where $N_{G}$ is the number of simple groups being considered. Then for each
type-T vertex
$\mathcal{v}_{{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}}$
we construct all $\mathcal{O}(N_{G})$ models incorporating this vertex and a
type-S vertex. The vertex
$\mathcal{v}_{{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}}$
is then removed from $\mathcal{G}$ if _all_ of these should be pruned, i.e.
have RHS of equation (19) larger than $273$. Having “pre-pruned” the vertices
of $\mathcal{G}$ in this way, we are left with
$\mathcal{V}_{{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}}=\mathcal{O}(20,\\!000)$
vertices and completing the construction of $\mathcal{G}$ by building the
$\mathcal{E}=\mathcal{O}(\mathcal{V}_{{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}}^{2})=\mathcal{O}(10^{8})$
hyperedges is within reach. In figure 2 only vertices which survive this
procedure are shown.

We can understand the huge number of vertices for $\operatorname{SU}(2)$,
$\operatorname{SU}(3)$ and $\operatorname{Sp}(2)$ even with reasonably small
$\Delta(\mathcal{v})$ in the following way. There are always collections of
hypermultiplets which can be interchanged without changing any of the data
relevant to anomaly-cancellation, i.e. the values of $b_{i}\cdot b_{i}$ and
$b_{0}\cdot b_{i}$. These can be read off from the various tensor product
decompositions for conjugate representations (recall that $R$ and
$\overline{R}$ are indistinguishable when it comes to anomaly-cancellation).
For example, the sums over all of $H_{R}$, $A_{R}$, $B_{R}$ and $C_{R}$ are
the same for both sides of

$\displaystyle\mathbf{\underline{1}}\oplus\mathbf{\underline{8}}\quad\longleftrightarrow\quad\mathbf{\underline{3}}\oplus\mathbf{\underline{6}}\,,$
(56)

for $\operatorname{SU}(3)$, which is clear from comparing
$\mathbf{\underline{3}}\otimes\mathbf{\underline{\overline{3}}}=\mathbf{\underline{1}}\oplus\mathbf{\underline{8}}$
and
$\mathbf{\underline{3}}\otimes\mathbf{\underline{3}}=\mathbf{\underline{\overline{3}}}\oplus\mathbf{\underline{6}}$.
These small-rank groups have many such replacement rules because they have
many representations of small dimension. For $\operatorname{SU}(3)$ the list
includes

$\displaystyle\mathbf{\underline{1}}\oplus\mathbf{\underline{8}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{3}}\oplus\mathbf{\underline{6}}\,,$
(57) $\displaystyle\mathbf{\underline{3}}\oplus\mathbf{\underline{15}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{8}}\oplus\mathbf{\underline{10}}\,,$
$\displaystyle\mathbf{\underline{1}}\oplus\mathbf{\underline{8}}\oplus\mathbf{\underline{27}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{6}}\oplus\mathbf{\underline{15}}\oplus\mathbf{\underline{15}^{\prime}}\,,$
$\displaystyle\mathbf{\underline{6}}\oplus\mathbf{\underline{24}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{15}}\oplus\mathbf{\underline{15}^{\prime}}\,,$
$\displaystyle\mathbf{\underline{3}}\oplus\mathbf{\underline{42}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{21}}\oplus\mathbf{\underline{24}}\,,$

as well as any sums or differences of these (e.g. from the first two lines it
is easy to see that we also have the replacement rule
$\mathbf{\underline{1}}\oplus\mathbf{\underline{15}}\longleftrightarrow\mathbf{\underline{6}}\oplus\mathbf{\underline{10}}$).
In contrast, such replacements for larger groups are few and far between: for
example the first two for $\operatorname{SU}(8)$ are

$\displaystyle\mathbf{\underline{1}}\oplus\mathbf{\underline{63}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{28}}\oplus\mathbf{\underline{36}}\,,$
(58) $\displaystyle\mathbf{\underline{8}}\oplus\mathbf{\underline{216}}$
$\displaystyle\longleftrightarrow\quad\mathbf{\underline{56}}\oplus\mathbf{\underline{168}}\,.$

These replacements can be used repeatedly to transform one vertex into another
as long as the multiplicity of charged hypermultiplets remains non-negative.
The changes are more than just cosmetic, since they have a real effect on the
possible edges a vertex can form. For example, there following four
$\operatorname{SU}(3)$ vertices $\mathcal{v}_{1}$, $\mathcal{v}_{2}$,
$\mathcal{v}_{3}$ and $\mathcal{v}_{4}$,

$\displaystyle\mathcal{H}(\mathcal{v}_{1})$
$\displaystyle=58\times\mathbf{\underline{3}}\oplus
3\times\mathbf{\underline{6}}\,,$ (59)
$\displaystyle\mathcal{H}(\mathcal{v}_{2})$
$\displaystyle=57\times\mathbf{\underline{3}}\oplus
2\times\mathbf{\underline{6}}\oplus\mathbf{\underline{8}}\,,$
$\displaystyle\mathcal{H}(\mathcal{v}_{3})$
$\displaystyle=56\times\mathbf{\underline{3}}\oplus\mathbf{\underline{6}}\oplus
2\times\mathbf{\underline{8}}\,,$ $\displaystyle\mathcal{H}(\mathcal{v}_{4})$
$\displaystyle=55\times\mathbf{\underline{3}}\oplus
3\times\mathbf{\underline{8}}\,,$

all have $b_{i}\cdot b_{i}=15$ and $b_{0}\cdot b_{i}=11$ and are related via
the replacements of (57). However, $\mathcal{v}_{2}$ and $\mathcal{v}_{3}$
each only have one self-edge (with bi-charged hypers
$15(\mathbf{\underline{3}},\mathbf{\underline{3}})$), whereas
$\mathcal{v}_{1}$ and $\mathcal{v}_{4}$ each have three self-edges (with bi-
charged hypers $15(\mathbf{\underline{3}},\mathbf{\underline{3}})$,
$10(\mathbf{\underline{3}},\mathbf{\underline{3}})\oplus(\mathbf{\underline{3}},\mathbf{\underline{6}})$
and
$5(\mathbf{\underline{3}},\mathbf{\underline{3}})\oplus(\mathbf{\underline{3}},\mathbf{\underline{6}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{3}})$
for $\mathcal{v}_{1}$, and similarly for $\mathcal{v}_{4}$).

##  Appendix C Bounds and pruning condition

In this appendix all statements only apply when $T\leq 1$ and
$\operatorname{U}(1)$, $\operatorname{SU}(2)$, $\operatorname{SU}(3)$ and
$\operatorname{Sp}(2)$ are omitted. Crucially, for the groups which remain all
non-trivial representations have $H_{R}\geq 3$ (saturated only by
$\tfrac{1}{2}\mathbf{\underline{6}}$ of $\operatorname{Sp}(3)$).

$a$ | $g_{a}(z)$ | $z_{a}^{\ast}$ |
---|---|---|---
$0,1,2$ | $56$ | $1$ |
$3$ | $\min\\{56,\,91-105z\\}$ | $\tfrac{13}{15}$ | $=0.8666\ldots$
$4,5$ | $\min\\{56,\,61-64z,\,91-105z\\}$ | $\tfrac{13}{15}$ | $=0.8666\ldots$
$6$ | $\min\\{56,\,61-64z,\,83-98z,\,93-108z\\}$ | $\frac{31}{36}$ | $=0.86111\ldots$
$7$ | $56-70z$ | $\tfrac{4}{5}$ | $=0.8$
$8,\ldots,15$ | $\min\\{56-70z,\,68-96z\\}$ | $\tfrac{17}{24}$ | $=0.708333\ldots$
$16,\ldots,25$ | $\min\\{56-70z,\,94-160z\\}$ | $\tfrac{47}{80}$ | $=0.5875$
$26$ | $\min\\{56-70z,\,78-130z,\,94-160z\\}$ | $\tfrac{47}{80}$ | $=0.5875$
$27$ | $\min\\{56-70z,\,84-162z\\}$ | $\tfrac{14}{27}$ | $=0.518518\ldots$
$28,\ldots,189$ | $\min\\{56-70z,\,91-224z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
$190,\ldots,209$ | $\min\\{56-70z,\,91-224z,\,251-650z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
$210,\ldots,230$ | $\min\\{56-70z,\,91-224z,\,253-693z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
$231,\ldots,252$ | $\min\\{56-70z,\,91-224z,\,254-737z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
$253,\ldots,275$ | $\min\\{56-70z,\,91-224z,\,254-782z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
$276,277,\ldots$ | $\min\\{56-70z,\,91-224z,\,253-828z\\}$ | $\tfrac{13}{32}$ | $=0.40625$
Table 5: Data for the bounds of equation (60).

Using the notation introduced in section 4.2, we claim that, for each
$a\in\\{0,1,2,\ldots\\}$,

$\displaystyle\inf_{\begin{subarray}{c}\mathcal{K}^{{\color[rgb]{0.31640625,0.34765625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.31640625,0.34765625,0.72265625}{{\mathrm{N}}}}{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}}\\\
\text{admissible}\end{subarray}}\Delta\big{(}\mathcal{K}^{{\color[rgb]{0.31640625,0.34765625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.31640625,0.34765625,0.72265625}{{\mathrm{N}}}}{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}};z,\mathcal{R}[a,3]\big{)}=\begin{cases}g_{a}(z)&z\in[0,z_{a}^{\ast}]\,,\\\
-\infty&z\in(z_{a}^{\ast},1]\,,\end{cases}$ (60)

for the concave, piece-wise linear functions $g_{a}(z)$ and corresponding
constants $z_{a}^{\ast}$ listed in table 5. These bounds were determined
empirically, as we describe further below. The models which set $g_{a}$ and
$z_{a}^{\ast}$ turn out to be quite simple:

$\displaystyle G$ $\displaystyle=G_{2}^{\kappa}\,,$
$\displaystyle\quad\mathcal{H}^{\mathrm{ch}}$
$\displaystyle=(10\times\mathbf{\underline{7}})^{\kappa}\,,$
$\displaystyle\quad a$ $\displaystyle\geq 0\,,$ (61)
$\displaystyle=\operatorname{Sp}(3)^{\kappa}\,,$
$\displaystyle=(\tfrac{35}{2}\mathbf{\underline{6}}\oplus\tfrac{1}{2}\mathbf{\underline{14}^{\prime}})^{\kappa}\,,$
$\displaystyle\in\\{3,4,5\\}\,,$
$\displaystyle=\operatorname{SU}(4)^{\kappa}\,,$
$\displaystyle=(16\times\mathbf{\underline{4}}\oplus
2\times\mathbf{\underline{6}})^{\kappa}\,,$ $\displaystyle\in\\{4,5,6\\}\,,$
$\displaystyle=\operatorname{SU}(6)^{\kappa}\,,$
$\displaystyle=(18\times\mathbf{\underline{6}}\oplus\mathbf{\underline{20}})^{\kappa}$
$\displaystyle=6\,,$ $\displaystyle=\operatorname{SU}(4)\,,$
$\displaystyle=20\times\mathbf{\underline{4}}\oplus
3\times\mathbf{\underline{6}}$ $\displaystyle=6\,,$
$\displaystyle=\operatorname{SO}(8)^{\kappa}\,,$
$\displaystyle=(4\times\mathbf{\underline{8}_{v}}\oplus
4\times\mathbf{\underline{8}_{c}}\oplus
4\times\mathbf{\underline{8}_{c}})^{\kappa}$
$\displaystyle\in\\{8,\ldots,15\\}\,,$
$\displaystyle=\operatorname{SO}(12)^{\kappa}\,,$
$\displaystyle=(8\times\mathbf{\underline{12}}\oplus
2\times\mathbf{\underline{32}})^{\kappa}\,,$
$\displaystyle\in\\{16,\ldots,26\\}\,,$ $\displaystyle=F_{4}^{\kappa}\,,$
$\displaystyle=(5\times\mathbf{\underline{26}})^{\kappa}\,,$
$\displaystyle=26\,,$ $\displaystyle=E_{6}^{\kappa}\,,$
$\displaystyle=(6\times\mathbf{\underline{27}})^{\kappa}\,,$
$\displaystyle=27\,,$ $\displaystyle=E_{7}^{\kappa}\,,$
$\displaystyle=(4\times\mathbf{\underline{56}})^{\kappa}\,,$
$\displaystyle\geq 28\,,$ $\displaystyle=\operatorname{SU}(20)\,,$
$\displaystyle=4\times\mathbf{\underline{20}}\oplus
3\times\mathbf{\underline{190}}$ $\displaystyle\in\\{190,\ldots,209\\}\,,$
$\displaystyle=\operatorname{SU}(21)\,,$
$\displaystyle=3\times\mathbf{\underline{21}}\oplus
3\times\mathbf{\underline{210}}$ $\displaystyle\in\\{210,\ldots,230\\}\,,$
$\displaystyle=\operatorname{SU}(22)\,,$
$\displaystyle=2\times\mathbf{\underline{22}}\oplus
3\times\mathbf{\underline{231}}$ $\displaystyle\in\\{231,\ldots,253\\}\,,$
$\displaystyle=\operatorname{SU}(23)\,,$
$\displaystyle=\mathbf{\underline{23}}\oplus 3\times\mathbf{\underline{254}}$
$\displaystyle\in\\{254,\ldots,275\\}\,,$
$\displaystyle=\operatorname{SU}(24)\,,$
$\displaystyle=3\times\mathbf{\underline{276}}$ $\displaystyle\geq 276\,.$

Many of the above models consist of taking $\kappa$ copies of a type-N vertex,
and these are what set the values of $z_{a}^{\ast}$ since as soon $z$ is large
enough so that
$\Delta(\mathcal{v}^{\color[rgb]{0.31640625,0.34765625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.31640625,0.34765625,0.72265625}{{\mathrm{N}}}};z,\mathcal{R}[a,3])$
is negative,
$\Delta(\mathcal{K}^{{\color[rgb]{0.31640625,0.34765625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.31640625,0.34765625,0.72265625}{{\mathrm{N}}}}{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}};z,\mathcal{R}[a,3])$
can be made arbitrarily negative by taking $\kappa\to\infty$.

Figure 7: Values of
$\Delta(\mathcal{K}^{{\color[rgb]{0.31640625,0.34765625,0.72265625}\definecolor[named]{pgfstrokecolor}{rgb}{0.31640625,0.34765625,0.72265625}{{\mathrm{N}}}}{\color[rgb]{0.87890625,0.40625,0.2109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.40625,0.2109375}{{\mathrm{T}}}}};z,\mathcal{R}[a,3])$
for $z=\frac{13}{15}$ and $a=5$ for the $\mathcal{O}(10^{6})$ admissible
models generated using vertices up to $\Delta(\mathcal{v})\leq\max\\{244,\dim
G(\mathcal{v})\\}$ and _without_ pruning. From these data one may read off
$g_{5}(z)$.

The functions $g_{a}(z)$ and constants $z_{a}^{\ast}$ were determined in the
following way. First, a multi-hypergraph $\mathcal{G}$ was build using only
type-N and type-T vertices with $\Delta(\mathcal{v})\leq\max\\{244,\dim
G(\mathcal{v})\\}$, keeping $\mathcal{G}$ relatively small. Then, _all_
admissible models with up to three vertices were build using the usual
branching procedure but with _no_ pruning, i.e. all admissible models were
kept regardless of how large their value of $\Delta(\mathcal{K})$.

From this subset of all admissible models one may empirically determine the
data of table 5. Although this does not comprise a minimization over all
admissible models as required, the bounds are satisfied by a wider-and-wider
margin as the number of type-T vertices in the models increases. For example,
figure 7 shows the bound in effect for $a=5$: only models with one or fewer
type-T vertices come anywhere close to saturating the bound and it is easy to
prove that the bounds hold for models with _only_ type-N vertices. With enough
case-work we suspect that one may be able to prove that the bounds hold
unequivocally.

##  Appendix D Automated Higgsing

As a non-trivial check of our results (both the code and the bounds discussed
in appendix C), we have checked that the ensembles of anomaly-free models for
both $T=0$ and $T=1$ are closed under Higgsing. By this we just mean the
group-theoretic process of picking a maximal subgroup and computing the
resulting hypermultiplet spectrum; we make no attempt at a detailed analysis
of the scalar potential for any of the models. In this section we outline some
of the details of this procedure.

Given a set of hypermultiplets $\mathcal{H}$ falling into representations of
the gauge group $G$, the hypermultiplets $\mathcal{H}^{\prime}$ which result
upon breaking to a subgroup $G^{\prime}\subset G$ are determined by

$\displaystyle\mathcal{H}^{\prime}=\varphi(\mathcal{H})\oplus(-1)\varphi(\mathrm{Adj}_{G})\oplus\mathrm{Adj}_{G^{\prime}}$
(62)

where $\varphi$ gives the branching rules. For example, $\operatorname{Sp}(7)$
has a maximal subgroup $\operatorname{Sp}(3)\times\operatorname{Sp}(4)$ for
which the branching rules include

$\displaystyle\varphi(\mathbf{\underline{1}})$
$\displaystyle=(\mathbf{\underline{1}},\mathbf{\underline{1}})\,,$ (63)
$\displaystyle\varphi(\mathbf{\underline{14}})$
$\displaystyle=(\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{8}})\,,$
$\displaystyle\varphi(\mathbf{\underline{90}})$
$\displaystyle=(\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{14}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{8}})\,,$
$\displaystyle\varphi(\mathbf{\underline{105}})$
$\displaystyle=(\mathbf{\underline{21}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{36}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{8}})\,,$

and using these on the $T=1$ anomaly-free model with

$G=\operatorname{Sp}(7)\,,\qquad\mathcal{H}=35\times\mathbf{\underline{1}}\oplus
16\times\mathbf{\underline{14}}\oplus\mathbf{\underline{90}}\,,$ (64)

results in the anomaly-free model with
$G^{\prime}=\operatorname{Sp}(3)\times\operatorname{Sp}(4)$ and
hypermultiplets given by

$\displaystyle\mathcal{H}^{\prime}$
$\displaystyle=\big{[}35\varphi(\mathbf{\underline{1}})\oplus
16\varphi(\mathbf{\underline{14}})\oplus\varphi(\mathbf{\underline{90}})\big{]}\oplus(-1)\varphi(\mathbf{\underline{105}})\oplus\big{[}(\mathbf{\underline{21}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{36}})\big{]}$
(65) $\displaystyle=35(\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus
16\big{[}(\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{8}})\big{]}\oplus\big{[}(\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{14}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{8}})\big{]}$
$\displaystyle\qquad\oplus(-1)\big{[}(\mathbf{\underline{21}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{36}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{8}})\big{]}\oplus\big{[}(\mathbf{\underline{21}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{36}})\big{]}$
$\displaystyle=36(\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus
16(\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus
16(\mathbf{\underline{1}},\mathbf{\underline{8}})\oplus(\mathbf{\underline{14}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{27}})\,.$

Generically, anomaly-free models have gauge groups containing more than one
simple factor and the procedure is more involved than simply checking maximal
subgroups of each gauge factor independently. There are models for which more
than one simple factor must break to a maximal subgroup simultaneously,
corresponding, e.g., to giving a bi-fundamental hypermultiplet a VEV: see
figure 6 for many such examples. For a model with gauge group
$G=\prod_{i=1}^{\kappa}G_{i}$ and hypermultiplets $\mathcal{H}$, introduce the
following notation. First, let $L=(L_{1},\ldots,L_{k})$ denote a choice of
subgroups for each $G_{i}$, where $L_{i}$ is either $G_{i}$ itself or one of
its maximal subgroups, and let $|L|$ be the number of $L_{i}$ for which
$L_{i}\neq G_{i}$ is a _proper_ subgroup. Write $G_{L}$ and $\mathcal{H}_{L}$
to denote the model which results upon Higgsing to the maximal subgroup of $G$
as specified by $L$ and write $G_{L}|$ and $\mathcal{H}_{L}|$ for the
restriction to those gauge factors for which $L_{i}\neq G_{i}$ (i.e. ignoring
gauge factors $G_{i}$ which were unaltered). Then we apply the following
procedure:

  1. 1.

Let $\mathcal{L}_{0}=\\{(G_{1},\ldots,G_{k})\\}$. That is, $\mathcal{L}_{0}$
consists only of the trivial Higgsing which leaves the model unaltered.

  2. 2.

For each $\ell=1,2,\ldots,\kappa$ in turn…

    1. (a)

Construct the set of candidates $\mathcal{L}_{\ell}$ consisting of $L$ with
$|L|=\ell$ which have the property that reverting any single $L_{i}\neq G_{i}$
in $L$ back to $G_{i}$ produces an element of $\mathcal{L}_{\ell-1}$.

    2. (b)

For each $L\in\mathcal{L}_{\ell}$, apply the Higgsing described by $L$ to the
full model, removing by hand all $\operatorname{U}(1)$,
$\operatorname{SU}(2)$, $\operatorname{SU}(3)$ and $\operatorname{Sp}(2)$
factors other than any $\operatorname{SU}(3)$ NHCs. If the resulting model is
anomaly-free (i.e. all $n_{R}\geq 0$ in $\mathcal{H}_{L}$), check if it
appears in the ensemble. Otherwise, if $\mathcal{H}_{L}|$ has $n_{R}<0$ for
any non-trivial $R$, remove $L$ from $\mathcal{L}_{\ell}$.

Following the above procedure finds all breaking patterns $G\to G^{\prime}$
which are minimal, in the sense that if Higgsing $G$ to the maximal subgroup
$G^{\prime}\subset G$ produces an anomaly-free model then there is no
intermediate subgroup $G^{\prime\prime}$ satisfying $G^{\prime}\subset
G^{\prime\prime}\subset G$ which also produces an anomaly-free model.

##  Appendix E Models with 3-charged hypermultiplets

Here we provide a complete list of anomaly-free models for $T\leq 1$ with
hypermultiplets charged under three gauge factors when the groups
$\operatorname{U}(1)$, $\operatorname{SU}(2)$, $\operatorname{SU}(3)$ and
$\operatorname{Sp}(2)$ are forbidden. There are exactly 20 for both $T=0$ and
$T=1$. The $T=0$ models are as follows:

$\;G$ | $\;\mathcal{H}$ | $b_{i}\in\mathbb{Z}$
---|---|---
$\operatorname{SU}(4)^{2}\times\operatorname{SO}(7)$ | $(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{35}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{7}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})$ | $2,6,4$
$\operatorname{SU}(4)^{2}\times G_{2}$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{7}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{7}})\end{aligned}$ | $2,6,4$
$\operatorname{SU}(4)^{2}\times\operatorname{SO}(8)$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{35}_{v}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}_{s}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}_{c}})$ | $4,4,4$
$\operatorname{SU}(4)^{2}\times\operatorname{SO}(8)$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{35}_{s}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}_{v}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}_{c}})$ | $4,4,4$
$\operatorname{SU}(4)\times\operatorname{SO}(7)^{2}$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{27}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{8}},\mathbf{\underline{8}})$ | $4,4,4$
$\operatorname{SU}(4)^{2}\times\operatorname{SO}(7)$ | $(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})$ | $4,4,4$
$\operatorname{SU}(4)^{3}$ | $2(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus 4(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})$ | $4,4,4$
$\operatorname{SU}(4)^{2}\times\operatorname{SU}(8)$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{70}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})$ | $4,4,2$
$\operatorname{SU}(4)\times\operatorname{SO}(7)\times\operatorname{Sp}(4)$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{27}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{42}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{8}},\mathbf{\underline{8}})$ | $4,4,2$
$\operatorname{SU}(4)^{2}\times\operatorname{Sp}(4)$ | $(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{42}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})$ | $4,4,2$
$\operatorname{SU}(4)\times\operatorname{Sp}(4)^{2}$ | $(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{42}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{42}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{8}},\mathbf{\underline{8}})$ | $4,2,2$
$\operatorname{SU}(4)^{2}\times\operatorname{Sp}(3)$ | $\begin{aligned} &2(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime\prime}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\oplus 3(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $2,5,1$
$\operatorname{SU}(4)^{2}\times\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime\prime}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\\\\[-5.0pt] &\quad{}\oplus 3(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{6}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $3,4,1$
$\operatorname{SU}(4)^{2}\times\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime\prime}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\oplus 3(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $3,4,1$
$\operatorname{SU}(4)^{2}\times\operatorname{SO}(7)^{2}$ | $\begin{aligned} &(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{7}},\mathbf{\underline{7}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{8}})\end{aligned}$ | $4,4,2,2$
$\operatorname{SU}(4)^{4}$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $2,2,4,4$
$\operatorname{SU}(4)^{3}\times\operatorname{SO}(7)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{8}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})\end{aligned}$ | $2,2,4,4$
$\operatorname{SU}(4)^{3}\times\operatorname{Sp}(4)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{42}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{8}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})\end{aligned}$ | $2,2,4,2$
$\operatorname{SU}(5)\times\operatorname{SU}(4)\times\operatorname{Sp}(3)^{2}$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{10}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{5}},\mathbf{\underline{1}},\mathbf{\underline{14}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{5}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{6}})\end{aligned}$ | $2,3,2,2$
$\operatorname{SU}(4)^{5}$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $2,2,2,2,4$

The $T=1$ models are as follows:

$\;G$ | $\;\mathcal{H}$ | $b_{i}\in\Gamma$
---|---|---
$\operatorname{SU}(4)^{2}\\!\times\\!G_{2}$ | $\begin{aligned} &(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{10}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{10}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{7}})\end{aligned}$ | $U:(4,4)^{2},(1,1)$
$\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{15}})\oplus(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $U:(3,3),(2,2)^{2}$
$\operatorname{SU}(4)\\!\times\\!\operatorname{Sp}(3)\\!\times\\!G_{2}$ | $\begin{aligned} &(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{21}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{4}},\mathbf{\underline{14}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{6}},\mathbf{\underline{7}})\end{aligned}$ | $U:(3,3),(2,2)^{2}$
$\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 4(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 4(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus 3(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\\\\[-5.0pt] &\quad{}\oplus 3(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $U:(3,3),(1,1)^{2}$
$\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{10}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 3(\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 4(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 3(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus 3(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $U:(3,3),(1,1)^{2}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{15}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{8}_{v}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}_{s}},\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}_{c}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $U:(1,1),(2,2)^{3}$
$\operatorname{SU}(4)^{2}\\!\times\\!G_{2}$ | $(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{20}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{20}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{7}})$ | $\Gamma_{1,1}:(5,3),(3,5),(1,1)$
$\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 3(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 5(\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{3}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\oplus 2(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 3(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $\Gamma_{1,1}:(1,1),(5,2),(2,1)$
$\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{3}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\oplus(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus 2(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus 5(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $\Gamma_{1,1}:(2,0),(4,2),(2,1)$
$\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{10}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{3}{2}(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\oplus\tfrac{3}{2}(\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{3}{2}(\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\oplus 5(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus\tfrac{1}{2}(\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $\Gamma_{1,1}:(3,1)^{2},(2,1)$
$\operatorname{SO}(7)\\!\times\\!\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{Sp}(3)$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}})\oplus\tfrac{3}{2}(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{14}^{\prime}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{1}{2}(\mathbf{\underline{7}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus(\mathbf{\underline{8}};\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}};\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 3(\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\oplus\tfrac{3}{2}(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus\tfrac{1}{2}(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{6}})\end{aligned}$ | $\Gamma_{1,1}:\scalebox{0.9}{(1,1),(2,0),(3,1),(2,1)}$
$G_{2}\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{7}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{10}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{10}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{10}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{6}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{6}},\mathbf{\underline{6}})\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\Gamma_{1,1}:(-1,-2),(4,2)^{3}$
$\operatorname{SU}(8)\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{36}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus 4(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\Gamma_{1,1}:(0,1),(4,0)^{3}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus 4(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(4,0)^{3}\\\\[-5.0pt] U&:(1,-2),(2,4)^{3}\end{aligned}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{SU}(4)^{2}\\!\times\\!\operatorname{SO}(7)$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{8}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(4,0)^{3}\\\\[-5.0pt] U&:(1,-2),(2,4)^{3}\end{aligned}$
$\operatorname{SO}(9)\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\\\\[-5.0pt] &\quad{}\oplus 4(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(4,0)^{3}\\\\[-5.0pt] U&:(1,-2),(2,4)^{3}\end{aligned}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{Sp}(4)\\!\times\\!\operatorname{SU}(4)^{2}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{42}},\mathbf{\underline{1}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{8}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(2,0),(4,0)^{2}\\\\[-5.0pt] U&:(1,-2),(1,2),(2,4)^{2}\end{aligned}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{SU}(4)^{4}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}})\oplus 2(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(2,0)^{2},(4,0)^{2}\\\\[-5.0pt] U&:(1,-2),(1,2)^{2},(2,4)^{2}\end{aligned}$
$\operatorname{SO}(8)\\!\times\\!\operatorname{SU}(4)^{2}\\!\times\\!G_{2}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{27}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus(\mathbf{\underline{1}};\mathbf{\underline{6}},\mathbf{\underline{6}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{15}},\mathbf{\underline{7}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{7}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:\scalebox{0.9}{$(0,2),(2,0),(6,0),(4,0)$}\\\\[-5.0pt] U&:\scalebox{0.9}{$(1,-2),(1,2),(3,6),(2,4)$}\end{aligned}$
$\operatorname{SO}(14)\\!\times\\!\operatorname{SU}(4)^{3}$ | $\begin{aligned} &(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}};\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{20}^{\prime}})\oplus(\mathbf{\underline{14}};\mathbf{\underline{6}},\mathbf{\underline{1}},\mathbf{\underline{1}})\\\\[-5.0pt] &\quad{}\oplus 4(\mathbf{\underline{1}};\mathbf{\underline{4}},\mathbf{\underline{4}},\mathbf{\underline{4}})\end{aligned}$ | $\begin{aligned} \Gamma_{1,1}&:(0,2),(4,-2),(4,0)^{2}\\\\[-5.0pt] U&:(1,-2),(4,-2),(2,4)^{2}\end{aligned}$

##  Appendix F Orbifold Models

In this appendix, we provide asymmetric orbifold models Dixon:1985jw ;
Dixon:1986jc ; Narain:1986qm ; Narain:1990mw  realizing the example anomaly-
free models of equations (32) and (35).

  * •

A model for (32):

Starting from the $E_{8}\times E_{8}$ heterotic theory and taking
$\Gamma^{4,4}(D_{4})$ lattice with

$\displaystyle\phi_{L}=(0,0)\,,\quad\phi_{R}=\tfrac{1}{2}(1,1)\,,\quad
V_{L}=\tfrac{1}{2}\left(1^{4},0^{4};0^{8}\right)\,,$ (66)

the invariant lattice and its dual are

$\displaystyle I$
$\displaystyle=\big{\\{}(p_{L},P_{L};0)\;\big{|}\;p_{L}\in\Lambda_{R}(D_{4})\,,\,P_{L}\in\Lambda_{R}(E_{8}\times
E_{8})\big{\\}}\,,$ (67) $\displaystyle I^{*}$
$\displaystyle=\big{\\{}(p_{L}^{*},P_{L};0)\;\big{|}\;p_{L}^{*}\in\Lambda_{W}(D_{4})\,,\,P_{L}\in\Lambda_{R}(E_{8}\times
E_{8})\big{\\}}\,.$

The spectrum of the model is

$\displaystyle G$ $\displaystyle=\mathrm{Spin}(8)\times\mathrm{Spin}(16)\times
E_{8}\,,$ (68) $\displaystyle\mathcal{H}$
$\displaystyle=(\mathbf{\underline{1}},\mathbf{\underline{128}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}_{v}},\mathbf{\underline{16}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}_{s}},\mathbf{\underline{16}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{8}_{c}},\mathbf{\underline{16}},\mathbf{\underline{1}}).$

  * •

A model for (35):

Starting from the $E_{8}\times E_{8}$ heterotic theory and taking
$\Gamma^{4,4}(D_{4})$ lattice with

$\displaystyle\phi_{L}=(0,0)\,,\quad\phi_{R}=\tfrac{1}{2}(1,1)\,,\quad
V_{L}=\tfrac{1}{2}\left(1^{4},0^{4};1^{4},0^{4}\right)\,,$ (69)

The lattices $I$ and $I^{*}$ are the same as above. The spectrum of the model
is

$\displaystyle G$
$\displaystyle=\mathrm{Spin}(8)\times\frac{\mathrm{Spin}(16)^{2}}{\mathbb{Z}_{2}}\,,$
(70) $\displaystyle\mathcal{H}$
$\displaystyle=(\mathbf{\underline{1}},\mathbf{\underline{128}},\mathbf{\underline{1}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{1}},\mathbf{\underline{128}})\oplus(\mathbf{\underline{1}},\mathbf{\underline{16}},\mathbf{\underline{16}})\,,$

where $\mathbb{Z}_{2}$ acts diagonally on $\mathrm{Spin}(16)^{2}$ and acts
non-trivially on the vector representation. Therefore, we get $H=512$. After
Higgsing, this model is dual to F-theory on elliptic $\mathbb{F}_{4}$.

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