An Abundance of Heterotic Vacua
AAn Abundance of Heterotic Vacua
Maxime Gabella ∗ , Yang-Hui He , † and Andre Lukas ‡ Rudolf Peierls Centre for Theoretical Physics,Oxford University, 1 Keble Road,Oxford, OX1 3NP, U.K. Merton College, Oxford, OX1 4JD, U.K.
Abstract
We explicitly construct the largest dataset to date of heterotic vacua arising fromstable vector bundles on Calabi-Yau threefolds. Focusing on elliptically fibered Calabi-Yau manifolds with spectral cover bundles, we show that the number of heterotic modelswith non-zero number of generations is finite. We classify these models according tothe complex base of their Calabi-Yau threefold and to the unification gauge group thatthey preserve in four dimensions. This database of the order of 10 models, which includespotential Standard Model candidates, is subjected to some preliminary statistical analyses.The additional constraint that there should be three net generations of particles gives adramatic reduction of the number of vacua. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] S e p ontents Heterotic string and M-theory remain promising approaches toward building phenomenologi-cally realistic models of four-dimensional particle physics. Since the beginning of superstringphenomenology in refs. [1, 2] two decades ago, much progress has been made. It is by nowwell-established that compactification of the E × E heterotic string on Calabi-Yau three-folds endowed with stable holomorphic SU ( n ) vector bundles leads to supersymmetric GrandUnified Theories (GUT) in four dimensions. Furthermore, introducing Wilson lines can breakthe GUT gauge group down to the Standard Model (SM) group.The advantage of this method is that very precise and succinct mathematical quan-tities, namely cohomology groups of the vector bundles, encode the particle spectrum andinteractions of the four-dimensional physics. A considerable amount of work has been de-voted to developing techniques for constructing vector bundles and computing the associatedcohomology groups [3–20]. 2uch of the literature has focused on finding specific vector bundles on select threefoldswhich lead to realistic theories. However, given the multitudes of Calabi-Yau threefolds andof potential vector bundles on them, it is important to have a view of the global picture,analyze the space of models, and especially determine how many models are quasi-realistic.Only recently has there been an effort to understand this heterotic landscape. In ref. [21], aspecial corner has been found which tends to produce realistic models (see ref. [22–24]).Indeed, with the advances of computing power and software in computational algebraicgeometry, a novel perspective on heterotic compactification has been proposed in ref. [13]. Inref. [14], so-called monad bundles are constructed over a large dataset of Calabi-Yau threefoldsknown as CICYs, or Complete Intersection Calabi-Yau manifolds [25]. One of the advantagesof this set is the embedding into a projective ambient space which facilitates standard tech-niques for calculating the requisite bundle cohomology groups. However, proving stability ingeneral for these bundles requires separate treatment and is rather difficult; a comprehensiveprocedure is still elusive [26].Luckily, a systematic technique for creating stable vector bundles does exist for a wideclass of Calabi-Yau manifolds, constructed by elliptic fibration [27, 28]. These ellipticallyfibered threefolds are tori fibered over a complex base surface which also have a zero section.Such manifolds have been completely classified [29, 30]. Stable vector bundles can be conve-niently constructed over them using the spectral cover method [27, 28]. An initial attempt atclassifying such vector bundles over this dataset of Calabi-Yau threefolds was undertaken inref. [6].The purpose of the current paper is to classify, as much as computer power allows,the spectral cover bundles over elliptically fibered threefolds and to examine some of theirproperties and statistical features in the light of basic physical constraints, such as the three-generation constraint. It turns out that a tremendous number is readily found. This consti-tutes the largest explicit dataset of stable vector bundles to date. For the Calabi-Yau manifoldsconstructed from Hirzebruch base spaces we find about 50 ,
000 inequivalent cases. After im-posing basic physical constraints this number is drastically reduced to about 1 , d P r , we find that the number of consistentvacua increases dramatically with r , and we are able to perform a complete classification for r ≤ r = 3 we find over 11 million models, of which about 400 ,
000 arestill compatible with basic physical constraints. For Calabi-Yau manifolds based on higherdel Pezzo surfaces, the classification is limited by computational power. We perform somestatistical analysis of the models and discuss model-building prospects.3he organization of the paper is as follows. In Section 2, we lay out the mathematicalconstruction of stable vector bundles over elliptically fibered Calabi-Yau threefolds and reviewhow physical aspects such as the number of generations, GUT gauge groups, and anomalycancellation are conveniently encoded. We summarize all requisite physical constraints asexplicit Diophantine inequalities in Section 3 and show that the number of solutions is finite.In Section 4, we proceed to classify stable SU ( n ) vector bundles from the spectral coverconstruction on the elliptically fibered Calabi-Yau threefolds. We conclude with discussionsand prospects in Section 5. Let us begin with a brief review of the compactification of heterotic string theory and, inthe non-perturbative regime, heterotic M-theory. This section serves as a reminder of themathematical constructions and physical constraints involved. We begin by motivating theneed for stable holomorphic vector bundles on Calabi-Yau threefolds and then specialize toa wide class of elliptically fibered Calabi-Yau threefolds. We then briefly review the spectralcover method for constructing stable bundles on these manifolds.
The “traditional” way to relate the ten-dimensional string theory to a four-dimensional space-time M is to start with the E × E heterotic string theory on a background M = M × X , (1)where X is a compact six-dimensional manifold (for a review see ref. [31]). Historically, thiswas the first approach toward string phenomenology [1, 2].The requirement of unbroken N = 1 supersymmetry in four dimensions further specifiesthe construction. The standard solution is that the non-compact (and maximally symmetric)space-time M is flat Minkowski space while the compact manifold X is a Calabi-Yau threefold,that is, a complex K¨ahler manifold with a metric of SU (3) holonomy. Equivalently, by Yau’stheorem, X is complex K¨ahler and has vanishing first Chern class of its tangent bundle, c ( X ) = 0.It is common to declare one of the E gauge groups to be the “visible sector,” which isto contain the particles of the SM, and the other E to be the “hidden sector.” Additionalhidden sectors can arise from the world-volumes of five-branes which may be included in the4ompactification. Throughout this paper we will not consider these hidden sectors explicitlybut we will ensure that a choice of data in the hidden sectors which leads to an overallconsistent model exists. In order to reduce the visible E group we allow a vector bundle V on X with gauge group G . The low-energy gauge group H will then be the commutant of G in E . Moreover, we will focus on the cases G = SU ( n ) for n = 3 , ,
5, so that H is oneof the standard grand unification groups E , SO (10), and SU (5), respectively. Subsequently,one needs to introduce appropriate Wilson lines in order to break H to the Standard Modelgauge group. Because G is a special unitary group the vector bundle V satisfies the condition c ( V ) = 0 . (2)In order to preserve N = 1 supersymmetry, the gauge connection F on X must satisfythe hermitian Yang-Mills equations F mn = F ¯ m ¯ n = g m ¯ n F m ¯ n = 0. These are rather difficultequations to solve. Luckily, theorems by Donaldson, Uhlenbeck and Yau [32] state that a holo-morphic vector bundle V on X will admit such a connection if and only if it is (poly-)stable. Acomplicated set of partial differential equations is thereby translated to a problem of pure alge-braic geometry. The proof of stability is still a difficult issue (see for example refs. [13,14,26]),but fortunately the spectral cover construction used in the present paper will automaticallyguarantee this.Now, in order to be consistent at the quantum level, we need to impose Green-Schwarzanomaly cancellation. This translates to a constraint on the second Chern classes of thetangent bundle of the compact manifold X , the visible sector bundle V , the hidden bundle˜ V , and the holomorphic curve W wrapped by five-branes [33]. This constraint reads (see forexample [15, 16]) c ( X ) − c ( V ) − c ( ˜ V ) = [ W ] , (3)where [ W ] is the homology class of W and provided that both V and ˜ V are vector bundleswith vanishing first Chern class. Since W is a holomorphic curve, its associated class [ W ]is effective, or, in other words, it is an element of the Mori cone of X . Given a Calabi-Yaumanifold X and a visible bundle V , a simple way to make sure that the anomaly conditioncan be satisfied is to demand that c ( X ) − c ( V ) ∈ Mori cone of
X . (4)In this case, one can always find a five-brane curve W such that the anomaly condition (3) isindeed satisfied for a trivial bundle ˜ V (although there may well be alternative choices whichinvolve a non-trivial hidden bundle ˜ V ). We will henceforth use the condition (4) for thepurpose of classifying spectral cover bundles. 5 .2 Elliptic Fibration As discussed above, the first object we need is a Calabi-Yau threefold X . In this paper, wewill focus on the rich data set of elliptically fibered Calabi-Yau threefolds since they allow fora convenient construction of stable vector bundles.An elliptically fibered Calabi-Yau threefold X is defined by a fibration X π → B (5)over a complex base surface B such that the fiber π − ( b ) is an elliptic curve for each genericpoint b ∈ B . We are referring to an elliptic curve rather than a torus because we require theexistence of a global section σ : B → X, (6)which associates to every point in B the zero element of the addition law on the elliptic curve.The existence of a global section is a surprisingly strong constraint [29], and as a resultthe complex base surface can only be one of the following [30] : Hirzebruch surfaces and theirblow-ups, del Pezzo surfaces, and Enriques surfaces. We will introduce these surfaces in detailin Section 4.One advantage of this fibered construction is that the Chern classes of X can be easilyexpressed in terms of those of the base surface B as [28] c ( X ) = 0 , (7) c ( X ) = c ( B ) + 11 c ( B ) + 12 σc ( B ) , (8) c ( X ) = − c ( B ) . (9)These formulae will be useful later. We remark that the Euler number of X is simply χ ( X ) = (cid:90) X c ( X ) = − c ( B ) . (10) As mentioned earlier, having background gauge fields which satisfy the hermitian Yang-Millsequations is equivalent to choosing a stable holomorphic vector bundle. The greatest advan-tage of elliptically fibered Calabi-Yau threefolds is that a systematic and relatively straight-forward method has been developed to construct holomorphic vector bundles on them whichare guaranteed to be stable. This is the so-called spectral cover construction [27, 28].6he idea is to first construct the bundles on individual elliptic fibers using a classicresult on stable bundles over elliptic curves due to Atiyah. Then these bundles over individualfibers are patched together over the base. In summary, an SU ( n ) bundle V over X is givenby the spectral data, consisting of the following two pieces : • The spectral cover C V : this is an n -fold cover of the base and is thus a divisor (alinear combination of hypersurfaces) in X with degree n over B . This implies that thecohomology class of C V in H ( X, Z ) (cid:39) H ( X, Z ) is of the general form[ C V ] = n σ + η , (11)where σ is the class of the zero section, and η is a curve class in H ( B, Z ). The class η must be effective in B , which means that it must be possible to express it as a linearcombination of effective classes S i ∈ H ( B, Z ) with non-negative coefficients : η = (cid:88) i a i S i , with a i ≥ . (12)The subset of effective classes forms a cone in H ( B, Z ) called the Mori cone . • The spectral line bundle N V : this is a line bundle on C V with first Chern class c ( N V ) = n ( 12 + λ ) σ + ( 12 − λ ) π ∗ η + ( 12 + nλ ) π ∗ c ( B ) . (13)The parameter λ has to be either integer or half-integer depending on the rank n of the SU ( n ) structure group : λ = (cid:40) m + 1 / n is odd ,m if n is even , (14)where m ∈ Z . When n is even, we must also impose η = c ( B ) mod 2, by which wemean that η and c ( B ) differ only by an even element of H ( B, Z ).The holomorphic SU ( n ) vector bundle V on X can be extracted from the above databy a so-called Fourier-Mukai transformation : ( C V , N V ) F M ←→ V (see refs. [34, 35] for someapplications of this transformation in string theory). The Chern classes of V are given interms of the spectral data as [18, 19, 28] c ( V ) = 0 , (15) c ( V ) = ησ − n − n c ( B ) + n (cid:18) λ − (cid:19) η · ( η − nc ( B )) , (16) c ( V ) = 2 λση · ( η − nc ( B )) . (17)7ne of the advantages of the spectral cover construction is that stability of V can beguaranteed by fairly simple algebraic conditions : the vector bundle V is stable if C V isirreducible. This will be the case if we impose the conditions (see for example [6])the linear system | η | is base-point free in B , (18) η − nc ( B ) is an effective curve in B . (19)We recall that the linear system | η | is the set of all effective curves linearly equivalent to η (thatis, which only differ from η by the divisor of a meromorphic function [36]). It is base-pointfree if its members have no common intersection. We will make these two rather technicalconditions more explicit for the surfaces we will encounter in Section 4.Finally, the five-brane class W can be split up into a curve class W B in the base surface B and the fiber class F of the elliptic fibration, so that W = W B + a f F, (20)with a f some integer. For most of the base spaces that we will consider, the class W is effectiveif and only if the following conditions hold : W B is effective , (21) a f ≥ . (22)There is an exception to this rule for Hirzebruch surfaces F r with r ≥ W B and a f in terms of the cohomology classes of the base B as W B = 12 c ( B ) − η, (23) a f = c ( B ) + (cid:18)
11 + n − n (cid:19) c ( B ) − n (cid:18) λ − (cid:19) η · ( η − n c ( B )) . (24) A salient feature of heterotic compactification is that the low-energy particles are given interms of the vector bundle cohomology groups for V [31]; these are well-defined mathematicalquantities to compute. In particular, for SU ( n ) bundles, we can count the net number ofgenerations, N (cid:48) gen , in the resulting Grand Unified Theory. This is a topological number andfrom the index theorem it can be expressed as N (cid:48) gen = 12 (cid:12)(cid:12)(cid:12) (cid:90) X c ( V ) (cid:12)(cid:12)(cid:12) . (25)8f we wish to further break the grand unified group to the Standard Model group, wenormally have to quotient the Calabi-Yau manifolds by a freely-acting discrete symmetry toobtain a non simply connected space and then turn on Wilson lines. Typically the symmetrygroup is a cyclic group Z k or a product thereof (for a recent discussion on potentially largediscrete symmetries, see ref. [37]). Let the order of this group be k . The net number ofgenerations on the quotient manifold, N g en , is then reduced by the order of this group andgiven by N gen = N (cid:48) gen /k . (26)For elliptically fibered threefolds it is usually not easy to find freely acting discrete symmetriesand we will not explicitly attempt this in the present paper. Instead, we will use some basicnecessary conditions for the existence of such a symmetry. First of all, eq. (26) implies thatthe “upstairs” number of generations, N (cid:48) gen , must be a multiple of three, N (cid:48) gen = 3 k , (27)so that the order of a discrete symmetry group which leads to three generations “downstairs”is given by k = N (cid:48) gen /
3. For the Calabi-Yau manifold X to allow for such a discrete symmetryits Euler number must, of course, be divisible by the order k , so χ ( X ) /k ∈ N . (28)Eqs. (27) and (28) are the two basic physical constraints which we will impose on the modelsfound in this paper. For practical calculations, they can be expressed in terms of the basesurface and the spectral data by using eqs. (17) and (9). In the previous section we have presented the rudiments of constructing stable, holomorphic SU ( n ) vector bundles on an elliptically fibered Calabi-Yau threefold. The requirement ofanomaly cancellation for a consistent heterotic vacuum and the physical condition of three netgenerations of low-energy particles lead to a set of constraints on these bundles. It is expedientto summarize these, now phrased in a succinct mathematical manner. In the following section,we will show how these constraints lead to a classification problem. For recent related work,the reader is also referred to refs. [10] and [11]. Combining eqs. (18), (19), (21), (22), (27),and (28), we gather the six following constraints. • Stability of the vector bundle V : 9.) | η | must be base-point free.2.) η − nc ( B ) must be effective. • Anomaly cancellation with five-branes (effectiveness of W ) :3.) W B = 12 c ( B ) − η must be effective.4.) a f = c ( B ) + (cid:16)
11 + n − n (cid:17) c ( B ) − n (cid:0) λ − (cid:1) η · ( η − n c ( B )) ≥ λ = m + 1 / n is odd, and λ = m if n is even, with m ∈ Z . • Three generations :5.) N (cid:48) gen = | λ η · ( η − n c ( B )) | = 3 k , with k ∈ N .6.) k divides χ ( X ) : 60 c ( B ) /k ∈ N .We recall that the rank n of the structure group SU ( n ) equals 3, 4 or 5, correspondingrespectively to low-energy gauge groups E , SO (10), or SU (5). Note that all the constraintsare conveniently expressed in terms of quantities on the base surface B . The Chern classes caneasily be computed for the various allowed base surfaces, and the curve η can be expanded intoa basis of second homology. A solution to these constraints will consists of a set of coefficientsthat specify the effective class η in terms of the generators of the Mori cone, as well as the valueof the arbitrary integer or half-integer parameter λ . We note that the above set of conditionsreally splits into two logically somewhat distinct parts. Conditions 1.) to 4.) guarantee theexistence of a consistent heterotic vacuum and our initial classification will, therefore, focuson these first four constraints. Constraints 5.) and 6.), on the other hand, are constraints ofa “phenomenological” nature and will only subsequently be imposed on the set of consistentvacua in order to filter out promising models. Hence, our classification problem can be statedas follows. Find all η (specified by non-negative integer coefficients of an expansion in thebasis of the Mori cone of B ) and λ (integer or half-integer according to the rank n ) such that the above constraints 1.) to 4.) are satisfied. Within this set find allcases which in addition satisfy constraints 5.) and 6.). We can immediately make some observations. First, note that the intersection number η · ( η − nc ( B )) appears in both constraints 4.) and 5.). Models with a zero net number of With an exception for F r ≥ , in which case a f ≥
96 + ar − a − b , see Section 4.1. λ (cid:54) = 0 , η · ( η − nc ( B )) (cid:54) = 0 . (29)These will be included with constraints 1.) to 4.) in our actual initial classification. Atechnical reason for demanding a non-zero number of generations has to do with the issue offiniteness which we will address shortly. On the so-obtained data set we will then impose thethree-generation constraints 5.) and 6.).Second, in all our constraints, λ only appears as a square or an absolute value; thus forevery solution with positive λ there is also a solution with negative λ . Since the third Chernclass of V depends explicitly on λ itself, as seen from (17), these two sets of solutions areactually different bundles.Finally, let us consider the issue of whether the number of solutions is finite or infinite.Let us examine conditions 2.) and 3.). Crucially, we see that these two conditions haveopposite signs in front of η . Effectiveness is a positivity condition and this means that 2.)and 3.) provide upper and lower bounds for the coefficients in the expression of η . If theMori cone is finitely generated, then this implies that there is only a finite number of possiblesolutions of η . As we will see below, all of our base surfaces have a finite-dimensional Moricone, except the ninth del Pezzo surface. Luckily, this particular surface will be ruled out bythe requirement of a non-zero net number of generations, that is by the condition (29).With a finite possible set of solutions for η , condition 4.) constitutes a quadratic in-equality for λ if the coefficient η · ( η − nc ( B )) does not vanish. This is precisely what wehave required in eq. (29) in order to have a non-vanishing number of families. As a result,the number of possible λ values is finite. This finiteness result is the technical reason for thenon-vanishing condition (29). In our detailed calculations below, it will turn out that thereare some cases for which η · ( η − nc ( B )) is indeed zero. They may lead to an infinite familyof stable vector bundles satisfying the anomaly constraints, although all of them with a zeronumber of generations. We will not presently address these bundles.Hence, since there is a finite number of solutions to our variables λ and η , we immediatelyhave a nice finiteness result. There is a finite number of solutions to constraints 1.) to 4.), together with thecondition (29) . That is, there is a finite number of spectral cover SU ( n ) vectorbundles on elliptically fibered Calabi-Yau threefolds which lead to anomaly-free het-erotic vacua with a non-vanishing number of generations. We have laid the foundation and presented the crux of our problem in the previous twosections. Now, let us perform a systematic study of the solutions to the six constraints foreach of the allowed bases for the elliptic fibration. Enriques base spaces have been shown to beruled out by effectiveness (see Section 6.1 of ref. [6]). This leaves us with only three possiblechoices : Hirzebruch surfaces, their blow-ups, and del Pezzo surfaces. We will address thespectral cover bundles on them case by case.
We begin with the Hirzebruch surfaces F r , which are P fibrations over P . We denote the classof the base P by S and that of the fiber by E . These classes have the following intersectionnumbers E · E = 0 , S · E = 1 , S · S = − r . (30)The self-intersection number r is an integer between 0 and 12 where the upper bound comesfrom a theorem in ref. [29]. Therefore, there are only 13 Hirzebruch surfaces to consider; wenote that this has not been thus far stressed in the literature.The curves in F r live in H ( B ; Z ), which is in fact spanned by S and E . Moreover, everyeffective curve can be expressed as a linear combination of these generators with non-negativeinteger coefficients, so we express our effective curve η as η = aS + bE , a, b ∈ Z ≥ . (31)Constraint 1.) requires that the linear system | η | be base-point free in F r . This is the case if η · S ≥ We thank Mark Gross for pointing this out to us. b ≥ r a . To compute the other constraints we need the Chern classes of F r which are given by c ( F r ) = 2 S + ( r + 2) E, (32) c ( F r ) = 4 . (33)Combining our above condition for effectiveness, constraints 2.) and 3.) become2.) a ≥ n, b ≥ n ( r + 2) ,3.) a ≤ , b ≤ r + 2) .Already, from these two conditions we see that the coefficients a, b are bounded and can onlyhave a finite number of solutions for η .Next, constraint 4.) becomes4.) 92+ n − n − n ( λ − ) (cid:0) ab − na − nb + nra − ra (cid:1) ≥ (cid:40) r < ,
96 + ar − a − b if r ≥ , The first case, for r <
3, corresponds to the standard situation, discussed in Section 2.3, wherea class W = W B + a f F in X is effective iff W B is effective and a f ≥
0. However, for Hirzebruchsurfaces F r with r ≥
3, the condition a f ≥ a f ≥
96 + ar − a − b, (34)and this leads to the more complicated constraint for this case. In any event, the abovecondition 4.) becomes a quadratic inequality for λ which leads to a finite number solutions.Finally, constraint 5.) amounts to5.) N (cid:48) gen = | λ (cid:0) ab − na − nb + nra − ra (cid:1) | = 3 k ,and from the above expression for the Chern classes and the intersection numbers we havethat c ( F r ) = 8 (interestingly, both this and c ( F r ) are independent of r ) so that the Eulercharacter (10) for X becomes − /k ∈ N . 13n all, the six constraints have therefore become very concrete inequalities in a, b, λ , and k , given r = 0 , . . . ,
12. Indeed, λ is integral or half-integral according to n , and a, b , and k arepositive integers. As discussed earlier, constraints 1.), 2.) and 3.) immediately give a finitenumber of possibilities for a and b which are simply lattice points in a polygon. Furthermore,condition 4.) restricts the possible values of λ . Hence, we have indeed a finite number ofsolutions. On this set, we can then impose the phenomenological conditions 5.) and 6.). Thiswill typically lead to a large reduction of the number of viable models.To explicitly solve the six equations is straightforward though tedious. A completelattice point search is implemented using Mathematica and C++. We present some illustrativeexamples of spectral bundles over some of the surfaces in Table 1, and a tally of all the solutionsin Table 2.The bundles in Table 1 are, according to Section 2.3, specified by the integers n, a, b ,and the (half-)integer λ . We see that we can produce quite small numbers of net generations.This should be contrasted with the results in refs. [13, 14]. One observation is that the smallerHirzebruch surfaces tend to produce models with fewer generations. Indeed, the minimumpossible number of generations achievable for each Hirzebruch F r decreases with r .Base n ( a, b ) λ F , F , F , F , F ,
26) 1 48Table 1:
Some examples of stable vector bundles on elliptically fibered Calabi-Yau threefolds over the firstfew Hirzebruch surfaces. The bundle is specified by integers n, a, b , and the integer or half-integer λ . Weshow examples in which the net number of GUT particle generations is equal to k for some natural number k , and such that k divides the Euler number of the Calabi-Yau threefold. We can think of k as the order ofa possible discrete group of symmetries. Table 2 gives the number of solutions of SU (3), SU (4), and SU (5) bundles on the ellipticthreefolds fibered over each of the Hirzebruch surfaces. For comparison we have also includedtwo additional sets of results. The first three column represent solutions to the constraints 1.)to 4.) only. Hence, these are stable bundles satisfying anomaly cancellation but their numberof generations is not necessarily a multiple of three. Interestingly, SU (4) bundles are the mostrare. The three middle columns count the number of solutions satisfying all six constraints.14his leads to an order 10 reduction in the number of bundles. Finally, in the rightmost threecolumns, we impose the extra condition that k ≤
10. This is a reasonable constraint becauseit is in general difficult to find discrete symmetries of very large order. A further reductionis thus seen. We also find some solutions (shown in parentheses) which gives exactly threegenerations without the need to quotient by any discrete group, that is k = 1. These aresolutions which correspond to three-generation Grand Unified Theories rather than StandardModel-like theories. These are quite uncommon (only 20 out of the 246), and are concentratedon the first three Hirzebruch surfaces.Constraints 1.) – 4.) 1.) – 6.) 1.) – 6.) and k ≤ SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) F
756 74 458 104 18 34 48(4) 6 18 F
878 108 602 140 32 58 56(6) 20 (4) 38 (4) F
740 40 454 68 10 24 24 (2) 4 10 F
666 16 352 66 4 14 12 0 2 F
650 4 306 36 2 10 6 0 0 F
660 0 280 40 0 2 2 0 0 F
682 0 266 28 0 8 0 0 0 F
710 0 258 30 0 8 0 0 0 F
740 0 252 16 0 6 0 0 0 F
774 0 250 24 0 8 0 0 0 F
810 0 250 18 0 6 0 0 0 F
846 0 250 22 0 4 0 0 0 F
882 0 250 18 0 4 0 0 0Total 9794 242 4228 610 66 196 148(12) 30(4) 68(2)Table 2:
The number of stable SU ( n ) vector bundles from the spectral construction over Calabi-Yauthreefolds fibered over the Hirzebruch surfaces and satisfying anomaly cancellation for n = 3 , , (corre-sponding respectively to gauge groups E , SO (10) , or SU (5) ) are given in the first three columns. Themiddle three columns tally those which also give rise to a number of net GUT particle generations divisibleby three (that is, they satisfy all our six constraints). The right-most three columns represent the bundleswhich also require k , the order of a possible discrete symmetry, to be less or equal to . The numbers inparentheses indicate models with exactly three net generations where they exist. Histograms of the net number of GUT particle generations (such that the number equals k for some natural number k and such that k divides the Euler number) for the elliptic Calabi-Yau threefoldfibered over the first Hirzebruch surface F for stable SU ( n ) -bundles at respectively (a) n = 3 , (b) n = 4 ,(c) n = 5 , and (d) combined. The vertical axis is the number of bundles, and the horizontal one the netnumber of generations. As a further illustration, let us examine the first Hirzebruch surface, which has a goodpopulation of solutions, and is also the only case which admits exactly three generations for all n = 3 , ,
5. To illustrate the distribution of the number of net generations, we plot a histogramin Figure 1. We see that most of the models arise at a small number of generations althoughmodels with a large number of generations do exist.To get an idea of the distribution over the entire family, we plot some three-dimensionalhistograms in Figure 2. As before, on the vertical axis we plot the number of solutions,and on the horizontal ones we plot the number of generations and the number r = 0 , . . . , Histograms of the net number of GUT particle generations (such that the number equals k for some natural number k and such that k divides the Euler number) for the elliptic Calabi-Yau threefoldsfibered over all the Hirzebruch surfaces F ,..., for stable SU ( n ) -bundles at respectively (a) n = 3 , (b) n = 4 , (c) n = 5 , and (d) combined. The vertical axis is the number of bundles, and one of the horizontalaxes is the net number of generations while the other, from 0 to 12, labels the specific Hirzebruch surfaces.Note that, from (b), there are no stable SU (4) bundles with a number of generations divisible by 3 for thefifth and higher Hirzebruch surfaces. .2 Blow-ups of Hirzebruch Surfaces Our next family of base surfaces is obtained by blowing up a point on the curve E of aHirzebruch surface F r for r = 0 , , , (cid:98) F r .The second homology is easy to describe. In addition to E and S described in eq. (30),there is now a new exceptional class G , corresponding to the blow-up. Now, if we define F + G = E , the intersection numbers are given by [36] E · E = 0 , S · S = − r, S · E = 1 , (35) F · F = G · G = − , S · F = 1 , S · G = 0 , G · F = 1 . (36)Here, an effective curve can be expressed as η = aS + bF + cG, (37)with a, b, c ∈ Z ≥ but not all 0, and the Chern classes are given by [17] c ( (cid:98) F r ) = 2 S + ( r + 2) F + ( r + 1) G, (38) c ( (cid:98) F r ) = 5 . (39)The base-point freeness condition can now be guaranteed by b ≥ a, b ≥ c, c ≥ b − a. (40)The remaining conditions are straightforward and so without much ado, we can explicitlysummarize the six constraints as1.) b ≥ ra and b ≥ a, b ≥ c, c ≥ b − a ,2.) a ≥ n, b ≥ ( r + 2) n, c ≥ ( r + 1) n ,3.) a ≤ , b ≤ r + 2) , c ≤ r + 1),4.) 82 + ( n − n ) − n ( λ − )( − ra + 2 ab − b + 2 bc − c + ( r − na − nb − nc ) ≥ | λ ( − ra + 2 ab − b + 2 bc − c + ( r − na − nb − nc ) | = 3 k ,6.) 420 /k ∈ N . We are grateful to Antonella Grassi for pointing this out to us.
18e proceed as before in solving these Diophantine inequalities by lattice-point searchand tally the solutions in Table 3. Again, we see that SU (4)-bundles are the most rare. Asbefore we show three data sets, those satisfying only constraints 1.) to 4.) , those satisfyingall six constraints, and those satisfying all six constraints and in addition having the order ofthe symmetry group k ≤
10. A dramatic reduction is seen in the number of solutions withthe imposition of these constraints.1.) – 4.) 1.) – 6.) 1.) – 6.) and k ≤ SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) (cid:98) F (cid:98) F (cid:98) F (cid:98) F The number of stable SU ( n ) vector bundles from the spectral construction over Calabi-Yauthreefolds fibered over the blow-ups of Hirzebruch surfaces and satisfying anomaly cancellation for n = 3 , , (corresponding respectively to gauge groups E , SO (10) , or SU (5) ) are given in the first three columns.The middle three columns tally those which also give rise to a number of net GUT particle generationsdivisible by three (that is, they satisfy all our six constraints). The right-most three columns represent thebundles which also require k , the order of a possible discrete symmetry, to be less or equal to . We are finally left with the del Pezzo family of surfaces. It will turn out that, because somehigher members of this family have a large number of generators for the Mori cone, these giverise to the most number of bundles. Indeed, elliptic fibrations over specific del Pezzo surfaceshave been favorable in constructing realistic models for the past few years [7, 8].Let us begin by introducing the geometry. The del Pezzo surfaces d P r is given by P ,the complex projective plane, blown-up at r generic points. There are only ten del Pezzosurfaces, with r = 0 , . . . ,
9, for which elliptic fibration is allowed. The first member, d P , isjust P , and the second, d P , is isomorphic to the first Hirzebruch surface, F .Again, we need the second homology group H ( d P r , Z ) to describe the curve classes.The generators are easy to obtain : they are simply the hyperplane class l in P as well asthe exceptional blow-up divisors E i with i = 1 , . . . , r . They have the following intersection19 Generators ( i < j < . . . ≤ r ) Number0 l E , l − E E i , l − E i − E j E i , l − E i − E j E i , l − E i − E j E i , l − E i − E j , l − E i − E j − E k − E l − E m E i , l − E i − E j , l − E i − E j − E k − E l − E m E i , l − E i − E j , l − E i − E j − E k − E l − E m ,3 l − E i − E j − E k − E l − E m − E n − E o E i , l − E i − E j , l − E i − E j − E k − E l − E m ,3 l − E i − E j − E k − E l − E m − E n − E o , . . . ∞ Table 4:
Generators of the Mori cone (of effective curves) for del Pezzo surfaces d P r . numbers (see for example ref. [6]) : l · l = 1 , l · E i = 0 , E i · E j = − δ ij . (41)The Chern classes are given by c ( d P r ) = 3 l − r (cid:88) i =1 E i , (42) c ( d P r ) = 3 + r. (43)The Mori cone for the del Pezzo surfaces is not as simple as the one for the previouscases; its generators are listed in Table 4. Every effective class can be written as linearcombinations of these generators with non-negative integer coefficients. It is a concern that d P has an infinite dimensional Mori cone. This may contradict our finiteness result. Luckily,as discussed in Section 6.2 of ref. [6], the generic d P surface is ruled out by the requirementof a non-zero number of generations, eq. (29). The basic reason is that this surface is itself anelliptic fibration over P with fiber class f . From effectiveness, η must be proportional to f and as a result the number of generations is zero. Therefore we need not consider this surface.However, we must point out that special d P surfaces, where additional isometries arefound in special points of moduli space, are allowed. Indeed, all the successful models in theliterature based on this surface are special d P [7, 8]. We will not consider these special caseshere. 20.) – 4.) 1.) – 6.) 1.) – 6.) and k ≤ SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) SU (3) SU (4) SU (5) d P
62 10 44 12 2 6 4 0 2 d P
878 108 602 140 32 58 56(6) 20 (4) 38 (4) d P d P , , , , , , d P . . . . . . . . . . . . . . . . . . . . . . . . . . .Table 5: The number of stable SU ( n ) vector bundles from the spectral construction over Calabi-Yauthreefolds fibered over the first four del Pezzo surfaces and satisfying anomaly cancellation for n = 3 , , (corresponding respectively to gauge groups E , SO (10) , or SU (5) ) are given in the first three columns.The middle three columns tally those which also give rise to a number of net GUT particle generationsdivisible by three (that is, they satisfy all our six constraints). The right-most three columns represent thebundles which also require k , the order of a possible discrete symmetry, to be less or equal to . Thenumbers in parentheses indicate those, where possible, with exactly three net generations. Finally, we need conditions for the linear system | η | to be base-point free. On d P r for2 ≤ r ≤
7, this is the case if the divisor η is such that η · E ≥ E whichsatisfies the two properties E · E = − E · c ( d P r ) = 1. Therefore, this condition is againtranslated into constraints on intersection numbers.Now we are ready to write our six constraints in terms of coefficients of η expandedinto the Mori cone, as well as λ and k . Indeed, from Table 4 we see that there is a veryrapidly increasing number of generators as r increases. Thus we have increasing numbers ofcoefficients to deal with and this complicates our algorithmic computations. We treat eachcase of r separately.As an example, let us discuss d P explicitly. There are three generators of the Mori coneand we can write η = aE + b ( l − E − E ) + cE . (44)Subsequently, the six constraints become :1.) b ≥ a, a + c ≥ b, b ≥ c ,2.) a ≥ n, b ≥ n, c ≥ n ,3.) a ≤ , b ≤ , c ≤
24, 21a) (b)(c) (d)Figure 3:
Histograms of the net number of grand unified particle generations (such that the number equals k for some natural number k and such that k divides the Euler number) for the elliptic Calabi-Yau threefoldfibered over the third del Pezzo surface d P for stable SU ( n ) -bundles at respectively (a) n = 3 , (b) n = 4 ,(c) n = 5 , and (d) combined. The vertical axis is the number of bundles, and the horizontal one the netnumber of generations. ( n − n ) − n ( λ − )( − a + 2 ab − b + 2 bc − c − na − nb − nc ) ≥ N (cid:48) gen = | λ ( − a + 2 ab − b + 2 bc − c − na − nb − nc ) | = 3 k ,6.) 420 /k ∈ N .Again, we can find all solutions via an exhaustive lattice-point search.In Table 5 we record the tally of solutions for the first four del Pezzo surfaces. We seethat the number of solutions grow exponentially, and d P r for 4 ≤ r ≤ d P , the richest so far, in Figure 3.22 Conclusion and Prospects
In this paper, inspired by recent advances in applying computer algebra and computationalalgebraic geometry to string phenomenology [13, 38], we initiated the construction and statis-tical analysis of the largest set of explicit stable bundles to date.
We carried out a classificationof spectral cover vector bundles, compatible with heterotic model-building constraints, overelliptically fibered Calabi-Yau manifolds. For both Hirzebruch and blown-up Hirzebruch basespaces we obtained a complete classification of anomaly-free bundles with about 30 , SU (3)cases, 20 , SU (5) cases, and only about 2 , SU (4) cases. It is interesting to note thedifference in numbers between SU (4) and the other two structure groups; this is related to acase distinction for the parameter λ in the spectral cover construction.We then imposed two physical constraints on these bundles, namely the three-generationconstraint and the requirement that the order of a possible discrete symmetry group is at most10. This led to a dramatic reduction in the number of viable models to about 1 , d P , where we found over million anomaly-free stable bundles. As before, the number of SU (4) bundles is relatively small with about 400 ,
000 cases. Weshould stress that among all these bundles the number of those that have different secondand third Chern classes is smaller by a factor 100 approximately. They could still however begenuinely different bundles, given that they have different coefficients for the expansion of η in terms of Mori cone generators or different values of λ .Imposing the physical constraints led to a reduction of the number of models by a factorof more than 10 but we are still left with about 400 ,
000 viable models at this stage. Wedid not explicitly classify bundles on del Pezzo surfaces d P r with r >
3, as this task exceedscurrent computer power.For Hirzebruch base spaces our approach led to a relatively small number of about 1 , U ( n ) structure groups over elliptically fibered Calabi-Yau man-ifolds. Discrete symmetries and Wilson lines are not required for such models and system-atically imposing detailed physical constraints might be a more straightforward task. Theseissues are currently under investigation. Acknowledgments
The authors would like to express their sincere gratitude to Antonella Grassi, Mark Grossand Tony Pantev for many helpful discussions. M. G. thanks the Berrow Foundation forsupporting his work. Y.-H. H is indebted to the UK STFC for an Advanced Fellowship aswell as the FitzJames Fellowship of Merton College, Oxford. A. L. is supported by the EC6th Framework Programme MRTN-CT-2004-503369.
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