The Geometry of the SU(2) G 2 -model.

Transcription

1 The Geometry of the SU() G -model. Mboyo Esole and Monica Jinwoo Kang arxiv:85.4v [hep-th] 8 May 8 Department of Mathematics, Northeastern University 6 Huttington Avenue, Boston, MA 5, U.S.A. Department of Physics, Harvard University 7 Oxford Street, Cambridge, MA 8, U.S.A j.esole@northeastern.edu, jkang@fas.harvard.edu Abstract We study elliptic fibrations that geometrically engineer an SU() G gauge theory realized by a Weierstrass model for the collision III+I ns. We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU() G -model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU() G -model to a five and six-dimensional supergravity with eight supercharges. By matching each crepant resolution with each Coulomb chamber of the fivedimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss counting number of hypermultiplets in presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory. Key words: Elliptic fibrations, crepant resolutions, flop transitions, Weierstrass models, Tate s algorithm, six-dimensional supergravity, five-dimensional supergravity, anomaly cancellations, Euler characteristic.

2 Contents Introduction. Canonical problems in F/M-theory Defining the SU() G -model Representations, Coulomb branches, hyperplane arrangements, and flops Non-Kodaira fibers Compactifications of F-theory and M-theory on an SU() G -model Outlook Geometric results 8. Geometric description Crepant resolutions Intersection theory Euler characteristics and Hodge numbers Triple intersection numbers Hyperplane arrangement Flops Connection to the SU() SU()-model The crepant resolutions and fiber structures 6. Resolution I Resolution II Resolution III Resolution IV Saturation of weights and representations Fiber enhancements d and 6d supergravity theories with eight supercharges 4. 5d N = supergravity physics Anomaly cancellations in 6d N = (, ) supergravity Counting hypermultiplets: numerical oddities

3 Introduction Semi-simple Lie groups appear naturally in compactifications of M-theory and F-theory on elliptic fibrations [, 4]. The Lie group is semi-simple when the discriminant of the elliptic fibration contains at least two irreducible components and such that the dual graph of the singular fiber over the generic point of i (i =, ) is reducible. These are called collisions of singularities and were first studied by Bershadsky and Johanson [4] as an application to physics of the work of Miranda on regularization of elliptic threefolds defined by singular Weierstrass models [55]. If we organize collision of singularities by the rank of the associated Lie algebra, the simplest collisions will correspond to the collision of two singular fibers with dual graphs Ã. The gauge group is simply Spin(4)=SU() SU() if the Mordell Weil group is trivial []. The next simplest cases to study involve an elliptic fibration with gauge group SU() G where G is a simple Lie group of rank two. There are three compact simple and simply connected Lie groups of rank two: SU(), Sp(4), and G. Thus, we get one of the following models: SU() SU(), SU() Sp(4), SU() G. The SU() SU()-model is interesting for its connection to the non-abelian sector of the Standard Model. The others are QCD-like theories obtained by replacing SU() by another simple and simply connected group of rank two. The group G is the smallest simply connected Lie group with a trivial center and all its representation are real. The SU() Sp(4)-model is studied in [6]. The G -model is considered in detail in []. The purpose of this paper is to study the geometry and physics of SU() G -models realized by the collision of singularities III + I ns. (.) The SU() G -model appears naturally in the study of elliptic threefolds where there are non- Higgsable clusters producing semi-simple groups that cannot be broken by charged matter without breaking supersymmetry [57]. The gauge group of smallest rank produced by a non-higgsable cluster with multiple factors is realized by a collision of type III + I ns with a trivial Mordell Weil group and give the gauge group G = SU() G. (.) The III + I ns -model appears as a non-higgsable cluster when the discriminant locus contains two rational curves of self-intersection and intersecting transversally or three rational curves form a chain of curves intersecting transversally at a point with self-intersections (,, ) [57] and [5,, 47]. The III + I ns -model was first discussed in the F-theory literature in [4] based on the work of Miranda [55]. In Miranda s regularization, the collision III + I ns is a collision of fibers for which the j-invariant is 78. The fiber over the generic point of the collision is a non-kodaira fiber composed of a chain of five rational curves intersecting transversally with multiplicities This fiber is a contraction of a Kodaira fiber of type III whose dual graph is the affine Dynkin diagram of type Ẽ7. Surprisingly, despite receiving a significant amount of attention in the last few years for their role in the study of superconformal field theories, many properties of the III + I ns model remain unknown. The type of questions that we consider are explained in section.. Throughout this paper, we work over the complex number C and use the conventions and notation of []. We work with an arbitrary base of dimension n and specialize to the case of Calabi-Yau threefolds only when necessary to connect with the physics. Since resolution of singularities is a local process, it does not care if the base B is compact or not. But the compactness of the base will matter when considering anomaly cancellations. Over non-compact bases, collisions of singularities are used

4 to classify 6d N = (, ) superconformal field theories using elliptic fibrations. These six-dimensional SCFTs are then compactified on a Riemann surface to yield N = (, ) SCFTs in four-dimensonal spacetime. Three rational curves that are transverse to each other and meet at the same point (such as the Kodaira fiber of type IV) are represented by three nodes connected to the same point. We write G and G t respectively for the affine Dynkin diagram of type G and its Langlands dual. In Kac s notation [5], they are denoted respectively G () and D () 4. The dual graph of the fiber I ns is of type G t and not G as often stated in the F-theory literature but clear from Figure. III dual graph of I ns G t G Figure : Conventions for dual graphs. Dynkin diagram. The black node represents the extra node of the affine. Canonical problems in F/M-theory A standard set of questions in F-theory and M-theory compactifications on an elliptically fibered Calabi-Yau threefold X are the following [4, 7, 6, 8]: (i) Coulomb branch and charged hypermultiplets. What is the structure of the Coulomb branch of the five-dimensional N = theory with Lie group g and representation R geometrically engineered by an elliptic fibration Y? How many hypermultiplets transform under each irreducible components of R? Can we completely fix the number of charged multiplets by comparing the triple intersection numbers and the prepotentials? (ii) Anomaly cancellations and uplift. Is the five-dimensional theory always compatible with an uplift to an anomaly free six-dimensional theory? What are the conditions to ensure cancellations of anomalies of a six-dimensional N = (, ) supergravity obtained by compactification of F-theory on Y? Can we fix the number of multiplets by the six-dimensional anomaly cancellation conditions? (iii) Fiber geometry. What is the fiber structure of the Y? What are the vertical curves that carry the weights of the representation R? What are the extremal flopping curves? (iv) Topological invariants. What is the Euler characteristic of a crepant resolution of a SU() G -model? What are the Hodge numbers of Y when Y is a Calabi-Yau threefold? These questions have been addressed recently for many geometries such as G, Spin(7), and Spin(8)- models []; F 4 -models [5]; SU(n)-models [8 4, 45]; and also for (non-simply connected) semisimple groups such as SO(4) and Spin(4)-models []; SU() SU(4), (SU() SU(4))/Z, SU() Sp(4), and the (SU() Sp(4))/Z -models [6]. We answer these canonical questions for the SU() G -model. We first do not restrict ourselves to Calabi-Yau threefolds, but discuss the resolutions and the Euler characteristic without fixing the

5 base of the fibration in the spirit of [,,4,9,4]. The computation of topological invariants such as the triple intersection numbers and the Euler characteristic are streamlined by recent pushforward theorems []. For recent work in F-theory and biraitonal geometry in physics, see for example [5, 6,, 6,,, 49, 5, 54] and reference within.. Defining the SU() G -model Given a morphism X B and an irreducible divisor S of B, the generic fiber over S is by definition the fiber over its generic point η. Such a fiber X η is a scheme over the residue field κ of η. The residue field κ is not necessarily geometrically closed. Some components of X η can be irreducible as a κ-scheme but will reduce to more irreducible after a field extension κ κ. In the case of a flat elliptic fibration, the Galois group of the minimal field extension that allows all the irreducible components of X η to be geometrically irreducible is Z for all non-split Kodaira fibers with the exception of I for which the Galois group could also be S or Z []. Kodaira fibers classify geometric generic fibers of an elliptic fibration (the fiber defined over the algebraic closure of the residue field). The generic fiber (defined over the residue field κ) are classified by the Kodaira type of the corresponding geometric generic fiber together with the Galois group of the minimal field extension necessary to make all irreducible components of the fiber geometrically irreducible. The Galois group is always Z unless in the case of the fiber I where it can also be Z or S. Thus, there are two distinct fibers with dual graph G t as the Galois group of an irreducible cubic can be the symmetric group S or the cyclic group Z []. When we specify the type of Galois group, we write I ns as I S or I Z. An I Z -model is very different from an I S -model already at the level of the fiber geometry as discussed in []. There are five different Kodaira fibers with dual graph à and thus producing an SU(), namely I ns, Is, III, IVns, and I ns. The SU() G could be realized by anyone of the following ten models I ns + I ns, I s + I ns, III + I ns, I ns + I ns, or IV ns + I ns, where I ns could be I Z or I S. For example, the non-higgsable models of type SU() G studied in the literature are typically of the type III + I S. In the rest of the paper, when we write I ns without further explanation, we always mean the generic I S. A Weierstrass model for the collision III + I S is [55] III + I S y z = x + fst xz + gs t z. (.) The discriminant locus is composed of three irreducible components S, T, and : = s t 6 (4f + 7g s), (.4) where S = V (s) and T = V (t) are two smooth Cartier divisors supporting respectively the fiber of type III and of type I ns. We assume that S and T intersect transversally. The fiber over the generic point of the leftover discriminant = 4f + 7g s is a nodal curve (Kodaira type I ). Following Tate s algorithm, the type of the decorated Kodaira fibers depends on the Galois group of the associated associated cubic polynomial P (q) = q + fsq + gs. (.5) Assuming that P (q) is irreducible, the Galois group is Z if the discriminant of P (q), (P ) = s (4f + 7g s), is a perfect square in the residue field of the generic point of T []. A simple way to have a Z Galois group is to increase the valuation of f along T []: III + I Z y z = x + fst xz + gs t z. (.6) 4

6 In this case, the j-invariant will be zero over the generic point of T in contrast to the case of equation (.8) where the j-invariant is 78 on both S and T. The fact that the Galois group of P (q) is Z is clear as it now takes the form: P (q) = q + gs. (.7) The elliptic fibrations of the type y z = x + afst xz + ags t z, (.8) is directly inspired from the non-higgsable model of type (,, ). It has a discriminant = a s t 6 (4af + 7g s). The generic fiber over V (a) is of Kodaira type II. In particiular, this elliptic fibration does not have a crepant resolution since it has Q-factorial terminal singularities at x = y = a = g =. This is a generic problem when the discriminant locus contains an irreducible component whose generic fiber is of Kodaira type II.. Representations, Coulomb branches, hyperplane arrangements, and flops When the elliptic fibration has dimension three or higher, we naturally associate the elliptic fibration to not only the Lie algebra g, but also a representation R of g. In the case of an SU() G -model, the representation R is the direct sum of the following irreducible representations (see section ) R = (, ) (, 4) (, 7) (, ) (, 7). (.9) The (, ) is the adjoint of SU(), the (, 4) is the adjoint of G. The (, 7) is the bifundamental representation of SU() G supported at the intersection S T of the two divisors supporting G and SU(). The representation (, ) (resp. (, 7)) is the fundamental of SU() (resp. G ) supported at the collision of the third component of the discriminant locus = (4f + 7g s) with the divisor S (resp. T ). In the case of a compactification on an elliptically fibered Calabi-Yau threefold, the adjoint representation is frozen when the curve supporting the corresponding group is a rational curve. In many study of the SU() G -model, the fundamental representations (, ) (, 7) are ignored because they are generated away from the collision of the curves supporting G and SU(). We will consider a full compact geometry that would require taking into account all the matter representations including those coming from the intersection with the left-over discriminant. From our point of view, we compute the weights as the minus of the intersection of vertical curves with fibral divisors. We then use the saturation of weights to find an irreducible representation containing them. This method is carefully explained in []. The use of weights via intersection numbers can be traced back to Gross and Aspinwall [7] and is based on the M-theory picture of M-branes wrapping chain of curves and becoming massless when the curve shrink to a point. It is also interesting to understand the representation via the traditional Katz-Vafa techniques that is purely rooted in representation theory [5]. The embedding of SU() G into E 7 can be described as follows: su g su so 7 su su 7 e 7. (.) We recall that G is the subgroup of SO(7) that preserves a chosen vector in its eight-dimensional real spinor representation. Moreover, SO(7) is a subgroup of SU(7) and we have the classic embedding A 6 E 7 by removing the appropriate external node of E 7. In this decomposition, the Levi subgroup We write the direct product of a representation r of SU() and r of G as (r, r ). We follow the usual tradition in physics of writing a representation by its dimension as there is no room for ambiguity with the representation we use in this paper. The and the of SU() are respectively the fundamental and the adjoint representation of SU(). The 7 and the 4 of G are respectively the fundamental and the adjoint representation of G. 5

7 is SU(7). We then embed G into SU(7) via its irreducible seven-dimensional representation and we notice that the centralizer is A. The study of the Coulomb branch of the gauge theory geometrically engineered by an elliptic fibration is the study of the minimal models over the Weierstrass model and how they flop to each other. This can also be described through the hyperplane arrangement I(g, R) with hyperplanes that are kernels of the weights of R restricted inside the dual fundamental Weyl chamber of g [8,9,9,4]. In the case of the SU() G -model, it is interesting to notice that the generic SU()-model and G -model do not have any flops [, 9]. However, the SU() G -model has the bifundamental representation (, 7) which contains several weights whose kernels are hyperplane intersecting the open dual Weyl chamber of A g and giving four chambers whose incidence graph is a chain illustrated in Figure. We see in this way that the hyperplane arrangement I(g, R) does not care of the fundamental and adjoint representations (, ) or (, 7) since only the weights ϖ of the bifundamental (, 7) define hyperplanes ϖ intersecting the interior of the dual Weyl chamber of g = A g and it is enough to study only I(A g, (, 7)). 4 4 Figure : The chamber structure of I(A g, (, 7)) and its adjacency graph. This also represents the structure of the extended Kähler cone of a SU() G -model. Replacing (, 7) with R = (, ) (, 4) (, 7) (, ) (, 7) does not change the adjacency graph since the adjoint and fundamental representations do not intersect the interior of the Weyl chamber of A g. The interior walls are given by the weights ϖ (,7) 5 = (;, ), ϖ (,7) 6 = (;, ), and ϖ (,7) 7 = (;, )..4 Non-Kodaira fibers There are many examples of non-kodaira fibers in the literature [9, 7, 4, 46, 55, 6]. Over the generic point of s = f = g =, the III-model has a non-kodaira fiber that are contractions of a fiber of type I. The G -model has non-kodaira fibers that are contractions of an I and a IV fiber []. At the collision III+I ns, we get a non-kodaira fiber that is an incomplete III with a dual graph that is a contraction of the dual graph of a Ẽ7 and specializes further to an incomplete II with a dual graph that is a contraction of the dual graph of a Ẽ8. The fibers found by Miranda at the collisionii+i ns matches the one we find in Resolution I for the generic fiber over S T. Miranda has already noticed in [55] that the non-kodaira fibers of Miranda s model were always contractions of Kodaira fibers. The same is true for Miranda s models of arbitrary dimension [6] and for flat elliptic threefolds [9]. A A D 4, G B 4 E 6 A G E 7 E 8 6

8 incomplete Ẽ7 incomplete Ẽ8 Res. I 4 Res. II 4 Res. III Res. IV Figure : Non-Kodaira Fibers for the SU() G -model..5 Compactifications of F-theory and M-theory on an SU() G -model We analyze the physics of the compactifications of M-theory and F-theory on elliptically fibered Calabi-Yau corresponding to SU() G -models. These give five and six-dimensional gauged supergravity theories with eight supercharges. We determine the matter content of these compactifications and study anomaly cancellations of the six-dimensional theory and their Chern-Simons terms. In the five dimensional theory, we also determine the structure of the Coulomb chambers. Each chamber corresponds to a specific crepant resolution that we determine explicitly. One complication is that two of the chambers are derived by blowups with a non-smooth center and require special care. The crepant resolutions of the Weierstrass model of a SU() G -model are listed in equation (.). The Euler characteristic of a SU() G -model obtained by one of these crepant resolutions is derived in Theorem.5. We determine the chamber structure of the hyperplane arrangement I(g, R) for the SU() G -model. There are four chambers whose adjacency graph is represented in Figure. For each chamber, we match an explicit crepant resolution of the Weierstrass model, so that the graph of flops matches the adjacency graph of the hyperplane arrangement. While three of the crepant resolutions are obtained by blowing up smooth centers, one requires a blowup with a non-smooth center. In order to connect with the physics, we study the compactification of M-theory and F-theory on a Calabi-Yau threefold given by an SU() G -model. For the five-dimensional supergravity theory [8,48], we compute the one-loop prepotential in the Coulomb branch, and determine in this way the Chern-Simons couplings, the number of vector multiplets, tensor multiplets, and hypermultiplets. The Chern-Simons couplings are computed geometrically as triple intersection numbers of fibral divisors (see Theorem.). We match the triple intersection polynomial with the prepotential to obtain constraints on the number of charged hypermultiplets (see equation 4.). In many cases, such a method will completely fix the number of multiplets, but here, they are only linear constraints. However, they are completely fixed by the anomaly equations of a six-dimensional uplift of the theory (see equation (4.)) or by using Witten s genus formula, which is a five-dimensional result. 7

9 Let S and T be the divisors supporting SU() and G respectively. Let be the third component of the discriminant locus. Then the number of charged hypermultiplets are (see section 4) n,7 = ST, n, = g(s), n, = S (8K + S + 7 T ), n,4 = g(t ), n,7 = T (5K + S + T ). The Hodge numbers of a Calabi-Yau threefold that is a SU() G -model are (see section.4) h, (Y ) = 4 K, h, (Y ) = 9K + 5KS + 4KT + S + 6ST + 6T + 4. In the six-dimensional supergravity theory, using Sadov s techniques [59], we check that anomalies are canceled explicitly by the Green Schwarz Sagnotti-West mechanism. The anomaly polynomial I 8 factors as a perfect square: I 8 = ( KtrR + Str F + T tr 7 F ), where F and F are the field strengths for SU() and G respectively. It is often the case that hypermultiplets in a fundamental representation of a unitary group is given by intersection of divisors. Interestingly, this is not the case for the number of fundamental in the representation (, ) if we want it to be consistent with the number found using triple intersection in the five-dimensional theory and anomaly cancellation in the six-dimensional theory. We discuss this subtle issues in section Outlook The rest of the paper is structured as follows. In section, we collect our geometric results. In section., we introduce the models that we study in this paper, define its Weierstrass model, its crepant resolutions, compute the Euler characteristic of the crepant resolutions and the triple intersection of the fibral divisors. In the case of a Calabi-Yau threefold, we also compute the Hodge numbers. We compute the adjacency graph of the hyperplane arrangement associated with an SU() G - model, and finally match the structure of the hyperplane arrangement with the flopping curves of the crepant resolutions. In section 4, we study the consequences of our geometric results for the physics of F-theory and M-theory compactified on an SU() G -model. We discuss the subtleties of counting the number of hypermultiplets in presence of singularities in section 4.. Geometric results In this section, we summarize the key results, and discuss some of their implications. The relative extended Kähler cone of the II+I ns -model is modeled by the hyperplane arrangement I(A G, (, 7)) and consists of four chambers arranged as illustrated in Figure. Each Each chamber corresponds to a crepant resolution given explicitly in equation (.).. Geometric description We consider the following Weierstrass model realizing an SU() G -model: III + I ns y z = x + fst xz + gs t z. (.) We assume that the coefficients f and g are algebraically independent and S = V (s) and T = V (t) are smooth divisors intersecting transversally. In each case, the Kodaira fiber over the generic point of S and T has respective dual graph à and G t. 8

10 . Crepant resolutions We use the following convention. Let X be a nonsingular variety. Let Z X be a complete intersection defined by the transverse intersection of r hypersurfaces Z i = V (g i ), where g i is a section of the line bundle I i and (g,, g r ) is a regular sequence. We denote the blowup of a nonsingular variety X along the complete intersection Z by X (g,, g r e ) X. 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 also do not write the obvious pullbacks. Each of the following four sequences of blowups is a different crepant resolution of the SU() G - model given by the Weierstrass model of equation (.). (x, y, s e ) (x, y, t w ) (y, w w ) Resolution I : X X X X, (x, y, p p ) (y, p, t w ) (p, t w ) Resolution II : X X X X (x, y, t w ) (x, y, s e ) (y, w w ) Resolution III: X X X X (x, y, t w ) (y, w w ) (x, y, s e ) Resolution IV: X X X X These are imbedded resolution with ambient space X = P[O B L L ].. Intersection theory,,. (.) All our intersection theory computations come down to the following three theorems. The first one is a theorem of Aluffi which gives the Chern class after a blowup along a local complete intersection. The second theorem is a pushforward theorem that provides a user-friendly method to compute invariant of the blowup space in terms of the original space. The last theorem is a direct consequence of functorial properties of the Segre class and gives a simple method to pushforward analytic expressions in the Chow ring of a projective bundle to the Chow ring of its base. Theorem. (Aluffi, [, Lemma.]). Let Z X be the complete intersection of d nonsingular hypersurfaces Z,..., Z d meeting transversally in X. Let f X X be the blowup of X centered at 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). (.) Theorem. (Esole Jefferson Kang, see []). Let the nonsingular variety Z X be a complete intersection of d nonsingular hypersurfaces Z,..., Z d meeting transversally in X. Let E be the class of the exceptional divisor of the blowup f X X centered at Z. Let Q(t) = a f Q a t a be a formal 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 d f Q(E) = Q(Z l )M l, where M l = l= 9 d m= m l Z m Z m Z l.

11 Theorem. (See [] and [,, 4, 4]). Let L be a line bundle over a variety B and π X = P[O B L L ] B a projective bundle over B. Let Q(t) = a π Q a t a be a formal power series in t such that Q a A (B). Define the auxiliary power series Q(t) = a Q a t a. Then Q(H) π Q(H) = H + Q(H) H= L H + Q() H= L 6L, where L = c (L ) and H = c (O X ()) is the first Chern class of the dual of the tautological line bundle of π X = P(O B L L ) B..4 Euler characteristics and Hodge numbers In the spirit of [], the Euler characteristic depends only on the sequence of blowups. Since the sequences that we consider are the same as those of the SU() SU()-models, they also share the same Euler characteristic. Moreover, since the rank are also the same, the Hodge numbers are identical as well. Using p-adic integration and the Weil conjecture, Batyrev proved the following theorem. Theorem.4 (Batyrev, []). Let X and Y be irreducible birational smooth n-dimensional projective algebraic varieties over C. Assume that there exists a birational rational map ϕ X Y that does not change the canonical class. Then X and Y have the same Betti numbers. Batyrev s result was strongly inspired by string dualities, in particular by the work of Dixon, Harvey, Vafa, and Witten [5]. As a direct consequence of Batyrev s theorem, the Euler characteristic of a crepant resolution of a variety with Gorenstein canonical singularities is independent on the choice of resolution. We identify the Euler characteristic as the degree of the total (homological) Chern class of a crepant resolution f Ỹ Y of a Weierstrass model Y B: ˆ χ(ỹ ) = c(ỹ ). We then use the birational invariance of the degree under the pushfoward to express the Euler characteristic as a class in the Chow ring of the projective bundle X. We subsequently push this class forward to the base to obtain a rational function depending upon only the total Chern class of the base c(b), the first Chern class c (L ), and the class S of the divisor in B: ˆ χ(ỹ ) = π f c(ỹ ). B In view of Theorem.4, this Euler characteristic is independent of the choice of a crepant resolution. Theorem.5. The generating polynomial of the Euler characteristic of an SU() G -model obtained by a crepant resolution of a Weierstrass model given in section.: χ(y ) = 6 S L SL + (S SL + S L)T + (S + )T ( + S)( + T )( 6L + S + T ) c(t B). Proof. The total Chern class of X = P B [O B L L ] is c(t X ) = ( + H)( + H + π L)( + H + π L)π c(t B).

12 where L = c (L ) and H = c (O X ()). The class of the Weierstrass equation is [Y ] = H + 6L. Since all the resolutions are crepant, it is enough to do the computation in one of them. We consider resolution I. We denote the blowups by f X X, f X X, and f X X, where E, W and W are respectively the classes of the first, second, and third blowups. The center of the three blowups have respectively classes: Z () = H + π L, Z () = H + π L, Z () = π S, Z () = f H + f π L E, Z () = f H + f π L E, Z () = f π T, Z () = f f H + f f π L f E W, Z () = W. The successive blowups give (see Theorem.) c(t X ) = ( + E )( + Z () E )( + Z () E )( + Z () ( + Z () )( + Z () )( + Z () ) c(t X ) = ( + W )( + Z () W )( + Z () W )( + Z () ( + Z () )( + Z () )( + Z () ) c(t X ) = ( + W )( + Z () W )( + Z () W ) ( + Z () )( + Z () f c(t X ) ) E ) f c(t X ) W ) f c(t X ) After the first blowup, the proper transform of Y is of class Y = f Y E. After the second blowup, the proper transform of Y is of class Y = f Y W. And finally after the third blowup, the proper transform of Y is Y = f Y W. Altogether, we have [Y ] = (f f f (H + 6π L) f f E f W W ) [X ]. We also have that c (X ) = f f f c (X ) f f E f W W. Hence c (Y ) = f f f c (Y ), which prove that the resolution is crepant. The total Chern class of Y is (see Theorem.) Then c(t Y ) [Y ] = f f f (H + 6π L) f f E f W W + f f f (H + 6π L) f f E f W W c(t X ) [X ]. ˆ χ(y ) = Y ˆ c(t Y ) [Y ] = B π f f f c(t Y ) [Y ] The final formula for the Euler characteristic follows directly from the pushforward Theorems. and.. By direct expansion and specialization, we have the following three lemmas: Lemma.6. For an elliptic threefold, the Euler characteristic of of an SU() G -model obtained by a crepant resolution of a Weierstrass model given in section. is: χ(y ) = 6( c L + L + S 5SL + ST 8LT + T ). By applying c = L = K, we have the following Lemma.

13 Lemma.7. In the case of a Calabi-Yau threefold, The Euler characteristic of an SU() G -model obtained by a crepant resolution of a Weierstrass model given in section. is: χ(y ) = 6(K + S + 5SK + ST + 8KT + T ). Lemma.8. The Euler characteristic for an elliptic fourfold, the Euler characteristic of an SU() G -model obtained by a crepant resolution of a Weierstrass model given in section. is given by χ(y 4 ) = 6 ( c L 7L + c L + c S 5c SL + c ST 8c LT + c T +S 5S L + 6S T + 54SL 44SLT + 9ST + 84L T 4LT + 4T ). Again, by the Calabi-Yau condition c = L = K, we have the following Lemma. Lemma.9. In the case of a Calabi-Yau foutfold, The Euler characteristic of an SU() G -model obtained by a crepant resolution of a Weierstrass model given in section. is χ(y 4 ) = 6 (c K + 6K + S + 4S K + 6S T + 49SK + 4SKT + 9ST + 76K T + KT + 4T ). Theorem.. In the Calabi-Yau case, the Hodge numbers of an SU() G -model given by the crepant resolution of a Weierstrass model given in section. are h, (Y ) = 4 K, h, (Y ) = 9K + 5KS + 4KT + S + 6ST + 6T + 4. There are three fibral divisors not touching the section of the elliptic fibration. This number is exactly the rank of SU() G. Hence, using the Shioda-Tata-Wazir theorem, we have h, (Y ) = + + K, h, (Y ) = h, (Y ) χ(y ). Theorem. (Shioda-Tate-Wazir; see Corollary 4.. of [6]). Let ϕ Y B be a smooth elliptic threefold, then ρ(y ) = ρ(b) + f + rank(mw(ϕ)) + where f is the number of geometrically irreducible fibral divisors not touching the zero section. Theorem.. Let Y be a smooth Calabi-Yau threefold elliptically fibered over a smooth variety B. Assuming the Mordell-Weil group of Y has rank zero, then h, (Y ) = h, (B) + f +, h, (Y ) = h, (Y ) χ(y ), where f is the number of geometrically irreducible fibral divisors not touching the zero section. In particular, if Y is a G-model with G a simple group, f is the rank of G..5 Triple intersection numbers Let Y be a crepant resolution of an SU() G -model defined by one of the crepant resolution f Y Y given in section.. Assuming that Y is a threefold, the triple intersection polynomial of Y is a polynomial containing the divisors (D a D b D c ) [Y ]. We express a triple intersection polynomial of the SU() G -models as a polynomial in ψ, ψ, φ, φ, and φ that couples respectively with the fibral divisors D s, Ds, Dt, Dt, and Dt. The pushforward is expressed in the base by pushing f is really the relative Picard number ρ(y /W ) of Y over the Weierstrass model W.

14 forward to the show ring of X and then to the base B. We recall that π X B is the projective bundle in which the Weierstrass model is defined. Then, ˆ ˆ F trip = [(ψ D s +ψ D s +φ D t +φ D t +φ D) t ] = π f [(ψ D s +ψ D s +φ D t +φ D t +φ D) t ]. Y Once the classes of the fibral divisors are determined, all is left is to compute the pushforward using the pushforward theorems of []. Theorem.. The triple intersection polynomial of an SU() G -model defined by one of the crepant resolutions in section. is Resolution I: F (I) trip = 9T φ φ (φ ( 6L + S + T ) Sψ ) + T φ (φ ( 9L + S + 6T ) Sψ ) Resolution II: B (T φ (9L S T ) + S (ψ ψ ) (ψ (S L) + ψ (L + S)) + 9ST ψ φ ) + T φ (φ (L S) + Sφ (ψ + φ ) S ((ψ ψ ) + φ )) + T φ (L S T ) + T φ (φ ( L + S + T ) Sψ ) F (II) trip =Sψ ( 4L S + T ) + T φ ( 6L + 7S + T ) + 4T φ (L T ) 5ST ψ φ ST ψ φ Resolution III: + T ( 9L + S + 6T )φ φ 6ST φ ψ + 9T φ φ (Sψ ( 6L + S + T )φ ) T φ ( 4L + S + 4T ) + T φ (φ ( L + S + T ) S (ψ + ψ + φ )) T φ (φ (S L) Sφ (ψ + φ ) + S ( ψ + ψ + φ ) ) + 6ST ψ ψ φ ST ψ φ ST ψ φ + Sψ ψ ( 4L + S + T ) + +Sψ ψ (4L T ) Sψ ( 4L + 4S + T ) F (III) trip = T (9L (S + T ))φ φ T φ φ ( 8L + 4S + 9T ) ST ψ φ ST ψ φ Resolution IV: + T φ (4L S 4T ) 4ST φ (9L S T ) ST ψ φ + ST ψ φ φ S(L + S T )ψ + T φ (φ (T L) Sψ ) + T φ φ (Lφ + Sψ ) ST ψ φ + 6ST ψ φ φ + 6ST ψ ψ φ ST ψ φ S( L + S + T )ψ (ψ ψ ψ + ψ ) S(L + S T )ψ (ψ ψ ψ ) 6ST ψ φ + 4T φ (L T ) F (IV) trip =S( 4L S + T )ψ + 4T (L T )φ + 9T (T L)φ φ + 7T (L T )φ φ + T (T L)φ 6ST ψ φ + Sψ (4L 4S T ) + 4T φ (L T ) + φ (T φ (T L) 6ST ψ ) + ψ (S( 4L + S + T )ψ 6ST φ ) + T φ φ (Lφ + Sψ ) + Sψ (ψ (4L T ) + 4T ψ φ T φ + 6T φ φ 6T φ )

15 Proof. We give the proof for the case of resolution I discussed in detail in section., the other cases follow the same pattern. ˆ F trip = [(ψ D s + ψ D s + φ D t + φ D t + φ D) t ] ˆY = [(ψ D s + ψ D s + φ D t + φ D t + φ D) t (H + 6L E W W )] X ˆ = f f f [(ψ D s + ψ D s + φ D t + φ D t + φ D) t (H + 6L E W W )] X ˆ = π f f f [(ψ D s + ψ D s + φ D t + φ D t + φ D) t (H + 6L E W W )]. B The classes of the fibral divisors in the Chow ring of X are [D] s = S E, [D] s = E, [D] t = T W, [D] t = W W, [D] t = W W. Denoting by M an arbitrary divisor in the class of the Chow ring of the base, The nonzero products intersection numbers of M, H, E, W, and W are ˆ ˆ ˆ ˆ H = 7L, E = S(L + S), W = T (L S + T ), W = T (5L S + T ), ˆY Y ˆ Y ˆ Yˆ W E = ST, W W = T ( L + S T ), W E = ST, W W = T ( L + S T ), ˆY ˆY ˆ Y Y H M = 9LM, HM = M, E W W = ST, ˆY ˆY Yˆ ˆ MH = 9LM, ME = SM, MW = T M, MW = T M. Y Y Y The triple intersection numbers of the fibral divisors follow from these by simple linearity. Y The triple intersection polynomials computed in Theorem. are very different from each other in chambers I, II, III, and IV. By looking at which monomial appear in each case, we can easily compare them with the triple intersections with the prepotentials..6 Hyperplane arrangement We consider the semi-simple Lie algebra g = A g. An irreducible representation of A g is the tensor product R R where R and R are respectively irreducible representations of A and g. Following a common convention in physics, we denote a representation by its dimension in bold character. The weights are denoted by ϖ I j where the upper index I denotes the representation R I and the lower index j denotes a particular weight of the representation R I. A weight of a representation of A g is denoted by a triple (a; b, c) such that (a) is a weight of A and (b, c) is a weight of g, all in the basis of fundamental weights. We use the same notation for coroots. Let φ = (ψ ; φ, φ ) be a vector of the coroot space of A g in the basis of fundamental coroots. Each weight ϖ defines a linear form φ ϖ defined by the natural 4

16 evaluation on a coroot. We recall that fundamental coroots are dual to fundamental weights. Hence, with our choice of conventions, ϖ ϖ is the usual Euclidian scalar product. We define the representation R as: R = (, ) (, 7) (, 7) (, ) (, 7), (.4) We would like to study the arrangement of hyperplanes perpendicular to the weights of the representation R inside the dual fundamental Weyl chamber of A g. We notice that the only representation that will contribute interior walls is the bifundamental (, 7). Theorem.4. The hyperplane arrangement I(A g, R) with R = (, 7) has four chambers whose sign vectors and twhose adjacency graph is given in Figure. A choice of a sign vector is (ϖ (,7) 5, ϖ (,7) 6, ϖ (,7) 7 ). With respect to it, the chambers are listed in Table. Proof. The open dual fundamental Weyl chamber is the half cone defined by the the positivity of the linear form induced by the simple roots: ψ >, φ φ >, φ + φ >. (.5) There are only three hyperplanes intersecting the interior of the fundamental Weyl chamber: ϖ (,7) 5, ϖ (,7) 6, ϖ (,7) 7. We use them in the order (ϖ (,7) 5, ϖ (,7) 6, ϖ (,7) 7 ), the sign vector is (ψ φ φ, ψ + φ φ, ψ φ ). First we consider when ψ φ >. Then ϖ,7 5 = ψ φ >, ϖ,7 6 = ψ +φ c = (ψ φ )+(φ c) >, 6 + ( φ + φ ) >. 6 = ψ + φ φ = (ψ φ ) + 6 >, it follows that 6 = ψ + φ φ = (ψ φ ) + (φ φ ) <, we can have 7 <. See Table. and ϖ,7 7 = ψ φ + φ = (ψ φ ) + (φ φ ) + ( φ + φ ) = ϖ,7 When ψ φ <, ϖ,7 5 = ψ φ <. Then we have either ϖ,7 (φ φ ) >, or ϖ,7 6 = ψ + φ φ = (ψ φ ) + (φ φ ) <. If ϖ,7 ϖ,7 7 = ϖ,7 6 + ( φ + φ ) >. When ϖ,7 both ϖ,7 7 > and ϖ,7.7 Flops In this section, we discuss the flops between the resolutions I, II, III, IV. Resolution I: Flopping curves Weight η A [;, ] (, 7) Resolution II: η [ ;, ] (, 7) ω (,7) 7 Resolution II: η [;, ] (, 7) Resolution III: η [ ;, ] (, 7) ω (,7) 6 Resolution III: η [;, ] (, 7) Resolution IV: η B [ ;, ] (, 7) ω (,7) 5 Table : The fibers that is the one that separates between the chambers and thus responsible for flops in the I s +Is -model. 5

17 Representation Weights (, ) ϖ (,) = (;, ) ϖ (,) = ( ;, ) ϖ (,7) = (;, ) ϖ (,7) = (;, ) ϖ (,7) = (;, ) (, 7) ϖ (,7) 4 = (;, ) ϖ (,7) 5 = (;, ) ϖ (,7) 6 = (;, ) ϖ (,7) 7 = (;, ) ϖ (,7) = (;, ) ϖ (,7) = (;, ) ϖ (,7) = (;, ) ϖ (,7) 4 = (;, ) (, 7) ϖ (,7) 5 = (;, ) ϖ (,7) 6 = (;, ) ϖ (,7) 7 = (;, ) ϖ (,7) 8 = ( ;, ) ϖ (,7) 9 = ( ;, ) ϖ (,7) = ( ;, ) ϖ (,7) = ( ;, ) ϖ (,7) = ( ;, ) ϖ (,7) = ( ;, ) ϖ (,7) 4 = ( ;, ) Table : Weights of the representations of A g. Subchambers ϖ (,7) 5 ϖ (,7) 6 ϖ (,7) 7 Explicit description < φ < φ < φ, φ < ψ + + < φ < φ < φ, φ φ < ψ < φ + < φ < φ < φ, φ φ < ψ < φ φ 4 < φ < φ < φ, < ψ < φ φ Table : Chambers of the hyperplane arrangement I(A G, R) with R = (, 7)..8 Connection to the SU() SU()-model The III + I ns -model shares a lot of common behavior with the SU() SU()-model studied in [] such as its Euler characteristic and Hodge numbers. This can be understood from the fact that the III + I ns -model is a deformation of the III + IV s -model that preserves the rank of the gauge group and the sequence of blowups that resolve the SU() G -model also resolve the SU() SU()-model. Even the Coulomb chamber of the SU() G -model can be described as a Z -collapse of the Coulomb chamber of the SU() SU()-model. The crepant resolutions and fiber structures In this section, we study the fibral structure of the elliptic fibrations obtained by the crepant resolutions of the SU() G -model given by the Weierstrass model Y y = x + fst x + gs t. (.) The resolutions are given by the sequence of blowups listed in section.. We analyze the fiber structure of each of these crepant resolutions and determine the weights of the rational curves produced by the degeneration over codimension-two points. These weights are important to determine the representation R. We denote the fibral divisors over S and T by Da s and Da t respectively. Their generic fibers are respectively written as Ca s and Ca. t We will focus on analyzing the collision III+I ns as we already know the behavior of the III-model and the G -model. 6

18 . Resolution I The resolution I is given by the following sequence of blow-ups: (x, y, s e ) (x, y, t w ) (y, w w ) X X X X (.) The proper transform of the Weierstrass model is The projective coordinates are then given by Y w y = w (e x + fst x + gs t ). (.) [e w w x ; e w w y ; z = ][w w x ; w w y ; s][x ; w y ; t][y ; w ]. (.4) The fibral divisors are given by se = for type III and tw w = for type I ns : D s III s = w y w e x =, D s e = w y st w (fx + gst) =. I ns D t t = w y w e x =, D t w = w =, D t w = e x + fst x + gs t =. (.5) On the intersection of S and T, we see the following curves: On S T D s Dt η s = t = w y w e x =, D s Dt η e = t = w =, η A e = t = y =, D s Dt η e = w = w =, D s Dt η e = w = t =, η e = w = w =, η e = w = fx + gst =. (.6) Hence we can deduce that the five fibral divisors split in the following way to produce the fiber in Figure 4, which is a fiber of type IV with contracted nodes. On S T C s η, C s η + ηa + η + η, C t C t η, η + η + ηa, C t η + η + η. (.7) η η A η η η Figure 4: Codimension-two Collision of SU() G -model, Resolution I In order to get the weights of the curves, the intersection numbers are computed between the codimension two curves and the fibral divisors. 7

19 D s D s D t D t D t Weight Representation η - [-;,] (, ) - - [;-,] (, 7) - [;,-] (, 7) (, 4) - [;-,] (, 4) η - [;,-] (, 7) (, 4) η A η η Table 4: Weights and representations of the components of the generic curve over S T in the resolution I of the SU() G -model. See section.5 for more information on the interpretation of these representations. The fiber of Figure 4 specializes further when f = : η η. (.8) This corresponds to the non-kodaira diagram in Figure 5, which is a fiber of type III with contracted nodes. 4 η η A η η Figure 5: Codimension-three enhancement of SU() G -model at S T V (f), Resolution I. Resolution II In this section, we study the resolution II in detail. The resolution II requires making a first blowup that does not have a smooth center; it is useful to rewrite equation (.) as Y y = x + fp tx + gp t, p = st. (.9) The resolution II is given by the following sequence of blowups (x, y, p p ) (y, t, p w ) (t, p w ) X X X X. (.) Where X = P[O B L L ]. The projective coordinates are then and the proper transform is [p w x p w y z = ][x w y p w ][y tw p ][t p ], (.) w y = p x + fp tw Y x + gp tw, p p = st. (.) 8

20 X = Bl (x,y,p )X has double point singularities at the ideal (p, p, s, t). Recall that we have two curves from III and three curves from I individually. We denote by Ds a and D t a the fibral divisors that project to S and T : III I D s s = p = w y p x = D s s = p = w y (p x + p tw (fx + gp w ) = D t w = p p st = w y p x = D t t = p = w = D t w = p p st = p x + p tw (fx + gp w ) = (.) At the intersection of S and T, the fiber enhances to a non-kodaira fiber presented in Figure 6, which is a fiber of type IV with contracted nodes. This is realized by the following splitting of curves. C s η + η On S T C s C t C t η η + η + η + η η + η (.4) The curves at the intersection are given by C t η + η + η + η On S T D η s = p = w = w y p x =, D s Dt η s = p = w = p =, D s η s = p = w = w =, D s Dt η s = p = t = w =, D s Dt η s = p = w = fx + gp w =, η s = p = w = p =, η = w = t =, η = w = w =. (.5) D s D s D t D t D t Weight Representation η - - [-;-,] (, 7) η - - [-;,] (, 7) η [;,-] (, 7) η - [;-,] (, 4) η - [;,-] (, 7) (, 4) Table 5: Weights and representations of the components of the generic curve over S T in the resolution II of the SU() G -model. See section.5 for more information on the interpretation of these representations. 9

21 η η η η η Figure 6: Codimension-two Collision of SU() G -model, Resolution II There is an enhancement when f = : η η + η. (.6) For this codimension three enhancement, we get a Non-Kodaira fiber corresponding to Figure 7, which is a fiber of type III with contracted nodes. η 4 η η η η Figure 7: Codimension-three enhancement of SU() G -model at S T V (f), Resolution II. Resolution III Now consider the following order of the blow-ups: (x, y, t w ) (x, y, s e ) (y, w w ) X X X X (.7) The projective coordinates are then given by [e w w x ; e w w y ; z = ][e x ; e w y ; t][x ; w y ; s][y ; w ]. (.8) The proper transform is identical to equation (.). It follows that the divisors are also identical to equation (.5). On the intersection of S and T, we see the following curves: On S T D t Ds η s = t = w y w e x =, D t Ds η s = w = w =, D t Ds η s = w = e =, η = w =, D t Ds η e = w = w =, D t Ds η e = w = fx + gst =, η = w = w =, η e = w = s =. (.9)

22 Hence, we can deduce that the five fibral divisors split in the following way to produce the fiber in codimension-two, which is presented in Figure 8. we observe that this is a fiber of type IV with contracted nodes. On S T C s C s C t η, η + η + η, η + η + η, C t η + η, C t η + η + η + η. (.) η η η η η Figure 8: Codimension-two Collision of SU() G -model, Resolution III In order to get the weights of the curves, the intersection numbers are computed between the codimension two curves and the fibral divisors. D s D s D t D t D t Weight Representation η - [;,-] (, 7) (, 4) η - - [-;-,] (, 7) η - - [;-,] (, 7) η - [;,-] (, 7) (, 4) η - [;,-] (, 7) (, 4) Table 6: Weights and representations of the components of the generic curve over S T in the resolution III of the SU() G -model. See section.5 for more information on the interpretation of these representations. Consider when f = for its codimension-three enhancement. We can observe the following change in the curve η only: η η. (.) This corresponds to the codimension three enhancement in Figure 9, which is a fiber of type III with contracted nodes.

23 η η η η Figure 9: Codimension-three enhancement of SU() G -model at S T V (f), Resolution III..4 Resolution IV The last crepant resolution of the collision of types III+I is given by the following order of the three blowups: (x, y, t w ) (y, w w ) (x, y, s e ) X X X X. (.) Its projective coordinates are then given by [e w w x ; e w w y ; z = ][e x ; e w y ; t][e y ; w ][x ; y ; s]. (.) The proper transform is identical to equation (.). It follows that the divisors are also identical to equation (.5). On the intersection of S and T, we see the following curves: On S T D s Dt η s = t = w y w e x =, D s Dt η s = w = w =, D s Dt ηb s = w = x =, η s = w = e =, η s = w = w =, D s Dt η e = w = fx + gst =, η e = w = s =. (.4) Hence we can deduce that the five fibral divisors split in the following way to produce the fiber in Figure, which is a fiber of type IV with contracted nodes. On S T C s C s η + η, C t η, C t η, η + η + ηb + η, C t η + ηb + η + η. (.5)

24 η η η η B η Figure : Codimension-two Collision of SU() G -model at S T, Resolution IV In order to get the weights of the curves, the intersection numbers are computed between the codimension two curves and the fibral divisors. D s D s D t D t D t Weight Representation η - [;,-] (, 7) (, 4) η - [;-,] (, 4) η B - - [-;,-] (, 7) η - [;,] (, ) (, 7) η - [;,] (, ) (, 7) Table 7: Weights and representations of the components of the generic curve over S T in the resolution IV of the SU() G -model. See section.5 for more information on the interpretation of these representations. For the codimension three fiber enhancement, consider when f =. Note that only the fiber η changes under this condition: η η. Even though only a single curve changed, we get a completely different fiber as a result. The codimension three enhancement is represented in Figure, which is a fiber of type III with contracted nodes. η η η B η Figure : Codimension-three enhancement of SU() G -model at S T V (f), Resolution IV.5 Saturation of weights and representations We determine representations attached to an elliptic fibrations using intersection of vertical curves with fibral divisors to compute weights and then we use the notion of saturation of weights [,5,6]. There are some subtleties that we would like to discuss in this subsection. In the resolution I, the curve C s degenerates as follows over S T : C s η + ηa + η + η,

25 where η and η have weights [;, ], ηa has weight [;, ], η has weights [;, ]. undergoes the following degeneration over S T : In the resolution IV, the curve C s C s η + η, where both η and η have weight [;, ], which is a weight of the representation (, ). There is another curve (η B) coming from the degeneration of Cs that carries the weight [,, ] of the bifundamental (, 7) of SU() G. To make sense of the representation we should attach to these degenerations, we recall first few fact about the weights involved. The adjoint representation of G consists of fourteen weights: two zero weights, six short roots that form an orbit of the Weyl group; and six weights that form another orbit of the Weyl group. The short roots of the adjoint of G form the nonzero weights of the fundamental representation 7 of G. The bifundamental representation of SU() G consists of the following Weyl orbit: the two weights of the representation (, ), the orbit of the weight [;, ] and the six weights of the orbit of [ ;, ]. Thus: The saturation of the weight [;, ] corresponds to the fundamental representation (, 7). the saturation of the weight [;, ] corresponds to the bifundamental representation (, 7). The saturation of the weight [;, ] corresponds to the adjoint representation (, 4). The saturation of the weight [;, ] corresponds to the fundamental representation (, ). The saturation of the set {[;, ], [;, ]} corresponds to the adjoint representation (, 4). The saturation of the set {[ ;, ], [ ;, ]} corresponds to the bifundamental rep. (, 7). We see that while taking the saturation of individual weights we might think that we get fundamental weights over S T, when we take saturations of set of weights we always get the adjoint and the bifundamental representation. 4