Compact, Singular G2-Holonomy Manifolds and M/Heterotic/F-Theory Duality
CCompact, Singular G -Holonomy Manifolds andM/Heterotic/F-Theory Duality Andreas P. Braun and Sakura Sch¨afer-Nameki Rudolf Peierls Centre for Theoretical Physics,University of Oxford, Oxford, OX1 3NP, UK andreas.braun physics.ox.ac.uk Mathematical Institute, University of OxfordWoodstock Road, Oxford, OX2 6GG, UK gmail: sakura.schafer.nameki
We study the duality between M-theory on compact holonomy G -manifolds and the het-erotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G -manifolds,called twisted connected sums, which lend themselves to an application of fiber-wise M-theory/Heterotic Duality. For a large class of such G -manifolds we are able to identify thedual heterotic as well as F-theory realizations. First we establish this chain of dualities forsmooth G -manifolds. This has a natural generalization to situations with non-abelian gaugegroups, which correspond to singular G -manifolds, where each of the K3-fibers degenerates.We argue for their existence through the chain of dualities, supported by non-trivial checks ofthe spectra. The corresponding 4d gauge groups can be both Higgsable and non-Higgsable,and we provide several explicit examples of the general construction. a r X i v : . [ h e p - t h ] A ug ontents G -manifolds 5 G -Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 M-theory on K3/Heterotic on T . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Fiber-wise Duality for Elliptic K3s . . . . . . . . . . . . . . . . . . . . . . . . 10 G -manifolds 12 G -manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 19 G with Non-Higgsable Gauge Groups 21 G -manifolds . . . . . . . . . . . . . . . . . . . . 22 G -Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 G -Manifolds for Non-abelian Theories . . . . . . . . . . . . . . . . 356.3.1 Dual Models with G = SU (2) . . . . . . . . . . . . . . . . . . . . . . . 356.3.2 Dual Models with G = SU (5) . . . . . . . . . . . . . . . . . . . . . . . 366.3.3 Dual Models with G = G . . . . . . . . . . . . . . . . . . . . . . . . . 376.3.4 Dual Models with a Gauge Group of Large Rank . . . . . . . . . . . . 37 Details of Geometric Constructions 40
A.1 Nef Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2 The Schoen Calabi-Yau from a Nef Partition . . . . . . . . . . . . . . . . . . . 41A.3 G Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.4 Calabi-Yau Fourfolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Exploring and developing tools to characterize minimally supersymmetric theories, in par-ticular super-conformal theories (SCFTs), is a challenging task. The successes of studying
N ≥ N = 1 theories. Here we would like to focus on 4d N = 1 theories. A powerful approach tostudying and classifying 6d N = (1 ,
0) SCFTs is provided by F-theory [1], which heavily re-lies on the purely geometric nature of the classification problem, complemented with anomalyconsiderations. In 4d F-theory vacua, the geometry is only one part of the story, and needsto be supplemented with four-form G -flux data. This somewhat complicates the F-theoreticapproach.An alternative way, which is largely geometric, and yields 4d N = 1 vacua, is M-theoryon 7-manifolds with G -holonomy. Here the geometry of the manifold determines both thegauge group (from codimension 6 singularities), and if present, the spectrum of chiral matter(codimension 7 singularities), without the necessity of turning on fluxes [2]. The shortcomingsof this approach are thus far the absence of any compact models, which have singularities toengineer chirality in 4d. String compactifications on G -manifolds result in either 3d N = 1(for heterotic) or 3d N = 2 (for Type II strings) theories [3], whose moduli space would indeedalso be of great interest to explore further. For a review of the works until the early 2000s werefer to [4].Recently progress in the construction of compact G -manifolds has been made withinmathematics [5, 6], based on earlier results by Kovalev [7]. These so-called twisted connectedsum (TCS) G -manifolds have a K3-fiberation over a three-manifold. In the physics literature,M-theory compactifications on such geometries were studied in [8–10] and in string theorycompactifications [11].The K3-fibration of the TCS geometries is particularly suggestive within the context of M-theory compactifications, as it is amenable to M-theory/heterotic duality. This duality statesthat M-theory on a K3-surface has the same low energy effective theory as the heterotic3tring on T [12]. Applying this duality fiberwise over a three-dimensional base, suggests theduality between M-theory on a TCS G -manifold and heterotic string theory on a Calabi-Yaumanifold, which has a Strominger-Yau-Zaslow (SYZ) fibration [13], i.e. a special Lagrangian T -fibration over a three-dimensional base:M-theory on TCS G -manifold ←→ Heterotic on SYZ-fibered Calabi-Yau three-fold . (1.1)This is the duality that we will explore in the present paper.The main result of our work is the construction of compact TCS G -manifolds, along withtheir heterotic and F-Theory duals. Our identification of dual geometries is substantiated bya general proof that the spectra of the dual compactifications agree. In particular, we are ableto extend this duality to non-abelian theories, i.e. TCS G -manifolds, which have singularK3-fibers over the entire three-manifold base. This does not yet realize the ever so elusivechiral fermions from a compact geometry, however it provides a construction of a singular G ,which has non-abelian gauge groups. The goal of this work is to setup the M-theory/Heteroticduality in the context of the TCS construction including singular K3-fibers, thus setting thestage for potentially modifying the geometry to include conical singularities such as proposedin the local modes of [2, 14].Among the non-abelian gauge theories we engineer, there are both Higgsable and non-Higgsable models. From an F-theoretic point of view, the non-Higgsable models constructedhere are examples of geometrically non-Higgsable clusters [15, 16] for Calabi-Yau fourfolds.In addition, we show that in the present case these theories are also non-Higgsable in the 4deffective theory.The plan of this paper is as follows. Section 2 reviews the TCS construction of G holonomymanifolds, as well as the duality between M-theory on K3 and heterotic on T , and outlines thefiber-wise duality which is applicable when the K3s in the M-theory geometry are ellipticallyfibered. In section 3 we then construct the heterotic dual to a smooth TCS G -manifold, wherethe geometry is the Schoen Calabi-Yau three-fold. Using the duality to heterotic we thenconstruct TCS fibrations with singular K3-fibers in section 4, which have non-abelian gaugegroups in 4d, that we show to be non-Higgsable. To solidify the study of these singular K3-fibered G compacitifications, we also determine the F-theory dual for each of these models insection 5. In section 6 this construction is generalized to M-theory on TCS G s whose buildingblock realize arbitrary not-necessarily non-Higgsable gauge groups. This setup provides amultitude of generalizations and extensions, some of which are discussed in section 7.4 M/Het Duality and TCS G -manifolds We study M-theory on manifolds with G holonomy, which are constructed as twisted con-nected sums with K3-fibered building blocks. We map this by fiber-wise application of the 7dheterotic/M-theory duality to the heterotic string on an SYZ-fibered Calabi-Yau three-fold.In this section we review the TCS construction for G -manifolds, as well as the standardheterotic on T to M-theory on K3 duality. G -Manifolds To begin with, we review the basics of the twisted connected sum (TCS) construction of G holonomy manifolds following [5–7]. Let Z ± = K π ± −→ P (2.1)be two holomorphic three-folds which are smooth and K3-fibered over P , and have first Chernclass c ( Z ± ) = [ S ± ] , (2.2)where [ S ± ] is the class of the generic K3-fiber. The Z ± are called building blocks. Note thatdue to (2.2) these are not Ricci-flat, as the K3-fiber is only twisted half as much as neededfor a Calabi-Yau three-fold. Furthermore, we require that H ( Z ± , Z ) is torsion free and thatthe image, N ± , of ρ ± : H ( Z ± , Z ) → H ( S ± , Z ) ∼ = Λ ≡ U ⊕ ⊕ ( − E ) ⊕ , (2.3)is primitive in H ( S ± , Z ), i.e. Λ /N ± has no torsion. The orthogonal complement of N ± in Λis denoted by T ± .From this we can construct two asymptotically cylindrical (acyl) Calabi-Yau (CY) three-folds X ± = Z ± \ S ± , (2.4)where S ± are smooth fibers above points p ± in the P bases of Z ± . A G -manifold is thenobtained as a twisted connected sum of X ± × S e ± as follows: in the vicinity of the excisedfiber S ± , X ± has an asymptotically cylindrical region in which it approaches S ± × S b ± × R + .In these asymptotic regions, the two building blocks X ± × S e ± are identified by exchanging S e ± with the circles S b ± , while the asymptotic K3-fibers are mapped to each other by ahyper-K¨ahler rotation, also called a Donaldson matching: g ∗ : ω S ± ←→ (cid:60) Ω S ∓ , (cid:61) (Ω S ± ) ←→ −(cid:61) (Ω S ∓ ) . (2.5) We will denote circles by S in the following. ω is the K¨ahler form, and Ω (2 , = (cid:60) Ω + i (cid:61) Ω the holomorphic two-form of the respectiveK3-fiber. The resulting 7-manifold J was shown to admit a G holonomy metric [5–7].The identification between S b ± and S e ± results in a three-sphere S glued together fromtwo solid tori. The G holonomy manifold then can be viewed as a K3-fibration over S K (cid:44) → J → S . (2.6)The K3-fibers are conjectured to be co-associative four-cycles in the G manifold, i.e. theyare calibrated with (cid:63) Φ , where Φ is the G form.By the Torelli theorem, the diffeomorphism g is induced by a lattice isometry g Λ : H ( S +0 , Z ) → H ( S − , Z ) , (2.7)and we can think of this lattice isometry as a being in turn induced from a common embeddingof N ± into Λ.Using the Mayer-Vietoris sequence then allows to compute the integral cohomology groupsof J in terms of the data of the building blocks and the Donaldson matching as [5, 6, 17] H ( J, Z ) = 0 H ( J, Z ) = N + ∩ N − ⊕ K ( Z + ) ⊕ K ( Z − ) H ( J, Z ) = Z [ S ] ⊕ Γ , / ( N + + N − ) ⊕ ( N − ∩ T + ) ⊕ ( N + ∩ T − ) ⊕ H ( Z + ) ⊕ H ( Z − ) ⊕ K ( Z + ) ⊕ K ( Z − ) H ( J, Z ) = H ( S ) ⊕ ( T + ∩ T − ) ⊕ Γ , / ( N − + T + ) ⊕ Γ , / ( N + + T − ) ⊕ H ( Z + ) ⊕ H ( Z − ) ⊕ K ( Z + ) ∗ ⊕ K ( Z − ) ∗ H ( J, Z ) = Γ , / ( T + + T − ) ⊕ K ( Z + ) ⊕ K ( Z − ) . (2.8)Here K ( Z ± ) = ker( ρ ± ) / [ S ± ] , (2.9)is the lattice, which is generated by the number of homologically independent components ofreducible K3-fibers in Z ± , modulo the class of the fibre. For orthogonal matchings, where N ± ⊗ R = ( N ± ⊗ R ) ∩ ( N ∓ ⊗ R ) ⊕ ( N ± ⊗ R ∩ T ∓ ⊗ R ) , (2.10)the two independent Betti numbers obey the simple equations: b = | K ( Z + ) | + | K ( Z − ) | + | N + ∩ N − | b = 23 + 2( h , + h , − ) + ( | K ( Z + ) | + | K ( Z − ) | ) − | N + ∩ N − | . (2.11)We will only consider such matchings in this paper.6uilding blocks may be realized from blowups of semi-Fano threefolds [6], but may also berealized as toric hypersurfaces, which in turn have an elegant combinatorial description [18].We will make extensive use of this method to construct and analyse building blocks withspecific properties, so what we summarize it here for completeness, however for a detailedexposition we refer the reader to [18]. We will provide detailed examples in the subsequentsections, which should make the method accessible.The starting point of the construction is a pair of four-dimensional lattice polytopes ♦ ◦ and ♦ in dual lattices N and M , respectively, obeying (cid:104) ♦ , ♦ ◦ (cid:105) ≥ − (cid:104) ♦ , ν e (cid:105) ≥ (cid:104) m e , ♦ ◦ (cid:105) ≥ m e = (0 , , ,
1) and ν e = (0 , , , − ♦ defines a compact but generally singulartoric variety through its normal fan Σ n ( ♦ ), together with a family of hypersurfaces Z s . Each k -dimensional face Θ of ♦ is associated with a (4 − k ) dimensional cone σ (Θ) of the normalfan Σ n ( ♦ ). The resulting variety Z s can be crepantly resolved into a smooth manifold Z byrefining the fan Σ → Σ n using all lattice points on ♦ ◦ as rays. Such manifolds have all of theproperties of building blocks. Concretely, the defining equation of the resolved hypersurfaceis Z : 0 = (cid:88) m ∈ ♦ c m z (cid:104) m,ν (cid:105) (cid:89) ν i ∈ ♦ ◦ z (cid:104) m,ν i (cid:105) +1 i . (2.13)Here, m are lattice points on ♦ , c m are generic complex coefficients, the z i are the homogeneouscoordinates associated with lattice points ν i on ♦ ◦ , and z is the homogeneous coordinateassociated with the ray through ν = (0 , , , − Z as defined in this way has firstChern class given by the fiber class, which equals to [ z ]. We can compute the Hodge numbers of Z as well the ranks of the lattices N and K in In the standard Calabi-Yau hypersurface case the power of z would be (cid:104) m, ν (cid:105) + 1. The absence of the+1 indicates that the first Chern class of Z is non-trivial and given by [ z ]. h i, ( Z ) = 0 and [11, 18] h , ( Z ) = (cid:96) ( ♦ ) − (cid:96) (∆ F ) + (cid:88) Θ [2] < ♦ (cid:96) ∗ (Θ [2] ) · (cid:96) ∗ ( σ n (Θ [2] )) − (cid:88) Θ [3] < ♦ (cid:96) ∗ (Θ [3] ) h , ( Z ) = − (cid:88) Θ [3] ∈ ♦ (cid:88) Θ [2] ∈ ♦ (cid:96) ∗ ( σ n (Θ [2] )) + (cid:88) Θ [1] ∈ ♦ (cid:96) ∗ ( σ n (Θ [1] ) + 1)( (cid:96) ∗ ( σ n (Θ [1] ))) | N | = (cid:96) (∆ ◦ F ) − (cid:88) Θ [2] F < ∆ ◦ F (cid:96) ∗ (Θ [2] F ) − (cid:88) ve Θ ◦ [1] F < ∆ ◦ F (cid:96) ∗ (Θ [1] F ) (cid:96) ∗ (Θ ◦ [1] F ) | K | = h , ( Z ) − | N | − (cid:96) ( ♦ ◦ ) − (cid:96) (∆ ◦ F ) + (cid:88) Θ ◦ [2] < ♦ ◦ (cid:96) ∗ (Θ ◦ [2] ) · (cid:96) ∗ ( σ n (Θ ◦ [2] )) − (cid:88) Θ ◦ [3] < ♦ ◦ (cid:96) ∗ (Θ ◦ [3] ) . (2.14)The quantities in these relations are defined as follows. ∆ F is the subpolytope of ♦ , which isorthogonal to ν e . It is reflexive and determines the algebraic family in which the K3 fibresare contained via the construction of [19]. The k -dimensional faces of ♦ are denoted by Θ [ k ] and σ n (Θ) is the cone in the normal fan associated with each Θ. The function (cid:96) counts thenumber of lattice points on a polytope, while the function (cid:96) ∗ counts the number of latticepoints in the relative interior. In the case of (cid:96) ∗ ( σ n (Θ [ k ] )) this refers to the points on ♦ ◦ inthe relative interior of σ n (Θ [ k ] ). Finally, in the expression for | N | , the sum only runs oververtically embedded (ve) faces Θ ◦ [1] of ∆ ◦ F , which are those faces bounding a face Θ ◦ [2] of ♦ ◦ which is perpendicular to F = ν ⊥ e , see [11, 18] for a detailed exposition. The faces Θ ◦ [1] andΘ [1] are the unique pair of faces on ∆ F , ∆ ◦ F obeying (cid:104) Θ [1] , Θ ◦ [1] (cid:105) .Note that exchanging the roles of ♦ and ♦ ◦ exchanges h , and | K | but keeps b + b invariant, which is relevant for G mirror symmetry [11]. T Let us review briefly the standard 7d duality with 16 supersymmetries between the heteroticstring theory and M-theory. For the remainder we will focus on the E × E heterotic stringfor concreteness. The first evidence for this duality is the observation that the moduli spaceof the heterotic string on T agrees with moduli space as M-theory on K3 [12] and is given by M = R + × M ≡ R + × [ SO (3 , , Z ) \ SO (3 , , R )/ SO (3 , R ) × SO (19 , R )] . (2.15)Here R + corresponds to the heterotic string coupling, and the volume of the K3, respectively,and the remaining part M is the Narain moduli space of the heterotic T compactification,i.e. the moduli space of signature (3 ,
16 + 3) of momentum and winding modes, modulo theaction of the T-duality group SO (3 , , Z ). On the dual side this corresponds to the moduli8pace of Einstein metrics on K3 with unit volume. Note that Einstein implies automaticallyRicci-flat on a K3 [20].By matching the effective actions, the fields are mapped as follows: λ het = e γ . (2.16)where λ het is the heterotic string coupling and e γ is the radius of the K3 surface. The otherfields are mapped as follows: the 10d heterotic string with G = E × E gauge symmetry, hasa metric g µν , anti-symmetric 2-form B µν , dilaton φ and gauge field A µ . Upon dimensionalreduction on T , the 7d fields are ( y i , i = 0 , · · · , R and ˆ α = 1 , · · · , e ˆ α the drei-beine and (cid:36) ( k ) denotes the harmonic k -forms along the torus) g µν dx µ dx ν = g ij dy i dy j + (cid:88) ˆ α =1 v ˆ αi dy i e ˆ α + (cid:88) ˆ α, ˆ β =1 ϕ ˆ α ˆ β e ˆ α e ˆ β B µν dx µ ∧ dx ν = b ij dy i ∧ dy j + (cid:88) α =1 b αi ∧ (cid:36) (1) α + (cid:88) a =1 β a (cid:36) (2) a A µ dx µ = A i dy i + (cid:88) a =1 16 (cid:88) I =1 a aI t Ia e φ = λ het . (2.17)The scalar fields are the dilaton, six scalars from the metric component ϕ ˆ α ˆ β along the torus,and three β from the B-field, expanded along the harmonic 2-forms on the T . The gaugefield gives rise to a aI , which are the three Wilson lines along the Cartan subalgebra u (1) of the gauge group. Overall, the dimension is 58, which agrees with that of M . This doesnot yet prove the coset structure, which follows by careful analysis of the points of symmetryenhancement in the moduli space [21].The gauge fields in 7d are from the unbroken gauge group, determined by the commutantof the Wilson lines inside G together with the gauge fields obtained from the metric, expandedalong the drei-bein of the torus, v ˆ αi , and the B -field components expanded along the harmonic1-forms: b αi . For generic points in the moduli space the gauge group is the Cartan subgroupof G , U (1) , together with the six abelian gauge fields from metric and B -field.On the other hand, starting with M-theory on a K3-surface, we have the metric ˜ g MN andthree-form C MNK . The metric deformations account for the moduli space M by a choice ofthree complex structures ω , ω , ω in Γ , ⊗ R , which precisely corresponds to the choice ofa point in the Grassmanian SO (3 , /SO (3) × SO (19). The lattice automorphisms, whichpreserve this oriented three-plane correspond to automorphisms of the K3-surface.9he gauge fields are obtained as follows: The K3 has 22 two-forms, along which the 3-formgives rise to abelian gauge fields a κi C = c ijk dy i ∧ dy j ∧ dy k + (cid:88) κ =1 a κi ω (2) κ . (2.18)These 22 U (1) gauge fields match with the heterotic spectrum at generic points in modulispace. The three-form c in 7d can be dualized to a two-form dc = (cid:63) db , (2.19)which matches with the 2-from in the heterotic compactification.For specific points in the moduli space, there are extra massless states which correspond toenhanced gauge symmetry. From the heterotic side, this happens whenever there is a latticepoint γ ∈ Γ , such that γ = −
2, which is perpendicular to the three-plane spanned by the ω i in Γ , ⊗ R . The set of all such γ span a root lattice of ADE type, which correspondsdirectly to the unbroken non-abelian gauge group of the theory. On the M-Theory side,these points in moduli space are precisely those ones for which the K3 surface develops thecorresponding ADE singularities. The gauge degrees of freedom arise from massless M2-braneswrapped on the collapsed rational curves. In a fiber-wise application of heterotic/M-theoryduality, it hence becomes natural to find a geometry for which every K3-fiber is singular. Itis precisely this behavior, which we will uncover for M-Theory on compact TCS G -manifoldswith heterotic duals. The M-theory/heterotic duality will now be applied fiberwise over a three-dimensional base,which on the M-theory side results in a G -manifold and on the heterotic in a Calabi-Yauthreefold compactification. The TCS-construction for G -manifolds of section 2.1 has a nat-ural such K3-fibration, see (2.6), which suggests that they are particularly suitable for theapplication of such a fiber-wise duality. We will show, that indeed the duality can be explicitlyrealized, when the K3-fibers of each TCS building block are elliptically fibered, i.e. E (cid:44) → K → P , (2.20)where the elliptic curve E is holomorphically fibered over the P . To implement this, it willbe useful to keep the following geometric point of view in mind: We should think about thehyper-K¨ahler structure, i.e. the three harmonic self-dual two-forms ω i , of K3 as being related10o the three S i of T . In particular, we may expand ω i in H ( K , Z ) = U ⊕ U ⊕ U ⊕ ( − E ) ⊕ (2.21)and we can think of the periods of ω i in U i as the radius of each S ( ω i in U j for i (cid:54) = j areoff-diagonal elements of the metric on T ) and the periods of ω i in ( − E ) ⊕ as the Wilsonlines around S i .Note that the case where ω has non-trivial periods on one of the U E ⊂ H ( K , Z )(together with some sublattice of E ⊕ ) but ω , ω are perpendicular to U E is particularlysimple as this gives a factorization of T = T f × S . If we think of ω as the K¨ahler form,this condition is nothing but demanding that the K3-surface S has an elliptic fibration with asection. If we fiber such an elliptic K3-surface S holomorphically over P to get a Calabi-Yauthree-fold, only Ω , = ω + iω varies and ω stays constant, so that in the heterotic dual, weonly fiber a T f ⊂ T non-trivially over P and we end up with heterotic on an elliptic K3 × S .Besides the twisting of T f = S × S over the base P , the K3-fibration also determines a flatvector bundle on T f via a spectral cover. This is encoded in the (varying) periods of Ω , within the E ⊕ lattice over the P base of the building block. Furthermore, the integrals ofthe K¨ahler form ω over cycles associated with the E ⊕ lattice correspond to a Wilson lineon S .The picture of the duality between M-Theory and heterotic string theory we have paintedhere lifts to the duality between F-theory and heterotic string theory in six dimensions andleads to (a subset of examples for) the duality between type IIA string theory and heteroticstring theory upon circle compactification.Although the global picture for a G -manifold realized as a twisted connected sum is morecomplicated, we can use the above logic for each K3-fibered building block separately, applythe duality, and then glue the ‘building blocks’ of the heterotic dual together as dictated by thematching (2.5). This will imply that both the geometry and the bundle data are consistentlyidentified on the heterotic side. Using only elliptic K3-surfaces as fibers on the buildingblocks of the G -manifold allows us to explicitely describe the dual heterotic geometry asit simplifies discriminating between geometry and bundle data. However, it is possible toapply the fiberwise duality for TCS building blocks more generally and consider non-ellipticK3-surfaces as fibers of building blocks. We leave an investigation of such models for thefuture. Due to the matching condition (2.5), the elliptic fiber on one side is identified with a non-algebraic toruson the other side. Heterotic Dual for smooth TCS G -manifolds We now apply the fiber-wise M-theory/heterotic duality to the TCS construction of G -manifolds for which the K3-fibers are elliptic, and determine a TCS-like construction of thedual heterotic compactification. The latter will be constructed from two building blocks,which are open hyper-K¨ahler manifolds, that are glued using the dual of the map in the G TCS construction. Under M-theory/heterotic duality, the K3-fibration of the TCS G -manifold maps to the SYZ T -fibration of the dual heterotic Calabi-Yau threefold. We showin particular, that the resulting Calabi-Yau three-fold is the Schoen manifold, i.e. the splitbi-cubic, for all such TCS G -manifolds. The starting point is M-Theory on a TCS G -manifold J . In the twisted connected sumconstruction for this G -manifold, we employ algebraic three-folds Z ± as building blocks, asexplained in section 2.1. These are K3 fibrations with fiber [ S ] over (cid:99) P , such that the canonicalclass of Z ± is given by a fiber [ S ]. The building blocks themselves are hence not Calabi-Yau(2.2).The starting point for the heterotic three-fold X het is as in section 2.3, where the dualityis applied to the setup with an elliptically fibered K3. Recall that for an elliptic K3, that isin addition fibered over (cid:98) P , only two of the three complex structures, ω and ω , of the K3vary. In the heterotic dual model, by fiberwise duality, therefore only a T f ⊂ T varies overthe base (cid:98) P , and the total space of the heterotic compactification is an elliptic K3 × S . Wenow apply the same ideas in the TCS construction to this Calabi-Yau three-fold.First we twist the fiber so as to have a non-trivial first Chern-class as in (2.2), i.e. theresulting geometry is dP × S with c ( dP ) = [ T f ] , (3.1)where [ T f ] is the class of the torus fiber T f = S × S . The next step in the TCS for G -manifolds is the removing a central fiber from Z ± to form X ± = Z ± \ S ± . Applying the sameprocess to the heterotic model, we define two building blocks V ± = dP \ T f . (3.2) We use the term building block here in analogy to the TCS construction, as V ± play a similar role as Z ± ,applied to the construction of the Calabi-Yau three-fold. S S S { T { T { V + { V - Figure 1: A cartoon of the Calabi-Yau three-fold X het = X , glued from two copies of V ± × T ± where T ± = S × S e . The elliptic fiber of V degenerates over 12 points of the open P base.From the duality, the V ± must asymptote to I × S b × S × S . In the G construction, theDonaldson matching identifies ω = ω − ω = − ω − ω = ω − , (3.3)which in the dual heterotic Calabi-Yau now becomes the identification S = S − S = S − S = S − . (3.4)Finally, the twisted connected sum uses X ± × S e ± and glues the S b ± in the open P \ p basesof X ± to S e ∓ , S e ± = S b ∓ , (3.5)which we may equally well apply to S × V ± . In conclusion, the Calabi-Yau three-fold isobtained by gluing M ± = V ± × T ± , T ± = S ± × S e ± , (3.6)along W = M + ∩ M − ∼ = I × ( S ) , (3.7)with the identifications in (3.4) and (3.5) – see figure 1 for a cartoon. We denote the resultingCalabi-Yau three-fold by X het . 13o identify the specific Calabi-Yau, first note that the Euler characteristic of X het is givenby χ ( X het ) = χ ( M + ) + χ ( M − ) − χ ( W ) = 0 , (3.8)which follows immediately, from additivity of the Euler characteristic. We used furthermorethat M + , M − and W all have an S -factor, and thereby vanishing Euler characteristic. Next,we may use the Mayer-Vietoris sequence to compute the Betti numbers of X het as H m ( X het , Z ) = ker( γ m ) ⊕ coker( γ m − ) , (3.9)where γ m : H m ( M + Z ) ⊕ H m ( M − , Z ) → H m ( W, Z ) . (3.10)It is not hard to see that ker( γ ) = 0, which implies that H is trivial. Futhermore, | coker( γ ) | = 3 and | ker( γ ) | = 16, so that h , ( X het ) = 19, and thereby h , ( X het ) = 19as well, i.e. X het = X , . This Calabi-Yau is known as the Schoen manifold [22] or the splitbi-cubic.Note that any M-theory compactification on a TCS G -manifold of the type consideredhere, i.e. with elliptic K3-fibers, is dual to a heterotic compactifications on X , . Themultitude of such TCS G -manifolds simply corresponds to different choices of vector bundleson X , .The SYZ-fibration of X het is known and has a simple structure [23], which we will recoverfrom the M-Theory duals: the discriminant locus consists of two sets of twelve disjoint S ’s.Any two S ’s from different groups have linking number 1 (Hopf link) and 0 otherwise. Thesame can be recovered from the Kovalev construction given above: the discriminant locus ofthe SYZ-fiber is S times the points over which the elliptic fiber of V degenerates. As the twobuilding blocks V ± are open versions of dP , this happens over 12 points each. The S in thediscriminant locus is the same as S e ± on both sides. As these are swapped with S b , the twogroups of 12 S ’s forming the discriminant locus are interlocked like the Hopf links shown infigure 2. In this section we give various constructions and limits of the heterotic Calabi-Yau three-fold X het = X , , i.e. the Schoen manifold, which will be useful in the following.14
2x 12x
Figure 2: The Hopf link. The discriminant locus of the TCS fibration is a collection of 12such Hopf links in the base of the fibration, which are retract to the above image.
The Calabi-Yau with Hodge numbers X , is a rather well-studied. It can be found as aresolution of the orbifold X o = T / ( Z × Z ) , (3.11)where the Z acts as α : ( z , z , z ) → ( z , − z , − z ) β : ( z , z , z ) → ( − z , z , − z + ) αβ : ( z , z , z ) → ( − z , − z , z − ) . (3.12)Let us try to see this explicitely. First note that we can think of V = dP \ T f as an ellipticfibration over an open P with two I ∗ fibers, i.e.( T f × S b × R ) / Z , (3.13)where the Z acts by giving a minus sign to all four directions. Note that the above gives aK3-surface when we replace R by S , which allows us to see that a K3-surface could be gluedfrom two copies of the building blocks V .We can now describe the above orbifold in detail. Let us choose coordinates z = x + ix z = x + ix z = x + ix , (3.14)so that the orbifold action is α :( x , x , x , x , x , x ) (cid:55)→ ( x , x , − x , − x , − x , − x ) β :( x , x , x , x , x , x ) (cid:55)→ ( − x , − x , x , x , − x , − x + ) . (3.15)We can pull apart this orbifold along the x direction. Note that any x can be mapped tothe closed interval [0 : ] by α and β , with x = 0 being fixed by α and x = fixed by β .15ence we can think of X o as being fibered over x = [0 : ] with generic fiber ( S ) and ‘ends’ M + = ( S ) × R / Z α and M − = ( S ) × R / Z β .The coordinates x i are associated with the various ingredients in figure 1 as follows. Inthe vicinity of x = 0, the involution α must act with a sign on T f = S × S and S b of M + ,together with the interval direction x . Similarly, in the vicinity of x = , the involution β must act with a sign on T f = S × S and S b of M − . This results in the identification M + S ∼ x ∼ S S ∼ x ∼ S S ∼ x ∼ S S b ∼ x ∼ S e S e ∼ x ∼ S b M − (3.16)and reproduces the gluing (3.4), so that X o is decomposed precisely in the same way as shownin figure 1. This decomposition has been considered by A.Kovalev (unpublished); althoughhe conjectured the existence of a Ricci-flat Calabi-Yau metric on this twisted connected sumCalabi-Yau, this has not yet been proven rigorously.The above identification allow us to see the holomorphic coordinates of X het as realizedby gluing M ± . In the identification above, the T in X het which replaces the K3 on the G -manifold J is given by (cid:61) ( z ) = (cid:61) ( z ) = (cid:61) ( z ) = 0, so it is special Lagrangian.Let us now describe how the classic presentation of the Schoen Calabi-Yau X a as thesplit bi-cubic is related to the orbifold X o . The Schoen Calabi-Yau is realized as a CICY in P × P × P with configuration matrix, which indicates the degrees of the hypersurfaces P P P . (3.17)This can be visualized in various ways. First note that, fixing a point on the P , the completeintersection (3.17) becomes a product of two elliptic curves. We may consider projectionswhich forget one of the P factors of the ambient space. These project to hypersurfaces ofdegree (3 ,
1) in P × P , which are nothing but a rational elliptic surfaces dP . The inverseimage of such a projection is an elliptic curve. Hence X a can be thought of as an ellipticfibration over dP in two different ways. Finally, note that both of these dP surfaces sharethe same base as elliptic surfaces, which implies that X a = dP × P dP . (3.18) Note that this choice is not unique, in fact x ↔ x and x ↔ x are symmetries of the whole configuration.We have made a choice for which our SYZ fiber is sLag. S S S { T { T { V + { V - Figure 3: The Kovalev limit of the Schoen Calabi-Yau three-fold. We have shown one ofthe two dP s in blue and green: it is the one arising from a projection to z , z . As the dP includes all of V + = dP \ T f , one can see the 12 singular fibers. On M − , the circle factorsexperience no more monodromies. Furthermore, the elliptic fiber of the dP , which becomesone of the elliptic fibers of X het is colored in green. It corresponds to the coordinate z of theorbifold. The other dP originating from projection to z , z is found by swapping M + ↔ M − .The manifold (3.17) has second Chern class c ( X a ) = 3 (cid:0) H + H + H · H + H · H (cid:1) , (3.19)where H , H are the hyperplane classes of the P factors and H is the hyperplane class ofthe P factor of the ambient space. As H fixes a unique point on the first P , which thengives a unique point on the P via the first equation, H corresponds to a single elliptic curve.Similarly, H · H gives three copies of the same elliptic curve. We hence find that we canrepresent c ( X a ) as 12 copies of each of the elliptic curves corresponding to the fibers of thetwo elliptic fibrations.The same structure can be seen from the orbifold version X o . Here, we identify z with thebase and the projection to each of the dP is given by forgetting the directions z or z . Theimage of the projection is ( T / Z k ) / Z s . Here, Z k acts by inverting all coordinates producinga Kummer surface, and Z s pairwise identifies the four I ∗ fibers of the Kummer surface see asan elliptic fibration. This produces a dP as an elliptic surface with two I ∗ fibers.The above allows to identify the two dP ’s and the elliptic fiber in our Kovalev picture,we have shown one such choice in figure 3.Although we should think of the T = S × S × S as an SYZ fiber T SY Z of the Calabi-Yau X het , the T f ⊂ T SY Z is seemingly holomorphic as the fiber of dP . However, the holomorphic17irections are different, it is z of M + which is the holomorphic coordinate on the fiber S × S b of one dP ( z of M + is the coordinate on the elliptic fiber S × S b of the other) and z is thecoordinate on S × x , which is the base. Recall that x is the direction of the interval alongwhich we do the Kovalev decomposition of X a . The Schoen Calabi-Yau X het can also be realized by a Weierstrass model over dP , whichcan itself be realized as an elliptic surface over P . This can be explicitely constructed as acomplete intersection in an ambient toric variety with weight systemˆ z ˆ x ˆ y x y w ˆ w ˆ z Σ of degrees W ˆ W W = − y + x + f (ˆ z , ˆ z ) xw + g (ˆ z , ˆ z ) w = 0ˆ W = − ˆ y + ˆ x + ˆ f (ˆ z , ˆ z )ˆ x ˆ w + ˆ g (ˆ z , ˆ z ) ˆ w = 0 , (3.21)where f, g, ˆ f , ˆ g are homogeneous polynomials of the indicated degrees in [ˆ z : ˆ z ]. Using [24](see Appendix A.1 and A.2), the Hodge numbers turn out to be h , ( X het ) = h , ( X het ) = 19.Projecting onto the coordinates ˆ y, ˆ x, ˆ w, ˆ z , ˆ z realizes one elliptic fibration of X het . Thebase B s of this elliptic fibration is a dP given by ˆ W = 0. The elliptic fibration of B s is inturn found by projecting to x, y, w, ˆ z , ˆ z , i.e. we forget the coordinates ˆ x, ˆ y, ˆ w . Note that thewhole three-fold X het can also be seen as being fibered by two elliptic curves E and (cid:98) E over the (cid:98) P with coordinates [ˆ z : ˆ z ] and that there is a second elliptic fibration found by swappingthe roles of ˆ y, ˆ x, ˆ w and x, y, w . A cartoon can be found on the right hand side of figure 4.We can understand h , ( X het ) = 19 as follows: there are 10 algebraic cycles in h , ( dP )for each of the dP s, but the two dP s share a common base (the (cid:98) P with coordinates [ˆ z : ˆ z ]).If we choose to consider the elliptic fibration associated with x, y, w , the base dP ˆ W = 0 inˆ y, ˆ x, ˆ w, ˆ z , ˆ z has h , ( B s ) = 10. Crucially, the equation W = 0 does not depend at all on thecoordinates ˆ x, ˆ y, ˆ w , i.e. the elliptic fibration is trivial over the elliptic curve (cid:98) E .In this presentation, the second Chern class of X het is given by c ( X het ) = 12 [ˆ z ] · ([ w ] + [ ˆ w ]) = 12( (cid:98) E + E ) . (3.22)Here E and (cid:98) E are the curve classes represented by the two elliptic curves.18 .3 Dual Smooth TCS G -manifolds In this section we study a simple compactification of heterotic E × E string theory on X het and its G dual. For this, we consider a generic E × E bundle V with ch ( V ) = 12 (cid:98) E inwhich the instantons are distributed as (6 ,
6) between the two E factors. In this case, each E bundle has 112 moduli and hence contributes 112 chiral multiplets to the 4d effective fieldtheory [25]. As we furthermore need to satisfy the anomaly condition ch ( X het ) = ch ( V ) + [NS5] (3.23)we need to introduce 12 NS5-branes wrapped on the elliptic curve E . Each of these NS5branes gives a U (1) vector multiplet and three chiral multiplets in four dimensions. Togetherwith the geometric moduli and the dilaton, we hence find n v = 12 n c = 1 + 2 ·
19 + 2 ·
112 + 36 = 299 (3.24)for the number of vector multiplets ( n v ) and chiral multiplets ( n c ).We now want to reproduce the above from a TCS G -manifold. As we want a genericbundle in E on one of the two elliptic curves E and (cid:98) E , it seems natural to construct a pairof building blocks for which the generic K3-fiber has the lattices T + = E ⊕ E ⊕ U ⊕ N + = UT − = U ⊕ N − = E ⊕ E ⊕ U . (3.25)We naturally have an orthogonal gluing with N + ∩ N − = 0 under the obvious identification.Furthermore, we expect to be able to choose the hyper-K¨ahler structure on both sides suchthat the matching (2.5) can be satisfied. While both K3 fibrations together give rise to thenon-trivial geometry of X het , only Z + carries the information of the E × E bundle!Let us be more concrete. For the building block Z + , let us take a hypersurface in a toricvariety with weight system y x w z z ˆ z ˆ z P , (3.26)defined by an algebraic equation P = − y + x + f , ( z, ˆ z ) xw + g , ( z, ˆ z ) w = 0 . (3.27)19ere y, x, w form a weighted projective space P and f, g are polynomials of the indicateddegrees in the coordinates [ z : z ] and [ˆ z : ˆ z ] on P × (cid:99) P . The building block Z + is then ageneric fibration of a Weierstrass elliptic K3-surface over (cid:98) P . In the language of tops, this isrecovered from ♦ ◦ + = − − − (3.28)using the construction reviewed in Section 2.1. With (2.14), we compute h , ( Z + ) = 3 h , ( Z + ) = 112 | N ( Z + ) | = 2 , (3.29)for this space, so that K ( Z + ) = 0 follows.For Z − , we use K3-surfaces in the family with N = U ⊕ ( − E ) ⊕ as the fibers. This canbe done by a choice ♦ ◦− = − − − (3.30)which realizes a generic fibration of a resolved Weierstrass K3 with two fibers of type II ∗ overa P . This space has h , ( Z − ) = 31 , h , ( Z − ) = 20 , | N ( Z − ) | = 18 , | K ( Z − ) | = 12 . (3.31)For an orthogonal gluing of the above building blocks, we hence find a smooth G -manifold J with Betti numbers b ( J ) = 12 b ( J ) = 23 + 2(112 + 20) + 12 = 299 , (3.32)which precisely reproduces the spectrum found for the heterotic compactification in (3.24),by identifying b ( M ) with n v and b ( M ) with n c !This identification of spectra is a very strong indication that for the smooth TCS G -manifolds with elliptic K3 building blocks we have identified the heterotic dual compactifi-cations. In the remainder of this paper we will generalize this construction of dual models,culminating in a general proof of the equivalence of spectra in section 6. This includesmodels with non-abelian gauge symmetries, which in M-theory correspond to singular TCS G -manifolds. 20 M-theory on G with Non-Higgsable Gauge Groups In this section, we will generalize the construction of dual models of the last section by con-sidering models with different distributions of instanton numbers. In particular, this will givemodels with non-Higgsable gauge groups, i.e. G -manifolds with non-deformable singularities.The generalization to include Higgsable gauge groups is provided in section 6. To prepare our discussion, let us examine the G -manifold discussed in the last section moreclosely. Over the base of each building block Z ± , the K3-fiber undergoes monodromies anddegenerates over a number of points. These monodromies correspond to the monodromies ofthe SYZ fibration of the Schoen Calabi-Yau, together with the E × E bundle on this geometry.A degeneration of the K3-fiber of a G -manifold into a (small) T which is almost constantover an interval (with non-trivial behavior only happening at the ends) corresponds to thelimit in which the SYZ T on the heterotic side becomes large and we have a semiclassicaldescription. It is precisely this limit which allows us to distinguish between geometry andbundles from the perspective of the K3-surface: the monodromies of the T in the bulk of theinterval give us the monodromies of the SYZ fiber and the monodromies affecting the cycles inthe two ends correspond to the bundle data [26]. As we have constructed our building blocksto be fibered by elliptic K3-surfaces, we can consistently split the T over the whole buildingblock into a T f (the elliptic curve) times an S , which sits in the base of the elliptic K3. Thedegeneration into a T over an interval can then be done in two separate steps, with only the T -part being non-trivial. This is, however, the same limit relevant for the duality betweenF-Theory and heterotic string theory, i.e. the well-known stable degeneration limit [27]. Here,the K3-surface degenerates into two dP surfaces which meet along a common elliptic curve,which is T f . It is this curve which tracks the non-trivial behavior of T ⊂ T (the SYZ fiber),whereas the two dP surfaces determine the two E bundles.Let us make the above explicit for the two building blocks constructed in the last section.For Z + , which is given by (3.27), the K3-fiber aquires A singularities over a number n µ ofpoints in the base (cid:98) P with coordinates [ˆ z , ˆ z ], without causing singularities in the three-fold Z + . Whereas a generic fiber contributes 24 to the Euler characteristic, such singular fiberscontribute only 23, so that we can compute n µ for any smooth K3-fibered manifold Z withoutreducible fibers from χ ( Z ) = 24(2 − n µ ) + 23 n µ = 48 − n µ . (4.1)As χ ( Z + ) = −
216 we find n µ ( Z + ) = 264. Applying the stable degeneration limit S + → P (cid:113) dP for every fiber of Z + , gives us a degeneration of Z + → ˇ Z + (cid:113) ˇ Z + . The elliptic curvein which the two dP surfaces meet has a discriminant∆ = 4 f (ˆ z ) + 27 g (ˆ z ) , (4.2)which is a homogeneous polynomial of degree 12. Hence there are 12 monodromy loci forthe SYZ fiber coming from Z + . Similarly, the degeneration S − → dP (cid:113) dP for every fiberof Z − produces a degeneration Z − → ˇ Z − (cid:113) ˇ Z − . Again, the T in which the two dP fibersmeet degenerates over 12 points of the (cid:98) P [ˆ z : ˆ z ]. In the associated G -manifold J , each ofthese degeneration loci becomes an S and, due to the gluing between Z + \ S + and Z − \ S − ,the 12 circles from Z + are interlocked with the 12 circles coming from Z − . This preciselyreproduces the structure of the SYZ fibration of the Schoen manifold reviewed in Section 3.2.In fact, it is not hard to see that this happens for any pair of building blocks fibered by ellipticK3-surfaces, which fits with the fact that they all correspond to compactifications of heteroticstrings on X het with different bundles.To find the degeneration loci of the K3-fiber associated with the E bundles, we have tofind the number ˇ n µ of degenerations of each of the two dP fibers in ˇ Z ± over [ˆ z , ˆ z ]. Thiscan again be done by computing their Euler characteristics and noting that a smooth dP fiber contributes 12 to the Euler characteristic while one with an A singularity contributes11. We can hence write χ ( ˇ Z ) = 12(2 − ˇ n µ ) + 11ˇ n µ = 24 − ˇ n µ . (4.3)It is straightforward to compute χ ( ˇ Z ) = −
96, so that ˇ n µ = 120 follows.Putting it all together, we have found that out of the 264 degenerations of the K3-fiber ofthe building block Z + , there are 120 corresponding to each of the two E bundles. The remain-ing 24 monodromy points pairwise coincide in the limit in which the K3-surface degeneratesinto two dP surfaces, these are the 12 degeneration points of the SYZ fibration.In the other building block Z − , each of the elliptic K3-surfaces it is fibered by has E × E in its Picard lattice. There are no monodromies corresponding to bundle data, which isconstant over the (cid:98) P base. There are only 24 monodromy loci, which again pair up to form12 points in the degeneration limit of the K3-surface. These 12 points correspond to the SYZfibration on the heterotic side. G -manifolds With the detailed understanding of the monodromies of the K3 fibration gained in the lastsection, we have gained more confidence that the G -manifold J constructed in Section 3.322ndeed corresponds to a heterotic model on X het with two E vector bundles V I that has adistribution of instantons such that ch ( V ) = ch ( V ) = 6 (cid:98) E .Note that the building block Z + we have employed is also elliptically fibered over F = P × (cid:98) P . From the similarity to the well-understood case of the duality between heteroticstring theory and F-theory, it is natural to assume that trading Z + for a building block Z + ,n which is elliptic over F n corresponds to a distribution of instanton numbersch ( V ) = (6 + n ) (cid:98) E ch ( V ) = (6 − n ) (cid:98) E (4.4)This means we are interested in construction building blocks as hypersurfaces in a toric varietywith weight system y x w z z ˆ z ˆ z P n n n n (4.5)for n = 0 , · · · ,
6. Let us first consider the case n = 1, which gives a smooth hypersurfaceof Euler characteristic χ ( Z +1 ) = −
216 as well. The 240 degeneration loci of the K3-fiberassociated with the two E bundles, however, are now distributed as 240 = 60 + 180. Asbefore, this is found by degenerating all K3-fibers into dP (cid:113) dP and couting the number ofdegenerations of each of the dP surfaces.For n ≥
2, a generic hypersurface P n = 0 in the ambient space defined by (4.5) above issingular and requires resolution. This can be seen as follows: the defining equation of Z + ,n isjust given by a Weierstrass model over F n P n = − y + x + f , n ( z, ˆ z ) xw + g , n ( z, ˆ z ) w = 0 . (4.6)where the subscripts indicate the weights of f and g under the scalings (4.5). This means that f and g necessarily vanish over z = 0 for n ≥ f, g and the discriminant ∆, as well as the ADE groupand Kodaira fiber type associated with the degeneration, are given by n f g ∆ fiber type G − I ∗ D IV ∗ E III ∗ E II ∗ E II ∗ E . (4.7)23s these models correspond to heterotic compactifications with the instanton distributions(4.4), we expect to find an unbroken gauge group whenever 6 − n becomes too small to fullybreak E . Although the computation of the unbroken gauge group together with the spectrumcan be done directly in the dual heterotic models, we find it more convenient to confirm theappearance of the above groups from the F-theory perspective. This is done in Section 5.Using the description in terms of projecting tops, the resolutions of these spaces can easilybe constructed and analysed, see Appendix A.3 for the details. The resulting smooth buildingblocks Z + ,n satisfy n | N ( Z + ,n ) | | K ( Z + ,n ) | h , ( Z + ,n )0 2 0 1121 2 0 1122 6 0 1283 8 0 1544 9 0 1825 10 1 2116 10 0 240 . (4.8)As we have engineered our TCS G models such that Z + ,n carries all the bundle data,the G -manifolds M n dual to heterotic models on X het with a distribution of instantons (4.4)are constructed as a twisted connected sum of the building blocks Z + ,n with the same Z − ,(3.30), throughout. In particular, the hyper-K¨ahler rotations which are used in the matchingdescend from the same lattice autmorphism for each of those models. Viewing the buildingblocks Z + ,n as K3 fibrations, the resolution of the singularities (4.7) results in fibers with alattice N = U ⊕ G . The identification of lattices we want to use to define a hyper-K¨ahlerrotation then results in N + ∩ N − = G . (4.9)This means that the matching (2.5) forces the K¨ahler forms in all of the K3-fibers for bothbuilding blocks to integrate to zero over the cycles contained in the lattice G . In other words,there is an ADE singularity of type G in every K3-fiber of the building blocks, and henceover every point of the S base of the glued G -manifold. Although there is no rigorousmathematical argument at present that these singularities will persist when the metric ofthe glued manifold is perturbed such that it becomes Ricci-flat, we conjecture based on theduality to the heterotic string (and also F-theory) that this is indeed the case.To find the spectra of M-theory on the resulting G -manifolds, we proceed as follows. Usingmirror symmetry for G -manifolds, the formulae (2.8) and (2.11) correctly reproduce the rankof the gauge group and the number of uncharged chiral multiplets even in the presence of thesingularities we encounter. The reason is that we can confidently determine b ( J ) by couting24he rank of the gauge group and that we have a smooth mirror J ∨ with N + ∩ N − = 0. Wecan hence find b via b ( J ∨ ) + b ( J ∨ ) = b ( J ) + b ( J ). It would be nice to have a rigorousmathematical argument or find the cohomology theory which reproduces these numbers inthe presence of singularities. In the present case, the number n v of U (1) vectors, neglectingthe Cartan of the non-abelian gauge group G , the non-abelian gauge group G and the numberof chiral multiplets can then be computed from (4.8) and (3.31) to be n G n v n c −
12 2991 −
12 2992 SO (8) 12 3273 E
12 3774 E
12 4325 E
13 4906 E
12 547 (4.10)We will recover this spectrum from the perspective of F-theory in the next section.
To lift this model to F-Theory, we simply note that the heterotic three-fold X het is an ellipticfibration over dP and replace the elliptic curve E by an elliptic K3-surface, with fiber E . Thismeans we consider K3-fibered Calabi-Yau fourfolds over dP as potential F-Theory duals.Generic fourfolds X F of this type were considered in [28]. To realize a K3-fibration over dP , we construct both the dP and the K3-surface as a Weier-strass model. The different instanton distributions (4.4) are then realized by fibering the baseof the K3-surface over the base of the dP in an approriate way. This means we realize theelliptic fourfolds X F,n as complete intersections in ambient spaces with weight systems x y w ˆ x ˆ y ˆ w z z ˆ z ˆ z Σ of degrees W ˆ W n n n n n n (5.1)The first equation has the formˆ W = − ˆ y + ˆ x + ˆ x ˆ w ˆ f (ˆ z ) + ˆ w ˆ g (ˆ z ) = 0 , (5.2)25 Z + dP dP X Figure 4: A sketch of the geometry X F,n . The two elliptic curves (cid:98) E and E are shown as wellas the non-trivial way they are fibered over the base B = (cid:98) P (cid:110) P . The fibration result inthe dP s, as well as the K3, as indicated by the boxes. For the general construction in section6, the base B is required to only have a projection to (cid:98) P and its fiber P is allowed to havedegenerations into multiple components over points in the base (cid:98) P .where ˆ f and ˆ g are of the indicated degree in ˆ z and ˆ z . If we project out the unhatted coor-dinates, this realizes a dP as an elliptic fibration over the (cid:98) P with homogeneous coordinatesˆ z : ˆ z . Together with the P with coordinates [ z : z ] it forms the base of our elliptic fourfold,which is now described by intersecting the above with W = − y + x + xw f , n ( z, ˆ z ) + w g , n ( z, ˆ z ) = 0 . (5.3)Here, f , ( z, ˆ z ) and g , ( z, ˆ z ) are homogeneous of the indicated weights.Note that this realization is very similar to our construction of the building blocks Z + ,n ,(4.5). In fact, X F,n is realized as the fiber product X F,n = Z + ,n × (cid:98) P dP , (5.4)where the common (cid:98) P is the one with coordinates [ˆ z : ˆ z ] . A sketch of the geometry isshown in figure 4. It is hence not surprising that the complete intersections presented aboveare also singular when n ≥
2, and we can repeat the same table as shown in (4.7) for theappearing singularities. The corresponding gauge groups are geometrically non-Higgsable inthe sense of [15, 16].The above presentation fits with the framework of [24], i.e. the complete intersection W = ˆ W = 0 is associated with a nef partition of − K of the ambient space and there is an We thank Dave Morrison for pointing this out. ◦ which can directly be found from (5.1) in the case n ≤
1. Thisis not the case for n ≥
2, which corresponds to the fact that these cases require resolution.These resolutions again have an elegant construction in terms of polytopes, see Appendix A.4for the details. After the resolution is performed with these methods, we can easily computethe Hodge numbers and hence the spectra of the F-theory compactification.In the case n = 5 there are loci over which the vanishing orders of f, g, ∆ are ord( f, g, ∆) =(4 , , X F, . These non-minimal pointscan be blown up, such that the F-Theory compactification is captured by an effective fieldtheory. This is the fourfold we are working with. In this case, the Hodge numbers of the base B of the elliptic fibration are hence not h , = 11 and h , = 0 as in all the other cases, but h , = 12 and h , = 1.In line with our general picture (5.4), this indicates the presence of tensionless strings forthe M-Theory compactification on the G -manifold J glued from Z + , and Z − , . In fact, theDonaldson matching forces to collapse the cycles in the K3 fibres of Z + , corresponding to E , as we have discussed above. In light of the existence of tensionless strings in the dualF-Theory model, we conjecture that this will force the collapse of a coassociative four-cyclewhich arises as a component of a reducible K3 fibre. The tensionless strings then arise fromwrapped M5-branes wrapped on this coassociative cycle. The 2d theories on the string are2d (0,2) supersymmetric [29, 30].For the comparision of the spectra, we perform the corresponding blowup at the level ofthe building block Z + , , which can be done by trading the top ♦ ◦ for ♦ ◦∗ , see (A.12). Thisdoes not change any of the relevant topological data for Z + , . From the point of view of theelliptic fibration, this can be understood as a flop which maps a non-flat fibration into a flatone, where the base in turn has been blown up. G -Flux An F-theory compactification to 4d is specified by the geometry of the elliptic fibration aswell as background G -flux. The tadpole cancellation condition in F-theory on an ellipticCalabi-Yau four-fold is χ ( X F )24 = N D + 12 (cid:90) X F G ∧ G , (5.5)where G is the four-form flux and N D the number of D3-branes. The number of D3-branestherefore depends on the flux turned on in the background. Furthermore, we need to impose27ux quantization [31] G + c ( X F )2 ∈ H ( X F , Z ) , (5.6)where X F is the smooth, resolved elliptic Calabi-Yau four-fold. If c is odd, the minimal fluxthat has to be switched on is therefore c ( X F ) /
2. Chirality is computed by integrating theflux over matter surfaces, however there are no codimension two enhancements whereby thereare no non-trivial matter loci in all our models.By direct computation of the second Chern classes in the models that we consider below,it in fact follows that c ( X F,n ) = 0 mod 2, and thus the minimal flux can be 0. In section 6we shall prove quite generally that c is even based on the structure of X F .In fact this can also be observed by considering the fluxes obtained in [32–34] from explicitresolutions and the so-called spectral divisor, which implies that the flux, which is bothproperly quantized and does not break the gauge group, will be proportional to the so-calledmatter-surface, i.e. the rational curves in the singular fiber above the codimension two matterloci in the base. Again in the case of X F,n , this flux vanishes by the absence of the matterloci. As G = 0, the table (4.7) directly tells us the non-abelian gauge group of the effective 4d fieldtheory, which matches the non-abelian gauge groups on the M-Theory side. As furthermore χ ( X F,n ) = 288 in all cases, we need to include 12 D3-branes in each model, due to tadpolecancellation. These correspond to the 12 NS5-branes wrapped on E in the dual heteroticmodels and together contribute 12 U (1) vector- and 36 chiral multiplets. Finally, there areno matter curves, so that the rest of the spectrum is given purely in terms of geometry. Thenumber of chiral multiplets in each model is hence n c = h , ( B ) + h , ( X F ) + h , ( X F ) − h , ( B ) + 3 N D = h , ( B ) + h , ( X F,n ) + h , ( X F,n ) − h , ( B ) + 36 (5.7)and the number of U (1) vectors neglecting the non-abelian gauge group content is n v = h , ( X F ) − h , ( B ) − h , ( B ) + N D − rank( G )= h , ( X F,n ) − h , ( B ) − h , ( B ) + 12 − rank( G ) . (5.8)28ogether with the Hodge numbers of the X F,n and the Hodge numbers of their base space wehence find n G h , ( X F,n ) h , ( X F,n ) h , ( X F,n ) χ ( X F,n ) n c n v −
12 112 140 288 299 121 −
12 112 140 288 299 122 SO (8) 16 128 152 288 327 123 E
18 154 176 288 377 124 E
19 182 203 288 432 125 E
21 212 231 288 490 136 E
20 240 260 288 547 12 (5.9)which precisely matches with the spectrum found from the G -manifolds on the M-Theoryside (4.10)! Besides a beautiful matching of the moduli counting, this also provides a dualdescription of non-Higgsable gauge groups in M-Theory. Finally let us comment on the non-Higgsability: the F-theory four-folds realize geometrically non-Higgsable gauge theories. Inmodels where there are non-trivial matter curves in the base of the F-theory four-fold, i.e.codimension two enhancements of the discriminant, we would expect to have vector-like mattereven in the absence of flux, unless the matter curves are rational. Such vector-like mattercould potentially Higgs the gauge group. In the present context, there are no matter loci, andthus the geometric non-Higgsability implies non-Higgsability of the 4d effective theory. In this section we generalize the construction made in earlier sections of this paper to includemore general non-abelian gauge groups. The idea is to replace the building block Z + to havea more general base B , as shown in figure 4. To motivate our generalization, consider the decomposition of the Calabi-Yau fourfolds X F constructed above. We may view the manifolds X F as elliptic fibrations over the buildingblocks Z + , where the elliptic curve (cid:98) E of this fibration only varies non-trivially over the base (cid:98) P of the K3-fibered threefold Z + . Note that the elliptic curve, whose complex structure isidentified with the axio-dilaton of F-theory, is E .This immediately leads us to conjecture that such a duality holds for any algebraic two-dimensional base B , which has a projection to (cid:98) P and the generic fiber of which is a P .In contrast to the simple examples of B = F n considered before, the fiber P may undergodegeneration into a bouquet of P s over points in (cid:98) P . From such a two-dimensional base29 , the building block Z + is then constructed as a Weierstrass elliptic fibration over B such that c ( Z + ) = [ ˆ H ], where ˆ H is the hyperplane class of (cid:98) P . As B has a projection to (cid:98) P , the same projection gives us a K3-fibration on Z + with generic fiber S + . We now have c ( Z + ) = [ ˆ H ] = [ S + ] as expected for a building block. From this, the F-Theory geometry X F is constructed as an elliptic fibration over Z + with fiber (cid:98) E , such that (cid:98) E degenerates over12 points of (cid:98) P but has no non-trivial dependence on S + . This ensures that X F is a Calabi-Yau fourfold. The geometry is again depicted in figure 4, with the suitable modificationsof the base B . Note that F-Theory compactifications in which the vanishing degrees of( f, g, ∆) are greater or equal to (4 , ,
12) give rise to tensionless strings. This implies thattensionsless strings also appear in M-Theory on TCS G -manifolds with the corresponding Z + and Z − . Here, they arise from M5-branes wrapped on collapsed coassociative cycles. Again,the theories can be studied along the lines of [29, 30].The G -manifold J , which gives the M-theory dual of F-Theory on X F is constructed asa twisted connected sum of Z + with Z − , where we take the same Z − as before, namely theone constructed via the tops (3.30), for every J . As before, the matching between Z + and Z − is such that we identify the two distinguished E lattices in H ( S + , Z ) and H ( S − , Z ). Theselattices are distinguished as we assume that both Z ± are constructed as Weierstrass models.In Z − the E ⊕ E lattice corresponds to the two II ∗ fibers. As Z + is also a Weierstrass model,there is a point in the moduli space of the fiber (as an algebraic family) in which it developstwo II ∗ fibers. This defines the distinguished E ⊕ E for Z + . Note that this implies that the G -manifold J will be forced to have ADE singularities whenever there are ( − E ⊕ E lattice, which sit in N . This happens whenever all of the S + over (cid:98) P have elliptic fibrations with reducible elliptic fibers.The elliptic fibration E (cid:44) → X F → B , where (cid:98) E (cid:44) → B → B , (6.1)of X F , which gives a dual description of M-Theory on J , is the same as the elliptic fibrationon Z + and constant over (cid:98) E . An elliptic fibration with reducible fibers on S + (present overall of (cid:98) P ), which gives us ADE singularities on J as discussed above, hence gives rise to thecorresponding ADE gauge groups on the F-Theory side. Futhermore, the matter loci occurat points on Z + . In F-theory, these are matter curves, (i.e. codimension two singular loci inthe base) of X F , each of which is a copy of (cid:98) E . In the M-theory geometry J these loci arecircles S e + , and their number and type agrees with the matter curves in F-theory.We conjecture that the singular TCS G -manifolds which underlie the M-theory compact-ifications exist and that they give rise to the same gauge theories on the F-Theory side. Note30hat while the ADE singularities arise in the limit in which E is collapsed on the F-theoryside, it is the Donaldson matching with Z − which forces to collapse the relevant cycle on theM-Theory side.To provide evidence for this conjecture, we will prove below that the spectra of unchargedparticles and the number of U (1) factors of the gauge group agree on both sides for any basegeometries B with the properties that B is a smooth algebraic surface and has a projection to (cid:98) P with generic fiber P . Our discussion does not rely on a specific construction of the buildingblock Z + or the Calabi-Yau fourfold X F . We will illustrate this method by constructing afew more examples below. To substantiate the construction of dual pairs of G -manifolds J and F-theory Calabi-Yaufour-folds X F , we now compute the spectra on both side, and check some global consistencyconditions.Let us first express the spectrum on the M-theory side in terms of the Hodge numbers ofthe building block Z + . From (2.11) and (3.31) we have n v = | N + ∩ N − | + | K ( Z + ) | + | K ( Z − ) | = | G | + | K ( Z + ) | + 12 n v + n c = 23 + 2( h , ( Z + ) + h , ( Z − ) + 2( K ( Z + ) + K ( Z − )) =2 (cid:0) h , ( Z + ) + | K ( Z + ) | (cid:1) + 87 , (6.2)where | G | is the rank of the non-abelian gauge group G , which may contain U (1) factorsfrom extra sections of the elliptic fibration of Z + by E . In our construction the Donaldsonmatching is such that N + = U ⊕ G and N + ∩ N − = G , (6.3)and such sections realizing U (1) symmetries enter in the same way as non-abelian factors.The lattice G has a root sublattice G root , generated by lattice vectors of length −
2, whichdetermines the non-abelian gauge group. The number of U (1)s in G is then given by rk( G ) − rk( G root ). To compare this to the spectrum on the F-theory side, (5.7) and (5.8), we needto relate the Hodge numbers of Z + with those of X F and B . Note that in our notation, n c counts the number of chiral multiplets neutral under G .The fourfold X F constructed above has two elliptic fibrations. Although the fibration by E defines the F-theory elliptic fibration of X F , i.e. the complex structure is identified withthe axio-dilaton, X F , is also elliptically fibered over Z + with fiber (cid:98) E (cid:98) E (cid:44) → X F → Z + . (6.4)31 generic elliptic Calabi-Yau X F with fiber (cid:98) E over Z + only has 12 singular fibers over (cid:98) P and no reducible fibers. This way of looking at X F is key to showing the equivalence of thespectra.We start by showing that the second Chern class of X F is always even, so that we do nothave to include any G -flux in order to have a well-defined model, as discussed in section 5.2.Using adjunction and ˆ σ · (ˆ σ + ˆ H ) = 0 for the section ˆ σ of the elliptic fibration by (cid:98) E and thehyperplane class ˆ H of (cid:98) P we find c ( X F ) = 11ˆ σ + 23 ˆ H · ˆ σ + c ( Z + )= 12 ˆ H · ˆ σ + c ( Z + ) , (6.5)so that c ( X F ) is even if and only if c ( Z + ) is even. This, however, is guaranteed for anyalgebraic threefold, see Lemma 5.10 of [6] for a discussion in the context of TCS buildingblocks. In fact, this argument is a special case of the one given in [35] for elliptic fourfoldswithout reducible elliptic fibers in codimension one in the base. Thus in summary, for anysuch model c ( X F )2 ∈ H ( X F , Z ) . (6.6)Next, we compute the Euler characteristic of X F using the elliptic fibration by (cid:98) E again. Tocompute the Euler characteristic, we consider the 12 points ˆ p i in (cid:98) P over which the fiber (cid:98) E is singular, separately from (cid:98) P with those twelve points excised and sum the results. Wemay further decompose ˆ P \ { ˆ p i } into simply connected patches, each of which is topologically (cid:98) E × U , for a disc U , and hence contributes zero to the Euler characteristic. This makes itclear that only the 12 points ˆ p i give a non-zero contribution. In a generic situation, fixingany ˆ p i on Z + gives a copy of a generic (i.e. irreducible and smooth) K3-fiber of Z + , times asingular (cid:98) E , which has χ = 1. We can hence compute χ ( X F ) = 12 × × χ ( K
3) = 288 , (6.7)for any fourfold X F of the form (6.4). Note that, together with the absence of G -flux, the D3-brane tadpole (5.5) implies that every such model requires χ ( X F ) /
24 = 12 D3-branes, whichcontribute 36 chiral multiplets and 12 vector multiplets in four dimensions. The number ofvector multiplets matches with the contribution of | K ( Z − ) | = 12 from Z − via (2.11), whichalso stays invariant for all of the models we consider.A similar argument shows that the base B of the elliptic fibration with fiber E always hasEuler characteristic 24. By construction, B is also elliptic with fiber (cid:98) E , which degeneratesover 12 points in (cid:98) P , i.e. over 12 copies of P in B . Hence χ ( B ) = 12 × χ ( P ) = 24 . (6.8)32he fact that X F is an elliptic fibration over Z + without singular fibers but with 8 distinctextra sections (in addition to the zero-section), which allows us to directly conclude that h , ( X F ) = h , ( Z + ) + 9 , (6.9)where one contribution is due to the fiber class (cid:98) E .To find h , ( X F ), we need to work a little bit harder. For this computation, it is advanta-geous to view X F as a fibration of (cid:98) E × S + over (cid:98) P . Conveniently, both (cid:98) E and S + degenerateover different points in (cid:98) P . This allows us to cut (cid:98) P into two overlapping open discs X (cid:98) E and X Z + such that X (cid:98) E = ( dP \ (cid:98) E ) × S + and X Z + = ( Z + \ S + ) × (cid:98) E , i.e. only one of the two fibersvaries non-trivially in each of the two halves of (cid:98) P . This allows to use the Mayer-Vietorissequence for X F = X (cid:98) E ∪ X Z + with X (cid:98) E ∩ X Z + retracting to (cid:98) E × S + × S . With γ k : H k ( X (cid:98) E ) ⊕ H k ( X Z + ) → H k ( X (cid:98) E ∩ X Z + ) (6.10)we then have H k ( X F ) = ker( γ k ) ⊕ coker( γ k − ) . (6.11)This allows us to rederive (6.9), but also gives b ( X F ) = ker( γ ) = 2 (cid:0) | K ( Z + ) | + h , ( Z + ) (cid:1) (6.12)as coker( γ ) = 0. As h , ( X F ) = 0 it follows that h , ( X F ) = | K ( Z + ) | + h , ( Z + ) . (6.13)We can now exploit χ ( X F ) = 288 together with h , ( X F ) = 2 (cid:0)
22 + 2 h , ( X F ) + 2 h , ( X F ) − h , ( X F ) (cid:1) (6.14)to find h , ( X F ) = h , ( Z + ) − h , ( Z + ) + | K ( Z + ) | + 31 . (6.15)For the base B of the elliptic fibration of X F with fiber E we find h , ( B ) = h , ( B ) + 9 = | K ( Z + ) | + 11 , (6.16)by exploiting its elliptic fibration with fiber (cid:98) E and base B , as well as h , ( B ) = | K ( Z + ) | + 2.Furthermore, as X F is an elliptic fibration over B and Calabi-Yau, h i, ( X F ) = 0 for i = 1 , , h , ( B ) = 0, as any non-trivial such class of the base would pull-back to X F .33e can now use adjunction for the Weierstrass model with elliptic fiber (cid:98) E to find that χ ( B ) = 124 (cid:90) B c ( B ) c ( B ) = 124 (cid:90) B c ( B ) · ˆ H = 124 (cid:90) B c ( B ) = χ ( B ) = 1 . (6.17)As χ ( B ) = 1 − h , ( B ) + h , ( B ) − h , ( B ) = 1 (6.18)and h , ( B ) = 0 we find h , ( B ) − h , ( B ) = 0, so that finally h , ( B ) = | K ( Z + ) | (6.19)follows from χ ( B ) = 24.We are now ready to compute the spectrum on the F-Theory side in terms of Z + . Firstwe compute the rank of the total gauge group n v = h , ( X F ) − h , ( B ) − h , ( B ) + 12= h , ( Z + ) + 9 − | K | − − | K | + 12= h , ( Z + ) + 9= | G | + | K ( Z + ) | + 12= h , ( X F ) , (6.20)where we have exploited the elliptic fibration on Z + with fiber E and h , ( B ) = | K ( Z + ) | + 2in the last line. The summand | G | counts the rank of the non-abelian gauge group fromreducible fibers of E together with extra sections of the elliptic fibration by E . Note that thenumber of sections of the elliptic fibration of X F by E is equal to the number of sections ofthe elliptic fibration of Z + by E , as E is constant over (cid:98) E . Next we check n v + n c = h , ( X F ) + h , ( X F ) + h , ( X F ) + 47= 9 + h , ( Z + ) + | K ( Z + ) | + h , ( Z + ) + | K ( Z + ) | + 31 + 47= 2 (cid:0) h , ( Z + ) + | K ( Z + )) | (cid:1) + 87 , (6.21)where n c again counts the number of chiral multiplets neutral under G . Comparing (6.20) and(6.21) with (6.2), we find a perfect agreement of the spectra between the dual compactificationsof M-theory on the G -manifold J and F-theory on the elliptic fourfold X F . Note that boththeories have the gauge group G × U (1) | K ( Z + ) | +12 , (6.22)where G contains non-abelian factors from G root and rk( G ) − rk( G root ) abelian gauge factors.34 .3 Some TCS G -Manifolds for Non-abelian Theories Let us bring the general proof given above to life and consider a few more examples of F-theory/M-theory dual pairs. In particular, let us consider situations in which there is aHiggsable non-abelian gauge group (we consider split and non-split models). G = SU (2)In this model we consider an elliptic building block Z + with N + = U ⊕ A . Such a model isfound from the dual pair of tops with vertices ♦ ◦ = x y z δ z ˆ z z − − − , ♦ = − − − − − − − − − − − . (6.23)The hypersurface (2.13) is obtained by first finding all the lattice points ν i on ♦ ◦ . Togetherwith ν these generate the rays of the fan of the toric variety P Σ in which the hypersurface Z + is embedded. In the present example, this gives rise to the blowup of the P , , bundleover the base B = (cid:98) P × P resolving the Kodaira I fiber. The correspondence between thehomogeneous coordinates of (cid:98) P × P and the rays of Σ is as follows: (cid:98) P × P Ray generator of P Σ z (2 , , , z (2 , , − , z (2 , , , z (0 , , , −
1) = ν (6.24)Note that the corresponding Tate model does not have a trivial canonical class, due to thedistinguished role of ν . This is responsible for c ( Z + ) = [ˆ z ] = ˆ H .The building block Z + is described by a Weierstrass model y + b xyw + b z yw = z δ x + x w b + xw z b + w z b , (6.25)where z δ is the section corresponding to the resolution divisor blowing up x = y = z = 0.The b i are sections of K − i P ⊗ K − i/ (cid:98) P ⊗ O ( H − n i ) where (cid:126)n = (0 , , , ,
2) are the vanishing ordersof the I Tate model. The I fiber degenerates further over∆ = ( b + 4 b (cid:0) ( b b − b ( b b + b ) + b (cid:0) b + 4 b (cid:1)(cid:1) = 0 . (6.26)These degeneration loci are present over points of Z + , so that the matter on the M-theoryside is localized on circles S e , see (2.1). On the F-theory side, the same degenerations of the35lliptic fibration happen over matter curves that are tori (the elliptic curves (cid:98) E ), and thus inthe absence of G -flux do not generate any chiral matter.Finally, let us compare the uncharged spectra of the dual models: From (2.14) it followsthat h , ( Z + ) = 4 h , ( Z + ) = 98 | N ( Z + ) | = 3 | K ( Z + ) | = 0 , (6.27)so that the Betti numbers of the G -manifold J are b = 13 , b = 270 . (6.28)The dual F-Theory model is compactified on a fourfold X F with Hodge numbers h , ( X F ) = 13 , h , ( X F ) = 98 , h , ( X F ) = 125 , (6.29)and h , ( B ) = 11 , h , ( B ) = 0 , (6.30)giving the same spectrum by (5.7) and (5.8). G = SU (5)Let us now consider an elliptic building block Z + with N + = U ⊕ A , realizing SU (5). Sucha model is found from the dual pair of tops with vertices ♦ ◦ = − − − ♦ = − − − − − − − − − − − − − − − − − . (6.31)The corresponding hypersurface is determined as before and is a resolved I Tate model. Thematter loci are again points in the Z + building block, and an equal number of elliptic curves (cid:98) E on the F-theory side. The comparison of the spectra follows from (2.14), whereby h , ( Z + ) = 7 h , ( Z + ) = 76 | N ( Z + ) | = 6 | K ( Z + ) | = 0 , (6.32)so that b = 16 , b = 223 . (6.33)36he dual F-theory geometry has h , ( X F ) = 16 , h , ( X F ) = 76 , h , ( X F ) = 100 , (6.34)as well as h , ( B ) = 11 , h , ( B ) = 0 , (6.35)again finding agreement. G = G Non-simply laced gauge groups are found in situations where G root (cid:54) = 0, but the individual P s of the resolution do not all become independent divisors on Z + and X F . We can engineera model with gauge symmetry G by an appropriate fibration of a K3 surface with a singleKodaira I ∗ fiber over P as Z + . This fibration is of non-split type, in the language of [36, 37],i.e. there is monodromy acting on some of the resolution divisors in the K3-surface, such thatthey form one irreducible divisor in Z + and X F . Such a geometry can be captured by a dualpair of tops [38] with vertices ♦ ◦ = − − − , ♦ = − −
10 6 6 − − − − . (6.36)From (2.14) it follows that h , ( Z + ) = 5 h , ( Z + ) = 90 | N ( Z + ) | = 4 | K ( Z + ) | = 0 , (6.37)so that b = 14 , b = 253 . (6.38)The dual F-theory geometry has h , ( X F ) = 14 , h , ( X F ) = 90 , h , ( X F ) = 116 , (6.39)as well as h , ( B ) = 11 , h , ( B ) = 0 . (6.40) A particularly interesting building block Z + can be found from the largest projecting top, ♦ ◦ = − − − , (6.41)37hich allows us to construct a model with a large rank K ( Z + ). It has h , ( Z + ) = 251 h , ( Z + ) = 0 | N ( Z + ) | = 10 | K ( Z + ) | = 240 , (6.42)so that matching with Z − gives b = 260 , b = 307 . (6.43)This model has gauge group E × U (1) . The dual F-theory geometry has h , ( X F ) = 260 , h , ( X F ) = 240 , h , ( X F ) = 20 , (6.44)and h , ( B ) = 251 , h , ( B ) = 240 . (6.45)It can be found from a nef partition of a polytope ∆ ◦ with vertices∆ = − − − − − − (6.46)This is an example with a highly non-trivial base B , where the P -fiber degenerates intomany reducible components, which is reflected in the large value for | K ( Z + ) | . The main goal of this paper was to initiate the M-theory/Heterotic duality in the con-text of G compactifications based on TCS constructions, and extending these to modelswith non-abelian gauge symmetry. These G -manifolds enjoy a K3-fibration and thereforeseem to be particularly amenable for the study of this duality. To explicitly realize the M-theory/Heterotic duality fiberwise, we require the K3-fibers of the G -building blocks to beelliptically fibered. For smooth TCS G -manifolds, we identify the dual SYZ-fibered Calabi-Yau three-fold as the Schoen manifold on the heterotic side. The key generalization however isto singular K3-fibered TCS geometries, which we motivated both from heterotic and F-theoryduals. The resulting models are 4d N = 1 supersymmetric gauge theories with non-abeliangauge groups and non-chiral matter. These gauge groups can be of Higgsable or non-Higgsabletype, and we provided constructions in both instances. There are numerous ways to utilizethis setup for extensions and generalizations. 38he first question concerns a more careful analysis of the duality to heterotic, and aspectral cover description of the models including a spectral line bundle. Determining this,would then yield a first principle derivation via heterotic/F-theory duality of the absence offlux, which we infered in this paper by the match of the spectra. Futhermore we argued thatdue to the integrality of c / G -flux.Looking ahead, a natural question is to utilize the M-theory/heterotic duality also toinclude conical singularities [2], which yield chiral matter in 4d. We were not able so far toidentify these codimension seven singularities within the TCS construction, where singularitiesare by construction codimension six. The setup presented in this paper gives alternative waysto try to extend the TCS construction to include chirality. For instance, starting with theorbifold description of the Schoen manifold on the heterotic side, one could use magnetizedtori instead, as in [39]. This generates chirality in the heterotic model, and understanding thedual in the M-theory construction may turn out to be insightful in guiding us towards realizinga chiral spectrum from G compactifications of M-theory. Likewise, chirality in F-theory islinked intimately with non-trivial G flux, threading through matter surfaces.Finally, we should note, that the TCS construction is suggestive of another limit, whichis related to the work [40] on a Higgs bundle description, a local model given in terms of anADE fibration over an associative cycle. We shall return to the connection of Higgs bundlesand TCS geometries elsewhere. Acknowledgments
We thank Jim Halverson, Alexei Kovalev, Magdalena Larfors, Dave Morrison, Timo Weigandand Michele del Zotto, for discussions. We thank the Aspen Center for Physics for hospi-tality during the completion of this work, and the working group on ‘Physics and Geometryof G -manifolds’ for discussions. The Aspen Center for Physics is supported by NationalScience Foundation grant PHY-1607611. This work was also partially supported by SimonsFoundation grant Details of Geometric Constructions
A.1 Nef Partitions
The combinatorial framework of [19] to construct families of Calabi-Yau hypersurfaces in toricvarieties has an elegant extension to complete intersections in toric varieties [24]. As in thecase of hypersurfaces, it starts with a pair of reflexive lattice polytopes which obey (cid:104) ∆ , ∆ ◦ (cid:105) ≥ − . (A.1)The polytope ∆ determines a toric variety via its normal fan, which can be refined using raysthrough lattice points on ∆ ◦ to find an appropriate desingularization of the ambient space A (which does not need to be a smooth space in general). A presentation of ∆ ◦ as the convexhull of the sum of n polytopes ∇ i ∆ ◦ = (cid:104)∇ , · · · , ∇ n (cid:105) conv (A.2)defines a nef partition if the dual polytopes ∆ i defined by (cid:104) ∆ i , ∇ j (cid:105) ≥ − δ ij (A.3)are lattice polytopes and recover ∆ as their Minkowski sum∆ = ∆ + · · · + ∆ n . (A.4)The corresponding nef partition is a decomposition[ − K A ] = (cid:88) i [ L i ] (A.5)into n nef divisors classes. These determine the n defining equations P i = 0 of the completeintersection as P i = (cid:88) m ∈ ∆ i c m (cid:89) ν k ∈ ∆ ◦ z (cid:104) m,ν k (cid:105) + δ ( ν k , ∇ i ) k = 0 , (A.6)for a choice of coefficients c m . Here, the sum runs over lattice points on ∆ i and the productruns over lattice points on ∆ ◦ . The function δ ( ν k , ∇ j ) is one if the lattice point ν k is on ∇ j and zero otherwise.The above framework allows a combinatorial computation of the Hodge numbers of theCalabi-Yau complete intersections. Swapping the roles of all of the ∆ i and the ∇ i realizes aconstruction of the mirror Calabi-Yau. 40 .2 The Schoen Calabi-Yau from a Nef Partition In this section we give some details about how the realization of the Schoen Calabi-Yau X het discussed in Section 3.2.2 as a Weierstrass double elliptic fibration over P is constructed as atoric complete intersection in the framework of Batyrev and Borisov. Consider the polytope∆ ◦ = − − − − − . (A.7)The lattice points of ∆ ◦ are the ray generators of a fan giving a toric variety with weightsystem ˆ z ˆ x ˆ y x y w ˆ w ˆ z Σof degrees W ˆ W . (A.8)One can confirm that there is a nef partition[ W ] = [ x ] + [ y ] + [ w ] + [ˆ z ][ ˆ W ] = [ˆ x ] + [ˆ y ] + [ ˆ w ] + [ˆ z ] (A.9)realizing (3.21). The vertices of the corresponding polytopes are ∇ W = − − ∇ ˆ W = − − − ∆ W = − −
21 1 − ∆ ˆ W = − −
21 1 − (A.10)From this, it follows that the non-trivial Hodge numbers are h , ( X het ) = h , ( X het ) = 19. A.3 G Building Blocks
In order to construct the building blocks Z + ,n , we start from the weight system (4.5). Next,we construct a polyhedron such that the linear relations between its vertices give rise to the41eights. We then neglect the coordinate ˆ z to find the associated top, which has the vertices˜ ♦ ◦ n = − − − n
10 0 0 1 0 . (A.11)Its dual ˜ ♦ n , given by (2.12), is not a projecting top for n ≥
2. The lattice points on ˜ ♦ n ,however, still correspond to all of the monomials appearing in the defining equation P n = 0.What is missing is a resolution of singularities, which can be achieved by a subdivision ofthe fan, which can be thought of as enlarging ˜ ♦ ◦ n . Correspondingly, the way ˜ ♦ n fails to be aprojecting top is by having vertices which are not lattice points. We can hence use the convexhull ♦ n of all lattice points of ˜ ♦ n instead and dualize back to find a polytope ♦ ◦ n enlarging ˜ ♦ ◦ n .If this process generates a projecting top, it gives us a smooth manifold which is a crepantresolution of the singularities we started with .In the present case, this process generates a projecting top for each n ≤
6. For n = 5, thereis a locus in the base of its elliptic fibration over which f, g, ∆ vanish with powers 4 , , ♦ ◦∗ . This blow-up does not change the Hodge numbers of Z + , . The same trick is of course applicable to Calabi-Yau hypersurfaces constructed from reflexive polytopes,and, as we will see in the next section, to Calabi-Yau complete intersections in toric varieties. ♦ ◦ = − − − ♦ ◦ = − − − ♦ ◦ = − − − ♦ ◦ = − − − ♦ ◦ = − − − ♦ ◦ = − − − ♦ ◦∗ = − − − ♦ ◦ = − − − (A.12)43 .4 Calabi-Yau Fourfolds To construct the Calabi-Yau fourfolds, we start from the weight system (5.1) and constructthe associated fan with ray generators x y w ˆ x ˆ y ˆ w z z ˆ z ˆ z −
10 0 0 0 0 0 1 − n
00 0 0 − − − − . (A.13)The convex hull ˜∆ ◦ n of these defines a reflexive polytope for n ≤ X F n for n = 0 , · · · , W = ˆ W = 0,i.e. − K A = W + ˆ WW = [ˆ z ] + [ z ] + [ z ] + [ x ] + [ y ] + [ w ]ˆ W = [ˆ z ] + [ˆ x ] + [ˆ y ] + [ ˆ w ] , (A.14)which determines a decomposition of ∆ ◦ into˜ ∇ W = − n
10 0 0 2 00 0 0 3 02 − − ˜ ∇ ˆ W = − − − . (A.15)For n ≥
2, ˜∆ ◦ is not reflexive, so that we are going to obtain a singular hypersurface. Thepolytopes of the dual nef partition, which are found from (cid:104) ˜∆ i , ˜ ∇ j (cid:105) ≥ − δ ij (A.16)are not lattice polytopes in this case. We can find a resolution by recalling that the ˜∆ i docontain all of the monomials appearing in the defining equations for X F n , but lack only a fewblow-ups to cure the singularities enforced by the equations. We hence replace the ˜∆ i by theconvex hull ∆ i of their integral points and dualize back to find polytopes ∇ i containing the˜ ∇ i using the above duality relations. As ∆ ◦ is the convex hull of the sum of the ∇ i , this alsoenlarges ˜∆ ◦ to ∆ ◦ , which corresponds to crepant (partial) resolutions of our singularities. Ifthis results in a nef parition of a reflexive polytope ∆ ◦ , we have achieved our goal of resolvingall singularities and can combinatorially compute the Hodge numbers.44n the present case, this method gives us nef partitions of reflexive polytopes for n ≤ n = 5, this gives a model with tensionless strings. We can blow up the base B to excise such loci which takes us to a model which has an effective field theory description.We denote the corresponding polytope by ∆ ◦∗ . The polytopes ∆ ◦ n are given by ∆ ◦ = − − − − − − ∆ ◦ = − − − − − − ∆ ◦ = − − − − − − ∆ ◦ = − − − − − − ∆ ◦∗ = − − − − − − ∆ ◦ = − − − − − − (A.17) References [1] J. J. Heckman, D. R. Morrison and C. Vafa,
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