Gauge Enhanced Quantum Criticality
Between Grand Unifications:
Categorical Higher Symmetry Retraction

Juven Wang1<EMAIL_ADDRESS>Yi-Zhuang You2<EMAIL_ADDRESS>
Dedicate to anniversaries of various researchers mentioned in: Related
presentation videos available online

1Center of Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138, USA
2Department of Physics, University of California, San Diego, CA 92093, USA

Prior work [1] shows that the Standard Model (SM) naturally arises near a
gapless quantum critical region between Georgi-Glashow (GG) $su(5)$ and Pati-
Salam (PS) $su(4)\times su(2)\times su(2)$ models of quantum vacua (in a phase
diagram or moduli space), by implementing a modified $so(10)$ Grand
Unification (GUT) with a Spin(10) gauge group plus a new discrete Wess-Zumino
Witten term matching a 4d nonperturbative global mixed gauge-gravity
$w_{2}w_{3}$ anomaly. In this work, we include Barr’s flipped $su(5)$ model
into the quantum landscape, showing these four GUT-like models arise near the
quantum criticality near SM. Highlights include: First, we find the precise GG
or flipped $u(5)$ gauge group requires to redefine a Lie group
U(5)${}_{\hat{q}}\equiv({\rm SU}(5)\times_{\hat{q}}{\rm
U}(1))/\mathbb{Z}_{\hat{q}}$ with $\hat{q}=2$ or $3$ (instead of non-
isomorphic analog $\hat{q}=1$ or 4), and different $\hat{q}$ are related by
multiple covering. Second, we show that the criticality between the GG and
flipped $u(5)$ is governed by the traditional symmetry-breaking Landau-
Ginzburg paradigm. Third, we show that analogous 3+1d deconfined quantum
criticalities, both between GG and PS, and between the flipped $u(5)$ and PS,
is governed by the beyond-Landau-Ginzburg paradigm. The SM and GG models can
have either 15 or 16 Weyl fermions per generation, while the PS, flipped
$u(5)$, and the modified $so(10)$ have 16n Weyl fermions; topological quantum
criticality occurs by tuning between the 15n vs 16n scenarios. Fourth, we
explore the higher global symmetries. Gauging the $\mathbb{Z}_{2}$ flip
symmetry between GG and flipped $u(5)$ models, leads to a potential
categorical higher symmetry that is a non-invertible global symmetry: within a
gauge group $\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$, the fusion rule of 2d
topological surface operator splits. Even if the mixed anomaly between
$\mathbb{Z}_{2}$ flip symmetry and two U(1) magnetic 1-symmetries at IR is
absent, the un-Higgs Spin(10) at UV retracts this categorical symmetry.

###### Contents

  1. 1 Introduction and Summary
    1. 1.1 Unity of Gauge Forces vs Many Dualities
    2. 1.2 Outline: The plan of the article
  2. 2 Refined ${\rm U}(5)_{\hat{q}}$ gauge theory
  3. 3 Various Grand Unification (GUT) Models as Vacuum Phases
    1. 3.1 Georgi-Glashow $u(5)$ vs Flipped $u(5)$ models
    2. 3.2 There are only two types of $u(5)$ GUTs
    3. 3.3 Pati-Salam $su(4)\times su(2)_{\text{L}}\times su(2)_{\text{R}}$ model
    4. 3.4 The Left-Right $su(3)\times su(2)_{\text{L}}\times su(2)_{\text{R}}\times u(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}$ model
    5. 3.5 The $so(10)$ Grand Unification and a DSpin-structure modification
    6. 3.6 Representations of SMs and Five GUT-like models in a unified Table
  4. 4 Quantum Landscape as Quantum Phase Diagram (Moduli Space)
    1. 4.1 Embedding Web by Internal Symmetry Groups
    2. 4.2 Quantum Phase Diagram and a Mother Effective Field Theory
      1. 4.2.1 Internal symmetries treated as global symmetries
      2. 4.2.2 Internal symmetries are dynamically gauged
  5. 5 Higher Symmetries of Standard Models and Grand Unifications
  6. 6 Categorical Symmetry and Its Retraction
    1. 6.1 Potential Non-Invertible Categorical Symmetry induced by $\mathbb{Z}_{2}^{\rm flip}$
    2. 6.2 Categorical symmetry retraction from two ${\rm U}(5)_{\hat{q}=2}$ to Spin(10)
    3. 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
  7. 7 Conclusion and Future Directions
  8. 8 Acknowledgements
  9. A Quantum Numbers and Representations of SMs and GUTs
    1. 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
  10. B Embedding
    1. B.1 Embedding Strategy
    2. B.2 Definition of Lie Groups
      1. B.2.1 Lie algebra and Lie Group of ${\rm U}(5)_{\hat{q}}$
      2. B.2.2 Lie algebra and Lie Group of ${\rm SO}(10)$
      3. B.2.3 Lie algebra and Lie Group of ${\rm Spin}(10)$
    3. B.3 Embedding and Projection Maps
      1. B.3.1 Embedding Map $r$
      2. B.3.2 Projection Map $p$
      3. B.3.3 Embedding Map $r^{\prime}$
    4. B.4 Obstruction to Lifting: Compatibility of $\mathbb{Z}_{5}$ Center
  11. C Flipping
    1. C.1 Flipping Isomorphism
    2. C.2 Intersection and Join
    3. C.3 Charge Lattice
  12. D Bibliography

##  1 Introduction and Summary

###  1.1 Unity of Gauge Forces vs Many Dualities

One of the open unsolved problems in fundamental physics and high-energy
physics (HEP) is:

“If the strong, electromagnetic, and weak forces of the Standard Model are
unified at high energies, (1.1) by which gauge group (of the gauged internal
symmetry) is this unification governed?”

We may quote this perspective as “Unity of Gauge Forces.” It is conventional
to regard our quantum vacuum in the 4-dimensional spacetime (denoted as 4d or
3+1d) governed by one of the candidate $su(3)\times su(2)\times u(1)$ Standard
Models (SMs) [2, 3, 4, 5], while lifting towards one of some Grand
Unification-like (GUT-like) structure [6, 7, 8, 9, 10, 11, 12] or String
Theory at higher energy scales.111Throughout our article, we denote $n$d for
$n$-dimensional spacetime, or $n^{\prime}+1$d as an $n^{\prime}$-dimensional
space and 1-dimensional time. We also denote the Lie algebra in the lower case
such as the $so(10)$ Lie algebra in the $so(10)$ GUT [8] , but denote their
Lie group in the capital case such as Spin(10). This perspective may be
schematically summarized as Fig. 1 (a).

(a) (b)

Figure 1: (a) “Unity of Gauge Forces” perspective seeks for a single unified
dynamically gauged internal symmetry at high energy, which is a more
_kinematic_ or _static_ issue of gauge theories towards higher energy. (b) Our
“quantum competing criticality” perspective [1] suggests that the SM is a low
energy quantum vacuum arising from the quantum competition of various neighbor
GUT-like vacua (which also appear at higher energy). Especially when there are
constraints from topological terms or nonperturbative global anomalies, the SM
arises near a gapless quantum critical region (schematically shown as the
shaded gray area). The gapless quantum critical region induces new beyond-SM
gapless modes, Dark Gauge forces, or excitations. This perspective is more on
the _dynamics, criticality,_ or _phase transition_ issue of gauge theories.

However, gauge symmetry is not a physical symmetry (unlike the global
symmetry) but only a gauge redundancy to describe interactions between
matters; thus the gauge group is not physical nor universal. Furthermore, it
is widely known that there are many different dual descriptions of the same
physical theories via different gauge groups. We may quote this perspective as
“Many Dualities.”

This raises the conflict between the above two perspectives: How could we ask
for the governing gauge group for the Unity of Gauge Forces, if gauge groups
are not universal, and if there are Many Dualities of possible different gauge
theory realizations of the same unification? Partly motivated to provide a
resolution of this conceptual conflict, our prior work [1] initiates an
alternative viewpoint: we propose that the SM vacuum may be a low energy
quantum vacuum arising from the quantum competition of various neighbor
higher-energy GUT or other unified theories’ vacua in an immense quantum phase
diagram, schematically summarized as Fig. 1 (b). Here we highlight and
summarize some of the viewpoints of Ref. [1]:

  1. 1.

When we treat the internal symmetry as a global symmetry (or in the weakly
gauged or ungauged limit), it is physically sensible to ask “what is the Unity
of the Governing Internal Symmetry Group ${{G}_{\text{internal}}}$?” In this
global symmetry limit, the immense quantum phase diagram in fact not only
contains many different GUTs in the same Hilbert space with same ’t Hooft
anomalies [13] of their internal global symmetries, but also give rise to the
SM near the quantum criticality222Let us clarify the terminology on
criticalities vs phase transitions.
$\bullet$ 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 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 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)
or a discontinuous phase transition (first order, without gapless modes, and
with a finite correlation length).
In the phase diagram, the spacetime dimensionality of the phase interface is
the same as that of the bulk phase. This is in contrast with the one-lower
spacetime dimensional physical interface or physical boundary of the bulk
phase.  between different GUTs. The quantum field theories (QFTs), sharing the
same ’t Hooft anomalies especially with the same global symmetries, are
believed to live in the same phase diagram with the same Hilbert space. These
QFTs are deformable to each other via symmetry-preserving interactions — known
as the _deformation class of QFTs_ , particularly advocated by Seiberg [15].
The _deformation class of quantum gravity theory_ is also proposed, by
McNamara-Vafa [16].

  2. 2.

Since the SM arises near the quantum criticality between different GUTs, it
makes sense to study the emergent (gauged or global) symmetries and dualities
of QFTs at this quantum criticality. We can further gauge the internal
symmetry to be a gauge theory, thus we can study possible Many Dualities of
these gauge theories.

  3. 3.

The above two different viewpoints highlight the validity of Unity of Internal
Symmetry and Many Dualities respectively, but now in the same quantum phase
diagram and same framework (Fig. 1 (b)). Moreover, various GUTs may encounter
_stable gapless quantum critical regions_ (the shaded gray area in Fig. 1 (b))
to enter other neighbor GUTs, if we apply the idea that the quantum
criticality is protected by the ’t Hooft anomalies of some spacetime-internal
symmetries:333The notation
$G_{1}\ltimes_{N_{\text{shared}}}G_{2}\equiv\frac{G_{1}\ltimes
G_{2}}{{N_{\text{shared}}}}$ means modding out their common normal subgroup
${N_{\text{shared}}}$. The $\ltimes$ is a semidirect product (as a
generalization of direct product) to specify a particular group extension.

$\displaystyle\bar{G}\equiv{{G_{\text{spacetime}}}\ltimes_{{N_{\text{shared}}}}{{G}_{\text{internal}}}}\equiv({\frac{{G_{\text{spacetime}}}\ltimes{{G}_{\text{internal}}}}{{N_{\text{shared}}}}}).$
(1.2)

The spacetime ${G_{\text{spacetime}}}$ and internal ${{G}_{\text{internal}}}$
symmetries may share a common normal subgroup ${N_{\text{shared}}}$. If there
is a ’t Hooft anomaly in $\bar{G}$ (here 4d, denoted as a one-higher
dimensional invertible topological quantum field theory [iTQFT] defined on
some $\bar{G}$-structure manifold in 5d with a partition function
${{\mathbf{Z}}}$), this iTQFT may be written schematically as

$\displaystyle{{\mathbf{Z}}}[{\cal B}_{\bar{G}}]={{\mathbf{Z}}}[{\cal
B}_{\bar{G}_{{\rm GUT}_{1}}}\smile{\cal B}_{\bar{G}_{{\rm
GUT}_{2}}}\smile\dots],$ (1.3)

where ${\cal B}_{\bar{G}}$ is the background field coupling to the spacetime-
internal symmetry ${\bar{G}}$.444The background field ${\cal B}_{\bar{G}}$ may
be organized as their dependence on various GUT subgroup’s background fields,
say ${\cal B}_{\bar{G}_{{\rm GUT}_{1}}},{\cal B}_{\bar{G}_{{\rm GUT}_{2}}}$,
etc. Here the cup product $\smile$ is for cohomology classes (analogous to the
wedge product $\wedge$ for differential forms). In general, the internal
symmetry group of various GUT models have subgroups overlapped, so their
background fields ${\cal B}_{\bar{G}}$ also overlapped — the expression (1.3)
is only schematic. When we have the symmetry breaking from ${\bar{G}}$ to
${\bar{G}_{{\rm GUT}_{j}}}$, etc., we are only left with the valid background
field ${\cal B}_{\bar{G}_{{\rm GUT}_{j}}}$ while other background fields must
be turned off — thus the ’t Hooft anomaly may be canceled to zero by this
symmetry breaking. In contrast, if we preserve the full ${\bar{G}}$ (happening
especially at the quantum critical region in between GUT phases), and if we
have other possibilities to cancel the ’t Hooft anomaly other than by symmetry
breaking, then we must have _no symmetric trivial gapped phase_ between
neighbor GUT phases, which likely resulting in _gapless quantum criticalities_
in many cases. In general, these phases are called _nontrivial phases_ : (1)
symmetry-preserving gapless, (2) symmetry-preserving gapped topologically
order with a low energy TQFT, (3) symmetry-breaking gapless or gapped, or (4)
their mixed combinations. In short, some nontrivial state of matter must be at
the phase transition or the quantum criticality to match the ’t Hooft anomaly.

In fact, the deconfined quantum criticality (DQC) in 2+1d in condensed matter
physics [17, 18] is the incarnation and reminiscence of this above idea of ’t
Hooft anomaly protected quantum criticality (See Ref. [18] and Appendix C of
Ref. [1] for a contemporary QFT overview of DQC). Recently the DQC phenomena
in 3+1d are explored in [19, 20, 21, 22], where Ref. [1] provides the first
3+1d DQC analog in the SM and beyond-the Standard Model (BSM) physics.

  4. 4.

In particular, when the internal symmetry is treated as a global symmetry,
Ref. [1] shows that between two GUT models, Georgi-Glashow (GG) $su(5)$ [6]
and Pati-Salam (PS) $su(4)\times su(2)\times su(2)$ [7], we can utilize a mod
2 class of nonperturbative global mixed gauge-gravity ’t Hooft anomaly in 4d
captured by the 5d iTQFT written in Stiefel-Whitney (SW) characteristic
classes:555The $w_{j}$ is the $j$-th Stiefel-Whitney (SW) characteristic
class. The $w_{j}(TM)$ is the SW class of spacetime tangent bundle $TM$ of
manifold $M$. The $w_{j}(V_{G})$ is the SW class of the principal $G$ bundle.

$\displaystyle{\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}w_{2}w_{3})}={\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}w_{2}(TM)w_{3}(TM))=}\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}w_{2}(V_{{\rm SO}(10)})w_{3}(V_{{\rm SO}(10)})).$ (1.4)

This 4d ’t Hooft anomaly requires the spacetime-internal global symmetry
$({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm Spin}(10))$ on a 4-manifold
$M^{4}$, captured by this 5d bulk invertible TQFT [23, 24] living on a
5-manifold $M^{5}$ with the anomaly-inflow bulk-boundary correspondence
$\partial M^{5}=M^{4}$.

This mixed gauge-gravitational anomaly is tightly related to the new SU(2)
anomaly [24] due to the bundle constraint $w_{2}w_{3}(TM)=w_{2}w_{3}(V_{G})$
with $G$ can be substituted by ${\rm SO}(3)\subset{\rm SO}(10)$ related to the
embedding ${\rm SU}(2)={\rm Spin}(3)\subset{\rm Spin}(10)$. This mod 2 class
$w_{2}w_{3}$ global anomaly has been checked to be absent in the conventional
$so(10)$ GUT by Ref. [23, 24]; thus the conventional $so(10)$ GUT is free from
4d anomaly classified by the $d=5$-th cobordism group defined in Freed-Hopkins
[25],

$\displaystyle\Omega^{d}_{\bar{G}}\equiv\mathrm{TP}_{d}(\bar{G}),$ (1.5)

here with $d=5$, $\bar{G}=({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
Spin}(10))$, and $\Omega^{5}_{({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
Spin}(10))}=\mathbb{Z}_{2}$ generated by ${\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}w_{2}w_{3})}$. However, Ref. [1] modifies the $so(10)$ GUT
by appending a new 4d Wess-Zumino-Witten (WZW) term with this $w_{2}w_{3}$
global anomaly in order to realize the SM vacuum as the quantum criticality
phenomenon between the neighbor GG SU(5) GUT and PS vacua:
$\bullet$ On either GG SU(5) or PS sides of the quantum phases, the
$w_{2}w_{3}$ anomaly is matched by breaking the internal Spin(10) symmetry
down to their GUT subgroups.
$\bullet$ But at the critical (gapless) region between GG and PS quantum
phases, the full Spin(10) symmetry can be preserved, while the $w_{2}w_{3}$
anomaly is matched by the BSM sector from the new 4d WZW topological term
(constructed out of the GUT-Higgs fields or their fractionalized partons)
living on a 4d boundary of a 5d bulk.
$\bullet$ Since the mod 2 class $w_{2}w_{3}$ anomaly 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. 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 field requires a constraint ${\rm{n}}=1\mod 2$ to
match the $w_{2}w_{3}$ anomaly, where ${\rm{n}}=3$ generation indeed works.
However, in general, we can just introduce a single or any odd number of GUT-
Higgs field sectors (independent from the ${\rm{n}}=3$ of SM) to match the
$1\mod 2$ class of $w_{2}w_{3}$ anomaly.

The dynamics of this modified $so(10)$ GUT with WZW term can be fairly
complicated, giving rise to many possible _gapless_ phases or _gapped_ TQFT
phases at low energy, enumerated in [1].

In this present work, we continue developing from Ref. [1] to include another
GUT model: the flipped $su(5)$ model (originally proposed by Barr [10] and
others [11]) and the left-right (LR) model [9] into our quantum landscape or
quantum phase diagram, exploring further the neighbors of the critical region
near SM. The GG model can have either choice of a gauge group of SU(5) or
U(5), but Barr’s flipped model must require a U(5). We should emphasize that
the U(5) Lie group of GG or Barr’s model requires a certain refined Lie group
that we name ${\rm U}(5)_{\hat{q}=2}$, see Sec. 2 for details. In many cases,
we do require the additional $u(1)$ gauge sector in addition to the $su(5)$
gauge sector of GG or Barr’s model, we thus call the corresponding models as
the GG $u(5)$ model and Barr’s flipped $u(5)$ model.

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. In this article, we mainly
focus on the scenarios all with 16n Weyl fermions with n the number of
generations.666 Readers can find Ref. [26, 27, 28, 29] and [23] for the
systematical studies on the nonperturbative global anomalies of various SM and
GUT via generalized cohomology or cobordism theories.
In contrast to the scenarios of SM or GUT with 16n Weyl fermions, Ref. [30,
31, 32] considers the SM or GUT with 15n Weyl fermions and with a discrete
variant of baryon minus lepton number ${\bf B}-{\bf L}$ symmetry preserved.
Ref. [30, 31, 32] then suggests the missing 16th Weyl fermions can be
substituted by additional symmetry-preserving 4d or 5d gapped topological
quantum field theories (TQFTs), or by the symmetry-preserving 4d gapless
interacting conformal field theories (CFTs), or other symmetry-breaking
sectors (e.g., the right-handed neutrinos), to saturate a certain
$\mathbb{Z}_{16}$ global anomaly. We will comment about the topological phase
transitions between the 15n to 16n Weyl fermions at the very end in Sec. 7.

###  1.2 Outline: The plan of the article

In Sec. 2, we clarify the U(5) Lie group structure of the GG or flipped
modles. They should be both refined as ${\rm U}(5)_{\hat{q}}$ gauge theories,
with ${\hat{q}}=2,3$, which is non-isomorphic to ${\hat{q}}=1,4$. For
applications, there we point out group theoretical facts like ${\rm
Spin}(10)\supset{\rm U}(5)_{\hat{q}=2,3}$ but ${\rm Spin}(10)\not\supset{\rm
U}(5)_{\hat{q}=1,4}$.

In Sec. 3, we clarify various GUT models as different vacua or different
phases of QFTs, and present their representations of SMs and five GUT-like
models in a unified Table 1. Additional details of quantum numbers and
representations of SMs and GUTs are provided in Appendix A.

In Sec. 4, we organize various SM or GUT models in a quantum landscape or in a
quantum phase diagram. The parameter space of the quantum phase diagram can be
specified for example via the GUT-Higgs condensation. Thus the parameter space
is also a moduli space. In Sec. 4.1, we provide the embedding web by their
internal symmetry groups. In Sec. 4.2, we derive a quantum phase diagram based
on the mother effective field theory of a modified $so(10)$ GUT of [1]. As a
toy model, we clarify the quantum phase structures when the internal
symmetries are treated as global symmetries in Sec. 4.2.1. Then we clarify the
quantum phase structures when the internal symmetries are dynamically gauged
in Sec. 4.2.2.

In Sec. 5, as the ordinary internal symmetry is dynamically gauged, the
outcome gauge theory can have the generalized global symmetries [33]. We
systematically explore these generalized global symmetries (the higher
symmetries) of SM and GUT.

In Sec. 6, we study the potential non-invertible global symmetries [34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44] (also known as categorical symmetries [45,
46, 47, 48]) of SM and GUT. We show that part of the gauge structure of the GG
${\rm U}(5)$ and the flipped ${\rm U}(5)$ gauge theories, with their
$\mathbb{Z}_{2}^{\rm flip}$ symmetry gauged, contains a gauge sector
$\big{[}({\rm U}(1)_{X_{1}}\times_{\mathbb{Z}_{4,X}}{\rm
U}(1)_{X_{2}})\rtimes\mathbb{Z}_{2}^{\rm flip}\big{]}$. There is a non-
invertible global symmetry exhibited by the topological 2-surface operators as
the gauge-invariant symmetry generators of their magnetic 1-symmetries. But we
show that this categorical symmetry is retracted thus disappears, when we
embed the theory in the modified $so(10)$ GUT of Spin(10).

In Sec. 7, we conclude and comment on future research directions.

In Appendix B, we provide various matrix representations of Lie algebras and
Lie groups of SU(5), ${\rm U}(5)_{\hat{q}}$, SO(10), and Spin(10). Then we
describe how they could embed each other properly. In Appendix C, we show the
flipping isomorphism between two ${\rm U}(5)_{\hat{q}=2}$ (the GG’s and the
flipped model’s) while both ${\rm U}(5)_{\hat{q}=2}$ can be embedded inside
the Spin(10). Then we show that the intersection of two ${\rm
U}(5)_{\hat{q}=2}$ contains the SM Lie groups, while the minimal Lie group
union of two ${\rm U}(5)_{\hat{q}=2}$ is exactly the Spin(10).

##  2 Refined ${\rm U}(5)_{\hat{q}}$ gauge theory

Here we point out there are in fact different non-isomorphic versions of U(5)
Lie groups (and their corresponding gauge theories) that we should refine and
redefine them as several ${\rm U}(5)_{\hat{q}}$ with ${\hat{q}}\in\mathbb{Z}$:

${\rm U}(5)_{\hat{q}}\equiv\frac{{\rm SU}(5)\times_{\hat{q}}{\rm
U}(1)}{\mathbb{Z}_{5}}\equiv{{\rm SU}(5)\times_{\mathbb{Z}_{5},\hat{q}}{\rm
U}(1)}\equiv\\{(g,\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta})\in{\rm SU}(5)\times{\rm U}(1)\big{|}(\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi n{\hat{q}}}{5}}),n\in\mathbb{Z}_{5}\\}.$
(2.1)

We use two data $(g,\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta})$ to label the ${\rm SU}(5)\times{\rm U}(1)$ group elements
respectively, while we identify $(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi n{\hat{q}}}{5}})$ for
$n\in\mathbb{Z}_{5}$, with a rank-5 identity matrix $\mathbb{I}$. Different
identifications of the generator of the center $Z({\rm SU}(5))=\mathbb{Z}_{5}$
with the ${\rm U}(1)$ charge ${\hat{q}}$ in principle give rise to different
Lie groups. All different ${\rm U}(5)_{\hat{q}}$ obey the group extension as
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), while different ${\hat{q}}$ identifications
specify different ${\rm U}(1)$ actions on the ${\rm SU}(5)$. But there are in
fact the following group isomorphisms

$\displaystyle{\rm U}(5)_{\hat{q}}\cong{\rm U}(5)_{-\hat{q}}\cong{\rm
U}(5)_{5m\pm\hat{q}}$ (2.2)

for any $m\in\mathbb{Z}$.777We can prove the group isomorphism by the
following. For ${\rm U}(5)_{\hat{q}}$ and ${\rm U}(5)_{-\hat{q}}$ defined in
(2.1), we can map $h_{j}=(g_{j},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta_{j}})\in{\rm U}(5)_{\hat{q}}$ and define the
group homomorphism map $f(h_{j})=(g_{j},\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta_{j}})$. Then we check that the group
homomorphism $f(h_{1})\cdot f(h_{2})=f(h_{1}\cdot h_{2})$ is true: On the
left-hand side, $f(h_{1})\cdot f(h_{2})=(g_{1}g_{2},\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt(\theta_{1}+\theta_{2})})$. On the right-hand side, $h_{1}\cdot
h_{2}=(g_{1}g_{2},\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(\theta_{1}+\theta_{2})})$ thus $f(h_{1}\cdot h_{2})=(g_{1}g_{2},\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt(\theta_{1}+\theta_{2})})$. In addition, it is injective (one-to-one) and
surjective (onto) thus a bijective group homomorphism. It would only be
bijective if the map $f(h_{j})=(g_{j},\hskip 1.0pt\mathrm{e}^{\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta_{j}})$, either a trivial identity map or the
outer automorphism on the U(1) part; thus ${\rm U}(5)_{\hat{q}}\cong{\rm
U}(5)_{-\hat{q}}$. The only exception of other isomorphism is when we shift
$\hat{q}$ to ${5m+\hat{q}}$ (which does not modify the identification in
(2.1)), thus we prove ${\rm U}(5)_{\hat{q}}\cong{\rm U}(5)_{-\hat{q}}\cong{\rm
U}(5)_{5m\pm\hat{q}}$.  Thus among general ${\hat{q}}\in\mathbb{Z}$, we have
_three distinct_ non-isomorphic types of ${\rm U}(5)_{\hat{q}}$ group for any
$m\in\mathbb{Z}$:

$\displaystyle(1)$ $\displaystyle{\rm U}(5)_{1}\cong{\rm U}(5)_{4}\cong{\rm
U}(5)_{5m+1}\cong{\rm U}(5)_{5m-1}.$ (2.3) $\displaystyle(2)$
$\displaystyle{\rm U}(5)_{2}\cong{\rm U}(5)_{3}\cong{\rm U}(5)_{5m+2}\cong{\rm
U}(5)_{5m-2}.$ (2.4) $\displaystyle(3)$ $\displaystyle{\rm U}(5)_{0}\cong{\rm
U}(5)_{5m}\cong{{\rm SU}(5)\times{\rm U}(1)}.$ (2.5)

We emphasize the distinctions of these three different non-isomorphic ${\rm
U}(5)_{\hat{q}}$ Lie groups (and their gauge theories) somehow seem not yet
been carefully examined in the previous high-energy particle physics
literature. Here we make attempts to address their Lie group differences in
the context of gauge theories and GUTs. Several comments are in order:888We
provide the detailed mathematical proofs of many statement listed here in a
separate work (jointly with Zheyan Wan et al) in [49].

  1. 1.

${\rm U}(5)_{\hat{q}}$ as a $k$-sheeted covering space of ${\rm
U}(5)_{k\hat{q}}$: If we compare the definition of ${\rm U}(5)_{\hat{q}}$ and
${\rm U}(5)_{k\hat{q}}$, we find that from (2.1), the ${\rm U}(5)_{\hat{q}}$
identifies $(\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi{\hat{q}}}{5}})$
while the ${\rm U}(5)_{k\hat{q}}$ identifies $(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
k{\hat{q}}}{5}})$. If the ${\rm U}(5)_{\hat{q}=1}$ has the periodicity of U(1)
as $\theta\in[0,2\pi)$, then the ${\rm U}(5)_{\hat{q}=k}$ has the periodicity
of U(1) as $\theta\in[0,\frac{2\pi}{k})$. Similarly, if the ${\rm
U}(5)_{\hat{q}}$ has the periodicity of U(1) as $\theta\in[0,2\pi)$, then the
${\rm U}(5)_{k\hat{q}}$ has the periodicity of U(1) as
$\theta\in[0,\frac{2\pi}{k})$. So importantly,
$\bullet$ the ${\rm U}(5)_{\hat{q}}$ is a $k$-sheeted covering space of ${\rm
U}(5)_{k\hat{q}}$.
$\bullet$ the ${\rm U}(5)_{\hat{q}=1}$ is a double covering space of ${\rm
U}(5)_{\hat{q}=2}$.
$\bullet$ the ${\rm U}(5)_{\hat{q}=2}$ is a double covering space of ${\rm
U}(5)_{\hat{q}=4}$.
$\bullet$ the ${\rm U}(5)_{\hat{q}=1}$ is a quadruple covering space of ${\rm
U}(5)_{\hat{q}=4}$,
but which goes back to itself because of the isomorphism ${\rm
U}(5)_{\hat{q}=1}\cong{\rm U}(5)_{\hat{q}=4}$.
$\bullet$ the ${\rm U}(5)_{\hat{q}=2}$ is a quadruple covering space of ${\rm
U}(5)_{\hat{q}=3}$,
but which goes back to itself because of the isomorphism ${\rm
U}(5)_{\hat{q}=2}\cong{\rm U}(5)_{\hat{q}=3}$.
$\bullet$ the ${\rm U}(5)_{\hat{q}}$ is a quadruple covering space of ${\rm
U}(5)_{4\hat{q}}\cong{\rm U}(5)_{4\hat{q}\mod 5}$,
but which goes back to itself because of the isomorphism ${\rm
U}(5)_{\hat{q}}\cong{\rm U}(5)_{4\hat{q}}\cong{\rm U}(5)_{4\hat{q}\mod 5}$.

  2. 2.

It can be shown that

${\rm U}(5)_{\hat{q}=1,4}\subset{\rm SO}(10)\text{ but }{\rm
U}(5)_{\hat{q}=1,4}\not\subset{\rm Spin}(10)$

because the homotopy group maps between $\pi_{1}({\rm
U}(5)_{\hat{q}=1,4})=\mathbb{Z}$ to $\pi_{1}({\rm SO}(10))=\mathbb{Z}_{2}$
cannot be lifted to the ${\rm SO}(10)$’s double-cover since ${\rm Spin}(10)$
has $\pi_{1}({\rm Spin}(10))=0$, which violates the lifting criterion (See
Proposition 1.33 of Hatcher [50]).

It can be shown that the ${\rm U}(5)_{\hat{q}=2,3}$ as a double cover version
of ${\rm U}(5)_{\hat{q}=1,4}$ satisfies:

${\rm U}(5)_{\hat{q}=2,3}\not\subset{\rm SO}(10)\text{ but }{\rm
U}(5)_{\hat{q}=2,3}\subset{\rm Spin}(10).$

Since $Z({\rm Spin}(10))=\mathbb{Z}_{4}$, $Z({\rm SO}(10))=\mathbb{Z}_{2}$,
and $Z(\frac{{\rm SO}(10)}{\mathbb{Z}_{2}})=0$, while ${\rm
U}(5)_{\hat{q}=2,3}$ is a quadruple cover of itself; overall we have:999Here
“$G_{1}\hookrightarrow G_{1}$” implies the inclusion thus also implies the
group embedding “$G_{1}\subset G_{2}$.” The vertical arrow “$\downarrow$”
implies the upper group $\tilde{G}$ is a double cover of the lower group $G$
such that we have a nontrivial extension $1\to\mathbb{Z}_{2}\to\tilde{G}\to
G\to 1$.

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66.72786pt\raise-71.18745pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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21.11671pt\raise-84.22356pt\hbox{\hbox{\kern 0.0pt\raise
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52.83896pt\raise-84.22356pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
52.83896pt\raise-84.22356pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\frac{{\rm
SO}(10)}{\mathbb{Z}_{2}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$ (2.12)

  3. 3.

Follow Atiyah-Bott-Shapiro [51], it is shown that the group homomorphism and
the embedding ${\rm SU}(5)\hookrightarrow{\rm SO}(10)$ can be lifted to ${\rm
SU}(5)\hookrightarrow{\rm Spin}(10)$. Similarly, the group homomorphism and
the embedding ${\rm U}(5)_{\hat{q}=1}\hookrightarrow{\rm
SO}(10)\times\frac{{\rm U}(1)}{\mathbb{Z}_{2}}=\frac{{\rm Spin}(10)\times{\rm
U}(1)}{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$ can be lifted to ${\rm
U}(5)_{\hat{q}=1}\hookrightarrow{\rm Spin}^{c}(10)\equiv\frac{{\rm
Spin}(10)\times{\rm U}(1)}{\mathbb{Z}_{2}}$. We can double cover or half-cover
of these results to obtain:

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which says not only the embedding ${\rm U}(5)_{\hat{q}=1,4}\subset{\rm
Spin}^{c}(10)$, but also the embedding ${\rm U}(5)_{\hat{q}=2,3}\subset{\rm
Spin}(10)\times{\rm U}(1)$ and ${\rm U}(5)_{\hat{q}=2,3}\subset\frac{{\rm
Spin}(10)\times{\rm U}(1)}{\mathbb{Z}_{4}}$.

  4. 4.

In a short summary, for our application on the SM and GUT physics, we shall
particularly focus on these two results:

$\displaystyle{\rm U}(5)_{\hat{q}=2,3}\subset{\rm Spin}(10),\quad\quad{\rm
U}(5)_{\hat{q}=2,3}\subset\frac{{\rm Spin}(10)\times{\rm
U}(1)}{\mathbb{Z}_{4}}.$ (2.20)

Here $\frac{{\rm Spin}(10)\times{\rm U}(1)}{\mathbb{Z}_{4}}$ can be
interpreted as $\frac{{\rm Spin}(10)\times{\rm U}(1)_{X}}{\mathbb{Z}_{4,X}}$
with $X\equiv 5({\mathbf{B}-\mathbf{L}})-4Y$, including the baryon minus
lepton number ${\mathbf{B}-\mathbf{L}}$ and the electroweak hypercharge $Y$,
is a good global symmetry respected by SM and the $su(5)$ GUT.

We provide a verification on (2.20) via exponential maps of the Lie algebras
into these Lie groups embedding in Appendix B.

  5. 5.

In order to study the ${\rm U}(5)_{\hat{q}}$ gauge theory, we should
understand the allowed Wilson line operators and their endpoint particle
charge representations (if the 1d line can be broken by the particle at open
ends). In particular, when the matter field $\psi$ is in the fundamental rep
${\bf 5}$ of SU(5), we can ask which U(1) charge representation $Q$ of $\psi$
is allowed. Because of the identification $(\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n}{5}}\mathbb{I},1)\sim(\mathbb{I},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi n{\hat{q}}}{5}})$ (for
$n\in\mathbb{Z}_{5}$) must act on the $\psi$ in $({\bf 5},Q)$ in the same way,
regardless of whether we consider
$\bullet$ $(\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi n}{5}}\mathbb{I},1)$ on $\psi$ of $({\bf 5},Q)$ which sends
$\psi$ to $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi n}{5}}\psi$.
or consider
$\bullet$ $(\mathbb{I},\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi n{\hat{q}}}{5}})$ on $\psi$ of $({\bf 5},Q)$ which gives
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n{\hat{q}}Q}{5}}\psi$.
The group element identification also means that the $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi n}{5}}=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
n{\hat{q}}Q}{5}}$, which is true if $\hat{q}Q=1\mod 5$. Thus we derive the
relation between the Lie group ${\rm U}(5)_{\hat{q}}$ and its corresponding
matter representation:

$\displaystyle\begin{tabular}[]{lcc c}\hline\cr&${\hat{q}}$&$Q$&matter rep
$({\bf 5},Q)$, $(\overline{\bf 5},-Q)$\\\ \hline\cr${\rm
U}(5)_{{\hat{q}}=1}$&1&1&$({\bf 5},1)$, $(\overline{\bf 5},-1)$\\\
\hline\cr${\rm U}(5)_{{\hat{q}}=2}$&2&3&$({\bf 5},3)$, $(\overline{\bf
5},-3)$\\\ \hline\cr${\rm U}(5)_{{\hat{q}}=3}$&3&2&$({\bf 5},2)$,
$(\overline{\bf 5},-2)$\\\ \hline\cr${\rm U}(5)_{{\hat{q}}=4}$&4&4&$({\bf
5},4)$, $(\overline{\bf 5},-4)$\\\ \hline\cr\end{tabular}.$ (2.26)

  6. 6.

If we want to choose the appropriate ${\rm U}(5)_{{\hat{q}}}$ Lie group for
the GG $su(5)$ or the flipped $su(5)$ models, we should consider the group
contains $(\overline{\bf{5}},-3)$ of $su(5)\times u(1)$. This (2.26) means
that the ${\rm U}(5)_{{\hat{q}}=2}$ is the correct choice. This matter
representation $(\overline{\bf 5},-3)$ is naturally included in ${\rm
U}(5)_{{\hat{q}}=2}$.

##  3 Various Grand Unification (GUT) Models as Vacuum Phases

In this article, we require the SM gauge group $G_{{\rm
SM}_{q}}\equiv\frac{{{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm L}\times{\rm
U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$ with $q=6$, where the 16 Weyl fermions are
in the following representation (see Fig. 2):

$\displaystyle\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus\bar{\nu}_{R}=(\overline{\bf
3},{\bf 1})_{2,L}\oplus({\bf 1},{\bf 2})_{-3,L}\oplus({\bf 3},{\bf
2})_{1,L}\oplus(\overline{\bf 3},{\bf 1})_{-4,L}\oplus({\bf 1},{\bf
1})_{6,L}\oplus{({\bf 1},{\bf 1})_{0,L}},$ (3.1)

written all in the left-handed ($L$) Weyl basis. Here we use the ${\rm
U}(1)_{\tilde{Y}}$ hypercharge instead of HEP phenomenology ${\rm U}(1)_{{Y}}$
hypercharge which is 1/6 of ${\rm U}(1)_{\tilde{Y}}$’s.101010Namely, $q_{{\rm
U}(1)_{{Y}}}=\frac{1}{6}q_{{\rm U}(1)_{\tilde{Y}}}$. If we use the hypercharge
${\rm U}(1)_{{Y}}$, then we have instead: $(\overline{\bf 3},{\bf
1})_{\frac{1}{3},L}\oplus({\bf 1},{\bf 2})_{-\frac{1}{2},L}\oplus({\bf 3},{\bf
2})_{\frac{1}{6},L}\oplus(\overline{\bf 3},{\bf
1})_{-\frac{2}{3},L}\oplus({\bf 1},{\bf 1})_{1,L}\oplus{({\bf 1},{\bf
1})_{0,L}}$. The $u$ and $d$ are the up and down quarks, the $\nu$ and $e$ are
the neutrino and electron.111111This matter content is for the first
generation of quarks and leptons. We can replace these quarks to the charm $c$
and strange $s$ for the second generation, or to the top $t$ and bottom $b$
for the third generation. We can replace these leptons to the muon $\mu$ and
tauon $\tau$ for the second and third generations. The $q_{L}$ and ${l}_{L}$
are both of the SU(2)L doublets; the $q_{L}$ contains the $(u_{L},d_{L})$
while the ${l}_{L}$ contains the $(\mu_{L},e_{L})$. We use the $L$ and $R$ to
specify the left/right-handed spacetime spinor of Spin(1,3). We use the L and
R to specify the left or right internal spinor representation, such as
$su(2)_{\text{L}}$ of the SM and the $su(2)_{\text{L}}\times su(2)_{\text{R}}$
of the Pati-Salam model. Conventionally, the spacetime $L$ and the internal L
are locked in the sense that the spacetime $L$-handed spinor is also the
internal ${\rm SU}(2)_{\text{L}}$ doublet, while the spacetime $R$-handed
spinor is the ${\rm SU}(2)_{\text{L}}$ singlet (or ${\rm SU}(2)_{\text{R}}$
doublet in the PS model). But we can regard the spacetime $R$-handed anti-
particle as the $L$-handed particle as written in (3.1).

Figure 2: Standard Model with 16n Weyl fermions and their
${{su(3)}_{c}\times{su(2)}_{\rm L}\times u(1)_{\tilde{Y}}}$ representation
(rep): $\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus\bar{\nu}_{R}=(\overline{\bf
3},{\bf 1})_{2,L}\oplus({\bf 1},{\bf 2})_{-3,L}\oplus({\bf 3},{\bf
2})_{1,L}\oplus(\overline{\bf 3},{\bf 1})_{-4,L}\oplus({\bf 1},{\bf
1})_{6,L}\oplus{({\bf 1},{\bf 1})_{0,L}}$. Figure 3: Georgi-Glashow $u(5)$ GUT
with 16n Weyl fermions and their $u(5)^{\rm 1st}=su(5)^{\rm 1st}\times
u(1)_{X}=su(5)^{\rm 1st}\times u(1)_{X_{1}}$ rep:
$(\bar{d}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})\oplus(\bar{\nu}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$.  Figure 4: Flipped $u(5)$ GUT
with 16n Weyl fermions and their $u(5)^{\rm 2nd}=su(5)^{\rm 2nd}\times
u(1)_{\chi}=su(5)^{\rm 1st}\times u(1)_{X_{2}}$ rep:
$(\bar{u}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R})\oplus(\bar{e}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$. Note that the charge of
$X_{1}=X\neq\chi=X_{2}$. Figure 5: Pati-Salam $su(4)\times su(2)_{\rm L}\times
su(2)_{\rm R}$ model and their rep: $(q_{L}\oplus l_{L})\oplus(q_{R}\oplus
l_{R})=(u_{L}\oplus d_{L}\oplus\nu_{L}\oplus
e_{L})\oplus(\bar{u}_{R}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R}\oplus\bar{e}_{R})=({\bf
4},{\bf 2},{\bf 1})\oplus(\overline{\bf 4},{\bf 1},{\bf 2})$. Figure 6: Left-
Right $su(3)\times su(2)_{\rm L}\times su(2)_{\rm
R}\times{u(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}}$ model and their rep:
$q_{L}\oplus l_{L}\oplus q_{R}\oplus l_{R}=(u_{L}\oplus
d_{L})\oplus(\nu_{L}\oplus
e_{L})\oplus(\bar{u}_{R}\oplus\bar{d}_{R})\oplus(\bar{\nu}_{R}\oplus\bar{e}_{R})=({\bf
3},{\bf 2},{\bf 1})_{\frac{1}{6}}\oplus({\bf 1},{\bf 2},{\bf
1})_{\frac{-1}{2}}\oplus(\overline{\bf 3},{\bf 1},{\bf
2})_{\frac{-1}{6}}\oplus({\bf 1},{\bf 1},{\bf 2})_{\frac{1}{2}}$. Figure 7:
The $so(10)$ GUT with 16 Weyl fermions in the rep ${\bf 16}^{+}$ of
$G_{so(10)}\equiv{\rm Spin}(10)$ gauge group.

###  3.1 Georgi-Glashow $u(5)$ vs Flipped $u(5)$ models

  1. 1.

_The $su(5)$ or $u(5)$ Grand Unification_ ($su(5)$ or $u(5)$ GUT): Georgi-
Glashow (GG) [6] hypothesized that the three SM gauge interactions merged into
a single electronuclear force at higher energy under a simple Lie algebra
$su(5)$, or precisely a Lie group $G_{\text{GG}}\equiv{\rm SU}(5)$ gauge
theory. The su(5) GUT works for 15n Weyl fermions, also for 16n Weyl fermions
(i.e., 15 or 16 Weyl fermions per generation). The Weyl fermions are in the
representation of $u(5)^{\rm 1st}=su(5)^{\rm 1st}\times u(1)_{X}=su(5)^{\rm
1st}\times u(1)_{X_{1}}$ as (see Fig. 3):

$\displaystyle(\bar{d}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})\oplus(\bar{\nu}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5},$ (3.2)

More precisely, they are in the representation of a refined ${\rm
U}(5)_{\hat{q}=2}$ group that we carefully define in Sec. 2:

$\displaystyle{\rm U}(5)_{\hat{q}=2}^{\text{1st}}\equiv\frac{{\rm
SU}(5)^{\text{1st}}\times_{\hat{q}=2}{\rm
U}(1)_{X}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{1st}}\times_{\hat{q}=2}{\rm U}(1)_{X_{1}}}{\mathbb{Z}_{5}}.$
(3.3)

The 16th Weyl fermion is an extra neutrino, sterile to the $su(5)$ but not
sterile to the $u(1)$ gauge force. The corresponding U(1) is typically called
the $X$ as ${\rm U}(1)_{X}\equiv{\rm U}(1)_{X_{1}}\equiv{\rm U}(1)_{5{({\bf
B}-{\bf L})}-4Y}\equiv{\rm U}(1)_{5{({\bf B}-{\bf L})}-\frac{2}{3}\tilde{Y}}$
which we also call $X_{1}$ because this corresponds to the U(1) subgroup of
the first type of $u(5)$ GUT.

  2. 2.

_The Barr’s flipped $su(5)$ or $u(5)$ GUT_ [10]:

The Weyl fermions are also in the representation of $u(5)^{\rm 2nd}=su(5)^{\rm
2nd}\times u(1)_{\chi}=su(5)^{\rm 2nd}\times u(1)_{X_{2}}$ (also precisely a
refined ${\rm U}(5)_{\hat{q}=2}$ group defined in Sec. 2) but arrange in a
different manner (see Fig. 4):

$\displaystyle(\bar{u}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R})\oplus(\bar{e}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}.$ (3.4)

More precisely, they are in the representation of a refined ${\rm
U}(5)_{\hat{q}=2}$ group defined in Sec. 2:

$\displaystyle{\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\equiv\frac{{\rm
SU}(5)^{\text{2nd}}\times_{\hat{q}=2}{\rm
U}(1)_{\chi}}{\mathbb{Z}_{5}}=\frac{{\rm
SU}(5)^{\text{2nd}}\times_{\hat{q}=2}{\rm U}(1)_{X_{2}}}{\mathbb{Z}_{5}}.$
(3.5)

The corresponding U(1) is typically called the $\chi$, as ${\rm
U}(1)_{\chi}\equiv{\rm U}(1)_{X_{2}}$ which we also call $X_{2}$ because this
corresponds to the U(1) subgroup of the second type of the $u(5)$ GUT (namely
the flipped model [10]).

###  3.2 There are only two types of $u(5)$ GUTs

Given the SM data and fermion representations, we can prove that there are
only two types of $u(5)$ GUTs (both embeddable inside a Spin(10) gauge group)
inside the largest possible internal U(16) group.

We sketch the proof below. The normalizer of this ${\rm SU}(5)$ in ${\rm
U}(16)$ is in fact

${\rm N}_{{\rm U}(16)}({\rm SU}(5))={\rm U}(5)\times{\rm U}(1)\times{\rm
U}(1)$

(precisely we need ${\rm U}(5)_{\hat{q}=2}\times{\rm U}(1)\times{\rm U}(1)$).
There are in fact the important four U(1) subgroups in ${\rm
U}(5)_{\hat{q}=2}\times{\rm U}(1)\times{\rm U}(1)$ listed below.

  1. 1.

The ${\rm U}(1)_{Y}$ which in our convention is also generated by the 24th
generator of the Lie algebra of $u(5)$ (precisely there are ${\rm
U}(1)_{Y_{1}}\equiv{\rm U}(1)_{\tilde{Y}}$ or ${\rm U}(1)_{Y_{2}}$ depending
on which $u(5)$ GUT models we choose). This ${\rm U}(1)_{Y}$ is inside the
SU(5). More precisely, the ${\rm U}(1)_{Y_{1}}\subset{\rm SU}(5)^{\text{1st}}$
of the GG model, and ${\rm U}(1)_{Y_{2}}\subset{\rm SU}(5)^{\text{2nd}}$ of
the Barr’s flipped model.

  2. 2.

The ${\rm U}(1)_{X}$ which in our convention is also generated by the 25th
generator of the Lie algebra of $u(5)$ (precisely there are ${\rm
U}(1)_{X_{1}}$ or ${\rm U}(1)_{X_{2}}$ depending on the which $u(5)$ GUT
models we choose). This ${\rm U}(1)_{X}$ is not inside the SU(5), but inside
the U(5) (precisely the ${\rm U}(5)_{\hat{q}=2}$ in Sec. 2).

  3. 3.

For two additional ${\rm U}(1)^{2}$ in the normalizer ${\rm N}_{{\rm
U}(16)}({\rm SU}(5))={\rm U}(5)\times{\rm U}(1)\times{\rm U}(1)$, we can
choose one U(1) to act on $\overline{\bf 5}$ alone, another U(1) to act on
${\bf 1}$ alone.

Naively, other than the two SM Weyl fermion representation combinations of the
$\overline{\bf 5}\oplus{\bf 10}\oplus{\bf 1}$ of SU(5) given in (3.2) and
(3.4), there shall be two more kinds of interpretations (thus totally four
kinds):

$\displaystyle(\bar{u}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{d}_{R}\oplus\bar{e}_{R})\oplus(\bar{\nu}_{R})$
$\displaystyle=$ $\displaystyle\overline{\bf 5}\oplus{\bf 10}\oplus{\bf 1},$
(3.6)
$\displaystyle(\bar{d}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{u}_{R}\oplus\bar{\nu}_{R})\oplus(\bar{e}_{R})$
$\displaystyle=$ $\displaystyle\overline{\bf 5}\oplus{\bf 10}\oplus{\bf 1}.$
(3.7)

All these four arrangements can establish the embedding ${\rm
SU}(5)\supset{{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm L}$. However, only the
(3.2) leads to

${\rm SU}(5)^{\text{1st}}\supset G_{{\rm SM}_{6}}$

and the (3.4) leads to

${\rm U}(5)^{\text{2nd}}_{\hat{q}=2}\supset G_{{\rm SM}_{6}}.$

The (3.6) and (3.7) both lead to unsuccessful embeddings:

${\rm N}_{{\rm U}(16)}({\rm SU}(5)^{\text{3rd}})={\rm
U}(5)^{\text{3rd}}_{\hat{q}=2}\times{\rm U}(1)\times{\rm U}(1)\not\supset
G_{{\rm SM}_{6}},\text{ and }{\rm N}_{{\rm U}(16)}({\rm
SU}(5)^{\text{4th}})={\rm U}(5)^{\text{4th}}_{\hat{q}=2}\times{\rm
U}(1)\times{\rm U}(1)\not\supset G_{{\rm SM}_{6}},$

because the linear combinations of their U(1) subgroups cannot give rise to
the SM’s ${\rm U}(1)_{Y}$ or ${\rm U}(1)_{\tilde{Y}}$. This concludes our
proof.

###  3.3 Pati-Salam $su(4)\times su(2)_{\text{L}}\times su(2)_{\text{R}}$
model

Pati-Salam (PS) [7] hypothesized that the lepton carries the fourth color,
extending SU(3) to SU(4). The PS also puts the left ${\rm SU}(2)_{\text{L}}$
and a hypothetical right ${\rm SU}(2)_{\text{R}}$ on the equal footing. The PS
gauge Lie algebra is $su(4)\times su(2)_{\text{L}}\times su(2)_{\text{R}}$,
and the PS gauge Lie group is

$G_{\text{PS}_{q^{\prime}}}\equiv\frac{{\rm SU}(4)_{c}\times({\rm
SU}(2)_{\text{L}}\times{\rm
SU}(2)_{\text{R}})}{\mathbb{Z}_{q^{\prime}}}=\frac{{\rm Spin}(6)\times{\rm
Spin}(4)}{\mathbb{Z}_{q^{\prime}}}$

with the mod $q^{\prime}=1,2$ depending on the global structure of Lie group.
We require $q^{\prime}=2$ in order to embed $G_{\text{PS}_{q^{\prime}}}$ into
the Spin(10) group. The particle excitations of this PS with 16n Weyl fermions
are constrained by the representation of $G_{\text{PS}_{q^{\prime}}}$as (see
Fig. 5):

$\displaystyle(q_{L}\oplus
l_{L})\oplus(\bar{q}_{R}\oplus\bar{l}_{R})=(u_{L}\oplus
d_{L}\oplus\nu_{L}\oplus
e_{L})\oplus(\bar{u}_{R}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R}\oplus\bar{e}_{R})=({\bf
4},{\bf 2},{\bf 1})\oplus(\overline{\bf 4},{\bf 1},{\bf 2}),$ (3.8)

written all in the left-handed ($L$) Weyl basis.121212To be clear, we have the
Weyl spacetime spinor ${\bf 2}_{L}$ of Spin(1,3) for $({\bf 4},{\bf 2},{\bf
1})\oplus(\overline{\bf 4},{\bf 1},{\bf 2})$ of $su(4)\times
su(2)_{\text{L}}\times su(2)_{\text{R}}$. Here we use the $L$ and $R$ to
specify the left/right-handed spacetime spinor of Spin(1,3). We use the L and
R to specify the left or right internal spinor representation of
$su(2)_{\text{L}}\times su(2)_{\text{R}}$.

###  3.4 The Left-Right $su(3)\times su(2)_{\text{L}}\times
su(2)_{\text{R}}\times u(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}$ model

Two version of internal symmetry groups for Senjanovic-Mohapatra’s Left-Right
(LR) model [9] are,

$G_{\text{LR}_{q^{\prime}}}\equiv\frac{{{\rm SU}(3)}_{c}\times{{\rm
SU}(2)}_{\rm L}\times{{\rm SU}(2)}_{\rm R}\times{\rm
U}(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}}{\mathbb{Z}_{3q^{\prime}}}$

with $q^{\prime}=1,2$. The particle excitations of this LR model with 16n Weyl
fermions are constrained by the representation of
$G_{\text{LR}_{q^{\prime}}}$as (see Fig. 6):

$q_{L}\oplus l_{L}\oplus\bar{q}_{R}\oplus\bar{l}_{R}=(u_{L}\oplus
d_{L})\oplus(\nu_{L}\oplus
e_{L})\oplus(\bar{u}_{R}\oplus\bar{d}_{R})\oplus(\bar{\nu}_{R}\oplus\bar{e}_{R})\\\
=({\bf 3},{\bf 2},{\bf 1})_{\frac{1}{6}}\oplus({\bf 1},{\bf 2},{\bf
1})_{\frac{-1}{2}}\oplus(\overline{\bf 3},{\bf 1},{\bf
2})_{\frac{-1}{6}}\oplus({\bf 1},{\bf 1},{\bf 2})_{\frac{1}{2}},$ (3.9)

written all in the left-handed ($L$) Weyl basis.

In general, there is a QFT embedding, the PS model
($G_{\text{PS}_{q^{\prime}}}$) $\supset$ the LR model
($G_{\text{LR}_{q^{\prime}}}$) $\supset$ the SM ($G_{{\rm
SM}_{q=3q^{\prime}}}$) for both $q^{\prime}=1,2$ via the internal symmetry
group embedding, see the details in Sec. 4.1.

###  3.5 The $so(10)$ Grand Unification and a DSpin-structure modification

Georgi and Fritzsch-Minkowski [8] hypothesized the $so(10)$ GUT (with a local
Lie algebra $so(10)$) that quarks and leptons as spacetime Weyl fermions (each
as a 2-component complex ${\bf 2}_{L}$ of ${{\rm Spin}(1,3)}$) become the
irreducible 16-dimensional spinor representation of Spin(10) (see Fig. 7):

$\displaystyle\text{${\bf 16}^{+}$ of $G_{so(10)}\equiv{\rm Spin}(10)$ gauge
group}.$ (3.10)

Thus, the 16n Weyl fermions can interact via the Spin(10) gauge fields at a
higher energy. In this case, the 16th Weyl fermion, previously a sterile
neutrino to the SU(5), is _no longer sterile_ to the Spin(10) gauge fields; it
also carries a charge 1, thus not sterile, under the gauged center subgroup
$Z({\rm Spin}(10))=\mathbb{Z}_{4}$.

In Ref. [1], there is a new sector involving either a new discrete torsion WZW
term or a fermion parton theory. The new sector can be manifested via the new
beyond-standard-model (BSM) Dirac fermions (each as a 4-component complex
${\bf 2}_{L}\oplus{\bf 2}_{R}$ of ${{\rm Spin}(1,3)}$) in the 10-dimensional
vector representation of $so(10)$ or Spin(10):

$\displaystyle\text{${\bf 10}$ of $G_{so(10)}\equiv{\rm Spin}(10)$ gauge
group}.$ (3.11)

This BSM fermion is _not compatible_ with the required spacetime-internal
symmetry group structure $({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
Spin}(10))$ that is necessary to realize the $w_{2}w_{3}$ anomaly. The
incompatibility is due to the spin-charge relation (i.e., the spacetime spin
and internal charge relation) imposed by $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm Spin}(10))$ constrains that the fermions
can be in the ${\bf 16}$ but not the ${\bf 10}$ of ${\rm Spin}(10)$.

Figure 8: The modified $so(10)$ GUT [1] with new Dirac fermions in the rep
${\bf 10}$ of $G_{so(10)}\equiv{\rm Spin}(10)$ gauge group. These fermions are
called colorons ($c$) and flavorons ($f$). These fermions can be regarded as
the fractionalizations of GUT-Higgs field (see $\Phi^{\rm{bi}}$ in Sec. 4.2)
in terms of the color-flavor separation, similar to the spin-charge separation
[52, 53, 54] in condensed matter phenomenon. Their representation $c\oplus
f\oplus{c^{\prime}}\oplus{f^{\prime}}=({\bf 3},{\bf 1})_{1,-2,-2}\oplus({\bf
1},{\bf 2})_{1,3,-2}\oplus(\overline{\bf 3},{\bf 1})_{1,2,2}\oplus({\bf
1},{\bf 2})_{1,-3,2}$ of $su(3)_{c}\times su(2)_{\rm
L}\times{u(1)^{\prime}}_{\text{gauge}}^{\text{dark}}\times u(1)_{Y_{1}}\times
u(1)_{X_{1}}$. They are $(c\oplus
f)\oplus({c^{\prime}}\oplus{f^{\prime}})={\bf 5}\oplus\overline{\bf 5}$ of
$su(5)$, and ${\bf 10}$ of $so(10)$.

The remedy, provided by Ref. [1], introduces two different fermion parities,
${\mathbb{Z}_{2}^{F}}$ and ${\mathbb{Z}_{2}^{F^{\prime}}}$, for the SM fermion
${\bf 16}$ and BSM fermion ${\bf 10}$ respectively, which together formally
forms a double spin structure called DSpin:

$\displaystyle\mathrm{DSpin}\equiv(\mathbb{Z}_{2}^{F}\times\mathbb{Z}_{2}^{F^{\prime}})\rtimes{\rm
SO}.$ (3.12)

Thus, the modified $so(10)$ GUT in Ref. [1] asks for a new spacetime-internal
structure:

$\displaystyle
G^{\text{modified}}_{\text{$so(10)$-GUT}}\equiv(\mathrm{DSpin}\times_{\mathbb{Z}_{2}^{F}}{\rm
Spin}(10))\times_{\mathbb{Z}_{2}^{F^{\prime}}}G_{\text{internal-fermionic-
parton}},$ (3.13)

where the internal symmetry of the fermionic parton theory $G_{\text{internal-
fermionic-parton}}$ can be implemented via ${\rm U}(1)^{\prime}$ or ${\rm
SU}(2)^{\prime}$, etc., see [1].

Several tables of representation data for the SM, the GG $su(5)$, the PS, and
the $so(10)$ models, can already be found in Appendix A of our previous work
[1]. In comparison, here we provide the flipped $su(5)$ or $u(5)$ model data
in Appendix A.

###  3.6 Representations of SMs and Five GUT-like models in a unified Table

For readers’ convenience to check the quantum numbers of various elementary
particles or field quanta of SMs and GUTs, we combine all the SM, the Georgi-
Glashow (GG) or the flipped ($su(5)$ or $u(5)$) models, the Pati-Salam (PS),
the left-right (LR), and the $so(10)$ models in a single Table 1. This Table 1
summarizes the more elaborated Appendix A of Ref. [1] and Appendix A of
present article in a brief but unified way.

$\begin{array}[]{c}\textbf{SM}\\\ \textbf{fermion}\\\ \textbf{spinor}\\\
\textbf{field}\end{array}$ ${{\rm SU}(3)}$ ${{\rm SU}(2)}_{\rm L}$ ${{\rm
SU}(2)}_{\rm R}$ ${\rm U}(1)_{\frac{\mathbf{B}-\mathbf{L}}{2}}$ ${\rm
U}(1)_{Y_{1}}$ ${\rm U}(1)_{\tilde{Y}_{R}}$ ${\rm U}(1)_{\rm{EM}}$ ${\rm
U}(1)_{X_{1}}$ $\mathbb{Z}_{4,X}$ $\mathbb{Z}_{2}^{F}$ ${\rm U}(1)_{X_{2}}$
${\rm U}(1)_{Y_{2}}$ SU(5)${}^{\text{1st}}$ SU(5)${}^{\text{2nd}}$
$G_{\text{PS}}$ Spin(10) $u_{L}$ ${{\mathbf{3}}}$ $q_{L}:\mathbf{2}$
$\mathbf{1}$ 1/6 1 4 2/3 1 1 1 1 1 (3,2) in 10 $\begin{array}[]{c}\bf{4},\\\
\bf{2},\\\ 1\end{array}$ $\bf{16}$ $d_{L}$ ${{\mathbf{3}}}$ $\mathbf{1}$ 1/6 1
$-2$ $-1/3$ 1 1 1 1 1 $\nu_{L}$ $\mathbf{1}$ $l_{L}:\mathbf{2}$ $\mathbf{1}$
$-1/2$ $-3$ 0 0 $-3$ 1 1 $-3$ $-3$ (1,2) in $\bar{\bf 5}$ $e_{L}$ $\mathbf{1}$
$\mathbf{1}$ $-1/2$ $-3$ $-6$ $-1$ $-3$ 1 1 $-3$ $-3$ $\bar{u}_{R}$
$\bar{\mathbf{3}}$ $\mathbf{1}$ $q_{R}:\mathbf{2}$ $-1/6$ $-4$ $-1$ $-2/3$ $1$
1 1 $-3$ 2 in ${\bf 10}$ in $\bar{\bf 5}$ $\begin{array}[]{c}\bar{\bf{4}},\\\
1,\\\ \bf{2}\end{array}$ $\bar{d}_{R}$ $\bar{\mathbf{3}}$ $\mathbf{1}$ $-1/6$
2 $-1$ 1/3 $-3$ 1 1 1 $-4$ in $\bar{\bf 5}$ in ${\bf 10}$
$\bar{\nu}_{R}={\nu}_{L}$ $\mathbf{1}$ $\mathbf{1}$ $l_{R}:\mathbf{2}$ 1/2 0 3
0 5 1 1 1 6 in ${\bf 1}$ in ${\bf 10}$ $\bar{e}_{R}=e_{L}^{+}$ $\mathbf{1}$
$\mathbf{1}$ 1/2 6 3 1 1 1 1 5 0 in ${\bf 10}$ in ${\bf 1}$

Table 1: Here we list down the Weyl fermion’s representations in the
$su(3)\times su(2)_{\text{L}}\times u(1)$ SM, the left-right ${su(3)\times
su(2)_{\text{L}}\times su(2)_{\text{R}}\times u(1)}$, the Pati-Salam
${su(4)\times su(2)_{\text{L}}\times su(2)_{\text{R}}}$, the Georgi-Glashow
${\rm U}(5)_{\hat{q}=2}^{\text{1st}}\equiv\frac{{\rm
SU}(5)^{\text{1st}}\times{\rm U}(1)_{X=1,\hat{q}=2}}{\mathbb{Z}_{5}}$, the
flipped ${\rm U}(5)_{\hat{q}=2}^{\text{2nd}}\equiv\frac{{\rm
SU}(5)^{\text{2nd}}\times{\rm U}(1)_{X_{2},\hat{q}=2}}{\mathbb{Z}_{5}}$ model,
and the $so(10)$ GUT of ${\rm Spin}(10)$ group.

The ${\rm U}(1)_{X_{1}}$ and ${\rm U}(1)_{X_{2}}$ Lie algebra generators in
(3.3) and (3.5) are also denoted as the 25th generators out of the 25
generators of the $u(5)^{\text{1st}}$ and $u(5)^{\text{2nd}}$ (the 1st for the
GG, and the 2nd for the flipped model). The ${\rm U}(1)_{Y_{1}}$ and ${\rm
U}(1)_{Y_{2}}$ Lie algebra generators are part of the $su(5)^{\text{1st}}$ and
$su(5)^{\text{2nd}}$ — they are the 24th generators out of the 25 generators
of the $u(5)^{\text{1st}}$ and $u(5)^{\text{2nd}}$. In short, we have these
relations between different expressions:

$\displaystyle\left\\{\begin{array}[]{lcl}{\rm U}(1)_{Y_{1}}\equiv{\rm
U}(1)_{\tilde{Y}}\equiv{\rm U}(1)_{6Y}\equiv{\rm
U}(1)^{\text{1st}}_{T_{24}}.\\\ {\rm U}(1)_{Y_{2}}\equiv{\rm
U}(1)^{\text{2nd}}_{T_{24}}.\\\ {\rm U}(1)_{X_{1}}\equiv{\rm
U}(1)_{X}\equiv{\rm U}(1)_{5{({\bf B}-{\bf L})}-4Y}\equiv{\rm U}(1)_{5{({\bf
B}-{\bf L})}-\frac{2}{3}{Y}_{1}}\equiv{\rm U}(1)^{\text{1st}}_{T_{25}}.\\\
{\rm U}(1)_{X_{2}}\equiv{\rm U}(1)_{\chi}\equiv{\rm
U}(1)^{\text{2nd}}_{T_{25}}.\\\ {\rm U}(1)_{X_{1}}\text{ and }{\rm
U}(1)_{X_{2}}\text{ share the same normal subgroup }Z({\rm
Spin}(10))=\mathbb{Z}_{4,X}.\end{array}\right.$ (3.19)

Here the compactification size of ${\rm U}(1)_{\tilde{Y}}$ is 1/6 of ${\rm
U}(1)_{Y}$, another way is identifying $\frac{{\rm
U}(1)_{\tilde{Y}}}{\mathbb{Z}_{6}}={\rm U}(1)_{Y}$. So their charge
quantizations are related via $q_{{\rm U}(1)_{Y_{1}}}\equiv q_{{\rm
U}(1)_{\tilde{Y}}}\equiv{6}q_{{\rm U}(1)_{Y}}$. We shall simply denote their
charge relation as $Y_{1}\equiv{\tilde{Y}}\equiv 6Y$. These U(1) factors of
$X_{1},Y_{1},X_{2}$ and $Y_{2}$ whose corresponding quantized charges span
some lattice subspace of a two-dimensional ${\mathbb{R}}^{2}$-plane. The
$X_{1},Y_{1},X_{2}$ and $Y_{2}$ charges have the following relations:

$\displaystyle\begin{pmatrix}X_{1}\\\
Y_{1}\end{pmatrix}=\frac{1}{5}\begin{pmatrix}1&4\\\
6&-1\end{pmatrix}\begin{pmatrix}X_{2}\\\ Y_{2}\end{pmatrix}\equiv
M_{\mathbb{Z}_{2}^{\rm flip}}\begin{pmatrix}X_{2}\\\ Y_{2}\end{pmatrix}$ ,
$\displaystyle\quad\begin{pmatrix}X_{2}\\\
Y_{2}\end{pmatrix}=\frac{1}{5}\begin{pmatrix}1&4\\\
6&-1\end{pmatrix}\begin{pmatrix}X_{1}\\\ Y_{1}\end{pmatrix}\equiv
M_{\mathbb{Z}_{2}^{\rm flip}}\begin{pmatrix}X_{1}\\\ Y_{1}\end{pmatrix}.\quad$
(3.20) $\displaystyle\begin{pmatrix}X_{1}\\\
X_{2}\end{pmatrix}=\frac{1}{6}\begin{pmatrix}1&5\\\
5&1\end{pmatrix}\begin{pmatrix}Y_{1}\\\ Y_{2}\end{pmatrix}\equiv
M_{XY}\begin{pmatrix}Y_{1}\\\ Y_{2}\end{pmatrix}$ ,
$\displaystyle\quad\begin{pmatrix}Y_{1}\\\
Y_{2}\end{pmatrix}=\frac{1}{4}\begin{pmatrix}-1&5\\\
5&-1\end{pmatrix}\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}\equiv
M_{YX}\begin{pmatrix}X_{1}\\\ X_{2}\end{pmatrix}.\;\quad$ (3.21)

$\bullet$ The (3.20) shows that $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$ can be
mapped onto each other via the $M_{\mathbb{Z}_{2}^{\rm
flip}}\equiv\frac{1}{5}(\begin{smallmatrix}1&4\\\ 6&-1\end{smallmatrix})$
where $M_{\mathbb{Z}_{2}^{\rm flip}}^{2}=1$, $\det(M_{\mathbb{Z}_{2}^{\rm
flip}})=-1$, and $M_{\mathbb{Z}_{2}^{\rm flip}}=M_{\mathbb{Z}_{2}^{\rm
flip}}^{-1}$ is itself a self-inverse. The $M_{\mathbb{Z}_{2}^{\rm flip}}$ is
part of the ${\mathbb{Z}_{2}^{\rm flip}}$ symmetry transformation that swaps
the GG $u(5)^{\text{1st}}$ model and the flipped $u(5)^{\text{2nd}}$ model.
$\bullet$ The (3.21) shows $M_{XY}M_{YX}=+1$, $\det(M_{XY})=-2/3$, and
$\det(M_{YX})=-3/2$; the $M_{XY}$ and $M_{YX}$ are inverse with respect to
each other.
$\bullet$ The two sets of charge lattices intersect at integer points that
match the charge assignment of the SM Weyl fermions, as shown in Fig.
9.131313In Fig. 9, the $(X_{1},Y_{1})=(X_{2},Y_{2})=(5,5)$ is also compatible
with two integral charge lattices, but this particle carries a hypercharge
$Y_{1}=5$ and a net EM charge $\frac{5}{6}$ which we do not observe in the
experiment. Other examples, such as the $(X_{1},Y_{1})=(5,-5)$ with
$(X_{2},Y_{2})=(-3,7)$ and $(X_{1},Y_{1})=(-3,7)$ with $(X_{2},Y_{2})=(5,-5)$,
also with nontrivial hypercharges or EM charges, that have no evidences yet
for their real-world existence.

Figure 9: Charge lattices of $(X_{1},Y_{1})$ (in blue as in Georgi-Glashow
[GG] Fig. 3) and $(X_{2},Y_{2})$ (in orange as in the flipped $u(5)$ GUT Fig.
4). The charge assignments of the SM Weyl fermions coincide with the
intersection (also the common integer points) of the two sets of charge
lattices. We mark the GG (3.2)’s
$(\bar{d}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R})\oplus(\bar{\nu}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$ of $su(5)^{\text{1st}}\times
u(1)_{X_{1}}$ highlighted in the three vertical blue bars. We mark the (3.4)’s
$(\bar{u}_{R}\oplus{l}_{L})\oplus(q_{L}\oplus\bar{d}_{R}\oplus\bar{\nu}_{R})\oplus(\bar{e}_{R})=\overline{\bf
5}_{-3}\oplus{\bf 10}_{1}\oplus{\bf 1}_{5}$ of $su(5)^{\text{2nd}}\times
u(1)_{X_{2}}$ highlighted in the three tilted horizontal orange bars.

Other than the above U(1) factors (i.e., $X_{1},Y_{1},X_{2}$ and $Y_{2}$), we
can find the following three U(1) factors of $T_{3,\rm{L}}$, $T_{3,{\rm R}}$,
and $\frac{\mathbf{B}-\mathbf{L}}{2}$. Some comments about these U(1) factors:

  1. 1.

$T_{3,\rm{L}}$ is generated by the third Lie algebra generator of the ${{\rm
SU}(2)}_{\rm L}$, while $T_{3,\rm{R}}$ is generated by the third Lie algebra
generator of the ${{\rm SU}(2)}_{\rm R}$. We take its $T_{3,\rm{L/R}}$’s
charge ($\pm\frac{1}{2}$) as the coefficient of its Lie algebra generator
$\frac{1}{2}\big{(}\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix}\big{)}$.

  2. 2.

The ${\mathbf{B}-\mathbf{L}}$ is the baryon ($\mathbf{B}$) minus lepton
($\mathbf{L}$) number.

  3. 3.

We have $T_{3,\rm{L}}+Y=Q_{\rm{EM}}$, the Lie algebra linear combination of
the third generator of ${{\rm SU}(2)}_{\rm L}$ and ${\rm U}(1)_{Y}$ gives the
${\rm U}(1)_{\rm{EM}}$ charge $Q_{\rm{EM}}$.

  4. 4.

We have $T_{3,\rm{R}}+Y=\frac{\mathbf{B}-\mathbf{L}}{2}$, the Lie algebra
linear combination of the third generator of ${{\rm SU}(2)}_{\rm R}$ and the
${\rm U}(1)_{Y}$ gives the ${\rm U}(1)_{\mathbf{B}-\mathbf{L}}$. We choose the
right-handed anti-particle to be in $\mathbf{2}$ of ${{\rm SU}(2)}_{\rm R}$
(so its right-handed particle to be in $\bar{\mathbf{2}}$ of ${{\rm
SU}(2)}_{\rm R}$) that makes a specific assignment on the $\pm$ sign of its
$T_{3,{\rm R}}$ charge. So we have the formula, $T_{3,{\rm L}}-T_{3,{\rm
R}}=Q_{\rm{EM}}-\frac{\mathbf{B}-\mathbf{L}}{2}$.

  5. 5.

We can introduce a new internal right hypercharge ${\rm
U}(1)_{{Y}_{\text{R}}}$ for ${{\rm SU}(2)}_{\rm R}$, as an analog of the
internal left electroweak hypercharge ${\rm U}(1)_{{Y}}={\rm
U}(1)_{{Y}_{\text{L}}}$ for ${{\rm SU}(2)}_{\rm L}$ sector, such that
$-T_{3,\rm{R}}+{{Y}_{\text{R}}}=Q_{\rm{EM}}$ and
$-T_{3,\rm{L}}+{{Y}_{\text{R}}}=\frac{\mathbf{B}-\mathbf{L}}{2}$.

  6. 6.

In fact, we can express all aforementioned U(1) in terms of some linear
combinations of three independent generators $T_{3,\text{L}}$,
$T_{3,\text{R}}$, and $\frac{\mathbf{B}-\mathbf{L}}{2}$. Here we list down the
relations of their charges via the linear combinations of $T_{3,\text{L}}$,
$T_{3,\text{R}}$, and $\frac{\mathbf{B}-\mathbf{L}}{2}$ charges:

$\displaystyle\left\\{\begin{array}[]{lcl}-T_{3,\text{R}}+\frac{\mathbf{B}-\mathbf{L}}{2}=-T_{3,\text{L}}+Q_{\rm{EM}}&=&Y_{\text{L}}\equiv
Y\equiv\frac{1}{6}Y_{1}.\\\
T_{3,\text{L}}+\frac{\mathbf{B}-\mathbf{L}}{2}=T_{3,\text{R}}+Q_{\rm{EM}}&=&Y_{\text{R}}\equiv\frac{1}{6}{\tilde{Y}}_{\text{R}}.\\\
T_{3,\text{L}}-T_{3,\text{R}}+\frac{\mathbf{B}-\mathbf{L}}{2}&=&Q_{\rm{EM}}.\\\
4T_{3,\text{R}}+6(\frac{\mathbf{B}-\mathbf{L}}{2})&=&X\equiv
X_{1}=5({\mathbf{B}-\mathbf{L}})-4Y=5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}Y_{1}.\end{array}\right.$
(3.26)

Since the appropriate linear combination of $T_{3,\text{R}}$ and
$(\frac{\mathbf{B}-\mathbf{L}}{2})$ in (3.26) contains the $X_{1}$ and
$Y_{1}$, thus which linear combination as in (3.20) contains also $X_{2}$ and
$Y_{2}$.

  7. 7.

The SM (Fig. 2) with a gauge group $G_{{\rm SM}_{q}}$ contains the U(1) Lie
algebra generators of $T_{3,\text{L}}$ and $Y_{1}$, thus also $Q_{\rm{EM}}$.
The SM does _not_ contain those of $T_{3,\text{R}}$,
$\frac{\mathbf{B}-\mathbf{L}}{2}$, $X_{1}$, $X_{2}$, or $Y_{2}$.

  8. 8.

The GG (Fig. 3) $su(5)$ GUT with a gauge group SU(5) contains the U(1) Lie
algebra generators of $T_{3,\text{L}}$ and $Y_{1}$, thus their linear
combinations include $Q_{\rm{EM}}$, but not the other U(1).

The GG $u(5)$ GUT with a gauge group ${\rm U}(5)_{\hat{q}=2}$ contains the
U(1) Lie algebra generators of $T_{3,\text{L}}$, $Y_{1}$ and $X_{1}$, thus
their linear combinations include everything else: $Q_{\rm{EM}}$,
$\frac{\mathbf{B}-\mathbf{L}}{2}$, $Y_{2}$ and $X_{2}$, also $T_{3,\text{R}}$
and $Y_{\text{R}}$.

  9. 9.

The flipped model (Fig. 4) $su(5)$ with only a gauge group SU(5) contains the
U(1) Lie algebra generators of $T_{3,\text{L}}$ and $Y_{2}$, thus their linear
combinations does not include $Q_{\rm{EM}}$, nor the other U(1). Thus the
flipped model with only a gauge group SU(5) is not enough to contain the SM
gauge group $G_{{\rm SM}_{q}}$.

The flipped $u(5)$ GUT with a gauge group ${\rm U}(5)_{\hat{q}=2}$ contains
the U(1) Lie algebra generators of $T_{3,\text{L}}$, $Y_{2}$ and $X_{2}$, thus
their linear combinations include everything else: $Q_{\rm{EM}}$,
$\frac{\mathbf{B}-\mathbf{L}}{2}$, $Y_{1}$ and $X_{1}$, also $T_{3,\text{R}}$
and $Y_{\text{R}}$.

  10. 10.

The PS model (Fig. 5) with a gauge group $G_{\text{PS}_{q^{\prime}}}$ contains
the U(1) Lie algebra generators of $T_{3,\text{L}}$, $T_{3,\text{R}}$, and
$\frac{\mathbf{B}-\mathbf{L}}{2}$. The LR model contains also all these
generators. Since the linear combinations of these three generators give all
the aforementioned U(1) Lie algebra generators in (3.26), the PS and LR models
contain all these.

  11. 11.

The $so(10)$ GUT (Fig. 7) with a gauge group Spin(10) contains the U(1) Lie
algebra generators of $T_{3,\text{L}}$, $Y_{1}$ and $X_{1}$. But it _not only_
contains the _discrete_ $\mathbb{Z}_{4,X}$ subgroup _but also_ the _continuous
Lie group_ ${\rm U}(1)_{X_{1}}$. Note that $Z({\rm
Spin}(10))=\mathbb{Z}_{4,X_{1}}=\mathbb{Z}_{4,X_{2}}=\mathbb{Z}_{4,X}$.

##  4 Quantum Landscape as Quantum Phase Diagram (Moduli Space)

We present the internal symmetry group embedding of various SMs and GUTs in
Sec. 4.1. Then we use this group embedding structure to explore a quantum
phase diagram containing these SMs and GUTs, and their quantum criticalities
(e.g., critical points or critical regions) in Sec. 4.2.

###  4.1 Embedding Web by Internal Symmetry Groups

For 16 Weyl fermion models, there is a maximal internal symmetry group U(16)
that rotates the 16 flavor of spacetime Weyl spinors in ${\bf 2}_{L}$. But
this U(16) again requires a refined definition, say ${\rm U}(16)_{\hat{q}}$,
as we did in Sec. 2. For our purpose, we just need some appropriate subgroup
${\rm Spin}(10)$ or ${{{\rm Spin}(10)\times_{\mathbb{Z}_{4,X}}{\rm U}(1)}}$ of
${\rm U}(16)_{\hat{q}}$, say ${\hat{q}}=1$, such that ${\rm
U}(16)_{\hat{q}}\supset{\rm Spin}(10)$ or ${\rm U}(16)_{\hat{q}}\supset{{{\rm
Spin}(10)\times_{\mathbb{Z}_{4,X}}{\rm U}(1)}}$. Then we can obtain the
following embedding web in Fig. 10 and Fig. 11. The arrow direction $G_{1}\to
G_{2}$ implies that $G_{1}$ internal symmetry can be broken down to $G_{2}$
(for example via appropriate scalar Higgs condensation), this also implies the
embedding $G_{1}\supset G_{2}$ and the inclusion $G_{1}\hookleftarrow G_{2}$.

Figure 10: Internal symmetry group embedding web for the $so(10)$ GUT, the
Georgi-Glashow (GG) and the flipped $u(5)$ models, and the Standard Model
(SM).

Some comments are in order:

  1. 1.

From (2.12), (2.19), and (2.20), we see that ${\rm Spin}(10)\supset{\rm
U}(5)_{\hat{q}=2}$, and $\frac{{\rm Spin}(10)\times{\rm
U}(1)}{\mathbb{Z}_{4}}\supset{\rm U}(5)_{\hat{q}=2}$. We provide a
verification via exponential maps of the Lie algebras into these Lie groups
embedding in Appendix B. Also there are two versions of ${\rm
U}(5)_{\hat{q}=2}$, the 1st for GG and the 2nd for the flipped model (see Sec.
3.2). Obviously, we also have ${\rm U}(5)_{\hat{q}}\supset{\rm SU}(5)$ for
both the 1st GG and the 2nd the flipped model

Figure 11: Follow Fig. 10, an internal symmetry group embedding web for the
$so(10)$ GUT, the GG and the flipped $u(5)$ models, and the SM; here we
include also the Pati-Salam (PS) and Left-Right (LR) models. We can use the
GUT-Higgs condensation to trigger the different routes of the symmetry
breaking patterns — we explain the realization of the vacuum expectation value
(vev) $\langle\Phi_{\bf{45}}^{\text{1st}}\rangle$,
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle$ and
$\langle\Phi_{\bf{54}}^{\text{}}\rangle$ in Sec. 4.2. This figure generalizes
the previous studies in [55] and in our prior work [1].

  2. 2.

We can explicitly check that the intersection and the union of two Lie groups
of GG and the flipped model:

$\displaystyle{{\rm U}(5)_{2}^{\text{1st}}\cap{\rm U}(5)_{2}^{\text{2nd}}}$
$\displaystyle=$ $\displaystyle{({{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm
L})\times_{{\mathbb{Z}_{6}}}{\rm U}(1)_{Y_{1}}\times_{{\mathbb{Z}_{6}}}{\rm
U}(1)_{Y_{2}}}.$ (4.1) $\displaystyle{{\rm U}(5)_{2}^{\text{1st}}\cup{\rm
U}(5)_{2}^{\text{2nd}}}$ $\displaystyle=$ $\displaystyle{\rm Spin}(10).$ (4.2)
$\displaystyle{{\rm SU}(5)_{2}^{\text{1st}}\cap{\rm SU}(5)_{2}^{\text{2nd}}}$
$\displaystyle=$ $\displaystyle{({{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm
L})}.$

This check is presented in Appendix C.2 based on the Lie algebra data in Table
8, then we can verify the exponential maps of the Lie algebras into these Lie
groups Also we show ${\rm SU}(5)^{\text{1st}}\supset G_{{\rm SM}_{6}}$ but
${\rm SU}(5)^{\text{2nd}}\not\supset G_{{\rm SM}_{6}}$, but ${\rm
U}(5)^{\text{2nd}}\supset G_{{\rm SM}_{6}}$ and ${\rm
SU}(5)^{\text{2nd}}\supset{({{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm
L})\times_{{\mathbb{Z}_{6}}}{\rm U}(1)_{Y_{2}}}$.

  3. 3.

Fig. 11 follows Fig. 10 by adding the PS and LR models. We have the PS model
$\supset$ the LR model $\supset$ the SM for both $q^{\prime}=1,2$ via the
internal symmetry group embedding:

$\displaystyle G_{\text{PS}_{q^{\prime}}}\supset
G_{\text{LR}_{q^{\prime}}}\supset G_{{\rm
SM}_{q=3q^{\prime}}}\equiv\frac{{{\rm SU}(3)}_{c}\times{{\rm SU}(2)}_{\rm
L}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q=3q^{\prime}}}.$ (4.3)

Namely, when $q^{\prime}=1$, we have $G_{\text{PS}_{1}}\supset
G_{\text{LR}_{1}}\supset G_{{\rm SM}_{3}}.$ Furthermore, only when
$q^{\prime}=2$, we can have the whole embedded into the Spin(10) for the
$so(10)$ GUT: ${{\rm Spin}(10)}\supset G_{\text{PS}_{2}}\supset
G_{\text{LR}_{2}}\supset G_{{\rm SM}_{6}}.$

###  4.2 Quantum Phase Diagram and a Mother Effective Field Theory

Now we follow Ref. [1] on the proposed parent effective field theory (EFT) as
a modified $so(10)$ GUT (with a Spin(10) gauge group) with a discrete torsion
class of WZW term. In Ref. [1], we had proposed that Georgi-Glashow (GG)
$su(5)$ and the Pati-Salam (PS) $su(4)\times su(2)\times su(2)$ models could
manifest different low energy phases of the same parent EFT, but overall all
of them share the same quantum phase diagram (see Figure 5 and 7 in [1]). By
_quantum phase diagram_ , we mean to find the governing _zero-temperature_
quantum ground states in the parameter spaces by tuning the QFT coupling
strengths.

In our present work, we further include the GG $u(5)$ and the flipped $u(5)$,
and the Left-Right (LR) model into this parent EFT (see Fig. 11, and Fig. 12
below). The parent EFT is basically the same modified $so(10)$ GUT in Ref. [1]
(except we need to refine the vev $\langle\Phi_{\bf{45}}^{\text{1st}}\rangle$,
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle$ and
$\langle\Phi_{\bf{54}}^{\text{}}\rangle$ below in Sec. 4.2.1), thus we follow
exactly the same notations and conventions there in [1]. This parent EFT
contains the following actions:

$\displaystyle S_{\text{YM-Weyl}}$ $\displaystyle=$
$\displaystyle\int_{M^{4}}{\mathrm{Tr}}(F\wedge\star F)+\hskip
1.0pt\mathrm{d}^{4}x\big{(}{\psi}^{\dagger}_{L}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\bar{\sigma}}^{\mu}{D}_{\mu,A})\psi_{L}\big{)},$ (4.4) $\displaystyle
S_{\text{Higgs}}$ $\displaystyle=$ $\displaystyle\int_{M^{4}}\hskip
1.0pt\mathrm{d}^{4}x\big{(}|{D}_{\mu,A}\Phi_{{\mathbf{R}}}|^{2}-{\rm
U}(\Phi_{{\mathbf{R}}})\big{)},$ (4.5) $\displaystyle S_{\text{Yukawa}}$
$\displaystyle=$ $\displaystyle\int_{M^{4}}\hskip
1.0pt\mathrm{d}^{4}x\Big{(}\frac{1}{2}\phi^{\intercal}\Phi^{\rm{bi}}\phi+\frac{1}{2}\sum_{a=1}^{5}\big{(}\psi_{L}^{\intercal}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}(\phi_{2a-1}\Gamma_{2a-1}-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\phi_{2a}\Gamma_{2a})\psi_{L}+\text{h.c.}\big{)}\Big{)},$ (4.6)
$\displaystyle S^{\text{WZW}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{M_{5}}{\cal B}({\Phi}_{\bf 54})\wedge\hskip
1.0pt\mathrm{d}{\cal C}({\Phi}_{\bf
45})\Big{|}_{{M^{4}=\partial{M^{5}}}}=\pi\int_{M^{5}}B(\tilde{\Phi}^{\rm{bi}})\wedge\hskip
1.0pt\mathrm{d}C(\hat{\Phi}^{\rm{bi}}))\Big{|}_{{M^{4}=\partial{M^{5}}}}.$
(4.7)

In summary, we have:
$\bullet$ the action of Yang-Mills field strength 2-form $F=\hskip
1.0pt\mathrm{d}A-\hskip 1.0pt\mathrm{i}\hskip 1.0pt{g}A\wedge A$ and Weyl
fermion $\psi_{L}$ coupling term $S_{\text{YM-Weyl}}$ in (4.4),
$\bullet$ the Higgs or GUT-Higgs $\Phi_{{\mathbf{R}}}$ action
$S_{\text{Higgs}}$ of some representation ${\mathbf{R}}$ in (4.5),
$\bullet$ the Yukawa-like action (4.6) coupling between the fermion $\psi_{L}$
in 16 of Spin(10), the GUT-Higgs $\phi$ in the vector 10 of Spin(10), and the
$\Phi^{\text{bi}}$ in the bivector ${\bf 100}={\bf 10}\times{\bf 10}$ of
Spin(10). The $\sigma^{2}$ matrix acts on the 2-component spacetime Weyl
spinor $\psi_{L}$. The $\Gamma_{a}$ (with $a\in\\{1,2,\dots,10\\}$) are ten
rank-16 matrices satisfying
$\\{\Gamma_{2a-1},\Gamma_{2b-1}\\}=2\delta_{ab},\\{\Gamma_{2a},\Gamma_{2b}\\}=2\delta_{ab},[\Gamma_{2a-1},\Gamma_{2b}]=0$
(for $a,b=1,2,\cdots,5$). So far all the terms listed above, (4.4), (4.5), and
(4.6), are within the framework of Landau-Ginzburg type of internal symmetry
breaking via the Higgs field.
$\bullet$ the WZW term. In order to go beyond the Landau-Ginzburg paradigm to
realize a richer quantum criticality between GG and PS models, we require to
add a WZW term written on a 4d manifold $M^{4}$ as a 5d $M^{5}$’s boundary.
The WZW term (4.7) is written in terms of ${\mathbb{R}}$-valued gauge fields,
2-form ${\cal B}({\Phi}_{\bf 54})$ and 2-form ${\cal C}({\Phi}_{\bf 45})$, or
$\mathbb{Z}_{2}$-valued cohomology class fields $B(\tilde{\Phi}^{\rm{bi}})$
and $C(\hat{\Phi}^{\rm{bi}})$. An ${\rm SO}(10)$ real bivector field
$\Phi^{\rm{bi}}\in{\mathbb{R}}$ is obtained from the tensor product of the two
$\phi$, in the ${\bf 10}\otimes{\bf 10}={\bf 1}_{\rm S}\oplus{\bf 45}_{\rm
A}\oplus{\bf 54}_{\rm S}$ of $so(10)$ also of Spin(10), with the anti-
symmetric (A) and symmetric (S) parts of tensor product:

$\Phi^{\rm{bi}}_{ab}=\phi_{a}\phi_{b}\text{
includes}\left\\{\begin{array}[]{l}{\mathrm{Tr}}\Phi^{\rm{bi}}=\sum_{a}\Phi^{\rm{bi}}_{aa}\text{
gives $\Phi_{{\mathbf{R}}}=\Phi_{{\mathbf{1}}}$ in ${\bf 1}_{\rm S}$}.\\\
\hat{\Phi}^{\rm{bi}}\equiv\Phi^{\rm{bi}}_{[a,b]}=\tfrac{1}{2}(\Phi^{\rm{bi}}_{ab}-\Phi^{\rm{bi}}_{ba})=\tfrac{1}{2}(\phi_{a}\phi_{b}-\phi_{b}\phi_{a})=\tfrac{1}{2}[\phi_{a},\phi_{b}]\text{
gives $\Phi_{{\mathbf{R}}}=\Phi_{{\mathbf{45}}}$ in ${\bf 45}_{\rm A}$}.\\\
\tilde{\Phi}^{\rm{bi}}\equiv\Phi^{\rm{bi}}_{\\{a,b\\}}=\tfrac{1}{2}(\Phi^{\rm{bi}}_{ab}+\Phi^{\rm{bi}}_{ba})=\tfrac{1}{2}(\phi_{a}\phi_{b}+\phi_{b}\phi_{a})=\tfrac{1}{2}\\{\phi_{a},\phi_{b}\\}\text{
gives $\Phi_{{\mathbf{R}}}=\Phi_{{\mathbf{54}}}$ in ${\bf 54}_{\rm
S}$}.\end{array}\right.$ (4.8)

We construct this WZW term (4.7) under a precise constraint to match the mod 2
class of 4d global gauge-gravitational anomaly captured by the 5d $w_{2}w_{3}$
term.

Figure 12: The quantum phase diagram (also the moduli space of vacuum
expectation values (vevs) of a set of scalar fields) is separated to eight
octants. The colors of quantum phases are designed to match the colors in Fig.
2 to Fig. 7. Here the real parameter $r_{\mathbf{R}}\in{\mathbb{R}}$ denotes
the coefficient of the effective quadratic potential (4.19)’s ${\rm
U}(\Phi_{{\mathbf{R}}})$ of $\Phi$ field in the representation ${\mathbf{R}}$.
The corresponding Higgs $\Phi$ field condenses in the
representation-${\mathbf{R}}$ if $r_{\mathbf{R}}<0$. The
$\langle\Phi_{\bf{54}}^{\text{}}\rangle\neq 0$ condenses when
$r_{\mathbf{54}}<0$. There are however two distinct
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle$ and
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle$ condensations for
$r_{\mathbf{45}}<0$; the two vevs are selected by $h>0$ and $h<0$
respectively. The quantum phase diagram contains the following phases in the
eight octants:
$\bullet$ the $so(10)$ GUT (with
$\langle\Phi_{\bf{45}}\rangle=\langle\Phi_{\bf{54}}\rangle=0$ in the first and
fifth octants, labeled by (I) and (V)),
$\bullet$ the Georgi-Glashow $u(5)$ GUT (GG, with
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$ but
$\langle\Phi_{\bf{54}}\rangle=0$ in the second octant (II)),
$\bullet$ the flipped $u(5)$ GUT (with
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$ but
$\langle\Phi_{\bf{54}}\rangle=0$ in the sixth octant (VI)),
$\bullet$ the $su(4)\times su(2)_{\mathrm{L}}\times su(2)_{\mathrm{R}}$ Pati-
Salam model (PS, with $\langle\Phi_{\bf{54}}\rangle\neq 0$ but
$\langle\Phi_{\bf{45}}\rangle=0$ in the fourth and eight octants, (IV) and
(VIII)),
$\bullet$ the $su(3)\times su(2)\times u(1)$ Standard Model (SM, with both
$\langle\Phi_{\bf{45}}\rangle\neq 0$ and $\langle\Phi_{\bf{54}}\rangle\neq 0$,
in the third and seventh octants, (III) and (VII)).
$\bullet$ the quantum critical region (around (0) in the white ball region)
occurs _if_ the criticality is enforced by the 4d boundary anomaly on of a 5d
$w_{2}w_{3}$ invertible TQFT, and _if_ we had added the WZW term into the
modified $so(10)$ GUT, and _if_ the ${{\rm
U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$ is deconfined, namely its fine
structure constant $g^{\prime 2}$ is below a certain critical value
$g_{c}^{\prime 2}$, and typically near the origin with small $r_{\mathbf{45}}$
and $r_{\mathbf{54}}$. The region outside (0) is shown in Fig. 13 (a), the
region (0) inside is shown in Fig. 13 (b). We summarize the nature of the
phase transitions in Sec. 7’s Table 2.

(a) (b)

Figure 13: (a) If ${{\rm U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$ is
confined, namely its fine structure constant $g^{\prime 2}$ is above a certain
critical value, then we call those eight octants as (I) to (VIII). (b) If
${{\rm U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$ is deconfined with
$g^{\prime 2}$ is below a certain critical value, then we call those eight
octants as (I)′ to (VIII)′, where they are altogether shown as the white ball
region (0) in Fig. 12, such that
(0)$=$(I)${}^{\prime}+$(II)${}^{\prime}+$(III)${}^{\prime}+$(IV)${}^{\prime}+$(V)${}^{\prime}+$(VI)${}^{\prime}+$(VII)${}^{\prime}+$(VIII)′.
We will also summarize the nature of the phase transitions in Sec. 7’s Table
2.

Manifestation of the WZW term in terms of a fermionic parton theory: Follow
Ref. [1], we have another realization of WZW term (4.7) by integrating out
some massive 5d fermionic parton theory ($|m|\gg 0$, we denote QED′ for it is
beyond the ordinary SM’s quantum electrodynamics QED sector):

$S^{\text{WZW}}_{\text{QED${}^{\prime}_{5}$}}[\xi,\bar{\xi},a,\Phi,{\cal
B},{\cal C}]=\int_{M^{5}}\bar{\xi}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\tilde{\gamma}^{\mu}D_{\mu}-m-\tilde{\gamma}^{5}\tilde{\Phi}^{\text{bi}}-\tilde{\gamma}^{6}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\hat{\Phi}^{\text{bi}}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\tilde{\gamma}^{5}\tilde{\gamma}^{\mu}\tilde{\gamma}^{\nu}{\cal
B}_{\mu\nu}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\tilde{\gamma}^{6}\tilde{\gamma}^{\mu}\tilde{\gamma}^{\nu}{\cal
C}_{\mu\nu})\xi\;\hskip 1.0pt\mathrm{d}^{5}x.$ (4.9)

The previously introduced two 2-form ${\mathbb{R}}$ gauge fields ${\cal
B}={\cal B}_{\mu\nu}\hskip 1.0pt\mathrm{d}x^{\mu}\wedge\hskip
1.0pt\mathrm{d}x^{\nu}$ and ${\cal C}={\cal C}_{\mu\nu}\hskip
1.0pt\mathrm{d}x^{\mu}\wedge\hskip 1.0pt\mathrm{d}x^{\nu}$ couple to the
8-component Dirac fermionic parton $\xi$ (or doubled version of 4-component
Dirac fermion) in 5d. While the 4d interface at $m=0$ appears in between the
5d bulk $m>0$ and $m<0$ phases:

$S^{\text{WZW}}_{\text{QED}^{\prime}_{4}}{[\xi,\bar{\xi},a,\Phi]}=\int_{M^{4}}\bar{\xi}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\gamma^{\mu}D_{\mu}-\tilde{\Phi}^{\text{bi}}-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\gamma^{\text{FIVE}}\hat{\Phi}^{\text{bi}})\xi\;\hskip
1.0pt\mathrm{d}^{4}x,$ (4.10)

now with the 4-component Dirac fermionic parton $\xi$ in 4d. Some explanations
below:

$\bullet$ In 4d, we can already define five gamma matrices
$\gamma^{0},\gamma^{1},\gamma^{2},\gamma^{3}$, and
$\gamma^{\text{FIVE}}\equiv(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})$, all are rank-4
matrices.141414Denote
$\sigma^{\mu\nu\cdots}\equiv\sigma^{\mu}\otimes\sigma^{\nu}\otimes\cdots$ as
the direct product of the standard Pauli matrices. Explicit matrix
representation of 4d gamma matrices are
$\gamma^{0}=\sigma^{10},\gamma^{1}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{21},\gamma^{2}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{22},\gamma^{3}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{23},\gamma^{\text{FIVE}}\equiv(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})=-\sigma^{30}.$ The 4d Dirac
fermion $\xi$ in (4.10) is a 4-component complex fermion ${\bf
2}_{L}\oplus{\bf 2}_{R}$ of ${\rm Spin}(1,3)$. The $\xi$ is also in the
10-dimensional vector representation of $so(10)$ or Spin(10). Namely, the 4d
Dirac fermion $\xi$ in (4.10) is overall in the following rep:

$\displaystyle\text{(${\bf 2}_{L}\oplus{\bf 2}_{R})$ of ${{\rm
Spin}(1,3)}$)}\times\text{(${\bf 10}$ of ${{\rm Spin}(10)}$)}.$ (4.11)

$\bullet$ However, the 4d parton theory has some extra symmetry that are not
presented in the original theory, including: (i) ${\rm U}(1)^{\prime}:\xi\to
e^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta}\xi$, (ii)
$\mathbb{Z}_{2}^{\text{CP}^{\prime}}:\xi(t,\vec{x})\to\gamma^{0}\gamma^{\text{FIVE}}\xi^{*}(t,-\vec{x})$,
(iii)
$\mathbb{Z}_{2}^{\text{T}^{\prime}}:\xi(t,\vec{x})\to\mathcal{K}\gamma^{0}\gamma^{\text{FIVE}}\xi(-t,\vec{x})$.
Note that the CP′ and T′ symmetries are for partons from the fractionalization
of the WZW term ((4.7) and (4.10)), which are unrelated to the CP and T
symmetries of the chiral fermions in the GUTs. The massless Dirac fermion are
two Weyl fermions: $\xi=\xi_{\text{L}}+\xi_{\text{R}}$.
(i) The ${\rm U}(1)^{\prime}$ symmetry forbids any Majorana mass of the form
$\xi_{\text{L/R}}\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\xi_{\text{L/R}}$
that potentially gaps out the Dirac fermion $\xi$.
(ii) Under the CP′ symmetry,
$\mathbb{Z}_{2}^{\text{CP}^{\prime}}:\bar{\xi}\xi\to-\bar{\xi}\xi$. The
$\bar{\xi}\xi$ mass is forbidden by the CP′ symmetry.
(iii) Under the T′ symmetry,
$\mathbb{Z}_{2}^{\text{T}^{\prime}}:\bar{\xi}\gamma^{\text{FIVE}}\xi\to-\bar{\xi}\gamma^{\text{FIVE}}\xi$.
The $\bar{\xi}\gamma^{\text{FIVE}}\xi$ mass is forbidden by the T′ symmetry.
Therefore the fermionic partons must be gapless in the presence of these ${\rm
U}(1)^{\prime}$, CP′, and T′ symmetries. This explains why the Dirac fermionic
parton $\xi$ in the vector rep ${\bf 10}$ of $so(10)$ can have a $w_{2}w_{3}$
anomaly [1] — although it looks that this vector rep can have quadratic mass
($\xi_{\text{L/R}}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}\xi_{\text{L/R}}$, $\bar{\xi}\xi$ or
$\bar{\xi}\gamma^{\text{FIVE}}\xi$) gapping all $\xi$ out (which seems näively
suggests they cannot have any anomaly), the enforced ${\rm U}(1)^{\prime}$,
CP′, and T′ symmetries can actually protect $\xi$ from adding those mass
terms. Because these ${\rm U}(1)^{\prime}$, CP′, and T′ symmetries are not
physical symmetries of the original WZW term (4.7), these symmetries must all
be dynamically gauged.

$\bullet$ In 5d, we define five gamma matrices
$\tilde{\gamma}^{0},\tilde{\gamma}^{1},\tilde{\gamma}^{2},\tilde{\gamma}^{3}$,
and $\tilde{\gamma}^{4}$. However, the 5d gamma matrices have different matrix
representations than the 4d gamma matrices — they are related by the
dimensional reduction on the domain wall normal to the $x_{1}$ direction. By
doubling the fermion content, we are able to introduce two more gamma
matrices, denoted $\tilde{\gamma}^{5}$ and $\tilde{\gamma}^{6}$, such that all
seven gamma matrices $\tilde{\gamma}^{0},\cdots,\tilde{\gamma}^{6}$ are rank-8
matrices satisfying the Clifford algebra relation
$\\{\tilde{\gamma}^{\mu},\tilde{\gamma}^{\nu}\\}=2\delta^{\mu\nu}$.151515In
contrast to footnote 14, explicit matrix representations of 5d gamma matrices
are  $\tilde{\gamma}^{0}=\sigma^{200},\tilde{\gamma}^{1}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{300},\tilde{\gamma}^{2}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{131},\tilde{\gamma}^{3}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{132},\tilde{\gamma}^{4}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{133},\tilde{\gamma}^{5}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{110},\tilde{\gamma}^{6}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{120}.$ So the 5d Dirac fermion $\xi$ in
(4.9) is a doubled version of 4-component complex fermion (${\bf 4}$ of ${{\rm
Spin}(1,4)}$), as the 8-component complex fermion. The $\xi$ is also in the
10-dimensional vector representation of $so(10)$ or Spin(10). Namely, the 5d
Dirac fermion $\xi$ in (4.9) is in the following rep:

$\displaystyle 2\times\text{(${\bf 4}$ of ${{\rm
Spin}(1,4)}$)}\times\text{(${\bf 10}$ of ${{\rm Spin}(10)}$)}.$ (4.12)

Notice that the 5d bulk fermions (4.12) have doubled components of the 4d
interface fermions (4.11).

$\bullet$ For the 5d bulk theory, if we define the $m>0$ as a trivial gapped
vacuum (say at $x>0$), then one can check that the $m<0$ side (say at $x<0$)
might be a nontrivial gapped vacuum with a low energy invertible TQFT
describing either gapped invertible topological order or gapped symmetry-
protected topological states (SPTs) in quantum matter. Indeed, the (4.7)’s
partition function $\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0ptS^{\text{WZW}})$
can match with $\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int w_{2}w_{3})$ in
a closed 5d bulk without a boundary, thus it describes the invertible TQFT
$w_{2}w_{3}$. The WZW term (4.7) also gives a 4d interface description, that
lives on a 4d boundary of a 5d bulk.

Below we explore quantum phases and their criticalities or phase transitions
in Fig. 12 by two aspects:
(1) when the internal symmetries are treated as global symmetries (as toy
models) in Sec. 4.2.1, and
(2) when the internal symmetries are dynamically gauged (as they are gauged in
our real-world quantum vacuum) in Sec. 4.2.2.

####  4.2.1 Internal symmetries treated as global symmetries

In the limit when we treat their internal symmetries as global symmetries
(then the Yang-Mills gauge field $A$ is only a non-dynamical background
field), the GG, the flipped, PS and LR models match the global gauge-
gravitational $w_{2}w_{3}(TM)=w_{2}w_{3}(V_{so(10)})$ anomaly via the internal
symmetry-breaking from Spin(10) to each individual subgroup (as the breaking
pattern in Fig. 11). The corresponding QFTs in Fig. 11 again manifest
different low energy phases of the same parent EFT, but overall all of them
share the same quantum phase diagram. By sharing the same quantum phase
diagram, we mean that they have the same Hilbert space and the same ’t Hooft
anomaly constraints at a deeper UV. In other words, they are in the same
_deformation class of QFTs_ , particularly advocated by Seiberg [15].161616In
fact, many works on gapping the anomaly-free chiral fermions are based on the
same logic: The chiral fermions that are free from ’t Hooft anomalies of the
chiral symmetry $G$, must be gappable without any chiral symmetry breaking in
$G$, via the $G$-symmetry preserving interaction deformations. See a series of
work along this direction and references therein: Fidkowski-Kitaev [56] in
0+1d, Wang-Wen [57, 58] for gapping chiral fermions in 1+1d, You-He-Xu-
Vishwanath [59, 60] in 2+1d, and notable examples in 3+1d by Eichten-Preskill
[61], Wen [62], You-BenTov-Xu [63, 64], BenTov-Zee [65], Kikukawa [66], Wang-
Wen [23], Razamat-Tong [67, 68], Catterall et al [69, 70], etc. The techniques
of gapping chiral fermions can be used in gapping the mirror sector.

  1. 1.

Starting from the $so(10)$ GUT (modified with WZW term or not), the
condensation of $\Phi_{\mathbf{45}}$ or/and $\Phi_{\mathbf{54}}$ will drive
the symmetry breaking transitions to various lower energy and lower internal
symmetry phases, summarized in Fig. 11. In particular, if $\Phi$ condenses to
a specific configuration $\langle\Phi\rangle$, the original Lie algebra
$\mathfrak{g}_{\text{large}}$ will be broken to its subalgebra that commutes
with $\langle\Phi\rangle$, given by

$\displaystyle\mathfrak{g}_{\text{small}}=\mathfrak{z}_{\mathfrak{g}_{\text{large}}}(\langle\Phi\rangle)\equiv\\{{\rm
T}\in\mathfrak{g}_{\text{large}}|[{\rm T},\langle\Phi\rangle]=0\\}.$ (4.13)

To realize the symmetry breaking from the Lie algebra
$\mathfrak{g}_{so(10)}\equiv so(10)$,

$\begin{split}\mathfrak{g}_{so(10)}&\to\mathfrak{g}_{\text{PS}}\equiv
su(4)\times su(2)_{\rm L}\times su(2)_{\rm R},\\\
\mathfrak{g}_{so(10)}&\to\mathfrak{g}_{\text{GG}}\equiv
su(5)^{\text{1st}}\times u(1)_{X_{1}},\\\
\mathfrak{g}_{so(10)}&\to\mathfrak{g}_{\text{flipped}}\equiv
su(5)^{\text{2nd}}\times u(1)_{X_{2}},\end{split}$ (4.14)

we must have the relations

$\begin{split}\mathfrak{z}_{so(10)}(\langle\Phi_{\mathbf{54}}\rangle)&=\mathfrak{g}_{\text{PS}},\\\
\mathfrak{z}_{so(10)}(\langle\Phi_{\mathbf{45}}^{\text{1st}}\rangle)&=\mathfrak{g}_{\text{GG}},\\\
\mathfrak{z}_{so(10)}(\langle\Phi_{\mathbf{45}}^{\text{2nd}}\rangle)&=\mathfrak{g}_{\text{flipped}},\end{split}$
(4.15)

whose solutions reads

$\begin{split}\langle\Phi_{\mathbf{54}}\rangle&\propto\left(\begin{smallmatrix}-3&&&&\\\
&-3&&&\\\ &&2&&\\\ &&&2&\\\
&&&&2\end{smallmatrix}\right)\otimes\left(\begin{smallmatrix}1&\\\
&1\end{smallmatrix}\right),\\\
\langle\Phi_{\mathbf{45}}^{\text{1st}}\rangle&\propto\left(\begin{smallmatrix}1&&&&\\\
&1&&&\\\ &&1&&\\\ &&&1&\\\
&&&&1\end{smallmatrix}\right)\otimes\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right),\\\
\langle\Phi_{\mathbf{45}}^{\text{2nd}}\rangle&\propto\left(\begin{smallmatrix}-1&&&&\\\
&-1&&&\\\ &&1&&\\\ &&&1&\\\
&&&&1\end{smallmatrix}\right)\otimes\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right).\\\ \end{split}$ (4.16)

$\bullet$ The $\langle\Phi_{\mathbf{54}}\rangle$ explicitly distinguishes the
first four-dimensional subspace from the last six-dimensional subspace of the
$so(10)$ vector, therefore breaking $\mathfrak{g}_{so(10)}=so(10)$ down to
$\mathfrak{g}_{\text{PS}}=so(6)\times so(4)$.
$\bullet$ The $\langle\Phi_{\mathbf{45}}^{\text{1st}}\rangle$ is proportional
to the $u(1)_{X_{1}}$ generator, which effectively requires the unbroken
generators to commute with $u(1)_{X_{1}}$ generator. This singles out
$\mathfrak{g}_{\text{GG}}=\mathfrak{z}_{so(10)}(u(1)_{X_{1}})=su(5)^{\text{1st}}\times
u(1)_{X_{1}}$.
$\bullet$ The $\langle\Phi_{\mathbf{45}}^{\text{2nd}}\rangle$ can be obtained
via the $\mathbb{Z}_{2}^{\text{flip}}$ transformation on the
$\langle\Phi_{\mathbf{45}}^{\text{1st}}\rangle$, where
$\mathbb{Z}_{2}^{\text{flip}}$ is described in Sec. 3.6 and Appendix C.1. The
$\langle\Phi_{\mathbf{45}}^{\text{2nd}}\rangle$ is proportional to the
$u(1)_{X_{2}}$ generator, which effectively requires the unbroken generators
to commute with $u(1)_{X_{2}}$ generator. This singles out
$\mathfrak{g}_{\text{GG}}=\mathfrak{z}_{so(10)}(u(1)_{X_{2}})=su(5)^{\text{2nd}}\times
u(1)_{X_{2}}$.
$\bullet$ Using (4.16), one can further verify that

$\begin{split}\mathfrak{z}_{\mathfrak{g}_{\text{PS}}}(\langle\Phi_{\mathbf{45}}^{\text{1st}}\rangle)=\mathfrak{z}_{\mathfrak{g}_{\text{PS}}}(\langle\Phi_{\mathbf{45}}^{\text{2nd}}\rangle)=\mathfrak{z}_{\mathfrak{g}_{\text{GG}}}(\langle\Phi_{\mathbf{54}}\rangle)=\mathfrak{z}_{\mathfrak{g}_{\text{flipped}}}(\langle\Phi_{\mathbf{54}}\rangle)=\mathfrak{g}_{\text{SM}},\end{split}$
(4.17)

which explicitly confirms that the simultaneous condensation of
$\Phi_{\mathbf{45}}$ (any of $\Phi_{\mathbf{45}}^{\text{1st}}$ and
$\Phi_{\mathbf{45}}^{\text{2nd}}$) and $\Phi_{\mathbf{54}}$ indeed breaks the
internal symmetry to $\mathfrak{g}_{\text{SM}}$. These results also agree with
the representation data and branching rules listed in [71, 72, 73].

  2. 2.

Quantum criticalities and phase transitions due to GUT-Higgs, with or without
WZW term:
In Fig. 12 and its caption, we have enumerated all the ground states in all
the eight octants of the phase diagram descended from the 4d parent theory. In
particular, the 4d phases (in the bulk portion of the phase diagram) maintain
regardless of whether we add the WZW term $S^{\text{WZW}}$ into the Landau-
Ginzburg type of 4d parent theory of the action:

$\displaystyle S_{\text{YM-Weyl}}+S_{\text{Higgs}}+S_{\text{Yukawa}}.$ (4.18)

The phase transition (between the eight octants in Fig. 12) is triggered by
the following GUT-Higgs potential ${\rm U}(\Phi_{{\mathbf{R}}})$ appeared in
(4.5), for example,171717Here we extract
${h\;\Phi_{\bf{45}}\cdot(\langle\Phi_{\bf{45}}^{\text{1st}}\rangle-\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle)}$
from
$h^{\prime}\Big{(}(\Phi_{\bf{45}}-\langle\Phi_{\bf{45}}^{\text{1st}}\rangle)^{2}-(\Phi_{\bf{45}}-\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle)^{2}\Big{)}+\dots$.

${\rm
U}(\Phi_{{\mathbf{R}}})=\Big{(}r_{{\mathbf{45}}}(\Phi_{{\mathbf{45}}})^{2}+\lambda_{{\mathbf{45}}}(\Phi_{{\mathbf{45}}})^{4}\Big{)}+\Big{(}r_{{\mathbf{54}}}(\Phi_{{\mathbf{54}}})^{2}+\lambda_{{\mathbf{54}}}(\Phi_{{\mathbf{54}}})^{4}\Big{)}+{h\;\Phi_{\bf{45}}\cdot(\langle\Phi_{\bf{45}}^{\text{1st}}\rangle-\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle)}+\dots.$
(4.19)

Although the 4d phases in all the eight octants are _not sensitive_ to the WZW
term, the quantum critical region (labeled as (0)) and the phase boundaries
between the eight octants are _highly sensitive_ to the WZW term. In those
critical regions, we must examine the nature of criticality by looking into
the full action

$\displaystyle S_{\text{YM-
Weyl}}+S_{\text{Higgs}}+S_{\text{Yukawa}}+S^{\text{WZW}}=S_{\text{YM-
Weyl}}+S_{\text{Higgs}}+S_{\text{Yukawa}}+S^{\text{WZW}}_{\text{QED${}^{\prime}_{5}$|QED${}^{\prime}_{4}$}}.$
(4.20)

In particular, in the right hand side of equality, we focus on a specific
scenario that the low energy physics of WZW is manifested by the 5d bulk/4d
interface QED′ written as
$S^{\text{WZW}}_{\text{QED${}^{\prime}_{5}$|QED${}^{\prime}_{4}$}}$ from (4.9)
and (4.10), with deconfined emergent dark gauge fields of ${\rm
U}(1)^{\prime}$ only near the critical region.

Below in Remark 4 and 5 respectively, we describe the Landau-Ginzburg type
criticalities (not sensitive to WZW term), and the beyond Landau-Ginzburg type
criticalities (sensitive to WZW term and anomaly constraints).

  3. 3.

How many distinct phase transitions there are in Fig. 12? We can enumerate
those occur in the $h>0$ side with the four quadrants (I), (II), (III), and
(IV) shown in Fig. 14. Then we can enumerate those occur in the $h<0$ side
with the last four quadrants (V), (VI), (VII), and (VIII) shown in Fig. 15.

In Fig. 14, there are four phase transitions between these four phases:
(I)-(II), (II)-(III), (III)-(IV), and (IV)-(I). There are also another four
phase transitions from either of these four phases into the quantum critical
region in (0): Namely, (I)-(0), (II)-(0), (III)-(0), and (IV)-(0). So there
are totally eight possible phase transitions in Fig. 14.

In Fig. 15, there are four phase transitions between these four phases:
(V)-(IV), (VI)-(VII), (VII)-(VIII), and (VIII)-(V). There are also another
four phase transitions from either of these four phases into the quantum
critical region in (0): Namely, (V)-(0), (VI)-(0), (VII)-(0), and (VIII)-(0).
So there are also totally eight possible phase transitions in Fig. 15.

Furthermore, the phase structures show many phases are the same: (I)=(V),
(III)=(VII), and (IV)=(VIII). The differences between (II) and (VI) are simply
two different Landau-Ginzburg symmetry-breaking vacua. Thus all phase
transitions in Fig. 14 have the exactly same nature as those in Fig. 15. There
is only one more phase transition between (II) and (VI), that is not shown in
Fig. 14 or Fig. 15. So totally there are nine distinct possible phase
transitions in Fig. 12 that we will enumerate.

Figure 14: A typical $h>0$ slice of Fig. 12’s quantum phase diagram. The
coordinates are potential tuning parameters $r_{{\mathbf{45}}}$ and
$r_{{\mathbf{54}}}$ of (4.19).  Figure 15: A typical $h<0$ slice of Fig. 12’s
quantum phase diagram. The coordinates are potential tuning parameters
$r_{{\mathbf{45}}}$ and $r_{{\mathbf{54}}}$ of (4.19).

Even more precisely, the stable quantum critical region (0) can also have
distinct symmetry breaking orders, so we can further precisely denote the dark
gauge ${\rm U}(1)^{\prime}$-deconfined critical region (0) as

$\displaystyle\text{(0) $=$
(I)${}^{\prime}+$(II)${}^{\prime}+$(III)${}^{\prime}+$(IV)${}^{\prime}+$(V)${}^{\prime}+$(VI)${}^{\prime}+$(VII)${}^{\prime}+$(VIII)${}^{\prime}$}.$
(4.21)

Each of them has the symmetry breaking orders from the ${\rm
U}(1)^{\prime}$-confined region (mentioned previously as (I) to (VIII)). Thus,
we also have the identifications of some deconfined critical phases:
(I)${}^{\prime}=$(V)′, (III)${}^{\prime}=$(VII)′, and
(IV)${}^{\prime}=$(VIII)′.

Here are some terminologies for the type of phase transitions:
(1) _Landau-Ginzburg_ type: Based on the original symmetry group (kinematic)
broken down to an unbroken symmetry group (dynamics) via a symmetry-breaking
order parameter $\langle{\cal O}_{\text{L}G}\rangle$.
(2) _Beyond Landau-Ginzburg_ type: Cannot be characterized via merely
symmetry-breaking order parameters. e.g., phase transitions involving
topological terms (e.g., WZW term), SPTs, intrinsic topological orders, or the
’t Hooft anomaly matching differently on two sides of phases, etc.
(3) _Wilson-Fisher_ type: Phase transition due to a scalar field condensation
$\langle\Phi\rangle\neq 0$, especially via $r\Phi^{2}+\lambda\Phi^{4}$ type
potential. However, Wilson-Fisher fixed point requires the beyond-mean-field
RG correction to the Gaussian fixed point.
(4) _Gross-Neveu type_ type: Typically the Wilson-Fisher type with additional
Yukawa coupling between fermions $\chi$ and scalars $\Phi$ as
$\chi^{\dagger}\Phi\chi$, again a phase transition due to a condensation
$\langle\Phi\rangle\neq 0$.
(5) Order of phase transition:
For a minimal positive $N$, if the $N$th derivative of the free energy (of
QFT) with respect to the driving parameter is discontinuous at the transition,
it is called the $N$th-order phase transition.
$\bullet$ First-order transition is also called a discontinuous transition,
while the correlation length remains finite and no additional gapless
excitations appear at the transition.
$\bullet$ Second-order and higher-order is called a continuous transition,
while the correlation length diverges to infinite181818Here the correlation
function for Landau-Ginzburg paradigm is typically the two-point correlator
$\langle{\cal O}(x_{1}){\cal O}(x_{2})\rangle$ of local order parameters of
individual spacetime points. In contrast, the correlation function for
topological order beyond the Landau-Ginzburg paradigm is the correlator of
strings or higher-dimensional extended operators.  and additional gapless
excitations (thus called critical, sometimes described by conformal field
theory) appear at the transition.
This above definition is applicable to Landau-Ginzburg as well as beyond
Landau-Ginzburg paradigm.

If the transition happens to be within Landau-Ginzburg paradigm, then:
$\bullet$ First-order means the order parameter has a discontinuous jump at
the transition.
$\bullet$ Second-order and higher-order means the order parameter changes
continuous without a jump at the transition.

  4. 4.

Landau-Ginzburg type criticalities and phase transitions:

$\bullet$ Phase transition between the octant (II) to (VI):

This is the phase transition between the GG $u(5)$ and flipped $u(5)$ GUTs. In
both the octant (II) and (VI), we have $r_{{\mathbf{45}}}<0$. The phase
transition between (II) and (VI) is triggered by $h>0$ and $h<0$, causing
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$ or
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$. In general, by tuning $h$,
these condensations jump from zero to nonzero, thus it is the first order
phase transition of traditional Landau-Ginzburg symmetry-breaking type with
the order parameter discontinuity. The WZW term does not play any role in the
phase transition. This also means there is no critical gapless mode directly
associated with this phase transition.

$\bullet$ Phase transition between the octant (II) to (III), similar to (VI)
to (III):

This is the phase transition between the GG $u(5)$ GUT and SM (similarly, the
transition between the flipped $u(5)$ GUT and SM). The phase transition is
triggered by tuning $r_{{\mathbf{54}}}>0$ to $r_{{\mathbf{54}}}<0$ of
$r\Phi^{2}+\lambda\Phi^{4}$ in (4.19). It is the continuous phase transition
of Wilson-Fisher type Landau-Ginzburg symmetry-breaking type.

$\bullet$ Phase transition between the octant (IV) to (III):

This is the phase transition between the PS and SM. The phase transition is
triggered by tuning $r_{{\mathbf{45}}}>0$ to $r_{{\mathbf{45}}}<0$ of
$r\Phi^{2}+\lambda\Phi^{4}$ in (4.19). It is again the continuous phase
transition of Wilson-Fisher type Landau-Ginzburg symmetry-breaking type.

  5. 5.

Beyond Landau-Ginzburg type criticalities and phase transitions: With the WZW
term, the criticality between the GG and the PS, and the criticality between
the flipped $u(5)$ and the PS, both of these criticalities are governed by the
Beyond Landau-Ginzburg paradigm. The critical regions are drawn in the white
region around the origin in Fig. 14 or Fig. 15.

$\bullet$ Phase transition between the octant (I) to (II), similar to (I) to
(VI):
The WZW term and the anomaly of 5d $w_{2}w_{3}$ invertible TQFT play a crucial
role in (I). The 4d phase transition from the modified $so(10)$ (I) to GG (II)
(similarly, the $so(10)$ (I) to the flipped (VI)) is a boundary phase
transition on the 5d bulk $w_{2}w_{3}$. The phase transition is triggered not
merely by tuning $r_{{\mathbf{45}}}>0$ to $r_{{\mathbf{45}}}<0$ of
$r\Phi^{2}+\lambda\Phi^{4}$ in (4.19), but also by the symmetry breaking to
cancel the anomaly on the 4d boundary of 5d invertible TQFT (when entering
from (I) to (II) or to (VI)). Overall, it is the continuous phase transition
of Wilson-Fisher type but Beyond-Landau-Ginzburg paradigm due to the anomaly
matching via the symmetry breaking on the 4d boundary of 5d invertible TQFT.

$\bullet$ Phase transition between the octant (I) to (IV):
The WZW term and the anomaly of 5d $w_{2}w_{3}$ invertible TQFT play a crucial
role in (I). The 4d phase transition from the modified $so(10)$ (I) to PS (IV)
is a boundary phase transition on the 5d bulk $w_{2}w_{3}$. The phase
transition is triggered not merely by tuning $r_{{\mathbf{54}}}>0$ to
$r_{{\mathbf{54}}}<0$ of $r\Phi^{2}+\lambda\Phi^{4}$ in (4.19), but also by
the symmetry breaking to cancel the anomaly on the 4d boundary of 5d
invertible TQFT (when entering from (I) to (IV)). Overall, it is the
continuous phase transition of Wilson-Fisher type but Beyond-Landau-Ginzburg
paradigm due to the anomaly matching via the symmetry breaking on the 4d
boundary of 5d invertible TQFT.

$\bullet$ Phase transition between the octant (II) to the critical region (0)
(the situation is similar to phase transitions of (III) to (0), (IV) to (0),
and (VI) to (0)):191919More precisely, we really mean to specify here the
phase transitions from the ${\rm U}(1)^{\prime}$ confined phase of GUT/SM to
the corresponding ${\rm U}(1)^{\prime}$ deconfined phase of GUT/SM, namely
(II) to (II)′, (III) to (III)′, (IV) to (IV)′, and (VI) to (VI)′. The critical
region (0) actually has children sub-phases including those in Fig. 13 (b).
The WZW term and the anomaly of 5d $w_{2}w_{3}$ invertible TQFT play a crucial
role in the critical region (0). The 4d phase transitions from the either
models of GUT/SM of (II), (III) and (IV) to the critical region (0) is a
boundary phase transition on the 5d bulk $w_{2}w_{3}$. These 4d phase
transitions are Gross-Neveu type because we also have Yukawa-Higgs
interactions $\chi^{\dagger}\Phi\chi$ in (4.10) (more than Wilson-Fisher of
$r\Phi^{2}+\lambda\Phi^{4}$). Moreover, there are deconfined dark gauge fields
of ${\rm U}(1)^{\prime}$ in the critical region (0), but the ${\rm
U}(1)^{\prime}$ is confined outside the critical region (0). Therefore,
overall it is the continuous phase transition of deconfined-confined
QED′-Gross-Neveu type beyond-Landau-Ginzburg paradigm. It is beyond Landau-
Ginzburg also due to two effects (1) WZW term and (2) anomaly matching via the
symmetry breaking on the 4d boundary of 5d invertible TQFT.

$\bullet$ Phase transition between the octant (I) to the critical region (0)
(or more precisely (I)′):
The WZW term and the anomaly of 5d $w_{2}w_{3}$ invertible TQFT play a crucial
role in both the critical region (I)′ and the modified $so(10)$ GUT (I).

The 4d phase transitions from the (I) to the critical region (I)′ is a
boundary phase transition that the 5d bulk $w_{2}w_{3}$ is always required
since the Spin(10) is preserved throughout the transition (if we regard the
Spin(10) global symmetry is realized locally onsite).

— If the _deconfined_ dark gauge fields ${\rm U}(1)^{\prime}$ in the critical
region (I)′ becomes _confined_ in the region (I), then overall it is the
continuous deconfined-confined phase transition of QED${}^{\prime}_{4}$
Beyond-Landau-Ginzburg paradigm, without any global symmetry-breaking. The
QED${}^{\prime}_{4}$ describes the ${\rm U}(1)^{\prime}$ dark gauge field
coupled to fermionic partons. When the gauge coupling is strong enough,
${g^{\prime}}^{2}>{g_{c}^{\prime}}^{2}$, it is possible to drive a confinement
transition, which gaps out all the fermionic partons and removes the ${\rm
U}(1)^{\prime}$ photon from the low-energy spectrum. However, this
nonperturbative nature of deconfined to confined phase transition of
QED${}^{\prime}_{4}$ cannot be captured easily by perturbative renormalization
group or Feynman diagram analysis.202020Ref. [74] studies the deconfined to
confined phase transition of QED${}^{\prime}_{4}$. Ref. [74] suggests that its
nature is a Berezinskii-Kosterlitz-Thouless (BKT) phase transition, as an
infinite order continuous phase transition.

— If the _deconfined_ dark gauge fields ${\rm U}(1)^{\prime}$ in the critical
region (I)′ remains _deconfined_ in the region (I), then overall there is no
phase transition. The critical region (I)′ and (I) are smoothly connected as
the same critical region.

The critical region (0) is further broken down to the different symmetry-
breaking orders from (I)′ to (VIII)′ shown in (4.21) with totally 5 refined
phases. Hence there are more refined versions of phases transitions than what
we had discussed above.

Overall, we need to enumerate all the possible phase transitions between these
regions: the ${{\rm U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$-confined
regions (I)′=(V)′, (II)′, (III)′=(VII)′, (IV)′=(VIII)′, and (VI)′, and the
${{\rm U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$-deconfined critical
regions (I)′=(V)′, (II)′, (III)′=(VII)′, (IV)′=(VIII)′, and (VI)′. We
summarize the nature of these phase transitions in Table 2.

Properties of Quantum Phase Transition Internal Global Symmetry
$\begin{array}[]{c}\text{Order of}\\\ \text{transition}\end{array}$
$\begin{array}[]{c}\text{Landau-Ginzburg}\\\ \text{order parameter}\\\
\langle{\cal O}_{\text{L}G}\rangle\\\ \end{array}$
$\begin{array}[]{c}\text{Critical}\\\ \text{theory}\end{array}$
$\begin{array}[]{c}\text{Anomaly}\\\ \text{Sym.Preserved}\\\
\text{Sym.Breaking}\end{array}$ $\begin{array}[]{c}\text{Fermionic}\\\
\text{parton}\end{array}$ $\begin{array}[]{c}\text{Deconfined}\\\ \text{${{\rm
U}(1)^{\prime}}_{\text{gauge}}^{\text{dark}}$}\end{array}$
$\begin{array}[]{c}\text{Beyond}\\\ \text{LGW}\end{array}$ (I)-(II)
$so(10)$-GG Cont. Condense $\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$
WF $\begin{array}[]{c}\text{SP to SB}\end{array}$
$\begin{array}[]{c}\text{gapped}\\\ \text{(confined)}\end{array}$
$\begin{array}[]{c}\text{No}\end{array}$ $\begin{array}[]{c}\text{Yes}\\\
\text{(anom.)}\end{array}$ (I)-(VI) $so(10)$-flipped Cont. Condense
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$ (I)-(IV) $so(10)$-PS Cont.
Condense $\langle\Phi_{\bf{54}}\rangle\neq 0$ WF $\begin{array}[]{c}\text{SP
to SB}\end{array}$ $\begin{array}[]{c}\text{gapped}\\\
\text{(confined)}\end{array}$ No $\begin{array}[]{c}\text{Yes}\\\
\text{(anom.)}\end{array}$ (II)-(III) GG-SM Cont. Condense
$\langle\Phi_{\bf{54}}\rangle\neq 0$ WF $\begin{array}[]{c}\text{SB to
SB}\end{array}$ $\begin{array}[]{c}\text{gapped}\\\
\text{(confined)}\end{array}$ $\begin{array}[]{c}\text{No}\end{array}$ No
(VI)-(III) flipped-SM (IV)-(III) PS-SM Cont. Condense
$\langle\Phi_{\bf{45}}\rangle\neq 0$ WF $\begin{array}[]{c}\text{SB to
SB}\end{array}$ $\begin{array}[]{c}\text{gapped}\\\
\text{(confined)}\end{array}$ No No (I)-(III) $so(10)$-SM Cont.
$\begin{array}[]{c}\text{Condense}\\\
\langle\Phi_{\bf{45}}\rangle,\langle\Phi_{\bf{54}}\rangle\neq 0\end{array}$ WF
$\begin{array}[]{c}\text{SP to SB}\end{array}$
$\begin{array}[]{c}\text{gapped}\\\ \text{(confined)}\end{array}$ No
$\begin{array}[]{c}\text{Yes}\\\ \text{(anom.)}\end{array}$ (II)-(IV) GG-PS
Cont. $\begin{array}[]{c}\text{Swap $\langle\Phi_{\bf{45}}\rangle\neq 0$}\\\
\text{and $\langle\Phi_{\bf{54}}\rangle\neq 0$}\end{array}$ WF
$\begin{array}[]{c}\text{SB to SB}\end{array}$
$\begin{array}[]{c}\text{gapped}\\\ \text{(confined)}\end{array}$
$\begin{array}[]{c}\text{No}\end{array}$
$\begin{array}[]{c}\text{No}\end{array}$ (VI)-(IV) flipped-PS (II)-(VI) GG-
flipped 1st $\begin{array}[]{c}\text{Swap
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$}\\\ \text{and
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$}\end{array}$ No
$\begin{array}[]{c}\text{SB to SB}\end{array}$
$\begin{array}[]{c}\text{gapped}\\\ \text{(confined)}\end{array}$ No No
(I)′-(II)′ $so(10)^{\prime}$-GG′ Cont. Condense
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$ QED${}^{\prime}_{4}$-GNY
$\begin{array}[]{c}\text{SP to SB}\end{array}$
$\begin{array}[]{c}\text{gapless}\\\ \text{(deconfined)}\end{array}$
$\begin{array}[]{c}\text{Yes}\end{array}$ $\begin{array}[]{c}\text{Yes}\\\
\text{(an.dcf.)}\end{array}$ (I)′-(VI)′ $so(10)^{\prime}$-flipped′ Condense
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$ (I)′-(IV)′
$so(10)^{\prime}$-PS′ Cont. Condense $\langle\Phi_{\bf{54}}\rangle\neq 0$
QED${}^{\prime}_{4}$-GNY SP to SB $\begin{array}[]{c}\text{gapless}\\\
\text{(deconfined)}\end{array}$ Yes $\begin{array}[]{c}\text{Yes}\\\
\text{(an.dcf.)}\end{array}$ (II)′-(III)′ GG′-SM′ Cont. Condense
$\langle\Phi_{\bf{54}}\rangle\neq 0$ WF $\begin{array}[]{c}\text{SB to
SB}\end{array}$ $\begin{array}[]{c}\text{gapped}\\\ \text{(Higgs
SSB)}\end{array}$ Yes $\begin{array}[]{c}\text{No}\end{array}$ (VI)′-(III)′
flipped′-SM′ (IV)′-(III)′ PS′-SM′ Cont. Condense
$\langle\Phi_{\bf{45}}\rangle\neq 0$ WF $\begin{array}[]{c}\text{SB to
SB}\end{array}$ $\begin{array}[]{c}\text{gapped}\\\ \text{(Higgs
SSB)}\end{array}$ Yes $\begin{array}[]{c}\text{No}\end{array}$ (I)′-(III)′
$so(10)^{\prime}$-SM′ Cont. $\begin{array}[]{c}\text{Condense}\\\
\langle\Phi_{\bf{45}}\rangle,\langle\Phi_{\bf{54}}\rangle\neq 0\end{array}$
QED${}^{\prime}_{4}$-GNY SP to SB $\begin{array}[]{c}\text{gapless}\\\
\text{(deconfined)}\end{array}$ Yes $\begin{array}[]{c}\text{Yes}\\\
\text{(an.dcf.)}\end{array}$ (II)′-(IV)′ GG′-PS′ Cont.
$\begin{array}[]{c}\text{Swap $\langle\Phi_{\bf{45}}\rangle\neq 0$}\\\
\text{and $\langle\Phi_{\bf{54}}\rangle\neq 0$}\end{array}$
QED${}^{\prime}_{4}$-GNY $\begin{array}[]{c}\text{SB to SB}\end{array}$
$\begin{array}[]{c}\text{gapless}\\\ \text{(deconfined)}\end{array}$
$\begin{array}[]{c}\text{Yes}\end{array}$ $\begin{array}[]{c}\text{Yes}\\\
\text{(decf.)}\end{array}$ (VI)′-(IV)′ flipped′-PS′ (II)′-(VI)′ GG′-flipped′
1st $\begin{array}[]{c}\text{Swap
$\langle\Phi_{\bf{45}}^{\text{1st}}\rangle\neq 0$}\\\ \text{and
$\langle\Phi_{\bf{45}}^{\text{2nd}}\rangle\neq 0$}\end{array}$ No
$\begin{array}[]{c}\text{SB to SB}\end{array}$
$\begin{array}[]{c}\text{gapped}\\\ \text{(Higgs SSB)}\end{array}$ Yes No
(I)′-(I) $so(10)^{\prime}$-$so(10)$ Cont. No
$\begin{array}[]{c}\text{deconfined-}\\\ \text{confined}\end{array}$ SP to SP
$\begin{array}[]{c}\text{gapless}\\\ \text{(deconfined)}\end{array}$ Yes
$\begin{array}[]{c}\text{Yes}\\\ \text{(decf.)}\end{array}$ ${(\rm
II)}^{\prime}$-(II) GG′-GG SB to SB ${(\rm III)}^{\prime}$-(III) SM′-SM SB to
SB ${(\rm IV)}^{\prime}$-(IV) PS′-PS SB to SB ${(\rm VI)}^{\prime}$-(VI)
flipped′-flipped SB to SB

Table 2: Here we summarize the properties of quantum phase transition in the
limit when the internal symmetries is treated as ungauged global symmetries in
Sec. 4.2.1. We abbreviate: Wilson-Fisher (WF), the QED${}^{\prime}_{4}$ with
Gross-Neveu-Yukawa (QED${}^{\prime}_{4}$-GNY), Symmetry Preserved (SP),
Symmetry Breaking (SB), and Landau-Ginzburg-Wilson (LGW). For beyond LGW,
(anom.) means due to anomaly, (decf) means due to deconfined-confined phase
transition. (an.dcf.) means due to both anomaly and deconfined-confined
effects.

####  4.2.2 Internal symmetries are dynamically gauged

When the internal symmetries are treated as global symmetries in Sec. 4.2.1,
the ${\rm U}(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$ is the only dynamical
gauge sector, which becomes deconfined only emergent near the quantum critical
region (the white region (0) in Fig. 12, in Fig. 13 (b), and in Fig. 14 and
Fig. 15). The emergent deconfined ${\rm
U}(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$ gauge field near the quantum
critical region is exactly the reason why we name the Gauged Enhanced Quantum
Criticality beyond the Standard Model in our prior work [1].

When internal symmetries are dynamically gauged as they are in our quantum
vacuum, then internal symmetry groups become gauge groups. We have additional
gauge sectors such as Spin(10), ${\rm U}(5)_{\hat{q}=2}$, $\frac{{\rm
SU}(4)_{c}\times({\rm SU}(2)_{\text{L}}\times{\rm
SU}(2)_{\text{R}})}{\mathbb{Z}_{2}}$, and $\frac{{{\rm SU}(3)}\times{{\rm
SU}(2)}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{6}}$, etc. We can ask the
dynamical fates of these gauge theories: confined or deconfined? When the
number of fermions are comparably small as in our quantum vacuum, the RG beta
function computation shows the asymptotic freedom at UV and the confinement at
IR for a non-abelian gauge theory. Other abelian gauge sectors, such as ${\rm
U}(1)_{X_{1}}$, ${\rm U}(1)_{X_{2}}$, and ${\rm U}(1)_{\rm EM}$, can stay
deconfined. We summarize their dynamics in Table 3.

Internal Symmetry Gauged
---
|  | confined sectors | deconfined sectors
| (I)=(V) | $so(10)$ |
| (II) | $su(5)$ | $u(1)_{X_{1}}$
| (III)=(VII) | $su(3)\times su(2)$ | $u(1)_{\text{EM}}$
| (IV)=(VIII) | $su(4)\times su(2)_{\rm L}\times su(2)_{\rm R}$ |
| (VI) | $su(5)$ | $u(1)_{X_{2}}$
(0) | (I)′=(V)′ | $so(10)$ | $u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$
(II)′ | $su(5)$ | $u(1)_{X_{1}}\times u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$
(III)′=(VII)′ | $su(3)\times su(2)$ | $u(1)_{\text{EM}}\times u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$
(IV)′=(VIII)′ | $su(4)\times su(2)_{\rm L}\times su(2)_{\rm R}$ | $u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$
(VI)′ | $su(5)$ | $u(1)_{X_{2}}\times u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$
Table 3: When internal symmetries are dynamically gauged as in our quantum
vacuum, we enlist the confined gauge groups and deconfined gauge groups for
each of the phases in Fig. 14 and Fig. 15. For simplicity, we only write down
the Lie algebra in the lower case. The full Lie group in the capital case can
be read from Fig. 11. The $u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$ only
emerges and deconfines in the quantum critical region (0), including (I)′ to
(VIII)′. The $u(1)_{\text{gauge}}^{{}^{\prime}\text{dark}}$ disappears and
confines in the (I) to (VIII).

When internal symmetries are dynamically gauged, the phase transitions
described in Table 2 also change or upgrade. For example, the Wilson-Fisher
transitions of scalar fields become the Anderson-Higgs transitions of scalar
fields interacting with gauge fields. Some of the QED${}^{\prime}_{4}$
transitions also need to take into account of the nonabelian gauge fields from
Spin(10) or its subgroups, which lead to various QCD${}^{\prime}_{4}$
transitions.

Moreover, once the ordinary internal symmetry (i.e., the 0-symmetry) is
dynamically gauged, the outcome gauge theory can have extended objects (e.g.,
Wilson or ’t Hooft 1d lines) which are the charged objects of the generalized
global symmetries [33] (i.e., here the 1-symmetry). We systematically explore
these generalized global symmetries of SM and GUT in the next Section 5.

##  5 Higher Symmetries of Standard Models and Grand Unifications

Here we point out that once the internal symmetry of SM and GUT ($G_{{\rm
SM}_{q=1,2,3,6}}$, $G_{\text{GG}}$, $G_{\text{PS}_{q^{\prime}=1,2}}$,
$G_{so(10)}$, $\dots$) are dynamically gauged (as they are in the dynamical
gauge theories), there are dynamical Wilson or ’t Hooft lines as charged
objects charged under the generalized global symmetries [33]. Pioneer works
[75, 76] have pointed out the Wilson line spectrum differences between
different versions of $G_{{\rm SM}_{q}}$ for $q=1,2,3,6$. Other prior works
also study the higher symmetries for these $G_{{\rm SM}_{q}}$ [77, 78, 79]. In
comparison, our present work will include not only the higher symmetries of
SMq in [77, 78, 79], but also other pertinent GUT models.

Recall [33], when the _charged objects_ are 1d line operators (say along a
closed curve ${\gamma^{1}}$), the _charge operators_ (also known as the
_symmetry generators_ or _symmetry defects_) are codimension-2 topological
operators (as 2d surface operators say on a closed surface $\Sigma^{2}$, in a
4d spacetime). As an schematic example, when the gauge group is abelian, the
path integral expectation value of the link configuration between the 1d line
and 2d surface operators with a linking number ${\rm
Lk}(\Sigma^{2},{\gamma^{1}})$ evaluated on a closed 4-manifold $M^{4}$ is
schematically given by:

$\displaystyle\langle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int_{\Sigma^{2}}B_{\text{charge}})\cdot\exp(\hskip
1.0pt\mathrm{i}\hskip
1.0ptq\oint_{\gamma^{1}}A_{\text{charged}})\rangle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0ptq\theta{\rm
Lk}(\Sigma^{2},{\gamma^{1}})}\cdot\langle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq\oint_{\gamma^{1}}A_{\text{charged}})\rangle\Big{|}_{M^{4}}.$ (5.1)

The expectation value is topological independent from the continuous
deformation of the codimension-2 topological operator of $B$, as long as it
does not cross the charged object of $A$. Hence this explains the meaning of
the name _topological operator_ : the system and the topological operator of
$B$ do _not_ have to be gapped, but its correlator is the same independent
from the topological deformation. However, we need to include
$\langle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq\oint_{\gamma^{1}}A_{\text{charged}})\rangle$ on the right hand side
whenever this operator $A$ is non-topological. If $A$ describes the 1d Wilson
line (or denoted as 1-line), the topological 2d surface operator of $B$ is
often called the topological Gukov-Witten 2-surface operator [80, 81]. See
further in-depth discussions on the topological operators in [42].

This expression is also the analogous Ward identity for the generalized 1-form
global symmetry, or simply denoted as the 1-symmetry. If we normalize the
expectation value properly, the proportionality ($\propto$) becomes the
equality ($=$) to the statistical Berry phase $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0ptq\theta{\rm Lk}(\Sigma^{2},{\gamma^{1}})}$. The
abelian phase $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0ptq\theta}$ means that we are focusing on the abelian 1-symmetry:
$\bullet$ When it is an abelian U(1) 1-symmetry denoted as ${\rm U}(1)_{[1]}$,
then $q\in\mathbb{Z}$ and $\theta\in[0,2\pi)$.
$\bullet$ When it is an abelian $\mathbb{Z}_{\rm N}$ 1-symmetry denoted as
$\mathbb{Z}_{{\rm N},[1]}$, then $q\in\mathbb{Z}_{\rm N}$ and
$\theta=\frac{2\pi k}{{\rm N}}$ with $k\in\mathbb{Z}_{\rm N}$.
For a gauge theory of gauge group $G_{g}$, the electric 1-symmetry is
associated with the unbroken center subgroup $Z(G_{g})$, the magnetic
1-symmetry is associated with the unbroken Pontryagin dual group of the first
homotopy group:
$\pi_{1}(G_{g})^{\vee}\equiv\operatorname{Hom}(\pi_{1}(G_{g}),{\rm U}(1))$.
Here are some familiar examples (results summarized in Table 4):

  1. 1.

For a 4d pure U(1) gauge theory, we have the electric 1-symmetry ${\rm
U}(1)_{[1]}^{e}$ and the magnetic 1-symmetry ${\rm U}(1)_{[1]}^{m}$, whose
1-symmetry measurements are characterized by the following two expectation
values:

$\displaystyle{\rm U}(1)_{[1]}^{e}$ $\displaystyle:$
$\displaystyle\langle\exp(\hskip 1.0pt\mathrm{i}\hskip