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...ta of six-dimensional F-theory backgrounds can be found in [15, 47]. More information on elliptic curves and the Mordell-Weil group can be found in standard introductory texts on the subject, such as =-=[48]-=-. Six-dimensional F-theory compactifications are defined for elliptically fibered CalabiYau threefolds with a section. Let us denote such a smooth elliptic fibration by X̂ and its base manifold by B. ...

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...-Weil group to the homology group H4. In other words, the group action — or addition — of sections carry over to into addition of the homology class of the corresponding sections under the Shioda map =-=[45,46]-=-. The dot product is the intersection product in the manifold X. π(C) is defined as the projection of a curve C to the base B. At the operational level, if we denote by Bα the pullback of the generato...

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... particular, the gravitational anomaly coefficient a corresponds to the canonical divisor while the coefficient bκ corresponds to the degeneration locus of the fiber that yields the Gκ gauge symmetry =-=[9, 10,13,14,44]-=-. The abelian anomaly coefficients can be obtained in the following way. Each abelian gauge field corresponds to a generator of the Mordell-Weil group of the elliptic fibration. The Mordell-Weil group...

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...take some first steps towards improving the current status. The abelian sector of six-dimensional F-theory backgrounds — obtained by compactification on an elliptically fibered Calabi-Yau threefold X =-=[13, 14]-=- — contains information about the Mordell-Weil group of the threefold X. In particular, the rank of the abelian gauge group is equal to the Mordell-Weil rank [14], while the anomaly coefficient matrix...

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...w w2 ❆ ❆ ❆ ❆ ❆ t t t t t ❞ Du Dv E3 Dw E1 E2 ❅ ❅ t t t t t t t ❞ Figure 5: Toric data for ambient spaces of some elliptic curve embeddings. These are among the 16 reflexive toric surfaces =-=[66]-=- (see also [67]). Since the variables z, x, and y have weights 1, 2, and 3, this can be regarded as a hypersurface in the weighted projective space P(1,2,3), which as a toric variety is illustrated in...

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... the theory. 3We note that the Mordell-Weil group has been studied in various contexts in string theory. The MordellWeil group of elliptically fibered surfaces has been studied using string junctions =-=[16, 17]-=- in [18, 19]. The torsion subgroup of the Mordell-Weil group has been studied for elliptically fibered threefolds in [20]. It is also possible to study the Mordell-Weil group of T 4 fibered manifolds ...

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...orm is taken with respect to an SO(1, T ) metric Ωαβ. λκ is the Dynkin index of the fundamental representation of Gκ. The explicit form of the anomaly equations have been written out, for example, in =-=[10,12,15,42,43]-=- and we do not reproduce them here. The anomaly coefficients, along with the modulus j determine important terms in the low-energy effective Lagrangian. The modulus j is a unit SO(1, T ) vector parame...

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... the theory. 3We note that the Mordell-Weil group has been studied in various contexts in string theory. The MordellWeil group of elliptically fibered surfaces has been studied using string junctions =-=[16, 17]-=- in [18, 19]. The torsion subgroup of the Mordell-Weil group has been studied for elliptically fibered threefolds in [20]. It is also possible to study the Mordell-Weil group of T 4 fibered manifolds ...

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... with projective coordinates (a, b, c, x, y). The a, b and c coordinates are the projective coordinates of the base manifold. We can resolve this manifold into a smooth degree 12 hypersurface X̂0 in P=-=[1, 1, 1, 3, 6]-=-. We denote the projective coordinates of this manifold by (a, b, c, v, w).13 Then, the birational map from X0 to X̂0 is given by v = y + f9 2(x− f6) , w = 1 2 (x+ f6 2 )− v2 . (5.7) We may rewrite (5...

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...solve these 108 points by blowing up a codimension-two locus to a single divisor in the ambient space, thereby recovering the smooth manifold X̂0. This transition can be described by a birational map =-=[14,59,60]-=-. In order to explain this birational map, it is useful to representX0 and X̂0 as hypersurfaces in projective varieties. X0 — represented by the equation (5.3) — can be thought of as a singular degree...

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...ions and its relation to the abelian sector of F-theory backgrounds. A more thorough description of the process of extracting the physical data of six-dimensional F-theory backgrounds can be found in =-=[15, 47]-=-. More information on elliptic curves and the Mordell-Weil group can be found in standard introductory texts on the subject, such as [48]. Six-dimensional F-theory compactifications are defined for el...

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...solve these 108 points by blowing up a codimension-two locus to a single divisor in the ambient space, thereby recovering the smooth manifold X̂0. This transition can be described by a birational map =-=[14,59,60]-=-. In order to explain this birational map, it is useful to representX0 and X̂0 as hypersurfaces in projective varieties. X0 — represented by the equation (5.3) — can be thought of as a singular degree...

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...nd break various U(1)’s in F-theory model building is essential to constructing models with desired properties.1 Meanwhile, from the point of view of addressing the question of 6D string universality =-=[7]-=-, a systematic understanding of what one could get in string theory — especially F-theory — is crucial. Such an understanding of the abelian sector of F-theory has yet to be gained. In this note, we a...

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...orm is taken with respect to an SO(1, T ) metric Ωαβ. λκ is the Dynkin index of the fundamental representation of Gκ. The explicit form of the anomaly equations have been written out, for example, in =-=[10,12,15,42,43]-=- and we do not reproduce them here. The anomaly coefficients, along with the modulus j determine important terms in the low-energy effective Lagrangian. The modulus j is a unit SO(1, T ) vector parame...

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...sections: z5, xz3, yz2, x2z, and xy. H0(6L) should only have six sections, but we know about seven: z6, xz4, yz3, x2z2, xyz, x3, y2. Thus, there must be a relation, and one argues — following Deligne =-=[65]-=- — that the coefficients of x3 and y2 must be units in the ring R and after an appropriate scaling, we get a Weierstrass equation of the form y2 + a1xyz + a3yz 3 = x3 + a2x 2z2 + a4xz 4 + a6z 6 . (B.1...

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...bound therefore restricts the representations allowed in F-theory models by bounding bκ. Such restrictions placed by the Kodaira bound have been explicitly demonstrated in the case of T = 0 models in =-=[11]-=-. Meanwhile, an analogous constraint on the abelian anomaly coefficient has yet to be found. Such a constraint would also restrict the allowed charge of matter in F-theory models. For example, by anom...

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...uge group is equal to the Mordell-Weil rank [14], while the anomaly coefficient matrix of the abelian gauge fields turns out to be the Néron-Tate height pairing matrix of the Mordell-Weil generators =-=[15]-=-. Therefore, in order to understand the abelian sector of supersymmetric F-theory backgrounds, one must study the Mordell-Weil group of elliptically fibered Calabi-Yau manifolds.3 The Mordell-Weil gro...

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...p of elliptically fibered surfaces has been studied using string junctions [16, 17] in [18, 19]. The torsion subgroup of the Mordell-Weil group has been studied for elliptically fibered threefolds in =-=[20]-=-. It is also possible to study the Mordell-Weil group of T 4 fibered manifolds — this has been done for certain T 4 fibered Calabi-Yau threefolds in [21]. 4The Mordell-Weil group of elliptically fiber...

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...· · , ŝVA}, the abelian anomaly coefficients are given by bij = −π(σ(ŝi) · σ(ŝj)) . (2.5) Some explanation of notation is due. σ is a map from the Mordell-Weil group to H4(X̂) defined by Shioda in =-=[45]-=-, which we refer to as the Shioda map. We explain this map in more detail in the subsequent section, but mention here that it is a homomorphism from the Mordell-Weil group to the homology group H4. In...

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...t, we frequently denote a rational section by a hatted lower-case roman letter and its homology class by the corresponding hatted upper-case letter throughout this note. The Shioda-Tate-Wazir theorem =-=[49]-=- states that {Ŝ1, · · · , ŜVA} along with the zero section Z, the vertical divisors Bα and the “fibral divisors” Tκ,I generate the homology group H4(X̂). The fibral divisors of X̂ are topologically ...

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...ng to see if one could find all such SU(2) theories at least in the case when T = 0. There is reason to be optimistic about this goal, given recent developments on the space of T = 0 theories such as =-=[11,61]-=-. A question that follows is whether there exist U(1) models in F-theory that cannot be enhanced to SU(2). We are not aware of any reason to believe that such models do not exist. If such models exist...

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... ❆ t t t t t ❞ Du Dv E3 Dw E1 E2 ❅ ❅ t t t t t t t ❞ Figure 5: Toric data for ambient spaces of some elliptic curve embeddings. These are among the 16 reflexive toric surfaces [66] (see also =-=[67]-=-). Since the variables z, x, and y have weights 1, 2, and 3, this can be regarded as a hypersurface in the weighted projective space P(1,2,3), which as a toric variety is illustrated in the first row ...

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... 3We note that the Mordell-Weil group has been studied in various contexts in string theory. The MordellWeil group of elliptically fibered surfaces has been studied using string junctions [16, 17] in =-=[18, 19]-=-. The torsion subgroup of the Mordell-Weil group has been studied for elliptically fibered threefolds in [20]. It is also possible to study the Mordell-Weil group of T 4 fibered manifolds — this has b...

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...orm is taken with respect to an SO(1, T ) metric Ωαβ. λκ is the Dynkin index of the fundamental representation of Gκ. The explicit form of the anomaly equations have been written out, for example, in =-=[10,12,15,42,43]-=- and we do not reproduce them here. The anomaly coefficients, along with the modulus j determine important terms in the low-energy effective Lagrangian. The modulus j is a unit SO(1, T ) vector parame...

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... 3We note that the Mordell-Weil group has been studied in various contexts in string theory. The MordellWeil group of elliptically fibered surfaces has been studied using string junctions [16, 17] in =-=[18, 19]-=-. The torsion subgroup of the Mordell-Weil group has been studied for elliptically fibered threefolds in [20]. It is also possible to study the Mordell-Weil group of T 4 fibered manifolds — this has b...