confirmed by resolving the $\text{I}_{2}$ singularities. Prior to resolution,
the generating section hits the singular point in the fiber at
$\\{f_{9}=\hat{f}_{12}=0\\}$. The resolution introduces a new exceptional
curve222222To ease the discussion, we are using language that assumes that the
elliptic fibration is a threefold, as is the case for 6D F-theory models. $c$,
and the generating section hits $c$ at $\\{f_{9}=\hat{f}_{12}=0\\}$. More
specifically, we have that

$\sigma(\hat{s})\cdot c=1$ (89)

at $\\{f_{9}=\hat{f}_{12}=0\\}$.

One can then examine the non-generating sections $q\hat{s}$, which realize
pseudo-charge $q$ at $\\{f_{9}=\hat{f}_{12}=0\\}$. We are particularly
interested in the orders of vanishing of the $\hat{x}$, $\hat{y}$, $\hat{z}$,
and $\hat{w}=3\hat{x}^{2}+f\hat{z}^{4}$ section components at
$\\{f_{9}=\hat{f}_{12}=0\\}$; since the singlet loci occur along a
codimension-one $\text{I}_{1}$ locus, the orders of vanishing at codimension
one are unimportant. Some of these orders of vanishing are listed in Table 7.
As noticed in Raghuram34 , these numbers follow a curious pattern. For a
section $q\hat{s}$, the orders of vanishing for
$(\hat{x},\hat{y},\hat{z},\hat{w})$ seem to be described by the expressions

$\displaystyle(2,3,1,4)$ $\displaystyle\times\frac{q^{2}}{4}\,,$
$\displaystyle\qquad\text{for even }q\,,$ (90) $\displaystyle(2,3,1,4)$
$\displaystyle\times\frac{q^{2}-1}{4}+(0,1,0,1)\,,$
$\displaystyle\qquad\text{for odd }q\,.$ (91)

According to the arguments in Section 5.1, these patterns should hold
regardless of the sign of $q$. These patterns have been verified up to $q=28$,
as seen in Table 7, and as discussed in Section 12, these same orders of
vanishing occur at singlet loci in models from the prior literature,
suggesting that the orders of vanishing described by Eqs. 90 and 91 are
universal features of the Weierstrass models at loci admitting charged
singlets.

$q$ | $\operatorname*{ord}(\hat{x})$ | $\operatorname*{ord}(\hat{y})$ | $\operatorname*{ord}(\hat{z})$ | $\operatorname*{ord}(\hat{w})$
---|---|---|---|---
$1$ | $0$ | $1$ | $0$ | $1$
$2$ | $2$ | $3$ | $1$ | $4$
$3$ | $4$ | $7$ | $2$ | $9$
$4$ | $8$ | $12$ | $4$ | $16$
$5$ | $12$ | $19$ | $6$ | $25$
$6$ | $18$ | $27$ | $9$ | $36$
$7$ | $24$ | $37$ | $12$ | $49$
$8$ | $32$ | $48$ | $16$ | $64$
$9$ | $40$ | $61$ | $20$ | $81$
$10$ | $50$ | $75$ | $25$ | $100$
$11$ | $60$ | $91$ | $30$ | $121$
$12$ | $72$ | $108$ | $36$ | $144$
$13$ | $84$ | $127$ | $42$ | $169$
$14$ | $98$ | $147$ | $49$ | $196$
$15$ | $112$ | $169$ | $56$ | $225$
$16$ | $128$ | $192$ | $64$ | $256$
$17$ | $144$ | $217$ | $72$ | $289$
$18$ | $162$ | $243$ | $81$ | $324$
$19$ | $180$ | $271$ | $90$ | $361$
$20$ | $200$ | $300$ | $100$ | $400$
$21$ | $220$ | $331$ | $110$ | $441$
$22$ | $242$ | $363$ | $121$ | $484$
$23$ | $264$ | $397$ | $132$ | $529$
$24$ | $288$ | $432$ | $144$ | $576$
$25$ | $312$ | $469$ | $156$ | $625$
$26$ | $338$ | $507$ | $169$ | $676$
$27$ | $364$ | $547$ | $182$ | $729$
$28$ | $392$ | $588$ | $196$ | $784$
Table 7: Orders of vanishing for the $\hat{x}$, $\hat{y}$, $\hat{z}$, and
$\hat{w}$ components of the $q\hat{s}$ sections at
$\\{f_{9}=\hat{f}_{12}=0\\}$. The generating section $\hat{s}$ is described by
Eq. 88, and the Weierstrass model is given in Section 6. This model supports
singlets with charge $q=\pm 1$ at $\\{f_{9}=\hat{f}_{12}=0\\}$, and the
section $q\hat{s}$ realizes pseudo-charge $q$ at this locus.

The analysis in Raghuram34  did not give an explanation for these expressions.
One aspect of the expressions, the $(0,1,0,1)$ term in Eqs. 90 and 91, has a
straightforward interpretation. When the unresolved elliptic fibration is
written in Weierstrass form, the singular points occur at $y=w=0$ on the
fiber. If the section hits the singular point on a fiber, the $\hat{y}$ and
$\hat{w}$ components should go to 0. The $(0,1,0,1)$ term therefore indicates
that the section hits the singular point on the fibers at loci supporting odd
charges. The $\frac{q^{2}}{4}$ and $\frac{q^{2}-1}{4}$ terms do not
immediately present explanations, but two pieces of evidence indicate that
Eqs. 90 and 91 are natural results. First, we can show that the formulas are
specializations of those proposed in Section 2.2. Second, these same
expressions appear in previous work on elliptic divisibility sequences,
suggesting that these formulas reflect deeper properties of elliptic
fibrations.

###  6.1 Connection to the proposed formulas

To see the connection between the observed singlet patterns and the formulas
in Section 2.2, it is helpful to rewrite the formulas in Eqs. 90 and 91 as

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{2}q^{2}-\frac{1}{2}\overline{q}_{2}}\right)=\mathopen{}\mathclose{{}\left\lfloor\frac{q^{2}}{4}}\right\rfloor$
(92)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(0,1,1}\right)\overline{q}_{2}\,.$
(93)

We have assumed that $q$ is an integer. Singlets occur at double points of the
discriminant locus where the singularity type enhances from
$\text{I}_{1}\times\text{I}_{1}$ at codimension one232323Since the singlet
locus is a double point of the discriminant locus, it resembles the
intersection of two $\text{I}_{1}$ loci for a sufficiently small neighborhood
around the singlet locus. to $\text{I}_{2}$ at codimension two. If we refer to
the singularity types by their corresponding ADE groups, the $\text{I}_{1}$
singularity type at codimension one can be associated with
$\operatorname{SU}(1)$, while the $\text{I}_{2}$ singularity type at
codimension two can be associated with $\operatorname{SU}(2)$. The center of
$\operatorname{SU}(n)$ has $d_{\operatorname{SU}(n)}=n$ elements. For
$\operatorname{SU}(1)$, there is only one element of the center denoted by
$\nu=0$, and according to Eq. 66

$T_{\operatorname{SU}(1)}(0)=0\,,\quad\vec{\tau}_{\operatorname{SU}(1)}(0)=(0,0,0)\,.$
(94)

As a result, all of the terms in the general proposals involving the
codimension-one behavior of the section are trivial for singlets. For
$\operatorname{SU}(2)$, there are two elements of the center, which are
denoted by $\mu=0$ and $\mu=1$. According to Eqs. 66 and 69,

$T_{\operatorname{SU}(2)}(\mu)=\frac{1}{2}\mu(2-\mu)=\frac{1}{2}\mu\,,\quad\vec{\tau}_{\operatorname{SU}(2)}(\mu)=(0,1,1)\times
u_{2}(\mu)=(0,1,1)\mu\,.$ (95)

If we set $\mu$ to $\overline{q}_{2}$ and plug this information into the
general formulas 18 and 19 from Section 2.2, we obtain

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{2}q^{2}-\frac{1}{2}\overline{q}_{2}}\right)$
(96)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(0,1,1}\right)u_{2}(q)\,.$
(97)

This exactly matches the rewritten form of the singlet orders of vanishing
above. Equations 90 and 91 therefore agree with the general formulas proposed
in Section 2.2.

The fact that $\mu=\overline{q}_{2}$ demonstrates that elliptic curve addition
respects the addition law for the centers of the groups in question, as
discussed in Section 5. In this case, we are interested in the
$\mathbb{Z}_{2}$ center of $\operatorname{SU}(2)$, the group corresponding to
the codimension-two singularity type. The generating section is associated
with the $\mu=1$ element of $\mathbb{Z}_{2}$. Based on the $\mathbb{Z}_{2}$
addition law, adding $\mu=1$ to itself an even number of times should give the
$\mu=0$ identity element of $\mathbb{Z}_{2}$, while adding $\mu=1$ to itself
an odd number of times should give the $\mu=1$ element of $\mathbb{Z}_{2}$. We
in fact see that even multiples of the generating section are associated with
$\mu=0$ and odd multiples of the generating section are associated with
$\mu=1$. We observe similar behaviors in the examples considered later.

###  6.2 The elliptic troublemaker sequence

Remarkably, the sequence $\lfloor\frac{q^{2}}{4}\rfloor$ seen in Eq. 92 also
appears in a different but somewhat related context: it is the elliptic
troublemaker sequence $R_{q}(1,2)$ defined in Stange’s work StangeEllTrouble
on the valuations of elliptic divisibility sequences. There in fact seem to be
direct parallels between Stange’s work and the problems explored in this
paper, suggesting that Stange’s results may guide us toward a firmer
understanding of the observed orders of vanishing. Therefore, let us briefly
review some of the relevant results from StangeEllTrouble  along with some
necessary background material.

Stange’s work considers elliptic curves over finite extensions of the $p$-adic
numbers $\mathbb{Q}_{p}$. The $p$-adic numbers are a completion of the
rational numbers $\mathbb{Q}$ just like the set of real numbers $\mathbb{R}$,
but they have a different notion of the “absolute value.” Suppose we consider
a nonzero rational number $x$, which we can uniquely write as
$x=\frac{r}{s}p^{n}$ for a prime integer $p$ and coprime integers $r,s$ both
coprime to $p$. We can then define the $p$-adic valuation of $x$ as

$v_{p}(x)=n\,.$ (98)

If $x$ is an integer, $v_{p}(x)$ gives the exponent of the largest power of
$p$ that divides $x$. We define $v_{p}(0)=\infty$. This $p$-adic valuation
satisfies the standard properties of a valuation:

  * •

$v_{p}(ab)=v_{p}(a)+v_{p}(b)$,

  * •

$v_{p}(a+b)\geq\min(v_{p}(a),v_{p}(b))$, with exact equality if $v_{p}(a)\neq
v_{p}(b)$,

  * •

$v_{p}(a)=\infty$ if and only if $a=0$.

Note that the order of vanishing also satisfies these properties, a fact that
will be exploited shortly. One can then define a $p$-adic absolute value
$\lvert x\rvert_{p}=p^{-v_{p}(x)}$.

There are two particular concepts regarding elliptic curves that are important
in Stange’s analysis. The first is the reduction of elliptic curves modulo a
prime $p$. Consider an elliptic curve with integral coefficients over a field
$K$ of characteristic 0 that is a finite extension of $\mathbb{Q}_{p}$. To be
concrete, one could consider this elliptic curve to be written in the Tate
form242424While this way of writing elliptic curves is known as the “Tate
form” in the F-theory literature, it is also known as the “Weierstrass form”
in other contexts. However, the F-theory literature typically reserves the
term “Weierstrass form” for the form $y^{2}=x^{3}+fx+g$. We therefore refer to
the form of the elliptic curve in Eq. 99 as the Tate form to be consistent
with the F-theory conventions.

$y^{2}+a_{1}xy+a_{3}x=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}\,,$ (99)

where the $a_{i}$ are integers. We can then form a reduced elliptic curve by
reducing the $a_{i}$ coefficients modulo the prime number $p$. The new reduced
elliptic curve is now over the residue field of $K$, where elements of $K$
that are equivalent modulo $p$ are identified. Even if the original curve is
smooth, its reduction may be singular. If the elliptic curve remains smooth
after reduction, one says that the original elliptic curve over $K$ has good
reduction; if the elliptic curve has a nodal singularity after reduction, then
the original elliptic curve is said to have multiplicative reduction; and if
the reduced elliptic curve has a cusp singularity, the original elliptic curve
is said to have additive reduction. Additionally, a rational point $P$ of the
original elliptic curve maps onto a rational point of the reduced elliptic
curve. We say that $P$ reduces to this new rational point.

The second important topic is that of elliptic divisibility sequences and
division polynomials. Suppose we have an elliptic curve $E$ such as that in
Eq. 99. Then we can define a sequence of polynomials in
$\mathbb{Z}[a_{1},a_{2},a_{3},a_{4},a_{6},x,y]$ known as the division
polynomials $\Psi_{m}$ for $E$. This sequence can be constructed using the
recursion relations

$\displaystyle\Psi_{2j+1}$
$\displaystyle=\Psi_{j+2}\Psi_{j}^{3}-\Psi_{j-1}\Psi_{j+1}^{3}\,,$
$\displaystyle\text{for }j\geq 2\,,$ (100) $\displaystyle\Psi_{2j}\Psi_{2}$
$\displaystyle=\Psi_{j-1}^{2}\Psi_{j}\Psi_{j+2}-\Psi_{j-2}\Psi_{j}\Psi_{j+1}^{2}\,,$
$\displaystyle\text{for }j\geq 3\,,$

and the initial values

$\displaystyle\Psi_{1}$ $\displaystyle=1\,,$ (101) $\displaystyle\Psi_{2}$
$\displaystyle=2y+a_{1}x+a_{3}\,,$ $\displaystyle\Psi_{3}$
$\displaystyle=3x^{4}+b_{2}x^{3}+3b_{4}x^{2}+3b_{6}x+b_{8}\,,$
$\displaystyle\Psi_{4}$
$\displaystyle=\Psi_{2}\mathopen{}\mathclose{{}\left(2x^{6}+b_{2}x^{5}+5b_{4}x^{4}+10b_{6}x^{3}+10b_{8}x^{2}+(b_{2}b_{8}-b_{4}b_{6})x+(b_{4}b_{8}-b_{6}^{2})}\right)\,.$

The $b_{i}$ parameters in these initial values are the same as those familiar
from, for instance, Tate’s algorithm in F-theory BershadskyEtAlSingularities :

$\displaystyle b_{2}$ $\displaystyle=a_{1}^{2}+4a_{2}\,,$ (102) $\displaystyle
b_{4}$ $\displaystyle=a_{1}a_{3}+2a_{4}\,,$ $\displaystyle b_{6}$
$\displaystyle=a_{3}^{2}+4a_{6}\,,$ $\displaystyle b_{8}$
$\displaystyle=b_{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{2}-a_{4}^{2}\,.$

The division polynomials can be used to describe the multiples of a point
under elliptic curve addition. If we describe a generic point on the elliptic
curve as $P=(x,y)$, then one can write the formula for a multiple of a point
as

$mP=\mathopen{}\mathclose{{}\left(\frac{\phi_{m}}{\Psi_{m}^{2}},\frac{\omega_{m}}{\Psi_{m}^{3}}}\right)\,,$
(103)

where $\phi_{m}$ and $\omega_{m}$ are polynomials in
$\mathbb{Z}[a_{1},a_{2},a_{3},a_{4},a_{6},x,y]$. Now consider a rational point
$P$. We can always scale the $a_{i}$ coefficients in Eq. 99 to convert $P$
into an integral point $P^{\prime}$. Evaluating the $\Psi_{m}$ for this
integral point $P^{\prime}$ and the scaled $a_{i}$ values gives a sequence of
integers $W_{n}$ known as an elliptic divisibility sequence (EDS). This
sequence, like all divisibility sequences, has the property that $W_{m}\mid
W_{n}$ if $m\mid n$.

With these ingredients, we can now describe the theorem in Stange’s work,
Theorem 28, that is relevant for understanding the singlet charges. Consider
an elliptic curve over an extension of $\mathbb{Q}_{p}$ with multiplicative
reduction modulo $p$ and a rational point $P$ on the elliptic curve that
reduces to the singular point. Then, the valuations of the $W_{m}$ associated
with $P$ are described by the formula

$v_{p}(W_{m})=R_{m}(a_{P},l_{P})+\begin{cases}S_{m/n_{P}}(p,p,v(p),0,s_{P},w_{P})&n_{P}\mid
m\\\ 0&n_{P}\nmid m\end{cases}\,,$ (104)

where $R_{m}(a_{P},l_{P})$ is the elliptic troublemaker sequence given
by252525In StangeEllTrouble , there are a variety of formulations of the
elliptic troublemaker sequence that are equivalent for integer $m$.

$R_{m}(a_{P},l_{P})=\frac{m^{2}-m}{2}a_{P}+\sum_{k=1}^{\lfloor\frac{ma_{P}}{l_{P}}\rfloor}\mathopen{}\mathclose{{}\left(kl_{P}-ma_{P}}\right)-m^{2}\sum_{k=1}^{\lfloor\frac{a_{P}}{l_{P}}\rfloor}\mathopen{}\mathclose{{}\left(kl_{P}-a_{P}}\right)\,.$
(105)

The integer $l_{P}$ is equivalent to $-v_{p}(j(E))$, the negative of the
valuation of the $j$-invariant of the elliptic curve. Meanwhile, $a_{P}$ is an
integer determined by $P$, which we can take to satisfy $0\leq a_{P}<l_{P}$;
it plays a role similar to our parameters $\nu$ and $\mu$.262626More
specifically, $a_{P}$ indicates the component of the Néron model special fiber
containing $P$. It is zero if and only if $P$ has non-singular reduction, and
since the theorem describes situations where $P$ has singular reduction,
$a_{P}$ will often be nonzero when we use this theorem. See StangeEllTrouble
for a more thorough explanation of $a_{P}$. In fact, when $0\leq a_{P}<l_{P}$
and $m$ is an integer, the elliptic troublemaker sequence can be written in
the useful form

$R_{m}(a_{P},l_{P})=\frac{a_{P}(l_{P}-a_{P})}{2l_{P}}m^{2}-\frac{\overline{(ma_{P})}_{l_{P}}(l_{P}-\overline{(ma_{P})}_{l_{P}})}{2l_{P}}\,.$
(106)

The integer $n_{P}$ is the smallest positive integer such that the reduction
of $n_{P}P$ equals the reduction of the identity point of the elliptic curve.
Finally $S$ is a correction term that is described in more detail in Appendix
B.

Even though this theorem is formulated for elliptic curves over extensions of
the $p$-adic numbers, it also seems give the correct orders of vanishing for
the sections described above. This is likely explained by the underlying
structure shared by both problems. The rational points of an elliptic curve
are analogues of the rational sections of an elliptic fibration, and the
$W_{m}$ elements of the elliptic divisibility sequence are roughly the
analogues of the $\hat{z}$ components of the sections considered
above.272727It is important to remember, however, that the $W_{m}$ are found
from the division polynomials. For particular choices of $P$ and $m$, the
$\Psi_{m}$, $\phi_{m}$, and $\omega_{m}$ in Eq. 103 may share common factors
that can be removed. In such cases, $W_{m}$ would be proportional to $\hat{z}$
but may have additional factors. This possibility is more relevant for the
cases with nonabelian gauge algebras described later. Meanwhile, investigating
the behavior of the sections at the locus $\\{f_{9}=\hat{f}_{12}=0\\}$
requires analyzing expressions for $f$, $g$, and the section components after
dropping terms proportional to $f_{9}$ and $\hat{f}_{12}$. This procedure
parallels reducing an elliptic curve modulo $p$. Since the elliptic fibration
has $\text{I}_{2}$ singularities at $\\{f_{9}=\hat{f}_{12}=0\\}$, the singular
fibers there have nodal singularities, just as one would find after reducing
an elliptic curve with multiplicative reduction. Additionally, the generating
section hits the singular point at $\\{f_{9}=\hat{f}_{12}=0\\}$, which
corresponds to a point $P$ having singular reduction. Finally, the order of
vanishing is itself a valuation and should play a role similar to the $p$-adic
valuation.

These analogies suggest that the orders of vanishing for the $\hat{z}$
components of the $q\hat{s}$ sections should be described by Stange’s theorem.
To make the connection more explicit, we can construct an elliptic curve
analogue of Eq. 87 by setting $f_{9}$ and $\hat{f}_{12}$, the parameters
corresponding to the matter locus, to an arbitrary prime integer $p$ (not
equal to 3) while setting $f_{6}$ to a number such as 1. These substitutions
lead to an elliptic curve

$y^{2}=x^{3}+(p-3)x+(p^{2}-p+2)$ (107)

with a discriminant

$\Delta=4f^{3}+27g^{2}=p^{2}\mathopen{}\mathclose{{}\left(27p^{2}-50p+99}\right)$
(108)

and a rational point

$(x,y)=(1,p)\,.$ (109)

This elliptic curve has multiplicative reduction modulo $p$. If we consider
the EDS associated with $P$, $l_{P}$ (which is generally given by
$-v_{p}(j(E))$) equals 2 for this situation, as would be expected since the
matter locus supports $\text{I}_{2}$ singularities. Since $P$ reduces to the
singular point, we can take $a_{P}$ to be 1, in agreement with the value of
$\mu$ for the generating section. If one calculates several multiples of $P$,
one finds that they never reduce to the identity point; evidently no multiples
of $P$ reduce to the identity, and $n_{P}$ can be taken to be infinitely
large. Stange’s theorem would then suggest that

$v_{p}(W_{q})=R_{q}(1,2)=\mathopen{}\mathclose{{}\left\lfloor\frac{q^{2}}{4}}\right\rfloor\,.$
(110)

In turn, one would expect the $\hat{z}$ section components for $q\hat{s}$ to
vanish to these same orders at the matter locus, as observed.

We therefore have a plausible explanation for the orders of vanishing observed
in the non-generating sections for the Weierstrass model considered above. In
turn, it is natural to expect that the orders of vanishing at loci genuinely
supporting charged singlets should obey the same pattern. Of course, since
Stange’s theorem is formulated in a different context, one would ideally have
a proof of a similar statement for elliptic fibrations. It would also be
beneficial to have a rigorous proof that the behavior of the $q\hat{s}$
sections matches that of generating sections admitting charge-$q$ matter. We
leave both of these questions to future work. Nevertheless, Stange’s theorem
seems to directly predict the observed orders of vanishing, and we will
shortly see that similar formulas agree with the observed orders of vanishing
when there are nonabelian gauge algebras as well.

##  7 Representations of $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$

Now that we have seen the connection between the orders of vanishing and
$\mathfrak{u}(1)$ charges for singlets, we can start investigating what
happens for matter simultaneously charged under a $\mathfrak{u}(1)$ and a
simple nonabelian gauge factor. We first focus on
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ gauge algebras, restricting our
attention to matter in the fundamental and antisymmetric representations of
$\mathfrak{su}(n)$. New complications arise when we include nonabelian gauge
factors. The global structure of the total gauge group can have a nontrivial
quotient involving both the $\mathfrak{u}(1)$ factor and an element of the
center of the nonabelian gauge factor. As a result, matter charged under the
nonabelian gauge factor can have fractional $\mathfrak{u}(1)$ charges in the
units specified by the Shioda map in Eq. 9. Additionally, there are now two
types of orders of vanishing to consider: those at the codimension-two locus
supporting the matter and those at the codimension-one locus supporting the
nonabelian gauge factor.

Despite these complications, many of the ideas seen in the singlet analysis
carry over to this new context. The proposals in Section 2.2 still seem to
correctly describe the orders of vanishing of the section components
supporting various $\mathfrak{u}(1)$ charges. Furthermore, we will still find
intriguing connections to the elliptic troublemaker sequences and the work in
StangeEllTrouble .

###  7.1 Fundamental representation of $\mathfrak{su}(n)$

As discussed in Section 4.1, an F-theory model with an
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ gauge algebra and global structure
described by the integer $\nu$ can support $\bm{n}_{q}$ matter with
$\mathfrak{u}(1)$ charge

$q=\frac{\nu}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (111)

In other words, a model supporting $\bm{n}_{q}$ matter should have a global
gauge group structure described by

$\nu=\overline{nq}_{n}\,.$ (112)

These observations allow us to adapt the strategy of Section 5 to fundamental
matter. We start with a seed model with an
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ gauge algebra, a global structure
described by $\nu=1$, and $\bm{n}_{1/n}$ matter. In F-theory, the
$\mathfrak{su}(n)$ gauge algebra is supported along a codimension-one locus in
the base with $\text{I}^{(s)}_{n}$ singularities, while the fundamental matter
is supported at a codimension-two locus where the singularity type enhances to
$\text{I}_{n+1}$. If this model has a generating section $\hat{s}$, the
multiples $m\hat{s}$ would have $\nu=\overline{m}_{n}$ and would realize
pseudo-charge $\frac{m}{n}$ at the codimension-two locus. We can therefore
obtain orders of vanishing corresponding to all allowed combinations of $\nu$
and the $\mathfrak{u}(1)$ charge by examining multiples of this generating
section.

To be more specific, we analyze three different seed models supporting
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ algebras. For $n=2k+1$, $k\geq 1$, we
consider the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\mathopen{}\mathclose{{}\left[-\frac{1}{48}\mathopen{}\mathclose{{}\left(b_{1,0}^{2}+4c_{2,1}\sigma}\right)^{2}+\frac{1}{2}b_{0,0}b_{1,0}c_{1,k}\sigma^{k}+c_{3,1}c_{1,k}\sigma^{k+1}+b_{0,0}^{2}\sigma^{2k}c_{0,2k}}\right]xz^{4}$
(113)
$\displaystyle\qquad\qquad\quad+\mathopen{}\mathclose{{}\left[\frac{1}{864}\mathopen{}\mathclose{{}\left(b_{1,0}^{2}+4c_{2,1}\sigma}\right)^{3}-\frac{1}{24}c_{1,k}\mathopen{}\mathclose{{}\left(b_{1,0}^{2}+4\sigma
c_{2,1}}\right)\mathopen{}\mathclose{{}\left(b_{0,0}b_{1,0}+2\sigma
c_{3,1}}\right)\sigma^{k}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{1}{12}b_{0,0}^{2}\mathopen{}\mathclose{{}\left(3c_{1,k}^{2}-b_{1,0}^{2}c_{0,2k}}\right)\sigma^{2k}$
$\displaystyle\qquad\qquad\qquad\qquad\mathopen{}\mathclose{{}\left.-\frac{1}{3}c_{0,2k}\mathopen{}\mathclose{{}\left(-2b_{0,0}^{2}c_{2,1}+3b_{1,0}b_{0,0}c_{3,1}+3\sigma
c_{3,1}^{2}}\right)\sigma^{2k+1}}\right]z^{6}\,.$

This Weierstrass model supports $\text{I}^{(s)}_{2k+1}$ singularities along
$\\{\sigma=0\\}$ and a generating section with components

$\displaystyle\hat{x}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{1}{2}b_{0,0}b_{1,0}+\sigma
c_{3,1}}\right)^{2}-\frac{2}{3}b_{0,0}^{2}\mathopen{}\mathclose{{}\left(\frac{b_{1,0}^{2}}{4}+\sigma
c_{2,1}}\right)\,,$ (114) $\displaystyle\hat{y}$
$\displaystyle=-\frac{\sigma}{2}\mathopen{}\mathclose{{}\left(b_{0,0}^{4}\sigma^{k-1}c_{1,k}+\mathopen{}\mathclose{{}\left(b_{0,0}b_{1,0}+2\sigma
c_{3,1}}\right)\mathopen{}\mathclose{{}\left(b_{1,0}b_{0,0}c_{3,1}-b_{0,0}^{2}c_{2,1}+\sigma
c_{3,1}^{2}}\right)}\right)\,,$ $\displaystyle\hat{z}$
$\displaystyle=b_{0,0}\,,$ $\displaystyle\hat{w}$
$\displaystyle=\frac{\sigma}{2}\Big{[}\mathopen{}\mathclose{{}\left(b_{1,0}b_{0,0}c_{3,1}-b_{0,0}^{2}c_{2,1}+\sigma
c_{3,1}^{2}}\right)\mathopen{}\mathclose{{}\left(b_{0,0}^{2}\mathopen{}\mathclose{{}\left(b_{1,0}^{2}-2\sigma
c_{2,1}}\right)+6\sigma b_{1,0}b_{0,0}c_{3,1}+6\sigma^{2}c_{3,1}^{2}}\right)$
$\displaystyle\phantom{=\frac{\sigma}{2}\Big{[}}+2b_{0,0}^{6}\sigma^{2k-1}c_{0,2k}+b_{0,0}^{4}\sigma^{k-1}c_{1,k}\mathopen{}\mathclose{{}\left(b_{0,0}b_{1,0}+2\sigma
c_{3,1}}\right)\Big{]}\,,$

implying that the gauge algebra is $\mathfrak{su}(2k+1)\oplus\mathfrak{u}(1)$.
The charged matter spectrum, summarized in Table 8, satisfies the gauge and
gauge–gravitational anomaly conditions for $b=[\sigma]$ and
$0pt=-2K_{B}+2[b_{0,0}]-\frac{2k}{2k+1}[\sigma]$. These models are equivalent
to the $\mathcal{Q}(2k,k,1,1,0,0,k)$ models in KuntzlerTateTrees .

Representation | Matter Locus
---|---
$\bm{\frac{n(n-1)}{2}}_{\mathrlap{2/n}}$ | $V_{a}=\\{\sigma=b_{1,0}=0\\}$
$\bm{n}_{\mathrlap{(n+1)/n}}$ | $V_{f,1}=\\{\sigma=b_{0,0}=0\\}$
$\bm{n}_{\mathrlap{(1-n)/n}}$ | $V_{f,2}=\\{\sigma=b_{0,0}c_{2,1}-b_{1,0}c_{3,1}=0\\}$
$\bm{n}_{\mathrlap{1/n}}$ | $V_{f,3}=\\{\sigma=b_{1,0}^{2}c_{0,2k}+c_{1,k}^{2}=0\\}$
$\bm{1}_{\mathrlap{2}}$ | $V_{s,2}=\\{b_{0,0}=c_{3,1}=0\\}$
$\bm{1}_{\mathrlap{1}}$ | $V_{s,1}=\\{\frac{\hat{y}}{\sigma}=\frac{\hat{w}}{\sigma}=0\\}\setminus\mathopen{}\mathclose{{}\left(V_{a}\cup V_{f,1}\cup V_{f,2}\cup V_{f,3}\cup V_{s,2}}\right)$
$\bm{(n^{2}-1)}_{\mathrlap{0}}$ | $\\{\sigma=0\\}$
Table 8:  Matter spectrum for the $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$
fundamental seed models for $n=2k+1$ and $k\geq 1$. Note that one can flip the
sign of all $\mathfrak{u}(1)$ charges and obtain an equally valid description
of the matter spectrum (with a different value of $\nu$). For $n=3$ ($k=1$),
the matter spectrum is similar but with two changes: the locus $V_{a}$ does
not support any matter, while the $\bm{3}_{-2/3}$ locus is described by
$\\{\sigma=b_{0,0}^{2}c_{1,k}-b_{1,0}b_{0,0}c_{2,1}+b_{1,0}^{2}c_{3,1}=0\\}$.

We can construct a seed model for $n=2k$, $k\geq 2$ by taking Eq. 113,
replacing $k$ with $k-1$, and setting

$c_{1,k-1}\to c_{1,k}\sigma+\frac{1}{2}b_{2,k-1}b_{1,0}\,,\quad c_{0,2k-2}\to
c_{0,2k-1}\sigma-\frac{1}{4}b_{2,k-1}^{2}\,.$ (115)

This Weierstrass model admits $\text{I}^{(s)}_{2k}$ singularities along
$\\{\sigma=0\\}$ and a generating section described by Eq. 114 with the above
substitutions. As a result, the gauge algebra is
$\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$. The matter spectrum, summarized in
Table 9, satisfies the gauge and gauge–gravitational anomaly conditions for
$b=[\sigma]$ and $0pt=-2K_{B}+2[b_{0,0}]-\frac{2k-1}{2k}[\sigma]$. These
models are equivalent to the $\mathcal{Q}(2k-1,k,1,1,0,0,k-1)$ models in
KuntzlerTateTrees .

Representation | Matter Locus
---|---
$\bm{\frac{n(n-1)}{2}}_{\mathrlap{2/n}}$ | $V_{a}=\\{\sigma=b_{1,0}=0\\}$
$\bm{n}_{\mathrlap{(n+1)/n}}$ | $V_{f,1}=\\{\sigma=b_{0,0}=0\\}$
$\bm{n}_{\mathrlap{(1-n)/n}}$ | $V_{f,2}=\\{\sigma=-2b_{0,0}c_{2,1}+2b_{1,0}c_{3,1}+b_{0,0}^{2}\sigma^{k-2}b_{2,k-1}=0\\}$
$\bm{n}_{\mathrlap{1/n}}$ | $V_{f,3}=\\{\sigma=2b_{1,0}\mathopen{}\mathclose{{}\left(b_{1,0}c_{0,2k-1}+b_{2,k-1}c_{1,k}}\right)-2c_{2,1}b_{2,k-1}^{2}+b_{0,0}b_{2,k-1}^{3}\sigma^{k-2}=0\\}$
$\bm{1}_{\mathrlap{2}}$ | $V_{s,2}=\\{b_{0,0}=c_{3,1}=0\\}$
$\bm{1}_{\mathrlap{1}}$ | $V_{s,1}=\\{\frac{\hat{y}}{\sigma}=\frac{\hat{w}}{\sigma}=0\\}\setminus\mathopen{}\mathclose{{}\left(V_{a}\cup V_{f,1}\cup V_{f,2}\cup V_{f,3}\cup V_{s,2}}\right)$
$\bm{(n^{2}-1)}_{\mathrlap{0}}$ | $\\{\sigma=0\\}$
Table 9: Matter spectrum for the $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$
fundamental seed models for $n=2k$ and $k\geq 2$. Note that one can flip the
sign of all $\mathfrak{u}(1)$ charges and obtain an equally valid description
of the matter spectrum (with a different value of $\nu$).

The $n=2$ case is somewhat unique, as one can obtain an $\mathfrak{su}(2)$
gauge factor (with $\text{I}_{2}$ fibers) without satisfying the split
condition. We therefore consider a separate Weierstrass model just for the
$n=2$ case:

$y^{2}=x^{3}+\mathopen{}\mathclose{{}\left(-\frac{\phi^{2}}{48}+f_{1}\sigma}\right)xz^{4}+\mathopen{}\mathclose{{}\left(-\frac{\phi^{3}}{864}+\frac{1}{12}f_{1}\sigma\phi+\gamma^{2}\sigma^{2}}\right)z^{6}\,.$
(116)

This model admits $\text{I}_{2}$ singularities along $\\{\sigma=0\\}$ and a
generating section of the form

$\mathopen{}\mathclose{{}\left[\hat{x}:\hat{y}:\hat{z}}\right]=\mathopen{}\mathclose{{}\left[-\frac{\phi}{12}:\gamma\sigma:1}\right]\,,\quad\hat{w}=f_{1}\sigma\,,$
(117)

implying that the gauge algebra is $\mathfrak{su}(2)\oplus\mathfrak{u}(1)$.
Matter in the $\bm{2}_{1/2}$ representation is supported at
$\\{\sigma=f_{1}^{2}+\phi\gamma^{2}=0\\}$, and $\bm{1}_{1}$ matter occurs at
$\\{\gamma=f_{1}=0\\}$. Because both the $\bm{2}$ and $\bm{1}$ representations
are self-conjugate, the sign of the $\mathfrak{u}(1)$ charge is unimportant.
(Additionally, there may be $\bm{3}_{0}$ matter that can propagate throughout
the $\\{\sigma=0\\}$ divisor.) For 6D models, the spectrum satisfies the
anomaly cancellation conditions with $0pt=-2K_{B}-\frac{1}{2}[\sigma]$.

As mentioned previously, we can calculate multiples of the models’ generating
sections and examine their behavior at the $\bm{n}_{1/n}$ loci, giving us
orders of vanishing for various pseudo-charges. In fact, our seed models also
admit fundamental matter with $\mathfrak{u}(1)$ charges larger than $1/n$,
allowing us to strengthen the analysis. First, some pseudo-charges can be
obtained in multiple ways. In an $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$
model, for example, the orders of vanishing for pseudo-charge $4/3$ can be
obtained from the behavior of $4\hat{s}$ at the $\bm{3}_{1/3}$ locus or from
the behavior of $\hat{s}$ at the $\bm{3}_{4/3}$ locus. We always obtain the
same orders of vanishing for a particular combination of $\nu$ and the pseudo-
charge even if we obtain the pseudo-charge in different ways, suggesting that
our strategy of looking at multiples of the generating section gives
reasonable results. Second, these higher-charge fundamental loci allow us to
probe some larger pseudo-charges without calculating larger multiples of the
generating section.

The orders of vanishing we obtain are summarized in Tables 10 and 11. One can
verify that the codimension-one orders of vanishing at the $\mathfrak{su}(n)$
locus are described by

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{SU}(n)}(\nu)=\mathopen{}\mathclose{{}\left(0,u_{n}(\nu),u_{n}(\nu)}\right)\,,\quad\operatorname{ord}_{1}(\hat{z})=0$
(118)

at the codimension-one locus supporting the $\mathfrak{su}(n)$ gauge factor.
Additionally, the codimension-two orders of vanishing at the fundamental locus
are given by

$\displaystyle\operatorname{ord}_{2}(\hat{z})$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{n+1}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SU}(n+1)}(\mu)}\right)$
(119)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{n+1}q^{2}+\frac{\nu(n-\nu)}{n}-\frac{\mu(n+1-\mu)}{n+1}}\right)$

and

$\displaystyle\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)$
$\displaystyle=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{SU}(n+1)}(\mu)$
(120)
$\displaystyle=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(0,u_{n+1}(\mu),u_{n+1}(\mu)}\right)$

for

$\mu=\overline{nq}_{n+1}\,.$ (121)

These formulas agree exactly with the proposals in Section 2.2.

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{2}$ | $1$ | $(0,1,0,1)$ | $(0,1,0,1)$
$1$ | $0$ | $(0,0,0,0)$ | $(0,1,0,1)$
$\frac{3}{2}$ | $1$ | $(0,1,0,1)$ | $(2,3,1,4)$
$2$ | $0$ | $(0,0,0,0)$ | $(2,4,1,5)$
$\frac{5}{2}$ | $1$ | $(0,1,0,1)$ | $(4,7,2,9)$
$3$ | $0$ | $(0,0,0,0)$ | $(6,9,3,12)$
$\frac{7}{2}$ | $1$ | $(0,1,0,1)$ | $(8,13,4,17)$
$4$ | $0$ | $(0,0,0,0)$ | $(10,16,5,21)$
$\frac{9}{2}$ | $1$ | $(0,1,0,1)$ | $(14,21,7,28)$
$5$ | $0$ | $(0,0,0,0)$ | $(16,25,8,33)$
(a) $\bm{2}_{q}$ representation of $\mathfrak{su}(2)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{2}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(0,2,0,2)$
$1$ | $0$ | $(0,0,0,0)$ | $(0,1,0,1)$
$\frac{4}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{5}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(2,4,1,5)$
$2$ | $0$ | $(0,0,0,0)$ | $(2,5,1,6)$
$\frac{8}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(6,9,3,12)$
$\frac{10}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(8,14,4,18)$
$4$ | $0$ | $(0,0,0,0)$ | $(12,18,6,24)$
$\frac{16}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(22,33,11,44)$
$\frac{20}{3}$ | $1\text{ or }2$ | $(0,1,0,1)$ | $(34,51,17,68)$
$8$ | $0$ | $(0,0,0,0)$ | $(48,72,24,96)$
(b) $\bm{3}_{q}$ representation of $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{1}{2}$ | $2$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{3}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(0,2,0,2)$
$1$ | $0$ | $(0,0,0,0)$ | $(0,1,0,1)$
$\frac{5}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{3}{2}$ | $2$ | $(0,2,0,2)$ | $(2,4,1,5)$
$\frac{9}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(4,7,2,9)$
$\frac{5}{2}$ | $2$ | $(0,2,0,2)$ | $(6,9,3,12)$
$3$ | $0$ | $(0,0,0,0)$ | $(6,11,3,14)$
$\frac{15}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(12,18,6,24)$
$\frac{9}{2}$ | $2$ | $(0,2,0,2)$ | $(16,26,8,34)$
$5$ | $0$ | $(0,0,0,0)$ | $(20,30,10,40)$
$\frac{25}{4}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(32,48,16,64)$
$\frac{15}{2}$ | $2$ | $(0,2,0,2)$ | $(46,69,23,92)$
(c) $\bm{4}_{q}$ representation of $\mathfrak{su}(4)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{2}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{3}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(0,3,0,3)$
$\frac{4}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(0,2,0,2)$
$1$ | $0$ | $(0,0,0,0)$ | $(0,1,0,1)$
$\frac{6}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{8}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(2,5,1,6)$
$\frac{12}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(6,9,3,12)$
$\frac{16}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(8,14,4,18)$
$\frac{18}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(12,18,6,24)$
$4$ | $0$ | $(0,0,0,0)$ | $(12,20,6,26)$
$\frac{24}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(20,30,10,40)$
$6$ | $0$ | $(0,0,0,0)$ | $(30,45,15,60)$
$\frac{36}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(44,66,22,88)$
(d) $\bm{5}_{q}$ representation of $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$

Table 10: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
various combinations of $\nu$ and pseudo-charge $q$ for fundamental matter of
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ with $2\leq n\leq 5$. The
$\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ data are found using the model in Eq.
116, the $\mathfrak{su}(3)\oplus\mathfrak{u}(1)$ and
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ data are found using the model in Eq.
113, and the $\mathfrak{su}(4)\oplus\mathfrak{u}(1)$ data are found using the
seed model for fundamentals with even $n$.

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{6}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{1}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{1}{2}$ | $3$ | $(0,3,0,3)$ | $(0,3,0,3)$
$\frac{2}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(0,3,0,3)$
$\frac{5}{6}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(0,2,0,2)$
$\frac{7}{6}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{5}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(2,6,1,7)$
$\frac{7}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(6,9,3,12)$
$\frac{5}{2}$ | $3$ | $(0,3,0,3)$ | $(6,10,3,13)$
$\frac{10}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(10,16,5,21)$
$\frac{7}{2}$ | $3$ | $(0,3,0,3)$ | $(12,18,6,24)$
$\frac{25}{6}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(14,24,7,31)$
$\frac{14}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(20,30,10,40)$
$\frac{35}{6}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(30,45,15,60)$
(a) $\bm{6}_{q}$ representation of $\mathfrak{su}(6)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{2}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{3}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(0,3,0,3)$
$\frac{4}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(0,4,0,4)$
$\frac{5}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(0,3,0,3)$
$\frac{6}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(0,2,0,2)$
$\frac{8}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{12}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(2,7,1,8)$
$\frac{16}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(6,9,3,12)$
$\frac{18}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(6,11,3,14)$
$\frac{24}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(12,18,6,24)$
$\frac{30}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(16,26,8,34)$
$\frac{32}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(20,30,10,40)$
$\frac{36}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(22,37,11,48)$
$\frac{40}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(30,45,15,60)$
$\frac{48}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(42,63,21,84)$
(b) $\bm{7}_{q}$ representation of $\mathfrak{su}(7)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{8}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{1}{4}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{3}{8}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(0,3,0,3)$
$\frac{1}{2}$ | $4$ | $(0,4,0,4)$ | $(0,4,0,4)$
$\frac{7}{8}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(0,2,0,2)$
$\frac{9}{8}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{7}{4}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(2,7,1,8)$
$\frac{9}{4}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(6,9,3,12)$
$\frac{21}{8}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(6,12,3,15)$
$\frac{27}{8}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(12,18,6,24)$
$\frac{7}{2}$ | $4$ | $(0,4,0,4)$ | $(12,19,6,25)$
$\frac{9}{2}$ | $4$ | $(0,4,0,4)$ | $(20,30,10,40)$
(c) $\bm{8}_{q}$ representation of $\mathfrak{su}(8)\oplus\mathfrak{u}(1)$

Table 11: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
various combinations of $\nu$ and pseudo-charge $q$ for fundamental matter of
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ with $6\leq n\leq 8$. The
$\mathfrak{su}(7)\oplus\mathfrak{u}(1)$ data are found using the model in Eq.
113, and the $\mathfrak{su}(6)\oplus\mathfrak{u}(1)$ and
$\mathfrak{su}(8)\oplus\mathfrak{u}(1)$ data are found using the seed model
for fundamentals with even $n$.

###  7.2 Antisymmetric representation of $\mathfrak{su}(n)$ for odd $n$

In F-theory models, matter in the antisymmetric representation is allowed to
have $\mathfrak{u}(1)$-charges of the form

$\frac{2\nu}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (122)

For odd $n$ (i.e., $n=2k+1$ with $k\geq 2$), each allowed $\mathfrak{u}(1)$
charge corresponds to a unique value of $\nu$:

$\nu=\overline{nq(k+1)}_{n}\,.$ (123)

Therefore, suppose we have a seed model with an
$\mathfrak{su}(2k+1)\oplus\mathfrak{u}(1)$ algebra, a global structure
described by $\nu=k+1$, and $\bm{\frac{n(n-1)}{2}}_{1/n}$ matter. The
antisymmetric matter is supported at a codimension-two locus along the
$\text{I}^{(s)}_{n}$ locus where the singularity type enhances to
$\text{I}^{*}_{n-4}$. If the model has a generating section $\hat{s}$, the
multiples $m\hat{s}$ would have $\nu=\overline{m(k+1)}_{n}$ and would realize
pseudo-charge $\frac{m}{n}$ at the codimension-two locus. Thus, multiples of
the generating section would give us all allowed combinations of $\nu$ and the
$\mathfrak{u}(1)$ charge.

We can consider the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\mathopen{}\mathclose{{}\left(-\frac{1}{48}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma}\right)^{2}+\frac{1}{2}\sigma^{k}\mathopen{}\mathclose{{}\left(b_{1}c_{1}-2c_{0}\sigma}\right)}\right)xz^{4}$
(124)
$\displaystyle\qquad+\frac{1}{864}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma}\right)^{3}-36\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma}\right)\mathopen{}\mathclose{{}\left(b_{1}c_{1}-2c_{0}\sigma}\right)\sigma^{k}+216c_{1}^{2}\sigma^{2k}}\right)z^{6}\,,$

which admits $\text{I}^{(s)}_{2k+1}$ singularities at $\\{\sigma=0\\}$ and a
generating section $\hat{s}$ with components

$[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[\frac{1}{12}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma}\right):\frac{1}{2}c_{1}\sigma^{k}:1}\right]\,,\quad\hat{w}=\frac{1}{2}\sigma^{k}\mathopen{}\mathclose{{}\left(b_{1}c_{1}-2c_{0}\sigma}\right)\,.$
(125)

The gauge algebra for this Weierstrass model is therefore
$\mathfrak{su}(2k+1)\oplus\mathfrak{u}(1)$. The matter spectrum includes
$\bm{\frac{n(n-1)}{2}}_{1/n}$ matter at $\\{\sigma=b_{1}=0\\}$,
$\bm{n}_{(k+1)/n}$ matter at $\\{\sigma=c_{1}=0\\}$, $\bm{n}_{-k/n}$ matter at
$\\{\sigma=b_{1}c_{0}-b_{2}c_{1}=0\\}$, $\bm{1}_{1}$ matter at
$\\{c_{0}=c_{1}=0\\}$, and delocalized $\bm{(n^{2}-1)}_{0}$ matter allowed to
propagate along $\\{\sigma=0\\}$. (We can freely flip the overall sign of the
$\mathfrak{u}(1)$ charges while replacing $\nu$ with $n-\nu$.) For 6D models,
this spectrum satisfies the gauge and gauge–gravitational anomaly conditions
for $0pt=-2K_{B}-\frac{k(k+1)}{n}[\sigma]$.

The multiples $m\hat{s}$ of the generating section support pseudo-charge
$\frac{m}{n}$ at the antisymmetric locus with $\nu=\overline{m(k+1)}_{n}$, so
we can use these multiples to find the orders of vanishing corresponding to
$\bm{\frac{n(n-1)}{2}}_{m/n}$ matter. The order of vanishing data are
summarized in Table 12. According to the formulas in Section 2.2, we expect
that the codimension-one orders of vanishing should be described by

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{SU}(n)}(\nu)=\mathopen{}\mathclose{{}\left(0,u_{n}(\nu),u_{n}(\nu)}\right)\,,\quad\operatorname{ord}_{1}(\hat{z})=0\,.$
(126)

The codimension-two orders of vanishing, meanwhile, should be given by

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+\frac{\nu(n-\nu)}{n}-T_{\operatorname{SO}(2n)}(\mu)}\right)$
(127)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{SO}(2n)}(\mu)$
(128)

for some value of $\mu$. Indeed, the data satisfy these relations with

$\mu=\overline{nq}_{4}\,.$ (129)

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(1,2,0,3)$
$\frac{2}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(1,2,0,2)$
$\frac{3}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(1,2,0,3)$
$\frac{4}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(2,3,1,4)$
$1$ | $0$ | $(0,0,0,0)$ | $(1,2,0,3)$
$\frac{6}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(3,5,1,6)$
$\frac{7}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(3,5,1,7)$
$\frac{8}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(4,6,2,8)$
$\frac{9}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(5,8,2,11)$
$2$ | $0$ | $(0,0,0,0)$ | $(5,8,2,10)$
$\frac{11}{5}$ | $2\text{ or }3$ | $(0,2,0,2)$ | $(7,11,3,15)$
$\frac{12}{5}$ | $1\text{ or }4$ | $(0,1,0,1)$ | $(8,12,4,16)$
(a) $\bm{10}_{q}$ representation of $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(1,3,0,4)$
$\frac{2}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(1,2,0,2)$
$\frac{3}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(1,3,0,4)$
$\frac{4}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(2,3,1,4)$
$\frac{5}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(1,3,0,4)$
$\frac{6}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(3,5,1,6)$
$1$ | $0$ | $(0,0,0,0)$ | $(1,3,0,4)$
$\frac{8}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(4,6,2,8)$
$\frac{9}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(3,6,1,8)$
$\frac{10}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(5,8,2,10)$
$\frac{11}{7}$ | $2\text{ or }5$ | $(0,2,0,2)$ | $(5,9,2,12)$
$\frac{12}{7}$ | $1\text{ or }6$ | $(0,1,0,1)$ | $(6,9,3,12)$
$\frac{13}{7}$ | $3\text{ or }4$ | $(0,3,0,3)$ | $(7,12,3,16)$
$2$ | $0$ | $(0,0,0,0)$ | $(7,11,3,14)$
(b) $\bm{21}_{q}$ representation of $\mathfrak{su}(7)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{9}$ | $4\text{ or }5$ | $(0,4,0,4)$ | $(1,4,0,5)$
$\frac{2}{9}$ | $1\text{ or }8$ | $(0,1,0,1)$ | $(1,2,0,2)$
$\frac{1}{3}$ | $3\text{ or }6$ | $(0,3,0,3)$ | $(1,4,0,5)$
$\frac{4}{9}$ | $2\text{ or }7$ | $(0,2,0,2)$ | $(2,3,1,4)$
$\frac{5}{9}$ | $2\text{ or }7$ | $(0,2,0,2)$ | $(1,4,0,5)$
$\frac{2}{3}$ | $3\text{ or }6$ | $(0,3,0,3)$ | $(3,5,1,6)$
$\frac{7}{9}$ | $1\text{ or }8$ | $(0,1,0,1)$ | $(1,4,0,5)$
$\frac{8}{9}$ | $4\text{ or }5$ | $(0,4,0,4)$ | $(4,6,2,8)$
$1$ | $0$ | $(0,0,0,0)$ | $(1,4,0,5)$
$\frac{10}{9}$ | $4\text{ or }5$ | $(0,4,0,4)$ | $(5,8,2,10)$
$\frac{11}{9}$ | $1\text{ or }8$ | $(0,1,0,1)$ | $(3,7,1,9)$
$\frac{4}{3}$ | $3\text{ or }6$ | $(0,3,0,3)$ | $(6,9,3,12)$
(c) $\bm{36}_{q}$ representation of $\mathfrak{su}(9)\oplus\mathfrak{u}(1)$

Table 12: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
various combinations of $\nu$ and pseudo-charge $q$ for antisymmetric matter
of $\mathfrak{su}(2k+1)\oplus\mathfrak{u}(1)$. The data are found using the
model in Eq. 124.

###  7.3 Antisymmetric representation of $\mathfrak{su}(n)$ for even $n$

Recall that matter in the antisymmetric representation of $\mathfrak{su}(n)$
is allowed to have $\mathfrak{u}(1)$ charges of the form $\frac{2\nu}{n}+j$
for integers $j$. When $n$ is even (i.e., when $n=2k$), the allowed $nq$ are
always even. More importantly, the same value of $q$ can occur for two
different values of $\nu$: $\nu=\overline{kq}_{n}$ or
$\nu=\overline{kq+k}_{n}$. Because of this fact, our strategy of using
multiples of the generating section becomes more difficult. Suppose we start
with a $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ seed model with $\nu=1$ and
$\bm{\frac{n(n-1)}{2}}_{2/n}$ matter. By considering multiples $m\hat{s}$ of
the generating section $\hat{s}$, we can obtain the pseudo-charges
$q=\frac{2m}{n}$ with

$\nu=\overline{m}_{n}=\overline{kq}_{n}\,.$ (130)

To obtain data for situations where $\nu=\overline{kq+k}_{n}$, we need a
different seed model. If we start with one admitting
$\bm{\frac{n(n-1)}{2}}_{2/n}$ matter with $\nu=k+1$, multiples $m\hat{s}$ of
the generating section $\hat{s}$ would give us pseudo-charges $\frac{2m}{n}$
with

$\nu=\overline{mk+m}_{n}\,.$ (131)

We can therefore use this seed model to obtain information about the
$\nu=\overline{kq+k}_{n}$ situations when $kq$ is odd. However, we would not
learn anything about the $\nu=\overline{kq+k}_{n}$ situations when $kq$ is
even. If we were to start with a seed model admitting
$\bm{\frac{n(n-1)}{2}}_{4/n}$ matter with $\nu=k+2$, we could obtain
information about the $\nu=\overline{kq+k}_{n}$ situations for some even
values of $kq$, but we would miss those values of $kq$ that are multiples of
$4$.

In order to get information about all of the $\nu=\overline{kq+k}_{n}$
situations, we would need an infinite number of seed models. This would
clearly be impractical, so we will consider only a limited number of $(\nu,q)$
combinations. Specifically, we will focus on the situations where
$\nu=\overline{kq}_{n}$ and the situations where $\nu=\overline{kq+k}_{n}$
with odd $kq$. Thus, we need two seed models. For the $\nu=\overline{kq}_{n}$
situations, we can consider the same Weierstrass model used for the
fundamentals of $\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$, which is described
in Section 7.1. This model has $\nu=1$ and admits the locus
$\\{\sigma=b_{1,0}=0\\}$ supporting $\bm{\frac{n(n-1)}{2}}_{2/n}$ matter.

For the $\nu=\overline{kq+k}_{n}$ cases with odd $kq$, we need a seed model
with $\nu=k+1$ and $\bm{\frac{n(n-1)}{2}}_{2/n}$ matter, at least for $k\geq
3$.282828The $\mathfrak{su}(4)\oplus\mathfrak{u}(1)$ with $k=2$ is somewhat
special, as $\nu=3$ is the inverse of $\nu=1$ in $\mathbb{Z}_{4}$. Thus, the
$\nu=3$ seed model would be equivalent to the $\nu=1$ seed model discussed
directly above, implying that we do not need a second seed model to obtain the
$\nu=\overline{kq+k}_{n}$ pseudo-charges with odd $kq$. We can use the
Weierstrass construction

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\mathopen{}\mathclose{{}\left[-\frac{1}{48}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma-4c_{1,0}\sigma^{k-1}}\right)^{2}+\sigma^{k}\mathopen{}\mathclose{{}\left(\frac{1}{3}b_{2}c_{1,0}-\frac{1}{2}b_{1}c_{1,1}+\sigma
c_{0,1}}\right)}\right]xz^{4}$ (132)
$\displaystyle\phantom{{}=x^{3}}+\Bigg{[}\frac{1}{864}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4\sigma
b_{2}-4\sigma^{k-1}c_{1,0}}\right)^{3}$
$\displaystyle\qquad\qquad\quad-\frac{1}{72}\sigma^{k}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4\sigma
b_{2}-4\sigma^{k-1}c_{1,0}}\right)\mathopen{}\mathclose{{}\left(6\sigma
c_{0,1}+2b_{2}c_{1,0}-3b_{1}c_{1,1}}\right)$
$\displaystyle\qquad\qquad\quad+\sigma^{2k}\mathopen{}\mathclose{{}\left(\frac{c_{1,1}^{2}}{4}-\frac{1}{3}c_{0,1}c_{1,0}}\right)+\frac{2}{27}c_{1,0}^{3}\sigma^{3k-3}\Bigg{]}z^{6}\,.$

It admits $\text{I}^{(s)}_{2k}$ singularities along $\\{\sigma=0\\}$ and a
generating section $\hat{s}$ with components

$\displaystyle=\mathopen{}\mathclose{{}\left[\frac{1}{12}\mathopen{}\mathclose{{}\left(b_{1}^{2}-4b_{2}\sigma}\right):\frac{1}{6}\sigma^{k-1}\mathopen{}\mathclose{{}\left(3\sigma
c_{1,1}-b_{1}c_{1,0}}\right):1}\right]\,,$ (133) $\displaystyle\hat{w}$
$\displaystyle=-\frac{1}{6}\sigma^{k-1}\mathopen{}\mathclose{{}\left(-b_{1}^{2}c_{1,0}+\sigma\mathopen{}\mathclose{{}\left(2b_{2}c_{1,0}+3b_{1}c_{1,1}}\right)-6\sigma^{2}c_{0,1}+2\sigma^{k-1}c_{1,0}^{2}}\right)\,.$

In addition to possible delocalized adjoint matter along $\\{\sigma=0\\}$, the
model supports $\bm{\frac{n(n-1)}{2}}_{2/n}$ matter at $\\{\sigma=b_{1}=0\\}$,
$\bm{n}_{(k+1)/n}$ matter at $\\{\sigma=c_{1,0}=0\\}$, $\bm{n}_{-(k-1)/n}$
matter at

$\\{\sigma=-\frac{1}{2}b_{1}^{2}c_{0,1}-\frac{1}{6}b_{2}^{2}c_{1,0}+\frac{1}{9}\sigma^{k-3}b_{1}^{2}c_{1,0}^{2}+\frac{1}{2}b_{1}b_{2}c_{1,1}=0\\}\,,$
(134)

and $\bm{1}_{1}$ matter at

$\mathopen{}\mathclose{{}\left\\{\frac{\hat{y}}{\sigma^{k-1}}=\frac{\hat{w}}{\sigma^{k-1}}=0}\right\\}\setminus\mathopen{}\mathclose{{}\left(\\{\sigma=b_{1}=0\\}\cup\\{\sigma=c_{1,0}=0\\}}\right)\,.$
(135)

(Of course, one can flip the signs of the $\mathfrak{u}(1)$ charges to obtain
an equally valid description of the spectrum with $\nu=k-1$.) One can verify
that, for 6D models, this spectrum satisfies the gauge and gauge–gravitational
anomalies with $0pt=-2K_{B}-\frac{(k-1)(k+1)}{n}[\sigma]$.

Table 13 summarizes the orders of vanishing obtained from these two seed
constructions. The codimension-one orders of vanishing satisfy the formulas

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{SU}(n)}(\nu)\,,\quad\operatorname{ord}_{1}(\hat{z})=0\,.$
(136)

The codimension-two orders of vanishing are given by

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SO}(2n)}(\mu)}\right)$
(137)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{SO}(2n)}(\mu)$
(138)

for

$\mu=\overline{\mathopen{}\mathclose{{}\left(q-\frac{2}{n}\nu}\right)}_{2}+\overline{2\nu}_{4}\,.$
(139)

These formulas exactly match those expected from the proposals in Section 2.2.

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(1,2,0,2)$
$1$ | $2$ | $(0,2,0,2)$ | $(2,3,1,4)$
$\frac{3}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(3,5,1,6)$
$2$ | $0$ | $(0,0,0,0)$ | $(4,6,2,8)$
$\frac{5}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(7,11,3,14)$
$3$ | $2$ | $(0,2,0,2)$ | $(10,15,5,20)$
$\frac{7}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(13,20,6,26)$
$4$ | $0$ | $(0,0,0,0)$ | $(16,24,8,32)$
$\frac{9}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(21,32,10,42)$
$5$ | $2$ | $(0,2,0,2)$ | $(26,39,13,52)$
$\frac{11}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(31,47,15,62)$
$6$ | $0$ | $(0,0,0,0)$ | $(36,54,18,72)$
$\frac{13}{2}$ | $1\text{ or }3$ | $(0,1,0,1)$ | $(43,65,21,86)$
$7$ | $2$ | $(0,2,0,2)$ | $(50,75,25,100)$
(a) $\bm{6}_{q}$ representation of $\mathfrak{su}(4)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{3}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(1,2,0,2)$
$\frac{1}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(1,3,0,3)$
$\frac{2}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(2,3,1,4)$
$1$ | $0$ | $(0,0,0,0)$ | $(1,3,0,3)$
$1$ | $3$ | $(0,3,0,3)$ | $(3,5,1,6)$
$\frac{4}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(4,6,2,8)$
$\frac{5}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(5,9,2,11)$
$\frac{5}{3}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(5,8,2,10)$
$2$ | $0$ | $(0,0,0,0)$ | $(6,9,3,12)$
$\frac{7}{3}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(9,14,4,18)$
$\frac{7}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(9,15,4,19)$
$\frac{8}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(12,18,6,24)$
$3$ | $0$ | $(0,0,0,0)$ | $(13,21,6,27)$
$3$ | $3$ | $(0,3,0,3)$ | $(15,23,7,30)$
$\frac{10}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(18,27,9,36)$
$\frac{11}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(21,33,10,43)$
$\frac{11}{3}$ | $1\text{ or }5$ | $(0,1,0,1)$ | $(21,32,10,42)$
$4$ | $0$ | $(0,0,0,0)$ | $(24,36,12,48)$
$\frac{13}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(29,45,14,59)$
$\frac{14}{3}$ | $2\text{ or }4$ | $(0,2,0,2)$ | $(34,51,17,68)$
(b) $\bm{15}_{q}$ representation of $\mathfrak{su}(6)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{4}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(1,2,0,2)$
$\frac{1}{4}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(1,4,0,4)$
$\frac{1}{2}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(2,3,1,4)$
$\frac{3}{4}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(3,5,1,6)$
$\frac{3}{4}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(1,4,0,4)$
$1$ | $4$ | $(0,4,0,4)$ | $(4,6,2,8)$
$\frac{5}{4}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(3,7,1,8)$
$\frac{5}{4}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(5,8,2,10)$
$\frac{3}{2}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(6,9,3,12)$
$\frac{7}{4}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(7,13,3,16)$
$\frac{7}{4}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(7,11,3,14)$
$2$ | $0$ | $(0,0,0,0)$ | $(8,12,4,16)$
$\frac{9}{4}$ | $1\text{ or }7$ | $(0,1,0,1)$ | $(11,17,5,22)$
$\frac{9}{4}$ | $3\text{ or }5$ | $(0,3,0,3)$ | $(11,19,5,24)$
$\frac{5}{2}$ | $2\text{ or }6$ | $(0,2,0,2)$ | $(14,21,7,28)$
(c) $\bm{28}_{q}$ representation of $\mathfrak{su}(8)\oplus\mathfrak{u}(1)$

Table 13: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ or
various combinations of $\nu$ and pseudo-charge $q$ for antisymmetric matter
of $\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$.

###  7.4 Connection to EDS valuations

So far, we have seen that the orders of vanishing obtained using the strategy
of Section 5 agree with the proposed formulas in Section 2.2. Just as was the
case with singlets, the appearance of these orders of vanishing can at least
be partially explained by the results in StangeEllTrouble  regarding the
$p$-adic valuations of elliptic divisibility sequences. However, there is one
new complication for the non-singlet cases. The formula for
$\operatorname{ord}_{2}(\hat{z})$ appears to be related to the difference in
the $p$-adic valuations of two different elliptic divisibility sequences: one
corresponding to the codimension-two behavior and the other corresponding to
the codimension-one behavior. Despite this additional difficulty, we can see a
clear connection between the proposals and the results in StangeEllTrouble .

##### Fundamental representation of $\mathfrak{su}(n)$

As described in Section 7.1, we could obtain all the allowed pseudo-charges by
starting with a seed model admitting $\bm{n}_{1/n}$ matter. In the F-theory
constructions we consider, this matter occurs at codimension-two loci where
the singularity type enhances from $\text{I}^{(s)}_{n}$ to $\text{I}_{n+1}$.
Moreover, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(0,1,0,1)$ at this
locus, indicating that $\mu=1$. Multiples $m\hat{s}$ of the generating section
support pseudo-charge $\frac{m}{n}$.

If we write this codimension-two locus as $\\{\sigma=\Delta_{1/n}=0\\}$ where
$\\{\sigma=0\\}$ is the $\mathfrak{su}(n)$ locus, we could imagine an elliptic
curve found by setting $\sigma$ and $\Delta_{1/n}$ to some prime $p$ and
setting other parameters to some arbitrary integer values. Because the
elliptic fibration supports singularity type $\text{I}_{n+1}$ at the
codimension-two locus, the elliptic curve analogue would have multiplicative
reduction modulo $p$. Meanwhile, consider the rational point $P$ on the
elliptic curve analogue found by making the appropriate substitutions into the
generating section expressions. As discussed in Section 6.2, the result in
StangeEllTrouble  states that the elliptic divisibility sequence $W^{(2)}_{m}$
associated with $P$ should have $p$-adic valuations given by an elliptic
troublemaker sequence:292929We are assuming that no multiples of $P$ reduce to
the identity point, but this seems to agree with the observed behaviors.

$v_{p}(W^{(2)}_{m})=R_{m}(a,l)\,.$ (140)

The integer $l$ should equal $v_{p}(\Delta)$, which in this case is $n+1$.
Meanwhile, $a$, which describes the component of the Neron fiber hit by $P$,
corresponds to $\mu$, which in this case is $1$. We would therefore expect
that

$v_{p}(W^{(2)}_{m})=R_{m}(1,n+1)=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{n+1}m^{2}-\frac{\overline{m}_{n+1}\mathopen{}\mathclose{{}\left(n+1-\overline{m}_{n+1}}\right)}{n+1}}\right)\,.$
(141)

However, $v_{p}(W^{(2)}_{m})$ overcounts $\operatorname{ord}_{2}(\hat{z})$.
Remember that if $\hat{x}$, $\hat{y}$, and $\hat{z}$ are proportional to
$A^{2}$, $A^{3}$ and $A$ for some factor $A$, one can scale the section
components and remove this factor of $A$. We always get rid of such factors
when calculating the $\operatorname{ord}_{2}(\hat{z})$ values. However, the
recursion formulas for the elliptic divisibility sequences, which are
analogous to formulas for the $\hat{z}$ components of multiples of the
generating section, do not remove such factors. In turn, the formulas from
StangeEllTrouble  count these extra factors as well. This did not cause any
issues for the case of singlets, in part because there was no special
codimension-one locus in the model. For the
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ models, the $\hat{y}$ and $\hat{w}$
components of the generating section are proportional to $\sigma$. As a
result, the formulas for the elliptic divisibility sequences would lead to
extra factors of $\sigma$ that can be scaled away. To recover
$\operatorname{ord}_{2}(\hat{z})$, we need to count these extra factors and
subtract the result from $v_{p}(W^{(2)}_{m})$.

We can in fact count these extra factors by considering a second EDS,
$W^{(1)}_{m}$, just for the factors of $\sigma$. Imagine we found an elliptic
curve by taking our seed model with $\bm{n}_{1/n}$ matter and setting $\sigma$
(but not $\Delta_{1/n}$) to a prime $p$. The resulting elliptic curve would
have multiplicative reduction modulo $p$. If we consider a point found by
making the appropriate substitutions into the generating section components,
the corresponding EDS valuations should be given by

$v_{p}(W^{(1)}_{m})=R_{m}(a,l)\,.$ (142)

The integer $l$ should equal $v_{p}(\Delta)=n$, while $a$ should be given by
the value of $\nu$ for the generating section, which is $1$. In the end, we
have

$v_{p}(W^{(1)}_{m})=R_{m}(1,n)=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n-1}{n}m^{2}-\frac{\overline{m}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m}_{n}}\right)}{n}}\right)\,.$
(143)

The difference of these two sequences of EDS valuations is given by

$v_{p}(W^{(2)}_{m})-v_{p}(W^{(1)}_{m})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{n(n+1)}m^{2}+\frac{\overline{m}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m}_{n}}\right)}{n}-\frac{\overline{m}_{n+1}\mathopen{}\mathclose{{}\left(n+1-\overline{m}_{n+1}}\right)}{n+1}}\right)$
(144)

Substituting $m=nq$, $\nu=\overline{m}_{n}$, and $\mu=\overline{m}_{n+1}$, we
can rewrite this as

$\displaystyle v_{p}(W^{(2)}_{m})-v_{p}(W^{(1)}_{m})$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n^{2}}{n(n+1)}q^{2}+\frac{\nu\mathopen{}\mathclose{{}\left(n-\nu}\right)}{n}-\frac{\mu\mathopen{}\mathclose{{}\left(n+1-\mu}\right)}{n+1}}\right)$
(145)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{\operatorname{SU}(n)}}{d_{\operatorname{SU}(n+1)}}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SU}(n+1)}(\mu)}\right)\,.$

This result agrees exactly with observed patterns in
$\operatorname{ord}_{2}(\hat{z})$ and with the proposals in Section 2.2.

##### Antisymmetric representation of $\mathfrak{su}(n)$ for odd $n$

We saw that all the allowed antisymmetric pseudo-charges for odd $n>4$ can be
obtained from multiples of the generating section in a model supporting
antisymmetric matter with charge $\frac{1}{n}$. Such matter occurs in models
with $\nu=\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil=k+1$. We
should therefore be able to obtain the formula for
$\operatorname{ord}_{2}(\hat{z})$ from those in StangeEllTrouble  describing
the EDS valuations. As with fundamental matter, we expect
$\operatorname{ord}_{2}(\hat{z})$ to be given by the difference of EDS
valuations corresponding to the codimension-two and codimension-one behavior.
For the codimension-one $\text{I}_{n}^{(s)}$ singularity type, consider the
Weierstrass model in Eq. 124, which admits an
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ model with $n=2k+1$. The locus
$\\{\sigma=0\\}$ supports the $\mathfrak{su}(n)$ factor, and the locus
$\\{\sigma=b_{1}=0\\}$ supports antisymmetric matter with $\mathfrak{u}(1)$
charge $\frac{1}{n}$ (up to sign). To construct the EDS associated with the
codimension-one behavior, we construct an analogous elliptic curve by setting
$\sigma$ to a prime integer $p$ and the remaining parameters to numbers such
that $f$ and $g$ of the resulting elliptic curve are integers. For instance,
we can consider the elliptic curve given by the following Weierstrass equation
(with $z$ set to 1):

$y^{2}=x^{3}-(p-3)\mathopen{}\mathclose{{}\left(2p^{k}+3p-9}\right)x-\mathopen{}\mathclose{{}\left(2(p-3)^{3}-p^{2k}+2(p-3)^{2}p^{k}}\right)\,.$
(146)

This elliptic curve has multiplicative reduction modulo $p$ with
$v(\Delta)=2k+1$, and it admits the rational point

$(x,y)=(3-p,p^{k})\,.$ (147)

The rational point reduces to the singular point modulo $p$, and no multiples
of this point reduce to the identity point. The EDS valuations associated with
this elliptic curve are given by

$v(W_{m}^{(1)})=R_{m}(k,2k+1)=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{k(k+1)}{n}m^{2}-\frac{\overline{mk}_{n}(n-\overline{mk}_{n})}{n}}\right)\,.$
(148)

For the codimension-two behavior, we need to find the EDS valuations
$v(W_{m}^{(2)})$ corresponding to $\text{I}^{*}_{n-4}$ loci with at
$\\{\sigma=b_{1}=0\\}$ with $\mu=1$ or $3$. The fibers at $\text{I}^{*}$ loci
have cusp singularities, so the analogous elliptic curves should have additive
reduction, unlike those with multiplicative reduction we encountered
previously. Fortunately, StangeEllTrouble  gives formulas for the valuations
of EDSs for elliptic curves of additive reduction. The key insight is that,
after a field extension by an element $p^{\prime}$, elliptic curves with
additive reduction can be converted to a more minimal form with either good or
multiplicative reduction. As an example, consider an elliptic curve $E$ over
$\mathbb{Q}_{p}$ of the form

$y^{2}=x^{3}+p^{2}f^{\prime}x+p^{3}g^{\prime}\,,$ (149)

where $p$ is a prime integer and $f^{\prime}$, $g^{\prime}$ are integers not
proportional to $p$ that satisfy

$\Delta=4\mathopen{}\mathclose{{}\left(p^{2}f^{\prime}}\right)^{3}+27\mathopen{}\mathclose{{}\left(p^{3}g^{\prime}}\right)^{2}\propto
p^{6+(n-4)}$ (150)

for an integer $n\geq 4$. This elliptic curve, the analogue of
$\text{I}^{*}_{n-4}$ singular fibers, has additive reduction modulo $p$.
Suppose we extend $\mathbb{Q}_{p}$ by $p^{\prime}=\sqrt{p}$, such that the
elliptic curve can be written as

$y^{2}=x^{3}+(p^{\prime})^{4}f^{\prime}x+(p^{\prime})^{6}g^{\prime}\,.$ (151)

The elliptic curve is now in a non-minimal form, and we can remove the factors
of ${p^{\prime}}$ by defining

$x=(p^{\prime})^{2}x^{\prime}\,,\quad y=(p^{\prime})^{3}y^{\prime}\,.$ (152)

After plugging in these redefinitions and removing the common factor of
$(p^{\prime})^{6}$, we obtain a new elliptic curve $E^{\prime}$ of the form

${y^{\prime}}^{2}={x^{\prime}}^{3}+f^{\prime}x^{\prime}+g^{\prime}\,.$ (153)

with a discriminant

$\Delta_{E^{\prime}}=\frac{\Delta_{E}}{(p^{\prime})^{12}}\,.$ (154)

$E^{\prime}$ no longer has additive reduction modulo $p^{\prime}$. If
$E^{\prime}$ has good reduction, we say that the original curve $E$ has
potential good reduction. If $E^{\prime}$ has multiplicative reduction, the
original curve $E$ has potential multiplicative reduction.

Now consider the elliptic divisibility sequence $W_{m}$ associated with a
point $P$ on $E$. The point $P$ maps onto a point $P^{\prime}$ on
$E^{\prime}$, and we can form an EDS $W_{m}^{\prime}$ associated with
$P^{\prime}$ and $E^{\prime}$. We can also have a valuation $v_{1}$ for the
extended field that is essentially a valuation with respect to $p^{\prime}$:
for an element $\zeta$ of the original field, $v_{1}(\zeta)=dv(\zeta)$, where
$d$ is the ramification degree of the extension. For the example in Eq. 149,
the ramification degree is $2$ because we extend the field by $\sqrt{p}$.

The valuations $v(W_{m})$ can be related to the valuations
$v_{1}(W_{m}^{\prime})$. According to Theorem 19 of StangeEllTrouble , if $E$
has potential good reduction, the valuations of $W_{m}$ are given by the
formula

$dv(W_{m})=\frac{m^{2}-1}{12}dv(\Delta_{E})+v_{1}(W_{m}^{\prime})\,.$ (155)

And according to Theorem 14 of StangeEllTrouble , the valuations
$v_{1}(W_{m}^{\prime})$ take the form

$v_{1}(W_{m}^{\prime})=\begin{cases}\frac{v_{1}(x^{\prime}_{P^{\prime}})}{2}m^{2}&n_{P^{\prime}}=1\\\
0&n_{P^{\prime}}\neq
1\end{cases}+\begin{cases}S_{m/n_{{P^{\prime}}}}\mathopen{}\mathclose{{}\left(p^{\prime},b,v_{1}(p^{\prime}),h,s,w}\right)&m\mid
n_{{P^{\prime}}}\\\ 0&m\nmid n_{P^{\prime}}\end{cases}\,.$ (156)

Here, $x^{\prime}_{P^{\prime}}$ is the $x^{\prime}$ coordinate of
$P^{\prime}$, and $n_{P^{\prime}}$ is the smallest integer such that
$n_{P^{\prime}}P^{\prime}$ reduces to the identity of $E^{\prime}$ modulo
$p^{\prime}$. The $S$ term in described in more detail in Appendix B, but as
argued there it can essentially be replaced by the constant $s$ for the cases
of interest in this paper.

If $E$ has potential multiplicative reduction, Theorem 29 of StangeEllTrouble
states that the valuations of $W_{m}$ are given by

$dv(W_{m})=\frac{m^{2}-1}{4}dv(f)+v_{1}(W_{m}^{\prime})\,,$ (157)

where $f$ can be read off from the equation for $E$.303030If $E$ is written in
Tate form, $f$ takes the standard form
$-((a_{1}^{2}+4a_{2})^{2}-24(a_{1}a_{3}+a_{4}))/48$. The quantity we refer to
as $f$ would more commonly be referred to as $c_{4}(E)$ in the mathematics
literature. If $P^{\prime}$ reduces to a point other than the singular point,
the $v_{1}(W_{m}^{\prime})$ are still given by Eq. 156. However, if
$P^{\prime}$ reduces to the singular point, the $v_{1}(W_{m}^{\prime})$ are
given by the multiplicative reduction expressions, and the $v(W_{m})$
valuations take the form

$v(W_{m})=\frac{m^{2}-1}{4}v(f)+\frac{1}{d}\mathopen{}\mathclose{{}\left(R_{m}(a,l)+\begin{cases}S_{m/n_{P^{\prime}}}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)&n_{P^{\prime}}\mid
m\\\ 0&n_{P^{\prime}}\nmid m\end{cases}}\right)$ (158)

for integers $a$, $l$, and $n_{P^{\prime}}$.

To apply this to the codimension-two $\text{I}^{*}_{n-4}$ singularities, we
consider an elliptic curve found by taking Eq. 124, setting $\sigma$ and
$b_{1}$ to integers $p$ and $6p$ (where $p$ is a prime integer not equal to 2
or 3), and letting other parameters be appropriate numbers. For a particular
choice of these numbers, we obtain the elliptic curve $E$

$y^{2}=x^{3}+p^{2}\mathopen{}\mathclose{{}\left(-3(1-3p)^{2}+4p^{k-1}}\right)x+p^{3}\mathopen{}\mathclose{{}\left(2(3p-1)^{3}-4(3p-1)p^{k-1}+p^{2k-3}}\right)\,,$
(159)

which admits a rational point

$P\colon(x,y)=(p(3p-1),p^{k})\,.$ (160)

This elliptic curve has additive reduction with $v(\Delta)=2k+3$. After
following the procedure outlined above, the resulting elliptic curve
$E^{\prime}$ has multiplicative reduction modulo $p^{\prime}$ with
$l=v_{1}(\Delta^{\prime})=4k-6$. Additionally, $P^{\prime}$ reduces to the
singular point on $E^{\prime}$. However, $4P^{\prime}$ reduces to the identity
point, implying that $n_{P^{\prime}}=4$. The $y^{\prime}$ coordinate of
$P^{\prime}$ is proportional to ${p^{\prime}}^{2k-3}$, and $a$ should be
$2k-3$. Therefore, the EDS valuations for the codimension-two behavior are
given by

$\displaystyle v(W_{m}^{(2)})$
$\displaystyle=\frac{m^{2}-1}{2}+\frac{1}{2}\mathopen{}\mathclose{{}\left(R_{m}(2k-3,4k-6)+\begin{cases}S_{m/4}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)&4\mid
m\\\ 0&4\nmid m\end{cases}}\right)$ (161)
$\displaystyle=\frac{n}{8}m^{2}-\frac{1}{2}\begin{cases}1-s&4\mid m\\\
1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{n}{4}&(2\nmid m)\end{cases}\,.$

Combining everything together, we expect that when the antisymmetric matter
has charge or pseudo-charge $q=\pm\frac{m}{n}$,
$\operatorname{ord}_{2}(\hat{z})$ should be given by

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{8n}m^{2}+\frac{\overline{mk}_{n}(n-\overline{mk}_{n})}{2n}-\frac{1}{2}\begin{cases}1-s&4\mid
m\\\ 1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{n}{4}&(2\nmid
m)\end{cases}\,.$ (162)

Demanding that this formula give the observed
$\operatorname{ord}_{2}(\hat{z})$ of 1 for antisymmetric matter of
$\mathfrak{su}(5)$ with $m=4$, we can set $s$ to 1, and if we plug in

$m=nq\,,\quad\nu=\overline{m(k+1)}_{n}=n-\overline{mk}_{n}\,,\quad\mu=\overline{m}_{4}=\overline{nq}_{4}\,,$
(163)

we find that

$\displaystyle v(W_{m}^{(2)})-v(W_{m}^{(1)})$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+\frac{\nu(n-\nu)}{n}-\begin{cases}0&\mu=0\\\
1&\mu=2\\\ \frac{n}{4}&\mu=1,3\end{cases}}\right)$ (164)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{SU(n)}}{d_{SO(2n)}}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SO}(2n)}(\mu)}\right)\,,$

in exact agreement with the proposals in Section 2.2 and the observed order-
of-vanishing data.

##### Antisymmetric representation of $\mathfrak{su}(n)$ for even $n$

For the antisymmetric representation with even $n$ (i.e, $n=2k$), we obtained
the order-of-vanishing data from two types of seed models admitting
$\bm{\frac{n(n-1)}{2}}_{2/n}$ matter: those with $\nu=1$ and those with
$\nu=\frac{n}{2}+1$. In order to establish a connection to the EDS valuations,
we need to analyze each seed model separately. Still, we can see that the
formulas from StangeEllTrouble  reproduce the observed order of vanishing data
and the expressions expected from the proposals.

We begin with the $\nu=1$ seed construction, which is described by the the
Weierstrass model in Eq. 113 with the substitutions in Eq. 115. The model’s
$\mathfrak{su}(2k)$ gauge factor is supported along the codimension-one locus
$\\{\sigma=0\\}$, which supports $\text{I}_{2k}^{(s)}$ singularities. The
$\bm{\frac{n(n-1)}{2}}_{2/n}$ matter is supported at the codimension-two locus
$\\{\sigma=b_{1,0}=0\\}$. As before, we need EDS valuations corresponding to
both the codimension-one and the codimension-two behavior. The codimension-one
EDS valuations, which were calculated above, are given by

$v(W^{(1)}_{m})=R_{m}(1,n)=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n-1}{n}m^{2}-\frac{\overline{m}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m}_{n}}\right)}{n}}\right)\,.$
(165)

We can calculate the codimension-two EDS valuations through a procedure
similar to that used for antisymmetric matter with odd $n$. This analysis is
performed in Section C.1.2, so as not to clutter the discussion here. The end
result is that

$v(W^{(2)}_{m})=\frac{1}{2}m^{2}-\frac{1}{2}\begin{cases}0&m\mid 2\\\ 1&m\nmid
2\end{cases}\,.$ (166)

Combining these two expressions we find that

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{n}m^{2}+\frac{\overline{m}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m}_{n}}\right)}{n}-\begin{cases}0&2\mid
m\\\ 1&2\nmid m\end{cases}}\right)\,.$ (167)

If we use the relations

$q=\frac{2}{n}m\,,\quad\nu=\overline{m}_{n}\,,\quad\mu=\overline{\mathopen{}\mathclose{{}\left(q-\frac{2\nu}{n}}\right)}_{2}+\overline{2\nu}_{4}=2\overline{m}_{2}$
(168)

for this seed model, our result can be rewritten as

$\displaystyle v(W_{m}^{(2)})-v(W_{m}^{(1)})$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+\frac{\nu\mathopen{}\mathclose{{}\left(n-\nu}\right)}{n}-\begin{cases}0&\mu=0\\\
1&\mu=2\end{cases}}\right)$ (169)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{\operatorname{SU}(n)}}{d_{\operatorname{SO}(2n)}}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SO}(n)}(\nu)}\right)\,.$

The $\nu=\frac{n}{2}+1$ seed construction is described by the Weierstrass
model in Eq. 132. The model supports $\text{I}^{(s)}_{2k}$ singularities along
$\\{\sigma=0\\}$ and a generating section with components given by Eq. 133.
The $\bm{\frac{n(n-1)}{2}}_{2/n}$ matter is supported at the codimension-two
locus $\\{\sigma=b_{1}=0\\}$. To find the EDS valuations corresponding to the
codimension-one behavior, we can construct an elliptic curve analogue by
setting $\sigma$ to some prime $p$ and the other parameters to some arbitrary
numbers. The resulting elliptic curve would have multiplicative reduction
modulo $p$ with $v_{p}(\Delta)=2k$, and the codimension-one EDS valuations
should be given by

$v(W^{(1)}_{m})=R_{m}(k+1,2k)=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{(k+1)(k-1)}{2k}m^{2}-\frac{\overline{m(k+1)}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m(k+1)}_{n}}\right)}{n}}\right)\,.$
(170)

Calculating the codimension-two EDS valuations requires using the procedure
for elliptic curves with additive reduction. The analysis is performed in
Section C.1.1, and the end result is

$v(W^{(2)}_{m})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}m^{2}-\begin{cases}0&2\mid
m\\\ \frac{n}{4}&2\nmid m\end{cases}}\right)\,.$ (171)

Combining the results, we find that

$v(W^{(2)}_{m})-v(W^{(1)}_{m})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{n}m^{2}+\frac{\overline{m(k+1)}_{n}\mathopen{}\mathclose{{}\left(n-\overline{m(k+1)}_{n}}\right)}{n}-\begin{cases}0&2\mid
m\\\ \frac{n}{4}&2\nmid m\end{cases}}\right)\,.$ (172)

If we use the relations

$q=\frac{2}{n}m\,,\quad\nu=\overline{(k+1)m}_{n}\,,\quad\mu=\overline{\mathopen{}\mathclose{{}\left(q-\frac{2\nu}{n}}\right)}_{2}+\overline{2\nu}_{4}=\mathopen{}\mathclose{{}\left(1+2\overline{(k+1)}_{2}}\right)\overline{m}_{2}\,,$
(173)

for this seed model, our result can be rewritten as

$\displaystyle v(W^{(2)}_{m})-v(W^{(1)}_{m})$
$\displaystyle=\frac{n}{4}q^{2}+\frac{\overline{\nu}_{n}\mathopen{}\mathclose{{}\left(n-\overline{\nu}_{n}}\right)}{n}-\begin{cases}0&\mu=0\\\
\frac{n}{4}&\mu=1\text{ or }3\end{cases}$ (174)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{d_{\operatorname{SU}(n)}}{d_{\operatorname{SO}(2n)}}q^{2}+T_{\operatorname{SU}(n)}(\nu)-T_{\operatorname{SO}(n)}(\nu)}\right).$

To summarize, we see that, regardless of which seed model for
$\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$ antisymmetric matter we use, the
expressions for $v(W^{(2)}_{m})-v(W^{(1)}_{m})$ calculated using the formulas
from StangeEllTrouble  agree with the proposals in Section 2.2. Moreover,
these formulas reproduce the orders of vanishing observed by calculating
multiples of the generating sections. This result gives us further confidence
in the proposals.

##  8 Representations of $\mathfrak{so}(2n)\oplus\mathfrak{u}(1)$

We now turn to models with $\mathfrak{so}(2n)\oplus\mathfrak{u}(1)$ gauge
algebras, focusing on matter charged in the vector and spinor representations
of $\mathfrak{so}(2n)$. Both these matter representations occur at
codimension-two loci where a codimension-one locus with $\text{I}^{*s}_{n-4}$
singular fibers intersects the residual $\text{I}_{1}$ discriminant locus. The
ideas and strategies used for the singlet and
$\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ situations carry over here, but the
analysis is more challenging due to the centers of the universal covering
groups. $\operatorname{Spin}(4k)$, the universal covering group of
$\mathfrak{so}(4k)$, has a $\mathbb{Z}_{4}$ center, but
$\operatorname{Spin}(4k+2)$, the universal covering group of
$\mathfrak{so}(4k+2)$, has a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ center. We
therefore analyze the $\mathfrak{so}(4k)$ and $\mathfrak{so}(4k+2)$
representations separately. The $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ center
poses the additional complication that it has more than one generating
element, as seen above when investigating the antisymmetric representation of
$\mathfrak{su}(n)$. We therefore need a larger number of seed models to
generate a representative sample of pseudo-charges.

###  8.1 Spinor representations $\mathfrak{so}(4k+2)$

We begin with matter charged in the spinor representations of
$\mathfrak{so}(4k+2)$, the simplest of the
$\mathfrak{so}(2n)\oplus\mathfrak{u}(1)$ situations considered here. As
discussed in Section 4.2, the two spinor representations of
$\mathfrak{so}(4k+2)$ are conjugate to each other, so a codimension-two locus
supporting $\mathfrak{so}(4k+2)$ spinor matter supports fields in both
representations. Here, we restrict our attention to the spinor representations
of $\mathfrak{so}(10)$ and $\mathfrak{so}(14)$, as attempts to realize spinor
matter in F-theory for $\mathfrak{so}(2n)$ algebras beyond $\mathfrak{so}(14)$
produce constructions with codimension-two loci where $f$ and $g$ vanish to
orders $(4,6)$. The spinors of $\mathfrak{so}(10)$ occur at codimension-two
loci where the singularity type enhances from type $\text{I}^{*s}_{1}$ to type
$\text{IV}^{*}$; in terms of the ADE groups, this is an enhancement from
$\operatorname{Spin}(10)$ to $\operatorname{E}_{6}$. The spinors of
$\mathfrak{so}(14)$ meanwhile occur at codimension-two loci with an
$\text{I}^{*s}_{3}\to\text{II}^{*}$ (or
$\operatorname{Spin}(14)\rightarrow\operatorname{E}_{8}$) enhancement.

Both $\operatorname{Spin}(10)$ and $\operatorname{Spin}(14)$ have
$\mathbb{Z}_{4}$ centers, and $\nu$ can take a value from $0$ to $3$. For
$\mathfrak{so}(10)$, the allowed $\mathfrak{u}(1)$ charges for the spinor with
highest weight $[0,0,0,0,1]$ are $\frac{4-\nu}{4}+\mathbb{Z}$. For
$\mathfrak{so}(14)$, the allowed $\mathfrak{u}(1)$ charges for the spinor with
highest weight $[0,0,0,0,0,0,1]$ are $\frac{\nu}{4}+\mathbb{Z}$. Suppose we
have a seed $\mathfrak{so}(10)\oplus\mathfrak{u}(1)$ or
$\mathfrak{so}(14)\oplus\mathfrak{u}(1)$ model with a codimension-two locus
supporting spinor matter with $\mathfrak{u}(1)$ charge $\pm\frac{1}{4}$. This
model would have either $\nu=1$ or $\nu=3$, and by the arguments in Section
5.1, we can freely choose our generating section such that we obtain either
value of $\nu$. Then, we can find multiples $m\hat{s}$ of the generating
section that would realize pseudo-charge $\pm\frac{m}{4}$ at the spinor locus
with either $\nu=\overline{m}_{4}$ (if the chosen generating section has
$\nu=1$) or $\nu=\overline{3m}_{4}$ (if the chosen generating section has
$\nu=3$). We can therefore generate all the allowed combinations of pseudo-
charge and $\nu$ from this seed model. Since the orders of vanishing of the
section components are insensitive to the sign of the pseudo-charge, the
orders of vanishing for pseudo-charge $-q$ are the same as those for pseudo-
charge $+q$. Based on the discussion in Section 5, we can use these data to
establish formulas relating the orders of vanishing to the $\mathfrak{u}(1)$
charge of the spinor matter.

##### $\mathfrak{so}(10)$

We consider the Weierstrass model given by

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{1}{3}\sigma^{2}\mathopen{}\mathclose{{}\left(b_{0,0}^{2}\mathopen{}\mathclose{{}\left(b_{2,1}^{2}+3\sigma
c_{0,3}}\right)+\sigma
b_{2,1}b_{0,0}c_{2,2}+\sigma\mathopen{}\mathclose{{}\left(\sigma
c_{2,2}^{2}-3c_{1,2}c_{3,1}}\right)}\right)xz^{4}$ (175)
$\displaystyle\qquad+\frac{\sigma^{3}}{108}\Big{[}\sigma\mathopen{}\mathclose{{}\left(27c_{3,1}^{2}\mathopen{}\mathclose{{}\left(b_{2,1}^{2}+4\sigma
c_{0,3}}\right)+8\sigma^{2}c_{2,2}^{3}-36\sigma c_{1,2}c_{3,1}c_{2,2}}\right)$
$\displaystyle\qquad\qquad\qquad-4b_{0,0}^{3}\mathopen{}\mathclose{{}\left(9\sigma
b_{2,1}c_{0,3}+2b_{2,1}^{3}}\right)+3\sigma
b_{0,0}^{2}\mathopen{}\mathclose{{}\left(9c_{1,2}^{2}-4c_{2,2}\mathopen{}\mathclose{{}\left(b_{2,1}^{2}+6\sigma
c_{0,3}}\right)}\right)$ $\displaystyle\qquad\qquad\qquad+6\sigma
b_{2,1}b_{0,0}\mathopen{}\mathclose{{}\left(2\sigma
c_{2,2}^{2}-3c_{1,2}c_{3,1}}\right)\Big{]}z^{6}\,.$

This model is equivalent to the $\mathcal{Q}(3,2,2,1,0,1,1)$ model from
KuntzlerTateTrees , which has an $\mathfrak{so}(10)\oplus\mathfrak{u}(1)$
gauge algebra. The $\mathfrak{so}(10)$ algebra is tuned along
$\\{\sigma=0\\}$, while the Weierstrass model admits a generating section
$\hat{s}$ with coordinates

$\displaystyle\hat{x}$ $\displaystyle=\sigma^{2}c_{3,1}^{2}-\frac{1}{3}\sigma
b_{0,0}^{2}\mathopen{}\mathclose{{}\left(b_{0,0}b_{2,1}+2\sigma
c_{2,2}}\right)\,,$ (176) $\displaystyle\hat{y}$
$\displaystyle=\frac{1}{2}\sigma^{2}\mathopen{}\mathclose{{}\left(b_{0,0}^{2}c_{3,1}\mathopen{}\mathclose{{}\left(b_{0,0}b_{2,1}+2\sigma
c_{2,2}}\right)-c_{1,2}b_{0,0}^{4}-2\sigma c_{3,1}^{3}}\right)\,,$
$\displaystyle\hat{z}$ $\displaystyle=b_{0,0}\,,$ $\displaystyle\hat{w}$
$\displaystyle=\sigma^{3}\Big{[}b_{0,0}^{4}\mathopen{}\mathclose{{}\left(\sigma
c_{2,2}^{2}+c_{1,2}c_{3,1}}\right)-4\sigma
b_{0,0}^{2}c_{2,2}c_{3,1}^{2}-c_{0,3}b_{0,0}^{6}$
$\displaystyle\qquad\quad+b_{2,1}b_{0,0}^{5}c_{2,2}-2b_{2,1}b_{0,0}^{3}c_{3,1}^{2}+3\sigma
c_{3,1}^{4}\Big{]}\,.$

As can be verified by comparison with the listed matter spectrum in
KuntzlerTateTrees , this model supports $\bm{16}_{-\frac{1}{4}}$ matter at
$\\{\sigma=b_{2,1}=0\\}$, $\bm{16}_{\frac{3}{4}}$ matter at
$\\{\sigma=b_{0,0}=0\\}$, and $\bm{10}_{\frac{1}{2}}$ matter at
$\\{\sigma=b_{0,0}c_{1,2}-b_{2,1}c_{3,1}=0\\}$. For this model, $\nu$ is $1$.
(Alternatively, we can flip the signs of the $\bm{16}$ charges and take $\nu$
to be $3$; since the $\bm{10}$ representation is real, the sign of its charge
is irrelevant as per the arguments in Section 5.1.) After determining the
spectrum of charged singlets and accounting for nonlocal adjoint matter
uncharged under the $\mathfrak{u}(1)$ algebra, one can verify that this matter
spectrum satisfies the 6D anomaly cancellation conditions with
$0pt=-2K_{B}+2[b_{0,0}]-\frac{5}{4}[\sigma]$.

We now calculate multiples of the generating section $\hat{s}$, examine their
behavior at the $\bm{16}_{-\frac{1}{4}}$ locus $\\{\sigma=b_{2,1}=0\\}$, and
find the orders of vanishing corresponding to various pseudo-charges. We can
also find the orders of vanishing at the $\bm{16}_{\frac{3}{4}}$ locus,
providing both an additional check on the data from the
$\bm{16}_{-\frac{1}{4}}$ locus and the ability to probe larger pseudo-charges
without calculating larger multiples of $\hat{s}$. The resulting data are
listed in Table 14(a). The observed codimension-one orders of vanishing
satisfy the equations

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(10)}(\nu)\,,$
(177)

And if we let

$\mu=\overline{4q}_{3}\,,$ (178)

where $q$ is the pseudo-charge, the codimension-two orders of vanishing
satisfy the equations

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{4}{3}q^{2}+T_{\operatorname{Spin}(10)}(\nu)-T_{\operatorname{E}_{6}}(\mu)}\right)\,.$
(179)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{E}_{6}}(\mu)\,.$
(180)

Since $\operatorname{Spin}(10)$ has a center with
$d_{\operatorname{Spin}(10)}=4$ elements and $\operatorname{E}_{6}$ has a
center with $d_{E_{6}}=3$ elements, the orders of vanishing satisfy the exact
patterns proposed in Section 2.2 for a
$\operatorname{Spin}(10)\to\operatorname{E}_{6}$ enhancement locus.

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(2,2,0,3)$
$\frac{1}{2}$ | $2$ | $(1,2,0,2)$ | $(2,2,0,3)$
$\frac{3}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(2,3,1,4)$
$1$ | $0$ | $(0,0,0,0)$ | $(2,2,0,3)$
$\frac{5}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(4,5,1,7)$
$\frac{3}{2}$ | $2$ | $(1,2,0,2)$ | $(4,6,2,8)$
$\frac{7}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(6,8,2,11)$
$2$ | $0$ | $(0,0,0,0)$ | $(6,8,2,11)$
$\frac{9}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(8,12,4,16)$
$\frac{5}{2}$ | $2$ | $(1,2,0,2)$ | $(10,14,4,19)$
$\frac{11}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(12,17,5,23)$
$3$ | $0$ | $(0,0,0,0)$ | $(12,18,6,24)$
$\frac{13}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(16,23,7,31)$
$\frac{7}{2}$ | $2$ | $(1,2,0,2)$ | $(18,26,8,35)$
$\frac{15}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(20,30,10,40)$
$\frac{9}{2}$ | $2$ | $(1,2,0,2)$ | $(28,42,14,56)$
$\frac{21}{4}$ | $1\text{ or }3$ | $(1,2,0,3)$ | $(38,57,19,76)$
$6$ | $0$ | $(0,0,0,0)$ | $(48,72,24,96)$
(a) $\bm{16}_{q}$ representation of $\mathfrak{so}(10)\oplus\mathfrak{u}(1)$

$\lvert q\rvert$ | $\nu$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---|---
$\frac{1}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(2,3,1,4)$
$\frac{1}{2}$ | $2$ | $(1,2,0,2)$ | $(2,3,1,4)$
$\frac{3}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(4,6,2,8)$
$1$ | $0$ | $(0,0,0,0)$ | $(4,6,2,8)$
$\frac{5}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(8,12,4,16)$
$\frac{3}{2}$ | $2$ | $(1,2,0,2)$ | $(10,15,5,20)$
$\frac{7}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(14,21,7,28)$
$2$ | $0$ | $(0,0,0,0)$ | $(16,24,8,32)$
$\frac{9}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(22,33,11,44)$
$\frac{5}{2}$ | $2$ | $(1,2,0,2)$ | $(26,39,13,52)$
$\frac{11}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(32,48,16,64)$
$3$ | $0$ | $(0,0,0,0)$ | $(36,54,18,72)$
$\frac{13}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(44,66,22,88)$
$\frac{7}{2}$ | $2$ | $(1,2,0,2)$ | $(50,75,25,100)$
$\frac{15}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(58,87,29,116)$
$4$ | $0$ | $(0,0,0,0)$ | $(64,96,32,128)$
$\frac{17}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(74,111,37,148)$
$\frac{9}{2}$ | $2$ | $(1,2,0,2)$ | $(82,123,41,164)$
$\frac{19}{4}$ | $1\text{ or }3$ | $(1,3,0,4)$ | $(92,138,46,184)$
(b) $\bm{64}_{q}\oplus\bm{14}_{2q}$ representation of
$\mathfrak{so}(14)\oplus\mathfrak{u}(1)$

Table 14: Orders of vanishing for various combinations of $\nu$ and pseudo-
charge $q$ and for spinor matter in $\mathfrak{so}(4k+2)\oplus\mathfrak{u}(1)$
models. The $\mathfrak{so}(10)\oplus\mathfrak{u}(1)$ data are found using the
model in Eq. 175, while the $\mathfrak{so}(14)\oplus\mathfrak{u}(1)$ data are
found using the model in Eq. 183.

##### $\mathfrak{so}(14)\oplus\mathfrak{u}(1)$

Matter charged in the spinor ($\bm{64}$) representation of $\mathfrak{so}(14)$
occurs at loci where the singularity type enhances from $\text{I}_{3}^{*s}$
(or $\operatorname{Spin}(14)$) to $\text{II}^{*}$ (or $\operatorname{E}_{8}$).
These codimension-two loci also support matter in the vector $(\bm{14})$
representation. In a 6D model, for instance, an irreducible codimension-two
$\text{I}_{3}^{*s}\to\text{II}^{*}$ locus supports a hypermultiplet of
$\bm{64}$ matter and a hypermultiplet of $\bm{14}$ matter. We should therefore
determine how the $\mathfrak{u}(1)$ charge of the $\bm{64}$ matter compares
with that of the $\bm{14}$ matter at the same locus. The $\bm{14}$
representation has a weight $[0,0,0,0,0,-1,1]$, and the $\bm{64}$
representation has weights $[0,0,0,0,0,0,1]$ and $[0,0,0,0,0,-1,0]$.313131We
follow the notation of Slansky  where the $\bm{64}$ representation has highest
weight $[0,0,0,0,0,0,1]$. Other sources may use alternative conventions. If
the curves in the $\operatorname{E}_{8}$ fiber supporting these two $\bm{64}$
weights are $c_{+}$ and $c_{-}$, then one can verify that $c_{+}+c_{-}$ is one
of the two curves in the fiber supporting the $\bm{14}$ weight.323232Because
the $\operatorname{E}_{8}$ adjoint representation decomposes as
$\bm{91}\oplus\bm{1}\oplus\bm{64}\oplus\overline{\bm{64}}\oplus\bm{14}\oplus\bm{14}$,
there are two curves corresponding to the $[0,0,0,0,0,-1,1]$ weight of the
$\bm{14}$ representation. We only need one of them for the purposes of this
argument. All the weights in the $\bm{64}$ representation should have the same
$\mathfrak{u}(1)$ charge $q$, implying that, for a section $\hat{s}$,

$\sigma(\hat{s})\cdot c_{+}=\sigma(\hat{s})\cdot c_{-}=q\,.$ (181)

But the curve $c_{+}+c_{-}$ should then satisfy

$\sigma(\hat{s})\cdot\mathopen{}\mathclose{{}\left(c_{+}+c_{-}}\right)=2q\,.$
(182)

The charge of the $[0,0,0,0,0,-1,1]$ weight should be the same as the charge
of the other weights in the same $\bm{14}$ representation. Therefore, the
charge of the $\bm{14}$ matter should be twice that of the $\bm{64}$ matter.
In other words, the codimension-two locus should support matter fields in the
$\bm{64}_{q}$ and $\bm{14}_{2q}$ representations, which are accompanied by
fields in the $\overline{\bm{64}}_{-q}$ and $\bm{14}_{-2q}$ representations.
In a 6D model, an irreducible $\text{I}_{3}^{*s}\to\text{II}^{*}$ locus would
support a $\bm{64}_{q}$ hypermultiplet and and a $\bm{14}_{2q}$
hypermultiplet. When we say that an $\text{I}_{3}^{*s}\to\text{II}^{*}$ locus
supports matter with $\mathfrak{u}(1)$ charge $q$, we implicitly take $q$ to
be the charge of the spinor matter; the charge of the vector matter can
automatically be determined from this information.

We now consider the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{\sigma^{2}}{3}\Big{[}4\beta_{2,1}^{2}b_{0,0}^{8}+8\sigma\beta_{2,1}b_{0,0}^{4}c_{2,2}$
(183) $\displaystyle\qquad\qquad\qquad+3\sigma
b_{0,0}^{2}\mathopen{}\mathclose{{}\left(\sigma
c_{0,4}-4\beta_{2,1}c_{3,1}^{2}}\right)+\sigma^{2}\mathopen{}\mathclose{{}\left(c_{2,2}^{2}-3c_{1,3}c_{3,1}}\right)\Big{]}xz^{4}$
$\displaystyle\qquad+\frac{\sigma^{3}}{108}\Big{[}-64\beta_{2,1}^{3}b_{0,0}^{12}-24\sigma^{2}\beta_{2,1}\Big{(}3b_{0,0}^{6}c_{0,4}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+b_{0,0}^{4}\mathopen{}\mathclose{{}\left(5c_{2,2}^{2}-3c_{1,3}c_{3,1}}\right)-12b_{0,0}^{2}c_{2,2}c_{3,1}^{2}+9c_{3,1}^{4}\Big{)}$
$\displaystyle\qquad\qquad\qquad+96\sigma\beta_{2,1}^{2}b_{0,0}^{6}\mathopen{}\mathclose{{}\left(3c_{3,1}^{2}-2b_{0,0}^{2}c_{2,2}}\right)+\sigma^{3}\Big{(}9b_{0,0}^{2}\mathopen{}\mathclose{{}\left(3c_{1,3}^{2}-8c_{0,4}c_{2,2}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+4\mathopen{}\mathclose{{}\left(2c_{2,2}^{3}-9c_{1,3}c_{3,1}c_{2,2}+27c_{0,4}c_{3,1}^{2}}\right)\Big{)}\Big{]}z^{6}\,.$

The $(f,g,\Delta)$ of this model vanish to orders $(2,3,9)$ at
$\\{\sigma=0\\}$, and the split condition for $\text{I}_{3}^{*}$ KatzEtAlTate
; GrassiMorrison ; MorrisonTaylorClusters  singularities is satisfied. The
model therefore supports an $\mathfrak{so}(14)$ gauge algebra along
$\\{\sigma=0\\}$. It also admits a generating section $\hat{s}$ with
coordinates

$\displaystyle\hat{x}$
$\displaystyle=-\frac{1}{3}\sigma\mathopen{}\mathclose{{}\left(2\beta_{2,1}b_{0,0}^{6}+2\sigma
b_{0,0}^{2}c_{2,2}-3\sigma c_{3,1}^{2}}\right)\,,$ (184)
$\displaystyle\hat{y}$
$\displaystyle=-\frac{1}{2}\sigma^{3}\mathopen{}\mathclose{{}\left(b_{0,0}^{4}c_{1,3}-2b_{0,0}^{2}c_{2,2}c_{3,1}+2c_{3,1}^{3}}\right)\,,$
(185) $\displaystyle\hat{z}$ $\displaystyle=b_{0,0}\,,$ (186)
$\displaystyle\hat{w}$
$\displaystyle=\sigma^{4}\mathopen{}\mathclose{{}\left(-b_{0,0}^{6}c_{0,4}+b_{0,0}^{4}\mathopen{}\mathclose{{}\left(c_{2,2}^{2}+c_{1,3}c_{3,1}}\right)-4b_{0,0}^{2}c_{2,2}c_{3,1}^{2}+3c_{3,1}^{4}}\right)\,,$
(187)

indicating that the full gauge algebra is
$\mathfrak{so}(14)\oplus\mathfrak{u}(1)$. At $\\{\sigma=b_{0,0}=0\\}$, the
singularity type enhances from $\text{I}_{3}^{*s}$ to $\text{II}^{*}$ (or from
$\operatorname{Spin}(14)$ to $\operatorname{E}_{8}$). This codimension-two
locus supports $\bm{64}_{\frac{1}{4}}\oplus\bm{14}_{\frac{1}{2}}$ matter,
along with the appropriate conjugate matter. At
$\\{\sigma=b_{0,0}^{4}c_{1,3}-2b_{0,0}^{2}c_{2,2}c_{3,1}+2c_{3,1}^{3}=0\\}$,
the singularity type enhances to $\text{I}_{4}^{*}$ (or
$\operatorname{Spin}(16)$), and the locus supports $\bm{14}_{\frac{1}{2}}$
matter. Based on these $\mathfrak{u}(1)$ charges, $\nu$ is $1$.
(Alternatively, we could say that the model has $\nu=3$ with
$\bm{64}_{-\frac{1}{4}}\oplus\bm{14}_{\frac{1}{2}}$ matter at
$\\{\sigma=b_{0,0}=0\\}$. Because the $\bm{14}$ representation is real, the
sign of its $\mathfrak{u}(1)$ charge is not important as per the arguments in
Section 5.1.) While $(f,g,\Delta)$ vanish to orders $(4,6,12)$ at
$\\{\sigma=\beta_{2,1}=0\\}$, the presence of such loci can be avoided by an
appropriate choice of divisor classes. For instance, if we take
$[\beta_{2,1}]$ to be trivial, there are no codimension-two $(4,6,12)$ loci.
After calculating the spectrum of charged singlets and $\mathfrak{so}(14)$
adjoints uncharged under the $\mathfrak{u}(1)$ algebra, one can show that this
spectrum satisfies the 6D anomaly cancellation conditions, at least for
situations without codimension-two $(4,6,12)$ loci.

As before, we can calculate multiples of $\hat{s}$ and find the orders of
vanishing of the section components at $\\{\sigma=b_{0,0}=0\\}$. These numbers
are listed in Table 14(b) along with the corresponding values of $\nu$ and of
the spinor pseudo-charge $q$. The codimension-one orders of vanishing satisfy
the equations

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(14)}(\nu)\,.$
(188)

The codimension-two orders of vanishing, meanwhile, satisfy the equations

$\displaystyle\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(4q^{2}+T_{\operatorname{Spin}(14)}(\nu)}\right)\,,$
(189)
$\displaystyle\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})\,.$
(190)

The center for $\operatorname{E}_{8}$, the group associated with the
codimension-two singularity type, is the trivial group. As such, the only
allowed value of $\mu$ is 0, and $T_{\operatorname{E}_{8}}(\mu)$ and
$\vec{\tau}_{\operatorname{E}_{8}}(\mu)$ are both trivial. Keeping this in
mind, we see that the observed orders of vanishing for the $\mathfrak{so}(14)$
spinors agree with the proposals in Section 2.2.

###  8.2 Spinor representations of $\mathfrak{so}(4k)$

We now turn to the spinor representations of $\mathfrak{so}(4k)$. We restrict
our attention to the spinors of $\mathfrak{so}(8)$ and $\mathfrak{so}(12)$, as
attempts to obtain spinors for larger values of $k$ produce models with
codimension-two loci where $(f,g)$ vanish to orders $(4,6)$. Both of these
algebras have two spinor representations. For $\mathfrak{so}(8)$, the two
spinor representations are $\bm{8_{\text{s}}}$, with highest weight
$[0,0,0,1]$, and $\bm{8}_{c}$, with highest weight $[0,0,1,0]$. Since these
two representations are both real, they occur at different codimension-two
loci in the F-theory model. For $\mathfrak{so}(12)$, the two spinor
representations are $\bm{32}$, with highest weight $[0,0,0,0,0,1]$, and
$\bm{32^{\prime}}$, with highest weight $[0,0,0,0,1,0]$.333333While this
convention agrees with YamatsuGroupTheory , the highest weights of the
$\bm{32}$ and $\bm{32^{\prime}}$ representation are switched in the
conventions of Slansky . They are pseudoreal representations that occur at
distinct codimension-two loci. While the $\bm{32}$ and $\bm{32^{\prime}}$
representations can in principle come in half-hypermultiplets in 6D models, we
are most interested in situations where matter in these representations also
has a nonzero charge under an $\mathfrak{u}(1)$ algebra. The total
representation is no longer pseudoreal after including this charge, so
$\mathfrak{so}(12)$ spinor matter with a nonzero $\mathfrak{u}(1)$ charge must
occur in full hypermultiplets. Because both the $\mathfrak{so}(8)$ and
$\mathfrak{so}(12)$ spinor representations are self-conjugate, the sign of the
$\mathfrak{u}(1)$ charge is unimportant for these cases.

The universal covering groups $\operatorname{Spin}(8)$ and
$\operatorname{Spin}(12)$ both have $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
centers. As discussed in Section 4, we label the four elements of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ with an integer $\nu$ ranging from 0 to
3. The allowed $\mathfrak{u}(1)$ charges are integral for all the spinor
representations when $\nu=0$ and are half-integral for all the spinor
representations when $\nu=2$. When $\nu=1$, the allowed charges for
$\bm{8_{\text{s}}}$ and $\bm{32^{\prime}}$ representations are half-integral,
while those for the $\bm{8_{\text{c}}}$ and $\bm{32}$ representations are
integral. When $\nu=3$, the roles are reversed: allowed charges for
$\bm{8_{\text{c}}}$ and $\bm{32}$ representations are half-integral, while
those for the $\bm{8_{\text{s}}}$ and $\bm{32}^{\prime}$ representations are
integral. Note that the $\mathfrak{so}(2n)$ fibers admit an automorphism that
exchanges the two nodes at the end of the Dynkin diagram; these are the two
nodes on the right hand side in Fig. 3. This automorphism exchanges the two
spinor representations for a given algebra. In turn, the automorphism sends
$\nu=1$ to $\nu=3$, and vice versa. The elements $\nu=2$ and $\nu=0$ are sent
to themselves under the automorphism. In F-theory models, there is an
ambiguity in the labeling of the curves exchanged under the automorphism, so
we can perform the automorphism without changing the underlying geometry. As a
result, we can often determine geometric information about one spinor
representation from information about the other spinor representation. For
instance, if we knew the orders of vanishing for $\bm{32}$ matter with some
$\mathfrak{u}(1)$ charge when $\nu=1$, we automatically know the orders of
vanishing for $\bm{32}^{\prime}$ matter with the same charge when $\nu=3$.

With this in mind, we can now describe how to generate various combinations of
the pseudo-charge and $\nu$. Starting with a $\nu=1$ model with
${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ or $\bm{32^{\prime}}_{\frac{1}{2}}$
matter, we can obtain all the half-integral $\nu=1$ pseudo-charges and all the
integral $\nu=0$ pseudo-charges. If we start with a $\nu=2$ model with
${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ or $\bm{32^{\prime}}_{\frac{1}{2}}$
matter, we can obtain all the half-integral $\nu=2$ pseudo-charges while
reproducing all the integral $\nu=0$ pseudo-charges. Obtaining the integral
$\nu=3$ pseudo-charges is more difficult. If we have a $\nu=3$ model with
${\bm{8_{\text{s}}}}_{,1}$ or $\bm{32^{\prime}}_{1}$ matter, we can generate
the odd $\nu=3$ pseudo-charges but not the even ones: the even multiples of
the generating section, which would realize the even pseudo-charges, would
have $\nu=0$ according to the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ group
structure. We could generate some of the even $\nu=3$ pseudo-charges from a
model with ${\bm{8_{\text{s}}}}_{,2}$ or $\bm{32^{\prime}}_{2}$ matter, but we
would miss those pseudo-charges that are multiples of 4. Even if we attempt to
fill in these gaps by starting with charges that are larger and larger powers
of $2$, we always miss some of the even $\nu=3$ pseudo-charges. We are
therefore unable to practically obtain order-of-vanishing data for all of the
even $\nu=3$ pseudo-charges in a particular range. Nevertheless, we find that
the order-of-vanishing data we are able to obtain follow the expected formulas
in Section 2.2, making it highly plausible that the missing pseudo-charges
follow these same patterns. Of course, once we have data for the
$\bm{8_{\text{s}}}$ and $\bm{32^{\prime}}$ representations, we can use the
automorphism to automatically obtain data for the $\bm{8_{\text{c}}}$ and
$\bm{32}$ representations. This allows us to cut down on the number of seed
models used. In fact, the triality of $\mathfrak{so}(8)$ models, discussed in
more detail below, implies that we can read off orders of vanishing for the
vector representation from the spinor data, and vice versa. As a result, we
can obtain a large portion of the $\mathfrak{so}(8)$ data from just a single
model, and the discussion of $\mathfrak{so}(8)$ spinors will also cover the
$\bm{8_{\text{v}}}$ representation.

##### $\mathfrak{so}(8)\oplus\mathfrak{u}(1)$

The two spinor representations of $\mathfrak{so}(8)$ and the vector
representation $\bm{8_{\text{v}}}$ all occur at codimension-two loci where the
singularity type enhances from $\text{I}_{0}^{*s}$ ($\operatorname{Spin}(8)$)
to $\text{I}_{1}^{*}$ ($\operatorname{Spin}(10)$), a fact that reflects the
triality of $\mathfrak{so}(8)$. As can be seen from the symmetries of its
Dynkin diagram, the $\mathfrak{so}(8)$ algebra admits automorphisms that
exchange the $\bm{8_{\text{s}}}$, $\bm{8_{\text{c}}}$, and the
$\bm{8_{\text{v}}}$ representations. These automorphisms also exchange the
$\nu=1$, $\nu=2$, and $\nu=3$ elements of the center in a way consistent with
the allowed $\mathfrak{u}(1)$ charges in Table 6. For example,
${\bm{8_{\text{c}}}}_{,1}$ matter (with $\mathfrak{u}(1)$ charge $q=1$) in a
$\nu=1$ model can alternatively be viewed as ${\bm{8_{\text{v}}}}_{,1}$ matter
in a $\nu=2$ model or as ${\bm{8_{\text{s}}}}_{,1}$ matter in a $\nu=3$ model.

Let us consider the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\frac{1}{3}\sigma^{2}\mathopen{}\mathclose{{}\left(-3b_{0,0}^{2}b_{2,1}^{2}+3\sigma
c_{1,2}c_{3,1}-c_{2,1}^{2}}\right)xz^{4}$ (191)
$\displaystyle\qquad\quad+\frac{1}{108}\sigma^{3}\Big{(}27\sigma
b_{0,0}^{2}c_{1,2}^{2}+108\sigma
b_{2,1}^{2}c_{3,1}^{2}-72b_{0,0}^{2}b_{2,1}^{2}c_{2,1}$
$\displaystyle\qquad\qquad\qquad\qquad\quad-36\sigma
c_{1,2}c_{3,1}c_{2,1}+8c_{2,1}^{3}\Big{)}z^{6}\,.$

The elliptic fibration has $\text{I}_{0}^{*s}$ singularities along
$\\{\sigma=0\\}$, signaling the presence of an $\mathfrak{so}(8)$ algebra. It
also admits a generating section $\hat{s}$ with coordinates

$[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[\frac{1}{3}\sigma\mathopen{}\mathclose{{}\left(3\sigma
c_{3,1}^{2}-2b_{0,0}^{2}c_{2,1}}\right):-\frac{1}{2}\sigma^{2}\mathopen{}\mathclose{{}\left(b_{0,0}^{4}c_{1,2}-2b_{0,0}^{2}c_{2,1}c_{3,1}+2\sigma
c_{3,1}^{3}}\right):b_{0,0}}\right]$ (192)

and

$\hat{w}=\sigma^{2}\mathopen{}\mathclose{{}\left(b_{0,0}^{4}\mathopen{}\mathclose{{}\left(\sigma
c_{1,2}c_{3,1}+c_{2,1}^{2}}\right)-4\sigma
b_{0,0}^{2}c_{2,1}c_{3,1}^{2}-b_{2,1}^{2}b_{0,0}^{6}+3\sigma^{2}c_{3,1}^{4}}\right)\,.$
(193)

Thus the total gauge algebra is $\mathfrak{so}(8)\oplus\mathfrak{u}(1)$. There
are four codimension-two loci where the singularity type enhances to
$\text{I}_{1}^{*}$: $\\{\sigma=b_{0,0}=0\\}$, $\\{\sigma=b_{2,1}=0\\}$,
$\\{\sigma=c_{2,1}-b_{0,0}b_{2,1}=0\\}$, and
$\\{\sigma=c_{2,1}+b_{0,0}b_{2,1}=0\\}$. The locus $\\{\sigma=b_{0,0}=0\\}$
supports matter with $\mathfrak{u}(1)$ charge $q=1$, the locus
$\\{\sigma=b_{2,1}=0\\}$ supports matter with $\mathfrak{u}(1)$ charge $q=0$,
and the remaining two loci support matter with $\mathfrak{u}(1)$ charge
$q=\frac{1}{2}$. Because of the triality of $\mathfrak{so}(8)$, there are
multiple ways of describing the $\mathfrak{so}(8)$ representations supported
at these loci, which are summarized in Table 15. Regardless of which
description of the matter representations is used, the spectrum satisfies the
6D anomaly cancellation conditions with $0pt=-2K_{B}+2[b_{0,0}]-[\sigma]$.

Locus | $\nu$
---|---
| $1$ | $1$ | $2$ | $2$ | $3$ | $3$
$\\{\sigma=b_{0,0}=0\\}$ | ${\bm{8_{\text{c}}}}_{,1}$ | ${\bm{8_{\text{c}}}}_{,1}$ | ${\bm{8_{\text{v}}}}_{,1}$ | ${\bm{8_{\text{v}}}}_{,1}$ | ${\bm{8_{\text{s}}}}_{,1}$ | ${\bm{8_{\text{s}}}}_{,1}$
$\\{\sigma=b_{2,1}=0\\}$ | ${\bm{8_{\text{c}}}}_{,0}$ | ${\bm{8_{\text{c}}}}_{,0}$ | ${\bm{8_{\text{v}}}}_{,0}$ | ${\bm{8_{\text{v}}}}_{,0}$ | ${\bm{8_{\text{s}}}}_{,0}$ | ${\bm{8_{\text{s}}}}_{,0}$
$\\{\sigma=c_{2,1}-b_{0,0}b_{2,1}=0\\}$ | ${\bm{8_{\text{v}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{c}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{c}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{v}}}}_{,\frac{1}{2}}$
$\\{\sigma=c_{2,1}+b_{0,0}b_{2,1}=0\\}$ | ${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{v}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{c}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{s}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{v}}}}_{,\frac{1}{2}}$ | ${\bm{8_{\text{c}}}}_{,\frac{1}{2}}$
Table 15: Matter representations supported at the four codimension-two
$\text{I}_{0}^{*s}\rightarrow\text{I}_{1}^{*}$ loci in the elliptic fibration
of Eq. 191. Because of the triality of $\mathfrak{so}(8)$, there are multiple
ways of describing the supported matter representations. Each column
corresponds to one choice of matter representations and $\nu$.

To generate the order-of-vanishing data, we calculate multiples of $\hat{s}$
and find the orders of vanishing of the section components at the various
loci. The results are listed in Table 16. Note that these results do not
include orders of vanishing for some combinations of $\nu$ and pseudo-charge.
For instance, we have only given orders of vanishing for the odd integer
pseudo-charges when $\nu=1,2,3$ in light of the difficulties in generating the
even integer pseudo-charges described above. We have also included only the
even integer pseudo-charges for $\nu=0$ when $\lvert q\rvert>8$; this simply
reflects the data we chose to generate, and there is no fundamental difficulty
in obtaining the odd pseudo-charges for $\nu=0$. Despite these missing pseudo-
charge values, one can see that the observed data follow the patterns proposed
in Section 2.2. The codimension-one orders of vanishing follow the pattern

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(8)}(\nu)\,,$
(194)

as expected, and if we let

$\mu=\begin{cases}\overline{(2q-2\nu)}_{4}&\text{for }\bm{8_{\text{s}}}\text{
or }\bm{8_{\text{c}}}\\\ \overline{(2q-\nu)}_{4}+\overline{2q}_{2}&\text{for
}\bm{8_{\text{v}}}\end{cases}\,,$ (195)

the codimension-two orders of vanishing satisfy the expressions

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{Spin}(8)}(\nu)-T_{\operatorname{Spin}(10)}(\mu)}\right)$
(196)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{Spin}(10)}(\mu)\,.$
(197)

Since $d_{\operatorname{Spin}(8)}=d_{\operatorname{Spin}(10)}=4$, the
codimension-two formulas agree with the expectations from Section 2.2.
Although we did not find orders of vanishing for the even pseudo-charges when
$\nu\neq 0$, Eq. 196 is guaranteed to give us integer
$\operatorname{ord}_{2}(\hat{z})$ for these cases. The expressions in Eq. 195
give $\mu=2$ whenever $q$ is even and $\nu$ takes the appropriate nonzero
value, and plugging this information into Eq. 196 always results in an
integer. These facts suggest that the expressions in Section 2.2 correctly
relate the $\mathfrak{u}(1)$ charge of $\mathfrak{so}(8)$ spinor or vector
matter to the orders of vanishing of the section components.

$\lvert q\rvert$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---
$1$ | $(1,2,0,2)$ | $(2,3,1,4)$
$3$ | $(1,2,0,2)$ | $(10,15,5,20)$
$5$ | $(1,2,0,2)$ | $(26,39,13,52)$
$7$ | $(1,2,0,2)$ | $(50,75,25,100)$
$9$ | $(1,2,0,2)$ | $(82,123,41,164)$
$11$ | $(1,2,0,2)$ | $(122,183,61,244)$
$13$ | $(1,2,0,2)$ | $(170,255,85,340)$
$15$ | $(1,2,0,2)$ | $(226,339,113,452)$
$17$ | $(1,2,0,2)$ | $(290,435,145,580)$
(a) $\nu=1$ for $\bm{8_{\text{c}}}$, $\nu=2$ for $\bm{8_{\text{v}}}$, $\nu=3$
for $\bm{8_{\text{s}}}$

$\lvert q\rvert$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---
$\frac{1}{2}$ | $(1,2,0,2)$ | $(1,2,0,3)$
$\frac{3}{2}$ | $(1,2,0,2)$ | $(3,5,1,7)$
$\frac{5}{2}$ | $(1,2,0,2)$ | $(7,11,3,15)$
$\frac{7}{2}$ | $(1,2,0,2)$ | $(13,20,6,27)$
$\frac{9}{2}$ | $(1,2,0,2)$ | $(21,32,10,43)$
$\frac{11}{2}$ | $(1,2,0,2)$ | $(31,47,15,63)$
$\frac{13}{2}$ | $(1,2,0,2)$ | $(43,65,21,87)$
$\frac{15}{2}$ | $(1,2,0,2)$ | $(57,86,28,115)$
$\frac{17}{2}$ | $(1,2,0,2)$ | $(73,110,36,147)$
(b) $\nu=2,3$ for $\bm{8_{\text{c}}}$, $\nu=1,3$ for $\bm{8_{\text{v}}}$,
$\nu=1,2$ for $\bm{8_{\text{s}}}$

$\lvert q\rvert$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---
$1$ | $(0,0,0,0)$ | $(1,2,0,2)$
$2$ | $(0,0,0,0)$ | $(4,6,2,8)$
$3$ | $(0,0,0,0)$ | $(9,14,4,18)$
$4$ | $(0,0,0,0)$ | $(16,24,8,32)$
$5$ | $(0,0,0,0)$ | $(25,38,12,50)$
$6$ | $(0,0,0,0)$ | $(36,54,18,72)$
$7$ | $(0,0,0,0)$ | $(49,74,24,98)$
$8$ | $(0,0,0,0)$ | $(64,96,32,128)$
$10$ | $(0,0,0,0)$ | $(100,150,50,200)$
$12$ | $(0,0,0,0)$ | $(144,216,72,288)$
$14$ | $(0,0,0,0)$ | $(196,294,98,392)$
$16$ | $(0,0,0,0)$ | $(256,384,128,512)$
(c) $\nu=0$ for $\bm{8_{\text{s}}}$, $\bm{8_{\text{c}}}$, and
$\bm{8_{\text{v}}}$

Table 16: Orders of vanishing of the section components for various
combinations pseudo-charges $q$ and $\nu$ for $\bm{8_{\text{s}}}$,
$\bm{8_{\text{c}}}$, and $\bm{8_{\text{v}}}$ matter in
$\mathfrak{so}(8)\oplus\mathfrak{u}(1)$ models. The data are obtained using
the $\mathfrak{so}(8)\oplus\mathfrak{u}(1)$ model in Eq. 191.

##### $\mathfrak{so}(12)\oplus\mathfrak{u}(1)$

The two spinor representations of $\mathfrak{so}(12)$ both occur at
codimension-two loci where the singularity type enhances from
$\text{I}_{2}^{*s}$ ($\operatorname{Spin}(12)$) to $\text{III}^{*}$
($\operatorname{E}_{7}$). After resolution, the codimension-two singularities
can either be incompletely or completely resolved MorrisonTaylorMaS ;
KanHalfHyper . In 6D models, an incomplete resolution locus supports a half-
hypermultiplet of spinor matter. A complete resolution locus supports a full
hypermultiplet of spinor matter and an additional singlet, as can be seen from
the decomposition of the $\operatorname{E}_{7}$ adjoint representation:

$\displaystyle\operatorname{E}_{7}$
$\displaystyle\to\operatorname{SO}(12)\times\operatorname{SU}(2)\,,$ (198)
$\displaystyle\bm{133}$
$\displaystyle\to(\bm{66},\bm{1})\oplus(\bm{32},\bm{2})\oplus(\bm{1},\bm{3})\,.$

The $(\bm{32},\bm{2})$ degrees of freedom combine to form a $\bm{32}$
hypermultiplet, while the positive and negative roots of the $(\bm{1},\bm{3})$
degrees of freedom form the extra singlet hypermultiplet. By the automorphism
of $\mathfrak{so}(12)$, there is a similar decomposition for the
$\bm{32}^{\prime}$, and this spinor is also accompanied by a singlet at
complete resolution loci. Since spinor matter with $\mathfrak{u}(1)$ charges
must come in full hypermultiplets, we are primarily interested in the complete
resolution loci. However, the spinor and singlet matter at these loci can both
be charged under the $\mathfrak{u}(1)$ algebra, raising the question of how
the singlet charge compares to the spinor charge. The charge of the singlet
matter is in fact twice that of the spinor matter.343434Recall that, because
the singlet representation is real, the sign of the singlet matter is
unimportant. Because the spinor matter occurs in full hypermultiplets, the
resolved $\operatorname{E}_{7}$ fiber contains two curves, $\gamma_{+}$ and
$\gamma_{-}$, representing the two highest weights of the spinor
representation in $(\bm{32},\bm{2})$ (or $(\bm{32^{\prime}},\bm{2})$). These
two curves would belong to the two different half-hypermultiplets that combine
to form the full hypermultiplet of spinor matter. The half-hypermultiplets
should have opposite $\mathfrak{u}(1)$ charges, implying that, for a
generating section $\hat{s}$,

$\sigma(\hat{s})\cdot\gamma_{+}=-\sigma(\hat{s})\cdot\gamma_{-}=q\,.$ (199)

But $\gamma_{+}$ and $\gamma_{-}$ also serve as the two weights of the
fundamental of $\mathfrak{su}(2)$. The difference $\gamma_{+}-\gamma_{-}$
between these two curves is the root in the $\mathfrak{su}(2)$ adjoint, and it
corresponds to the curve supporting the singlet. The charge of the singlet
matter should be given by

$\sigma(\hat{s})\cdot(\gamma_{+}-\gamma_{-})=2q\,.$ (200)

The $\mathfrak{u}(1)$ charge of the singlet matter is therefore twice that of
the spinor matter, as claimed above. While we gave this argument for the
$\bm{32}$ spinor, the same argument carries through for the $\bm{32^{\prime}}$
spinor without significant modifications.

The $\mathfrak{so}(12)$ algebra does not admit the triality of the
$\mathfrak{so}(8)$ algebra. We would therefore need multiple Weierstrass
models to obtain a representative set of $q$ and $\nu$ combinations, but we
have not been able to obtain models that support $\bm{32^{\prime}}_{1}$ matter
with $\nu=3$ or $\bm{32^{\prime}}_{\frac{1}{2}}$ matter with $\nu=2$ (or the
$\bm{32}$ matter related to this $\bm{32^{\prime}}$ matter by the
automorphism). Whenever we attempt to construct such models, $(f,g,\Delta)$
end up vanishing to orders $(4,6,12)$ at the would-be spinor locus. However,
there is the $\mathfrak{so}(12)\oplus\mathfrak{u}(1)$ Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{1}{3}\sigma^{2}\mathopen{}\mathclose{{}\left[3\sigma
b_{0,0}^{2}c_{0,3}+\sigma\mathopen{}\mathclose{{}\left(\sigma
c_{2,2}^{2}-3c_{3,1}\mathopen{}\mathclose{{}\left(b_{2,1}c_{3,1}+\sigma
c_{1,3}}\right)}\right)+b_{2,1}^{2}b_{0,0}^{4}}\right]xz^{4}$ (201)
$\displaystyle\qquad\quad+\frac{\sigma^{3}}{108}\Big{[}9\sigma
b_{0,0}^{2}\mathopen{}\mathclose{{}\left(4\sigma
b_{2,1}\mathopen{}\mathclose{{}\left(c_{2,2}^{2}+c_{1,3}c_{3,1}}\right)+4b_{2,1}^{2}c_{3,1}^{2}+3\sigma^{2}c_{1,3}^{2}-8\sigma
c_{0,3}c_{2,2}}\right)$
$\displaystyle\qquad\qquad\qquad\quad+4\sigma^{2}\mathopen{}\mathclose{{}\left(-9c_{3,1}c_{2,2}\mathopen{}\mathclose{{}\left(2b_{2,1}c_{3,1}+\sigma
c_{1,3}}\right)+2\sigma c_{2,2}^{3}+27c_{0,3}c_{3,1}^{2}}\right)$
$\displaystyle\qquad\qquad\qquad\quad-36\sigma
b_{2,1}b_{0,0}^{4}c_{0,3}-8b_{2,1}^{3}b_{0,0}^{6}\Big{]}z^{6}\,,$

which has $\text{I}_{2}^{*s}$ fibers along $\\{\sigma=0\\}$ and a generating
section $\hat{s}$ with components

$\displaystyle\hat{x}$
$\displaystyle=-\frac{1}{3}\sigma\mathopen{}\mathclose{{}\left(2\sigma
b_{0,0}^{2}c_{2,2}+b_{2,1}b_{0,0}^{4}-3\sigma c_{3,1}^{2}}\right)\,,$ (202)
$\displaystyle\hat{y}$
$\displaystyle=-\frac{1}{2}\sigma^{3}\mathopen{}\mathclose{{}\left(b_{0,0}^{4}c_{1,3}-2b_{0,0}^{2}c_{2,2}c_{3,1}+2c_{3,1}^{3}}\right)\,,$
$\displaystyle\hat{z}$ $\displaystyle=b_{0,0}\,,$ $\displaystyle\hat{w}$
$\displaystyle=\frac{\sigma^{3}}{3}\Big{[}3b_{0,0}^{4}\mathopen{}\mathclose{{}\left(c_{3,1}\mathopen{}\mathclose{{}\left(\sigma
c_{1,3}-b_{2,1}c_{3,1}}\right)+\sigma c_{2,2}^{2}}\right)$
$\displaystyle\qquad\qquad-12\sigma
b_{0,0}^{2}c_{2,2}c_{3,1}^{2}+b_{0,0}^{6}\mathopen{}\mathclose{{}\left(4b_{2,1}c_{2,2}-3c_{0,3}}\right)+9\sigma
c_{3,1}^{4}\Big{]}\,.$

The model, which has $\nu=1$, supports
$\bm{32^{\prime}}_{\frac{1}{2}}\oplus\bm{1}_{1}$ matter at
$\\{\sigma=b_{0,0}=0\\}$, $\bm{32}_{0}$ matter at $\\{\sigma=b_{2,1}=0\\}$,
and $\bm{12}_{\frac{1}{2}}$ matter at
$\\{\sigma=3b_{0,0}^{2}c_{0,3}-4b_{2,1}b_{0,0}^{2}c_{2,2}+3b_{2,1}c_{3,1}^{2}=0\\}$.
In 6D models, $\\{\sigma=b_{0,0}=0\\}$ supports full hypermultiplets of
$\bm{32^{\prime}}_{\frac{1}{2}}$ matter, while $\\{\sigma=b_{2,1}=0\\}$
supports half-hypermultiplets of $\bm{32}_{0}$ matter. This same construction
can alternatively be viewed as a $\nu=3$ model with
$\bm{32}_{\frac{1}{2}}\oplus\bm{1}_{1}$ matter at $\\{\sigma=b_{0,0}=0\\}$,
$\bm{32^{\prime}}_{0}$ matter at $\\{\sigma=b_{2,1}=0\\}$, and
$\bm{12}_{\frac{1}{2}}$ matter at
$\\{\sigma=3b_{0,0}^{2}c_{0,3}-4b_{2,1}b_{0,0}^{2}c_{2,2}+3b_{2,1}c_{3,1}^{2}=0\\}$.
For either description, the massless charged spectrum, after accounting for
additional singlets and $\mathfrak{so}(12)$ adjoints, satisfies the 6D anomaly
cancellation conditions with
$\tilde{b}=-2K_{B}+2[b_{0,0}]-\frac{3}{2}[\sigma]$.

We can therefore find the orders of vanishing for at least some of the spinor
pseudo-charges by calculating the components of multiples of $\hat{s}$ and
finding their orders of vanishing at $\\{\sigma=b_{0,0}=0\\}$. The resulting
data are given in Table 17. The codimension-one orders of vanishing satisfy
the expression

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(12)}(\nu)\,.$
(203)

And if we let353535For all of the orders of vanishing in Table 17, $\mu$ would
be 0. While it might seem arbitrary to use this more complicated expression
for $\mu$, it can be motivated by the requirement that Eq. 205 give integer
$\operatorname{ord}_{2}(\hat{z})$ for the $(q,\nu)$ combinations that were not
generated.

$\mu=\overline{(2q+\nu)}_{2}\,,$ (204)

the codimension-two orders of vanishing satisfy

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(2q^{2}+T_{\operatorname{Spin}(12)}(\nu)-T_{\operatorname{E}_{7}}(\mu)}\right)$
(205)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{E}_{7}}(\mu)\,.$
(206)

Recalling that $d_{\operatorname{Spin}(12)}=4$ and
$d_{\operatorname{E}_{7}}=2$, we see that these expressions agree exactly with
those expected from Section 2.2. Note that we can rewrite the relationship
between $q$, $\nu$, and $\mu$ as

$\mu=\begin{cases}1&\nu=2,3\\\ 0&\nu=0,1\end{cases}\;\text{ for
}\bm{32}_{q}\oplus\bm{1}_{2q}\,,\quad\mu=\begin{cases}1&\nu=1,2\\\
0&\nu=0,3\end{cases}\;\text{ for }\bm{32^{\prime}}_{q}\oplus\bm{1}_{2q}\,.$
(207)

$\lvert q\rvert$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---
$1$ | $(0,0,0,0)$ | $(2,3,1,4)$
$2$ | $(0,0,0,0)$ | $(8,12,4,16)$
$3$ | $(0,0,0,0)$ | $(18,27,9,36)$
$4$ | $(0,0,0,0)$ | $(32,48,16,64)$
$5$ | $(0,0,0,0)$ | $(50,75,25,100)$
$6$ | $(0,0,0,0)$ | $(72,108,36,144)$
$7$ | $(0,0,0,0)$ | $(98,147,49,196)$
$8$ | $(0,0,0,0)$ | $(128,192,64,256)$
(a) $\nu=0$ for both $\bm{32}_{q}\oplus\bm{1}_{2q}$ and
$\bm{32^{\prime}}_{q}\oplus\bm{1}_{2q}$ matter

$\lvert q\rvert$ | $\operatorname{ord}_{(1)}$ | $\operatorname{ord}_{(2)}$
---|---|---
$\frac{1}{2}$ | $(1,3,0,3)$ | $(2,3,1,4)$
$\frac{3}{2}$ | $(1,3,0,3)$ | $(6,9,3,12)$
$\frac{5}{2}$ | $(1,3,0,3)$ | $(14,21,7,28)$
$\frac{7}{2}$ | $(1,3,0,3)$ | $(26,39,13,52)$
$\frac{9}{2}$ | $(1,3,0,3)$ | $(42,63,21,84)$
$\frac{11}{2}$ | $(1,3,0,3)$ | $(62,93,31,124)$
$\frac{13}{2}$ | $(1,3,0,3)$ | $(86,129,43,172)$
$\frac{15}{2}$ | $(1,3,0,3)$ | $(114,171,57,228)$
$\frac{17}{2}$ | $(1,3,0,3)$ | $(146,219,73,292)$
(b) $\nu=1$ for $\bm{32^{\prime}}_{q}\oplus\bm{1}_{2q}$ matter, $\nu=3$ for
$\bm{32}_{q}\oplus\bm{1}_{2q}$ matter

Table 17: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
various combinations of $\nu$ and pseudo-charges $q$ for $\mathfrak{so}(12)$
spinor matter. This matter occurs at codimension-two loci with
$\text{I}_{2}^{*s}\to\text{III}^{*}$ enhancements. The data are obtained using
the $\mathfrak{so}(12)\oplus\mathfrak{u}(1)$ model in Eq. 201.

Of course, these conclusions for the $\mathfrak{so}(12)$ spinors were reached
with only a subset of the allowed $(q,\nu)$ combinations. While we have not
been able to find seed constructions for the missing combinations, it is
unclear whether such models exist or whether some more fundamental constraint
limits our ability to obtain these models. Perhaps some argument shows that,
for instance, $\bm{32^{\prime}}_{1}$ matter cannot occur in a $\nu=3$ F-theory
model without codimension-two (4,6,12) loci. It would be worthwhile to explore
these questions in future work.

###  8.3 Vector representation of $\mathfrak{so}(4k+2)$

We now consider the vector representation of $\mathfrak{so}(4k+2)$, which
occurs at codimension-two loci where the singularity type enhances from
$\text{I}^{*s}_{2k-3}$ (or $\operatorname{Spin}(4k+2)$) to
$\text{I}^{*}_{2k-2}$ (or $\operatorname{Spin}(4k+4)$). Specifically, an
irreducible $\text{I}^{*s}_{2k-3}\rightarrow\text{I}^{*}_{2k-2}$ locus in a 6D
model would support a hypermultiplet of vector ($\bm{4k+2}$) matter. As per
the discussion in Section 4.2, the allowed $\mathfrak{u}(1)$ charges are half-
integral for $\nu=1$ and $3$ and integral for $\nu=0$ and $2$. Since the
vector representation is real, the sign of the $\mathfrak{u}(1)$ charge is
unimportant. The automorphism previously mentioned that exchanges the two
spinor representations is still applicable. It sends the vector representation
to itself, but as before it sends $\nu$ to its inverse element in
$\mathbb{Z}_{4}$. Thus, the automorphism essentially has no impact on matter
in the vector representation, even though it may change the value of $\nu$ for
the model.

We need multiple seed models to generate the pseudo-charges corresponding to
the allowed $\mathfrak{u}(1)$ charges. If we start with a $\nu=1$ model with
$\bm{(4k+2)}_{\frac{1}{2}}$ matter, we can generate the
$\frac{1}{2}+2\mathbb{Z}$ pseudo-charges for $\nu=1$, the
$\frac{3}{2}+2\mathbb{Z}$ pseudo-charges for $\nu=3$, the odd integral pseudo-
charges for $\nu=2$, and the even integral pseudo-charges for $\nu=0$. By the
automorphism, this same model also gives us the $\frac{3}{2}+2\mathbb{Z}$
pseudo-charges for $\nu=1$ and the $\frac{1}{2}+2\mathbb{Z}$ pseudo-charges
for $\nu=3$. This leaves us with the odd $\nu=0$ and even $\nu=2$ pseudo-
charges. The odd $\nu=0$ pseudo-charges can be obtained from a $\nu=0$ model
admitting $\bm{(4k+2)}_{1}$ matter. In the process, we can also obtain the
even $\nu=0$ pseudo-charges, providing us an additional check on some of the
data generated from the $\nu=1$ model.

Obtaining the even $\nu=2$ pseudo-charges is more difficult. Starting with a
$\nu=2$ model admitting $\bm{(4k+2)}_{2}$ matter, we could obtain the even
$\nu=2$ pseudo-charges that are not multiples of $4$. One could use a $\nu=2$
model with $\bm{(4k+2)}_{4}$ vector matter to obtain some pseudo-charges that
are multiples of 4, but we would still miss those pseudo-charges that are
multiples of 8. In order to obtain pseudo-charges proportional to larger and
larger powers of 2, we need seed models admitting vector matter with larger
and larger $\mathfrak{u}(1)$ charges. Even obtaining a model admitting
$\bm{(4k+2)}_{2}$ matter is challenging, let alone models with higher
charges.363636If we take the formulas in Section 2.2 at face value, the
$\hat{z}$ component of the section should vanish to order 2 at the
$\bm{(4k+2)}_{2}$ matter locus. The algebraic structure of such models is more
complicated, since it makes use of non-UFD cancellations Raghuram34 . We
therefore will not generate any of the even $\nu=2$ pseudo-charges here.

For the $\nu=1$ seed models, we use Weierstrass constructions of the form

$y^{2}=x^{3}-v^{2}\mathopen{}\mathclose{{}\left(3\phi^{2}+f_{k+1}v^{k-1}}\right)xz^{4}+v^{3}\mathopen{}\mathclose{{}\left(2\phi^{3}+\phi
f_{k+1}v^{k-1}+\gamma^{2}v^{2k-3}}\right)$ (208)

and consider models where $k$ varies from 2 to 6. To avoid the additional
complications of a spinor locus, we construct these seed models over an
$\mathbb{F}_{4}$ base and let $\\{v=0\\}$ be the curve of self-intersection
$-4$. This model admits $\text{I}_{2k-3}^{*s}$ singularities along $\\{v=0\\}$
and a generating section $\hat{s}$ with coordinates

$[\hat{x}:\hat{y}:\hat{z}]=[\phi v:\gamma