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...the weights by Li, i = 1 · · ·n, and we have shown that the u(n) theory has phases given by the signs (+ · · ·+) , (+ · · ·+−) , · · · , (+− · · ·−) , (− · · ·−) . (3.2) 7Our conventions are those of =-=[40]-=-. 23 III III IV Figure 4: Phases of the su(5) theory with fundamental representation 5 matter. The blue/yellow boxes correspond to decorating with ±. The green lines and red dots will play a role late...

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...ny cases, the network of phases or, equivalently, small resolutions form a so-called minuscule 1Dyck paths are staircase paths on a (representation) graph, which are not allowed to cross the diagonal =-=[34]-=-. 7 representation of g̃, where the representation structure is exactly given by the flop transitions. I.e. not only is the structure of the fibers determined in terms of representation data, but also...

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...ity which corresponds to the phase Φρ+···+, the length of the element w increases by one when one performs a Weyl reflection with respect to a root corresponding to the weights of R. 6See for example =-=[37]-=- for some more details on Bruhat order and related matters. 13 35 4 6 7 8 9 10 11 1213 2 1 14 16 15 Figure 1: Example for Bruhat ordering of the phases or equivalently Weyl group quotients, for the u(...

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...is case. A minuscule representation of a Lie algebra is defined as an irreducible representation with the property that the Weyl group acts transitively on all weights occurring in the representation =-=[38]-=-. For all the simply-laced Lie algebras these are listed in table 1, where the $i are the fundamental weights 〈α∨i , ωj〉 = δij . (2.19) Furthermore, a quasi-minuscule representation is one such that t...

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...as a chance of being an effective curve, since 〈φ, [C]〉 = − ∫ C φ < 0 (7.3) for an effective curve C (i.e., the pairing gives the negative of the area). Conversely, deep results of Kleiman [45], Mori =-=[46]-=-, and others tell us that on Calabi–Yau varieties of low dimension, classes whose area is positive will be effective classes (up to a rational multiple). We will focus on curves C which have nonzero i...

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...aces associates to each singular fiber a decorated affine Dynkin diagram corresponding to a simple Lie algebra g, where the decoration indicates the multiplicities of the irreducible fiber components =-=[1, 2]-=-. When the nonsingular elliptic surface is the resolution of a (singular) Weierstrass model, the Dynkin diagram can be associated with the singularity. For higher-dimensional elliptically fibered geom...

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...ination of the nonabelian part of the (geometric19) gauge algebra of the corresponding F-theory model. The singularities are enhanced in codimension two, which we discuss following [23, 25] (see also =-=[48, 49, 26, 50]-=-). Let Σα ⊂ B be an irreducible subvariety of codimension two along which some enhancement occurs: necessarily, Σα ⊂ ∆. We make a local model for the 18Recently, a generalization to genus-one fibratio...

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...whether c has a chance of being an effective curve, since 〈φ, [C]〉 = − ∫ C φ < 0 (7.3) for an effective curve C (i.e., the pairing gives the negative of the area). Conversely, deep results of Kleiman =-=[45]-=-, Mori [46], and others tell us that on Calabi–Yau varieties of low dimension, classes whose area is positive will be effective classes (up to a rational multiple). We will focus on curves C which hav...

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...e in the Coulomb branch [18, 19]. A crepant resolution of the Calabi-Yau variety then corresponds to a Coulomb phase of the three-dimensional theory. The study of this correspondence was initiated in =-=[9,11]-=- in the case of Calabi-Yau threefolds, and further pursued in the case of Calabi-Yau fourfolds in [12,20–22]. More concretely, any crepant resolution will resolve the codimension-one Kodaira fibers an...

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...e in the Coulomb branch [18, 19]. A crepant resolution of the Calabi-Yau variety then corresponds to a Coulomb phase of the three-dimensional theory. The study of this correspondence was initiated in =-=[9,11]-=- in the case of Calabi-Yau threefolds, and further pursued in the case of Calabi-Yau fourfolds in [12,20–22]. More concretely, any crepant resolution will resolve the codimension-one Kodaira fibers an...

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...roups and their representations Let G be a compact Lie group. It is known that finite-dimensional complex representations of such a group are always the direct sum of irreducible representations (see =-=[70]-=-, for example). For simplicity, we assume that G is connected. The complex representations of G can be analyzed by means of a Cartan subgroup H ⊂ G, which is a maximal torus contained in G; H is itsel...

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... upshot is a determination of the nonabelian part of the (geometric19) gauge algebra of the corresponding F-theory model. The singularities are enhanced in codimension two, which we discuss following =-=[23, 25]-=- (see also [48, 49, 26, 50]). Let Σα ⊂ B be an irreducible subvariety of codimension two along which some enhancement occurs: necessarily, Σα ⊂ ∆. We make a local model for the 18Recently, a generaliz...

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...urfold with fiber type G in codimension one realizes the Coulomb branch with gauge group broken to U(1)r, where r is the rank of G; inclusion of matter introduces a substructure in the Coulomb branch =-=[18, 19]-=-. A crepant resolution of the Calabi-Yau variety then corresponds to a Coulomb phase of the three-dimensional theory. The study of this correspondence was initiated in [9,11] in the case of Calabi-Yau...

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...aces associates to each singular fiber a decorated affine Dynkin diagram corresponding to a simple Lie algebra g, where the decoration indicates the multiplicities of the irreducible fiber components =-=[1, 2]-=-. When the nonsingular elliptic surface is the resolution of a (singular) Weierstrass model, the Dynkin diagram can be associated with the singularity. For higher-dimensional elliptically fibered geom...

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...tation for the representation theory of the exceptional compact simple Lie groups E6, E7, and E8. We use the simply-connected form of each of these. For En, n = 6, 7, 8, we follow the presentation of =-=[71]-=-. We begin with the vector space spanned by n + 1 vectors L0, L1, . . . , Ln as well as the dual space spanned by e0, e1, . . . , en. The root space for En will be Ker(3e0− ∑n j=1 ej). In particular, ...

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...to have a standard type II∗ fiber in codimension two. 23There are more examples involving non-simply-laced Lie algebras, which we will not consider here, as well as higher rank ADE examples. See e.g. =-=[55]-=-. 61 8.2 Fibers of E6 type with monodromy An example of phases with non-trivial monodromy was discussed in section 6.2, for e6 → su(6) ⊕ su(2) and non-trivial Γ = Z2. In this case, the fibers are not ...

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...n particular in cases with monodromy. We now give the geometric setup and an example for realizing both of these aspects. Elliptic Calabi–Yau varieties with multiple sections were studied recently in =-=[58]-=- 27 We shall restrict ourselves to SU(5) models with one extra rational section. In [68] the Tate forms for SU(5) were obtained for SU(5)×U(1) models realized in P112, or more precisely in the blowup ...

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...ination of the nonabelian part of the (geometric19) gauge algebra of the corresponding F-theory model. The singularities are enhanced in codimension two, which we discuss following [23, 25] (see also =-=[48, 49, 26, 50]-=-). Let Σα ⊂ B be an irreducible subvariety of codimension two along which some enhancement occurs: necessarily, Σα ⊂ ∆. We make a local model for the 18Recently, a generalization to genus-one fibratio...

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...dition ensures the absence of tensionless strings in the associated M-theory compactification by ensuring that all fibers are one-dimensional; it is well-understood for elliptic Calabi–Yau threefolds =-=[51]-=- but not for elliptic Calabi–Yau fourfolds [52]. 58 are obtained from X and X using some embedding {|u| < } → U . Moreover, the classes of algebraic curves on X are generated by the Γ-invariant subla...

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...elliptic Calabi-Yau varieties, where it has been known that the fibers in higher codimension need not belong to Kodaira’s list [26, 27, 24, 28] (for earlier examples illustrating a related issue, see =-=[29]-=-). Detailed studies of such resolutions for Calabi-Yau fourfolds, mostly focusing on an SU(5) gauge group, i.e., I5 Kodaira fiber in codimension one, appeared in [27, 24, 28] using algebraic methods, ...

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...urfold with fiber type G in codimension one realizes the Coulomb branch with gauge group broken to U(1)r, where r is the rank of G; inclusion of matter introduces a substructure in the Coulomb branch =-=[18, 19]-=-. A crepant resolution of the Calabi-Yau variety then corresponds to a Coulomb phase of the three-dimensional theory. The study of this correspondence was initiated in [9,11] in the case of Calabi-Yau...

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...(x, y, z; ζ1) (x, y, ζ1; ζ2) (y, z; δ0) (y, ζ1; δ1) , (9.4) where the simple roots are associated to the divisors as follows 28 (α0, α1, α2, α3, α4)↔ (z, ζ1, ζ2, δ1, δ0) . (9.5) The notation is as in =-=[28]-=-, i.e., (x1, x2, x3; ζ) stands for the resolution xi → xiζ and [x1, x2, x3] are projective coordinates of the blowup P2. This resolution realizes the phase I in table 28Note that the weights/roots ass...

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... group of an elliptic fibration So far we studied the Coulomb phases of three-dimensional N = 2 gauge theories. We now move on to the corresponding geometric analysis. The basic setup closely follows =-=[11,42]-=-. The Lie group associated to an elliptic fibration pi : X → B is determined via the compactification of F-theory to M-theory. Let X be a resolution of singularities of the total space of X with trivi...

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...ious observations that require further investigation. In the mathematics literature, considerations of Weyl group actions as flops in the context of the Minimal Model program have appeared in Matsuki =-=[35]-=-. The main difference with the present work is in that we do not restrict our attention to normal crossing singularities and address global issues of the resolution. Furthermore, our main object of st...

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...hose class gives the corresponding root). Thus, any divisors on the coweight lattice beyond the coroots must arise from elements of the Mordell-Weil group, i.e., additional rational sections of X → B =-=[56]-=-. In particular, we can determine from global properties of the model whether or not there is a u(1) factor in the gauge group, and if not, there are relations among the local divisors near the codime...

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...ssibility of gauge groups whose root systems are not simply-laced) is spelled out in detail for the classical groups in [11] (with some further explanation in [42]), and for the exceptional groups in =-=[44]-=-. 7.2 Representation associated to an elliptic fibration The representations given by other curves can be worked out as well. One thing that is important to remember is that the total representation i...

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...restrict ourselves to SU(5) models with one extra rational section. In [68] the Tate forms for SU(5) were obtained for SU(5)×U(1) models realized in P112, or more precisely in the blowup Bl[0,1,0]P112=-=[4]-=-. The singularities along z = 0 in the base can be characterized in terms the equation Q(i1, i2, i3, i4, i5, i6, i7) : sy2 + b0,i5zi5yx2 + b1,i6zi6sywx+ b2,i7zi7s2yw2 = c0,i1z i1s3w4 + c1,i2z i2s2w3x+...

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...0, 0) to (n/2, n/2), which are staircase paths that do not cross the diagonal, but are allowed to touch it. Two examples of Dyck paths are shown in figure 14. These are counted by the Catalan numbers =-=[41]-=- #Dyck paths from (0, 0) to (k, k) = Ck = (2k)! k! (k + 1)! . (5.6) We can now prove that bn = ( n− 1 n 2 − 1 ) . (5.7) The induction starting point is b2 = 1. The induction step is 4bn − an,1 = 4 ( n...

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...reducible subvariety of codimension two along which some enhancement occurs: necessarily, Σα ⊂ ∆. We make a local model for the 18Recently, a generalization to genus-one fibrations has been discussed =-=[47]-=-, but we have no need of that generalization here. 19For F-theory models in four dimensions, part of this “geometric” gauge algebra may be lifted by a superpotential, resulting in the actual gauge alg...

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...ination of the nonabelian part of the (geometric19) gauge algebra of the corresponding F-theory model. The singularities are enhanced in codimension two, which we discuss following [23, 25] (see also =-=[48, 49, 26, 50]-=-). Let Σα ⊂ B be an irreducible subvariety of codimension two along which some enhancement occurs: necessarily, Σα ⊂ ∆. We make a local model for the 18Recently, a generalization to genus-one fibratio...

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...gs in the associated M-theory compactification by ensuring that all fibers are one-dimensional; it is well-understood for elliptic Calabi–Yau threefolds [51] but not for elliptic Calabi–Yau fourfolds =-=[52]-=-. 58 are obtained from X and X using some embedding {|u| < } → U . Moreover, the classes of algebraic curves on X are generated by the Γ-invariant sublattice of Φ̃. For each curve in the central fibe...

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...ata (i.e., the roots and coroots) are determined by considering which curves C move in families that sweep out divisors D. For such a curve, by Witten’s analysis of the quantization of wrapped branes =-=[5]-=- (see also [43]), the spectrum contains a massive vector with the same gauge charges as the curve. In the limit where this curve has zero area, the vector becomes massless and we get nonabelian gauge ...

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...e the $i are the fundamental weights 〈α∨i , ωj〉 = δij . (2.19) Furthermore, a quasi-minuscule representation is one such that the Weyl group acts transitively on the nonzero weights (see for instance =-=[39]-=-). In fact there is a unique quasi-minuscule representation for the simply-laced Lie algebras, which has as highest weight the unique dominant root. The zero weights of this representation are one-to-...