Heterotic-F-theory duality with Wilson line symmetry-breaking

We begin with an E8 x E8 Heterotic model broken to an SU(5)gauge and a mirror SU(5)gauge, where one SU(5) and its spectrum is identified as the visible sector while the other can be identified as a hidden mirror world. In both cases we obtain the minimal supersymmetric standard model spectrum after Wilson-line symmetry-breaking enhanced by a low energy R-parity enforced by a local (or global) U(1)x-symmetry. Using Heterotic/F-theory duality, we show how to eliminate the vector-like exotics which were obtained in previous constructions. In these constructions, the Calabi-Yau [CY] four-fold was defined by an elliptic fibration with section over a base B3 and a GUT surface given by K3/ℤ2 = Enriques surface. In the present paper we construct a quotient CY four-fold fibered by tori with two elliptic structures given by a pair of sections fibered over the Enriques surface. Using Heterotic/F-theory duality we are able to define the cohomologies used to derive the massless spectrum. Our model for the 'correct' F-theory dual of a Heterotic model with Wilson-line symmetry-breaking builds on prior literature but employs the stack-theoretic version of the dictionary between the Heterotic semi-stable Es-bundles with Yang-Mills connection and the dP9-fibrations used to construct the F-theory dual.


The physics
Supersymmetric grand unified theories [SUSY GUTs] [17,18,29] have many nice properties. These include an explanation of the family structure of quarks and leptons with the requisite charge assignments under the Standard Model [SM] gauge group SU(3) C × SU(2) L × U(1) Y and a prediction of gauge coupling unification at a scale of order 10 16 GeV. The latter is so far the only direct hint for the possible observation of supersymmetric particles at the LHC. UV completions of SUSY GUTs in string theory also provide a consistent quantum mechanical description of gravity. As a result of this golden confluence, many groups have searched for SUSY GUTs in string theory. In fact, it has been shown that by demanding SUSY GUTs in string constructions one can find many models with features much like that of the minimal supersymmetric Standard Model [MSSM] [2,3,8,32,[36][37][38].
The past several years have seen significant attention devoted to the study of supersymmetric GUTs in F -theory [5,6,9,20,21,24,52]. Both local and global SU(5) F -theory GUTs have been constructed where SU (5) is spontaneously broken to the SM via JHEP12(2019)016 non-flat hypercharge flux. One problem with this approach for GUT breaking is that large threshold corrections are generated at the GUT scale due to the non-vanishing hypercharge flux [9,10,21,44]. An alternative approach to breaking the GUT group is using a Wilson line in the hypercharge direction, i.e. a so-called flat hypercharge line bundle. In this case it is known that large threshold corrections are not generated at the GUT scale (or, in fact, leading to precise gauge coupling unification at the compactification scale in orbifold GUTs) [33,49] and [1,27,50,53].
In a previous paper, the present authors and collaborators constructed a global SU(5) F -theory model with Wilson line breaking [43]. The model contained the vector multiplets for the MSSM gauge group, 3 families of quarks and leptons, 4 pairs of Higgs doublets, and in addition, a vector-like pair of chiral multiplets in the representation (3, 2) −5/6 ⊕ (3, 2) +5/6 . In terms of the model defined on an elliptically fibered CY 4-fold with GUT surface defined as an Enriques surface, K 3 /Z 2 , the massless spectrum is given in terms of cohomologies of the flux line bundle (or twists of the flux line bundle) on the GUT surface. It was then shown that the holomorphic Euler character of any flat bundle on S GUT is equivalent to its Todd genus [6,20,43], through so we have that Since all h m (S GUT , L Y ) cannot be vanishing, we are guaranteed to get some massless vector or chiral states, (3, 2) −5/6 's and (3, 2) +5/6 's. We emphasize that the presence of some kind of vector-like exotic matter is not a specific issue with this Enriques model but rather a general property of any model that breaks SU(5) GUT → SU(3) × SU(2) × U(1) Y with a flat U(1) Y bundle on a holomorphic surface S GUT . We note that this derivation is, however, only valid if the F -theory is compactified on a CY 4-fold with section. In the present paper we show how to eliminate the vector-like exotics and evade this theorem. We need to understand how to modify the description of the exotics (and the matter content) in a situation where there is no single distinguished section, or, more precisely, where the torus fibration has two sections 'on equal footing.' The prescription we use gives exactly such a description. The purpose of this paper is to present a model for Heterotic/F -theory duality in which SU(5) symmetry is broken (on both sides) by Wilson lines. The work derives from the previous global F -theory model with Wilson line symmetry-breaking [43]. It modifies the previous model so as to allow the construction of a Heterotic dual. It adapts previously known Heterotic techniques for eliminating undesirable features of the model, such as vector-like exotics, by constructing the torus-fibration (on both sides) with two sections [22]. This allows us to use the Heterotic technique of translation by the difference of the two sections to form the requisite Z 2 -action in order to evade the above-mentioned theorem and eliminate the vector-like exotics. The model has some very nice features. It contains the gauge group SU(5) broken via a Wilson line in the hypercharge direction to SU(3) × SU(2) × U(1). It contains three families of quarks JHEP12(2019)016 and leptons and one pair of Higgs doublets. Furthermore our F -theory model exhibits a Z 4 R-symmetry in the semi-stable limit of the F -theory model.
Some problems for this construction are as follows. The gauge symmetry U(1) X also forbids Majorana masses for the right-handed neutrinos. In addition, in subsection 8.7 we calculate the D-term for U(1) X (see eq. 3.30 in [24,25]), requiring D U(1) X = 0 so that this symmetry is not spontaneously broken by fields derived from the adjoint representation of E 8 . In addition, U(1) X is not sufficient to prevent dimension-5 baryon and lepton number violating operators. It is possible that the local U(1) X symmetry is broken down to a global U(1) X symmetry via a Stueckelberg mechanism [24,52], but this is beyond the scope of the present paper. The U(1) X symmetry may also be broken to a Z 2 matter parity by a non-perturbative effect at the GUT scale. This would then allow for right-handed neutrino Majorana masses at the GUT scale.
Finally, a very novel feature of the model is that it contains a twin/mirror SU(5) symmetry broken to a mirror SM with three families of mirror quarks and leptons and a pair of mirror Higgs multiplets. 1 This is a direct consequence of the fact that the GUT surface, S ∨ GUT = Enriques, is a branched (therefore irreducible) double cover of the base B 2 . The gauge and Yukawa couplings in the visible and mirror worlds are determined by volume moduli which must still be stabilized and supersymmetry broken. As a result mirror matter does not necessarily have the same mass as visible matter or the same value of their gauge and Yukawa couplings. This mirror sector is a possible candidate for the dark matter in the universe.

The mathematics
We next give an idea of the mathematics of our model for Heterotic/F -theory duality in which SU(5)-symmetry is broken (on both sides) by a Wilson line construction. As mentioned, the mathematical model derives from a previous global F -theory model with Wilson line symmetry-breaking [43]. It modifies the previous model so as to allow the construction of a Heterotic dual. It also adapts previously known Heterotic techniques for eliminating undesirable features of the model, such as vector-like exotics, by constructing the torusfibration (on both sides) that admits two sections. More properly, the two sections, taken together, should be thought of as determining an invariant g 1 2 linear series on the torus fiber. 1 Mirror world defined as the parity transform of the Standard Model has been reviewed in the paper by L.B. Okun [48]. This paper has many references which we refer to the reader. Some of these references however include another related definition of the mirror world given by a generalized Z2-symmetry which takes the Standard Model into the twin or mirror sector with states having identical charges but different masses and couplings. Papers in this genre include, for example, [7,13], and [4].

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Our goal for this paper is to present a global phenomenonologically consistent Heterotic model V ∨ 3 /B ∨ 2 and F -theory dual W ∨ 4 /B ∨ 3 . These will be constructed as equivariant Z 2 -quotients of another set of dual Heterotic/F -theory models V 3 /B 2 and W 4 /B 3 with respective involutionsβ are our ultimate goal. However much of the work will center on V 3 /B 2 and W 4 /B 3 and their respective involutions, passing to the quotients only late in the story. Furthermore much of the work has already been completed in our companion papers [14,15], including in [15] the complete construction of and the involution β 3 and calculation of their numerical invariants. The groundwork contained in those papers will be referred to as needed in what follows.
In short, the fundamental challenge is to construct compatible involutions (1.3) and (1.4) so that both leave their top-degree holomorphic forms invariant, that is, so that their respective quotients are Calabi-Yau manifolds. Because of other necessary characteristics of the Heterotic and F -theory models, we showed in [14]  on the relative (Weierstrass) one-form on W 4 /B 3 . We showed this necessity on the F -theory side in [14] by tracing the Tate form back to its E 8 -origins, namely We showed in [15] that the coefficients a j of the Tate form, as well as z and y/x must go to minus themselves under the Z 2 -action. One identifies the configuration of exceptional curves in the crepant resolution of (1.8) with the configuration of the positive simple roots in the E 8 -Dynkin diagram. A consequence of this identification is that (1.6) sends each positive simple root to its negative. We preserve E 8 -symmetry by counteracting this reversal of roots by the operation of complex conjugation on the complex algebraic group E C 8 , an operation that leaves untouched the compact real form E 8 . This last is reflected in the fact that tracing real roots back to E 8 requires that (1.7) and (1.8) be rescaled by dividing both sides by a 6 0 . This rescales both zand y/x by a −1 0 and y/a 0 x indeed becomes invariant under the Z 2 -action. JHEP12(2019)016

The organization of the paper
In section 2 we derive the Weierstrass equation for the Tate form. We introduce the construction of a second section τ of W 4 /B 3 in addition to the tautological section ζ 'at infinity. ' We discuss the action of the involutionβ 4 /β 3 as reflecting complex conjugation on the complex algebraic groups SL(5, C) and E C 8 whose compact real forms are SU(5) and E 8 respectively.
In section 3 we derive the spectral variety from the Tate form and discuss its decomposition into components of degree four and one respectively.
In sections 4 and 5 we discuss the semi-stable limit and the relation to the Heterotic dual. In particular we show how to build the normal-crossing K3 from an elliptic curve with two flat E-bundles.
In section 6 we project the fourfold W 4 /B 3 to the P 1 -bundle Q/B 3 whose fiber is the degree-2 linear series determined by the two points τ (b 3 ) and ζ (b 3 ) where the two sections intersect the fiber. We do this by projecting each fiber of W 4 /B 3 from the third point of intersection with the line joining τ (b 3 ) and ζ (b 3 ). This projection, that we name W 4 blows up the singular locus of W 4 . In this way we obtain a commutative diagram where the top horizontal map is crepant partial resolution, the left-hand vertical map is 2-1 and the bottom horizontal map is a P 1 -fibration whose fibers correspond to the degree-2 linear system on the fibers of the right-hand vertical map determined by the lifts of the two sections (τ ) and (ζ) to W 4 /B 3 . The Calabi-Yau fourfold W 4 is a branched double cover of Q. The affine fiber coordinate ϑ 0 of Q/B 3 is a section of K −1 B 3 as are the coefficients a j in the Tate form and t := y/x. We next construct a full crepant resolution of W 4 to obtain a smooth modelW 4 . This is accomplished in section 7. The first step is the partial resolution W (1) 4 /B 3 of W 4 /B 3 that will create an divisor D 0 connecting the divisors D 1 (the inherited component from W 4 /B 3 ) and D 4 (the exceptional divisor of W 4 /W 4 ). Together these three divisors comprise 4 . This will be followed by a third partial desingularization W 4 /B 3 extending over a general point of S GUT . Its exceptional divisor will be reducible so that that configure themselves over a general point of S GUT as an extended A 4 -Dynkin diagram. That will in turn be followed by a final codimension-2 desingularizationW 4 /B 3 of W  ⊆ S GUT . It should be noted that, in the process of putting (ζ) and (τ ) on equal footing as the first step in the desingularization, neither can be given preference as the one passing JHEP12(2019)016 through the inherited component The 'inherited' role is assumed by D 0 while, over a general point of S GUT , the proper transform ζ of (ζ) intersects D 1 and the proper transform (τ ) of (τ ) intersects D 4 .
In section 8 we calculate the Higgs line bundle that will govern the computation of the chiral spectrum. In subsections 8.5 and 8.6 we compute the G-flux. In subsection 8.7 we establish the vanishing of the D-term for a suitably chosenβ-symmetric Kähler metric.
In section 9 we discuss the symmetry-breaking induced by wrapping the Wilson line on the involutionβ 4 onW 4 . The Wilson line breaks SU(5)-symmetry to the SU It is in section 10 that we calculate the complete spectrum of the theory. The desired invariants follow rather directly from the results of [15] where the toric presentation of B 3 and its involution β 3 were explored in detail. The present paper demonstrates a method for eliminating vector-like exotics when breaking the GUT symmetry with the Wilson line. In this section we discuss just one SU(5) gauge × SU(5) Higgs sector in the semi-stable limit of the F -theory model, however, the massless spectrum in the hidden SU(5) gauge × SU(5) Higgs is identical.
Finally in section 11 we discuss the Z 4 R-symmetry on the semi-stable limit of our F -theory model. Remark 1. Throughout this paper, we will employ the following notational convention. Projections with image space A will in general be denoted as π A . This notation will be employed regardless of the domain of the map, which (hopefully) will be clear from the context.
Remark 2. Throughout this paper, we will let where B 2 is the D 2 del Pezzo surface studied in section 6 of [15]. B 2 is a double cover of the projective plane

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branched along a specific smooth quartic curve admitting a Z 4 -action. Denote We form with homogeneous fiber coordinates [w, x, y] and canonical bundle For a j , z, y we write the Tate form (1.7) for an elliptically fibered fourfold W 4 /B 3 in determinantal form as with respect to the induced involution on W 4 /B 3 . The equation z = 0 defines a smooth surface that, by the adjunction formula, must be a K3-surface.

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Referring to (2.4) we change the equation of W 4 /B 3 into Weierstrass form based at the section ζ in the standard way. Namely we complete the square with respect to y as follows.
Then we eliminate the x 2 -term by setting finally yielding the Weierstass form for our Calabi-Yau fourfold W 4 . The discriminant of (2.6) is given by Expanding the discriminant in powers of z the coefficient of z 4 becomes

The second section
Beside the standard section ζ : B 3 → W 4 we require a second section Substituting this section of P/B 3 into Tate form (1.7), as in [14,15] one concludes that the condition that it lies in W 4 is This section allows a change of the group structure on the fibers of W 4 /B 3 by a translation.
Intertwining this translation with the action ofβ 4 allows us both to eliminate vector-like exotics from the F -theory model W ∨ 4 /B ∨ 3 and to introduce a (4 + 1)-split in the spectral divisor giving the U(1) X discussed in the Introduction. Translation of fibers by this section of course leaves the Weierstrass form on smooth elliptic fibers and I 1 -fibers invariant, that is, all fibers over (B 3 − S GUT ).
Notice that, over a general point b 3 ∈ B 3 , τ − ζ is not of finite order on Pic 0 of the cuspidal curve wy 2 = x 3 since the parameter t = y x = z takes all values. So τ − ζ is not of finite order in Pic 0 π −1 (b 3 ) if the a j are sufficiently small. Furthermore, if a 5 = −a 0 are small and a 2 = a 3 = a 4 = 0, the same argument gives So these same assertions hold for a general allowable choice of the coefficients a j . This fact is essential to the proof in [15] of lemma 6ii) below. It says that our F -theory model has a single Higgs doublet. Over is given by the equation x − z 2 w = 0. Letting a jkl := a j + a k + a l and using that a 420 = −a 53 we obtain by (2.7) that the third point of intersection with W 4 is given by substituting x = z 2 w in (2.4) to obtain w z 6 w 2 + a 420 z 5 w 2 1 y 2 + a 420 z 2 wy 1 = 0, that is, y 2 + a 420 z 2 wy − z 5 w 2 (z + a 420 ) = y − z 3 w y + (z + a 420 ) z 2 w = 0.
We denote the third section as Finally, we therefore have a fourth section defined by the third point of intersection of the tangent line to Taken together these calculations yield the following.

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Lemma 2. The sections ζ, τ , υ and µ of W 4 /B 3 satisfy the following relations in The last two relations imply that ζ + τ ≡ υ + µ, that is in classical language, the two divisors are members of the same distinguished g 1 2 on W 4 /B 3 .

The quotient Calabi-Yau manifolds
In order to wrap a Wilson line, we must require that involutionβ 4 /β 3 in (1.4) with quotient W ∨ 4 /B ∨ 3 be such that β 3 acts fixpoint-freely on the smooth anti-canonical divisor S GUT ⊆ B 3 yielding an Enriques surface and that the induced involutionβ 3 /β 2 on the dual Heterotic with quotient V ∨ 3 /B ∨ 2 be such thatβ 3 acts freely. In order that W ∨ 4 be Calabi-Yau, the involution β 3 on B 3 = P [u 0 ,v 0 ] ×B 2 must have only finite fixpoint-set since S GUT is an ample divisor in the Fano manifold B 3 . This forces β 3 to act as (−1) on the meromorphic two-form on B 3 with pole on S GUT . That in turn forcesβ 4 with quotient W ∨ 4 /B ∨ 3 to act as (−1) on the relative one-form dx/y on W 4 /B 3 since otherwise W ∨ 4 would not be Calabi-Yau. On the other hand, the induced involutionβ 3 /β 2 on the Heterotic threefold V 3 /B 2 must act as (+1) on the relative oneform dx/y since otherwise V ∨ 3 would not be Calabi-Yau. The possibility, even necessity, of the sign-reversal is explained in [14].

Three Calabi-Yau fourfolds related by quotienting
Following [15] we use root systems and toric geometry to actually define three base threefolds and associated Calabi-Yau fourfolds that we denote by B ∧ 3 is defined to be the resolution of the graph of the Cremona involution on P h SU(5) with respect to a basis given by the choice of a system of simple roots of SU(5) balanced between positive and negative Weyl chambers. We have reserved the least cumbersome notation for the intermediate one B 3 because, as we have already mentioned, it is computationally most convenient to work in that setting.
On the F -theory side our ultimate target is the (orbifold) Calabi-Yau fourfold W ∨ 4 with smooth Heterotic dual Calabi-Yau threefold V ∨ 3 having two bundles with Yang-Mills connections with structure group symmetry-breaking

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as described in [15]. Notice that the initial fourfolds and the involutions do not involve any choice of Weyl chamber. It is only the crepant resolutions of the Calabi-Yau fourfolds and threefolds that imply such choices. This is explained in [14].
In order to wrap a Wilson line, we require that W 4 /B 3 given in (1.7) admit an equivariant involutionβ 4 /β 3 with quotient W ∨ 4 /B ∨ 3 such that β 3 acts fixpoint-free on the smooth anti-canonical divisor S GUT ⊆ B 3 yielding an Enriques surface In order that W ∨ 4 be Calabi-Yau, the involution β 3 on B 3 = P [u 0 ,v 0 ] × B 2 must have only finite fixpoint-set since S GUT is an ample divisor in the Fano manifold B 3 . As explained in [14] this forces β 3 to act as (−1) on the meromorphic two-form on B 3 with pole on S GUT .

Unfolding the E 8 -singularity
A basic principle in the mathematics of String Theory is that the geometric model (1.7) of F -theory must be considered as having evolved according to the unfolding of the E 8 -surface singularity In [14] we have observed that principle to the letter, tracing the equivariant crepant resolution implicit in the Tate form (1.7) back to the Brieskorn-Grothendieck equivariant crepant resolution [11,51] of the semi-universal deformation of the rational double point singularity (1.8) by requiring that the section defining S GUT be given by a formula The assumption (2.7) will, as we will see, reduces the maximal subgroup decomposition so that, on the F -theory side, one begins with the identification of maximal tori compatible with the three-dimensional commutative diagram obtained by pasting the top and bottom morphisms ofṠ

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to the top and bottom morphisms, respectively, of the commuting diagraṁ of real analytic outer complex conjugation involutions. [14] and [15] are built around the necessity of a choice of a positive-negative pair of Weyl chambers of E 8 with the requirement that every step of constructions must commute with the passage between these two chambers. In particular, as we have shown in [14], the equivariant crepant resolutioñ W 4 /B 3 of W 4 /B 3 depends on the choice of Weyl chamber. This means that we will have two copies of the crepantly resolvedW 4 /B 3 that we designate by lettingẆ 4 /B 3 denote the F -theory model with a choice of positive chamber andẄ 4 /B 3 with its negative as the choice of positive Weyl chamber. The action ofβ 4 /β 3 onẆ 4 /B 3 will be a holomorphic involution that acts on roots as the longest element of the Weyl group W SU(5) yielding a Calabi-Yau quotient, and similarly for the quotient of the action onẄ 4 /B 3 . However these quotients are only real-analytically equivalent, not complex-analytically equivalent. As explained in [15] the exceptional curves over the quotient S ∨ GUT are 'flopped' when passing from one to the other. The flop is essentially invisible on the Heterotic side since it tracks only the real E 8 -bundles and the 'flop' becomes the passage between the two possible complexifications of the same real E 8 -bundle.
Identifying exceptional components over S GUT with positive simple roots forces the involutionβ 4 to act as the non-trivial involution on the A 4 -Dynkin diagram, that is, by the longest element of the Weyl group W SU(5) on the exceptional divisors of the crepant resolution of W 4 /B 3 . Again as shown in [14], it is the commutativity of the geometric involutionsβ 4 /β 3 andβ 3 /β 2 with the complex conjugate involution ι in (2.10) that allows us to incorporate both in the simultaneous quotienting on both the Heterotic and F -theory models that preserves initial E 8 -symmetry and subsequent SU(5)-symmetry since (2.10) acts trivially on SU(5) and E 8 .

The spectral divisor
The crepant resolutionW 4 /B 3 of W 4 /B 3 will have I 5 -type fibers over generic points of This I 5 -fibration over S GUT carries the SU(5) gauge -symmetry. On the other hand, SU(5) Higgs -symmetry is broken on a five-sheeted branched covering of B 3 given by the lift of to a divisorC Higgs ⊆W 4 . Its symmetry is broken by assigning non-trivial eigenvalues to the fundamental representation SU(5) Higgs using the spectral construction with respect to the push-forward to B 3 of a line bundle L Higgs onC Higgs . We see this as follows.

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We formP using the fiber coordinate t for O B 3 (N ). The natural projection ) is defined except along the section where x = y = w = 0.P contains the smooth five-dimensional incidence hypersurface Y given by the equation thereby forming a smooth quadric hypersurface over B 3 with distinguished section given by x = y = w = 0. So the restriction of (3.2) to Y is defined except along the section where it spreads the exceptional locus over the linear locus {x = 0} in P . The result is a birational morphismŶ → P that maps the exceptional locus over the section isomorphically onto {x = 0} ⊆ P and blows down the linear loci Ignoring what happens over {w = 0}, that is, setting w = 1 and using t as the affine fiber we obtain the equation the divisor given by the intersection withŴ 4 then has equation This is the equation on the affine set w = 1 of the intersection of the proper transform of W 4 with the locus given by and is called the spectral divisor. The spectral divisor, in particular, contains the singular locus of W 4 . The condition (2.7) implies that homogeneous form in (3.3) is divisible by z − t, that is, the spectral divisor admits a (4 + 1) factorization. The and leaves (3.3) invariant. Said otherwise, since W 4 is smooth except over {z = 0} ⊆ B 3 , the proper transform W 4 ⊆Ŷ of W 4 blows up the codimension-2 subvariety with exceptional locus P [t,z] × S GUT . The proper transformŴ 4 of W 4 intersects this exceptional locus in the hypersurface given by the equation It is immediate to check that the sections ζ and τ both lie in W 4 /B 3 and both lie in For fiber coordinate t for SP with is given by One sees easily that W 4 and (3.4) have contact of order 4 along (ζ) and order 1 along (τ ). Given b 3 ∈ (B 3 − S GUT ), we denote by SP (4) that since 0 = a 0 + a 2 + a 3 + a 4 + a 5 has equation where a jk = a j + a k , etc. 2 Thus the spectral divisor Higgs + C is the image of D = D (4) + D (1) inŴ 4 . 3 Thus the involutionβ 4 /β 3 preserves (3.9). Using that a 54320 = 0, the Higgs curve, that will be important throught this paper, is defined as the image in S GUT of the common solutions to the two equations Writing (3.10) as two equations in the variablet 2 /z 2 , the solution set doubly covers the surface in B 3 defined by the resultant equation obtained by substituting in the first equation to obtain that, again using a 54320 = 0, reduces to with branch locus defined by the restriction of the divisor class N .

Adjusting the E 8 -evolution
Because the image (τ ) of our second section τ actually generates with C Higgs in our F -theory model SU(5) gauge × SU(5) Higgs will be actually replaced by with maximal torus comprising a maximal torus in E 8 and so a vector space isomorphism of Cartan subalgebras and associated commutative diagram

Semi-stable degeneration and
as follows. Using the affine parameters we define the affine family That is

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In this way we identifyW as an elliptic fibration over the closure . Also as in [15] as coefficients in the Tate form (1.7) for W 4 /B 3 we require that (that is, anti-invariant under the involution β 3 ), in other words they must be linear combinations of the forms listed in table 2 in [15]. Furthermore also as described via table 2 in [15] the four-dimensional family of forms spanned by a 2 , a 3 , a 4 , where the a j,0 lie in the three-dimensional (−i)-eigenspace of the Z 4 R-symmetry T u,v , again as given in table 2 in [15]. Finally As we again showed in [15], is a particular degree-2 del Pezzo surface on which the involution β 2 on B 2 acts with four fixpoints. Thus B 2,δ = B 2 for all δ and and applying the relations (4.4). a 2 , a 3 , a 4 , a 5 are required to have no common zeros on B 3,δ for δ on the half-open real interval (0, 1]. The Tate form (1.7) and the action β 3 on B 3 as described in the tables in section 4 of [15] then determine equivariant involutionsβ 4 /β 3 on W 4 /B 3 . For generic choice of z, S GUT will not contain any of the eight fixpoints of β 3 acting on B 3 so β 3 will act freely on S GUT yielding a smooth Enriques surface as quotient. The sections a j,δ , z δ ∈ H 0 K −1 B 3 in the Tate form will be allowed to vary under the contraction (4.6) and in particular the discriminant component S GUT,δ ⊆ B 3,δ varies as defined by JHEP12(2019)016

Degeneration of a single K3-surface
Over each point (a, b) ∈ C 2 associate the Weierstrass form where g 2 is homogeneous of total degree 4 and g 3 homogeneous of total degree 6. Restrict the Weierstrass form to the locus so that for δ = 0, the discriminant has degree 24. As shown below compactifying at infinity yields a K3-surface elliptically fibered over the closure of Γ δ and setting δ = 0 yields the union of two dP 9 's meeting over Since g 2 and g 3 in (4.12) must be homogeneous forms of degree 4 and 6 respectively in [u 0 , v 0 ], dividing (4.12) by (u 0 + v 0 ) 6 gives the affine equation On the P 1 -fiber over b 2 ∈ B 2 we obtain the Weierstrass form where g 2 (b 2 , a) is a rational function of a with denominator of degree 4 and g 3 (b 2 , a) can be expressed as a rational function of a with denominator of degree 6. Thus one can write a global decomposition of (4.10) with Then, letting b = a −1 , we can equivalently write where the g 2 and g 3 are the functions on the curve {ab = 1} over the point b 2 in B 2 .

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Then for any b 2 ∈ B 2 and any (a, b) ∈ C 2 we can write the Weierstrass form 4.2 dP 9 -bundles over B 2 giving a singular Calabi-Yau fourfold As in section 4.1 we consider The union of our two dP 9 -bundles is then given as the subspace of defined by (5.11) and the equation Returning to our fibration for each fiber π −1 B 3 (b 3 ) ofW 4 /B 3 we will associate two copies of the Weierstrass equation, namely the one distinguished by designatingζ (b 3 ) as the identity element of the group structure and the other distinguished by designatingτ (b 3 ) as the identity element of the group structure. The two fibers π −1 ) are identified under the isomorphism induced by the involutionβ 4 induced by the involution β 3 on B 3 . However, as will become clear in section 6 the line bundle is not trivial over any b 3 ∈ S GUT . As we have seen in (2.6), the identification acts on sheaves F on the elliptic curve in Weierstrass form can be obtained as the smoothing the union of two dP 9 's described in subsubsection 4.1. That is, we realizeW 4 /B 2 as the smoothing of two dP 9 -bundles dP a ∪ dP b (4.14) over B 2 .

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Simultaneously, via we move S GUT to the reducible quadric z 0 given by where q (u 1 , v 1 , u 2 , v 2 ) is invariant under the action of the involution β 2 on B 2 . So at δ = 0 S GUT splits into two 'horizontal' components given by u 0 − v 0 = 0 and u 0 + v 0 = 0 and a 'vertical' component given by is independent of δ. From the Weierstrass forms just above, we read off thatW 4 is the smoothing over ab = 1 of the union of the two dP 9 -bundles given over each point b 2 ∈ B 2 by and These two dP 9 's over b 2 contain the common fiber of V 3 /B 2 whose Weierstass form is The spectral data on the Heterotic side, namely the two E 8 -bundles on the fiber of V 3 /B 2 over b 2 , are given via the Friedman-Morgan-Witten classification [23] by the two dP 9bundles. Namely in section 4.2 of [23] Friedman-Morgan-Witten give a classifying space for imbeddings of an elliptic fiber E b 2 of V 3 /B 2 into a rational elliptic surface dP 9 (b 2 ), each such corresponding canonically by a theorem of E. Looijenga [41] to an isomorphism class of flat E C 8 -bundles F over E b 2 . Considered as fibrations over B 2 , fibers are so-called dP 9 -surfaces. Setting [s, t] = [a ′ , a ′′ ], respectively [s, t] = [b ′ , b ′′ ] for s, t as in [23], fibers are given uniquely in P 3 1,1,2,3 by an equation weighted as per their respective indices.

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So to realize the semi-stable degeneration geometrically, the crepant resolutionW 4 /B 3 of W 4 /B 3 is given over the locus The deformation in this section is given by restricting the Weierstrass form to the locus Γ δ = {(a, b) : ab = δ} as δ goes to 0. Furthermore S GUT deforms with δ via the formula We thereby obtain the family W 4,δ of fourfolds over the affine line C * δ as in subsection 4.1. Then the Heterotic model (V 3 , F a , F b ) over B 2 canonically corresponds to a normal crossing Calabi-Yau 4-fourfold with two components dP a and dP b obtained by making the construction described just above equivariantly over B 2 .
Thus the family W 4,δ /B 3,δ defined by the Tate form on B 3,δ degenerates as δ approaches zero to a reducible Calabi-Yau 4-fold giving the Heterotic model V 3 /B 2 defined by a = b = 0. Finally there are common distinguished sections on every W 4,δ /B 3,δ . defined equivariantly by The Calabi-Yau threefold is elliptically fibered over B 2 × {a = b = 0}, the general fiber of dP a /B 2 has an I 5 -fiber at a = ∞ and the general fiber of dP b /B 2 has an A 4 -fiber at b = ∞.

The involution
Then by construction the involutionβ 4 /β 3 induces involutionsβ 4,δ /β 3,δ on W 4,δ /B 3,δ for all δ ≥ 0. Furthermore the above asssumptions force and (4.21) The action of these involutions over the fixpoints of the action of β 2 on B 2 is treated in detail in section 2.4 of [14].

Passing from Heterotic theory to F-theory
Essentially one passes from the Heterotic model to the F -theory model by reading the subsections of the previous section in reverse order and from bottom to top. We can paste one of these dP 9 along E at s = 0 and the other along E at s = 0 to obtain a normal crossing elliptic K3-surface with unobstructed deformation space thereby joining dP ∨ a to dP ∨ b to form a normal crossing Calabi-Yau fourfold. Having that, a theorem of Kawamata-Namikawa [31] guarantees that the normal crossing elliptic Calabi-Yau fourfold has an unobstructed deformation theory. B 3 = P [u 0 ,v 0 ] × B 2 so that our only choices are the section of K −1 B 2 in the definition of z 0 and the smooth section z of K −1 B 3 in (4.19).

Initial data
In Heterotic theory, we begin with the smooth degree-2 del Pezzo del Pezzo (B 2 , β 2 ) with involution β 2 with 4 fixpoints constructed in [15]. Letting B ∨ 2 denote the quotient, we are given a smooth, elliptically fibered Calabi-Yau threefold with Weierstrass form in order that V ∨ 3 have trivial canonical bundle. Has fundamental group Z 2 .

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with Yang-Mills connections. Pulling back via we have two β 2 -invariant E 8 -bundles, F a and F b each with a Yang-Mills connection with fiber over b 2 ∈ B 2 . The smooth, torus-fibered Calabi-Yau threefold V ∨ 3 /B ∨ 2 with fundamental group Z 2 is also endowed with two disjoint sections The pull-back of the two sections under becomes the union of two disjoint sections ζ, τ : Also, following section 2.4 of [14], the fiber of the smooth threefold V ∨ 3 /B ∨ 2 over the four orbifold points of B ∨ 2 must be acquired with multiplicity two, being doubly covered by the elliptic fibers E b 2 of V 3 /B 2 over the β 2 -fixpoints b 2 ∈ B 2 . The covering must be unbranched via translation by a distinguished half-period. Given the prior conditions imposed on our model that half-period must be O E b 2 (ζ (b 2 ) − τ (b 2 )). The role of the logarithmic transform over a neighborhood of each β 2 -fixpoint, especially how it induces the action (b 2 , (x, y)) → (β 2 (b 2 ) , (x, y)) ofβ 3 /β 2 on (Weierstrass) fibers of V 3 /B 2 , as well as the transition from the β 2 -pull-backs is explained in [14]. The Yang-Mills connections on the bundles (5.3) restrict to sums of eight flat line bundles on each fiber of V 3 /B 2 and are completely determined by that family of restrictions.

5.2
Building a normal-crossing K3 from an elliptic curve with two flat E 8bundles As in (4.18) the two flat E 8 -bundles on the elliptic fiber
That is, over B 2 we have the union dP a ∪ dP b of two dP 9 -bundles with dP a ∩ dP b = V 3 .
Since the canonical bundle of this normal-crossing variety is trivial, the theorem of Kawamata-Namikawa cited above establishes that its deformation space is unobstructed. In fact we employ the very specific smoothing determined by (4.2) and (4.6).
Finally for the homogeneous coordinates [u 0 , v 0 ] in [15] and the curve (4.3), the equa- allow us as in (4.2) to form the family of hypersurfaces with fibers given by The forms (4.6) then yield our F -theory model W 4 /B 3 at δ = 1.

In section 4.2 of [23]
Friedman-Morgan-Witten gives a classifying space for imbeddings of an elliptic curve E in Weierstrass form into dP 9 's, each corresponding canonically to an isomorphism class of flat E 8 -bundles The flat E 8 -bundle F is given as the sum of eight flat line bundles, each given by the divisor of the form p − e where e = ζ ∩ E is the identity element of E considered as an abelian group and p is a geometrically given point on the torus E. Again we follow Friedman-Morgan-Witten in [14] and consider the family of dP 9 -hypersurfaces . − α 6 s 6 (5.5)
ii) The action of the involutionβ 3 on the intersection V 3 of the two dP 9 -bundles is given by the map where addition is with respect to the addition law on the elliptic curve and the (identical) Weierstass forms iii) Over fixpoints b 2 of the action of β 2 on B 2 , τ (b 2 ) − ζ (b 2 ) is a non-trivial half-period on the fiber E b 2 of V 3 /B 2 but the action ofβ 3 on the intersection cycle of length nine determining the E 8 -bundles on the fiber is given by tensoring with the canonical section of the line bundle associated to the divisor ζ (b 2 ) − τ (b 2 ) .

Proof.
i) This assertion is immediate from the definition ofβ 4 .

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ii) First of all the involution is given by but the fiber of V 3 /B 2 is given in (4.16) and (4.17) as so that it does not reflect the sign change of a and we showed in [14] how the sign change of y is absorbed in the process of taking residue. Secondly the equations of the two K3-surfaces fibersW respectively, ii) is proved.
iii) Choosing a 5 = 0 at the fixpoints b 3 of the action ofβ on B 3 separates ζ (b 3 ) from τ (b 3 ). The assertion is forced by the fact that, in the canonical identification of an elliptic curve E with Pic 0 (E), the trivial bundle is associated to ζ (b 3 ) for the Weierstrass form based at ζ (b 3 ) and is associated to τ (b 2 ) for the Weierstrass form based at τ (b 3 ).

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so that there is a unique translation such that the composition takes the trivial bundle to itself under the change of basepoint of the torus E from e to e ′ . Using the decomposition of the moduli space of semi-stable E 8 -bundles on E, the functorial diagram is such that the bottom horizonal map is the identity map. This is what allows us to induce a semi-stable E 8 -bundle on the quotient See [14] for a more detailed analysis.

SU(5) gauge -roots and the semi-stable limit
The action ofβ 3 on V 3 /B 2 = dP a ∩ dP b is explained in [14] through the lens of its local action over a neighborhood of β 2 -fixpoints, as is the compatibility of this action with the action induced byβ 4 on the semi-stable degenerationW One checks directly that this action is compatible with the semi-stable degeneration of S GUT to z 0 = q · u 2 0 − v 2 0 = 0 . z 0 deposits an I 5 -fiber at the point {a ′′ = 0} on each fiber of dP a /B 2 and at the point {b ′′ = 0} on each fiber of dP b /B 2 . By construction,ζ andτ can only meet over {z δ = 0}. If

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is any of the 4 fixpoints of the involution β 2 , then when δ = 0 the fixpoints of the action of β 3 on the fiber P [u 0 ,v 0 ] × {b 2 } occur at a = 0, ∞ since β takes a to −a. We choose the in the definition of z 0 so as not to vanish on any of the fixpoints of the action of β 2 on B 2 .
6 Geometric model-double cover form 6.1 Putting the sections ζ and τ of W 4 /B 3 on equal footing As mentioned earlier, the Z 2 -action will ultimately incorporate a translation in the fiber direction that will interchange the two distinguished sections ζ and τ , and so will interchange the two distinguished divisors, (ζ) and (τ ), to which they correspond. In particular, the Weierstrass forms for the elliptic fibration determined by either of the two sections must be 'on equal footing with the other one' throughout. So we will have to begin from another birational model for W 4 /B 3 that achieves the desired 'equal footing'. On each fiber of W 4 /B 3 , the intersection with the sections (ζ)+(τ ) distinguishes a g 1 2 (that is, a linear series of projective dimension one and degree two). Taken together the g 1 2 's yield a P 1 -bundle Since we are eventually going to change the elliptic group structure on the torus fibers of W 4 /B 3 from the one given by (ζ) that does not pass through the set of singular points of W 4 to the one given by (τ ) that actually contains the set of singular points of W 4 , we first want to define a crepant partial resolution of our F -theory model W 4 /B 3 that becomes a branched double cover of Q/B 3 and has the property that the proper transform of (τ ) misses the singular set of the resulting fourfold entirely. We achieve this by projecting each torus fiber from the third point υ (b 3 ) of intersection of the line between ζ (b 3 ) and τ (b 3 ) with the fiber of W 4 /B 3 over b 3 .

Line between the sections
We have denoted the third point of intersection of this line with Thus we can modify the defining equation (Notice that this is the identity matrix over a first-order neighborhood of S GUT .) Recalling again that −a 53 = a 420 , we obtain as the equation for W 4 in the projective coordinates [w, X, Y ].

Fundamental projection and modified Weierstrass form
By considering (6.3) as a quadratic equation in w, the Tate form then yields a birationally equivalent double-cover where the branched double cover W 4 of Q is given by the equation where the fiber coordinates of Q/B 3 are given by (6.4) also allows us to see that W 4 is birationally a double cover where

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and with branch locus ∆ given by the zeros of the homogeneous polynomial (6.4). Notice that (6.5) is invariant under the transformations   [(a 0 , a 2 , a 3 , a 4 , a 5 so both are possible but only one will leave the holomorphic four-form on W 4 invariant, namely the one that is compatible with the action ofβ 4 /β 3 that transforms the relative one-form by dx y → − dx y . To see which one, we divide (6.5) by X 4 and define We have the affine equation on the relative one-form so we conclude that w 0 must be invariant. Furthermore, since the sections ζ and τ are now given by {X = 0}, so, referring to (6.5), their equation becomes and so each of the two sections must be taken to itself underβ 4 /β 3 . The canonical bundle of Q is where (X 0 ) denotes the divisor {X 0 = 0} whereas the branch locus ∆ has divisor class 4 (X 0 ) + 4N.
We therefore replace W 4 with the birationally equivalent double cover branched over ∆ and W 4 is Calabi-Yau. Furthermore is the union of the proper transforms of the two original sections of W 4 /B 3 .

The branch locus
The branch locus ∆ is defined by the equation the total space of the line bundle O B 3 (N ). Then the equation for can be rewritten as Again rescaling the a j and appealing to Bertini's theorem, we will have for general choices of the a j ∈ A that singularities of ∆ are supported on the locus Thus S GUT × B 3 W 4 has two components, each isomorphic to S GUT × B 3 Q. They are given by One of these components, that we will denote as D 1 , intersects (ζ), while the other, that we will denote as D 4 , intersects (τ ). We write D 1 =: {G 1 = 0} and D 4 =: {G 4 = 0} so that on W 4 . Over {z = 0} the two components coincide as the component {ϑ(ϑ 0 − a 5 ) = 0} locus of the reducible double cover

The standard P P P 112 -formulation
To relate the presentation (6.4) to the more standard P 112 -notation used for this type of model, letP with coordinates (w, x, y) and write so that that can be rewritten as (6.11) We include appendix A.1 by Sakura Schäfer-Nameki containing a brief overview of elliptically-fibered Calabi-Yau manifolds symmetric with respect to two sections.

Localizing at the singularities of W 4
We first rearrange terms of (6.7) in increasing order of total degree in the variables (z, ϑ 0 ) so that the branch locus ∆ is given by (7.1) ∆ will have ordinary nodal singularities along {z = ϑ 0 = 0} except where the quadratic normal cone (a 5 ϑ 0 + a 4 z) 2 + 4a 5 a 420 z 2 = 0 is not of maximal rank, namely where a 5 a 420 = 0.
If a 5 a 420 = 0, the reduced quadratic cone is given by on which the cubic cone evaluates as a 2 5 (a 420 a 4 + (2a 4 + a 2 + a 5 ) a 5 ) = 0.

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So the quadratic and cubic cone both vanish identically over {a 5 = 0} but both only vanish over {a 420 = a 53 = 0} when additionally a 3 5 (a 4 − a 0 + a 5 ) = 0, (7.2) a locus that has only finite intersection with S GUT . We next rearrange terms of (6.7) in increasing order of total degree in the variables (z, a 5 − ϑ 0 ). The branch locus ∆ is then given by

First modification of ∆
We first make the modification of ∆ and W 4 at {ϑ = z = 0}. Namely, inside we write The proper transform ∆ (1) of ∆ in Q (1) is given by the equation of the quadratic equation in ϑ 1 Z 14 is singular, that is, where Z 0 = a 5 = a 420 = 0. Substituting in (7.5) we conclude that, in addition, a 4 = 0. Since ∂ ∂Z 0 applied to (7.5) and evaluated at such singular points would also have to vanish, we would also have a 2 = 0 and so a 0 = 0 contradicting the assumption of a generic allowable selection of the a j and z in the linear system |A|.
The assumption that the curve {a 5 = z = 0} ⊆ B 3 is smooth can be weakened to allow nodal singularities. Since a 4 = 0 there, the potential nodal singularities in ∆ (1) thereby created are in fact already resolved by the fact that the entire fiber of Z 0 already lies in ∆ (1) over any point where a 5 = a 4 = 0. In fact, in section 6.3 of [15] we impose a normal-crossing assumption on The proper transform ∆ (1) of ∆ in Q (1) is the total transform minus twice the exceptional divisor, so that the proper transform is again twice the anticanonical divisor of Q (1) . So the branched double cover W (1) 4 given by the equation is again Calabi-Yau. We retain the notation D 1 and D 4 for the proper transforms in W

Second modification of ∆
Recalling that z = Z 0 Z 14 we must next attend to the singularities of ∆ (1) lying in {Z 14 = 0}. As we have seen in the last subsection, these lie on the locus {a 5 − ϑ 1 Z 0 = Z 14 = 0} that only intersects D 0 over {a 5 = Z 0 = 0}.

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First, referring to (7.5), we rewrite the equation for ∆ (1) in terms of the variables ((a 5 − ϑ 1 Z 0 ) , Z 14 ) as Next define and The equation for the proper transform ∆ (2) of ∆ (1) becomes The proper transform ∆ (2) in Q (2) is the total transform minus twice the exceptional divisor, so that the proper transform is again twice the anticanonical divisor of Q (2) . Therefore we have a Calabi-Yau fourfold W Over the exceptional locus, given by {Z 23 = 0} ⊆ Q (2) , the equation for ∆ (2) is a perfect square so that W (2) 4 × B 3 S GUT splits into five components. Two of these are new components lying over {Z 23 = 0} ⊆ Q (2) that we call D 2 and D 3 . In addition we have lifted components, the two components that we continue to call D 1 and D 4 lying over Z 14 = 0 ⊆ Q (2) , and finally the lifted component over {Z 0 = 0} that we continue to call D 0 . We number things so that, over a general point of S GUT , D 2 intersects D 1 and D 3 intersects D 4 . The incidence of the five components over a general point of S GUT is that of the extended Dynkin diagramÃ 4 . By lemma 4 singular points of ∆ (2) can only occur where Z 23 = 0, soZ 14 = 0 there and these points can be singular points of ∆ (2) only if ϑ 1 ϑ 2 Z 14 = −a 4 and

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Multiplying both sides of this last equation by ϑ 1 and substituting, then recalling that at these points a 5 − ϑ 1 Z 0 = 0 and that a 54320 = 0, we obtain a 5 a 0 + a 3 (a 5 + a 4 ) = 0 that can be rewritten as the relation given in (7.4).
Lemma 5. Singularities of the branch locus ∆ (2) only occur over and then only where Over a general point of {z = a 5 = 0}, D 0 splits into the two components D 01 + D 04 that reconfigure with the specialization of D 2 + D 3 over {z = a 5 = 0} to form the extended Dynkin diagram for D 5 as is shown by the relation Over a general point of {z = a 420 = a 53 = 0} again D 0 splits into two components, augmenting the extended Dynkin diagramÃ 4 to the extended Dynkin diagramÃ 5 .

Singularities of higher codimension
We have seen in lemma 5 that, at singular points of ∆ (2) , Z 23 = 0 andZ 14 = 0 and they lie over In fact, if one rewrites (7.9) in the form Recalling thatZ 14 and ϑ 1 ϑ 2 cannot vanish simultaneously, one sees that is given by the vanishing of all four entries in the 2 × 2 matrix (7.11) is a simple nodal locus of W 4 . SinceZ 14 is never zero along C we conclude that The entries in (7.11) are all invariant under the action ofβ 4 /β 3 . Therefore we can take either small blow-up to complete the crepant resolution.

The smooth modelW 4
After taking the small resolution of the nodal curve C where∆ is non-singular.
We retain the notations D j , j = 0, . . . , 4, for the proper transforms inW 4 of the corresponding divisors in W (2) 4 . We will let G j denote the canonical section of the line bundle determined by the divisor D j , that is

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on the exceptional fibers ofW 4 . As explained in [14], the action ofβ 4 on the roots D 1 , . . . , D 4 reverses the choice of positive Weyl chamber used in making the dictionary between exceptional divisors ofW 4 and the SU(5) gauge -roots. This reversal of positive Weyl chamber exactly reverses the non-trivial involution (7.14) thereby preserving the SU(5) gauge -symmetry of the quotient W

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where (X) denotes the divisor given by X 0 = 0, etc. The Picard group ofW 4 is generated by The cohomology class of∆ ⊆Q is given by 4 ((X 2 ) + N ) − 2 ((Z 0 ) + (Z 23 )) (7.26) and where, as before, N =π * (c 1 (N )) and N is the line bundle on B 3 whose sections include z and the a j . Thus the Picard group of∆ admits the effective square root of the branch locus of πQ. Also Higgs + image (τ ) ≡ 5 (X 2 ) + 5N ∈ Pic Q . (7.27) We have the following linear equivalences onQ: Finally, since none of the blow-ups in the resolution over B 3 touch (X 0 ), from now on we will simply identify (X) := (X 0 ) = (X 1 ) = (X 2 ) .

Intersections in Q (2)
We compute push-forwards to B 3 of intersections inQ as follows. From the fact that (π B 3 ) * ((Z 0 ) · (Z 23 )) is supported on the curve {z = a 5 = 0} we conclude that

Physical interpretation
We begin with the projection map Recall that E 8 has the subgroup SU(5) gauge × SU(4) Higgs × U(1) X and the Higgs operator is a non-trivial element of the center of the enveloping algebra of SU(4) Higgs . We use it to break E 8 -symmetry of the subgroup SU(4) Higgs . To accomplish this we must identify a line bundle

The Higgs bundle
So we begin with any line bundle L on D (4) . By the Grothendieck Riemann-Roch theorem, where R is the ramification divisor of ψ.
To construct L Higgs we proceed as follows. Applying the formula for the discriminant of a fourth-degree equation (7.1) we obtain that the discriminant has class 6 · N on B 3 . Thus we must have so that we must choose an effective divisor class on D (4) whose push-forward to B 3 has class c 1 N 3 . One obvious line bundle to use is We denote the line bundle on D (4) given by this divisor as = {t/z = 1} · D (4) and, as in [15] the Higgs curve is given by a branched double coverΣ We will need the skew component of the push-forward to Σ Since the degree of the canonical bundle of a curve is 2g − 2, the genus of Σ As we showed just above L is nothing more than the theta-characteristic Z 2 ·Z in section 5.1 of [15] for Z 2 = Σ

. Thus
We therefore have the desired Euler characteristics for the Higgs bundle on the matter and Higgs curves. However the ranks of the relevant spaces of sections are not quite right.
We will rectify the undesired outcome by modifying L  The G-flux GW 4 is defined as the push-forward intoW 4 of the Chern class of L Higgs modified by aβ 4 -invariant two cycle in (X) so that it has intersection number zero with π * B 3 (C) for any curve C in S GUT . Since we can essentially identify C Higgs ⊆Q with its lifting C Higgs ⊆W 4 we will first work to compute the class of the G-flux GQ ⊆Q. By (7.27) Higgs ≡ 4 (X) + 5N ∈ Pic Q .

The D-term
We next consider the U(1) factor of the (4 + 1)-decomposition of the spectral variety. Following [24] this D term must vanish, otherwise we might break R-parity in the low energy theory. After resolution,W is a branched double cover, so thatW 4 will inherit ample divisors from sufficiently ample divisors onQ. So, to compute the D-term we must first adjust the divisor

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so that its intersections with all curves of the form (X) ·Q D and π * B 3 (point) are zero. Now so that we must adjust by −4N . But the D-term is computed by identifying an ample divisor onQ that has zero intersection number with the two-class But GQ ·Q C Higgs − 4 X −Ŝ GUT = GQ ·Q 0 = 0 so any ample divisor trivially has zero intersection with (8.12).
9 Wilson line: symmetry-breaking to the standard model (5) denote the hypercharge direction when acting on the fundamental represesentation. The Wilson line is the flat rank-5 vector bundle L Y given by the homomorphism that takes the generator to exp (6πi · Y ) when viewed as acting on the fundamental representation of SU (5)  permutation group S 4 ⊆ W SU(5) as in [15]. Now on the Heterotic side dP a ∪dP b is simply the blow-up of W 4,0 with center (υ).
Since (x, y) → (x, −y) before we introduce the translation into the definition ofβ 4 we have that whereas composing with the translation we have that allowing the translation to descend to the quotient. Previous construction utilized a single section that was invariant under the involution, thereby forcing the existence of vector-like exotics. The existence of two sections and the incorporation of the translation between them into the Z 2 -action allows the elimination of the vector-like exotics.
It is (υ) on which the non-contractible cycle of dP a ∪ dP b is wrapped, while (ζ) and (τ ) are interchanged. Thus it is (υ) that endows the quotient of the action byβ 4 with a U(1) X -symmetry.
10 Bulk and chiral spectra with Wilson Line on W ∨ 4

Bulk spectrum with Wilson line
Since the canonical bundle of S GUT is trivial, the introduction of the translation into the definition ofβ 4 will allow us to replace the trivial line bundle by in the computation of the bulk spectrum. More specifically, the semi-stable degeneration is the geometric bridge between Heterotic theory and F -theory. The introduction of the translation into the definition ofβ 4,0 affects the dictionary (4.18) that is given in terms of the Weierstrass form on the dP 9 -bundles coming from the F -theory side and flat line bundles on the elliptic fibers on the Heterotic side. Change of basepoint from the one given by ζ to the one given by τ does not change the uniquely given Weierstrass form of the elliptic fiber however on the other side it does change the sum of eight flat line bundles given by the restriction of the E 8 -bundle since they are given by the differences between each of the eight points marked by the (asymptotic) Tate form and the identity element of the elliptic fiber as a group and the involution changes this last from the intersection of the torus fiber with (ζ) to the intersection of the torus fiber with (τ ). Therefore referring to tables 1 and 2 in section 7 of [15], the flux distribution associated to the Z 2 -action given by the involutionβ 4 and the Wilson line that is wrapped by that involution is presented in the tables below. Namely the distribution of MSSM matter fields is as follows: In this paper we have constructed an SU(5) GUT F -theory model. We have shown how to break the GUT group with a non-local Wilson line. Thus we are able to identify the GUT scale with the compactification scale of the GUT surface. Our model includes three families of quarks and leptons and one pair of Higgs doublets. The price to pay for this result is that we have a mirror world where the mass scales and couplings of the mirror states may be different than for the MSSM. This mirror world can, in principle, be the dark matter of the universe. There may or may not also be direct couplings of the mirror and MSSM sectors of the theory. These give interesting physics as in [4,7,13]. There are no vector-like exotics in the bulk spectrum or on the matter curves, neither are there chiral exotics.
The existence of the Z 4 R-symmetry generated by the automorphism T u,v that reverses the sign on the holomorphic 4-form addresses the issue of dimension-5 proton decay operators and forbids a µ-term [39]. The charges of the matter states under the Z 4 R-symmetry are given in table 3. Finally, Wilson line symmetry-breaking is addressed in sections 9 and 10.
However there are several issues which are not resolved in this paper. Moduli stabilization is not addressed. We have not generated a µ-term. In principle the Z 4 R-symmetry can be broken by non-perturbative physics down to matter parity which then allows for a µ-term of order the weak scale and severely suppressed dimension-5 proton decay operators. We have not discussed the possible Yukawa interactions needed to give quarks and leptons mass. We may or may not also have right-handed neutrinos which would be useful for a see-saw mechanism of neutrino masses. Finally, the U(1) X gauge symmetry may be broken to Z 2 matter parity via non-perturbative effects at the GUT scale or by a Stueckelberg mechanism. This would then allow for right-handed neutrino Majorana masses near the GUT scale.
A P P P 112 Formulations by Sakura Schäfer-Nameki A.1 RealizingW 4 in P P P 112 An alternative formulation of the elliptic fibrations W 4 can be given in terms of the P 112fibration in [45]. Let This elliptic fibration has two rational sections w = y = 0 w = y + b 0 x 2 = 0 that can be made symmetric by the shift Denote by γ i = ord z (c i ) and let Q (γ 0 , γ 1 , γ 3 , γ 0 , β 0 , β 1 , β 2 ) denote the quartic (A.1) for the given values. Using Tate's algorithm, there are various ways to degenerate this to an I 5 fiber about the locus {z = 0} [34], both by specifying the vanishing orders in (A.1) without further relations among the leading order coefficients c i and b j ('canonical Tate models'), or by imposing relations among the coefficients ('noncanonical models').
The I 5 singular fiber enhances to I 6 along −a 53 = a 420 = 0 and a 4 −a 5 a 0 +a 3 a 3 = 0, and we have I * 1 enhancement along a 5 = 0. The sections intersect the I 5 fiber along D 0 and D 2 , where, as above, the rational curves in the I 5 associated to the simple roots α i are denoted by D i , with the extended node corresponding to D 0 . This type of I 5 fiber with two rational sections was denoted by I (0||1) 5 , where the separation of the sections is #| − 1. [34,35] A.2 Other models with similar fiber type A different set of I (0||1) 5 non-canonical models were determined in [34] This model is based on the I 4 fiber Q (3, 2, 1, 0, 0, 0, 1) given by c 0 z 3 w 4 + c 1 z 2 w 3 x + c 2 zw 2 x 2 + c 3 zwx 3 = y 2 + b 0 x 2 y + b 1 ywx + b 2 zw 2 y.
where in addition we impose This last is solved as follows: This fibration Q(3, 2, 1, 1, 0, 0, 1)| P 0 =0 has generically I 5 fiber, with same intersection pattern with the two rational sections as W 4 , but it provides more matter loci, shown in the table, with the following expressions for the matter curves