfaces may avoid the singularities of the ambient space [10]. This process corresponds to triangulating the (dual) polytope in a special way, and is called an MPCP-triangulation. 3 As for the statistics, a total of 473, 800, 776 reflexive polytopes in R 4 have been classified [23][24][25], each of which gives rise to a toric four-fold A as well as a family of Calabi-Yau three-folds X. It turns out that only 124 out of them describe smooth manifolds, for which no MPCP-triangulations are required. In the physics literature, especially in the context of heterotic string phenomenology, construction of vector bundles have been attempted in several ways. They include spectral cover construction [35][36][37][38][39][40][41], bundle extension [43][44][45] and the mixture thereof [42]. In many of them, it was essential for the base three-folds to have a torus-fibration structure. On the other hand, monad construction [11] does not assume any extra structure and has proved particularly useful for algorithmically scanning a vast number of vector bundles [15,16,[47][48][49].
A monad vector bundle is essentially the quotient of two Whitney sums of line bundles. More precisely, a monad bundle V over a Calabi-Yau three-fold X is defined by the short exact sequence of the form: where b i and c j are integer vectors of length h 1,1 (X), representing the first Chern classes of the is the rank of V .
From Eq. (3.1), one can readily read off the Chern class of V : where J r ∈ H 1,1 (X, R) represent the harmonic (1, 1)-forms c 1 (O X (e r )), the d rst are the triple intersection numbers defined by

Phenemenological Constraints on the Vacua
With the geometric constraints so far explained, one would only be able to guarantee the right number of supersymmetry at low energy, that is, N = 1 at D = 4. Since the goal of string phenomenology is to obtain (supersymmetric versions of) the Standard Model, more criteria should further be imposed on the N = 1 vacua. To make things clear, let us emphasize that in this paper the terminology "Standard-like" model will imply the following: • Three generations of quarks and leptons, and no exotics; • Cancellation of heterotic anomaly.
Here we translate the above three phenomenological constraints into the conditions on the vacuum topology.

Gauge Group
As explained in the Introduction, the structure group G of the visible sector bundle sits in E 8 and its commutant H becomes the low-energy gauge group. In order to obtain H = SO(10) and SU (5), one must choose G = SU (4) and SU (5), respectively. In particular, the rank of the bundle should be either 4 or 5, and hence, by Eq. (3.2), where r b and r c are the ranks of the two vector bundles in the defining sequence Eq. (3.1) of V .
What is more, since the structure group should be "special" unitary, the first Chern class c 1 (V ) of V is to vanish. By Eq. (3.4), this corresponds to We still have to break the GUT group further down to a Standard-like one, and this second breaking will require π 1 (X)-Wilson-lines. However, given the observation that most of the torically constructed Calabi-Yau three-folds have a trivial first fundamental group [50], they must be quotiented out by freely-acting discrete symmetries so that we may turn on appropriate Wilson-lines.
Therefore, we will eventually have to look for a discrete symmetry group G that acts freely on X, and make a quotient spaceX = X/G, which will then have a non-trivial first fundamental group π 1 (X) G.

Cancellation of Heterotic Anomaly
Heterotic models need to satisfy a well-known anomaly condition. So far we have only mentioned one holomorphic vector budle V for the visible sector, but the theory has another bundleṼ for the hidden sector. Heterotic vacua can also have five-branes whose strong-coupling origin is M5-branes.
In order to keep the four-dimensional Lorentz symmetry, their world-volumes must stretch along the external Minkowski M 4 . The remaining two dimensions should then wrap holomorphic two-cycles in X for unbroken supersymmetry. Thus, the homology classes associated with these two-cycles must be effective and hence, belong to the Mori cone in H 2 (X, Z). In other words, the corresponding four-fourms must belong to the corresponding cone in H 4 (X, Z).
In this most general set-up, heterotic anomaly cancellation imposes a topological constraint relating the Calabi-Yau three-fold, the two vector bundles and the five-brane classes. When c 1 (T X) = c 1 (V ) = c 1 (Ṽ ) = 0, the anomaly condition can be expressed, at the level of cohomology, as where W = i W i is the sum of the five-brane classes. Note that W itself should also belong to the Mori cone of X as all the summands W i do. In our discussion, however, without mentioning the second bundleṼ we presume a trivial bundle for the hidden sector. Thus, the anomaly constraint in Eq. (4.3) says that c 2 (T X) − c 2 (V ) is effective. Table 1 shows how the low-energy particle spectra are determined from various bundle-valued cohomology groups. Assuming that V is a stable bundle 4 implies that h 0 (X, V ) = 0 = h 3 (X, V ), and hence, to obtain three net-generations of quarks and leptons, we must have

Particle Spectra
where the Aiyah-Singer index theorem [51] has been applied to the differential operator / ∂ V on X, and |G| is the order of the discrete symmetry group G, with which we will have to quotient the "upstairs" three-folds X.

Discrete System for Standard-like Vacua
In this sub-section, we briefly summarise the model-building requirements that have so far been discussed: • Calabi-Yau three-fold A reflexive polytope ∆ ⊂ R 4 describes a Calabi-Yau three-fold X. In the computer package PALP [34], by inserting the list of lattice vertices of ∆, or equivalently, the corresponding "weight system" [24], one can obtain all the topological invariants of X relevant to the heterotic compactification.
• Monad vector bundle The r b + r c integer vectors b i and c j of length h 1,1 (X), each labelling a line bundle summand, parameterise our monad bundle V .
• Standard-like constraints The internal backgrounds are also constrained by the Standard-like phenomenology. It turns out that given a Calabi-Yau three-fold X, that is, for a fixed topology of X, the monad parameters b i and c j must obey the algebraic equations of degree 0, 1, 2 and 3 shown in Table 2.
Physics Origin Background Geometry Algebraic constraints on b i and c j Gauge Group a. G = SU (n), for n = 4, 5 b. X has a discrete free action G a1. r b − rc = 4 or 5 a2.
Anomaly Calabi-Yau three-folds X and a G-bundle V thereon; the integers b r i and c r j parameterise the bundle V as in Eq. (3.1), where i = 1, . . . , r b , j = 1, . . . , r c and r = 1, . . . , h 1,1 (X). The second column states the constraints on the background geometry and the third column expresses the corresponding algebraic equations that the parameters b r i and c r j must obey, where d rst = X J r ∧ J s ∧ J t are the intersection numbers for a given basis {J r } 1≤r≤h 1,1 (X) for H 1,1 (X, R).
Note that the integer combinatorics under the algebraic constraints of small degrees has formed a simple discrete system for Standard-like heterotic vacua. However, this system of vacua is far from being finite yet. To begin with, there are no upper bounds on r b and r c that count the number of monad parameters. Before initiating an exploration of the landscape, one first needs to add more constraints to make the system finite, and those extra constraints had better be related to preferred phenomenology. Now, if the line bundle summands in Eq. (3.1) are ample, or equivalently, if all the monad parameters b r i and c r j are positive, 5 then by Kodaira's vanishing theorem [51,52], the cohomology group H 2 (X, V ) vanishes and hence, the low-energy effective theory acquires no anti-generations. Of course this is a phenomenologically preferred feature, although not necessary.
We call a monad positive if it is only parameterised by positive integers, and semi-positive if all its parameters are either positive or zero. Secondly, one can also constrain the relative size of these monad parameters so that each entry of the vector b i − c j may be non-negative, for all i and j.
In this case, we will call the monad generic since the monad map f in Eq. (3.1), thought of as an r c × r b matrix of polynomials, may generically have all the entries nonzero.

Exploring a Region of the Landscape
As for the first step, one can think of exploring generic, positive monads over a "small" class of Calabi-Yau three-folds. As was mentioned in sub-section 2.2, the total dataset of torically constructed Calabi-Yau three-folds are way too large to grasp altogether. Therefore, at the initial stage, those in "smooth" ambient spaces have first been considred amongst the total of 500 million [15]. It turns out that over these O(100) manifolds, the generic, positive monads are finite in number under the constraints in Table 2, and the Standard-like vacua have indeed been classified, resulting in 61 candidate models.
Based on this experience, one can become more ambitious and extend the vacuum search, both bundle-wise and Calabi-Yau-wise. Firstly, with the positivity condition a bit relaxed, the generic, semi-positive monads have been explored over the same O(100) Calabi-Yau three-folds [15]. The Standard-like vacua with the monads of this type turn out to form an infinite class and hence, they have been explored under an artificial upper bound on the monad parameters, resulting in 85 models. Secondly, the programme has also been extended to include singular ambient manifolds with small h 1,1 [16]. A total of O(300) torically constructed Calabi-Yau three-folds have the Hodge number h 1,1 ≤ 3, and the generic, positive monads have been classified thereon, giving rise to new candidate models.

Summary and Outlook
In this paper, we have discussed a systematic approach towards Standard-like heterotic vacua.
The proposed algorithms have indeed been implemented in a computer package [17]. Simplicity of the integer combinatorics for the N = 1 heterotic vacua was the essential feature that made this approach a tractable programme. It was motivated by the general observation that any carefullychosen single model is likely to fail the detailed structure of the Standard Model. Thus, the spirit of the programme is to obtain a large number of Standard-like models, on which further constraints should be imposed later on to refine the set of candidates, eventually reaching the "genuin" Standard Model(s).
The combinatorics of toric geometry has been invaluable for constructing toric ambient fourfolds, to which Calabi-Yau three-folds have been embedded as hypersurfaces, and for computing their topological invariants relevant to the four-dimensional phenomenology. Smooth ambient fourfolds have been considered as a starter [15], and general ambient four-folds have also been dealt with [16] by partially resolving the singularities, if they bear any, so that the smoothness of the hypersurface Calabi-Yau three-folds are guaranteed. In both cases, the generic, positive monads (and some semi-positive ones, too, in the former case) have been probed under the Standard-like criteria. We have thus obtained a set of canidate models, that are anomaly-free and that have a chance to produce three generations of quarks and leptons without any anti-generations.
To guarantee the three-generation property of these candidates, further study of discrete symmetries of the manifolds are essential. Braun has recently classified the free group actions on complete intersection Calabi-Yau three-folds in multi-projective spaces [53] and his algorithm can in principle be generalised to toric cases. The line-bundle cohomologies on the torically constructed Calabi-Yau three-folds are also an essential part of the model building. The starting point would be to work out the cohomologies on the ambient toric varieties, which have already been investigated in the mathematics and physics literatures [29,[54][55][56]. Practical conversion of this information into the line-bundle cohomologies on the hypersurfaces is a rewarding work along the line of monad bundles and heterotic strings. As for the completion of the detailed particle spectra, the cohomologies of the monads in different representations are also to be revealed.