Computational Algebraic Geometry in String and Gauge TheoryView this Special Issue
Combinatorics in Heterotic Vacua
We briefly review an algorithmic strategy to explore the landscape of heterotic vacua, in the context of compactifying smooth Calabi-Yau threefolds with vector bundles. The Calabi-Yau threefolds 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.
Compactifications of heterotic theory [1, 2] and heterotic M-theory [3–7] on smooth Calabi-Yau threefolds provide a simple and compelling way to reach supersymmetry at four dimensions. A Calabi-Yau threefold necessarily admits a Ricci-flat metric , where are, respectively, the holomorphic and antiholomorphic indices. One also turns on an internal gauge field, in a subalgebra of the full , resulting in the reduction of the four-dimensional gauge group down to the commutant of . To preserve supersymmetry, the gauge field should satisfy the Hermitian Yang-Mills equations: where is the associated field strength. Although these equations cannot be solved analytically, the Donaldson-Uhlenbeck-Yau theorem [8, 9] states that, on a holomorphic (poly) stable bundle, there exists a unique connection that solves (1.1).
So each of the heterotic vacua comes in two pieces: a Calabi-Yau threefold and a holomorphic stable vector bundle thereon. Studying the detailed geometry, however, is not an easy task. To begin with, we do not even know the Ricci-flat metrics on Calabi-Yau threefolds. Fortunately, as will be seen shortly, it turns out that the topology of a vacuum already determines many interesting features of the four-dimensional effective theory.
In order for the heterotic models to be “Standard-like,” they must give rise to the correct gauge group, , possibly with an extra factor, as well as a correct spectrum for light particles coming in three generations. Firstly, the choices and for the structure group of reduce the to the four-dimensional gauge groups and , respectively, which are desirable in the viewpoint of Grand Unification1. The light particles then arise from the branching of the adjoint of into , and the spectrum is determined by various bundle-valued cohomologies on the Calabi-Yau threefold , as summarised in Table 1. Of course, the gauge group should be further broken down to a standard-like one and discrete Wilson-lines are made use of, if there ever exists any, for this second breaking.
In this paper, we will make it clear how the construction of standard-like heterotic vacua turns into the integer combinatorics for a discrete system. Specifically, the Calabi-Yau threefolds will be torically constructed and described by the combinatorics of reflexive lattice polytopes .2 Next, monad vector bundles  will be constructed thereon, equivalent of turning on internal gauge fluxes over the Calabi-Yau threefolds.
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 heterotic vacua. Next, in Section 4, further constraints will be imposed on the internal geometry so that the resulting four-dimensional effective theory may mimic the standard model. We will conclude in Section 5 with a summary and outlook.
2. Toric Construction of Calabi-Yau Threefolds
Soon after the famous 7890 Calabi-Yau threefolds were realised as complete intersections of hypersurfaces in multiprojective spaces [12–16], Kreuzer and Skarke have classified the Calabi-Yau threefolds that arise as codimension-one hypersurfaces in toric fourfolds, comprising a much bigger dataset [17–19]. This construction, first proposed by Batyrev , 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 [20–23] or to the excellent, introductory reviews for physicists [24, 25].
2.1. Ambient Toric Fourfolds
The Calabi-Yau threefolds are embedded in toric fourfolds as hypersurfaces and, therefore, we will start with the description of these ambient toric varieties. A toric fourfold is described by the combinatorial data called a fan in , which is a collection of convex cones in with their common apex at the origin . For the sake of Calabi-Yau subvarieties, however, every fan is not appropriate. We first define a certain class of convex polytopes in , of which fans of a special kind are made.
The polytopes considered here must contain the origin as the unique interior lattice point and all the vertices must lie in the lattice . Such polytopes are called reflexive. It can be shown that, for a given reflexive polytope in , the dual polytope defined by also has all its vertices on the lattice , 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 fourfold .
Now, as for the construction of toric fourfold from a given fan in , several equivalent methods are known. What best suits our purpose amongst them is Cox’s homogenous-coordinate approach , where a complex homogeneous coordinate is associated to each one-dimensional cone in the fan. Thus, if the fan has edges, then there are homogeneous coordinates for . The next task is to identify a certain measure zero subset of which should be removed. Let be a set of edges that do not span any cone in the fan and let be the linear subspace defined by setting . Now, let be the union of the subspaces for all such . Then, the toric fourfold is constructed as a quotient of by the following action: where the coefficients are defined by the linear relations amongst the edges. Hence, form a matrix which is often referred to as a charge matrix . The identification rule in (2.2) can be schematically written as Note that the construction of toric fourfolds in (2.3) naturally generalises that of projective space , the simplest toric fourfold, in which case and ; that is,
2.2. Calabi-Yau Threefolds
A Calabi-Yau hypersurface to the toric fourfold is constructed in a straightforward manner without requiring any further data: as long as the polytope is reflexive, it also defines . Note that, in this case, is also a reflexive polytope since . To a reflexive polytope in , we can associate a family of Calabi-Yau threefolds defined as the vanishing loci of the polynomials of the form where are the homogeneous coordinates of associated to the lattice vertices of and are numerical coefficients parameterising the complex structure of .
Heterotic compactifications ask for compact Calabi-Yau threefolds that are smooth. However, a toric fourfold constructed by (2.3) usually bears singularities and they in general descend to the hypersurfaces too. In order to make nonsingular, we partially desingularise so that the hypersurfaces may avoid the singularities of the ambient space . 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 reflexive polytopes in have been classified [17–19], each of which gives rise to a toric fourfold as well as a family of Calabi-Yau threefolds . It turns out that only 124 out of them describe smooth manifolds, for which no MPCP-triangulations are required.
3. Monad Construction of Vector Bundles
In the physics literature, especially in the context of heterotic string phenomenology, construction of vector bundles has been attempted in several ways. They include spectral cover construction [28–34], bundle extension [35–37], and the mixture thereof . In many of them, it was essential for the base threefolds to have a torus-fibration structure. On the other hand, monad construction  does not assume any extra structure and has proved particularly useful for algorithmically scanning a vast number of vector bundles [39–43].
A monad vector bundle is essentially the quotient of two Whitney sums of line bundles. More precisely, a monad bundle over a Calabi-Yau threefold is defined by the short exact sequence of the form where and are integer vectors of length , representing the first Chern classes of the summand line bundles and . The bundle is a holomorphic -bundle, where is the rank of .
From (3.1), one can readily read off the Chern class of : where represent the harmonic -forms , the are the triple intersection numbers defined by and the are the 4-forms furnishing the dual basis to the Kähler generators , subject to the duality relation As can be seen from (3.3), the Chern class of only depends on the integer parameters and , as well as the topology of the base manifold . Choosing an appropriate morphism in the defining sequence (3.1) corresponds to the tuning of more refined invariants of .
4. Towards the Standard Model
Sections 2 and 3 have shown that the vacuum topology is essentially described by lattice vertices and integer parameters, both of which are discrete and combinatorial in nature. One can therefore attempt to construct heterotic vacua in an algorithmic way. Torically constructed Calabi-Yau threefolds form a dataset of reflexive polytopes represented by the lattice vertices, and monad bundles are explored on each of the base manifolds by varying the integer parameters.
4.1. Phenomenological 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, at . Since the goal of string phenomenology is to obtain (supersymmetric versions of) the standard model, more criteria should further be imposed on the vacua. To make things clear, let us emphasize that in this paper the terminology “standard-like” model will imply the following: (i)Gauge invariance under , possibly with an extra factor; (ii)three generations of quarks and leptons, and no exotics; (iii)cancellation of heterotic anomaly.
Here, we translate the above three phenomenological constraints into the conditions on the vacuum topology.
4.1.1. Gauge Group
As explained in Section 1, the structure group of the visible sector bundle sits in and its commutant becomes the low-energy gauge group. In order to obtain and , one must choose and , respectively. In particular, the rank of the bundle should be either 4 or 5 and, hence, by (3.2), where and are the ranks of the two vector bundles in the defining sequence (3.1) of . What is more, since the structure group should be “special” unitary, the first Chern class of is to vanish. By (3.4), this corresponds to where and are the -tuples of integers labelling the summand line bundles and, hence, parameterising the monad .
We still have to break the GUT group further down to a standard-like one, and this second breaking will require -Wilson-lines. However, given the observation that most of the torically constructed Calabi-Yau threefolds have a trivial first fundamental group , 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 that acts freely on and make a quotient space , which will then have a nontrivial first fundamental group .
4.1.2. Cancellation of Heterotic Anomaly
Heterotic models need to satisfy a well-known anomaly condition. So far, we have only mentioned one holomorphic vector bundle for the visible sector but the theory has another bundle 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 . The remaining two dimensions should then wrap holomorphic two cycles in for unbroken supersymmetry. Thus, the homology classes associated with these two cycles must be effective and, hence, belong to the Mori cone in. In other words, the corresponding four-forms must belong to the corresponding cone in .
In this most general setup, heterotic anomaly cancellation imposes a topological constraint relating the Calabi-Yau threefold, the two vector bundles, and the five-brane classes. When , the anomaly condition can be expressed, at the level of cohomology, as where is the sum of the five-brane classes. Note that itself should also belong to the Mori cone of as all the summands do. In our discussion, however, without mentioning the second bundle , we presume a trivial bundle for the hidden sector. Thus, the anomaly constraint in (4.3) says that is effective.
4.1.3. Particle Spectra
Table 1 shows how the low-energy particle spectra are determined from various bundle-valued cohomology groups. Assuming that is a stable bundle4 implies that , and, hence, to obtain three net generations of quarks and leptons, we must have where the Aiyah-Singer index theorem  has been applied to the differential operator on and is the order of the discrete symmetry group , with which we will have to quotient the “upstairs” threefolds .
4.2. Discrete System for Standard-Like Vacua
In this subsection, we briefly summarise the model-building requirements that have so far been discussed. (i)Calabi-Yau threefold: a reflexive polytope describes a Calabi-Yau threefold . In the computer package PALP , by inserting the list of lattice vertices of or, equivalently, the corresponding “weight system” , one can obtain all the topological invariants of relevant to the heterotic compactification.(ii) Monad vector bundle: the integer vectors and of length , each labelling a line bundle summand, parameterise our monad bundle .(iii) Standard-like constraints: the internal backgrounds are also constrained by the standard-like phenomenology. It turns out that given a Calabi-Yau threefold , that is, for a fixed topology of , the monad parameters and must obey the algebraic equations of degree 0, 1, 2, and 3 shown in Table 2.
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 and 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 (3.1) are ample or, equivalently, if all the monad parameters and are positive,5 then, by Kodaira's vanishing theorem [45, 47], the cohomology group vanishes and, hence, the low-energy effective theory acquires no antigenerations. 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 semipositive 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 may be nonnegative, for all and . In this case, we will call the monad generic since the monad map in (3.1), thought of as an matrix of polynomials, may generically have all the entries nonzero.
4.3. 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 threefolds. As was mentioned in Section 2.2, the total dataset of torically constructed Calabi-Yau threefolds are way too large to grasp altogether. Therefore, at the initial stage, those in “smooth” ambient spaces have first been considered amongst the total of 500 million . It turns out that over these 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, semipositive monads have been explored over the same Calabi-Yau threefolds . 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 . A total of torically constructed Calabi-Yau threefolds have the Hodge number , and the generic, positive monads have been classified thereon, giving rise to new candidate models.
5. 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 . Simplicity of the integer combinatorics for the heterotic vacua was the essential feature that made this approach a tractable programme. It was motivated by the general observation that any carefully chosen 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 “genuine” standard model(s).
The combinatorics of toric geometry has been invaluable for constructing toric ambient fourfolds, to which Calabi-Yau threefolds 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 , and general ambient fourfolds have also been dealt with  by partially resolving the singularities, if they bear any, so that the smoothness of the hypersurface Calabi-Yau threefolds is guaranteed. In both cases, the generic, positive monads (and some semipositive ones, too, in the former case) have been probed under the standard-like criteria. We have thus obtained a set of candidate models, that are anomaly free and that have a chance to produce three generations of quarks and leptons without any antigenerations.
To guarantee the three-generation property of these candidates, further study of discrete symmetries of the manifolds is essential. Braun has recently classified the free group actions on complete intersection Calabi-Yau threefolds in multiprojective spaces , and his algorithm can in principle be generalised to toric cases. The line-bundle cohomologies on the torically constructed Calabi-Yau threefolds 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 [23, 50–52]. 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.
The author is grateful to Yang-Hui He and Andre Lukas for collaborations and to Lara Anderson and James Gray for discussion, on the projects which this paper is based upon. He would especially like to thank Maximilian Kreuzer, who passed away in the middle of collaborations on part of the work reviewed here, for invaluable correspondence and advice.
- The choice also gives rise to GUT models. However, they have an inherent trouble in doublet-triplet splitting of Higgs multiplet (see, for a recent example, ) and, hence, we will not address the models of this type here.
- In algebraic geometry, Calabi-Yau threefolds 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.
- MPCP is a short for maximal, projective, crepant and partial. A triangulation is said to be maximal if all lattice points of the polytope are involved, projective if the Kähler cone of the resolved manifold has a nonempty interior, and crepant if no points outside the polytope are taken. In practice, all possible MPCP-triangulations of a given reflexive polytope are searched by the computer package PALP .
- Testing the bundle stability is indeed one of the crucial steps for our model construction. However, it is not at all an easy task to check if a given bundle is stable. So, our strategy is first to make use of some consequences of stability and then to check the validity at the very end of the story. In this paper, focusing on the discrete combinatorics for the vacua, we will not say more about the issue of stability.
- To be precise, the Kodaira vanishing assumes that the vectors and lie in the Kähler cone of . In case the Kähler cone does not coincide with the positive region, one may redefine the standard basis vectors of to be the Kähler cone generators. For this to work, however, the cone generators should form a linearly independent basis and hence, we implicitly restrict ourselves to the Calabi-Yau threefolds of this type.
P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, “Vacuum configurations for superstrings,” Nuclear Physics B, vol. 258, pp. 46–74, 1985.View at: Google Scholar
M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, vol. 2 of Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1987.
A. Lukas, B. A. Ovrut, K. S. Stelle, and D. Waldram, “The Universe as a domain wall,” Physical Review D, vol. 59, no. 8, pp. 1–9, 1999.View at: Google Scholar
A. Lukas, B. A. Ovrut, and D. Waldram, “Non-standard embedding and five-branes in heterotic M theory,” Physical Review D, vol. 59, no. 10, pp. 1–17, 1999.View at: Google Scholar
C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Birkhäuser, 1980.
M. Kreuzer and H. Skarke, “Complete classification of reflexive polyhedra in four dimensions,” Advances in Theoretical and Mathematical Physics, vol. 4, p. 1209, 2002.View at: Google Scholar
M. Kreuzer, “Toric geometry and Calabi-Yau compactifications,” Ukrainian Journal of Physics, vol. 55, no. 5, pp. 613–625, 2010.View at: Google Scholar
W. Fulton, Introduction to Toric Varieties, Princeton University Press, 1993.
T. Oda, Convex Bodies and Algebraic Geometry, Springer, 1988.
D. Cox, J. Little, and H. Schenck, Toric Varieties, American Mathematical Society, 2011.
K. Hori et al., Mirror Symmetry, American Mathematical Society, 2003.
B. Andreas and D. Hernandez-Ruiperez, “Comments on heterotic string vacua,” Advances in Theoretical and Mathematical Physics, vol. 7, pp. 751–786, 2004.View at: Google Scholar
Y. H. He, M. Kreuzer, S. J. Lee, and A. Lukas, “Heterotic bundles on Calabi-Yau manifolds with small Picard number,” In preparation.View at: Google Scholar
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.
R. Hartshorne, Algebraic Geometry (Graduate Texts in Mathematics), Springer, 1977.
A. Lukas,, L. Anderson, J. Gray, Y. H. He, and S. J. Lee, CICY package, based on methods described in Refs. [15, 47-49].