Tall sections from non-minimal transformations

In previous work, we have shown that elliptic fibrations with two sections, or Mordell-Weil rank one, can always be mapped birationally to a Weierstrass model of a certain form, namely, the Jacobian of a ℙ112\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{P}}}^{112} $$\end{document} model. Most constructions of elliptically fibered Calabi-Yau manifolds with two sections have been carried out assuming that the image of this birational map was a “minimal” Weierstrass model. In this paper, we show that for some elliptically fibered Calabi-Yau manifolds with Mordell-Weil rank-one, the Jacobian of the ℙ112\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{P}}}^{112} $$\end{document} model is not minimal. Said another way, starting from a Calabi-Yau Weierstrass model, the total space must be blown up (thereby destroying the “Calabi-Yau” property) in order to embed the model into ℙ112\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{P}}}^{112} $$\end{document}. In particular, we show that the elliptic fibrations studied recently by Klevers and Taylor fall into this class of models.


Introduction
F-theory [1][2][3] has been a powerful tool for engineering a wide range of string vacua in diverse dimensions. While F-theory vacua that have already been found are extremely rich in the range of physical data -such as gauge groups or matter representations -that it can have, exploration of F-theory vacua are still well underway with expectations of more possibilities being in store.
A geometric F-theory compactification on an elliptically fibered Calabi-Yau (CY) manifold X over the base B can be thought of as a type IIB compactification on B with a varying axio-dilaton profile. The value of the axio-dilaton is given by the complex structure of the elliptic fiber over each point. This information is summarized by a Weierstrass equation of the elliptic fibration y 2 = x 3 + f x + g , (1.1) where x, y, f and g are sections of the bundles −2K, −3K, −4K and −6K, respectively. Here, K is the canonical bundle of the base manifold B.
Recently, there has been an interesting development in constructing F-theory models with abelian gauge symmetry with novel matter representations. In [4], the authors study a family of F-theory vacua, first discovered in [5], whose gauge algebra is given by u(1) and has matter with charge three-times of the minimal charge.
For X with complex dimension ≥ 3, in order for an F-theory vacua compactified on X to have gauge algebra u(1), the elliptic fibration must have two sections. In other words, the elliptic fibration must have Mordell-Weil (MW) rank-one. In [6], it was shown that an elliptically fibered manifold with two sections must be birationally equivalent to a Weierstrass model with a particular structure, that is that there must exist some sections b and c i of appropriate line bundles such that the coefficients of the Weierstrass model are given by: x ′ + c 0 c 2 3 − (This structure is obtained in [6] by first birationally embedding the elliptic fibration into a P 112 bundle over the base, and then computing the relative Jacobian of the resulting fibration.) What is interesting about the models investigated in [4], which we henceforth refer to as Klevers-Taylor (KT) models, is that their Weierstrass form deviates from this general form, i.e., there does not exist b and c i such that f and g can be written as (1.2) in general. 1 An easy way to see that this is the case is to compute the height of the generator of the MW groups of the elliptic fibrations X → P 2 in [4]. Some of the heights exceed the upper bound on the height that the generating section of a Calabi-Yau model, whose Weierstrass equation takes the form (1.2), must satisfy. We thus arrive at an apparent contradiction. The resolution of the problem is that the models of [4] are birationally equivalent to Weierstrass models of the form (1.2) by a non-minimal transformation, i.e., a map that does not preserve the canonical class of the manifold. 2 More concretely, assuming that the Weierstrass form of a Klevers-Taylor model is given by equation (1.1), we can find a section a ∈ O(D) for some effective line bundle D such that the Weierstrass model can be put into the form (1.2), i.e., there exist b and c i such that Note that this elliptic fibration is no longer Calabi-Yau, since f a 4 ∈ O(−4K + 4D) and ga 6 ∈ O(−6K + 6D). Thus we find that new classes of CY manifolds with MW rank-one, such as the KT models, may be generated via non-minimal maps from non-CY manifolds. Any such model can be obtained by the following steps in principle: 1. Begin with a non-CY Weierstrass model (1.2) where x ∈ O(−2K + 2D) and y ∈ O(−3K + 3D), or with its "parent" non-CY model in P 112 , which uses the same sections b and c i as coefficients.
2 For another occurance of non-minimal Weierstrass fibrations in F-theory, see [17]. 3 It is interesting to note that over the points of the base B other than those on the divisor a = 0, the elliptic fibers of the Weierstrass models (1.1) and (1.2) have the same complex structure. To put it differently, away from the codimension-one locus a = 0, the j-functions of the axio-dilaton of the models (1.1) and (1.2) have the same value.

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Note that this last step is part of the usual procedure of obtaining a "minimal" Weierstrass model from a "non-minimal" one, and involves a blowdown in the total space (see sections II.3 and III.3 in [18], eq. (15) in [19], as well as eq. (2.6) and footnote 10 in [20]). The blowdown restores the Calabi-Yau property, since the holomorphic n-form vanishes on the divisor which is being blown down.
While the possibility of constructing more general classes of MW rank-one CY manifolds in this manner was hinted in [6], the KT models are, to our knowledge, the first examples where this scenario is realized. It would be interesting to understand how to systematically implement these steps to find new CY fibrations with MW rank-one.
Arguably the most interesting aspect of models obtained this way is that their sections are "taller," whose meaning we define in section 2, than those that are birationally equivalent to (1.2) by a minimal map. A generating section being tall has a direct physical consequence when the elliptic fibration is a CY threefold -the taller the generating section of the Mordell-Weil group, the bigger the maximum charge of the matter under the corresponding u(1). As mentioned, the Klevers-Taylor models have matter with u(1) charge 3, while the matter in models constructed in [6] all have charge ≤ 2.
This paper is organized as follows. In section 2, we review some background material. We first review basic bounds on the height of the rational section of a Weierstrass model of the form (1.2). We then show how such bounds get looser by allowing the model (1.2) to be non-CY, and discuss the property of CY models obtained by a non-minimal map. We also define a convenient metric to compare heights of rational sections, and discuss its physical relevance. We present our main result in section 3 by explicitly identifying b, c i and a by which the Weierstrass coefficients of the KT models can be constructed via equation (1.4). We conclude with some remarks in section 4. There are several unwieldy expressions that are collected in the appendix.

Background
Let us review some salient points about elliptic fibrations with two sections. It was shown in [6] that any elliptically fibration over a base B with MW rank-one, is birationally equivalent to the Weierstrass model X ′ (1.2) whose generating section is given by for some b and c i . We have introduced the projective coordinates X ′ , Y ′ and Z ′ for which

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to make the equations less cluttered. The entries for X ′ , Y ′ and Z ′ of (2.1) are assumed to be mutually prime in the sense that there does not exist p such that Given that x ′ and y ′ are sections of O(−2K + 2D) and O(−3K + 3D) for some effective D, we find that where we use brackets [b] to denote the line bundle of which b is a section. We find that where in discussing line bundles or divisors, For the purpose of this paper, we further assume that for some x and y. 4 By the birational transformation we arrive at a Calabi-Yau fibration (1.1) with MW rank-one with generating rational section The transformation (2.8) is non-minimal when D is non-trivial, i.e., D > 0. 4 It would be interesting to understand the consequences of relaxing this condition.

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The height h(S) of a rational section S is a divisor in the base manifold B obtained by first taking the image σ(S) of S under the Shioda map [21][22][23] and projecting its selfintersection down to the base B [6,23]: (2.10) When the base B is a surface, and there are no additional gauge group factors, i.e., there are no codimension-one singularities beyond type I 1 , h(S) is given by [6] h We thus see that h(S) is bounded by Note that in the examples studied in [6], it was assumed that the elliptic fibration (1.  .7) for some a, f , g, x and y.
The height of the generating section S has the physical interpretation as the corresponding u(1) anomaly coefficient when B is a surface [6,23]. 5 The anomaly cancellation conditions then imply that where I labels the hypermultiplets charged under the u(1) algebra. When the gauge algebra is given solely by u(1), the charges in (2.13) are integrally quantized. Inspired by the anomaly equations, we define the additional measure of tallness associated to a section S. While the height h(S) is only partially ordered with respect to the relation defined in (2.6), t(S) is ordered since it is a number. We say that a section S 1 is taller than S 2 when t(S 1 ) > t(S 2 ) .

Klevers-Taylor models
In this section, we find a, b and c i that relate to the f and g of the Klevers-Taylor models (1.1)

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for a i that can be explicitly written in terms of s i . The explicit formulae for a i and b are given in appendix A. Since the section (3.1) should be given by equation (2.9), we have the additional conditions which are key in obtaining the desired a, b and c i .
Before going further, we make a few assumptions, that turn out to be true for the Klevers-Taylor models. We first assume that a is irreducible, and that a does not divide b. We also assume that c 3 is not divisible by a. If so, it follows that a 2 |c 2 , a 3 |c 1 and a 4 |c 0 . We do not wish to consider this trivial case. We need to make one more technical assumption, which we explain shortly.
Let us start with the first equation of (3.2). We can arrange Without loss of generality, we find that for some κ, where we have assumed that b and a do not have a common divisor. We then arrive at Meanwhile, Now the right-hand-side of this equation must be divisible by a. Here we make the assumption that each of the two terms on the right-hand-side of this equation are divisible by a. This implies that for some k. Also, a must divide a 3 d. a being irreducible, it must either divide a 3 or d.
When a divides d, c 3 is divisible by a due to equation (3.6), contrary to our assumption. It thus follows that a 3 is divisible by a.

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In the KT models, a 3 is irreducible for generic complex structure. In fact, a 3 is the SU(2) divisor at the point of enhanced gauge symmetry [4]. Hence, a 3 and a must agree up to a constant factor. This constant can be absorbed trivially into the definition of the birational map (2.8). We can therefore identify a 3 with a. Plugging this into the above equations, we arrive at While we have explicit expressions for a i and b, the equations written above alone do not determine k and d uniquely. Note that once d is determined, so is k.
Upon plugging what we have into the second equation of (3.2), we arrive at It follows that there exists some M such that Now using the explicit expressions of the coefficients a i of the KT model, we find that for a polynomial L of s i . Using the expression for a 2 and a 3 , and equation (3.12), we find that M = L + 3 4 bk 2 . (3.14) Plugging this into the first equation of (3.12), we arrive at By the first equation in (3.9), and the fact that which is another convenient miracle, c 1 is guaranteed to be well-defined. The non-trivial condition on k and d comes from the fact that c 1 , c 2 and c 3 as given by equations (3.9) and (3.15), must be such that c 0 , as can be computed from the first equation of equation (1.4), is a polynomial of the sections s i . That is,

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where the expression for the coefficient f for the KT models is written out explicitly in the appendix of [4]. For example, equation (3.17) is not satisfied by the solution (k, d) = (0, −a 2 /3) of the first equation of (3.9). We can find k and d with a 2 = − 3 2 (bk + 2d) (3.18) such that d has the minimal number of terms as a polynomial of the sections s i . d and k are given by By the way we have obtained c i , it is not entirely clear that the second equation of (1.4) should hold, but it does. Now (3.19) and (3.20) are not the only polynomials of s i such that (3.19) has the minimal number of terms. There are, in fact nine such pairs of (k, d) that are part of a two-parameter family of solutions of (3. (3. 22) In the solution given by (3.19) and (3.20), ℓ 1 and ℓ 2 are set to

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To summarize, we have found that the Klevers-Taylor models (1.1) are birationally equivalent to the models (1.2) by the map (2.8). The x ′ and y ′ of the Weierstrass model X ′ given by (1.2)  c 0 and c 1 are given by equations (3.24): Explicit

Comments and future directions
In this paper, we have shown that the Klevers-Taylor models can be obtained via nonminimal transformations from non-Calabi-Yau manifolds of the form (1.2) constructed in [6]. We conclude by commenting on our results and speculating on future directions to pursue.
Non-genericity of polynomials. It is crucial that the polynomials a i and b of s i involved in constructing the KT models satisfy intricate relations. For example, the relation (3.13) played an important role in discovering the birational map from (1.2) to the KT models. Equation (3.13) is possible because the ring of functions on a 3 is not a universal factorization domain (UFD), as pointed out in [4]. 6 For this reason, we expect that 6 The fact that the functions on a3 is not a UFD does not come as a surprise. After all, a3 becomes the su(2) locus as the u(1) gauge symmetry is un-Higgsed by taking b → 0. The charge-three hypermulitplets then organize themselves as part of the spin-3/2 representation of su (2). If a3 had been a generic smooth curve, such exotic representations of su(2) cannot appear. See [4] and [25] for more discussions.

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constructing u(1) theories exotic matter following the steps laid out in the introduction to be a very hard in general.
Generalization of KT models. It would be interesting to generalize the KT models in a meaningful way. Now the fact that the rational section of the KT models had the form (3.1) was important in finding the rational map between them and the the Weierstrass models of the form (1.2). A systematic understanding and generalization of the form (3.1) would be desirable. Do the KT models saturate all Weierstrass models with sections of the form (3.1)? Is there a nice generalization of the form (3.1) which would lead to a new class of models? Understanding the answer to these questions would be imperative to either constructing new F-theory models with abelian gauge symmetry, or explaining why those constructions should not be possible.
Bounds on u(1) charges and anomaly coefficients. An important physical question is what kind of theories are constructible in string theory. It would be desirable to gain a better understanding which values of physical observables such as abelian charges and anomaly coefficients of u(1) gauge groups are allowed in F-theory models. A good first step would be to understand the constraints on the line bundle D = [a] defined in the introduction in order for the Calabi-Yau fibration obtained by the birational map (2.8) to have a smooth resolution.
Non-enhanceable u(1)s. As noted in [26], the u(1) gauge symmetry for any F-theory model with the Weierstrass form (1.2) can be enhanced to a non-abelian gauge symmetry by taking b → 0, unless a codimension-one (4, 6, 12) singularity is induced by doing so. There exist many elliptically fibered CY threefolds that give rise to u(1)s that are non-enhanceable in the sense that they cannot be enhanced without inducing a codimension-one (4, 6, 12) singularity [27]. The possibility of constructing Calabi-Yau manifolds by non-minimal transformations suggests that there might exist models with u(1) gauge symmetry that are non-enhanceable for a different reason -because they simply do not have moduli that can be tuned to enhance the u(1) into anything else.
Let us explain this point. Suppose there exists some Weierstrass model (1.2) that can be mapped to a Calabi-Yau manifold by the minimal transition (2.8). In order for such a map to be possible, i.e., in order for the equations (1.4) to hold for some f and g, the b and c i of equation (1.2) must be tuned. It is imaginable that perhaps for some D, b may be completely fixed in order for there to exist a section a ∈ O(D) such that If this were the case, it would be impossible to enhance the u(1) gauge symmetry further as b cannot be taken to zero. It would be extremely interesting to see if this scenario can be realized.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.