The Geometry of SO(3), SO(5), and SO(6) models
aa r X i v : . [ h e p - t h ] M a y The Geometry of SO( ), SO( ), andSO( )-models Mboyo Esole ♢ and Patrick Jefferson ♣ ♢ Department of Mathematics, Northeastern University360 Huntington Avenue, Boston, MA 02115, USA ♣ Jefferson Laboratory, Harvard University
17 Oxford Street, Cambridge, MA 02138, U.S.A
Abstract
SO( ), SO( ), and SO( )-models are singular elliptic fibrations with Mordell–Weil torsion Z / Z and singular fibers whose dual fibers correspond to affine Dynkin diagrams of type A , C , and A respectively, where we emphasize the distinction between SO ( n ) and its universal cover Spin ( n ) .While the SO( )-model has been studied before, the SO( ) and SO( )-models are studied here forthe first time. By computing crepant resolutions of their Weierstrass models, we study their fiberstructures and topological invariants. In the special case that the SO ( n ) -model is an ellipticallyfibered Calabi-Yau threefold, we compute the Chern-Simons couplings and matter content of a 5D N = supergravity theory with gauge group SO ( n ) , which is related to M-theory compactified onthis Calabi-Yau threefold. We also verify the 6D lift of the 5D matter content is necessary andsufficient for anomaly cancellation in 6D ( , ) supergravity theories geometrically engineered byF-theory compactified on the same threefold. We find that the associated 5D and 6D supergravitytheories with SO ( n ) gauge symmetry indeed differ from their Spin ( n ) cousins, with one strikingconsequence of this distinction being that all such theories must include adjoint matter. ontents G -models and crepant resolutions of Weierstrass models . . . . . . . . . . . . . . . . . 72.5 Understanding G -models for orthogonal groups of small rank . . . . . . . . . . . . . . 72.6 Spin( n )-models versus SO( n )-models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.7 Weierstrass models with Mordell–Weil group Z / Z . . . . . . . . . . . . . . . . . . . . 92.8 Canonical forms for SO( ), SO( ), SO( ), and SO( )-models . . . . . . . . . . . . . . . 112.9 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 N = ( , ) supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.2 5D N = supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337.3 M-theory on a Calabi-Yau threefold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.4 F-theory on a Calabi-Yau threefold and anomaly cancellation . . . . . . . . . . . . . . 357.4.1 Generalized Green-Schwarz mechanism in F-theory . . . . . . . . . . . . . . . . 357.4.2 Anomaly cancellations for SO( ), SO( ), and SO( )-models . . . . . . . . . . . 381 Introduction
The utilization of gauge theory as a tool to study low-dimensional topology has dramatically in-creased our understanding of three-dimensional and four-dimensional manifolds. Similarly, gaugetheories geometrically engineered by elliptic fibrations are nowadays the major engine of progressin our mathematical understanding of elliptic fibrations, with key developments being motivated byideas and questions from theoretical physics which have helped shape our understanding of theirtopological invariants [5,30,34,35], degenerations [2,3,12,17,18,39,73], the structure of their singularfibers [13, 16, 25, 43, 61], their links with the study of Higgs bundles [4], and the geography of theirnetworks of flops [53, 80] as discussed in [26–29, 31, 41, 42, 50]. Conversely, M-theory and F-theorycompactifications on elliptic fibrations continue to play a key role in the study of supersymmetricgauge theories with eight supercharges in six-dimensional spacetime [21,46,49,51,60,62,67,69,75,76]and five-dimensional spacetime [20, 47, 48, 54, 55, 57, 81]. For reviews of F-theory, see [22, 51, 67, 79].Interestingly, the perspective that gauge theories bring to the study of elliptic fibrations is notonly influenced by the physics of local operators associated with the gauge algebra, but is alsoshaped by subtler physics considerations related to the global structure of the gauge group. Anillustrative example is the distinction between the Lie groups Spin( n ) and SO( n ), which despitebeing isomorphic as Lie algebras have different global structures. The global structures can typicallyonly be distinguished in gauge theories by studying the dynamics of operators associated to non-localobjects such as Wilson lines and instantons. For instance, the simple observation that the volume ofSO( ) is half of the volume of SU( ) has important consequences for the structure of their respectiveinstantons as beautifully explained in [23, §4.3]. The key point here is that while every SU( )-bundlenaturally gives rise to an SO( )-bundle, not all SO( )-bundles can be lifted to SU( )-bundles. Thisintrinsic difference is also reflected in the properties of instanton solutions, as the minimal charge ofan SO( )-instanton can be a fraction of the minimal charge of an SU( )-instanton [23].It turns out that the distinction between Spin ( n ) and SO ( n ) can be seen clearly in the setting ofelliptic fibrations, as the center of any gauge group geometrically engineered by an elliptic fibrationis conjectured to be isomorphic to the Mordell–Weil group of the elliptic fibration [59], and moreoverthe Z / Z that appears in the exact sequence connecting Spin( n ) and SO( n ) is the torsion subgroup ofthe Mordell–Weil group of the elliptic fibration. Thus, a precise understanding of the Mordell–Weiltorsion subgroup is crucial for a thorough understanding of physics related to the global structureof the gauge group.The subject of this paper is the SO ( n ) -model, and its distinction from the Spin ( n ) -model. Whilehigher rank SO ( n ) groups are related to I ∗ k fibers, the low rank examples n = , , are associatedto Kodaira fibers which do not belong to the same infinite family and hence require a case-by-caseanalysis. The simple groups SO ( n ) with n = , , are each subject to one of the four accidentalisomorphisms of Lie algebras:A ≅ B ≅ C , B ≅ C , D ≅ A ⊕ A , D ≅ A , where the semi-simple, connected and simply-connected, compact Lie groups corresponding to theabove Lie algebras are the low dimensional spin groupsSpin ( ) ≅ SU ( ) ≅ USp ( ) , Spin ( ) = SU ( ) × SU ( ) , Spin ( ) = USp ( ) , Spin ( ) = SU ( ) . The quotients of the above spin groups by Z / Z are the special orthogonal groupsSO ( ) , SO ( ) , SO ( ) SO ( ) . ( n ) -models with n = , , have received comparatively lessattention. While SO( )-models are well understood [59], SO( )-models were only recently studied[37], and much less is known about SO( ) and SO( )-models in F-theory; one goal of this paper isto fill that gap.We define SO( ), SO( ), and SO( )-models and study their properties through the eyes of M-theory and F-theory compactifications to five-dimensional (5D) [14, 53] and six-dimensional (6D)supergravity theories with eight supercharges [44, 63, 67, 69]. We use the geometry of the SO ( n ) -models to explore the Coulomb branch of a 5D N = supergravity theory with gauge group G = SO ( n ) for n = , , and hypermultiplets in the representation R , where R is determined by thedegeneration of the elliptic fibration over codimension two points in the base of the elliptic fibration.In particular, we study the one-loop exact prepotential of this 5D theory in terms of the intersectionring of the SO ( n ) -model. We also analyze the details of anomaly cancellation in the 6D ( , ) theory engineered by F-theory compactified on the same SO ( n ) -model; by enumerating the chargedhypermultiplets in terms of geometric quantities, we verify the same 5D content (uplifted to 6D) isboth necessary and sufficient for the cancellation of anomalies in the 6D supergravity theory.The computations described above are strong indications that the SO ( n ) -models studied in thispaper ( n = , , ) geometrically engineer consistent 5D and 6D supergravity theories. Unsurprisingly,it turns out that these theories are indeed distinct from their related Spin ( n ) cousins, as theirmassless spectra differ in various ways. One distinction is that the geometry of the SO ( n ) -modelsprevents the existence of 5D and 6D hypermultiplets transforming in the spinor representation ofSO ( n ) . Another distinction, somewhat more subtle, is that the structure of these SO ( n ) -modelsrequires the existence of 5D and 6D hypermultiplets in the adjoint representation, which implies thatthe non-gravitational sectors of these theories cannot be described as gauge theories with ultravioletfixed points.The remainder of the paper is structured as follows. In Section 2, we specify our conventions, dis-cuss some preliminary notions, and summarize the results of our computations. Section 3 contains adetailed description of the resolution, fiber structure, and topological invariants of the SO ( ) -model.Similar details are presented in Sections 4 and 5 for the SO ( ) and SO ( ) -models, respectively.The Hodge numbers of the SO ( n ) models are computed and tabulated in Section 6. Finally, thegeometric and topological aspects computed in the prior sections are used to study F/M-theorycompactifications on special cases of SO ( n ) -models corresponding to elliptically-fibered Calabi-Yauthreefolds in Section 7. 3 O( )-model Weierstrass equation y z = x + a x z + a xz Discriminant ∆ = a ( a − a ) Singular fibersMatter representation adjoint ∶ ( ) , n = + K Euler characteristic L + L c ( T B ) Triple intersections F = L (− α + α α + α α − α ) SO( )-model Weierstrass equation y z = x + a x z + s xz Discriminant ∆ = s ( s − a ) Singular fibers Matter representation adjoint+vector ∶ ( ⊕ ) , n = + K , n = K Euler characteristic L ( + L )( + L ) c ( T B ) Triple intersections F = L (− α + α α − α + α α − α ) SO( )-model Weierstrass equation y z + a xyz = x + mtx z + s xz Discriminant ∆ = s (( a + ms ) − s ) Singular fibers Matter representation adjoint+vector: ( ⊕ ) , n = + K , n = K Euler characteristic L + L c ( T B ) Triple intersections F − = L (− α + α α + α α + α α α )+ L (− α − α − α − α α + α α + α α + α α α ) , F + = F − + L ( α − α ) Table 1: Summary of results for the geometry of the SO(3), SO(5) and SO(6)-models.4
Preliminaries and summary of results
In this section, we introduce our conventions and some basic definitions. We mostly follow thepresentation of [28] with some basic adaptations to the case of a simple, connected compact group G . Definition 2.1 (Weierstrass model) . Consider a variety B endowed with a line bundle L → B . AWeierstrass model Y → B over B is a hypersurface cut out by the zero locus of a section of theline bundle of O ( ) ⊗ π ∗ L ⊗ in the projective bundle P ( O B ⊕ L ⊗ ⊕ L ⊗ ) → B . We denote by O ( ) the dual of the tautological line bundle of the projective bundle, and denote by O ( n ) ( n > )its n th-tensor product O ( ) ⊗ n . The relative projective coordinates of the P bundle are denoted by [ x ∶ y ∶ z ] . In particular, x is a section of O ( ) ⊗ π ∗ L ⊗ , y is a section of O ( ) ⊗ π ∗ L ⊗ , and z is a section of O ( ) . Following Tate and Deligne’s notation, the defining equation of a Weierstrassmodel is Y ∶ zy ( y + a x + a z ) − ( x + a x z + a xz + a z ) = , where the coefficient a i ( i = , , , , ) is a section of L ⊗ i on B . Such a hypersurface is an ellipticfibration since over the generic point of the base, the fiber is a nonsingular cubic planar curve witha rational point ( x = z = ). We use the convention of Deligne’s formulaire [19] and introduce thefollowing definitions: b = a + a , b = a a + a , b = a + a , b = b a − a a a + a a − a ,c = b − b , c = − b + b b − b , ∆ = − b b − b − b + b b b , j = c ∆ . (2.1)The above quantities are subject to the relations b = b b − b , = c − c . The discriminant locus is the subvariety of B cut out by the equation ∆ = , and is the locusof points p of the base B such that the fiber over p (i.e. Y ∣ p ) is singular. Over a generic point of ∆ , the fiber is a nodal cubic that degenerates to a cuspidal cubic over the codimension two locus c = c = . Up to isomorphism, the j -invariant j = c / ∆ uniquely characterizes nonsingular ellipticcurves. The intersection theory discussed in this paper essentially relies on three theorems from [30], where weuse the fact that each crepant resolution is expressed by a sequence of blowups. The first theorem,due to Aluffi, expresses the Chern class of a blowup along a local complete intersection. Thesecond theorem describes the pushforward associated to a blowup whose center is a local completeintersection. The third and final theorem provides an explicit map of an analytic expression in theChow ring of a projective bundle to the Chow ring of its base.
Theorem 2.2 (Aluffi, [1, Lemma 1.3]) . Let Z ⊂ X be the complete intersection of d nonsingularhypersurfaces Z , . . . , Z d meeting transversally in X . Let f ∶ ̃ X Ð→ X be the blowup of X centeredat Z . We denote the exceptional divisor of f by E . The total Chern class of ̃ X is then: c ( T ̃ X ) = ( + E ) ( d ∏ i = + f ∗ Z i − E + f ∗ Z i ) f ∗ c ( T X ) . (2.2)5 heorem 2.3 (Esole–Jefferson–Kang, see [30]) . Let the nonsingular variety Z ⊂ X be a completeintersection of d nonsingular hypersurfaces Z , . . . , Z d meeting transversally in X . Let E be theclass of the exceptional divisor of the blowup f ∶ ̃ X Ð→ X centered at Z . Let ̃ Q ( t ) = ∑ a f ∗ Q a t a be aformal power series with Q a ∈ A ∗ ( X ) . We define the associated formal power series Q ( t ) = ∑ a Q a t a ,whose coefficients pullback to the coefficients of ̃ Q ( t ) . Then the pushforward f ∗ ̃ Q ( E ) is f ∗ ̃ Q ( E ) = d ∑ ℓ = Q ( Z ℓ ) M ℓ , where M ℓ = d ∏ m = m ≠ ℓ Z m Z m − Z ℓ . Theorem 2.4 (Esole–Jefferson–Kang, see [30]) . Let L be a line bundle over a variety B and π ∶ X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] Ð→ B a projective bundle over B . Let ̃ Q ( t ) = ∑ a π ∗ Q a t a be a formalpower series in t such that Q a ∈ A ∗ ( B ) . Define the auxiliary power series Q ( t ) = ∑ a Q a t a . Then π ∗ ̃ Q ( H ) = − Q ( H ) H ∣ H =− L + Q ( H ) H ∣ H =− L + Q ( ) L , where L = c ( L ) and H = c ( O X ( )) is the first Chern class of the dual of the tautological linebundle of π ∶ X = P ( O B ⊕ L ⊗ ⊕ L ⊗ ) → B . Notation 2.5 (Blowups) . Let X be a nonsingular variety. Let Z ⊂ X be a complete intersectiondefined by the transverse intersection of r hypersurfaces Z i = V ( g i ) , where ( g , ⋯ , g r ) is a regularsequence. We denote the blowup of a nonsingular variety X with center the complete intersection Z by X ̃ X. ( g , ⋯ , g r ∣ e ) The exceptional divisor is E = V ( e ) . We abuse notation and use the same symbols for x , y , s , e i and their successive proper transforms. We do not write the obvious pullbacks. Definition 2.6 (Resolution of singularities) . A resolution of singularities of a variety Y is a properbirational morphism ϕ ∶ ̃ Y Ð→ Y such that ̃ Y is nonsingular and ϕ is an isomorphism away from thesingular locus of Y . In other words, ̃ Y is nonsingular and if U is the singular locus of Y , ϕ maps ϕ − ( Y ∖ U ) isomorphically onto Y ∖ U . Definition 2.7 (Crepant birational map) . A birational map ϕ ∶ ̃ Y → Y between two algebraicvarieties with Q -Cartier canonical classes is said to be crepant if it preserves the canonical class, i.e. K ̃ Y = ϕ ∗ K Y . Following a common convention in physics, we denote an irreducible representation R of a Lie algebra g by its dimension (in boldface.) The weights are denoted by ̟ Ij where the upper index I denotesthe representation R I and the lower index j denotes a particular weight of the representation R I .Let φ be a vector of the coroot space of g in the basis of the fundamental coroots. Each weight ̟ defines a linear form ( φ, ̟ ) by the natural evaluation on a coroot. We recall that fundamentalcoroots are dual to fundamental weights. Hence, with our choice of conventions, ( φ, ̟ ) is the usualEuclidian scalar product. Definition 2.8 (Hyperplane arrangement I ( g , R ) ) . The hyperplane arrangement I ( g , R ) is definedinside the dual fundamental Weyl chamber of g and its hyperplanes are the kernel of the weights ofthe representation R [26, 27, 50]. 6 crepant resolution over a Weierstrass model is always a minimal model over the Weierstrassmodel. Distinct minimal models are connected by a sequence of flops. The geography of flops (orequivalently of its extended Kähler-cone) of a G -model is conjectured to be given by the chamberstructure of the hyperplane arrangement I ( g , R ) , where g is the Lie algebra whose type is dual to thedual graph defined by the fibral divisors of the G -model and R is the representation whose weightsare given by minus the intersection numbers of rational curves comprising singular fibers locatedover codimension-two points in the base [53]. G -models and crepant resolutions of Weierstrass models One of the fundamental insights of F-theory is that the geometry of an elliptic fibration ϕ ∶ Y → B naturally determines a triple ( R , G, g ) where R is a representation of a reductive Lie group G with asemi-simple Lie algebra g . Moreover, the fundamental group of the Lie group G is isomorphic to theMordell–Weil group of the elliptic fibration. We call such an elliptic fibration a G -model. G -modelsare used in M-theory and F-theory compactifications on elliptically fibered varieties to geometricallyengineer gauge theories with gauge group G and matter transforming in the representation R ofthe gauge group. Under mild assumptions, a G -model is birational to a singular Weierstrass model[19,64,65]. Starting from a singular Weierstrass model, we can retrieve a smooth elliptic fibration viaa resolution of singularities. In the best situation, we can ask the resolution to be crepant. Whencrepant resolutions are not possible, we settle for a partial resolution with Q -factorial terminalsingularities. In both cases, the (partial)-resolution is a minimal model over the Weierstrass modelin the sense of Mori’s theory. In the physically relevant cases, we are mostly interested in crepantresolutions. Starting from threefolds, minimal models are not unique but connected by a finitesequence of flops. The network of flops of a given G -model is described by a hyperplane arrangementI( g , R ) defined in terms of the Lie algebra g and the representation R . Two chambers of a hyperplanearrangement sharing a common boundary wall of one dimension lower are said to be adjacent. Theadjacent graph of the chambers of the hyperplane arrangement I( g , R ) of a G -model with a triple ( R , G, g ) is conjectured to be isomorphic to the network of flops of the minimal models derivedby crepant resolutions of the Weierstrass model birational to the G -model. Each of the chambersalso coincides with a Coulomb phase of a 5D N = supergravity theory with gauge group G andhypermultiplets in the representation R , related to compactification of M-theory on the G -modelassuming that the G -model is a Calabi-Yau threefold. G -models for orthogonal groups of small rank Several G -models have been studied from the perspective of M-theory and F-theory compactifica-tions. The SU( ), SU( ), SU( ), and SU( )-models are studied in [41–43]. Aspects of the generalSU( n )-models are discussed in [40]. The G , Spin( ), and Spin( )-models are studied in [28], theF -models in [32], the E -models in [33]. Semi-simple Lie groups of rank two or three have beenstudied recently: Spin( )=SU( ) × SU( ) and SO( ) in [37], SU( ) × G in [36], SU( ) × SU( ) in [29],and SU( ) × SU( ) in [38], The Euler characteristics of G -models defined by crepant resolutions ofWeierstrass models resulting from Tate’s algorithm have been computed recently in [30] for G aconnected and simple group for characteristic numbers of other type of ellliptic fibrations [38, 39].The Hodge numbers for G -models that are Calabi-Yau threefolds are also computed in [30]. Forconnected compact simple Lie groups, there are few cases left to study, with one subtle case beingthat of special orthogonal groups of small rank.Special orthogonal groups SO( + n ) or SO( + n ) require an elliptic fibration with a Mordell–Weil group Z / Z and a discriminant locus containing an irreducible component S such that the7eneric fiber over S is respectively of Kodaira type I ∗ ns n and I ∗ s n while the generic fibers over otherirreducible components of the discriminant locus are irreducible curves. In particular, SO( m )-modelswith small rank ( ≤ m ≤ ) cannot be achieved with fibers of type I ∗ n as n would have to be negative.The aim of this paper is to study the geometry of SO( n )-models in the case of n = , , :SO ( ) , SO ( ) , or SO ( ) . The SO( )-model is studied in [37], and is more complicated as SO( ) is not a simple group and thusrequires a collision of singularities. Moreover, the dual graph ̃ A can be realized by several Kodairatypes, which implies that there are several distinct ways to construct an SO( )-model [37].The geometric engineering of SO( ), SO( ), and SO( )-models relies on the accidental isomor-phisms so ≅ su so ≅ sp so ≅ su . The SO( ), SO( ), and SO( )-models are respectively derived by imposing the existence of a Z / Z Mordell–Weil group in the SU( ), Sp( ), and SU( )-models. The corresponding Kodaira fibers arerespectively I or III , I ns , and I s , and their dual graphs are respectively the affine Dynkin diagrams (see Figure 1) ̃ A , ̃ C t , and ̃ A . ̃ A ̃ C t ̃ A Figure 1: Affine Dynkin diagrams corresponding to the respective Lie groups SO(3), SO(5), andSO(6). In the above graphs, the black node represents the affine node; deleting this node producesthe corresponding finite Dynkin diagrams of type A ≅ B , C ≅ B and A ≅ D . n )-models versus SO( n )-models By applying Tate’s algorithm in reverse one can construct a G -model for any of the following simplyconnected and simple Lie groups:SU ( n ) ( n ≥ ) , Sp ( n ) ( n ≥ ) , Spin ( n ) ( n ≥ ) , G , F , E , E , E Spin groups of low rank are retrieved by the following accidental isomorphisms:Spin ( ) ≅ SU ( ) , Spin ( ) ≅ SU ( ) × SU ( ) , Spin ( ) ≅ USp ( ) , Spin ( ) ≅ SU ( ) . On the other hand, orthogonal groups are much more complicated to handle than Spin groups sincethey require the Mordell–Weil group to be isomorphic to Z / Z . For n ≥ , Spin( n ) is a simply-connected double-cover of SO( n ): Ð→ Z / Z Ð→ Spin( n ) π Ð→ SO( n ) Ð→ .
8t follows that an SO( n )-model with n ≥ is given by an elliptic fibration with a Mordell–Weilgroup Z / Z and a discriminant locus which contains an irreducible component S such that the fiberover the generic point of S is of type I ∗ k and the fibers over the remaining generic points of ∆ areirreducible curves (type I or II). More specifically, a fiber of type I ∗ s k ≥ with a Mordell–Weil torsion Z / Z gives an SO( + k )-model while a fiber of type I ∗ ns k ≥ gives an SO( + k )-model. An SO( )-modelrequires a fiber of type I ∗ ss and Mordell–Weil torsion Z / Z . These points are summarized in Table2. G Generic fiber over S Mordell–Weil groupSpin( ) I ∗ ss trivialSpin( + k ) I ∗ s k trivialSpin( + k ) I ∗ ns + k trivialSO( ) I ∗ ss Z / Z SO( + k ) I ∗ sk Z / Z SO( + k ) I ∗ ns + k Z / Z Table 2: Spin( n )-models vs SO( n )-models. Z / Z A genus-one fibration ϕ ∶ Y → B is a proper surjective morphism ϕ between two algebraic varieties Y and B such that the generic fiber is a smooth projective curve of genus one. A genus-one fibration ϕ ∶ Y → B is said to be an elliptic fibration if ϕ is endowed with a choice of rational section. Arational section of a morphism ϕ ∶ Y → B is morphism σ ∶ U → Y such that U a Zariski open subsetof B and ϕ ○ σ restricts to the identity on U . The rational section turns the generic fiber into abona fide elliptic curve: a genus-one curve with a choice of a rational point. Given an elliptic curve E defined over a field k , the Mordell–Weil group of E is the group of k -rational points of E . TheMordell–Weil theorem states that the Mordell–Weil group is an abelian group of finite rank. Thisis a theorem proven by Mordell in the case of an elliptic curve defined over the rational numbers Q and later generalized to number fields and abelian varieties by Weil. For an elliptic fibration, theMordell–Weil group is the Mordell–Weil of its generic fiber or equivalently, the group of its rationalsections. Throughout this paper, we work over the complex numbers. We write V ( f , ⋯ , f n ) for thealgebraic set defined by the solutions of f = ⋯ = f n = .An important model of an elliptic fibration is the Weierstrass model, which expresses an ellipticfibration as a certain hypersurface in a P -projective bundle with the generic fibers being a planarcubic curve in the P -fiber. Weierstrass models are very convenient as they have ready-to-useformulas which compute important data such as the j -invariant and the discriminant locus. Tate’salgorithm provides a simple procedure to identify the type of a singular fiber by manipulating thecoefficients of the Weierstrass model [66,74]. Intersection theory on an elliptic fibration is also mademuch easier in a Weierstrass model as the ambient space is a projective bundle. On a Weierstrassmodel, the Mordell–Weil group law can be expressed geometrically by the chord-tangent algorithm.Weierstrass models are written in the notation of Deligne and Tate [19] reviewed in Section 2.1.The Weierstrass model is expressed as the locus zy + a xyz + a xyz = x + a x z + a xz + a z , in a projective P -bundle with projective coordinates [ x ∶ y ∶ z ] , as is discussed in more detail inSection 2.1. The fundamental line bundle of a Weierstrass model is denoted by L and the coefficient9 i ( i = , , , , ) is a section of L ⊗ i . If the Weierstrass model is Calabi-Yau, its canonical class istrivial and the line bundle L is equal to the anti-canonical line bundle of the base B of the fibration.Since we work in characteristic zero, we can shift y to eliminate a and a and write the Weierstrassmodel as follows. zy = x + a x z + a xz + a z . The zero section is x = z = and the involution defined by sending a rational point [ x ∶ y ∶ z ] toits opposite with respect to the Mordell–Weil group takes the following simple form: [ x ∶ y ∶ z ] ↦ [ x ∶ − y ∶ z ] . In particular, if a is identically zero, there is a new rational section x = y = , which is its owninverse and it follows that the Mordell–Weil group is generically Z / Z . An elliptic fibration withMordell–Weil group Z / Z can always be put in the form zy = x ( x + a xz + a z ) . The discriminant locus of this elliptic fibration is ∆ = a ( a − a ) . One can check that the fiber over the locus V ( a ) is of Kodaira type I while the fiber over theother component of the reduced discriminant is I . It follows that the generic elliptic fibration witha Z / Z Mordell–Weil group is an SO( )-model and the class of the divisor supporting the groupSO( ) is necessarily a section of the line bundle L ⊗ .A generic Weierstrass model with a Mordell–Weil torsion subgroup Z / Z is given by the followingtheorem which is a direct consequence of a classic result in the study of elliptic curves in numbertheory (see for example [52, §5 of Chap 4]) and was first discussed in a string theoretic setting byAspinwall and Morrison [7]. Theorem 2.9.
An elliptic fibration over a smooth base B and with Mordell–Weil group Z / Z isbirational to the following (singular) Weierstrass model. zy = x ( x + a xz + a z ) , The section x = y = is the generator of the Z / Z Mordell–Weil group and x = z = is theneutral element of the Mordell–Weil group. The discriminant of this Weierstrass model is ∆ = a ( a − a ) . This model has a fiber of type I ns over V ( a ) , and fibers of type I over V ( a − a ) . As we willsee after a crepant resolution of singularities, at the collision of these two components, namely at V ( a , a ) , we get a fiber of type III. The dual graph of I ns is the affine Dynkin diagram ̃ A . Itfollows that the Lie algebra g associated with this elliptic fibration is A , the Lie algebra of thesimply connected group SU( ). The generic Weierstrass model with a Z / Z Mordell–Weil groupautomatically gives an SO( )-model since we have the exact sequence Ð→ Z / Z Ð→ SU( ) Ð→ SO( ) Ð→ . The SO( ) and the SU( )-models are rather different. For example, an SO( )-model cannot havematter charged in a spin representation of A while an SU( )-model can always have matter inthe representation of SU( ) [41], which is a spin representation. Generically, the SO( )-modelonly has matter in the adjoint representation while the SU( )-model has matter in both the adjointand fundamental representations. Geometrically, this is due to the existence of fibers of type I in codimension two for SU( )-model; by contrast, the SO( )-model only has fibers of type III overcodimension-two points. 10 .8 Canonical forms for SO( ), SO( ), SO( ), and SO( )-models The SO( )-model is the generic case of a Weierstrass model with Mordell–Weil group Z / Z as givenin Theorem 2.9 and has been studied in [7, 59]. The SO( )-model was constructed in [37]. To theauthors’ knowledge, the SO( ) and SO( )-models have not been constructed explicitly before andwere announced in [30] where we computed their Euler characteristics [30] and additional charac-teristic numbers were computed in [34]. The SO( )-model is obtained from the SO(3)-model via abase change that converts the section a to a perfect square a = s of an irreducible and reducedCartier divisor so that the fiber over V ( s ) is of type I ns . If instead we had a = as , the group wouldbe semi-simple. The SO(6)-model is subsequently derived from the SO(5)-model by requiring thatthe section a is a perfect square modulo s so that the fiber type over S = V ( s ) is of type I s .The Weierstrass models for the SO( ), SO( ), SO( ), and SO( )-models are as follows:SO ( ) -model ∶ zy = x ( x + a xz + a z ) , (2.3)SO ( ) -model ∶ zy = x ( x + a xz + stz ) , (2.4)SO ( ) -model ∶ zy = x ( x + a xz + s z ) , (2.5)SO ( ) -model ∶ zy + a yxz = x ( x + msxz + s z ) , m ≠ , ± . (2.6)The SO( )-model is derived from the SO( )-model by a base change a → st where S = V ( s ) and T = V ( t ) . The SO( )-model is derived from the SO( )-model by a base change a → s where S = V ( a ) for SO( ) and S = V ( s ) for SO( )-models. The SO( )-model is derived from the SO( )-model by forcing the fiber over the generic point of S to be type I s rather than type I ns . For SO( )and SO( ), a cannot be identically zero, otherwise the Mordell–Weil group becomes Z / Z × Z / Z .Moreover, in the SO( )-model, if a is identically zero, the fiber over the generic point S = V ( s ) becomes I ∗ s and the Lie algebra becomes D . In the SO( )-model, we can complete the square in y and end up with zy = x ( x + a xz + s z ) where a = a / + ms . This shows that the SO( )-modelis a limiting case of the SO( )-model in which a is a perfect square modulo s , hence the fiber over S is of type I s instead of I ns .SO ( ) ≅ SO ( ) × SO ( ) is the product of two simple groups and therefore requires two irreduciblecomponents S = V ( s ) and T = V ( t ) , which also implies that their product st is a section of L ⊗ [37].For the SO( ), SO( ), and SO( )-models, the class of the divisor S supporting the gauge group iscompletely fixed in terms of the fundamental line bundle L of the Weierstrass model. The divisor S is a section of L ⊗ for SO( )-models and of L ⊗ for SO( ) and SO( )-models. If we denote by L = c ( L ) the first Chern class of L , then the class of S in the Chow ring of the base is respectively L for SO( )-models; and L for SO( ) and SO( )-models. In the Calabi-Yau case, L = − K where K is the canonical class of the base. In the case where the elliptic fibration is a Calabi-Yau threefold,the curve S supporting the gauge group always has genus g = + K for the SO( )-model and + K for the SO( ) and SO( )-model. Since the number of hypermultiplets transforming in the adjointrepresentation is given by the genus of the curve S , this implies that these models cannot be definedwith S a rational curve and hence always has matter transforming in the adjoint representation. In physics, a G -model provides a geometric engineering of gauge theories, related to F-theory andM-theory compactifications, yielding a gauge group G and matter fields transforming in the repre- The same is true for G = SU( n )/( Z / Z ) and G = Sp( + n )/( Z / Z ) where Sp( m ) is the compact connected Liegroup with Lie algebra C m . In particular, it follows that in the Calabi-Yau case, the class of the divisor S dependsonly on the canonical class of the base. R of G . The chambers of the hyperplane arrangement I( g , R ) correspond to the Coulombbranches of a five-dimensional N = (eight supercharges) gauge theory with gauge group G andhypermultiplets transforming in the representation R . The flats of the hyperplane arrangement arethe mixed Coulomb-Higgs branches of the gauge theory. The faces of the hyperplane arrangementare identified with partial resolutions of the Weierstrass model corresponding to the G -model.For each of the G -models that we consider in this paper, we do the following:1. We discuss their crepant resolutions and a classification of the singular fibers of the resolvedgeometries. The SO( )-model is the only one that has non-Kodaira fibers, namely, a fiberof type 1-2-1, that is a contraction of a Kodaira fiber of type I ∗ . There is a unique crepantresolution for the SO( )-model and the SO( )-model; and two crepant resolutions connectedby a flop for the SO( )-model. While it is known that Spin( + n )-models that are crepantresolutions of Weierstrass models are usually not flat , Spin( ) and Spin( )-models can begiven by flat fibrations that are crepant resolutions of Weierstrass models [28]. We show thatthe crepant resolution of the Weierstrass models defining SO( ), SO( ), and SO( )-models alsogive flat fibrations.2. The understanding of the crepant resolutions allows us to compute topological data such asthe Euler characteristic of these varieties [30] and other characteristic invariants [34], the tripleintersection numbers of their fibral divisors, and in the Calabi-Yau threefold case, the Hodgenumbers.3. The discriminant locus has an irreducible component S over which the generic fiber is reduciblewhile the fibers away from S are irreducible. The generic fiber over S is respectively I ns orIII for SO( ), I ns for SO( ), and I s for SO( )-models. The discriminant of the SO( )-modelcontains two irreducible components, one of type I ns and one of type I . The discriminant ofthe SO( ) model (resp. the SO( )-model) contains three irreducible components meeting atthe same locus, one of type I ns (resp. I s ) and the two others of type I ; all three componentsmeet at the same codimension-two locus.The singular fibers over the intersections of these components are responsible for the rep-resentation R . The representation is determined by computing intersection numbers of thecomponents of the codimension two fibers with the fibral divisors. These intersection numbersare interpreted as minus the weights of a representation. The full representation is derivedfrom few weights only by using the notion of saturated set of weights . The adjoint represen-tation is always present and we determine the non-adjoint components of R geometrically.The SO( )-model does not have any representation other than the adjoint representation. Incontrast, the SU ( ) -model will have a fundamental representation . For the SO( )-model,the collision I ns + I gives a non-Kodaira fiber of type 1-2-1, which is an incomplete fiber oftype I ∗ ss and corresponds to the vector representation .I ns + I + I → − − . For SO( )-model, the collision I s + I gives a Kodaira fiber of type I ∗ s whose dual graph is ̃ D and corresponds to the vector representation .I s + I → I ∗ s . While the Weierstrass models of the SO( ) or SO( )-model have unique crepant resolutions,the SO( )-model has two crepant resolutions connected by an Atiyah-like flop. Non-flatness here means that the fiber can have higher dimensional components.
12. We identify the representation R from codimension two degenerations of the generic singularfiber using intersection theory. We then study the hyperplane arrangement I( g , R ). In theSO( )-model, the representation R is the adjoint, thus I( g , R ) has only one chamber that isthe full dual fundamental Weyl chamber of A . In the SO( )-model, the representation R isthe direct sum of the adjoint and the vector representation of SO( ), since all the weights ofthe vector representations are also weights of the adjoint representation, I( g , R ) has only onechamber that is again the full dual fundamental Weyl chamber of A . In the SO( )-model,the representation R is also the direct sum of the adjoint and the vector representation ofSO( ). However, the vector representation of SO( ) has two weights whose kernel intersectsthe interior of the dual fundamental Weyl chamber of A . It follows that we have two chambersseparated by the wall orthogonal to the weight ( , , − ) .5. In the case where the elliptic fibration is a Calabi-Yau threefold, we study compactificationsof M-theory; these compactifications can be used to geometrically engineer a 5D N = (i.e.eight supercharges) supergravity theory with gauge group G and hypermultiplets in the rep-resentation R . This N =
5D supergravity theory has a Coulomb branch, and the dynamicsof the theory on the Coulomb branch are controlled by a one-loop exact prepotential F IMS .We determine the number of hypermultiplets transforming in each of the irreducible compo-nents of R by comparing the exact prepotential of a 5D gauge theory with gauge group G and an undetermined number of hypermultiplets transforming in the representation R , withthe triple intersection numbers of the fibral divisors of the elliptic fibration. By making thiscomparison, we determine explicitly the multiplicity of each irreducible component R i in the5D hypermultiplet representation R in terms of the intersection ring of the elliptic fibration.6. The elliptic fibrations studied here can also be used to define F-theory compactifications geo-metrically engineering 6D ( , ) supergravity when the elliptic fibration is a Calabi-Yau three-fold. Since 6D ( , ) theories are chiral, the cancellation of anomalies is an important consis-tency condition. We determine the number of charged hypermultiplets in the 6D theory byexpressing the anomaly cancellation conditions in terms of geometric quantities and verifyingthat the allowed matter content matches what we find in the 5D N = theory. The SO( )-model is given by a crepant resolution of the generic Weierstrass model with Mordell–Weil torsion Z / Z . Without loss of generality, we can put the generator of the Mordell–Weil groupat the origin x = y = . The Weierstrass model of an SO( )-model can then always be defined by theequation Y ∶ y z − x ( x + a xz + a z ) = . (3.1)We assume that the varieties V ( a ) and V ( a ) are smooth irreducible varieties intersecting transver-sally. In particular, a is not a perfect square.The Mordell–Weil group of the elliptic fibration ϕ ∶ Y → B is isomorphic to Z / Z and generatedby the section Σ ∶ x = y = . The neutral element of the Mordell–Weil group is the section Σ ∶ x = z = . Both elements of the Mordell–Weil group are on the line x = tangent to the genericfiber at x = y = .Using equations (2.1), the short form of the Weierstrass equation is specified by c = ( a − a ) , c = − a ( a − a ) . ∆ = a ( a − a ) . (3.2)The following theorem describes the fiber geometry of a minimal crepant resolution of an SO(3)-model. Theorem 3.1.
The reduced discriminant of a SO(3)-model consists of two smooth divisors V ( a − a ) and V ( a ) . The intersection V ( a , a ) of these two divisors is non-transverse and correspondsto the cuspidal locus of the elliptic fibration. The singular Weierstrass model Y has three types ofsingular fibers:1. The generic fiber over V ( a ) is of type I ns (two rational curves meeting transversally along adivisor of degree 2 such that a quadratic field extension is needed to split the divisor into twoclosed rational points).2. The generic fiber over V ( a − a ) is of type I (a nodal cubic).3. The generic fiber over the collision V ( a , a ) is of Kodaira type III (two rational curves inter-section along a double point): Locus Fiber V ( a ) I ns V ( a − a ) I V ( a , a ) III
Proof.
The cuspidal locus is by definition V ( c , c ) . A direct computation gives b = a , c = ( a − a ) , c = − a ( a − a ) , and thus it follows that the cuspidal locus is V ( a , a ) . Sincethe valuations of ( c , c , ∆ ) over V ( a ) and V ( a − a ) are respectively ( , , ) and ( , , ) , itfollows from Tate’s algorithm that the generic fiber over the divisor V ( a ) is Kodaira type I andthe generic fiber over V ( a − a ) is Kodaira type I (a nodal elliptic curve). (A fiber of type I ischaracterized by two rational curves intersecting transversally at two closed points—Kodaira fibersof type I n are described in Step 2 of Tate’s algorithm.) In Tate’s algorithm one classifies geometricfibers, and in order to describe the geometric fiber, it is necessary to work in the splitting field of thepolynomial s − a T + a whose discriminant is b = a − a . If b is a perfect square on the divisorover which we have the I n fiber, the fiber is split; otherwise, the fiber is non-split. In the presentcase, a = and b = − a is not a perfect square modulo a , hence the generic fiber over V ( a ) isa non-split I fiber, i.e. type I ns . We also expect to see the fiber degeneates at the intersection ofthe two components of the discriminant locus since that intersection is supported on the cuspidallocus of the fibration. We prove in the next Section 3.3 that the fiber over V ( a , a ) is Kodaira typeIII (a type III fiber is composed of two rational curves meeting at a double point.) This is a resultof the fact that the type I fiber, which is located over V ( a ) , degenerates to a type III fiber over V ( a ) . Corrollary 3.2. If a is a perfect square, then the fiber over V ( a ) is type I s . An SO(3)-model withan I s fiber over V ( a ) can always be described as:SO(3)-model with I s ∶ Y ∶ zy + a xyz − x − a xz = (3.3) Corrollary 3.3. If a is identically zero, then the fiber over V ( a ) is type III. An SO( ) -model witha type III fiber can always be described as:SO(3)-model with III ∶ Y ∶ zy − x − a xz = (3.4) with j -invariant, j = over V ( a ) . .1 Singularities of the Weierstrass model A Weierstrass model can only be singular away from its zero section x = z = . For that reason, wediscuss the singularities in the open patch z ≠ . The Weierstrass equation (3.1) is then a doublecover of the base B branched along y = x ( x + a x + a ) = . The branch locus consists of twoirreducible divisors V ( y, x ) and V ( y, x + a x + a ) meeting transversally in codimension two along V ( y, x, a ) . We notice that the divisor V ( y, x ) is the generator of the Mordell–Weil group on eachsmooth curve. Since the singularities of a double cover are those of its branch locus, it is immediateto see that the singular scheme is supported on the locusSing ( Y ) ∶ V ( x, y, a ) . (3.5) The minimal crepant resolution of the SO(3)-model requires only a single blowup. We can takethe center of this blowup to be either of the ideals ( x, y ) or ( x, y, a ) . We denote by X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] the original ambient space, which is the P bundle. We obtain a crepant resolution byblowing up along the ideal ( x, y ) . We denote by X the blowup of X along the ideal ( x, y ) . Theblowup can be implemented explicitly by the substitution ( x, y ) = ( x e , y e ) where V ( e ) = E isthe exceptional divisor, which is a P bundle with fiber parametrized by [ x ∶ y ] . The projectivecoordinates describing X are [ x = e x ∶ y = e y ∶ z ] , [ x ∶ y ] . (3.6)The fibers of X Ð→ B are Hirzerbuch surfaces of degree one. Theorem 3.4.
The blowup f ∶ X X ( x, y ∣ e ) (3.7) provides a crepant resolution of the SO(3)-model given by the Weierstrass equation (3.1) .The proper transform of the Weierstrass model Y under the blowup is the elliptic fibration ϕ ∶ Y → B given as the following hypersurface in X : Y ∶ e y z − x ( e x + e a x z + a z ) = . (3.8)The section Σ = V ( x, z ) = V ( x , z ) . The generating section of the Mordell–Weil group is Σ = V ( e , x ) . Theorem 3.5.
The fiber I ns over V ( a ) is composed of the following two curves: C ∶ a = y z − x ( e x + a z ) = , [ e x ∶ e y ∶ z ][ x ∶ y ] . (3.9a) C ∶ a = e = , [ ∶ ∶ z ][ x ∶ y ] . (3.9b)The curve C is the proper transform of the original elliptic fiber and corresponds to the usual nor-malization of a nodal curve . C is the exceptional curve coming from the blowup and is parametrized For the generic curve C , z and x are units. To show that C is a rational curve, we show that is has a rationalparametrization in the patch ze ≠ . Then fixing z = , introducing the variable t = y / x , and solving for e x , wefind a rational parametrization of C given by t ↦ [ e x ∶ e y ∶ ][ ∶ y / x ] = [ t − a ∶ t ( t − a ) ∶ ][ ∶ t ] . [ x ∶ y ] —notice that C corresponds to the zero section of the Hirzebruch surface (i.e. F ) fiberof X . Both C and C are smooth rational curves parametrized by [ x ∶ y ] . When we restrict to V ( a ) , we see that the section V ( x , z ) is on C while the section V ( x , e ) is located on C . Thegenerator of the Mordell–Weil group touches the curve C but not C [59]. The intersection of theirreducible components C and C is a zero-cycle of degree two: C ∩ C ∶ a = e = y − a x = [ ∶ ∶ z ][ x ∶ y ] , (3.10)As we move over V ( a ) , C ∩ C defines a double double cover of V ( a ) branched at V ( a , a ) , whichsplits into two distinct divisors in the special case that a is a perfect square. Over the branch locus V ( a , a ) , the intersection (3.10) collapses to a double point (i.e. a tangent) and hence the generictype I ns fiber degenerates to a type III fiber—see Figure 2.We now turn our attention to the divisors D and D swept out by C and C as they move overthe base. Notice that C is both smooth and birationally-equivalent to P , while C is clearly a P .Hence, both D and D are projective bundles of the type P [ O S ⊕ L ] where S = V ( a ) : D ≅ P S [ O S ⊕ L ] , D ≅ P S [ O S ⊕ L ] . (3.11) B V ( a ) V ( a − a ) C C C C Weierstrass model y z = x + a x z + a xz Discriminant ∆ = a ( a − a ) Matter representation adjointRepresentation multiplicity n = + K Euler characteristic L + L c ( T B ) Triple intersections F = L ( − α + α α + α α − α ) Figure 2: Summary of geometry for the resolution of the SO( )-model given by (3.8). Note that theKähler cone only consists of a single chamber. 16 im B Euler characteristic Calabi-Yau case L c ( c − L ) L − c L ( c − c L + L ) c ( c + c ) L ( c − c L + c L − L ) c ( c − c + − c c ) Table 3: Euler characteristic of the SO(3)-model for bases of dimension up to . The i th Chern classof the base is denoted c i . The Calabi-Yau cases are obtained by imposing L = c . In this section, we compute the generating function for the Euler characteristic along with thetriple intersections D i D j D k . The Euler characteristic is the degree of the zero component of the(homological) total Chern class. The result is given by the following theorem. Theorem 3.6 (See [30]) . The generating function for the Euler characteristic of an SO( )-model is ϕ ∗ c ( Y ) = Lt + Lt c t ( T B ) , (3.12) where c t ( T B ) = ∑ c i ( T B ) t i is the Chern polynomial of the base B . The Euler characteristic of an SO(3)-model over a base B of dimension d is the coefficient t d for d = dim B of the generating function. See Table 3 for some examples. We note that the generatingfunction is the same as that of an elliptic fibration of type E studied in [3].We now turn our attention to the triple intersections D a D b D c . Theorem 3.7.
Assume that the base B is a smooth surface. The triple intersection numbers for thefibral divisors D and D for an SO(3)-model are F = ( α D + α D ) = L ( − α + α α + α α − α ) . (3.13a) Proof.
The first step in computing intersection products is to identify the classes of the divisors D and D . We exploit the linear relations relating D and D with the sections Σ and Σ comingfrom Mordell–Weil group: Σ = V ( z, x ) , Σ = V ( e , x ) . (3.14) Σ is the divisor corresponding to the zero section V ( x , z ) and Σ = V ( e , x ) is the divisor corre-sponding to the generator of the Mordell–Weil group. Using the equation of the elliptic fibration wefind that: ( z ) = , ( x ) = Σ + Σ . (3.15)From these linear relations we get: Σ = H , Σ = − E + H + L, (3.16)where we have used ( x ) = L + H − E and ( z ) = H . According to the defining equation (3.8) for Y , D , D must satisfy the linear relations E = Σ + D , D + D = L. (3.17)17hat is D = E − H − L, D = − E + H + L. (3.18)We compute the triple intersection numbers by using two pushforwards [30]: D a D b D c = π ∗ f ∗ ( D a D b D c ) . (3.19) Given a G -model with representation R , knowledge of the triple intersection numbers provides aderivation of the number of matter multiplets for a 5D supergravity theory with gauge group G andhypermultiplets in the representation R in terms of geometric data. Since we know from looking atthe weights of the vertical curves of the elliptic fibration that the only possible matter representa-tions are in the adjoint representation, we compute the Intriligator-Morrison-Seiberg (IMS) potentialfor a gauge theory with Lie algebra A and n adj hypermultiplets transforming in the adjoint rep-resentation. The one loop quantum correction to the prepotential is (with mass parameters set tozero): F IMS = ( ∑ α ∣( α, φ )∣ − n adj ∑ α ∣( α, φ )∣ ) = ( − n adj ) φ , (3.20)where φ > is in the the dual of the fundamental Weyl chamber and α are the weights of the adjointrepresentation of A . This is supposed to match the computation of the triple intersection number D = − K . By a direct comparison, we get: n adj = + K . (3.21)In particular, the number n adj is never zero as K is an integral number for any smooth surface B .Assuming the Calabi-Yau condition, the curve V ( a ) has Euler characteristic χ and genus g with χ = ( − K + K )( − K ) = − K , g = − χ = + K . (3.22)We can then express the number of adjoints in terms of the genus of the curve V ( a ) : n adj = g. (3.23) Remark 3.8.
We can derive the above result geometrically as well. In a Calabi-Yau, for a P fibration over a curve of genus g , D = K D ∩ [ D ] = ( − g ) . For the second equality, see [45][Chap.V, Corollary 2.11]. Assuming that this is equal to the coefficient of F , we have n adj = g . Definition 4.1.
The SO(5)-model is a Weierstrass model with defining equation Y ∶ y z = ( x + a x z + s xz ) . (4.1) We will take α in the basis of fundamental weights and φ in the basis of simple coroots. ssumptions : The divisors V ( a ) and V ( s ) of B are generic. In particular, they are smooth, notproportional to each other nor is a a perfect square modulo s . Remark 4.2.
The SO(5)-model is is obtained from the SO(3)-model by way of the substitution a Ð→ s where s is a section of L ⊗ . This specialization does not alter the Mordell–Weil group,which is still Z / Z and generated by the point x = y = of the generic fiber. Lemma 4.3.
The discriminant and the cuspidal locus of the elliptic fibration (4.1) are respectively ∆ = s ( s + a )( s − a ) , V ( c , c ) = V ( s, a ) . (4.2)We denote the divisor V ( s ) of B as S . The three components of the reduced discriminantintersect pairwise transversely along the cuspidal locus V ( s, a ) . Tate’s algorithm indicates thatover the generic point of S = V ( s ) the generic geometric fiber is type I , while the geometric fiberover the generic points of the two other components of the discriminant is a Kodaira fiber of typeI (i.e. a nodal curve). Since by assumption a is not a perfect square modulo s , the generic fiberover S is actually not geometrically irreducible; rather, the fiber type is I ns . Such a fiber has as itsdual graph the extended Dynkin diagram ̃ C t . The singular locus of the elliptic fibration defined in (4.1) is supported onSing ( Y ) = V ( x, y, s ) . (4.3) Theorem 4.4.
A crepant resolution f ∶ Y Ð→ Y of the elliptic fibration (4.1) is given by thefollowing sequence of blowups along smooth centers: X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] X X ( x, y, s ∣ e ) ( x, y, e ∣ e ) (4.4) The relative projective coordinates of X , X , and X are parametrized as follows: [ e e x ∶ e e y ∶ z ] , [ e x ∶ e y ∶ s ] , [ x ∶ y ∶ e ] . (4.5) The proper transform of the elliptic fibration (4.1) is Y ∶ y z − e e x − a x z − e s xz = . (4.6) Proof.
We first blow up X = P [ O B ⊕ L ⊗ ⊕ L ⊗ ] along the ideal ( x, y, s ) and we denote the blowupspace X . The exceptional divisor E = V ( e ) is a P projective bundle over V ( x, y, s ) . We abusenotation and implement the blowup by way of the substitution ( x, y, s ) ↦ ( e x , e y, e s ) . Whenthese substitutions are performed on the defining equation of Y , we can factor out two powersof e which ensures that the blowup is crepant for the elliptic fibration. For the second blowup X = Bl ( x,y,e ) X . The exceptional divisor E = V ( e ) is a P projective bundle over V ( x , y , e ) .Once again we can factor out two powers of e to obtain a crepant blowup. By studying the Jacobiancriterion, we see that there are no singularities left if V ( a ) is a smooth divisor.19 im B Euler characteristic Calabi-Yau case L c ( c − L ) L − c ( L − Lc + c ) L c ( c + c ) ( − L + L c − Lc + c ) L c ( c − c − c c ) Table 4: Euler characteristic of SO(5)-model for bases of dimension up to . The i th Chern class ofthe base is denoted c i . The Calabi-Yau cases are obtained by imposing L = c . In this section, we compute the pushforward of the generating function for the Euler characteristicand triple intersection form of the SO(5)-model to B . The SO( )-model refers to the elliptic fibration Y → B defined by the crepant resolution given in Theorem 4.4. Theorem 4.5.
Assume that the base B is a smooth surface. The triple intersection polynomial forthe fibral divisors D and D for a SO( )-model is F = ( α D + α D + α D ) = L ( − α + α α − α + α α − α ) . (4.7a) Proof.
Pushforward D a D b D c [ Y ] ∩ [ X ] with D = L − E , D = E − E , D = E , [ Y ] = H + L − E − E . (4.8) Theorem 4.6 (See [30]) . The generating function for the Euler characteristic of a SO( )-model is ϕ ∗ c ( Y ) = Lt ( + Lt )( + Lt ) c t ( T B ) , (4.9) where c t ( T B ) = ∑ c i ( T B ) t i is the Chern polynomial of the base B . Corrollary 4.7.
The elliptic fibration defined by the crepant resolution described in Theorem 4.4 isa flat elliptic fibration ϕ ∶ Y Ð→ B . The fibral divisors are: D ∶ s = y z − x ( e e x + a z ) = , [ e e x ∶ e e y ∶ z ] , [ e x ∶ e y ∶ ] , [ x ∶ y ∶ e ] (4.10) D ∶ e = y − a x = , [ ∶ ∶ z ] , [ e x ∶ e y ∶ s ] , [ x ∶ y ∶ ] (4.11) D ∶ e = y − x ( a x + e s z ) = , [ ∶ ∶ z ] , [ ∶ ∶ s ] , [ x ∶ y ∶ e ] (4.12) Remark 4.8.
The total transform of the divisor S = V ( s ) in Y is V ( te e ) composed of threeirreducible and reduced components. We denote the irreducible divisors D = V ( s ) , D = V ( e ) ,and D = V ( e ) in Y . We denote their generic fibers over S as (resp.) C , C , and C . The fibers C and C are geometrically irreducible while C is not. After a quadratic field extension C splitsinto rational curves, namely C + and C − with fibers C + and C − . The curves C , C + , C , C − definea Kodaira fiber of type I with dual graph the (untwisted) affine Dynkin diagram ̃ A . The curves C , C , and C have as their dual graphs the twisted affine Dynkin diagram ̃ C t (in the notation ofCarter), also denoted ̃ D ( ) in the notation of Kac—see Figure 3.20ver the degeneration locus V ( a ) , we find Y ∶ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ C Ð→ C C Ð→ C ′ C Ð→ C (4.13)where Y ∶ ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ C ∶ s = y z − e e x = , [ e e x ∶ e e y ∶ z ] , [ e x ∶ e y ∶ ] , [ x ∶ y ∶ e ] C ′ ∶ e = y = , [ ∶ ∶ z ] , [ e x ∶ ∶ s ] , [ x ∶ ∶ ] C ∶ e = y − e s xz = , [ ∶ ∶ z ] , [ ∶ ∶ s ] , [ x ∶ y ∶ e ] . (4.14)Note that the conic C remains non-degenerate in over the locus V ( a ) . Theorem 4.9.
The fiber over the generic point of the divisor S = V ( s ) is of type I ns and degeneratesalong V ( s, a ) to a non-Kodaira fiber of type − − . The divisors D and D are isomorphic tothe projective bundle P S [ O S ⊕ L ] while D is a double cover of S × P and a branch locus overthe cuspidal locus V ( s, a ) . Defining S ′ as the double cover of S branched over V ( s, a ) , D isisomorphic to P S ′ [ O S ⊕ L ] : D ≅ P S [ O S ⊕ L ] , D ≅ P S ′ [ O S ′ ⊕ L ] , D ≅ P S [ O S ⊕ L ] . (4.15) Proof.
Since the center of the second blowup is away from D , it is enough to stop at the first blowupto understand the geometry of C and D . By solving for e , we see that C is the normalizationof a nodal curve. In the patch x ≠ , an affine parameter of the rational curve C is y / x . It followsthat the divisor D with generic fiber C is isomorphic to P S [ O S ⊕ L ] .The curve C is a conic defined in P with coordinates [ x ∶ y ∶ e ] . The discriminant of the conicis s / which is a unit. Hence we again get a P -bundle over S . Working in the patch x ≠ , aftersolving for e , we can parametrize the conic by y / x . It follows that the divisor D with fiber C isisomorphic to P S [ O S ⊕ L ] .To understand the geometry of D , we consider the proper morphism π ∶ D → S . Since thefibers of π are not connected, we consider the Stein factorization π = f ○ ρ with f ∶ D → S ′ amorphism with connected fibers and ρ ∶ S ′ → S the double cover of S branched over V ( s, a ) . TheEuler characteristic of S ′ is χ ( S ′ ) = χ ( S ) − [ S ][ a ] = ( − g ) − L = − L − L = − L and wealso retrieve D = ( − g ′ ) = χ ( S ′ ) = − L .we go to a field extension where we can take the square root of a . It is then clear that C is adouble cover (branched along V ( a ) ) of a P bundle over S with projective fiber paramatrized by [ e x ∶ s ] . In particular, since e x and s are both sections of L ⊗ over S , the P bundle is trivial.Thus D is the double cover of S × P branched along the divisor V ( a ) of S , that is along thecuspidal locus of the elliptic fibration.The geometric fiber of D is composed of two non-intersecting projective bundle that coincideinto a double line over the cuspidal locus V ( s, a ) in B .The geometric fiber over the generic point of S is a Kodaira fiber of type I . The descriptionof the geometric fiber I requires at least a quadratic field extension for the square root of a to bewell defined. The fiber over the generic point of S has an affine Dynkin diagram of type ̃ C , whichcorresponds to a type I ns fiber. The fiber degenerates over the cuspidal locus V ( a ) ∩ S where thecurve C degenerates into two coinciding lines giving a fiber of type − − which we can think ofas an incomplete I ∗ . 21 heorem 4.10. The intersection numbers between the divisors D a and their generic fibers C a givethe following intersection matrix ( D a C b ) = C C C D D D ⎡⎢⎢⎢⎢⎢⎣ − − − ⎤⎥⎥⎥⎥⎥⎦ (4.16) More generally, we have ϕ ∗ ( D a D b ϕ ∗ M ∩ [ Y ]) = D D D D D D ⎡⎢⎢⎢⎢⎢⎣ − − − ⎤⎥⎥⎥⎥⎥⎦ ( SM ) ∩ [ B ] , where M ∈ A ∗ ( B ) . (4.17) Proof.
The first equation is obtained from the second one by choosing M to be the generic point ofthe divisor S of B . The second equation is a direct pushforward computation with D = L − E , D = E − E , D = E , [ Y ] = ( H + L − E − E ) We do the pushforward in three steps since ϕ = π ○ f ○ f . We first compute the pushforward f ∗ to the Chow ring of X , then we compute the pushforward f ∗ to the Chow ring of X , and finallywe compute the pushforward π ∗ to the Chow ring of B .When the divisors D i are not all geometrically irreducible, we notice that the quadratic intersec-tion numbers correspond to a twisted affine Dynkin diagram. In the present case, we get the twisteddiagram ̃ C t in the notation of Carter or ̃ D ( ) in the notation of Kac—see Figure 3. It is possible toread off the Cartan matrix for the ordinary Dynkin diagram from the incidence matrix by rescalingthe a th row by D a ⋅ C a , and then deleting the affine node corresponding to C and examining thecomponents of the remaining × block. We obtain ( D a ⋅ C b ) = C C D D [ − − ] . (4.18)The above matrix is minus the C Cartan matrix.
We now consider the matrix of weight vectors we obtained from the curves over the cuspidal locus V ( a , s ) . We delete the first row and column (corresponding to the affine node): ( w a ( C b )) = [ ̟ ̟ ] = C ′ C D D [ − − ] . (4.19)While the second column vector ̟ is a root of C , the first column ̟ = − , is a weight ofthe representation of C . The saturation of the singleton { ̟ } is precisely the set of weights ofthe representation of C , with highest weight , . The of C is defined as the non-trivialirreducible component ⋀ in the decomposition of the antisymmetric representation ⋀ V C = ⋀ ⊕ C .Equivalently, the representation is the vector representation of so ( ) . This representation is quasi-minuscule and self-dual. The adjoint representation is with highest weight , .22 ( s + a ) V ( s − a ) B V ( s ) C C ′ C C C + C − C Weierstrass model y z = x + a x z + s xz Discriminant ∆ = s ( s − a ) Matter representations adjoint + vector w/ geometric weight ( − , ) Representation multiplicities n = + L ; n = L Euler characteristic L ( + L )( + L ) c ( T B ) Triple intersection form F = − L ( α + α + α ) + L α ( α + α ) Figure 3: Summary of the geometry of the resolution (4.6) of the SO( )-model. Note that all thecrepant resolutions of Weierstrass model are isomorphic to each other. Thus, its extended Kählercone consists of a single chamber. The variable φ ∈ h (where h ⊂ g is the Cartan subalgebra) is expressed in the basis of simplecoroots. The weights ̟ and the roots α are expressed in the canonically dual basis (i.e. the basisof fundamental weights).The relevant part of the prepotential for a 5D gauge theory with gauge algebra C and hyper-multiplets in the representations and is (with mass parameters set equal to zero) [53]: F IMS = ( ∑ α ∣( φ, α )∣ − ∑ R n R ∑ ̟ ∣( φ, ̟ )∣ ) . (4.20)The open fundamental Weyl chamber is the subset of h with positive intersection with all thesimple roots. The simple roots of so are: , − − , . (4.21)It follows that the (dual) open fundamental Weyl chamber is the cone of h defined by: φ > φ > φ > . (4.22)23
10 4 , , − , − , , − , , − , , − , , , − − , , − − , , − , , − − , Table 5: Weights of the representations , , and of C expanded in the basis of fundamentalweights. Remark 4.11.
A weight defines a hyperplane through the origin that intersect the open fundamentalWeyl chamber if and only if both ̟ and − ̟ are not dominant weights.For the SO(5)-model, the representations that we consider are the adjoint and the vector rep-resentation. The nonzero weights of the representation are all dominant up to an overall sign.Hence, I ( C , ⊕ ) has only a unique chamber [27]. Proposition 4.12.
The prepotential for the Lie algebra so with n adj matter multiplets transformingin the adjoint representation and n matter multiplets transforming in the vector representationconsists of a single phase. Explicitly, the prepotential is: F IMS = ( − n − n + ) φ + ( n − n + ) φ φ + ( − n + n − ) φ φ + ( − n ) φ . Proposition 4.13.
Under the identification φ i = α i , the prepotential F ± IMS matches F ± ∣ α = if andonly if n = n − = L , n = + L . (4.23) If we impose the Calabi-Yau condition and assume the base is a surface, + K is the genus of thecurve V ( s ) of class L in the base and we have: n = g − = K , n = g = + K . (4.24) In particular, the genus cannot be zero.Proof.
We can compare the potential F IMS with the triple intersection form F after setting α = .To have a match of the types of monomials present in the potential, we have to eliminate thecoefficient of the term α α . This condition gives n = ( n − ) . We then get a perfect match F IMS = F by imposing n = + L . 24 The SO(6)-model
Definition 5.1.
The SO(6)-model is specified by the Weierstrass equation Y ∶ y z + a xyz = x + msx z + s xz , (5.1)where m is a constant number different from − , , and . The coefficient a is a generic section of L , and s is a smooth section of L ⊗ . Remark 5.2.
The SO(6)-model is obtained from the SO(5)-model by making the substitution a = a + ms . This allows for the generic fiber to be of type I s and the discriminant to not haveother reducible fiber types in codimension one. The Mordell–Weil group is still Z / Z .The discriminant is ∆ = − s ( a − s + ms )( a + s + ms ) . (5.2) ∆ vanishes at order 4 at s = , while c and c are non-zero there. It follows from Tate’s algorithmthat the geometric generic fiber over V ( s ) is of type I . Since b restricted to V ( s ) is a perfectsquare, it follows that the generic fiber over V ( s ) is type I s —this is the main difference betweenan SO(6)-model and an SO(5)-model. As is clear from the discriminant, we should impose m ≠ ± to avoid introducing a type I fiber over V ( a ) . More specifically, we impose ≠ m ≠ ± to avoidintroducing new sections that will modify the Mordell–Weil group. There are two isomorphic resolutions connected to each other by an Atiyah flop, induced by theinverse map of the Mordell–Weil group: Y Y Y Y − Y + ( x, y, s ∣ e ) ( y , e ∣ e ) ( y + a x , e ∣ e ) flop (5.3)We first consider the resolution Y − . The proper transform of Y is Y − ∶ y ( e y + a x ) z = e x ( x + msxz + s z ) , [ e e x ∶ e e y ∶ z ] , [ x ∶ e y ∶ s ] , [ y ∶ e ] . (5.4)The fibral divisors are: D ∶ s = y ( e y + a x ) z − e x = , [ e e x ∶ e e y ∶ z ] , [ x ∶ e y ∶ ] , [ y ∶ e ] (5.5) D ∶ e = e y + a x = , [ ∶ ∶ z ] , [ x ∶ e y ∶ s ] , [ y ∶ ] (5.6) D ∶ e = x = , [ ∶ ∶ z ] , [ ∶ ∶ s ] , [ y ∶ e ] (5.7) D ∶ e = a yz − e ( x + msxz + s z ) = , [ ∶ ∶ z ] , [ x ∶ ∶ s ] , [ y ∶ e ] . (5.8)25ver the degeneration locus V ( a ) , we find Y − ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ C Ð→ C C Ð→ C C Ð→ C C Ð→ C + C + + C − Y + ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ C Ð→ C C Ð→ C + C + + C − C Ð→ C C Ð→ C (5.9)where Y − ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ C ∶ s = e y z − e x = , [ e e x ∶ e e y ∶ z ] , [ x ∶ e y ∶ ] , [ y ∶ e ] C ∶ e = e = , [ ∶ ∶ z ] , [ x ∶ ∶ s ] , [ y ∶ ] C ∶ e = x = , [ ∶ ∶ z ] , [ ∶ ∶ s ] , [ y ∶ e ] C ± ∶ e = x + ( − m ± √ m − ) sz = , [ ∶ ∶ z ] , [ x ∶ ∶ s ] , [ y ∶ e ] (5.10)and we note that Y + differs from the case of Y − described above only by the substitution C ± ↔ C ± . Y ± is nonsingular provided m ≠ . The geometry of the singular fibers is displayed in Figure 4. Theorem 5.3.
The fiber over the generic point of S , which is Kodaira type I s , degenerates alongthe cuspidal locus V ( s, a ) of the elliptic fibration to a type I ∗ fiber. We have D ≅ P S [ O S ⊕ L ] D ≅ P S [ O S ⊕ L ] D ≅ P S [ O S ⊕ O S ] D ≅ Bl I ( P S [ O S ⊕ O S ]) , I = ( a , x + msxz + s z ) , (5.11) where for D , the fibers of the P -bundle are paramatrized by [ x ∶ s ] and the ideal I represents K distinct points, namely, two distinct points on [ S ] ⋅ [ a ] = K distinct fibers of P S [ O S ⊕ O S ] .Proof. First, following the same argument as for the SO( )-model, we have D ≅ P S [ O S ⊕ L ] . Next,the defining equation for C is e y = − a x , which gives the parametrization [ ∶ ∶ z ][ x ∶ − a x ∶ s ][ y ∶ ] . Since the divisor is defined in the patch zy ≠ , we see that C is parametrized by [ x / z ∶ s ] .We recall that both x / z and s are of class L − E . Thus, D ≅ P S [ O S ⊕ O S ] . Similarly, projectivescaling implies C is parametrized by homogeneous coordinates [ y / s ∶ e ] , whence D ≅ P S [ O S ⊕ L ] .Finally observe that C , can be parametrized by a y ′ − e ( x ′ + msx ′ + s ) = , [ x ′ ∶ s ] , [ y ′ ∶ e ] , ( x ′ , y ′ ) = ( x / z, y / z ) . (5.12)This can be viewed as the blowup of the trivial projective bundle S × P [ x ′ ∶ s ] along the ideal I = ( a , x ′ + msx ′ + s ) . Rescaling x ′ , y ′ by the unit z , we recover precisely the hypersurface equation ( . ) . Since x ′ and s ′ are of the same class, the equation x ′ + msx ′ + s ) gives two distinct pointsin the P fiber parametrized by [ x ′ ∶ s ′ ] over S ∩ V ( a ) . That means that D is obtained from aprojective bundle P S [ O S ⊕ L ] by blowing up two points (corresponding to x ′ + msx ′ + s = oneach of the fiber at the intersection of S and V ( a ) , that is a total of [ S ][ a ] = L points.It follows from this theorem that D = D = D = ( − g ) = − L , and D = ( − g ) − K = − L . Given a variety X and an ideal I , we denote by Bl I X the blowup of X centered at I . ( p + ) V ( p − ) B V ( s ) C C + C C − C C C C C Weierstrass model y z + a xyz = x + msx z + s xz Discriminant ∆ = s p + p − = s (( a + ms ) − s ) Matter representation adjoint + vector w/ geometric weight ( , , − ) Representation multiplicities n = + L ; n = L Euler characteristic L + L c ( T B ) Triple intersections F − = L ( − α + α α + α α α + α α − α + α α − α α + α α α − α + α α − α ) , F + − F − = L ( α − α ) Figure 4: Summary of SO(6)-model geometry.
Theorem 5.4.
The intersection numbers between the divisors D a and their generic fibers C a givethe following intersection matrix: ( D a C b ) = C C C C D D D D ⎡⎢⎢⎢⎢⎢⎢⎢⎣ − − − − ⎤⎥⎥⎥⎥⎥⎥⎥⎦ (5.13)27 ore generally, φ ∗ ( D a D b ⋅ φ ∗ M ∩ [ Y − ]) = C C C C D D D D ⎡⎢⎢⎢⎢⎢⎢⎢⎣ − − − − ⎤⎥⎥⎥⎥⎥⎥⎥⎦ ( S ⋅ M ) ∩ [ B ] (5.14) where M ∈ A ∗ ( B ) and φ = Bl ( x,e ) ○ φ . (5.15) Proof.
The divisor class group of X is not generated by the classes H, L, E , E . To circumventthis complication, we therefore blow up along I ( D ) = ( x, e ) and factor out a single copy of theexceptional divisor E ; we denote the proper transform of this blowup by Y ′− . The second equationabove is then a direct pushforward computation with [30] D = L − E D = E − E D = E D = E − E [ Y ′− ] = ( H + L − E − E − E ) . (5.16)Note that the equation (5.13) is obtained from (5.14) by choosing M to be the generic point of thedivisor S of B .The lower right × block of the intersection matrix (5.13) determines (minus) the Cartan matrixof A : D a C b = C C C ⎡⎢⎢⎢⎢⎢⎣ − − − ⎤⎥⎥⎥⎥⎥⎦ D D D . (5.17) We now the matrix of weight vectors one obtains from the curves over the codimension two degen-eration locus V ( t, a ) : ( w a ( C b )) Y − = C C C C + C − ⎡⎢⎢⎢⎢⎢⎢⎢⎣ − − − − − ⎤⎥⎥⎥⎥⎥⎥⎥⎦ D D D D (5.18) ( w a ( C b )) Y + = C C + C − C C ⎡⎢⎢⎢⎢⎢⎢⎢⎣ − − − − − ⎤⎥⎥⎥⎥⎥⎥⎥⎦ D D D D (5.19)28 im B Euler characteristic Calabi-Yau case L c ( c − L ) L − c ( L − Lc + c ) L c ( c + c ) ( − L + L c − Lc + c ) L c ( − c − c c + c ) Table 6: Euler characteristic of SO(6)-model for bases of dimension up to . The i th Chern class ofthe base is denoted c i . The Calabi-Yau cases are obtained by imposing L = c .We use the above intersection matrices to determine the matter representation. (In fact, only oneof the two is required for this purpose, so we will focus on Y − .) Deleting the affine node, we obtainthe following matrix of weight vectors: ( w a ( C b )) = [ ̟ ̟ ̟ ̟ ] = C C C + C − ⎡⎢⎢⎢⎢⎢⎣ − − − − ⎤⎥⎥⎥⎥⎥⎦ D D D . (5.20)The new weight is third (and identically, the fourth) column of the above matrix, namely ̟ = ( , , − ) .The saturation of { ̟ } is the of SO(6), hence the matter supported on V ( a ) is in the vectorrepresentation V D = ⋀ A . We now compute the generating function for the Euler characteristic, which is the same for bothresolutions.
Theorem 5.5.
The generating function for the Euler characteristic of a SO(6)-model is, φ ∗ c ( Y ± ) = Lt + Lt c t ( T B ) . (5.21) where c t ( T B ) is the Chern polynomial of B and the coefficient of t n gives the Euler characteristicof a SO(6)-model over a base of dimension n . Theorem 5.6.
The triple intersection polynomial for the crepant resolution Y − is: F − = ( ∑ α i D i ) = L ( − α + α α + α α + α α α ) + L ( − α − α − α + ( α + α + α α ) α − α α ) . (5.22) The triple intersection polynomial for Y + is F + = ( ∑ α i D i ) = L ( − α + α α + α α + α α α ) + L ( − α − α − α + ( α + α + α α ) α − α α ) . (5.23)Observe that F ± are related to each other by the involution α ↔ α , and the difference betweenthe two polynomials is: F − − F + = L ( α − α ) . (5.24)29 .5 Counting 5D matter multiplets The element φ of the Cartan subalgebra h is expressed in a basis of simple coroots. The weights ̟ and the roots α are expressed in the basis canonically dual to the basis of simple coroots, namelythe basis of fundamental weights. , , , − , , , − − , , − , , − , − , , , − , , , , − − , , − , − , , − , , , , , , , − , , , , − , − , − , − , , − , − − , , − Table 7: Weights of the representations , of A expanded in the basis of fundamental weights.The relevant part of the prepotential is F IMS = ( ∑ α ∣( φ, α )∣ − ∑ R n R ∑ ̟ ∣( φ, ̟ )∣ ) . (5.25)The open fundamental Weyl chamber is the subset of elements of the Cartan subalgebra h withpositive intersection with all the simple roots. The simple roots of A are: , − , − , , − , − , . (5.26)It follows that the open fundamental Weyl chamber is the cone of h defined by: φ − φ > − φ + φ − φ > − φ + φ > . (5.27) Remark 5.7.
A weight defines a hyperplane through the origin that intersect the open fundamentalWeyl chamber if and only if both w and − w are not dominant weights. This basis is also referred to as the
Dynkin basis in some references. , , − with hyperplane φ − φ = . It follows thatthe hyperplane arrangement I ( A , ) has two chamberslabeled by the sign of the linear form − φ + φ . (5.28) Proposition 5.8.
The prepotential for so with n matter multiplets transforming in the adjointrepresentation and n matter multiplets transforming in the vector representation can be seen todepend on two phases, corresponding to the sign of the linear form − φ + φ . The correspondingprepotentials are: F + IMS = ( − n − n ) φ + ( − n ) φ + ( − n ) φ + n ( φ + φ ) φ − n φ φ + ( − + n − n )( φ + φ ) φ (5.29a) F − IMS = ( − n − n ) φ + ( − n ) φ + ( − n ) φ + n ( φ + φ ) φ − n φ φ + ( − + n − n )( φ + φ ) φ . (5.29b) where the ± superscript appearing in the symbol F ± IMS is correlated with the sign ( − φ + φ ) = ± . Theprepotentials F ± IMS are related to each other by the transposition φ ↔ φ . Remark 5.9.
The difference between the two F ± IMS is proportional to the cube of the linear form: F − IMS − F + IMS = − n ( − φ + φ ) (5.30) Remark 5.10.
The prepotential F ± IMS have monomials of the type φ φ and φ φ that are notpresent in F + nor F − . Thus, matching the prepotential and the triple intersection numbers willforce their coefficients to be zero, namely n = n − . Proposition 5.11.
Under the identification φ i = α i , the prepotential F ± IMS matches the triple inter-section form F ± ∣ α = if and only if n = n − = L , n = + L . (5.31) If we impose the Calabi-Yau condition and assume the base is a surface, + K is the genus of thecurve V ( s ) of class L in the base and we have: n = g − = K , n = g = + K . (5.32) In particular, the genus cannot be zero.Proof.
We can compare the prepotential F ± IMS with the triple intersection form F ± after setting α = . To have a match of the types of monomials present in the potential, we have to eliminatethe coefficients of the terms φ φ and φ φ in F ± IMS . This condition imposes n = n − . We thenget a perfect match F ± IMS = F ± ∣ α = by imposing n = + L .31 Hodge numbers
In this section, we compute the Hodge numbers of the SO(3), SO(5), and SO(6)-models assumingthat the base B is a smooth rational surface and the resolved elliptic fibration is a Calabi-Yauthreefold. As usual for these models, the Mordell–Weil group has trivial rank and torsion Z / Z . Theorem 6.1.
Let B be a smooth compact rational surface with canonical class K . Let Y Ð→ B bethe crepant resolution of an SO(3), SO(5), or SO(6)-model over B . If Y is a Calabi-Yau threefoldthen the non-zero Hodge numbers of Y are h , = h , = h , = h , = , and h , , h , = h , = dim H ( Ω Y ) given by Table 8. The Euler characteristic of a Calabi-Yau threefold is χ ( Y ) = ( h , − h , ) . (6.1)Model χ ( Y ) g h , ( Y ) h , ( Y ) SO(3) − K + K − K + K SO(5) − K + K − K + K SO(6) − K + K − K + K n adj n V g g ( g − ) g ( g − ) Table 8: Euler characteristic and Hodge numbers for SO(3), SO(5), and SO(6)-models in the caseof a Calabi-Yau threefold over a compact rational surface B of canonical class K . The divisor S iscurve of genus g . The number of multiplets transforming in the adjoint and vector representationsare respectively n adj and n V . We explore a particular application of our geometric results to compactifications of F/M-theory onelliptically-fibered Calabi-Yau threefolds. In particular, we determine the 6D gauge theoretic de-scriptions associated to the SO( n )-models described in this paper in the special case that they areelliptically fibered Calabi-Yau threefolds. The low energy effective description of F-theory compact-ified on an elliptically fibered Calabi-Yau threefold is 6D ( , ) supergravity. However, supergravitytheories in 6D have gravitational, gauge, and mixed anomalies at one loop due to the presence ofchiral matter, and therefore anomaly cancellation places strong constraints on the matter spectrum.Determining the possible 6D matter content consistent with anomaly cancellation is therefore a taskof primary importance for stringy compactifications of this sort.We begin this section by reviewing relevant aspects of supergravity theories with 8 superchargesin 5D and 6D. We then determine the number of hypermultiplets charged in a given representationby solving the anomaly cancellation conditions and checking that they match the results of the 5Dcomputations described earlier. The match between the 6D and 5D matter spectra is essentially dueto the fact that F-theory compactified on an elliptically fibered Calabi-Yau threefold times a circleis dual to M-theory compactified on the same threefold, which implies that the gauge theory sectorof the 6D ( , ) theory compactified on a circle admits a description as a 5D N = theory. N = ( , ) supergravity We collect some useful facts about 6D theories [67, 69]. Six-dimensional (gauged) supergravity with8 supercharges has SU(2) R-symmetry. The fermions of the theory can be formulated as sym-plectic Majorana–Weyl spinors, which transform in the fundamental representation of the SU(2)32-symmetry group. There are four types of massless on-shell supermultiplets: a graviton multi-plet, n T tensor multiplets, n ( ) V vector multiplets characterized by a choice of gauge group, and n H hypermultiplets transforming in a representation of the gauge group.We will call an antisymmetric p -form with self-dual (resp. anti-self-dual) field strength a self-dual (resp. anti-self-dual) p -tensor field. In addition, we will simply refer -tensor fields as ‘tensors’.The graviton multiplet contains an anti-self-dual tensor, while each tensor multiplet includes a self-dual tensor. The self-duality properties of these tensor fields cannot be derived consistently froma known action principle, and consequently 6D N = ( , ) supergravity does not at present have aconventional Lagrangian formulation for n T > .The tensor multiplet scalars parametrize the homogeneous symmetric space SO ( , n T )/ SO ( n T ) [68]. The quaternionic scalars parametrize locally a non-compact quaternionic-Kähler manifold. Thechiral tensor multiplets can induce local anomalies that have to be cancelled.Multiplet FieldsGraviton ( g µν , B − µν , ψ − µ ) Vector ( A µ , λ − ) Tensor ( B + µν , φ, χ + ) Hyper ( q, ζ + ) Table 9: Supermultiplets of
N = ( , ) six-dimensional supergravity. The indices µ and ν referto the six-dimensional spacetime coordinates. The tensor g µν is the metric of the six0dimensionalspacetime. The fields ψ − µ , λ − , χ + , ζ + are symplectic Majorana–Weyl spinors. The field ψ − µ is thegravitino. The chirality of fermions is indicated by the ± superscript. The tensor B + µν is a two-form with self-dual field strength, while B − µν is a two-form with anti-self-dual field-strength. Thescalar field φ is a pseudo-real field. The hypermultiplet scalar q is a quaternion composed of fourpseudo-real fields. N = supergravity Five-dimensional (Yang-Mills-Einstein) supergravity with 8 supercharges has SU(2) R-symmetry.All spinors are symplectic Majorana spinors and transform in the fundamental representation of theSU(2) R-symmetry group. There are three types of massless on-shell supermultiplets: a gravitonmultiplet, n ( ) V vector multiplets, and n ( ) H hypermultiplets. The graviton multiplet contains a vectorfield called the graviphoton.The scalar fields of the hypermultiplets are called hyperscalars, and transform in the fundamentalrepresentation of SU(2). Each hyperscalar is a complex doublet, giving altogether four real fields.The hyperscalars collectively parametrize a quaternionic-Kähler manifold of real dimension n ( ) H .The vector multiplet scalars φ parametrize a real n ( ) V dimensional manifold called a very specialreal manifold which can described in terms of very special coordinates as a hypersurface F = of anaffine real space of dimension n ( ) V + .The dynamics of the gravity and vector fields at the two-derivative level are completely deter-mined by a real cubic polynomial F whose coefficients are the Chern-Simons couplings appearing inthe 5D action. In particular, the cubic potential F determines both the matrix of gauge couplingsand the metric on the real manifold parametrized by the vector multiplet scalars.33ultiplet FieldsGraviton ( g µν , A µ , ψ µ ) Vector ( A µ , φ, λ ) Hyper ( q, ζ ) Table 10: Supermultiplets for
N = five-dimensional supergravity. The indices µ and ν refer tothe five-dimensional spacetime coordinates. The tensor g µν is the metric of the five-dimensionalspacetime. The fields ψ µ , λ, ζ are symplectic Majorana spinors. The field ψ µ is the gravitino and A µ is the graviphoton. The hyperscalar is a quaternion. Kaluza-Klein reduction from 6D to 5D
If we compactify F-theory on Y to 6D and then further compactify on a circle S , we anticipate theeffective description will include the same field content as the compactification of M-theory on Y to5D. This is summarized in Table 12.In this section we summarize the Kaluza Klein (KK) reduction of 6D N = ( , ) on a circle and therelation of this theory to 5D N = supergravity. To facilitate a comparison between the dual F/Mtheory compactifications on a smooth threefold, we study these 5D theories on the Coulomb branch,where the non-Cartan vector fields and charged hypermultiplets acquire masses due to spontaneousgauge symmetry breaking and are subsequently integrated out.It is possible to cast the massless fields of the 6D KK reduction in the canonical 5D framework.First, the neutral 6D hypermultiplets descend to 5D hypermultiplets, n ( ) H = n H , (7.1)where we use n H to denote neutral hypermultiplets in 6D.We next turn our attention to the vector fields. Dimensional reduction of the 6D tensor fieldsproduces both 5D tensor and vector fields. However, the 6D self-duality condition also descends to a5D constraint that can be imposed at the level of the action to dualize all of the 5D massless tensorsto vectors. Thus we are free to assume that on top of the graviton multiplet, the field contentof our 5D KK theory is comprised solely of vector multiplets and hypermultiplets. The 5D KKreduced vector fields include the graviphoton, n T + tensors, and n ( ) V vectors, making for a total of n T + n ( ) V + vector fields. Accounting for the fact that one of the vector fields must belong to thegraviton multiplet, the total number of 5D vector multiplets is given by [14]: n ( ) V = n T + n ( ) V + . (7.2)The next step in our analysis is to understand the geometric origin of the field theoretic data viathe F/M theory compactifications. Compactification of 11D supergravity on a resolved Calabi-Yau threefold Y leads to 5D N = supergravity on the Coulomb branch, coupled to n ( ) V = h , ( Y ) − vector multiplets and n H = h , ( Y ) + hypermultiplets [14]. The Coulomb branch of the 5D gauge sector is parametrized by We emphasize here that when a comparison is made between the 6D theory compactified on a circle and the 5Dtheory on the Coulomb branch, the number n ( ) V is equal to the number of uncharged 6D vectors. N = ( , ) sugra on R , × S ↓ N = sugra on R , n ( ) V = n ( ) V + n T + n ( ) H = n H Table 11: Identification between multiplets from KK reduction of 6D
N = ( , ) and 5D N = supergravity multiplets on the Coulomb branch.F-theory on Y M-theory on Y F-theory on Y × S ↓ ↓ ↓ D=6
N = ( , ) sugra D=5 N = sugra D=5 N = sugra n ( ) V + n T = h , ( Y ) − n ( ) V = h , ( Y ) − n ( ) V = h , ( Y ) − n H = h , ( Y ) + n H = h , ( Y ) + n H = h , ( Y ) + n T = h , ( B ) − Table 12: Compactification of F-theory and M-theory on a smooth Calabi-Yau threefold Y . Weassume that all two-forms in the five-dimensional theory are dualized to vector fields. The numbersof neutral hypermultiplets are the same in 6D and 5D. By contrast, the number of 5D vectormultiplets is n ( ) V = n ( ) V + n T + , where we emphasize that the 5D theory is on the Coulomb branch.the n ( ) V vector multiplet scalars, and corresponds to the extended Kähler cone of Y restricted tothe unit volume locus. The Chern-Simons couplings determining the one-loop quantum correctedprepotential on the Coulomb branch are identified as the triple intersection numbers of the effectiveirreducible divisors of Y , appearing as coefficients of the triple intersection polynomial.Given a G -model with representation R , we consider the triple intersection polynomial F of the G -model and the prepotential F IMS of a 5D gauge theory with gauge group G and an undeterminednumber of hypermultiplets in the representation R . We can determine which values of the numbersof charged hypermultiplets are necessary to get a perfect match between F ∣ α = and F IMS . In this section, we prove that the SO(3), SO(5), and SO(6)-models define anomaly free 6D
N = ( , ) supergravity theories. The low energy effective description of F-theory compactified on an elliptically fibered Calabi-Yauthreefold Y with a base B is six-dimensional N = ( , ) supergravity coupled to n T = h , ( B ) − tensor multiplets, n H = h , ( Y ) + neutral hypermultiplets, and n ( ) V massless (Cartan) vectormultiplets such that n ( ) V + n T = h , ( Y ) − [63]. That is: h , ( Y ) = n ( ) V + h , ( B ) + . (7.3) There is an additional Kähler modulus controlling the overall volume of Y which belongs to the universal hyper-multiplet and is thus not counted among the 5D Coulomb branch parameters. In this case, we expand a Kähler classin a basis of h , ( Y ) − irreducible effective divisors identified with the coroots of the affine Dynkin diagram ̃ g t .
35e assume that Y is a simply-connected elliptically fibered Calabi-Yau threefold with holonomySU( ). The restriction on the holonomy is stronger than the condition that Y has a trivial canonicalclass, in particular, it implies that h , ( Y ) = h , ( Y ) = . Then we also have h , ( B ) = h , ( B ) = and the Enriques–Kodaira classification identifies B to be a rational surface or an Enriques surface.However, the condition that Y is simply connected also requires that B is simply connected and thisrules out the Enriques surface [10, Chap VI]. Thus B is a rational surface. The only non-trivialHodge number of B is h , ( B ) , its Euler characteristic is c ( T B ) = + h , ( B ) and its signature τ ( T B ) = − h , ( B ) . Moreover, since h , ( B ) = , its holomorphic Euler characteristic χ ( O B ) = h , − h , is . Thus, Noether’s theorem ( χ ( O B ) = K + c ( T B ) ) implies that h , ( B ) = − K .The Hodge index theorem states that the intersection of two-forms in B has signature ( , h , ( B ) − ) .Consider a 6D N = ( , ) supergravity theory with n T tensor multiplets, n ( ) V vector multiplets,and n H hypermultiplets. We assume that the gauge group is simple. We distinguish n H neutralhypermultiplets and n R i hypermultiplets transforming in the representation R i of the gauge group.CPT invariance requires that R i is a quaternionic representation [72], but if R i is pseudo-real, butwhen R i is pseudo-real, we can have half-hypermultiplets transforming under R i , which can givehalf-integer values for n R i . Following [46], we count as neutral any hypermultiplet whose charge isgiven by the zero weight of a representation. We denote by dim R i, the number of zero weights inthe representation R i . The total number of charged hypermultiplets is then [46] n chH = ∑ i ( dim R i − dim R i, ) n R i , (7.4)and the total number of hypermultiplets is n H = n H + n chH . The pure gravitational anomaly is cancelledby the vanishing of the coefficient of tr R in the anomaly polynomial [71, Footnote 3]: n H − n ( ) V + n T − = . (7.5)Using the duality between F-theory on an elliptically fibered Calabi-Yau threefold with base B andtype IIB on B , Noether’s formula implies the following for the number of tensor multiplets [69]: n T = h , ( B ) − = − K . (7.6)If the gauge group is a simple group G , the remaining part of the anomaly polynomial is [72]: I = − n T ( tr R ) + X ( ) tr R − X ( ) , (7.7)where X ( n ) = tr adj F n − ∑ i n R i tr R i F n . (7.8)Choosing a reference representation F , we have after some trace identities: The trace identities fora representation R i of a simple group G are tr R i F = A R i tr F F , tr R i F = B R i tr F F + C R i ( tr F F ) (7.9) A rational surface is a surface birational to the complex projective plane P . Any smooth rational surface is P ,the Hirzebruch surface F n ( n ≠ ), or derived from them by a finite sequence of blowups [11, Theorem V.10]. AnEnriques surface is a surface with h , = , K B = but K B ≠ O B . An Enriques surface has h , = , h , = h , = . F for each simple component G of the gauge group. Thecoefficients A R i , B R i , and C R i depends on the gauge groups and are listed in [9, 24, 78]. We thenhave X ( ) = ( A adj − ∑ i n R i A R i ) tr F F (7.10) X ( ) = ( B adj − ∑ i n R i B R i ) tr F F + ( C adj − ∑ i n R i C R i )( tr F F ) . (7.11)If G does not have two independent quartic Casimir invariants, we take B R i = [69]. In a gaugetheory with at least two quartic Casimirs, to have a chance to cancel the anomaly, the coefficient of tr F F must vanish: B adj − ∑ i n R i B R i = . (7.12)We are then left with: I = − n T ( tr R ) + ( A adj − ∑ i n R i A R i ) ( tr F F ) ( tr R ) − ( C adj − ∑ i n R i C R i ) ( tr F F ) . (7.13)If the simple group G is supported on a divisor S and K is the canonical class of the base of theelliptic fibration, we can factor I as a perfect square following Sadov’s analysis [69, 70]: I = ( K tr R − λ S tr F F ) . (7.14)Sadov showed this factorization matches the general expression of I if and only the followinganomaly cancellation conditions hold [69] (see also [46, 67]): n H − n ( ) V + n T − = , (7.15a) n T = − K , (7.15b) ( B adj − ∑ i n R i B R i ) = , (7.15c) λ ( A adj − ∑ i n R i A R i ) = KS, (7.15d) λ ( C adj − ∑ i n R i C R i ) = − S . (7.15e)The coefficient λ is a normalization factor chosen such that the smallest topological charge of anembedded SU( ) instanton in G is one [ ? , 58, 67]. This forces λ to be the Dynkin index of thefundamental representation of G as summarized in Table 13 [67].Using adjunction ( KS + S = g − ), the last two anomaly equations give an expression for thegenus of S : λ ( A adj − ∑ i n R i A R i ) − λ ( C adj − ∑ i n R i C R i ) = ( g − ) . (7.16) We denote this representation by F as we have chosen the fundamental representation(s) for convenience. However,any representation can be used as a reference representation. A n ( n ≥ ) B n ( n ≥ ) C n ( n ≥ ) D n ( n ≥ ) E E E F G λ Table 13: The normalization factors for each simple gauge algebra. See [58]. ), SO( ), and SO( )-models We will now consider the specificity of the SO( ), SO( ), and SO( )-model. Since we use the vectorrepresentation of SO( ), SO( ), and SO( ) as the reference representation, λ takes the respectivelythe same value as for SU( ), B n and D n . g λ A adj B adj C adj A V B V C V S g ( S ) so (3) / / -4K + K so (5) − -2K + K so (6) − -2K + K Table 14: Coefficients for the trace identities in the case of SO( ), SO( ), and SO( ). In all casesthe reference representation F is the vector representation—namely, the of SO( ), the of SO( ),and the of SO( ).We will need the following trace identities [70, 72, 77]:SO ( ) ∶ tr adj F = tr vec F , tr adj F = ( tr vec F ) , (7.17)SO ( n ) n ≥ ∶ tr adj F = ( n − ) tr vec F , tr adj F = ( n − ) tr vec F + ( tr vec F ) . (7.18)For the SO ( ) -model the only representation that we consider is in the adjoint ( ) and we also usethis representation as our reference representation. For the SO ( ) and SO ( ) -model, we find matterin the adjoint and the vector representations. We use the vector representation as the fundamentalrepresentation. Thus, our choice of λ follows from the trace identities B = A ∶ tr F = tr F , (7.19) B = C ∶ tr F = tr F , (7.20) D = A ∶ tr F = tr F . (7.21)We first ignore the condition for the cancellation of the gravitational anomaly, namely equation(7.15a). After fixing the conventions for the trace identities and the coefficient λ , we are left withlinear equations that have a unique solution. For SO( ), all the remaining equation gives n adj = g .For SO( n ) with n = , , equation (7.15c) gives n V = ( − n )( n adj − ) . Feeding this in equation(7.16) gives n adj = g and all the remaining equations are satisfied:SO ( ) ∶ n adj = + K = g, (7.22)SO ( ) ∶ n adj = + K = g, n V = K = ( g − ) , (7.23)SO ( ) ∶ n adj = + K = g, n V = K = ( g − ) . (7.24)We are left with the pure gravitational anomaly, which requires checking equation (7.15a). Sincewe have explicit expressions for the number of charged hypermultiplets, this is a straightforward38omputation. We recall that: n T = − K , n H = n H + n chH , n H = h , ( Y ) + , n chH = ∑ i ( dim R i − dim R i, ) n R i (7.25)For all adjoint representations, the number of zero weights is the number of simple roots, which isthe rank of the Lie algebra. The vector representation of SO( ) does not have any zero weights, butthe vector representation of SO( ) has exactly one zero weight.For SO( ), we have n ( ) V = dim ( adj ) = , (7.26) n H = h , ( Y ) + = + K , (7.27) n chH = ( dim so ( ) − rk so ( )) n ( ) V = + K . (7.28)For SO( ), we have n ( ) V = dim ( adj ) = , (7.29) n H = h , ( Y ) + = + K , (7.30) n chH = ( dim so ( ) − rk so ( )) n adj + ( − ) n = + K . (7.31)In computing n chH , we use ( − ) n instead of n since the representation has one zero weight.For SO( ), we have n ( ) V = dim ( adj ) = , (7.32) n H = h , ( Y ) + = + K , (7.33) n chH = ( dim so ( ) − rk so ( )) n ( ) V + n = + K . (7.34)We then see immediately that in all three models, equation (7.15a) is also satisfied. Acknowledgements
The authors are grateful to Chris Beasley, James Halverson, Ravi Jagadeesan, Monica Kang, CodyLong, Sabrina Pasterski, and Julian Salazar for discussions. P.J. would like to thank the organizersof the 2018 Summer Workshop at the Simons Center for Geometry and Physics for their hospitalityand support during part of this work. M.E. is supported in part by the National Science Foundation(NSF) grant DMS-1406925 and DMS-1701635 “Elliptic Fibrations and String Theory”. P.J. is sup-ported by the Harvard University Graduate Prize Fellowship. P.J. would like to extend his gratitudeto Cumrun Vafa for his tutelage and continued support.
References [1] P. Aluffi. Chern classes of blowups.
Math. Proc. Cambridge Philos. Soc. , 148(2):227–242, 2010.[2] P. Aluffi and M. Esole, Chern class identities from tadpole matching in type IIB and F-theory.
JHEP , 03:032, 2009.[3] P. Aluffi and M. Esole, New Orientifold Weak Coupling Limits in F-theory.
JHEP , 02:020,2010. 394] L. B. Anderson, M. Esole, L. Fredrickson and L. P. Schaposnik, Singular Geometryand Higgs Bundles in String Theory, SIGMA (2018) 037. doi:10.3842/SIGMA.2018.037[arXiv:1710.08453 [math.DG]].[5] P. Arras, A. Grassi and T. Weigand, Terminal Singularities, Milnor Numbers, and Matter inF-theory, J. Geom. Phys. , 71 (2018) doi:10.1016/j.geomphys.2017.09.001[6] P. S. Aspinwall and M. Gross, The SO(32) heterotic string on a K3 surface, Phys. Lett. B ,735 (1996)[7] P. S. Aspinwall and D. R. Morrison, Nonsimply connected gauge groups and rational points onelliptic curves, JHEP , 012 (1998)[8] P. S. Aspinwall, S. H. Katz and D. R. Morrison, Lie groups, Calabi-Yau threefolds, and Ftheory, Adv. Theor. Math. Phys. , 95 (2000)[9] S. D. Avramis and A. Kehagias, A Systematic search for anomaly-free supergravities in sixdimensions, JHEP , 052 (2005).[10] W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, (2004) The Enriques Kodaira Classifi-cation. In: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete /A Series of Modern Surveys in Mathematics (3. Folge), vol 4. Springer, Berlin, Heidelberg[11] A. Beauville, Complex algebraic surfaces, Cambridge University Press, 1996.[12] M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, Geometricsingularities and enhanced gauge symmetries. Nucl. Phys. , B481:215–252, 1996.[13] A. P. Braun and T. Watari, On Singular Fibres in F-Theory, JHEP , 031 (2013)doi:10.1007/JHEP07(2013)031 [arXiv:1301.5814 [hep-th]].[14] A. C. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara, Eleven-dimensional supergravity com-pactified on Calabi-Yau threefolds, Phys. Lett. B , 76 (1995)[15] R. W. Carter.
Lie algebras of finite and affine type , Cambridge Studies in Advanced Mathe-matics 96. Cambridge University Press, Cambridge, 2005.[16] A. Cattaneo. Crepant resolutions of Weierstrass threefolds and non-Kodaira fibers. arXiv:1307.7997v2 [math.AG] .[17] A. Clingher, R. Donagi and M. Wijnholt, The Sen Limit, Adv. Theor. Math. Phys. , no. 3,613 (2014).[18] A. Collinucci, F. Denef, and M. Esole. D-brane Deconstructions in IIB Orientifolds. JHEP ,02:005, 2009.[19] P. Deligne, Courbes élliptiques: formulaire d’après J. Tate. (French)
Modular functions of onevariable , IV (
Proc. Internat. Summer Schoo l, Univ. Antwerp, Antwerp, 1972), pp. 53–73. Lec-ture Notes in Math., Vol. 476, Springer, Berlin, 1975.[20] M. Del Zotto, J. J. Heckman and D. R. Morrison, 6D SCFTs and Phases of 5D Theories, JHEP , 147 (2017) 4021] M. Del Zotto, J. Gu, M. X. Huang, A. K. Kashani-Poor, A. Klemm and G. Lockhart, TopologicalStrings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs, JHEP , 156 (2018)[22] F. Denef, Les Houches Lectures on Constructing String Vacua, Les Houches , 483 (2008)[arXiv:0803.1194 [hep-th]].[23] R. Dijkgraaf and E. Witten. Topological gauge theories and group cohomology. Communicationsin Mathematical Physics, 129(2), 393?429 (1990). doi:10.1007/bf02096988[24] J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. , 1819 (1994)[25] M. Esole, J. Fullwood, and S.-T. Yau. D elliptic fibrations: non-Kodaira fibers and neworientifold limits of F-theory. Commun. Num. Theor. Phys. , no. 3, 583 (2015).[26] M. Esole, S. G. Jackson, R. Jagadeesan, and A. G. Noël. Incidence Geometry in a Weyl ChamberI: GL n , arXiv:1508.03038 [math.RT].[27] M. Esole, S. G. Jackson, R. Jagadeesan and A. G. Noël, Incidence Geometry in a Weyl ChamberII: SL n , arXiv:1601.05070 [math.RT].[28] M. Esole, R. Jagadeesan and M. J. Kang, The Geometry of G , Spin(7), and Spin(8)-models,arXiv:1709.04913 [hep-th].[29] M. Esole, R. Jagadeesan and M. J. Kang, 48 Crepant Paths to SU ( ) × SU ( ) , arXiv:1905.05174[hep-th].[30] M. Esole, P. Jefferson and M. J. Kang, Euler Characteristics of Crepant Resolutions of Weier-strass Models, arXiv:1703.00905 [math.AG].[31] M. Esole, M. J. Kang and S. T. Yau, A New Model for Elliptic Fibrations with a Rank OneMordell–Weil Group: I. Singular Fibers and Semi-Stable Degenerations, arXiv:1410.0003 [hep-th].[32] M. Esole, P. Jefferson and M. J. Kang, The Geometry of F -Models, arXiv:1704.08251 [hep-th].[33] M. Esole and S. Pasterski, D -flops of the E -model, arXiv:1901.00093 [hep-th].[34] M. Esole and M. J. Kang, Characteristic numbers of elliptic fibrations with non-trivial Mordell–Weil groups, arXiv:1808.07054 [hep-th].[35] M. Esole and M. J. Kang, Characteristic numbers of crepant resolutions of Weierstrass models,arXiv:1807.08755 [hep-th].[36] M. Esole and M. J. Kang, The Geometry of the SU(2) × G -model, JHEP (2019) 091doi:10.1007/JHEP02(2019)091 [arXiv:1805.03214 [hep-th]].[37] M. Esole and M. J. Kang, Flopping and Slicing: SO(4) and Spin(4)-models, arXiv:1802.04802[hep-th].[38] M. Esole, M. J. Kang and S. T. Yau, Mordell–Weil Torsion, Anomalies, and Phase Transitions,arXiv:1712.02337 [hep-th].[39] M. Esole and R. Savelli, ‘Tate Form and Weak Coupling Limits in F-theory, JHEP , 027(2013) 4140] M. Esole and S. H. Shao, “M-theory on Elliptic Calabi-Yau Threefolds and 6d Anomalies,”arXiv:1504.01387 [hep-th].[41] M. Esole, S.-H. Shao, and S.-T. Yau. Singularities and Gauge Theory Phases. Adv. Theor.Math. Phys. , 19:1183–1247, 2015.[42] M. Esole, S. H. Shao and S. T. Yau, Singularities and Gauge Theory Phases II, Adv. Theor.Math. Phys. 20, 683 (2016)[43] M. Esole and S. T. Yau, Small resolutions of SU(5)-models in F-theory, Adv. Theor. Math.Phys. 17, no. 6, 1195 (2013)[44] S. Ferrara, R. Minasian and A. Sagnotti, Low-energy analysis of M and F theories on Calabi-Yauthreefolds, Nucl. Phys. B , 323 (1996)[45] R. Hartshorne,
Algebraic Geometry , Graduate Texts in Mathematics 52, Springer-Verlag, 1977.[46] A. Grassi and D. R. Morrison. Group representations and the Euler characteristic of ellipticallyfibered Calabi-Yau threefolds.
J. Algebraic Geom. , 12(2):321–356, 2003.[47] T. W. Grimm and H. Hayashi, F-theory fluxes, Chirality and Chern-Simons theories. JHEP , 027 (2012)[48] T. W. Grimm and A. Kapfer, “Anomaly Cancelation in Field Theory and F-theory on a Circle,”JHEP , 102 (2016) .[49] B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch.Phys. , 294 (2015)[50] H. Hayashi, C. Lawrie, D. R. Morrison and S. Schafer-Nameki, Box Graphs and Singular Fibers,JHEP , 048 (2014)[51] J. J. Heckman and T. Rudelius, Top Down Approach to 6D SCFTs, J. Phys. A , no. 9,093001 (2019)[52] D. Husemöller, Elliptic curves . Second edition. With appendices by Otto Forster, Ruth Lawrenceand Stefan Theisen. Graduate Texts in Mathematics, 111. Springer-Verlag, New York, 2004.[53] K. A. Intriligator, D. R. Morrison, and N. Seiberg. Five-dimensional supersymmetric gaugetheories and degenerations of Calabi-Yau spaces.
Nucl.Phys. , B497:56–100, 1997.[54] P. Jefferson, S. Katz, H. C. Kim and C. Vafa, “On Geometric Classification of 5d SCFTs,” JHEP , 103 (2018).[55] P. Jefferson, H. C. Kim, C. Vafa and G. Zafrir, Towards Classification of 5d SCFTs: SingleGauge Node.[56] K. Kodaira. On compact analytic surfaces. II, III.
Ann. of Math. (2) 77 (1963), 563–626; ibid. ,78:1–40, 1963.[57] Y. Kimura, Discrete gauge groups in certain F-theory models in six dimensions,arXiv:1905.03775 [hep-th]. 4258] V. Kumar, D. R. Morrison and W. Taylor, Global aspects of the space of 6D N = 1 supergrav-ities, JHEP , 118 (2010).[59] C. Mayrhofer, D. R. Morrison, O. Till and T. Weigand, Mordell–Weil Torsion and the GlobalStructure of Gauge Groups in F-theory, JHEP , 16 (2014)[60] S. Monnier, G. W. Moore and D. S. Park, “Quantization of anomaly coefficients in 6D
N = ( , ) supergravity,” JHEP , 020 (2018)[61] D. R. Morrison and W. Taylor, “Matter and singularities,” JHEP , 022 (2012)[62] D. R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. , 1072 (2012)[63] D. R. Morrison and C. Vafa. Compactifications of F theory on Calabi-Yau threefolds. 2. Nucl.Phys. , B476:437–469, 1996.[64] D. Mumford and K. Suominen,
Introduction to the theory of moduli , in Algebraic Geometry,Oslo 1970, Proceedings of the 5th Nordic summer school in Math, Wolters-Noordhoff, 1972,171-222.[65] N. Nakayama. On Weierstrass models. In Algebraic geometry and commutative algebra, Vol.II, pp. 405–431. Kinokuniya, Tokyo, 1988.[66] I. Papadopoulos,
Sur la classification de Néron des courbes elliptiques en caractéristique résidu-elle et (French), Journal of Number Theory 44, 119-152 (1993).[67] D. S. Park, Anomaly Equations and Intersection Theory, JHEP , 093 (2012)[68] L. J. Romans, Selfduality for Interacting Fields: Covariant Field Equations for Six-dimensionalChiral Supergravities, Nucl. Phys. B , 71 (1986).[69] V. Sadov, Generalized Green-Schwarz mechanism in F theory, Phys. Lett. B , 45 (1996).[70] A. Sagnotti, A Note on the Green-Schwarz mechanism in open string theories, Phys. Lett. B , 196 (1992)[71] S. Randjbar-Daemi, A. Salam, E. Sezgin and J. A. Strathdee, An Anomaly Free Model inSix-Dimensions, Phys. Lett. , 351 (1985).[72] J. H. Schwarz, Anomaly - free supersymmetric models in six-dimensions, Phys. Lett. B ,223 (1996)[73] A. Sen, “Orientifold limit of F theory vacua,” Phys. Rev. D , R7345 (1997)[74] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil. Modularfunctions of one variable , IV (
Proc. Internat. Summer School , Univ. Antwerp, Antwerp, 1972),pp. 33–52. Lecture Notes in Mathematics 476, Springer-Verlag, Berlin, 1975.[75] W. Taylor and A. P. Turner, Generic matter representations in 6D supergravity theories, JHEP , 081 (2019)[76] J. Tian and Y. N. Wang, E-string spectrum and typical F-theory geometry, arXiv:1811.02837[hep-th]. 4377] P. van Nieuwenhuizen, Anomalies in Quantum Field Theory: Cancellation of Anomalies ind=10 Supergravity, Leuven University Press, Leuven (1988).[78] T. van Ritbergen, A. N. Schellekens and J. A. M. Vermaseren, Group theory factors for Feynmandiagrams, Int. J. Mod. Phys. A , 41 (1999)[79] T. Weigand, F-theory, PoS TASI , 016 (2018)[80] E. Witten, Phase transitions in M theory and F theory, Nucl. Phys. B , 195 (1996).[81] D. Xie and S. T. Yau, Three dimensional canonical singularity and five dimensional N = 1SCFT, JHEP1706