1.0pt\theta_{e}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}A_{\;\text{charge}-{\rm U}(1)_{[1]}^{e}})\cdot\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{e}\oint_{\gamma^{1}}A_{\;\text{charged}-{\rm
U}(1)_{[1]}^{e}})\rangle=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{e}\theta_{e}{\rm Lk}(\Sigma^{2},{\gamma^{1}})}\cdot\langle\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{e}\oint_{\gamma^{1}}A)\rangle\Big{|}_{M^{4}}.$
(5.2) $\displaystyle{\rm U}(1)_{[1]}^{m}$ $\displaystyle:$
$\displaystyle\langle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\hskip
1.0pt\mathrm{d}A_{\;\text{charge}-{\rm U}(1)_{[1]}^{m}})\cdot\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{m}\oint_{\gamma^{1}}V_{\;\text{charged}-{\rm
U}(1)_{[1]}^{m}})\rangle=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{m}\theta_{m}{\rm Lk}(\Sigma^{2},{\gamma^{1}})}\cdot\langle\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{m}\oint_{\gamma^{1}}V)\rangle\Big{|}_{M^{4}}.$

The Wilson line of a 1-form gauge field $A$ is the ${\rm U}(1)_{[1]}^{e}$
electric charged object, and ’t Hooft line’s of a dual 1-form gauge field $V$
is the ${\rm U}(1)_{[1]}^{m}$ magnetic charged object; they are related by the
Hodge dual $\star$ as $\hskip 1.0pt\mathrm{d}A=\star\hskip 1.0pt\mathrm{d}V$.
An open boundary of the magnetic 2-surface $\hskip 1.0pt\mathrm{d}A$ gives
rise to 1d object closely related to an _improperly quantized_ electric Wilson
1-line of $A$. Vice versa, an open boundary of the electric 2-surface $\hskip
1.0pt\mathrm{d}V$ gives rise to 1d object closely related to an _improperly
quantized_ magnetic ’t Hooft 1-line of $V$.

$\bullet$ The _improperly quantized_ operators refer to the continuous
unquantized $\theta$, thus these operators live on the boundary of one-higher
dimensional operator.
$\bullet$ In contrast, the _genuine_ operators (e.g., those Wilson or ’t Hooft
lines specified by the quantized $q_{e}.q_{m}\in\mathbb{Z}$) do not require to
live on the boundary of one-higher dimensional operator.

  2. 2.

For a 4d pure SU(N) gauge theory, we have the electric 1-symmetry
$\mathbb{Z}_{{\rm N},[1]}^{e}$ characterized by

$\langle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi}{{\rm
N}}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\Lambda)\;{\mathrm{Tr}}_{\text{R}}(\text{P}\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\oint_{\gamma^{1}}a))\rangle=\exp\left(\frac{\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi}{{\rm
N}}{\text{Lk}({\Sigma^{2}},{\gamma^{1}})}\right)\cdot\langle{\mathrm{Tr}}_{\text{R}}(\text{P}\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\oint_{\gamma^{1}}a))\rangle,$ (5.3)

where gauge field $a$ is Lie algebra su(N) valued. The $\text{P}\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\oint a)$ specifies a SU(N) group element where P
is the path ordering. Tr is the trace in the representation R of SU(N). Here
we take the fundamental representation for R. The
$\Lambda\in\operatorname{H}^{2}(M^{4},\mathbb{Z}_{{\rm N}})$ as a
$\mathbb{Z}_{{\rm N}}$-cohomology class, tightly related to the generalized
second Stiefel-Whitney class $w_{2}(V_{{\rm PSU}({\rm
N})})\in\operatorname{H}^{2}(M,\mathbb{Z}_{\rm N})$, as the obstruction of
promoting the PSU(N) bundle to SU(N) bundle. This becomes obvious when we
promote the SU(N) gauge theory to a U(N) gauge theory with additional
constraints, here and below following [33]. The U(N) gauge theory also has the
benefits to go to the PSU(N) gauge theory. In the U(N) gauge theory, we
introduce this constraint to the path integral,

$\displaystyle\int[{\cal D}\Lambda]\dots\exp\big{(}\frac{\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi}{{\rm
N}}\int_{M^{4}}\Lambda\cup(c_{1}-B)\big{)}$ (5.4)

with the gauge bundle constraint $c_{1}=w_{2}(V_{{\rm PSU}({\rm
N})})=B\mod{\rm N}$ where the first Chern class $c_{1}=c_{1}(V_{{\rm U}({\rm
N})})\in\mathbb{Z}$ is from the U(1) part of U(N). By staring at the two
expressions in (5.3) and (5.4), it becomes also clear that an open boundary of
the magnetic 2-surface $w_{2}(V_{{\rm PSU}({\rm N})})=B$ gives rise to 1d
object closely related to the _improperly quantized_ electric Wilson 1-line of
$a$. Vice versa, an open boundary of the electric 2-surface $\Lambda$ gives
rise to 1d object closely related to the _improperly quantized_ magnetic ’t
Hooft 1-line of PSU(N) gauge theory.

  3. 3.

Thus, for a 4d pure PSU(N) gauge theory, we have the magnetic 1-symmetry
$\mathbb{Z}_{{\rm N},[1]}^{m}$ characterized by the magnetic 2-surface
operator $\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi}{{\rm
N}}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}B)$
linking with the magnetic ’t Hooft 1-line of PSU(N) gauge theory.

  4. 4.

For a 4d pure U(N) gauge theory (or the refined U(N)${}_{\hat{q}}$ gauge
theory discussed in Sec. 2), the electric 1-symmetry is given by the center
$Z({\rm U}({\rm N}))={\rm U}(1)$, while the magnetic 1-symmetry is given by
the Pontryagin dual group of the first homotopy group $\pi_{1}({\rm U}({\rm
N}))^{\vee}=\operatorname{Hom}(\pi_{1}({\rm U}({\rm N})),{\rm
U}(1))=\operatorname{Hom}(\mathbb{Z},{\rm U}(1))={\rm U}(1)$. This means that:
$\bullet$ 4d pure U(N) gauge theory without matter kinematically has ${{\rm
U}(1)}_{[1]}^{e}$ and ${{\rm U}(1)}_{[1]}^{m}$ 1-symmetries.
$\bullet$ 4d pure U(N) (say the refined U(N)${}_{\hat{q}=1}$) gauge theory
with the gauge-charged matter in the fundamental of SU(N) and in the unit
charge of U(1) written as the $(\mathbf{N},1)$ representation, kinematically
reduces the electric 1-symmetry to none — because the charged Wilson line of
${{\rm U}(1)}_{[1]}^{e}$ of the earlier pure U(N) gauge theory now becomes
breakable with two open ends attached the gauge-charged matter
$(\mathbf{N},1)$ and $(\overline{\mathbf{N}},-1)$, which nullify ${{\rm
U}(1)}_{[1]}^{e}$ to zero. But the magnetic 1-symmetry ${{\rm
U}(1)}_{[1]}^{m}$ maintains.

Higher symmetries of 4d pure gauge theories
---
QFT | $Z(G_{g})$ |  $\pi_{1}(G_{g})$. $\pi_{1}(G_{g})^{\vee}$ | 1-form $e$ sym $G_{[1]}^{e}$ | 1-form $m$ sym $G_{[1]}^{m}$ |
${\rm U}(1)$ | ${\rm U}(1)$ |  $\mathbb{Z}$. ${\rm U}(1)$ | ${{\rm U}(1)}_{[1]}^{e}$ | ${{\rm U}(1)}_{[1]}^{m}$ |
${\rm SU}({\rm N})$ | $\mathbb{Z}_{\rm N}$ | 0\. 0 | ${\mathbb{Z}}_{{\rm N},[1]}^{e}$ | 0 |
${\rm PSU}({\rm N})$ | 0 |  $\mathbb{Z}_{\rm N}$. $\mathbb{Z}_{\rm N}$ | 0 | ${\mathbb{Z}}_{{\rm N},[1]}^{m}$ |
${\rm U}({\rm N})$ | ${\rm U}(1)$ |  $\mathbb{Z}$. ${\rm U}(1)$ | ${{\rm U}(1)}_{[1]}^{e}$ | ${{\rm U}(1)}_{[1]}^{m}$ |
Higher symmetries of 4d SMs or GUTs with SM matters
QFT | $Z(G_{g})$ |  $\pi_{1}(G_{g})$. $\pi_{1}(G_{g})^{\vee}$ | 1-form $e$ sym $G_{[1]}^{e}$ | 1-form $m$ sym $G_{[1]}^{m}$ |
$G_{{\rm SM}_{q}}\equiv\frac{{{\rm SU}(3)}\times{{\rm SU}(2)}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$ | $\mathbb{Z}_{6/q}\times{\rm U}(1)$ |  $\mathbb{Z}$. ${\rm U}(1)$ | $\mathbb{Z}_{6/q,[1]}^{e}$ | ${{\rm U}(1)}_{[1]}^{m}$ |
$G_{{\rm SM}_{6}}\equiv\frac{{{\rm SU}(3)}\times{{\rm SU}(2)}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{6}}$ | ${\rm U}(1)$ |  $\mathbb{Z}$. ${\rm U}(1)$ | 0 | ${{\rm U}(1)}_{[1]}^{m}$ |
SU(5) (GG or flipped) | $\mathbb{Z}_{5}$ | 0\. 0 | 0 | 0 |
U(5)${}_{\hat{q}}$ (GG or flipped) | ${\rm U}(1)$ |  $\mathbb{Z}$. ${\rm U}(1)$ | 0 | ${{\rm U}(1)}_{[1]}^{m}$ |
$G_{\text{PS}_{q^{\prime}}}\equiv\frac{{\rm SU}(4)\times{\rm SU}(2)_{\text{L}}\times{\rm SU}(2)_{\text{R}}}{\mathbb{Z}_{q^{\prime}}}$ | ${\mathbb{Z}_{4}\times_{\mathbb{Z}_{q^{\prime}}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})}$ |  $\mathbb{Z}_{q^{\prime}}$. $\mathbb{Z}_{q^{\prime}}$ | $\mathbb{Z}_{2/q^{\prime},[1]}^{e}$ | $\mathbb{Z}_{q^{\prime},[1]}^{m}$ |
$G_{\text{PS}_{2}}\equiv\frac{{\rm SU}(4)\times{\rm SU}(2)_{\text{L}}\times{\rm SU}(2)_{\text{R}}}{\mathbb{Z}_{2}}$ | ${\mathbb{Z}_{4}\times_{\mathbb{Z}_{2}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})}$ |  $\mathbb{Z}_{2}$. $\mathbb{Z}_{2}$ | 0 | $\mathbb{Z}_{2,[1]}^{m}$ |
Spin(10) | $\mathbb{Z}_{4}$ | 0\. 0 | 0 | 0 |
Table 4: For an internal symmetry $G_{\text{internal}}=G_{g}$ as a gauge
group, we list down its center subgroup $Z(G_{g})$, its first homotopy group
$\pi_{1}(G_{g})$ and its Pontryagin dual $\pi_{1}(G_{g})^{\vee}$. We also list
down the 1-form $e$ sym $G_{[1]}^{e}$ and 1-form $m$ sym $G_{[1]}^{m}$
(without matter for the pure gauge theory, and with SM matter for the SMs and
GUTs). For ${\rm SM}_{q}$, there is a choice of $q=1,2,3,6$. Here the SM
matters are the 15 of 16 left-handed ($L$) Weyl fermions: $(\overline{\bf
3},{\bf 1})_{2,L}\oplus({\bf 1},{\bf 2})_{-3,L}\oplus({\bf 3},{\bf
2})_{1,L}\oplus(\overline{\bf 3},{\bf 1})_{-4,L}\oplus({\bf 1},{\bf
1})_{6,L}\oplus{({\bf 1},{\bf 1})_{0,L}}$ of $G_{{\rm
SM}_{q}}\equiv\frac{{{\rm SU}(3)}\times{{\rm SU}(2)}\times{\rm
U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$; or $\overline{\bf 5}\oplus{\bf 10}\oplus
1$ of SU(5); or the $({\bf 4},{\bf 2},{\bf 1})\oplus(\overline{\bf 4},{\bf
1},{\bf 2})$ of the $G_{\text{PS}_{q^{\prime}}}$ with $q^{\prime}=1,2$; or the
${\bf 16}$ of the Spin(10).

Now we are ready to provide some systematic applications to SM or GUT (results
summarized in Table 4):212121Part of these results presented here follow
Section 1.5 of [77] and some unpublished notes of the first author (J. Wang)
with Miguel Montero [78]. JW thanks Miguel Montero on the related discussions
and collaborations.

  1. 1.

Higher Symmetry for $G_{{\rm SM}_{q}}$ gauge theory with $q=1,2,3,6$:
The electric 1-symmetry is related to the center $Z(G_{{\rm
SM}_{q}})=\mathbb{Z}_{6/q}\times{\rm U}(1)$, while the magnetic 1-symmetry is
related to the Pontryagin dual group of homotopy group $\pi_{1}(G_{{\rm
SM}_{q}})^{\vee}={\rm U}(1)$. This means:
$\bullet$ without the SM fermionic matter of quarks and leptons, we have the
corresponding $\mathbb{Z}_{6/q,[1]}^{e}\times{{\rm U}(1)}_{[1]}^{e}$ and
${{\rm U}(1)}_{[1]}^{m}$ 1-symmetries.
$\bullet$ with the SM fermionic matter for $G_{{\rm SM}_{6}}$, we are left
with no electric 1-symmetry, because the gauged charge matter $({\bf 3},{\bf
2})_{1,L}$ explicitly can open up thus break the minimal charged object Wilson
line of ${{\rm U}(1)}_{[1]}^{e}$ with two open ends. But the magnetic
1-symmetry ${{\rm U}(1)}_{[1]}^{m}$ maintains.
$\bullet$ with the SM fermionic matter for $G_{{\rm SM}_{q}}$, we are left
with electric 1-symmetry $\mathbb{Z}_{6/q,[1]}^{e}$ and magnetic 1-symmetry
${{\rm U}(1)}_{[1]}^{m}$.

  2. 2.

Higher Symmetry for SU(5) of the Georgi-Glashow (GG) or the Barr’s flipped
$su(5)$ models:
$\bullet$ without the SM fermionic matter, the center $Z({\rm
SU}(5))=\mathbb{Z}_{5}$ gives rise to the electric 1-symmetry
$\mathbb{Z}_{5,[1]}^{e}$, while $\pi_{1}({\rm SU}(5))=0$ gives no magnetic
1-symmetry.
$\bullet$ with SM fermionic matter such as $\overline{\bf 5}$ of SU(5) breaks
the electric 1-symmetry to none.

  3. 3.

Higher Symmetry for U(5)${}_{\hat{q}=2}$ of the GG or the flipped $su(5)$
models:
$\bullet$ without the SM fermionic matter, the center $Z({\rm U}(5))={\rm
U}(1)$ gives rise to the electric 1-symmetry ${{\rm U}(1)}_{[1]}^{e}$, while
$\pi_{1}({\rm U}(5))^{\vee}={\rm U}(1)$ gives the magnetic 1-symmetry ${{\rm
U}(1)}_{[1]}^{m}$.
$\bullet$ with SM fermionic matter such as $\overline{\bf 5}_{-3}$ of
U(5)${}_{\hat{q}=2}$ breaks the electric 1-symmetry to none. But the magnetic
1-symmetry ${{\rm U}(1)}_{[1]}^{m}$ maintains.

  4. 4.

Higher Symmetry for Pati-Salam $G_{\text{PS}_{q^{\prime}}}$ gauge theory with
$q^{\prime}=1,2$:
$\bullet$ without the SM fermionic matter, the center
$Z(G_{\text{PS}_{q^{\prime}}})={\mathbb{Z}_{4}\times_{\mathbb{Z}_{q^{\prime}}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})}$
gives rise to the electric 1-symmetry
$({\mathbb{Z}_{4}\times_{\mathbb{Z}_{q^{\prime}}}(\mathbb{Z}_{2}\times\mathbb{Z}_{2})})_{[1]}^{e}$,
while the
$\pi_{1}(G_{\text{PS}_{q^{\prime}}})^{\vee}={\mathbb{Z}_{q^{\prime}}}$ gives
the magnetic 1-symmetry $\mathbb{Z}_{q^{\prime},[1]}^{m}$.
$\bullet$ with the matter $({\bf 4},{\bf 2},{\bf 1})\oplus(\overline{\bf
4},{\bf 1},{\bf 2})$, the electric 1-symmetry becomes
$\mathbb{Z}_{2/q^{\prime},[1]}^{e}$, while the magnetic 1-symmetry
$\mathbb{Z}_{q^{\prime},[1]}^{m}$ remains.

  5. 5.

Higher Symmetry for the $so(10)$ GUT and Spin(10) gauge group:
$\bullet$ without any SM matter (no ${\bf 16}$ of Spin(10)), the center
$Z({\rm Spin}(10))=\mathbb{Z}_{4}$ gives rise to the electric 1-symmetry
$\mathbb{Z}_{4,[1]}^{e}$, while $\pi_{1}({\rm Spin}(10))^{\vee}=0$ gives no
magnetic 1-symmetry.
$\bullet$ with ${\bf 16}$ of Spin(10), there are no electric nor magnetic
1-symmetries left.

##  6 Categorical Symmetry and Its Retraction

Table 4 summarizes various higher symmetries of SMs and GUTs. In particular,
for those embeddable into the Spin(10) group (e.g., only $G_{{\rm SM}_{q}}$
with $q=6$ and $G_{\text{PS}_{q^{\prime}}}$ with $q^{\prime}=2$, listed in
Fig. 10 and Fig. 11), we only have 1-form symmetries for them:

$\displaystyle G_{{\rm SM}_{6}}$ $\displaystyle:$ $\displaystyle{{\rm
U}(1)}_{[1]}^{m},$ (6.1) GG $su(5)$ GUT with SU(5) $\displaystyle:$
$\displaystyle\text{none},$ (6.2) GG or flipped $u(5)$ GUT with ${\rm
U}(5)_{\hat{q}=2}$ $\displaystyle:$ $\displaystyle{{\rm U}(1)}_{[1]}^{m},$
(6.3) PS model with $G_{\text{PS}_{2}}$ $\displaystyle:$
$\displaystyle\mathbb{Z}_{2,[1]}^{m},$ (6.4) $so(10)$ or modified $so(10)$ GUT
with Spin(10) $\displaystyle:$ $\displaystyle\text{none}.$ (6.5)

Some curious facts are:

  1. 1.

All electric 1-symmetries are broken by gauged charged fermionic matter. We
are only left with either magnetic 1-symmetries or none in (6.5).

  2. 2.

Regardless of which lower energy GUTs that we start with, when we approach to
the (modified) $so(10)$ GUT as a mother unified EFT at the deeper UV, all
1-symmetries are gone.

In Sec. 5, we had investigated the higher symmetries that are invertible
global symmetries. By invertible global symmetries, we mean that the fusion
algebras of symmetry generators (i.e., charge operators) follow the group law.
For any symmetry generator (say $U_{1},U_{2},\dots$) of an invertible global
symmetry, its fusion algebra is a binary operation (say “$\times$” for the
fusion) which must obey:
(1) the closure,
(2) the associativity $U_{1}\times(U_{2}\times U_{3})=(U_{1}\times
U_{2})\times U_{3}$,
(3) the identity operator 1 existence, so $U\times 1=1\times U=U$,
(4) the inverse operator operator $U^{-1}$ existence so that $U\times
U^{-1}=1$.

In this Section 6, we investigate any potential non-invertible global
symmetries in the SM or GUT models. The non-invertible global symmetries
correspond to the fusion algebras of symmetry generators (i.e., charge
operators) do not follow the group law. In particular, we will search for the
existence of a symmetry generator $U$ such that the fusion rule of $U$ with
any operator $U^{\prime}$ can never produce only the identity operator. Namely

$\displaystyle\text{ generally }U\times U^{\prime}$ $\displaystyle=$
$\displaystyle\sum_{j}U_{j}$ (6.6) $\displaystyle\text{ or at most }U\times
U^{\prime}$ $\displaystyle=$ $\displaystyle 1+\dots$ (6.7)

where the $\dots$ may be nonzero, including other operators (e.g.,
$U^{\prime\prime}$, etc.) The formal form of fusion algebra sometimes uses
“$\otimes$” for the fusion, and uses “$\oplus$” for the splitting on the right
hand side. Here we simply uses the product “$\times$” and the sum “$+$”
because these relations in (6.7) hold in the correlation function computation,
both in the QFT path integral formulation or in the quantum matter lattice
regularization formulation. Namely, we indeed have this relation holds in the
expectation value form

$\displaystyle\langle U\times U^{\prime}\rangle=\langle\sum_{j}U_{j}\rangle.$
(6.8)

Non-invertible global symmetries have appeared long ago in the 2-dimensional
CFTs [34, 35, 36, 37, 38, 39]. They also appeared recently under the name of
_algebraic higher symmetry_ or _categorical symmetry_ [46, 47], and _fusion
category symmetry_ [45, 48].222222Note that Wen et al’s usage of categorical
symmetry [82] is different from other research groups’ usage of categorical
symmetry. Instead, Wen et al’s usage of algebraic higher symmetry [46] is the
same as other research groups’ usage of categorical symmetry. From now on,
follow the recent development [40, 41, 42, 43, 44], we shall call these types
of symmetries as _non-invertible global symmetries_ or _categorical
symmetries_ interchangeably.

###  6.1 Potential Non-Invertible Categorical Symmetry induced by
$\mathbb{Z}_{2}^{\rm flip}$

Given the SM’s ${\rm SU}(3)_{c}\times{\rm SU}(2)_{\rm L}$, there are actually
two ways to embed it inside an SU(5). The first one is the Georgi-Glashow (GG)
SU(5) that we denoted ${\rm SU}(5)^{\text{1st}}$; the second one is the Barr’s
flipped SU(5) that we denoted ${\rm SU}(5)^{\text{2nd}}$. We can define a
${\rm U}(5)_{\hat{q}=2}\equiv\frac{{\rm SU}(5)\times{\rm
U}(1)_{\hat{q}=2}}{\mathbb{Z}_{5}}$ for both versions of SU(5), which we call
the two versions ${\rm U}(5)_{\hat{q}=2}^{\text{1st}}$ and ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$.

We denote the GG’s ${\rm U}(5)$ as the first kind ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}$ embedded inside the Spin(10), where

${\rm U}(5)_{\hat{q}=2}^{\text{1st}}\equiv\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X,\hat{q}=2}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X_{1},\hat{q}=2}}{\mathbb{Z}_{5}}.$

We denote ${\rm U}(1)_{X}$ also the ${\rm U}(1)_{X_{1}}$ which is also
generated by the 25th Lie algebra generator $T_{25}$ of this ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}$. The $\hat{q}=2$ specifies the identification
between the ${\rm SU}(5)$’s center and the normal subgroup of ${\rm U}(1)$ via
our definition in (2.1).

We denote the Barr’s flipped ${\rm U}(5)$ as the second kind ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$ embedded inside the Spin(10), where

${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\equiv\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm
U}(1)_{\chi,\hat{q}=2}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X_{2},\hat{q}=2}}{\mathbb{Z}_{5}}.$

We denote ${\rm U}(1)_{\chi}$ also the ${\rm U}(1)_{X_{2}}$ which is also
generated by the 25th Lie algebra generator $T_{25}$ of this ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$.

There is a magnetic 1-symmetry ${{\rm U}(1)}_{[1]}^{m_{X_{1}}}$ from the GG’s
$\pi_{1}({\rm U}(5)_{\hat{q}=2}^{\text{1st}})^{\vee}=\pi_{1}({\rm
U}(1)_{X_{1}})^{\vee}={\rm U}(1)$. There is a magnetic 1-symmetry ${{\rm
U}(1)}_{[1]}^{m_{X_{2}}}$ from the Barr’s flipped model’s $\pi_{1}({\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})^{\vee}=\pi_{1}({\rm U}(1)_{X_{2}})^{\vee}={\rm
U}(1)$. There is a $\mathbb{Z}_{2}^{\rm flip}$ transformation, swapping
between ${\rm U}(5)_{\hat{q}=2}^{\text{1st}}$ and ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$. Naively if we study a gauge theory including
the union of the gauge group ${\rm U}(5)_{\hat{q}=2}^{\text{1st}}$ and ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$, denoted as ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm U}(5)_{\hat{q}=2}^{\text{2nd}}$
together with the outer automorphism exchanging these two U(5)s, as “$({\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})\rtimes\mathbb{Z}_{2}^{\rm flip}$,” then we
expect to find a potential categorical symmetry for this 4d gauge theory.
However, such a potential categorical symmetry is not realized, for various
reasons that we explore in this section.

###  6.2 Categorical symmetry retraction from two ${\rm U}(5)_{\hat{q}=2}$ to
Spin(10)

  1. 1.

Need Spin(10) to contain both of the two ${\rm U}(5)_{\hat{q}=2}$: Dynamically
gauging the union $({\rm U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})$ already inevitably brings us to the full
gauge group Spin(10) (checked in Appendix C and (C.6)). Thus, needless to say
whether we gauge the $\mathbb{Z}_{2}^{\rm flip}$ or not, when the two U(5)s
are gauged, we are already at the full Spin(10). For a pure Spin(10) gauge
theory without matter fields, there is only a 1-form electric global symmetry
$\mathbb{Z}_{4,[1]}$, and no categorical symmetry. For the Spin(10) gauge
theory with fermions in the 16, there are neither higher symmetries (according
to Table 4) nor categorical symmetries.

  2. 2.

GUT-Higgs scale eliminates some electric 1-symmetries: We hope to rationalize
and characterize why 1-symmetries are gone at the deeper UV in the Spin(10)
gauge group, but there seems to have 1-symmetries in the ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}$ and ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}$
gauge theories. The closer look at the Higgs mechanism from the $so(10)$ GUT
model with Spin(10) gauge group to the U(5) gauge theories, we already add
GUT-Higgs with the rep ${\bf 45}$ of Spin(10), so we can Higgs Spin(10) down
to U(5). But this rep ${\bf 45}$ has the branching rule from Spin(10) to SU(5)
as ${\bf 45}\sim{\bf 1}\oplus{\bf 10}\oplus{\overline{\bf 10}}\oplus{\bf 24}$
or to the ${\rm U}(5)_{\hat{q}=2}$ ${\bf 45}\sim{\bf 1}_{0}\oplus{\bf
10}_{4}\oplus{\overline{\bf 10}}_{-4}\oplus{\bf 24}_{0}$. These branching
rules tell us that the electric 1-symmetries, if any, are broken at the GUT-
Higgs scale. Thus, any electric 1-symmetries (such as the
${\mathbb{Z}}_{5,[1]}^{e}$ for the pure SU(5) gauge theory, or the ${{\rm
U}(1)}_{[1]}^{e}$ for the pure U(5) gauge theory) could be emergent at much
lower energy at IR far below the GUT-Higgs scale.

How are the two magnetic 1-symmetries ${{\rm U}(1)}_{[1]}^{m_{X_{1}}}$ and
${{\rm U}(1)}_{[1]}^{m_{X_{2}}}$ disappear when we go to deeper energy from
two ${\rm U}(5)_{\hat{q}=2}$ to the Spin(10)? After all the GUT-Higgs are in
the rep of the electric sector not the magnetic sector, so the GUT-Higgs does
not remove the magnetic 1-symmetries. In order to understand how magnetic
1-symmetries disappear or are removed, we study a toy model involving only the
two crucial U(1) gauge sector: the $({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})$ gauge theory.

###  6.3 Categorical symmetries in a toy model $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory

We introduce a toy model of ${\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}}$ gauge theory, in terms of two U(1) sectors: ${\rm
U}(1)_{X_{1}}\equiv{\rm U}(1)_{X}^{\text{1st}}$ and ${\rm
U}(1)_{X_{2}}\equiv{\rm U}(1)_{\chi}^{\text{2nd}}$, while ${\rm
U}(1)_{Y_{1}}\equiv{\rm U}(1)_{\tilde{Y}}$ and ${\rm U}(1)_{Y_{2}}\equiv{\rm
U}(1)_{T_{24}}^{\text{2nd}}$, we have the following spacetime-internal
structures with the spacetime Spin group:

$\displaystyle{\rm Spin}\times_{\mathbb{Z}_{2}^{\rm F}}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})$ (6.9)
$\displaystyle={\rm Spin}\times_{\mathbb{Z}_{2}^{\rm F}}({\rm
U}(1)_{Y_{1}}\times_{\mathbb{Z}_{5}}{\rm U}(1)_{X_{1}})=({\rm
Spin}\times_{\mathbb{Z}_{2}^{\rm F}}{\rm
U}(1)_{X_{1}})\times_{\mathbb{Z}_{5}}{\rm U}(1)_{Y_{1}}\supset({\rm
Spin}\times_{\mathbb{Z}_{2}^{\rm
F}}\mathbb{Z}_{4,X_{1}})\times_{\mathbb{Z}_{5}}{\rm U}(1)_{Y_{1}}$ (6.10)
$\displaystyle={\rm Spin}\times_{\mathbb{Z}_{2}^{\rm F}}({\rm
U}(1)_{Y_{2}}\times_{\mathbb{Z}_{5}}{\rm U}(1)_{X_{2}})=({\rm
Spin}\times_{\mathbb{Z}_{2}^{\rm F}}{\rm
U}(1)_{X_{2}})\times_{\mathbb{Z}_{5}}{\rm U}(1)_{Y_{2}}\supset({\rm
Spin}\times_{\mathbb{Z}_{2}^{\rm
F}}\mathbb{Z}_{4,X_{2}})\times_{\mathbb{Z}_{5}}{\rm
U}(1)_{Y_{2}}.\quad\quad\quad$ (6.11)

Dynamically gauge the internal symmetry group $({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})=({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})=({\rm
U}(1)_{Y_{2}}\times_{\mathbb{Z}_{5}}{\rm U}(1)_{X_{2}})$, we obtain their
gauge theory.

There is also a $\mathbb{Z}_{2}^{\rm flip}$ as an _outer automorphism_ of
$({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})$ exchanging
the two ${\rm U}(1)$ subgroup, which we can define on this spacetime-internal
structures

$\displaystyle{\rm Spin}\times_{\mathbb{Z}_{2}^{\rm F}}\big{(}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{)}.$ (6.12)

Several comments about their higher symmetries and categorical symmetries, and
their potential obstructions:

  1. 1.

Higher symmetries without gauge charged matter:

For a pure 4d $G_{g}=({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})$ gauge theory without gauge charged matter, we have the center

$Z(G_{g})=({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm U}(1)_{X_{2}})$

itself inducing the product of the two U(1) electric 1-symmetries, denoted as
${{{\rm U}(1)}_{[1]}^{e_{X_{1}}}\times_{\mathbb{Z}_{4,X[1]}}{{\rm
U}(1)}_{[1]}^{e_{X_{2}}}}$ modding out the shared common ${\mathbb{Z}_{4,X}}$
1-symmetry. We have

$\pi_{1}(G_{g})^{\vee}=\operatorname{Hom}(\pi_{1}(G_{g}),{\rm
U}(1))=\operatorname{Hom}(\mathbb{Z}_{X_{1}}\times\mathbb{Z}_{X_{2}},{\rm
U}(1))={\rm U}(1)_{X_{1}}\times{\rm U}(1)_{X_{2}}$

inducing the magnetic 1-symmetry ${{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}$.

  2. 2.

$\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory:

If we furthur gauge the $\mathbb{Z}_{2}^{\rm flip}$ symmetry, we get a
nonabelian gauge theory of a gauge group

$\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}.$

If such a 4d gauge theory $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ exists as part of the
GUT or BSM physics, we have a potential categorical symmetry that we elaborate
below in the Remark 5.232323Follow the discussion in [42], a similar example
is gauging the $\mathbb{Z}_{2}$ outer automorphism exchanging the two ${\rm
E}_{8}$ in an $({\rm E}_{8}\times{\rm E}_{8})$ gauge theory. The outcome is a
disconnected compact Lie group gauge theory  $({\rm E}_{8}\times{\rm
E}_{8})\rtimes\mathbb{Z}_{2}$ referred to as the gauge symmetry of the $({\rm
E}_{8}\times{\rm E}_{8})$ heterotic string theory [83]. But the center $Z({\rm
E}_{8})=0$ and the $\pi_{1}({\rm E}_{8})^{\vee}=0$, so there are no higher nor
categorical symmetries in this gauge theory.

  3. 3.

Electric higher symmetries:

For a 4d $G_{g}=({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})$ gauge theory with SM gauged electrically charged matter, the
1-form $e$ symmetry ${{{\rm
U}(1)}_{[1]}^{e_{X_{1}}}\times_{\mathbb{Z}_{4,X[1]}}{{\rm
U}(1)}_{[1]}^{e_{X_{2}}}}$ is explicitly broken to only a subgroup ${{\rm
U}(1)}_{[1]}^{e}$. Furthermore, the remained ${{\rm U}(1)}_{[1]}^{e}$ is gone
when embedding $({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})$ into the ${\rm U}(5)_{\hat{q}=2}$ subgroup. Thus, we are not
interested in pursuing the electric higher symmetries furthur since they are
all broken at the GG and flipped $su(5)$ models with matter.

  4. 4.

Magnetic higher symmetries:

For a 4d $G_{g}=({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})$ gauge theory with SM gauged electrically charged matter, but
we are still left with two 1-form $m$ symmetries preserved: ${{{\rm
U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm U}(1)}_{[1]}^{m_{X_{2}}}}$, together with
the SO structure and the $\mathbb{Z}_{2}^{\rm flip}$ symmetry. The full
spacetime-internal symmetry for this gauge theory requires at least the
structure:

${\rm SO}\times\Big{(}\big{(}{{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}\big{)}\rtimes\mathbb{Z}_{2}^{\rm flip}\Big{)}$

where the $\mathbb{Z}_{2}^{\rm flip}$ belongs to an _outer automorphism_ of
${{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm U}(1)}_{[1]}^{m_{X_{2}}}}$
exchanging the two ${\rm U}(1)_{[1]}$.

If we gauge $\mathbb{Z}_{2}^{\rm flip}$ to obtain the 4d gauge theory
$\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$, naively we only
require the spacetime-internal symmetry structure:

${\rm SO}\times\Big{(}\big{(}{{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}\big{)}\Big{)}.$

But there is a potential categorical symmetry that we elaborate below in the
Remark 5.

  5. 5.

Potential categorical symmetry in $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory:

Just like ${\rm O}(2)={\rm U}(1)\rtimes\mathbb{Z}_{2}$ or ${\rm U}(1)\rtimes
S_{N}$ gauge theories has categorical symmetries [42, 40], we can look at the
potential categorical symmetry of the $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory. As
mentioned since the electric higher symmetries are explicitly broken by gauged
charged matter, we focus on the potential categorical symmetry involving
magnetic higher symmetries. Each of the $U(1)_{X_{1}}$ and $U(1)_{X_{2}}$ has
their own 1d ’t Hooft line operators and 2d magnetic topological surface
operators:

$T^{X_{j}}_{q_{m_{j}}}\equiv\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{m_{j}}\oint_{\gamma^{1}}V_{X_{j}}),\quad\quad
U^{X_{j}}_{\theta_{m_{j}}}\equiv\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta_{m_{j}}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{j}}),$

for $j=1,2$ along some closed 1-curve and some closed 2-surfaces
${\Sigma^{2}}$. They are labeled by magnetic charges $q_{m_{j}}$ and magnetic
angles $\theta_{m_{j}}\in[0,2\pi)$. But once the $\mathbb{Z}_{2}^{\rm flip}$
is dynamically gauged, these operators are not gauge invariant. Instead, the
gauge invariant ’t Hooft line is242424It is possible to use the representation
theory of certain dual gauge group to describe these ’t Hooft lines. For
example, the ’t Hooft line in the 4d ${\rm O}(2)={\rm
U}(1)\rtimes\mathbb{Z}_{2}$ gauge theory can be understood as the Wilson line
in the representation of the 4d $\tilde{{\rm O}}(2)=\frac{{\rm
U}(1)\rtimes\mathbb{Z}_{4}}{\mathbb{Z}_{2}}=\mathrm{Pin}^{-}(2)$ gauge theory.
A naive expectation is the following: the ’t Hooft line in our 4d
$\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory can be
understood as the Wilson line in the representation of the 4d
$\big{[}\frac{({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{4}^{\rm flip}}{\mathbb{Z}_{2}^{\rm
flip}}\big{]}$ gauge theory. All these semi-direct product should be defined
via proper group extensions. We leave this subtle topic on the “S-duality” of
these theories for future work.

$\displaystyle T^{\rm flip}_{q_{m}}\equiv
T^{X_{1}}_{q_{m}}+T^{X_{2}}_{q_{m}}\equiv\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{m}\oint_{\gamma^{1}}V_{X_{1}})+\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{m}\oint_{\gamma^{1}}V_{X_{2}}).$ (6.13)

The gauge invariant topological magnetic surface operator is

$\displaystyle U^{\rm flip}_{\theta_{m}}\equiv
U^{X_{1}}_{\theta_{m}}+U^{X_{2}}_{\theta_{m}}\equiv\exp(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\theta_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{1}})+\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{2}}).$ (6.14)

Under the $\mathbb{Z}_{2}^{\rm flip}$ gauged transformation, the operators
labeled by $X_{1}$ and $X_{2}$ are swapped.

The ’t Hooft lines in (6.13) are non-topological, in the sense that their
deformations introduce propagations of dual photons. The fusions of these non-
topological operators may introduce short-distance singularities and other
non-universal contributions in the operator product expansion (OPE).
Nevertheless, there is still a fusion rule following from the topological
contribution

$\displaystyle T^{\rm flip}_{q_{m}}\times T^{\rm flip}_{q^{\prime}_{m}}=T^{\rm
flip}_{q_{m}+q^{\prime}_{m}}+T^{X_{1}}_{q_{m}}\times
T^{X_{2}}_{q^{\prime}_{m}}+T^{X_{1}}_{q^{\prime}_{m}}\times
T^{X_{2}}_{q_{m}}.$ (6.15)

The topological magnetic 2-surface operators are topological of quantum
dimension 2, with the following fusion rule:

$\displaystyle U^{\rm flip}_{\theta_{m}}\times U^{\rm
flip}_{\theta^{\prime}_{m}}=\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt{({\theta_{m}}+{\theta^{\prime}_{m}})}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{1}})+\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt{({\theta_{m}}+{\theta^{\prime}_{m}})}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{2}})$ (6.16) $\displaystyle\quad\quad+\exp(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\theta_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{1}})\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta^{\prime}_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{2}})+\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta^{\prime}_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{1}})\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta_{m}\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}\star\hskip
1.0pt\mathrm{d}V_{X_{2}})\quad\quad\quad\quad$ (6.17)
$\displaystyle\quad\quad=U^{\rm
flip}_{\theta_{m}+\theta^{\prime}_{m}}+(U^{X_{1}}_{\theta_{m}}\times
U^{X_{2}}_{\theta^{\prime}_{m}}+U^{X_{1}}_{\theta^{\prime}_{m}}\times
U^{X_{2}}_{\theta_{m}}).$ (6.18)

The fusion rule splits, which indicates the magnetic 2-surface operators are
non-invertible. Only the first operator $U^{\rm
flip}_{\theta_{m}+\theta^{\prime}_{m}}$ of quantum-dimension 2 is the same
type of magnetic 2-surface operator (6.14) that we start with. The remained
term is a more general type of 2-surface operator

$\displaystyle U^{\rm flip}_{\theta_{m},\theta^{\prime}_{m}}\equiv U^{\rm
flip}_{\theta^{\prime}_{m},\theta_{m}}\equiv(U^{X_{1}}_{\theta_{m}}\times
U^{X_{2}}_{\theta^{\prime}_{m}}+U^{X_{1}}_{\theta^{\prime}_{m}}\times
U^{X_{2}}_{\theta_{m}})$ (6.19)

carrying the dependence of $({\theta_{m},\theta^{\prime}_{m}})$ that still
gauge invariant under the $\mathbb{Z}_{2}^{\rm flip}$’s swapping
$X_{1}\leftrightarrow X_{2}$. The earlier topological surface operator in
(6.14) is the special kind of the more general case:

$U^{\rm flip}_{\theta_{m}}\equiv U^{\rm flip}_{\theta_{m},0}\equiv U^{\rm
flip}_{0,\theta_{m}},\quad U^{\rm flip}_{\theta^{\prime}_{m}}\equiv U^{\rm
flip}_{\theta^{\prime}_{m},0}\equiv U^{\rm flip}_{0,\theta^{\prime}_{m}}.$

The fusion rule between two such general operators also splits

$U^{\rm flip}_{\theta_{m},\theta^{\prime}_{m}}\times U^{\rm
flip}_{\vartheta_{m},\vartheta^{\prime}_{m}}=(U^{X_{1}}_{\theta_{m}+\vartheta_{m}}\times
U^{X_{2}}_{\theta^{\prime}_{m}+\vartheta^{\prime}_{m}}+U^{X_{1}}_{\theta^{\prime}_{m}+\vartheta^{\prime}_{m}}\times
U^{X_{2}}_{\theta_{m}+\vartheta_{m}})+(U^{X_{1}}_{\theta_{m}+\vartheta_{m}}\times
U^{X_{2}}_{\theta^{\prime}_{m}+\vartheta^{\prime}_{m}}+U^{X_{1}}_{\theta^{\prime}_{m}+\vartheta^{\prime}_{m}}\times
U^{X_{2}}_{\theta_{m}+\vartheta_{m}})\\\ =U^{\rm
flip}_{{\theta_{m}+\vartheta_{m}},{\theta^{\prime}_{m}+\vartheta^{\prime}_{m}}}+U^{\rm
flip}_{{\theta_{m}+\vartheta^{\prime}_{m}},{\theta^{\prime}_{m}+\vartheta_{m}}}$
(6.20)

The global symmetry generated by (6.14) is thus _non-invertible_. It is a
_non-invertible global symmetry_ , or a _categorical symmetry_.

  6. 6.

Categorical symmetry retraction: Where does the categorical symmetry (6.20) of
the $\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory go in the
end, after embedding into the GUTs?

    1. (a)

Follow Sec. 6.2, the categorical symmetry is retracted when we embed ${\rm
U}(1)_{X_{1}}\subset{\rm U}(5)_{\hat{q}=2}^{\text{1st}}$ and ${\rm
U}(1)_{X_{2}}\subset{\rm U}(5)_{\hat{q}=2}^{\text{2nd}}$. Because the union of
gauge group, either “$({\rm U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})$” or “$({\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})\rtimes\mathbb{Z}_{2}^{\rm flip}$,” require to
gauge the full Spin(10) group with Weyl fermions in the 16, which has no
higher symmetries nor category symmetries.

    2. (b)

Another possible explanation at IR of categorical symmetry breaking, is
checking the possible mixed anomaly between the magnetic 1-form symmetries
$\big{(}{{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}\big{)}$ and the $\mathbb{Z}_{2}^{\rm
flip}$-symmetry. If there is any such mixed anomaly, when the
$\mathbb{Z}_{2}^{\rm flip}$-symmetry is dynamically gauged, then at least part
of the magnetic 1-symmetry needs to be broken. If so, the categorical symmetry
must also be broken.

For example, based on the calculation in [84, 77], the 5th bordism group
gives:

$\Omega^{{\rm SO}\times\mathbb{Z}_{2}\times\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm
U}(1)^{m_{X_{2}}}}_{5}=\mathbb{Z}_{2}^{5},\\\ \text{generated by
}w_{2}w_{3},\;\tau_{5}^{m_{X_{1}}}=\mathrm{Sq}^{2}\tau_{3}^{m_{X_{1}}},\;\tau_{5}^{m_{X_{2}}}=\mathrm{Sq}^{2}\tau_{3}^{m_{X_{2}}},\;a^{5},\;aw_{2}^{2}.$
(6.21) $\Omega^{{\rm O}\times\mathbb{Z}_{2}\times\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm
U}(1)^{m_{X_{2}}}}_{5}=\mathbb{Z}_{2}^{11},\\\ \text{generated by
}w_{2}w_{3},a^{5},a^{3}w_{1}^{2},aw_{1}^{4},aw_{2}^{2},a^{2}\tau_{3}^{m_{X_{1}}},a^{2}\tau_{3}^{m_{X_{2}}},w_{1}^{2}\tau_{3}^{m_{X_{1}}},w_{1}^{2}\tau_{3}^{m_{X_{2}}},\tau_{5}^{m_{X_{1}}},\tau_{5}^{m_{X_{2}}}.$
(6.22)

The particular cobordism classification of 4d anomalies that fits our theory
is this [85]:

$\displaystyle\Omega^{{\rm SO}\times\mathbb{Z}_{2}\ltimes(\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm U}(1)^{m_{X_{2}}})}_{5}.$ (6.23)

However, the twisted cobordism calculation is more difficult, Ref. [85]
predicts that the $\Omega^{{\rm SO}\times\mathbb{Z}_{2}\ltimes(\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm U}(1)^{m_{X_{2}}})}_{5}$ is either the
same as $\mathbb{Z}_{2}^{5}$ in (6.21), or as $\mathbb{Z}_{2}^{4}$ where the
two generators $\tau_{5}^{m_{X_{1}}}=\mathrm{Sq}^{2}\tau_{3}^{m_{X_{1}}}$ and
$\tau_{5}^{m_{X_{2}}}=\mathrm{Sq}^{2}\tau_{3}^{m_{X_{2}}}$ in $\Omega^{{\rm
SO}\times\mathbb{Z}_{2}\times\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm U}(1)^{m_{X_{2}}}}_{5}$ reduce to a
single one $\tau_{5}^{m_{X_{1}}}+\tau_{5}^{m_{X_{2}}}$ in $\Omega^{{\rm
SO}\times\mathbb{Z}_{2}\ltimes(\mathrm{B}{\rm
U}(1)^{m_{X_{1}}}\times\mathrm{B}{\rm U}(1)^{m_{X_{2}}})}_{5}$. Thus we can
use the data from (6.21) and (6.22) to deduce whether our $({{{\rm
U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}})\rtimes\mathbb{Z}_{2}^{\rm flip}$ symmetry has any
’t Hooft anomaly or not. Several comments are in order:

      1. i.

Here $w_{j}\equiv w_{j}(TM)$ is the $j$-th Stiefel-Whitney (SW) characteristic
class of spacetime tangent bundle $TM$ of manifold $M$.

      2. ii.

The $\tilde{\tau}_{3}$ is the generator of
$\operatorname{H}^{3}(\mathrm{B}^{2}{\rm U}(1),{\rm U}(1))=\mathbb{Z}$ and
$\tau_{3}=(\tilde{\tau}_{3}\mod 2)$ is the generator of
$\operatorname{H}^{3}(\mathrm{B}^{2}{\rm
U}(1),\mathbb{Z}_{2})=\mathbb{Z}_{2}$. The $\mathrm{Sq}^{2}$ is from the
Steenrod square, here mapping the 3rd cohomology group to the 5th cohomology
group.

      3. iii.

The upper labels ${m_{X_{1}}}$ and ${m_{X_{2}}}$ are for specifying magnetic
1-symmetries from either sector of ${{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}$.

      4. iv.

The $a$ is the generator of
$\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2}^{\rm
flip},\mathbb{Z}_{2})=\mathbb{Z}_{2}$.

      5. v.

In particular, the classification in (6.22) indicates the
$a^{2}\tau_{3}^{m_{X_{j}}}=\mathrm{Sq}^{1}(a\tau_{3}^{m_{X_{j}}})=w_{1}a\tau_{3}^{m_{X_{j}}}$
term, which specifies the potential mixed anomalies in 4d between the
$\mathbb{Z}_{2}^{\rm flip}$ symmetry and any of the magnetic U(1) 1-symmetries
(for both $j=1,2$).

      6. vi.

To check whether this $a^{2}\tau_{3}^{m_{X_{j}}}$ anomaly is present in our
theory, we can couple the magnetic 1-symmetry ${{\rm U}(1)}_{[1]}^{m_{X_{j}}}$
to the 2-form magnetic background $B^{m_{X_{j}}}$ field, and couple the
$\mathbb{Z}_{2}^{\rm flip}$ symmetry to the 1-form or 1-cochain gauge field
$a$. Suppose in the presence of $B^{m_{X_{j}}}$ field, under the larger gauge
transformation of $\mathbb{Z}_{2}^{\rm flip}$ symmetry, the partition function
is not fully invariant but obtains only a $(-1)$ phase, then it signals that
the theory has a mod 2 global anomaly.

      7. vii.

Under the $\mathbb{Z}_{2}^{\rm flip}$ symmetry, the ${\rm U}(1)_{X_{1}}$ and
${\rm U}(1)_{X_{2}}$ are swapped. Thus their associated gauged charges of the
quarks and leptons are also swapped. Out of the 16 Weyl fermions per
generation, the conventional left-handed Weyl spinors ($u_{L}$, $d_{L}$,
$\nu_{L}$, and $e_{L}$, those also coupled to the SU(2)L) maintain their ${\rm
U}(1)_{X_{1}}={\rm U}(1)_{X_{2}}$ gauged charges; but only the conventional
right-handed Weyl spinors ($\bar{u}_{R}$, $\bar{d}_{R}$, $\bar{\nu}_{R}$, and
$\bar{e}_{R}$, those can be coupled to the SU(2)R, which we also flip them to
the left-handed particle as the right-handed anti-particle) swap their ${\rm
U}(1)_{X_{1}}$ and ${\rm U}(1)_{X_{2}}$ gauged charges:

$\displaystyle\begin{tabular}[]{lcc c}\hline\cr&${\rm U}(1)_{X_{1}}\equiv{\rm
U}(1)_{X}^{\text{1st}}$&${\rm U}(1)_{X_{2}}\equiv{\rm
U}(1)_{\chi}^{\text{2nd}}$&\\\ \hline\cr$\bar{u}_{R}$&1&$-3$\\\
\hline\cr$\bar{d}_{R}$&$-3$&1\\\ \hline\cr$\bar{\nu}_{R}={\nu}_{L}$&5&1\\\
\hline\cr$\bar{e}_{R}=e_{L}^{+}$&1&5\\\ \hline\cr\end{tabular}.$ (6.29)

Although we do not yet know the full classification of anomalies from (6.22),
but we have enough informations to deduce that actually our $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory is:

$\text{{\bf mixed anomaly free} within $\mathbb{Z}_{2}^{\rm flip}$ and
magnetic 1-symmetries $\big{(}{{{\rm U}(1)}_{[1]}^{m_{X_{1}}}\times{{\rm
U}(1)}_{[1]}^{m_{X_{2}}}}\big{)}\rtimes\mathbb{Z}_{2}^{\rm flip}$}.$

The reasoning is that the 4d anomaly is captured by the large gauge
transformation of $\mathbb{Z}_{2}^{\rm flip}$-background field
$a\in\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2}^{\rm
flip},\mathbb{Z}_{2})=\mathbb{Z}_{2}$ and the ${{\rm
U}(1)}_{[1]}^{m}$-background field
$\tau_{3}\in\operatorname{H}^{3}(\mathrm{B}^{2}{\rm
U}(1),\mathbb{Z}_{2})=\mathbb{Z}_{2}$. To trigger a non-vanishing 4d anomaly,
we must turn on the 3-dimensional background field $\tau_{3}$, while the
possible anomalies term (involving 1-symmetries) based on dimensional analysis
counting into a 5d iTQFT can be: $a^{2}\tau_{3}^{m_{X_{j}}}$ or
$w_{1}a\tau_{3}^{m_{X_{j}}}$ or
$\tau_{5}^{m_{X_{j}}}=\mathrm{Sq}^{2}\tau_{3}^{m_{X_{j}}}$. Only
$a^{2}\tau_{3}^{m_{X_{j}}}$ or $w_{1}a\tau_{3}^{m_{X_{j}}}$ or some linear
form of $a$ as $\tilde{w}a\tau_{3}^{m_{X_{j}}}$ (where $\tilde{w}$ can be some
twisted 1-cochain specified by the structure of (6.22)) involve the desired
mixed anomalies. But the anomaly coefficients seem to be zero by any
consideration.252525If the anomaly coefficient depends on the number of
fermions are swapped under $\mathbb{Z}_{2}^{\rm flip}$, then it is the two
fermions swapped with the other two fermions in (6.29), then the anomaly
coefficient seems to be 0 mod 2. If the anomaly coefficient depends on their
${\rm U}(1)_{X_{1}}-{\rm U}(1)_{X_{2}}$ charge differences, we still have
$((-3-1)+(1-(-3))+(1-5)+(5-1))=0$ and
$((-3-1)^{2}+(1-(-3))^{2}+(1-5)^{2}+(5-1)^{2})=4^{3}=0\mod 64.$ Given this
data in (6.29), these anomaly coefficients suggest that it is highly
impossible to have any mod 2 global anomaly.

      8. viii.

Thus we propose the categorical symmetry (6.20) of the $\big{[}({\rm
U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ gauge theory actually
is sensible and well-defined kinematically, as long as if we do not embed the
theory into the full combined theory of GG and flipped $su(5)$ GUTs. The
categorical symmetry in (6.20) is only retracted after embedding into the
GUTs.

##  7 Conclusion and Future Directions

Here are some concluding remarks and future directions:

  1. 1.

Three generations of SM quarks and leptons: Our work so far had focused on 1
generation of SM quarks and leptons. We can ask whether the number n of
generations of quarks and leptons modify any story we had discussed
previously.

In the conventional $so(10)$ GUT without a WZW term, there is no constraint on
n generation since we can take n copies of theories.

In the modified $so(10)$ GUT with a WZW term, the mod 2 class $w_{2}w_{3}$
anomaly of (1.4) is matched by the sector of GUT-Higgs fields (or their
fractionalized partons) and their 4d WZW term alone, we just need to ensure
the anomaly index from GUT-Higgs WZW sector contributes 1 mod 2.

$\bullet$ If each generation of 16 SM Weyl fermions associates with its own
GUT-Higgs field, then the generation number n times of 16 SM Weyl fermions
with n GUT-Higgs WZW sector requires a constraint ${\rm{n}}=1\mod 2$ to match
the $w_{2}w_{3}$ anomaly, where ${\rm{n}}=3$ generation indeed works.

$\bullet$ However, in general, we can just introduce a single (or any odd
number) of GUT-Higgs WZW sector to match the 1 mod 2 class of $w_{2}w_{3}$
anomaly. After all, we may only need a single GUT-Higgs to achieve the gauge
symmetry-breaking from Spin(10) to other subgroups, regardless of the number
of n. In this case, our discussion on quantum criticalities can be applied to
any n of SM or GUT.

Of course, an open question is which model fits the best to the HEP
phenomenology and experiment.

  2. 2.

What is mass?

We come back to a philosophical or metaphysical question: What is mass? From a
modern quantum matter perspective, we can address this issue with a physical
and mathematical answer. The mass more generally describes a massive or gapped
energy spectrum – gapped with respect to the ground state(s). Here the mass is
defined as the correlation function (of the corresponding
operators/excitations/states) decaying exponentially. Then we can further
address what are the known mechanisms for providing a massive or gapped
spectrum.

$\bullet$ Traditionally, the free/single-particle/mean-field mechanism
includes Anderson-Higgs [86, 87, 88] or chiral symmetry breaking [89] with a
bilinear quadratic mass term turned on in a Lagrangian or Hamiltonian theory.

$\bullet$ Modern quantum systems provide other many-body interacting
mechanisms to generate a non-quadratic non-mean-field mass, see Table 5.

In our work [1], and in Ultra Unification [30, 31, 32], these many-body
interacting mechanisms ((4)-(6)) are used for the new BSM physics.

Mass mechanism | $\begin{array}[]{c}\text{Symmetry Property}\\\ \end{array}$ | $\begin{array}[]{c}\text{Topological Order}\\\ \text{with low energy TQFT}\end{array}$ | $\begin{array}[]{c}\text{Description:}\\\ \text{Free/Single-Particle/Mean-Field}\\\ \text{or Many-Body Interacting}\\\ \text{(examples in references)}\end{array}$
---|---|---|---
(1) Anderson-Higgs | Symmetry Breaking | ✗ | $\begin{array}[]{c}\text{Mean-Field}\\\ \text{\cite[cite]{[\@@bibref{}{Anderson1963, EnglertBrout1964, Higgs1964}{}{}]}}\end{array}$
(2) $\begin{array}[]{c}\text{Confinement:}\\\ \text{Chiral SB}\end{array}$ | Symmetry Breaking | ✗ | $\begin{array}[]{c}\text{Mean-Field}\\\ \text{\cite[cite]{[\@@bibref{}{NambuJonaLasinio1961}{}{}]}}\end{array}$
(3) $\begin{array}[]{c}\text{Confinement:}\\\ \text{s confinement}\end{array}$ | $\begin{array}[]{c}\text{Symmetry Preserving}\end{array}$ | ✗ | Many-Body Interacting
(4) $\begin{array}[]{c}\text{Symmetric}\\\ \text{Mass}\\\ \text{Generation}\\\ \text{(Anomaly-Free)}\end{array}$ | $\begin{array}[]{c}\text{Symmetry Preserving}\end{array}$ | ✗ | $\begin{array}[]{c}\text{Many-Body Interacting}\\\ \text{\cite[cite]{[\@@bibref{}{FidkowskifSPT2,Wang2013ytaJW1307.7480,Wang2018ugfJW1807.05998, YouHeXuVishwanath1705.09313, YouHeVishwanathXu1711.00863, Eichten1985ftPreskill1986, Wen2013ppa1305.1045, You2014oaaYouBenTovXu1402.4151, YX14124784,BenTov2015graZee1505.04312, Kikukawa2017ngf1710.11618,WangWen2018cai1809.11171, RazamatTong2009.05037, Tong2104.03997, Catterall2020fep, CatterallTogaButt2101.01026}{}{}]}}\end{array}$
(5) $\begin{array}[]{c}\text{Symmetic}\\\ \text{Gapped}\\\ \text{Topological Order}\\\ \text{(Anomalous)}\end{array}$ | $\begin{array}[]{c}\text{Symmetry Preserving}\end{array}$ | ✓ | $\begin{array}[]{c}\text{Many-Body Interacting}\\\ \text{\cite[cite]{[\@@bibref{}{VishwanathSenthil1209.3058, Senthil1405.4015, Wang2017locWWW1705.06728}{}{}]}}\end{array}$
(6) $\begin{array}[]{c}\text{Symmety}\\\ \text{Extension}\\\ \text{Gapped}\\\ \text{(Anomaly}\\\ \text{Trivialized)}\end{array}$ | $\begin{array}[]{c}\text{Symmetry Extension}\\\ \text{$K\to\tilde{G}\overset{\iota}{\to}G$}\end{array}$ | $\begin{array}[]{c}\text{\char 51: TQFT if $K$ is gauged,}\\\ \text{and if spacetime dim $d\geq 3$}\\\\[5.69054pt] \text{\char 55: no TQFT if $\tilde{G}$}\\\ \text{remains ungauged.}\end{array}$ | $\begin{array}[]{c}\text{Many-Body Interacting}\\\ \text{\cite[cite]{[\@@bibref{}{Wang2017locWWW1705.06728}{}{}]}}\end{array}$
Table 5: Some mechanisms to generate the mass or energy gap, both in the
traditional free/single-particle/mean-field level or the quantum many-body
interacting systems. Chiral SB for Chiral Symmetry Breaking. The s confinement
is the smooth confinement. ✓ for Yes and ✗ for no. The symmetry-extension
mechanism and its short exact sequence or fibration
$K\to\tilde{G}\overset{\iota}{\to}G$ to construct $G$-symmetric gapped phase,
are summarized in [92].

  3. 3.

What is transition? What is criticality?

Following footnote 2, now we can revisit the terminology on criticalities vs
phase transitions with examples in Fig. 12-Fig. 15.
$\bullet$ The criticality means the system with gapless excitations (gapless
thus critical, sometimes conformal) and with an infinite correlation length,
it can be either (i) a continuous phase transition (many examples in Table 2)
as an unstable critical point/line/etc. as an unstable renormalization group
(RG) fixed point which has at least one relevant perturbation in the phase
diagram, or (ii) a critical phase (the white region (0) in Fig. 12-Fig. 15) as
a stable critical region controlled by a stable RG fixed point which does not
any relevant perturbation in the phase diagram.
$\bullet$ The phase transition [14] means the phase interface between two (or
more) bulk phases in the phase diagram. The phase transition can be a
continuous phase transition (second order or higher order, with gapless modes;
many examples in Table 2) or a discontinuous phase transition (first order,
without gapless modes, and with a finite correlation length; e.g., (II)-(VI)
and (II)′-(VI)′ Fig. 12-13).

  4. 4.

Boundary criticality or phase transition: The modified $so(10)$ GUT with a 4d
WZW term lives on a boundary of 5d invertible TQFT
$w_{2}w_{3}(TM)=w_{2}w_{3}(V_{{\rm SO}(10)})$ of (1.4). When the internal
Spin(10) symmetry is treated as a global symmetry (not yet dynamically
gauged), the modern condensed matter viewpoint is that

(1) If we impose any regularization (e.g., lattice) such that the internal
Spin(10) symmetry acts _onsite_ (i.e., the global symmetry acts on a local
site), then we must realize the 4d criticality as a boundary criticality on
the 4d boundary of the 5d bulk invertible TQFT. Thus the ’t Hooft anomaly in
4d is manifested by the boundary physics of 5d invertible TQFT via the anomaly
inflow [93]. The internal onsite Spin(10) symmetry can be dynamically gauged
if we gauge altogether the full 4d boundary-5d bulk coupled system (e.g., the
_long-range entangled_ gauged boundary-bulk coupled systems in Sec. 7 of
[94]).

(2) In contrast, if we are allowed to realize the internal Spin(10) symmetry
acts _non-onsite_ , then the ’t Hooft anomaly in 4d is realized as the
obstruction to gauge this non-onsite symmetry in 4d.

Since we aim to gauge the internal symmetry at the end, we shall take the
viewpoint (1), and we can interpret our criticalities or phase transitions in
Table 2 as 4d boundary criticalities or phase transitions of a 5d gapped bulk.

  5. 5.

Bulk criticality or phase transition: When the internal Spin(10) symmetry is
dynamically gauged, the 5d bulk turns from a gapped (1.4) to a gapless system
(because at the Gaussian fixed point where the pure Yang-Mills kinetic term
$(\hskip 1.0pt\mathrm{d}A)^{2}$ in 5d is marginal, the $A^{2}\hskip
1.0pt\mathrm{d}A$ and $A^{4}$ become weak and irrelevant at IR, hence the
confined gauge fields in 4d becomes deconfined and gapless in the 5d bulk).
The purple regions of the phases with the gauged Spin(10) in Fig. 14 and Fig.
15 thus have a 5d gapless bulk.

$\bullet$ The Spin(10) preserving system (purple regions) can be regarded as
the GUT-Higgs field _disordered_ phase, which has the GUT-Higgs field
_disordered_ both in the 4d boundary and 5d bulk.

$\bullet$ The Spin(10) breaking system (regions other than purple) can be
regarded as the GUT-Higgs field _ordered_ phase on the 4d boundary, but the
GUT-Higgs field can be either _ordered_ (breaking) or _disordered_
(preserving) in the 5d bulk (e.g., see a recent study [95] in condensed
matter, and references therein).

So the 5d bulk phases (in Fig. 14 and Fig. 15) moving from the purple region
to non-purple regions, can go through either —

$\bullet$ 5d bulk phase transition: a _disordered_ (Spin(10) preserving and
gauge boson _gapless_) phase to an _ordered_ (Spin(10) breaking to subgroups
and gauge boson _partially gapped_) phase, which happens together with the
boundary disorder-order transition. The presence of Spin(10) anomaly on the 4d
boundary (and the SPT order in the 5d bulk) could modify the critical
exponents and scaling dimensions of the critical GUT-Higgs fields at the
transition, which was discussed as gapless SPT states in condensed matter
literature [96, 97, 98, 99, 100, 101, 102, 103].

$\bullet$ 5d bulk no phase transition: maintain a _disordered_ (Spin(10)
preserving and gauge boson gapless) phase. In this case, the _disorder_
-_order_ transition is only a boundary transition that happens only in the 4d
boundary [95].

The 5d bulk phases (in Fig. 14 and Fig. 15) moving between different non-
purple regions, can introduce another possibility —

$\bullet$ 5d bulk phase transition: between different _ordered_ (Spin(10)
breaking to subgroups and gauge boson _partially gapped_) phases.

  6. 6.

16n vs 15n Weyl fermions, and topological criticality or topological phase
transition:

The SM and GG $su(5)$ models can have either choice of 15 or 16 Weyl fermions
for each generation. In contrast, in order to be consistent with the SM data
constraint, the PS, the GG $u(5)$, the flipped $u(5)$, and the modified
$so(10)$ have 16 Weyl fermions per generation. Thus, in all the figures from
Fig. 12-Fig. 13 and Fig. 14-Fig. 15, we see that the hyperplane/line set at
$r_{\mathbf{45}}=0$ separates one side $r_{\mathbf{45}}>0$ which has models
with 16n Weyl fermions, while the other side $r_{\mathbf{45}}<0$ can have
models with either choice of 15n or 16n Weyl fermions (except the flipped
$u(5)$ can only have 16n Weyl fermions).

$\bullet$ On the GG $u(5)$ model, if we are able to break down the continuous
${\rm U}(1)_{X_{1}}$ variant of the baryon minus lepton number $X_{1}\equiv
5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}Y_{1}$ symmetry down to a discrete
variant of $\mathbb{Z}_{4,X_{1}}=\mathbb{Z}_{4,X}$ around the hyperplane at
$r_{\mathbf{45}}=0$, then the $\mathbb{Z}_{4,X}$-symmetry gives rise to a
$\mathbb{Z}_{16}$ class global anomaly [27, 29], such that we can introduce
new sectors in addition to 15n Weyl fermions to go beyond the SM: via adding
gapped TQFTs or gapless CFTs to cancel the global anomaly (known as Ultra
Unification [30, 31, 32]).

$\bullet$ On the flipped $u(5)$ model, there is also a continuous ${\rm
U}(1)_{X_{2}}$ symmetry, where
$X_{2}\equiv\frac{1}{5}X_{1}+\frac{4}{5}Y_{1}=({\mathbf{B}-\mathbf{L}})+\frac{2}{3}Y_{1}$,
but this ${\rm U}(1)_{X_{2}}$ _cannot_ be broken down to
$\mathbb{Z}_{4,X_{2}}=\mathbb{Z}_{4,X}$ if we want to preserve the SM’s ${\rm
U}(1)_{Y_{1}}$. So the $\mathbb{Z}_{16}$ class global anomaly of
$\mathbb{Z}_{4,X}$-symmetry does not occur to constrain the flipped $u(5)$
model.

So over the 8 octants in Fig. 12-Fig. 15, only the regions of the blue (II)
and the red (III)=(VII) can have the 15n Weyl fermion scenario with new BSM
TQFT/CFT sectors. Also their ${{\rm
U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$-deconfined analogs (II)′ and
(III)′=(VII)′ can have ${{\rm
U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$-_deconfined criticality_ as well
as _topological criticality_ simultaneously.

An interesting future direction is that: Under what mechanism without fine-
tuning, can the 16n to 15n Weyl fermion _topological phase transition_
(involving with 4d TQFTs/CFTs on the 15n fermion side) occur as _the same
phase transition simultaneously_ as those phase transitions described
previously in Table 2?

  7. 7.

Completeness Hypothesis vs Absence of Global Symmetries vs Absence of
Topological Operators:

We have found that the potential categorical symmetry in the low energy sector
of $\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$-gauge theory
(presumably if we Higgs down the $so(10)$ GUT’s Spin(10) down to this
restricted subgroup of the two U(5) GUTs: GG and flipped models). But the
categorical symmetry in fact disappears to none, even when we try to embed
this $\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$ to the dynamically
gauged union of $({\rm U}(5)_{\hat{q}=2}^{\text{1st}}\cup{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}})$, which already inevitably requires the full
gauge group Spin(10) (see (C.6)).

Although our present work only studies QFT and GUT models (not quantum
gravity), we find the _absence_ of generalized global symmetries (neither
higher symmetries nor categorical symmetries) in the UV model. It will be
interesting to know whether this phenomenon is relatable to several
conjectures on the universal features of quantum gravity (QG):
$\bullet$ Completeness Hypothesis about the spectrum [104, 105, 106]: every
representation of any gauge group must be occupied by particle states.
$\bullet$ Absence of Global Symmetries [107, 108, 109, 110, 111, 112]: All
global symmetries (including the generalization of higher symmetries of
extended objects, or non-invertible categorical symmetries, etc.) in QG must
be either explicitly broken or dynamically gauged.
$\bullet$ Absence of Topological Operators: Take examples in 4d (or general
$d$d) following [42, 113],
— the completeness of gauge theory particle state spectrum is equivalent to no
topological Gukov-Witten 2-surface (or more general $(d-2)$d) operators (if
and only if Wilson 1-line operators are endable with the 0d particles);
— the completeness of twist vortices (cosmic strings, or string states) is
equivalent to no topological Wilson 1-line operators (if and only if Gukov-
Witten 2-surface operators [or generally $(d-2)$d] are endable with the twist
vortices [or generally $(d-3)$d]).
— the completeness of magnetic monopole is equivalent to no topological
magnetic 2-surface operators (if and only if ’t Hooft 1-line operators [or
generally $(d-3)$d] are endable with the 0d magnetic monopoles [or generally
$(d-4)$d]).
— the completeness of magnetic twist vortices is equivalent to no topological
’t Hooft 1-line operators [or generally $(d-3)$d] (if and only if magnetic
2-surface operators are endable with the 1d magnetic twist vortices).

It will be illuminating to learn whether the absence of higher and categorical
symmetries in our model has implications on the completion of the spectrum. It
will be interesting to understand how our model on the modified $so(10)$ GUT
has any interpretations/connections to the above QG conjectures.

##  8 Acknowledgements

JW deeply appreciates Pierre Deligne, Jun Hou Fung, Yuta Hamada, Jacob
McNamara, Miguel Montero, Shu-Heng Shao, Ryan Thorngren, Cumrun Vafa, Zheyan
Wan, and Yunqin Zheng for helpful comments. We thank Zheyan Wan for the
collaborations on upcoming works. YZY is supported by a startup fund at UCSD.
This work is supported by NSF Grant DMS-1607871 “Analysis, Geometry and
Mathematical Physics” and Center for Mathematical Sciences and Applications at
Harvard University.

##  Appendix A Quantum Numbers and Representations of SMs and GUTs

Follow the setup of Appendix A of Ref. [1], here we summarize the
representations of “elementary” chiral fermionic particles of quarks and
leptons of SMs and GUTs in Tables. In particular, we had already discussed the
representations of the GG $su(5)$ GUT and the PS model, and their embedding
into the $so(10)$ GUT in Appendix A.1 and A.2 of Ref. [1]; so we shall skip
those. We focus on the flipped $su(5)$ GUT.

##### Spacetime symmetry representation

Weyl fermions are spacetime Weyl spinors, which we prefer to write all Weyl
fermions as ${\bf 2}_{L}\text{ of }{{\rm Spin}(1,3)}=\rm{SL}(2,{\mathbb{C}})$
with a complex representation in the 4d Lorentz signature. On the other hand,
the Weyl spinor is ${\bf 2}_{L}\text{ of }{\rm Spin}(4)={\rm
SU}(2)_{L}\times{\rm SU}(2)_{R}$ with a pseudoreal representation in the 4d
Euclidean signature.

##### Internal symmetry representation

Below we provide Table 6 to organize the internal symmetry representations of
particle contents of the flipped $su(5)$ GUT with the ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$ gauge group, embedding into the $so(10)$ GUT
with the Spin(10) gauge group.

###  A.1 Embed the SM into the flipped $u(5)$ GUT with ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$, then into the $so(10)$ GUT

Let us compare the GG and the flipped $su(5)$ GUT embedding:

  1. 1.

The gauge theory embedding $so(10)$ GUT $\supset$ ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}$ GUT $\supset$ the $su(5)$ GUT $\supset$ the
SM6 only contains the $G_{{\rm SM}_{q=6}}$ via an internal symmetry group
embedding:

$\displaystyle{{\rm Spin}(10)}\supset{\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\supset G_{\text{GG}}\equiv{\rm SU}(5)\supset
G_{{\rm SM}_{6}}\equiv\frac{{{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm
L}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{6}}.$ (A.1)

The representations of quarks and leptons for these models are organized in
Table 2 of Ref. [1]. Here the ${\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\equiv\frac{{\rm SU}(5)^{\text{1st}}\times{\rm
U}(1)_{X,\hat{q}=2}}{\mathbb{Z}_{5}}=\frac{{\rm SU}(5)^{\text{1st}}\times{\rm
U}(1)_{X_{1},\hat{q}=2}}{\mathbb{Z}_{5}}$ contains the ${\rm
U}(1)_{X}\equiv{\rm U}(1)_{X_{1}}$ subgroup.

  2. 2.

The gauge theory embedding $so(10)$ GUT $\supset$ ${\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}$ GUT $\supset$ the SM6, only contains the
$G_{{\rm SM}_{q=6}}$ via an internal symmetry group embedding:

$\displaystyle{{\rm Spin}(10)}\supset{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}\supset\underset{\equiv{\rm
U}(5)_{\hat{q}=2}^{\text{1st}}\cap{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}}{{\Big{(}({{\rm SU}(3)}_{c}\times{{\rm
SU}(2)}_{\rm L})\times_{{\mathbb{Z}_{6}}}{\rm
U}(1)^{\text{1st}}_{\tilde{Y}}\times_{{\mathbb{Z}_{6}}}{\rm
U}(1)^{\text{2nd}}_{T_{24}}\Big{)}}}\supset G_{{\rm SM}_{6}}.$ (A.2)

In contrast, if we break ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}$ to ${\rm
SU}(5)^{\text{2nd}}$, this route does not contain $G_{{\rm SM}_{6}}$:

$\displaystyle{{\rm Spin}(10)}\supset{\rm
U}(5)_{\hat{q}=2}^{\text{2nd}}\supset{\rm SU}(5)^{\text{2nd}}\supset({{\rm
SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm L})\times_{{\mathbb{Z}_{6}}}{\rm
U}(1)^{\text{2nd}}_{T_{24}}\supset\underset{\equiv{\rm
SU}(5)^{\text{1st}}\cap{\rm SU}(5)^{\text{2nd}}}{({{\rm SU}(3)}_{c}\times{{\rm
SU}(2)}_{\rm L})}.$ (A.3)

The representations of quarks and leptons for these models are organized in
Table 6. Here the ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\equiv\frac{{\rm
SU}(5)^{\text{2nd}}\times{\rm
U}(1)_{\chi,\hat{q}=2}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X_{2},\hat{q}=2}}{\mathbb{Z}_{5}}$
contains the ${\rm U}(1)_{\chi}\equiv{\rm U}(1)_{X_{2}}$ subgroup.

$\begin{array}[]{c}\textbf{SM}\\\ \textbf{fermion}\\\ \textbf{spinor}\\\
\textbf{field}\end{array}$ ${{\rm SU}(3)}$ ${{\rm SU}(2)}$ ${\rm
U}(1)_{X_{2}}$ ${\rm U}(1)_{Y_{2}}$
$\underset{\frac{6X_{2}}{5}-\frac{Y_{2}}{5}}{{\rm U}(1)_{{Y_{1}=}}}$ ${\rm
U}(1)_{\rm{EM}}$ ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$ ${\rm U}(1)_{X_{1}}$
$\mathbb{Z}_{5,X}$ $\mathbb{Z}_{4,X}$ $\mathbb{Z}_{2}^{F}$ SU(5) $\bar{u}_{R}$
$\overline{\mathbf{3}}$ $\mathbf{1}$ $-3$ 2 $-4$ $\frac{-2}{3}$ $-1/3$ 1 $1$ 1
1 $\overline{\bf{5}}$ $l_{L}$ $\mathbf{1}$ ${\mathbf{2}}$ $-3$ $-3$ $-3$ 0 or
$-1$ $-1$ $-3$ $-3$ 1 1 $q_{L}$ ${{\mathbf{3}}}$ $\mathbf{2}$ 1 1 $1$
$\frac{2}{3}$ or $\frac{-1}{3}$ 1/3 1 1 1 1 ${\bf{10}}$ $\bar{d}_{R}$
$\overline{\mathbf{3}}$ $\mathbf{1}$ 1 $-4$ $2$ $\frac{1}{3}$ $-1/3$ $-3$ $-3$
1 1 $\bar{\nu}_{R}={\nu}_{L}$ $\mathbf{1}$ $\mathbf{1}$ 1 6 $0$ 0 1 5 0 1 1
$\bar{e}_{R}=e_{L}^{+}$ $\mathbf{1}$ $\mathbf{1}$ 5 0 $6$ 1 1 1 1 1 1
${\bf{1}}$

Table 6:  The $G_{{\rm SM}_{q=6}}$ cannot be embedded into ${\rm
SU}(5)^{\text{2nd}}$ given the way their representations are chosen as (3.4).
The $G_{{\rm SM}_{q=6}}$ can be embedded into the flipped $u(5)$ GUT only if
we include the full ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\equiv\frac{{\rm
SU}(5)^{\text{2nd}}\times{\rm
U}(1)_{\chi,\hat{q}=2}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X_{2},\hat{q}=2}}{\mathbb{Z}_{5}}$ gauge
group. Then we can embed this into the $so(10)$ GUT with a Spin(10) gauge
group. The ${\rm SU}(5)^{\text{2nd}}\not\supset{\rm U}(1)_{Y}$ or ${\rm
U}(1)_{Y_{1}}$, but we have ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\supset{\rm
U}(1)_{Y}$ or ${\rm U}(1)_{Y_{1}}$.

##  Appendix B Embedding

###  B.1 Embedding Strategy

The strategy to embed ${\rm SU}(5)$ in ${\rm Spin}(10)$ is to first embed
${\rm SU}(5)$ in ${\rm SO}(10)$ by identifying ${\mathbb{C}}^{5}$ with
${\mathbb{R}}^{10}$ via the complex-to-real mapping $x+\hskip
1.0pt\mathrm{i}\hskip 1.0pty\xrightarrow{r}(x,y)$, and then lift the embedding
from ${\rm SO}(10)$ to ${\rm Spin}(10)$ (from $r$ to $r^{\prime}$ in the
following diagram).

Furthermore, we ask “how to embed ${\rm U}(5)$, or more precisely ${\rm
U}(5)_{\hat{q}}$ (several non-isomorphic versions ${\rm
U}(5)_{\hat{q}}\equiv\frac{{\rm SU}(5)\times_{\hat{q}}{\rm
U}(1)}{\mathbb{Z}_{5}}$ defined in (2.1), via the identification ${\rm
U}(5)_{\hat{q}}\equiv\\{(g,\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta})\in{\rm SU}(5)\times{\rm
U}(1)\big{|}(\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi n}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n{\hat{q}}}{5}}),n\in\mathbb{Z}_{5}\\}$), into ${\rm SO}(10)$ or ${\rm
Spin}(10)$?” As we shall show, it turns out that ${\rm
U}(5)_{\hat{q}}\subset{\rm SO}(10)$ for ${\hat{q}}=1,4$, while ${\rm
U}(5)_{\hat{q}}\subset{\rm Spin}(10)$ for ${\hat{q}}=2,3$.

${{\mathbb{Z}_{5}}}$${{\tilde{\rm U}(1)}}$${{{\rm SU}(5)}}$${{{\rm
U}(5)_{\hat{q}}}\equiv\frac{{\rm SU}(5)\times_{\hat{q}}{\rm
U}(1)}{\mathbb{Z}_{5}}}$${{{\rm U}(1)^{\prime}}}$${{\mathbb{Z}_{2}}}$${{{\rm
Spin}(10)}}$${{{\rm
SO}(10)}}$$\scriptstyle{p}$$\scriptstyle{r}$$\scriptstyle{\det}$$\scriptstyle{r^{\prime}}$$\scriptstyle{z=\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0ptn}{5}}}$$\scriptstyle{z^{5\hat{q}}}$ (B.1)

  1. 1.

Here we carefully distinguish several different U(1): (i) the ${\rm U}(1)$ in
${\rm U}(5)_{\hat{q}}\equiv\frac{{\rm SU}(5)\times_{\hat{q}}{\rm
U}(1)}{\mathbb{Z}_{5}}$, (ii) the $\tilde{\rm U}(1)$ as the center of ${{\rm
U}(5)_{\hat{q}}}$ such that $\tilde{\rm U}(1)=Z({{\rm U}(5)_{\hat{q}}})\supset
Z({{\rm SU}(5)})=\mathbb{Z}_{5}$, (iii) the ${\rm U}(1)^{\prime}$ as the
quotient $\frac{{{\rm U}(5)_{\hat{q}}}}{{\rm SU}(5)}$. All these three groups
are all isomorphic to any ordinary $S^{1}$, but they have different roles that
we explain below.

  2. 2.

$\bullet$ The “$\hookrightarrow$” means the inclusion, the former group can be
embedded into the later group. The “$\to$” means the group homomorphism.

$\bullet$ Here ${{\rm SU}(5)}\hookrightarrow{{\rm
U}(5)_{\hat{q}}}\overset{\det}{\to}{{\rm U}(1)^{\prime}}$ is part of the short
exact sequence $1{\to}{{\rm SU}(5)}{\to}{{\rm
U}(5)_{\hat{q}}}\overset{\det}{\to}{{\rm U}(1)^{\prime}}{\to}1$ where ${{\rm
U}(1)^{\prime}}\equiv\frac{{\rm U}(1)}{\mathbb{Z}_{5{\hat{q}}}}$ is related to
modding out ${\mathbb{Z}_{5{\hat{q}}}}$ of the ${\rm U}(1)$ defined in ${{\rm
U}(5)_{\hat{q}}}$ in (2.1). The inclusion ${{\rm SU}(5)}\hookrightarrow{{\rm
U}(5)_{\hat{q}}}$ indicates that ${{\rm SU}(5)}$ is a normal subgroup of
${{\rm U}(5)_{\hat{q}}}$, while the ${{\rm U}(1)^{\prime}}$ is a quotient
group of ${{\rm U}(5)_{\hat{q}}}$. So

${{\rm U}(1)^{\prime}}=\frac{{\rm U}(5)_{\hat{q}}}{{\rm SU}(5)}=\frac{{\rm
U}(1)}{\mathbb{Z}_{5{\hat{q}}}}.$

$\bullet$ Here ${\mathbb{Z}_{2}}\hookrightarrow{{\rm
Spin}(10)}\overset{p}{\to}{{\rm SO}(10)}$ is part of the short exact sequence
$1\to{\mathbb{Z}_{2}}\to{{\rm Spin}(10)}\to{{\rm SO}(10)}\to 1$, so ${{\rm
Spin}(10)}/{{\rm SO}(10)}={\mathbb{Z}_{2}}$.

$\bullet$ Here ${\mathbb{Z}_{5}}\overset{\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0ptn}{5}}}{\hookrightarrow}{\tilde{\rm
U}(1)}\overset{z^{5\hat{q}}}{\to}{{\rm U}(1)^{\prime}}$ is part of the short
exact sequence $1\to{\mathbb{Z}_{5}}\to{\tilde{\rm
U}(1)}\overset{z^{5\hat{q}}}{\to}{{\rm U}(1)^{\prime}}\to 1$.

$\bullet$ There are no other short exact sequences in this (B.1), other than
the above relations that we report.

  3. 3.

For ${\mathbb{Z}_{5}}\overset{z=\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0ptn}{5}}}{\hookrightarrow}{\tilde{\rm
U}(1)}\overset{z^{5\hat{q}}}{\to}{{\rm U}(1)^{\prime}}$, the first map
${\mathbb{Z}_{5}}\overset{z=\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0ptn}{5}}}{\hookrightarrow}{\tilde{\rm U}(1)}$
inclusion specifies the identification of the center $Z({\rm
SU}(5))={\mathbb{Z}_{5}}\equiv\\{{z=\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0ptn}{5}}}|n\in\\{0,1,2,3,4\\}\\}$ and its map to the
total group $\tilde{\rm U}(1)\equiv\\{z=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta}|\theta\in[0,2\pi)\\}$. The second map
${\tilde{\rm U}(1)}\overset{z^{5\hat{q}}}{\to}{{\rm U}(1)^{\prime}}=\frac{{\rm
U}(5)_{\hat{q}}}{{\rm SU}(5)}=\frac{{\rm U}(1)}{\mathbb{Z}_{5{\hat{q}}}}$ says
that the $z\in{\tilde{\rm U}(1)}$ maps to the ${z^{5\hat{q}}}\in{{\rm
U}(1)^{\prime}}$, thus also says that the $\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0ptn}{5}}\in{\tilde{\rm U}(1)}$ becomes the kernel of the map $\tilde{\rm
U}(1)\overset{z^{5\hat{q}}}{\to}{\rm U}(1)$ that maps to the identity $\hskip
1.0pt\mathrm{e}^{{2\pi\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\hat{q}}}}=1\in{{\rm
U}(1)}$.

All these facts coincide with our definition of ${\rm U}(5)_{\hat{q}}$ in
(2.1), which also justifies (B.1)’s second line: ${{\rm
SU}(5)}\hookrightarrow{{\rm U}(5)_{\hat{q}}}\overset{\det}{\to}{{\rm
U}(1)^{\prime}}$.

  4. 4.

The ${{\rm U}(5)_{\hat{q}}}\overset{r}{\to}{{\rm SO}(10)}$ and the ${{\rm
U}(5)_{\hat{q}}}\overset{r^{\prime}}{\to}{{\rm Spin}(10)}$ only imply that the
maps are group homomorphisms. The map may or may not be inclusion or
embedding, which depends on the value of ${\hat{q}}$.

If ${{\rm U}(5)_{\hat{q}}}\subset{\rm SO}(10)$ (in particular ${\hat{q}}=1$),
then a non-trivial fact is that the lifting is only possible, if the lifted
${{\rm U}(5)_{\hat{q}^{\prime}}}\subset{\rm Spin}(10)$ has the ${\rm SU}(5)$
fundamental representation carries an ${\hat{q}^{\prime}}=2q\quad(\text{mod
}5)$ charge under ${\rm U}(1)$. Because the isomorphism (2.2), we can deduce
that

${{\rm U}(5)_{\hat{q}=1,4}}\subset{\rm SO}(10)$ while ${{\rm
U}(5)_{\hat{q}^{\prime}=2,3}}\subset{\rm Spin}(10)$.

In the following subsections, we provide some nontrivial checks of these
facts.

###  B.2 Definition of Lie Groups

####  B.2.1 Lie algebra and Lie Group of ${\rm U}(5)_{\hat{q}}$

Let $\\{{|\psi_{i}\rangle}|i=1,2,\cdots,5\\}$ be a set of orthonormal basis of
${\mathbb{C}}^{5}$, i.e. ${\langle\psi_{i}|\psi_{j}\rangle}=\delta_{ij}$ given
the dual basis ${\langle\psi_{i}|}={|\psi_{i}\rangle}^{\dagger}$. The ${\rm
U}(5)$ group is the group of isometries of ${\mathbb{C}}^{5}$, which is
canonically identified with the group of $5\times 5$ unitary matrices. They
are generated by 25 generators $T_{a}$ ($a=1,2,\cdots,25$):

$U\in{\rm U}(5):U=\exp\Big{(}\sum_{a=1}^{25}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta_{a}T_{a}\Big{)}.$ (B.2)

The generators $T_{a}$ are basis of $5\times 5$ Hermitian matrices, and can be
chosen as

$\begin{array}[]{l}\begin{array}[]{lll}T_{1}={|\psi_{1}\rangle}{\langle\psi_{2}|}+{|\psi_{2}\rangle}{\langle\psi_{1}|},&T_{11}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{1}\rangle}{\langle\psi_{2}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{2}\rangle}{\langle\psi_{1}|},\\\
T_{2}={|\psi_{1}\rangle}{\langle\psi_{3}|}+{|\psi_{3}\rangle}{\langle\psi_{1}|},&T_{12}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{1}\rangle}{\langle\psi_{3}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{3}\rangle}{\langle\psi_{1}|},\\\
T_{3}={|\psi_{1}\rangle}{\langle\psi_{4}|}+{|\psi_{4}\rangle}{\langle\psi_{1}|},&T_{13}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{1}\rangle}{\langle\psi_{4}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{4}\rangle}{\langle\psi_{1}|},\\\
T_{4}={|\psi_{1}\rangle}{\langle\psi_{5}|}+{|\psi_{5}\rangle}{\langle\psi_{1}|},&T_{14}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{1}\rangle}{\langle\psi_{5}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{5}\rangle}{\langle\psi_{1}|},\\\
T_{5}={|\psi_{2}\rangle}{\langle\psi_{3}|}+{|\psi_{3}\rangle}{\langle\psi_{2}|},&T_{15}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{2}\rangle}{\langle\psi_{3}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{3}\rangle}{\langle\psi_{2}|},\\\
T_{6}={|\psi_{2}\rangle}{\langle\psi_{4}|}+{|\psi_{4}\rangle}{\langle\psi_{2}|},&T_{16}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{2}\rangle}{\langle\psi_{4}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{4}\rangle}{\langle\psi_{2}|},\\\
T_{7}={|\psi_{2}\rangle}{\langle\psi_{5}|}+{|\psi_{5}\rangle}{\langle\psi_{2}|},&T_{17}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{2}\rangle}{\langle\psi_{5}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{5}\rangle}{\langle\psi_{2}|},\\\
T_{8}={|\psi_{3}\rangle}{\langle\psi_{4}|}+{|\psi_{4}\rangle}{\langle\psi_{3}|},&T_{18}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{3}\rangle}{\langle\psi_{4}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{4}\rangle}{\langle\psi_{3}|},\\\
T_{9}={|\psi_{3}\rangle}{\langle\psi_{5}|}+{|\psi_{5}\rangle}{\langle\psi_{3}|},&T_{19}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{3}\rangle}{\langle\psi_{5}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{5}\rangle}{\langle\psi_{3}|},\\\
T_{10}={|\psi_{4}\rangle}{\langle\psi_{5}|}+{|\psi_{5}\rangle}{\langle\psi_{4}|},&T_{20}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{4}\rangle}{\langle\psi_{5}|}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt{|\psi_{5}\rangle}{\langle\psi_{4}|},\\\
\end{array}\\\ \\\
\begin{array}[]{l}T_{21}={|\psi_{1}\rangle}{\langle\psi_{1}|}-{|\psi_{2}\rangle}{\langle\psi_{2}|},\\\
T_{22}=2{|\psi_{3}\rangle}{\langle\psi_{3}|}-{|\psi_{4}\rangle}{\langle\psi_{4}|}-{|\psi_{5}\rangle}{\langle\psi_{5}|},\\\
T_{23}={|\psi_{4}\rangle}{\langle\psi_{4}|}-{|\psi_{5}\rangle}{\langle\psi_{5}|},\\\
T_{24}=-3{|\psi_{1}\rangle}{\langle\psi_{1}|}-3{|\psi_{2}\rangle}{\langle\psi_{2}|}+2{|\psi_{3}\rangle}{\langle\psi_{3}|}+2{|\psi_{4}\rangle}{\langle\psi_{4}|}+2{|\psi_{5}\rangle}{\langle\psi_{5}|},\\\
T_{25}=\hat{q}({|\psi_{1}\rangle}{\langle\psi_{1}|}+{|\psi_{2}\rangle}{\langle\psi_{2}|}+{|\psi_{3}\rangle}{\langle\psi_{3}|}+{|\psi_{4}\rangle}{\langle\psi_{4}|}+{|\psi_{5}\rangle}{\langle\psi_{5}|}).\\\
\end{array}\end{array}$ (B.3)

In particular, the last five generators $T_{21},T_{22},\cdots,T_{25}$ are the
Cartan generators. Among them, the last one $T_{25}$ is the ${\rm U}(1)$
subgroup generator. The overall coefficient $\hat{q}=1,2,3,4$ in $T_{25}$
labels the ${\rm U}(1)$ charge carried by the ${\rm SU}(5)$ fundamental
representation, which is to be determined later.

The ${\rm U}(1)$ subgroup contains a $\mathbb{Z}_{5}$ subgroup that can be
shared with the center of ${\rm SU}(5)$,

$\mathbb{Z}_{5}=\langle\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{25}}\rangle=\\{\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}nT_{25}}|n=0,1,2,3,4\\}.$ (B.4)

Assigning $T_{25}$ with different charge numbers $\hat{q}$ will lead to
different embeddings of the $\mathbb{Z}_{5}$ subgroup in ${\rm SU}(5)$, which
lead to different ${\rm U}(5)_{\hat{q}}={\rm
SU}(5)\times_{\mathbb{Z}_{5},{\hat{q}}}{\rm U}(1)$ group structures,

$\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}T_{25}}=\left\\{\begin{array}[]{ll}\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}\frac{1}{2}(T_{24}-5T_{21})}&\hat{q}=1,\\\ \hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}T_{24}}&\hat{q}=2,\\\ \hskip 1.0pt\mathrm{e}^{-\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{24}}&\hat{q}=3,\\\ \hskip
1.0pt\mathrm{e}^{-\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}\frac{1}{2}(T_{24}-5T_{21})}&\hat{q}=4.\\\ \end{array}\right.$ (B.5)

The difference will become important when embedding ${\rm U}(5)_{\hat{q}}$
into ${\rm SO}(10)$ or ${\rm Spin}(10)$.

####  B.2.2 Lie algebra and Lie Group of ${\rm SO}(10)$

Let $\\{{|e_{i}\rangle}|i=1,2,\cdots,10\\}$ be a set of orthonormal basis of
${\mathbb{R}}^{10}$, i.e. ${\langle e_{i}|e_{j}\rangle}=\delta_{ij}$ given by
the dual basis ${\langle e_{i}|}={|e_{i}\rangle}^{\intercal}$. The ${\rm
SO}(10)$ group is the group of rotations in ${\mathbb{R}}^{10}$, which is
canonically identified with the group of $10\times 10$ orthogonal matrices.
They are generated by 45 generators:

$O\in{\rm SO}(10):O=\exp\bigg{(}\sum_{\\{i,j\\}\subset\\{1,\cdots,10\\}}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta_{i\wedge j}\;e_{i\wedge j}\bigg{)}.$ (B.6)

The generators $e_{i\wedge j}$ are basis of $10\times 10$ pure-imaginary anti-
symmetric Hermitian matrices, given by

$e_{i\wedge j}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt({|e_{i}\rangle}{\langle
e_{j}|}-{|e_{j}\rangle}{\langle e_{i}|})$ (B.7)

for all 2-subsets $\\{i,j\\}$ in $\\{1,2,\cdots,10\\}$. Note that the basis is
anti-symmetric $e_{i\wedge j}=-e_{j\wedge i}$ by definition.

####  B.2.3 Lie algebra and Lie Group of ${\rm Spin}(10)$

The construction of the ${\rm Spin}(10)$ group starts with the Clifford
algebra ${\rm Cl}(10)=M_{16}(\mathbb{H})$, which is isomorphic to the algebra
of $16\times 16$ matrices over the quaternion field $\mathbb{H}$. Let
$\Gamma_{1},\Gamma_{2},\cdots,\Gamma_{10}$ be the generators of ${\rm
Cl}(10)$, satisfying $\\{\Gamma_{i},\Gamma_{j}\\}=2\delta_{ij}$. For the
purpose of spelling out the matrix representations, these $16\times 16$
quaternion matrices can be written as $32\times 32$ complex matrices, by
embedding $M_{16}(\mathbb{H})$ into $M_{32}({\mathbb{C}})$. One explicit
choice of the complex matrix representation can be

$\begin{array}[]{ll}\Gamma_{1}\bumpeq\sigma^{11000},&\Gamma_{2}\bumpeq\sigma^{30100},\\\
\Gamma_{3}\bumpeq\sigma^{13000},&\Gamma_{4}\bumpeq\sigma^{30300},\\\
\Gamma_{5}\bumpeq\sigma^{12012},&\Gamma_{6}\bumpeq\sigma^{30221},\\\
\Gamma_{7}\bumpeq\sigma^{12020},&\Gamma_{8}\bumpeq\sigma^{30202},\\\
\Gamma_{9}\bumpeq\sigma^{12032},&\Gamma_{10}\bumpeq\sigma^{30223},\end{array}$
(B.8)

where
$\sigma^{\mu\nu\cdots}\equiv\sigma^{\mu}\otimes\sigma^{\nu}\otimes\cdots$
denotes the direct product of Pauli matrices. The symbol $\bumpeq$ (reads “can
be represented as”) indicates that the equality is only a basis dependent
statement.

The ${\rm Spin}(10)$ group is a Lie group, whose Lie algebra corresponds to
the grade-2 subspace of ${\rm Cl}(10)$, which is spanned by 45 basis elements
(as Lie group generators),

$\Gamma_{i\wedge j}=\frac{\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{2}[\Gamma_{i},\Gamma_{j}],$ (B.9)

such that the Lie group elements are generated by

$\mathsf{O}\in{\rm
Spin}(10):\mathsf{O}=\exp\bigg{(}\sum_{\\{i,j\\}\subset\\{1,\cdots,10\\}}\frac{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta_{i\wedge j}}{2}\Gamma_{i\wedge j}\bigg{)}.$
(B.10)

The Spin group is fully contained in the even-graded subspace ${\rm
Cl}^{\text{even}}={\rm Cl}^{0}\oplus{\rm Cl}^{2}\oplus{\rm
Cl}^{4}\oplus\cdots$ of the associated Clifford algebra. For ${\rm
Cl}(10)=M_{16}(\mathbb{H})$, the even-graded subspace ${\rm
Cl}^{\text{even}}(10)=M_{16}({\mathbb{C}})\oplus M_{16}({\mathbb{C}})$ splits
into two independent algebras of $16\times 16$ matrices over the complex field
${\mathbb{C}}$. The two $M_{16}({\mathbb{C}})$ subspaces are specified by the
following projection operators (projected down from $M_{32}({\mathbb{C}})$),

$P_{\pm}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt\prod_{j=1}^{10}\Gamma_{j}}{2}.$ (B.11)

This means that the irreducible fundamental representation of ${\rm Spin}(10)$
is only 16-dimensional, even though the associated ${\rm Cl}(10)$ requires a
32-dimensional representation. By choosing one of the $M_{16}({\mathbb{C}})$
subspace (say the subspace of that survives the $P_{+}$ projection),
$\Gamma_{i\wedge j}$ can be represented as $16\times 16$ matrices as follows
(substituting Eqn. (B.8) into Eqn. (B.9) followed by the projection
$\Gamma_{i\wedge j}\to P_{+}\Gamma_{i\wedge j}P_{+}$)

$\begin{array}[]{lllll}\Gamma_{1\wedge 2}\bumpeq\sigma^{1100},&\Gamma_{1\wedge
3}\bumpeq\sigma^{2000},&\Gamma_{1\wedge
4}\bumpeq\sigma^{1300},&\Gamma_{1\wedge
5}\bumpeq-\sigma^{3012},&\Gamma_{1\wedge 6}\bumpeq\sigma^{1221},\\\
\Gamma_{1\wedge 7}\bumpeq-\sigma^{3020},&\Gamma_{1\wedge
8}\bumpeq\sigma^{1202},&\Gamma_{1\wedge
9}\bumpeq-\sigma^{3032},&\Gamma_{1\wedge
10}\bumpeq\sigma^{1223},&\Gamma_{2\wedge 3}\bumpeq-\sigma^{3100},\\\
\Gamma_{2\wedge 4}\bumpeq\sigma^{0200},&\Gamma_{2\wedge
5}\bumpeq-\sigma^{2112},&\Gamma_{2\wedge
6}\bumpeq-\sigma^{0321},&\Gamma_{2\wedge
7}\bumpeq-\sigma^{2120},&\Gamma_{2\wedge 8}\bumpeq-\sigma^{0302},\\\
\Gamma_{2\wedge 9}\bumpeq-\sigma^{2132},&\Gamma_{2\wedge
10}\bumpeq-\sigma^{0323},&\Gamma_{3\wedge
4}\bumpeq\sigma^{3300},&\Gamma_{3\wedge
5}\bumpeq\sigma^{1012},&\Gamma_{3\wedge 6}\bumpeq\sigma^{3221},\\\
\Gamma_{3\wedge 7}\bumpeq\sigma^{1020},&\Gamma_{3\wedge
8}\bumpeq\sigma^{3202},&\Gamma_{3\wedge
9}\bumpeq\sigma^{1032},&\Gamma_{3\wedge
10}\bumpeq\sigma^{3223},&\Gamma_{4\wedge 5}\bumpeq-\sigma^{2312},\\\
\Gamma_{4\wedge 6}\bumpeq\sigma^{0121},&\Gamma_{4\wedge
7}\bumpeq-\sigma^{2320},&\Gamma_{4\wedge
8}\bumpeq\sigma^{0102},&\Gamma_{4\wedge
9}\bumpeq-\sigma^{2332},&\Gamma_{4\wedge 10}\bumpeq\sigma^{0123},\\\
\Gamma_{5\wedge 6}\bumpeq\sigma^{2233},&\Gamma_{5\wedge
7}\bumpeq-\sigma^{0032},&\Gamma_{5\wedge
8}\bumpeq\sigma^{2210},&\Gamma_{5\wedge
9}\bumpeq\sigma^{0020},&\Gamma_{5\wedge 10}\bumpeq-\sigma^{2231},\\\
\Gamma_{6\wedge 7}\bumpeq-\sigma^{2201},&\Gamma_{6\wedge
8}\bumpeq-\sigma^{0023},&\Gamma_{6\wedge
9}\bumpeq\sigma^{2213},&\Gamma_{6\wedge
10}\bumpeq\sigma^{0002},&\Gamma_{7\wedge 8}\bumpeq\sigma^{2222},\\\
\Gamma_{7\wedge 9}\bumpeq-\sigma^{0012},&\Gamma_{7\wedge
10}\bumpeq\sigma^{2203},&\Gamma_{8\wedge
9}\bumpeq-\sigma^{2230},&\Gamma_{8\wedge
10}\bumpeq-\sigma^{0021},&\Gamma_{9\wedge 10}\bumpeq\sigma^{2211}.\end{array}$
(B.12)

The representation space is ${\mathbb{C}}^{16}$.

###  B.3 Embedding and Projection Maps

As laid out in Eqn. (B.1), some of the ${\rm U}(5)_{\hat{q}}$ group can be
embedded into ${\rm SO}(10)$ by an embedding map $r$, and the ${\rm Spin}(10)$
group can be projected to ${\rm SO}(10)$ by a projection map $p$. This
potentially enables us to lift the embedding $r$ to $r^{\prime}$, then
$r^{\prime}$ will tell us how to embed which ${\rm U}(5)_{\hat{q}^{\prime}}$
in ${\rm Spin}(10)$. However, we will show that the lifting is not always
possible. It crucially depends on the choice of the charge number $\hat{q}$ or
$\hat{q}^{\prime}$.

####  B.3.1 Embedding Map $r$

The embedding map $r:{\mathbb{C}}^{5}\to{\mathbb{R}}^{10}$ sends the
representation space ${\mathbb{C}}^{5}$ to ${\mathbb{R}}^{10}$ by

$r:\sum_{i=1}^{5}z_{i}{|\psi_{i}\rangle}\mapsto\sum_{i=1}^{5}({\rm
Re}z_{i}{|e_{2i-1}\rangle}+\operatorname{Im}z_{i}{|e_{2i}\rangle}).$ (B.13)

It induces a functorial map $r:{\rm U}(5)\to{\rm SO}(10)$ by

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}\pgfsys@endscope\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.411pt}{-1.30695pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\pgfsys@color@rgb@stroke{0}{0}{1}\pgfsys@invoke{
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}\hbox{$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}r}$}
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which is the embedding map $r$. The embedding map between Lie groups also
implies the embedding map of the corresponding Lie algebras, as
$r:\mathfrak{u}(5)\to\mathfrak{so}(10)$, which allows us to establish the
relation among generators:

$\begin{array}[]{l}\begin{array}[]{ll}r(T_{1})=e_{1\wedge 4}-e_{2\wedge
3},&r(T_{11})=e_{1\wedge 3}+e_{2\wedge 4},\\\ r(T_{2})=e_{1\wedge
6}-e_{2\wedge 5},&r(T_{12})=e_{1\wedge 5}+e_{2\wedge 6},\\\
r(T_{3})=e_{1\wedge 8}-e_{2\wedge 7},&r(T_{13})=e_{1\wedge 7}+e_{2\wedge
8},\\\ r(T_{4})=e_{1\wedge 10}-e_{2\wedge 9},&r(T_{14})=e_{1\wedge
9}+e_{2\wedge 10},\\\ r(T_{5})=e_{3\wedge 6}-e_{4\wedge
5},&r(T_{15})=e_{3\wedge 5}+e_{4\wedge 6},\\\ r(T_{6})=e_{3\wedge
8}-e_{4\wedge 7},&r(T_{16})=e_{3\wedge 7}+e_{4\wedge 8},\\\
r(T_{7})=e_{3\wedge 10}-e_{4\wedge 9},&r(T_{17})=e_{3\wedge 9}+e_{4\wedge
10},\\\ r(T_{8})=e_{5\wedge 8}-e_{6\wedge 7},&r(T_{18})=e_{5\wedge
7}+e_{6\wedge 8},\\\ r(T_{9})=e_{5\wedge 10}-e_{6\wedge
9},&r(T_{19})=e_{5\wedge 9}+e_{6\wedge 10},\\\ r(T_{10})=e_{7\wedge
10}-e_{8\wedge 9},&r(T_{20})=e_{7\wedge 9}+e_{8\wedge 10},\end{array}\\\ \\\
\begin{array}[]{l}r(T_{21})=e_{1\wedge 2}-e_{3\wedge 4},\\\
r(T_{22})=2e_{5\wedge 6}-e_{7\wedge 8}-e_{9\wedge 10},\\\ r(T_{23})=e_{7\wedge
8}-e_{9\wedge 10},\\\ r(T_{24})=-3e_{1\wedge 2}-3e_{3\wedge 4}+2e_{5\wedge
6}+2e_{7\wedge 8}+2e_{9\wedge 10},\\\ r(T_{25})=\hat{q}(e_{1\wedge
2}+e_{3\wedge 4}+e_{5\wedge 6}+e_{7\wedge 8}+e_{9\wedge
10}).\end{array}\end{array}$ (B.15)

Given that $r$ is a linear map (i.e. $r(\theta_{a}T_{a})=\theta_{a}r(T_{a})$),
Eqn. (B.15) automatically specifies the map $r$ for any element in the Lie
algebra.

####  B.3.2 Projection Map $p$

The projection map $p:{\rm Spin}(10)\to{\rm SO}(10)$ is defined by the short
exact sequence

$1\to{\mathbb{Z}_{2}}\hookrightarrow{{\rm Spin}(10)}\xrightarrow{p}{{\rm
SO}(10)}\to 1.$ (B.16)

By comparing Eqn. (B.10) and Eqn. (B.6), the projection map simply identifies
the Lie group generators

$p\Big{(}\frac{\Gamma_{i\wedge j}}{2}\Big{)}=e_{i\wedge j}$ (B.17)

for all pairs $\\{i,j\\}\subset\\{1,2,\cdots,10\\}$. The mapping
$p:\mathfrak{spin}(10)\xrightarrow{\cong}\mathfrak{so}(10)$ is one-to-one
(hence invertible) on the Lie algebra level (but not on the Lie group level),
which allows us to define
$p^{-1}:\mathfrak{so}(10)\xrightarrow{\cong}\mathfrak{spin}(10)$,

$p^{-1}(e_{i\wedge j})=\frac{\Gamma_{i\wedge j}}{2}.$ (B.18)

####  B.3.3 Embedding Map $r^{\prime}$

The invertibility of $p$ on the Lie algebra level allows us to lift the map
$r$ to $r^{\prime}$ on the Lie algebra level by defining
$r^{\prime}=p^{-1}\circ r$, i.e. $r^{\prime}(T_{i})=p^{-1}(r(T_{i}))$.

${{\mathfrak{u}(5)}}$${{\mathfrak{spin}(10)}}$${{\mathfrak{so}(10)}}$$\scriptstyle{r}$$\scriptstyle{r^{\prime}}$$\scriptstyle{p}$$\scriptstyle{p^{-1}}$
(B.19)

Using Eqn. (B.15) and Eqn. (B.18), we obtain

$\begin{array}[]{l}\begin{array}[]{ll}r^{\prime}(T_{1})=\frac{1}{2}(\Gamma_{1\wedge
4}-\Gamma_{2\wedge 3}),&r^{\prime}(T_{11})=\frac{1}{2}(\Gamma_{1\wedge
3}+\Gamma_{2\wedge 4}),\\\ r^{\prime}(T_{2})=\frac{1}{2}(\Gamma_{1\wedge
6}-\Gamma_{2\wedge 5}),&r^{\prime}(T_{12})=\frac{1}{2}(\Gamma_{1\wedge
5}+\Gamma_{2\wedge 6}),\\\ r^{\prime}(T_{3})=\frac{1}{2}(\Gamma_{1\wedge
8}-\Gamma_{2\wedge 7}),&r^{\prime}(T_{13})=\frac{1}{2}(\Gamma_{1\wedge
7}+\Gamma_{2\wedge 8}),\\\ r^{\prime}(T_{4})=\frac{1}{2}(\Gamma_{1\wedge
10}-\Gamma_{2\wedge 9}),&r^{\prime}(T_{14})=\frac{1}{2}(\Gamma_{1\wedge
9}+\Gamma_{2\wedge 10}),\\\ r^{\prime}(T_{5})=\frac{1}{2}(\Gamma_{3\wedge
6}-\Gamma_{4\wedge 5}),&r^{\prime}(T_{15})=\frac{1}{2}(\Gamma_{3\wedge
5}+\Gamma_{4\wedge 6}),\\\ r^{\prime}(T_{6})=\frac{1}{2}(\Gamma_{3\wedge
8}-\Gamma_{4\wedge 7}),&r^{\prime}(T_{16})=\frac{1}{2}(\Gamma_{3\wedge
7}+\Gamma_{4\wedge 8}),\\\ r^{\prime}(T_{7})=\frac{1}{2}(\Gamma_{3\wedge
10}-\Gamma_{4\wedge 9}),&r^{\prime}(T_{17})=\frac{1}{2}(\Gamma_{3\wedge
9}+\Gamma_{4\wedge 10}),\\\ r^{\prime}(T_{8})=\frac{1}{2}(\Gamma_{5\wedge
8}-\Gamma_{6\wedge 7}),&r^{\prime}(T_{18})=\frac{1}{2}(\Gamma_{5\wedge
7}+\Gamma_{6\wedge 8}),\\\ r^{\prime}(T_{9})=\frac{1}{2}(\Gamma_{5\wedge
10}-\Gamma_{6\wedge 9}),&r^{\prime}(T_{19})=\frac{1}{2}(\Gamma_{5\wedge
9}+\Gamma_{6\wedge 10}),\\\ r^{\prime}(T_{10})=\frac{1}{2}(\Gamma_{7\wedge
10}-\Gamma_{8\wedge 9}),&r^{\prime}(T_{20})=\frac{1}{2}(\Gamma_{7\wedge
9}+\Gamma_{8\wedge 10}),\end{array}\\\ \\\
\begin{array}[]{l}r^{\prime}(T_{21})=\frac{1}{2}(\Gamma_{1\wedge
2}-\Gamma_{3\wedge 4}),\\\ r^{\prime}(T_{22})=\frac{1}{2}(2\Gamma_{5\wedge
6}-\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10}),\\\
r^{\prime}(T_{23})=\frac{1}{2}(\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10}),\\\
r^{\prime}(T_{24})=\frac{1}{2}(-3\Gamma_{1\wedge 2}-3\Gamma_{3\wedge
4}+2\Gamma_{5\wedge 6}+2\Gamma_{7\wedge 8}+2\Gamma_{9\wedge 10}),\\\
r^{\prime}(T_{25})=\frac{1}{2}{\hat{q}}(\Gamma_{1\wedge 2}+\Gamma_{3\wedge
4}+\Gamma_{5\wedge 6}+\Gamma_{7\wedge 8}+\Gamma_{9\wedge
10}),\end{array}\end{array}$ (B.20)

which applies to the whole Lie algebra given that $r^{\prime}$ is a linear map
(i.e. $r^{\prime}(\theta_{a}T_{a})=\theta_{a}r^{\prime}(T_{a})$). However,
there could be obstruction to further lift the Lie algebra embedding
$r^{\prime}$ to the Lie group level. The key relies on whether the
$\mathbb{Z}_{5}$ subgroup shared between ${\rm SU}(5)$ and ${\rm U}(1)$ can be
consistently defined after lifting ${\rm U}(5)_{\hat{q}}$ to ${\rm Spin}(10)$.
This depends on the choice of the charge number $\hat{q}$.

###  B.4 Obstruction to Lifting: Compatibility of $\mathbb{Z}_{5}$ Center

Recall Eqn. (B.5) that the charge number $\hat{q}$ affects how the
$\mathbb{Z}_{5}$ generator $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{25}}\in{\rm U}(1)$ is identified with a
group element inside ${\rm SU}(5)$. The key is to check if any of these
identifications will be broken under the embedding map $r^{\prime}$. Since
Eqn. (B.5) only involves three generators $T_{21}$, $T_{24}$ and $T_{25}$, we
will only focus on their images under the $r^{\prime}$ map. Further more,
because $T_{21}$, $T_{24}$ and $T_{25}$ are commuting Cartan generators (so
will be $r^{\prime}(T_{21})$, $r^{\prime}(T_{24})$ and $r^{\prime}(T_{25})$),
we can find a good basis of ${\mathbb{C}}^{16}$ to simultaneously diagonalize
$r^{\prime}(T_{21})$, $r^{\prime}(T_{24})$ and $r^{\prime}(T_{25})$. By
explicit calculation (by substituting Eqn. (B.12) to Eqn. (B.20) and
diagonalize the $16\times 16$ matrices), the eigenvalues of
$r^{\prime}(T_{21})$, $r^{\prime}(T_{24})$ and $r^{\prime}(T_{25})$ in their
common eigenbasis are listed in Table 7.

Table 7: Eigenvalues of $r^{\prime}(T_{21})$, $r^{\prime}(T_{24})$ and $r^{\prime}(T_{25})$ in their common eigenbasis. ${\rm SU}(5)$ | $r^{\prime}(T_{21})$ | $r^{\prime}(T_{24})$ | $r^{\prime}(T_{25})$
---|---|---|---
$\overline{\mathbf{5}}$ | $-1$ | $-3$ | $-3\hat{q}/2$
$1$ | $-3$ | $-3\hat{q}/2$
$0$ | $2$ | $-3\hat{q}/2$
$0$ | $2$ | $-3\hat{q}/2$
$0$ | $2$ | $-3\hat{q}/2$
$\mathbf{10}$ | $0$ | $-4$ | $\hat{q}/2$
$0$ | $-4$ | $\hat{q}/2$
$0$ | $-4$ | $\hat{q}/2$
$-1$ | $1$ | $\hat{q}/2$
$-1$ | $1$ | $\hat{q}/2$
$-1$ | $1$ | $\hat{q}/2$
$1$ | $1$ | $\hat{q}/2$
$1$ | $1$ | $\hat{q}/2$
$1$ | $1$ | $\hat{q}/2$
$0$ | $6$ | $\hat{q}/2$
$\mathbf{1}$ | $0$ | $0$ | $5\hat{q}/2$

Because of the group isomorphism given in (2.2), we will only have to check
two cases, $\hat{q}=1$ and $\hat{q}=2$:

  * •

For $\hat{q}=1$, $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{25}}$ is identified with $\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}(T_{24}-5T_{21})/2}$ in ${\rm U}(5)$, but $\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}(T_{25})}$ can not be identified with $\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}((T_{24}-5T_{21})/2)}$ in ${\rm Spin}(10)$, because

$\begin{split}r^{\prime}((T_{24}-5T_{21})/2)-r^{\prime}(T_{25}|_{\hat{q}=1})&\bumpeq{\rm
diag}(\tfrac{5}{2},-\tfrac{5}{2},\tfrac{5}{2},\tfrac{5}{2},\tfrac{5}{2},-\tfrac{5}{2},-\tfrac{5}{2},-\tfrac{5}{2},\tfrac{5}{2},\tfrac{5}{2},\tfrac{5}{2},-\tfrac{5}{2},-\tfrac{5}{2},-\tfrac{5}{2},\tfrac{5}{2},-\tfrac{5}{2})\\\
&\overset{\mod 5}{=}\tfrac{5}{2}\mathbb{1},\end{split}$ (B.21)

meaning $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}((T_{24}-5T_{21})/2)}=-\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}(T_{25})}$, which posts an obstruction to lift $r^{\prime}$
out of the exponent to the Lie group level.

  * •

For $\hat{q}=2$, $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{25}}$ is identified with $\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip 1.0pt}{5}T_{24}}$ in
${\rm U}(5)$, while $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}r^{\prime}(T_{25})}$ can also be identified
with $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}(T_{24})}$ in ${\rm Spin}(10)$ consistently, because

$\begin{split}r^{\prime}(T_{24})-r^{\prime}(T_{25}|_{\hat{q}=2})&\bumpeq{\rm
diag}(0,0,5,5,5,-5,-5,-5,0,0,0,0,0,0,5,-5)\\\ &\overset{\mod
5}{=}0,\end{split}$ (B.22)

meaning $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{5}r^{\prime}(T_{24})}=\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{5}r^{\prime}(T_{25})}$, which posts no
obstruction to lift $r^{\prime}$ out of the exponent to the Lie group level.

In conclusion, the $\mathbb{Z}_{5}$ center can be compatibly defined and
shared between ${\rm SU}(5)$ and ${\rm U}(1)$, if and only if $\hat{q}=2$.
With $\hat{q}=2$, there is no obstruction to further lift $r^{\prime}$ to the
Lie group level and hence the embedding map $r^{\prime}:{\rm U}(5)\to{\rm
Spin}(10)$ is well-defined.

##  Appendix C Flipping

###  C.1 Flipping Isomorphism

The standard GG ${\rm U}(5)$ and flipped ${\rm U}(5)$ (denoted as ${\rm
U}(5)_{\text{1st}}$ and ${\rm U}(5)_{\text{2nd}}$ in the following) can be
both embedded in the same ${\rm Spin}(10)$ group as long as their charge
numbers are taken to be the same even integer. We will implicitly assume the
$q=2$ case in the following discussion. ${\rm U}(5)_{\text{1st}}$ and ${\rm
U}(5)_{\text{2nd}}$ are related by the flipping isomorphism, denoted as $f$,
which is an _outer_ automorphism of ${\rm U}(5)$ and also an _inner_
automorphism of ${\rm SO}(10)$ as well as ${\rm Spin}(10)$, as shown in the
diagram below.

${{{\rm U}(5)_{\text{1st}}}}$${{{\rm Spin}(10)}}$${{{\rm SO}(10)}}$${{{\rm
U}(5)_{\text{2nd}}}}$$\scriptstyle{r^{\prime}_{1}}$$\scriptstyle{r_{1}}$$\scriptstyle{p}$$\scriptstyle{r^{\prime}_{2}}$$\scriptstyle{r_{2}}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{f}$
(C.1)

The flipping isomorphism $f$ can be specified as an inner automorphism of
$f:{\rm SO}(10)\to{\rm SO}(10)$, such that for $O\in{\rm SO}(10)$:
$f(O)=F^{-1}OF$ with $F=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi e_{2\wedge 4}}$. This can also be interpreted as a basis
transformation of ${\mathbb{R}}^{10}$,

$f({|e_{i}\rangle})=F{|e_{i}\rangle}=\left\\{\begin{array}[]{ll}-{|e_{i}\rangle}&i\in\\{2,4\\}\\\
{|e_{i}\rangle}&i\notin\\{2,4\\}\end{array}\right.\equiv(-1)^{\delta_{i\in\\{2,4\\}}}{|e_{i}\rangle},$
(C.2)

where $\delta_{i\in\\{2,4\\}}$ is the Kronecker delta symbol that equals 1
when $i\in\\{2,4\\}$ and equals 0 when $i\notin\\{2,4\\}$. The flipping
isomorphism is a duality, as $F^{2}=1$, or $f\circ f=\operatorname{id}$.

The flipping isomorphism can be lifted to $f:{\rm Spin}(10)\to{\rm Spin}(10)$,
such that for $\mathsf{O}\in{\rm Spin}(10)$:
$f(\mathsf{O})=\mathsf{F}^{-1}\mathsf{O}\mathsf{F}$ with $\mathsf{F}=\pm\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(\pi/2)\Gamma_{2\wedge
4}}=\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma_{2\wedge
4}=\mp\Gamma_{2}\Gamma_{4}$. The $\pm$ sign ambiguity arise from the
$\mathbb{Z}_{2}$ subgroup freedom when embedding ${\rm SO}(10)$ in ${\rm
Spin}(10)$. Nevertheless, this sign ambiguity does not affect the definition
of the flipping isomorphism $f(\mathsf{O})$ because $\mathsf{F}$ always
appears twice. The flipping morphism can also be translated to an inner
automorphism of ${\rm Cl}(10)$, as

$f(\Gamma_{i})=\mathsf{F}^{-1}\Gamma_{i}\mathsf{F}=(-1)^{\delta_{i\in\\{2,4\\}}}\Gamma_{i}$
(C.3)

which applies to the ${\rm Spin}(10)$ generators (as grade-2 elements of ${\rm
Cl}(10)$) as

$f(\Gamma_{i\wedge j})=\mathsf{F}^{-1}\Gamma_{i\wedge
j}\mathsf{F}=(-1)^{\delta_{i\in\\{2,4\\}}+\delta_{j\in\\{2,4\\}}}\Gamma_{i\wedge
j}.$ (C.4)

Again, one can see $f\circ f=\operatorname{id}$.

Given the embedding map $r^{\prime}_{1}:{\rm U}(5)_{\text{1st}}\to{\rm
Spin}(10)$ in Eqn. (B.20) and the flipping isomorphism $f$, the flipped
embedding map $r^{\prime}_{2}:{\rm U}(5)_{\text{2st}}\to{\rm Spin}(10)$ can be
defined as $r^{\prime}_{2}=f\circ r^{\prime}_{1}$, i.e.
$r^{\prime}_{2}(T_{i})=f(r^{\prime}_{1}(T_{i}))$, which enables us to compare
the two embeddings $r^{\prime}_{1}$ and $r^{\prime}_{2}$ as in Table 8.

$T_{i}$ |  $r^{\prime}_{1}(T_{i})$ (as $r^{\prime}_{1}:\mathfrak{u}(5)_{\text{1st}}\to\mathfrak{spin}(10)$) |  $r^{\prime}_{2}(T_{i})$ (as $r^{\prime}_{2}:\mathfrak{u}(5)_{\text{2nd}}\to\mathfrak{spin}(10)$)
---|---|---
$T_{1}$ | $\frac{1}{2}(\Gamma_{1\wedge 4}-\Gamma_{2\wedge 3})$ | $\frac{1}{2}(-\Gamma_{1\wedge 4}+\Gamma_{2\wedge 3})$
$T_{2}$ | $\frac{1}{2}(\Gamma_{1\wedge 6}-\Gamma_{2\wedge 5})$ | $\frac{1}{2}(\Gamma_{1\wedge 6}+\Gamma_{2\wedge 5})$
$T_{3}$ | $\frac{1}{2}(\Gamma_{1\wedge 8}-\Gamma_{2\wedge 7})$ | $\frac{1}{2}(\Gamma_{1\wedge 8}+\Gamma_{2\wedge 7})$
$T_{4}$ | $\frac{1}{2}(\Gamma_{1\wedge 10}-\Gamma_{2\wedge 9})$ | $\frac{1}{2}(\Gamma_{1\wedge 10}+\Gamma_{2\wedge 9})$
$T_{5}$ | $\frac{1}{2}(\Gamma_{3\wedge 6}-\Gamma_{4\wedge 5})$ | $\frac{1}{2}(\Gamma_{3\wedge 6}+\Gamma_{4\wedge 5})$
$T_{6}$ | $\frac{1}{2}(\Gamma_{3\wedge 8}-\Gamma_{4\wedge 7})$ | $\frac{1}{2}(\Gamma_{3\wedge 8}+\Gamma_{4\wedge 7})$
$T_{7}$ | $\frac{1}{2}(\Gamma_{3\wedge 10}-\Gamma_{4\wedge 9})$ | $\frac{1}{2}(\Gamma_{3\wedge 10}+\Gamma_{4\wedge 9})$
$T_{8}$ | $\frac{1}{2}(\Gamma_{5\wedge 8}-\Gamma_{6\wedge 7})$ | $\frac{1}{2}(\Gamma_{5\wedge 8}-\Gamma_{6\wedge 7})$
$T_{9}$ | $\frac{1}{2}(\Gamma_{5\wedge 10}-\Gamma_{6\wedge 9})$ | $\frac{1}{2}(\Gamma_{5\wedge 10}-\Gamma_{6\wedge 9})$
$T_{10}$ | $\frac{1}{2}(\Gamma_{7\wedge 10}-\Gamma_{8\wedge 9})$ | $\frac{1}{2}(\Gamma_{7\wedge 10}-\Gamma_{8\wedge 9})$
$T_{11}$ | $\frac{1}{2}(\Gamma_{1\wedge 3}+\Gamma_{2\wedge 4})$ | $\frac{1}{2}(\Gamma_{1\wedge 3}+\Gamma_{2\wedge 4})$
$T_{12}$ | $\frac{1}{2}(\Gamma_{1\wedge 5}+\Gamma_{2\wedge 6})$ | $\frac{1}{2}(\Gamma_{1\wedge 5}-\Gamma_{2\wedge 6})$
$T_{13}$ | $\frac{1}{2}(\Gamma_{1\wedge 7}+\Gamma_{2\wedge 8})$ | $\frac{1}{2}(\Gamma_{1\wedge 7}-\Gamma_{2\wedge 8})$
$T_{14}$ | $\frac{1}{2}(\Gamma_{1\wedge 9}+\Gamma_{2\wedge 10})$ | $\frac{1}{2}(\Gamma_{1\wedge 9}-\Gamma_{2\wedge 10})$
$T_{15}$ | $\frac{1}{2}(\Gamma_{3\wedge 5}+\Gamma_{4\wedge 6})$ | $\frac{1}{2}(\Gamma_{3\wedge 5}-\Gamma_{4\wedge 6})$
$T_{16}$ | $\frac{1}{2}(\Gamma_{3\wedge 7}+\Gamma_{4\wedge 8})$ | $\frac{1}{2}(\Gamma_{3\wedge 7}-\Gamma_{4\wedge 8})$
$T_{17}$ | $\frac{1}{2}(\Gamma_{3\wedge 9}+\Gamma_{4\wedge 10})$ | $\frac{1}{2}(\Gamma_{3\wedge 9}-\Gamma_{4\wedge 10})$
$T_{18}$ | $\frac{1}{2}(\Gamma_{5\wedge 7}+\Gamma_{6\wedge 8})$ | $\frac{1}{2}(\Gamma_{5\wedge 7}+\Gamma_{6\wedge 8})$
$T_{19}$ | $\frac{1}{2}(\Gamma_{5\wedge 9}+\Gamma_{6\wedge 10})$ | $\frac{1}{2}(\Gamma_{5\wedge 9}+\Gamma_{6\wedge 10})$
$T_{20}$ | $\frac{1}{2}(\Gamma_{7\wedge 9}+\Gamma_{8\wedge 10})$ | $\frac{1}{2}(\Gamma_{7\wedge 9}+\Gamma_{8\wedge 10})$
$T_{21}$ | $\frac{1}{2}(\Gamma_{1\wedge 2}-\Gamma_{3\wedge 4})$ | $\frac{1}{2}(-\Gamma_{1\wedge 2}+\Gamma_{3\wedge 4})$
$T_{22}$ | $\frac{1}{2}(2\Gamma_{5\wedge 6}-\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10})$ | $\frac{1}{2}(2\Gamma_{5\wedge 6}-\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10})$
$T_{23}$ | $\frac{1}{2}(\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10})$ | $\frac{1}{2}(\Gamma_{7\wedge 8}-\Gamma_{9\wedge 10})$
$T_{24}$ | $\frac{1}{2}(-3\Gamma_{1\wedge 2}-3\Gamma_{3\wedge 4}+2\Gamma_{5\wedge 6}+2\Gamma_{7\wedge 8}+2\Gamma_{9\wedge 10})$ | $\frac{1}{2}(3\Gamma_{1\wedge 2}+3\Gamma_{3\wedge 4}+2\Gamma_{5\wedge 6}+2\Gamma_{7\wedge 8}+2\Gamma_{9\wedge 10})$
$T_{25}$ | $\Gamma_{1\wedge 2}+\Gamma_{3\wedge 4}+\Gamma_{5\wedge 6}+\Gamma_{7\wedge 8}+\Gamma_{9\wedge 10}$ | $-\Gamma_{1\wedge 2}-\Gamma_{3\wedge 4}+\Gamma_{5\wedge 6}+\Gamma_{7\wedge 8}+\Gamma_{9\wedge 10}$
Table 8: Embeddings of $\mathfrak{u}(5)_{\text{1st}}$ and
$\mathfrak{u}(5)_{\text{2nd}}$ (both of $q=2$) in $\mathfrak{spin}(10)$.

###  C.2 Intersection and Join

Treat $\mathfrak{u}(5)_{\text{1st}}$ and $\mathfrak{u}(5)_{\text{2nd}}$ as Lie
subalgebra of $\mathfrak{spin}(10)$. Their intersection is (see the
highlighted rows in Table 8)

$\begin{split}\mathfrak{u}(5)_{\text{1st}}\cap\mathfrak{u}(5)_{\text{2nd}}&=\mathrm{span}\\{T_{1},T_{8},T_{9},T_{10},T_{11},T_{18},T_{19},T_{20},T_{21},T_{22},T_{23},T_{24},T_{25}\\}\\\
&=\mathrm{span}\\{T_{8},T_{9},T_{10},T_{18},T_{19},T_{20},T_{22},T_{23}\\}\oplus\mathrm{span}\\{T_{1},T_{11},T_{21}\\}\\\
&\hskip
12.0pt\oplus\mathrm{span}\\{T_{24}\\}\oplus\mathrm{span}\\{T_{25}\\}\\\
&=\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)_{Y}\oplus\mathfrak{u}(1)_{X}.\end{split}$
(C.5)

The join of $\mathfrak{u}(5)_{\text{1st}}$ and $\mathfrak{u}(5)_{\text{2nd}}$
(the $\mathfrak{u}(5)_{\text{1st}}$ and $\mathfrak{u}(5)_{\text{2nd}}$
together generate the minimal Lie algebra as their union, which should be
understood as the minimal vector space generated by both Lie algebras together
closed under any of their Lie brackets) is

$\begin{split}\mathfrak{u}(5)_{\text{1st}}{\cup}\mathfrak{u}(5)_{\text{2nd}}&=(\mathfrak{u}(5)_{\text{1st}}\cap\mathfrak{u}(5)_{\text{2nd}}){\cup}\mathrm{span}\\{\Gamma_{i\wedge
j}|i\in\\{1,2,3,4\\},j\in\\{5,6,7,8,9,10\\}\\}\\\
&=\mathrm{span}\\{\Gamma_{i\wedge j}|\\{i,j\\}\subset\\{1,2,\cdots,10\\}\\}\\\
&=\mathfrak{spin}(10).\end{split}$ (C.6)

On the Lie group level, we have

${{{\rm Spin}(10)}}$${{{\rm U}(5)_{\text{1st}}}}$${{{\rm
U}(5)_{\text{2nd}}}}$${{G_{\text{SM}}\times_{{\mathbb{Z}_{5}}}{\rm
U}(1)_{X}}}$$\scriptstyle{r^{\prime}_{1}}$$\scriptstyle{r^{\prime}_{2}}$$\scriptstyle{f}$
(C.7)

where $G_{\text{SM}}=({\rm SU}(3)\times{\rm SU}(2))\times_{\mathbb{Z}_{6}}{\rm
U}(1)_{Y}$. We have ${G_{\text{SM}}\times_{{\mathbb{Z}_{5}}}{\rm U}(1)_{X}}$
because that the ${\rm U}(1)_{Y}\times_{{\mathbb{Z}_{5}}}{\rm U}(1)_{X}$
structure has a shared
${\mathbb{Z}_{5}}={\mathbb{Z}_{5,X}}={\mathbb{Z}_{5,Y}}$, see Table 1.

###  C.3 Charge Lattice

Define the ${\rm U}(1)$ charges $Y_{1}:=r^{\prime}_{1}(T_{24})$,
$X_{1}:=r^{\prime}_{1}(T_{25})$ in ${\rm U}(5)_{\text{1st}}$, and
$Y_{2}:=r^{\prime}_{2}(T_{24})$, $X_{2}:=r^{\prime}_{2}(T_{25})$ in ${\rm
U}(5)_{\text{2nd}}$. Based on Table 8, these charges are related by the
flipping isomorphism $f$:

$\begin{pmatrix}X_{1}\\\ Y_{1}\end{pmatrix}=\frac{1}{5}\begin{pmatrix}1&4\\\
6&-1\end{pmatrix}\begin{pmatrix}X_{2}\\\
Y_{2}\end{pmatrix},\quad\begin{pmatrix}X_{2}\\\
Y_{2}\end{pmatrix}=\frac{1}{5}\begin{pmatrix}1&4\\\
6&-1\end{pmatrix}\begin{pmatrix}X_{1}\\\ Y_{1}\end{pmatrix}.$ (C.8)

See also the discussions in Sec. 3.6.

The two sets of charge lattices intersect at points that matches the charge
assignment of fundamental fermions in the SM, as shown in Fig. 9. Moreover,
${\rm U}(1)_{X_{1}}$ and ${\rm U}(1)_{X_{2}}$ shares a $\mathbb{Z}_{4}$
subgroup, because

$\begin{split}X_{1}-X_{2}&=r^{\prime}_{1}(T_{25}|_{q=2})-r^{\prime}_{2}(T_{25}|_{q=2})\\\
&\bumpeq{\rm diag}(0,0,-4,-4,-4,4,4,4,0,0,0,0,0,0,-4,4)\\\ &\overset{\mod
4}{=}0,\end{split}$ (C.9)

meaning that $\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{4}X_{1}}=\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{4}X_{2}}$, which generates a $\mathbb{Z}_{4}$ group

$\mathbb{Z}_{4}=\langle\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{4}X_{1}}\rangle=\\{\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{4}mX_{1}}|m=0,1,2,3\\}.$ (C.10)

This is also the $\mathbb{Z}_{4}$ center of ${\rm Spin}(10)$.

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