Abelian F-theory models with charge-3 and charge-4 matter

Journal of High Energy Physics, May 2018

Abstract This paper analyzes U(1) F-theory models admitting matter with charges q = 3 and 4. First, we systematically derive a q = 3 construction that generalizes the previous q = 3 examples. We argue that U(1) symmetries can be tuned through a procedure reminiscent of the SU(N ) and Sp(N ) tuning process. For models with q = 3 matter, the components of the generating section vanish to orders higher than 1 at the charge-3 matter loci. As a result, the Weierstrass models can contain non-UFD structure and thereby deviate from the standard Morrison-Park form. Techniques used to tune SU(N ) models on singular divisors allow us to determine the non-UFD structures and derive the q = 3 tuning from scratch. We also obtain a class of a q=4 models by deforming a prior U(1) × U(1) construction. To the author’s knowledge, this is the first published F-theory example with charge-4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:


Abelian F-theory models with charge-3 and charge-4 matter

JHE Abelian F-theory models with charge-3 and charge-4 Nikhil Raghuram 0 1 0 77 Massachusetts Avenue , Cambridge, MA 02139 , U.S.A 1 Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology , USA This paper analyzes U(1) F-theory models admitting matter with charges q = 3 and 4. First, we systematically derive a q = 3 construction that generalizes the previous We argue that U(1) symmetries can be tuned through a procedure reminiscent of the SU(N ) and Sp(N ) tuning process. For models with q = 3 matter, the components of the generating section vanish to orders higher than 1 at the charge3 matter loci. As a result, the Weierstrass models can contain non-UFD structure and thereby deviate from the standard Morrison-Park form. Techniques used to tune SU(N ) models on singular divisors allow us to determine the non-UFD structures and derive the q = 3 tuning from scratch. We also obtain a class of a q = 4 models by deforming a prior U(1) construction. To the author's knowledge, this is the rst published F-theory example with charge-4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4. F-Theory; Superstring Vacua - HJEP05(218) 1 Introduction 2 Overview of abelian gauge groups in F-theory Elliptic curve group law Rational sections, the abelian sector, and the Mordell-Weil group HJEP05(218) Charged matter 2.4 Anomaly cancellation 3 Charge-3 models Tuning abelian models Non-UFD tunings and the normalized intrinsic ring 3.3 Tuning models with q = 3 3.3.1 3.3.2 Canceling terms up to fourth order Finding f and g 3.4 Structure of the charge-3 construction 3.5 Matter spectra 3.6 UnHiggsings of the q = 3 construction 2.1 2.2 2.3 3.1 3.2 4.1 4.3 { i { 4 Charge-4 models Higgsing the U( 1 ) U( 1 ) construction 4.2 Structure of the charge-4 construction Matter spectra 5 Comments on q > 4 6 Conclusions and future directions A Mathematica les C Charge-4 expressions C.1 P2 form C.2 Weierstrass form D U( 1 ) U( 1 ) expressions Introduction A key objective of the F-theory program is determining which charged matter representations can arise in F-theory models, a task with important implications for the landscape and swampland. Clearly, we cannot characterize the full landscape of F-theory models without knowing all of the representations that can be realized in F-theory. At the same time, one may nd that certain representations cannot be obtained in F-theory, even when the corresponding matter spectra satisfy the known low-energy conditions. This scenario would inspire a variety of questions, such as whether these representations could be attained through other string constructions or whether some previously unknown low-energy ular, there are open questions regarding the construction of models with charges q > 2 (in appropriately quantized units). The goal of this work is to provide new insights into F-theory models admitting q = 3 and q = 4 matter, with the hope that these ideas can inform our understanding of models with arbitrary charges. The reason for the more challenging nature of abelian F-theory models lies in the different manifestations of non-abelian and abelian symmetries. F-theory models in 12 2d dimensions are constructed using a Calabi-Yau d-fold that is an elliptic bration over a base B. Non-abelian gauge symmetries occur when the ber becomes singular along a codimension one locus in B, while charged matter often occurs at codimension two loci with singular bers. The codimension one singularity types and their corresponding non-abelian gauge algebras have already been classi ed [33{36], and in many cases, one can relate the codimension two singularity types to di erent charged matter representations [1, 2, 37]. These dictionaries provide a strategy for constructing an F-theory model admitting a particular gauge group and charged matter spectrum. One rst reads o the singularity types and loci that produce the desired gauge data. Then, one determines the algebraic conditions that make the elliptic bration support the appropriate singularities. This process, known as tuning, has been used to systematically construct a variety of non-abelian gauge groups and charged matter [2, 6, 38]. In contrast, abelian gauge groups are not associated with elliptic curve singularities along codimension one loci. They instead arise when there are additional rational secgroup [11, 12, 34]. Thus, the usual procedures for obtaining non-abelian groups do not carry over to abelian groups in an immediately obvious way, making the construction of F-theory models with abelian gauge symmetries more di cult. Take, for example, the question of how to construct an F-theory model with a single U( 1 ) gauge group and no additional non-abelian groups. There is a well known U( 1 ) construction, the MorrisonPark model [12], but it admits only q = 1 and q = 2 matter. [22] presented a construction supporting q = 3 matter, which was found within a set of toric models. However, this construction was found somewhat by chance, raising the question of whether it could be systematically derived from scratch. That is, instead of looking within a set of models, could someone start with the goal of nding a q = 3 model and follow a series of steps to obtain this construction? The Weierstrass model also has a structure quite di erent from the Morrison-Park form, posing the related question of whether we can understand how and why the structures di er. While there has been some discussion of F-theory models with q = 4 matter [39], there is, to the author's knowledge, no published U( 1 ) model with charges q 4. This makes an understanding of q = 3 models all the more important, as the features that distinguish the q = 3 construction from the Morrison-Park form would likely play a role in q 4 models as well. This work presents a systematic method for tuning a q = 3 construction and presents a class of models admitting q = 4 matter. A central theme is that the presence of q 3 matter is tied to the order of vanishing of the section components. As is well known from [12], q = 2 matter occurs when the components of the section vanish on some codimension two locus; in Weierstrass form, the z^, x^, and y^ components vanish to orders 1, 2, and 3. In the models discussed here, the section components vanish to higher orders at the q 3 loci, directly a ecting the structure of the Weierstrass model. For instance, the z^ component of the q = 3 construction vanishes to order 2 on the q = 3 locus, reminiscent of a divisor with double point singularities. As discussed in section 3, one can build abelian F-theory models through a process similar to the SU(N ) and Sp(N ) tuning procedure. Instead of making the discriminant proportional to a divisor supporting a non-abelian symmetry, we tune quantities to be proportional to the z^ component of the section. When z^ vanishes to orders larger than 1, the tuning process allows for structures associated with rings that are not unique factorization domains (UFDs); these structures can be derived using the normalized intrinsic ring technique of [6]. Following the procedure leads to a generalization of the previous q = 3 construction in [22], with a direct link between the speci c structures in the q = 3 Weierstrass model and the singular nature of z^. We also obtain a q = 4 F-theory construction by deforming a previous U( 1 ) U( 1 ) construction from [4]. To the author's knowledge, this is the rst published F-theory example admitting q = 4 matter. While we do not derive this construction using the normalized intrinsic ring, the section components of the q = 4 construction vanish to higher orders as well, and the Weierstrass model contains structures suggestive of non-UFD behavior. The rest of this paper is organized as follows. Section 2 reviews some aspects of abelian groups in F-theory that are important for the discussion. Section 3 describes how abelian symmetries can be tuned and uses the process to systematically derive a q = 3 construction. In section 4, we construct and analyze a construction admitting q = 4 matter. Section 5 { 2 { includes some comments about q > 4 models, while section 6 summarizes the ndings and mentions some directions for future work. There are accompanying Mathematica les containing expressions for the constructions derived here; details about these Mathematica les are given in appendix A. 2 Overview of abelian gauge groups in F-theory In this section, we review those aspects of F-theory that are necessary for the rest of the discussion. We will not be too detailed here, instead referring to the mentioned references for further details. More general reviews of F-theory can be found in [40{42]. F-theory can be described from either a Type IIB perspective or an M-theory perspective. In the Type IIB view, an F-theory model can be thought of as a Type IIB compacti cation in which the presence of 7-branes causes the axiodilaton to vary over the compacti cation space. The axiodilaton is represented as the complex structure of an elliptic curve, and the F-theory compacti cation involves an elliptic bration X over a compacti cation base B. In this paper, we will assume that the base B is smooth. Mathematically, the elliptic bration can be described using the global Weierstrass equation [x : y : z] refer to the coordinates of a P2;3;1 projective space in which the elliptic curve is embedded, and f and g are sections of line bundles over B. To guarantee a consistent compacti cation that preserves some supersymmetry, we demand that the total elliptic bration X is a Calabi-Yau manifold by imposing the Kodaira constraint: f and g must respectively be sections of O( 4KB) and O( 6KB), where KB is the canonical class of the base B. The Weierstrass equation is often written in a chart where z 6= 0, in which case the x; y; z coordinates can be rescaled so that z = 1. This procedure leads to the local Weierstrass form y2 = x3 + f x + g commonly seen in the F-theory literature. Note that the elliptic ber is allowed to be singular along loci in the base. Codimension one loci with singular bers are associated with non-abelian gauge groups, while codimension two loci with singular bers are associated with charged matter. F-theory can also be understood via its duality with M-theory. To illustrate the idea, let us rst consider M-theory on T 2. Shrinking one of the cycles in the T 2 leads to Type IIA compacti ed on S1, which is dual to Type IIB on S1. The radii of the circles in the dual Type II theories are inverses of each other, and if we shrink the Type IIA circle, the circle dimension on the Type IIB side decompacti es. Similarly, we can consider M-theory on a smooth, elliptically bered CY d-fold. Roughly, applying the above shrinking procedure berwise gives a Type IIB theory on the base B with a varying axiodilaton . This Type IIB model can then be thought of as an F-theory model on an elliptically bered CY d-fold. Of course, the full duality involves several subtleties not captured in the discussion above, particularly with regards to singularities and the details of the shrinking procedure. While (2.1) (2.2) { 3 { these issues are not too crucial for the discussion here, readers interested in further details can consult, for instance, [43, 44]. This subsection is largely based on [45], to which we refer for further details. The points of an elliptic curve form an abelian group under an addition operation that we denote [+]. To describe the addition law, we rst identify a particular point Z as the be the point [x : y : z] = [1 : 1 : 0]. Note that, in Weierstrass form, Z is a ex point,1 as the tangent line at Z intersects the elliptic curve at this point with multiplicity 3; in other words, the tangent line at Z does not intersect the elliptic curve at any point other than Z. Given two points P = [xP : yP : zP ] and Q = [xQ : yQ : zQ], P [+]Q has coordinates2 x = xP zP2 x2Q + f zQ4 + xQzQ2 x 2P + f zP4 2zP zQ yP yQ gzP3 zQ3 y = yP2 yQzQ3 3xQx2P yQzQzP2 + 3xP x2QyP zP zQ2 + yQ2yP zP3 3gzP3 zQ3 yQzP3 yP zQ3 f zP zQ xQyQzP5 + 2xP yQzP3 zQ2 2xQyP zQ3zP2 xP yP zQ5 z = xQzP2 xP zQ2: Meanwhile, the point P [+]P = 2P has the coordinates (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) x = 3x2P + f zP4 2 8xP yP2 y = z = 2yP zP : 3x2P + f zP4 3 + 12xP yP2 3x2P + f zP4 8yP4 1While Z is a ex point in Weierstrass form, the identity element may not be a ex point when an elliptic curve is written in other forms. This subtlety is particularly relevant for the P2 form of the q = 4 elliptic bration in section 4. 2If desired, one could use the Weierstrass equation to eliminate f and g and rewrite (2.3) through (2.5) entirely in terms of the P and Q coordinates. Additionally, the elliptic curve addition formula is typically written in a chart where z = 1. After setting zP and zQ to 1 in the expressions and eliminating f and g, one recovers the standard form given in, for example, appendix A of [12]. { 4 { Note that the 2P expressions do not follow directly from plugging zQ = zP ; xQ = xP ; yQ = yP into (2.3) through (2.5), as all of the section components in (2.3){(2.5) vanish with this substitution. For a point P = [xP ; yP ; zP ], the inverse P is simply [xP : yP : zP ]. Rational sections, the abelian sector, and the Mordell-Weil group Unlike the non-abelian sector, the abelian sector of the gauge group is not associated with codimension one loci in the base with elliptic curve singularities. Instead, the abelian sector is associated with rational sections of the elliptic bration. For our purposes, an F-theory construction will always have at least one rational section, the zero section o^.3 If the model is written in the global Weierstrass form of equation (2.1), the zero section is o^ : [x^ : y^ : z^] = [1 : 1 : 0]: But an elliptic bration may have additional rational sections. In fact, these rational sections form a group, known as the Mordell-Weil group, under the addition operation described in section 2.1, with o^ serving as the identity [50]. According to the Mordell-Weil theorem [51], the group is nitely generated and takes the form Z r G : G is the torsion subgroup, with every element of G having nite order; the torsion group will not be important for the purposes of this paper. r meanwhile is called the Mordell-Weil rank. If an elliptic bration has Mordell-Weil rank r, the abelian sector of the corresponding F-theory model includes a U( 1 )r gauge algebra [11, 12, 34]. The justi cation for this statement is most easily seen in the dual M-theory picture, as discussed in [11]. For concreteness, let us restrict ourselves to 6D F-theory models, although similar arguments apply in 4D. Additionally, we assume there are no codimension one singularities apart from the standard I1 singularity, as we are not interested in situations with non-abelian symmetry. Consider M-theory compacti ed on a resolved elliptically bered Calabi-Yau threefold X~ . M-theory on X~ is a 5D model that, in the F-theory limit, leads to a 6D N = 1 F-theory model. According to Poincare duality, there is a harmonic two-form ! for every four-cycle in X~ . The two-forms serve as zero-modes for the M-theory three-form C3, and we can expand C3 using a basis of two-forms. In other words, we write C3 as a sum of terms of the form A ^ !; the one-forms A represent vectors in the 5D theory. Thus, to nd the vectors of the 6D F-theory model, we consider a basis of four-cycle homology classes of X~ , nd the corresponding 5D vectors A, and track the sources of these 5D vectors in the 6D F-theory model. When there are no codimension one singularities (apart from I1 singularities), there are three types4 of four-cycle homology classes that are of interest: the homology class 3See [46{49] for discussions of situations without a zero section. 4When there are codimension one singularities, there is a fourth type of four-cycle homology class that corresponds to the Cartan gauge bosons of a non-abelian gauge group in the F-theory model. Since we are gauge bosons for the U( 1 )r gauge group. Z associated with the zero section, the homology classes S1 through Sr associated with the r generators of the Mordell-Weil group, and the homology classes B that come from bering the elliptic curve over two-cycles in the base. 5D vectors associated with Z and B do not correspond to gauge bosons in the 6D F-theory model. Instead, they arise from the KK reduction of either the metric or tensors in the 6D F-theory model. But 5D vectors associated to S1 through Sr come from vector multiplets in the 6D model. These are the However, the 5D vectors do not directly correspond to the Si but are rather associated with combinations of Si with Z and the B . At least informally, we must isolate the part of the Si that is orthogonal to the other four-cycles. This is done using the Tate-Shioda map , which is a homomorphism from the Mordell-Weil group to the homology group of four-cycles. For a situation with no codimension one singularities, the Tate-Shioda map is given by [12] (s^) = S Z ( S Z B KB) B ; where KB are the coordinates of the canonical class of the base written in the basis B . Thus, the U( 1 ) gauge bosons are actually associated with the homology class (s^i), and the Tate-Shioda map plays an important role in physical expressions. An important property of a rational section s^, particularly for anomalies, is its height h(s^). The height is a divisor in the base given by [12] h(s^) = ( (s^) (s^)) ; where is a projection onto the base. For a 6D F-theory model with no codimension one singularities apart from I1 singularities, the height can be expressed in a simpler form [12, 26]: h(s^) = 2 ( KB + ( S Z)) ; (2.11) (2.12) (2.13) HJEP05(218) where S is the homology class of the section s^. This expression can often be simpli ed further. Suppose that, in global Weierstrass form, the section has coordinates [x^ : y^ : z^]. Additionally, assume that the coordinates have been scaled so that they are all holomorphic and that there are no common factors between x^, y^ and z^ that could be removed by rescalings. We can consider a curve z^ = 0 in the base, and we denote the homology class of this curve [z^]. s^ coincides with the zero section at loci in the base where z^ = 0, so the height is given by [12, 26] h(s^) = 2 ( KB + [z^]) : (2.14) Since the height is written entirely in terms of homology classes of the base, this expression is useful for calculations, particularly those related to anomaly cancellation. Note that if there are multiple generators, one may be interested in a height matrix, which includes entries such as ( (s^i) (s^j )) for distinct generators s^i and s^i. Here, we are primarily interested in situations with a rank-one Mordell Weil group, so this generalized form will not be too important. { 6 { Even though the abelian gauge symmetry is not associated with codimension one singularities, charged matter still occurs at codimension two loci with singular bers, as discussed in [11]. Again, we restrict ourselves to a model with an abelian gauge group but no additional non-abelian gauge groups. The model has various codimension two loci with I2 singularities. After these singularities are resolved, the bers at these codimension two loci consist of two P1s which intersect each other at two points. One of the components, the one containing the zero section, can be thought of as the main elliptic curve, with the other component being the extra P1 introduced to resolve the singularity. In the M-theory picture, charged matter arises from M2 and anti-M2 branes wrapping this extra component. To calculate the charge of this matter, we must examine the M2 brane world-volume action. The action contains a term of the form R C3, where the integral is over the M2 brane world-volume. For the situation at hand, the M2 brane wraps a component c of the singular ber. C3 meanwhile has an expansion involving terms of the form A ^ !, where ! is a harmonic two-form of the resolved CY manifold X~ . Integrating over the c component leads to a term in the action of the form R A over a world-line, thereby giving the action for charged matter. The charge comes from integrating the two-form ! associated with the U( 1 ) gauge boson A. However, for a CY n-fold, each ! is dual to a (2n 2)-cycle , and Therefore, the charges supported at an I2 locus are given by The sign corresponds to whether c is wrapped by an M2 brane or an anti-M2 brane. In situations without additional non-abelian symmetries, the charge formula reduces to [11, 12] For a generating section s^ = [x^ : y^ : z^], charged matter occurs at [12, 15] Clearly, the above condition is satis ed if all of the components of the section vanish at some codimension two locus. Not only is the elliptic ber singular when this happens, but the section itself is ill-de ned. Analyzing such situations requires that we resolve the section, a process described in [12]. Afterwards, the section appears to \wrap" one of the P1's of the I2 ber. Rational sections typically behave this way at loci supporting q matter. At q = 2 loci, the z^, x^, and y^ components (in Weierstrass form) vanish to orders 1,2, and 3. As described later, the components vanish to higher orders at loci supporting q 3 matter. For instance, z^ vanishes to order 2 for q = 3 loci and order 4 for q = 4 loci. This higher order of vanishing likely a ects the way the section wraps components, but we will not signi cantly investigate resolutions of the q = 3 and q = 4 models here. However, it would be interesting to better understand the wrapping behavior in models with q matter in future work. Z c ! = c : q = (s^) c: q = (S Z) c: y^ = 3x^2 + f z^4 = 0: { 7 { (2.15) (2.16) (2.17) (2.18) 2 3 from supergravity. Since 6D is the largest dimension in which supergravity theories can admit charged matter, the 6D anomaly cancellation conditions will be particularly important here as a consistency check on the models. In 6D supergravity models, anomalies are typically canceled through the Green-Schwarz mechanism. However, not all models are anomaly free; in order for anomalies to cancel, the massless spectrum must obey particular conditions. While the anomaly cancellation conditions come from low-energy considerations, they do have a geometric interpretation in F-theory [11], and the conditions can be written in terms of parameters describing the F-theory compacti cation. The general anomaly cancellation conditions for models with abelian gauge groups are given in [10, 11, 52]. Here, we restrict our attention to the case of a single U( 1 ) gauge group with no additional gauge symmetries. In the F-theory model, the Mordell-Weil group is generated by a single section, which we refer to as s^. Suppose the model has a base B with canonical class KB. Then, the gauge and mixed gravitational-gauge anomaly conditions are KB h(s^) = 6 1 X q I 2 I 3 1 X qI4: I h(s^) h(s^) = (2.19) The index I runs over the hypermultiplets, with qI denoting the charge of the Ith hypermultiplet. h(s^) meanwhile is the height of the section s^, as described in 2.2. There are also the pure gravitational anomaly conditions H V + 29T = 273 KB KB = 9 T; (2.20) where H, V , and T denote the total number of hypermultiplets, vector multiplets, and tensor multiplets, respectively. Again, the anomaly conditions can be viewed as fully lowenergy supergravity constraints, even though they are phrased here in terms of F-theory parameters. The anomaly conditions can be used to derive two relations that are particularly useful for q 3 models. The rst is the tallness constraint [26] 5While this work was being completed, the author became aware of the upcoming work [53], which detailed analysis of this relation along with analogues for situations with multiple U( 1 ) factors. This constraint suggests that a section with large enough h(s^) is forced to have some higher charge matter. But the anomaly equations in (2.19) also imply that5 Specializing to situations where (2.14) applies, this relation can be rewritten as h(S) h(S) 2KB h(S) max qI2: I h(s^) (h(s^) + 2KB) = [z^] ( KB + [z^]) = 3 1 X q I I2 q 2 I 1 : 1 12 X q I2 q 2 I I 1 (2.21) (2.22) (2.23) Note that q2(q2 1)=12 is 0 for q = 0; 1 and is a positive integer for q therefore directly determine the number of q = 2 hypermultiplets given h(s^), KB, and the number of q 3 multiplets; importantly, the q = 2 multiplicity can be determined without any information about the q = 1 hypermultiplets. As discussed in section 3.5 and section 4.3, this anomaly relation seems to have a direct F-theory realization: it describes the loci where the three components of the section vanish, leaving the section ill-de ned. Moreover, every term in the sum on the right-hand side is non-negative, allowing us to conclude that h(s^) (h(s^) + 2KB) max q I 31 I2 q 2 I 1 : (2.24) This bound in some sense has the opposite e ect as the tallness constraint: if we wish to obtain a model admitting a certain charge q, we must have a su ciently large h(s^). The relation resembles the genus condition [54] for SU(2) F-theory models, although we leave an in-depth exploration of any connection to future work. 3 Charge-3 models While there is a previous F-theory construction admitting q = 3 matter [22], there are still open questions regarding its intricate structure. On the one hand, the construction in [22], which we henceforth refer to as the KMOPR model, was not purposefully constructed with the goal of realizing q = 3 matter. Instead, it was found somewhat by chance in a class of toric constructions. But if we wish to understand ways of obtaining q > 3 models, it behooves us to determine whether we can construct q = 3 models from scratch. That is, rather than searching through a set of constructions with the hope of nding a q = 3 model, could we use general principles and mathematical conditions to directly construct a q = 3 model? Moreover, [5] argued that the structure of the KMOPR model di ers from that of the well-known Morrison-Park construction [12]. In [26], it was shown that the KMOPR Weierstrass model is birationally equivalent to one in Morrison-Park form, although the Morrison-Park form Weierstrass model does not satisfy the Calabi-Yau condition. Nevertheless, the analysis in [26] depended on unexpected cancellations between expressions in the KMOPR model. [5, 26] hinted that the cancellations could be explained using rings that are not unique factorization domains (UFDs), but they did not describe how to understand or derive the construction's speci c structures. This section describes a method for systematically deriving a q = 3 construction. One can construct a Weierstrass model with non-trivial Mordell-Weil rank through a process similar to tuning SU(N ) and Sp(N ) singularities. However, instead of tuning the discriminant to be proportional to some power of a divisor in the base, we tune quantities to be proportional to a power of the z^ component of the section. In non-abelian contexts, models with gauge groups tuned on singular divisors can have non-UFD structure, which can be derived using the normalized intrinsic ring technique discussed in [6]. For the q = 3 construction, z^ has a singular structure, and the quotient ring R=hz^i is not a UFD. Starting with an ansatz for z^, we can use the normalized intrinsic ring to derive a generalization of the KMOPR model. The intricate structure of the q = 3 construction is therefore directly { 9 { linked to the singular nature of z^. Moreover, the normalized intrinsic ring provides a new perspective on the birational equivalence of the q = 3 and Morrison-Park models. We rst describe the tuning process for abelian models and illustrate the procedure by rederiving the Morrison-Park form. We then brie y review the normalized intrinsic ring technique before using it to derive the q = 3 construction and analyze its structure. This section concludes with some comments on the matter spectrum and on ways of unHiggsing the U( 1 ) symmetry to non-abelian groups. For a single U( 1 ) group, we need a section [x^ : y^ : z^] (other than the zero section) such that This expression is simply a rewriting of the global Weierstrass form in (2.1), with the x; y; z coordinates replaced with components of the section. The left-hand side has a similar structure to the expression for the discriminant = 4f 3 + 27g2. Moreover, the equation shows that y^2 x^ 3 must be proportional to z^4, reminiscent of the conditions for an I4 singularity. These observations suggest that a U( 1 ) can be tuned using a method similar to that used for tuning SU(N ) or Sp(N ) gauge groups: 1. We rst expand x^ and y^ as series in z^. We assume that z^, x^ and y^ are all holomorphic. y^ 2 x^3 / z^4: (3.1) (3.2) 4. Finally, we can read o f and g from the expression for y^2 3 x^ . While the process outlined above is similar to the In tuning process, note that, unlike f and g in a standard non-abelian tuning, x^ and y^ can vanish to orders 4 and 6 on some codimension two locus. In fact, this seems to generally happen for U( 1 ) models with q 3. To illustrate this procedure, we rst consider a situation in which z^ is equal to a generic parameter b. We expand x^ and y^ as series in b: x^ = x0 + x1b + x2b2 + : : : y^ = y0 + y1b + y2b2 + : : : : (3.3) Note that we are only interested in expressions for the xi and yi up to terms proportional to b; for instance, a term proportional to b in xi can be shifted to xi+1 without loss of generality. Said another way, the important properties of xi and yi are their images in the quotient ring R=hbi, in which elements that di er only by terms proportional to b are identi ed. Here, R refers to the coordinate ring of (an open subset of) the base B. Since b is a generic parameter, we assume that R=hbi is a unique factorization domain (UFD). 2. We tune x^ and y^ so that tional to either z^6 or x^. This step bears the most resemblance to the In tuning process. 3. If necessary, we perform additional tunings so that y^2 x^3 is a sum of terms proporPlugging the expansions of x^ and y^ gives To perform the tuning, we work order by order, imposing relations such as y^ 2 We now need to tune the xi and yi so that (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) x^ = The right-hand side of this equation already matches the right-hand side of equation (3.2), so no further tunings are required. We can thus read o that f = 34 x22 + 2 y4 + f2b2 g = 14 x32 + x2y4 + y42b2 f2 2 + x2b2 Notice that we have added and subtracted an f2x^b2 term from x^ 3 y^2, leading to the inclusion of f2 terms in both f and g. If we rede ne parameters as coe cients adjusted:6 and so on. Since all of the constraints involve congruence relations modulo b, we are essentially considering the conditions to be equations in the quotient ring R=hbi. But the solutions for xi and yi that ensure y^ 2 x^ 3 / b4 are already known for situations where R=hbi is a UFD. We should use the UFD non-split I4 tuning [2, 36], only with the numerical HJEP05(218) 2 x2 = = c3 f2 = c0; y4 = 1 These are exactly the f and g for the Morrison-Park U( 1 ) form [12]. The section, meanwhile, is now given by x^ = c32 2 which agrees with the expressions in [12] up to an unimportant negative sign in y^.7 6The order one terms in the standard I4 tuning can be removed by a rede nition of , x2, and y4. 7To address the negative sign discrepancy, one can let b ! b, which changes the sign of z^ but leaves x^ and y^ unchanged. Then, one can scale (x^; y^; z^) by (( 1 )2; ( 1 )3; ( 1 )) and obtain the exact form of the section in [12]. Given that the Morrison-Park form seems to arise from the UFD solutions to the tuning conditions, a natural next step is to consider situations in which R=hz^i is not a UFD. In these cases, there are alternative solutions to the tuning constraints, allowing for deviations from the Morrison-Park form. For example, suppose that which is a possible solution even if R=hz^i is a UFD. For this solution, y2 identically. However, one could also let x21 vanishes z^ = 2 B 2 : y2 x 2 1 For this z^, R=hz^i is not a UFD, as explained in more detail below. A constraint such as can be solved in multiple ways. We can let so this second possibility is also a solution. Note that this second solution depends on x21 is an expression that happens to be proportional to the This example raises two questions: when are multiple solutions possible? And how can we determine the form of the other solutions? Multiple solutions are allowed when R=hz^i is not a UFD and polynomials may have multiple factorizations up to terms proportional to z^. In the example above, x21 and y2 represent two distinct ways of factoring the same polynomial in R=hz^i, as x 21 and y2 di er only by a term proportional to z^. As noted in [6], the quotient ring R=I for an ideal I is non-UFD if the variety V corresponding to I is singular. For the abelian tuning process, we can have a non-UFD R=hz^i if the divisor z^ = 0 in the base is singular. This is the case for the KMOPR model: the z^ component is given by z^ = s7s82 s6s8s9 + s5s92; and the divisor z^ = 0 has double point singularities at s8 = s9 = 0. The q = 3 and q = 4 models derived here have a singular z^ as well. We can obtain the alternative solutions by using the normalized intrinsic ring [6], which we brie y review here. Even if z^ = 0 is singular, it has a normalization that is smooth in codimension one. The normalized intrinsic ring describes functions on this normalized variety. Consider the ring R=hz^i, where R refers to the coordinate ring of (an open subset of) the base B. Because the variety z^ = 0 is singular, R=hz^i is not a UFD. However, the eld of fractions of R=hz^i is a UFD. The normalized intrinsic ring, written as R^=hz^i, is de ned as the integral closure of this eld of fractions, and we can take R^=hz^i to be a UFD.8 To construct it explicitly, we add elements from the eld of fractions that satisfy a monic polynomial with coe cients in R=hz^i. In the z^ = 2 B 2 example, we know that 2 B = 0: We therefore add an element H~ satisfying intrinsic ring can formally written as H~ = 0 and H~ 2 = B. Thus, the normalized (3.19) (3.20) R^=hz^i = R[H~ ]=h H~ ; B H~ 2i: We follow the notation in [6], in which all parameters in the normalized intrinsic ring (that are not well-de ned in the quotient ring) are capitalized and marked with a tilde. Since we take the normalized intrinsic ring to be a UFD, the solutions to the constraints should be the UFD solutions when we work in the normalized intrinsic ring. For instance, the solution for (3.14) would take the form x1 ~ 1 y2 ~2; 1 to remove all instances of H~ . Then, and for simplicity we let ~1, an element of the normalized intrinsic ring, be H~ . But in the tuning process, x1 and y2 appear in the expansion of the section components, and since we are interested in situations where x^ and y^ are holomorphic, x1 and y2 should be well-de ned as elements of R=hz^i. We therefore need to use the equivalence relations implied by (3.20) ~1 = H~ ! and we recover the alternative tuning. In general, nding the non-UFD solutions involves starting with the UFD solutions in the normalized intrinsic ring and determining how to make these expressions well-de ned in R=hz^i. 3.3 Tuning models with q = 3 We now describe how to systematically derive a U( 1 ) construction admitting q = 3 matter. The goal is to demonstrate that the normalized intrinsic ring techniques can generate q = 3 models, not to nd the most general construction. As such, we will not focus on whether the algebraic tunings used here are the most general possibilities. However, the tuning presented here is more general than the KMOPR construction, as discussed later. Our starting point is the assumption that z^ = b(2) a2 + 2b( 1 ) a b + b(0) b : 2 8If z^ is one-dimensional (as would be the case for 6D theories), R^=hz^i is automatically a UFD; see section 2.4 (particularly Theorem 2.14) of [55] for further details. In 4D, z^ = 0 would be complex two-dimensional, and even after normalization there may be singularities at codimension two. Thus, R^=hz^i may not be a UFD in 4D. To derive the models considered here, we will assume that, regardless of dimension, R^=hz^i is a UFD. This form for z^ is equivalent to that in the KMOPR model but with di ering symbols. Note that the divisor z^ = 0 in the base would have double point singularities on a = b = 0, and R=hz^i is not a UFD. The tuning for x^ and y^ can therefore have non-UFD structure, which we derive using the normalized intrinsic ring. For this particular z^, we form the normalized intrinsic ring by adding a new element B~ that satis es the relations ~ bB B~2 b(2) a + b( 1 ) b = 0 aB~ + b( 1 ) a + b(0) b = 0 This normalized intrinsic ring is essentially the same as that used for the symmetric matter models in [6]. We then expand x^ and y^ as power series in z^. x^ = x0 + x1z^ + x2z^ 2 y^ = y0 + y1z^ + y2z^2 + y3z^3 + y4z^4: (3.27) The series can be truncated at orders 2 and 4; if included, higher order terms can be absorbed into other parameters once the tuning is completed. For convenience, we de ne the quantity to be the left-hand side of (3.1): := y^2 3 x^ : In general, we choose notations that agree with the SU(2) model discussed in [6]. The symbol indicates that expressions are equivalent when viewed as elements of the normalized intrinsic ring. For instance, an expression such as x1 x1 is proportional to B~ in the normalized intrinsic ring; however, since x1 should be wellde ned in the quotient ring, the expression tB~ must be converted to a well-de ned quotient tB~ would suggest that ring expression. 3.3.1 Canceling terms up to fourth order Order 0 cancellation. We need y 2 0 x 3 0 in R=hz^i.9 Thus, we can set If R=hz^i were a UFD, the only way to satisfy this constraint would be to have x0 and y0 be proportional to the square and cube of some parameter, respectively. This parameter is the equivalent of the c3 parameter in the Morrison Park tuning. For the case at hand, R=hz^i is not a UFD, but R^=hz^i is a UFD. In principle, we can therefore let x0 and y0 be proportional to the square and cube of some parameter T~ in R^=hz^i. However, x0 and y0 are elements of the coordinate ring and must have well-de ned expressions in R=hz^i. In fact, for the z^ considered here, T~2 and T~3 are well-de ned in R=hz^i only if T~ is well-de ned x0 := t2 y0 := t3; (3.30) where t is well-de ned in R=hz^i. With these de nitions, y02 x30 vanishes identically, and is proportional to z^. 9See section 5 of [6] for a more detailed discussion. (3.24) (3.25) (3.26) (3.28) (3.29) HJEP05(218) Order 1 cancellation. The condition for / z^2 is that This condition can be satis ed by setting is now proportional to z^2. Order 2 cancellation. The condition for If we work in R^=hz^i, which is a UFD, the only way to satisfy this condition (without forcing t to be a perfect square) is to have x1 1 t~ 6 y2 3 8 t 36 ~ is an element of R^=hz^i, which we can write as10 ~ = ~ B However, x1 and y2 are elements of the coordinate ring, and the above tunings involving B~ must be rewritten as expressions that are well-de ned in R=hz^i. To obtain a non-trivial tuning, we should not tune in a way that makes ~ well-de ned in R=hz^i. Therefore, in order for both x1 and t to be well-de ned, t must take the form t := ( 1 ) a + (0) b: 1 6 Using (3.24) and (3.25) to replace B~ b and B~ a with expressions in R=hz^i, we de ne x1 to be x1 := Meanwhile, (3.26) implies that y2 should be de ned to be y2 := 3 8 t 36 With these tunings, is proportional to z^3. For convenience, we de ne the quantity 2;rem to be 2;rem := 48 1 t2 2 b(2) (20) 2 2b( 1 ) (0) ( 1 ) + b(0) ( 1 ) : 10One could use the more general expression ~ = ( 1 ) a + (0) b + B~. However, after the full tuning is completed, (0) and ( 1 ) can be removed by rede nitions of the other parameters in the Weierstrass model. We therefore drop (0) and ( 1 ) from the beginning to simplify the discussion. (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) (3.37) (3.38) (3.39) (3.40) Order 3 cancellation. The condition for / z^4 is that 2;rem In R^=hz^i, this condition can be written as 1 2 b(2) (0) 2 2;rem cannot be canceled without further tunings: for instance, the other terms within the square brackets are proportional to either a or b, while the contributions from 2;rem are not. We should not use tunings that change the form of z^ or tune in a way that removes the non-UFD structure. But we can introduce a and b factors by tuning ( 1 ) and (0). In particular, we can let HJEP05(218) ( 1 ) = (2) a + ( 1 ) b (0) = (01) a + (0) b: (3.43) 2 Additionally, b(2) (0) 2b( 1 ) (0) ( 1 ) + b(20) (21) should be the sum of two terms: one proportional to t, and the other proportional to z^. This is not the case after the tunings done so far, but we can satisfy this condition by letting ( 1 ) = (01). We therefore de ne (0) and ( 1 ) as Now, and t is quadratic in a and b : ( 1 ) := (2) a + ( 1 ) b (0) := ( 1 ) a + (0) b; The 2;rem terms can now be canceled by letting y3 := y30 + 96 at least up to terms proportional to z^. The third order cancellation condition now reads t ~3 is not well de ned in R=hz^i, so we cannot use y30 to cancel this term. But working in R^=hz^i, we can cancel the remaining terms using tunings that, in R^=hz^i, take the form x2 1 6 + 1 ~2 432 (3.44) (3.45) (3.47) (3.48) (3.49) We can immediately convert the x2 expression into a well-de ned quantity in R=hz^i, giving the following de nition for x2: The tuning . ~ term in the y30 expression, however, cannot be written in R=hz^i without further must be well-de ned in R=hz^i, so it should take the form Then, y30 should be de ned as To summarize, we have performed the following tunings: 1 48 1 6 1 48 ( 1 ) := (2) a + ( 1 ) b (0) := ( 1 ) a + (0) b x2 := y3 := 96 t i is therefore proportional to z^4. 3.3.2 Finding f and g Ultimately, we need to extract f and g from the relation y^ 2 x^3 = z^4 f x^ + gz^2 Now that y^2 x^3 is proportional to z^4, we can start extracting portions of f and g. Unlike in the Morrison-Park case, we need to further tune parameters in x^ and y^ to extract f and g. As in the previous steps, we will work order by order. If we let we have the condition that Our goal is now to cancel the order 4 and order 5 terms on the left-hand side of the above equation. Order 4 cancellation. The condition for the order 4 terms to cancel is that HJEP05(218) 3;rem + y22 3x12x2 + 3tx1y3 where 3;rem is given by (3.58). There are several terms in the above expression that are explicitly proportional to t2. Such terms can fairly easily be canceled by tuning f0 to take the form f0 := b (0) + a ( 1 ) The cancellation condition now takes the form tx1 h +x12 h 12 1 16 (0) b( 1 ) b +b(2) a ( 1 ) b(0) b +b( 1 ) a b (0) + a ( 1 ) 1 144 i Working in R^=hz^i, this condition is equivalent to B~2 3 1728 (0) b + ( 1 ) a f00 + B~ b(2) (0) 2b( 1 ) ( 1 ) +b(0) (2) If all the terms in square brackets were well-de ned in R=hz^i, we could immediately read o to b(21) an expression for f00 that would cancel terms. However, this is not currently the case. The terms that have even powers of B~ are already well-de ned in R=hz^i, since B~2 is equivalent b(0)b(2). But the B~ term in the square brackets is currently not well-de ned in R=hz^i. Without modifying , which would lead to a trivial tuning, the only way to x this term is to force (b(2) (0) 2b( 1 ) ( 1 ) + b( 1 ) (2)) to be a sum of terms proportional to a or b. This can be accomplished with the ansatz that (0), ( 1 ), and (2) take the form (0) := t(0) b + t( 1 ) a ( 1 ) := t( 1 ) b + t(2) a (2) := t(2) b + t(3) a: (3.67) Like the q = 3 z^ component, the q = 4 z^ seems to admit a quadratic structure. However, the expressions and , which play the role of a and b, are themselves quadratic in a1 and b1. From the discussion in section 3.5, the U( 1 ) symmetry in the q = 3 construction can be unHiggsed to an SU(3) symmetry tuned on either a or b . At the same time, an SU(3) model with matter charged in the symmetric representation (6) can be Higgsed down to a U( 1 ) model with q = 4 matter [4]. SU(3) gauge groups supporting 6 matter are tuned on divisors with double point singularities [2, 37], so for the q = 4 model, a and b should be replaced with some expressions having double point structure. This is exactly what is seen in (4.11), as and have the requisite quadratic structure. In fact, the height of the generating section is 6[ ] + 2([d1] [a1] [b1]), which displays the expected factor of 6 discussed in section 3.6. The ((s6a1 coe cients, meanwhile, are simply expressions for the normalized intrinsic ring parameters of and .19 In fact, we can obtain the f and g for the q = 4 Weierstrass model by starting with f and g for the q = 3 model and making the replacements given in table 5. This observation provides further evidence that our construction supports q = 4 matter, as the U( 1 ) U( 1 ) Higgsings that give q = 4 matter also lead to q = 3 matter. If a1 is constant (allowing us to divide freely by a1), the highest charge supported by the model is q = 3, and the two models should match. But the dictionary between the q = 3 and q = 4 constructions also suggests that the two Weierstrass models are birationally equivalent. In particular, a14f and a61g can be written in the form of a q = 3 model without division by a1. Since a41f 2 4KB + 4[a1] and a61g 2 6KB + 6[a1], the Weierstrass model with q = 3 structure is not a Calabi-Yau manifold unless [a1] is trivial. Thus, the q = 4 model is birationally equivalent to the q = 3 model, although the model in q = 3 form does not satisfy the Calabi-Yau condition. This result seems to be a q = 4 analogue of the statement in [26] that the Morrison-Park and the q = 3 Weierstrass models are birationally equivalent. It is tempting to speculate that U( 1 ) models with q > 4 should also be birationally equivalent to lower charge models; we leave a thorough investigation of this conjecture for future work. In summary, the q = 3 and q = 4 models seem to be related, but the q = 4 construction has some additional normalized intrinsic ring structure. It would be interesting to further examine the connections between the two constructions and use these patterns to obtain a more general q = 4 form. 4.3 Matter spectra We now determine the codimension two I2 singularities of the q = 4 construction and the corresponding matter content. The results of this analysis are summarized in tables 6 and 7. There are two important aspects of the matter content analysis: the type of charge supported at an I2 locus, and the multiplicity of matter elds with a particular charge. While the actual charge values are typically determined by resolving singularities, we instead use indirect methods to determine the charges, leaving a full resolution analysis 19For example, compare these coe cients to (3.24) and (3.25). q = 3 Parameter Expression to obtain q = 4 Model a b b(2) b( 1 ) b(0) t(3) t(2) t( 1 ) t(0) h(0), h( 1 ), h(2) (0), ( 1 ) f2 b1s2 a1s5 a1 are required for the conversions. Each locus is written as a variety V associated to an ideal I generated by two equations. and are de ned in (4.12), while t is given in (4.15). Charge Charge 4 3 2 1 4 3 2 1 I2 Locus V (Iq=4) = fa1 = b1 = 0g V (Iq=3) = f = = 0g=V (Iq=4) V (Iq=2) = ft = z^ = 0g=(V (Iq=4) [ V (Iq=3)) V (Iq=1) = fy^ = 3x^2 + f z^4 = 0g=(V (Iq=4) [ V (Iq=3) [ V (Iq=2)) Multiplicity m4 = [a1] [b1] m3 = ([z^] + 2KB [a1] [b1]) ( KB + [a1] + [b1]) for future work. However, we present more detailed calculations of the matter multiplicities. As in the q = 3 matter analysis, we assume that we are working in six dimensions. The codimension two I2 loci are supported at the intersection of the divisors y^ = 0 3x^2 + f z^4 = 0: (4.13) In principle, we could directly calculate the resultant of these two expressions and read t = s1 3 + (s2s6b1 s2s8a1 s5s3b1) 2 + 2s3s8 + a1s6 (d0s8b1 s3d2b1 + s6d2a1 a1d1s8) : 2 (s8d0 s6d1 + s3d2) HJEP05(218) The section is therefore ill-de ned at t = z^ = 0. This locus includes the loci a1 = b1 = 0 and = = 0. By the homomorphism argument in section 4.1, the a1 = b1 = 0 locus should support q = 4 matter, with a q = 4 multiplicity of m4 = [a1] [b1]. Meanwhile, we know that if a1 is a constant, we recover a q = 3 model with q = 3 matter supported on the = = 0 locus. This locus should still contribute q = 3 matter even when a1 is not a constant, as long as we exclude the a1 = b1 = 0 locus. The q = 3 locus is therefore = 0g=fa1 = b1 = 0g. To count the q = 3 multiplicities, we note that the resultant and with respect to a1 takes the form Resa1 ( ; ) = b14r3; o information about the matter spectrum. However, calculating this resultant is computationally complex, so we rst consider the simpler problem of determining loci at which the section becomes ill-de ned. Matter with q 2 is supported at such loci, so this trick allows us to more quickly determine information about the matter content. The important starting observation is that x^ The b41 factor in the resultant suggests that a1 = b1 = 0 is a degree 4 root of = = 0. In total, there are [ ] [ ] points in the = = 0 locus, so the q = 3 multiplicity should be m3 = [ ] [ ] 4[a1] [b1] = ([z^] + 2KB [a1] [b1]) ( KB + [a1] + [b1]) 4[a1] [b1]: (4.18) Note that if we undo the deformations in (4.3), factorizes as (a1b2 a2b1)(a1b3 a3b1). This unHiggsing therefore splits the = = 0 locus into two loci: (a1b2 a2b1) = = 0, and (a1b3 a3b1) = = 0. In the original U( 1 ) U( 1 ) model in [4], these two loci support (2; 1) and (1; 2) matter, which are the types of charged matter that eld theory considerations suggest should become q = 3 matter after Higgsing. The match between these matter loci before and after Higgsing is further evidence that the = = 0 locus supports q = 3 matter. The q = 2 locus consists of the t = z^ = 0 points that do not support q = 4 or q = 3 matter. To calculate the q = 2 multiplicity, we start with the [t] [z^] intersection points and 20One can actually obtain this t expression by starting with the t of the q = 3 construction, making the appropriate substitutions from table 5, and adding a term proportional to z^ to remove all fractional terms. t vanishes to order 4 on a1 = b1 = 0. exclude those points corresponding to q = 4 or q = 3 matter. We therefore must examine the resultant of t and z^ with respect to a1, which is given by Resa1 (t; z^) = b120r36d23r2: r2 is a complicated, irreducible polynomial that we do not give here. The b210 factor suggests that the q = 4 locus is an degree 20 root of the system, while the r36 suggests that the q = 3 locus is a degree 6 root of the system.21 Intriguingly, these numbers exactly match the 1)=12 factors appearing in (2.23). After removing the contributions from the q = 4 and q = 3 loci, we nd that the q = 2 multiplicity is given by 6m3 20m4: (4.19) (4.20) (4.21) (4.22) (4.23) (4.24) HJEP05(218) This result is in exact agreement with (2.23), suggesting an F-theory interpretation of this anomaly equation. The [z^] ( KB + [z^]) re ects the fact that q 2 matter is supported at places where the section components vanish; since x^ y^ Resa1 (y^; 3x^2 + f z^4) / b1256r216; the loci where the section components vanish are simply the t = z^ = 0 loci. Meanwhile, the q2(q2 1)=12 factors represent the degree of the roots of the t = z^ = 0 system. q = 1 matter is supported at the y^ = 3x^2 + f z^4 = 0 loci that do not support q 2 matter. y^ and 3x^2 + f z^4 intersect at 12( KB + [z^])2 points, but we must account for the q 2 loci before we can read o the q = 1 multiplicity. We therefore need to calculate the multiplicities of the q 2 loci within the locus described by (4.22). As in the q = 2 and q = 3 analyses, this information can be read o from the resultant with respect to a1. In this case, calculating the resultant is computationally intensive if all parameters are allowed to be generic. We therefore evaluate the resultant for special cases in which some of the parameters are set to speci c integer values. First, consider a situation where all parameters except a1 and b1 are set to speci c where r2 is the same factor appearing in Resa1 (t; z^) with the appropriate values for the parameters plugged in. This result suggests that the a1 = b1 = 0 locus, which supports q = 4 matter, has multiplicity 256 within the (4.22) locus, while the q = 2 locus has multiplicity 16. r3, which corresponds to the q = 3 locus, does not depend on b1, and when all parameters except a1 and b1 are set to integers, r3's contribution to the resultant is simply an integer factor. To read o the q = 3 multiplicity, we consider an alternative scenario in which all parameters except a1 and s8 are set to integers. Then, Resa1 (y^; 3x^2 + f z^4) / r381r216; 21The d23 factor is due to fact that the highest order a1 terms in z^ and t are both proportional to d2. However, this does not correspond to a true locus at which z^ and t both vanish. suggesting that the q = 3 multiplicity is 81 and that the q = 2 multiplicity is 16. With these two results, we can now read o that 12( KB + [z^]) ( KB + [z^]) 16m2 81m3 256m4; (4.25) exactly in agreement with the anomaly conditions in (2.19). Finally, let us examine some possible ways of unHiggsing the q = 4 construction. Of course, the U( 1 ) symmetry can be unHiggsed back to U( 1 ) U(1) by undoing the deformations in (4.3). The model can then be further unHiggsed to an SU(3) model supporting symmetric matter [4]. But there are other ways of unHiggsing the U(1) symmetry to non abelian gauge groups. As with the Morrison-Park and q = 3 constructions, the general strategy is to tune parameters so that the generating section becomes vertical, coinciding with the zero section [1 : 1 : 0]. We therefore need to tune z^ to vanish; x^ and y^ will then be a square and a cube of some expression, which can be scaled so that the generating section becomes [1 : 1 : 0]. In particular, let us restrict ourselves to unHiggsings in which a1 is set to 0. Already, the discriminant is proportional to b21, suggesting there is an SU(2) tuned on b1 = 0. The z^ component (after rescaling the section coordinates by powers of b1) takes the form d1s32 s6d0s3 + s2b1d02; which is quadratic in s3 and d0. To make the section vertical, we should tune the above expression to zero. We cannot let both d0 and s3 be zero, as the discriminant then vanishes exactly. However, sending s3 and s2 to zero makes the discriminant proportional to b41d30s21. f and g are not proportional to b1, d0, or s1,implying that the enhanced model has an SU(4) tuned on b1 = 0, an SU(3) tuned on d0 = 0, and an SU(2) tuned on s1 = 0. The homology classes in table 4 imply that h(s^) = 12[b1] + 6[d0] + 2[s1]: The coe cients for the homology classes supporting SU(N ) are given by N (N agreement with the results from section 3.6 and the expectations from [56]. An interesting question is whether the q = 4 models admit unHiggsings to just an SU(2) gauge group, like the Morrison-Park and q = 3 constructions. This unHiggsing procedure would involve setting z^ to be zero while keeping a1 and b1 generic. Presumably, the SU(2) would be tuned on t = 0, which has a quadruple point singularity at a1 = b1 = 0. So far, the author has not identi ed a way of actually performing this unHiggsing; in all cases considered, t factorizes, indicating the gauge group is product of non-abelian groups rather than a single SU(2). However, a systematic investigation of all possible unHiggsings has not been performed. This issue has important implications for the F-theory swampland, which we discuss further in section 6. (4.26) (4.27) 1), in We have seen that, in models with q = 3 and q = 4 matter, the components of the section vanish to higher orders at the loci supporting q = 3 or q = 4 matter. It is natural to speculate that similar behavior should occur for q > 4 models. Without an explicit Weierstrass model, it is di cult to make de nitive claims about q > 4 matter. However, one can make conjectures about q > 4 models by considering the behavior of the sections in a model admitting q = 1 matter. Suppose that an F-theory model has a rank-one MordellWeil group with no additional non-abelian gauge groups. Let us denote the generating section as s^. If this F-theory model supports q = 1 matter, there is some codimension two I2 locus in which the elliptic curve splits into two components. One of these components, which we denote c, will not intersect the zero section, and because this locus supports q = 1 matter, Using the elliptic curve addition law, we can construct sections ms^, where m is some integer. From the homomorphism property of the Tate-Shioda map, the ms^ sections should satisfy (s^) c = 1: (ms^) c = m: The matter at this I2 locus seems to have \charge" m under the section ms^. Of course, ms^ does not generate the Mordell-Weil group for jmj 6= 1, and the matter supported at this locus does not truly have charge m. Nevertheless, the local behavior of ms^ likely mimics that of the generating section in a genuine q = m model. We can therefore obtain some speculative insights into q = m matter by examining the behavior of ms^. This strategy was used in [12] to anticipate the behavior of models supporting q = 2 matter, and we use it here to conjecture about the behavior of sections admitting q > 2 matter. We start with a simpli ed form of the Morrison-Park model that only supports q = 1 matter [12]. The Weierstrass model (in a chart where z = 1) takes the form (y + f9) (y f9) = (x f6) x2 + f6x + f^12 2f62 ; while the generating section is s^ : [x : y : z] = [f6 : f9 : 1]: There are I2 singularities at f9 = f^12 = 0 that, according to the analysis in [12], support q = 1 matter. Our goal here is to use the elliptic curve addition law to calculate the ms^ sections and examine their behavior at f9 = f^12 = 0. For example, the 2s^ section takes the form 2s^ : [x : y : z] = [f^122 8f6f92 : f^132 + 12f6f^12f92 8f94 : 2f9]: The (z^, x^, y^) components vanish to orders (1; 2; 3) at f9 = f^12 = 0, in agreement with the known behavior of sections at q = 2 loci. (5.1) (5.2) (5.3) (5.4) (5.5) z^ Order of Vanishing x^ Order of Vanishing y^ Order of Vanishing 2 3 4 5 6 7 8 9 10 11 1 2 4 6 9 12 16 20 25 30 2 4 8 12 18 24 32 40 50 60 3 7 12 19 27 37 48 61 75 91 m2 2m2 3m2 4 4 4 4 { 37 { 4 : the simpli ed Morrison-Park model described by equations (5.3) and (5.4). The orders of vanishing for m = 2, 3, and 4 agree with the behavior of the generating sections in models admitting q = 2, 3, and 4 matter. One can therefore conjecture that the components of sections in models supporting q = m matter would vanish to the orders listed here at the q = m loci. For m = 3, we again see behavior in line with the known q = 3 models. The components of 3s^, given by z^ = f^122 12f6f92 x^ = f^142f6 +8f^132f92 24f^122f6 f9 2 2 96f^12f6f94 +144f63f94 +64f96 y^ = 3f^162f9 60f^142f6f93 +96f^132f95 +144f^122f6 f9 2 5 1152f^12f6f97 +64 27f63f97 +8f99 ; vanish to orders (2; 4; 7) at f9 = f^12 = 0, just like the components of the generating section for the q = 3 construction in section 3. The (z^; x^; y^) components for 4s^ vanish to orders (4; 8; 12). The generating section for the q = 4 construction vanishes to these same orders at the q = 4 loci, giving further credence to the idea that this construction truly supports q = 4 matter. Table 8 summarizes the orders of vanishing for the ms^ sections at f9 = f^12 = 0. As expected, the m > 2 section components show singular behavior, with the z^, x^,and y^ components vanishing to orders greater than 1. Given that the behavior of the m = 2; 3; 4 sections agrees with the behavior of the known q = 2; 3; 4 models, one can conjecture that the generating section components for q > 4 models will also vanish to these orders at the q = m loci. In fact, the cases presented in table 8 suggest patterns in the orders of vanishing. For even m, the orders of vanishing for (z^; x^; y^) seem to be given by Meanwhile, the (z^; x^; y^) orders of vanishing for odd values of m seem to be given by These patterns have been veri ed for the ms^ sections with m 1 2(m2 1) 3(m2 1) + 1 (5.6) (5.7) (5.8) (5.9) (5.10) HJEP05(218) It would be interesting to investigate whether the patterns hold for all values of m, both in the simpli ed Morrison-Park form and in actual q = m models. Perhaps the expressions could be proven with a better understanding of the resolutions at the f9 = f^12 = 0 loci. These questions are left for future work. But if these orders of vanishing are correct, this information may be useful for inferring features of the q = m Weierstrass models. Recall that in section 3, the q = 3 Weierstrass model could be derived with the knowledge that z^ vanishes to order 2 on the q = 3 loci. In the same way, one might hope that the orders of vanishing determine the Weierstrass model's structure in a predictable fashion, allowing for a systematic derivation of q > 4 models. These patterns could also give a quick way of detecting the presence of q > 2 matter. Regardless of the type of charge supported, charged matter in a U( 1 ) model (that is not also charged under some additional nonabelian symmetry) occurs at an I2 locus, so examining the discriminant does not provide an immediate way of reading o the charge. But if the behavior of the z^ component can distinguish between the di erent charges, one may be able to at least guess the charge content of a model without the need for an explicit resolution. 6 Conclusions and future directions To summarize, we have constructed U( 1 ) F-theory models admitting both q = 3 and q = 4 matter. In both cases, all of the section components vanish to orders higher than 1 at the q = 3; 4 matter loci. As a result, the Weierstrass models have non-UFD structure that deviates from the standard Morrison-Park form. With the aid of the normalized intrinsic ring, we were able to nd the appropriate non-UFD structures for the q = 3 matter and systematically derive a generalization of the q = 3 construction described in [22]. A class of q = 4 constructions were also constructed, although the models were found by deforming the earlier U( 1 ) U( 1 ) construction in [4]. Nevertheless, the q = 4 construction shows signs of normalized intrinsic ring structure as well. We nally discussed some conjectures regarding models with q > 4 matter. A natural direction for future work is to search for models admitting q > 4 matter. There are a few di erent strategies that may give new insights into this issue. Just as deforming a U( 1 ) U( 1 ) model led to q = 4 matter, deforming models with multiple U( 1 ) factors could lead to larger charges. This process would likely require an initial model with somewhat exotic matter charged under multiple U( 1 ) factors. For instance, the possible Higgsings of the U( 1 )3 construction in [19] cannot give q > 4 matter, although they can produce q = 3 and q = 4 matter. Alternatively, one could obtain large charges by Higgsing models with non-abelian symmetry. [56] gives examples of the eld-theoretic Higgsing processes that could produce q > 4 matter. However, it can be di cult to identify the deformations of F-theory models corresponding to a speci c Higgsing. A better understanding of the F-theory realizations of Higgsing processes, particularly Higgsing on adjoints, would be helpful to develop concrete methods for q > 4 models. There is the possibility of building q > 4 models from scratch, although the algebraic complexity of the models discussed here suggests this approach may be unwieldy. Based on the q = 3 derivation in section 3, we would likely need some knowledge of the q > 4 singularity structures. Analyses similar to section 5.1 of [12] or section 5 here could provide the necessary insights to construct q > 4 Weierstrass models. At the very least, such e orts could illustrate the local behavior of sections at loci supporting arbitrary charges. But there are interesting questions about q = 3 and q = 4 models as well. On the one hand, neither of the Weierstrass models discussed here admit the full range of matter spectra consistent with the anomaly equations in (2.19), suggesting that there may be generalizations of these constructions. In particular, the q = 4 construction can almost certainly be extended in some way. The models should also be subjected to a more thorough resolution analysis. Resolutions of the q = 3 construction should be similar to resolutions of the KMOPR construction in [22], and the analysis of the q = 4 matter loci in section 4.1 paints a rough picture of the behavior of the section there. Nevertheless, a more rigorous analysis of the codimension two singularities would be helpful for con rming the matter analysis presented here. It would also be useful to count the uncharged hypermultiplets in these models, possibly with the techniques used in [57]. Meanwhile, the q = 3 and q = 4 sections discussed here (as well as the q > 4 sections in section 5) have components that exhibit singular behavior, raising the question of whether the sections themselves are singular. Preliminary indications suggest that the sections are indeed singular. One can describe the section using a system of equations: in addition to equations describing the elliptic bration, the system would include equations such as xz^2 x^z2 = 0, where x^ and z^ refer to the section components. One can then use the Jacobian condition to determine loci where the section is singular. Of course, the elliptic bration needs to be resolved at the relevant codimension two singularities, and the section needs to be resolved to account for loci where the section components vanish. After the resolution procedure, the section may wrap a component of an I2 ber, as described previously. An initial analysis indicates that, at the q 3 loci, many of the sections described here are singular at the intersections between the I2 ber components. This information would be important for comparing the models presented here to the results in [25]. However, a more thorough analysis should be performed to understand any possible singularities in these sections. It would be interesting to explore these issues in future work, possibly in the context of a broader analysis of singular sections. The q = 3 and q = 4 models also o er avenues to explore F-theory physics. When a model has q > 2 matter, the anomaly cancellation conditions (2.19) do not uniquely determine the spectrum, even if one xes h(s^) and KB. In non-abelian contexts where this situation occurs, there are matter transitions connecting the vacua with di erent matter spectra [38]. Abelian F-theory models should also exhibit such transitions, which would change the charge content of the theory without changing the gauge group or other parts of the spectrum. Because the SU(2) construction in [6] admits matter transitions, the q = 3 construction here, which can be unHiggsed to this same SU(2) model, should admit matter transitions as well. Seeing transitions involving q = 4 matter would probably require some generalization of the construction given here. Because abelian symmetries manifest themselves di erently than non-abelian symmetries in F-theory, U( 1 ) transitions would likely give a new understanding of these models. Matter transitions could also be used to derive q > 4 models. For instance, an SU(4) gauge group with 10 matter (and a suitable number of adjoints) can be Higgsed down to a U( 1 ) with q = 6 matter through a process similar to the SU(4) ! U( 1 ) Higgsing discussed in section 4.3. SU(4) models have matter transitions that change the amount of 10 matter, implying that the corresponding U( 1 ) models should have transitions that change the amount of q = 6 matter. In particular, one can start with an SU(4) model without 10 matter and use the transitions to grow 10 matter [6]. Thus, the explicit SU(4) transition could potentially be used to reverse engineer a U( 1 ) matter transition that generates a q = 6 model from a known q = 4 model. Unhiggsing q = 4 models could also be an important check of the swampland statement in [6] that certain non-abelian representations, including the 5 representation of SU(2), cannot be realized in F-theory. Field theoretically, if an SU(2) symmetry is Higgsed down to U( 1 ), the presence of 5 matter would lead to q = 4 matter after Higgsing. An examination of unHiggsings of the q = 4 construction is therefore important, as an enhancement to an SU(2) model with 5 would invalidate the statement. However, it is crucial to note that the existence of a q = 4 F-theory model does not by itself guarantee the existence of an SU(2) model with 5 matter. For instance, the would-be SU(2) divisor may factor into multiple components, much like the situation observed in [58]. Alternatively, the resulting SU(2) Weierstrass model may have codimension two (4,6) singularities [6]. So far, the author has not identi ed a way of achieving this SU(2) enhancement, but a complete investigation of all possible unHiggsings has not been done. Finally, the investigations here hint at a deeper interpretation of the section components that should be understood better. The z^ component seems to be the de ning feature of the q = 3 construction, and the anomaly equation (2.23) seems to manifest itself through the section components. Understanding the physical meaning of the section components may provide new insights into abelian F-theory models. For instance, [27, 59] analyze U( 1 ) models, including the Morrison-Park construction and the original q = 3 construction in [22], in the Sen limit. Similar Type IIB investigations could elucidate the role played by the section components. In any case, a more physical description of the models discussed here may inform e orts to nd U( 1 ) models admitting larger charges. Acknowledgments I would especially like to thank Washington Taylor, not only for his numerous comments on the technical aspects of this work, but also for his incredible support and encouragement while this work was being completed. Additionally, I would like to thank Mirjam Cvetic, Denis Klevers, Craig Lawrie, Dave Morrison, Paul Oehlmann, and Andrew Turner for helpful discussions. I am also grateful to Samuel Monnier, Gregory Moore, and Daniel Park for sharing work prior to publication. This work is supported by the O ce of High Energy Physics of U.S. Department of Energy under Contract Number DE-SC0012567. A Mathematica les There are two Mathematica les, Charge3Model.nb and Charge4Model.nb, that respectively contain expressions for the q = 3 and q = 4 constructions. These les are contained Parameter a b b(0) b( 1 ) b(2) t(0) t( 1 ) t(2) t(3) h(0) h( 1 ) h(2) (0) ( 1 ) f2 a1 b1 d0 d1 d2 s1 s2 s3 s5 s6 s8 a b b0 b1 b2 ' t0 t1 t2 t3 h0 h1 h2 0 1 f2 a1 b1 d0 d1 d2 s1 s2 s3 s5 s6 s8 as ancillary les in the arXiv submission and can be obtained by downloading the gzipped source of the submission. Each le contains the f and g of the Weierstrass model (assigned to the variables f and g) and the x^, y^, and z^ components of the generating section (assigned to the variables x, y, and z). Because some of the parameters have typographical features, such as subscripts or macrons, that are not easily used in Mathematica, the Mathematica variable names may be slightly di erent than the parameter names used here. Tables 9 and 10 give the dictionaries between the model parameters and Mathematica variables. z^= b(2) a2+2b( 1 ) a b+b(0) b; 2 x^ = t2 6 72 h(2) a2+2h( 1 ) a b+h(0) b 2 2 2 b( 1 ) b(0)b(2) z^ i 2 b(2) a+b( 1 ) b t(2) a2+2t( 1 ) a b+t(0) b 2 b( 1 ) a+b(0) b t(3) a2+2t(2) a b+t( 1 ) b2 iz^; The components of the section (in Weierstrass form) are given by y^= t3 t 18 h(2) a2+2h( 1 ) a b+h(0) b 2 2 2 b( 1 ) b(0)b(2) iz^2+ 1 2 4 ( 1 ) a+ (0) b z^ where 1 h 432 1 h 72 h h 2 48 1 1728 1 12 Meanwhile, g is given by + t b(2) a+b( 1 ) b t(2) a2+2t( 1 ) a b+t(0) b 2 b( 1 ) a+b(0) b t(3) a2+2t(2) a b+t( 1 ) b2 iz^; h(2) a+h( 1 ) b b( 1 ) a+b(0) b 3 h( 1 ) a+h(0) b b(2) a+b( 1 ) b z^ 3 + b(2) t( 1 ) a+t(0) b 2b( 1 ) t(2) a+t( 1 ) b +b(0) t(3) a+t(2) b z^ ; t = t(3) a3 + 3t(2) a b + 3t( 1 ) a b2 + t(0) b: 2 3 For the Weierstrass model, f is given by f = h(2) a2 + 2h( 1 ) a b + h(0) b2 + t(2) a + t( 1 ) b 2 t( 1 ) a + t(0) b t(3) a + t(2) b 3h b(2)t(0) 2b( 1 )t( 1 ) + b(0)t(2) b(2) a + b( 1 ) b b(2)t( 1 ) 2b( 1 )t(2) + b(0)t(3) b( 1 ) a + b(0) b + ( 1 ) a + (0) b t h(2)b(0) 2h( 1 )b( 1 ) + h(0)b(2) z^ (0) b(2) a + b( 1 ) b ( 1 ) b( 1 ) a + b(0) b z^ + f2z^2: 1 2 576 g = g0 + g1z^ + g2z^2; #2 2 i i i (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) with g0 = and 3 864 2 h sq + 1 864 h3 + 3 864 1 12 2 !2 144 3 (144)2 1 4 6 1 3 1 1 3 288 144 288 1 72 ( 1 ) a + 2 2 !2 6 48 2 2h 3 3 3456 6912 h 1 4 1 48 g1 = (0) b b(2) a + b( 1 ) b (2) a + ( 1 ) b b( 1 ) a + b(0) b f2 h b(2)h(0) 2b( 1 )h( 1 ) + b(0)h(2) b(2) t( 1 ) 2 + b(0) t(2) 2 (0) b(2) a + b( 1 ) b ( 1 ) b( 1 ) a + b(0) b ( 1 ) a + (0) b b(2) (0) (0) b( 1 ) (2) a + ( 1 ) b (0) b ( 1 ) b(0) (2) a + ( 1 ) b (0) b b(2) a + b( 1 ) b b( 1 ) a + b(0) b h(2) a + h( 1 ) b b(2)t( 1 ) 2b( 1 )t(2) + b(0)t(3) h( 1 ) a + h(0) b (0) h(2) a + h( 1 ) b ( 1 ) h( 1 ) a + h(0) b 2 + 1 6 f2 h 2 24 g2 = ( 1 ) a + (0) b 2 b(0)b(2) 2 i i i : 2 2304 h( 1 ) a +h(0) b (2) a + ( 1 ) b h(2) a +h( 1 ) b ( 1 ) a + (0) b 1 864 3 cu + 2 576 h(2) (0) 2h( 1 ) ( 1 ) +h(0) (2) t ( 1 ) a + (0) b t 1 12 (2) a + ( 1 ) b ( 1 ) a + (0) b t f2t 2 b(2) (0) 2b( 1 ) ( 1 ) +b(0) (2) 2 2 b(0)b(2) sq b(2) a +b( 1 ) b 2b( 1 )t(2) +b(0)t(3) b( 1 ) a +b(0) b (B.7) (B.8) (B.9) In the gi expressions, we have used and cu = 1 2 C C.1 Charge-4 expressions P2 form In the P2 form of the elliptic embedding in P2, the q = 4 model is h = h(2) a2 + 2h( 1 ) a b + h(0) b2 + (2) = t(3) a + t(2) b; ( 1 ) = t(2) a + t( 1 ) b; (0) = t( 1 ) a + t(0) b; sq = (21) (2) (0); (B.10) (B.11) (B.12) (B.13) (B.14) HJEP05(218) 2t(32) 3t( 1 )t(2)t(3) +t(0)t(3) a3 3 t( 1 )t(2) 2t(21)t(3) +t(0)t(2)t(3) a b 2 2 2 +3 t(21)t(2) 2t(0)t(2) +t(0)t( 1 )t(3) a b2 + 2t(31) 3t(0)t( 1 )t(2) +t(20)t(3) b3 : (B.15) 2 bration, in which the elliptic ber is described via an where [u : v : w] are the P2 coordinates. The zero section has components p u s1u2 + s2uv + s3v2 + s5uw + s6vw + s8w2 + (a1v + b1w) d0v2 + d1vw + d2w2 = 0; [u : v : w] = [0 : b1 : a1]; while the generating section has components [u : v : w] with h (s2b1 s5a1) 2 s3d1b12 d2s6a12 2d0s8a1b1 + s8d1a12 i; v = s1b1 3 + s5 2 + d2 (s3b1 s6a1) 2 s8 (d0b1 d1a1) 2; w = s1a1 3 s2 2 s3 (d1b1 d2a1) 2 + d0 (s6b1 s8a1) 2: and are de ned as = d2a12 d1a1b1 + d0b12 = s8a12 s6a1b1 + s3b12: (C.1) (C.2) (C.3) (C.4) (C.5) The f and g in Weierstrass form are given by g = 1 s d0d23a14 +b1d0 d1 3 3 2 3d0d1d2 a1 s1 f = 2 2 2 3s1s8 3s1s3 2 2 a1 d1 3d0d2 a1b1d0d1 + b d 2 2 1 0 a d 2d0d2 3s1s3 + b1d2s3(2s3s5 s5s6s8 2 1 48 2 3s5s6) + s2s6 6 4s3s8 2 d0d22a13 +b1d0 d1 2 2 2d0d2 a1 4s1s8 + b1d2s5 9s1s8 2s52 2 1 2d0d2 b12 +a1d2 2 2 8s1s8 s32 + 4s22s8 3s6(2s2s5 +s1s6) s3 +2s22s26 2 b1d2s3 6b1d2s3 s2 4s1s3 +(s2s6 2s3s5) s 4s3s8 2 6 h d0d2a12 +b1d0d1a1 3d0d1d2 a12 +b1d0d1 3 2 3s1s8 3s1s5s6 i 2 9 1 1 27 1 18 1 18 1 72 h 1 18 2 9 1 36 1 24 1 36 1 54 1 3 1 3 1 6 1 6 1 6 1 2 (C.6) 6d02d22 2d0d2 2d0d2 1 3 1 3d0d1d2 b13 +a13d23 s2 9s1s3 2s22 d1d22a13 +b1 d1 2 3 3 2 3d0d1d2 a1 +b1d0 d1 2d0d2 2s22 2s53 2 i 3 1 3d0d1d2 a13 +b1d0 3 3 4s53 +9s1(3b1d2s1 2s5s8) 1 72 a1d0h16b12s2 9s1s3 2s22 d22 +3s8(s5s6 2s2s8) s62 4s3s8 +6b1 s6 6s3s52 +s6(9s1s6 8s2s5) +2 3 s22 +2s1s3 s6 8s2s3s5 s8 d2 1 18 1 1 36 72 (b1d1 +a1d2) 12b1d2s3 s2s3s5 + s22 6s1s3 s6 1 72 (b1d0 +a1d1)h2b1d2 26 4s3s8 s5 s62 +2s3s8 3s2s6s8 + s 26 4s3s8 s2s62 +s3(2s2s8 3s5s6) 21 2d0d2 a12 +b12d20 h2 s62 +2s3s8 s52 6s2s6s8s5 d0d1a12 +b1d0a1 2b1d2 s2 2s52 3s1s8 2 3s1s5s6 6 s52 +2s1s8 s32 + 2s8s22 +18s5s6s2 33s1s62 s3 +s22s62 The z^ component of the generating section is z^ = (s2b1 a1s5) 2 de ned as in (C.5). The x^ and y^ components are lengthy and are not given here. However, they are included in the Mathematica notebooks described in appendix A. D U( 1 ) U( 1 ) expressions The below formulas are for the U( 1 ) U( 1 ) model of [4], with some minor typos corrected. For the Weierstrass model, the f and g are given by f = 1 48 26 4s3s8 2 + 12 b1b2b3s3(2s3s5 s2s6) 1 3 a23b12b22 +a22b12b32 +a12b22b32 s 22 3s1s3 1 6 1 1 6 1 3 + (a1b2b3 +a2b1b3 +a3b1b2) s2s62 +s3(2s2s8 3s5s6) + 6 (b1a2a3 +b2a1a3 +b3a1a2) 2b1b2b3 s22 3s1s3 3s2s6s8 +s5 s62 +2s3s8 a2a3b1b2 +a22a3b1b3 +a1a3b1b22 +a12a3b2b3 +a1a2b1b32 +a12a2b2b32 (2s2s5 3s1s6) 2 2 2 2 2 2 a22a23b21 +a12a22b32 +b22a21a23 a1a2a32b1b2 a1a22a3b1b3 a12a2a3b2b3 s 25 3s1s8 (D.1) 2 (b2b3a1 +b3b1a2 +b1b2a3) 2 2 12b1b2b3s3 s2s3s5 + s 6s1s3 s6 + s 4s3s8 2 6 s2s62 +s3(2s2s8 3s5s6) b b a (b1a2a3 +b2a1a3 +b3a1a2) 2 6 4s3s8 s5 s62 +2s3s8 3s2s6s8 b b a 31 32 33 +b13b33a23 +b23b33a13 s2 9s1s3 2s22 b1b2a2a32 +b1b3a2a3 +a1a2b1b32 +a1a2b2b32 +a1a3b2b3 +a1a3b1b2 2 2 2 2 2 2 2 2 2 4b1b2b3s2 2s22 a1a2a3 16b12b22b23s2 9s1s3 2s22 2b1b2b3 6 s25 +2s1s8 s32 + 2s8s22 +18s5s6s2 33s1s62 s3 +s2s6 2 2 +6b1b2b3 s6 6s3s52 +s6(9s1s6 8s2s5) +2 3 s22 +2s1s3 s6 8s2s3s5 s8 4s3s8 72 1 36 1 b1b2a1a2a33 +b1b3a1a3a23 +b2b3a2a3a1 2 2 2 2 2 2 3 3s1s5s6 +s2 3s1s8 2s52 b1a1a22a32 +b2a2a12a32 +b3a3a1a2 2 2 2b1b2b3 s2 2s52 3s1s8 3s1s5s6 b b a a 41 22 22 43 +b14b32a24a32 +b22b34a14a22 +b12b34a24a12 +b24b32a14a32 +b12b24a34a12 s21 2 2 2 a1a2a3b2b3 +a2a1a3b1b3 +a3a1a2b1b2 4b1b2b3 2s22 3s1s3 s5 3s1s2s6 + s26 +2s3s8 s5 2 +18s2s6s8s5 6 s22 +2s1s3 s8 2 33s1s6s8 2 i 3s1s8 3s1s5s6 b2b33a13a22 +b1b33a23a12 +b2b3a13a32 +b1b23a33a12 +b1b3a23a32 +b1b2a3a2 3 3 3 3 2 2s22 a2a3b13b32 +a3a2b13b22 +a3a1b12b23 +a1a3b23b32 +a1a2b22b33 +a2a1b1b3 3 3 3 3 3 3 2 3 b a a 21 22 23 +b22a12a32 +b3a1a2 2 2 2 h 8b1b2b3 +2 s26 +2s3s8 s5 2 6s2s6s8s5 3s8 s1 s62 +8s3s8 3s22s8 i 6 2 2 2 xQ = b1b2s3 3 2 2 1 2 1 2 2 1 3 1 2 2 2 2 2 1 4 4 1 2 3 1 2 1 1 2 1 3 2 1 2 2 1 2 1 4 2 1 3 3 2 1 3 2 2 1 54 1 18 a31a32b33 +a23a33b13 +a1a3b2 3 3 3 9s1(3b1b2b3s1 2s5s8)+4s53 b1b2a1a22a33 +b1b2a2a12a33 +b1b3a1a32a23 +b3b1a1a3a23 +b2b3a2a32a13 +b2b3a3a2a1 2 2 2 2 2 2 2 2 3 9b1b2b3s12 +2s53 9s1s8s5 2 2 3 b21b22b23s21 + 2 9 b1b2b3s5 9s1s8 2s52 + 1 s28 s25 4 4s1s8 : There are two generating sections, Q and R. The Weierstrass components of Q are The components of the R section can be found by taking the Q components and performing a2 $ a3, b2 $ b3. a22b21 +a1b2 2 2 s26 +8s3s8 + a1a2b1b2 5s62 +4s3s8 2 2 2 a1a2(a2b1 +a1b2)s6s8 +a1a2s8; a2b2b3s5s8a14 + 2 2 a3b1b24(s3s5 +s2s6)a13 +a2a3b1b23(s2s8 s5s6)a13 + b2s3(b1b2b3s2 +s6s8)a13 2 3 a2a3b1b2s5s8a1 2 3 a2b2 b1b2b3(s5s6 +s2s8) 3s6s8 a1 + a2b22 2b1b2b3(s3s5 s2s6) s8 s62 +2s3s8 a 3 1 2 4 2 a3b1b2s2s3a1 a2a3b12b23(s3s5 +s2s6)a12 b1b2s3 s26 +2s3s8 a1 3 2 + a2b1b22 s36 +5s3s8s6 b1b2b3s2s3 a12 +a2a3b12b22(2s5s6 s2s8)a12 2 1 3 2 2 2 a2a3b1b2s5s8a1 4 2 2 a2b1b3s5s8a1 2 2 a2b1 b1b2b3(s5s6 +s2s8) 3s6s8 a1 a2b1b2 b1b2b3(s3s5 2s2s6)+s8 2s62 +s3s8 2 a21 +2a2a3b1b2s2s3a1 + b1b2s3s6a1 3 3 a2b1b2s3 2s62 +s3s8 a1 + a2a3b1s5s8a1 + a2b1b2 s36 +5s3s6s8 b1b2b3s2s3 a1 2 2 3 1 2 2 2 + a b 1 32 21 2b1b2b3(s3s5 s2s6) s8 s62 +2s3s8 2 4 2 a1 a2a3b1b2s2s3 3 2 3 2 2 4 a2b1b3s3s5 + a2b1b2s3s6 + a2a3b1b2(s3s5 +s2s6) a2b1b2s3 s62 +2s3s8 + a2b1s3(b1b2b3s2 +s6s8) 1 4 2 4 a2a3b1s2s8 1 4 2 1 2 1 2 2 1 4 3 2 1 6 1 2 3 2 3 2 2 (D.3) (D.5) 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. HJEP05(218) Phys. 17 (2013) 1195 [arXiv:1107.0733] [INSPIRE]. [4] M. Cvetic, D. Klevers, H. Piragua and W. Taylor, General U( 1 ) U( 1 ) F-theory compacti cations and beyond: geometry of unHiggsings and novel matter structure, JHEP 11 (2015) 204 [arXiv:1507.05954] [INSPIRE]. [5] D. Klevers and W. Taylor, Three-Index Symmetric Matter Representations of SU(2) in F-theory from Non-Tate Form Weierstrass Models, JHEP 06 (2016) 171 [arXiv:1604.01030] [INSPIRE]. [6] D. Klevers, D.R. Morrison, N. Raghuram and W. Taylor, Exotic matter on singular divisors in F-theory, JHEP 11 (2017) 124 [arXiv:1706.08194] [INSPIRE]. [7] T.W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [INSPIRE]. [8] J. Marsano, N. Saulina and S. Schafer-Nameki, A Note on G- uxes for F-theory Model Building, JHEP 11 (2010) 088 [arXiv:1006.0483] [INSPIRE]. [9] M.J. Dolan, J. Marsano, N. Saulina and S. Schafer-Nameki, F-theory GUTs with U( 1 ) Symmetries: Generalities and Survey, Phys. Rev. D 84 (2011) 066008 [arXiv:1102.0290] [10] D.S. Park and W. Taylor, Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry, JHEP 01 (2012) 141 [arXiv:1110.5916] [INSPIRE]. [11] D.S. Park, Anomaly Equations and Intersection Theory, JHEP 01 (2012) 093 [arXiv:1111.2351] [INSPIRE]. [12] D.R. Morrison and D.S. Park, F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds, JHEP 10 (2012) 128 [arXiv:1208.2695] [INSPIRE]. [13] M. Cvetic, T.W. Grimm and D. Klevers, Anomaly Cancellation And Abelian Gauge Symmetries In F-theory, JHEP 02 (2013) 101 [arXiv:1210.6034] [INSPIRE]. [14] C. Mayrhofer, E. Palti and T. Weigand, U( 1 ) symmetries in F-theory GUTs with multiple sections, JHEP 03 (2013) 098 [arXiv:1211.6742] [INSPIRE]. [15] M. Cvetic, D. Klevers and H. Piragua, F-Theory Compacti cations with Multiple U( 1 )-Factors: Constructing Elliptic Fibrations with Rational Sections, JHEP 06 (2013) 067 [arXiv:1303.6970] [INSPIRE]. [16] V. Braun, T.W. Grimm and J. Keitel, Geometric Engineering in Toric F-theory and GUTs with U( 1 ) Gauge Factors, JHEP 12 (2013) 069 [arXiv:1306.0577] [INSPIRE]. [17] M. Cvetic, A. Grassi, D. Klevers and H. Piragua, Chiral Four-Dimensional F-theory Compacti cations With SU(5) and Multiple U( 1 )-Factors, JHEP 04 (2014) 010 [arXiv:1306.3987] [INSPIRE]. [18] J. Borchmann, C. Mayrhofer, E. Palti and T. Weigand, SU(5) Tops with Multiple U( 1 )s in F-theory, Nucl. Phys. B 882 (2014) 1 [arXiv:1307.2902] [INSPIRE]. [19] M. Cvetic, D. Klevers, H. Piragua and P. Song, Elliptic brations with rank three Mordell-Weil group: F-theory with U( 1 ) U( 1 ) U( 1 ) gauge symmetry, JHEP 03 (2014) 021 [arXiv:1310.0463] [INSPIRE]. [20] I. Antoniadis and G.K. Leontaris, F-GUTs with Mordell-Weil U( 1 )'s, Phys. Lett. B 735 (2014) 226 [arXiv:1404.6720] [INSPIRE]. Mordell-Weil group, arXiv:1406.5174 [INSPIRE]. [21] M. Kuntzler and S. Schafer-Nameki, Tate Trees for Elliptic Fibrations with Rank one [22] D. Klevers, D.K. Mayorga Pena, P.-K. Oehlmann, H. Piragua and J. Reuter, F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches, JHEP 01 (2015) 142 [arXiv:1408.4808] [INSPIRE]. [23] M. Esole, M.J. Kang and S.-T. Yau, A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations, arXiv:1410.0003 [24] C. Lawrie and D. Sacco, Tate's algorithm for F-theory GUTs with two U( 1 )s, JHEP 03 [INSPIRE]. [INSPIRE]. (2015) 055 [arXiv:1412.4125] [INSPIRE]. [25] C. Lawrie, S. Schafer-Nameki and J.-M. Wong, F-theory and All Things Rational: Surveying U( 1 ) Symmetries with Rational Sections, JHEP 09 (2015) 144 [arXiv:1504.05593] [26] D.R. Morrison and D.S. Park, Tall sections from non-minimal transformations, JHEP 10 (2016) 033 [arXiv:1606.07444] [INSPIRE]. [27] D.K. Mayorga Pena and R. Valandro, Weak coupling limit of F-theory models with MSSM spectrum and massless U( 1 )'s, JHEP 03 (2018) 107 [arXiv:1708.09452] [INSPIRE]. [28] M. Cvetic and L. Lin, The Global Gauge Group Structure of F-theory Compacti cation with U( 1 )s, JHEP 01 (2018) 157 [arXiv:1706.08521] [INSPIRE]. [29] W. Buchmuller, M. Dierigl, P.K. Oehlmann and F. Ruhle, The Toric SO(10) F-theory Landscape, JHEP 12 (2017) 035 [arXiv:1709.06609] [INSPIRE]. [30] M.J. Dolan, J. Marsano and S. Schafer-Nameki, Uni cation and Phenomenology of F-theory GUTs with U( 1 )P Q, JHEP 12 (2011) 032 [arXiv:1109.4958] [INSPIRE]. [31] V. Braun, T.W. Grimm and J. Keitel, New Global F-theory GUTs with U( 1 ) symmetries, JHEP 09 (2013) 154 [arXiv:1302.1854] [INSPIRE]. [32] S. Krippendorf, S. Schafer-Nameki and J.-M. Wong, Froggatt-Nielsen meets Mordell-Weil: A Phenomenological Survey of Global F-theory GUTs with U( 1 )s, JHEP 11 (2015) 008 [arXiv:1507.05961] [INSPIRE]. [33] D.R. Morrison and C. Vafa, Compacti cations of F-theory on Calabi-Yau threefolds. 1, Nucl. [34] D.R. Morrison and C. Vafa, Compacti cations of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [INSPIRE]. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE]. [35] M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 JHEP 08 (2011) 094 [arXiv:1106.3854] [INSPIRE]. [hep-th/9606008] [INSPIRE]. 080 [arXiv:1512.05791] [INSPIRE]. [39] P.-K. Oehlmann, personal communication. [40] F. Denef, Les Houches Lectures on Constructing String Vacua, Les Houches 87 (2008) 483 [arXiv:0803.1194] [INSPIRE]. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE]. arXiv:1104.2051 [INSPIRE]. (2011) 48 [arXiv:1008.4133] [INSPIRE]. [41] T. Weigand, Lectures on F-theory compacti cations and model building, Class. Quant. Grav. [42] W. Taylor, TASI Lectures on Supergravity and String Vacua in Various Dimensions, [43] T.W. Grimm, The N = 1 e ective action of F-theory compacti cations, Nucl. Phys. B 845 [44] F. Bonetti and T.W. Grimm, Six-dimensional (1; 0) e ective action of F-theory via M-theory on Calabi-Yau threefolds, JHEP 05 (2012) 019 [arXiv:1112.1082] [INSPIRE]. [45] J.H. Silverman, The Arithmetic of Elliptic Curves, Springer, (1986). [46] V. Braun and D.R. Morrison, F-theory on Genus-One Fibrations, JHEP 08 (2014) 132 [arXiv:1401.7844] [INSPIRE]. arXiv:1404.1527 [INSPIRE]. [hep-th/9304104] [INSPIRE]. [47] D.R. Morrison and W. Taylor, Sections, multisections and U( 1 ) elds in F-theory, [48] L.B. Anderson, I. Garc a-Etxebarria, T.W. Grimm and J. Keitel, Physics of F-theory compacti cations without section, JHEP 12 (2014) 156 [arXiv:1406.5180] [INSPIRE]. [49] M. Cvetic, R. Donagi, D. Klevers, H. Piragua and M. Poretschkin, F-theory vacua with Z3 gauge symmetry, Nucl. Phys. B 898 (2015) 736 [arXiv:1502.06953] [INSPIRE]. [50] R. Wazir, Arithmetic on elliptic threefolds, Compos. Math. 140 (2004) 567 [math/0112259]. [51] S. Lang and A. Neron, Rational points of abelian varieties over function elds, Am. J. MAth. [52] J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [53] S. Monnier, G.W. Moore and D.S. Park, Quantization of anomaly coe cients in 6D N = (1; 0) supergravity, JHEP 02 (2018) 020 [arXiv:1711.04777] [INSPIRE]. [54] V. Kumar, D.S. Park and W. Taylor, 6D supergravity without tensor multiplets, JHEP 04 (2011) 080 [arXiv:1011.0726] [INSPIRE]. [55] S.D. Cutkosky, Resolution of Singularities, American Mathematical Society, (2004). JHEP 07 (2014) 028 [arXiv:1402.4054] [INSPIRE]. [1] S.H. Katz and C. Vafa , Matter from geometry, Nucl. Phys. B 497 ( 1997 ) 146 [2] D.R. Morrison and W. Taylor , Matter and singularities, JHEP 01 ( 2012 ) 022 [3] M. Esole and S.-T. Yau, Small resolutions of SU(5)-models in F-theory, Adv . Theor. Math. [37] V. Sadov , Generalized Green-Schwarz mechanism in F-theory , Phys. Lett. B 388 ( 1996 ) 45 [38] L.B. Anderson , J. Gray , N. Raghuram and W. Taylor , Matter in transition, JHEP 04 ( 2016 ) [56] W. Taylor and A.P. Turner , An in nite swampland of U(1) charge spectra in 6D supergravity [57] Y.-N. Wang , Tuned and Non-Higgsable U(1)s in F-theory , JHEP 03 ( 2017 ) 140 [58] F. Baume , M. Cvetic , C. Lawrie and L. Lin , When rational sections become cyclic: Gauge enhancement in F-theory via Mordell-Weil torsion , JHEP 03 ( 2018 ) 069 [59] A.P. Braun , A. Collinucci and R. Valandro , The fate of U(1)'s at strong coupling in F-theory,

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP05%282018%29050.pdf

Nikhil Raghuram. Abelian F-theory models with charge-3 and charge-4 matter, Journal of High Energy Physics, 2018, 50, DOI: 10.1007/JHEP05(2018)050