11institutetext: Department of Physics
Robeson Hall, 0435
Virginia Tech
850 West Campus Drive
Blacksburg, VA 24061, USA22institutetext: Department of Physics and Astronomy,
University of Pennsylvania,
Philadelphia, PA 19104, USA

# Orders of Vanishing and U(1) Charges in F-theory

Nikhil Raghuram 2 and Andrew P. Turner  raghuram.nikhil at gmail.com turnerap
at sas.upenn.edu

###### Abstract

Many interesting questions about F-theory models, including several concerning
the F-theory swampland, involve massless matter charged under U(1) gauge
symmetries. It is therefore important to better understand the geometric
properties of F-theory models realizing various U(1) charges. We propose that,
for F-theory models described by elliptic fibrations in Weierstrass form, the
U(1) charge of light matter is encoded in the orders of vanishing of the
section components corresponding to the U(1) gauge symmetry. We give specific
equations relating the U(1) charges to the orders of vanishing that seem to
hold for both U(1)-charged singlets and for matter additionally charged under
a simply-laced nonabelian gauge algebra. Our formulas correctly describe
properties of F-theory models in the prior literature, and we give an argument
that they should describe the orders of vanishing for arbitrarily high U(1)
charges. They also resemble formulas for the $p$-adic valuations of elliptic
divisibility sequences developed by Stange StangeEllTrouble . These proposals
could serve as a U(1) analogue of the Katz–Vafa method, allowing one to
determine U(1) charges without resolution. Additionally, they predict
geometric information about F-theory models with general U(1) charges, which
may be useful for exploring the F-theory landscape and swampland.

##  1 Introduction

A major goal of the string theory program is understanding how to construct a
compactification of string theory realizing a desired massless spectrum. This
problem is important for both characterizing the landscape of string
compactifications and finding stringy realizations of our observed universe.
F-theory VafaF-theory ; MorrisonVafaI ; MorrisonVafaII  has emerged as a
powerful tool for constructing string compactifications in large part because
it geometrizes several physical features. Much of the information about an
F-theory model’s massless spectrum, particularly regarding its unbroken gauge
symmetry and its light charged matter, is encoded in the mathematical
properties of its corresponding elliptic fibration. To be more specific, an
F-theory compactification down to $12-2d$ real dimensions is described by a
complex $d$-dimensional elliptically fibered Calabi–Yau manifold. Massless
nonabelian gauge bosons are supported along divisors in the base with singular
elliptic fibers. After resolution, the components in the fibers at codimension
one intersect in the pattern of an affine Dynkin diagram, in line with the
Kodaira classification of singularity types KodairaII . When supplemented with
information about monodromy TateMath , this affine Dynkin diagram tells us
which nonabelian gauge algebra is supported along this locus MorrisonVafaI ;
MorrisonVafaII ; BershadskyEtAlSingularities . Light charged matter,
meanwhile, is typically localized at codimension-two loci in the base where
the singularity type enhances. The representation of this charged matter can
be determined by resolving the singularities and determining how the
introduced fiber components at codimension two intersect the resolution
divisors for the codimension-one singularities.

While properly determining the nonabelian gauge algebra and matter
content111Since this paper focuses on massless matter, we typically omit the
word “massless” and use the term “matter” to refer to massless matter. with
this approach requires resolving singularities,222The string junction program
offers a valid method for determining nonabelian gauge groups and matter
representations that does not require resolutions. (See, for instance,
GrassiHalversonShanesonPhysics ; GrassiHalversonShanesonMath .) We do not
significantly discuss this approach here, although it would be interesting to
investigate how the results obtained here tie into these methods. one can
often read off singularity types and their associated physical data through
easier methods. If the elliptic fibration is in the Weierstrass form

$y^{2}=x^{3}+fxz^{4}+gz^{6}\,,$ (1)

the Kodaira table (given in Table 4) relates the singularity types and their
corresponding Lie algebras to the orders of vanishing of $f$, $g$, and
$\Delta=4f^{3}+27g^{2}$ at a particular divisor in the base. Furthermore, even
though the Kodaira classification strictly holds only at codimension one, we
can often heuristically use the Kodaira table to find singularity types at
codimension two in the base as well. One can then determine matter
representations with the Katz–Vafa method KatzVafa , in which one breaks the
adjoint of the enhanced singularity type’s corresponding Lie algebra to
representations of the nonabelian gauge algebra. These techniques allow one to
calculate the gauge group and charged matter content of a model simply by
considering orders of vanishing, making the process of constructing and
analyzing nonabelian F-theory models significantly easier. In addition to
letting us quickly determine the massless spectrum of a given model, they can
guide the process of constructing an F-theory model with a desired nonabelian
gauge algebra and charged matter spectrum by describing expected features of
the elliptic fibration.

However, many interesting questions in the F-theory program involve abelian
gauge algebras and light matter charged under them. Clearly, $\mathfrak{u}(1)$
algebras are important for phenomenological model building in F-theory: the
Standard Model gauge algebra has a $\mathfrak{u}(1)$ factor, and many
proposals for extending the Standard Model include extra $\mathfrak{u}(1)$
algebras RizzoZprime ; LangackerZprime ; GrimmWeigand ; HalversonTASI . They
also offer an interesting arena for questions about the landscape and
swampland VafaSwamp  of F-theory models. As an example, consider 6D F-theory
models with a $\operatorname{U}(1)$ gauge group. It has not yet been
definitively determined which $\operatorname{U}(1)$ charges of massless matter
can and cannot occur in such models, even though there has been significant
progress on this problem and more general questions regarding constraints on
$\operatorname{U}(1)$ theories ParkTaylor ; ParkIntersection ; MorrisonParkU1
; KleversEtAlToric ; MonnierMoorePark ; RaghuramTaylorLargeCharge ;
CianciMayorgaPenaValandroHighU1 ; LeeWeigandAbelianSwamp ;
CollinucciEtAlHighCharge ; AndersonGrayOehlmannQuotients . However, there are
infinite families of charged matter spectra with unbounded
$\operatorname{U}(1)$ charges that are consistent with the known 6D low-energy
constraints ParkTaylor ; TaylorTurnerSwamp . Only a finite number of these
matter spectra can be realized in F-theory models
KumarMorrisonTaylorGlobalAspects , presenting the possibility of an infinite
swampland of such models. (See RaghuramTaylorTurnerEnhancement  for recent
developments regarding some of these infinite families.)

For these reasons, much work has focused on $\mathfrak{u}(1)$ charged matter
in F-theory constructions, both for matter charged only under
$\mathfrak{u}(1)$ algebras and matter additionally charged under nonabelian
algebras GrimmWeigand ; DolanMarsanoSaulinaSchaferNameki ;
MarsanoSaulinaSchaferNamekiU1 ; MorrisonParkU1 ; CveticGrimmKlevers ;
MayrhoferPaltiWeigand ; BorchmannSU5TopSummary ; CveticKleversPiraguaMultU1 ;
GrimmKapferKeitelRational ; BraunGrimmKeitel ; CveticGrassiKleversPiragua ;
BorchmannSU5Top ; CveticKleversPiraguaSong ; KrippendorfEtAlGUT ;
BraunCollinucciValandro ; AntoniadisLeontaris ; KuntzlerTateTrees ;
KleversEtAlToric ; EsoleKangYau ; LawrieSacco ; LawrieEtAlRational ;
CveticKleversPiraguaTaylor ; GrimmKapferKleversArithmetic ; CveticEtAlHetU1 ;
MorrisonParkTall ; MorrisonParkTaylor ; WangU1s ; CveticLinU1 ;
MayorgaPenaValandro ; BuchmullerEtAlSO10 ; BaumeEtAlTorsion ; Raghuram34 ;
KimuraK3U1a ; LeeRegaladoWeigand ; CianciMayorgaPenaValandroHighU1 ;
TaylorTurnerGeneric ; KimuraK3U1b ; LeeWeigandAbelianSwamp ;
CollinucciEtAlHighCharge ; KimuraZ4 ; KimuraHalf3Fold ; KimuraHalfFourfold ;
OehlmannSchimannek ; RaghuramTaylorTurnerSM ; KnappScheideggerSchimannek .
However, there are still open questions regarding $\mathfrak{u}(1)$ algebras
in F-theory, in part because, unlike nonabelian algebras, they are not
associated with codimension-one singularities of the fibration. Instead, they
are associated with extra rational sections of the elliptic fibration
MorrisonVafaII . In more concrete terms, a rational section of a fibration in
Weierstrass form is described by a solution

$[x:y:z]=[\hat{x}:\hat{y}:\hat{z}]$ (2)

of the Weierstrass equations, where $\hat{x}$, $\hat{y}$, $\hat{z}$ are
sections of appropriate line bundles on the base. An elliptic fibration may
admit several or even an infinite number of such sections. They form a
finitely generated group under an operation known as elliptic curve addition.
There is a $\mathfrak{u}(1)$ algebra corresponding to each generating section
of this group with infinite order. Matter charged under a $\mathfrak{u}(1)$
algebra still occurs at codimension-two loci in the base with singular fibers,
but the $\mathfrak{u}(1)$ charge is determined by how the section intersects
the resolved singular fiber at the matter locus. Since $\mathfrak{u}(1)$
algebras involve sections of the elliptic fibration, much of the technology
for nonabelian gauge algebras does not carry over. The Katz–Vafa method, for
instance, cannot be used to read off $\mathfrak{u}(1)$ charges, as matter with
different $\mathfrak{u}(1)$ charges can occur at codimension-two loci with the
same singularity type. This can be seen, for instance, in the Morrison–Park
model MorrisonParkU1 , which has a single $\mathfrak{u}(1)$ gauge factor: in
this model, the $\bm{1}_{1}$ loci and $\bm{1}_{2}$ loci are all of type
$\text{I}_{2}$, where $(f,g,\Delta)$ vanish to orders $(0,0,2)$.

To the authors’ knowledge, the prior F-theory literature does not present any
systematic procedure analogous to the Katz–Vafa method for reading off
$\mathfrak{u}(1)$ charges from simple features such as orders of vanishing.
However, there were tentative indications in Raghuram34  that, at least for
singlets, the $\mathfrak{u}(1)$ charge is encoded in orders of vanishing of
the $\hat{x}$, $\hat{y}$, and $\hat{z}$ section components at a matter locus.
It was noted there that in Weierstrass models with a $\operatorname{U}(1)$
gauge symmetry admitting $\bm{1}_{3}$ and $\bm{1}_{4}$ matter, the section
components, particularly $\hat{z}$, vanish to orders larger than 1 at the
corresponding matter loci. The work also argued that the complicated non-UFD
structure KleversEtAlExotic  of these models directly reflects these higher
orders of vanishing. These observations naturally suggest some correlation
between the singlet charge and the orders of vanishing of the section
components; in fact, Raghuram34  hypothesized a specific relation between
these quantities.

Our goal here is to develop these observations into a systematic framework
that could serve as a Katz–Vafa analogue for $\mathfrak{u}(1)$ charges.
Specifically, we aim to find rules relating $\mathfrak{u}(1)$ charges to the
orders of vanishing of the $\hat{x}$, $\hat{y}$, $\hat{z}$, and
$\hat{w}=3\hat{x}^{2}+f\hat{z}^{4}$ components333When an elliptic fibration is
in Weierstrass form, singularities occur at points on the fiber where
$y=3x^{2}+fz^{4}=0$. Thus, $\hat{w}$ provides valuable information about where
and how a section hits a singular point on a fiber. of the generating section.
In addition to singlet matter, we consider $\mathfrak{u}(1)$-charged matter
that is also charged under a semi-simple nonabelian gauge algebra. To make the
scope of the analysis more manageable, we assume that the nonabelian part of
the gauge algebra is simply-laced.444Some speculative thoughts on non-simply-
laced situations are presented in Section 13. Additionally, we only consider
matter in generic TaylorTurnerGeneric  representations of the nonabelian gauge
factors, and we focus on matter loci where the elliptic fiber singularity type
undergoes a rank-one enhancement.555There are notable examples even for
$\mathfrak{u}(1)$-charged singlet matter where the singularity type undergoes
higher-rank enhancement, such as Grassi:2021wii , where the
$\mathfrak{u}(1)$-charged singlets are all supported at codimension-two
$\text{II}\to\text{IV}$ loci. Even with these simplifying assumptions, we
consider several matter representations commonly found in F-theory
models:666We do not significantly analyze matter supported along type III or
IV loci. It would be interesting to investigate these situations in future
work.

  * •

the singlet representation occurring at codimension-two
$\text{I}_{1}\to\text{I}_{2}$ loci;

  * •

the fundamental and antisymmetric representations of $\mathfrak{su}(n)$
occurring, respectively, at codimension-two
$\text{I}_{n}^{s}\to\text{I}_{n+1}$ and
$\text{I}_{n}^{s}\to\text{I}^{*}_{n-4}$ loci;

  * •

the vector representations of $\mathfrak{so}(2n)$ occurring at codimension-two
$\text{I}_{n-4}^{*s}\to\text{I}_{n-3}^{*}$ loci;

  * •

the spinor representations of $\mathfrak{so}(8)$, $\mathfrak{so}(10)$,
$\mathfrak{so}(12)$, and $\mathfrak{so}(14)$ occurring, respectively, at
codimension-two $\text{I}_{0}^{*s}\to\text{I}_{1}^{*}$,
$\text{I}_{1}^{*s}\to\text{IV}^{*}$, $\text{I}_{2}^{*s}\to\text{III}^{*}$, and
$\text{I}_{3}^{*s}\to\text{II}^{*}$ loci;

  * •

the $\bm{27}$ representation of $\mathfrak{e}_{6}$ occurring at codimension-
two $\text{IV}^{*s}\to\text{III}^{*}$ loci;

  * •

and the $\bm{56}$ representation of $\mathfrak{e}_{7}$ occurring at
codimension-two $\text{III}^{*}\to\text{II}^{*}$ loci.

We propose a set of formulas, described in Section 2, that seem to correctly
relate $\mathfrak{u}(1)$ charges and orders of vanishing for all of these
cases. The most important of these formulas describes the order of vanishing
of $\hat{z}$ at a codimension-two matter locus. To illustrate the basic idea,
consider matter that occurs at the intersection of a gauge divisor supporting
a simple nonabelian gauge algebra $\mathfrak{g}$ with the residual
$\text{I}_{1}$ discriminant locus. Suppose that, at this matter locus, the
singularity type enhances from one associated with $\mathfrak{g}$ to one
associated with an enhanced Lie algebra $\mathfrak{h}$. Then, if $G$ and $H$
are the universal covering groups of $\mathfrak{g}$ and $\mathfrak{h}$, the
order of vanishing for $\hat{z}$ at the matter locus,
$\operatorname{ord}_{2}(\hat{z})$, is schematically given by

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{G}}{d_{H}}q^{2}+\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{G}}\right)_{\mathcal{I}\mathcal{I}}-\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{H}}\right)_{\mathcal{J}\mathcal{J}}}\right)\,,$
(3)

where $q$ is the $\mathfrak{u}(1)$ charge,
$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{G}}\right)_{\mathcal{I}\mathcal{I}}$
and
$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{H}}\right)_{\mathcal{J}\mathcal{J}}$
are particular diagonal elements of the inverse Cartan matrices for $G$ and
$H$, and $d_{G}$ and $d_{H}$ are the orders (number of elements) of the
centers of $G$ and $H$.

We present three lines of evidence in support of these proposals. First, they
are satisfied by independently derived F-theory models from the prior
literature with $\mathfrak{u}(1)$ gauge algebras, as discussed in Section 12.
Yet these previous F-theory models support only relatively small
$\mathfrak{u}(1)$ charges, and we would like to test the formulas over a large
range of charges. Fortunately, a previously used strategy MorrisonParkU1 ;
Raghuram34 , which is reviewed in Section 5, allows us to probe the behavior
of models supporting large charges without explicitly constructing them.
Roughly, the $\mathfrak{u}(1)$ charge of matter is determined by how a
generating section $\hat{s}$ of an elliptic fibration behaves at a matter
locus in the base. One can also consider sections $m\hat{s}$ that are
multiples of the generating section under elliptic curve addition. If a model
admits matter with $\mathfrak{u}(1)$ charge $q$, then at the matter locus, the
non-generating section $m\hat{s}$ behaves as though it were a generating
section supporting charge $mq$. Therefore, if we wish to obtain the orders of
vanishing for generating section components in models supporting large
charges, we can start with a model admitting smaller charges and examine
multiples of the generating section. When we perform this analysis for the
various types of charged matter mentioned above, the resulting orders of
vanishing agree exactly with the proposed formulas. Finally, similar formulas
describe the $p$-adic valuations of elliptic divisibility sequences (EDSs)
associated with singular elliptic curves StangeEllTrouble , although these
formulas are written in a somewhat different format. Elliptic divisibility
sequences are closely related to the procedure above involving multiples of
generating sections, suggesting that the formulas from StangeEllTrouble
should resemble our proposals, as observed.

These three points provide strong evidence in favor of the proposals, but they
do not constitute a formal proof. We will not attempt to give such a proof
here. One might in fact expect that the proposals hold only heuristically,
given that the analogous Katz–Vafa method (at least as described above) is
itself somewhat heuristic KatzVafa ; GrassiMorrisonGroupReps ;
MorrisonTaylorMaS ; GrassiMorrison ; EsoleKangSpell ; Grassi:2018rva . In
particular, both methods involve applying the Kodaira classification at
codimension two in the base; even though this often gives correct results, the
Kodaira classification strictly holds only at codimension one. It would be
important in future work to more properly establish our proposals and
determine their exact range of validity.

However, even if they only hold heuristically, the proposals offer many
potential benefits. Just as the Katz–Vafa method allows one to quickly
determine nonabelian representations without resolution, these proposals would
allow one to easily determine $\mathfrak{u}(1)$ charges. Moreover, if one
wishes to find an F-theory model realizing particular $\mathfrak{u}(1)$
charges, these formulas predict properties of the model’s generating section.
Said another way, the formulas provide properties to aim for when constructing
an F-theory model with desired $\mathfrak{u}(1)$ charges. Since the proposals
give the orders of vanishing for arbitrary charges, they could be an
invaluable tool for exploring the F-theory landscape and swampland. Of course,
they are intrinsically interesting for the mathematics of elliptic fibrations,
particularly given their connection to $p$-adic valuations of elliptic
divisibility sequences.

The formulas also exemplify the general theme that simple representations of
light matter are easier to realize in string models than more complicated ones
IbanezUrangaSimpleReps ; TaylorTurnerGeneric . This contrasts with the
situation in quantum field theory: while one must still satisfy conditions
such as anomaly cancellation, the process of writing down a quantum field
theory with complicated matter representations is not significantly more
difficult than that for simpler representations. One can observe these ideas
when working with nonabelian gauge symmetries, for which the more complicated
representations tend to have larger dimensions. F-theory models with
$\mathfrak{su}(2)$ algebras, for instance, almost automatically support light
matter in the fundamental ($\bm{2}$) and adjoint ($\bm{3}$) representations.
By contrast, the $\bm{4}$ representation only occurs when the
$\mathfrak{su}(2)$ algebra is tuned in an intricate way KleversTaylor ;
KleversEtAlExotic . Even though matter representations with larger
$\mathfrak{u}(1)$ charges do not necessarily have larger dimensions, one would
still expect that it is easier to obtain small $\mathfrak{u}(1)$ matter
charges in F-theory models than large ones. Indeed, our proposals suggest that
the orders of vanishing for the section components increase with
$\mathfrak{u}(1)$ charge. According to the observations in Raghuram34 , these
larger orders of vanishing are associated with complicated structures in the
Weierstrass models. The proposals therefore give a concrete explanation as to
how large $\mathfrak{u}(1)$ charges are more difficult to realize than small
$\mathfrak{u}(1)$ charges.

The rest of this paper is organized as follows. Section 2 summarizes our
notations and results and shows how to apply the proposals in a particular
example. Given the length of this paper, we have attempted to make this
section as self-contained as possible. As such, a reader with sufficient
background knowledge of F-theory should be able to understand and apply our
formulas after reading only Section 2, at least at a mechanical level. In
Section 3, we review those aspects of $\mathfrak{u}(1)$ gauge algebras in
F-theory that are used in this paper. Because the centers of compact Lie
groups are important for our results, Section 4 discusses these centers and
their connection to $\mathfrak{u}(1)$ charges in more detail. This section
reviews ideas in CveticLinU1 , but it also explains in more detail our
notations for the centers first mentioned in Section 2. Additionally, we list
the allowed $\mathfrak{u}(1)$ charges for the various matter representations
considered here. Section 5 describes the general strategy for our analysis,
particularly the tactic of using multiples of generating sections to derive
information about models supporting large $\mathfrak{u}(1)$ charges. Section
5.1 discusses how the signs of $\mathfrak{u}(1)$ charges fit into our
strategies and our proposed formulas.

Sections 6, 7, 8, 9, 10 and 11, which are the bulk of this paper, contain the
detailed investigations of the specific types of charged matter we consider
here. Section 6 focuses on singlet matter with $\mathfrak{u}(1)$ charges,
while Sections 7, 8, 9 and 10 focus on $\mathfrak{u}(1)$-charged matter that
is additionally charged in representations of $\mathfrak{su}(n)$,
$\mathfrak{so}(2n)$, $\mathfrak{e}_{6}$, and $\mathfrak{e}_{7}$, respectively.
For each type of matter, we list explicit order-of-vanishing data for various
$\mathfrak{u}(1)$ charges and show that they satisfy the proposed formulas. We
also relate the expressions for each case to similar formulas for EDS
valuations in StangeEllTrouble . In Section 11, we consider matter charged
under multiple $\mathfrak{u}(1)$ and simple Lie algebras, although we do not
perform an exhaustive analysis of these situations. While these sections may
not be necessary for readers simply interested in the final results, they
provide important evidence in favor of the formulas in Section 2. In Section
12, we discuss how previous models in the F-theory literature follow our
proposed formulas. Section 13 provides some concluding thoughts and future
directions.

##  2 Summary of results

This section summarizes our results and describes our conventions. We also
provide an example of how to use these results to determine the
$\mathfrak{u}(1)$ charges in a particular model. We have attempted to make
this section as self-contained as possible, such that a reader solely
interested in the final results can consult this section without referring to
the rest of this paper. Nevertheless, such readers may benefit from Sections 3
and 4, which review the background material underlying these results. Section
5.1, which discusses signs of $\mathfrak{u}(1)$ charges, may also be helpful.
Finally, Appendix A summarizes how the results given in this section are
specialized for particular gauge algebras and representations. While Appendix
A does not contain any results that cannot be obtained using the material in
this section, some readers may find the specialized expressions given there to
be useful.

###  2.1 Notations and conventions

##### Weierstrass form

We typically work with elliptic fibrations in the global Weierstrass form

$y^{2}=x^{3}+fxz^{4}+gz^{6}\,.$ (4)

Here, $[x:y:z]$ are the homogeneous coordinates of a $\mathbb{P}^{2,3,1}$
space, and $f$, $g$ are holomorphic sections of line bundles on the base. The
discriminant of this elliptic fibration is

$\Delta\equiv 4f^{3}+27g^{2}\,.$ (5)

Since we focus on elliptically fibered Calabi–Yau manifolds, $f$ and $g$ are
respectively sections of $\mathcal{O}(-4K_{B})$ and $\mathcal{O}(-6K_{B})$,
where $K_{B}$ is the canonical class of the base $B$ of the elliptic
fibration. We may also write elliptic fibrations in the local Weierstrass form

$y^{2}=x^{3}+fx+g$ (6)

more commonly found in the F-theory literature. The global Weierstrass form
and the local form are related by going to a chart where $z=1$. However, the
zero section, which occurs at $[x:y:z]=[1:1:0]$, is more clearly visible in
the global Weierstrass form.

##### Section components

Rational sections of the elliptic fibration are described as
$[\hat{x}:\hat{y}:\hat{z}]$, where the section components $\hat{x}$,
$\hat{y}$, and $\hat{z}$ solve the Weierstrass form above. In line with the
embedding of the elliptic fiber in $\mathbb{P}^{2,3,1}$, we are free to
rescale the section components as

$[\hat{x}:\hat{y}:\hat{z}]\cong[\lambda^{2}\hat{x}:\lambda^{3}\hat{y}:\lambda\hat{z}]\,.$
(7)

Thus, even though the section components can in principle be rational, we can
clear denominators through a rescaling. Throughout this work, we assume that
the section components have been rescaled to clear denominators and to remove
any common factors that can be scaled away. Therefore, we take the section
components $\hat{x}$, $\hat{y}$, and $\hat{z}$ to be holomorphic sections of
line bundles on the base that solve the global Weierstrass form above. We also
define the section component

$\hat{w}=3\hat{x}^{2}+f\hat{z}^{4}\,.$ (8)

Under the rescaling above, $\hat{w}$ becomes $\lambda^{4}\hat{w}$.

##### Singularity types

As described more fully in Section 3, each singularity type in the Kodaira
classification is associated with an ADE Lie algebra,777As we are interested
in simply-laced gauge algebras in this paper, we focus on the split versions
of the singularity types. and in turn, a universal covering group for the ADE
algebra. Therefore, in an abuse of language, we often refer to the singularity
types by their associated universal covering groups. For instance, we may
refer to a split $\text{I}_{n}$ singularity as an $\operatorname{SU}(n)$
singularity. While this can be done unambiguously for most singularity types,
there are a few cases that require clarifications. The $\text{I}_{2}$ and III
singularity types are both associated with $\operatorname{SU}(2)$, while the
$\text{I}_{3}$ and IV singularity types are both associated with
$\operatorname{SU}(3)$. In this paper, the terms “$\operatorname{SU}(2)$
singularity” and “$\operatorname{SU}(3)$ singularity” will correspond to
$\text{I}_{2}$ and $\text{I}_{3}$ singularities, respectively; we always refer
to the III and IV singularity types using the Kodaira notation. Meanwhile, we
take the group for the $\text{I}_{1}$ and II singularity types to be
$\operatorname{SU}(1)$.

##### Shioda map

The Shioda map is defined to be ParkIntersection ; MorrisonParkU1 ;
GrimmKapferKeitelRational

$\sigma(\hat{s})=\mathcal{S}-\mathcal{Z}-\pi^{*}\mathopen{}\mathclose{{}\left(D_{B}}\right)+\sum_{\kappa,I,J}\mathopen{}\mathclose{{}\left(\mathcal{S}\cdot\alpha_{\kappa,I}}\right)\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}\mathcal{T}_{\kappa,J}\,.$
(9)

Here, $\mathcal{S}$ is the divisor corresponding to the section $\hat{s}$,
$\mathcal{Z}$ is the divisor corresponding to the zero section, and
$\pi^{*}(D_{B})$ is the pullback of a divisor $D_{B}$ in the base $B$. For a
6D F-theory model, described by an elliptically fibered threefold, the
$\pi^{*}(D_{B})$ term can be written as

$\sum_{\alpha}\mathopen{}\mathclose{{}\left((\mathcal{S}-\mathcal{Z})\cdot\mathcal{Z}\cdot
B^{\alpha}}\right)B_{\alpha}\,,$ (10)

where the $B_{\alpha}$ are pullbacks of the basis divisors of $H_{2}(B)$ and
the $\alpha$ indices are lowered and raised using the symmetric bilinear form
$\Omega_{\alpha\beta}$ for $H_{2}(B)$. The index $\kappa$ labels the simple
nonabelian gauge factors making up the gauge algebra, and $I,J$ run from $1$
to the rank of the $\kappa$th gauge factor. Finally,
$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)$ is the
inverse Cartan matrix for the $\kappa$th gauge factor. Each nonabelian gauge
factor is associated with a codimension-one locus in the base where the fiber
(after resolution) consists of irreducible curves forming an affine Dynkin
diagram. The $\alpha_{\kappa,I}$ are the irreducible curves corresponding to
the simple roots of the $\kappa$th Lie algebra, and the
$\mathcal{T}_{\kappa,I}$ are the fibral divisors formed by fibering the
$\alpha_{\kappa,I}$ over the codimension-one locus. Note that we will always
assume the zero section is holomorphic; our primary focus is on elliptic
fibrations in Weierstrass form, for which this assumption is valid.

#####  $\mathfrak{u}(1)$ charge units

We choose units for charges consistent with the definition of the Shioda map
in Eq. 9, as proposed in CveticLinU1 . In these units, singlet
$\mathfrak{u}(1)$ charges are integers, and the lattice of singlet charges has
unit spacing. The $\mathfrak{u}(1)$ charge of matter that is also charged
under a nonabelian gauge algebra may be fractional, owing to the term in
Shioda map involving the inverse Cartan matrix. However, our formulas can
easily be adapted for alternative charge normalizations: if the lattice of
singlet charges has a spacing of $n$, one simply replaces $q$ in Eqs. 18 and
20 with $q/n$.

##### Matter representations

When we refer to matter in some representation $\bm{\mathrm{R}}$ of a gauge
algebra, we often implicitly mean the matter supported at a particular
codimension-two locus in the F-theory base, which may involve matter fields in
the representations $\bm{\mathrm{R}}$ and its conjugate
$\overline{\bm{\mathrm{R}}}$. In 6D F-theory models, which have
$\mathcal{N}=(1,0)$ supersymmetry, codimension-two loci typically support full
hypermultiplets888If $\bm{\mathrm{R}}$ is a pseudoreal representation, matter
can occur in half-hypermultiplets. However, we are most interested in
representations that have nonzero $\mathfrak{u}(1)$ charges, so cases where
the entire representation (including the $\mathfrak{u}(1)$ charge) is
pseudoreal are not too important here. of $\bm{\mathrm{R}}$ matter, which
contain fields in both $\bm{\mathrm{R}}$ and $\overline{\bm{\mathrm{R}}}$. In
4D F-theory models, a chiral multiplet in a representation $\bm{\mathrm{R}}$
contains fields only in the $\bm{\mathrm{R}}$ representation, but there must
be an accompanying CPT-conjugate antichiral multiplet in the
$\overline{\bm{\mathrm{R}}}$ representation. Both of these multiplets are
supported at the same codimension-two locus. Thus, while we often describe
matter as being in a representation $\bm{\mathrm{R}}$ as a shorthand, one
should remember that, due to supersymmetry considerations, there actually may
be fields in both $\bm{\mathrm{R}}$ and $\overline{\bm{\mathrm{R}}}$.

##### Residues

We denote the least non-negative residue of $a$ modulo $b$ as
$\overline{a}_{b}$:

$\overline{a}_{b}=b\mathopen{}\mathclose{{}\left(\frac{a}{b}-\mathopen{}\mathclose{{}\left\lfloor\frac{a}{b}}\right\rfloor}\right)\,.$
(11)

For instance,

$\overline{11}_{6}=5\,,\quad\overline{12}_{6}=0\,,\quad\overline{13}_{6}=1\,.$
(12)

We also define $u_{b}(a)$ as

$u_{b}(a)=\min(\overline{a}_{b},b-\overline{a}_{b})\,.$ (13)

Roughly, $u_{b}(a)$ gives the distance from an integer $a$ to the nearest
multiple of $b$. For instance,

$u_{6}(11)=1\,,\quad u_{6}(12)=0\,,\quad u_{6}(13)=1\,.$ (14)

##### Orders of vanishing

We denote the order of vanishing of an expression $y$ at a locus $\\{x=0\\}$
as

$\operatorname*{ord}_{x=0}(y)\,.$ (15)

We also denote the order of vanishing of $y$ at a codimension-two locus
$\\{x_{1}=x_{2}=0\\}$ as

$\operatorname*{ord}_{x_{1},x_{2}}(y)\,.$ (16)

Since we are giving general proposals about the behavior at loci supporting
$\mathfrak{u}(1)$ charges, we often want to describe the orders of vanishing
at a codimension-one or codimension-two locus supporting a gauge group or
charged matter without specifying a particular locus. Therefore, we denote the
order of vanishing of $y$ at an unspecified codimension-one locus as
$\operatorname{ord}_{1}(y)$ and the order of vanishing at an unspecified
codimension-two locus as $\operatorname{ord}_{2}(y)$.

##### Gauge group centers

At various points, we discuss elements of the center $Z(G)$ of a simple Lie
group $G$. We refer to the order of $Z(G)$ (the number of elements in the
center of $G$) as $d_{G}$. This number also equals the determinant of the
Cartan matrix of $G$. We label elements of $Z(G)$ with the integer $\nu$; we
also use $\nu$ to denote the discrete quotient involving the $\nu$ element of
the center. For most of the gauge groups discussed here, the center is
$\mathbb{Z}_{m}$ for some $m$, and we let $\nu$ run from $0$ to $m-1$. For
$\operatorname{Spin}(4k)$, however, the center is
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Since there are four elements in this
center, we let $\nu$ run from 0 to 3, with $\nu=0$ referring to the identity
element of the center. At codimension-two loci in the base supporting charged
matter, the singularity type enhances, and one can associate a Lie group with
the enhanced singularity type. It is useful to consider the center of this
enhanced group, even though the enhanced group does not represent a physical
gauge group. We label elements of the enhanced group’s center with the integer
$\mu$.

It is also useful to define the function $T_{G}(\nu)$ and the triplet of
functions $\vec{\tau}_{G}(\nu)$, where $G$ is a simple Lie group. The
expressions for $T_{G}(\nu)$ and $\vec{\tau}_{G}(\nu)$, which are given in
Table 1, depend on the group $G$ in question. As discussed in Section 4, the
$T_{G}(\nu)$ are essentially diagonal elements of the inverse Cartan matrix of
$G$. The $\vec{\tau}_{G}(\nu)$ provide information about the orders of
vanishing of the $\hat{x}$, $\hat{y}$, and $\hat{w}$ section components.

Singularity Type | $G$ | $T_{G}(\nu)$ | $\vec{\tau}_{G}(\nu)$ | Valid Values of $\nu$
---|---|---|---|---
$\text{I}_{1}$ | — | $0$ | $(0,0,0)$ | 0
II | — | $0$ | $\mathopen{}\mathclose{{}\left(0,0,0}\right)$ | 0
$\text{I}_{n}$ | $\operatorname{SU}(n)$ | $\frac{\nu(n-\nu)}{n}$ | $\mathopen{}\mathclose{{}\left(0,1,1}\right)u_{n}(\nu)$ | $0,1,\ldots,n-1$
III | $\operatorname{SU}(2)$ | $\frac{\nu(2-\nu)}{2}$ | $\mathopen{}\mathclose{{}\left(1,1,1}\right)u_{2}(\nu)$ | $0,1$
IV | $\operatorname{SU}(3)$ | $\frac{\nu(3-\nu)}{3}$ | $\mathopen{}\mathclose{{}\left(1,1,2}\right)u_{3}(\nu)$ | $0,1,2$
$\text{I}_{n-4}^{*}$ | $\operatorname{Spin}(2n)$ | $\begin{cases}0&\nu=0\\\ 1&\nu=2\\\ \frac{n}{4}&\nu=1,3\end{cases}$ | $\begin{cases}\mathopen{}\mathclose{{}\left(0,0,0}\right)&\nu=0\\\ \mathopen{}\mathclose{{}\left(1,2,2}\right)&\nu=2\\\ \mathopen{}\mathclose{{}\left(1,\mathopen{}\mathclose{{}\left\lfloor\frac{n}{2}}\right\rfloor,\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil}\right)&\nu=1,3\end{cases}$ | $0,1,2,3$
$\text{IV}^{*}$ | $\operatorname{E}_{6}$ | $\frac{2\nu(3-\nu)}{3}$ | $\mathopen{}\mathclose{{}\left(2,2,3}\right)u_{3}(\nu)$ | $0,1,2$
$\text{III}^{*}$ | $\operatorname{E}_{7}$ | $\frac{3}{2}\nu$ | $\mathopen{}\mathclose{{}\left(2,3,3}\right)\nu$ | $0,1$
$\text{II}^{*}$ | $\operatorname{E}_{8}$ | $0$ | $\mathopen{}\mathclose{{}\left(0,0,0}\right)$ | $0$
Table 1: Expressions for $T_{G}(\nu)$ and $\vec{\tau}_{G}(\nu)$. While the
singularity types $\text{I}_{1}$ and II do not have listed gauge group, $G$
can be thought of as $\operatorname{SU}(1)$ in these situations. In this
paper, we focus on the split versions of these singularity types.

###  2.2 Proposal

Suppose a codimension-one locus $\\{\sigma=0\\}$ in the base supports a
simply-laced gauge algebra $\mathfrak{g}$ whose universal covering group is
$G$. The elliptic fibers along this locus are singular, and the singularity
type is that corresponding to $G$ in the Kodaira classification. If there is
also a $\mathfrak{u}(1)$ gauge algebra, the section components of its
associated generating section should vanish to orders

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{G}(\nu)$
(17)

for some value of $\nu$ denoting an element of the center of $G$. As described
in more detail in Section 4, $\nu$ encodes information about which component
of the resolved singular fibers along $\\{\sigma=0\\}$ is hit by the
generating section. This in turn (at least partially) describes the global
structure of the gauge group CveticLinU1 .

Now suppose there is some matter with $\mathfrak{u}(1)$ charge $q$ occurring
at a codimension-two locus in the base with an enhanced singularity type. At
least locally, this codimension-two locus can be thought of as the
intersection of the codimension-one loci $\\{\sigma_{i}=0\\}$,999Some
situations involve matter localized at nodal singularities of irreducible
codimension-one loci in the base. The most notable examples are
$\mathfrak{u}(1)$ charged singlets, which occur at double points of the
discriminant locus $\\{\Delta=0\\}$. Locally, such situations still look like
the intersection of multiple codimension-one loci, even though these loci may
be identified globally. For the case of singlets, the singularity type can be
thought of as enhancing from $\text{I}_{1}\times\text{I}_{1}$ to
$\text{I}_{2}$, or from $\operatorname{SU}(1)\times\operatorname{SU}(1)$ to
$\operatorname{SU}(2)$. with $i$ running from $1$ to $N$.101010In almost all
cases of interest, $N$ will be 2. However, we leave open the possibility that
$N$ can be greater than 2 to account for more exotic matter types, such as
trifundamental representations. Each of these loci has singular fibers of type
$G_{i}$, where we are referring to singularity types by their corresponding
ADE groups in the Kodaira classification.111111For the singularity types
$\text{I}_{1}$ and II, whose corresponding group can roughly be thought of as
$\operatorname{SU}(1)$, the only allowed value of $\nu$ is 0. As indicated in
Table 1, $T(\nu)$ and $\vec{\tau}(\nu)$ are 0 for these situations. Each
$\\{\sigma_{i}=0\\}$ locus also has a corresponding $\nu_{i}$ consistent with
the codimension-one orders of vanishings described above. At the codimension-
two locus, the singularity type enhances from $G_{1}\times\ldots\times G_{N}$
to some singularity type $H$, where $H$ is again an ADE group. The generating
section components for the $\mathfrak{u}(1)$ should then vanish to orders

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{\prod_{i}d_{G_{i}}}{d_{H}}q^{2}+\sum_{i}T_{G_{i}}(\nu_{i})-T_{H}(\mu)}\right)$
(18)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{H}(\mu)\,.$
(19)

at the codimension-two locus. Here $d_{G}$ represents the order (the number of
elements) of the center of $G$, while $\mu$ is an integer representing a
particular element of the center of $H$.

An important special case is matter charged under the gauge algebra
$\mathfrak{g}\oplus\mathfrak{u}(1)$, where $\mathfrak{g}$ is a simple Lie
algebra. We again assume that $\mathfrak{g}$ is simply-laced. There is a
codimension-one locus $\\{\sigma=0\\}$ along which the elliptic fibers are
singular with singularity type $G$. The orders of vanishing at
$\\{\sigma=0\\}$ are still given by Eq. 17 for some $\nu$ in the center of
$G$. Now suppose that matter charged under $\mathfrak{g}\oplus\mathfrak{u}(1)$
occurs at the intersection of $\\{\sigma=0\\}$ with the residual discriminant
locus, which has singularity type $\text{I}_{1}$. For the residual
discriminant locus, whose ADE group can be thought of as
$\operatorname{SU}(1)$, the associated $\nu$, $T(\nu)$, and $\vec{\tau}(\nu)$
are 0. If the enhanced singularity type at the codimension-two locus is $H$,
the orders of vanishing at the codimension-two locus are

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{G}}{d_{H}}q^{2}+T_{G}(\nu)-T_{H}(\mu)}\right)$
(20)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{H}(\mu)\,.$
(21)

These expressions provide a procedure for determining the $\mathfrak{u}(1)$
charge of some matter given the orders of vanishing of the section components,
at least up to sign. From $f$, $g$, and the discriminant $\Delta$, one can
determine the $G_{i}$ and $H$, which in turn determine the $d_{G_{i}}$ and
$d_{H}$. One can also use Equations 17 and 19 to read off $\nu$ and $\mu$ for
the matter in question.121212There may be two values of $\nu$ (or $\mu$)
corresponding to a particular value of $\vec{\tau}_{G}(\nu)$ (or
$\vec{\tau}_{H}(\mu)$). However, $T_{G}(\nu)$ will be the same for these two
values of $\nu$, so this ambiguity does not cause any problems when using Eq.
18. This information can then be plugged into Eq. 18, and one can solve for
$q$ up to a sign.

To use these relations to predict the orders of vanishing for matter with a
particular $\mathfrak{u}(1)$ charge, one must determine the value of $\mu$
corresponding to $\nu$ and $q$. The relations between these parameters, which
depend on the specific groups and representations in question, are described
in Table 2. Alternatively, one can try all values of $\mu$ allowed for the
codimension-two singularity type $H$ and determine which values of $\mu$ give
sensible orders of vanishing for a desired $\nu$ and $q$. However, note that
even if the formulas predict reasonable orders of vanishing for a choice of
$\nu$ and $q$, it does not imply that one can actually find an F-theory model
that realizes the appropriate matter with this $\mathfrak{u}(1)$ charge. The
proposals should be thought of as giving the orders of vanishing assuming that
a model supporting the appropriate matter exists. In fact, one likely could
use these formulas to argue that certain $\mathfrak{u}(1)$ charges cannot be
realized in F-theory. One would first calculate the orders of vanishing
predicted by the formulas for the $\mathfrak{u}(1)$ charge in question. If
attempts to construct F-theory model with a rational section admitting these
orders of vanishing always lead to obstructions, it would hint that the
corresponding $\mathfrak{u}(1)$ charge cannot be realized.

For all the models discussed here, including several examples previously seen
in the literature, these formulas are satisfied exactly. However, these
formulas may more generally give the lowest orders of vanishing of the section
components. As a helpful analogy, consider the Kodaira table in Table 4, which
lists the orders of vanishing of $f$, $g$, and $\Delta$ for various
singularity types. In some elliptic fibrations, the orders of vanishing at a
locus supporting a particular singularity type may exceed those listed in the
table, so long as these larger orders of vanishing do not correspond to an
enhanced singularity type. For instance, while $f$, $g$, and $\Delta$
typically vanish to orders $(3,4,8)$ at an $\text{IV}^{*}$ locus, they could
alternatively vanish to orders $(4,4,8)$. However, if they vanish to orders
$(3,5,9)$, the singularity type would be $\text{III}^{*}$ rather than
$\text{IV}^{*}$. One might expect that the orders of vanishing of the section
components behave in a similar way. As an example, our formulas predict that,
for an $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ model with $\nu=1$,
$(\hat{x},\hat{y},\hat{z},\hat{w})$ should vanish to orders $(0,1,0,1)$ at the
codimension-one $\mathfrak{su}(5)$ locus. An
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ model with $\nu=1$ and codimension-one
orders of vanishing such as $(0,1,0,2)$ may still be consistent with our
proposals. However, orders of vanishing such as $(0,2,0,2)$ would correspond
to different value of $\nu$, so the proposals would suggest
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ models with such codimension-one
orders of vanishing would have a different global gauge group structure. Since
we do not analyze F-theory models exhibiting such behaviors here, these
thoughts on models with higher orders of vanishing are somewhat speculative.
It would therefore be important to more properly establish whether such
situations can occur and how to handle them in future work.

Gauge Factor | Enhancement | Representation | Allowed $q$ | $\mu$
---|---|---|---|---
— | $\text{I}_{1}\to\text{I}_{2}$ | $\bm{1}_{q}$ | $\mathbb{Z}$ | $\overline{q}_{2}$
$\operatorname{SU}(n)$ | $\text{I}_{n}^{(s)}\to\text{I}_{n+1}$ | $\bm{n}_{q}$ | $\frac{\nu}{n}+\mathbb{Z}$ | $\overline{nq}_{n+1}$
$\operatorname{SU}(n)$, odd $n$ | $\text{I}_{n}^{(s)}\to\text{I}^{*}_{n-4}$ | $\bm{\frac{n}{2}(n-1)}_{q}$ | $\frac{2\nu}{n}+\mathbb{Z}$ | $\overline{nq}_{4}$
$\operatorname{SU}(n)$, even $n$ | $\text{I}_{n}^{(s)}\to\text{I}^{*}_{n-4}$ | $\bm{\frac{n}{2}(n-1)}_{q}$ | $\frac{2\nu}{n}+\mathbb{Z}$ | $\overline{(q-\frac{2}{n}\nu)}_{2}+\overline{2\nu}_{4}$
$\operatorname{Spin}(2n)$ | ${\text{I}_{n-4}^{*(s)}}\to\text{I}^{*}_{n-3}$ | $\bm{2n}_{q}$ | $\frac{\nu}{2}+\mathbb{Z}$ | $\overline{(2q-\nu)}_{4}+\overline{2q}_{2}$
$\operatorname{Spin}(8)$ | ${\text{I}_{0}^{*}}^{(s)}\to\text{I}^{*}_{1}$ | ${\bm{8_{\text{s}}}}_{,q}$ | $\begin{cases}\frac{1}{2}+\mathbb{Z}&\nu=1,2\\\ \mathbb{Z}&\nu=0,3\end{cases}$ | $\overline{(2q-2\nu)}_{4}$
$\operatorname{Spin}(8)$ | ${\text{I}_{0}^{*}}^{(s)}\to\text{I}^{*}_{1}$ | ${\bm{8_{\text{c}}}}_{,q}$ | $\begin{cases}\frac{1}{2}+\mathbb{Z}&\nu=2,3\\\ \mathbb{Z}&\nu=0,1\end{cases}$ | $\overline{(2q-2\nu)}_{4}$
$\operatorname{Spin}(10)$ | ${\text{I}_{1}^{*}}^{(s)}\to{\text{IV}^{*}}$ | $\bm{16}_{q}$ | $\frac{4-\nu}{4}+\mathbb{Z}$ | $\overline{4q}_{3}$
$\operatorname{Spin}(12)$ | ${\text{I}_{2}^{*}}^{(s)}\to{\text{III}}^{*}$ | $\bm{32}_{q}\oplus\bm{1}_{2q}$ | $\begin{cases}\frac{1}{2}+\mathbb{Z}&\nu=2,3\\\ \mathbb{Z}&\nu=0,1\end{cases}$ | $\overline{(2q+\nu)}_{2}$
$\operatorname{Spin}(12)$ | ${\text{I}_{2}^{*}}^{(s)}\to{\text{III}}^{*}$ | $\bm{32^{\prime}}_{q}\oplus\bm{1}_{2q}$ | $\begin{cases}\frac{1}{2}+\mathbb{Z}&\nu=1,2\\\ \mathbb{Z}&\nu=0,3\end{cases}$ | $\overline{(2q+\nu)}_{2}$
$\operatorname{Spin}(14)$ | ${\text{I}_{3}^{*}}^{(s)}\to{\text{II}}^{*}$ | $\bm{64}_{q}\oplus\bm{14}_{2q}$ | $\frac{\nu}{4}+\mathbb{Z}$ | 0
$\operatorname{E}_{6}$ | ${\text{IV}^{*}}^{(s)}\to{\text{III}}^{*}$ | $\bm{27}_{q}$ | $\frac{\nu}{3}+\mathbb{Z}$ | $\overline{3q}_{2}$
$\operatorname{E}_{7}$ | ${\text{III}^{*}}\to{\text{II}}^{*}$ | $\bm{56}_{q}\oplus\bm{1}_{2q}$ | $\frac{\nu}{2}+\mathbb{Z}$ | 0
Table 2: Relations between $\nu$, $\mu$, and the $\mathfrak{u}(1)$ charge $q$
for various representations. Note that, in our conventions, the $\bm{32}$
representation of $\operatorname{Spin}(12)$ has highest weight
$[0,0,0,0,0,1]$, while the $\bm{32^{\prime}}$ representation has highest
weight $[0,0,0,0,1,0]$. This convention differs from Slansky  but agrees with
YamatsuGroupTheory .

###  2.3 Example

As an example of how to read off $\mathfrak{u}(1)$ charges, consider the
F-theory model described by a Weierstrass equation of the form

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\mathopen{}\mathclose{{}\left(c_{1}c_{3}-b^{2}c_{0}-\frac{1}{3}c_{2}^{2}}\right)xz^{4}$
(22)
$\displaystyle\qquad\quad+\mathopen{}\mathclose{{}\left(c_{0}c_{3}^{2}-\frac{1}{3}c_{1}c_{2}c_{3}+\frac{2}{27}c_{2}^{3}-\frac{2}{3}b^{2}c_{0}c_{2}+\frac{1}{4}b^{2}c_{1}^{2}}\right)z^{6}$

with

$\displaystyle c_{0}$
$\displaystyle=\sigma\Big{[}8c_{1,1}c_{3,0}^{3}+\sigma\mathopen{}\mathclose{{}\left(2c_{3,0}^{2}\mathopen{}\mathclose{{}\left(2b^{2}c_{1,1}^{2}+c_{1,2}}\right)-8bc_{1,1}c_{3,0}c_{2,1}+c_{2,1}^{2}}\right)$
(23)
$\displaystyle\qquad\quad+\sigma^{2}\mathopen{}\mathclose{{}\left(c_{2,1}\mathopen{}\mathclose{{}\left(4b^{3}c_{1,1}^{2}-bc_{1,2}}\right)+2c_{3,0}\mathopen{}\mathclose{{}\left(-4b^{4}c_{1,1}^{3}+b^{2}c_{1,1}c_{1,2}+c_{1,3}}\right)}\right)-\sigma^{3}c_{0,4}\Big{]}\,,$
$\displaystyle c_{1}$
$\displaystyle=\sigma\Big{[}2c_{3,0}\mathopen{}\mathclose{{}\left(2bc_{1,1}c_{3,0}+c_{2,1}}\right)+b\sigma\mathopen{}\mathclose{{}\left(c_{1,2}c_{3,0}-2bc_{1,1}c_{2,1}}\right)+b\sigma^{2}c_{1,3}+\sigma^{3}c_{1,4}\Big{]}\,,$
$\displaystyle c_{2}$ $\displaystyle=c_{3,0}^{2}+b\sigma
c_{2,1}+\sigma^{4}c_{2,4}\,,$ $\displaystyle c_{3}$
$\displaystyle=bc_{3,0}+\sigma^{4}c_{3,4}\,.$

We assume that the base of the elliptic fibration is complex two-dimensional,
giving us a 6D F-theory model. The parameters $b$ and $c_{i,j}$ are
holomorphic sections of line bundles over the base, with the line bundles
chosen such that $f$ and $g$ are holomorphic sections of
$\mathcal{O}(-4K_{B})$ and $\mathcal{O}(-6K_{B})$. This model is a tuned
version of the Morrison–Park form MorrisonParkU1 , and it admits a generating
section with components

$\displaystyle\hat{x}$
$\displaystyle=\frac{1}{3}b^{2}c_{3,0}^{2}-\frac{2}{3}\sigma\mathopen{}\mathclose{{}\left(b^{3}c_{2,1}}\right)+\sigma^{4}\mathopen{}\mathclose{{}\left(2bc_{3,0}c_{3,4}-\frac{2}{3}b^{2}c_{2,4}}\right)+\sigma^{8}c_{3,4}^{2}\,,$
(24) $\displaystyle\hat{y}$
$\displaystyle=-2\sigma\mathopen{}\mathclose{{}\left(b^{5}c_{1,1}c_{3,0}^{2}}\right)+\sigma^{2}\mathopen{}\mathclose{{}\left(b^{6}c_{1,1}c_{2,1}-\frac{1}{2}b^{5}c_{1,2}c_{3,0}}\right)-\frac{1}{2}\sigma^{3}\mathopen{}\mathclose{{}\left(b^{5}c_{1,3}}\right)$
$\displaystyle\qquad-\frac{1}{2}\sigma^{4}\mathopen{}\mathclose{{}\left(b^{2}\mathopen{}\mathclose{{}\left(b^{2}c_{1,4}-2bc_{2,4}c_{3,0}+4c_{3,0}^{2}c_{3,4}}\right)}\right)+b^{3}\sigma^{5}c_{2,1}c_{3,4}$
$\displaystyle\qquad+b\sigma^{8}c_{3,4}\mathopen{}\mathclose{{}\left(bc_{2,4}-3c_{3,0}c_{3,4}}\right)-\sigma^{12}c_{3,4}^{3}\,,$
$\displaystyle\hat{z}$ $\displaystyle=b\,,$ $\displaystyle\hat{w}$
$\displaystyle=-4\sigma\mathopen{}\mathclose{{}\left(b^{6}c_{1,1}c_{3,0}^{3}}\right)+b^{6}\sigma^{2}c_{3,0}\mathopen{}\mathclose{{}\left(6bc_{1,1}c_{2,1}-c_{3,0}\mathopen{}\mathclose{{}\left(4b^{2}c_{1,1}^{2}+c_{1,2}}\right)}\right)$
$\displaystyle\qquad+b^{6}\sigma^{3}\mathopen{}\mathclose{{}\left(bc_{2,1}\mathopen{}\mathclose{{}\left(c_{1,2}-4b^{2}c_{1,1}^{2}}\right)+c_{3,0}\mathopen{}\mathclose{{}\left(8b^{4}c_{1,1}^{3}-2b^{2}c_{1,1}c_{1,2}-c_{1,3}}\right)}\right)$
$\displaystyle\qquad+\sigma^{4}\mathopen{}\mathclose{{}\left(b^{6}c_{0,4}+b^{3}c_{3,0}\mathopen{}\mathclose{{}\left(b^{2}c_{1,4}-2bc_{2,4}c_{3,0}+4c_{3,0}^{2}c_{3,4}}\right)}\right)$
$\displaystyle\qquad+2b^{4}\sigma^{5}\mathopen{}\mathclose{{}\left(bc_{2,1}c_{2,4}+c_{3,0}c_{3,4}\mathopen{}\mathclose{{}\left(2bc_{1,1}c_{3,0}-3c_{2,1}}\right)}\right)$
$\displaystyle\qquad+b^{5}\sigma^{6}c_{3,4}\mathopen{}\mathclose{{}\left(c_{1,2}c_{3,0}-2bc_{1,1}c_{2,1}}\right)+b^{5}\sigma^{7}c_{1,3}c_{3,4}$
$\displaystyle\qquad+b^{2}\sigma^{8}\mathopen{}\mathclose{{}\left(b^{2}c_{2,4}^{2}+bc_{3,4}\mathopen{}\mathclose{{}\left(bc_{1,4}-8c_{2,4}c_{3,0}}\right)+14c_{3,0}^{2}c_{3,4}^{2}}\right)-4\sigma^{9}b^{3}c_{2,1}c_{3,4}^{2}$
$\displaystyle\qquad-\sigma^{12}c_{3,4}^{2}\mathopen{}\mathclose{{}\left(4b^{2}c_{2,4}-12bc_{3,0}c_{3,4}-3c_{3,4}^{2}\sigma^{4}}\right)\,.$

This indicates that the model has a $\mathfrak{u}(1)$ gauge algebra.
Additionally, the discriminant is proportional to $\sigma^{5}$:

$\Delta=2bc_{1,1}c_{3,0}^{4}\Delta^{\prime}\sigma^{5}+\mathcal{O}(\sigma^{6})\,,$
(25)

where

$\displaystyle\Delta^{\prime}$
$\displaystyle=-112b^{8}c_{1,1}^{3}c_{2,1}c_{3,0}^{2}-4b^{4}c_{1,3}c_{2,1}c_{3,0}^{2}+4b^{3}c_{0,4}c_{3,0}^{3}+128b^{9}c_{1,1}^{4}c_{3,0}^{3}+8b^{2}c_{1,4}c_{3,0}^{4}$
(26)
$\displaystyle\qquad-16bc_{2,4}c_{3,0}^{5}-4b^{6}c_{1,1}c_{2,1}\mathopen{}\mathclose{{}\left(c_{2,1}^{2}-3c_{1,2}c_{3,0}^{2}}\right)+12b^{7}c_{1,1}^{2}c_{3,0}\mathopen{}\mathclose{{}\left(3c_{2,1}^{2}-2c_{1,2}c_{3,0}^{2}}\right)$
$\displaystyle\qquad+b^{5}\mathopen{}\mathclose{{}\left(-2c_{1,2}c_{2,1}^{2}c_{3,0}+\mathopen{}\mathclose{{}\left(c_{1,2}^{2}+8c_{1,1}c_{1,3}}\right)c_{3,0}^{3}}\right)+32c_{3,0}^{6}c_{3,4}.$

One can verify that the split condition GrassiMorrison

$\displaystyle\frac{9g}{2f}\Bigg{|}_{\sigma=0}=-\psi^{2}$ (27)

is satisfied, indicating that the singularity type along $\\{\sigma=0\\}$ is
$\text{I}_{5}^{s}$ and that the supported gauge algebra is $\mathfrak{su}(5)$.
There are no other codimension-one loci supporting nonabelian gauge algebras
in the model, and the total gauge algebra is
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$.

There are two sources of charged matter in this model. If $\\{\sigma=0\\}$ is
a curve of genus $g$, there are $g$ hypermultiplets of adjoint ($\bm{24}$)
matter not localized at a codimension-two locus. The $\mathfrak{u}(1)$ charge
of this matter is 0, and this matter is therefore not of significant interest
to us; however, we must account for it to properly satisfy the 6D anomaly
cancellation conditions. The remaining matter is localized at codimension-two
loci. Along $\\{\sigma=0\\}$, the singularity type enhances to type
$\text{I}_{1}^{*}$ (or $\operatorname{Spin}(10)$) at $\\{\sigma=c_{3,0}=0\\}$
and to type $\text{I}_{6}$ (or $\operatorname{SU}(6)$) at $\\{\sigma=b=0\\}$,
$\\{\sigma=c_{1,1}=0\\}$, and $\\{\sigma=\Delta^{\prime}=0\\}$. By the
Katz–Vafa method, the $\text{I}_{1}^{*}$ locus supports hypermultiplets of
two-index antisymmetric ($\bm{10}$) matter, while the $\text{I}_{6}$ loci
support hypermultiplets of fundamental ($\bm{5}$) matter. There are also
codimension-two loci not along $\\{\sigma=0\\}$ where the singularity type
enhances to $\text{I}_{2}$: $\\{b=c_{3,4}=0\\}$ and

$\displaystyle V_{q=1}$
$\displaystyle=\\{\hat{y}/\sigma=\hat{w}/\sigma=0\\}\setminus$ (28)
$\displaystyle\qquad\Big{(}\\{\sigma=c_{3,0}=0\\}\cup\\{\sigma=b=0\\}\cup\\{\sigma=c_{1,1}=0\\}\cup\\{b=c_{3,4}=0\\}\Big{)}\,.$

These loci support singlets uncharged under the $\mathfrak{su}(5)$ algebra.

To determine the $\mathfrak{u}(1)$ charges, we should look at the orders of
vanishing of the section components. First, along $\\{\sigma=0\\}$, the
section components, listed in Eq. 24, vanish to orders

$\operatorname*{ord}_{\sigma=0}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname*{ord}_{\sigma=0}(\hat{x}),\operatorname*{ord}_{\sigma=0}(\hat{y}),\operatorname*{ord}_{\sigma=0}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(0,1,1}\right)\,.$
(29)

Comparing to Eq. 17 and noting that

$\vec{\tau}_{\operatorname{SU}(5)}(\nu)=(0,1,1)\times u_{5}(\nu)\,,$ (30)

we see that $\nu$ is either 1 or 4. Either can be chosen without affecting the
end results of our calculation, so we take $\nu$ to be 1. We additionally know
that the center of $\operatorname{SU}(5)$ is $\mathbb{Z}_{5}$, so
$d_{SU(5)}=5$.

The codimension-two orders of vanishing at the matter loci are listed in Table
3. This information allows us to determine the $\mathfrak{u}(1)$ charges up to
sign, which we demonstrate for three of the matter loci.

  * •

First, consider the locus $\\{\sigma=c_{3,0}=0\\}$ locus, which supports
$\bm{10}$ matter. From Table 3, we see that

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)-(2,3,4)\times\operatorname{ord}_{2}(\hat{z})=\vec{\tau}_{\operatorname{Spin}(10)}(\mu)=(1,2,3)\,.$
(31)

According to Table 1, $\mu$ is therefore either $1$ or $3$ for this matter
locus. Additionally, the center of $\operatorname{Spin}(10)$ is
$\mathbb{Z}_{4}$, and $d_{\operatorname{Spin}(10)}$ is 4. Plugging this
information into either Eq. 18 or Eq. 20 leads to

$q^{2}=\frac{d_{\operatorname{Spin}(10)}}{d_{\operatorname{SU}(5)}}\Big{(}2\operatorname{ord}_{2}(\hat{z})+T_{\operatorname{Spin}(10)}(\mu)-T_{\operatorname{SU}(5)}(\nu)\Big{)}=\frac{4}{5}\Big{(}2\times
0+\frac{5}{4}-\frac{4}{5}\Big{)}=\frac{9}{25}\,.$ (32)

Therefore, $\lvert q\rvert$ is $\frac{3}{5}$ for the matter supported here.

  * •

Second, consider the locus $\\{\sigma=b=0\\}$, which supports $\bm{5}$ matter.
From Table 3,

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)-(2,3,4)\times\operatorname{ord}_{2}(\hat{z})=\vec{\tau}_{\operatorname{SU}(6)}(\mu)=(0,3,3)\,.$
(33)

According to Table 1, $\mu$ is $3$ for this matter locus, and since the center
of $\operatorname{SU}(6)$ is $\mathbb{Z}_{6}$, $d_{\operatorname{SU}(6)}$ is
6. Plugging this information into either Eq. 18 or Eq. 20 leads to

$q^{2}=\frac{d_{\operatorname{SU}(6)}}{d_{\operatorname{SU}(5)}}\Big{(}2\operatorname{ord}_{2}(\hat{z})+T_{\operatorname{SU}(6)}(\mu)-T_{\operatorname{SU}(5)}(\nu)\Big{)}=\frac{6}{5}\Big{(}2\times
1+\frac{9}{6}-\frac{4}{5}\Big{)}=\frac{81}{25}\,.$ (34)

Therefore, $\lvert q\rvert$ is $\frac{9}{5}$ for the matter supported here.

  * •

Third, consider the locus $\\{b=c_{3,4}=0\\}$, which supports singlet matter,
uncharged under $\operatorname{SU}(5)$, with a potentially nonzero
$\mathfrak{u}(1)$ charge. This locus consists of double points of the
discriminant curve. The singularity type enhances from
$\text{I}_{1}\times\text{I}_{1}$ to $\text{I}_{2}$, or, in terms of ADE
groups, from $\operatorname{SU}(1)\times\operatorname{SU}(1)$ to
$\operatorname{SU}(2)$. For the $\operatorname{SU}(1)$ loci, $\nu$ is 0, and
there are no nontrivial contributions from either
$\vec{\tau}_{\operatorname{SU}(1)}(\nu)$ or $T_{\operatorname{SU}(1)}(\nu)$.
We still need to determine $\mu$, however. According to the codimension-two
orders of vanishing,

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)-(2,3,4)\times\operatorname{ord}_{2}(\hat{z})=\vec{\tau}_{\operatorname{SU}(2)}(\mu)=(0,0,0)\,,$
(35)

implying that $\mu$ is 0. Additionally, $d_{SU(2)}$ is 2. Plugging this
information into either Eq. 18 or Eq. 20 leads to

$q^{2}=d_{SU(2)}\Big{(}2\operatorname{ord}_{2}(\hat{z})+T_{\operatorname{SU}(2)}(\mu)\Big{)}=2\Big{(}2\times
1+0\Big{)}=4\,.$ (36)

Therefore, $\lvert q\rvert$ is $2$ for the matter supported here.

The values of $\lvert q\rvert$ for the other loci can be found by similar
procedures, and the results are summarized in the penultimate column of Table
3.

Locus | Enhancement | $\operatorname{ord}_{2}(\hat{x},\hat{y},\hat{z},\hat{w})$ | $\lvert q\rvert$ |  $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ Rep.
---|---|---|---|---
$\\{\sigma=c_{3,0}=0\\}$ | $\text{I}_{5}\times\text{I}_{1}\rightarrow\text{I}_{1}^{*}$ | (1,2,0,3) | $\frac{3}{5}$ | $\bm{10}_{-\frac{3}{5}}$
$\\{\sigma=b=0\\}$ | $\text{I}_{5}\times\text{I}_{1}\rightarrow\text{I}_{6}$ | (2,6,1,7) | $\frac{9}{5}$ | $\bm{5}_{-\frac{9}{5}}$
$\\{\sigma=c_{1,1}=0\\}$ | $\text{I}_{5}\times\text{I}_{1}\rightarrow\text{I}_{6}$ | (0,2,0,2) | $\frac{4}{5}$ | $\bm{5}_{-\frac{4}{5}}$
$\\{\sigma=\Delta^{\prime}=0\\}$ | $\text{I}_{5}\times\text{I}_{1}\rightarrow\text{I}_{6}$ | (0,1,0,1) | $\frac{1}{5}$ | $\bm{5}_{\frac{1}{5}}$
$\\{b=c_{3,4}=0\\}$ | $\text{I}_{1}\times\text{I}_{1}\rightarrow\text{I}_{2}$ | (2,3,1,4) | $2$ | $\bm{1}_{2}$
$V_{q=1}$ | $\text{I}_{1}\times\text{I}_{1}\rightarrow\text{I}_{2}$ | (0,1,0,1) | $1$ | $\bm{1}_{1}$
Table 3: Matter loci for $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ example along
with the codimension-two orders of vanishing of the section components. For
typographical reasons, we describe the singularity types using the Kodaira
notation rather than the associated ADE gauge groups.

We have not yet found the signs of the charges, which cannot be directly
determined from our relations. For singlets, the sign of the charge is
irrelevant, as the hypermultiplets contain fields in both the $\bm{1}_{q}$ and
$\bm{1}_{-q}$ representations. For the other representations, the relative
sign of the $\mathfrak{u}(1)$ charges must be consistent with the global
structure of the gauge group, as described in more detail in CveticLinU1  and
in Section 4. In particular, the $\mathfrak{u}(1)$ charges for matter in a
specific representation of $\mathfrak{su}(5)$ must be separated by integers,
even if the charges themselves are fractional. Therefore, if we take the
$\mathfrak{u}(1)$ charge at $\\{\sigma=\Delta^{\prime}=0\\}$ to be
$+\frac{1}{5}$, the $\mathfrak{u}(1)$ charges at $\\{\sigma=b=0\\}$ and
$\\{\sigma=c_{1,1}=0\\}$ should be $-\frac{9}{5}$ and $-\frac{4}{5}$,
respectively. By similar arguments, the charge at $\\{\sigma=c_{3,0}=0\\}$
should be $-\frac{3}{5}$.131313A quick way of seeing this is to consider the
decomposition $\bm{5}\otimes\bm{5}=\bm{15}\oplus\bm{10}$. If
$\bm{5}_{\frac{1}{5}}$ and $\bm{5}_{-\frac{4}{5}}$ are valid representations,
then $\bm{5}_{\frac{1}{5}}\otimes\bm{5}_{-\frac{4}{5}}$ would have
$\mathfrak{u}(1)$ charge $-\frac{3}{5}$.

This analysis leads to the representations listed in the last column of Table
3. If one calculates the multiplicities of the matter loci and includes the
$\bm{24}_{0}$ adjoint matte, one can verify that this matter spectrum
satisfies the 6D gauge and gauge–gravitational anomaly cancellation conditions
ErlerAnomaly ; ParkTaylor ; ParkIntersection  for the generating section
height

$0pt=-2K_{B}+2[b]-\frac{4}{5}[\sigma]\,.$ (37)

Note that we have determined the spectrum without performing any resolutions
of the singular Weierstrass model, illustrating the power of this approach.

##  3 Review of F-theory

In this section, we will review some basic aspects of F-theory that will be
important for the following discussion. Further information on F-theory can be
found in DenefLesHouches ; TaylorTASI ; WeigandTASI ; ParkIntersection ;
CveticLinReview .

F-theory can be thought of most straightforwardly as a non-perturbative
version of Type IIB string theory that allows for consistent compactifications
in the presence of 7-branes. These 7-branes cause the axiodilaton field to
vary over the compactification space, leading the theory to be strongly-
coupled in some regions. We track the variation of the axiodilaton by encoding
it as the complex structure of an elliptic curve; this leads to the structure
of an elliptic fibration $X$ over the compactification space $B$. We assume
here that the base $B$ is smooth.

The elliptic fibration associated with an F-theory model can be described
mathematically by a Weierstrass model 4,

$y^{2}=x^{3}+fxz^{4}+gz^{6}\,,$ (38)

where $[x:y:z]$ are the homogeneous coordinates of $\mathbb{P}^{2,3,1}$ and
$f,g$ are holomorphic sections of line bundles on the base $B$. We will always
consider elliptically fibered Calabi–Yau manifolds, in which case $f$ and $g$
are respectively sections of the line bundles $\mathcal{O}(-4K_{B})$ and
$\mathcal{O}(-6K_{B})$, with $K_{B}$ the canonical class of the base.

The elliptic fiber of this fibration may become singular over certain loci in
the base. Fiber singularities at codimension one in the base signal that
nonabelian gauge groups are supported at these loci (in the context of Type
IIB, this indicates the presence of 7-branes wrapping these loci), while fiber
singularities at codimension two in the base herald the presence of localized
charged matter. The possible codimension-one fiber singularities were
classified by Kodaira and Néron KodairaI ; KodairaII ; NeronClassification ,
and are listed in Table 4. Using the Kodaira classification, one can read off
the singularity type at a codimension-one locus by examining the orders of
vanishing of $f$, $g$, and the discriminant $\Delta=4f^{3}+27g^{2}$.

Singularity |  |  |  |  |
---|---|---|---|---|---
Type | $\operatorname*{ord}(f)$ | $\operatorname*{ord}(g)$ | $\operatorname*{ord}(\Delta)$ |  Dynkin |
Diagram | $UniversalCovering\newline Group$ |  |  |  |
$\text{I}_{0}$ | $\geq 0$ | $\geq 0$ | $0$ |  — |  —
$\text{I}_{1}$ | $0$ | $0$ | $1$ |  — |  —
II | $\geq 1$ | $1$ | $2$ |  — |  —
III | $1$ | $\geq 2$ | $3$ |  $A_{1}$ |  $\operatorname{SU}(2)$
IV | $\geq 2$ | $2$ | $4$ |  $A_{2}$ |  $\operatorname{SU}(3)$
$\text{I}_{n}$ | $0$ | $0$ | $N$ |  $A_{n-1}$ |  $\operatorname{SU}(n)$
$\text{I}^{*}_{0}$ | $\geq 2$ | $\geq 3$ | $6$ |  $D_{4}$ |  $\operatorname{Spin}(8)$
$\text{I}^{*}_{n}$ | $2$ | $3$ | $N+6$ |  $D_{n+4}$ |  $\operatorname{Spin}(2n+8)$
$\text{IV}^{*}$ | $\geq 3$ | $4$ | $8$ |  $\operatorname{E}_{6}$ |  $\operatorname{E}_{6}$
$\text{III}^{*}$ | $3$ | $\geq 5$ | $9$ |  $\operatorname{E}_{7}$ |  $\operatorname{E}_{7}$
$\text{II}^{*}$ | $\geq 4$ | $5$ | $10$ |  $\operatorname{E}_{8}$ |  $\operatorname{E}_{8}$
Non-minimal | $\geq 4$ | $\geq 6$ | $\geq 12$ | Not valid in F-theory at codim.-one
Table 4: The Kodaira classification of codimension-one singular fibers. The
orders of vanishing of $f,g,\Delta$ at the singular locus in the base are
given. The Dynkin diagram entry gives the intersection pattern of the
exceptional $\mathbb{P}^{1}$s introduced by the resolution procedure; these
are potentially subject to Tate monodromy, though we will only consider ADE
cases here. The gauge algebra entries list the universal cover of the gauge
algebras associated with the given singularity type; which gauge algebra is
actually realized by a given singularity is determined via the Tate split
conditions mentioned in the text. Note that the final row lists non-minimal
singularities, which cannot be resolved crepantly such that the fibration
remains flat.

If one resolves the codimension-one fiber singularities via a sequence of
blowups, new $\mathbb{P}^{1}$ curves are introduced in the fiber over the
codimension-one locus. The $\mathbb{P}^{1}$ curves (including the
$\mathbb{P}^{1}$ hit by the zero section that exists prior to resolution)
intersect in the pattern of an affine ADE Dynkin diagram, with the
$\mathbb{P}^{1}$ hit by the zero section serving as the affine node. Thus,
each singularity type can be associated with an ADE Lie algebra. In
compactifications to six or fewer dimensions, the $\mathbb{P}^{1}$ curves can
undergo monodromy when traversing a closed path along the discriminant locus,
identifying them as being associated with the same exceptional divisor. This
corresponds to a folding of the Dynkin diagram and thus a reduction of the
associated Lie algebra to one of non-ADE type. Whether this occurs can be
determined by considering certain algebraic conditions on $f,g,\Delta$ known
as the Tate split conditions, which we will not elaborate on further here. In
this paper, we will always consider split fibers, in which case the associated
Lie algebra is always the ADE algebra corresponding to the given Kodaira
singularity. This Lie algebra is precisely the nonabelian gauge algebra
supported at the given codimension-one locus in the F-theory model.141414It is
worth noting that, in some situations, the true gauge algebra of the theory
can differ from that indicated by the geometry due to T-branes
CecottiCordovaHeckmanVafaTBranes ; AndersonHeckmanVafaTBranes . In this paper,
we assume that T-brane effects can be ignored and that the geometry accurately
reflects the gauge algebra and light charged matter of the model. In the
discussion that follows, it is important to consider the exceptional divisors
found by fibering the new $\mathbb{P}^{1}$ curves over the codimension-one
locus, which we denote by the symbol $\mathcal{T}$. The number of such
exceptional divisors associated with a codimension-one locus equals the rank
of the supported gauge algebra.

As mentioned above, matter is supported at codimension-two loci where the
singularity type enhances. In the Type IIB picture, this occurs at the
intersection of the 7-brane stacks associated with the codimension-one
singular loci. While the Kodaira classification provides a complete
classification of all possible codimension-one singularities, there is as yet
no complete classification of the singularity types at codimension two;
nonetheless, we often refer to codimension-two singularities using the Kodaira
singularity type associated to the orders of vanishing of $(f,g,\Delta)$ at
the given codimension-two locus.

In most situations, the matter representations supported at a given
codimension-two locus can be read off without performing a full resolution to
codimension two using what is known as the Katz–Vafa method KatzVafa .
Consider a given codimension-one singular locus $\\{\sigma=0\\}$ supporting a
gauge algebra $\mathfrak{g}$. At a codimension-two locus where the singularity
enhances further, we can naively associate a larger gauge algebra
$\mathfrak{g}^{\prime}\supset\mathfrak{g}$ with the enhanced singularity type
using the Kodaira classification (note, however, that there is not actually an
enhanced gauge algebra supported at the codimension-two locus). We can then
consider the branching rule for the adjoint representation of
$\mathfrak{g}^{\prime}$ to representations of $\mathfrak{g}$. This branching
rule will always include the adjoint representation of $\mathfrak{g}$ as well
as some number of singlets; the remaining representations, which always appear
in conjugate pairs, are the Katz–Vafa prediction for the charged light matter
supported at the codimension-two locus.

The Katz–Vafa method, while convenient, does not always produce the correct
result; in order to properly determine the matter representations supported at
a codimension-two locus, one must carry out a resolution of the singular
Calabi–Yau manifold to codimension-two. The charged matter representations can
then be read off from the dual M-theory picture. In this context, nonabelian
charged matter arises from M2 branes wrapping particular combinations of the
$\mathbb{P}^{1}$ curves making up the resolved fiber over the codimension-two
locus; the group theoretic weight associated with a particular combination of
curves is determined by computing its intersection with the various
exceptional divisors. More concretely, suppose we have a model with a
semisimple gauge algebra $\mathfrak{g}$. We can expand $\mathfrak{g}$ as a
direct sum of $K$ simple Lie algebras labeled by the index $\kappa$:

$\mathfrak{g}=\bigoplus_{\kappa=1}^{K}\mathfrak{g}_{\kappa}\,.$ (39)

If a codimension-two locus supports matter in a representation $\bm{R}$ of
$\mathfrak{g}$, then the resolved fiber at that locus consists of a set of
irreducible curves. A (possibly reducible) curve $c$ that supports the weight
$\vec{w}$ for the representation $\bm{R}$ satisfies

$\mathcal{T}_{\kappa,I}\cdot c=w_{\kappa,I}\,.$ (40)

Here, $\mathcal{T}_{\kappa,I}$ is the exceptional divisor corresponding to the
$I$th simple root of $\mathfrak{g}_{\kappa}$, while $w_{\kappa,I}$ is the
element of $\vec{w}$ corresponding to the $I$th Cartan generator of
$\mathfrak{g}_{\kappa}$ in the Dynkin basis (the basis of fundamental
weights).

###  3.1 Abelian gauge algebras in F-theory and the Mordell–Weil group

Abelian $\mathfrak{u}(1)$ gauge factors arise in F-theory in a different
fashion, and are not associated with codimension-one fiber singularities.
Rather, abelian gauge factors are associated with additional rational sections
of the elliptic fibration MorrisonVafaII .

F-theory constructions described by a Weierstrass model 38 will always have at
least one rational section, the zero section $\hat{o}$, given by
$[\hat{x}:\hat{y}:\hat{z}]=[1:1:0]$. However, an elliptic fibration may have
additional rational sections. These rational sections form a group, known as
the Mordell–Weil group. We discuss the addition operation of this group in the
following section. According to the Mordell–Weil theorem, this group is
finitely generated, taking the form LangNeron

$\mathbb{Z}^{r}\oplus\Gamma\,.$ (41)

The group $\Gamma$ is the torsion subgroup, which is associated in the
F-theory context with the global structure of the gauge group; we will not
discuss the torsional part of the Mordell–Weil group further here. The integer
$r$ is known as the Mordell–Weil rank, and the $r$ independent sections other
than the zero section are known as generating sections.

An F-theory model associated with an elliptic fibration that has Mordell–Weil
rank $r$ has an abelian sector including a $\mathfrak{u}(1)^{r}$ gauge
algebra. The motivation for this most naturally comes from the dual M-theory
picture. To each generating section $\hat{s}$ there is associated a divisor
class $\sigma(\hat{s})$ whose Poincaré dual acts as an additional zero mode
for the M-theory 3-form $C_{3}$, indicating the presence of an additional
$\operatorname{U}(1)$ gauge boson associated with $\hat{s}$. The divisor
$\sigma(\hat{s})$ is given by the Shioda map, which takes the form 9:

$\sigma(\hat{s})=\mathcal{S}-\mathcal{Z}-\sum_{\alpha}\mathopen{}\mathclose{{}\left((\mathcal{S}-\mathcal{Z})\cdot\mathcal{Z}\cdot
B^{\alpha}}\right)B_{\alpha}+\sum_{\kappa,I,J}\mathopen{}\mathclose{{}\left(\mathcal{S}\cdot\alpha_{\kappa,I}}\right)\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}\mathcal{T}_{\kappa,J}\,.$
(42)

This is a homomorphism from the Mordell–Weil group to the Néron–Severi group
with rational coefficients. As discussed above, weights of nonabelian matter
representations are associated with particular combinations of fibral curves
in the resolved threefold; the $\operatorname{U}(1)$ charges of the matter
associated with a fibral curve $c$ are given by

$q=\sigma(\hat{s}_{i})\cdot c\,,$ (43)

with $\hat{s}_{i}$ being the generating sections of the Mordell–Weil group.

One final important ingredient is the Néron–Tate height pairing between
rational sections $\hat{s}_{i},\hat{s}_{j}$ (with $i,j$ possibly equal), given
by

$\displaystyle 0pt_{ij}$
$\displaystyle=-\pi_{*}(\sigma(\hat{s}_{i})\cdot\sigma(\hat{s}_{i}))$ (44)
$\displaystyle=-\pi_{*}(\mathcal{S}_{i}\cdot\mathcal{S}_{j})-K_{B}+n_{i}+n_{j}-\sum_{\kappa}\mathopen{}\mathclose{{}\left(\mathcal{S}_{i}\cdot\alpha_{\kappa,I}}\right)\mathopen{}\mathclose{{}\left(\mathcal{R}_{\kappa}^{-1}}\right)_{IJ}\mathopen{}\mathclose{{}\left(\mathcal{S}_{j}\cdot\alpha_{\kappa,J}}\right)b_{\kappa}\,,$

with $\pi_{*}$ the pushforward to the homology lattice of the base,
$n_{i}=\pi_{*}(\mathcal{S}_{i}\cdot\mathcal{Z})$ the locus along which the
section $\hat{s}_{i}$ intersects the zero section, and $\mathcal{R}_{\kappa}$
the normalized root matrix for the $\kappa$th nonabelian gauge factor. The
height matrix $0pt_{ij}$ plays the role of the 6D anomaly coefficients for the
abelian gauge factors.

###  3.2 The elliptic curve group law

As we have seen, the rational sections of an elliptic fibration form a group,
called the Mordell–Weil group. The addition operation of rational sections is
a fiberwise extension of the group law for elliptic curves, which we will
summarize here. For more information, see Silverman:1338326 .

The addition operation of the Mordell–Weil group of rational points of an
elliptic curve is induced by the standard addition in $\mathbb{C}$ via the
definition of the elliptic curve as a complex torus $\mathbb{C}/\Lambda$.
Under the map from this description of the elliptic curve to that given by the
Weierstrass equation, the induced operation $[+]$ is a well-defined addition
on rational points of the elliptic curve with identity element $Z$, typically
chosen to be the point $[1:1:0]$ when the elliptic curve is written in global
Weierstrass form. This addition operation is defined by the property that if
$P,Q,R$ are the three points of intersection of the elliptic curve with a line
(counted with multiplicity), then $P[+]Q[+]R=Z$. Algorithmically, given two
points $P$ and $Q$, we can find their sum $P[+]Q$ by forming the line that
passes through both $P$ and $Q$ (if $P=Q$, then we instead use the tangent
line to the elliptic curve at $P$). This line intersects the elliptic curve in
a third point $R$; the line passing through $R$ and $Z$ again intersects the
elliptic curve in a third point, which is the result $P[+]Q$.

As we will make frequent use of elliptic curve addition in this paper, we
include here explicit expressions for the addition law when the elliptic curve
is written in global Weierstrass form, with identity $Z=[1:1:0]$. Given two
distinct points $P_{1}=[x_{1}:y_{1}:z_{1}]$ and $P_{2}=[x_{2}:y_{2}:z_{2}]$,
$P_{3}=P_{1}[+]P_{2}$ has coordinates

$\displaystyle x_{3}$
$\displaystyle=x_{1}z_{1}^{2}\mathopen{}\mathclose{{}\left(x_{2}^{2}+fz_{2}^{4}}\right)+x_{2}z_{2}^{2}\mathopen{}\mathclose{{}\left(x_{1}^{2}+fz_{1}^{4}}\right)-2z_{1}z_{2}(y_{1}y_{2}-gz_{1}^{3}z_{2}^{3})\,,$
(45) $\displaystyle y_{3}$
$\displaystyle=-y_{1}^{2}y_{2}z_{2}^{3}-3x_{2}x_{1}^{2}y_{2}z_{2}z_{1}^{2}+3x_{1}x_{2}^{2}y_{1}z_{1}z_{2}^{2}+y_{2}^{2}y_{1}z_{1}^{3}-3gz_{1}^{3}z_{2}^{3}\mathopen{}\mathclose{{}\left(y_{2}z_{1}^{3}-y_{1}z_{2}^{3}}\right)$
$\displaystyle\qquad-
fz_{1}z_{2}\mathopen{}\mathclose{{}\left(x_{2}y_{2}z_{1}^{5}+2x_{1}y_{2}z_{1}^{3}z_{2}^{2}-2x_{2}y_{1}z_{2}^{3}z_{1}^{2}-x_{1}y_{1}z_{2}^{5}}\right)\,,$
$\displaystyle z_{3}$ $\displaystyle=x_{2}z_{1}^{2}-x_{1}z_{2}^{2}\,.$

Given a point $P=[x:y:z]$, the point $2P=P[+]P$ has coordinates

$\displaystyle x_{2P}$
$\displaystyle=\mathopen{}\mathclose{{}\left(3x^{2}+fz^{4}}\right)^{2}-8xy^{2}\,,$
(46) $\displaystyle y_{2P}$
$\displaystyle=-\mathopen{}\mathclose{{}\left(3x^{2}+fz^{4}}\right)^{3}+12xy^{2}\mathopen{}\mathclose{{}\left(3x^{2}+fz^{4}}\right)-8y^{4}\,,$
$\displaystyle z_{2P}$ $\displaystyle=2yz\,.$

The inverse of a point $P=[x:y:z]$ is simply $-P=[x:-y:z]$.

##  4 Gauge group centers and allowed charges

As a first step towards relating $\mathfrak{u}(1)$ charges to the orders of
vanishing of section components, we should examine the types of
$\mathfrak{u}(1)$ charges that can occur in F-theory models. The allowed
$\mathfrak{u}(1)$ charges of singlet matter are quantized. In line with the
definition of the Shioda map in Eq. 9, it is natural to normalize the charges
such that the singlet charges are integral and that the lattice of singlet
charges has a spacing of 1 CveticLinU1 . However, matter that is also charged
under a nonabelian gauge factor may have fractional charges in this
normalization. The allowed fractional charges in an F-theory model depend on
the global structure of the gauge group, the description of which involves the
center of the gauge group’s universal cover.

Since it plays a key role in this paper, let us review the relation between
fractional charges and elements of the center first described in CveticLinU1 .
Suppose we have a model with a gauge algebra
$\mathfrak{g}\oplus\mathfrak{u}(1)$, where $\mathfrak{g}$ is a semisimple Lie
algebra given by

$\mathfrak{g}=\bigoplus_{\kappa=1}^{K}\mathfrak{g}_{\kappa}\,.$ (47)

Recall from Section 3 that matter in a representation $\bm{R}_{q}$ of
$\mathfrak{g}\oplus\mathfrak{u}(1)$ comes from M2 branes wrapping curves in
the resolved fiber of a codimension-two locus in the base. Specifically, a
weight $\vec{w}$ of the representation $\bm{R}_{q}$ comes from wrapping an M2
brane on a curve $c$ satisfying,

$\mathcal{T}_{\kappa,I}\cdot c=w_{\kappa,I}\,,$ (48)

where $\mathcal{T}_{\kappa,I}$ is the fibral divisor corresponding to the
$I$th simple root of $\mathfrak{g}_{\kappa}$ and $w_{\kappa,I}$ is the element
of $\vec{w}$ corresponding to the $I$th Cartan generator of
$\mathfrak{g}_{\kappa}$ The charge of this matter under the $\mathfrak{u}(1)$
algebra is

$q=\sigma(\hat{s})\cdot c\,.$ (49)

Equation 9 states that the Shioda map is given by

$\sigma(\hat{s})=\mathcal{S}-\mathcal{Z}-\pi^{*}\mathopen{}\mathclose{{}\left(D_{B}}\right)+\sum_{\kappa,I}l_{\kappa,I}\mathcal{T}_{\kappa,I}\,,$
(50)

where

$l_{\kappa,I}=\sum_{J}(\mathcal{S}\cdot\alpha_{\kappa,J})\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}\,.$
(51)

In rough terms, the numbers $l_{\kappa,I}$ describe which of the components in
the ADE fibers along codimension-one loci are hit by the section. They
therefore depend on the behavior of the section at codimension one. However,
these numbers are allowed to be fractional, as the inverse Cartan matrix
$\mathcal{C}^{-1}_{\kappa}$ can have fractional entries. In fact, the only
fractional contributions to the charge $q$ come from the $l_{\kappa,I}$ term
in the Shioda map, implying that

$q-\sum_{\kappa,I}l_{\kappa,I}\mathcal{T}_{\kappa,I}\cdot c\in\mathbb{Z}\,.$
(52)

This statement is true for any weight of any $\bm{R}_{q}$ representation
realized in the F-theory model. Note that
$\sum_{\kappa,I}l_{\kappa,I}\mathcal{T}_{\kappa,I}\cdot c$ may differ by
integers for different weights in an irreducible representation $\bm{R}_{q}$.

If $\mathcal{Q}$ is the generator of the $\mathfrak{u}(1)$ algebra and
$E_{\kappa,I}$ are the elements of the Cartan sub-algebra of
$\mathfrak{g}_{\kappa}$, then
$\Upsilon=\mathcal{Q}-\sum_{\kappa,I}l_{\kappa,I}E_{\kappa,I}$ is an element
of the Cartan subalgebra of $\mathfrak{g}\oplus\mathfrak{u}(1)$.
Exponentiating this gives us an element $C=\exp[2\pi i\Upsilon]$ of
$G\times\operatorname{U}(1)$, the universal covering group for the algebra
$\mathfrak{g}\oplus\mathfrak{u}(1)$. $G$ is the simply connected Lie group
associated with $\mathfrak{g}$. For any representation $\bm{R}_{q}$ of
$\mathfrak{g}\oplus\mathfrak{u}(1)$, $C$ acts on a weight $\vec{w}$ of the
representation as

$C\vec{w}=\exp\mathopen{}\mathclose{{}\left[2\pi
i\mathopen{}\mathclose{{}\left(q-\sum_{\kappa,I}l_{\kappa,I}w_{\kappa,I}}\right)}\right]\vec{w}\,.$
(53)

The group element $C$ essentially multiplies each weight by a complex number
of magnitude 1. This complex number is the same for all weights in an
irreducible representation $\bm{R}_{q}$, as
$\sum_{\kappa,I}l_{\kappa,I}w_{\kappa,I}$ differs only by integers for the
different weights of this irreducible representation.151515One way of seeing
this is to note that two different weights for the same irreducible
representation differ by a root. The term
$\sum_{\kappa,I}l_{\kappa,I}w_{\kappa,I}$ essentially takes the form
$\sum_{\kappa,I,J}n_{J}\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}w_{\kappa,I}$,
where the $n_{J}$ are integers. The inverse Cartan matrices convert weight and
root vectors from the Dynkin basis to the basis of simple roots, and root
vectors have integral elements in the latter basis. Therefore, the difference
in
$\sum_{\kappa,I,J}n_{J}\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}w_{\kappa,I}$
for two different weights, which differ by a root, should be integral. Thus,
$C$ is proportional to the identity element and commutes with every element of
$G\times\operatorname{U}(1)$. In other words, $C$ is in the center of
$G\times\operatorname{U}(1)$. Moreover, Eqs. 48 and 52 imply that $C$ acts
trivially on any representation $\bm{R}_{q}$ realized in the F-theory model.
The gauge group of the F-theory model is therefore
$(G\times\operatorname{U}(1))/\Xi$, where $\Xi$ is a subgroup of the center of
$G\times\operatorname{U}(1)$ that includes $C$.

In summary, the $l_{i}$ coefficients in $\sigma(\hat{s})$, which control the
allowed fractional charges in the F-theory model and affect the global
structure of the gauge group, can be associated with an element of the gauge
group’s center. It is therefore worth describing the elements of the center in
more detail. If the gauge algebra is $\mathfrak{g}\oplus\mathfrak{u}(1)$ as
before, we are interested in the center $Z(G\times\operatorname{U}(1))$, where
$G$ is the simply connected Lie group associated with $\mathfrak{g}$. Since
$\mathfrak{g}$ is a sum of simple Lie algebras
$\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\ldots\oplus\mathfrak{g}_{K}$,
the group $G$ is a product $G_{1}\times G_{2}\times\ldots G_{K}$ of the
universal covers of the $\mathfrak{g}_{\kappa}$. The center of
$G\times\operatorname{U}(1)$ is then

$Z\mathopen{}\mathclose{{}\left(G\times\operatorname{U}(1)}\right)=Z(G)\times
Z\mathopen{}\mathclose{{}\left(\operatorname{U}(1)}\right)=\mathopen{}\mathclose{{}\left(\prod_{\kappa}Z(G_{\kappa})}\right)\times\operatorname{U}(1)\,.$
(54)

An element of
$Z\mathopen{}\mathclose{{}\left(G\times\operatorname{U}(1)}\right)$ can be
written as $(\nu_{1},\nu_{2},\ldots,\nu_{K},\gamma)$, where $\nu_{\kappa}$ is
an element $Z(G_{\kappa})$ and $\gamma$ is an element of
$\operatorname{U}(1)$. We want elements of
$Z\mathopen{}\mathclose{{}\left(G\times\operatorname{U}(1)}\right)$ analogous
to $C$ above, so we let $\gamma$ be $\exp[2\pi i\mathcal{Q}]$. The
$l_{\kappa,I}$ are then encoded in a choice of the $\nu_{\kappa}$. The centers
of some of the compact simple Lie groups are given in Table 5. Many of these
centers are $\mathbb{Z}_{n}$ groups, so it is convenient to let the
$\nu_{\kappa}$ be integers corresponding to elements of the $\mathbb{Z}_{n}$.
Even for the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ center of
$\operatorname{Spin}(4k)$, we label the elements of the center with integers
ranging from 0 to 3. The notations for each group are discussed in more detail
in the following subsections.

Algebra | Group | Center
---|---|---
$\mathfrak{su}(n)$ | $\operatorname{SU}(n)$ | $\mathbb{Z}_{n}$
$\mathfrak{so}(2n+1)$ | $\operatorname{Spin}(2n+1)$ | $\mathbb{Z}_{2}$
$\mathfrak{sp}(n)$ | $\operatorname{Sp}(n)$ | $\mathbb{Z}_{2}$
$\mathfrak{so}(4k)$ | $\operatorname{Spin}(4k)$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
$\mathfrak{so}(4k+2)$ | $\operatorname{Spin}(4k+2)$ | $\mathbb{Z}_{4}$
$\mathfrak{e}_{6}$ | $\operatorname{E}_{6}$ | $\mathbb{Z}_{3}$
$\mathfrak{e}_{7}$ | $\operatorname{E}_{7}$ | $\mathbb{Z}_{2}$
$\mathfrak{e}_{8}$ | $\operatorname{E}_{8}$ | —
$\mathfrak{f}_{4}$ | $\operatorname{F}_{4}$ | —
$\mathfrak{g}_{2}$ | $\operatorname{G}_{2}$ | —
Table 5: Centers of various groups. If a center is not specified, the center
of the group is trivial. $n$ and $k$ are integers.

We can describe the relation between the $\nu_{\kappa}$ and the $l_{\kappa,I}$
more explicitly: because

$l_{\kappa,I}=\sum_{J}\mathopen{}\mathclose{{}\left(\mathcal{S}\cdot\alpha_{J}}\right)\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}\,,$

the $l_{\kappa,I}$ are determined by the behavior of the section along the
codimension-one loci supporting the $\mathfrak{g}_{\kappa}$ gauge algebras.
Let us first focus on the situation where $K=1$, such that the gauge algebra
is $\mathfrak{g}_{1}\oplus\mathfrak{u}(1)$. We additionally assume that
$\mathfrak{g}_{1}$ is simply-laced, as we primarily consider only simply-laced
algebras in this paper. At the codimension-one locus supporting
$\mathfrak{g}_{1}$, the fiber consists of irreducible curves forming an affine
ADE diagram corresponding to $\mathfrak{g}_{1}$; again, the curve representing
the affine node is the one hit by the zero section. Fig. 1 illustrates this
fiber for an example where $\mathfrak{g}_{1}=\mathfrak{so}(10)$. The
generating section for the $\mathfrak{u}(1)$ should intersect one of these
curves transversally at generic points along the codimension-one
locus.161616We should not expect the section to exhibit more extreme
behaviors, such as wrapping a curve, at generic points along the codimension-
one locus. Such behavior would require all the section components to vanish
simultaneously, at least prior to resolution. The section components would
then be proportional to appropriate powers of the codimension-one locus, and
these factors can be scaled away to remove the wrapping behavior. The numbers
$l_{\kappa,I}$ essentially indicate which of these curves is intersected by
the section. Each element of the center should in turn correspond to a curve
in the codimension-one fiber that may be hit by the section, and vice
versa.171717Along these lines, note that the different values of $\nu$
distinguish between, for instance, the $\text{I}_{n}^{(01)}$,
$\text{I}_{n}^{(0|1)}$, $\text{I}_{n}^{(0||1)}$, … models in the language of
KuntzlerTateTrees ; LawrieEtAlRational . This statement naively seems to be
problematic in the $\mathfrak{so}(10)$ example: while the codimension-one
fiber has six irreducible curves, there are only four elements of
$Z(\operatorname{Spin}(10))=\mathbb{Z}_{4}$. However, the irreducible curves
in the fiber may have multiplicities greater than one; these multiplicities
are given by the dual Kac labels (or comarks) of the highest root. Because a
section should intersect the fiber only once, it can only intersect one of the
irreducible curves with multiplicity $1$ KuntzlerTateTrees ;
LawrieEtAlRational . For the $\mathfrak{so}(10)\oplus\mathfrak{u}(1)$ example,
only the four curves at the ends of the fiber have multiplicity $1$. These
four curves exactly match the four elements of
$Z(\operatorname{Spin}(10))=\mathbb{Z}_{4}$, in line with the expectation that
the curve hit by the section at codimension one should specify an element of
the center. If the section hits the affine component, we say that $\nu=0$, the
identity element of the center. If the section hits the upper-left component
in Fig. 1, then we say that $\nu=2$, and if the section hits the upper-right
or lower-right curves, then we say that $\nu$ is $1$ or $3$, respectively.

Figure 1: Fiber at generic points along a codimension-one locus supporting an
$\mathfrak{so}(10)$ algebra. The shaded curve corresponds to the affine node
of the $\hat{D}_{5}$ Dynkin diagram. The numbers within each circle denote the
multiplicity of the curve.

The $l_{\kappa,I}$ numbers also provide a way to determine the allowed
fractional charges for an F-theory model. Again, suppose that the gauge
algebra is $\mathfrak{g}_{1}\oplus\mathfrak{u}(1)$. If the section hits the
affine node of the fiber at codimension one, the $l_{1,I}$ are $0$ and the
$\mathfrak{u}(1)$ charges for matter in any representation of
$\mathfrak{g}_{1}$ must be integral. If the section hits a curve corresponding
to the simple root $J$ of $\mathfrak{g}_{1}$, the $l_{1,I}$ are given by
$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}$. To
find the allowed $\mathfrak{u}(1)$ charges for matter in an irreducible
representation $\bm{R}_{q}$ of $\mathfrak{g}_{1}\oplus\mathfrak{u}(1)$, we
consider any weight $\vec{w}$ of $\bm{R}_{q}$ in the Dynkin basis. The allowed
$\mathfrak{u}(1)$ charges are then given by

$\mathopen{}\mathclose{{}\left(\sum_{I}l_{1,I}w_{I}}\right)+j\,,\quad
j\in\mathbb{Z}\,.$ (55)

Of course, we can relate the curve hit by the section to an element of
$Z(G_{1})$, providing us with a way to determine the $\mathfrak{u}(1)$ charges
consistent with the global structure of the gauge group. Eq. 55 shows that the
allowed $\mathfrak{u}(1)$ charges for $\overline{\bm{R}}$ are the negative of
those for $\bm{R}$, since the weights of $\overline{\bm{R}}$ are the negative
of those of $\bm{R}$. This ensures that we can form $\bm{R}_{q}$
hypermultiplets by combining fields in the $\bm{R}_{q}$ and
$\overline{\bm{R}}_{-q}$ representations.

To illustrate these ideas, consider the
$\mathfrak{so}(10)\oplus\mathfrak{u}(1)$ example above and assume that, at
codimension-one, the section hits the upper-right curve in Fig. 1
corresponding to $\nu=1$. The Cartan and inverse Cartan matrices for
$\mathfrak{so}(10)$ are

$\mathcal{C}_{\mathfrak{so}(10)}=\begin{pmatrix}2&-1&0&0&0\\\ -1&2&-1&0&0\\\
0&-1&2&-1&-1\\\ 0&0&-1&2&0\\\
0&0&-1&0&2\end{pmatrix}\,,\qquad\mathcal{C}^{-1}_{\mathfrak{so}(10)}=\begin{pmatrix}1&1&1&\frac{1}{2}&\frac{1}{2}\\\
1&2&2&1&1\\\ 1&2&3&\frac{3}{2}&\frac{3}{2}\\\
\frac{1}{2}&1&\frac{3}{2}&\frac{5}{4}&\frac{3}{4}\\\
\frac{1}{2}&1&\frac{3}{2}&\frac{3}{4}&\frac{5}{4}\end{pmatrix}\,,$ (56)

and the upper-right curve corresponds to $J=4$. Therefore,

$l_{1,I}=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{I4}\,.$
(57)

For the $\bm{10}$ representation of $\mathfrak{so}(10)$, the highest weight is
$[1,0,0,0,0]$, and

$\sum_{I=1}^{5}\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{I4}w_{I}=\frac{1}{2}\,.$
(58)

The allowed $\bm{10}$ charges are therefore $j+\frac{1}{2}$ for integer $j$.
The $\bm{16}$ representation, meanwhile, has a highest weight of
$[0,0,0,0,1]$, and

$\sum_{I=1}^{5}\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{I4}w_{I}=\frac{3}{4}\,.$
(59)

The allowed $\bm{16}$ charges are therefore $j-\frac{1}{4}$ for integer $j$.

It is useful to define a few functions of $\nu$. While the specific form of
these functions depends on the simple gauge algebra in question, as described
in detail below, we can state the general idea behind these functions without
reference to a particular gauge algebra. As mentioned above, each possible
value of $\nu$ is identified with a curve hit by the section at generic points
along the codimension-one locus supporting a simple gauge algebra
$\mathfrak{g}$. We define

$T_{G}(\nu)=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{g}}}\right)_{II}\,,$
(60)

where $I$ denotes the curve identified with $\nu$. If $\nu$ is 0, we take
$T_{G}(\nu)$ to be 0 as well. In the $\mathfrak{so}(10)$ example, $\nu=1$
corresponds to the upper-right curve in Fig. 1, which is specified by $I=4$.
Since

$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{44}=\frac{5}{4}\,,$
(61)

we have

$T_{\operatorname{Spin}(10)}(1)=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{44}=\frac{5}{4}\,.$
(62)

Along similar lines,

$\displaystyle T_{\operatorname{Spin}(10)}(0)$ $\displaystyle=0\,,$ (63)
$\displaystyle T_{\operatorname{Spin}(10)}(2)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{11}=\frac{1}{2}\,,$
$\displaystyle T_{\operatorname{Spin}(10)}(3)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{so}(10)}}\right)_{55}=\frac{5}{4}\,.$

We also define a function $\vec{\tau}_{G}(\nu)$ whose output is a triplet of
integers. Roughly, it gives the values of

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)$

such that the section hits the fibral curve associated with $\nu$, at least at
generic points along the codimension-one locus supporting
$\mathfrak{g}$.181818Intriguingly, the values of $\vec{\tau}_{G}(\nu)$ for
various $G$, $\nu$ also appear in the Tate’s algorithm table
BershadskyEtAlSingularities  as the orders of vanishing of
$a_{2}+\frac{1}{4}a_{1}^{2}$, $a_{3}$, and $a_{4}$ for various singularity
types. This observation may be related to the shifts in $x$ and $y$ performed
when converting the elliptic fibration from Tate form to Weierstrass form. We
will describe $\vec{\tau}_{G}(\nu)$ in more detail when we individually
discuss the different gauge algebras. Even though we have developed these
ideas based on the codimension-one behavior of the sections, it will be useful
to use these same functions at codimension two as well. When these expressions
are used at codimension two, $\nu$ is replaced with $\mu$.

These ideas generalize naturally for models where $\mathfrak{g}$ consists of
more than one simple algebra. In particular, let $\mathfrak{g}$ be the direct
sum of simple algebras $\mathfrak{g}_{\kappa}$. Each $\mathfrak{g}_{\kappa}$
simple algebra is supported on a codimension-one locus in the base. The fiber
at this locus consists of a set of curves forming the affine Dynkin diagram of
$\mathfrak{g}_{\kappa}$. We have a set of numbers $l_{\kappa,I}$ for each
$\mathfrak{g}_{\kappa}$ that describe which of the fibral curves is hit by the
section, at least at generic points along the codimension-one locus for
$\mathfrak{g}_{\kappa}$. If the section hits the affine node, then
$l_{\kappa,I}=0$. If the section hits one of the other curves, then

$l_{\kappa,I}=\sum_{J}\mathopen{}\mathclose{{}\left(\mathcal{S}\cdot\alpha_{\kappa,J}}\right)\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\kappa}}\right)_{IJ}\,,$
(64)

where $\mathcal{C}^{-1}_{\kappa}$ is the inverse Cartan matrix for
$\mathfrak{g}_{\kappa}$. Of course, the section can only hit those curves with
multiplicity 1 in the fiber. As before, the $l_{\kappa,I}$ can alternatively
by described using a specific element of $Z(G\times\operatorname{U}(1))$. For
each $\kappa$, we can associate $l_{\kappa,I}$ with an element $\nu_{\kappa}$
of the center of $G_{\kappa}$, the universal covering group for
$\mathfrak{g}_{\kappa}$. Then, the gauge group has a quotient by a group
containing the element $(\nu_{1},\ldots,\nu_{K},\gamma)$ of
$Z(G\times\operatorname{U}(1))$, where $K$ is the total number of simple,
nonabelian gauge algebras. We can then find $T_{G_{\kappa}}(\nu_{\kappa})$ and
$\vec{\tau}_{G_{\kappa}}(\nu_{\kappa})$ for each $\mathfrak{g}_{\kappa}$.

We now describe how the ideas above are applied for the types of gauge
algebras discussed here. We focus only on the simply-laced algebras. For each
gauge algebra, we describe how the possible values of $\nu$ match with curves
in the resolved fibers and give explicit expressions for $T_{G}(\nu)$ and
$\vec{\tau}_{G}(\nu)$. We also discuss the allowed $\mathfrak{u}(1)$ charges
for matter in representations of these gauge algebras.

###  4.1 $\mathfrak{su}(n)$

The affine Dynkin diagram for $\mathfrak{su}(n)$, shown in Fig. 2, has $n$
nodes. The curves are labeled by $I$, which runs from 0 to $n-1$ with the
$I=0$ curve serving as the affine node of the diagram. All of the curves in
the resolved $\mathfrak{su}(n)$ fiber have multiplicity $1$, so any of them
can be hit by a section. This agrees with the fact that the center of
$\operatorname{SU}(n)$ is $\mathbb{Z}_{n}$, and each curve should correspond
to an element of the center. We take $\nu$ to be $n-I$, where $I$ refers to
the curve hit by the section, or 0 if the section hits the $I=0$ affine node
curve. In the language of KuntzlerTateTrees ; LawrieEtAlRational , $\nu=0$
corresponds to the $\text{I}_{n}^{(01)}$ fiber, $\nu=1$ corresponds to the
$\text{I}_{n}^{(0|1)}$ fiber, and in general $\nu$ corresponds to an
$\text{I}_{n}^{(0|\ldots|1)}$ fiber with $u_{\nu}(n)$ bars between the 0 and
the 1.

12$n-2$$n-1$0$\cdots$ Figure 2: Dynkin diagram for $\mathfrak{su}(n)$. The
numbers give the value of $I$ corresponding to each node; the affine node has
$I=0$.

The inverse Cartan matrix for $\operatorname{SU}(n)$ takes the form
YamatsuGroupTheory

$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{su}(n)}}\right)_{IJ}=\frac{1}{n}\min(I,J)\mathopen{}\mathclose{{}\left(n-\max(I,J)}\right)\,.$
(65)

Therefore, we define

$T_{\operatorname{SU}(n)}(\nu)=\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{su}(n)}}\right)_{n-\nu,n-\nu}=\frac{\nu(n-\nu)}{n}\,.$
(66)

Since the highest weight of the fundamental representation is
$[1,0,\ldots,0]$, Eq. 55 implies that the allowed $\mathfrak{u}(1)$ charges
for the fundamental representation are

$\frac{\nu}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (67)

The two-index antisymmetric representation has a highest weight of
$[0,1,0,\ldots,0]$, and its allowed $\mathfrak{u}(1)$ charges are

$\frac{2\nu}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (68)

Note that the allowed charges for $\nu^{\prime}=n-\nu$ are the negative of
those for $\nu$. This fact agrees with the group structure of the center, as
the inverse of $\nu$ in $\mathbb{Z}_{n}$ is $n-\nu$.

For the antisymmetric representation of $\mathfrak{su}(n)$, the same charge
can occur for different values of $\nu$ when $n$ is even. When the gauge
algebra is $\mathfrak{su}(6)\oplus\mathfrak{u}(1)$, for instance,
$\bm{15}_{1}$ matter can occur when $\nu=0$ or when $\nu=3$. Because $\nu$
describes the global structure of the gauge group, a situation involving
$\bm{15}_{1}$ with $\nu=0$ is distinct from a situation involving
$\bm{15}_{1}$ with $\nu=3$. We should therefore expect that the orders of
vanishing for $\bm{15}_{1}$ should be different for $\nu=0$ and $\nu=3$. This
does not pose any issue if one is reading off the charges from a model, as one
can determine $\nu$ from the codimenson-one orders of vanishing. However, when
trying to predict the orders of vanishing for a certain charge in a spectrum,
one may also need to choose a value of $\nu$. The need for this extra
information is not a shortcoming of our approach; rather, it reflects the
unavoidable fact that models with different global gauge groups are distinct.

We also define

$\vec{\tau}_{\operatorname{SU}(n)}(\nu)=\mathopen{}\mathclose{{}\left(0,1,1}\right)\times
u_{n}(\nu)$ (69)

for $n\geq 2$, where $u_{n}(\nu)$ is defined in Section 2.1. These numbers
describe the orders of vanishing for $\hat{x}$, $\hat{y}$, and $\hat{w}$ such
that the section hits the appropriate curve in the $\text{I}_{n}$ fiber.
However, $\operatorname{SU}(2)$ and $\operatorname{SU}(3)$ can also be
realized by type III and type IV fibers. To account for these special cases,
we also define

$\vec{\tau}_{\text{III}}(\nu)=\mathopen{}\mathclose{{}\left(1,1,1}\right)\times
u_{2}(\nu)\,,\quad\vec{\tau}_{\text{IV}}(\nu)=\mathopen{}\mathclose{{}\left(1,1,2}\right)\times
u_{3}(\nu)\,.$ (70)

###  4.2 $\mathfrak{so}(2n)$

While the affine Dynkin diagram for $\mathfrak{so}(2n)$, shown in Fig. 3, has
$n$ nodes, only four of them occur with multiplicity 1 in the resolved fiber.
These are the nodes at the ends of the diagram, namely the affine node and the
three nodes labeled by $I=1$, $n-1$, and $n$. The center
$Z(\operatorname{Spin}(2n))$ is $\mathbb{Z}_{4}$ for odd $n$ and
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ for even $n$. In either case,
$Z(\operatorname{Spin}(2n))$ has four elements, in agreement with the number
of multiplicity-one curves in the resolved fiber. For odd $n$, we label the
elements of $\mathbb{Z}_{4}$ by $\nu=0,1,2,3$. We then identify $\nu=0$ with
the affine node, $\nu=1$ with the $I=n-1$ node, $\nu=2$ with the $I=1$ node,
and $\nu=3$ with the $I=n$ node. For even $n$, we still label the elements of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ by $\nu=0,1,2,3$. We also identify
$\nu=0$ with the affine node, $\nu=1$ with the $I=n-1$ node, $\nu=2$ with the
$I=1$ node, and $\nu=3$ with the $I=n$ node. In the language of
KuntzlerTateTrees ; LawrieEtAlRational , $\nu=0$ corresponds to the
${\text{I}^{*}_{n-4}}^{(01)}$ fiber, $\nu=2$ corresponds to the
${\text{I}^{*}_{n-4}}^{(0|1)}$ fiber, and $\nu=1,3$ correspond to the
${\text{I}^{*}_{n-4}}^{(0||1)}$ fiber.

The inverse Cartan matrix for $\mathfrak{so}(2n)$ takes the form
YamatsuGroupTheory

$\mathcal{C}^{-1}_{\mathfrak{so}(2n)}=\frac{1}{2}\begin{pmatrix}2&2&2&\cdots&2&1&1\\\
2&4&4&\cdots&4&2&2\\\ 2&4&6&\cdots&6&3&3\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
2&4&6&\cdots&2(n-2)&(n-2)&(n-2)\\\ 1&2&3&\cdots&(n-2)&n/2&(n-2)/2\\\
1&2&3&\cdots&(n-2)&(n-2)/2&n/2\end{pmatrix}\,.$ (71)

Therefore,

$T_{\operatorname{Spin}(2n)}(\nu)=\begin{cases}0&\nu=0\\\ 1&\nu=2\\\
\frac{n}{4}&\nu=1,3\end{cases}\,.$ (72)

The allowed charges for various representations depend on the value of $n$
modulo 4. The vector representation of $\mathfrak{so}(2n)$ has a highest
weight of $[1,0,\ldots,0]$. There are also two spinor representations for all
$n$, which have highest weights $[0,0,\ldots,0,1]$ and $[0,0,\ldots,1,0]$. For
odd $n$, these representations are conjugates of each other, and spinor
hypermultiplets contain fields in both of these representations. However, for
even $n$, these two spinors are self-conjugate and can therefore be supported
at distinct loci ( and can belong to distinct hypermultiplets in 6D contexts).
The allowed $\mathfrak{u}(1)$ charges for these representations are summarized
in Table 6. As for the antisymmetric representation of $\mathfrak{su}(n)$,
matter in particular representations can occur with the same charge for
different values of $\nu$. Note that, for odd $n$, the allowed charges for
$\nu=1$ and $\nu=3$ are negatives of each other, in line with the fact that
$\nu=1$ and $\nu=3$ are each other’s inverses in $\mathbb{Z}_{4}$. Finally, we
define

$\vec{\tau}_{\operatorname{Spin}(2n)}(\nu)=\begin{cases}(0,0,0)&\nu=0\\\
(1,2,2)&\nu=2\\\
(1,\mathopen{}\mathclose{{}\left\lfloor\frac{n}{2}}\right\rfloor,\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil)&\nu=1,3\end{cases}$
(73)

12$n-2$$n-1$$n$0$\cdots$ Figure 3: Dynkin diagram for $\mathfrak{so}(2n)$. The numbers give the value of $I$ corresponding to each node, with the affine node having $I=0$. Nodes marked with two concentric circles occur with multiplicity 2 in the fiber. Group | Irrep | $\nu$
---|---|---
0 | 1 | 2 | 3
$\mathfrak{so}(8\ell)$ | V | 0 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$
S | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0
C | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$
$\mathfrak{so}(8\ell+2)$ | V | 0 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$
S | 0 | $\frac{3}{4}$ | $\frac{1}{2}$ | $\frac{1}{4}$
C | 0 | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{3}{4}$
$\mathfrak{so}(8\ell+4)$ | V | 0 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$
S | 0 | 0 | $\frac{1}{2}$ | $\frac{1}{2}$
C | 0 | $\frac{1}{2}$ | $\frac{1}{2}$ | 0
$\mathfrak{so}(8\ell+6)$ | V | 0 | $\frac{1}{2}$ | 0 | $\frac{1}{2}$
S | 0 | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{3}{4}$
C | 0 | $\frac{3}{4}$ | $\frac{1}{2}$ | $\frac{1}{4}$
Table 6: Fractional parts of the allowed $\mathfrak{u}(1)$ charges for various
representations of $\mathfrak{so}(2n)$. V refers to the vector representation
with highest weight $[1,0,\ldots,0]$. S refers to the spinor representation
with highest weight $[0,\ldots,0,1]$. C refers to the spinor representation
with highest weight $[0,\ldots,1,0]$.

###  4.3 $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$, and $\mathfrak{e}_{8}$

123$\times 3$$4$$5$$6$0 (a) $\mathfrak{e}_{6}$

012$\times 3$3$\times 4$$4$$\times 3$$5$$6$$7$ (b) $\mathfrak{e}_{7}$

12$\times 4$3$\times 6$$4$$\times 5$$5$$\times 4$$6$$\times 3$$7$0$8$$\times
3$ (c) $\mathfrak{e}_{8}$

Figure 4: Dynkin diagrams for $\mathfrak{e}_{6}$,$\mathfrak{e}_{7}$ and
$\mathfrak{e}_{8}$. The numbers within each circle give the value of $I$
corresponding to each node, with the affine node having $I=0$. Nodes marked
with two concentric circles occur with multiplicity greater than 1 in the
fiber. If the multiplicity is greater than 2, the multiplicity is indicated
just outside the node. Nodes with concentric circles without a given
multiplicity have multiplicity 2.

The affine Dynkin diagram for $\mathfrak{e}_{6}$, shown in Fig. 4(a), has
seven nodes. Three of these nodes—the $I=0$, $1$, and $5$ nodes—occur with
multiplicity 1 in the resolved fiber, in agreement with the three elements of
$Z(\operatorname{E}_{6})=\mathbb{Z}_{3}$. Labeling the elements of
$\mathbb{Z}_{3}$ with $\nu=0,1,2$, we associate $\nu=0$ with the $I=0$ affine
node, $\nu=1$ with the $I=1$ curve, and $\nu=2$ with the $I=5$ curve. In the
language of KuntzlerTateTrees ; LawrieEtAlRational , $\nu=0$ corresponds to
the ${\text{IV}^{*}}^{(01)}$ fiber, and $\nu=1,2$ correspond to the
${\text{IV}^{*}}^{(0|1)}$ fiber. The inverse Cartan matrix of
$\mathfrak{e}_{6}$ is

$\mathcal{C}^{-1}_{\mathfrak{e}_{6}}=\frac{1}{3}\begin{pmatrix}4&5&6&4&2&3\\\
5&10&12&8&4&6\\\ 6&12&18&12&6&9\\\ 4&8&12&10&5&6\\\ 2&4&6&5&4&3\\\
3&6&9&6&3&6\end{pmatrix}\,.$ (74)

Based on its diagonal entries for $I=1$ and $I=5$, we define

$T_{\operatorname{E}_{6}}(\nu)=\frac{2}{3}\nu(3-\nu)\,.$ (75)

We also define

$\vec{\tau}_{\operatorname{E}_{6}}(\nu)=\mathopen{}\mathclose{{}\left(2,2,3}\right)\times
u_{3}(\nu)$ (76)

In this paper, we are primarily interested in matter in the $\bm{27}$
representation, with highest weight $[1,0,0,0,0,0]$, and its conjugate. From
the form of the inverse Cartan matrix, the allowed $\mathfrak{u}(1)$ charges
for the $\bm{27}$ representation are

$\frac{\nu}{3}+j\text{ for }j\in\mathbb{Z}\,.$ (77)

The allowed charges for the $\overline{\bm{27}}$ representation are the
negatives of those for the $\bm{27}$ representation.

The affine Dynkin diagram for $\mathfrak{e}_{7}$, shown in Fig. 4(b), has two
nodes that occur with multiplicity $1$: the $I=0$ affine node and the $I=6$
node. The center of $\operatorname{E}_{7}$ is $\mathbb{Z}_{2}$, which has two
elements labeled by $\nu=0$ and $\nu=1$. We associate $\nu=0$ with the affine
$I=0$ node and $\nu=1$ with the $I=6$ node. In the language of
KuntzlerTateTrees ; LawrieEtAlRational , $\nu=0$ corresponds to the
${\text{III}^{*}}^{(01)}$ fiber, and $\nu=1,2$ correspond to the
${\text{III}^{*}}^{(0|1)}$ fiber. Since the inverse Cartan matrix of
$\mathfrak{e}_{7}$ has

$\mathopen{}\mathclose{{}\left(\mathcal{C}^{-1}_{\mathfrak{e}_{7}}}\right)_{I6}=\frac{1}{2}\mathopen{}\mathclose{{}\left(2,4,6,5,4,3,3}\right)\,,$
(78)

we define

$T_{\operatorname{E}_{7}}(\nu)=\frac{3}{2}\nu\,.$ (79)

We also define

$\vec{\tau}_{\operatorname{E}_{7}}(\nu)=\mathopen{}\mathclose{{}\left(2,3,3}\right)\times
u_{2}(\nu)=\mathopen{}\mathclose{{}\left(2,3,3}\right)\times\nu\,.$ (80)

We are primarily interested in matter in the pseudoreal $\bm{56}$
representation of $\operatorname{E}_{7}$, which has highest weight
$[0,0,0,0,0,1,0]$. The allowed $\mathfrak{u}(1)$ charges for $\bm{56}$ matter
are therefore

$\frac{\nu}{2}+j\text{ for }j\in\mathbb{Z}\,.$ (81)

Finally, the affine Dynkin diagram for $\mathfrak{e}_{8}$, shown in Fig. 4(c)
has only one node that occurs with multiplicity 1: the affine node. The center
of $\operatorname{E}_{8}$ is trivial, so we identify the affine node with
$\nu=0$. We will not consider matter charged under an $\mathfrak{e}_{8}$ gauge
algebra, as it is difficult if not impossible to find localized
$\mathfrak{e}_{8}$ matter without $f$ and $g$ vanishing to orders $(4,6)$ at
codimension two.191919Such loci do not admit an easy interpretation in terms
of charged matter, and they are in fact associated with tensionless non-
critical strings and superconformal field theories. See the review article
HeckmanRudeliusSCFT  and references contained therein for more information.
However, we will encounter situations where the singularity type enhances to
$\operatorname{E}_{8}$ at codimension two. As evidenced by Eq. 18 and 19, the
formulas we develop here use $T(\nu)$ and $\vec{\tau}(\nu)$ for the group
corresponding to the codimension-two singularity type. Therefore, we define

$T_{\operatorname{E}_{8}}(\nu)=0\,,\quad\vec{\tau}_{\operatorname{E}_{8}}(\nu)=\mathopen{}\mathclose{{}\left(0,0,0}\right)\,.$
(82)

##  5 General strategy

The ultimate goal of this work is to argue that information about the
$\mathfrak{u}(1)$ charge spectrum of an F-theory model is encoded in the
orders of vanishing of the section components. In particular, we would like to
establish explicit relations between the orders of vanishing at a codimension-
two locus and the charge supported at that locus. Ideally, one would probe
this problem by examining the generating sections in F-theory models that
realize various types of charges. However, the currently known F-theory
constructions realize only a small subset of the possible charges. As an
example, consider F-theory models with just a $\operatorname{U}(1)$ gauge
group. Explicit constructions of this type have realized charges up to $q=\pm
6$ MorrisonParkU1 ; KleversEtAlToric ; Raghuram34 ; CollinucciEtAlHighCharge ;
OehlmannSchimannek ; KnappScheideggerSchimannek , even though indirect
arguments suggest F-theory $\operatorname{U}(1)$ models should be able to
admit charges at least as large as $q=\pm 21$ RaghuramTaylorLargeCharge .
Since only a limited number of charges have actually been observed in F-theory
models, one might imagine it would be difficult to make statements about
arbitrary charges.

Fortunately, as first pointed out in MorrisonParkU1 , there is a way to at
least conjecture about charges not seen in the currently known F-theory
constructions. To simplify the discussion, we assume that we are working with
a 6D F-theory model, although the arguments should carry over to lower-
dimensional models as well. Suppose our F-theory model has a $\mathfrak{u}(1)$
gauge algebra and that there is a codimension-two locus in the base supporting
matter with $\mathfrak{u}(1)$ charge $q$. At the geometric level, the
singularity type of the elliptic fiber enhances at this locus. After
resolution, there is at least one exceptional curve at this locus supporting
charged matter, which we refer to as $c$. Since the locus supports matter with
charge $q$, the generating section $\hat{s}$ behaves at this locus in a way
such that

$\sigma(\hat{s})\cdot c=q\,.$ (83)

The elliptic fibration also admits sections that are multiples of the
generating section. Since the Shioda map is a homomorphism, a section
$m\hat{s}$ satisfies

$\sigma(m\hat{s})\cdot c=m\sigma(\hat{s})\cdot c=mq$ (84)

at this same codimension-two locus in the base. It naively appears as if the
matter has “charge” $mq$ according to the section $m\hat{s}$. In the rest of
this paper, we will refer to the quantity $\sigma(\hat{s}^{\prime})\cdot c$ as
the _pseudo-charge_ when $\hat{s}^{\prime}$ is not a generating section. More
specifically, we will say that a section $\hat{s}^{\prime}$, which may not
necessarily be a generating section, “realizes a pseudo-charge $q^{\prime}$”
at a particular locus when $\sigma(\hat{s}^{\prime})\cdot c=q^{\prime}$. Of
course, since $m\hat{s}$ is not a generating section, it does not truly
correspond to a $\mathfrak{u}(1)$ gauge symmetry, and the model does not
genuinely have charge $mq$ matter supported at this locus. Nevertheless, we
expect that the section $m\hat{s}$ behaves as a generating section would in a
model that genuinely supported charge $mq$ matter, at least near the
codimension-two locus.

Thus, our strategy is to use non-generating sections realizing pseudo-charge
$q$ to glean information about the generating sections in models genuinely
admitting charge $q$ matter. Specifically, we start with a model admitting
some set of relatively small charges, which we refer to as the “seed” model.
Then, we consider multiples of the generating section, allowing us to find the
orders of vanishing for the section components for non-generating sections
admitting higher pseudo-charges. These data can then be used to establish and
test the expressions in Section 2.2 relating $\mathfrak{u}(1)$ charges to the
orders of vanishing. While we described this strategy in the context of a
model with just a $\mathfrak{u}(1)$ gauge algebra, it is equally applicable
whenever the gauge algebra includes a $\mathfrak{u}(1)$ factor, even if there
are other nonabelian or abelian gauge factors. In addition to admitting larger
pseudo-charges than the generating section, a non-generating section can be
associated with a different element $\nu$ of the center. If a generating
section $\hat{s}$ is associated with a center element $\nu$, then
$\hat{s}^{\prime}=m\hat{s}$ is associated with the center element
$\nu^{\prime}=m\nu$; in other words, the center element for $m\hat{s}$ is
found by adding together $m$ copies of $\nu$ according to the group law of the
center. This ensures that the pseudo-charges realized by $m\hat{s}$ are
allowed for $\nu^{\prime}=m\nu$ according to the rules in Section 4.

Key to this strategy is the assertion that a non-generating section realizing
pseudo-charge $q$ and a generating section admitting charge $q$ behave the
same way at the loci in question. This is a reasonable claim, since the
behavior near a codimension-two locus is a local property of the section. The
question of whether a section is a generating section, on the other hand,
involves global properties of the model; the local behavior of the section
near a particular locus would presumably not depend on such global properties.
We will also see more concrete, albeit circumstantial, evidence in favor of
this assertion. First, we can verify this claim in cases where we have an
explicit F-theory model genuinely admitting charge $q$. Additionally, there
are often multiple ways of obtaining a pseudo-charge $q$. For instance, a
pseudo-charge $q=4$ can be obtained from the section $4\hat{s}$ in a model
admitting $q=1$ matter or from the section $2\hat{s}$ in a model admitting
$q=2$ matter. If our assertion is true, then we should see the same local
behavior for a particular pseudo-charge regardless of how we obtain a
particular pseudo-charge. In fact, our analysis gives the same orders of
vanishing for each given pseudo-charge even when that pseudo-charge is
produced in independent ways.202020More precisely, we obtain the same orders
of vanishing whenever we produce a pseudo-charge with the same value of $\nu$.
See Section 4, particularly Section 4.1, for more details. This suggests that
our expressions relating the charge to the orders of vanishing hold more
broadly.

###  5.1 Signs

At this point, we should discuss how the sign of the charge fits into our
proposal. Using only the orders of vanishing of the $\hat{x}$, $\hat{y}$,
$\hat{z}$, and $\hat{w}$ section components, one should not be able to
determine the sign of the charge. When an elliptic fibration has a generating
section $\hat{s}$, we are equally free to choose the inverse section
$-\hat{s}$ as the generating section. If we take the generating section to be
$-\hat{s}$, the $\mathfrak{u}(1)$ charges should be the negative of those when
$\hat{s}$ is the generating section. However, for an elliptic fibration in
Weierstrass form, one can find the inverse of a section by flipping the sign
of the $\hat{y}$ component while leaving the other section components
unchanged. As a result, the orders of vanishing of the section components at a
particular locus should be the same for $\hat{s}$ and $-\hat{s}$. A formula
relating the orders of vanishing to the charge should therefore be insensitive
to the sign of the charge. Additionally, the question of whether to use
$-\hat{s}$ instead of $\hat{s}$ as the generating section is equivalent to
question of whether to use $\nu$ or its inverse in the center. Therefore,
formulas for the orders of vanishing should not distinguish between $\nu$ (or
$\mu$) and its inverse. These ideas are demonstrated by the proposed formulas
in Section 2.2: they only depend on the square of the charge $q$, and the
$T_{G}(\nu)$ and $\vec{\tau}_{G}(\nu)$ are the same for $\nu$ and its inverse
element.

This ambiguity between $\hat{s}$ and $-\hat{s}$ reflects the idea that the
overall sign of the charges is not important physically. We should be able to
flip the sign of all the $\mathfrak{u}(1)$ charges without changing the
physics. The more relevant property is the sign of a charge relative to other
charges in the theory. There are a few different types of relative signs we
should consider. First, if the gauge algebra has multiple $\mathfrak{u}(1)$
factors, matter in a representation of the gauge algebra has a (possibly zero)
charge for each $\mathfrak{u}(1)$ factor. The relative sign of the charges in
this representation can be important. In a 6D model with a
$\mathfrak{u}(1)^{2}$ gauge algebra, for example, one might wish to determine
if the representation of some hypermultiplet is $\bm{1}_{1,1}$ or
$\bm{1}_{1,-1}$. These representations are not conjugates of each other, and
$\bm{1}_{1,1}$ hypermultiplets should be distinguished from $\bm{1}_{1,-1}$
hypermultiplets. One can determine these sorts relative signs by applying the
proposed rules for the orders of vanishing to linear combinations of the
generating sections. For the $\mathfrak{u}(1)^{2}$ example, suppose the
generating sections are $\hat{s}_{1}$ and $\hat{s}_{2}$. Then, $\bm{1}_{1,-1}$
hypermultiplets would appear to be uncharged under the section
$\hat{s}_{1}+\hat{s}_{2}$, whereas $\bm{1}_{1,1}$ hypermultiplets would have
pseudo-charge $2$ according to $\hat{s}_{1}+\hat{s}_{2}$. One could therefore
distinguish between $\bm{1}_{1,1}$ and $\bm{1}_{1,-1}$ hypermultiplets by
applying the formulas in Section 2.2 to the section components of
$\hat{s}_{1}+\hat{s}_{2}$.

Second, one may want to determine the sign of one matter multiplet’s charge
relative to that of another matter multiplet. For example, let us consider a
model with a $\mathfrak{g}\oplus\mathfrak{u}(1)$ algebra, where $\mathfrak{g}$
is a nonabelian Lie algebra. Suppose that the model has two matter loci
supporting matter in the representations $\bm{R}_{q_{1}}$ and $\bm{R}_{q_{2}}$
for some irreducible representation $\bm{R}$ of $\mathfrak{g}$. The overall
sign of $q_{1}$ and $q_{2}$ is unimportant: we are free to flip the overall
sign of the $\mathfrak{u}(1)$ charges, after which the matter representations
would be $\bm{R}_{-q_{1}}$ and $\bm{R}_{-q_{2}}$. However, the relative sign
between $q_{1}$ and $q_{2}$ is unchanged by this flip and may therefore be a
meaningful property of the model. The equations in Section 2.2, particularly
Eq. 18, are insensitive to this relative sign.

While the inability to distinguish this relative sign is admittedly a
shortcoming of the proposals, this information can, in many cases, be obtained
relatively easily by alternative means. It is first important to recall that
matter fields in a representation $\bm{R}_{q}$ are typically accompanied by
matter fields in the representation $\overline{\bm{R}}_{-q}$. In 6D, for
example, full hypermultiplets in a representation $\bm{R}_{q}$ have fields
transforming in both representations $\bm{R}_{q}$ and
$\overline{\bm{R}}_{-q}$. If $\bm{R}$ is either a real or pseudoreal
representation of $\mathfrak{g}$, the $\bm{R}$ representation is isomorphic to
the $\overline{\bm{R}}$ representation, and a hypermultiplet of $\bm{R}_{q}$
matter can equivalently be viewed as a hypermultiplet of $\bm{R}_{-q}$ matter.
Even the relative signs of the charges are therefore unimportant when $\bm{R}$
is either real or pseudoreal. An important example is singlet matter in the
$\bm{1}_{q}$ representation, which is uncharged under the nonabelian gauge
algebra. The singlet representation is real, and thus even the relative sign
of the $\mathfrak{u}(1)$ charge is unimportant.

This leaves us with the situations where $\bm{R}$ is not self-conjugate. For
the charge normalization conventions used here, the allowed $\mathfrak{u}(1)$
charges for matter in the $\bm{R}_{q}$ representation, which may be
fractional, are separated by integers. Therefore, if one knows $\lvert
q_{1}\rvert$ and $\lvert q_{2}\rvert$, one can often use this condition to
find the relative sign difference between $q_{1}$ and $q_{2}$. As an example,
suppose that $\mathfrak{g}$ is $\mathfrak{su}(5)$ and $\bm{R}$ is the
fundamental ($\bm{5}$) representation. If $\lvert q_{1}\rvert$ is
$\frac{1}{5}$ and $\lvert q_{2}\rvert$ is $\frac{4}{5}$, one can automatically
deduce that $q_{1}$ and $q_{2}$ must have opposite signs.

This trick fails, however, when the allowed charges are either integral or
half-integral. Depending on the context, there are other strategies that may
be used to determine the relative signs. In 6D models with
$\mathfrak{g}\oplus\mathfrak{u}(1)$ gauge algebras, the massless spectrum must
satisfy the anomaly condition ErlerAnomaly ; ParkTaylor ; ParkIntersection

$\sum_{i}x_{\bm{R},q_{i}}E_{\bm{R}}q_{i}=0\,,$ (85)

where $x_{\bm{R},q_{i}}$ is the number of hypermultiplets in the
representation $\bm{R}_{q_{i}}$. The anomaly coefficient $E_{\bm{R}}$ is
defined by

$\operatorname{tr}_{\bm{R}}F^{3}=E_{\bm{R}}\operatorname{tr}F^{3}\,,$ (86)

where $F$ is the field strength for $\mathfrak{g}$,
$\operatorname{tr}_{\bm{R}}$ is the trace in the $\bm{R}$ representation, and
$\operatorname{tr}$ is the trace in the fundamental representation.212121For
$\mathfrak{su}(n)$, $E_{\bm{R}}$ is $1$ for the fundamental representation and
$n-4$ for the two-index antisymmetric representation. For all the other
representations considered in this paper, $E_{\bm{R}}$ is 0. Since this
formula is clearly sensitive to the sign of $q_{i}$, one can sometimes glean
information about the relative signs of the charges from this condition. In 4D
models, Yukawa couplings correspond to codimension-three loci in the base
where the singularity type enhances, which can often be viewed as the
intersection of codimension-two loci supporting the matter participating in
the Yukawa interaction. The Yukawa term in the Lagrangian must be invariant
under gauge transformations, so the $\mathfrak{u}(1)$ charges of matter fields
participating in the interaction must sum to 0. Thus, one can determine some
information about the relative signs by knowing that the matter at certain
loci admit a Yukawa interaction. In fact, one can often construct 6D and 4D
models with the same Weierstrass tuning, potentially allowing both of these
strategies to be used.

To summarize, the proposed formulas in Section 2.2 are not sensitive to the
sign of the charge, but the physically relevant signs can often be determined
without too much difficulty. There are some cases where certain signs are both
meaningful and difficult to determine, however. While one may need to perform
resolutions to fully determine the charges in such cases, the proposed
formulas still provide an easy method of determining the absolute value of the
charge, making them an invaluable tool. Of course, when attempting to find the
orders of vanishing corresponding to a given charge, one can use the formulas
in Section 2.2 for either choice of sign and still obtain the same result.

##  6 Charged singlets

The primary focus for the rest of this paper is applying the strategy outlined
in Section 5 to models admitting various types of matter. For each example, we
generate order-of-vanishing data and show that they satisfy the proposals. Let
us first consider loci supporting charged singlets. We can narrow our focus to
F-theory models where the total gauge algebra is simply $\mathfrak{u}(1)$,
although our results appear to be valid for singlets in more general contexts.
The matter in such models is supported at codimension-two loci in the base
where the elliptic curve singularity type enhances to $\text{I}_{2}$; in 6D
models, each such irreducible locus supports a single hypermultiplet of
matter. The corresponding $\mathfrak{u}(1)$ charges for this matter are
integral, as implied by the form of the Shioda map in Eq. 9. Typically, one
would determine the charge of the matter supported at a particular locus by
resolving the $\text{I}_{2}$ singularities and examining the behavior of the
section in the smooth model. In line with the proposals of Section 2.2,
however, we now ask whether one can determine these $\mathfrak{u}(1)$ charges
from orders of vanishing of the section components without performing a
resolution.

This question was already discussed in Raghuram34 , and the analysis there
suggested a particular relation between the charge and the orders of vanishing
in Weierstrass form. The work employed the strategy in Section 5 for an
F-theory construction described by the Weierstrass model MorrisonParkU1

$y^{2}=x^{3}+\mathopen{}\mathclose{{}\left(\hat{f}_{12}-3f_{6}^{2}}\right)xz^{4}+\mathopen{}\mathclose{{}\left(f_{9}^{2}+2f_{6}^{3}-\hat{f}_{12}f_{6}}\right)z^{6}\,.$
(87)

This elliptic fibration has Mordell–Weil rank $1$, and the generating section
$\hat{s}$ can be written as

$\hat{s}\colon[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[f_{6}:f_{9}:1}\right]\,.$
(88)

There are no additional nonabelian gauge symmetries in this model, and the
gauge algebra for this model is therefore $\mathfrak{u}(1)$. The only matter
in this model, supported at $\\{f_{9}=\hat{f}_{12}=0\\}$, has charge $q=1$, as