Non-flat elliptic four-folds, three-form cohomology and strongly coupled theories in four dimensions
UUUITP–11/21
Non-flat elliptic four-folds, three-form cohomology andstrongly coupled theories in four dimensions
Paul-Konstantin Oehlmann
Department of Physics and Astronomy, Uppsala University, Regementsvägen 1, 57120Uppsala, Sweden
Abstract
In this note we consider smooth elliptic Calabi-Yau four-folds whose fiber ceases tobe flat over compact Riemann surfaces of genus g in the base. These non-flat fiberscontribute Kähler moduli to the four-fold but also add to the three-form cohomology for g > . In F-/M-theory these sectors are to be interpreted as compactifications of six/fivedimensional N = (1 , superconformal matter theories. The three-form cohomologyleads to additional chiral singlets proportional to the dimension of five dimensionalCoulomb branch of those sectors. We construct explicit examples for E-string theoriesas well as higher rank cases. For the E-string theories we further investigate conifoldtransitions that remove those non-flat fibers. First, we show how non-flat fibers canbe deformed from curves down to isolated points in the base. This removes the chiralsinglet of the three-forms and leads to non-perturbative four-point couplings amongmatter fields which can be understood as remnants of the former E-string. Alternatively,the non-flat fibers can be avoided by performing birational base changes, analogous to6D tensor branches. For compact bases these transitions alternate all Hodge numbersbut leave the Euler number invariant. a r X i v : . [ h e p - t h ] F e b ontents ( E × U (1)) / Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Example II: ( SO (10) × U (1) ) / Z . . . . . . . . . . . . . . . . . . . . . . . . . 18 E × SU (2) × SU (3)
245 Conclusion and Outlook 27A Toric resolution of the ( E × U (1)) / Z model 28 A.1 The resolved Tate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28A.2 Explicit Toric three-and four-fold . . . . . . . . . . . . . . . . . . . . . . . . . 30
B Review: Hodge numbers from polytopes 32 Introduction
In recent times, our understanding of strongly coupled super symmetric theories (possiblycoupled to gravity) in various dimensions has made great progress by means of geometricmethods. Those theories are generically hard to control within the framework of regularquantum field theories. Six dimensional superconformal field theories (SCFT) for exampleare genuine strongly coupled and were believed to contain massless string excitations in thespectrum [18, 32–35]. Evidence for their existence could be made in explicit constructionmade in string theory. The string construction uses of compactification geometries thatprovide certain divisors that are wrapped by branes and lead to massive BPS strings inthe non-compact directions. If those divisors are collapsed to points these strings becometensionless and support the SCFT [56]. For 6D theories with minimal supersymmetry themost flexible tool of choice has been F-theory [49]. F-theory itself geometrizes the typeIIB axio-dilation into the complex structure of an auxiliary elliptic fiber which is put ontop of the physical compactification space. The total geometry becomes that of an ellipticCalabi-Yau(CY) three-fold which features the consistent description e.g. of branes withlarge IIB string coupling. Classifying all non-compact elliptic-threefolds that can be shrunkto a (possibly singular) point has lead to an extensive list of 6D SCFTs [36–38] (see [39] fora review). This direction is also fruitful to further classify 5D SCFTs using M-theory on thesame class of geometries, related to F-theory by a circle compactification [53, 54]. To accessM-theory though the CY geometry must be fully resolved. This can be done by performingbirational base changes until the elliptic fiber admits at most minimal singularities. Suchfiber singularities then admit a crepant resolution according to the Tate-algorithm. This res-olution strategy naturally reflects the 6D tensor branch followed by the 5D coulomb branchupon the circle reduction. However, there is yet another way to resolve the geometry. Thisresolution is crepant and hence Calabi-Yau but does not respect the dimension of the ellipticfiber, called non-flat resolution. In such cases a non-minimal fiber singularity in codimensiontwo can be replaced by surfaces E i of complex dimension two. Shrinking those surfaces backto points reaches the respective 5D SCFT point. This point of view has proven to be veryefficient to characterize 5D SCFTs and its phase structure [15–17, 40]. In addition it alsoallows to study the non-minimal singularities of elliptic three-folds and their F-theory liftsdirectly, without the need to change the base [3, 5, 50]. Extending this geometric approachto 4D is of course highly desirable. Related geometric approaches e.g. those pioneered bySeiberg and Witten [41, 42] have lead to many new insights in N = 2 , , SCFTs and theirconstruction [43–46]. Clearly, one would like to extend this program further to minimalsupersymmetric theories in 4D. The reduced amount of supersymmetry though, does notprotect the moduli spaces from non-perturbative corrections anymore which can obstructthe SCFT point.This was also observed e.g. in [29] where it was the goal to construct non-trivial SCFTsusing F-theory on non-compact elliptic four-folds similar extending the approach of super-conformal matter [37]. Using birational base changes loci of non-minimal singularities incodimension three could be removed which introduced new Kähler moduli. However, theexistence of this SCFT point at the origin of the Kähler moduli space might be obstructed2y the aforementioned quantum corrections. The corrections originate from Euclidean D3instantons that wrap the collapsing cycle and mix Kähler and complex structure modulispace of the geometry.This note wants to take a similar approach to [29] and investigate the (classical) defor-mation space of elliptic four-folds with non-minimal singularities but in codimension two.Opposed to [29] we do not want to focus solely on the resolution phase via a birational basechange but also on the non-flat resolution and (partial) Higgs branches. A major differenceof elliptic four-folds as opposed to three-folds is that they can have non-trivial three-formcohomology. As we will show in the following, these cohomology classes are naturally re-lated to non-flat fibers in codimension two. The dimension of this moduli space is givenby the additional independent Hodge number h , . Due to their absence in three-folds andthe fact that their dimension is self-dual under mirror symmetry [10] makes the three-foldcohomology interesting in their own right. Unfortunately, these contributions have not beengiven much attention in the literature apart from [8, 9]. Moreover, in the application to 4DSCFTs, this contribution yields yet another part in the moduli space of the four-fold whichmight mix with Kähler and complex structure moduli under quantum corrections.The non-flat contributions we are considering in this work appear in codimension twowhich makes them very analogous to the superconformal matter in three-folds. Indeed,one might construct four-folds simply from a three-fold fibered over another P which is aperspective we will large use in this work. Due to this analogy one might wonder weatherit is possible to characterize the geometric moduli space of those non-flat fibrations via the6D/5D SCFT data.Moreover, we want to show that these fibers and hence their non-perturbative contributionscan dynamically be created/destroyed by studying conifold transition between four-folds.These transitions generically change the h , contribution but in fact also all other Hodgenumbers. However we will find that transition related to base changes keep the Euler numberinvariant, which is important in order to connecting 4D F-theory vacua [13, 14].Another class of conifold transitions can be interpreted as a (partial) Higgs branch thatkeeps the gauge group but removes the non-flat fiber from codimension two. This transitiondoes not fully avoid the non-flat fiber though. Instead they are pushed down to codimensionthree and hence correspond to points in the base of the four-fold. Scenarios like those wereconsidered in [4] where they were shown to lead to non-perturbative four-point couplings.These superpotential terms are mediated by D1 string instantons between matter curvesthat meet at the respective points. Those couplings are also present in our cases andinvolve matter curves that are remnants of the original E-string curve. Moreover, sincethe same transition admits an origin from a 6D partial Higgs branch the involved mattercurve representations of the four-point coupling are enforced geometrically and dictated by6D anomaly cancellation.This work is structured as follows: In Section 2 we give an overview of the geometryof elliptic four-folds. We infer the Hodge numbers for non-flat fibrations and link thosein Section 2.1 to the 6D/5D SCFT data. Those contribute additional chiral singlets in afour-fold which we review in Section 2.2. In Section 2.3 we also show how (partial) Higgsbranches of those SCFTs naturally to the prediction of non-perturbative four-point couplings.30 00 h , h , h , h , h , h , Figure 1:
The Hodge half-diamond of a CY four-fold and its non-trivial entries. h , is notindependent but related to the other three entries. In Section 3 we present our two main examples which are families of four-folds with E-stringtheories that exhibit a ( E × U (1)) / Z and ( SO (10) × U (1) ) / Z gauge group. Our maintool to study and construct these compact CY geometries is toric geometry. However wealso expect our arguments to be valid beyond those constructions and for non-compactgeometries as we. In Section 4 we also give examples with higher rank SCFTs that have an E × SU (3) × SU (2) gauge group, to show the validity of our proposal. More details of thetoric resolutions are presented in the Appendices A. One of the main goals of this note is to give a physical explanation to certain Hodge numbersof elliptic Calabi-Yau four-fold X with non-flat fibers in the context of F/M-theory. For thiswe start by reviewing the independent Hodge numbers of a Calabi-Yau four-fold. These aredepicted in the upper half of the Hodge diamond in Figure 1. A four-fold X admits Kählerand complex structure parameters whose dimensions are counted by h , ( X ) and h , ( X ) analogous to those found in three-folds. Unlike three-folds, four-folds can also exhibit non-trivial three-form cohomology counted by h , ( X ) which is self-dual under mirror symmetry.The dimension of h , ( X ) of a four-fold is not independent but related to the other Hodgenumbers as h , = 2(22 + 2 h , + 2 h , − h , ) . (2.1)The Euler number can be computed as χ = 6(8 + h , + h , − h , ) . (2.2)This quantity will be important later when discussing transitions among smooth four-folds.In addition we are interested in elliptic four-folds with non-flat fibers in codimension two.In order to investigate the contribution of non-flat fibers to the Kähler moduli we firstturn to smooth three-folds X . For a smooth elliptic three-fold X we are able to splitup the Kähler moduli contributions into pieces that directly connect with the F-theoryinterpretation. This split is due to the Shioda-Tate-Wazir theorem and given as h , ( X ) = 1 + rank ( M W ( X )) + n fibral + h , ( B ) . (2.3)The first contribution is that of the Mordell-Weil group and the number of n fibral fibraldivisors that resolve some ABCDEFG type singularity. Taken together, both parts count4he rank of the full F-theory gauge group. Second there is the contribution from base divisorsthat contribute the universal hypermultiplet and the 6D tensor multiplets. In [5] it was firstproposed that non-flat fibrations generally require the addition of new divisor classes in thatdecomposition. In practice there can be k = 1 . . . n p points in the two-fold base B wherethe fiber decomposes into E i,k surface components with i = 1 . . . n non-flat ,k [3, 5, 17] as wewill further explain in Section 2.1.Having reviewed the three-fold case, we now turn to four-folds. Inspired from eqn. 2.3we propose a similar type of Shioda-Tate-Wazir decomposition for four-folds given by h , ( X ) = 1 + rank ( M W ( X )) + h , ( B ) + n fibral + h , ( X ) non-flat . (2.4)The general decomposition is analogous to the three-fold case but we explicitly added con-tributions from non-flat fibers. In the case of four-folds these are n c codimension two lociin B and as such irreducible Riemann surfaces C α of genus g α with α = 1 . . . n c . Just asfor three-folds, these curves admit complex two dimensional surfaces E i,α as their fiber. Inanalogy to three-folds these non-flat fibers E i,α can be shrunk to singular curves C α in thefour-fold geometry controlled by a volume modulus of the surface. Hence these surfacescontribute a additional Kähler parameters to those of the four-fold. Note that there canbe additional non-flat degenerations in codimension three on the base B . These types ofnon-flat fibers are rather generic and do not need to contribute to h , ( X ) in the decompo-sition of Equation 2.4. The degenerations in codimension three that we are going to studyhere e.g. in Sections 3 and throughout this paper are relatively mild and indeed do notcontribute to h , ( X ) .In the following we want to deduce how non-flat fibers contribute to h , ( X ) . Non-flatfibers will in general not be the only source to the three-form cohomology but the one wewant to focus on in this work. As we have just extended the non-flat surfaces E i,α to befibered over a curve their total space is a divisor divisors D i,α with E i,α → D i,α ↓ π C α . (2.5)Note that in general, there can be several divisors D i,α labeled with i = 1 . . . n nf ,α thatrestrict to the same α = 1 . . . n c base curves C α . This fibration structure allows to deduce thecohomology of (2.5) from those of fiber and base using the Leray-Hirsch theorem [9,52]. Sincethe fibers E i,α are compact and connected its Hodge numbers are trivial but h , ( E i,α ) = h , ( E i,α ) = 1 . This data is enough to show that the divisors D i,α support a non-trivialone-form cohomology whose dimensions is given by h , ( D i,α ) = h , ( C g α ) · h , ( E i,α ) = g α for i = 1 . . . n α . (2.6)This non-trivial one-form cohomology embeds non-trivially into the four-fold X and is themain source for the three-form cohomology. This fact has been shown in [9, 52] via the socalled Gysin isomorphism ι . For a smooth toric hypersurface X and under the assumptionthat D is toric, the Gysin spectral sequence has been evaluated in [52]. The result is that5hose divisors that admit a non-trivial one-form cohomology essentially inject a non-trivialthree-form cohomology in X of that same dimensions ⊕ i,α H , ( D i,α ) ⊕ i,α ι : −−−→ H , ( X ) . (2.7)Putting the pieces together we deduce the contribution of non-flat fibrations to the three-form cohomology as h , non-flat ( X ) = (cid:88) α g α · n α . (2.8)The Formula (2.8) can be used in general to also construct bases B that exhibit themselvesnon-trivial three-forms h , ( B ) as has been done in [9]. The trick is simply too look forbases that themselves admit divisors with non-trivial one-form cohomology. This however isnot the focus of study for this work. Moreover, although eqn. (2.7) has been proven in theframework of toric hypersurfaces, it is plausible to be of general validity. In the end froman intuitive point we might simply think of those three-forms to come from the sections ofthe genus g surfaces twisted by the surface component.As this work mainly focuses on the contribution of non-flat fibers, we will always considerbases with only trivial fibration structure in 3 and 4. Analogous to three-folds and alsoarguing from a physics perspective in Section 2.1 each of those surface E i,α contributes anadditional Kähler parameter . We summarize the contributions of all non-flat surfaces using(2.8) as h , ( X ) non-flat = n c (cid:88) α n nf,α , h , ( X ) non-flat = n c (cid:88) α g α · n nf ,α . (2.9)We can now turn to implications for the physics of F-theory compactifications. The concreteexamples presented in the sections below all make use of the toric description in terms of theBatyrev construction. For those exists a nice way to compute the Hodge numbers from thecombinatorial data of the polytope ∆ which we review in Appendix B. This allows first toexplicitly construct non-flat four-folds via toric geometry and to explicitly show the validityof the expressions above. The structures of non-flat fibers appear naturally when one considers elliptic fibrations thatexhibit non-minimal singularities at codimension two (or higher). In the Weierstrass modelthis amounts to loci where the functions f, g and discriminant ∆ admits a vanishing orderin the window (4 , , ≤ ord van ( f, g, ∆) < (8 , , . (2.10)In order to discuss those cases, we consider three-folds and the occurrence of those singu-larities in some detail first. In a threefold X these singularities occur over smooth points The exact identification is given again via the Gysin spectral sequence [52] that reads ⊕ i,α H , ( D i,α ) ⊕ i,α ι : −−−−→ H , ( X ) . . In codimension three [29], this window is bounded by the vanishing orders (12 , , from above.
6f the base B and signal the presence of non-perturbative objects that become light in thesix dimensional F-theory. The simplest example are those of E-string theories that exist atthe lowest end of the window (2.10). These theories are typically engineered by collisions of e and su branes in a point of B . In a threefold X , there are three possibilities to removethose singularities and make the geometry smooth :1. Performing a complex structure deformation that removes the singularity. This defor-mation corresponds to the Higgs branch of the 6D strongly coupled theory.2. Blowing up the intersection point(s) in B until the fiber becomes regular. This reso-lution corresponds to the tensor branch of the 6D strongly coupled theory.3. Resolving the fiber in a non-flat way without changing the base. In M-theory thisrepresents a point in the 5D Coulomb branch of the circle reduced 6D SCFT, relatedvia the usual F/M-theory duality.Note that the two different resolutions (2) and (3) should not be thought of largely differenttheories as they collapse down to the same singularity. Instead these resolutions should bethought of as different points in the extended Kähler cone of the resolved threefold X [15–17]related by flop transitions. Hence their Kähler moduli spaces are the same and thus in 5Dthe dimension of the Coulomb branches are the same too. Since the 5D Coulomb branch isthe sum of the 6D tensor branch plus the rank of the gauge algebra factors we can write h , non-flat ( X ) = dim ( Coulomb D ) = dim ( Tensor D ) + rank ( G D ) . (2.11)For the simplest case of an E-string theory the (4 , , collision can be avoided by a singleblow-up in B which yields dim(Tensor D ) = 1 and no gauge symmetry. In the same way thenon-flat resolution can be performed by a single surface over that point with h , non-flat = 1 asexpected. Other non-trivial higher rank examples are discussed in Section 4.Having clarified the geometric and physics implications of non-flat fibrations in three-folds we can move to a four-fold. The main point for geometry is that the non-flat resolutionover a codimension two locus do not depend on whether it happens over points in a three-fold base B or curves in a four-fold base B . The idea is analogous to the resolutionof ADE singularities at codimension one in an elliptic K3 and those at codimension onein an elliptic three-fold. The general resolution procedure is exactly the same and theycontribute with the same amount of fibral divisors in the Shioda-Tate-Wazir decompositioneqn. 2.3. An important difference in a three-fold though is whether a fiber singularity is(semi-)split or non-split. Such effects are caused by some additional monodromy effectsalong the codimension one curve in B . Their presence affects the resolution of the fibersingularities by identifying fibral curves and effectively folds the ADE singularity by anouter automorphism to a non-simply laced algebra. Similar to those, one might expectmonodromies to also be present in four-folds that act non-trivial on the non-flat fiber. Suchcases are not discussed in this work but left for future investigations. In the absence ofsuch monodromies, the number of non-flat resolution surfaces in a four-fold can directly be We assume a resolution of all codimension one
ADE type of singularities here. X . From a physics perspective this makes sense as thetheories might simply be viewed as compactifications of the 6D/(5D) theories on Riemann-surfaces C g α . This physics interpretation allows us to express the contributions of non-flat fibers to h , ( X ) and h , ( X ) in terms of 6D/5D SCFT data from eq. (2.11) and eq. (2.8) as h , non-flat ( X ) = n c (cid:88) α dim ( Coulomb D ) α , (2.12) h , non-flat ( X ) = n c (cid:88) α dim ( Coulomb D ) α g α , (2.13)when summing over the n c curves C α in the base. In Section 3 we consider several examplesof four-folds with E-string curves as well as higher rank examples in Section 4. In this section we review the contribution of three-form cohomology in the F/M-theory,following [1]. In order to do so we consider M-theory on a four-fold X and consider thecontributions that lead to neutral chiral fields in 4D. We first focus on those contributionsthat solely come from the base B and lift to neutral singlets in the 4D F-theory. There arecomplex scalars T κ with β = 1 . . . h , ( B ) + 1 that come from the expansion of the Kählerform J and RR four-form C of IIB string theory. In a basis of ω κ ∈ H , ( B ) we expandthose as J = v κ ω κ , C = B κ ∧ ω κ + . . . . (2.14)Dualizing the two-forms B κ in 4d gives rise to axions ρ κ that combine with v κ to complexifiedKähler moduli T κ . These axions can be gauged and are important for anomaly cancellation.Further chiral singlets originate from complex structure moduli h , ( X ) of the full four-fold.Our main interest though are the chiral singlets that are inherited from h , ( X ) that donot come from the base and in our case are given by h , ( Y ) − h , ( B ) = h , ( X ) non-flat . (2.15)These are obtained from the expansion of the M-theory C -form and are denoted by N β with β = 1 . . . h , ( X ) non-flat . These singlets are unfortunately not very well understood.I.e. it is not clear whether the axion parts σ of the N β couple to curvature terms of theform σF ∧ F and σR ∧ R in the 4D effective action. These singlets though appear in aninteresting way [47] in the Kähler moduli, given as T κ = 12 w κ,β,γ v β v γ + 14 d κ,β,γ ( N + N ) β ( N + N ) γ + iρ κ , (2.16)with w the base intersection form on B and d κ,β,γ a holomorphic function on the com-plex structure moduli space. The curious observation though is, that the N β enjoys of an Cases with monodromy might directly incorporate possible twisted circle reductions [53] in the geometry. N → − N that the other chiral singlet fields do not have.Understanding these fields and their couplings is beyond the scope of this note. Instead thisreview should serve as a motivation of why these singlets are interesting and that non-flatfibrations naturally produce them. Having discussed the role of the resolutions of non-minimal singularities in three-and four-folds in Section 2.1 we now want to consider their deformations. These deformations can beof very general type e.g. they can fully remove all singular fibers resulting in a broken gaugegroup. The specific kind of deformations we want to consider here are those which keep thegauge group G but only create/remove (4 , , loci at codimension two. For concretenesswe fix such a locus to be the vanishing of the ideal I E-string = { z, p } . For this we consider theinverse problem by starting with a three-fold geometry that does not posses any (4 , , yet. This geometry should have some non-Abelian gauge algebra G localized over z = 0 and possibly additional Abelian factors. In general these gauge group factors are expectedto lead to massless hypers in 6D that carry non-trivial representations R i . These matterrepresentations are found at loci where the vanishing order of the discriminant ∆ in theWeierstrass form enhances. Such loci we denote by the vanishing ideals I matter ,i which mightbe very complicated and not necessarily of complete intersection type. For simplicity we fixone of them to be I matter , = { z, p } . At next we perform a complex structure deformation,such that one polynomial factors out one power of z as a → z b , (2.17)where b itself is some other polynomial. For some of the matter loci I matter ,i we now requirethat these are themselves non-trivial in { a, p } which we denote as I matter ,k ( a, z, p ) . Thoseideals we require to become reducible upon the factorization eqn. (2.17) as I matter ,k ( a, z, p ) a → z b −−−→ I matter , ⊕ ˆ I matter ,k . (2.18)The deformation eqn. (2.17) forces matter ideals to be moved onto the I matter , locus. Sincethe loci themselves are identified with matter loci, the Weierstrass coefficients ( f, g, ∆) admita non-trivial vanishing order over them before tuning. The factorization in eqn. (2.18) there-fore increases the vanishing order of the I matter , to the E-string loci I E-string . From the fieldtheory side, such E-string transitions decrease the hypermultiplet sector S and turns theminto non-perturbative E-string sector. The way we have set up those transitions above, mightseem odd and artificial at first glance. However, those transitions are highly constrained bythe 6D anomaly conditions due to the fact that all E-string points should admit a tensorbranch where the field theory anomalies are properly canceled. Hence physics tells us thatthe 6D matter content must change in such a transition and hence the requirement (2.18)is actually very natural to appear. In particular when the gauge group G stays fixed suchtransitions are essentially unique [3, 55]. One obvious constrained is given by the gravita-tional anomaly which fixes the dimensions of representations in S to (cid:80) k dim ( R k ) = 29 asit is the same contribution of a 6D tensor multiplet. Hence the transition can never involve9 representation R with dimension larger than . Consistency of the physics and hencethe geometry allows therefore to deduce the subset of matter ideals I matter ,k in I matter ,i thatadmits the correct factorization properties as dictated by the 6D anomalies.These structures generalize in a straight forward manner to four-folds. Indeed, thevarious polynomials important in the discussion can simply be taken to be sections on thebase B . I.e. the same kind of complex structure deformation creates (4 , , curvesspecified by the very same codimension two ideals that lead to the factorization of mattercurves I matter ,k . However we can use this structure in order to make an additional observationin four-folds: The factorization property (2.18) under the deformation (2.17) guarantees the matter loci I matter , and I matter ,k ( a, z, p ) to all vanish at the codimension three point(s) z = a = p = 0 before that transition. The prescribed deformation simply ensured that thepolynomial a vanishes to first order at z = p = 0 in order to obtain the (4 , , loci. Henceeven when the (4 , , loci are absent at z = p = 0 , it is present at the codimension threelocus a = z = p = 0 . Indeed this effect is ensured by the reducibility of the matter ideals I matter ,k enforced by the 6D E-string transition.As a result we see the (4 , , curve in the four-fold base B was simply deformed to onecodimension lower where all curves of the ideals I matter ,k lie on. Enhanced singularities inthe elliptic fiber at codimension three are generically interpreted as Yukawa couplings inthe 4D superpotential W . In the IIB picture these are induced from intersections of threematter curves in the respective point that is systematically been tracked by the F-theorytorus. However when the fiber becomes of (4 , , type one expects non-perturbativeeffects to be present similar to the E-string theories in 6D. Indeed codimension three pointsof such non-minimal type have been studied in [4] where they were shown to lead to gaugeinvariant four-point couplings in the 4D superpotential. In the IIB picture these couplingsare generated by D1 instanton strings that stretch between the involved matter mattercurves that meet in the (4 , , point.The main point of our geometric construction is, that it allows to interpret those codi-mension three (4 , , points naturally in terms of the 6D E-string transition. As the 6Danomalies have dictated the matter in the transition they also fix the very same curves in4D that meet in the (4 , , point at codimension three. These points and the mattercurves I matter ,k that meet them are therefore naturally interpreted as the remnants of the6D E-string transitions which lead to the non-perturbative four-point coupling of type W (cid:51) (cid:89) i ∈ k R k , (2.19)where representations R k may occur multiple times. E-string transitions that preserve thetotal gauge group have been classified in [3] including the respective change in the 6D matterspectrum. Hence knowledge can be used to infer the induced 4D non-perturbative couplings.A simple example is that of an SU (7) × U (1)
6D gauge theory. Anomalies of the 6D tensorbranch force the following change in the hypermultiplet sector ∆ S = (cid:16) − q ⊕ q ⊕ (cid:17) inan E-string transition. By the arguments above, the very same representations must be If R is (pseudo)real this condition can be relaxed by a factor / . Moreover note that the gauge group G must be subgroup of E the flavor symmetry of the E-string theory. (4 , , point inthe four-fold. Indeed, the lowest order gauge invariant coupling [57] in the superpotentialappears at fourth order as W (cid:51) − q · − q · − q · q . (2.20)In Section 3 geometry and physics of similar examples are discussed in detail. One way to obtain three-folds with non-flat fibers is to perform conifold transitions [3, 5]that originate from those that are flat. We will adopt the very same strategy here for four-folds. Concretely, we want to perform conifold transitions among three types of (compact)four-folds X ,a , X ,b and X ,c as summarized in Figure 2. To stay close to the analogy ofthree-folds, we actually construct the four-folds as three-fold fibrations over another P .This allows us to simply extract the three-fold conifold to four-folds. In this way, we areable to use the 6D anomalies of E-string transitions and to extend those to 4D. Note thatthis is just an auxiliary construction for illustrational purposes i.e. the logic also works forgeneral four-folds.In order to investigate the non-flat fiber structure and its underlying physics, we performconifold transitions that do not change the codimension one and Mordell-Weil structure i.ethe 4D gauge group. We start in X ,a which admits a non-flat fiber in codimension threewhich is of course absent in the analogous three-fold X ,a . We then perform a complexstructure deformation to enhance this non-flat fiber to curves in one codimension higher.Resolving the geometry fully leads to X ,b . These four-folds will exhibit in general a non-trivial three-form cohomology corresponding to the non-flat surface. In this section weconsider E-strings curves only and hence the three-form contribution corresponds directlyto the genus of the base curve. The final transition corresponds to a birational base changethat removes all non-flat fibers. Keeping track of all Hodge numbers of both three andfour-folds allows the computation of the Euler number too. Both in three-and four-folds wewill observe how the first transition changes the Euler numbers while the second one doesnot. Our main example is going to be a ( E × U (1)) / Z gauge theory. Here we focus on thedetails of the singular model and present details of the resolution in Appendix A. Section 3.2concerns similar configurations with gauge group ( SO (10) × U (1) × U (1)) / Z . Besides beingricher in structure which we present more technical details of the resolved geometry directly.All four-folds of consideration are compact and all Hodge numbers are computed via theBatyrev formulas. ( E × U (1)) / Z The starting theory we want to engineer is that of an ( E × U (1)) / Z
6D gauge theory whosespectrum admits massless hypermultiplets in the representations , , . (3.1)11 χ (cid:54) = 0 ∆ χ = 0 (transition 1) (transition 2) T T T X ,a B X ,a P B P B B P (cid:99) B (cid:99) B X ,b X ,c X ,b X ,c Non-flat in codim 2Non-flat in codim 3 Flat
Figure 2:
Summary of a chain of conifold transitions in elliptic four-folds. The startingfour-fold X ,a is constructed from a three-fold X ,a fibered over a P . We then performconifolds in four-folds inherited from the three-folds. Conifold 1 renders X ,b non-flat andconifold two removes the non-flat fiber again by a birational base change (partially). In order to engineer this theory in F-theory we start with the Tate-model , given as Y = X + a XY Z + a X Z + a Y Z + a XZ + a Z . (3.2)The affine coordinates [ Y, X, Z ] describe the elliptic fiber in an P , , ambient space. Thedivisor Z = 0 is the zero-section and a i sections of line bundles of the base, fixed to bepowers in the first Chern class of the base [ a i ] ∈ c ( B ) i . The Weierstrass coefficients caneasily be computed by completing the square and cube in X and Y which allows a mappinginto the simpler Weierstrass form Y = X + f XZ + gZ , (3.3)where Weierstrass coefficients and discriminant is related to the Tate-form as f = 148 ( − ( a + 4 a ) + 24( a a + 2 a )) ,g = 1864 (( a + 4 a ) − a + 4 a )( a a + 2 a ) + 216( a + 4 a )) , ∆ =4 f + 27 g . (3.4)We start by engineering an u gauge factor by setting a = 0 globally which leads to anadditional holomorphic section s of the torus fiber at s : { X ; Y ; Z } = { − a ; 1 } , (3.5)and hence a non-trivial Mordell-Weil group. Matter that is charged under this gauge factoris found at the locus a = a = 0 where the fiber becomes of I type. The e factor is12ngineered over Z : { z = 0 } according to the Tate-classification (e.g. see [20]) by employingthe factorization of the Tate-coefficients { a , a , a , a } → { za , , z a , , z a , , z a , } , (3.6)which can be shown to lead to type IV split fibers and hence an e over z = 0 . Matter inthe representation is found at the locus z = a , = 0 . Up to now we have not specifiedthe dimension of the base B and hence the CY manifold. We start with a three-fold X ,a where the matter loci are just points that are counted by intersecting their divisor classesin the base cohomology. For some generic base this leads to the multiplicities n = 6(1 − g ) + Z , n = [ a , ] · [ a , ] = 34( g − − Z + 12 c . (3.7)where g denotes the genus of the curve Z . The resulting threefold geometry X ,a admitsonly minimal singularities. Hence the geometry X ,a can be fully resolved in a flat mannerwhich we demonstrate in Appendix A. An explicit geometry Y ,a with F base and e placedover the − curve is also given in Appendix A. Using (3.7) one can compute the spectrumwhich admits two -plets, 106 charged- and 163 uncharged singlets. The Hodge numbersof this geometry are given as ( h , , h , ) χ ( Y ,a ) = (10 , − . (3.8)The Kähler parameters are exactly as expected from the Shioda-Tate-Wazir theorem andthe complex structure parameters precisely contribute the amount of neutral singlets neededfor anomaly cancellation.Next we want to tune E-strings along the e gauge group by further performing the factor-ization a , → z b , . (3.9)Counting the degrees of freedom we found this change to come at the cost of ∆ h , ( X ) = (3 c − Z ) Z = 6(1 − g ) + Z , (3.10)complex structures in X ,b . When plugging the factorization into (3.4) one finds that onestill obtains an e type of gauge group over { z = 0 } . However now the former matter locus z = a , = 0 got enhanced from a (3 , , to a non-minimal (4 , , singularity. Hence in X ,b , the -plets got exchanged to E-string theories. Note that the tuning also involved achange in classes of the matter loci. The full charged matter spectrum of the 6D modelis given as n E-string = 6(1 − g ) + Z , n = [ a , ] · [ b , ] = 40( g −
1) + 12 c − Z . (3.11)We summarize the full change in the hypermultiplet sector S in the transition as ∆ S = n E-string · (cid:16) ⊕ ⊕ (cid:17) → n E-string · [ E-string ] . (3.12) The Abelian charges are evaluated in the Appendix A. In order not to induce more singularities in the Weierstrass model, we restrict to bases where at mostthe divisor Z can admit negative self-intersections with Z < − but no other. u − e E-string
Figure 3:
Two ( E × U (1)) / Z geometries and the superconformal transition where an singlet is tuned onto a locus which merges the local su and e enhancement into anon-flat fiber. As already implicitly used in the considerations of anomalies before, a well defined SUGRAtheory is only obtained when we get rid of those n E-string
E-string contributions. This canbe done blowing up the base exactly n E-string times, which brings us to geometry X ,c . Sinceboth theories X ,a and X ,c are consistent SUGRA theories one can analyze the possiblerepresentations that get lost in such an E-string transitions on very general grounds [3].From the change of the 6D anomaly lattice one can derive the change of the spectrum in an ( E × U (1)) Z gauge group which in general must be given by the representations ∆( q + q + ) , q = 9 q , (3.13)as it is the case in the geometry above. Note that the transition requires a neutral singletfor each E-string it produces. This can be directly checked when considering the smoothnon-flat resolution of X ,b . The exact toric resolution is given in Appendix A as well esfor the concrete three-fold Y ,b in detail. A schematic depiction of the fiber properties isshown in Figure 3. There the non-flat surface component is highlighted as a red box, aswell as its reducible components. In the example one observes these reducible componentsto arrange in a closed loop that is attached to other fibral divisors [15–17]. Computing theHodge numbers for Y ,b we find ( h , , h , ) χ ( Y ,b ) = (12 , − . (3.14)These Hodge numbers are consistent with the considerations made earlier. We rememberthat we have created two E-string type non-flat fibers that contribute one Kähler class each.This tuning though came at the cost of one complex structure for each of them. To removethe non-flat fibers/E-string points we can blow-up the − curve which results in a − curve.In the toric setup we demonstrate this in Appendix A for geometry Y ,c and show that itadmits the same Hodge numbers (3.17) as three-fold Y ,b . The four-fold compactification
Having clarified the conifold transitions in the three-fold geometries and their associated F-theory physics, we can perform the same strategy in four-folds and consider their associated4D physics. To keep matters close to 6D we construct the four-fold as a three-fold fibration14ver another P . In general we have lots of freedom to fiber X over the new P . Forsimplicity we chose this fibration to be trivial, which extends to the four-fold base as B = B × P . (3.15)The matter loci of X ,a are readily obtained from X ,a . These codimension two loci becomeRiemann-surfaces whose genus can be computed explicitly as g (cid:48) = 1 + 24(1 − g ) + 5 Z , g (cid:48) = 1 + 216(3( g −
1) + c ) − Z . (3.16)These quantities are expressed in terms of the classes the B inherited classes Z , c andtheir genus g . Again the explicit resolved geometry Y ,a is given in Appendix A for whichwe compute the Hodge numbers ( h , , h , , h , ) χ ( Y ,a ) = (11 , , . (3.17). In comparison to Y ,a there is exactly one more Kähler class of the additional base P .Contributions to h , ( Y ,a ) are trivial, as the fibration is flat and the base with h , ( F × P ) =0 does not contribute either. When moving to the matter sector we find the two pletsin 6D to be localized over the genus g (cid:48) = 5 curve in the B base and similarly the singletsover a curve of genus g (cid:48) = 1893 .Before performing the first conifold transition, we consider interaction terms between thecharged fields present in the 4D superpotential W . A key feature of F-theory which makes italso relevant for phenomenological model building is the presence of Yukawa couplings thatare suppressed in perturbative IIB string theory [21–26]. An example is that of the top quarkin an su theory. This coupling is schematically given by couplings of type · · . InF-theory these points lead to type IV ∗ fiber degenerations. Those degenerations highlightlocal IIB string couplings of order one and hence strongly coupled in nature. Similarly,when one further unhiggses the su to e there are couplings of type · · . Over thesepoints, the fiber enhances to a type II ∗ singularity, i.e. a point of e . This fact can bereadily checked when adding the term Z z a , to the Tate-model which breaks the u gaugesymmetry. In the following we will call this four-fold with only E gauge group, X , . For afully resolved model we have depicted fiber structure of the e point in Figure 4. Note thatthe exact fiber topology in a smooth model might not be of precise e shape. Instead thefiber structure might only be a bouquet which misses a couple of e nodes [27]. This effecthowever does not to obstruct the phenomenological implications of the Yukawa couplingsin [11, 28]. By setting the polynomial a , in the Tate-model to zero restores the global u gaugefactor under which the -plets are non-trivially charged. From a field theory perspectiveone therefore expects the Yukawa coupling to be forbidden. Therefore one might come tothe conclusion that those intersection points should be absent in a given geometry. However,what we actually observe is, that the e Yukawa points enhance to (4 , , as discussed in An idea to infer the above coupling is by viewing the local enhancement as that of e and decomposethe fundamentals representation into those of e to infer the Yukawa coupling (cid:51) . The effect on the fibers can be explained as an IIB orientifold effect that removes certain modes thatcorrespond to the respective nodes in the diagram. ( ) U (1) − ( ) − Figure 4:
Two depictions of the e divisor in blue and its fiber structure in the four-fold X , and X ,a . In X , the Yukawa point leads to a (reduced) E fiber bouquet. Uponadding a u brane the E fiber becomes non-flat fiber and leads to a four-point coupling. Section 2.1. From the general discussion of Section 2.1 we therefore a four-point couplingthat involves matter representation of the 6D tensor transition eqn. (3.12). Indeed in oursituation the (4,6,12) singularity precisely occurs over the points z = a , = a , = 0 where and − matter curves meet . The resulting four-point coupling is thereforeschematically given as W (cid:51) · · · − . (3.18)From the perspective of the global tuning, this point is very analogous to an E-string tran-sition in which we have tuned an extra U (1) onto the e Yukawa point as shown in Figure 4.Note also that generically those points come with a multiplicity when counting the intersec-tion points as N Yukawa ( X ,a ) = Z · [ a , ] · [ a , ] = 2(48(1 − g ) − Z ) , (3.19)that is N Yukawa ( Y ,a ) = 152 respectively. Finally we comment on the fiber structure of theresolved model X ,a . Here the non-flat Yukawa point is depicted in Figure 4. As opposed tothe codimension one non-flat fibers, these do not contribute additional Kähler parametersas they simply come from the degeneration of one of the exceptional e resolution divisors. Enhancing to non-flat fibers in codimension two
In the next step we want to perform the matter-transition we did in the three-fold givenby (3.9) now in the four-fold geometry X ,b . From the four-fold perspective, this tuningpushes the codimension three (4,6,12) singularity one codimension higher and enhances the -plet curve into that of an E-string. On the other hand, from a 6D perspective we caninterpret this curve as the compactification of g −
1) + Z E-strings points on a P whichmakes them a single curve of genus g (cid:48) E-string = 1 + 24(1 − g ) + 5 Z . Similarly to the three-fold X ,b in the four-fold X ,b this E-string curve by obtaining contributions of the singletcurve. Indeed, the singlet curve changes it genus by ∆ g = 9(8( − g − Z ) to ˜ g = 1 + 4(180( g −
1) + 54 c − Z ) . (3.20) This exact example has been anticipated in [4].
16n the concrete example of four-fold Y ,b that is detailed in Appendix A we compute theHodge numbers as ( h , , h , , h , ) χ ( Y ,b ) = (12 , , . (3.21)These Hodge numbers are to be understood from the general discussion in Section 2: Firstthere is the E-string theory whose non-flat surface which contributes a single new Kählerclass as its 6D rank is one. Note that we have started with two D E-string theories, thatboth contributed an individual Kähler class. Since both contributions are merged along thesame genus-five surface in the base, they now contribute a single Kähler class. Second thereis the contribution of the five h , three-forms. These are inherited from the rank one theorymultiplied by the genus of its curve.Finally, we can discuss the removal of these non-flat surfaces by changing the B baseanalogous to the 6D case. We take two equivalent ways by either blowing up the base B to get X ,c and then compactify this configuration over another P or by blowing up thefour-fold base in X ,b in the B direction to get X ,c . In both cases we end up with a base (cid:98) B = ( BLB ) × P . Notably X ,c admits no E-string points anymore which might not thecase for the four-fold X ,c . This can be seen from realizing that the E-string curve class inthe four-fold is given by Z · [ a , ] = 3 c ( B ) · Z − Z + 6 H · Z . (3.22)When we pullback c ( B ) · Z ∼ Z from B this is enough to require absence of E-stringpoints in a three-fold but not sufficient for the four-fold anymore. In order to do so we alsoneed to require the relation H · Z ∼ by e.g. perform additional blow-ups in the base.Hence even when started with a non-higgsable cluster in 6D, i.e. E on a − curve thatadmits neither matter and is fully flat, the new P direction introduce them. The sameis true for the explicit non-flat resolution in Y ,c where we have performed a base changesolely in the B direction. Performing this resolution has split the respective genus five in B into six genus-zero curves as depicted in Figure 5. The full four-fold geometry is givenin Appendix A and admits the Hodge numbers ( h , , h , , h , ) χ ( Y ,c ) = (19 , , . (3.23)The Hodge numbers make sense as we have introduced two more resolution divisors to B . This change in the base has split up the single non-flat surface into six disconnectedgenus zero curves. As all of them host a non-flat fiber, they all contribute additional Kählerparameters to Y ,c . None of the non-flat surfaces contribute to h , ( Y ,c ) anymore. In orderto obtain the fully flat four-fold, six more base blowups are required to split to remove allgenus-zero curves. The exact polytope of Y (cid:48) ,c is again given in the appendix and it is fullyflat. Moreover it admits exactly the same Hodge numbers as Y ,c given in (3.23).Finally we are in the position to compare the transition between three-folds and thosebetween four-folds. While removing non-flat fibers of Y ,b via a base change to Y ,c has notchanged the Hodge numbers at all, this was not the case in the analogous transition of four-folds. Here in fact virtually all Hodge numbers where involved in the transition. However17 E-string e E-string
Figure 5:
Depiction of the fiber structure of the e divisor in blue. The left shows anE-string fiber over a genus five curve resolved as a non-flat fiber. On the right a base changehas deformed this curve into six genus zero curves that are removed by additional blow-ups.All transitions change the Hodge numbers but not the Euler number of the compact four-fold. an important observations is that all those transitions left the Euler number invariant. Thisobservation is particularly important when it comes to the inclusion of G flux which we havenot addressed in this work. The relevant point is that it makes it easier to satisfy the changein the tadpole cancellation conditions eqn. (5.1) when a consistent G flux configuration hasbeen found in either phase. We will comment on this point again in the conclusion. ( SO (10) × U (1) ) / Z The second example works similar as the one before but admits a more rich matter structure.Moreover we use the chance to directly give more attention to the explicit toric resolutionof the geometry. Again we use E-string transitions in 6D to understand four-point Yukawacouplings and the general change of the matter spectrum in 4D. The model at hand admitstwo u gauge factors that are engineered with the most general elliptic model with threesections [30]. This model is given by the elliptic curve in an dP which we parametrizeby the coordinates { u, v, w } and the two exceptional coordinates e , e . An additional so singularity over Z : x = 0 can be engineered via a toric top [5, 6] which we resolve directlyvia the exceptional coordinates f , f , f , g , g and f being the affine node. The resolvedhypersurface is given as p = d e e f f f g u + d e e f f f f g g u v + d e f f f g uv + d e e f f u w + d e e f f f f g g uvw + d e f f v w + d e f f f g g uw + d e f f g vw . (3.24)The d i are to be interpreted as generalized Tate-coefficients that are sections of the base.This form of the elliptic curve can be mapped into Weierstrass form by blowing down allexceptional divisors and use the Arten-Tate algorithm. The Stanley-Reisner ideal for thechosen fiber triangulation is given by SRI : { e w, e e , e v, e f , e f , e g , e g , uw, uv, f u, f u, f u, g u, g u, f w, f v, f f ,f g , f w, e f , f v, f f , f f , e w, f w, g w, g v, g v, e f , e g , f g , e g } . (3.25) One can return to the singular model by setting all exceptional coordinates to one and replaces f with x . SO (10) × U (1) T I : x = d = 0 − / , − / : 2 I : x = d = 0 − / , / : 2 I : x = ( d d − d d ) = 0 / , : 4 I : x = d = 0 / , : 2 I : d = d = 0 1 , − I : d = d = 0 − , − I : d = d = 0 , I / { I , I , I , I , I } : , I : , I / { I , I } : − , − Table 1:
Summary of the 6D matter spectrum. We also give the respective codimension twoloci and their multiplicities using eqn. (3.26) . The longer (quotient) ideals I , , are givenin eqn. (3.27) . This model has been analyzed in detail in [5] where more details of the exact spectrumcomputation are given. We choose a simple two-fold base as B = F and a fibration withbundle choice of the base sections d i : Z : ( D x ) : H , [ d ] : H + 2 H , [ d ] : H + 2 H , [ d ] : H + 2 H , [ d ] : 2 H , [ d ] : 2 H + 2 H , [ d ] : 2 H , [ d ] : 2 H + 2 H , [ d ] : 2 H + 2 H , (3.26)where again Z is the so divisor and H and H the two classes of F . With the help ofthe Weierstrass model one can find the reducible fiber components which give rise to the 6Dmatter. In Table 1 we have summarized multiplicity and matter representations. I i singletloci that are given as: I : { ( d d d d − d d d d + d d d d + d d d + d d x − d d d d x + d d d d x + 2 d d d d x − d d d d d x + d d d x ) , ( − d d d − d d x + d d d d x − d d d d x + d d d x + d d x ) } ,I : { ( − d d d + d d + d d ) , − ( − d d d + d d + d d x ) } ,I : { ( d d + d d x − d d d x ) , d d + d d x − d d d x } , (3.27)The multiplicities can be computed by using the choices of classes (3.26) and the basicintersections on F , H · H = 1 and H i = 0 . This is enough to show 6D gauge anomaly For the quotient ideals, the contained loci have to be subtracted with multiplicities that are determinedusing the resultant. (1,0,0,0)w (0,1,0,0)u (1,-1,0,0)e (0,-1,0,0)v (-1,0,0,0) F basey (0,0,0,1)y (0,0,0,-1)x (0,0,-1,0) so topf (0,0,1,0)f (1,0,1,0)f (0,1,1,0)f (1,1,1,0)g (1,1,2,0)g (1,2,2,0) , ∆ = { Fiber, Base, Top } :( h , , h , ) χ ( Y ,a ) = (10 , − . (3.28)Note that the elliptic fibration structure is inherited from a dP fibration of the ambientspaces. This allows also to directly obtain the projection π from the ambient variety givenvia a projection onto the two last coordinates in ∆ . Using the
42 + 1 neutral singlets alsoallows to show cancellation of the gravitational anomaly. Next we want to perform an E-string transition which we do by factorizing another power of the so divisor x out of thebase section d . This tuning effectively merges two / , -plets and two − / , − / as wellas two , and , -plets as one can explicitly see from Table (1) and eqn. (3.27). After thistransition we have two non-minimal singularities over d = x = 0 . Lets again summarizethe total change in the spectrum S ∆ S = 2 · (cid:0) − / , − / ⊕ / , ⊕ − , − ⊕ , ⊕ , (cid:1) → · [ E-strings ] . (3.29)This transition is fully consistent with the expectation of a tensor-matter transitions derivedin [3] The geometry that admits the two (4 , , points can be resolved using toric geometry.This is done by simply adding the vertex vertex f = (0 , , , to the polytope (3.28). Theaddition of the vertex respects the fibration and does not change the base while also leavingthe new polytope reflexive. Hence the the anti-canonical hypersurface is still Calabi-Yau,which we denote by Y ,b with Hodge numbers: ( h , , h , ) χ ( Y ,b ) = (12 , − . (3.30)In comparison to Y ,a we find the expected loss of two complex structure parameters thatare traded for the two Kähler parameters that come fro the non-flat fibers. In the toricrealization, those are given by the ambient divisor f = 0 that intersect the base twice. Thiscan be seen from the new hypersurface which is explicitly given as p = d e e f f f f g g u + d e e f f f f g g u v + d e f f f g uv + d e e f f u w + d e e f f f f f g g uvw + d e f f f v w + d e f f f f g g uw + d e f f f g vw . A comment from the perspective of the toric geometry is in order. Here the generic fiberstructure parametrized by u, v, w, e , e has not changed at all but the modification appearsonly at codimension two over d = 0 . There the f coordinate factors out globally andcontributes a non-flat surface. Intersections of the non-flat surface can be computed via atriangulation that leads to the Stanley-Reisner ideal SRI : { e w, e e , e v, e f , e f , e f , e g , uw, uv, f u, f u, f u, f u, g u, f w, f v, f f ,f g , f w, e f , f v, f f , f f , e w, f w, f w, f v, g v, e f , e g , f g , e f } . (3.31)20 o − , − , , , so E-string
Figure 6:
The fiber structure of the ( SO (10) × U (1) ) / Z model. On the left a complexstructure deformation tunes four matter representations onto the same point while keepingcodimension one fibers inert. Over this point the resulting fiber becomes non-flat which isshown via the red surface on the right. The tuning process and the resulting fiber structures are summarized in Figure 6. From theperspective of the 6D F-theory compactification, the E-string points can be avoided by goingto the tensor branch. The blow-up of the ambient variety can be done explicitly by addingthe vertices xb : (0 , , , , xb : (0 , , , to the polytope which gives a new three-fold Y ,c with Hodge numbers ( h , , h , )( Y ,c ) = (12 , . (3.32)These Hodge numbers are identical to those, of the non-flat model as expected. The resultingfibration has been made fully flat and anomalies can be shown to cancel as expected. The four-fold compactification
We follow the same strategy as in Section 3.1 and fiber the three-fold over another P toobtain the four-fold Y ,a . Torically this is done by simply adding a fifth direction in thepolytope (3.28) and by adding the vertices z : (0 , , , , , z : (0 , , , , − to fill out thefull lattice Z . The fibration structure is still inherited from the ambient variety, where theprojection is given onto the last three columns resulting in the base B = ( P ) . Again we donot expect to find any three-form cohomology in Y ,a as the fibration is flat in codimensiontwo and the base B too simple to admit h , contributions. This is readily checked bycomputing the Hodge numbers via the Batyrev prescription ( h , , h , , h , ) χ ( X ,a ) = (11 , , . (3.33)We focus again on the structure of the so divisor x = 0 and the − , − curve over d = 0 which will become the (4 , , curve upon deformation. . Since d ∈ c ( B ) wefind the − , − matter curve to be of genus g ( − , ) = 1 + 12 [ x ] · [ d ] · ([ x ] + [ d ] − K − b ) = 1 . (3.34) From the direct product structure of the base, we simply add to all base classes in d i in eqn. 3.26 acontribution of H . x = 0 as f = x ( d (2 d d − d d d + 2 d d ) + x R ) + O ( x ) ,g = x ( d d d + d x R ) + x d ) + O ( x ) , ∆ = x ( d d d + x d R i + x d ) + d O ( x ) , (3.35)with R i being some residual polynomials. The discriminant can be used to find the per-turbative Yukawa and non-perturbative four-point couplings. E.g. there are (3 , , pointswhen d = 0 and d = 0 which give rise to the expected trilinear Yukawa couplings of thetype { x = d = d = 0 } Y : − / , − / · − / , − / · / , , (3.36) { x = d = d = 0 } Y : − / , − / · − / , / · / , . (3.37)These couplings are all fully consistent the expected gauge symmetry. In the following wewant to focus on the non-perturbative couplings that involve the − , − matter curve.The only point that involve this curve is localized at x = d = d = 0 . From comparingwith the general classes of the matter loci in Table 1 and (3.27) we find the matter curves { x = d = d = 0 } (cid:51) − / , − / , / , , , , − , − , to intersect at this point. Similar as in the sections before, these loci are to be interpreted asnon-perturbative four-point coupling points [4] generated by D1 instantons. Interestingly,this locus allows for two independent four-point couplings that are possible via the involvedrepresentations. The two possible coupling are schematically depicted as W (cid:51) − / , − / · − / , − / · / , · , + − / , − / · − / , − / · − / , · − , − . (3.38)From the perspective of the smooth geometry, given via (3.24), one finds that it is the f = 0 fibral divisor that splits into a non-flat component over d = d = 0 . This comesas no surprise as it is going to be the same (ambient divisor) that is pushed to a non-flatsurface at codimension two, when performing the conifold transition. Enhancing to non-flat fibers in codimension two
Now we perform the four-fold analog of the 6D tensor transition which enhances the − / , − / curve over x = d = 0 to a (4,6,12) curve via the factorization d → d x , resulting in thefour-fold Y ,b This transition again simply merges parts of the -plet and singlet mattercurves into that of -over which the fiber attains a non-minimal singularity. In the smoothgeometry the non-minimal singularity over that curve is resolved by the f non-flat surface,analogous to the three-fold case. In this regard it makes again sense to interpret the non-flatmatter curve as the compactification of 6D E-string theories.From the perspective of toric geometry the four-fold Y ,b is simply obtained by addingthe same toric vertex f = (0 , , , , to the 5d polytope with CY hypersurface X ,b which22dmits the Hodge numbers ( h , , h , , h , ) χ ( Y ,b ) = (12 , , . (3.39)The above data is exactly as expected: First there is the new class that contributes thatnon-flat surface at f = 0 over the genus-one curve x = d = 0 which also contributes in h , ( Y ,b ) . Finally we want to get rid of this non-flat fiber again be performing a base changeof B = ( P ) . In the first step we want to deform the curve x = d = 0 by performing acomplex structure deformation in the d followed by a blow-up. For this we remember that [ d ] ∼ c ( B ) . Hence by a complex structure deformation we enforce first the factorization d → y y z z ˆ d , (3.40)with [ ˆ d ] ∼ [2 x ] and hence [ x ] · [ ˆ d ] ∼ . Therefore the only non-trivial curve classes arewhen x = 0 intersects the other four components y i = 0 and z i = 0 . By performing thefirst blow-up in the base, that is inherited from the three-fold Y ,c we are adding the rays xb : (0 , , , − , , xb : (0 , , , , , (3.41)to the full 5D polytope. Note that the base has become B = ( BL F ) × P and admits atriangulation with Stanley-Reisner ideal: SRI : { x x , x y , x y , z z , x xb , y xb , xb xb , x xb , y xb , y y } . (3.42)This base change has turned the genus one curve x = d = 0 into two P s over x = z = 0 and x = z = 0 that do not intersect and host a non-flat surface each. The resultingfourfold Y ,c admits the Hodge numbers ( h , , h , , h , ) χ ( Y ,c ) = (15 , , . (3.43)The change in the Hodge numbers is explained as before: First there are two more classesfrom the two blow-ups which has removed all h , contributions. This genus one curve hassplit into two P with a non-flat fiber over each of them. Both of these fibers contribute anindependent non-flat fiber, which is responsible for the third additional Kähler parameter.If we want to also get rid of those we need a base that forbids the intersections x = z i = 0 as well which can be done with yet two more blow-ups. The two additional rays that are tobe added to the polytope that do the job are given as xb : (1 , , , , , xb : (1 , , , , − . (3.44)The resulting four-fold Y (cid:48) ,c is finally fully flat and admits the very same Hodge numbers as Y ,c ( h , , h , , h , ) χ ( Y (cid:48) ,c ) = ( h , , h , , h , ) χ ( Y ,c ) = (15 , , , (3.45)We find no change in the Hodge numbers at all, which can be explained by the fact thatthe non-flat fibers are simply exchanged for the two new base classes. The blown-up baseadmits a regular start triangulation with the following SRI SRI B : { x y , x y , x z , x z , x x , , x xb , y xb , y xb , z xb , xb xb , x xb , y xb , xb xb , x xb , y xb , y xb , z xb , x xb , y xb , y y , z z } . (3.46)23n the above ideal we have underlined the components that forbid the non-flat curve classeswhen compared to the blow-down. We conclude by making the same important observationas in the example from Section 3.1: In the transition from Y ,b to Y (cid:48) ,c all Hodge numbersand in particular h , change. However the change is always such that the Euler numbersstay inert. This observation is again important when including G fluxes and to satisfy thecondition in eqn. (5.1) which we will comment on in Section 5. E × SU (2) × SU (3) In this section we want to consider more complicated examples that admit multiple curveswith different superconformal matter curves over each to demonstrate the validity of eqn. (2.12).For this we engineer a generalized Tate model, with an e , an su and an su divisor via p = b Y + b X + a XY Z + a Z X + a Z Y + a Z X + a Z . (4.1)Here, the Arten Tate-algorithm can be used to bring the above model into Weierstrass formwhich makes it easier to read off the singularity structure. The Weierstrass coefficients aregiven as f = −
148 ( a − a b ) + 12 b ( − a a + 2 a b ) b ,g = 1864 (( a − a b ) + 36 b ( a − a b )( a a − a b ) b + 216 b ( a − a b ) b ) . (4.2)The zero-section is given by Z = 0 in the generalized Tate-model is not a trivial section of B anymore. This results in the pre-factors b and b to be non-trivial divisors of the basewhose zero-locus gives the additional su and su singularities respectively. The base classesof b and b parametrize two new line bundle classes denoted by Z su and Z su . Engineeringthe additional e singularity over Z e : z = 0 in Tate form is standard and can be takenfrom the literature (e.g. [20]) by factorizing { a , a , a , a , a } → { za , , z a , , z a , , z a , , z a , } . (4.3)The option to allow for Z su and Z su shifts the classes a i such that one ends up with [ b ] ∼ Z su , [ b ] ∼ Z su , [ a , ] ∼ c − Z e , [ a , ] ∼ c − Z su − Z e , [ a , ] ∼ c − Z su − Z su − Z e , [ a , ] ∼ c − Z su − Z su − Z e , [ a , ] ∼ c − Z su − Z su − Z e . (4.4)The model above admits three types of superconformal matter collisions at codimension twowhich can be read off from eqn. (4.2) at [ E-string ] : z = a , = 0 , [ e su ] : z = b = 0 , [ e su ] : z = b = 0 , (4.5)as e.g. discussed in [15,16]. In general [ e su n ] conformal matter gives rise to a ord van ( f, g, ∆) =(4 , ,
12 + n ) collisions in the Weierstrass model. The tensor branches of those models have24een analyzed in detail in [37]. They are given via a linear chain of P ’s with self-intersection ( − n ) given as e su n su su su n-1 -1 -2 -2 -2 . . . . (4.6)The superscript denotes additional su k gauge algebra factors hat are enforced in the Weier-strass model upon the base change. Note that bi fundamental matter lies in between thesenodes. These codimension two collisions can be resolved by a non-flat fiber. From the n -dimensional 6D tensor branch and upon collecting the contributions of the gauge algebrafactors the full 5D coulomb branch dimension is given asdim ( Coulomb D ([ e su n ])) = 1 + 12 n ( n − . (4.7)As argued in Section 2.1 the non-flat fiber resolution must include the same amount ofsurface contributions with their own Kähler classes. In order to compute the contributionto h , as well, requires the computation of the genus of those curves in B , given as g E-string =1 + 12 (6 c − Z su − Z su − Z e ) · (5 c − Z su − Z su − Z e ) · Z e ,g e su =1 + 12 Z su · Z e · ( Z su + Z e − c ) ,g e su =1 + 12 Z su · Z e · ( Z su + Z e − c ) . (4.8)Putting all pieces together, the contributions to h , ( X ) via eqn. (2.12) and eqn. (2.11) andusing the knowledge of the tensor branches in eqn. (4.7) is given as h , non-flat ( X ) = g E-string + 2 g [ e su ] + 4 g [ e su ] . (4.9)This can be explicitly checked by constructing a family of four-folds that exhibit a simpletoric description. The simplest base to take is B = P . In terms of the base hyperplaneclass H we can fix all line bundle choices as Z e ∼ H , Z su ∼ n su H , Z su ∼ n su H . (4.10)Here the n su n are positive integers bounded by effectiveness of all b i , a i,k in (4.4). For thoseconstructions, the base is again too simple to contribute h , . It is also important to notethat this particular geometry does not come from a direct compactification of an alreadypresent 6D theory. Hodge numbers of the family of four-folds X ( n su , n su ) are given as h , ( X ( n su , n su )) =10 + (1 − δ (19 − n su − n su , ) + 3(1 − δ ( n su , ) + 6(1 − δ ( n su , ) ,h , ( X ( n su , n su )) = g E-string (1 − δ (19 − n su − n su , )+2 g e su (1 − δ ( n su , ) + 4 g e su (1 − δ ( n su , ) . (4.11)This is double checked via the Batyrev construction. The 5D polytope ∆( n su , n su ) thatrealizes the two-parameter family of four-folds X ( n su , n su ) is summarized in Table 2.25ber ray Z (-2, -1, 0, 0, 0) X (1, -1, 0, 0, 0) Y (0, 1, 0, 0, 0) e (0, -1, 0, 0, 0) e (-1, -1, 0, 0, 0) e (-1, 0, 0, 0, 0) E -coord ray m (-2, -1, 0, 6, 0) l (-2, -1, 0, 5, 0) k (-2, -1, 0, 4, 0) k (-1, -1, 0, 4, 0) h (-2, -1, 0, 3, 0) h (-1, 0, 0, 3, 0) g (-2, -1, 0, 2, 0) g (0, -1, 0, 2, 0) f (-2, -1, 0, 1, 0) surface ray h (-1, -1, 0, 3, 0) f (-1, -1, 0, 1, 0) f (0, -1, 0, 1, 0) g (-1, -1, 0, 2, 0) f (-1, 0, 0, 1, 0) g (-1, 0, 0, 2, 0) f (0, 0, 0, 1, 0)Base ray x ( n su + n su -2, n su -1, -1, -1, -1) x (-2, -1, 1, 0, 0) x (-2, -1, 0, 0, 1) h , h , h , χ n su n su
11 153 1982 11088 0 017 105 1300 7320 0 117 66 801 4560 0 217 40 459 2664 0 317 27 250 1488 0 417 27 150 888 0 517 40 135 720 0 614 120 1522 8544 1 020 78 960 5460 1 120 45 563 3276 1 220 25 305 1848 1 320 18 162 1032 1 420 24 110 684 1 514 91 1147 6468 2 020 55 693 3996 2 120 28 386 2316 2 220 14 200 1284 2 320 13 111 756 2 419 24 95 588 2 514 68 850 4824 3 0 h , h , h , χ n su n su
20 38 492 2892 3 120 17 263 1644 3 220 9 137 936 3 320 14 90 624 3 414 51 625 3576 4 020 27 351 2112 4 120 12 188 1224 4 220 10 110 768 4 314 40 466 2688 5 020 22 264 1620 5 120 13 155 1020 5 219 16 113 744 5 314 35 367 2124 6 020 23 225 1380 6 120 20 158 996 6 214 36 322 1848 7 020 30 228 1356 7 114 43 325 1824 8 019 42 267 1512 8 114 56 370 2016 9 0
Table 2:
Summary of the rays of the polytope ∆( n su , n su ) that leads to smooth ellipticfour-fold X ( n su , n su ) with e , su and su fibers over a base P . The Hodge numbers ofall consistent inequivalent four-folds are summarized below.
26n the same table also Hodge and Euler numbers are computed for all reflexive polytopesconsistent with the expectation from eqn. (4.11). The fully resolved hypersurface is given as p = b e f f g h Y + b e e f f f g g h k X + a , e e e f f f f f g g g g h h h k k l m XY Z + a , e e e f f f f f g g g g h h h k k l m X Z + a , e e e f f f f f g g g g h h h k k l m Y Z + a , e e e f f f f f g g g g h h h k k l m XZ + a , e e e f f f f g g g h h k l Z , (4.12)and has been analyzed for various (fiber) triangulations e.g. in [15] with f being the affinecomponent of e . The split into the various non-flat loci at (4.5) can be readily verified atthe loci given in eqn. (4.5). This note has considered a systematic analysis of smooth elliptic four-folds that exhibitnon-flat fibers. First we have shown how non-flat fibers in codimension two contribute tothe Hodge numbers and in particular to the three-form cohomology. Via the M-theoryduality such three-forms lead to additional chiral singlets. In F/M-theory these non-flatconfigurations are to be interpreted as compactifications of 6D/5D superconformal mattertheories on a Riemann-surface. This allowed us to identify their contributions to the Hodgenumbers in terms of the 6D/5D tensor(coulomb) branch dimensions and the genus of theRiemann-surface. The validity of this proposal is checked for several examples which include6D SCFTs of various ranks.Furthermore we have investigated conifold transitions among four-folds that remove thosenon-flat fibers and hence these specific chiral singlet fields. The first branch of these tran-sitions is analogous to a 6D tensor branch and corresponds to a birational base change.This transition changes all Hodge numbers but most notably does not change the Eulernumber of the compact model. Note that this work has focused primarily on the geometricaspects and not the inclusion of G flux. The fact that the Euler number does not changein such transitions simplifies the matching of 4D SUSY vacua substantially [13, 14] due tothe vanishing of the right hand side of ∆ ( n D ) + 12 ∆ (cid:18)(cid:90) G ∧ G (cid:19) = ∆ (cid:16) χ (cid:17) . (5.1)Euler number preserving transitions therefore do not require a to change the G flux (norm)or number of D branes during the transition. The second type of transitions we haveconsidered is analogous to a (partial) 6D Higgs branch that keeps the total gauge groupbut moves the non-flat fiber from curves down to points of the base B . These points leadto non-perturbative four-point matter couplings, mediated by D1 instantons and do changethe Euler number. The existence of these non-perturbative interaction points is enforcedgeometrically and can be interpreted as a remnant of the 6D E-string theory. In fact we27ave argued that the matter representations involved in such couplings can be deduced dueto the anomalies of the 6D E-string transition.This work serves as a first step towards the investigation of 4D/3D theories obtainedfrom F/M-theory on non-flat elliptic four-folds. From here on there are several directionsto go in the future. First it is important to fully include G fluxes. As argued, the rightstarting point might be the phase without those fibers compute a consistent configurationand perform the conifold to the non-flat configuration. Moreover in order to fully understandthe 4D theory and its possible non-perturbative effects, a better understanding of the 4Deffective action is desirable. As analyzed in [29], Euclidean D3 instantons can lead to amixing of the complex structure and Kähler moduli. This work shows that three-formsare a potential third contribution to those moduli sectors that are characteristic to non-flatresolution. Understanding all those effects is important to clarify how quantum correctionsmight obstruct 4D SCFT points in the IR. As the four-fold is fully smooth those contributionsare best analyzed in the 3D M-theory. There are also further generalizations possible frompure geometric point of view. These include the addition of monodromies that act on thesuperconformal matter curves, analogous to to split fibers in elliptic three-folds. Such effectwould naturally incorporate the folding action familiar from twisted compactifications [53]into the four-dimensional picture. Finally it would also be very interesting to explicitlyconstruct the heterotic duals of those theories and match the contributions of h , non-flat ( X ) to that of the NS5 branes [31]. Acknowledgements
The author thanks Markus Dierigl, Mohsen Karkheiran, Magdalena Larfors, Fabian Ruehleand Thorsten Schimannek for discussions. P.O. thanks in particular Markus Dierigl andThorsten Schimannek for reading an earlier version of the draft and their useful comments.The author would also like to acknowledge the hospitality of the SCGP during the program“Geometry and Physics of Hitchin Systems” where initial parts of this work. The work ofP.K.O. is supported by a grant of the Carl Trygger Foundation for Scientific Research.
A Toric resolution of the ( E × U (1)) / Z model In this appendix we look at some geometric details that were left out in the main section.These include in particular the fully resolved three-and fourfolds and their intersections.
A.1 The resolved Tate model
Starting from the tuning of the Tate model in Section 3.1 we need to resolve the I via theexceptional divisor e = 0 the e with the set { f , f , f , g , g , g , h } that can be engineeredas blow-ups of the P , , fiber ambient space. The fully resolved U(1) restricted Tate modelis given by p = e f g Y + e f f g X + a , e f f f g g g h XY Z + a , e f f f g g g h X Z + a , f g Y Z + a , f f g g h XZ (A.1)28here are two sections given by Z = 0 that intersects the affine node f and e = 0 thatintersects f in a phase that employs the SRI: SRI : { Ze , Zf , Zf , Zg , Zg , Zg , Zh , Y X, Y f , Y g , Y h , e f , f f , f g , f g ,f h , Xf , Xg , Xg , Xg , Xh , e g , e g , e g , e h , f g , f h , f g , f h } (A.2)Graphically we give the intersection of the e (1)6 fiber as [ f ][ g ][ f ] e Z [ g ][ h ][ g ][ f ] . The intersections allow us to compute the U (1) charges by using the smooth geometry inM-theory and lift it back to F-theory. For this we employ the Shioda map, that is the divisorthat gives the U (1) generator in the M-theory expansion of the C form. For simplicity wefocus here on the fibral part, which is given in the difference of the two sections. By thendemanding orthogonality with respect to all other fibral divisors leaves us with the followinglinear combination σ ( s ) = [ e ] − [ Z ] + 13 (4 f + 5 g + 6 h + 4 g + 2 f + 3 g ) . (A.3)The above concrete form allows us to compute the explicit charges of matter multiplets byintersecting it with the reducible fibral curves at codimension two. E.g. first there are the over a , = a , = 0 . Here one can see, that the coordinate e = 0 becomes reducible.The fiber topology is that of an I curve as given in Figure 4. Since e = 0 is a − curve onthe fiber ambient space which coincides with the U (1) charge of the singlets using eqn.(A.3).Then there are the -plets over z = a , = 0 . This can be seen by noting, that the fiberhere becomes of e type. From the reducible fiber components one can deduce the e chargesusing (A.3). For four-folds there is an additional enhancement locus that is of interest. Acodimension three by first setting a , = a , = 0 , where the hypersurface becomes p = e f ( g Y + e f g X + a , f f g g g h XY Z + a , f f g g g h X Z ) , (A.4)This is also the singlet locus, but note that the f e component also factors out. As f = 0 restricts onto the z = 0 locus in the base, setting it to zero is a codimension threelocus. Moreover, coordinates { f , g , g , Y, e } are fully unrestricted and they parametrizea two-dimensional surface. In Figure 4 we have computed all intersections of the fibralcomponents over that locus. In the red box we have computed the intersections of thefive components that sit inside the non-flat surface f = 0 . This is unlike the shape aregular Dynkin diagram shaped fiber should have but it is characteristic for a non-flatfiber. Secondly we the enhanced model with a non-flat fiber at codimension two. This isdone, by further factorizing one power of z out of a , as zb , . This leads to the codimension In codimension three it appears often, that the fiber obtains a bouquet shape form, where certain nodesare deleted. This is however z = a , = 0 . Note that we use the toric resolution of the fiber asgiven via the top construction that we are using to construct full three-and four-folds in thenext section. We have added the new fibral coordinate f which removes f as one of the e fibral divisors. Notably, it is still present in the ambient space and will be introduced as anon-flat fiber momentarily. The new resolved fibration is given as p = e f f g Y + e f f g X + a , e f f f f g g g h XY Z + a , e f f f g g g h X Z + , f g Y Z + b , f f f g g g h XZ . (A.5)Intersections can be computed via a triangulation of the toric ambient space, that leads tothe SRI: SRI : { Ze , Zf , Zf , Zg , Zg , Zg , Zh , Zf , Y X, Y f , Y f , Y g , Y h , e f , f f , f g , f g , f h , f f , Xg , Xg , Xg , Xh , Xf , e f , e g , e g , e g ,e h , f f , g f , g f , h f , f g , f h , f g , f h , g g g } (A.6)All intersections of the e fiber components as well as the singlets can be computed asusual and their structure has not changed at all. Notably, over a , = 0 the fiber becomes P = f ( e f g X + e f g Y + a , e f f f g g g h XY Z + a , e f f g g g h X Z + b , f f g g g h XZ ) . (A.7)We have exactly the same behavior as before, that is f = 0 is a non-flat fiber componentbut this time it happens over codimension two. Note also that it is exactly the sameambient space coordinate the appears here. Using the SRI above, one can also compute theintersections of the fibral components as given in Figure 5. There we observe again a similarintersection picture as in codimension three, that is a loop of components that sit inside thenon-flat surface. A.2 Explicit Toric three-and four-fold
We give the explicit toric polytope for the three-folds and four-folds that are consideredin Section 3. The toric rays of the polytope are given in the tables below. The threefold X is constructed from a regular fine star triangulation of the toric fan associated to theBatyrev polytope ∆ . These polytopes are a combination of various rays as given below.The projection onto the bases B is inherited from the toric ambient space and we fix it tobe onto the last two coordinates. We then give the various polytope ingredients below. TheU(1) restricted Tate-model is a combination of the e top and completed with the choice ofa base. The e resolution divisor becomes non-flat upon adding the ray f and the non-flatfibers can be avoided by adding the two − curves that correspond to the toric rays b and b to the base. Note that these are next to the − curve on which the e is localized andthus reduces its self-intersection by one each time.30: U (1) -Tate fiberX (1,-1,0,0)Y (0,1,0,0)Z (-2,-1,0,0) e (1,0,0,0) E1: e Fiber f (-2,-1,0,-1) f (0,0,0,-1) f (0,-1,0,-1) g (-2,-1,0,-2) g (-1,-1,0,-2) g (-1,0,0,-2) g (-2,-1,0,-3) E2: non-flat- e Fiber f (-2,-1,0,-1) f (0,0,0,-1) f (0,-1,0,-1) g (-2,-1,0,-2) g (-1,-1,0,-2) g (-1,0,0,-2) g (-2,-1,0,-3) f (0,1,0,-1) B1: F z (-2,-1,0,1) x (-2,-1,1,0) x (-2,-1,-1,-4)B2: BL F z (-2,-1,0,1) x (-2,-1,1,0) x (-2,-1,-1,-4) b (-2,-1,1,-1) b (-2,-1,-1,-5)The constituents above are constructed to toric polytopes from which we can compute theHodge numbers via the Batyrev construction. The results in addition to flatness are givenbelow: ∆ CY h , h , Flat(F,E1,B1) X
10 162 (cid:88) (F,E2,B1) ˆ X
12 160 X (F,E2,B2) (cid:98) X
12 160 (cid:88) (F,E1,B1) (cid:98) X
12 160 (cid:88) (A.8)Similarly, the four-folds are constructed by enlarging the toric ambient space and adding a P direction. These are given via two new toric rays y and y in the following. Using thetwo base blow-ups b , b from the thee-folds degenerates the curve with the non-flat fiberin codimension two from a genus five curve to six genus zero curves in the base. Thesetransitions change the Hodge numbers but not the Euler number of the fourfold as can beseen directly. To fully resolve those six non-flat P ’s one needs to introduce the six blow-ups b ... . These blow-ups destroy the direct product structure of the base but make the fibrationfully flat. Note that these last six blow-ups keep all Hodge numbers invariant in full analogyto its 6D cousins. The various polytope ingredients are given below: F: U (1) -Tate fiberX (1,-1,0,0,0)Y (0,1,0,0,0)Z (-2,-1,0,0,0) e (1,0,0,0,0) E1: e Fiber f (-2,-1,0,-1,0) f (0,0,0,-1,0) f (0,-1,0,-1,0) g (-2,-1,0,-2,0) g (-1,-1,0,-2,0) g (-1,0,0,-2,0) g (-2,-1,0,-3,0)E2: e Fiber f (-2,-1,0,-1,0) f (0,0,0,-1,0) f (0,-1,0,-1,0) g (-2,-1,0,-2,0) g (-1,-1,0,-2,0) g (-1,0,0,-2,0) g (-2,-1,0,-3,0) f (0,1,0,-1,0) B1: F × P base z (-2,-1,0,1,0) x (-2,-1,1,0,0) x (-2,-1,-1,-4,0) y (-2,-1,0,0,1) y (-2,-1,0,0,-1)B2: ( BL F ) × P ) z (-2,-1,0,1,0) x (-2,-1,1,0,0) x (-2,-1,-1,-4,0) b (-2,-1,1,-1,0) b (-2,-1,-1,-5,0) y (-2,-1,0,0,1) y (-2,-1,0,0,-1) B3: Bl ( BL F ) × P ) z (-2,-1,0,1,0) x (-2,-1,1,0,0) x (-2,-1,-1,-4,0) b (-2,-1,1,-1,0) b (-2,-1,-1,-5,0) y (-2,-1,0,0,1) y (-2,-1,0,0,-1) b (-2,-1,0,-1,1) b (-2,-1,0,-1,-1) b (-2,-1,0,-2,1) b (-2,-1,0,-2,-1) b (-2,-1,0,-3,-1) b (-2,-1,0,-3,-1) ∆ can be composed that allows an easy computation ofHodge numbers via the Batyrev formula as: ∆ CY h , h , h , χ Flat(F,E1,B1) X
11 0 1447 8769 (cid:88) (F,E2,B1) ˆ X
12 6 1281 7776 X (F,E2,B2) (cid:98) X
19 0 1269 7776 X (F,E2,B3) (cid:98) X (cid:48)
19 0 1269 7779 (cid:88) (F,E1,B3) (cid:98) X (cid:48)
19 0 1269 7779 (cid:88) (A.9)
B Review: Hodge numbers from polytopes
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