Abelian Ftheory models with charge3 and charge4 matter
JHE
Abelian Ftheory models with charge3 and charge4
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) Ftheory 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 nonUFD structure and thereby deviate from the standard MorrisonPark form. Techniques used to tune SU(N ) models on singular divisors allow us to determine the nonUFD 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 Ftheory example with charge4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4.
FTheory; Superstring Vacua

HJEP05(218)
1 Introduction
2 Overview of abelian gauge groups in Ftheory
Elliptic curve group law
Rational sections, the abelian sector, and the MordellWeil group
HJEP05(218)
Charged matter
2.4 Anomaly cancellation
3 Charge3 models
Tuning abelian models
NonUFD 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 charge3 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 Charge4 models
Higgsing the U(
1
) U(
1
) construction
4.2 Structure of the charge4 construction
Matter spectra
5 Comments on q > 4
6 Conclusions and future directions
A Mathematica les
C Charge4 expressions
C.1 P2 form
C.2 Weierstrass form
D U(
1
) U(
1
) expressions
Introduction
A key objective of the Ftheory program is determining which charged matter
representations can arise in Ftheory models, a task with important implications for the landscape
and swampland. Clearly, we cannot characterize the full landscape of Ftheory models
without knowing all of the representations that can be realized in Ftheory. At the same
time, one may nd that certain representations cannot be obtained in Ftheory, even when
the corresponding matter spectra satisfy the known lowenergy 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 lowenergy
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
Ftheory 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 Ftheory models lies in the
different manifestations of nonabelian and abelian symmetries. Ftheory models in 12
2d
dimensions are constructed using a CalabiYau dfold that is an elliptic
bration over a
base B. Nonabelian 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 nonabelian
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 Ftheory 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 nonabelian 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 nonabelian groups do not
carry over to abelian groups in an immediately obvious way, making the construction of
Ftheory models with abelian gauge symmetries more di cult. Take, for example, the
question of how to construct an Ftheory model with a single U(
1
) gauge group and no
additional nonabelian 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 MorrisonPark form, posing the related question of whether we can understand how
and why the structures di er. While there has been some discussion of Ftheory 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 MorrisonPark 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 Ftheory
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 nonabelian 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
Ftheory construction by deforming a previous U(
1
)
U(
1
) construction from [4]. To the
author's knowledge, this is the rst published Ftheory 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 nonUFD behavior.
The rest of this paper is organized as follows. Section 2 reviews some aspects of abelian
groups in Ftheory 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 Ftheory
In this section, we review those aspects of Ftheory 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 Ftheory can be found in [40{42].
Ftheory can be described from either a Type IIB perspective or an Mtheory
perspective. In the Type IIB view, an Ftheory model can be thought of as a Type IIB
compacti cation in which the presence of 7branes causes the axiodilaton to vary over
the compacti cation space. The axiodilaton is represented as the complex structure of
an elliptic curve, and the Ftheory 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 CalabiYau 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 Ftheory 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
nonabelian gauge groups, while codimension two loci with singular bers are associated
with charged matter.
Ftheory can also be understood via its duality with Mtheory. To illustrate the idea,
let us rst consider Mtheory 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 Mtheory on a
smooth, elliptically
bered CY dfold. 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 Ftheory model on an elliptically bered CY dfold.
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 MordellWeil group
Unlike the nonabelian 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 Ftheory 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 MordellWeil group, under the addition operation
described in section 2.1, with o^ serving as the identity [50]. According to the MordellWeil
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
MordellWeil rank.
If an elliptic bration has MordellWeil rank r, the abelian sector of the corresponding
Ftheory model includes a U(
1
)r gauge algebra [11, 12, 34]. The justi cation for this
statement is most easily seen in the dual Mtheory picture, as discussed in [11]. For
concreteness, let us restrict ourselves to 6D Ftheory 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 nonabelian
symmetry. Consider Mtheory compacti ed on a resolved elliptically
bered CalabiYau
threefold X~ . Mtheory on X~ is a 5D model that, in the Ftheory limit, leads to a 6D N = 1
Ftheory model. According to Poincare duality, there is a harmonic twoform ! for every
fourcycle
in X~ . The twoforms serve as zeromodes for the Mtheory threeform C3,
and we can expand C3 using a basis of twoforms. In other words, we write C3 as a sum of
terms of the form A ^ !; the oneforms A represent vectors in the 5D theory. Thus, to nd
the vectors of the 6D Ftheory model, we consider a basis of fourcycle homology classes of
X~ , nd the corresponding 5D vectors A, and track the sources of these 5D vectors in the
6D Ftheory model.
When there are no codimension one singularities (apart from I1 singularities), there
are three types4 of fourcycle 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 fourcycle homology class that
corresponds to the Cartan gauge bosons of a nonabelian gauge group in the Ftheory 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 MordellWeil group, and the homology classes B
that come from
bering the elliptic curve over twocycles in the base. 5D vectors associated with Z and
B do not correspond to gauge bosons in the 6D Ftheory model. Instead, they arise from
the KK reduction of either the metric or tensors in the 6D Ftheory 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 fourcycles. This is done using the TateShioda
map , which is a homomorphism from the MordellWeil group to the homology group of
fourcycles. For a situation with no codimension one singularities, the TateShioda 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 TateShioda 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 Ftheory 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 rankone 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 nonabelian 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 Mtheory
picture, charged matter arises from M2 and antiM2 branes wrapping this extra component.
To calculate the charge of this matter, we must examine the M2 brane worldvolume
action. The action contains a term of the form R C3, where the integral is over the M2
brane worldvolume. 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 twoform 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 worldline, thereby giving the action
for charged matter. The charge comes from integrating the twoform ! associated with the
U(
1
) gauge boson A. However, for a CY nfold, 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 antiM2 brane. In
situations without additional nonabelian 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 illde 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 GreenSchwarz 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 lowenergy
considerations, they do have a geometric interpretation in Ftheory [11], and the conditions can be
written in terms of parameters describing the Ftheory 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 Ftheory model, the MordellWeil 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 gravitationalgauge 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 Ftheory
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 Ftheory realization: it describes
the loci where the three components of the section vanish, leaving the section illde ned.
Moreover, every term in the sum on the righthand side is nonnegative, 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) Ftheory models, although we leave
an indepth exploration of any connection to future work.
3
Charge3 models
While there is a previous Ftheory 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 wellknown MorrisonPark construction [12]. In [26], it was shown
that the KMOPR Weierstrass model is birationally equivalent to one in MorrisonPark
form, although the MorrisonPark form Weierstrass model does not satisfy the CalabiYau
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 nontrivial MordellWeil 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 nonabelian contexts, models
with gauge groups tuned on singular divisors can have nonUFD 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 MorrisonPark models.
We rst describe the tuning process for abelian models and illustrate the procedure by
rederiving the MorrisonPark 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 nonabelian 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 lefthand 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 nonabelian 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 righthand side of this equation already matches the righthand 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 nonsplit 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 MorrisonPark 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 MorrisonPark 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 MorrisonPark 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 nonUFD if the variety V corresponding to
I is singular. For the abelian tuning process, we can have a nonUFD 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 wellde 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 wellde 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 nonUFD solutions involves
starting with the UFD solutions in the normalized intrinsic ring and determining how to
make these expressions wellde 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 onedimensional (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 twodimensional,
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 nonUFD 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 lefthand 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 wellde 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 wellde ned expressions in R=hz^i. In
fact, for the z^ considered here, T~2 and T~3 are wellde ned in R=hz^i only if T~ is wellde ned
x0 := t2
y0 := t3;
(3.30)
where t is wellde 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 wellde ned in R=hz^i. To obtain a nontrivial
tuning, we should not tune
in a way that makes ~ wellde ned in R=hz^i. Therefore, in
order for both x1 and t to be wellde 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 nonUFD 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 wellde 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 wellde 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 MorrisonPark 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 lefthand 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 wellde 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 wellde 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 wellde 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 CalabiYau 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
CalabiYau condition. This result seems to be a q = 4 analogue of the statement in [26]
that the MorrisonPark 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 illde 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 illde 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 Ftheory 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 MorrisonPark 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 MorrisonPark 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 nonabelian 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 Ftheory 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 Ftheory model has a rankone
MordellWeil group with no additional nonabelian gauge groups. Let us denote the generating
section as s^. If this Ftheory 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 TateShioda 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 MordellWeil 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 MorrisonPark 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 MorrisonPark 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 MorrisonPark 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
) Ftheory 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 nonUFD structure that
deviates from the standard MorrisonPark form. With the aid of the normalized intrinsic
ring, we were able to
nd the appropriate nonUFD 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 nonabelian symmetry. [56] gives examples of the eldtheoretic
Higgsing processes that could produce q > 4 matter.
However, it can be di cult to
identify the deformations of Ftheory models corresponding to a speci c Higgsing. A better
understanding of the Ftheory 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 Ftheory 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 nonabelian contexts where this
situation occurs, there are matter transitions connecting the vacua with di erent matter
spectra [38]. Abelian Ftheory 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 nonabelian symmetries in Ftheory, 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 nonabelian representations, including the 5 representation of SU(2),
cannot be realized in Ftheory. 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 Ftheory model does not by itself guarantee the existence of an SU(2)
model with 5 matter. For instance, the wouldbe 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 Ftheory models. For instance, [27, 59] analyze
U(
1
) models, including the MorrisonPark 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 DESC0012567.
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
Charge4 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 (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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