v^{k}:1]\,,\quad\hat{w}=-f_{k+1}v^{k+1}\,.$ (209)

The gauge algebra is therefore $\mathfrak{so}(4k+2)\oplus\mathfrak{u}(1)$. The
locus $\\{v=\gamma=0\\}$ supports $\bm{(4k+2)}_{\frac{1}{2}}$ matter, while
$\\{f_{k+1}=\gamma=0\\}$ supports $\bm{1}_{1}$ singlet matter. Together, this
matter spectrum satisfies the 6D anomaly cancellation conditions for
$0pt=-2K_{B}-\frac{2k+1}{4}[v]$. This model can be viewed as either a $\nu=1$
model or as a $\nu=3$ model; these two options are related by the automorphism
discussed previously.

For the $\nu=0$ seed models, we use Weierstrass constructions of the form

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\sigma^{2}\mathopen{}\mathclose{{}\left(-\frac{1}{3}c_{2,1}^{2}+c_{3,0}\sigma^{k-1}c_{1,k+1}-\frac{1}{4}\sigma^{2k-2}\mathopen{}\mathclose{{}\left(b_{2,k}^{2}+4c_{0,2k+1}\sigma}\right)}\right)xz^{4}$
(210)
$\displaystyle\qquad\quad+\frac{\sigma^{3}}{108}\Big{[}8c_{2,1}^{3}+9\sigma^{2k-3}\mathopen{}\mathclose{{}\left(3\sigma^{2}c_{1,k+1}^{2}-\mathopen{}\mathclose{{}\left(2\sigma
c_{2,1}-3c_{3,0}^{2}}\right)\mathopen{}\mathclose{{}\left(b_{2,k}^{2}+4c_{0,2k+1}\sigma}\right)}\right)$
$\displaystyle\qquad\qquad\qquad\quad-36c_{2,1}c_{3,0}c_{1,k+1}\sigma^{k-1}\Big{]}z^{6}\,,$

and we let $k$ vary from 2 to 6. We no longer assume that the base is
$\mathbb{F}_{4}$.373737In fact, if the base is $\mathbb{F}_{4}$ and
$\\{\sigma=0\\}$ has self-intersection -4, the parameter $c_{3,0}$ is forced
to be proportional to $\sigma$, and the gauge algebra enhances. Again, there
are $\text{I}_{2k-3}^{*s}$ singularities along $\\{\sigma=0\\}$, and the model
admits a generating section $\hat{s}$ with coordinates

$\displaystyle[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[c_{3,0}^{2}-\frac{2}{3}\sigma
c_{2,1}:-c_{3,0}^{3}+\sigma
c_{2,1}c_{3,0}-\frac{1}{2}\sigma^{k+1}c_{1,k+1}:1}\right]\,,$ (211)
$\displaystyle\hat{w}=3c_{3,0}^{4}-4\sigma
c_{2,1}c_{3,0}^{2}+\sigma^{2}c_{2,1}^{2}+c_{3,0}\sigma^{k+1}c_{1,k+1}-\frac{1}{4}\sigma^{2k}\mathopen{}\mathclose{{}\left(b_{2,k}^{2}+4c_{0,2k+1}\sigma}\right)\,.$
(212)

The codimension-two locus $\\{\sigma=c_{3,0}=0\\}$ supports $\bm{(4k+2)}_{1}$
matter, while the codimension-two locus $\\{\sigma=b_{2,k}=0\\}$ supports
$\bm{(4k+2)}_{0}$ matter. When $k=2$, the locus $\\{\sigma=c_{2,1}=0\\}$
supports $\bm{16}_{0}$ matter, and after including the contributions from
singlets and uncharged adjoints, the spectrum satisfies the 6D anomaly
cancellation conditions with $0pt=-2K_{B}$. When $m>2$, $(f,g,\Delta)$ vanish
at least to orders $(4,6,12)$ at $\\{\sigma=c_{2,1}=0\\}$, making it difficult
to determine the matter associated with this locus and its contribution to the
$\mathfrak{so}(4k+2)$ anomaly conditions. However, if we assume that any
matter contributions at $\\{\sigma=c_{2,1}=0\\}$ are uncharged under the
$\mathfrak{u}(1)$, just as was the case for $k=2$, the rest of the spectrum
(including singlets) satisfies the $\mathfrak{u}(1)$ anomaly cancellation
conditions with $0pt=-2K_{B}$. Since we are only interested in the
$\\{\sigma=c_{3,0}=0\\}$ vector locus, we proceed with this construction in
spite of the codimension-two $(4,6,12)$ locus. As confirmation that this
$(4,6,12)$ locus does not cause significant issues, we can verify that the
results for the even pseudo-charges found from the $\nu=0$ construction agree
with those found from the $\nu=1$ construction.

We now use these constructions to generate
$\mathfrak{so}(4k+2)\oplus\mathfrak{u}(1)$ models for $k$ ranging from $2$ to
$6$ and find the orders of vanishing associated with the various vector
pseudo-charges. We do not list the orders of vanishing at codimension one for
the sake of brevity, but they are all given by

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(4k+2)}(\nu)\,.$
(213)

The orders of vanishing at codimension two are listed in Table 18. If we let

$\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}\,,$ (214)

all of these numbers follow the formulas

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{Spin}(4k+2)}(\nu)-T_{\operatorname{Spin}(4k+4)}(\mu)}\right)$
(215)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{Spin}(4k+4)}(\mu)\,.$
(216)

Since the center of $\operatorname{Spin}(2n)$ has
$d_{\operatorname{Spin}(2n)}=4$ elements, these formulas exactly agree with
the expectations from Section 2.2.

$\nu$ | $\lvert q\rvert$ | $\mathfrak{so}(10)$ | $\mathfrak{so}(14)$ | $\mathfrak{so}(18)$ | $\mathfrak{so}(22)$ | $\mathfrak{so}(26)$
---|---|---|---|---|---|---
0 | $1$ | $(1,2,0,2)$ | $(1,2,0,2)$ | $(1,2,0,2)$ | $(1,2,0,2)$ | $(1,2,0,2)$
| $2$ | $(4,6,2,8)$ | $(4,6,2,8)$ | $(4,6,2,8)$ | $(4,6,2,8)$ | $(4,6,2,8)$
| $3$ | $(9,14,4,18)$ | $(9,14,4,18)$ | $(9,14,4,18)$ | $(9,14,4,18)$ | $(9,14,4,18)$
| $4$ | $(16,24,8,32)$ | $(16,24,8,32)$ | $(16,24,8,32)$ | $(16,24,8,32)$ | $(16,24,8,32)$
| $5$ | $(25,38,12,50)$ | $(25,38,12,50)$ | $(25,38,12,50)$ | $(25,38,12,50)$ | $(25,38,12,50)$
| $6$ | $(36,54,18,72)$ | $(36,54,18,72)$ | $(36,54,18,72)$ | $(36,54,18,72)$ | $(36,54,18,72)$
| $7$ | $(49,74,24,98)$ | $(49,74,24,98)$ | $(49,74,24,98)$ | $(49,74,24,98)$ | $(49,74,24,98)$
| $8$ | $(64,96,32,128)$ | $(64,96,32,128)$ | $(64,96,32,128)$ | $(64,96,32,128)$ | $(64,96,32,128)$
| $9$ | $(81,122,40,162)$ | $(81,122,40,162)$ | $(81,122,40,162)$ | $(81,122,40,162)$ | $(81,122,40,162)$
| $10$ | $(100,150,50,200)$ | $(100,150,50,200)$ | $(100,150,50,200)$ | $(100,150,50,200)$ | $(100,150,50,200)$
| $11$ | $(121,182,60,242)$ | $(121,182,60,242)$ | $(121,182,60,242)$ | $(121,182,60,242)$ | $(121,182,60,242)$
| $12$ | $(144,216,72,288)$ | $(144,216,72,288)$ | $(144,216,72,288)$ | $(144,216,72,288)$ | $(144,216,72,288)$
1 or 3 | $\frac{1}{2}$ | $(1,3,0,3)$ | $(1,4,0,4)$ | $(1,5,0,5)$ | $(1,6,0,6)$ | $(1,7,0,7)$
| $\frac{3}{2}$ | $(3,6,1,7)$ | $(3,7,1,8)$ | $(3,8,1,9)$ | $(3,9,1,10)$ | $(3,10,1,11)$
| $\frac{5}{2}$ | $(7,12,3,15)$ | $(7,13,3,16)$ | $(7,14,3,17)$ | $(7,15,3,18)$ | $(7,16,3,19)$
| $\frac{7}{2}$ | $(13,21,6,27)$ | $(13,22,6,28)$ | $(13,23,6,29)$ | $(13,24,6,30)$ | $(13,25,6,31)$
| $\frac{9}{2}$ | $(21,33,10,43)$ | $(21,34,10,44)$ | $(21,35,10,45)$ | $(21,36,10,46)$ | $(21,37,10,47)$
| $\frac{11}{2}$ | $(31,48,15,63)$ | $(31,49,15,64)$ | $(31,50,15,65)$ | $(31,51,15,66)$ | $(31,52,15,67)$
| $\frac{13}{2}$ | $(43,66,21,87)$ | $(43,67,21,88)$ | $(43,68,21,89)$ | $(43,69,21,90)$ | $(43,70,21,91)$
| $\frac{15}{2}$ | $(57,87,28,115)$ | $(57,88,28,116)$ | $(57,89,28,117)$ | $(57,90,28,118)$ | $(57,91,28,119)$
| $\frac{17}{2}$ | $(73,111,36,147)$ | $(73,112,36,148)$ | $(73,113,36,149)$ | $(73,114,36,150)$ | $(73,115,36,151)$
| $\frac{19}{2}$ | $(91,138,45,183)$ | $(91,139,45,184)$ | $(91,140,45,185)$ | $(91,141,45,186)$ | $(91,142,45,187)$
| $\frac{21}{2}$ | $(111,168,55,223)$ | $(111,169,55,224)$ | $(111,170,55,225)$ | $(111,171,55,226)$ | $(111,172,55,227)$
| $\frac{23}{2}$ | $(133,201,66,267)$ | $(133,202,66,268)$ | $(133,203,66,269)$ | $(133,204,66,270)$ | $(133,205,66,271)$
| $\frac{25}{2}$ | $(157,237,78,315)$ | $(157,238,78,316)$ | $(157,239,78,317)$ | $(157,240,78,318)$ | $(157,241,78,319)$
2 | $1$ | $(2,3,1,4)$ | $(2,3,1,4)$ | $(2,3,1,4)$ | $(2,3,1,4)$ | $(2,3,1,4)$
| $3$ | $(10,15,5,20)$ | $(10,15,5,20)$ | $(10,15,5,20)$ | $(10,15,5,20)$ | $(10,15,5,20)$
| $5$ | $(26,39,13,52)$ | $(26,39,13,52)$ | $(26,39,13,52)$ | $(26,39,13,52)$ | $(26,39,13,52)$
| $7$ | $(50,75,25,100)$ | $(50,75,25,100)$ | $(50,75,25,100)$ | $(50,75,25,100)$ | $(50,75,25,100)$
| $9$ | $(82,123,41,164)$ | $(82,123,41,164)$ | $(82,123,41,164)$ | $(82,123,41,164)$ | $(82,123,41,164)$
| $11$ | $(122,183,61,244)$ | $(122,183,61,244)$ | $(122,183,61,244)$ | $(122,183,61,244)$ | $(122,183,61,244)$
Table 18: Codimension-two orders of vanishing of
$(\hat{x},\hat{y},\hat{z},\hat{w})$ for various pseudo-charges $q$ of
$\mathfrak{so}(4k+2)\oplus\mathfrak{u}(1)$ vector matter. This matter is
supported at $\text{I}_{2k-3}^{*s}\to\text{I}_{2k-2}^{*s}$ loci. The data are
found using the constructions in Eqs. 208 and 210.

###  8.4 Vector representation of $\mathfrak{so}(4k)$

We now turn to the vector representation of $\mathfrak{so}(4k)$. Previously,
we used the triality of $\mathfrak{so}(8)$ to determine the orders of
vanishing for $\bm{8_{\text{v}}}$ matter from those for $\bm{8_{\text{s}}}$
and $\bm{8_{\text{c}}}$ matter. We can no longer use this triality for $k>2$,
so we must separately analyze vector matter for these cases. The vector
representation of $\mathfrak{so}(4k)$ occurs at codimension-two loci where the
singularity type enhances from $\text{I}^{*s}_{2k-4}$ (or
$\operatorname{Spin}(4k)$) to $\text{I}^{*}_{2k-3}$ (or
$\operatorname{Spin}(4k+2)$). In 6D models, an irreducible locus of this type
supports a hypermultiplet of $\bm{4k}$ matter. Just as for the
$\mathfrak{so}(4k+2)$ vector representation, the allowed $\mathfrak{u}(1)$
charges for the $\mathfrak{so}(4k)$ vector representation are half-integral
for $\nu=1,3$ and integral for $\nu=0,2$. And because the vector
representation is real, the sign of the $\mathfrak{u}(1)$ charge is
unimportant for vector matter. The automorphism for $\mathfrak{so}(4k)$ does
not affect the vector matter but exchanges $\nu=1$ and $\nu=3$.

We again need multiple models to generate a representative sample of pseudo-
charges. If we start with a $\nu=1$ model with $\bm{4k}_{\frac{1}{2}}$ matter,
one can obtain all the half-integral $\nu=1$ pseudo-charges and all the
integral $\nu=0$ pseudo-charges by examining multiples of the generating
section. By the automorphism, this same model can be viewed as a $\nu=3$ model
with $\bm{4k}_{\frac{1}{2}}$ matter, and the $\nu=1$ pseudo-charges can
equivalently be viewed as $\nu=3$ pseudo-charges. Therefore, we can obtain all
of the half-integral $\nu=3$ pseudo-charges from this same construction. This
leaves us with the $\nu=2$ pseudo-charges. Starting with a $\nu=2$ seed model
with $\bm{4k}_{1}$ matter, which can be constructed relatively easily, we can
get all of the odd $\nu=2$ pseudo-charges. This also gives us an alternative
method of generating the even $\nu=0$ pseudo-charges. Obtaining the even
pseudo-charges with $\nu=2$ is more difficult: in order to get a pseudo-charge
proportional to $2^{j}$, we would need a seed model admitting
$\bm{4k}_{2^{j}}$ matter. Therefore, we only generate the odd $\nu=2$ pseudo-
charges here.

For the $\nu=1$ seed models, we consider constructions over an
$\mathbb{F}_{4}$ base given by the Weierstrass model

$y^{2}=x^{3}-v^{2}\mathopen{}\mathclose{{}\left(3\phi^{2}+f_{k}v^{k-2}}\right)xz^{4}+v^{3}\mathopen{}\mathclose{{}\left(2\phi^{3}+\phi
f_{k}v^{k-2}+\gamma^{2}v^{2k-3}}\right)\,,$ (217)

where $\\{v=0\\}$ is a curve in $\mathbb{F}_{4}$ of self-intersection $-4$. We
let $k$ vary from 3 to 5.383838Since we previously determined the orders of
vanishing for $\mathfrak{so}(8)$ vector matter, we do not need to consider the
$k=2$ version of this construction. The model has $\text{I}^{*s}_{2k-4}$
singular fibers along $\\{v=0\\}$ and admits a generating section of the form

$[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[v\phi:v^{k}\gamma:1}\right]\,,\quad\hat{w}=-v^{k}f_{k}\,,$
(218)

implying that the gauge algebra is $\mathfrak{so}(4k)\oplus\mathfrak{u}(1)$.
The codimension-two locus $\\{v=f_{k}=0\\}$ supports $\bm{4k}_{\frac{1}{2}}$
matter, while the codimension-two locus $\\{f_{k}=\gamma=0\\}$ supports
$\bm{1}_{1}$ matter. The resulting spectra satisfy the 6D anomaly cancellation
conditions with $0pt=-2K_{B}-\frac{k}{2}[v]$. By the automorphism, these
constructions can alternatively be viewed as $\nu=3$ model with the same
matter spectra.

For the $\nu=2$ seed models, we use constructions given by the Weierstrass
model

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{v^{2}}{3}\mathopen{}\mathclose{{}\left(c_{2,1}^{2}+3b_{0,0}^{2}c_{0,2k-1}v^{2k-3}+3b_{0,0}^{2}b_{2,k-1}^{2}v^{2k-4}-3c_{3,1}c_{1,k}v^{k-1}}\right)xz^{4}$
(219)
$\displaystyle\qquad\quad-\frac{v^{3}}{108}\Big{[}-8c_{2,1}^{3}-9v^{2k-4}\Big{(}b_{0,0}^{2}\mathopen{}\mathclose{{}\left(3vc_{1,k}^{2}-8c_{2,1}\mathopen{}\mathclose{{}\left(b_{2,k-1}^{2}+vc_{0,2k-1}}\right)}\right)$
$\displaystyle\qquad\qquad\qquad\quad+12vc_{3,1}^{2}\mathopen{}\mathclose{{}\left(b_{2,k-1}^{2}+vc_{0,2k-1}}\right)\Big{)}+36c_{2,1}c_{3,1}v^{k-1}c_{1,k}\Big{]}z^{6}\,.$

Again, we construct these elliptic fibrations over an $\mathbb{F}_{4}$ base
and take $\\{v=0\\}$ to be a curve of self-intersection $-4$. We also let $k$
run from $3$ to $5$. There are $\text{I}^{*s}_{2k-4}$ singularities along
$\\{v=0\\}$, and the Weierstrass model admits a generating section with
components

$\displaystyle\hat{x}$
$\displaystyle=\frac{v}{3}\mathopen{}\mathclose{{}\left(3vc_{3,1}^{2}-2b_{0,0}^{2}c_{2,1}}\right)\,,$
(220) $\displaystyle\hat{y}$
$\displaystyle=\frac{v^{2}}{2}\mathopen{}\mathclose{{}\left(2b_{0,0}^{2}c_{2,1}c_{3,1}-b_{0,0}^{4}c_{1,k}v^{k-2}-2vc_{3,1}^{3}}\right)\,,$
$\displaystyle\hat{z}$ $\displaystyle=b_{0,0}\,,$ $\displaystyle\hat{w}$
$\displaystyle=-v^{2}\Big{(}-b_{0,0}^{4}c_{2,1}^{2}+4vb_{0,0}^{2}c_{2,1}c_{3,1}^{2}-3v^{2}c_{3,1}^{4}-b_{0,0}^{4}c_{3,1}v^{k-1}c_{1,k}$
$\displaystyle\qquad\quad\qquad+b_{0,0}^{6}v^{2k-4}b_{2,k-1}^{2}+b_{0,0}^{6}c_{0,2k-1}v^{2k-3}\Big{)}\,.$

Thus, the gauge algebra is $\mathfrak{so}(4k)\oplus\mathfrak{u}(1)$.393939For
$k=5$, $[c_{0,2m-1}]$ is ineffective. As a result, there is an extra
generating section, and the gauge algebra is
$\mathfrak{so}(20)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$. However, the
extra $\mathfrak{u}(1)$ gauge factor does not affect the discussion and can be
safely ignored. The model supports $\bm{4k}_{1}$ matter at $\\{v=b_{0,0}=0\\}$
and $\bm{4k}_{0}$ matter at $\\{v=b_{2,k-1}=0\\}$. After accounting for
charged singlets, this spectrum satisfies the anomaly cancellation conditions
for $0pt=-2K_{B}+2[b_{0,0}]-[v]$.

We now construct multiples of the generating sections in these models and find
the orders of vanishing for various pseudo-charges $q$ of $\bm{4k}$ vector
matter. We do not list the codimension-one orders of vanishing here for
brevity, but the generated data for $k=2$ through $5$ satisfy the equations

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{Spin}(4k)}(\nu)\,.$
(221)

The codimension-two orders of vanishing for $k=3$ through $5$ are listed in
Table 19; for $k=2$, we can use the previously determined $\mathfrak{so}(8)$
data in Table 16. If we let

$\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}\,,$ (222)

all of the codimension-two orders of vanishing satisfy the equations

$\displaystyle\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{Spin}(4k)}(\nu)-T_{\operatorname{Spin}(4k+2)}(\mu)}\right)\,,$
(223)
$\displaystyle\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{Spin}(4k+2)}(\mu)\,.$
(224)

Since $\operatorname{Spin}(2n)$ has a center with
$d_{\operatorname{Spin}(2n)}=4$ elements, the observed orders of vanishing
agree with the expectations from Section 2.2.

$\nu$ | $\lvert q\rvert$ | $\mathfrak{so}(12)$ | $\mathfrak{so}(16)$ | $\mathfrak{so}(20)$
---|---|---|---|---
0 | $1$ | $(1,2,0,2)$ | $(1,2,0,2)$ | $(1,2,0,2)$
| $2$ | $(4,6,2,8)$ | $(4,6,2,8)$ | $(4,6,2,8)$
| $3$ | $(9,14,4,18)$ | $(9,14,4,18)$ | $(9,14,4,18)$
| $4$ | $(16,24,8,32)$ | $(16,24,8,32)$ | $(16,24,8,32)$
| $5$ | $(25,38,12,50)$ | $(25,38,12,50)$ | $(25,38,12,50)$
| $6$ | $(36,54,18,72)$ | $(36,54,18,72)$ | $(36,54,18,72)$
| $7$ | $(49,74,24,98)$ | $(49,74,24,98)$ | $(49,74,24,98)$
| $8$ | $(64,96,32,128)$ | $(64,96,32,128)$ | $(64,96,32,128)$
| $9$ | $(81,122,40,162)$ | $(81,122,40,162)$ | $(81,122,40,162)$
| $10$ | $(100,150,50,200)$ | $(100,150,50,200)$ | $(100,150,50,200)$
| $11$ | $(121,182,60,242)$ | $(121,182,60,242)$ | $(121,182,60,242)$
| $12$ | $(144,216,72,288)$ | $(144,216,72,288)$ | $(144,216,72,288)$
| $13$ | $(169,254,84,338)$ | $(169,254,84,338)$ | $(169,254,84,338)$
1 or 3 | $\frac{1}{2}$ | $(1,3,0,4)$ | $(1,4,0,5)$ | $(1,5,0,6)$
| $\frac{3}{2}$ | $(3,6,1,8)$ | $(3,7,1,9)$ | $(3,8,1,10)$
| $\frac{5}{2}$ | $(7,12,3,16)$ | $(7,13,3,17)$ | $(7,14,3,18)$
| $\frac{7}{2}$ | $(13,21,6,28)$ | $(13,22,6,29)$ | $(13,23,6,30)$
| $\frac{9}{2}$ | $(21,33,10,44)$ | $(21,34,10,45)$ | $(21,35,10,46)$
| $\frac{11}{2}$ | $(31,48,15,64)$ | $(31,49,15,65)$ | $(31,50,15,66)$
| $\frac{13}{2}$ | $(43,66,21,88)$ | $(43,67,21,89)$ | $(43,68,21,90)$
| $\frac{15}{2}$ | $(57,87,28,116)$ | $(57,88,28,117)$ | $(57,89,28,118)$
| $\frac{17}{2}$ | $(73,111,36,148)$ | $(73,112,36,149)$ | $(73,113,36,150)$
| $\frac{19}{2}$ | $(91,138,45,184)$ | $(91,139,45,185)$ | $(91,140,45,186)$
| $\frac{21}{2}$ | $(111,168,55,224)$ | $(111,169,55,225)$ | $(111,170,55,226)$
| $\frac{23}{2}$ | $(133,201,66,268)$ | $(133,202,66,269)$ | $(133,203,66,270)$
| $\frac{25}{2}$ | $(157,237,78,316)$ | $(157,238,78,317)$ | $(157,239,78,318)$
2 | $1$ | $(2,3,1,4)$ | $(2,3,1,4)$ | $(2,3,1,4)$
| $3$ | $(10,15,5,20)$ | $(10,15,5,20)$ | $(10,15,5,20)$
| $5$ | $(26,39,13,52)$ | $(26,39,13,52)$ | $(26,39,13,52)$
| $7$ | $(50,75,25,100)$ | $(50,75,25,100)$ | $(50,75,25,100)$
| $9$ | $(82,123,41,164)$ | $(82,123,41,164)$ | $(82,123,41,164)$
| $11$ | $(122,183,61,244)$ | $(122,183,61,244)$ | $(122,183,61,244)$
| $13$ | $(170,255,85,340)$ | $(170,255,85,340)$ | $(170,255,85,340)$
Table 19: Codimension-two orders of vanishing of
$(\hat{x},\hat{y},\hat{z},\hat{w})$ for various pseudo-charges $q$ of
$\mathfrak{so}(4k)\oplus\mathfrak{u}(1)$ vector matter. This matter is
supported at $\text{I}_{2k-4}^{*s}\to\text{I}_{2k-3}^{*}$ loci. The data are
found using the constructions in Equations 217 and 219.

###  8.5 Connection to EDS valuations

We can now show that the formulas for EDS valuations in StangeEllTrouble
reproduce the patterns for $\mathfrak{so}(4k)$ and $\mathfrak{so}(4k+2)$
models proposed in Section 2.2 and observed in this section. As described in
Section 7.4, for each seed model, we should subtract the EDS valuations for
the codimension-one behavior from those for the codimension-two behavior. The
resulting numbers should agree with the observed
$\operatorname{ord}_{2}(\hat{z})$ values for multiples of the seed model’s
generating section admitting the appropriate pseudo-charges.

There are five singularity types that need to be considered for the
$\mathfrak{so}$ representations discussed above. Some of the EDS valuations
for $\text{I}^{*}_{2k-3}$ singularities were calculated in Section 7.4 when
analyzing the antisymmetric representation of $\mathfrak{su}(2k+1)$; the
remaining ones are calculated in Section C.2. Those for the
$\text{I}^{*}_{2k-4}$, $\text{IV}^{*}$, $\text{III}^{*}$, and $\text{II}^{*}$
singularity types are respectively calculated in Section C.1, Section C.3,
Section C.4, and Section C.5. Having found the EDS valuations for these
different singularity types, we can show how to reproduce the
$\operatorname{ord}_{2}(\hat{z})$ formulas for the $\mathfrak{so}(2n)$
representations.

##### Vectors of $\mathfrak{so}(4k+2)$

We considered two different seed models for the $\bm{(4k+2)}_{q}$
representation of $\mathfrak{so}(4k+2)\oplus\mathfrak{u}(1)$, so we must
determine the EDS valuations for these two models separately. We start with
the $\nu=1$ seed model. The generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,k,0,k+1)$ along the
codimension-one locus supporting the $\mathfrak{so}(4k+2)$. The EDS valuations
corresponding to the codimension-one behavior are given by Eq. 161 with $n$
replaced with $2k+1$ and $s$ set to 1:

$v(W_{m}^{(1)})=\frac{2k+1}{8}m^{2}-\frac{1}{2}\begin{cases}0&4\mid m\\\
1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{2k+1}{4}&(2\nmid m)\end{cases}$
(225)

At the codimension-two locus supporting the seed model’s
$\bm{(4k+2)}_{\frac{1}{2}}$ matter, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,k+1,0,k+1)$. The EDS
valuations corresponding to the codimension-two behavior are therefore given
by Eq. 353 with $n$ replaced by $2k+2$:

$v(W_{m}^{(2)})=\frac{2k+2}{8}m^{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\
\frac{2k+2}{4}&2\nmid m\end{cases}\,.$ (226)

The difference between these two sequences of EDS valuations is

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{4}m^{2}+\begin{cases}0&4\mid
m\\\ 1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{2k+1}{4}&(2\nmid
m)\end{cases}-\begin{cases}0&2\mid m\\\ \frac{2k+2}{4}&2\nmid
m\end{cases}}\right)$ (227)

Using the relations

$q=\frac{m}{2}\,,\quad\nu=\overline{m}_{4}\,,\quad\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}=\overline{m}_{2}\,,$
(228)

we can rewrite this as

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{SO}(4k+2)}(\nu)-T_{\operatorname{SO}(4k+4)}(\mu)}\right)\,.$
(229)

The right-hand side of this expression exactly matches the expected formula
for $\operatorname{ord}_{2}(\hat{z})$ from the proposals, and the formula
agrees with the observed orders of vanishing.

For the $\nu=0$ seed model, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(0,0,0,0)$ at the
codimension-one locus supporting the $\mathfrak{so}(4k+2)$ factor. Because all
the generating section components vanish to order 0, the EDS valuations
$v(W_{m}^{(1)})$ are $0$. At the codimension-two locus supporting the
$\bm{(4k+2)}_{1}$ matter, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,2,0,2)$, and the
corresponding EDS valuations $v(W_{m}^{(2)})$ are given by Eq. 358. We
therefore have

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=v(W_{m}^{(2)})=\frac{m^{2}}{2}-\frac{1}{2}\begin{cases}0&2\mid
m\\\ 1&2\nmid m\end{cases}\,.$ (230)

Using the relations

$q=m\,,\quad\nu=0\,,\quad\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}=2\overline{m}_{2}\,,$
(231)

this can be rewritten as

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{SO}(4k+2)}(\nu)-T_{\operatorname{SO}(4k+4)}(\mu)}\right)\,.$
(232)

Again, the right-hand side agrees with expression for
$\operatorname{ord}_{2}(\hat{z})$ expected from the proposals and reproduces
the observed orders of vanishing.

##### Vectors of $\mathfrak{so}(4k)$

We considered two different seed models for the $\bm{4k}_{q}$ representation
of $\mathfrak{so}(4k)\oplus\mathfrak{u}(1)$, so we need to perform separate
EDS analyses for each of these models. We start with the $\nu=1$ seed model.
Along the codimension-one locus supporting the $\mathfrak{so}(4k)$ gauge
factor, the generating section components $(\hat{x},\hat{y},\hat{z},\hat{w})$
vanish to orders $(1,k,0,k)$. Therefore, the EDS valuations corresponding to
the codimension-one behavior are given by Eq. 353 with $n$ set to $2k$:

$v(W^{(1)}_{m})=\frac{2k}{8}m^{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\
\frac{2k}{4}&2\nmid m\end{cases}\,.$ (233)

At the codimension-two locus supporting the $\bm{4k}_{1/2}$ matter, the
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,k,0,k+1)$, and the
EDS valuations corresponding to the codimension-two behavior are given by Eq.
161 with $n$ set to $2k+1$ and $s$ set to $1$:

$v(W^{(2)}_{m})=\frac{2k+1}{8}m^{2}-\frac{1}{2}\begin{cases}0&4\mid m\\\
1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{2k+1}{4}&(2\nmid m)\end{cases}\,.$
(234)

Using the relations404040We are free to use either
$\nu=\overline{m}_{2},\mu=\overline{m}_{4}$ or
$\nu=3\overline{m}_{2},\mu=\overline{3m}_{4}$, as the two options are related
by the automorphism. One obtains similar results with either choice.

$q=\frac{m}{2}\,,\quad\nu=\overline{m}_{2}\,,\quad\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}=\overline{m}_{4}\,,$
(235)

one can write the difference of the two EDS valuations as

$v(W^{(2)}_{m})-v(W^{(1)}_{m})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{SO}(4k)}(\nu)-T_{\operatorname{SO}(4k+2)}(\mu)}\right)\,.$
(236)

The right-hand side agrees with the expected formula for
$\operatorname{ord}_{2}(\hat{z})$ from the proposals and reproduces the
observed order of vanishing data.

For the $\nu=2$ seed model, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,2,0,2)$ along the
codimension-one locus supporting the $\mathfrak{so}(4k)$ gauge algebra. The
EDS valuations for the codimension-one behavior are therefore given by Eq.
358:

$v(W_{m}^{(1)})=\frac{m^{2}}{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\ 1&2\nmid
m\end{cases}\,.$ (237)

At the codimension-two locus supporting the $\bm{4k}_{1}$ matter,
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(2,3,1,4)$. The
corresponding EDS valuations for this situations are given by Eq. 364:

$v(W_{m}^{(2)})=m^{2}\,.$ (238)

If we use the relations

$q=m\,,\quad\nu=2\overline{m}_{2}\,,\quad\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}=0\,,$
(239)

the difference between these two EDS valuations can be written as

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+T_{\operatorname{SO}(4k)}(\nu)-T_{\operatorname{SO}(4k)}(\mu)}\right)\,,$
(240)

in agreement with the proposals.

##### Spinors of $\mathfrak{so}(8)$

While we considered only one seed model, given by Eq. 191, for the spinors of
$\mathfrak{so}(8)$, we generated data using two different matter loci with
distinct charges. We need to perform the EDS analysis separately for each of
these loci. The codimension-one EDS valuations are the same for both these
loci and are given by Eq. 358. The codimension-two EDS valuations, however,
are different for the two cases.

At the loci $\\{\sigma=c_{2,1}\pm b_{0,0}b_{2,1}=0\\}$, which support matter
with $\mathfrak{u}(1)$ charge $\frac{1}{2}$, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,2,0,3)$. Therefore,
the codimension-two orders of vanishing are described by Eq. 161 with $n=5$
and $s=1$, and we have

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{4}m^{2}-\begin{cases}0&4\mid
m\\\ 1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{5}{4}&m\nmid
2\end{cases}+\begin{cases}0&2\mid m\\\ 1&(2\nmid m)\end{cases}}\right)\,.$
(241)

If we note that $q=m/2$ and use the appropriate expressions for $\nu$ and
$\mu$, this formula reproduces the $\operatorname{ord}_{2}(\hat{z})$ given in
Tables 16(b) and 16(c) and agrees with the proposed formulas in Section 2.2.

At the locus $\\{\sigma=b_{0,0}=0\\}$, which supports matter with
$\mathfrak{u}(1)$ charge $1$, the generating section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(2,3,1,4)$. The EDS
valuations for this situation, calculated in Section C.2, are given by
$v(W_{m}^{(2)})=m^{2}$. Therefore

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(m^{2}+\begin{cases}0&2\mid
m\\\ 1&(2\nmid m)\end{cases}}\right)\,.$ (242)

Noting that $q=m$ and that $\mu=0$ for this seed model, this expression again
agrees with the proposals and the observed $\operatorname{ord}_{2}(\hat{z})$
values.

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

For the spinors of $\mathfrak{so}(10)$, we considered a single seed model
given by Eq. 176. The section components $(\hat{x},\hat{y},\hat{z},\hat{w})$
vanish to orders $(1,2,0,3)$ at the codimension-one locus supporting the
$\text{I}_{1}^{*s}$ singularities and to orders $(2,2,0,3)$ at the
codimension-two locus supporting the $\bm{16}_{-\frac{1}{4}}$ matter. To show
the connection to EDS valuations, we should subtract the EDS valuations for
$\text{I}^{*}_{1}$ singularities in Eq. 161 (with $n=5$, $s=1$) from those for
$\text{IV}^{*}$ singularities in Eq. 370:

$\displaystyle v(W_{m}^{(2)})-v(W_{m}^{(1)})$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{2}{3}m^{2}-\begin{cases}0&3\mid
m\\\ \frac{2}{3}&3\nmid m\end{cases}}\right)$ (243)
$\displaystyle\qquad\qquad-\mathopen{}\mathclose{{}\left(\frac{5}{8}m^{2}-\frac{1}{2}\begin{cases}0&4\mid
m\\\ 1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{5}{4}&(2\nmid
m)\end{cases}}\right)$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{12}m^{2}+\begin{cases}0&\overline{m}_{4}=0\\\
1&\overline{m}_{4}=2\\\
\frac{5}{4}&\overline{m}_{4}=1,3\end{cases}-\begin{cases}0&\overline{m}_{3}=0\\\
\frac{4}{3}&\overline{m}_{3}=1,2\end{cases}}\right)\,.$

Then, if we let

$m=-4q\,,\quad\nu=\overline{m}_{4}\,,\quad\mu=\overline{(-m)}_{3}=\overline{4q}_{3}\,,$
(244)

we find that

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{4}{3}q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=2\\\ \frac{5}{4}&\nu=1,3\end{cases}-\begin{cases}0&\mu=0\\\
\frac{4}{3}&\mu=1,2\end{cases}}\right)\,,$ (245)

exactly in agreement with the proposed formula for
$\operatorname{ord}_{2}(\hat{z})$.

##### Spinors of $\mathfrak{so}(12)$

For the spinors of $\mathfrak{so}(12)$, we considered only one seed model
given by Eq. 201. At the codimension-one locus supporting the
$\text{I}_{2}^{*}$ singularities, the section components
$(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(1,3,0,3)$. Therefore,
the EDS valuations corresponding to the codimension-one behavior are given by
Eq. 353 with $n=6$:

$v(W_{m}^{(1)})=\frac{6}{8}m^{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\
\frac{6}{4}&2\nmid m\end{cases}\,.$ (246)

At the codimension-two $\text{III}^{*}$ locus supporting the
$\bm{32^{\prime}}_{\frac{1}{2}}\oplus\bm{1}_{1}$ matter, the section
components vanish to orders $(2,3,1,4)$. The EDS valuations corresponding to
the codimension-two behavior, which are calculated in Section C.4.1, are given
by $v(W_{m}^{(2)})=m^{2}$. The difference of the EDS valuations is given by

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{1}{2}m^{2}+\begin{cases}0&2\mid
m\\\ \frac{3}{2}&2\nmid m\end{cases}}\right)\,.$ (247)

After noting that $q=\frac{m}{2}$, $\nu=\overline{m}_{2}$ or
$3\overline{m}_{2}$, and $\mu=0$, we can rewrite this expression as

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(2q^{2}+T_{\operatorname{SO}(12)}(\nu)-T_{\operatorname{E}_{7}}(\mu)}\right)\,.$
(248)

Thus, we have recovered the expected expression for
$\operatorname{ord}_{2}(\hat{z})$ from Section 2.2.

##### Spinors of $\mathfrak{so}(14)$

For the spinors of $\mathfrak{so}(14)$, we considered a single seed model
described by Eq. 183. The $(\hat{x},\hat{y},\hat{z},\hat{w})$ sections
components vanish to orders $(1,3,0,4)$ at the codimension-one
$\mathfrak{so}(14)$ locus and to orders $(2,3,1,4)$ at the codimension-two
$\text{II}^{*}$ matter locus. We therefore subtract the EDS valuation for
$\text{I}^{*}_{3}$ singularities given in Eq. 161 (with $n=7$ and $s=1$) from
those for $\text{II}^{*}$ singularities in Eq. 392:

$\displaystyle v(W_{m}^{(2)})-v(W_{m}^{(1)})$
$\displaystyle=m^{2}-\mathopen{}\mathclose{{}\left(\frac{7}{8}m^{2}-\frac{1}{2}\begin{cases}0&4\mid
m\\\ 1&(2\mid m)\text{ and }(4\nmid m)\\\ \frac{7}{4}&(2\nmid
m)\end{cases}}\right)$ (249)
$\displaystyle=\frac{1}{8}m^{2}+\frac{1}{2}\begin{cases}0&\overline{m}_{4}=0\\\
1&\overline{m}_{4}=2\\\ \frac{7}{4}&\overline{m}_{4}=1,3\end{cases}\,.$

Letting

$m=4q\,,\quad\nu=\overline{m}_{4}\,,\quad\mu=0\,,$ (250)

we find that

$v(W_{m}^{(2)})-v(W_{m}^{(1)})=\frac{1}{2}\mathopen{}\mathclose{{}\left(4q^{2}+\frac{1}{2}\begin{cases}0&\nu=0\\\
1&\nu=2\\\ \frac{7}{4}&\nu=1,3\end{cases}}\right)\,,$ (251)

exactly in agreement with the formula for $\operatorname{ord}_{2}(\hat{z})$
expected from the proposals.

##  9 Representations of $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$

For models with $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$ gauge algebras, we are
primarily interested in matter in the $\bm{27}$ representation of
$\mathfrak{e}_{6}$. Such matter occurs at the intersection of the codimension-
one $\text{IV}^{*}$ locus supporting the $\mathfrak{e}_{6}$ gauge algebra with
the residual $\text{I}_{1}$ discriminant locus. The singularity type enhances
to $\text{III}^{*}$ at this codimension-two locus, or $\operatorname{E}_{7}$
in the terms of ADE groups. The center of $\operatorname{E}_{6}$ is
$\mathbb{Z}_{3}$, whose elements we denote by $\nu=0,1,2$. For a gauge group
with a quotient involving the $\nu$ element of the center, the allowed
$\mathfrak{u}(1)$ charges for $\bm{27}$ matter are then

$\frac{\nu}{3}+j\text{ for }j\in\mathbb{Z}\,.$ (252)

In other words, a model with an $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$ gauge
algebra can support $\bm{27}_{q}$ matter when the gauge group quotient
involves the element

$\nu=\overline{3q}_{3}\,.$ (253)

of the $\operatorname{E}_{6}$ center. The allowed charges for $\nu=1$ and
$\nu=2$ are equivalent up to a sign, as discussed in Section 4.3.

To find the orders of vanishing for the $\bm{27}$ charges, we can start with
the Weierstrass model

$y^{2}=x^{3}+\sigma^{3}f_{3}xz^{4}+\sigma^{4}\mathopen{}\mathclose{{}\left(y_{2}^{2}-f_{3}\sigma
x_{2}-x_{2}^{3}\sigma^{2}}\right)z^{6}\,,$ (254)

which admits the section $\hat{s}$ with coordinates

$[\hat{x}:\hat{y}:\hat{z}]=\mathopen{}\mathclose{{}\left[\sigma^{2}x_{2}:\sigma^{2}y_{2}:1}\right]\,,\quad\hat{w}=\sigma^{3}\mathopen{}\mathclose{{}\left(f_{3}+3x_{2}^{2}\sigma}\right)\,.$
(255)

The $f$ and $g$ of the Weierstrass model are respectively proportional to
$\sigma^{3}$ and $\sigma^{4}$, and the order $\sigma^{4}$ term in $g$ is a
perfect square. We therefore have an $\mathfrak{e}_{6}$ gauge factor tuned
along $\\{\sigma=0\\}$. Including the $\mathfrak{u}(1)$ generated by the
section above, the gauge algebra is $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$.
This model is in fact equivalent to the $\mathcal{Q}(3,2,2,1,0,1,2)$ model of
KuntzlerTateTrees  with $b_{0,0}$ set to 1. By comparison to the spectrum
given there, we can determine that the model supports $\bm{27}_{1/3}$ matter
at $\\{\sigma=y_{2}=0\\}$ and that $\nu=1$. (Alternatively, we can take $\nu$
to be 2 and say that $\\{\sigma=y_{2}=0\\}$ supports $\bm{27}_{-1/3}$ matter.)
There is also $\bm{1}_{1}$ matter supported at
$\\{y_{2}=f_{3}+3x_{2}^{2}\sigma=0\\}$. For a 6D F-theory model constructed
from this elliptic fibration, this matter spectrum, together with the
$1+\frac{1}{2}[\sigma]\cdot(K_{B}+[\sigma])$ hypermultiplets of $\bm{78}_{0}$
matter, satisfies the anomaly cancellation conditions with
$0pt=-2K_{B}-\frac{4}{3}[\sigma]$. The gauge group is therefore
$(\operatorname{E}_{6}\times\operatorname{U}(1))/\mathbb{Z}_{3}$.

The sections $m\hat{s}$ realize pseudo-charge $m/3$ at the $\bm{27}$ locus
$\\{\sigma=y_{2}=0\\}$ with $\nu=\overline{m}_{3}$, giving us information
about the orders of vanishing at $\bm{27}_{q}$ loci for all of the allowed
charges. Table 20 lists the orders of vanishing at $\\{\sigma=0\\}$ at
$\\{\sigma=y_{2}=0\\}$ for some of these pseudo-charges. The codimension-one
orders of vanishing for a pseudo-charge $q$, which would correspond to a
quotient involving the $\nu=\overline{3q}_{3}$ element of the center, are
described by the formula

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{E}_{6}}(\nu)\,.$
(256)

And if we let

$\mu=\overline{3q}_{2}\,,$ (257)

the codimension-two orders of vanishing follow the expressions

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{3}{2}q^{2}+T_{\operatorname{E}_{6}}(\nu)-T_{\operatorname{E}_{7}}(\mu)}\right)$
(258)

and

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{E}_{7}}(\mu)\,.$
(259)

Since $d_{\operatorname{E}_{6}}=3$ and $d_{\operatorname{E}_{7}}=2$, these
formulas are in exact agreement with the proposals from Section 2.2. We expect
that the generating section components should vanish to these same orders at
genuine $\bm{27}_{q}$ loci in $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$ models.

$\lvert q\rvert$ | $\nu$ | Codimension One | Codimension Two
---|---|---|---
$\frac{1}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(2,3,0,3)$
$\frac{2}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(2,3,1,4)$
$1$ | $0$ | $(0,0,0,0)$ | $(2,3,0,3)$
$\frac{4}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(4,6,2,8)$
$\frac{5}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(6,9,2,11)$
$2$ | $0$ | $(0,0,0,0)$ | $(6,9,3,12)$
$\frac{7}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(10,15,4,19)$
$\frac{8}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(12,18,6,24)$
$3$ | $0$ | $(0,0,0,0)$ | $(14,21,6,27)$
$\frac{10}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(18,27,9,36)$
$\frac{11}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(22,33,10,43)$
$4$ | $0$ | $(0,0,0,0)$ | $(24,36,12,48)$
$\frac{13}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(30,45,14,59)$
$\frac{14}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(34,51,17,68)$
$5$ | $0$ | $(0,0,0,0)$ | $(38,57,18,75)$
$\frac{16}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(44,66,22,88)$
$\frac{17}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(50,75,24,99)$
$6$ | $0$ | $(0,0,0,0)$ | $(54,81,27,108)$
$\frac{19}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(62,93,30,123)$
$\frac{20}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(68,102,34,136)$
$7$ | $0$ | $(0,0,0,0)$ | $(74,111,36,147)$
$\frac{22}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(82,123,41,164)$
$\frac{23}{3}$ | $1\text{ or }2$ | $(2,2,0,3)$ | $(90,135,44,179)$
$8$ | $0$ | $(0,0,0,0)$ | $(96,144,48,192)$
Table 20: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
combinations of $\nu$ and $\bm{27}$ pseudo-charges $q$ at codimension one and
codimension two. These data are found by calculating multiples of the
generating section 255 for the $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$
Weierstrass model in Eq. 254.

###  9.1 Connection to EDS valuations

Because all of the allowed combinations of $\nu$ and pseudo-charge can be
obtained from multiples of the generating section $\hat{s}$, we expect that
Eq. 258 should agree with formulas in StangeEllTrouble  describing the
valuations of elliptic divisibility sequences. As with our previous examples,
we must consider two sets of EDS valuations: $v(W_{m}^{(1)})$ for the
codimension-one singularity type and $v(W_{m}^{(2)})$ for the codimension-two
singularity type. The EDS valuations for the codimension-one $\text{IV}^{*}$
singularity type are calculated in Section C.3 and are given by Eq. 370:

$v(W_{m}^{(1)})=\frac{2}{3}m^{2}-\begin{cases}0&3\mid m\\\ \frac{2}{3}&3\nmid
m\end{cases}\,.$ (260)

The EDS valuations for the codimension-two $\text{III}^{*}$ singularities,
meanwhile, are calculated in Section C.4.2, and are given by Eq. 384:

$v(W_{m}^{(2)})=\frac{3}{4}m^{2}-\begin{cases}0&2\mid m\\\ \frac{3}{4}&2\nmid
m\end{cases}\,.$ (261)

The orders of vanishing of $\hat{z}$ at codimension-two for multiples
$m\hat{s}$ of the generating section $\hat{s}$ should then be given by
$v(W_{m}^{(2)})-v(W_{m}^{(1)})$:

$\operatorname{ord}_{2}(\hat{z})=\frac{3}{4}m^{2}-\frac{2}{3}m^{2}-\begin{cases}0&2\mid
m\\\ \frac{3}{4}&2\nmid m\end{cases}+\begin{cases}0&3\mid m\\\
\frac{2}{3}&3\nmid m\end{cases}\,.$ (262)

Since $m\hat{s}$ admits pseudo-charge $q=\frac{m}{3}$ for the $\bm{27}$
matter, this expression can be rewritten as

$\operatorname{ord}_{2}(\hat{z})=\frac{3}{4}q^{2}-\begin{cases}0&2\mid(3q)\\\
\frac{3}{4}&2\nmid(3q)\end{cases}+\begin{cases}0&3\mid(3q)\\\
\frac{2}{3}&3\nmid(3q)\end{cases}\,.$ (263)

Recalling that $\nu=\overline{3q}_{3}$ and that $\mu=\overline{3q}_{2}$, this
formula exactly matches Eq. 258, as expected.

##  10 Representations of $\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$

For models with $\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ gauge algebras, we are
mostly interested in matter in the $\bm{56}$ representation. The center of
$\operatorname{E}_{7}$ is $\mathbb{Z}_{2}$, whose elements we denote as
$\nu=0,1$. As discussed in Section 4.3, when the quotient involves the $\nu$
element of the center, the allowed $\mathfrak{u}(1)$ charges for $\bm{56}$
matter are

$\frac{\nu}{2}+j\text{ for }j\in\mathbb{Z}\,.$ (264)

In other words, a model with an $\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ gauge
algebra can support $\bm{56}_{q}$ matter when the quotient involves the
element of the center corresponding to

$\nu=\overline{2q}_{2}\,.$ (265)

In F-theory models, $\bm{56}$ matter occurs at codimension-two loci where the
singularity type enhances from from $\text{III}^{*}$ to $\text{II}^{*}$.
Phrased in the ADE language, the singularity type enhances from
$\operatorname{E}_{7}$ to $\operatorname{E}_{8}$. The $\bm{56}$ representation
is pseudoreal, and if the supported $\bm{56}$ matter is uncharged under the
$\mathfrak{u}(1)$, it can be realized as half-hypermultiplets in 6D models.
The fibers at such loci would take the shape of an incompletely resolved
$\operatorname{E}_{8}$ diagram MorrisonTaylorMaS ; KanHalfHyper . However, if
the $\bm{56}$ matter has a nonzero $\mathfrak{u}(1)$ charge, the total
representation, which includes the $\mathfrak{u}(1)$ charge, is no longer
pseudoreal. Therefore, $\bm{56}_{q}$ matter can only come in full
hypermultiplets in 6D when $q$ is nonzero, and the fibers at the corresponding
codimension-two loci form completely resolved $\operatorname{E}_{8}$ diagrams.
Since we are almost exclusively interested in $\mathfrak{u}(1)$-charged
$\bm{56}$ matter, we primarily deal with full hypermultiplets. The pseudoreal
nature of the $\bm{56}$ representation also implies that, in line with the
arguments in Section 5.1, the sign of the $\mathfrak{u}(1)$ charge is
unimportant for $\bm{56}$ matter.

The codimension-two $\text{II}^{*}$ loci admitting complete resolutions
support both $\bm{56}$ and singlet matter. Specifically, in 6D models, an
irreducible codimension-two locus with a completely resolvable $\text{II}^{*}$
singular fiber supports a full hypermultiplet of $\bm{56}$ matter and a full
hypermultiplet of singlet matter. Just as was the case for the spinors of
$\mathfrak{so}(12)$, the appearance of this extra singlet can be understood
using the Katz–Vafa method and the following branching rule:

$\displaystyle\operatorname{E}_{8}$
$\displaystyle\to\operatorname{E}_{7}\times\operatorname{SU}(2)\,,$ (266)
$\displaystyle\bm{248}$
$\displaystyle\to(\bm{133},\bm{1})\oplus(\bm{56},\bm{2})\oplus(\bm{1},\bm{3})\,.$

The $(\bm{56},\bm{2})$ degrees of freedom combine to form the full
hypermultiplet of $\bm{56}$ matter. The singlet hypermultiplet is formed by
the degrees of freedom corresponding to the positive and negative roots of the
$(\bm{1},\bm{3})$ in this decomposition.

This additional singlet can also have a $\mathfrak{u}(1)$ charge, raising the
question of how this singlet’s charge compares to the charge of the $\bm{56}$
matter. In the resolved $\operatorname{E}_{8}$ fiber at the codimension-two
locus, there are two curves, $\gamma_{+}$ and $\gamma_{-}$, which represent
the highest $\bm{56}$ weights in the $(\bm{56},\bm{2})$ of the decomposition.
The $\mathfrak{u}(1)$ charges for these $\bm{56}$ weights should be negatives
of each other, implying that

$q_{\bm{56}}=\sigma(\hat{s})\cdot\gamma_{+}=-\sigma(\hat{s})\cdot\gamma_{-}\,,$
(267)

where $\hat{s}$ is the generating section for the $\mathfrak{u}(1)$. These two
weights are also the highest and lowest weights of an $\mathfrak{su}(2)$
fundamental, so the curves are related by

$\gamma_{+}=\gamma_{-}+c\,.$ (268)

The curve $c$ represents a root of the $(\bm{1},\bm{3})$ part of the
decomposition, and it corresponds to the singlet matter. The charge of this
singlet is

$q_{\bm{1}}=\sigma(\hat{s})\cdot
c=\sigma(\hat{s})\cdot\gamma_{+}-\sigma(\hat{s})\cdot\gamma_{-}=2\sigma(\hat{s})\cdot\gamma_{+}=2q_{56}\,.$
(269)

In other words, the charge of the singlet supported at the $\text{II}^{*}$
locus should be twice that of $\bm{56}$ matter. Since the singlet
representation is real, the arguments in Section 5.1 suggest that the sign of
the singlet charge does not matter.

We can now investigate how the orders of vanishing of the section components
encode the charges of the $\bm{56}$ matter and the associated singlet. We
start with the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{1}{3}\sigma^{3}\mathopen{}\mathclose{{}\left(3b^{2}c_{0,3}+\sigma\mathopen{}\mathclose{{}\left(c_{2,2}^{2}-3c_{1,3}c_{3,1}}\right)}\right)xz^{4}$
(270)
$\displaystyle\quad+\frac{1}{108}\sigma^{5}\mathopen{}\mathclose{{}\left(9b^{2}\mathopen{}\mathclose{{}\left(3c_{1,3}^{2}\sigma-8c_{0,3}c_{2,2}}\right)+4\mathopen{}\mathclose{{}\left(27c_{0,3}c_{3,1}^{2}-9c_{1,3}c_{2,2}c_{3,1}\sigma+2c_{2,2}^{3}\sigma}\right)}\right)z^{6}\,,$

which admits a generating section $\hat{s}$ with components

$z\begin{gathered}=\mathopen{}\mathclose{{}\left[c_{3,1}^{2}\sigma^{2}-\frac{2}{3}b^{2}c_{2,2}\sigma^{2}:-\frac{1}{2}\sigma^{3}\mathopen{}\mathclose{{}\left(b^{4}c_{1,3}-2b^{2}c_{2,2}c_{3,1}+2c_{3,1}^{3}}\right):b}\right]\,,\\\
\hat{w}=-\sigma^{3}\mathopen{}\mathclose{{}\left(b^{6}c_{0,3}-b^{4}c_{2,2}^{2}\sigma-b^{4}c_{1,3}c_{3,1}\sigma+4b^{2}c_{2,2}c_{3,1}^{2}\sigma-3c_{3,1}^{4}\sigma}\right)\,.\end{gathered}$
(271)

This model is equivalent to the $\mathcal{Q}(3,3,2,1,0,1,2)$ model of
KuntzlerTateTrees  after trivial redefinitions of the parameters. It supports
an $\mathfrak{e}_{7}$ gauge algebra along $\\{\sigma=0\\}$. In 6D F-theory
models constructed from this elliptic fibration, the codimension-two locus
$\\{\sigma=b=0\\}$ supports full hypermultiplets of
$\bm{56}_{1/2}\oplus\bm{1}_{1}$ matter, in line with the observations in
KuntzlerTateTrees . The model also supports $\bm{1}_{2}$ hypermultiplets at
$\\{b=c_{3,1}=0\\}$ and $\bm{1}_{1}$ hypermultiplets at

$\mathopen{}\mathclose{{}\left\\{\frac{\hat{y}}{\sigma^{3}}=\frac{\hat{w}}{\sigma^{3}}=0}\right\\}\textbackslash\\{b=c_{3,1}=0\\}\,.$
(272)

While $f$ and $g$ vanish to orders $4$ and $6$ at the codimension-two locus
$\\{\sigma=c_{0,3}=0\\}$, one can avoid the problems created by this locus by
choosing the classes of the parameters such that $[\sigma]\cdot[c_{0,3}]=0$.
For instance, one can choose the divisor classes such that $[c_{0,3}]=0$, in
which case $[b]=-2K_{B}{}-\frac{3}{2}[\sigma]$. In this situation, one can
verify that the charged matter spectrum satisfies the gauge and mixed
gauge–gravitational anomaly cancellation conditions with
$0pt=-2K_{B}{}+2[b]-\frac{3}{2}[\sigma]$. From the $\mathfrak{u}(1)$ charge of
the $\bm{56}$ matter, one can see that the generating section, and therefore
the model as a whole, has $\nu=1$.

We now consider multiples of the generating section $\hat{s}$. The $\bm{56}$
matter at $\\{\sigma=b=0\\}$ would have pseudo-charge $q=\frac{1}{2}m$ for the
section $m\hat{s}$ with $\nu=\overline{m}_{2}$. The $m\hat{s}$ sections
therefore give us analogues for all of the possible $\bm{56}$ charges
described Eq. 264. Table 21 lists the orders of vanishing of the $m\hat{s}$
section components for various values of $m$, which follow a clear pattern.
The codimension-one orders of vanishing are described by the equations

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\vec{\tau}_{\operatorname{E}_{7}}(\nu)\,.$
(273)

The codimension-two orders of vanishing are given by

$\displaystyle\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(2q^{2}+T_{\operatorname{E}_{7}}(\nu)}\right)\,,$
(274)
$\displaystyle\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})\,.$
(275)

The center of $\operatorname{E}_{8}$ is trivial, and $\mu$,
$T_{\operatorname{E}_{8}}(\mu)$, and $\vec{\tau}_{\operatorname{E}_{8}}(\mu)$
are all trivial as well. These formulas therefore agree with the expectations
from Section 2.2.

$\lvert q\rvert$ | $\nu$ | Codimension One | Codimension Two
---|---|---|---
$\frac{1}{2}$ | $1$ | $(2,3,0,3)$ | $(2,3,1,4)$
$1$ | $0$ | $(0,0,0,0)$ | $(2,3,1,4)$
$\frac{3}{2}$ | $1$ | $(2,3,0,3)$ | $(6,9,3,12)$
$2$ | $0$ | $(0,0,0,0)$ | $(8,12,4,16)$
$\frac{5}{2}$ | $1$ | $(2,3,0,3)$ | $(14,21,7,28)$
$3$ | $0$ | $(0,0,0,0)$ | $(18,27,9,36)$
$\frac{7}{2}$ | $1$ | $(2,3,0,3)$ | $(26,39,13,52)$
$4$ | $0$ | $(0,0,0,0)$ | $(32,48,16,64)$
$\frac{9}{2}$ | $1$ | $(2,3,0,3)$ | $(42,63,21,84)$
$5$ | $0$ | $(0,0,0,0)$ | $(50,75,25,100)$
$\frac{11}{2}$ | $1$ | $(2,3,0,3)$ | $(62,93,31,124)$
$6$ | $0$ | $(0,0,0,0)$ | $(72,108,36,144)$
$\frac{13}{2}$ | $1$ | $(2,3,0,3)$ | $(86,129,43,172)$
$7$ | $0$ | $(0,0,0,0)$ | $(98,147,49,196)$
$\frac{15}{2}$ | $1$ | $(2,3,0,3)$ | $(114,171,57,228)$
$8$ | $0$ | $(0,0,0,0)$ | $(128,192,64,256)$
$\frac{17}{2}$ | $1$ | $(2,3,0,3)$ | $(146,219,73,292)$
$9$ | $0$ | $(0,0,0,0)$ | $(162,243,81,324)$
Table 21: Orders of vanishing of $(\hat{x},\hat{y},\hat{z},\hat{w})$ for
combinations of $\nu$ and the $\bm{56}$ pseudo-charge $q$ at codimension one
($\\{\sigma=0\\}$) and codimension two ($\\{\sigma=b=0\\}$). These data are
found by calculating multiples of the generating section 271 for the
$\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ Weierstrass model in Eq. 270; pseudo-
charge $q$ is realized by the section $2q\hat{s}$. One should observe the same
orders of vanishing in Weierstrass models supporting
$\bm{56}_{q}\oplus\bm{1}_{2q}$ matter.

###  10.1 Connection to EDS valuations

Since all of the allowed combinations of $\nu$ and pseudo-charge $q$ can be
obtained from multiples of the generating section $\hat{s}$, we expect that
the formulas for the orders of vanishing above should agree with the formulas
for EDS valuations in StangeEllTrouble . As before, we start by finding the
EDS valuations corresponding to the codimension-one behavior and the
codimension-two behavior of Eq. 270. For the $\text{III}^{*}$ singularities at
codimension-one, the generating section 271 vanishes to orders $(2,2,0,3)$ at
$\\{\sigma=0\\}$. The corresponding EDS valuations are calculated by a
procedure similar to that in Section C.4.2 and are given by Eq. 384:

$v(W_{m}^{(1)})=\frac{3}{4}m^{2}-\begin{cases}0&2\mid m\\\ \frac{3}{4}&2\nmid
m\end{cases}\,.$ (276)

For the $\text{II}^{*}$ singularities at codimension-two, the generating
section 271 vanishes to orders $(2,3,1,4)$ at $\\{\sigma=b=0\\}$. The
codimension-two EDS valuations are calculated by a procedure similar to that
in Section C.5 and are given by Eq. 392:

$v(W_{m}^{(2)})=m^{2}\,.$ (277)

The formula for $\operatorname{ord}_{2}(\hat{z})$ of the $m\hat{s}$ sections
should be given by the difference

$v(W^{(2)}_{m})-v(W^{(1)}_{m})=\frac{1}{4}m^{2}+\begin{cases}0&2\mid m\\\
\frac{3}{4}&2\nmid m\end{cases}\,.$ (278)

Noting that the section $m\hat{s}$ realizes pseudo-charge $q=\frac{m}{2}$ for
the $\bm{56}$ matter, we should expect that

$\operatorname{ord}_{2}(\hat{z})=q^{2}+\frac{3}{4}\overline{2q}_{2}=\frac{1}{2}\mathopen{}\mathclose{{}\left(2q^{2}+\frac{3}{2}\overline{2q}_{2}}\right)\,,$
(279)

exactly in agreement with the observed order of vanishing data, Eq. 274, and
the proposals in Section 2.2.

##  11 Matter charged under multiple gauge algebra factors

Up to this point, we have only analyzed situations where the gauge algebra is
either $\mathfrak{u}(1)$ or $\mathfrak{g}\oplus\mathfrak{u}(1)$, where
$\mathfrak{g}$ is a simple Lie algebra. However, F-theory models can have more
complicated gauge algebras consisting of multiple $\mathfrak{u}(1)$ and
nonabelian factors. Matter in such models can be charged under many of these
factors. For instance, an F-theory model might have an
$\mathfrak{su}(m)\oplus\mathfrak{su}(n)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$
gauge algebra and matter in the $(\bm{m},\overline{\bm{n}})_{q_{1},q_{2}}$
representation. This section focuses on these situations. We divide the
analysis into two parts. We first discuss how to approach situations with
matter charged under multiple $\mathfrak{u}(1)$ factors. We then turn to
matter charged under multiple simple nonabelian factors and (at least) one
$\mathfrak{u}(1)$. Despite the added complications in these examples, we will
see that the connections between the $\mathfrak{u}(1)$ charges and orders of
vanishing proposed in Section 2.2 still hold.

###  11.1 Multiple $\mathfrak{u}(1)$ factors

If there are $L$ $\mathfrak{u}(1)$ factors, the Mordell–Weil rank of the
corresponding elliptic fibration should be $L$, and each $\mathfrak{u}(1)$
factor corresponds to a particular generating section. The formulas in Section
2.2 should hold without modification for each generating section. More
specifically, if the nonabelian part of the gauge algebra is simple and
nontrivial, each generating section has an associated integer $\nu$ that
specifies which component of the resolved fiber at codimension-one is hit by
the section.414141If the nonabelian part of the gauge algebra is semisimple
with $K$ simple nonabelian factors, there are a total of $K\times L$ integers
playing the role of $\nu$, one for each combination of generating section and
simple nonabelian factor. Along the codimension-one locus supporting this
nonabelian gauge algebra, the section components for a particular generating
section should vanish to the orders dictated by its $\nu$ integer. If some
matter has a certain charge under a particular $\mathfrak{u}(1)$, the section
components for the corresponding generating section should vanish to the
orders proposed in Section 2.2 at the matter locus. To determine some matter
field’s charge under all of the $L$ $\mathfrak{u}(1)$ factors, one would apply
the procedures outlined in Section 2.2 to each generating section.

Note that there can be different choices for the basis of these
$\mathfrak{u}(1)$’s when there are multiple $\mathfrak{u}(1)$ factors. The
$\mathfrak{u}(1)$ charges of matter can change if one uses a different
$\mathfrak{u}(1)$ basis. In the elliptic fibration, the basis of
$\mathfrak{u}(1)$ algebras corresponds to the basis of generating sections. If
one changes the $\mathfrak{u}(1)$ basis, the generating section basis should
change as well, and the new $\mathfrak{u}(1)$ charges should agree with the
orders of vanishing for the new basis sections as per the rules in Section
2.2. Thus, even though the charge spectrum changes with the $\mathfrak{u}(1)$
basis, the corresponding change in the basis of sections should ensure that
the formulas in Section 2.2 still hold. As long as one maintains a consistent
relation between the $\mathfrak{u}(1)$ basis and the basis of generating
sections, changing the basis of $\mathfrak{u}(1)$ algebras should not pose any
problems.

Given that the formulas in Section 2.2 are essentially unchanged when there
are multiple $\mathfrak{u}(1)$ factors, we will not extensively discuss
examples of this type here. However, Section 12 lists some models in the prior
F-theory literature with multiple $\mathfrak{u}(1)$ factors, all of which
agree with the ideas discussed here and in Section 2.2.

###  11.2 Multiple nonabelian factors

Now let us consider the case where the nonabelian part of the gauge algebra
has multiple simple factors. Specifically, suppose the gauge algebra is

$\bigoplus_{\kappa=1}^{K}\mathfrak{g}_{\kappa}\oplus\mathfrak{u}(1)\,,$ (280)

where each $\mathfrak{g}_{\kappa}$ is a simple Lie algebra. As before, we
assume that the $\mathfrak{g}_{\kappa}$ are all simply-laced. Because the
argument above suggests that the presence of multiple $\mathfrak{u}(1)$
factors does not significantly change the analysis, we assume for simplicity
that there is only a single $\mathfrak{u}(1)$ associated with a generating
section $\hat{s}$. Each $\mathfrak{g}_{\kappa}$ is supported along a
codimension-one locus $\\{\sigma_{\kappa}=0\\}$ in the base with singular
fibers of the appropriate type. We also have integers $\nu_{\kappa}$
describing which resolved fiber component along $\\{\sigma_{\kappa}=0\\}$ is
hit by $\hat{s}$. According to Section 2.2, the section components for
$\hat{s}$ should vanish to the orders described by Eq. 17 at each of the
$\mathfrak{g}_{\kappa}$ loci:

$\operatorname*{ord}_{\sigma_{k}=0}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname*{ord}_{\sigma_{k}=0}(\hat{x}),\operatorname*{ord}_{\sigma_{k}=0}(\hat{y}),\operatorname*{ord}_{\sigma_{k}=0}(\hat{w})}\right)=\vec{\tau}_{G_{\kappa}}(\nu_{\kappa})\,.$
(281)

Here, $G_{\kappa}$ is the universal covering group for
$\mathfrak{g}_{\kappa}$. Now consider matter that is charged under multiple
$\mathfrak{g}_{\kappa}$ and that is supported at the codimension-two
intersection locus of the corresponding $\\{\sigma_{\kappa}=0\\}$. If the
singularity type enhances at this matter locus to one associated with the
group $H$, the section components should vanish to the orders specified by
Equations 18 and 19 for some $\mu$ in the center of $H$. The index $i$ in Eq.
18 runs over those simple nonabelian algebras under which the matter is
charged.

To illustrate these ideas, let us focus on models with
$\mathfrak{su}(m)\oplus\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ gauge algebras
and matter in the bifundamental $(\bm{m},\overline{\bm{n}})_{q}$
representation. At the codimension-two locus supporting the bifundamental
matter, the singularity type enhances from
$\text{I}_{m}^{s}\times\text{I}_{n}^{s}$ (or
$\operatorname{SU}(m)\times\operatorname{SU}(n)$) to $\text{I}_{m+n}^{s}$ (or
$\operatorname{SU}(m+n)$). If we have integers $(\nu_{m},\nu_{n})$ describing
the fiber components hit by $\hat{s}$, then the allowed $\mathfrak{u}(1)$
charges for this bifundamental matter are given by

$\frac{\nu_{m}}{m}-\frac{\nu_{n}}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (282)

According to the formulas in Section 2.2, the section components of $\hat{s}$
should vanish to the orders

$\begin{gathered}\operatorname*{ord}_{\sigma_{m}=0}(\hat{z})=0\,,\\\
\mathopen{}\mathclose{{}\left(\operatorname*{ord}_{\sigma_{m}=0}(\hat{x}),\operatorname*{ord}_{\sigma_{m}=0}(\hat{y}),\operatorname*{ord}_{\sigma_{m}=0}(\hat{w})}\right)=\vec{\tau}_{\operatorname{SU}(m)}(\nu_{m})=(0,1,1)\times
u_{m}(\nu_{m})\end{gathered}$ (283)

at the codimension-one locus $\\{\sigma_{m}=0\\}$ supporting the
$\mathfrak{su}(m)$ algebra and to the orders

$\begin{gathered}\operatorname*{ord}_{\sigma_{n}=0}(\hat{z})=0\,,\\\
\mathopen{}\mathclose{{}\left(\operatorname*{ord}_{\sigma_{n}=0}(\hat{x}),\operatorname*{ord}_{\sigma_{n}=0}(\hat{y}),\operatorname*{ord}_{\sigma_{n}=0}(\hat{w})}\right)=\vec{\tau}_{\operatorname{SU}(n)}(\nu_{n})=(0,1,1)\times
u_{n}(\nu_{n})\end{gathered}$ (284)

at the codimension-one locus $\\{\sigma_{n}=0\\}$ supporting the
$\mathfrak{su}(n)$ algebra. At the matter locus, the $\hat{z}$ component of
the generating section should vanish to order

$\displaystyle\operatorname{ord}_{2}(\hat{z})$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{mn}{m+n}q^{2}+T_{\operatorname{SU}(m)}(\nu_{m})+T_{\operatorname{SU}(n)}(\nu_{n})-T_{\operatorname{SU}(m+n)}(\mu)}\right)$
(285)
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{mn}{m+n}q^{2}+\frac{\nu_{m}(m-\nu_{m})}{m}+\frac{\nu_{n}(n-\nu_{n})}{n}-\frac{\mu(m+n-\mu)}{m+n}}\right)\,,$

where $\mu$ is an integer representing an element of the center of
$\operatorname{SU}(m+n)$. Additionally, we would expect that

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=(2,3,4)\times\operatorname{ord}_{2}(\hat{z})+\vec{\tau}_{\operatorname{SU}(m+n)}(\mu)$
(286)

One can verify that this formula gives integral
$\operatorname{ord}_{2}(\hat{z})$ if

$\mu=\overline{\mathopen{}\mathclose{{}\left(mq+\nu_{n}+\frac{m}{n}\nu_{n}}\right)}_{m+n}\,.$
(287)

This expression for $\mu$ also reduces to the $\mathfrak{su}(m)$ fundamental
$\mu$ expression when $n=1$.

We can check these proposals for the bifundamentals by looking at a model with
an $\mathfrak{su}(4)\oplus\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ gauge
algebra. This example is given by the Weierstrass model

$\displaystyle y^{2}$
$\displaystyle=x^{3}+\mathopen{}\mathclose{{}\left(-\frac{c_{2}^{2}}{3}+c_{1}c_{3}-c_{0}b^{2}}\right)xz^{4}$
(288)
$\displaystyle\qquad\quad+\mathopen{}\mathclose{{}\left(\frac{2}{27}c_{2}^{3}-\frac{2}{3}b^{2}c_{0}c_{2}+\frac{1}{4}b^{2}c_{1}^{2}-\frac{1}{3}c_{1}c_{3}c_{2}+c_{0}c_{3}^{2}}\right)z^{6}\,,$

where

$\begin{gathered}c_{0}=\frac{1}{4}\sigma_{4}^{2}\sigma_{5}^{4}\mathopen{}\mathclose{{}\left({b_{2}^{\prime}}^{2}+4{c_{0}^{\prime}}\sigma_{4}}\right)\,,\\\
c_{1}=\frac{1}{2}\sigma_{4}\sigma_{5}^{2}({b_{1}^{\prime}}{b_{2}^{\prime}}+2{c_{1}^{\prime}}\sigma_{4})\,,\\\
c_{2}=\frac{1}{4}\mathopen{}\mathclose{{}\left(4\sigma_{4}\sigma_{5}c_{2}^{\prime}+{b_{1}^{\prime}}^{2}}\right)\,,\\\
c_{3}=\frac{b{b_{1}^{\prime}}}{2}+c_{3}^{\prime}\sigma_{4}\sigma_{5}\,.\end{gathered}$
(289)

Because the discriminant is proportional to $\sigma_{4}^{4}\sigma_{5}^{5}$ and
the split conditions are satisfied, this model supports an $\mathfrak{su}(4)$
algebra on $\\{\sigma_{4}=0\\}$ and an $\mathfrak{su}(5)$ algebra on
$\\{\sigma_{5}=0\\}$. Additionally, it admits a generating section $\hat{s}$
with section components

$\displaystyle\hat{x}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{b{b_{1}^{\prime}}}{2}+c_{3}^{\prime}\sigma_{4}\sigma_{5}}\right)^{2}-\frac{1}{6}b^{2}\mathopen{}\mathclose{{}\left({b_{1}^{\prime}}^{2}+4c_{2}^{\prime}\sigma_{4}\sigma_{5}}\right)\,,$
(290) $\displaystyle\hat{y}$
$\displaystyle=-\frac{1}{4}\sigma_{4}\sigma_{5}\Big{(}b^{4}{b_{1}^{\prime}}{b_{2}^{\prime}}\sigma_{5}+2b^{4}{c_{1}^{\prime}}\sigma_{4}\sigma_{5}-2b^{3}{b_{1}^{\prime}}{c_{2}^{\prime}}+2b^{2}{b_{1}^{\prime}}^{2}{c_{3}^{\prime}}$
$\displaystyle\qquad\qquad\qquad\quad-4b^{2}{c_{2}^{\prime}}{c_{3}^{\prime}}\sigma_{4}\sigma_{5}+6b{b_{1}^{\prime}}{c_{3}^{\prime}}^{2}\sigma_{4}\sigma_{5}+4{c_{3}^{\prime}}^{3}\sigma_{4}^{2}\sigma_{5}^{2}\Big{)}\,,$
$\displaystyle\hat{z}$ $\displaystyle=b\,,$ $\displaystyle\hat{w}$
$\displaystyle=-\frac{1}{4}\sigma_{4}\sigma_{5}\Big{(}b^{6}{b_{2}^{\prime}}^{2}\sigma_{4}\sigma_{5}^{3}+4b^{6}{c_{0}^{\prime}}\sigma_{4}^{2}\sigma_{5}^{3}-b^{5}{b_{1}^{\prime}}^{2}{b_{2}^{\prime}}\sigma_{5}-2b^{5}{b_{1}^{\prime}}{c_{1}^{\prime}}\sigma_{4}\sigma_{5}+2b^{4}{b_{1}^{\prime}}^{2}{c_{2}^{\prime}}$
$\displaystyle\qquad\qquad\qquad\quad-2b^{4}{b_{1}^{\prime}}{b_{2}^{\prime}}{c_{3}^{\prime}}\sigma_{4}\sigma_{5}^{2}-4b^{4}{c_{1}^{\prime}}{c_{3}^{\prime}}\sigma_{4}^{2}\sigma_{5}^{2}-4b^{4}{c_{2}^{\prime}}^{2}\sigma_{4}\sigma_{5}$
$\displaystyle\qquad\qquad\qquad\quad-2b^{3}{b_{1}^{\prime}}^{3}{c_{3}^{\prime}}+16b^{3}{b_{1}^{\prime}}{c_{2}^{\prime}}{c_{3}^{\prime}}\sigma_{4}\sigma_{5}-14b^{2}{b_{1}^{\prime}}^{2}{c_{3}^{\prime}}^{2}\sigma_{4}\sigma_{5}$
$\displaystyle\qquad\qquad\qquad\quad+16b^{2}{c_{2}^{\prime}}{c_{3}^{\prime}}^{2}\sigma_{4}^{2}\sigma_{5}^{2}-24b{b_{1}^{\prime}}{c_{3}^{\prime}}^{3}\sigma_{4}^{2}\sigma_{5}^{2}-12{c_{3}^{\prime}}^{4}\sigma_{4}^{3}\sigma_{5}^{3}\Big{)}\,.$

The complete gauge algebra is therefore
$\mathfrak{su}(4)\oplus\mathfrak{su}(5)\oplus\mathfrak{u}(1)$. The charged
matter spectrum, which is summarized in Table 22, satisfies the anomaly
cancellation conditions with
$0pt=-2K_{B}+2[b]-\frac{3}{4}[\sigma_{4}]-\frac{4}{5}[\sigma_{5}]$. As can be
seen from the $\mathfrak{u}(1)$ charges, we can take $\nu_{4}$ and $\nu_{5}$
to both be 1; this agrees with the orders of vanishing of
$(\hat{x},\hat{y},\hat{z},\hat{w})$ at $\\{\sigma_{4}=0\\}$ and
$\\{\sigma_{5}=0\\}$. The codimension-two orders of vanishing at the matter
loci in Table 22 also agree with the formulas discussed previously.

Matter | Locus | $\operatorname{ord}_{(2)}$
---|---|---
$(\bm{4},\overline{\bm{5}})_{\mathrlap{1/20}}$ | $\\{\sigma_{4}=\sigma_{5}=0\\}$ | $(0,2,0,2)$
$(\bm{4},\bm{1})_{\mathrlap{5/4}}$ | $\\{\sigma_{4}=b=0\\}$ | $(2,3,1,4)$
$(\bm{4},\bm{1})_{\mathrlap{-3/4}}$ | $\\{\sigma_{4}=b^{2}{b_{2}^{\prime}}\sigma_{5}-2b{c_{2}^{\prime}}+2{b_{1}^{\prime}}{c_{3}^{\prime}}=0\\}$ | $(0,2,0,2)$
$(\bm{4},\bm{1})_{\mathrlap{1/4}}$ | $\\{\sigma_{4}=b{b_{2}^{\prime}}^{3}\sigma_{5}^{2}-2{b_{1}^{\prime}}^{2}{c_{0}^{\prime}}+2{b_{1}^{\prime}}{b_{2}^{\prime}}{c_{1}^{\prime}}-2{b_{2}^{\prime}}^{2}{c_{2}^{\prime}}\sigma_{5}=0\\}$ | $(0,1,0,1)$
$(\bm{6},\bm{1})_{\mathrlap{1/2}}$ | $\\{\sigma_{4}=b_{1}^{\prime}=0\\}$ | $(1,2,0,2)$
$(\bm{1},\bm{5})_{\mathrlap{6/5}}$ | $\\{\sigma_{5}=b=0\\}$ | $(2,3,1,4)$
$(\bm{1},\bm{5})_{\mathrlap{-4/5}}$ | $\\{\sigma_{5}=b{c_{2}^{\prime}}-{b_{1}^{\prime}}{c_{3}^{\prime}}=0\\}$ | $(0,2,0,2)$
$(\bm{1},\bm{5})_{\mathrlap{1/5}}$ | $\\{\sigma_{5}={b_{1}^{\prime}}^{2}{c_{0}^{\prime}}-{b_{1}^{\prime}}{b_{2}^{\prime}}{c_{1}^{\prime}}-{c_{1}^{\prime}}^{2}\sigma_{4}=0\\}$ | $(0,1,0,1)$
$(\bm{1},\bm{10})_{\mathrlap{2/5}}$ | $\\{\sigma_{5}=b_{1}^{\prime}=0\\}$ | $(1,2,0,2)$
$(\bm{1},\bm{1})_{\mathrlap{2}}$ | $\\{b=c_{3}^{\prime}=0\\}$ | $(1,2,3,4)$
Table 22: Matter loci for the
$\mathfrak{su}(4)\oplus\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ model, along
with the codimension-two orders of vanishing of
$(\hat{x},\hat{y},\hat{z},\hat{w})$. In addition to the matter loci listed
here, there are $(\bm{1},\bm{1})_{1}$ singlets located at loci where
$\hat{y}/(\sigma_{4}\sigma_{5})$ and $\hat{w}/(\sigma_{4}\sigma_{5})$
simultaneously vanish that are not part of the matter loci above. The model
may also have delocalized $(\bm{6},\bm{1})_{0}$ and $(\bm{1},\bm{10})_{0}$
adjoint matter that can propagate along the $\\{\sigma_{4}=0\\}$ and
$\\{\sigma_{5}=0\\}$ gauge divisors.

Importantly, the model admits $(\bm{4},\overline{\bm{5}})_{1/20}$ matter at
the locus $\\{\sigma_{4}=\sigma_{5}=0\\}$, where the section components vanish
to orders $(0,2,0,2)$. As with the previous examples, we can use multiples of
the generating section to predict the orders of vanishing for at least some
combinations of $\nu_{4}$, $\nu_{5}$, and bifundamental $\mathfrak{u}(1)$
charge $q$. Specifically, we can obtain the combinations of the form

$\nu_{4}=\overline{r}_{4}\,,\quad\nu_{5}=\overline{r}_{5}\,,\quad
q=\frac{r}{20}\,,$ (291)

where $r$ is an integer. The resulting orders of vanishing, which are listed
in Table 23, satisfy Equations 283 through 287. Thus, the proposals in Section
2.2 hold for the $(\bm{4},\overline{\bm{5}})$ pseudo-charges considered here,
suggesting that they should hold generally for
$(\bm{4},\overline{\bm{5}})_{q}$ matter with genuine $\mathfrak{u}(1)$ charge
$q$.

$\lvert q\rvert$ | $\nu_{4}$ | $\nu_{5}$ | $\operatorname*{ord}_{\sigma_{4}=0}$ | $\operatorname*{ord}_{\sigma_{5}=0}$ | $\operatorname{ord}_{(2)}$
---|---|---|---|---|---
$\frac{1}{20}$ | $1$ | $1$ | $(0,1,0,1)$ | $(0,1,0,1)$ | $(0,2,0,2)$
$\frac{1}{10}$ | $2$ | $2$ | $(0,2,0,2)$ | $(0,2,0,2)$ | $(0,4,0,4)$
$\frac{3}{20}$ | $3$ | $3$ | $(0,1,0,1)$ | $(0,2,0,2)$ | $(0,3,0,3)$
$\frac{1}{5}$ | $0$ | $4$ | $(0,0,0,0)$ | $(0,1,0,1)$ | $(0,1,0,1)$
$\frac{1}{4}$ | $1$ | $0$ | $(0,1,0,1)$ | $(0,0,0,0)$ | $(0,1,0,1)$
$\frac{3}{10}$ | $2$ | $1$ | $(0,2,0,2)$ | $(0,1,0,1)$ | $(0,3,0,3)$
$\frac{7}{20}$ | $3$ | $2$ | $(0,1,0,1)$ | $(0,2,0,2)$ | $(0,4,0,4)$
$\frac{2}{5}$ | $0$ | $3$ | $(0,0,0,0)$ | $(0,2,0,2)$ | $(0,2,0,2)$
$\frac{9}{20}$ | $1$ | $4$ | $(0,1,0,1)$ | $(0,1,0,1)$ | $(2,3,1,4)$
$\frac{1}{2}$ | $2$ | $0$ | $(0,2,0,2)$ | $(0,0,0,0)$ | $(0,2,0,2)$
$\frac{11}{20}$ | $3$ | $1$ | $(0,1,0,1)$ | $(0,1,0,1)$ | $(0,4,0,4)$
$\frac{3}{5}$ | $0$ | $2$ | $(0,0,0,0)$ | $(0,2,0,2)$ | $(0,3,0,3)$
$\frac{13}{20}$ | $1$ | $3$ | $(0,1,0,1)$ | $(0,2,0,2)$ | $(2,4,1,5)$
$\frac{7}{10}$ | $2$ | $4$ | $(0,2,0,2)$ | $(0,1,0,1)$ | $(2,4,1,5)$
$\frac{3}{4}$ | $3$ | $0$ | $(0,1,0,1)$ | $(0,0,0,0)$ | $(0,3,0,3)$
$\frac{4}{5}$ | $0$ | $1$ | $(0,0,0,0)$ | $(0,1,0,1)$ | $(0,4,0,4)$
$\frac{17}{20}$ | $1$ | $2$ | $(0,1,0,1)$ | $(0,2,0,2)$ | $(2,5,1,6)$
Table 23: Orders of vanishing for $(\hat{x},\hat{y},\hat{z},\hat{w})$ at
codimension one and two for combinations of $\nu_{4}$, $\nu_{5}$, and pseudo-
charge $q$ for $(\bm{4},\overline{\bm{5}})$ bifundamental matter. These data
are found using the
$\mathfrak{su}(4)\oplus\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ construction
described by Equations 288 and 289.

These same ideas should be applicable for representations beyond the
$(\bm{m},\overline{\bm{n}})$ bifundamentals. One can also have
$(\bm{m},\bm{n})_{q}$ bifundamentals of
$\mathfrak{su}(m)\oplus\mathfrak{su}(n)\oplus\mathfrak{u}(1)$, whose allowed
$\mathfrak{u}(1)$ charges are given by

$\frac{\nu_{m}}{m}+\frac{\nu_{n}}{n}+j\text{ for }j\in\mathbb{Z}\,.$ (292)

These bifundamentals also occur at the intersection of codimension-one
$\text{I}_{m}^{*s}$ and $\text{I}_{n}^{*s}$ loci where the singularity type
enhances to $\text{I}_{m+n}^{*s}$. Therefore, we expect that, for instance,

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{mn}{m+n}q^{2}+T_{\operatorname{SU}(m)}(\nu_{m})+T_{\operatorname{SU}(n)}(\nu_{n})-T_{\operatorname{SU}(m+n)}(\mu)}\right)$
(293)

for some $\mu$ representing an element of $\mathbb{Z}_{m+n}$, the center of
$\operatorname{SU}(m+n)$. This formula gives integer
$\operatorname{ord}_{2}(\hat{z})$ for

$\mu=\overline{\mathopen{}\mathclose{{}\left(mq-\nu_{n}-\frac{m}{n}\nu_{n}}\right)}_{m+n}\,,$
(294)

and the expression expression for $\mu$ also reduces to that for
$\mathfrak{su}(m)$ fundamentals when $n=1$. The proposals in Section 2.2
should also cover other types of matter charged under multiple nonabelian
gauge algebras, such as $(\bm{m},\bm{n})_{q}$ matter charged under
$\mathfrak{so}(m)\oplus\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ algebras.
However, there are also matter representations involving only a simple
nonabelian algebra that occur at singularities of a gauge divisor. If the
gauge algebra is $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$, for instance, one
can have matter in the adjoint or symmetric representations of
$\mathfrak{su}(n)$ localized at double point singularities of the gauge
divisor SadovGreenSchwarz ; MorrisonTaylorMaS ; CveticKleversPiraguaTaylor ;
AndersonGrayRaghuramTaylorMiT ; KleversTaylor ; KleversEtAlExotic . If we
focus on a sufficiently small neighborhood near the matter locus, these
situations more closely resemble the bifundamental matter situations
considered above: the matter locus locally appears to be the intersection of
two codimension-one $\text{I}_{n}^{*s}$ loci, even though these two components
are actually connected due to global properties of the gauge divisor. If such
matter also has a $\mathfrak{u}(1)$ charge, the orders of vanishing of the
section components may be described by formulas akin to those for
bifundamental matter. These more exotic matter representations lie outside of
the scope of this paper, so we do not rigorously analyze them here.
Nevertheless, they pose an interesting direction for future work.

##  12 Examples in explicit constructions

We have so far performed somewhat indirect tests of the proposals in Section
2.2. Specifically, we have used multiples of generating sections admitting
relatively small $\mathfrak{u}(1)$ charges as proxies for genuine generating
sections admitting larger $\mathfrak{u}(1)$ charges. Underlying this strategy
is the assumption that both types of sections should behave similarly at the
appropriate loci. Since the $\mathfrak{u}(1)$ charge only depends on the local
behavior of sections near a matter locus, this assumption is reasonable. This
strategy of course lets us circumvent the difficulties in finding F-theory
models admitting large charges.

However, we can also test the proposals by examining previous F-theory models
with $\mathfrak{u}(1)$ algebras. We have verified that, for the following
models in the prior literature, the massless charged matter spectrum agrees
with the formulas in Section 2.2:

  * •

The Morrison–Park construction in MorrisonParkU1 , which has a
$\operatorname{U}(1)$ gauge group and admits $\bm{1}_{1}$ and $\bm{1}_{2}$
matter.

  * •

The $\operatorname{U}(1)$ constructions in KleversEtAlToric  and Raghuram34
that admit $\bm{1}_{1}$, $\bm{1}_{2}$, and $\bm{1}_{3}$ matter.

  * •

The $\operatorname{U}(1)$ construction in Raghuram34  that admits
$\bm{1}_{1}$, $\bm{1}_{2}$, $\bm{1}_{3}$, and $\bm{1}_{4}$ matter.

  * •

The $\operatorname{U}(1)\times\operatorname{U}(1)$ models in
CveticKleversPiraguaMultU1  and CveticKleversPiraguaTaylor . By looking at
linear combinations of the generating sections, we can also recover the
relative signs of the charges under the two $\mathfrak{u}(1)$ factors.

  * •

The $\operatorname{U}(1)\times\operatorname{U}(1)\times\operatorname{U}(1)$
model in CveticKleversPiraguaSong . Again, we can also recover the relative
signs of charges under the three $\mathfrak{u}(1)$ factors by considering
linear combinations of the generating sections.

  * •

The models in KleversEtAlExotic  with $\mathfrak{u}(1)$ algebras, namely the
$F_{2}$, $F_{3}$, $F_{5}$, $F_{6}$, $F_{7}$, $F_{8}$, $F_{9}$, $F_{12}$,
$F_{14}$, and $F_{15}$ models. Note that the charge spectra are derived using
non-Weierstrass descriptions of the elliptic fibrations, so when there are
multiple $\mathfrak{u}(1)$ gauge algebras, once must be careful in matching
the basis of Weierstrass generating sections to the basis of $\mathfrak{u}(1)$
charges.

  * •

The $\mathfrak{su}(5)\oplus\mathfrak{u}(1)$,
$\mathfrak{so}(10)\oplus\mathfrak{u}(1)$,
$\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$, and
$\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ models in KuntzlerTateTrees . While
KuntzlerTateTrees  includes models with other gauge algebras, matter spectra
are not listed for these alternative models. Note that the $\mathfrak{u}(1)$
charges are given in an alternative set of units where all charges, including
those for matter also charged under a nonabelian algebra, are integral. The
lattice spacing for singlet charges in this convention may be an integer
larger than 1. While these models are tunings of elliptic fibrations with
fibers embedded $\mathbb{P}^{1,1,2}$, we can form the equivalent Weierstrass
models by performing the same tunings in the Morrison–Park Weierstrass model
given in MorrisonParkU1 .

  * •

The $\mathfrak{su}(5)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$,
$\mathfrak{su}(4)\oplus\mathfrak{u}(1)\oplus\mathfrak{u}(1)$,
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$, and
$\mathfrak{su}(4)\oplus\mathfrak{u}(1)$ models in BorchmannSU5Top . Again, the
$\mathfrak{u}(1)$ charges in these models are given in an alternative set of
units where all charges are integral. The
$\mathfrak{su}(5)\oplus\mathfrak{u}(1)$ and
$\mathfrak{su}(4)\oplus\mathfrak{u}(1)$ models given there are either
equivalent to or specializations of models in KuntzlerTateTrees .

  * •

The $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ model in
RaghuramTaylorTurnerSM , which is a generalization of the
$\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$ model in
KleversEtAlToric . The $\mathfrak{su}(5)$ enhancement discussion in
RaghuramTaylorTurnerSM  also makes use of a “would-be”
$(\bm{3},\bm{2})_{-5/6}$ matter locus. At this would-be locus, the section
components vanish to the orders appropriate for $(\bm{3},\bm{2})_{-5/6}$
matter according to the equations in Sections 2.2 and 11.2.

  * •

The models involving $\mathfrak{u}(1)$ factors in
RaghuramTaylorTurnerEnhancement , including the
$\mathfrak{su}(18)\oplus\mathfrak{u}(1)$ and
$\mathfrak{su}(19)\oplus\mathfrak{u}(1)$ models. While these models were not
analyzed using resolutions, the matter spectra given there, which can be
recovered with the techniques presented here, agree with the anomaly
cancellation conditions and the expectations from Higgsing related nonabelian
models.

These models provide a broad set of gauge algebras, covering many of the
representations discussed here. Many of them also have multiple
$\mathfrak{u}(1)$ factors and matter charged under multiple nonabelian
factors, in further support of the ideas in Section 11. The fact that a
variety of models in independent works follow the formulas in Section 2.2,
together with the evidence presented in previous sections, suggests that our
proposals should hold more broadly.

##  13 Conclusions

To summarize, we proposed a series of formulas, outlined in Section 2.2, that
relate the $\mathfrak{u}(1)$ charge of matter supported at a particular locus
to the orders of vanishing of the generating section’s components at that
locus. As discussed in Section 12, these formulas accurately reproduce the
$\mathfrak{u}(1)$ charge spectrum of various F-theory models from the prior
literature. They also correctly describe the results of the indirect method of
Section 5 for analyzing large $\mathfrak{u}(1)$ charges, and they appear to be
related to formulas from StangeEllTrouble  for the $p$-adic valuations of
elliptic divisibility sequences. In addition to offering a way of quickly
determining the $\mathfrak{u}(1)$ charges of light matter supported in an
F-theory model, the proposals can aid explorations of the F-theory landscape
by predicting the geometric properties of models admitting desired charge
spectra.

There are several natural directions to pursue in future work. First, it is
important to determine how these proposals would apply in models with gauge
algebras and nonabelian matter representations beyond those covered here. For
instance, our analysis focused on matter charged under abelian and simply-
laced nonabelian gauge factors, but one might expect similar formulas to hold
when non-simply-laced gauge factors are present as well. One possibility is
that the proposals are essentially unchanged even when the gauge algebra is
non-simply laced. Even if matter is charged under a $\mathfrak{u}(1)$ factor
and a non-simply-laced nonabelian factor, one would take the Lie group $G$ for
the nonabelian factor to be the simply-laced Lie group associated with the
codimension-one singularity type. The only practical difference would then be
that monodromy effects would restrict the allowed values of $\nu$ and $\mu$.
This is just a tentative suggestion, however, and more work should be done to
systematically investigate how the formulas may change for matter charged
under non-simply-laced factors.

Furthermore, the proposals should be checked in models admitting
$\mathfrak{u}(1)$-charged matter additionally charged in exotic nonabelian
representations, such as the adjoint,424242While the adjoint representation by
itself is not typically viewed as an exotic representations, it is difficult
to find F-theory constructions with adjoint matter that also has a
$\mathfrak{u}(1)$ charge. Such situations would likely require adjoint matter
localized at double point singularities MorrisonTaylorMaS ; KleversEtAlExotic
. symmetric, and three-index antisymmetric representations of
$\mathfrak{su}(n)$. Situations where the codimension-two singularity type
enhances by a rank greater than one are also worthy of examination. An
intriguing question is whether our proposals apply for $\mathfrak{u}(1)$
models with superconformal matter loci, where $f$ and $g$ vanish to orders
$(4,6)$ or greater at codimension two. If so, our formulas could provide
interesting insights into $\operatorname{U}(1)$ flavor symmetries of 6D SCFTs
LeeRegaladoWeigand ; ApruzzietalU1SCFT .

A second important direction is clarifying the formulas’ exact range of
validity. At the same time, we should better understand why the proposals work
from both mathematical and physical perspectives. The seemingly related
formulas in StangeEllTrouble  describing EDS valuations may provide some
insights into geometric explanations for the proposals. In fact, the parallels
between these two contexts suggest an intriguing connection between topics in
physics and number theory. This possibility makes understanding the physics
behind the proposals all the more important. In particular, one might hope
that a physical understanding of the proposals may translate to new
perspectives on elliptic divisibility sequences and related mathematical
topics. Alternatively, it may be possible to definitively establish the
proposals by considering intersection numbers in the full elliptic fibration.
The charge itself is given by $\sigma(\hat{s})\cdot c$, suggesting that the
proposals could be understood in terms of similar intersection numbers. This
approach would likely require a more concrete understanding of how the
wrapping behavior of sections at codimension-two loci is encapsulated in the
section components.

While the analysis here focused on models in Weierstrass form, there may be
similar relations between $\mathfrak{u}(1)$ charges and orders of vanishing
for elliptic fibrations in alternative forms, such as those in Tate form
BershadskyEtAlSingularities ; KatzEtAlTate  or those described as
hypersurfaces in $\mathbb{P}^{1,1,2}$ MorrisonParkU1 ; KuntzlerTateTrees . It
is oftentimes more convenient to work with these alternative forms, and an
easy method for reading off $\mathfrak{u}(1)$ charges in such situations could
simplify several analyses.

Finally, it would be interesting to use the proposals to characterize the
landscape and swampland of F-theory $\mathfrak{u}(1)$ models. A variety of
important open problems involve understanding which models with “large”
$\mathfrak{u}(1)$ charges can occur in F-theory. Even for the simplest case of
models with just a $\mathfrak{u}(1)$ gauge algebra, determining a bound on the
$\mathfrak{u}(1)$ charges of singlet matter and constructing F-theory models
with singlet charges larger than 6 (in appropriately scaled units) remain key
goals in the F-theory program. Because they seem to work for arbitrarily large
charges, the proposals would allow us to glean information about models with
larger charges without directly constructing them. Thus, they may show the way
towards constructions with new types of $\mathfrak{u}(1)$ charges, such as
singlet charges larger than $6$. Alternatively, the proposals may enable one
to anticipate geometric obstructions to realizing particular $\mathfrak{u}(1)$
charges. Such observations could, for instance, lead to new bounds on the
$\mathfrak{u}(1)$ charges and charge spectra that can occur in F-theory. A
better physical understanding of the proposals may also reveal other
constraints on $\mathfrak{u}(1)$ algebras in F-theory. In general, a clearer
knowledge of the geometric properties of $\mathfrak{u}(1)$-charged matter can
offer new insights into F-theory.

###### Acknowledgements.

We would like to thank Washington Taylor for helpful discussions, and Timo
Weigand and Antonella Grassi for useful comments. The work of APT was
supported by DOE grant DE-SC00012567 and DOE (HEP) Award DE-SC0013528. The
work of NR was supported by NSF grant PHY-1720321.

##  Appendix A Expressions for specific gauge algebras and representations

###  A.1 Singlets

Charged singlets occur at codimension-two loci in the base of the elliptic
fibration with $\text{I}_{2}$ singular fibers. In units corresponding to the
definition of Shioda map in Eq. 9, charged singlets should have integer
charges. At a codimension-two locus supporting singlets with charge $q$, the
$\hat{z}$ component vanishes to order

$\operatorname{ord}_{2}(\hat{z})=\frac{q^{2}}{4}-\frac{\overline{q}_{2}}{4}=\mathopen{}\mathclose{{}\left\lfloor\frac{q^{2}}{4}}\right\rfloor\,,$
(295)

while the $\hat{x}$, $\hat{y}$, and $\hat{w}$ components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(0,1,1}\right)\overline{q}_{2}$
(296)

Based on these equations, if one knows the orders of vanishing of the section
components at a codimension-two singlet locus, one can determine the charge
$q$ of the supported matter (up to sign) as

$q^{2}=\operatorname{ord}_{2}(\hat{y})+\operatorname{ord}_{2}(\hat{z})=\operatorname{ord}_{2}(\hat{w})\,.$
(297)

###  A.2 $\mathfrak{su}(n)$

For models with an $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ gauge algebra, we
label elements of the center of $\operatorname{SU}(n)$ with the parameter
$\nu$, which can run from 0 to $n-1$. The section components in a model with a
quotient corresponding to $\nu$ vanish to the following orders at the
codimension-one locus supporting the $\mathfrak{su}(n)$ algebra:

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(0,1,1}\right)\times
u_{n}(\nu)\,.$ (298)

##### Fundamentals

An $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ model can have fundamental matter
with charge $q=\frac{\nu}{n}+j$ for integers $j$. In other words, fundamental
matter with charge $q$ can occur when $\nu$ is $\overline{nq}_{n}$. At the
codimension-two locus supporting the $\bm{n}_{q}$ matter, the $\hat{z}$
component should vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{n+1}q^{2}+\frac{\nu(n-\nu)}{n}-\frac{\mu\mathopen{}\mathclose{{}\left(n+1-\mu}\right)}{(n+1)}}\right)\,,$
(299)

where

$\mu=\overline{nq}_{n+1}\,.$ (300)

The other section components should vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(0,1,1}\right)\times
u_{n+1}(\mu)\,.$ (301)

Based on these equations, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{su}(n)$
factor and the codimension-two fundamental locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported fundamental matter (up to sign)
as

$q^{2}={\frac{2(n+1)}{n}\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{z})+\frac{\tilde{\mu}\mathopen{}\mathclose{{}\left(n+1-\tilde{\mu}}\right)}{2(n+1)}-\frac{\tilde{\nu}\mathopen{}\mathclose{{}\left(n-\tilde{\nu}}\right)}{2n}}\right)}\,,$
(302)

where

$\begin{gathered}\tilde{\nu}=\operatorname{ord}_{1}(\hat{y})=\operatorname{ord}_{1}(\hat{w})\,,\\\
\tilde{\mu}=\operatorname{ord}_{2}(\hat{y})-3\operatorname{ord}_{2}(\hat{z})=\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})\,.\end{gathered}$
(303)

##### Antisymmetric matter for odd $n$

An $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ model with discrete quotient $\nu$
can have antisymmetric matter with charge $q=\frac{2\nu}{n}+j$ for integers
$j$. For odd $n$ (i.e., $n=2k+1$), this means that antisymmetric matter with
charge $q$ can occur when

$\nu=\overline{nq(k+1)}_{n}\,.$ (304)

At the codimension-two locus supporting the antisymmetric matter, the
$\hat{z}$ component should vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+\frac{\nu(n-\nu)}{n}-\begin{cases}0&\mu=0\\\
1&\mu=2\\\ \frac{n}{4}&\mu=1,3\end{cases}}\right)$ (305)

with

$\mu=\overline{nq}_{4}\,.$ (306)

The other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)\\\
=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\begin{cases}(0,0,0)&\mu=0\\\
(1,2,2)&\mu=2\\\
\mathopen{}\mathclose{{}\left(1,\mathopen{}\mathclose{{}\left\lfloor\frac{n}{2}}\right\rfloor,\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil}\right)&\mu=1,3\end{cases}$
(307)

If one knows the orders of vanishing at the codimension-one $\mathfrak{su}(n)$
locus and at the codimension-two antisymmetric locus, the $\mathfrak{u}(1)$
charge of the supported matter can be determined (up to sign) as

$q^{2}=\frac{8}{n}\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{z})-\frac{\tilde{\nu}(n-\tilde{\nu})}{2n}+\frac{1}{2}\begin{cases}0&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=0\\\
1&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=2\\\
\frac{n}{4}&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil\end{cases}}\right)$
(308)

where $\tilde{\nu}=\operatorname{ord}_{1}(\hat{y})$.434343If we let
$\tilde{\mu}=\operatorname{ord}_{2}(\hat{y})-3\operatorname{ord}_{2}(\hat{z})$
and
$\tilde{\mu}^{\prime}=\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})$,
the last term in parentheses in Eq. 308 can be written as
$\frac{\tilde{\mu}+\tilde{\mu}^{\prime}}{8}$.

##### Antisymmetric matter for even $n$

As before, an $\mathfrak{su}(n)\oplus\mathfrak{u}(1)$ model with discrete
quotient $\nu$ can have antisymmetric matter with charge $q=\frac{2\nu}{n}+j$
for integers $j$. For even $n$ (i.e., $n=2k$), this means that antisymmetric
matter with charge $q$ can occur when $\nu$ is either $\overline{(kq)}_{n}$ or
$\overline{(kq+k)}_{n}$. At the codimension-two locus supporting the
antisymmetric matter, the $\hat{z}$ component vanishes to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{n}{4}q^{2}+\frac{\nu(n-\nu)}{n}-\begin{cases}0&\mu=0\\\
1&\mu=2\\\ \frac{n}{4}&\mu=1,3\end{cases}}\right)\,,$ (309)

where

$\mu=\overline{\mathopen{}\mathclose{{}\left(q-\frac{2}{n}\nu}\right)}_{2}+\overline{2\nu}_{4}$
(310)

The other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\begin{cases}(0,0,0)&\mu=0\\\
(1,2,2)&\mu=2\\\
\mathopen{}\mathclose{{}\left(1,\frac{n}{2},\frac{n}{2}}\right)&\mu=1,3\end{cases}\,.$
(311)

If one knows the orders of vanishing at the codimension-one $\mathfrak{su}(n)$
locus and at the codimension-two antisymmetric locus, the $\mathfrak{u}(1)$
charge of the supported matter can still be determined (up to sign) using Eq.
308.

###  A.3 $\mathfrak{so}(2n)$

For models with an $\mathfrak{so}(2n)\oplus\mathfrak{u}(1)$ gauge algebra, we
label the elements of the center of $\operatorname{Spin}(2n)$ with the
parameter $\nu$, which can run from 0 to $4$. The section components in a
model with the quotient corresponding to $\nu$ vanish to the following orders
at the codimension-one locus supporting the $\mathfrak{so}(2n)$:

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\begin{cases}(0,0,0)&\nu=0\\\
(1,2,2)&\nu=2\\\
(1,\mathopen{}\mathclose{{}\left\lfloor\frac{n}{2}}\right\rfloor,\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil)&\nu=1,3\end{cases}\,.$
(312)

The allowed $\mathfrak{u}(1)$ charges for the various $\mathfrak{so}(2n)$
representations considered here are summarized in Table 6.

##### Vector matter

At a codimension-two locus supporting $\bm{2n}_{q}$ matter, the $\hat{z}$
component should vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=2\\\ \frac{n}{4}&\nu=1,3\end{cases}-\begin{cases}0&\mu=0\\\ 1&\mu=2\\\
\frac{n+1}{4}&\mu=1,3\end{cases}}\right)\,,$ (313)

where

$\mu=\overline{(2q-\nu)}_{4}+\overline{2q}_{2}\,.$ (314)

The other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\begin{cases}(0,0,0)&\mu=0\\\
(1,2,2)&\mu=2\\\
(1,\mathopen{}\mathclose{{}\left\lfloor\frac{n+1}{2}}\right\rfloor,\mathopen{}\mathclose{{}\left\lceil\frac{n+1}{2}}\right\rceil)&\mu=1,3\end{cases}\,.$
(315)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{so}(2n)$
algebra and the codimension-two $\bm{2n}$ locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) with the
formula

$q^{2}=2\operatorname{ord}_{2}(\hat{z})-\begin{cases}0&\operatorname{ord}_{1}(\hat{w})=0\\\
1&\operatorname{ord}_{1}(\hat{w})=2\\\
\frac{n}{4}&\operatorname{ord}_{1}(\hat{w})=\mathopen{}\mathclose{{}\left\lceil\frac{n}{2}}\right\rceil\end{cases}+\begin{cases}0&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=0\\\
1&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=2\\\
\frac{n+1}{4}&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=\mathopen{}\mathclose{{}\left\lceil\frac{n+1}{2}}\right\rceil\end{cases}\,.$
(316)

##### Spinor matter for $\mathfrak{so}(8)$

At a codimension-two locus supporting ${\bm{8_{\text{s}}}}_{,q}$ or
${\bm{8_{\text{c}}}}_{,q}$ matter, the $\hat{z}$ component should vanish to
order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=1,2,3\end{cases}-\begin{cases}0&\mu=0\\\ 1&\mu=2\\\
\frac{5}{4}&\mu=1,3\end{cases}}\right)\,,$ (317)

where

$\mu=\overline{(2q-2\nu)}_{4}\,.$ (318)

The other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\begin{cases}(0,0,0)&\mu=0\\\
(1,2,2)&\mu=2\\\ (1,2,3)&\mu=1,3\end{cases}\,.$ (319)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{so}(8)$
algebra and the codimension-two $\bm{8_{\text{s}}}$ or $\bm{8_{\text{c}}}$
locus, one can determine the $\mathfrak{u}(1)$ charge $q$ of the supported
matter (up to sign) with the formula

$q^{2}=2\operatorname{ord}_{2}(\hat{z})-\begin{cases}0&\operatorname{ord}_{1}(\hat{w})=0\\\
1&\operatorname{ord}_{1}(\hat{w})=2\end{cases}+\begin{cases}0&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=0\\\
1&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=2\\\
\frac{5}{4}&\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})=3\end{cases}$
(320)

##### Spinor matter for $\mathfrak{so}(10)$

At a codimension-two locus supporting $\bm{16}_{q}$ matter (along with the
conjugate $\overline{\bm{16}}_{-q}$ matter), the $\hat{z}$ component should
vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{4}{3}q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=2\\\ \frac{5}{4}&\nu=1,3\end{cases}-\frac{2}{3}\mu(3-\mu)}\right)\,,$
(321)

where

$\mu=\overline{4q}_{3}\,.$ (322)

The other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(2,2,3}\right)\times
u_{3}(\mu)\,.$ (323)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{so}(10)$
algebra and the codimension-two $\bm{16}$ locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) with the
formula

$q^{2}=\frac{3}{2}\operatorname{ord}_{2}(\hat{z})-\frac{3}{4}\times\begin{cases}0&\operatorname{ord}_{1}(\hat{w})=0\\\
1&\operatorname{ord}_{1}(\hat{w})=2\\\
\frac{5}{4}&\operatorname{ord}_{1}(\hat{w})=3\end{cases}+\frac{1}{3}\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})}\right)\,.$
(324)

##### Spinor matter for $\mathfrak{so}(12)$

At a codimension-two locus supporting $\bm{32}_{q}\oplus\bm{1}_{2q}$ matter or
$\bm{32^{\prime}}_{q}\oplus\bm{1}_{2q}$ matter, the $\hat{z}$ component should
vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(2q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=2\\\
\frac{3}{2}&\nu=1,3\end{cases}-\frac{3}{2}\overline{(2q+\nu)}_{2}}\right)\,,$
(325)

while the other components should vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(2,3,3}\right)\times\overline{(2q+\nu)}_{2}\,.$
(326)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{so}(12)$
algebra and the codimension-two spinor locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) as

$q^{2}=\operatorname{ord}_{2}(\hat{z})-\begin{cases}0&\operatorname{ord}_{1}(\hat{w})=0\\\
\frac{1}{2}&\operatorname{ord}_{1}(\hat{w})=2\\\
\frac{3}{4}&\operatorname{ord}_{1}(\hat{w})=3\end{cases}+\frac{3}{8}\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x})-2\operatorname{ord}_{2}(\hat{z})}\right)\,.$
(327)

##### Spinor matter for $\mathfrak{so}(14)$

At a codimension-two locus supporting $\bm{64}_{q}\oplus\bm{14}_{2q}$ matter
(and matter in the conjugate representation), the $\hat{z}$ component should
vanish to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(4q^{2}+\begin{cases}0&\nu=0\\\
1&\nu=2\\\ \frac{7}{4}&\nu=1,3\end{cases}}\right)\,,$ (328)

and the other section components should vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})\,.$
(329)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{so}(14)$
algebra and the codimension-two spinor locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) as

$q^{2}=\frac{1}{2}\operatorname{ord}_{2}(\hat{z})-\frac{1}{4}\times\begin{cases}0&\operatorname{ord}_{1}(\hat{w})=0\\\
1&\operatorname{ord}_{1}(\hat{w})=2\\\
\frac{7}{4}&\operatorname{ord}_{1}(\hat{w})=4\end{cases}\,.$ (330)

###  A.4 $\mathfrak{e}_{6}$

For models with an $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$ gauge algebra, we
label the elements of the center of $\operatorname{E}_{6}$ with the parameter
$\nu$, which can run from $0$ to $2$. The section components in a model with
the quotient corresponding to $\nu$ vanish to the following orders at the
codimension-one locus supporting the $\mathfrak{e}_{6}$ algebra:

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,2,3}\right)\times
u_{3}(\nu)\,.$ (331)

#####  $\bm{27}$ matter

An $\mathfrak{e}_{6}\oplus\mathfrak{u}(1)$ model can have $\bm{27}$ matter
with charge $q=\frac{\nu}{3}+k$ for integer $k$. In other words, $\bm{27}_{q}$
matter can occur when $\nu$ is $\overline{3q}_{3}$. At the codimension-two
locus supporting that $\bm{27}_{q}$ matter, the $\hat{z}$ component vanishes
to order

$\operatorname{ord}_{2}(\hat{z})=\frac{1}{2}\mathopen{}\mathclose{{}\left(\frac{3}{2}q^{2}+\frac{2}{3}\nu(3-\nu)-\frac{3}{2}\overline{3q}_{2}}\right)\,,$
(332)

The other section components vanish to orders

$\displaystyle\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})+\mathopen{}\mathclose{{}\left(2,3,3}\right)\times\overline{3q}_{2}\,.$
(333)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{e}_{6}$
algebra and the codimension-two $\bm{27}$ locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) as

$q^{2}={\frac{4}{3}\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{z})-\frac{\tilde{\nu}(3-\tilde{\nu})}{3}+\frac{\tilde{\mu}}{4}}\right)}\,,$
(334)

where

$\displaystyle\begin{gathered}\tilde{\nu}=\operatorname{ord}_{1}(\hat{x})=\operatorname{ord}_{1}(\hat{y})\,,\\\
\tilde{\mu}=\operatorname{ord}_{2}(\hat{y})-3\operatorname{ord}_{2}(\hat{z})=\operatorname{ord}_{2}(\hat{w})-4\operatorname{ord}_{2}(\hat{z})\,.\end{gathered}$
(337)

###  A.5 $\mathfrak{e}_{7}$

For models with an $\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ gauge algebra, we
label the elements of the center of $\operatorname{E}_{7}$ with the parameter
$\nu$, which can be either $0$ or $1$. The section components in a model with
the quotient corresponding to $\nu$ vanish to the following orders at the
codimension-one locus supporting the $\mathfrak{e}_{7}$ algebra:

$\operatorname{ord}_{1}(\hat{z})=0\,,\quad\mathopen{}\mathclose{{}\left(\operatorname{ord}_{1}(\hat{x}),\operatorname{ord}_{1}(\hat{y}),\operatorname{ord}_{1}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,3}\right)\times\nu\,.$
(338)

#####  $\bm{56}$ matter

An $\mathfrak{e}_{7}\oplus\mathfrak{u}(1)$ model can have $\bm{56}$ matter
with $\mathfrak{u}(1)$ charge $q=\frac{\nu}{2}+k$ for integer $k$. In other
words, $\bm{56}_{q}$ matter can occur when $\nu$ is $\overline{2q}_{2}$. At
the codimension-two locus supporting this $\bm{56}_{q}$ matter, the $\hat{z}$
component should vanish to order

$\operatorname{ord}_{2}(\hat{z})=q^{2}+\frac{3}{4}\nu=q^{2}+\frac{3}{4}\times\overline{2q}_{2}\,,$
(339)

while the other section components vanish to orders

$\mathopen{}\mathclose{{}\left(\operatorname{ord}_{2}(\hat{x}),\operatorname{ord}_{2}(\hat{y}),\operatorname{ord}_{2}(\hat{w})}\right)=\mathopen{}\mathclose{{}\left(2,3,4}\right)\times\operatorname{ord}_{2}(\hat{z})\,.$
(340)

Based on these formulas, if one knows the orders of vanishing of the section
components at the codimension-one locus supporting the $\mathfrak{e}_{7}$
algebra and the codimension-two $\bm{56}$ locus, one can determine the
$\mathfrak{u}(1)$ charge $q$ of the supported matter (up to sign) as

$q^{2}={\operatorname{ord}_{2}(\hat{z})-\frac{1}{4}\operatorname{ord}_{1}(\hat{y})}={\operatorname{ord}_{2}(\hat{z})-\frac{1}{4}\operatorname{ord}_{1}(\hat{w})}\,.$
(341)

##  Appendix B The $S$ correction term

The results in StangeEllTrouble  often use the sequence $S_{n}$ of numbers in
$\mathbb{Z}\cup\\{\infty\\}$, which is defined as

$S_{n}\mathopen{}\mathclose{{}\left(p,b,d,h,s,w}\right)=\begin{cases}b^{j}s+\frac{b^{j}-1}{b-1}h+d\mathopen{}\mathclose{{}\left(v_{p}(n)-j}\right)+w&v_{p}(n)>j\\\
b^{v_{p}(n)}s+\frac{b^{v_{p}(n)}-1}{b-1}h&v_{p}(n)\leq j\end{cases}\,.$ (342)

Here, $v_{p}$ is a $p$-adic valuation on $\mathbb{Q}$ associated with $p$,
while $b$, $d$, $h$, $s$, and $w$ are non-negative integers in particular
sets:

$\displaystyle b\in$ $\displaystyle p\mathbb{Z}^{>0}\cup\\{1\\}$
$\displaystyle d$ $\displaystyle\in\mathbb{Z}^{>0}$ $\displaystyle h\in$
$\displaystyle\mathbb{Z}^{\geq 0}$ $\displaystyle s\in$
$\displaystyle\mathbb{Z}^{>0}\cup\\{\infty\\}$ $\displaystyle w$
$\displaystyle\in\mathbb{Z}^{\geq 0}\cup\\{\infty\\}\,.$ (343)

The integer $j$ is $0$ if $b=1$; if $b$ is a multiple of $p$, then $j$ is the
smallest non-negative integer such that

$d\leq
b^{j}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(b-1}\right)s+h}\right)\,.$
(344)

For the examples encountered here, $d$ is $v_{p}(p)$, which is $1$. This
implies that $j$ is $0$ regardless of the value of $b$, and the definition of
$S_{n}$ simplifies to

$S_{n}\mathopen{}\mathclose{{}\left(p,b,v_{p}(p),h,s,w}\right)=s+v_{p}(n)+\begin{cases}w&v_{p}(n)>0\\\
0&v_{p}(n)\leq 0\end{cases}\,.$ (345)

The variable $w$ is allowed to be 0; in fact, the ideas discussed in Section 5
of StangeEllTrouble  suggest that, for all of the cases considered here, $w$
is expected to be 0. When this is the case, we are left with

$S_{n}\mathopen{}\mathclose{{}\left(p,b,v_{p}(p),h,s,0}\right)=s+v_{p}(n)$
(346)

If we are interested in using these formulas to explain the order of vanishing
patterns, the $v_{p}(n)$ term may seem somewhat strange. While the valuation
of an integer is a meaningful quantity, one might expect that the order of
vanishing of an integer at some locus should be thought of as 0. It therefore
seems that, for the purposes of understanding the order-of-vanishing patterns,
the $v_{p}(n)$ in $S_{n}$ should be dropped.

To see if this is a reasonable prescription, it is worth understanding why
this $v_{p}(n)$ occurs. Recall that $S_{n}$ appears in formulas describing the
valuations of elliptic divisibility sequences. Therefore, consider an example
EDS with the initial values

$W_{1}=1\,,\quad W_{2}=2p^{3}\,,\quad W_{3}=8p^{8}-4p^{4}-1\,,\quad
W_{4}=4p^{3}\mathopen{}\mathclose{{}\left(8p^{12}-6p^{4}-1}\right)\,,$ (347)

where $p$ is some prime integer. The $W_{n}$ with even $n$ are proportional to
$p$, implying that $n_{P}=2$ for this situation. One can then calculate that

$W_{6}=-2p^{3}\mathopen{}\mathclose{{}\left(8p^{8}-4p^{4}-1}\right)\mathopen{}\mathclose{{}\left(512p^{24}-768p^{20}+96p^{12}+120p^{8}+36p^{4}+3}\right)\,.$
(348)

For most choices of $p$, $W_{6}$ is proportional to $p^{3}$, and
$v_{p}(W_{6})$ is 3. If, however, we choose $p$ to be 3, the last factor in
the above expression is a factor of 3, and $v_{p}(W_{6})$ enhances to $4$.
Similar sorts of $v_{p}(W_{n})$ enhancements may occur in other elliptic
divisibility sequences, and the $v_{p}(n)$ term accounts for this fact. Based
on Eq. 342, this effect only occurs for particular choices of $n$ and $p$.
However, this term is not particularly relevant for the analogous situations
in elliptic fibrations where valuations are replaced with orders of vanishing;
an integer $n$ would not vanish at a particular locus in the base.
Additionally, note that the examples considered here are formed by starting
with an elliptic fibration, setting the parameters involved at a particular
locus to some arbitrary prime $p$, setting the other parameters to some
integers, and analyzing an EDS corresponding to the resulting elliptic curve.
We are interested in the behavior for arbitrary $p$, but for a given value of
$n$, the “accidental” increases in $v_{p}(W_{n})$ only occur for particular
choices of $p$. This effect should therefore be ignored.

Thus, while the $v_{p}(n)$ is important for characterizing $p$-adic valuations
of elliptic divisibility sequences, it seems that it can be dropped for the
purposes of establishing analogies to the orders of vanishing. In other words,
if we are looking to justify the expressions we obtain for the order of
vanishing, one should be able to replace $S_{n}$ with $s$, a positive
constant. However, it would be ideal to have a more formal understanding of
how to adapt the formulas in StangeEllTrouble , particularly the $S$
correction term, to scenarios involving the orders of vanishing of section
components in elliptic fibrations.

##  Appendix C Additional calculations of EDS valuations

In this appendix, we calculate various EDS valuations corresponding to various
singularity types using the formulas in StangeEllTrouble .

###  C.1 $\text{I}^{*}_{2k-4}$ singularities

####  C.1.1 $\nu=1,3$ or $\mu=1,3$

For models with an $\text{I}^{*}_{2k-4}$ locus where $\nu$ or $\mu$ for the
generating section equals 1 or 3, we are interested in the EDS valuations when
the components $(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to order
$(1,k,0,k)$. As an example, we can consider the Weierstrass model in Eq. 132,
the $\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$ seed model with $\nu=k+1$ used to
find order of vanishing data for the antisymmetric representation. The
$\text{I}^{*}_{2k-4}$ singularities are supported at $\\{\sigma=b_{1}=0\\}$,
so we construct an elliptic curve $E$ by setting $\sigma$ and $b_{1}$ to a
prime $p$ and setting the remaining parameters to arbitrary numbers. As an
example, $E$ can take the form

$\displaystyle y^{2}$
$\displaystyle=x^{3}-\frac{p^{2}}{48}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(-12+p-4p^{k-2}}\right)^{2}-48p^{k-2}\mathopen{}\mathclose{{}\left(1+p}\right)}\right]x$
(349)
$\displaystyle\qquad\quad+\frac{p^{3}}{864}\mathopen{}\mathclose{{}\left[(p-12)^{3}-12(p-12)p^{k-2}(7p-6)+48p^{2k-4}(13p-6)}\right]\,,$

in which case it admits the rational point $P$ given by

$P\colon(x,y)=\mathopen{}\mathclose{{}\left(\frac{p}{12}(p-12),\frac{5}{6}p^{k}}\right)\,.$
(350)

Since this elliptic curve has additive reduction modulo $p$, we extend the
field by the element $p^{\prime}=(p^{1/2})$, let
$x\rightarrow{p^{\prime}}^{2}x^{\prime},y\rightarrow{p^{\prime}}^{3}y^{\prime}$,
and divide through by ${p^{\prime}}^{6}$. This gives us an elliptic curve
$E^{\prime}$ of the form

$\displaystyle{y^{\prime}}^{2}$
$\displaystyle={x^{\prime}}^{3}+\mathopen{}\mathclose{{}\left[-\frac{1}{48}\mathopen{}\mathclose{{}\left({p^{\prime}}^{2}-12-4{p^{\prime}}^{2k-4}}\right)^{2}+{p^{\prime}}^{2k-4}\mathopen{}\mathclose{{}\left(1+{p^{\prime}}^{2}}\right)}\right]{x^{\prime}}$
(351)
$\displaystyle\qquad\quad+\Big{[}\frac{1}{864}\mathopen{}\mathclose{{}\left({p^{\prime}}^{2}-12-4{p^{\prime}}^{2k-4}}\right)^{3}-\frac{1}{12}\mathopen{}\mathclose{{}\left({p^{\prime}}^{2}-12-4{p^{\prime}}^{2k-4}}\right){p^{\prime}}^{2k-4}\mathopen{}\mathclose{{}\left(1+{p^{\prime}}^{2}}\right)$
$\displaystyle\qquad\qquad\qquad+\frac{1}{3}{p^{\prime}}^{4k-6}+\frac{2}{27}{p^{\prime}}^{6k-12}\Big{]}\,,$

which admits a rational point

$P^{\prime}\colon(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left(\frac{{p^{\prime}}^{2}}{12}-1,\frac{5}{6}{p^{\prime}}^{2k-3}}\right)\,.$
(352)

with $3{x^{\prime}}^{2}+f^{\prime}$ proportional to ${p^{\prime}}^{2k-4}$. The
new elliptic curve $E^{\prime}$ has multiplicative reduction modulo
$p^{\prime}$ with $v_{p^{\prime}}(\Delta^{\prime})=4k-8$. Meanwhile, the point
$P^{\prime}$ has singular reduction, implying that the EDS valuations are
given by Eq. 158 with $a=2k-4$.444444The quantity
$w^{\prime}=3{x^{\prime}}^{2}+f^{\prime}$ is proportional to
${p^{\prime}}^{2k-4}$, whereas $y^{\prime}$ is proportional to
${p^{\prime}}^{2k-3}$. The integer $a$ should equal the smaller of these two
exponents, which is $2k-4$. Even multiples of $P^{\prime}$ reduce to the
identity point with the $(x^{\prime},y^{\prime})$ coordinates of $2P^{\prime}$
being proportional to $({p^{\prime}}^{-2},{p^{\prime}}^{-3})$. Therefore,
$n_{P^{\prime}}$ equals $2$, and the $S$ term, which is essentially a constant
according to Appendix B, should be set to $1$. Combining these observations
together, we have

$\displaystyle v(W_{m})$
$\displaystyle=\frac{m^{2}-1}{2}+\frac{1}{2}\mathopen{}\mathclose{{}\left(R_{m}(2k-4,4k-8)+\begin{cases}1&2\mid
m\\\ 0&2\nmid m\end{cases}}\right)$ (353)
$\displaystyle=\frac{n}{8}m^{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\
\frac{n}{4}&2\nmid m\end{cases}\,.$

####  C.1.2 $\nu=2$ or $\mu=2$

For models with an $\text{I}^{*}_{2k-4}$ locus where the generating section
has $\nu=2$ or $\mu=2$, we are interested in EDS valuations when the
generating section components $(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to
orders $(1,2,0,2)$. As an example, we can consider the
$\mathfrak{su}(2k)\oplus\mathfrak{u}(1)$ seed model with $\nu=1$, which is
used to find order of vanishing data for the fundamental and antisymmetric
representations of $\mathfrak{su}(2k)$. The Weierstrass model is given by Eq.
113 with the substitutions described around Eq. 115. The section components
are given by Eq. 114 with the same substitutions performed. The
$\text{I}^{*}_{2k-4}$ singularities are supported at $\\{\sigma=b_{1,0}=0\\}$,
so we construct an elliptic curve $E$ by setting $\sigma$ and $b_{1,0}$ to a
prime $p$ and the remaining parameters to some arbitrary numbers. For example,
we can consider an elliptic curve $E$ taking the form

$\displaystyle y^{2}$
$\displaystyle=x^{3}+p^{2}\mathopen{}\mathclose{{}\left(-3(1+3p)^{2}+32p^{k-1}+p^{2k-4}(p-1)}\right)x$
(354)
$\displaystyle\quad+p^{3}\mathopen{}\mathclose{{}\left(2(1+3p)^{3}-32(1+3p)p^{k-1}-(p-1)(7p-2)p^{2k-4}+(19-3p)p^{2k-3}}\right)\,,$

which admits a rational point

$P\colon(x,y)=\mathopen{}\mathclose{{}\left(2p(5p-1),4p^{2}(3-7p-p^{k-2})}\right)\,.$
(355)

This elliptic curve has additive reduction modulo $p$. We therefore extend the
field by an element $p^{\prime}=(p)^{1/2}$, let
$x\to{p^{\prime}}^{2}x^{\prime},y\to{p^{\prime}}^{3}y^{\prime}$, and divide
through by ${p^{\prime}}^{6}$. The resulting elliptic curve $E^{\prime}$ is
given by

$\displaystyle{y^{\prime}}^{2}$
$\displaystyle={x^{\prime}}^{3}+\mathopen{}\mathclose{{}\left[-3\mathopen{}\mathclose{{}\left(1+3{p^{\prime}}^{2}}\right)^{2}-{p^{\prime}}^{4k-8}+{p^{\prime}}^{4k-6}+32{p^{\prime}}^{2k-2}}\right]{x^{\prime}}$
(356)
$\displaystyle\qquad+\mathopen{}\mathclose{{}\left[2\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(1+3{p^{\prime}}^{2}}\right)^{3}-16{p^{\prime}}^{2k-2}\mathopen{}\mathclose{{}\left(1+3{p^{\prime}}^{2}}\right)+{p^{\prime}}^{4k-8}\mathopen{}\mathclose{{}\left(-1+14{p^{\prime}}^{2}-5{p^{\prime}}^{4}}\right)}\right)}\right]\,,$

and it admits a point $P^{\prime}$ given by

$P^{\prime}\colon(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left(2\mathopen{}\mathclose{{}\left(5{p^{\prime}}^{2}-1}\right),4p^{\prime}\mathopen{}\mathclose{{}\left(3-7{p^{\prime}}^{2}-{p^{\prime}}^{2k-4}}\right)}\right)\,.$
(357)

We can now calculate the EDS valuations according to the formulas in
StangeEllTrouble . The $k=2$ case has potential good reduction, and the EDS
valuation calculation uses Eqs. 155 and 156. Even multiples of $P^{\prime}$
reduce to the identity point, and the $(x^{\prime},y^{\prime})$ coordinates of
$2P^{\prime}$ are respectively proportional to
$\mathopen{}\mathclose{{}\left(p^{\prime}(7{p^{\prime}}^{2}-2)}\right)^{-2}$
and
$\mathopen{}\mathclose{{}\left(p^{\prime}(7{p^{\prime}}^{2}-2)}\right)^{-3}$.
Therefore $n_{P^{\prime}}=2$ and the $S$ term, which is essentially a constant
according to Appendix B, should be replaced with 1.

The $k>2$ cases, like those encountered for the antisymmetrics of
$\mathfrak{su}(2k+1)$, have potential multiplicative reduction. However, the
point $P^{\prime}$ does not reduce to the singular point, and the EDS
valuation calculation therefore involves Eqs. 157 and 156. Even multiples of
$2P^{\prime}$ reduce to the identity point, and the $(x^{\prime},y^{\prime})$
coordinates of $2P^{\prime}$ are proportional to
$\mathopen{}\mathclose{{}\left(p^{\prime}\mathopen{}\mathclose{{}\left(-3+7{p^{\prime}}^{2}+{p^{\prime}}^{2k-4}}\right)^{2}}\right)^{-2}$
and
$\mathopen{}\mathclose{{}\left(p^{\prime}\mathopen{}\mathclose{{}\left(-3+7{p^{\prime}}^{2}+{p^{\prime}}^{2k-4}}\right)^{2}}\right)^{-3}$.
Therefore, $n_{P^{\prime}}=2$, and the $S$ term should be replaced with 1.

Putting everything together, we find that the EDS valuations for $k\geq 2$ are
given by

$v(W_{m})=\frac{m^{2}}{2}-\frac{1}{2}\begin{cases}0&2\mid m\\\ 1&2\nmid
m\end{cases}\,.$ (358)

###  C.2 $\text{I}^{*}_{2k-3}$ singularities with $\nu=0$ or $\mu=0$

For $\text{I}^{*}_{2k-3}$ singular fibers with $\nu=0$ or $\mu=0$, we are
primarily interested in two scenarios: one where the generating section
components $(\hat{x},\hat{y},\hat{z},\hat{w})$ vanish to orders $(0,0,0,0)$
and one where they vanish to orders $(2,3,1,4)$. When the section components
vanish to orders $(0,0,0,0)$, the associated EDS valuations are $0$, as can be
verified using the formulas in StangeEllTrouble . The $(2,3,1,4)$ situation,
however, has nontrivial EDS valuations.

To analyze this situation, we first use the seed model in Eq. 219 to construct
an elliptic curve $E$. Specifically, the $\text{I}^{*}_{2k-3}$ singularities
occur at $\\{v=b_{0,0}=0\\}$, so we set $v$ and $b_{0,0}$ to a prime $p$ and
set the other parameters to some arbitrary numbers. However, the $\hat{z}$
component of the generating section would then be equal to $p$, whereas
$\hat{z}$ was not proportional to $p$ in the other situations we have
encountered so far. To account for this, we absorb the $\hat{z}^{4}$ and
$\hat{z}^{6}$ factors in the Weierstrass model into the $f$ and $g$ of $E$.
For example, we can take $E$ to be given by the equation

$\displaystyle y^{2}$
$\displaystyle=x^{3}-p^{6}\mathopen{}\mathclose{{}\left(3-2p^{k-1}+p^{2k-2}+p^{2k-1}}\right)x$
(359)
$\displaystyle\qquad\quad-p^{9}\mathopen{}\mathclose{{}\left(-2+2p^{k-1}-p^{2k-3}+p^{2k-2}+p^{2k-1}}\right)\,.$

This elliptic curve admits the rational point

$P\colon(x,y)=\mathopen{}\mathclose{{}\left(-p^{2}(2p-1),-p^{3}\mathopen{}\mathclose{{}\left(1-3p+p^{k+1}}\right)}\right)\,.$
(360)

This elliptic curve is non-minimal and has additive reduction modulo $p$. We
therefore extend the field by the element $p^{\prime}=p^{1/2}$, let $x\to
x^{\prime}{p^{\prime}}^{6},y\to y^{\prime}{p^{\prime}}^{9}$, and divide both
sides of the elliptic curve’s equation by ${p^{\prime}}^{18}$. This procedure
give us a new elliptic curve $E^{\prime}$ taking the form

$\displaystyle{y^{\prime}}^{2}$
$\displaystyle={x^{\prime}}^{3}+\mathopen{}\mathclose{{}\left(-3+2{p^{\prime}}^{2k-2}-{p^{\prime}}^{4k-4}-{p^{\prime}}^{4k-2}}\right)x$
(361)
$\displaystyle\qquad\quad+\mathopen{}\mathclose{{}\left(2-2{p^{\prime}}^{2k-2}+{p^{\prime}}^{4k-6}-{p^{\prime}}^{4k-4}-{p^{\prime}}^{4k-2}}\right)\,,$

which admits the rational point

$P^{\prime}\colon(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left(\frac{1-2{p^{\prime}}^{2}}{{p^{\prime}}^{2}},-\frac{1-3{p^{\prime}}^{2}+{p^{\prime}}^{2k+2}}{{p^{\prime}}^{3}}}\right)\,.$
(362)

The elliptic curve $E^{\prime}$ has multiplicative reduction modulo
$p^{\prime}$, and hence $E$ has potential multiplicative reduction modulo $p$.
The rational point $P^{\prime}$, meanwhile, reduces to the identity modulo
$p^{\prime}$. Therefore, the EDS valuations should be described by Eqs. 156
and 157 with $v(f)=6$, $n_{P^{\prime}}=1$ and
$v_{1}(x^{\prime}_{P^{\prime}})=-2$. Putting everything together, we have

$v(W_{m})=\frac{3}{2}\mathopen{}\mathclose{{}\left(m^{2}-1}\right)+\frac{1}{2}\mathopen{}\mathclose{{}\left(-m^{2}+S_{m/n_{{P^{\prime}}}}\mathopen{}\mathclose{{}\left(p^{\prime},b,v_{1}(p^{\prime}),h,s,w}\right)}\right)\,.$
(363)

The $S$ term should be a constant according to Appendix B. Since the $\hat{z}$
component of the generating section vanishes to order 1, $S$ should be set to
$3$ such that $v(W_{1})=1$. This leaves us with

$v(W_{m})=m^{2}\,.$ (364)

###  C.3 $\text{IV}^{*}$ singularities

For $\text{IV}^{*}$ singular fibers, we are primarily interested in the
analogous EDS valuations when the generating section is described by $\nu=1,2$
or $\mu=1,2$. We first find an elliptic curve $E$ by taking the construction
in Eq. 175, setting $\sigma$ and $b_{2,1}$ to a prime integer $p$, and
replacing the other parameters with numbers such that $f$ and $g$ are
integers. For instance, we can consider an elliptic curve $E$ over
$\mathbb{Q}_{p}$ of the form

$y^{2}=x^{3}+p^{3}x+p^{4}(1-p-p^{2})\,,$ (365)

which has additive reduction modulo $p$. After making the same substitutions
into the generating section components in Eq. 176, we can see that $E$ admits
a rational point $P$ with coordinates

$P\colon(x,y)=(p^{2},p^{2})\,.$ (366)

We now extend $\mathbb{Q}_{p}$ by $p^{\prime}=p^{1/3}$, let
$x={p^{\prime}}^{4}x^{\prime}$ and $y={p^{\prime}}^{6}y^{\prime}$, and remove
an overall factor of ${p^{\prime}}^{12}$ from the expression for $E$. This
procedure gives us a new elliptic curve $E^{\prime}$ of the form

${y^{\prime}}^{2}={x^{\prime}}^{3}+p^{\prime}x+(1-{p^{\prime}}^{3}-{p^{\prime}}^{6})\,,$
(367)

which admits a rational point $P^{\prime}$ with coordinates

$P^{\prime}\colon(x^{\prime},y^{\prime})=({p^{\prime}}^{2},1)\,.$ (368)

The elliptic curve $E^{\prime}$ has good reduction modulo $p^{\prime}$, and
the original elliptic curve $E$ therefore has potential good reduction modulo
$p$. In turn, Eq. 155 (from Theorem 19 in StangeEllTrouble ) suggests that the
EDS valuations corresponding to $E$ and $P$ are given by

$v(W_{m})=\frac{2}{3}\mathopen{}\mathclose{{}\left(m^{2}-1}\right)+\frac{1}{3}v_{1}(W_{m}^{\prime})\,.$
(369)

Here, $v_{1}(W_{m}^{\prime})$ represents the EDS valuations corresponding to
$E^{\prime}$ and $P^{\prime}$ with respect to $p^{\prime}$, which should be
given by Eq. 156. If one calculates multiples of $P^{\prime}$, one finds that
$3P^{\prime}$ reduces to the identity point modulo $p^{\prime}$ and that
$v_{1}(W_{3})$ is $2$. Therefore, $n_{P}$ in Eq. 156 is $3$, and the $S$ term,
which should be a constant according to Appendix B, can be replaced with $2$.
Combining all of this information leads to the formula

$v(W_{m})=\frac{2}{3}\mathopen{}\mathclose{{}\left(m^{2}-1}\right)+\begin{cases}\frac{2}{3}&3\mid
m\\\ 0&3\nmid m\end{cases}=\frac{2}{3}m^{2}-\begin{cases}0&3\mid m\\\
\frac{2}{3}&3\nmid m\end{cases}\,.$ (370)

###  C.4 $\text{III}^{*}$ singularities

####  C.4.1 $\nu=0$ or $\mu=0$

For $\text{III}^{*}$ singularities with either $\nu$ or $\mu$ equal to 0, we
are mostly interested in the EDS valuations corresponding to generating
sections whose $(\hat{x},\hat{y},\hat{z},\hat{w})$ components vanish to orders
$(2,3,1,4)$ at a particular locus. As an example, we can consider the
Weierstrass model in Eq. 201 and the locus $\\{\sigma=b_{0,0}=0\\}$. The
$\hat{z}$ component in this example, which is equivalent to $b_{0,0}$,
vanishes to order one at this locus. To account for this fact, we construct
the elliptic curve analogue $E$ of the fibration given by

$y^{2}=x^{3}+fb_{0,0}^{4}+gb_{0,0}^{6}\,,$ (371)

where $f$ and $g$ are taken from Eq. 201. At the end of the calculation, we
will address the fact that these factors of $b_{0,0}$ come from $\hat{z}$. As
with the other examples, the parameters should be set to appropriate numbers.
We set $\sigma$ and $b_{0,0}$ to a prime $p$ and let the other parameters be
numbers such that $f$ and $g$ are integers. For a particular choice of these
numbers, we obtain an elliptic curve $E$ of the form

$y^{2}=x^{3}-p^{7}\mathopen{}\mathclose{{}\left(12p^{3}+p^{2}+p-6}\right)x-p^{11}\mathopen{}\mathclose{{}\left(16p^{4}+p^{3}-20p^{2}-12p+11}\right)\,,$
(372)

which admits a rational point

$P\colon(x,y)=\mathopen{}\mathclose{{}\left(-p^{2}\mathopen{}\mathclose{{}\left(2p^{3}+2p^{2}-1}\right),-p^{3}\mathopen{}\mathclose{{}\left(p^{4}-3p^{2}+1}\right)}\right)\,.$
(373)

The discriminant is proportional to $p^{21}$, and $E$ has additive reduction
modulo $p$. We then extend the field with the element $p^{\prime}=p^{1/4}$ of
ramification degree $d=4$, let $x={p^{\prime}}^{14}{x^{\prime}}$ and
$y={p^{\prime}}^{21}{y^{\prime}}$, and divide both sides of the equation for
the elliptic curve by ${p^{\prime}}^{42}$. This procedure leads to a new
elliptic curve $E^{\prime}$ given by

${y^{\prime}}^{2}={x^{\prime}}^{3}+\mathopen{}\mathclose{{}\left(6-{p^{\prime}}^{4}-{p^{\prime}}^{8}-12{p^{\prime}}^{12}}\right){x^{\prime}}-{p^{\prime}}^{2}\mathopen{}\mathclose{{}\left(11-12{p^{\prime}}^{4}-20{p^{\prime}}^{8}+{p^{\prime}}^{12}+16{p^{\prime}}^{16}}\right)\,.$
(374)

$E^{\prime}$ has good reduction modulo $p^{\prime}$, and $E$ therefore has
potential good reduction. Additionally, $E^{\prime}$ admits a rational point

$P^{\prime}\colon(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left({p^{\prime}}^{-6}\mathopen{}\mathclose{{}\left(1-2{p^{\prime}}^{8}-2{p^{\prime}}^{12}}\right),{p^{\prime}}^{-9}\mathopen{}\mathclose{{}\left(-1+3{p^{\prime}}^{8}-{p^{\prime}}^{16}}\right)}\right)$
(375)

that reduces to the identity point on $E^{\prime}$ modulo $p^{\prime}$.
According to Eq. 155, which comes from Theorem 19 of StangeEllTrouble , the
EDS valuations associated with $(E,P)$ should be given by the formula

$v(W_{m})=\frac{21}{12}(m^{2}-1)+\frac{1}{4}v_{1}(W_{m}^{\prime})\,,$ (376)

where $v_{1}(W_{m}^{\prime})$ represents the EDS valuations for
$(E^{\prime},P^{\prime})$. Since $P^{\prime}$ reduces to the identity point,
Theorem 14 of StangeEllTrouble  states that

$\displaystyle v_{1}(W_{m}^{\prime})$
$\displaystyle=\frac{v_{1}(x^{\prime}_{P^{\prime}})}{2}m^{2}+S_{m}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)$
(377)
$\displaystyle=-3m^{2}+S_{m}\mathopen{}\mathclose{{}\left(p^{\prime},p^{\prime},v_{1}(p^{\prime}),0,s,w}\right)\,.$

The discussion in Appendix B suggests that the $S$ term can be replaced with a
constant $s$. If we put these equations together, we find that

$v(W_{m})=\frac{21}{12}(m^{2}-1)+\frac{1}{4}\mathopen{}\mathclose{{}\left(-3m^{2}+s_{P}}\right)=m^{2}+\frac{1}{4}(s-7)\,.$
(378)

Now we can address the fact that we included extra factors of $b_{0,0}$ in $f$
and $g$. Because $\hat{z}$ for the generating section actually vanishes to
order 1 at the locus in question, we set $s$ to 7 such that $v(W_{1})$ is 1.
In the end, we have

$v(W_{m})=m^{2}\,.$ (379)

####  C.4.2 $\nu=1$ or $\mu=1$

For the EDS valuations when the generating section components vanish to orders
$(2,2,0,3)$, we can consider the Weierstrass model in Eq. 254 and section
components in Eq. 255. We then obtain an elliptic curve $E$ and a point $P$ by
setting $\sigma$ and $y_{2}$ to a prime $p$ and letting the other parameters
be integers. As an example, we could take $(E,P)$ to be

$\displaystyle E:$ $\displaystyle\,y^{2}=x^{3}+p^{3}x-p^{5},$ $\displaystyle
P:$ $\displaystyle\,(x,y)=(p^{2},p^{3})\,.$ (380)

This elliptic curve has additive reduction modulo $p$. We can obtain a new
elliptic curve by extending the field by $p^{\prime}=p^{1/4}$, defining
$x=(p^{\prime})^{6}x^{\prime}$ and $y=(p^{\prime})^{9}y^{\prime}$, and
removing factors of $(p^{\prime})^{18}$:

${y^{\prime}}^{2}={x^{\prime}}^{3}+x^{\prime}-(p^{\prime})^{2}\,.$ (381)

This new curve has good reduction, and the original elliptic curve $E$
therefore has potential good reduction. The point $P$ on $E$ corresponds to a
point $P^{\prime}$ on $E^{\prime}$ with coordinates

$(x^{\prime},y^{\prime})=\mathopen{}\mathclose{{}\left((p^{\prime})^{2},(p^{\prime})^{3}}\right)$
(382)

According to Eqs. 155 and 156, the valuations of an EDS corresponding to a
point on the original curve follow the formula

$v_{p}(W_{m})=\frac{3}{4}\mathopen{}\mathclose{{}\left(m^{2}-1}\right)+\frac{1}{4}\begin{cases}S_{m/n_{P^{\prime}}}(p^{\prime},b,v_{p^{\prime}}(p^{\prime}),h,s,w)&n_{P^{\prime}}\mid
m\\\ 0&n_{P^{\prime}}\nmid m\end{cases}\,.$ (383)

One can show that $2P^{\prime}$ reduces to the identity point modulo
$p^{1/4}$, implying that $n_{P^{\prime}}$ is $2$. Additionally, the
coordinates of $2P^{\prime}$ show that $v(W_{2}^{\prime})$ is 3. The $S$ term,
which should essentially be a constant according to Appendix B, should
therefore be replaced with 3, leading to the formula

$v_{p}(W_{m})=\frac{3}{4}\mathopen{}\mathclose{{}\left(m^{2}-1}\right)+\begin{cases}\frac{3}{4}&2\mid
m\\\ 0&2\nmid m\end{cases}=\frac{3}{4}m^{2}-\begin{cases}0&2\mid m\\\
\frac{3}{4}&2\nmid m\end{cases}\,.$ (384)

###  C.5 $\text{II}^{*}$ singularities

For the $\text{II}^{*}$ singularity type, we are primarily interested in
situations where the generating section components