G_4 flux, chiral matter and singularity resolution in F-theory compactifications
GG flux, chiral matter and singularity resolution inF-theory compactifications Sven Krause, Christoph Mayrhofer and Timo Weigand , Institut f¨ur Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 19, D-69120 Heidelberg Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China
E-mail:
[email protected] , [email protected] , [email protected]
Abstract:
We construct a set of chirality inducing G -fluxes in global F-theory compactifica-tions on Calabi-Yau four-folds. Special emphasis is put on models with gauge group SU (5) × U (1) X relevant in the context of F-theory GUT model building, which are de-scribed in terms of a U (1)-restricted Tate model. A G -flux arises in a manner completelyanalogous to the U (1) X gauge potential. We describe in detail the resolution by blow-upof the various singularities responsible for the U (1) X factor and the standard SU (5) gaugegroup and match the result with techniques applied in the context of toric geometry. Thisprovides an explicit identification of the structure of the resolved fibre over the mattercurves and over the enhancement points relevant for Yukawa couplings. We compute theflux-induced chiral index both of SU (5) charged matter and of SU (5) singlets chargedonly under U (1) X localised on curves which are not contained in the SU (5) locus. Wefurthermore discuss global consistency conditions such as D3-tadpole cancellation, D-termsupersymmetry and Freed-Witten quantisation. The U (1) X gauge flux is a global extensionof a class of split spectral cover bundles. It constitutes an essential ingredient in the con-struction of globally defined F-theory compactifications with chiral matter. We exemplifythis in a three-generation SU (5) × U (1) X model whose flux satisfies all of the above globalconsistency conditions. We also extend our results to chiral fluxes in models without U (1)restriction. a r X i v : . [ h e p - t h ] O c t ontents G fluxes from U (1)-restricted Tate models 31.3 Summary of results 5 U (1) -restricted Tate models and their resolution 73 SU (5) × U (1) X Models, resolution and matter curves 10 P structure 143.3 SU (5) matter curves from co-dimension two enhancements 163.4 U (1) X generator, matter charges and SU (5) singlets 19 G -Flux in U (1) -restricted Tate models 23 G -fluxes 234.2 Global constraints: D3-tadpole, D-term and St¨uckelberg masses 254.3 Comparison with split spectral cover bundles 274.4 A three-generation model 294.5 Generalising the flux to chiral non-restricted SU (5)-models 30 P -fibre structure of U (1) -restricted SU (5) -models 36 A.1 GUT Surface 38A.2 Enhancements Curves 38A.3 Enhancements Points 41A.4 Generic Structure on C P -fibre structure for non-restricted SU (5)-models 46 B Intersection Properties 47
B.1 List of intersection numbers 47B.2 Derivation of Intersection Properties from the Stanley-Reisner Ideal 48
C Fibre Ambient Space 49 – 1 –
Introduction
F-theory [1] provides an elegant framework to study a very broad class of string vacua. Itspower and its beauty are rooted in the geometrisation of the back-reaction of physical ob-jects, here seven-branes of Type IIB string theory, on the ambient space. This is achievedby means of a non-trivial fibration of an auxiliary elliptic curve over the physical space-time;its complex structure represents the varying axio-dilaton sourced by the seven-branes. Theholomorphic nature of the relevant geometric data — seven-branes wrap divisors of the baseof the fibration upon compactification to four dimensions — makes the study of the associ-ated string vacua amenable to techniques of algebraic geometry. These geometric methodsgive us insights into systems beyond the perturbative realm such as mutually non-local[p,q]-seven-branes. The resulting marriage between the concept of brane localised gaugedegrees of freedom and the appearance of exceptional gauge groups is largely responsiblefor the revived recent interest, triggered by [2–6], in F-theory also from a phenomenologicalperspective (see [7–9] for reviews on F-theory and its recent applications).Motivated by the prospects of local F-theory model building in the context of GUTphenomenology, a great deal of recent effort has gone into the construction of globallyconsistent four-dimensional F-theory vacua. From the start, it has been clear that thekey to the construction of such vacua and to understanding their properties is having ahandle on the singularity structure of elliptic four-folds. This is because the non-abeliangauge groups, the matter spectrum and the Yukawa interactions of a model are in one-to-one correspondence with the singularities in the fibre of the Calabi-Yau four-fold overloci of, respectively, complex co-dimension one, two and three on the base (see [10] for adescription of the relevant Tate algorithm and [11, 12] for more recent extensions thereof).In order to make sense of the four-dimensional effective action via dimensional reduction ofthe dual M-theory, discussed in detail in [13], it is necessary to work not with this singularfour-fold Y , but rather with a resolved Calabi-Yau ˆ Y . Mathematically, the singular pointsin the fibre are replaced by a collection of P s whose intersection structure reproduces theDynkin diagram of the simple group associated with the singularity. Physically, resolvingthis singularity corresponds to moving in the Coulomb branch of the non-abelian gaugegroups in the dual M-theory. In the F-theory limit of vanishing fibre volume, the resolvedspace ˆ Y and the singular Y are indistinguishable. However, it is in terms of the smoothand well-defined ˆ Y that all computations are performed. The techniques for resolution of singular elliptic fibrations were applied to F-theory soonafter its discovery, starting mainly in compactifications to six dimensions. Most notably, us-ing the powerful tools of toric geometry, an efficient algorithm was developed to completelyresolve singular Calabi-Yau three-folds that are hypersurfaces of toric spaces [14, 15]. Inthe context of F-theory GUT model building the first complete resolutions of Calabi-Yaufour-folds with SU (5) gauge group, as required in the spirit of [2–5], were constructedin [16, 17]. This was done likewise in the framework of toric geometry, generalising themethods of [14, 15] to four-folds constructed as complete intersections of toric ambient– 2 –paces. As demonstrated in [18, 19], the efficiency of the toric approach allows for a sys-tematic construction and study of a large set of four-dimensional F-theory GUT vacuawhich, in particular, comprises the full four-fold associated with the base space construc-ted previously in [20], see also [21]. It is important to stress that the toric resolutionautomatically takes care not only of the co-dimension one singularities, corresponding toseven-branes, but also of the higher co-dimensional singularities along matter curves andYukawa points. What the construction provides is the blow-up of the singularities over thedivisors where they appeared. Thereby, the blow-up introduces a set of rk( G ) extra blow-up divisors fibred over the base divisor associated with gauge group G . This automaticallyresolves also the higher co-dimension singularities. In particular, one has full computa-tional control over the intersection properties of the resolution divisors, the complete setof Hodge numbers and important topological invariants such as the Euler characteristic or c ( ˆ Y ). These invariants enter phenomenologically relevant constraints such as the three-brane tadpole [22] or the flux quantisation condition [23]. On the other hand, the structure of the singularity enhancements over the matter curves and Yukawa points is rather im-plicitly contained in the toric data, see e.g. [24]. For practical computations, it is oftendesirable to have more direct access to this information.More recently, the resolution of singular four-folds with SU (5) GUT symmetries hasbeen re-addressed in [25] using a different method corresponding to a small resolution, asopposed to a blow-up, with special emphasis on the matter curves and Yukawa points.Indeed this analysis has confirmed the general philosophy of higher singularities of SO (10)and SU (6) type over matter curves in generic SU (5) models as well as the appearance of SO (12) and E enhancement points. At a technical level, however, the structure especiallyof the E point is more complicated than usually anticipated. There, the singularity cor-responds to the non-extended E Dynkin diagram or T − , , [25]. In the recent work of [26],amongst other things, the consequences of these technical subtleties were analysed. Theauthors found that the expected structure both of matter states and, in particular, of theircouplings at the enhancement points is unaffected. G fluxes from U (1) -restricted Tate models The geometry of the four-fold and its resolution, important as it is, makes only half of thestory in constructing F-theory vacua. The second, equally crucial ingredient is G -flux. ViaF/M-theory duality, G -fluxes are known to describe both what corresponds in the TypeIIB limit to background flux F − τ H and gauge flux F along the seven-branes. Both ofthem are key players in moduli stabilisation, but the latter are, in addition, indispensablein order to produce a chiral matter spectrum.By F/M-theory duality, specifying G -fluxes amounts to choosing suitably quantisedelements of H ( ˆ Y ) subject to the condition that the four-form has ’precisely one leg alongthe fibre’. Taking into account F-term conditions the flux must eventually be of (2 , The analogue of the Type IIB closed fluxes H and F are given by G -flux whichcan locally be written as a wedge product of a three-form on the base with one of the two If necessary, this F-term condition will fix some of the complex structure moduli. – 3 –-forms along the non-singular fibre. Gauge fluxes on the other hand have one leg alongthe singular fibres or rather along the resolution of the singular fibre in ˆ Y .One type of such gauge flux that is particularly easy to understand is flux associatedwith the Cartan generators of the non-abelian gauge group G along a divisor W . Suchfluxes can be written as G = F ∧ ν , where by ν ∈ H , ( ˆ Y ) we denote the two-forms dualto the resolution divisors introduced when resolving the non-abelian singularity over W and F ∈ H , ( W ). Cartan fluxes of course break the gauge group G . If we are interestedin a chiral spectrum with unbroken non-abelian gauge group we thus need another type offlux. In this paper we are interested in a type of G gauge flux that does produce chiral-ity without breaking any non-abelian gauge symmetry. We approach the construction ofchirality inducing G -fluxes in the context of F-theory compactifications with explicit U (1)gauge symmetries, for the following two reasons: First, we will exploit the fact that in thepresence of such U (1)s there exists a particularly natural candidate for a special elementin H , ( ˆ Y ) with one leg along the fibre [27] lending itself to the construction of G -flux.Second, U (1) symmetries play a prominent rˆole in concrete phenomenological applicationsof F-theory thanks to their selection rules in the matter coupling sector (see e.g. [28–30]for some early references, followed by many others). It is therefore of particular interest toconstruct fluxes in models with abelian gauge symmetries.The appearance of massless U (1)s depends on the full global geometric data of thecompactification and cannot be determined in any local approach to F-theory model build-ing [31],[27]. In the context of the so-called U (1)-restricted Tate model [27], an explicitconstruction of models with U (1) gauge symmetries was given. The idea is to start fromF-theory on an elliptic Calabi-Yau four-fold Y with no massless U (1) gauge potentialsand to restrict the complex structure moduli such as to unhiggs a U (1) gauge symmetry.The elliptic fibre acquires a Kodaira I , or in other words an SU (2) singularity over acurve C on the base space B . Note that this happens in co-dimension one in complexstructure moduli space. The arising curve of SU (2) singularities is the self-intersectionlocus of the I -component of the discriminant ∆ of the four-fold. The singularity can beresolved by a blow-up procedure similar to the resolution of singularities in co-dimensionone. This gives rise to an exceptional divisor { s = 0 } . Its dual two-form leads to an extra U (1) gauge potential upon expanding the M-theory three-form C as C = A ∧ w X withw X = − [ S ] + [ Z ] + c ( B ) [27]. Obviously, this construction provides a natural candidatefor the U (1) X flux G = F ∧ w X [32]. Note that this type of flux is special in that it isgiven by a four-form that can be written as the wedge product of two harmonic two-forms.From general arguments [2, 6], this flux leads to a chiral index for matter states chargedunder U (1) X by integration of G over the associated matter surfaces. The reason why this In particular note that the hypercharge Cartan flux used to break SU (5) → SU (3) × SU (2) × U (1) Y as in [4, 5] must be chosen such as not to produce chirality as otherwise U (1) Y would become massive. We denote the two-form dual to the divisor in class S by the symbol [ S ] ∈ H , ( ˆ Y ). Furthermore, oneneeds to subtract the fibre class [ Z ] and the first Chern class of the tangent bundle of the base in order tomake sure that the resulting two-form indeed has only one leg along the fibre [27]. For models involvingextra non-abelian singularities, the definition of w X will be modified as specified in the sequel. – 4 –onclusion could not be checked explicitly in [27] was an insufficient understanding of thefibre structure over the matter curves as arise e.g. in models with SU (5) gauge symmetry.In this paper, inspired by the explicit description of fibres over the matter curves [25], weare able to explicitly compute the chiral spectrum induced by U (1) X flux.In fact, quite recently the same type of gauge flux was independently discussed indetail in the beautiful work [33], albeit with slightly different methods. This analysis startswith the construction of a gauge flux which cannot be decomposed into the wedge of two-forms. In a second step the complex structure of the four-fold is restricted leading toa U (1)-restricted Tate model. Under this deformation the original gauge flux turns intoflux that can be written as a wedge product. It completely agrees with the constructionoutlined above. At a technical level the construction of the flux differs slightly from oursin that the authors of [33] perform a small resolution as opposed to a blow-up of the SU (2)singularity. When the dust has settled, though, the two resolution procedures turn out tobe completely equivalent. Among the consistency checks performed in [33] is a successfulcomputation of the chiral index in models with SU (2) × U (1) X and Sp (2) × U (1) X gaugesymmetry and a match of the D3-brane tadpole with perturbative results in the type IIBlimit. We perform the construction of U (1) X flux, following our logic spelled out above, forGUT models with gauge group SU (5) × U (1) X . The choice of this gauge group is of coursemotivated by the aim of constructing globally defined F-theory GUT models. The technicalcore of the present paper is the resolution of the SU (2) curve responsible for the U (1) X factor together with the resolution of the SU (5) singularity in a way that gives full access tothe resolved fibre over the matter curves. In this regard, our work has considerable overlapwith the recent analysis in [26] which independently used a blow-up procedure to resolvean SU (5) model, however, without a further U (1) restriction. At a phenomenological level,apart from serving as a welcome selection rule that forbids dimension four proton decay,the U (1) symmetry enhancement leads to a set of GUT singlets with the correct quantumnumbers to be interpreted as right-handed neutrinos. The computation of the chirality ofsuch GUT singlets, which are sensitive to the full details of the global compactification,has largely remained elusive in the ”semi-local” approach to F-theory GUT model buildingvia spectral covers [16, 35].The blow-up procedure, performed in this article, reproduces the toric weights of theresolution divisors as appearing in the toric examples of [16–19, 27]. The main pointof our analysis, however, is to make visible the structure of the fibre above the mattercurves as a prerequisite for computing the chiral index. Along the way, we give an explicitprocedure to derive the U (1) X charges of the matter states on purely geometric grounds.These charges identify U (1) X as the abelian subgroup in the breaking SO (10) → SU (5) × U (1) X . In particular, we identify the − and as states descending from the spinorial Note that [26] focuses on the global realisation of gauge fluxes in the spirit of the so-called spectraldivisor construction [34], which is a different approach to gauge fluxes than the one pursued here. In common abuse of noation, here and in the sequel we say SO (10), but mean ”Spin(10)”. – 5 –epresentation of SO (10) while the Higgs − + c.c. is of the type present also in perturbativemodels. This explains why the
10 10 5 coupling is present in generic F-theory models butnot at the perturbative level in Type IIB orientifolds: While the U (1) selection rulesoperate in exactly the same manner, the charge assignments differ due to the possibility ofmulti-pronged strings states in F-theory.The final result for the chiral index is then extremely simply: The chiral index of astate of U (1) X charge q that is localised along a matter curve C on the base space is q (cid:82) C F .In fact, this result matches the formula derived in the context of the S [ U (4) × U (1) X ]split spectral covers for the special case of zero non-abelian SU (4) bundle part – however,it matches only for matter states charged under SU (5). These are the fields localised onthe GUT brane to which the local philosophy of the spectral cover construction applies.The SU (5) singlets on the curve of SU (2) enhancement away from the GUT braneon the other hand are sensitive to the global details of the compactification. Indeed,the chirality formula derived from the proper G -flux corrects the spectral cover formulaaccordingly. With the explicit resolution at hand it is a simple matter to compute the D3-tadpole (cid:82) ˆ Y G ∧ G and to evaluate the D-term supersymmetry condition for the gaugeflux. Again we find global corrections compared to the spectral cover expressions. Thisdemonstrates that the G -flux can be viewed as a global extension of the split spectralcover fluxes and that the latter cannot be trusted except for the chirality of the SU (5)matter states, which constitutes a truly local observable.In addition to the simple form for the chirality in SU (5) × U (1) X models, we arriveat slightly more involved expressions for the chiral indices in non-restricted SU (5)-models.To this end we partially define G in terms of 4-cycles that cannot be represented as thedual of the intersection of two divisors in the four-fold, as was recently described in [33],where, however, the chirality was not computed. Upon extending this we find that therecombination process implicit in moving away from the U (1)-locus in complex structuremoduli space is nicely reflected in the chirality formula for those curves which are affectedby the recombination. In the other cases, the chiral index takes the same form as in the U (1)-restricted case.The remainder of this paper is organised as follows: In section 2 we review the detailsof the U (1)-restricted Tate model with special emphasis on singularity resolution via blow-up and the appearance of the U (1) X gauge potential. Section 3 is devoted to a detailedresolution of the SU (5) × U (1) X model. We begin in 3.1 by discussing the blow-up procedurefor the combined resolution of singularities associated with the non-abelian and the abelianpart of the gauge group and relate this to the singularity resolution in the framework of toricgeometry. In section 3.2 we outline the general procedure to deduce the fibre structure inco-dimension one, two and three. The details of this computation are collected in appendixA. We proceed in section 3.3 with an in-depth analysis of the fibre structure over the mattercurves. In section 3.4 we study the geometric realisation of the U (1) X gauge symmetryand compute the U (1) X charges of the charged matter fields. The G -flux is the subjectof section 4. After stating the Freed-Witten quantisation condition, whose derivation willbe presented in the upcoming [36], we compute in 4.1 the chiral index of charged matter– 6 –tates including SU (5) singlets, making heavy use of the geometric structure found inthe previous section. Section 4.2 is devoted to the global consistency conditions such asD3-tadpole and D-term supersymmetry. In section 4.3 we compare the G -flux to thesplit spectral cover construction. In section 4.4 we illustrate the use of G gauge fluxesin F-theory compactifications by constructing a three-generation SU (5) × U (1) X modelon a Calabi-Yau four-fold which meets the D3-brane tadpole, the D-term supersymmetryas well as the Freed-Witten quantisation condition. Finally, in section 4.5 we considerthe deformation of the U (1)-restricted model and its G -flux by brane recombination andidentify the deformed chiral flux. Many details of the computations of section 3 and 4 arerelegated to the appendices. Our conclusions are contained in section 5. U (1) -restricted Tate models and their resolution To set the stage, we present in this section the details of the U (1)-restricted Tate modeland describe in detail its resolution via blow-ups. This is important in order to understandthe resulting U (1) gauge symmetry.We consider an F-theory compactification on the elliptically fibred Calabi-Yau four-fold Y described by a Weierstrass model in Tate form. The Tate polynomial cuts out Y as the hypersurface P T = { y + a xyz + a yz = x + a x z + a xz + a z } (2.1)of a P , , bundle over a base space B . As usual ( x, y, z ) denote the homogeneous coordin-ates of the P , , fibre, which can therefore not vanish simultaneously so that xyz lies inthe Stanley-Reisner ideal. The a i depend on the base coordinates in such a way as to formsections of K − iB , powers of the canonical bundle on B . Note that the intersection of P T with the divisor { z = 0 } leads to y = x . Together with the linear relation from P , , this fixes a point in the fibre. Thus { z = 0 } represents a copy of the base B , the uniquesection of the generic Weierstrass model.While we are at it, let us fix some notation: Unless stated otherwise we denote a divisordefined as the vanishing locus of some coordinate or polynomial t by { t = 0 } or sometimesshort { t } . Its homology class in H ( Y ), or H ( B ), will be referred to as T , with Poincar´edual two-form [ T ] ∈ H ( Y ), or H ( B ). Furthermore, the first Chern class of the tangentbundle of B will often be abbreviated as c := c ( B ).The singular fibres of a Weierstrass model Y lie over the discriminant locus ∆ in thebase B , famously known to be given by the vanishing of∆ = 4 f + 27 g (2.2)with f = − (cid:0) b − b (cid:1) , g = (cid:0) b − b b + 216 b (cid:1) , (2.3) b = 4 a + a , b = 2 a + a a , b = 4 a + a . – 7 –s will be exploited in more detail in the next section, specification of the vanishing ordersof the sections a i as prescribed by the Tate algorithm [10] along co-dimension one loci inthe base B engineers non-abelian singularities in the fibre above the respective divisors.Independently of these non-abelian gauge groups along divisors, the presence of a U (1)gauge group not arising as the Cartan of a non-abelian gauge group is associated with asingularity in complex co-dimension two . The simplest type of such models was workedout in [27] and involves setting the section a ≡
0. In this case the U (1)-restricted Tatehypersurface embedded by (2.1) becomes singular at x = y = 0 = a = a . The concreteform of the singularity follows by inspection of the discriminant locus∆ = (cid:16) a a a (cid:2) a + 8 a ˜ a + 16( a − a ) (cid:3) + a a ( a + 36 a ) (2.4) − a (cid:2) a a + 8 a a + 2(8 a + 15 a a ) − a a (cid:3) (2.5)+ a (cid:2) a + 8 a a + 16( a − a ) (cid:3) − a (cid:17) . (2.6)From the vanishing degree of order two one infers a curve of SU (2) singularities located at x = y = 0 in the fibre over the curve C : { a = 0 } ∩ { a = 0 } (2.7)on B .The singularity over C in the four-fold Y must be resolved explicitly. Let us assumefor now that C is the only singularity of Y and thus consider a model without non-abelian gauge groups, reserving the implementation of further non-abelian singularities forthe next section. The discriminant locus is thus a single connected I -locus with a co-dimension two-singularity along its curve of self-intersection C . The probably simplestpossible type of resolution, which is the one applied in [27] and which we will also explorein this article, is by a standard blow-up procedure. In this process one introduces a newhomogeneous blow-up coordinate s along with the proper transform ˜ x, ˜ y of the originalcoordinates x, y , y = ˜ ys, x = ˜ xs. (2.8)Furthermore, one introduces the extra scaling relation(˜ x, ˜ y, s ) (cid:39) ( λ − ˜ x, λ − ˜ y, λs ) , (2.9)which follows by requiring that x and y be unchanged under rescaling s . There are also other types of geometries realising massless U (1)s. Most notably, consider two seven-branes wrapping homologous divisors with vanishing mutual intersection. For seven-branes of the same[p,q]-type, the same S pinches in the elliptic fibre over both seven-branes. Fibreing this S between theseven-branes also gives rise to an element in H , ( Y ) that is associated with the U (1) gauge potentialof the relative U (1), as discussed more recently in [37–39]. This non-generic case is the four-dimensionalanalogue of the situation for F-theory on K
3, where the seven-branes are points on the base of K massive U (1)s as a consequence of certainnon-harmonic two-forms was argued for in [39]. Note that together with the P , , relation for ( x, y, z ) this can also be brought into the form ( z, ˜ y, s ) (cid:39) ( λz, λ ˜ y, λ s ) used in [27]. – 8 –he effect of this blow-up is that, where before the fibre was given by a degree-6polynomial in P , , , it is now given by a degree-(6 , −
1) polynomial in the following space(relabeling ˜ x → x and ˜ y → y ): x y z sZ · S − − · { xy, zs } and the proper transform of the Tate polynomial becomes P T = { y s + a xyzs + a yz = x s + a x z s + a xz } . (2.10)On the new, resolved Calabi-Yau four-fold ˆ Y thus created, { z = 0 } still defines a sectionof the fibre, giving a copy of the base. Furthermore, the divisor { s = 0 } also gives a copyof the base with an additional P over the curve C defined in (2.7). This can be seen byconsidering the restriction of the Tate polynomial to { s = 0 } , which is a y = a x (2.11)after setting z to 1 as sz is in the SR-ideal. Over all base points away from C this fixesthe fibre entirely; however, on this curve one is left with a P parametrised by [ x : y ].In other words, above the curve C , { s = 0 } and the Tate polynomial do not intersecttransversally, whereas they do over every other base point. Thus the singularity is replacedby a P .If the original four-fold Y is realised as a hypersurface or complete intersection of atoric space, one can arrive at the same scaling relations and proper transform followingthe toric algorithm of [14, 15] as applied more recently to the resolution of Calabi-Yaufour-folds in [16–19]. This method, which is described in more detail at the end of 3.1, iscomputationally very powerful and thus particularly well-suited for an efficient treatmentof more complicated models, e.g. in the presence of extra non-abelian singularities. In [27]it was used to resolve U (1)-restricted Tate models describing certain SU (5) × U (1) F-theoryGUT models using the geometries of [16, 17].Finally, we stress that our blow-up procedure differs at a technical level from the smallresolution performed for U (1)-restricted Tate models in the recent work of [33], even thoughthe final results of both approaches match perfectly.The crucial property of the resolved space ˆ Y is that by construction h , ( ˆ Y ) hasincreased by one compared to h , ( Y ) due to the new resolution divisor class S . Thissignals the appearance of a new massless gauge symmetry. Recall that in the language ofF/M-theory duality massless brane U (1) symmetries arise from expansion of the M-theorythree-form C in terms of elements ν i of H , ( ˆ Y ) ‘with one leg along the fibre and one legalong the base B ’. This means that (cid:90) ˆ Y ν i ∧ π ∗ [ D a ] ∧ π ∗ [ D b ] ∧ π ∗ [ D c ] = 0 , (cid:90) ˆ Y ν i ∧ [ Z ] ∧ π ∗ [ D b ] ∧ π ∗ [ D c ] = 0 , (2.12)– 9 –here π ∗ [ D i ] denotes the pull back of the two-form dual to the divisor D i ⊂ B and [ Z ] isthe two-form dual to the section { z = 0 } with divisor class Z . In fact, a natural candidatefor such a two-form is the combination [27] w X = − [ S ] + c + [ Z ] . (2.13)To verify that the first requirement in (2.12) is fulfilled we note that the topologicalintersection numbers of S with three base divisors are the same as those of Z . This isbecause the additional P in S over C does not occur for generic curves in the productof the divisor classes c ( K − B ) and c ( K − B ), but only for the vanishing locus of the specificrepresentatives a = 0 = a . Furthermore, in a Weierstrass model the section Z is knownto satisfy (cid:82) ˆ Y [ Z ] ∧ ([ Z ] + c ) ∧ . . . = 0, which is just right for w X to also satisfy the secondrequirement in (2.12).Note that the construction of w X automatically allows us to write down the G -fluxassociated with the U (1) X . Instead of elaborating on this flux in the simple U (1) model,though, we proceed to an in-depth analysis of SU (5) × U (1) X models. SU (5) × U (1) X Models, resolution and matter curves
In this section we extend the U (1)-restricted model by an SU (5) singularity in the fibre overa divisor { w = 0 } in the base B . The standard procedure to generate an SU (5) singularityover a divisor W is to fix the vanishing orders of the sections a i of the Tate polynomialon { w = 0 } according to Tate’s algorithm [10]. In addition, we must set a ≡
0, since weare interested in the U (1)-restricted version thereof. In summary, the Tate sections arerestricted as a = a , a = a , w, a = a , w , a = a , w . (3.1)The discriminant now takes the form∆ = w (cid:0) P + Q w + R w + S w + T w (cid:1) , (3.2)where P = a a , ( − a a , + a , a , ) ,Q = a ( − a a , − a a , a , a , − a a , + 8 a , a , ) ,R = − a a , a , + 30 a a , a , − a a , a , a , − a a , a , + 16 a , a , ,S = 96 a a , a , − a , a , − a , a , a , + 27 a , ,T = 64 a , . Comparison with [10] confirms the presence of the SU (5)-singularity over w and further-more suggests the following enhancement loci: co-dimension two enhancements occur onthe intersection of { w = 0 } with This is true for models without non-abelian gauge groups. As we will see later, in presence of extranon-abelian singularities this expression receives modifications (see also [33]). In the non-restricted case, we would impose the vanishing behaviour a = a , w . – 10 – { a } , where ∆ vanishes to order 7, indicating SO (10)-enhancement, • { a , } , and • { a a , − a , a , } .In the latter two cases ∆ vanishes to order 6, indicating SU (6)-enhancement. Co-dimensionthree enhancements occur on • { a } ∩ { a , } , • { a } ∩ { a , } , where in both cases ∆ vanishes to order 8, indicating SO (12)- or E -enhancement, and • { a , } ∩ { a , } , where ∆ vanishes to order 7, indicating SU (7)-enhancement.Note that the splitting of the SU (6)-enhancement curve, as well as the appearance of the SU (7)-enhancement point, are features of the U (1)-restricted model that do not occur innon-restricted SU (5)-models. Both are of course intimately related to the presence of a U (1) X gauge symmetry as we will see. We next describe in detail the blow-up procedure to resolve the singularities of the abovemodel. To this aim four exceptional divisors are introduced to take care of the SU (5)singularity in addition to the resolution divisor S from the U (1) restriction. The resolu-tion process can be motivated by the Tate algorithm as is described in section 7 of [10].Restricting ourselves to the I n -branch for the moment and denoting, as before, the divisordefining the GUT surface by { w = 0 } , one can summarise the procedure as follows: Define x = x , y = y on the original, singular manifold. Then at each step of the resolutionprocess, a new variable e i is introduced such that only those monomials with the lowestorder in ( x k , y l , w ) remain in P T | e i =0 for the current k , l . If the remaining polynomialfactorises into either x k ˜ P or y l ˜ P , one defines new coordinates on the blow-up by x k +1 , y l or x k , y l +1 respectively, where x k +1 = x k /w , y l +1 = y l /w . With the new coordinates theprocess is then repeated. The algorithm terminates when P T | e i =0 does not factorise anyfurther.Then the resolution process turns out as follows, where from each line to the next arelabeling is implicit, losing the˜over the resolution coordinates in each case, ( x, y, w ) → (˜ xe , ˜ ye , ˜ we ) ,y → y w ( x, y , w ) → (˜ xe , ˜ y e , ˜ we ) ,x → x w ( x , y , w ) → ( ˜ x e , ˜ y e , ˜ we ) ,y → y w ( x , y , w ) → ( ˜ x e , ˜ y e , ˜ we ) . The order of the labels of the e i is chosen such that their intersection structure coincides with thestandard root intersection structure (see below). – 11 –his may be summarised as( x, y, w ) → (˜ xe e e e , ˜ ye e e e , ˜ we e e e ) . If we were considering a generic SU (5) model, in which a is not set to zero, this wouldbe sufficient. For a = 0, one has to perform the additional resolution required by the U (1)restriction. As before, this amounts to( x, y ) → (˜ xs, ˜ ys ) . The total resolution process for the SU (5) × U (1)-restricted Tate model can thus be sum-marised as ( x, y, w ) → (˜ xse e e e , ˜ yse e e e , e e e e e ) . (3.3)Here ˜ w was relabeled e , motivated by the fact that it now denotes the divisor definingthe remaining P fibred over the GUT surface. Whereas, before the blow-up it defined theentire (singular) torus fibration of that surface. The proper transform of the Tate equationnow becomes y s e e + a x y z s + a , y z e e e = x s e e e + a , x z s e e e + a , x z e e e e . (3.4)Each of the blow-ups induces a new scaling relation by requiring charge invariance ofthe resolution routine. As discussed already for the simple U (1) model of the previoussection, the blow-up ( x, y ) → (˜ xs, ˜ ys ) induces the divisor class S, and both x and y arenot charged under this class as it only appears on the resolved ambient space. Thencharge invariance requires ˜ x and ˜ y to obtain a charge of − x, ˜ y, s ) ∼ ( λ − ˜ x, λ − ˜ y, λs ) is induced. Combining therelations from all five blow-ups one arrives at a structure for the ambient space of the formdisplayed in Table 1. Modulo base triangulations this structure allows for 36 triangulations.It is also possible to arrive at the summarised resolution process (3.3) via differentblow-up routes, such as e.g.:(1) :( x, y, e ) → ( xe , ye , e e ) , ( y, e ) → ( ye , e e ) , ( x, e ) → ( xe , e e ) , ( y, e ) → ( ye , e e ) , ( x, y ) → ( xs, ys ); (2) :( x, y ) → ( xs, ys ) , ( y, s, e ) → ( ye , se , e e ) , ( s, e ) → ( se , e e ) , ( s, e ) → ( se , e e ) , ( s, e ) → ( se , e e ) . (3.5)The induced set of scaling relations will be different in each case; however, each set is alinear combination of each other set. The above choice, motivated by the Tate algorithm,is used here because in this case only x , y , e i and e are charged under each of the E i .While the various possible resolution routes induce equivalent scaling relations, theylead to partially inequivalent triangulations. Each triangulation leads to a different set of– 12 – y z s e e e e e P T W · · · · · · · · · ¯ K · · · · · · · Z · · · · · · S − − · · · · · · − E − − · · · · · − − E − − · · · · · − − E − − · · · · · − − E − − · · · · · − − − − − − v v v v v Table 1 . Divisor classes and coordinates of the ambient space, not including part of the basecoordinates and classes. Here x, y, z, s are coordinates of the “fibre ambient space” of the Calabi-Yau four-fold. Furthermore, within the CY-four-fold, each of the zero loci of the e i consists of oneof the 5 P s fibred over the GUT surface. For completeness the base classes W and ¯ K = [ c ( B )]are included. The bottom of the table is only relevant to torically embedded Calabi-Yau four-folds.It lists a choice for the vectors corresponding to the one-cones of the toric fan. Their relevance isexplained below. coordinates which are not allowed to vanish simultaneously, i.e. a different Stanley-Reisnerideal. As is well-known, these constraints can be deduced e.g. from the requirement thatthe Fayet-Iliopoulos D-terms of the underlying linear sigma model is positive. Then thedifferences between the triangulations become clear by considering the Stanley-Reisnerideal. Its generator set includes the following elements for all triangulations { xy, xe e , xe e , xe , ye e , ye , ye , zs, ze e , ze e , ze , se , se , se , e e } (3.6)along with one of the following 36 options, (cid:40) ye ze (cid:41) ⊗ xe , xe xe , ze ze , ze ⊗ (cid:40) se xe (cid:41) ⊗ e e , e e e e , e e e e , e e . (3.7)One notes however that these only lead to six different Calabi-Yau four-folds: From theproper transform of the Tate polynomial (3.4) and the SR-ideal elements that occur forall triangulations (3.6), it is clear that all elements of the first two columns of the abovelist vanish on the Calaby-Yau four-fold. For example, even if Z may intersect E in theambient space, it never intersects any of the E i on the four-fold. This allows us to fix theelements from those two columns for the future analysis, and we make the canonical choice ze and ze , ze . Let the remaining 6 triangulations be denoted by T ij with i ∈ { , } and j ∈ { , , } , so i runs over the third and j over the last column.– 13 –e conclude this discussion with the following comparison with resolutions in thecontext of toric geometry: If the Calabi-Yau four-fold is embedded in a toric ambientspace, there exists an alternative resolution method, described in [14–17, 19]. In such acase, the generators of the one-dimensional cones of the toric variety form a rational strictlyconvex polytope in d real dimensions, where d is the complex dimension of the toric ambientspace. Monomials of the Tate polynomial (with the a i expanded) then correspond to pointsin the polar dual polytope, the M-lattice polytope (if the Calabi-Yau is a hypersurface) orto points in one element of the nef-partition of the polar dual polytope (if the Calabi-Yauis a complete intersection). Restricting the Tate polynomial coefficients to a i = a i,k w k therefore corresponds toremoving points from this dual polytope and constructing the dual of the remainder. Aswas shown in [15], for the canonical choice of one-cones: x = ( − , , , y = (0 , − , z = (2 , , e = (2 , , v ) (3.8)the above algorithm determines the exceptional variables in the SU (5) case to take theform e = (1 , , v ) , e = (0 , , v ) , e = (1 , , v ) , e = (0 , , v ) . (3.9)Then the scaling relations induced by these one-cones are precisely the ones obtained inthe above-mentioned resolution process motivated by the Tate algorithm, and vice versa.This nicely connects the two algorithms and provides a consistency check for the resolutionprocess.Let us note that the structure of the above resolution ambient space is that of a toricallydescribed ambient fibre fibred over a possibly non-toric base three-fold. We have furtherseen that the blow-up procedure of [10], which holds for general models, produces the sameadditional scaling relations as the alogrithm of [15] for torically embedded models. Thenfor any potential gauge group inducing singularity, one can use Figure 3.2 and Table 3.1of the toric paper [15] to immediately read off the scaling relations for the fibre ambientspace of the resolution, regardless of whether this resolution is toric or not. For the reader’sconvenience the structure of those scaling relations is summarised in Appendix C. P structure With the fibre ambient space of the resolution at hand, the natural next step is to investigatethe fibre itself. This is well-known to be generically a torus, which splits into 5 P ’s overthe GUT surface, and is expected to split into more than 5 P ’s over enhancement curvesand points on said surface. In addition, in the U (1)-restricted model the torus also splitsinto 2 P s over the locus a , = a , = 0. The intersection structure of the GUT- P s isexpected to be that of the extended Dynkin diagram ˜ A associated with SU (5) on theGUT-surface, ˜ A or ˜ D on the enhancement curves and ˜ A , ˜ D or a degenerate versionof ˜ E on the enhancement points. The degeneration of the latter was recently noticedin [25] in the framework of a small resolution process, as opposed to blow-up. Indeed, this The latter only holds if such a partition exists. – 14 –with correct P -multiplicities for the ˜ D i -cases) is precisely the structure one finds in theframework described thus far (and also in the independent [26]).Determining the fibre structure is rather technical and we relegate the computationsto appendix A. Here we merely present the general idea and summarise the results of thisanalysis.First of all the 5 generic P s in the fibre over the GUT surface are given by the fibresof the divisors { e i = 0 } , i ∈ { , . . . , } inside the Calabi-Yau manifold. Put differently, the P s are given by the intersections P i = { e i } ∩ { P T | e i =0 } ∩ { y a } ∩ { y b } , i = 0 , . . . , Here { y a } , { y b } denote divisors corresponding to base coordinates which are neither { e } nor any of the enhancement loci, and we assume that their intersection on the GUT-surface is 1. For general intersection number n , (3.10) defines the formal sum of n P i s.A standard and well-known property of the resolution P s is that they intersect inthe fibre according to the extended Dynkin diagram of the gauge group G , which in thiscase is SU (5). More precisely, the intersection pattern of the P -fibred resolution divisors i = 0 , . . . , rk( G ) is (cid:90) ˆ Y π ∗ γ ∧ [ E i ] ∧ [ E j ] = − C ij (cid:90) W γ ∀ γ ∈ H ( B ) (3.11)with C ij the Cartan matrix of G . We use the sign conventions that C ij has a +2 on thediagonal. Indeed these intersection numbers are derived in detail in the appendix.To see the P -fibre structure above enhancement curves, one notes that P T | e i =0 mayfactorise above certain loci. For example, consider the case i = 1. Generically, the first P denoted by P is given by { e } ∩ { y s e e + a x y z s } ∩ { y a } ∩ { y b } . (3.12)Since ye , se , and ze are in the SR-ideal for all triangulations, this can be simplified to { e } ∩ { e e + a x } ∩ { y a } ∩ { y b } . (3.13)On a (cid:54) = 0 one can further use the SR-ideal elements xe and xe e to express P as { e } ∩ { a x } ∩ { y a } ∩ { y b } . (3.14)On the other hand, on a = 0 P splits into two P s, namely P = { e } ∩ { e } ∩ { a } ∩ { y a } , and P = { e } ∩ { e } ∩ { a } ∩ { y a } . (3.15) In the case of toric Calabi-Yau hypersurfaces, e.g. this is simply the total ambient space; in the caseof CICYs (of the form P T ∩ P B ∩ ... ∩ P B n ) this is the space given by the intersection of the various P B i inside the overall ambient space. – 15 –gain, this is to be understood as a complete intersection on the five-fold. The key todescribing the P s over the curves is to realise that the Tate constraint, which is the secondintersection, may factorise over the higher co-dimension loci.In some triangulations e e is in the SR-ideal and no splitting occurs for P ; however,for all triangulations some of the P i split above the locus a = 0, such that a total of 6different P s appears. Their intersection structure changes to ˜ D , the expected extendedDynkin diagram associated to SO (10). We note that in this framework the correct multipli-cities of the ˜ D -diagram appear, which differs from the analysis in [25]. Similarly, abovethe loci a , = 0 and a , a , − a a , = 0 the P s generically split to form ˜ A -structures.The details of the splitting processes for each curve are collected in A.2.Upon inspection of the enhancement points, one further finds ˜ A -enhancement on a , = a , = 0 and ˜ D -enhancement on a = a , = 0 - again with the correct multiplicities.The appearance of the ˜ A -enhancement is a speciality of the U (1)-restricted model and inagreement with the field theoretic expectations, given the localisation of SU (5) singletsalong the curve a , = 0 = a , .On the locus a = a , = 0 on the other hand, 2 pairs of triangulations ( T i and T i ) lead to (two different) almost- E -structures, where one of the multiplicities is not asexpected from the E -Dynkin diagram, while the remaining pair of triangulations ( T i )leads to the non-Dynkin type structure, which in [25] was named T − , , . Again, the detailscan be found in A.3.As an example the P -structure and splitting process for triangulation T (in thenotation introduced after (3.7)) is depicted in Figure 1. SU (5) matter curves from co-dimension two enhancements Having understood in detail the P -structure of the resolved fibre, we can address thephysical interpretation of the co-dimension two loci as matter surfaces [40] and of theenhancement points as point of Yukawa interactions [2, 4]. For definiteness the follow-ing analysis is carried out for the triangulation T . The remaining cases are covered inappendix A, to which we refer again for most of the technical details.Let us first recall the general picture expected to emerge from well-known argumentsby F/M-theory duality: We start with the fields charged under the non-abelian gauge group G = SU (5), beginning in co-dimension one, i.e. over the surface W on the base B . Thereare two sources for the gauge bosons in the adjoint representation. The Cartan generatorsof the adjoint representation of G are obtained from expansion of the three-form potentialinto the two-forms dual to the resolution divisors E i , i = 1 , . . . , rk( G ). In addition, M2-branes wrapping the P ’s of the degenerated fibre can join in all possible ways to form,together with the opposite orientation, the complete set of roots of the Lie algebra. In thispicture, the M2-branes wrapping a single P i are the simple roots α i of the Lie algebra of G . On co-dimension two loci the singularity enhances further to ˜ G = SO (10) or SU (6).Along these curves some of the P ’s in the fibre split and fuse to form new P ’s with T.W. thanks Thomas Grimm for pointing this out. – 16 –
A 01 2 3 401 2 3G 3H 44D 14 24 2B3C 01 2E 3x 3F 401 2E 3x 3L 43s0A4D 14 24 3x3K
10 5 5 C C H C m
10 10 5 5 5 1GUT surface
Figure 1 . The P -structure and splitting process for triangulation T . The dashed lines encirclingone or several P s in the 2nd and 3rd row correspond to the ones of the top diagram, each identifiedby their colour. Those P s which are marked by a double index with two numbers always havemultiplicity 2, except P in the
10 10 5 -diagram, which has multiplicity 3. All other P s havemultiplicity 1. a different intersection pattern. While M2-branes wrapping the ‘original’ P ’s are stillpresent, there are extra massless states from M2-branes wrapping those new combinationsof P ’s. Again, these states include the adjoint representation of G . The additional M2-branes wrapping a split P can join with the M2-branes wrapping the roots to make upfurther representations of G .In co-dimension three there arises yet another enhancement of the singularity in thefibre, and hence, in the resolved manifold additional spheres over these points. Furthersplittings and fusions occur such that we obtain even more states and, therefore, extrarepresentations at these loci. These points are at the intersection of two enhancementcurves. According to the previous argument, the representations before the enhancementalways have to be included in the representation at the enhancement. Hence, at these pointswe have the representations of both curves. The factorisation of spheres gives us a splittingof states into different representations. Put differently, at these points M2-branes of twopossibly different representations can join and form a state in another representation ofthe group G . These gives us the Yukawa couplings to matter localised at the enhancementcurves.After these general remarks, we turn in greater detail to the matter representations inco-dimension two and exemplify how the above picture is realised. In our notation C R is the– 17 –atter surface in the four-fold ˆ Y associated with representation R of G . The projectionof C R to the base B is denoted by C R . For example, for R = of SU (5), it turns outthat C = { w = 0 } ∩ { a = 0 } on B . The representation R is characterised by its highestweight vector (cid:126)β R . Its descendants (cid:126)β kR , k = 2 , . . . , dim( R ), are obtained by acting with theroot vectors. Each of the dim( R ) components of the representation R corresponds to asurface C kR given by fibreing suitable combinations of P s over C R . The associated physicalstate is described by a M2-brane wrapping the fibre of C kR . From the above we see that C R splits into various components of this representation.To see what kind of new representations appear it is most convenient to calculate theCartan charges of the new states and compare them with the charges as given in weighttables of the representations of G , e.g. [41]. Note that this very general procedure has beenapplied in various places in the M-theory literature, in particular also in the recent [26].Since the gauge bosons in the Cartan of G corresponding to Cartan generator H i areassociated with the two-forms dual to the resolution divisors E i , the Cartan charges ofa state are obtained by integrating these two-forms over the two-cycle wrapped by theM2-brane. Indeed the integrals become the intersection numbers of the curve associatedwith the state and the E i ’s.Let us therefore intersect the new curves P αβ that accrue on the enhancement lociwith the divisors E i . To do so, we use the fact that the six-form | T | , dual to the genericelliptic curve, is a form entirely on B , | T | · E i = 0 , (3.16)and that the sum of the split curves has to add up to the class of the original one, | P i | = (cid:88) α, β m αβ | P αβ | and in particular | T | = (cid:88) i =0 | P i | . (3.17)Here m αβ is the multiplicity of the split components and | C | is the class to the curve C . With (3.16) and (3.17) we can formulate all products | P αβ | · E i in terms of effectiveintersections which we can read off from the equations in appendix A. As an example, letus consider P in the notation of (3.10) and its splitting on the -curve. First of all, fromthe intersection structure derived in Appendix A.1, we obtain | P | · ( E , E , E , E ) = (1 , , ,
1) (3.18)for the intersections of P . Together with (3.16) and (3.17), this gives the Cartan charges | P | · ( E , E , E , E ) = (0 , , , − , (3.19)which corresponds to the root α of SU (5). Similarly, we may obtain the Cartan chargesof P , P , P which represent, in the obvious way, the other simple roots of SU (5). Fromappendix A.2, we see that, for the triangulation we are using here, P splits into P , P and P D . Furthermore, from tables A.16 and A.18 we observe that P has the sameCartan charges as P . To calculate the Cartan charges of P , we use | P | = | P | + | P B | . (3.20)– 18 –he Cartan charges of P B are obtained in the same way as those of P . With (3.17), wethen not only identify the Cartan charges of P but also those of P D , | P || P D | (cid:41) · ( E , E , E , E ) = (cid:40) (1 , − , , − , , , − . (3.21)By comparing these vectors with the tables of the irreducible representations of SU (5)as listed e.g. in [41], we find that M2-branes wrapping P =: − P C , P D =: P C or P B =: P C (3.22)are states of the -representation. Indeed, it is possible to identify all of the mattersurfaces C k , the result of which is listed in table A.19. Finally, the same type of analysisapplied to the two -matter curves in appendix A.2 reveals the matter surfaces associatedwith the fundamental representations, see tables A.27 and A.33. U (1) X generator, matter charges and SU (5) singlets Let us now address the extra U (1) gauge group factor which appears in the U (1)-restrictedmodel. We will generalise the construction of the specific two-form w X of eq. (2.13) leadingto a U (1) X gauge potential A X via C = A X ∧ w X from the pure U (1) X model of section 2.It is sensible to define U (1) X to be orthogonal to the Cartan U (1)s within G = SU (5).To this end let us first recall some elementary group theoretic facts concerning the Cartan U (1)s in a non-abelian gauge group.Since it will turn out that G = SU (5) and the properly defined U (1) X enjoy anembedding into a higher group ˜ G , we phrase this discussion in the language of some non-abelian group ˜ G containing the original G as a subgroup according to ˜ G → G × U (1) X for U (1) X in the Cartan of ˜ G . To describe U (1) X ⊂ ˜ G one specifies a linear combination X = rk( ˜ G ) (cid:88) I =1 t I H I , with H I the Cartan generators of ˜ G. (3.23)Under the decomposition ˜ G → G × U (1) X the irreducible representations R (cid:48) of ˜ G de-composes into a direct sum of irreducible representations R q of G with U (1) X charge q , R (cid:48) = ⊕ q R q . The weight vector (cid:126)β kR (cid:48) of the states of an irreducible representation R (cid:48) can beexpanded in terms of the simple roots of ˜ G , (cid:126)β kR q = [ β kR q ] I α I , I = 1 , . . . , rk( ˜ G ) . (3.24)Under the decomposition of R (cid:48) , we obtain the U (1) X charge q of R q by q = t I C IJ [ β kR q ] J =: t I [ β kR q ] I ∀ k ≤ rk( R q ) . (3.25)Here C IJ is the Cartan matrix of ˜ G and [ β kR q ] I , with indices downstairs, are the ˜ G -Cartancharges of the state corresponding to (cid:126)β kR q .– 19 – SU (2)-curve
10 5 m H Figure 2 . Schematic drawing of the intersection of the generic and degenerate elliptic fibre withthe divisors S and Z in the triangulation T . The green and blue crosses indicate the intersectionpoints of Z and S , respectively, with the fibre. On the SU (2)-curve S itself becomes a P . Notethat enhancement points (Yukawa interactions) are ignored in this picture. To understand how the extra U (1) due to the resolution divisor S fits in, we firstconsider the combination E := S − Z − [ c ( B )] , (3.26)which is the na¨ıve analogue of the two-form defined in (2.13) that described the U (1)generator in the U (1)-restricted model without extra gauge group. In appendix B wederive the intersection numbers of the resolution divisor S and the remaining divisors ofˆ Y . In Figure 2 we further depict the intersections of S and Z with the various P s. Fromthese we find that, again in triangulation T for definiteness, (cid:90) ˆ Y π ∗ γ ∧ [ E i ] ∧ [ E ] = | P i | · E (cid:90) W γ = δ ,i (cid:90) W γ ∀ γ ∈ H ( B ) . (3.27)– 20 –hen only α , the M2-brane wrapping P , obtains a non-zero charge under the additional U (1). More precisely, we can formally extend the set of SU (5) Cartan divisors { E i , i =1 , . . . , } by E into the set { E I , I = 1 , . . . , } and observe the relations, valid for all γ ∈ H ( B ), (cid:90) ˆ Y π ∗ γ ∧ [ E i ] ∧ [ E J ] = | P i | · E J (cid:90) W γ = − − − − (cid:90) W γ (3.28)and (cid:90) ˆ Y π ∗ γ ∧ [ E ] ∧ [ E ] = − (cid:90) B γ ∧ c . (3.29)The matrix in (3.28) is, up to the last row and a sign, the Cartan matrix C IJ of SO (10).The fact that the integral involving 2 factors of E does not localise on the SU (5) surface W is just as expected, as otherwise we would encounter a fifth root α and, therefore,gauge group SO (10) on W , not SU (5). Nonetheless, the hidden appearance of SO (10) ata group theoretic level is now apparent. Indeed, we will see momentarily that the chargesof all matter representations do allow for an interpretation in terms of the embedding of G = SU (5) into ˜ G = SO (10). Note that we have derived this structure exploiting solelythe intersection properties of the resolution divisors computed in detail in the appendix.From group theory it is known that the U (1) X generator X = (cid:88) I =1 t I H I = 4 (cid:88) I =1 C I H I with C IJ C JK = δ IK (3.30)breaks SO (10) to SU (5) by ‘removing’ the simple root α . All the other roots, { α i , i =1 , . . . , } , are uncharged under this U (1) X .From this anew group theory interlude, it is obvious that the divisorW X = − (cid:88) I =1 t I E I with (cid:126)t = (2 , , , , T (3.31)is orthogonal to all the (simple) roots of SU (5), (cid:90) ˆ Y w X ∧ [ E i ] ∧ π ∗ γ = | P i | · W X (cid:90) B γ ∧ [ W ] = 0 ∀ γ ∈ H ( B ) . (3.32)By w X we denote the dual two-form to the divisor W X . Up to an overall factor thatdetermines the normalisation for the U (1) X charges of the matter states, as will be seenmomentarily.Having W X puts us into a position to compute the U (1) X charge of the SU (5) matterstates in the various representations R q . From the described way how the group theoreticproperties are encoded in the geometry, it is clear that the charge q of a state of therepresentation R q can be computed by evaluating | P C kRq | · W X = (cid:88) I =1 (cid:18) | P C kRq | · E I (cid:19) t I = q . (3.33)– 21 –o calculate this charge we need, besides the already known weights [ β kR q ] i , the in-tersection of P C kRq with E . The computation of this intersection proceeds in a manneranalogous to the computation of the intersection of the matter P s with the SU (5) divisors E i , detailed in the previous section. It uses in particular the intersection numbers (3.27).If we add this value as a fifth entry to the Cartan charges of the weight vector we canwrite the U (1) X charge as q R = t I [ β kR q ] I . (3.34)For the representation this procedure leads to q = t I [ β k ] I = − . (3.35)What we should note here is that the extended weights are those of the -representationof SO (10), to be more specific, those of the − part when SO (10) is broken to SU (5) × U (1) X .Repeating the above steps for the two -curves { a , = w = 0 } and { a a , − a , a , = w = 0 } curve, we obtain for the U (1) X -charge q = − q = 2 , (3.36)respectively. Here the − comes again from the -representation of SO (10) and fromits -representation.Note that the states descending from the spinorial representation of SO (10) haveno perturbative analogue. In Type IIB language they arise from multi-pronged [p,q]-stringsas discussed e.g. in [42]. The on the other hand does have a perturbative description interms of fundamental strings. This of course is in agreement with, and moreover explains,the fact that the − − coupling is present in F-theory and not perturbatively inIIB. While, as pointed out in [39, 43], the U (1) selection rules operate in exactly the sameway as in Type IIB the charge of the states may differ because non-perturbative states arepresent.What is still left are the states on the SU (2)-curve, { a , = a , = 0 } . Along this co-dimension two locus the fibre degenerates to two P ’s intersecting each other in two points,see appendix A.4 for further details. Since the SU (2)-curve is normal to the GUT-divisor W , the M2-branes wrapping these spheres will not be charged under the Cartans of the SU (5). However, they have a non-zero weight under E . Using the fact that the genericelliptic fibre is pierced once, both from S and Z , we find( S − Z ) · | T | = ( S − Z ) · ( | P − | + | P | ) = 0 . (3.37)From this relation we can deduce the E -weight of these states. Their total weight vectorsare | P − | · E I = (0 , , , , −
1) and | P | · E I = (0 , , , , . (3.38)This is the missing singlet state plus conjugate of the decomposition of the -representationof SO (10). Consistently, we find that its U (1) X -charge is q = − . (3.39)– 22 –he spectrum of the SU (5) × U (1) X model along with the particle interpretation inthe context of an SU (5) GUT model is summarised in table 3.40. Matter curve C R R q SO (10) origin GUT interpretation { a = w = 0 } − ( Q L , U cR , e cR ) { a , = w = 0 } ( D cR , L ) { a a , − a , a , = w = 0 } + − Higgs { a , = a , = 0 } − N cR (3.40) G -Flux in U (1) -restricted Tate models G -fluxes After this discussion of resolutions and the structure of the matter curves, we are in theposition to approach the construction of a class of chirality inducing gauge fluxes. Whatwe have achieved towards this aim so far is the construction of a two-form w X ∈ H , ( ˆ Y ),explicitlyw X = − t ([ S ] − [ Z ] − c ( B )) − (cid:88) i =1 t i [ E i ] = − (cid:88) I t I [ E I ] , t I = (2 , , , , , (4.1)which satisfies the constraint (2.12). Upon expansion of the M-theory three-form as C = A X ∧ w X this realises a U (1) X gauge potential A X . Now, in view of C = A X ∧ w X + . . . = ⇒ dC = dA X ∧ w X (4.2)it is clear that this construction of a massless U (1) X potential automatically yields a naturalcandidate for the associated gauge flux [32]. The flux is obtained, as usual, by replacingthe three- respectively the four-dimensional field strength dA X by an internal two-form F X ∈ H , ( ˆ Y ), G = F X ∧ w X . (4.3)In fact, the well-known condition for an element of H ( ˆ Y ) to yield a suitable gauge fluxis, similar to (2.12), (cid:90) ˆ Y G ∧ π ∗ [ D a ] ∧ π ∗ [ D b ] = 0 , (cid:90) ˆ Y G ∧ [ Z ] ∧ π ∗ [ D a ] = 0 , (4.4)which is clearly satisfied as long as F X ∈ H , ( B ). In this case also the F-term supersym-metry condition G ∈ H , ( ˆ Y ) holds automatically. The matter states arising in the U (1)-restricted case of [33] can similarly be summarised as states inthe triplet representation of SU (3). However, the charge normalisation of this paper differs by a factor of . – 23 –he two-form F X must be quantised in agreement with the M-theory version [23] ofthe Type IIB Freed-Witten quantisation condition [44], G + 12 c ( ˆ Y ) ∈ H ( ˆ Y , Z ) . (4.5)For an analysis of this constraint in the recent F-theory literature see [45]. To evaluatethis quantisation condition in the case at hand we must compute c ( ˆ Y ) for the resolvedspace and analyse its divisability properties mod 2. The details of this computation will bepresented in [36]. Since we will need it for concrete applications, though, we display thefinal result already here: c ( ˆ Y ) = c ( Y ) + ∆ c , (4.6) c ( Y ) = c ( B ) − c + 12[ Z ] , (4.7)∆ c = − [ W ] w X + 2 [ W ] (cid:8) [ Z ] + [ ¯ K ] − [ S ] + [ E ] + [ E ] (cid:9) − X ] [ E ] (4.8)+2 [ ¯ K ] (cid:8) (cid:0) [ Z ] + [ ¯ K ] − [ S ] (cid:1) − (3 u i + 2 v i − i ) [ E i ] (cid:9) In the above, c ( Y ) is the expression referring to the smooth Weierstrass model and ∆ c represents the corrections, computed in [36], from blow-up of the codimension-one SU (5)and codimension-two SU (2) singularities. The entries of (cid:126)u = (1 , , ,
1) and (cid:126)v = (1 , , , x and y with respectto the resolution divisors E i as displayed in Tabel 1 (and (cid:126) , , , c = [ W ] ∧ w X mod 2, which, combined with c ( Y ) ∈ H ( Y , Z ) [45], yields thequantisation condition π ∗ F X + 12 [ W ] ∈ H ( ˆ Y , Z ) . (4.9)To conclude, the construction of the massless U (1) X via the U (1)-restricted Tate model,as pursued in [27] and in this paper, gives for free a special type of G gauge flux. Thiswas independently realised and worked out in great detail also in the recent [33], whichhas substantial overlap with our work in this regard, even though the logic of the approachand the explicit techniques differ. Note that flux of the type (4.3) is special in that it canbe written as the product of two harmonic forms. We will have more to say about fluxesnot sharing this property in section 4.5.Of course the main motivation to consider gauge fluxes in compactifications is the factthat they can give rise to a chiral spectrum of charged matter modes. According to thegeneral expectation (see e.g. [2, 6]) the chiral index of an N = 1 chiral multiplet localisedon a matter surface should be given by an appropriate integral of G over the correspondinglocus. To the best of our knowledge no derivation from first principles, i.e. involving thephysics of wrapped M2-branes, of this intuitive assertion has been given. However, it ispossible to compare this ansatz with known expressions in dual heterotic or Type IIBsetups. Indeed, this is the route we will follow (see also [33] and, for a similar treatmentof the so-called spectral divisor proposal for gauge fluxes, [26]).Consider a matter multiplet in representation R q localised on the matter surface C R q .In keeping with the notation of the previous section, we denote by (cid:126)β kR the weight vector– 24 –ssociated with R q and by C kR the corresponding component of the matter surface. Fromgeneral arguments [2, 6] the chiral index is given by integrating G over C kR q , where eachvalue of k should give the same result. Since we have the full geometric structure of thematter surfaces C kR at our disposal, we can explicitly compute this quantity and check ifthis is the case. Since the flux splits as in (4.3) the integral reduces to χ ( R q ) = (cid:90) C kRq π ∗ F X ∧ w X = | P C kRq | · W X (cid:90) C Rq ı ∗ F X . (4.10)Now we use the result of (3.33), where we showed that for each value of k the pre-factorjust gives the U (1) X charge q of the state. To summarise, we have found that χ ( R q ) = q (cid:90) C Rq ı ∗ F X . (4.11)This matches precisely the chiral index derived from the Hirzebruch-Riemann-Roch the-orem for matter at the intersection curves of two seven-branes in perturbative Type IIBorientifolds (see e.g. [46] for a discussion and references). As we will discuss in section 4.3it is also consistent with heterotic duality and spectral covers.Note that the final expression (4.11) is of course straightforward to evaluate given aconcrete Calabi-Yau four-fold Y as it only involves an integral of F X over the curve C R inthe base. For the case of the SU (5) × U (1) X model the U (1) X charges and the curves C R of the various matter representations are summarised in (3.40). In this section we describe the global consistency conditions that must be satisfied by the G -flux. In fact, it is these global aspects for which a full understanding of fluxes in termsof explicit four-forms becomes particularly important.Turning on G -flux leads to a contribution in the M2/D3-tadpole condition of theform [22] N D + 12 (cid:90) ˆ Y G ∧ G = χ ( ˆ Y )24 (4.12)with χ ( ˆ Y ) the Euler characteristic of the resolved elliptic four-fold. Given the concreteexpression for G in (4.3) it is a simple matter to compute the flux contribution explicitly,12 (cid:90) ˆ Y G ∧ G = 12 (cid:90) ˆ Y π ∗ F X ∧ π ∗ F X ∧ (cid:16) (cid:88) I t I [ E I ] (cid:17) ∧ (cid:16) (cid:88) J t J [ E J ] (cid:17) . (4.13)This is straightforwardly evaluated with the help of the intersection numbers collected inappendix B, see also (3.28). The result is12 (cid:90) ˆ Y G ∧ G = (cid:90) B F X ∧ F X ∧ (cid:16) Q W ] − ( t ) c ( B ) (cid:17) (4.14)– 25 –ith a group theoretic factor Q = − (cid:88) ( I,J ) (cid:54) =(5 , t J C IJ t J = 30 (4.15)in terms of the SO (10) Cartan matrix C IJ . The numerical result for this case is then12 (cid:90) ˆ Y G ∧ G = (cid:90) B F X ∧ F X ∧ (15 [ W ] − c ( B )) . (4.16)Switching on U (1) X gauge flux of the form (4.3) generates a K¨ahler moduli dependentD-term in the four-dimensional N = 1 effective action. This D-term is rooted in the gaugingof the shift symmetry of some of the N = 1 chiral multiplets corresponding to the K¨ahlermoduli. As a result, the U (1) X symmetry receives a G -flux dependent St¨uckelberg massand only remains as a global symmetry in the low-energy effective action. Both the factthat U (1) X is not present as a massless gauge symmetry and its persistence as a selectionrule broken only by M5-instantons is of course rather important for phenomenologicalapplications.The G -induced gauging, the resulting flux-induced U (1) X mass and the D-term canbe computed very explicitly via F/M-theory duality. Their origin in the eleven-dimensionalsupergravity is the Chern-Simons coupling S CS = (cid:82) C ∧ G ∧ G . As shown in detailin [13, 39], dimensional reduction of the M-theory action on ˆ Y including G flux leads to anaction in 3 dimensions which can be brought into the standard form of three-dimensional N = 2 gauged supergravity. Upon uplifting this three-dimensional action to 4 dimensions,one recovers, amongst other things, precisely the form of a D-term potential as well as theflux induced St¨uckelberg masses.While the reader is referred to [13, 39] for the details of this dimensional reduction,we here sketch the main ideas for completeness. One ingredient in the D-term piece of thethree-dimensional scalar potential is the moduli dependent function T , which in the caseat hand takes the form T = 14 V (cid:90) ˆ Y J ∧ J ∧ G . (4.17)Here J is the K¨ahler form of the Calabi-Yau four-fold ˆ Y and V its volume. The vec-tor multiplet content in 3 dimensions associated with the seven-brane U (1)s follows fromexpansion of the K¨ahler form and M-theory three-form as J = v Λ w Λ + . . . , C = A Λ w Λ + . . . (4.18)with w Λ the set of all resolution two-forms introduced by resolution of the various fibresingularities. The bosonic degrees of freedom of the vector multiplets are ( ξ Λ , A Λ ) uponrescaling ξ Λ = v Λ V [13]. The three-dimensional D-term associated with a specific U (1) Λ with potential A Λ is now given by the expression D Λ = ∂ ξ Λ T | ξ Λ =0 . (4.19) Recent investigations of M5/D3-instantons in this context include [34, 43, 47–51]. – 26 –pplied to the current framework of the SU (5) × U (1) X model with U (1) X flux G weare interested in the D-term associated with the multiplet ( ξ X , A X ). From the above wededuce that the D-term of the three-dimensional supergravity is given by V (cid:82) ˆ Y w X ∧ J ∧ G .This form of the D-term for U (1) X flux was already anticipated in [27].To evaluate the appearing integral one performs the same type of computation thatleads to the expression for (cid:82) ˆ Y G ∧ G . Furthermore one still has to uplift the D-term tothe four-dimensional effective action as corresponding to the F-theory limit. This requiresrescaling the linear multiplets describing the K¨ahler moduli as detailed in [13]. The effectof this rescaling is to replace the prefactor V by V B , where V B is the volume of the basespace B . Taking this into account and combining it with (4.16) the result for the D-termof the four-dimensional effective action is D X = − V B (cid:90) B J ∧ F X ∧ (15 [ W ] − c ( B )) . (4.20)In the presence of N = 1 chiral matter multiplets Φ i charged under U (1) X the fullD-term potential takes the usual form V D (cid:39) (cid:16) (cid:88) i q i | φ i | + D X (cid:17) . (4.21)If we insist on unbroken gauge symmetry the VEV of the charged matter fields must vanishterm by term and the gauge flux must satisfy the D-term supersymmetry condition (cid:90) B J ∧ F X ∧ (15 [ W ] − c ( B )) = 0 . (4.22)This constraint must be met inside the K¨ahler cone. In absence of any mass terms for thematter fields, however, the D-term D X = 0 of course fixes only one linear combinationof matter field VEVs and K¨ahler moduli. Finally note that the SU (5) singlets + c.c. localised at the matter surface C of SU (2) enhancement correspond to recombinationmoduli whose VEV breaks U (1) X without affecting the SU (5) symmetry. We will comeback to this point in section 4.5. It is interesting to compare the globally defined U (1) X G -flux with the bundles constructedin the spectral cover approach [6, 52]. The spectral covers encode the geometry of theneighbourhood of the GUT brane W . It is therefore expected that they correctly capturelocal quantities such as the chiral index of SU (5) charged matter, but might miss certaincorrections that are sensitive to global details of the four-fold away from the GUT brane. Aproposal for a global completion of spectral covers has been made in [53] and, for the caseof non-split spectral covers, further subjected to global tests in [26]. However, since we areinterested here in models with U (1) X symmetry the analogous spectral cover is of the so-called split type [28], [29],[16], generalising the construction of S [ U ( N ) × U (1)] spectral coverbundles from the heterotic string [54–56]. According to the general arguments of [27, 31]such split spectral covers are much more sensitive to the global details of the model andwe do not expect to obtain quantitative match at all levels.– 27 –o be explicit we compare our fluxes to the S [ U (4) × U (1) X ] bundles in the formdescribed in [16]. Such bundles are constructed in terms of a U (4) bundle V and a linebundle L with c ( V ) + c ( L ) = 0. The U (1) X G -flux is analogous only to a special typeof such bundles where the non-abelian data of V - associated with the SU (4) piece of thestructure group - are switched off. In the notation of [16], eq. (85), the corresponding splitspectral cover bundle is achieved by setting λ = 0. The flux is then effectively describedonly by a two-form on W , given in the notation of [16] by the quantity ζ ∈ H , ( W )subject to the quantisation condition, eq. (87),14 ζ + 12 c ( W ) ∈ H ( W, Z ) . (4.23)Our claim is that the U (1) X G -flux is the precise global completion of this type oflocal split spectral cover flux. In particular one must identify ζ/ F X that appears in G = F X ∧ w X . From the start it is clear that the G -flux is more generalas F X need not be an element of H ( W ) but rather of H ( B ). Indeed, inspection of thechirality formulae for the SU (5) charged matter eq. (90), (92), (93) of [16], confirms thatthey precisely match our global result (4.11) if we identify F X and ζ/
4. On the other hand,consider the chiral index of the GUT singlets localised on the curve C away from theGUT brane. In [16], eq. (97), a conjecture was made for the chiral index of these singlets(see also [35]), generalising the arguments that had lead to the chiral index of the GUTmatter, which in our case reads χ ( ) | sp . cover = 5 (cid:82) B ζ ∧ W ∧ c ( W ). In the globally defined U (1) X model, the matter curve C of the singlets is in the class − c ( B ) ∧ c ( B ). Fora general GUT surface W , the two expressions do therefore not match. This does not comeas a surprise, given the local limitations of the split spectral cover.Finally consider the D3-brane tadpole, given for the U (1) X G -flux of this paper by(4.16). The corresponding formula in the split spectral cover, eqn. (100) of [16], is (cid:82) ˆ Y G ∧ G ↔ − (cid:82) B ζ ∧ ζ ∧ W . The reason for the mismatch with the globally correct result(4.16) is that the latter receives contributions away from W given by the second term. Tounderstand this better note that heuristically we can think of c ( B ) as the class of thepart of the discriminant locus that meets the GUT brane in the SO (10) curve a = 0. In the spectral cover approach the spectral cover is viewed as a local deformation of theGUT brane obtained by tilting some of its components in the normal direction. Thismorally describes the component of the I discriminant locus in the neighbourhood ofthe GUT brane. At a computational level the two ”branes” - { w = 0 } and { a = 0 } - are not distinguished properly. Interestingly, if we indeed identified the classes W and c ( B ) in (4.16) we would precisely recover the wrong spectral cover result. Similarlywe find a mismatch in the quantisation condition (4.23) of the spectral cover flux and theresult (4.9) for the global G -flux. Restricted to the GUT surface W , the latter gives[ F X + ( c ( B ) + c ( W ))] | W ∈ H ( W, Z ) by adjunction, which again differs by globalcorrections due to c ( B ). Recall that a is a section of K − B . The same logic almost works - up to an overall factor of 3 - for comparison of the chiral indices for theGUT singlets. – 28 –n conclusion we have demonstrated that the U (1) X gauge flux is perfectly consistentwith the interpretation of a proper globalisation of the split spectral cover flux obtainedby setting λ = 0. It agrees, where it should, with the spectral cover results and correctsthese where the latter are no longer applicable. As an illustration we now present an F-theory SU (5) × U (1) X model based on a well-definedCalabi-Yau four-fold and incorporate the G -flux analysed in this paper to achieve 3 chiralgenerations of GUT matter. The example geometry we pick is the model of section 4.1.5of [18]. The base B is P [3] blown-up over two curves and one point. The scaling relationsof the homogeneous coordinates on the basis are: y y y y y y y y H H H H c ( B )] = 2 H + H + 2 H + H . The intersection form is I = 2 H H H + H + H H − H + H H − H H − H H − H H − H − H H + 2 H H H − H H . (4.25)We take w ≡ y as the GUT coordinate and enforce the SU (5) × U (1) X restrictedTate model. This choice is motivated by the fact that the brane { y = 0 } is a del Pezzo 4surface, which makes it suitable for GUT breaking via hypercharge flux as in [4, 5]. Sincethe four-fold is realised as a complete intersection in a toric space, we can perform allcomputations directly in the framework of toric geometry. In particular, this allows usto compute the Euler characteristic of the resolved ˆ Y as χ ( ˆ Y )24 = 452 . (4.26)In order to construct a well-defined G -flux we must satisfy the quantisation condition(4.9). Consistently, this results in a half-integer number for the flux induced three-tadpole (cid:82) ˆ Y G ∧ G , which, together with (4.26), guarantees an integer number N D of D3-branesaccording to (4.12).It is now a simple matter to search for three-generation solutions. For example, theflux choice F X = 12 ([ H ] + [ H ] + 10 [ H ] − H ]) (4.27)results in the desirable values χ ( ) = − (cid:90) C F X = − , χ ( m ) = 3 (cid:90) C m F X = − , χ ( H ) = 2 (cid:90) C H F X = 0 . (4.28) The unresolved ambient space of the four-fold is given in (146) of [18]. Note that we are counting the chiral index of m as opposed to m . – 29 –he number of SU (5) singlets comes out rather large, 5 (cid:82) C F X = − m H . Clearly a detailed phenomenologicalinvestigation of this toy model is not what we are aiming for here. Rather we do note thatthe D3-brane tadpole induced by the gauge flux is12 (cid:90) ˆ Y (w X ∧ F X ) ∧ (w X ∧ F X ) = 152 . (4.29)The D3-brane tadpole cancellation condition can thus well be satisfied with our flux withoutthe need for anti-D3 branes. Importantly, the flux induced D-term has a solution D X = 0inside the K¨ahler cone. Explicitly if we expand the K¨ahler form in terms of the K¨ahlercone generators as J = (cid:80) i r i v with r i ∈ R + and v = H + H , v = H + H + H , v = H + H ,v = H + H + H , v = 2 H + 2 H − H , (4.30)the D-term condition D X (cid:39) r + 190 r + 75 r − r = 0 (4.31)can be solved on a co-dimension one locus inside the K¨ahler cone, as required.Thus we have found a globally consistent three-generation SU (5) GUT model withsupersymmetric flux for which three-brane tadpole cancellation can be achieved by intro-ducing an integer number of D3-branes. More refined model building involving globallydefined G -fluxes along these lines is left for future work. SU (5) -models The G -flux considered so far is special in the following sense: For a Calabi-Yau four-fold H , ( ˆ Y ) splits into the so-called vertical and horizontal subspaces H , ( ˆ Y ) and H (2 , ( ˆ Y ) [57]. The difference is that only an element of H , ( ˆ Y ) can be written asthe wedge product of two two-forms. As noted several times, our U (1) X flux in the re-stricted Tate model is precisely of this form. However, it is related to a chirality-inducingflux not sharing this property upon brane recombination. To see this we recall that alsothe U (1)-restricted Tate model itself is a rather special construction as it involves settingthe Tate section a = 0 such as to create a curve of SU (2) singularities along C . Themassless charged matter states in representation + c.c. localised on C are precisely themassless recombination moduli which appear as the U (1) X symmetry is unhiggsed. Whilethis mechanism operates for the most general F-theory models, it has a particularly intuit-ive interpretation in models with a Type IIB orientifold limit as discussed in detail in [27]:In this Sen limit the U (1) restriction leads to a split of a single brane brane invariant underthe involution (and thus of the type of a Whitney umbrella [58]) into a brane-image branepair in the same homology class of the double cover X of the F-theory base B . The SU (2) curve C is the intersection locus of brane and image-brane not contained in the– 30 –rientifold plane and indeed hosts the massless recombination moduli charged under thecombination U (1) − U (1) (cid:48) of the brane-image pair.Now, Higgsing the U (1) X symmetry by moving away from the locus a = 0 does notremove the gauge flux completely, nor does it destroy the chirality. Rather, the deformationresults in an SU (5) model with G -flux that cannot be written as F X ∧ w X any longer -after all the harmonic two-form w X exists only for a = 0. However, from a field theoreticperspective it is expected that under this deformation the chiral index for the respresent-ation along the curve { a = 0 = w } is unchanged. The two curves hosting and,respectively, − in the U (1)-restricted case, on the other hand, join into a single objectas a (cid:54) = 0. This corresponds to the fact that as U (1) X is higgsed there is no distinctionbetween both types of matter any more. The chiral index for matter in representation will therefore be the sum of the values for and − .It is possible to describe the G -flux for a (cid:54) = 0 quite explicitly. In fact, in [33] suchtype of flux was the starting point from which the factorisable G -flux for a = 0 wasapproached. Here we treat the problem in the reverse order. As far as the fluxes in thenon-restricted model are concerned, our SU (5) setup covers a situation where the non-factorisable flux does induce a chiral spectrum, a case which was not considered explicitlyin [33].To begin with, we rewrite the flux in the restricted model by introducing the objectW R , with “R” for remainder,W R = t S + W X = t ( Z + [ c ]) − (cid:88) i =1 t i E i . (4.32)For U (1)-restricted models we have found G to be (partially) given by G = [( − t S + W R ) · F ] in ˆ Y (4.33)= [ P T · ( − t S + W R ) · F ] in X . (4.34)Here X denotes the ambient five-fold of the resolved Calabi-Yau four-fold in the U (1)-restricted case and we rewrite the flux as the class dual to the intersection of the variousdivisor classes S, W R , F . In non-restricted models there is no class S and in particularthere is no second class with the same intersections numbers as Z which could be used toconstruct the flux in a similar way as above. However, considering the divisor { x = 0 } onefinds that P T | x =0 = (cid:8) y ( y + a z ) = a (cid:9) . (4.35)Much in the spirit of [33], this allows us to write a bona fide G -flux provided a factorisesas a = ρ τ . Let us call the four- and five-folds in the case a (cid:54) = 0 ˜ Y and ˜ X . One obtainstwo combinations of divisor classes which can be used to define a four-form in ˜ Y ,[ X · Y · P ] , (4.36)[ X · Y · T ] . (4.37) Clearly nobody would ever confuse the divisor class T with the object appearing in (4.17). – 31 –ticking with P for the moment, a consistent type of G -flux is given by G = [( − t X · Y + P T · W R ) · P ] in ˜ X . (4.38)Comparison with (4.34) shows that the general form of the flux for a (cid:54) = 0 is related to theform of the flux in the U (1)-restricted case by replacing F ↔ P , P T · S ↔ X · Y. (4.39)We stress that unlike for a = 0, the G -flux (4.38) factorises only on the ambient ˜ X butnot on the four-fold ˜ Y .The first of the formal identifications in (4.39) implies that, while in the U (1) restrictedmodel one is free to choose any F ∈ H , ( B ), if a (cid:54) = 0 only 2-forms P with the property0 < P < A can appear. The second identification guarantees that the expression in termsof P for the chiral index of all states except for the recombination modes along C is thesame as the one in terms of F if a = 0. Indeed, since s = 0 is the P pasted into thepoint x = y = 0 in the fibre ambient space (over every base point) , the intersections of S with the various P s after the U (1)-resolution are the same as those of X ∩ Y before theresolution - provided X ∩ Y lies on the Calabi-Yau. The latter is the case on the locus { ρ = 0 } , which explains the further restriction of F = P . More concretely one has (cid:90) X [ P T · S · P · D a · D b ] = (cid:90) ˜ X [ X · Y · P · D a · D b ] . (4.40)where D a , D b (cid:54) = S as S does not exist on ˜ X .In particular consider the P s in the fibre over the various enhancement curves C R ,where R = , H , m . These are given by intersections of the type e i ∩ P T | e i =0 ∩ D R (4.41)where e.g. D = a . Then the integral of the flux part X · Y · P over a P fibred over C R is given by [ e i ∩ P T | e i =0 ∩ D R ] · X · Y · P , (4.42)which is the intersection of 6 divisors on a five-fold. Note that on generic points of the GUTsurface P T | e i =0 becomes redundant as it does not intersect x ∩ y ∩ ρ transversally. However,over the enhancement curves, P T | e i =0 splits in some cases - as analysed in detail in previoussections - and therefore does not always become redundant. Then the flux contributionfrom [ X · Y · P ] may be nonzero only for those P s for which the second defining equationdoes indeed become redundant.To see this more explicitly, consider the P -fibre structure of the non-restricted model.As is summarised in Appendix A.5, this is very similar to the structure of the restrictedmodel: For each of the co-dimension-one and co-dimension-two singular loci, the definingequations for the fibre- P s only change for one of the P s, whilst the others are defined in thesame manner. The P s which differ turn out to be precisely those, which are intersected Upon intersection with P T , the P remains in ˆ Y only over the curve C . – 32 –y the divisor s = 0 in the U (1)-restricted case. Then from the structure of the other P s it is clear that over generic points along the SU (5) locus only P is intersected by x ∩ y ∩ ρ , while over the -curve only P C is intersected (for all other P s , x , y or bothare contained in the list of variables that have to be non-zero). We thus obtain that foreach weight component C k of the matter surface χ ( ) = (cid:90) C k G = − (cid:90) C [ P ] . (4.43)For the analysis of the recombined -curve consider the intersection of ρ with the -curve. Even though the -curve is a single connected object in the generic non-restrictedmodel, this intersection splits into the two (co-dimension three) loci ρ ∩ a , and ρ ∩ a , a , − a a , . One notes that P splits differently above the two loci, as is shown in AppendixA.5. In both cases x ∩ y ∩ ρ intersects only one P , namely the one which is structurallythe same as the one intersected by s = 0 for restricted models. Then the integral of G over the matter surface C k , for each k corresponding to one of the weights, is the sum ofthe integrals over the two loci, and the resulting chirality formula becomes χ ( ) = (cid:90) C k G = (cid:90) C k A G + (cid:90) C k B G = 2 (cid:90) C A [ P ] − (cid:90) C B [ P ] , (4.44)where C A = { a , = 0 } ∩ { w = 0 } and C B = { a , a , − a a , = 0 } ∩ { w = 0 } . Thisis the geometric incarnation of our previous field theoretic statement that the and the − states of the restricted model pair up once U (1) X is broken.Let us conclude this discussion with some general remarks. As stressed before, for a (cid:54) = 0 the G -flux is an element of H , ( ˜ Y ) hor . . For flux of this type, the D-term potential,which was V (cid:82) ˆ Y w X ∧ J ∧ G in the a = 0-case, now automatically vanishes identicallyand does not put any restrictions any longer on the K¨ahler moduli. This is of course inagreement with the Higgsing of the U (1) X . On the other hand, G -flux in H , ( ˜ Y ) hor . does induce a superpotential of Gukov-Vafa-Witten type (cid:82) ˜ Y Ω ∧ G , whose associated F-term supersymmetry condition famously fixes some of the complex structure moduli of theCalabi-Yau four-fold. This is reflected in the necessity to factorise a = ρτ [33] in order forthe G -flux (4.38) to exist. It is interesting to compare the situation with the descriptionof gauge fluxes in the weak coupling limit corresponding to Type IIB orientifolds on aCalabi-Yau three-fold X . As always the I -locus of the F-theory model splits into theorientifold plane together with a single seven-brane. In models without non-abelian gaugeenhancement, for a = 0 this latter seven-brane has the topology of a brane-image branepair [27], while for a (cid:54) = 0 it corresponds to a single invariant brane of Whitney type [58].This is expected to hold also in more complicated setups. In Type IIB, gauge flux F canin general be decomposed into a sum of two types of fluxes f + f where f ∈ ι ∗ H , ( X ),i.e. it is the pullback of a 2-form on X , and f is dual to 2-cycle of the brane which is– 33 –omologically trivial on X . Only fluxes of type f can induce a chiral spectrum, while onyfluxes of type f can induce a superpotential. Note that for invariant branes, the allowedgauge flux must be anti-invariant under the orientifold involution, F ∈ H , − ( D ). In thiscase the D-term vanishes identically as observed also in F-theory. It is therefore natural tosuspect that chirality inducing G -fluxes of the type (4.38) for a = ρτ (cid:54) = 0 corresponds, inan orientifold limit, to gauge flux F with non-trivial components both of the type f and f . It will be interesting to investigate this more quantitatively in the future. Finally westress that while conceptually very rewarding the generic SU (5)-model without any U (1)restriction is phenomenologically less relevant because of dangerous dimension-four protondecay operators. In this article, we have constructed globally defined gauge fluxes in F-theory compactifica-tions based on the U (1)-restricted Tate model of [27]. While the construction of such fluxesis more general, we have focused on the special case of an SU (5) × U (1) X model. These arethe models that have recently appeared in the context of phenomenology inspired F-theoryGUT compactifications. A special restriction of the complex structure moduli of the el-liptically fibred Calabi-Yau four-fold induces a curve of SU (2) singularities in addition tothe SU (5) singularities along the GUT divisor. We have detailed the resolution of this sin-gularity via a blow-up procedure resulting in an extra harmonic two-form. This two-formis used to constructed the form w X . The G -flux is then simply G = π ∗ F X ∧ w X with F X ∈ H , ( B ). Such fluxes had been proposed already in [32] and were also consideredindependently in the recent [33]. The technical core of the present work is an analysis ofthe resolved fibres over the matter curves and the Yukawa points, taking into account boththe SU (5) singularity resolution and its interplay with the resolution divisor associatedwith the U (1) X gauge factor. Our results precisely match the toric resolution techniquesapplied to elliptic four-folds in [16–19]. In particular, we derive from first principles the U (1) X charges of the various matter fields and identify these charges as consistent withthe branching SO (10) → SU (5) × U (1) X . Part of the matter arises from the spinor andthe remaining part from the vector representation of SO (10). Along the way we confirmthe general structure of enhancements which had been found — albeit with different meth-ods from ours and for the case of the non-restricted SU(5) model without extra U (1) X symmetry — recently in [25] and [26].As a consequence of the geometric structure of the matter surfaces, we derive thesimple and intuitive formula χ ( R q ) = q (cid:82) C Rq F X for the chiral index of states of U (1) X charge q along the curve C R q on the base. This formula holds, in particular, for the chiralindex of SU (5) singlets. As these are not localised on the GUT brane, they are especiallysensitive to the global details of the four-fold. Further, important quantities that canonly be determined reliably in a global context are the flux induced D3-brane tadpole, theSt¨uckelberg mass for the U (1) X gauge boson and the D-term supersymmetry condition onthe G -flux, each of which we discuss in detail.– 34 –he technology of the U (1)-restricted Tate model and G -fluxes of the type discussedhere and, independently, in [33] paves the way for truly global F-theory model buildingfree of any assumptions concerning the validity of a spectral cover approach. In fact, wehave shown that the U (1) X fluxes of this paper represent the global extension of a certaintype of split spectral cover fluxes. The expressions for the chiral index for SU (5) GUTmatter agree in both cases, while those for the index of the GUT singlets, the D3-branetadpole and the D-term receive non-local corrections. Indeed, given an explicit Calabi-Yaufour-fold, e.g. of the type as the models in [16–19], it is a simple matter to search forconsistent G -fluxes that lead to a chiral spectrum. We have exemplified this by providinga fully consistent supersymmetric, tadpole canceling three-generation SU (5) GUT modelbased on a geometry of [19].In providing this model we have anticipated the explicit form, derived in detail inthe upcoming [36], of the flux quantisation condition. In this work we will also derive ananalytic expression for the Euler characteristic of the U (1) X restricted model.Furthermore, we have given a form of the gauge flux for more general, non-restrictedmodels in the spirit of [33], which directly relates to the flux in the U (1)-restricted case.The resulting flux remains to be chiral but cannot be understood as a wedge of two two-forms. It will be interesting to analyse in more detail the relation to Type IIB fluxes inthe future. Acknowledgements
We are grateful to Andreas Braun, Andr´es Collinucci, Thomas Grimm, Arthur Hebecker,Max Kerstan, Eran Palti and Roberto Valandro for discussions. T.W. thanks ThomasGrimm for initial collaboration and for informing us about a related project, and ac-knowledges hospitality of the Max-Planck-Institut f¨ur Physik, M¨unchen. S.K. thanks theKlaus-Tschira-Stiftung for financial support. This work was furthermore supported by theTransregio TR33 ”The Dark Universe”. – 35 –
The P -fibre structure of U (1) -restricted SU (5) -models In this appendix we collect several aspects of the P -fibre structure of U (1)-restricted SU (5)-models such as • the defining equations (partially in inhomogeneous form) of the various P s, • the splitting structure, • the intersection structure of the P s, • the root assignment for each P , • the P -combination for – each state of the - or the -representation on each enhancement curve, – the roots α i and the highest weights µ , µ on the enhancement points.The general idea for how to find the P s is spelled out in section 3.2. To keep this appendixself-contained, however, we reproduce parts of this description here.Away from enhancement loci, the P s in the fibre over the GUT surface are givenby the intersection of the following four divisors inside the five-fold consisting of the fibreambient space fibred over the base, P i = { e i } ∩ { P T | e i =0 } ∩ { y a } ∩ { y b } , i = 0 , . . . , . (A.1)Here { y a } , { y b } denote base divisors which are neither { e } nor any of the enhancement loci,and we assume that their intersection on the GUT-surface is 1. (For general intersectionnumber n , the above defines the formal sum of n P i s.) To keep this appendix as clear aspossible, in the following we leave out the “divisor brackets”, {} , and lose the y k , whichare to be thought of as being implicitly present. Then the generic 5 P s are given inhomogeneous form by P : e ∩ y s e e + a x y z s − x s e e e , P : e ∩ y s e e + a x y z s, P : e ∩ y s e e + a x y z s + a , y z e e e , P : e ∩ a x y z s + a , y z e e e − a , x z s e e e − a , x z e e e e , P : e ∩ a x y z s − x s e e e − a , x z s e e e . Let us proceed by considering P to illustrate some concepts. Since ye , se and ze are in the SR-ideal for all triangulations, the defining equations can be simplified to e ∩ e e + a x. – 36 –n a (cid:54) = 0 it is convenient to further use the SR-ideal elements xe and xe e to re-express P and to collect the variables which were set to 1, e ∩ a x, ( y, z, s, e , e ) = 1 . This is what we call the partially inhomogeneous form. From this one can immediatelyread off that P cannot intersect P or P as those require e or (respectively) e to vanish.This reduces the number of possible intersections one needs to consider when determiningthe intersection structure. The other major property important for the latter aspect is thefact that P i ∩ P j should be given by the intersection of at most 5 divisors inside the five-fold,two of which are { y a } and { y b } . Then in the above notation, if the four polynomials one ob-tains considering two P s do not contain a redundant element, then the two do not intersect.Staying in the above example, on a = 0 P T | e =0 factorises and therefore P splits into P : e ∩ e , ( x, y, z, s ) = 1 , P : e ∩ e , ( x, y, z, s ) = 1 . In some triangulations e e is in the SR-ideal and no splitting occurs for P ; however, forall triangulations some of the P i split above the locus a = 0, such that a total of 6 P sappears. In the later subsections of this appendix, for each enhancement locus, we list all P s occurring in some triangulation, summarise the splitting process for each triangulationand determine the intersection structure in each case.With this information at hand one can calculate the Cartan charge of each of the P s.The Cartan charges are minus the intersection numbers with the divisors E i obtained byfibreing P i over the GUT divisor W . Again, to stay in the above example, for triangulationsin which P exists, one has P → P + P . Then the first Cartan charge for each P is given by the sum of the intersections with P and P . On the other hand, fortriangulations where P → P , it is given by the intersection with P only. The Cartancharge then allows one to find an expression for each of the P s as a linear combination of theroots, α i , and the highest weights of the - and the -representations, µ , µ respectively.To see how the Cartan charges and the weights are related we refer e.g. to [41].In turn this can be used to determine which P s an M2-brane must wrap in order tolead to a state in a certain representation. We therefore list, for each enhancement curve,the root combination for each P and the P -combination for each state of the appropriaterepresentation. In the case of the enhancement points, states of both representations occurand for compactness we only list the P -combinations for the roots, α i , and the highestweights, µ , µ . – 37 – .1 GUT Surface The P s, in partially inhomogeneous form, are given by P : e ∩ e + a z − e , ( x, y, s, e , e ) = 1 , (A.2) P : e ∩ a x, ( y, z, s, e , e ) = 1 , (A.3) P : e ∩ e + a x + a , e , ( y, z, s, e , e ) = 1 , (A.4) P : e ∩ a y s + a , y e − a , s e − a , e e , ( x, z, e , e ) = 1 , (A.5) P : e ∩ a y − e − a , e , ( x, z, s, e , e ) = 1 . (A.6)Their intersection structure is P − P − P − P − P ( − P ) , (A.7)which is the intersection structure of the extended Dynkin diagram ˜ A , as expected for an SU (5)-singularity.The connection between roots and P s is trivial in the generic case: Each P i corres-ponds to − α i . A.2 Enhancements CurvesRepresentation 10 on the Curve { a } On the locus a = 0 several of the P s split and the following P s appear: P : e ∩ e , ( x, y, z, s, e ) = 1 , (A.8) P A : e ∩ e − e , ( x, y, s, e ) = 1 , (A.9) P : e ∩ e , ( x, y, z, s ) = 1 , (A.10) P : e ∩ e , ( x, y, z, s ) = 1 , (A.11) P : e ∩ e , ( x, y, z, s, e ) = 1 , (A.12) P B : e ∩ e + a , e , ( y, z, s, e ) = 1 , (A.13) P C : e ∩ a , y e − a , s e − a , e e e , ( x, z, e ) = 1 , (A.14) P D : e ∩ e + a , e , ( x, z, s, e ) = 1 . (A.15)Depending on the triangulation T ij , i = 1 , , j = 1 , , T i T i T i P P A P A P + P A P P P + P P P P + P B P B P B P P C P + P C P + P + P C P P + P + P D P + P D P D (A.16)– 38 –he intersection structure is P A P D > P / P − P / P < P B P C . (A.17)Note that P / P − P / P is short notation for one of the following three cases: P − P , P − P , or P − P . In particular, P and P never occur in the same triangulation.Then for each triangulation the above gives the structure of the extended Dynkin diagram˜ D associated to SO (10).The root representation of each P becomes, again depending on the triangulation: P i Root ( T i ) Root ( T i ) Root ( T i ) P · · µ − α − α P A α + α + α + α α + α + α + α − µ + α + 2 α + 2 α + α P · µ − α − α − α − α P − α − µ + α + α · P − µ + α + α + α · · P B µ − α − α − α − α − α P C − α − µ + α + α − µ + α + α P D µ − α − α − α µ − α − α − α − α (A.18) The P -combination for the various states of the -representation are: Weight P i − Combination ( T i , T i , T i ) P P A P P P P B P C P D · , · ,
2) (1 , ,
1) ( · , ,
2) (2 , , · ) (2 , · , · ) (1 , ,
1) (1 , ,
1) (1 , , µ ( · , · ,
2) (1 , ,
1) ( · , ,
1) (1 , , · ) (0 , · , · ) (0 , ,
0) (0 , ,
0) (1 , , µ − α ( · , · ,
2) (1 , ,
1) ( · , ,
1) (1 , , · ) (1 , · , · ) (1 , ,
1) (0 , ,
0) (1 , , µ − α − α ( · , · ,
2) (1 , ,
1) ( · , ,
2) (2 , , · ) (1 , · , · ) (1 , ,
1) (0 , ,
0) (1 , , µ − α − α ( · , · ,
1) (1 , ,
0) ( · , ,
0) (1 , , · ) (1 , · , · ) (1 , ,
0) (1 , ,
0) (1 , , µ − α − α − α ( · , · ,
1) (1 , ,
0) ( · , ,
1) (2 , , · ) (1 , · , · ) (1 , ,
0) (1 , ,
0) (1 , , µ − α − α − α ( · , · ,
1) (0 , ,
0) ( · , ,
0) (0 , , · ) (0 , · , · ) (0 , ,
0) (0 , ,
0) (1 , , µ − α − α − α ( · , · ,
1) (0 , ,
0) ( · , ,
1) (0 , , · ) (0 , · , · ) (1 , ,
1) (0 , ,
0) (0 , , µ − α − α − α − α ( · , · ,
1) (0 , ,
0) ( · , ,
1) (1 , , · ) (0 , · , · ) (0 , ,
0) (0 , ,
0) (1 , , µ − α − α − α − α ( · , · ,
1) (0 , ,
0) ( · , ,
1) (1 , , · ) (1 , · , · ) (1 , ,
1) (0 , ,
0) (1 , , µ − α − α − α − α ( · , · ,
2) (0 , ,
0) ( · , ,
0) (1 , , · ) (1 , · , · ) (1 , ,
1) (1 , ,
1) (1 , ,
1) (A.19)
Note that the overall torus has Cartan charge 0 so the first line gives the multiplicities ofthe various P s for each triangulation. – 39 – epresentation 5 on the Curve { a , } On a , = 0, the following additional P s appear: P s : e ∩ s, ( y, z, e , e , e ) = 1 , (A.20) P E : e ∩ e + a x, ( y, z, e , e ) = 1 , (A.21) P x : e ∩ x, ( y, z, e , e , e ) = 1 , (A.22) P F : e ∩ a y s − a , x s e − a , e e , ( z, e , e ) = 1 . (A.23)The splitting procedure becomes, depending on the triangulation:Original T j T j P P E P s + P E P P x + P F P F (A.24)The intersection structure is P − P − P E − P s / P x − P F − P ( − P ) , (A.25)which for each triangulation gives the structure of the extended Dynkin diagram ˜ A asso-ciated with SU (6).The root representation of each P becomes, again depending on the triangulation: P i Roots for T j Roots for T j P s · µ − α − α P E − α − µ + α P x − µ + α + α · P F µ − α − α − α − α (A.26)The P -combination for the various states of the -representation are: Weight P i -Combination ( T j , T j ) P P P s P E P x P F P ,
1) (1 ,
1) ( · ,
1) (1 ,
1) (1 , · ) (1 ,
1) (1 , µ (1 ,
1) (0 ,
0) ( · ,
1) (0 ,
0) (0 , · ) (1 ,
1) (1 , µ − α (1 ,
1) (1 ,
1) ( · ,
1) (0 ,
0) (0 , · ) (1 ,
1) (1 , µ − α − α (1 ,
0) (1 ,
0) ( · ,
1) (1 ,
0) (0 , · ) (1 ,
0) (1 , µ − α − α − α (0 ,
0) (0 ,
0) ( · ,
1) (0 ,
0) (0 , · ) (1 ,
1) (0 , µ − α − α − α − α (0 ,
0) (0 ,
0) ( · ,
1) (0 ,
0) (0 , · ) (1 ,
1) (1 ,
1) (A.27)
Representation 5 − on the Curve { a , a , − a a , } On a , a , − a a , , the following additional P s occur: P G : e ∩ a x + a , , ( z, s, e , e , e ) = 1 . (A.28) P H : e ∩ a y − a , s, ( x, z, e , e , e ) = 1 . (A.29)– 40 –he splitting process is simply the following:Original T ij P P G + P H (A.30)The intersection structure is P − P − P − P G − P H − P ( − P ) , (A.31)which is the structure of the extended Dynkin diagram ˜ A associated with SU (6).The root representation of each P becomes P i Roots for T ij P G − µ + α + α P H µ − α − α − α (A.32)The P -combinations for the various states of the -representation are: Weight P i -Combination P P P P G P H P µ µ − α µ − α − α µ − α − α − α µ − α − α − α − α A.3 Enhancements PointsYukawa Coupling 10 10 5 on a ∩ a , On a = a , = 0, the following P appear in addition to those appearing on a = 0 P : e ∩ e , ( x, z, s ) = 1 , (A.34) P J : e ∩ a , y − a , , ( x, z, e , e , e ) = 1 . (A.35)Now there are two splitting processes: One starting from the SO (10)-curve on a = 0, theother starting from the SU (5)-curve on a , a , − a a , . For the first of these, the splittingis relatively simple:Original T i T i T i P C P + P J P + P J P + P + P J P D P + P P P (A.36)– 41 –ith all other P s invariant. The second splitting process is slightly more involved anddepends again on the triangulation. In particular, note that P H becomes trivial:Original T i T i T i P P A P A P + P A P P P + P P P P + P B P B P B P G P + P J P + P + P J P + P + P + P J P H · · · P P + 2 P + P P + P P (A.37)The intersection structure becomes P J − P − < P / P − P A P / P − P B , (A.38)where, depending on the triangulation: P − < P / P P / P = T i P | P − P T i P P − | P T i P − P | P These diagrams have the structure of E , T , , , E respectively, i.e. they are not extendedDynkin diagrams and in particular not ˜ E . This nicely reproduces the result of [25], how-ever with the correct multiplicities ( i ∈ { , } ).Nevertheless, it is possible to express the P s in terms of the simple roots and thehighest weights of the - and the representations: P i Root ( T i ) Root ( T i ) Root ( T i ) P · · µ − α − α P A α + α + α + α α + α + α + α − µ + α + 2 α + 2 α + α P · µ − α − α − α − α P − α − µ + α + α · P − µ + α + α + α · · P B µ − α − α − α − α − α P − µ + α + α µ − α − α − α − α P J µ − α − α − α µ − α − α − α µ − α − α − α (A.39) – 42 –he P -combination for the various roots and highest weights take the following form: Root P i -Combination ( T i , T i , T i ) P P A P P P P B P P J · , · ,
3) (1 , ,
1) ( · , ,
2) (2 , , · ) (3 , · , · ) (1 , ,
1) (2 , ,
2) (1 , , µ ( · , · ,
1) (1 , ,
0) ( · , ,
1) (1 , , · ) (1 , · , · ) (0 , ,
1) (1 , ,
1) (0 , , µ ( · , · ,
1) (1 , ,
1) ( · , ,
0) (1 , , · ) (2 , · , · ) (0 , ,
0) (1 , ,
1) (1 , , − α ( · , · ,
1) (1 , ,
1) ( · , ,
0) (0 , , · ) (0 , · , · ) (0 , ,
0) (0 , ,
0) (0 , , − α ( · , · ,
0) (0 , ,
0) ( · , ,
1) (1 , , · ) (0 , · , · ) (0 , ,
0) (0 , ,
0) (0 , , − α ( · , · ,
0) (0 , ,
0) ( · , ,
2) (1 , , · ) (1 , · , · ) (1 , ,
1) (0 , ,
0) (0 , , − α ( · , · ,
2) (0 , ,
0) ( · , ,
1) (0 , , · ) (0 , · , · ) (0 , ,
0) (1 , ,
1) (1 , , − α ( · , · ,
0) (0 , ,
0) ( · , ,
0) (1 , , · ) (2 , · , · ) (0 , ,
0) (1 , ,
1) (0 , ,
0) (A.40)
Yukawa Coupling 10 5 5 on a ∩ a , On a = a , = 0, the only P occurring that has not appeared before is P K : e ∩ a , s + a , e , ( x, z, e , e ) = 1 . (A.41)The splitting process becomes Original T T T P P A P A P + P A P P P + P P P P + P P P P P + P x + P K P + P + P x + P K P + P + P + P x + P K P P + P + P D P + P D P D Original T T T P P A P A P + P A P P P + P P P P + P + P s P + P s P + P s P P + P K P + P + P K P + P + P + P K P P + P + P D P + P D P D (A.42) The intersection structure is P A P D > P / P − P / P − P < P s / P x P K , (A.43)which for each triangulation gives the structure of the extended Dynkin diagram ˜ D asso-ciated with SO (12). – 43 –he root representation of each P becomes, again depending on the triangulation: P i Root ( T i ) Root ( T i ) Root ( T i ) P · · µ − α − α P A α + α + α + α α + α + α + α − µ + α + 2 α + 2 α + α P · µ − α − α − α − α P − α − µ + α + α · P − µ + α + α + α · · P K − µ + α + α − µ + α + α − µ + α + α P D µ − α − α − α µ − α − α − α − α P i Root ( T ) Root ( T ) Root ( T ) P µ − α − α − α − α − α P x − µ + α + α − µ + α + α − µ + α + α P i Root ( T ) Root ( T ) Root ( T ) P µ − α − α − α − µ + α − µ + α P s µ − α − α µ − α − α µ − α − α (A.44) The P -combination for the various roots and highest weights take the following form,where the order of triangulations is ( T , T , T , T , T , T ): µ µ µ P (0 , , , · , · ,
2) ( · , · , , · , · ,
2) ( · , · , , · , · ,
2) ( · , · , , · , · , P A (1 , , , , ,
1) (1 , , , , ,
1) (1 , , , , ,
1) (1 , , , , , P ( · , , , · , ,
2) ( · , , , · , ,
1) ( · , , , · , ,
1) ( · , , , · , , P (2 , , · , , , · ) (1 , , , , ,
0) (1 , , , , ,
0) (1 , , , , , P (2 , · , · , , · , · ) (0 , · , · , , · , · ) (1 , · , · , , · , · ) (1 , · , · , , · , · ) P (2 , , , , ,
2) (0 , , , , ,
0) (1 , , , , ,
1) (1 , , , , , P s ( · , · , · , , ,
1) ( · , · , · , , ,
0) ( · , · , · , , ,
1) ( · , · , · , , , P x (1 , , , · , · , · ) (0 , , , · , · , · ) (1 , , , · , · , · ) (0 , , , · , · , · ) P K (1 , , , , ,
1) (0 , , , , ,
0) (0 , , , , ,
1) (1 , , , , , P D (1 , , , , ,
1) (1 , , , , ,
1) (1 , , , , ,
1) (1 , , , , , − α − α − α − α P ( · , · , , · , · ,
0) ( · , · , , · , · ,
0) ( · , · , , · , · ,
0) ( · , · , , · , · , P A (0 , , , , ,
0) (0 , , , , ,
0) (0 , , , , ,
0) (0 , , , , , P ( · , , , · , ,
1) ( · , , , · , ,
0) ( · , , , · , ,
1) ( · , , , · , , P (1 , , · , , , · ) (0 , , · , , , · ) (0 , , · , , , · ) (1 , , · , , , · ) P (0 , · , · , , · , · ) (1 , · , · , , · , · ) (0 , · , · , , · , · ) (1 , · , · , , · , · ) P (0 , , , , ,
0) (1 , , , , ,
1) (1 , , , , ,
1) (0 , , , , , P s ( · , · , · , , ,
0) ( · , · , · , , ,
1) ( · , · , · , , ,
0) ( · , · , · , , , P x (0 , , , · , · , · ) (0 , , , · , · , · ) (1 , , , · , · , · ) (0 , , , · , · , · ) P K (0 , , , , ,
0) (0 , , , , ,
0) (1 , , , , ,
1) (0 , , , , , P D (0 , , , , ,
0) (0 , , , , ,
0) (0 , , , , ,
0) (1 , , , , ,
1) (A.45) – 44 – ukawa Coupling 5 5 1 on a , ∩ a , Starting from the -locus a = 0, only one additional P appears: P L : e ∩ a y − a , ( x, z, e , e , e ) = 1 (A.46)The splitting procedure becomes: Original T ij P F P s · P L (A.47)The intersection structure is: P − P − P E − P s / P x − P s − P L − P ( − P ) , (A.48)which for each triangulation gives the structure of the extended Dynkin diagram ˜ A asso-ciated to SU (7).The root representation of each P becomes, depending on the triangulation: P i Roots for T j Roots for T j P E − α − µ + α P s · µ − α − α P x − µ + α + α · P s µ − µ + α + α P L µ − α − α − α − α (A.49) The P -combination for the various roots and highest weights take the following form: Root P i − Combination ( T j , T j ) P P P E P s P x P F P ,
1) (1 ,
1) (1 ,
1) ( · ,
1) (1 , · ) (1 ,
1) (1 , µ (1 ,
1) (0 ,
0) (0 ,
0) ( · ,
1) (0 , · ) (1 ,
1) (1 , µ (0 ,
0) (0 ,
0) (0 ,
0) ( · ,
1) (0 , · ) (1 ,
1) (0 , α (1 ,
1) (0 ,
0) (0 ,
0) ( · ,
0) (0 , · ) (0 ,
0) (0 , α (0 ,
0) (1 ,
1) (0 ,
0) ( · ,
0) (0 , · ) (0 ,
0) (0 , α (0 ,
0) (0 ,
0) (1 ,
1) ( · ,
1) (0 , · ) (0 ,
0) (0 , α (0 ,
0) (0 ,
0) (0 ,
0) ( · ,
0) (1 , · ) (1 ,
1) (0 , α (0 ,
0) (0 ,
0) (0 ,
0) ( · ,
0) (0 , · ) (0 ,
0) (1 ,
1) (A.50)
A.4 Generic Structure on C Over the curve C , the Tate polynomial splits again and consequently the P s over thiscurve are given by a , ∩ a , ∩ y a intersected with one of the following two equations s = 0 , ( z = 1) , (A.51) y e e + a x y z = x s e e e + a , x z e e e . (A.52)– 45 –heir intersection structure is P A == P B , (A.53)which is the extended Dynkin diagram ˜ A , associated to SU (2). A.5 Differences to the P -fibre structure for non-restricted SU (5) -models In this section we summarise the differences of the P -fibre structure of non-restrictedmodels to the one described above. In general the P s are very similar as only thosechange for which an additional a , -term appears or for which removing s has any relevance.Further, the elements of the Stanley-Reisner ideal { xy, zs } are replaced by { xyz } , whichmay have an effect on which variables have to be nonzero in the partially inhomogeneousform used above.In particular, in the fibre over generic points along the SU (5) locus as well as overenhancement curves there is only one P which changes in each case. In the following welist those P s that change their structure, restricting ourselves to the fibre over co-dimensionone and -two singular loci, P → e ∩ a y x + a , y e − a , x e − a , x e e − a , e e e , ( z, e , e ) = 1 , P C → e ∩ a , y e − a , x e − a , x e e e − a , e e e , ( z, e ) = 1 , P H → e ∩ a a , y − a , a , x − a a , e , ( z, e , e , e ) = 1 . Here P is the relevant P over generic points along the SU (5) curve and P C occursover the -curve a = 0. Finally, P H lies, in the U (1)-restricted mode, over thelocus a , a , − a a , , which, in the non-restricted case, takes the more general form a , ( a , a , − a a , ) + a a , .For the analysis of the recombined -curve it is also convenient to consider the inter-section of a , with the -curve. This intersection splits into the two loci a , ∩ a , and a , ∩ a , a , − a a , . It is then of interest to consider the splitting of P above these loci.Above a , ∩ a , one obtains P x → e ∩ x, ( z, e , e , e ) = 1 , P F → e ∩ a y − a , x − a , e , ( z, e , e , e ) = 1 , while above a , ∩ a , a , − a a , one obtains P G → e ∩ a x + a , , ( z, e , e , e ) = 1 . P H → e ∩ a a , y − a , a , x, ( z, e , e , e ) = 1 . Of course for all P s, s is removed from the list of variables that are set to 1, if it is present. – 46 – Intersection Properties
B.1 List of intersection numbers
Here we collect some useful intersection numbers involving the resolution divisors of the SU (5) × U (1) X model: (cid:90) ˆ Y D a ∧ D b ∧ E i ∧ E j = − C ij (cid:90) B D a ∧ D b ∧ W, (B.1) (cid:90) ˆ Y D a ∧ D b ∧ D c ∧ E i = 0 , (B.2) (cid:90) ˆ Y D a ∧ D b ∧ Z ∧ E i = δ i (cid:90) B D a ∧ D b ∧ W, (B.3) (cid:90) ˆ Y D a ∧ D b ∧ D c ∧ Z = (cid:90) B D a ∧ D b ∧ D c , (B.4) (cid:90) ˆ Y D a ∧ D b ∧ Z ∧ Z = − (cid:90) B D a ∧ D b ∧ c ( B ) , (B.5) (cid:90) ˆ Y D a ∧ D b ∧ D c ∧ S = (cid:90) B D a ∧ D b ∧ D c , (B.6) (cid:90) ˆ Y D a ∧ D b ∧ Z ∧ S = 0 , (B.7) (cid:88) i (cid:90) ˆ Y D a ∧ D b ∧ S ∧ E i = (cid:90) B D a ∧ D b ∧ W, (B.8) (cid:90) ˆ Y D a ∧ D b ∧ S ∧ E = (cid:90) ˆ Y D a ∧ D b ∧ S ∧ E = 0 , (B.9) (cid:90) ˆ Y D a ∧ D b ∧ S ∧ S = − (cid:90) B D a ∧ D b ∧ c ( B ) . (B.10)Eq. (B.1) is the standard implementation of the intersection structure of the Dynkindiagram in the resolution divisors E i of a non-abelian singularity, with C ij the Cartanmatrix for, in this case, SU (5). Eq. (B.2) follows from the fact that the two-forms dualto the resolution divisors have only ’one leg along the fibre’. The rationale for (B.3) is therelation E + (cid:80) i =1 E i = W with W the pullback of the SU (5) divisor in the base B , alongwith (B.2). This reflects the homological relation (cid:80) i =0 P i = [ T ] for the resolution P inthe fibre, with [ T ] the smooth fibre class. Eq. (B.5) is a consequence of Z ( Z + c ( B )) = 0together with (B.4). The relation (B.6) follows from the observation made after eq. (2.13)that the intersection of S with any 3 divisor classes pulled back from B equals their inter-section with Z . The intersection of the sections Z and S , however, vanishes, as indicatedin (B.7) since { zs } is in the SR-ideal. The intersections (B.8) and (B.9) follow from theconsiderations of the SR ideal, while the last relation, (B.10), follows from considerationsanalogous to those, which lead to (B.5) (see below).– 47 – .2 Derivation of Intersection Properties from the Stanley-Reisner Ideal The above relations can also be explicitly derived from the Stanley-Reisner ideal of theresolution manifold. From the optional elements (cid:40) xe se (cid:41) ⊗ (cid:40) ye ze (cid:41) ⊗ xe , xe xe , ze ze , ze ⊗ e e , e e e e , e e e e , e e one can choose ze and ze , ze from columns two and three - since on the Calabi-Yaumanifold Z never intersects any of the E i , the possibility of them intersecting in the ambientspace is irrelevant to the analysis. Then these three properties along with the ones followingfrom elements of the SR-ideal which appear for all triangulations can be used to derive theabove relations. First of all one obtains Z E = Z E = Z E = Z E = 0 , S E = S E = 0 . (B.11)Furthermore one finds: se : S P = S ( E + E ) ,ye : 3 c E = (1 , , , k E k E ,ye : (3 c − S ) E = (1 , , , k E k E ,xe : (2 c − S ) E = (1 , , , k E k E ,xy : 6 ( Z + c ) − c S + S = − S (5 P − E ) + c (2 , , , k E k , − ( E + 2 E ) (2 E − E ) − E − E E . The first column of the above SR-ideal options leads to one of the following intersectionproperties: se : S E = 0 ,xe : (2 c − S ) E = (1 , , , i E i E . In addition to these properties one trivially finds that the intersection of four basedivisors classes is zero, as is the intersection of three base divisor classes with one of theexceptional classes. Combining these properties and noting that [ P W ] = 6( Z + c ) − S − (2 , , , k E k , one finds[ P W ]( Z − S ) =6( Z + c ) Z − c S + S + S (5 P − E )= − Z + c ) c − c S + 2 S E + c (2 , , , k E k − ( E + 2 E ) (2 E − E ) − E − E E ⇒ [ P W ]( Z − S ) D a D b D c = 0– 48 –howing that S gives the same intersection numbers (with three base divisors) as thesection defined by [ z = 0]. This section adheres to the usual relation, present already innon-resolved, non-restricted models: xyz : ( Z + c ) Z = 0 , ⇒ [ P W ] Z ( Z + c ) = 0 . To find a similar property for S , one first considers the term xys and then substitutesthis into the corresponding expression for S : xys : S [5 c − S − P + E ] = S (cid:2) c − c (12 P − E ) + 2 P ( P − E ) (cid:3) . ⇒ [ P W ] S ( S + c ) = c S (6 c − P + E ) + S (5 c − S − P + E ) . = S (cid:2) c − P c + 3 E c + 2 P − P E (cid:3) . ⇒ [ P W ] S ( S + c ) D a D b = 0 , where D a , D b are proper transforms of base divisor classes. C Fibre Ambient Space
For the reader’s convenience in this appendix we collect some aspects of the fibre ambientspace that arises in the blow-up resolution process described in 3.1. First of all, let us focuson the scaling relation induced by each blow-up. For the new blow-up coordinate e ni thesewill be x y w / e e ni n − a n − b − n a, b ) is an element of the following ordered list:[(2 , , (1 , , (1 , , (0 , , (0 , , ( − , , (0 , − i (C.2)which are the points of the toric diagram for P . Note that e = w / e . The sets of e ni necessary to resolve a particular singularity above the GUT surface are encoded for varioussingularities in table 3.1 in [15], part of which we reproduce in the following (the formatbeing n : i in each entry): SU (5) 1 : 2 , , , SU (6) 1 : 2 , , , , SO (10) 1 : 3 , ,
5; 2 : 1 , SO (12) 1 : 3 , ,
6; 2 : 1 , , E ,
5; 2 : 1 , ,
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