Ultra Unification

Juven Wang<EMAIL_ADDRESS>http://idear.info/

Center of Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138, USA

Strong, electromagnetic, and weak forces were unified in the Standard Model
(SM) with spontaneous gauge _symmetry breaking_. These forces were further
conjectured to be unified in a simple Lie group gauge interaction in the Grand
Unification (GUT). In this work, we propose a theory beyond the SM and GUT by
adding new gapped topological sectors consistent with the nonperturbative
global anomaly matching and cobordism constraints (especially from the baryon
minus lepton number ${\bf B}-{\bf L}$ and the mixed gauge-gravitational
anomaly). Gapped topological sectors are constructed via _symmetry extension_
, whose low energy contains unitary topological quantum field theories
(TQFTs): either 3+1d non-invertible TQFT (long-range entangled gapped phase),
or 4+1d invertible or non-invertible TQFT (short-range or long-range entangled
gapped phase), or right-handed neutrinos, or their combinations. We propose
that a new high-energy physics frontier beyond the conventional 0d particle
physics relies on the new Topological Force and Topological Matter including
gapped extended objects (gapped 1d line and 2d surface operators or defects,
etc., whose open ends carry deconfined fractionalized anyonic particle or
string excitations). Physical characterizations of these gapped extended
objects require the mathematical theories of cohomology, cobordism, or
category. Although weaker than the weak force, Topological Force is infinite-
range or long-range which does not decay in the distance, and mediates between
the linked worldvolume trajectories via fractional or categorical statistical
interactions.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Logic lead to Ultra Unification
    1. 2.1 Assumptions
    2. 2.2 Anomaly and Cobordism Constraints
    3. 2.3 Consequences lead to Ultra Unification
    4. 2.4 Symmetry Breaking vs Symmetry Extension: Dirac or Majorana masses vs Topological mass
    5. 2.5 Gauging a discrete Baryon B, Lepton L, and Electroweak Hypercharge $Y$
    6. 2.6 Topological Sector and Topological Force
  3. 3 Ultra Unification Path Integral
    1. 3.1 Standard Model and the $su(5)$ GUT Path Integral coupled to $X\equiv 5({\mathbf{B}-\mathbf{L}})-4Y$
    2. 3.2 Topological Sector and TQFT Path Integral coupled to $X\equiv 5({\mathbf{B}-\mathbf{L}})-4Y$
      1. 3.2.1 Symmetry extension $[\mathbb{Z}_{2}]\to{\rm Spin}\times{\mathbb{Z}_{4,X}}\to{\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$ and a 4d $[\mathbb{Z}_{2}]$ gauge theory
    3. 3.3 General Principle
    4. 3.4 Detect Topological Forces/Sectors and the Essence
    5. 3.5 Summary of Ultra Unification in Drawings
  4. 4 Acknowledgements and Bibliography

“Die einzig wahre Unsterblichkeit liegt in den eigenen Kindern.”
“The only true immortality lies in one’s children.”

“Meine Arbeit ist Ihnen gewidmet. Sie können dies sicherlich in Ihrem Herzen
fühlen.”
“My work is dedicated to you. You can surely feel this in your heart.”

Intermezzo in A major. Andante teneramente, Six Pieces for Piano, Op. 118
Johannes Brahms in 1893

##  1 Introduction and Summary

Unification is a central theme in theoretical physics. In 1864-1865, Maxwell
[1] unified the electricity and magnetism into the electrodynamics theory,
where the derived electromagnetic wave manifests the light phenomena. In
1961-1967, Glashow-Salam-Weinberg (GSW) [2, 3, 4, 5] made landmark
contributions to the electroweak theory of the unified electromagnetic and
weak forces between elementary particles, including the prediction of the weak
neutral current. The GSW theory together with the strong force is now known as
the Standard Model (SM), which is verified to be theoretically and
experimentally essential to describe the subatomic high energy physics (HEP).
In 1974, Georgi-Glashow [6, 7] hypothesized that at a higher energy, the three
gauge interactions of the SM would be merged into a single electronuclear
force under a simple Lie group gauge theory, known as the Grand Unification or
Grand Unified Theory (GUT).

Figure 1: Unification of forces and interactions in fundamental physics. In
fact, the proposed Ultra Unification can still play a role of unification even
if Grand Unification (GUT) is not favored by Nature or verified by
experiments. Namely, based on the anomaly and cobordism constraint, we can
still propose the Standard Model (SM) + gapped Topological Sector +
Topological Force in a consistent way without GUT. The _dark_ gray area
schematically shows the possible energy scales for various GUT scenarios, such
as the $su(5),so(10),\dots,so(18)$ GUT around $10^{16}$G$e$V. The _light_ gray
area schematically suggests that the possible energy gap $\Delta_{\rm TQFT}$
for Topological Sector can range from as low energy as the SM, to as high
energy to somewhere within the GUT scales (e.g., below the $so(10)$ GUT
scale). The dashed lines mean hypothetical unifications that have not yet been
confirmed by experiments. Forces are arranged from the strongest to the
weakest (horizontally from the left to the right) in the Higgs vacuum.

In this work, follow our previous investigations based on nonperturbative
global anomalies and cobordism constraints [8, 9, 10],111Related global
anomalies and cobordism constraints on SM, GUT, and BSM are explored also in
[11, 12, 13, 14]. Freed gave a generalized cohomology description of the
$su(5)$ GUT anomalies [11]. Garcia-Etxebarria-Montero [12] and Davighi-
Gripaios-Lohitsiri [14] used Atiyah-Hirzebruch spectral sequence (AHSS) to
compute the classification of global anomalies. The author with Wen [13] and
with Wan [8] used Adams spectral sequence (ASS) [15], Thom-Madsen-Tillmann
spectra [16, 17], and Freed-Hopkins theorem [18] to obtain the classification
of all invertible quantum anomalies. In addition, there in Ref. [8], we can
also fully characterize the $d$d ’t Hooft anomaly [19] of global symmetry $G$
as $(d+1)$d cobordism invariants, precisely as topological terms of cohomology
classes and fermionic topological invariants. The cobordism invariants can be
read from the Adams chart in Ref. [8]. Thus we focus on employing Ref. [8]
result. By classifying all the invertible quantum anomalies, we must include
• all _local anomalies_ (_perturbative anomalies_):
captured by perturbative Feynman diagram loop calculations, classified by the
integer $\mathbb{Z}$ classes (the free classes).
e.g., Adler-Bell-Jackiw (ABJ) anomalies [20, 21], perturbative gravitational
anomalies[22]. • all _global anomalies_ (_nonperturbative anomalies_):
classified by finite abelian groups as a product of $\mathbb{Z}_{n}$ (the
torsion classes) for some positive integer $n$.
e.g., Witten SU(2) [23] and the new SU(2) anomalies [24], global gravitational
anomalies [25]. More examples of $d$d anomalies characterized by $(d+1)$d
cobordism invariants can be found in [26, 27].  we propose an Ultra
Unification that a new Topological Force comes into a theme of unification
joining with three known fundamental forces and other hypothetical GUT forces.
(See Figure 1.) More concretely, there is a new gapped Topological Sector
whose underlying dynamical gauge interactions are the Topological Forces. In a
modern perspective, we should view the SM and GUT all as effective field
theories suitable below certain energy scales. Likewise, Ultra Unification
should be viewed as an effective field theory which contains SM and GUT but
also additional gapped Topological Sectors with low energy Lorentz invariant
unitary topological quantum field theories (TQFTs) of Schwarz type (which is
the 4d analog of the 3d Chern-Simons-Witten theories [28, 29, 30]).
Topological Force and the gapped Topological Sector here have specific
physical and mathematical meanings, which we will clarify in Sec. 2.6. Before
digging into Topological Sector, we should state the assumptions and the logic
lead to the assertion of Ultra Unification, in Sec. 2.222Conventions: We
follow the conventions of Ref. [8, 9, 10]. We denote $n$d for $n$-dimensional
spacetime. We also follow the modern condensed matter terminology on the
interacting phases of quantum matter [31, 32]. For example, a long-range
entangled gapped topological phase whose low energy describes the
noninvertible TQFT is known as an intrinsic topological order, which includes
examples of fractional quantum Hall states. A short-range entangled gapped
topological phase protected by some global symmetry $G$ whose low energy
describes the invertible TQFT is known as a symmetry-protected topological
state (SPTs), which includes examples of topological insulators and
topological superconductors [33, 34].
$\bullet$ By a noninvertible TQFT, it means that the absolute value partition
function $|{\bf Z}(M)|\neq 1$ on a generic spacetime manifold $M$ with
nontrivial topology (e.g., cycles or homology classes).
$\bullet$ By an invertible TQFT (iTQFT), it means that the absolute value
partition function $|{\bf Z}(M)|=1$ on any spacetime manifold $M$ with any
topology. Thus ${\bf Z}(M)$ implies the existence of an inverted phase ${\bf
Z}^{\prime}(M)\equiv{\bf Z}(M)^{-1}$ which defines another iTQFT ${\bf
Z}^{\prime}(M)$ that can cancel with the original iTQFT ${\bf Z}(M)$, as the
stacking of two iTQFTs become a trivial vacuum ${\bf Z}(M)\cdot{\bf
Z}^{\prime}(M)=1$ for any $M$.
$\bullet$ By a trivial gapped vacuum with no or trivial TQFT, it means that
the partition function ${\bf Z}(M)=1$ on any spacetime manifold $M$ with any
topology.

##  2 Logic lead to Ultra Unification

###  2.1 Assumptions

Our logic lead to Ultra Unification starts with the three Assumptions mostly
given by Nature and broadly confirmed by experiments:

  1. 1.

Standard Model gauge group $G_{\text{SM}_{q}}$: The Standard Model gauge
theory has a local Lie algebra $su(3)\times su(2)\times u(1)$, but the global
structure of Lie group $G_{\text{SM}_{q}}$ has four versions

$\displaystyle G_{\text{SM}_{q}}\equiv\frac{{\rm SU}(3)\times{\rm
SU}(2)\times{\rm U}(1)}{\mathbb{Z}_{q}},\quad\text{ with }q=1,2,3,6.$ (2.1)

All the quantum numbers of quarks and leptons are compatible with the
representations of any version of $q=1,2,3,6$. To confirm which version is
used by Nature, it requires the experimental tests on extended objects such as
1d line or 2d surface operators (see recent expositions in [35, 36, 37, 38]).

SU(5) and Spin(10) gauge group: Conventionally, people write the Georgi-
Glashow model [6] as the $su(5)$ GUT and Fritzsch-Minkowski model [7] as the
$so(10)$ GUT because they have the local Lie algebra $su(5)$ and $so(10)$
respectively. However, they have the precise global Lie group SU(5) and
Spin(10) respectively.

  2. 2.

Observed 15 Weyl fermions per generation by experiments: So far the HEP
experiments only confirmed the 15 Weyl fermions per generation of SM.333Here
Weyl fermions are spacetime Weyl spinors, which is ${\bf 2}_{L}$ of ${{\rm
Spin}(3,1)}=\rm{SL}(2,{\mathbb{C}})$ with a complex representation in the
Lorentz signature. On the other hand, the Weyl spinor is ${\bf 2}_{L}$ of
${\rm Spin}(4)={\rm SU}(2)_{L}\times{\rm SU}(2)_{R}$ with a pseudoreal
representation in the Euclidean signature. The experimentalists have _not_ yet
confirmed the existence of 16th Weyl fermion for each generation. The 16th
Weyl fermion is also known as the right-handed neutrino $\nu_{R}$ or the
sterile neutrino. Namely, given the number of generation (or family)
$N_{\text{gen}}=3$, we have $15\times N_{\text{gen}}=45$ Weyl fermions
confirmed in the SM. Similarly, we consider $15\times N_{\text{gen}}=45$ Weyl
fermions applicable to the $su(5)$ GUT.444However, the $so(10)$ GUT requires
16 Weyl fermions per generation due to the fermions sit at the 16 of the
Spin(10). This fact is used to argue the possibility of topological quantum
phase transition between the energy scale of the 15n Weyl-fermion $su(5)$ GUT
and 16n Weyl-fermion $so(10)$ GUT in Ref. [10]. (The quantum numbers and
representations of the elementary particles in SM and GUT can be found in
Table 1 and 2 of [9].)

  3. 3.

A variant discrete Baryon minus Lepton number (${\mathbf{B}-\mathbf{L}}$) is
preserved at high energy: We hypothesize a discrete ${X}$ symmetry, which is a
modified version of $({\mathbf{B}-\mathbf{L}})$ number up to some electroweak
hypercharge $Y$ [39] is preserved (preserved at least at a higher
energy):555Follow [8, 9, 10], we choose the convention that the ${\rm
U}(1)_{\rm{EM}}$ electromagnetic charge is $Q_{\rm{EM}}=T_{3}+Y$. The ${\rm
U}(1)_{\rm{EM}}$ is the unbroken (not Higgsed) electromagnetic gauge symmetry
and $T_{3}=\frac{1}{2}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ is a generator
of SU(2)${}_{\text{weak}}$.

$\displaystyle X$ $\displaystyle\equiv$ $\displaystyle
5({\mathbf{B}-\mathbf{L}})-4Y.$ (2.2)

The importance of this discrete symmetry $\mathbb{Z}_{4,X}$ (as a mod 4
symmetry of ${\rm U}(1)_{X}$), in the context of global anomalies for SM and
GUT is emphasized by Garcia-Etxebarria-Montero [12].

It is easy to check that (e.g., see the Table 1 and 2 of [9]):
$\bullet$ all the particles from $\bar{\bf 5}$ of $su(5)$ GUT has a ${\rm
U}(1)_{X}$ charge $-3$.
$\bullet$ all the particles from 10 of $su(5)$ GUT has a ${\rm U}(1)_{X}$
charge $+1$.
$\bullet$ the singlet right-handed neutrino (if any) is in 1 of $su(5)$ GUT
with a ${\rm U}(1)_{X}$ charge $+5$.
$\bullet$ all the fermions of SM has a $\mathbb{Z}_{4,X}$ charge $+1$.
$\bullet$ the electroweak Higgs $\phi$ has a ${\rm U}(1)_{X}$ charge $-2$,
thus a $\mathbb{Z}_{4,X}$ charge $+2$.
The $\mathbb{Z}_{4,X}$ also contains the fermion parity $\mathbb{Z}_{2}^{F}$
(whose operator $(-1)^{F}$ gives $(-1)$ to all fermions) as a normal subgroup:

$\displaystyle{\rm U}(1)_{X}\supset\mathbb{Z}_{4,X}\supset\mathbb{Z}_{2}^{F}.$
(2.3)

By looking at these consistent $\mathbb{Z}_{4,X}$ quantum number of SM
particles, it is natural to hypothesize the discrete $X$ symmetry plays an
important role at a higher energy.

In Sec. 2.2, we review the anomaly and cobordism constraints given in [8, 9,
10]. Readers can freely skip the technical discussions on anomalies, and
directly go to the final logic step lead to Ultra Unification in Sec. 2.3.

###  2.2 Anomaly and Cobordism Constraints

Based on the three mild and widely accepted assumptions listed in Sec. 2.1, we
then impose the constraints from all invertible quantum anomalies via the
cobordism calculation on SM and GUT models. The purpose is to check the
consistency of the 15n Weyl fermion SM and GUT models:

Check: Perturbative local and nonperturbative global anomalies via cobordism.

The classification of $d$d ’t Hooft anomalies of global symmetries $G$ is
equivalent to the classification of $(d+1)$d invertible TQFTs with
$G$-symmetry defined on a $G$-structure manifold,666For the QFT setup, we only
require the category of smooth, differentiable, and triangulable manifolds.
given by the cobordism group data $\Omega^{d}_{G}\equiv\mathrm{TP}_{d}(G)$
defined in Freed-Hopkins [18]. The symmetry

$\displaystyle
G\equiv({\frac{{G_{\text{spacetime}}}\ltimes{{G}_{\text{internal}}}}{{N_{\text{shared}}}}})\equiv{{G_{\text{spacetime}}}\ltimes_{{N_{\text{shared}}}}{{G}_{\text{internal}}}}$
(2.4)

contains the spacetime symmetry ${G_{\text{spacetime}}}$ and the internal
symmetry ${{G}_{\text{internal}}}$.777$\bullet$ The ${G_{\text{spacetime}}}$
is the spacetime symmetry, such as the spacetime rotational symmetry ${\rm
SO}\equiv{\rm SO}(d)$ or the fermionic graded spacetime rotational Spin group
symmetry ${\rm Spin}\equiv{\rm Spin}(d)$.
$\bullet$ The ${{G}_{\text{internal}}}$ is the internal symmetry, such as
${{G}_{\text{internal}}}$ in the SM as $G_{\text{SM}_{q}}\equiv\frac{{\rm
SU}(3)\times{\rm SU}(2)\times{\rm U}(1)}{\mathbb{Z}_{q}}$ with $q=1,2,3,6.$ We
also have ${{G}_{\text{internal}}}={\rm SU}(5)$ in the SU(5) GUT and
${{G}_{\text{internal}}}={\rm Spin}(10)$ in the Spin(10) GUT. The
${N_{\text{shared}}}$ is the shared common normal subgroup symmetry between
${G_{\text{spacetime}}}$ and ${{G}_{\text{internal}}}$. The “semi-direct
product $\ltimes$” extension is due to a group extension from
${{G}_{\text{internal}}}$ by ${G_{\text{spacetime}}}$. For a trivial
extension, the semi-direct “$\ltimes$” becomes a direct product “$\times$.”
Our perspective is that:
$\bullet$ We can treat the spacetime-internal $G$ as a global symmetry, and we
view the anomaly associated with $G$ as ’t Hooft anomalies [19] of $G$
symmetry.
$\bullet$ Then, we can ask all obstructions to dynamically gauging the
${{G}_{\text{internal}}}$ as a gauge group, which give rise to all the
dynamical gauge anomaly cancellation conditions that any consistent gauge
theory must obey.
To proceed, we follow the results of [8, 9, 40], the relevant total spacetime-
internal symmetry $G$ for the SMq is $G={\rm
Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm SM}_{q}}$
with $q=1,2,3,6$, and for the $su(5)$ GUT is $G={\rm
Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times{\rm SU}(5)$. Only
the $q=6$ case of ${\rm SM}_{6}$ can be embedded into the $su(5)$ GUT.

##### Standard Models

Ref. [8] considers the classification of $G$-anomalies in 4d given by the
$d$=5 cobordism group $\Omega^{d}_{G}\equiv\mathrm{TP}_{d}(G)$ for all
Standard Models of ${\rm SM}_{q}$ with an extra discrete $\mathbb{Z}_{4,X}$
symmetry:

$\displaystyle\mathrm{TP}_{d=5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm
SM}_{q}})=\left\\{\begin{array}[]{ll}\mathbb{Z}^{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}\times\mathbb{Z}_{16},&\quad
q=1,3.\\\
\mathbb{Z}^{5}\times\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{16},&\quad
q=2,6.\end{array}\right.$ (2.7)

Here we summarize the anomaly classification for the Standard Model $G_{{\rm
SM}_{q}}\equiv\frac{{\rm SU}(3)\times{\rm SU}(2)\times{\rm
U}(1)_{Y}}{\mathbb{Z}_{q}}$ obtained in [8, 9, 40]. Below we write the 4d
anomalies in terms of the 5d cobordism invariants or invertible topological
quantum field theories (iTQFTs). For perturbative local 4d anomalies, we can
also write them customarily as the 6d anomaly polynomials:888Here we follow
the conventions of [8, 9, 10].
$\bullet$ We can characterize anomalies via (perturbative) _local_ anomalies
or (nonperturbative) _global_ anomalies.
$\bullet$ We can also characterize anomalies via their induced fields: _pure
gauge_ , _mixed gauge-gravity_ , or _gravitational_ anomalies.
$\bullet$ The $c_{j}(G)$ is the $j$th Chern class of the associated vector
bundle of the principal $G$-bundle.
$\bullet$ We will use $\text{CS}_{2n-1}^{V}$ to denote the Chern-Simons
$2n-1$-form for the Chern class (if $V$ is a complex vector bundle) or the
Pontryagin class (if $V$ is a real vector bundle). The relation between the
Chern-Simons form and the Chern class is $c_{n}(V)=\hskip
1.0pt\mathrm{d}\text{CS}_{2n-1}^{V}$ where the $\hskip 1.0pt\mathrm{d}$ is the
exterior differential and the $c_{n}(V)$ is regarded as a closed differential
form in de Rham cohomology.
$\bullet$ The $\mu$ is the 3d Rokhlin invariant. If $\partial M^{4}=M^{3}$,
then $\mu(M^{3})=(\frac{\sigma-{\rm F}\cdot{\rm F}}{8})(M^{4})$, thus
$\mu(\text{PD}(c_{1}({\rm U}(1))))$ is related to $\frac{c_{1}({\rm
U}(1))(\sigma-{\rm F}\cdot{\rm F})}{8}$. Here $\cdot$ is the intersection form
of $M^{4}$. The ${\rm F}$ is the characteristic 2-surface in a 4-manifold
$M^{4}$, it obeys the condition ${\rm F}\cdot x=x\cdot x\mod 2$ for all
$x\in\operatorname{H}_{2}(M^{4},\mathbb{Z})$. By the Freedman-Kirby theorem:
$(\frac{\sigma-{\rm F}\cdot{\rm F}}{8})(M^{4})=\text{Arf}(M^{4},{\rm F})\mod
2$.
$\bullet$ The PD is defined as the Poincaré dual.
$\bullet$ The $\tilde{\eta}$ is a mod 2 index of 1d Dirac operator as a
cobordism invariant of $\Omega_{1}^{{\rm Spin}}=\mathbb{Z}_{2}$.
$\bullet$ The ${\eta}^{\prime}$ is a mod 4 index of 1d Dirac operator as a
cobordism invariant of $\Omega_{1}^{{\rm
Spin}\times\mathbb{Z}_{4}}=\mathbb{Z}_{4}$.
$\bullet$ The Arf invariant [41] is a 2d cobordism invariant of
$\Omega_{2}^{{\rm Spin}}=\mathbb{Z}_{2}$. The Arf appears to be the low energy
iTQFT of a 1+1d Kitaev fermionic chain [42], whose boundary hosts a single
0+1d real Majorana zero mode on each of open ends.
$\bullet$ We use the notation “$\sim$” to indicate the two sides are equal in
that dimension up to a total derivative term.
$\bullet$ Because of the $\mathbb{Z}_{4,X}\supset\mathbb{Z}_{2}^{F}$, we have
$\displaystyle
0\to\mathbb{Z}_{2}^{F}\to\mathbb{Z}_{4,X}\to\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}=\mathbb{Z}_{2}\to
0.$ (2.8) We define the cohomology classes of background gauge field ${\cal
A}_{{\mathbb{Z}_{4}}}\in\operatorname{H}^{1}(M,{\mathbb{Z}_{4,X}})$ is the
generator from the cohomology group
$\operatorname{H}^{1}(\mathrm{B}{\mathbb{Z}_{4,X}},\mathbb{Z}_{4})$ of ${{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}}$. We also define the
background gauge field $({\cal A}_{{\mathbb{Z}_{2}}})\equiv({\cal
A}_{{\mathbb{Z}_{4}}})\mod 2$, where ${\cal
A}_{{\mathbb{Z}_{2}}}\in\operatorname{H}^{1}(M,\mathbb{Z}_{2})$ is the
generator from
$\operatorname{H}^{1}(\mathrm{B}({\mathbb{Z}_{4,X}}/\mathbb{Z}_{2}^{F}),\mathbb{Z}_{2})$
of ${{\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}}$.

  1. 1.

U(1)${}_{Y}^{3}$: 4d $\mathbb{Z}$ class local pure gauge anomaly from 5d
$\text{CS}_{1}^{{\rm U}(1)}c_{1}({\rm U}(1))^{2}$ and 6d $c_{1}({\rm
U}(1))^{3}$.

  2. 2.

U(1)Y-SU(2)2: 4d $\mathbb{Z}$ class local pure gauge anomaly from 5d
$\text{CS}_{1}^{{\rm U}(1)}c_{2}({\rm SU}(2))$ and 6d $c_{1}({\rm
U}(1))c_{2}({\rm SU}(2))$.

  3. 3.

U(1)Y-SU(3)2: 4d $\mathbb{Z}$ class local pure gauge anomaly from 5d
$\text{CS}_{1}^{{\rm U}(1)}c_{2}({\rm SU}(3))$ and 6d $c_{1}({\rm
U}(1))c_{2}({\rm SU}(3))$.

  4. 4.

U(1)Y-(gravity)2: 4d $\mathbb{Z}$ class local mixed gauge-gravity anomaly from
5d $\mu(\text{PD}(c_{1}({\rm U}(1))))$ and 6d $\frac{c_{1}({\rm
U}(1))(\sigma-{\rm F}\cdot{\rm F})}{8}$.

  5. 5.

SU(3)3: 4d $\mathbb{Z}$ class local pure gauge anomaly from 5d
$\frac{1}{2}{\text{CS}_{5}^{{\rm SU}(3)}}$ and 6d $\frac{1}{2}{c_{3}({\rm
SU}(3))}$.

  6. 6.

Witten SU(2) anomaly: 4d $\mathbb{Z}_{2}$ class global mixed gauge-gravity
anomaly from 5d $c_{2}({\rm SU}(2)\tilde{\eta}\equiv\tilde{\eta}({\rm
PD}(c_{2}({\rm SU}(2))))$ and 6d $c_{2}({\rm
SU}(2))\text{Arf}\equiv\text{Arf}({\rm PD}(c_{2}({\rm SU}(2))))$. However, the
original Witten $\mathbb{Z}_{2}$ global anomaly becomes _mutated_ :
$\bullet$ When $q=1$ or $3$, Witten anomaly _mutated_ to a 4d $\mathbb{Z}_{4}$
class global mixed gauge-gravity anomaly, given by a 5d cobordism invariant
$c_{2}({\rm SU}(2))\eta^{\prime}\equiv\eta^{\prime}({\rm PD}(c_{2}({\rm
SU}(2))))$. There is a short exact sequence
$0\to\mathbb{Z}_{2}\to\mathbb{Z}_{4}\to\mathbb{Z}_{2}\to 0$ where the original
Witten anomaly $c_{2}({\rm SU}(2)\tilde{\eta}$ sits at the $\mathbb{Z}_{2}$
quotient and the $({\cal A}_{{\mathbb{Z}_{2}}})c_{2}({\rm SU}(2))$ sits at the
$\mathbb{Z}_{2}$ normal subgroup, while the mutated $c_{2}({\rm
SU}(2))\eta^{\prime}$ sits at the $\mathbb{Z}_{4}$ total group.
$\bullet$ When $q=2$ or $6$, Witten anomaly _mutated_ to become part of 4d
$\mathbb{Z}$ class local mixed gauge-gravity anomaly (firstly explained in
[43]), given by a 5d cobordism invariant $\frac{1}{2}\text{CS}_{1}^{{\rm
U}(2)}c_{2}({\rm U}(2))\sim\frac{1}{2}c_{1}({\rm U}(2))\text{CS}_{3}^{{\rm
U}(2)}$. There is a short exact sequence
$0\to\mathbb{Z}\overset{2}{\to}\mathbb{Z}\to\mathbb{Z}_{2}\to 0$ where the
original Witten anomaly $c_{2}({\rm SU}(2)\tilde{\eta}$ sits at the
$\mathbb{Z}_{2}$ quotient and the $\text{CS}_{1}^{{\rm U}(1)}c_{2}({\rm
SU}(2))\sim c_{1}({\rm U}(1))\text{CS}_{3}^{{\rm SU}(2)}$ sits at the
$\mathbb{Z}$ normal subgroup, while the mutated
$\frac{1}{2}\text{CS}_{1}^{{\rm U}(2)}c_{2}({\rm
U}(2))\sim\frac{1}{2}c_{1}({\rm U}(2))\text{CS}_{3}^{{\rm U}(2)}$ sits at the
$\mathbb{Z}$ total group.

  7. 7.

$({\cal A}_{{\mathbb{Z}_{2}}})c_{2}({\rm SU}(2))$: 4d $\mathbb{Z}_{2}$ global
gauge anomaly given by a 5d cobordism invariant $({\cal
A}_{{\mathbb{Z}_{4}}}\mod 2)c_{2}({\rm SU}(2))$.
$\bullet$ When $q=1$ or $3$, as explained earlier, it fuses with Witten
anomaly to become a $\mathbb{Z}_{4}$ global anomaly.
$\bullet$ When $q=2$ or $6$, this $\mathbb{Z}_{2}$ global anomaly occurs.

  8. 8.

$({\cal A}_{{\mathbb{Z}_{2}}})c_{2}({\rm SU}(3))$: 4d $\mathbb{Z}_{2}$ global
gauge anomaly.
$\bullet$ When $q=1$ or $2$, this is given by a 5d cobordism invariant $({\cal
A}_{{\mathbb{Z}_{4}}}\mod 2)c_{2}({\rm SU}(3))$.
$\bullet$ When $q=3$ or $6$, this is given by a 5d cobordism invariant $({\cal
A}_{{\mathbb{Z}_{4}}}\mod 2)c_{2}({\rm U}(3))$.

  9. 9.

$c_{1}({\rm U}(1))^{2}\eta^{\prime}$: 4d $\mathbb{Z}_{4}$ global mixed gauge-
gravity anomaly.
$\bullet$ When $q=1$, this is given by a 5d cobordism invariant $c_{1}({\rm
U}(1))^{2}\eta^{\prime}\equiv\eta^{\prime}({\rm PD}(c_{1}({\rm U}(1))^{2}))$.
$\bullet$ When $q=2$, this is given by a 5d cobordism invariant $c_{1}({\rm
U}(2))^{2}\eta^{\prime}\equiv\eta^{\prime}({\rm PD}(c_{1}({\rm U}(2))^{2}))$.
$\bullet$ When $q=3$, this is given by a 5d cobordism invariant $c_{1}({\rm
U}(3))^{2}\eta^{\prime}\equiv\eta^{\prime}({\rm PD}(c_{1}({\rm U}(3))^{2}))$.
$\bullet$ When $q=6$, this is given by a 5d cobordism invariant $c_{1}({\rm
U}(2))^{2}\eta^{\prime}\sim c_{1}({\rm U}(3))^{2}\eta^{\prime}$.

  10. 10.

$\eta(\text{PD}({\cal A}_{{\mathbb{Z}_{2}}}))$: 4d $\mathbb{Z}_{16}$ global
mixed gauge-gravity anomaly is given by a 5d cobordism invariant
$\eta(\text{PD}({\cal A}_{{\mathbb{Z}_{4}}}\mod 2))$. The
$\eta(\text{PD}({\cal A}_{{\mathbb{Z}_{2}}}))$ is the value of Atiyah-Patodi-
Singer (APS [44]) eta invariant ${\eta}\in\mathbb{Z}_{16}$ on the Poincaré
dual (PD) submanifold of ${\cal A}_{{\mathbb{Z}_{2}}}$. It descends from the
bordism group calculation of $\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$.
Many previous works had also explored these 5d $\mathbb{Z}_{16}$ fermionic
invariants [45, 46, 47]. The relation between 5d $\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))$ and 4d $\eta$ is given by the Smith homomorphism [48,
49].

##### Gerogi-Glashow $su(5)$ Grand Unification

Ref. [8, 40] consider the classification of $G$-anomalies in 4d given by the
$d$=5 cobordism group $\Omega^{d}_{G}\equiv\mathrm{TP}_{d}(G)$ for the $su(5)$
GUT with an extra discrete $\mathbb{Z}_{4,X}$ symmetry:

$\displaystyle\mathrm{TP}_{d=5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times{\rm
SU}(5))=\mathbb{Z}\times\mathbb{Z}_{2}\times\mathbb{Z}_{16}.$ (2.9)

Below we also write down the 4d anomalies in terms of the 5d cobordism
invariants or iTQFTs, or the 6d anomaly polynomials:

  1. 1.

SU(5)3: 4d $\mathbb{Z}$ class local gauge anomaly. It is given by a 5d
cobordism invariant $\frac{1}{2}\text{CS}_{5}^{{\rm SU}(5)}$, or more
precisely $\frac{1}{2}({({\cal A}_{{\mathbb{Z}_{2}}})^{2}\text{CS}_{3}^{{\rm
SU}(5)}+\text{CS}_{5}^{{\rm SU}(5)}})$ including the ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ contribution.

  2. 2.

$({\cal A}_{{\mathbb{Z}_{2}}})c_{2}({\rm SU}(5))$: 4d $\mathbb{Z}_{2}$ class
global gauge anomaly. It is given by a 5d cobordism invariant $({\cal
A}_{{\mathbb{Z}_{4}}}\mod 2)c_{2}({\rm SU}(5))$.

  3. 3.

$\eta(\text{PD}({\cal A}_{{\mathbb{Z}_{2}}}))$: 4d $\mathbb{Z}_{16}$ global
mixed gauge-gravity anomaly is given by a 5d cobordism invariant
$\eta(\text{PD}({\cal A}_{{\mathbb{Z}_{4}}}\mod 2))$. This is again the same
$\mathbb{Z}_{16}$ global anomaly from $\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$.

###  2.3 Consequences lead to Ultra Unification

Ref. [9, 10] checked all the above local and global anomalies enlisted in Sec.
2.2 vanished for the SMq with $q=1,2,3,6$ and for the $su(5)$ GUT with 15 Weyl
fermions per generation, except the 4d $\mathbb{Z}_{16}$ class global anomaly
(the mixed gauge-gravitational anomaly probed by the discrete
$\mathbb{Z}_{4,X}$ symmetry and fermionic spacetime rotational symmetry ${\rm
Spin}(d)$ background fields) may _not_ be completely matched.999We should
briefly compare the perspectives of Garcia-Etxebarria-Montero [12], Davighi-
Gripaios-Lohitsiri [14], and our previous result [9, 10], and those with Wen
[13] or with Wan [8]. In terms of the relevancy to the 4d anomalies of SM and
$su(5)$ GUT given the spacetime-internal symmetry $G$,
$\bullet$ Garcia-Etxebarria-Montero [12] checked: $G={\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4},\;{\rm Spin}\times{\rm
SU}(n),\;{\rm Spin}\times{\rm Spin}(n)$.
$\bullet$ Wang-Wen [13] checked: ${G=\frac{{\rm Spin}\times{\rm
Spin}(n)}{{\mathbb{Z}_{2}^{F}}},\frac{{\rm Spin}\times{\rm
Spin}(10)}{{\mathbb{Z}_{2}^{F}}},\;{\rm Spin}\times{\rm SU}(5)}$.
$\bullet$ Davighi-Gripaios-Lohitsiri [14] checked: $G={\rm
Spin}\times{G_{\text{SM}_{q}}}$, ${{\rm Spin}\times{\rm Spin}(n)}$, and other
GUTs.
$\bullet$ Wan-Wang [8, 40] checked: $G={\rm Spin}\times{G_{\text{SM}_{q}}}$,
${\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{G_{\text{SM}_{q}}}$,
${\rm Spin}\times{\rm SU}(5)$, ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{\rm SU}(5)$, ${{\rm
Spin}\times{\rm Spin}(n)}$, $\frac{{\rm Spin}\times{\rm
Spin}(n)}{{\mathbb{Z}_{2}^{F}}}$ e.g., $n=10,18$, and other GUTs.
Thus, given by the starting assumption in Sec. 2.1, only Wan-Wang [8, 40]
contains the complete anomaly classification data for $G={\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{G_{\text{SM}_{q}}}$ and ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{\rm SU}(5)$ that we need for
completing the argument. So we must have to employ the results of Ref. [8].
Although Ref. [12, 14] checked several global anomalies, but they do not
exhaust checking all anomalies that we need for $G={\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{G_{\text{SM}_{q}}}$ and ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{\rm SU}(5)$.
Indeed, specifically for $G={\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{G_{\text{SM}_{q}}}$ and ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}\times{\rm SU}(5)$, only Ref. [8,
40] exhausted the cobordism classifications for all their possible anomalies,
and only Ref. [9, 10] completed the anomaly-matching checks.  The cobordism
invariant for any $\upnu\in\mathbb{Z}_{16}$ corresponds to a 5d iTQFT
partition function

$\displaystyle\Big{(}{\bf
Z}_{{\text{$5$d-iTQFT}}}\Big{)}^{\upnu}\equiv\exp(\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{16}\cdot\upnu\cdot\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))\bigg{\rvert}_{M^{5}}),$ $\displaystyle\text{ with
}\eta\equiv\eta_{\text{Pin}^{+}}\in\mathbb{Z}_{16},$
$\displaystyle\quad\upnu\in\mathbb{Z}_{16}.$ (2.10)

Given a $G\supseteq{{\rm Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}}$
structure, the cohomology class ${\cal
A}_{{\mathbb{Z}_{2}}}\in\operatorname{H}^{1}(M,\mathbb{Z}_{2})$ is the
generator from
$\operatorname{H}^{1}(\mathrm{B}(\mathbb{Z}_{4,X}/\mathbb{Z}_{2}^{F}),\mathbb{Z}_{2})$.
So $({\cal A}_{{\mathbb{Z}_{2}}})\equiv({\cal A}_{{\mathbb{Z}_{4}}})\mod 2$ is
the quotient for the $\mathbb{Z}_{4,X}$ gauge field ${\cal
A}_{{\mathbb{Z}_{4}}}$, which is the background field for the symmetry
$\mathbb{Z}_{4,{X}}\subset{\rm U}(1)_{X}$. The $\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))$ is the value of $\eta\in\mathbb{Z}_{16}$ on the
Poincaré dual (PD) submanifold of the cohomology class ${\cal
A}_{{\mathbb{Z}_{2}}}$. This PD takes ${\cap{\cal A}_{{\mathbb{Z}_{2}}}}$ from
5d to 4d. The APS eta invariant
$\eta\equiv\eta_{\text{Pin}^{+}}\in\mathbb{Z}_{16}$ is the cobordism invariant
of $\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$. The notation
“$\mid_{M^{5}}$” means the evaluation on this invariant on a 5-manifold
${M^{5}}$. We summarize the consequences and implications of this
$\mathbb{Z}_{16}$ anomaly mismatching in Sec. 2.3.

$\bullet$ Ref. [12] used the existence of $\mathbb{Z}_{16}$ global anomaly and
its anomaly matching to verify the conventional lore: the 16 Weyl fermions per
generation scenario, by introducing a right-handed neutrino per generation.

$\bullet$ Ref. [9, 10] take the $\mathbb{Z}_{16}$ anomaly as a secrete
entrance to find hidden new sectors beyond the Standard Model, given the fact
the HEP experiments only have detected 15 Weyl fermions per generation thus
far.

##### Consequences

Given the assumptions in Sec. 2.1, the $\mathbb{Z}_{16}$ anomaly index for the
SMq ($q=1,2,3,6$) and the $su(5)$ GUT both have the $(N_{\text{gen}}=3)$ and
the $(15=-1\mod 16)$ per generation. So we have the following anomaly matching
condition to match the $\mathbb{Z}_{16}$ anomaly:

$\displaystyle\boxed{(-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(N_{\text{gen}}=3)}+n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}+\text{new
hidden sectors})=0\mod 16.}$ (2.11)

The question is: How to match the $\mathbb{Z}_{16}$ anomaly? We enlist as many
ways as possible.

  1. 1.

Standard Lore: We can introduce the right-handed neutrino (the 16th Weyl
fermion) number $n_{\nu_{R}}=1$ for each generation (so
$n_{\nu_{e,R}}=n_{\nu_{\mu,R}}=n_{\nu_{\tau,R}}=1$ for electron, muon, and tau
neutrinos). In this case, there is no new hidden sector.

    1. (1a).

Massless: $\mathbb{Z}_{4,X}$ can be preserved if fermions are gapless.

    2. (1b).

Dirac mass: $\mathbb{Z}_{4,X}$ is also preserved by Yukawa-Higgs-Dirac
Lagrangian, but $\mathbb{Z}_{4,X}$ is spontaneously broken by the Higgs
condensate to give a Dirac mass gap.

    3. (1c).

Majorana mass: $\mathbb{Z}_{4,X}$ is broken explicitly by Majorana mass term.

  2. 2.

Proposals in Ref. [9, 10]: Ref. [9, 10] proposed other novel ways to match the
$\mathbb{Z}_{16}$ anomaly. Consequently, we can introduce new hidden sectors
beyond the SM and the $su(5)$ GUT:

    1. (2a).

$\mathbb{Z}_{4,X}$-symmetry-preserving anomalous gapped 4d topological quantum
field theory (TQFT).101010In the context of 3d boundary and 4d bulk, a novel
surface topological order was firstly pointed out by Vishwanath-Senthil in an
insightful work [50]. Later on many people follow up on developing the surface
topological order constructions (see overviews in [32, 31] and [51]). Ref. [9]
generalizes this condensed matter idea to find an anomalous symmetric 4d TQFT
living on the boundary of 5d fermionic SPTs (2.10). We also call the finite
_energy gap_

$\displaystyle\Delta_{\rm TQFT}\equiv E_{\text{excited}}-E_{\text{ground
states}}$ (2.12)

for the first excitation(s) above the ground state sectors of this 4d TQFT as
Topological mass gap. The underlying quantum system has a 4d intrinsic
topological order. We name the anomaly index $\upnu_{\rm 4d}$ for this
anomalous 4d TQFT.

    2. (2b).

$\mathbb{Z}_{4,X}$-symmetry-preserving 5d invertible topological quantum field
theory (iTQFT) given by the 5d cobordism invariant in (2.10). The underlying
quantum system has a 5d symmetry-protected topological state (SPTs) with an
_extra bulk 5th dimension_ whose 4d boundary can live the 4d Standard Model
world. We name the anomaly index $\upnu_{\rm 5d}$ to specify the boundary 4d
anomaly of this 5d iTQFT.

    3. (2c).

${\mathbb{Z}_{2}^{F}}$-symmetry-preseving 5d bulk
$[\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}]$-gauged TQFT and 4d boundary
$[\mathbb{Z}_{4,X}]$-gauged TQFT. Overall the spacetime rotational symmetry is
the fermionic Spin group ${\rm Spin}(d)$ graded the bosonic rotation special
orthogonal ${\rm SO}(d)$ by the fermion parity ${\mathbb{Z}_{2}^{F}}$.

    4. (2d).

5d bulk
$[{\mathbb{Z}_{2}^{F}}\times\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}]$-gauged
TQFT and 4d boundary $[{\mathbb{Z}_{2}^{F}}\times\mathbb{Z}_{4,X}]$-gauged
TQFT. Overall the spacetime rotational symmetry is the bosonic special
orthogonal ${\rm SO}(d)$.

    5. (2e).

$\mathbb{Z}_{4,X}$-symmetry-breaking gapped phase (e.g. Ginzburg-Landau
paradigm phase or 4d TQFT).

    6. (2f).

$\mathbb{Z}_{4,X}$-symmetry-preserving or $\mathbb{Z}_{4,X}$-symmetry-breaking
gapless phase, e.g., extra massless theories, free or interacting conformal
field theories (CFTs).

Scenarios 2(2a) and 2(2b), and their linear combinations, are the _root
phases_ for new hidden gapped Topological Sector. For example, _breaking_ part
of the global symmetries in the linear combined 2(2a) and 2(2b) would give
rise to Scenario 2(2e). For another example, _gauging_ part of the global
symmetries in the linear combined 2(2a) and 2(2b) would give rise to other
Scenarios:
$\bullet$ Gauging $[\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}]$ in 5d and
gauging $[\mathbb{Z}_{4,X}]$ in 4d gives rise to Scenario 2(2c).
$\bullet$ Gauging
$[{\mathbb{Z}_{2}^{F}}\times\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}]$ in
5d and gauging $[{\mathbb{Z}_{2}^{F}}\times\mathbb{Z}_{4,X}]$ in 4d gives rise
to Scenario 2(2d).
The underlying quantum systems in Scenario 2(2a) and 2(2c) have the symmetry-
enriched topologically ordered state (SETs) in a condensed matter terminology.

We name the combination of the above anomaly matching scenarios, including the
standard lore (right-handed neutrinos in 1) and the new proposals (all
enlisted in 2) beyond the SM and the GUT, as the Ultra Unification. Although
introducing additional CFTs (Scenario 2(2e)) to match the anomaly is equally
fascinating, we instead mostly focus on introducing the gapped Topological
Sector due to high-energy physics phenomenology (HEP-PH) constraints (see a
summary in [9]). A central theme of Ultra Unification suggesting a new HEP
frontier is that

Ultra Unification: HEP-PH provides Gapped Extended Objects beyond Particle
Physics.  (2.13)

Some more comments:
$\bullet$ These gapped extended objects (of 1d, 2d, 3d, $\dots$) are
formulated mathematically in terms of _gapped_ TQFT extended operators (1d
line, 2d surface, 3d brane, etc., topological operators), beyond the 0d
particles.
$\bullet$ These extended operators are heavy in the sense that they sit at the
energy scale above the TQFT energy gap $\Delta_{\rm TQFT}$ (so at or above the
scale of $E_{\text{excited}}$ in (2.12)).
$\bullet$ These extended operators are heavy in the sense that they have
Topological mass and they can interact with dynamical gravity. So these gapped
extended objects may be the Dark Matter candidate.
$\bullet$ There are fractionalized anyonic excitations at the open ends of 1d
line, 2d surface, 3d brane, etc., topological operators. In other words, the
anyonic particle 1d worldline is the 1d line topological operator. The anyonic
string 2d worldsheet is the 2d surface topological operator.111111Of course we
know that in above 3d spacetime (such as 4d and 5d that we concern), a 0d
particle by itself can only have bosonic or fermionic statistics. A 0d
particle does not have fractional or anyonic statistics [52] in above 3d
spacetime. However, 1d worldine and 2d worldsheet can be linked in 4d. Triple
and quadruple 2d worldsheets can be linked in 4d, etc (See [53, 54, 55] and
[56, 57, 58, 59]). These give rise to “anyonic statistics” to multi
excitations of gapped particles or gapped strings in a 4d spacetime and above.

In summary, based on the anomaly matching and cobordism constraints, we
propose that the SM and Georgi-Glashow SU(5) GUT (with 15 Weyl fermions per
generation, and with a discrete baryon minus lepton number $\mathbb{Z}_{4,X}$
preserved) contains a new hidden sector that can be a linear combination of
above Scenarios [9, 10]. In particular we can focus on the new hidden sectors
given by a 4d TQFT (with the anomaly index $\upnu_{\rm 4d}$) and a 5d iTQFT
(with the 4d boundary’s anomaly index $\upnu_{\rm 5d}$), so (2.11) becomes

$\displaystyle\boxed{(-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(N_{\text{gen}}=3)}+n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}+\upnu_{\rm
4d}-\upnu_{\rm 5d})=0\mod 16.}$ (2.14)

###  2.4 Symmetry Breaking vs Symmetry Extension:
Dirac or Majorana masses vs Topological mass

The distinctions between Dirac mass, Majorana mass, and Topological mass are
already explored in Ref. [9]. They represent the Scenarios 1(1b), 1(1c), and
2(2a) respectively in Sec. 2.3. Here we summarize their essences that:

$\bullet$ Symmetry breaking: Dirac mass and Majorana mass are induced by
_symmetry breaking_ — either global symmetry breaking or gauge symmetry
breaking, for example via the Anderson-Higgs mechanism or through Yukawa-Higgs
term.

$\bullet$ Symmetry extension: Topological mass as the energy gap above a TQFT
with ’t Hooft anomaly (of a spacetime-internal symmetry $G$) can be induced by
_symmetry extension_ [51]. The symmetry extension mechanism extended the
original Hilbert space (with nonperturbative global anomalies) to an enlarged
Hilbert space by adding extra degrees of freedom to the original quantum
system. The enlarged Hilbert space is meant to trivialize the ’t Hooft-anomaly
in $G$ in an extended $\tilde{G}$. The pullback $r^{*}$ of the map

$\tilde{G}\overset{r}{\longrightarrow}G$

can be understood as part of the group extension in an exact sequence [51],
while in general this can be generalized as the fibrations of their
classifying spaces and higher classifying spaces [60, 61, 62].

###  2.5 Gauging a discrete Baryon B, Lepton L, and Electroweak Hypercharge
$Y$

We provide some more logical motivations why we should preserve
$\mathbb{Z}_{4,X}$ and dynamically gauge $\mathbb{Z}_{4,X}$ with $X\equiv
5({\mathbf{B}-\mathbf{L}})-4Y$, at a higher energy (Scenario 2(2c) and 2(2d)
in Sec. 2.3):

  1. 1.

First, as stated before, the $\mathbb{Z}_{4,X}$ is a good global symmetry read
from the quantum numbers of SM particles and SM path integral kinematically.
It is a global symmetry that has _not yet_ been dynamically gauged in the
$G_{\text{SM}_{q}}$ nor in the SU(5) of the $su(5)$ GUT.

  2. 2.

The $\mathbb{Z}_{4,X}=Z({\rm Spin}(10))$ sits at the center subgroup
$\mathbb{Z}_{4}$ of the Spin(10) for the $so(10)$ GUT [12]. Thus the
$\mathbb{Z}_{4,X}$ must be dynamically gauged, if the $so(10)$ GUT is the
correct path to unification _at a higher energy_. It is natural to consider
the following group embedding from the GUT to the SM [8, 9, 10]:

$\displaystyle{\frac{{\rm Spin}(d)\times{\rm
Spin}(10)}{{\mathbb{Z}_{2}^{F}}}\supset{\rm
Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times{\rm
SU}(5)\supset{\rm
Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\frac{{\rm
SU}(3)\times{\rm SU}(2)\times{\rm U}(1)}{\mathbb{Z}_{6}}}.$ (2.15)

$\bullet$ It is worthwhile mentioning that the $\mathbb{Z}_{16}$ global
anomaly (occurred in the cobordism group for the $G={\rm
Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times{\rm SU}(5)$ and
${\rm Spin}(d)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm
SM}_{q}}$) disappear in the case of $G=\frac{{\rm Spin}(d)\times{\rm
Spin}(10)}{{\mathbb{Z}_{2}^{F}}}$.

So the $\mathbb{Z}_{4,X}$ can be an anomalous symmetry in the ${{\rm SM}_{q}}$
and the $su(5)$ GUT, but the $\mathbb{Z}_{4,X}$ is an anomaly-free symmetry in
the $so(10)$ GUT.121212The anomalous symmetry means a non-onsite symmetry in
the condensed matter terminology, that cannot be realized acting only locally
on a 0-simplex (a point). The anomaly-free symmetry means an onsite symmetry
in the condensed matter terminology, that acts only locally on a 0-simplex (a
0d point), which can be easily gauged by coupling dynamical variables living
on 1-simplices (1d line segments).

$\bullet$ The 15n Weyl fermion SMq or the 15n Weyl fermion $su(5)$ GUT alone
may have a $\mathbb{Z}_{16}$ global anomaly, while they can become anomaly-
free at a higher-energy 16n Weyl fermion $so(10)$ GUT [13]. This fact
motivates Ref. [10] to propose an analogous concept of topological quantum
phase transition happens between two energy scales: (1) above the energy scale
of the SMq or the $su(5)$ GUT, (2) below the energy scale of the $so(10)$ GUT.

  3. 3.

Global symmetry must be gauged or broken in quantum gravity. If for the above
reasons, we ask the $\mathbb{Z}_{4,X}$-symmetry to be preserved, then the
$\mathbb{Z}_{4,X}$ must be dynamically gauged at a higher energy for the sake
of quantum gravity.131313String theory landscape and swampland develop the
similar concepts of the use of cobordism for quantum gravity, see [63] and
References therein. This can be understood as the deformation classes of
quantum gravity. The deformation classes of quantum field theory is also
proposed by Seiberg in [64].

###  2.6 Topological Sector and Topological Force

Topological Force and the gapped Topological Sector have specific physical and
mathematical meanings in our context. We should clarify what they are, and
then what they are not in the next:

  1. 1.

Topological Sector: In Sec. 2.3 and (2.14), we propose a Topological Sector
beyond the Standard Model (BSM) includes an appropriate linear combination of
the following theories (selecting the best scenario to fit into the HEP
phenomenology):

    1. (a).

4d long-range entangled gapped topological phase with an energy gap (named the
gap $\Delta_{\rm TQFT}$) whose low energy physics is characterized by a 4d
noninvertible topological quantum field theories. This 4d TQFT is a Schwarz
type unitary TQFT (which is the 4d analog of the 3d Chern-Simons-Witten
theories [29, 30]).

$\bullet$ This 4d TQFT has a ’t Hooft anomaly [19] of a global symmetry $G$.
We named the anomaly index $\upnu_{\rm 4d}$ for this 4d TQFT.

$\bullet$ Proper mathematical tools to study this 4d TQFT requires the
category or higher category theories.

    2. (b).

5d short-range entangled gapped topological phase with an energy gap whose low
energy physics is characterized by a 5d invertible topological quantum field
theory (iTQFT). This 5d iTQFT is also a unitary TQFT. But the iTQFT is
nontrivial distinct from a trivial gapped vacuum only in the presence of a
global symmetry $G$, see Footnote 2. A $G$-symmetric iTQFT is mathematically
given by a $G$-cobordism invariant, classified by an appropriate cobordism
group $\Omega^{d}_{G}\equiv\mathrm{TP}_{d}(G)$, defined in the Freed-Hopkins
classification of invertible topological phases (TP) [18].

$\bullet$ The boundary of this 5d iTQFT has a 4d ’t Hooft anomaly of a global
symmetry $G$. We named the anomaly index $\upnu_{\rm 5d}$ for this 5d iTQFT.

$\bullet$ Proper mathematical tools to study this 4d TQFT requires
characteristic classes, cohomology, and cobordism theories.

    3. (c).

5d long-range entangled gapped topological phase with an energy gap whose low
energy physics is characterized by a 5d TQFT. This is the case when the
discrete $X$ symmetry is dynamically gauged, stated in Scenario 2(2c) and
2(2d) in Sec. 2.3.

  2. 2.

Topological Force: In the context of Sec. 2.3, Topological Force is a discrete
gauge force mediated between the linked worldvolume trajectories (1d
worldlines, 2d worldsheets from gapped extended operators) via fractional or
categorical statistical interactions (See Sec. 5 and 6 of [9]).

$\bullet$ Bosonic finite group gauge theory: The conventional discrete gauge
theories are bosonic types of finite group gauge theories [65, 66]. Bosonic
types mean that their ultraviolet (UV) completion only requires a local tensor
product Hilbert space of local (gauge-invariant) bosonic operators; the UV
completion does not require local (gauge-invariant) fermionic operators. The
underlying TQFT does not require the spin structures and can be defined on
non-Spin manifolds (such as the oriented SO structures). The TQFTs are known
as bosonic or non-Spin TQFTs.

$\bullet$ Fermionic finite group gauge theory: The discrete gauge theories for
our gapped topological sectors for the beyond SM hidden sector are fermionic
types of finite group gauge theories [61, 47]. Fermionic types mean that their
UV completion must require local (gauge-invariant) fermionic operators. The
underlying TQFT requires the additional spin structures defined on Spin
manifolds. The TQFTs are known as Spin TQFTs. In fact the 4d TQFT in Sec. 2.3
requires the ${\rm Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ structure
and can be defined on the ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ manifolds (including both
Spin manifolds and some non-Spin manifolds).

We should emphasize that our Topological Sector and Topological Force are
_not_ the kinds of Chern class topological terms which are already summed over
in the continuous Lie group gauge theory. Namely, our Topological Sector and
Topological Force are not the followings:

— The $\theta$-term with or without a dynamical $\theta$-axion [67, 68], well-
known as $\theta F\wedge F$ or $\theta F\tilde{F}$ in the particle physics, is
in fact related to the second Chern class $c_{2}(V_{G})$ and the square of the
first Chern class $c_{1}(V_{G})$ of the associated vector bundle of the gauge
group $G$:

$\displaystyle\frac{\theta}{8\pi^{2}}{\mathrm{Tr}}({F}\wedge{F})=\frac{\theta}{2}c_{1}(V_{G})^{2}-\theta\;c_{2}(V_{G}).$
(2.16)

In particular, here we consider $G$ as the U(N) or SU(N) gauge group, so we
can define the Chern characteristic classes associated with complex vector
bundles. The $V_{G}$ is the associated vector bundle of the principal $G$
bundle. This $\theta$-term is a topological term, but it is summed over as a
weighted factor to define a Yang-Mills gauge theory partition function [35,
69, 70]. This $\theta$-term does not define a quantum system or a quantum
phase of matter by itself, distinct from our 4d TQFT (with intrinsic
topological order) and 5d iTQFT (with SPTs) as certain unitary quantum phases
of matter by themselves.

— The instantons [71, 72] or the sphalerons [73], are also not the Topological
Sector and Topological Force in our context. Instantons and sphalerons are
again the objects with nontrivial Chern class integrated over the spacetime
manifold. Those objects are already defined as part of the SM and GUT
continuous group gauge theories.

As we will mention in Sec. 3.1, we can also include (1) the $\theta$-term with
or without a dynamical $\theta$-axion, (2) instantons, and (3) sphalerons,
into the Standard Model and the $su(5)$ GUT path integral. These objects are
already in the old paradigm of the SM and GUT models. These objects do not
affect the anomaly matching and cobordism constraints (especially the
$\mathbb{Z}_{16}$ global anomaly) discussed in Sec. 2.2.141414However, those
$\theta$ terms in Yang-Mills gauge theory may affect the higher anomalies
involving higher generalized global symmetries, see for example [69, 70].
These objects belong to the Standard Model and the $su(5)$ GUT path integral
(Sec. 3.1), not to the Topological Sector (TQFT) path integral (Sec. 3.2), but
they all can be included as part of the Ultra Unification path integral (Sec.
3).

##  3 Ultra Unification Path Integral

In this section, we provide the functional path integral (i.e., partition
function) ${\mathbf{Z}}_{{\rm UU}}$ of Ultra Unification, which includes the
standard paradigm of the Standard Model path integral ${\mathbf{Z}}_{{\rm
SM}}$ or the Georgi-Glashow $su(5)$ GUT path integral ${\mathbf{Z}}_{{\rm
GUT}}$ in Sec. 3.1. Then we provide the Topological Sector TQFT path integral
${\mathbf{Z}}_{{\rm TQFT}}\equiv{\bf Z}_{\text{5d-iTQFT}}\cdot{\bf
Z}_{\text{4d-TQFT}}$ in Sec. 3.2

###  3.1 Standard Model and the $su(5)$ GUT Path Integral coupled to $X\equiv
5({\mathbf{B}-\mathbf{L}})-4Y$

##### Standard Model Path Integral coupled to $X$

Now we describe the Standard Model (SM) path integral:

$\displaystyle{\mathbf{Z}}_{{\rm SM}}[{\cal
A}_{{\mathbb{Z}_{4}}}]\equiv\int[{\cal D}{\psi}][{\cal D}\bar{\psi}][{\cal
D}A][{\cal D}\phi]\dots\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\left.S_{\text{SM}}[\psi,\bar{\psi},A,\phi,\dots,{\cal
A}_{\mathbb{Z}_{4}}]\right\rvert_{M^{4}})$ (3.1)

The $\dots$ depends on the details of which variant versions of SM that we
look at (e.g., adding axions or not). In the schematic way, we have the
action:

$S_{\text{SM}}=-\int_{M^{4}}\sum_{I=1,2,3}{\mathrm{Tr}}(F_{I}\wedge\star
F_{I})+\frac{\theta_{I}}{8\pi^{2}}{\mathrm{Tr}}\Big{(}F_{I}\wedge
F_{I}\Big{)}+\int_{M^{4}}\Big{(}\bar{\psi}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\not{D}_{A,{\cal A}_{\mathbb{Z}_{4}}})\psi\\\ +|{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}}\phi|^{2}-{\rm
U}(\phi)-({\psi}^{\dagger}_{L}\phi\psi_{R}+{\rm h.c.})\Big{)}\,\hskip
1.0pt\mathrm{d}^{4}x.$ (3.2)

But more precisely we really need more details in the Lagrangian with Weyl
fermions:

${\cal L}_{\rm SM}={\cal L}_{\text{YM}}+{\cal L}_{\theta\text{-Chern}}+{\cal
L}_{\text{Weyl}}+{\cal L}_{\text{Higgs}}+{\cal L}_{\text{Yukawa-Higgs}}\\\
=-\frac{1}{4}\sum_{I=1,2,3}F_{I,\mu\nu}^{\mathrm{a}}F_{I}^{\mathrm{a}\mu\nu}+\frac{\theta}{8\pi^{2}}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{I,\mu\nu}^{\mathrm{a}}F_{I,\mu^{\prime}\nu^{\prime}}^{\mathrm{a}}+{\psi}^{\dagger}_{L}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\sigma}^{\mu}{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}})\psi_{L}+{\psi}^{\dagger}_{R}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{\mu}{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}})\psi_{R}\\\ +|{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}}\phi|^{2}-{\rm
U}(\phi)-({\psi}^{\dagger}_{L}\phi\psi_{R}+{\rm h.c.}).$ (3.3)

Here come some Remarks:

  1. 1.

Yang-Mills gauge theory [74] has the action
$S_{\text{YM}}=\int{\mathrm{Tr}}(F\wedge\star F)$ and Lagrangian ${\cal
L}_{\text{YM}}=-\frac{1}{4}F_{\mu\nu}^{\mathrm{a}}F^{\mathrm{a}{\mu\nu}}$.
The $F$ is the Lie algebra valued field strength curvature 2-form $F=\hskip
1.0pt\mathrm{d}A-\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\rm g}A\wedge A$, with its
Hodge dual $\star$, all written in differential forms. In the trace “Tr” we
pick up a Lie algebra representation ${\bf R}$ whose generators labeled by
“$\mathrm{a}$.” There is an overall constant can be absorbed into the field
$A$ and coupling ${\rm g}$ redefinition.

The $\theta$-term and dynamical $\theta$-axion: We can also introduce the
Chern class topological $\theta$-term
$S=\int\frac{\theta}{8\pi^{2}}{\mathrm{Tr}}\Big{(}F\wedge F\Big{)}$ and ${\cal
L}_{\theta\text{-Chern}}=\frac{\theta}{8\pi^{2}}\epsilon^{\mu\nu\mu^{\prime}\nu^{\prime}}F_{\mu\nu}^{\mathrm{a}}F_{\mu^{\prime}\nu^{\prime}}^{\mathrm{a}}$.
Given a ${{\rm U}({\rm N})}$ or SU(N) bundle $V_{G}$ and its field strength
$\widehat{F}$, the first and second Chern classes are given by
$c_{1}(V_{G})=\frac{{\mathrm{Tr}}{F}}{2\pi}$ and
$c_{2}(V_{G})=-\frac{1}{8\pi^{2}}{\mathrm{Tr}}({F}\wedge{F})+\frac{1}{8\pi^{2}}({\mathrm{Tr}}{F})\wedge({\mathrm{Tr}}{F})$,
so that
$\frac{\theta}{8\pi^{2}}{\mathrm{Tr}}({F}\wedge{F})=\frac{\theta}{2}c_{1}(V_{G})^{2}-\theta\;c_{2}(V_{G})$.
If a dynamical $\theta$-axion [67, 68] is introduced, it requires a summation
of the compact $\theta$ in the path integral measure $\int[{\cal D}\theta]$.

The path integral $\int[{\cal D}A]$ for continuous Lie group gauge field
theory (here U(1), SU(N), U(N) for the SM and $su(5)$ GUT), really means (1)
the summation of all inequivalent principal gauge bundles $P_{A}$, and then
(2) the summation of all inequivalent gauge connections $\tilde{A}$ (under a
given specific principal gauge bundles $P_{A}$), where $\tilde{A}$ is a
(hopefully globally defined physically) $1$-form gauge connection. So we
physically define:

$\int[{\cal D}A]\dots\equiv\sum_{\text{gauge bundle $P_{A}$}}\int[{\cal
D}\tilde{A}]\dots.$

  2. 2.

Dirac fermion theory has $S_{\text{Dirac}}=\int\bar{\psi}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\not{D}_{A})\psi\,\hskip 1.0pt\mathrm{d}^{4}x$ and
${\cal L}_{\text{Dirac}}=\bar{\psi}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\not{D})\psi$ with the Dirac spinor $\psi$ defined as a section of the
spinor bundles. The $\bar{\psi}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\not{D})\psi$ is an inner product in the complex vector space with the
Dirac operator $\not{D}_{A}$ as a natural linear operator in the vector space.
The path integral $\int[{\cal D}{\psi}][{\cal D}\bar{\psi}]$ is (1) the
summation of all inequivalent spinor bundles, and then (2) the summation of
all inequivalent sections (as spinors) of spinor bundles (under a given
specific spinor bundle). We requires spin geometry and spin manifold, in
particular we require the ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}={\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ structure.

In fact preferably we present not in the Dirac spinor basis, but we present
all of (3.3) in the Weyl spinor basis (below).

  3. 3.

Weyl fermion theory has $S_{\text{Weyl}}=\int{\cal L}_{\text{Weyl}}\,\hskip
1.0pt\mathrm{d}^{4}x=\int{\psi}^{\dagger}_{L}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\sigma}^{\mu}{D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}})\psi_{L}\,\hskip
1.0pt\mathrm{d}^{4}x$. Weyl spinor bundle splits the representation of the
Dirac spinor bundle. Weyl spinor again is defined as the section of Weyl
spinor bundle. The Weyl spacetime spinor is in ${\bf 2}_{L}$ of ${{\rm
Spin}(3,1)}=\rm{SL}(2,{\mathbb{C}})$ with a complex representation in the
Lorentz signature, or ${\bf 2}_{L}$ of ${\rm Spin}(4)={\rm
SU}(2)_{L}\times{\rm SU}(2)_{R}$ with a pseudoreal representation in the
Euclidean signature. We also write the analogous right-handed Weyl fermion
theory. The $\sigma^{\mu}$ and $\bar{\sigma}^{\mu}$ are the standard spacetime
spinor rotational $su(2)$ Lie algebra generators. We will emphasize and
illuminate the meanings of covariant derivative ${D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}}$ altogether in Remark 5.

  4. 4.

Higgs theory has $S_{\text{Higgs}}=\int{\cal L}_{\text{Higgs}}\,\hskip
1.0pt\mathrm{d}^{4}x=\int\Big{(}|{D}_{\mu,A,{\cal A}}\phi|^{2}-{\rm
U}(\phi)\Big{)}\,\hskip 1.0pt\mathrm{d}^{4}x$. The Higgs field bundle is
typically a trivial complex line bundle.151515In Higgs theory, people in
general do not consider nontrivial complex line bundles for Higgs field. But
it may be amusing to consider the alternative. The Higgs scalar field is the
section of a field bundle. The electroweak Higgs is in complex value
${\mathbb{C}}$ and also in ${\bf 2}$ of SU(2) gauge field. Again by doing
summation $\int[{\cal D}\phi]$ we (1) sum over the field bundles, and (2) sum
over the section of each field bundle. We illuminate the meanings of covariant
derivative ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ altogether in Remark 5.

  5. 5.

Covariant derivative operator ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ in (3.3)
is defined as:

$\displaystyle{D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}\equiv\nabla_{\mu}-\hskip
1.0pt\mathrm{i}\hskip 1.0ptg\,q_{{\bf R}}\,A_{\mu}-\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{{X}}{\cal A}_{\mathbb{Z}_{4},\mu}.$ (3.4)

Placed on a curved spacetime (with a non-dynamical metric, only with
background gravity) requires a covariant derivative $\nabla_{\mu}$, and a spin
connection for the spinors.
Comments about the term $g\,q_{{\bf R}}\,A$ in a differential form (e.g.,
quantum numbers read from Table 1 in [9]):

$\displaystyle g\,q_{{\bf
R}}\,A\equiv(q_{e}A_{\mu}+g_{su(2)}\sum_{\mathrm{a}=1}^{3}\frac{\varsigma^{\mathrm{a}}}{2}A_{{su(2)},\mu}^{\mathrm{a}}+g_{su(3)}\sum_{\mathrm{a}=1}^{8}\frac{\tau^{\mathrm{a}}}{2}A_{{su(3)},\mu}^{\mathrm{a}})\hskip
1.0pt\mathrm{d}x^{\mu}.$ (3.5)

The ${\varsigma^{\mathrm{a}}}$ and ${\tau^{\mathrm{a}}}$ are the rank-2 and
rank-3 Lie algebra generator matrix representations for ${su(2)}$ and
${su(3)}$ respectively. The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acting on
$\psi_{L}$ contains the $su(2)$ gauge field. The ${D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}}$ acting on $\psi_{R}$ does not contain the $su(2)$ gauge
field, because the $su(2)$ weak interaction is a maximally parity violating
chiral gauge theory. The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acting only
on quarks (both $\psi_{L}$ and $\psi_{R}$) contain the $su(3)$ gauge field.
The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acting on $\phi$ contains the
$su(2)$ gauge field.

Comments about the term $q_{{X}}{\cal A}_{\mathbb{Z}_{4}}$ (e.g., quantum
numbers read from Table 1 and 2 in [9]):
$\bullet$ The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acts on all left-handed
SM Weyl fermion $\psi_{L}$ via its ${\cal A}_{\mathbb{Z}_{4}}$ charge
$q_{{X}}=1$.
$\bullet$ The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acts on all right-handed
SM Weyl fermion $\psi_{R}$ via its ${\cal A}_{\mathbb{Z}_{4}}$ charge
$q_{{X}}=-1$.
$\bullet$ The ${D}_{\mu,A,{\cal A}_{\mathbb{Z}_{4}}}$ acts on the electroweak
Higgs $\phi$ via its ${\cal A}_{\mathbb{Z}_{4}}$ charge $q_{{X}}=2$.

The subtle part is that ${\cal A}_{\mathbb{Z}_{4}}$ should be treated as a
cohomology class, such as a cohomology or cochain gauge field. The ${\cal
A}_{{\mathbb{Z}_{4}}}\in\operatorname{H}^{1}(M,\mathbb{Z}_{4})$ is the
generator from
$\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{4,X},\mathbb{Z}_{4})$. In physics,
for the continuum QFT theorists who prefer to think
$\mathbb{Z}_{4,{X}}\subset{\rm U}(1)_{X}$ as a continuum gauge field breaking
down to a discrete $\mathbb{Z}_{4,{X}}$, we can introduce an extra
$\mathbb{Z}_{4}$ charge new Higgs field $\varphi$ and its potential ${\rm
V}(\varphi)$:

$\displaystyle{\psi}^{\dagger}_{L}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\sigma}^{\mu}{D}_{\mu,{\cal
A}_{\mathbb{Z}_{4}}})\psi_{L}+{\psi}^{\dagger}_{R}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{\mu}{D}_{\mu,{\cal
A}_{\mathbb{Z}_{4}}})\psi_{R}+|(\partial_{\mu}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt2{\cal A}_{\mathbb{Z}_{4},\mu})\phi|^{2}+|(\partial_{\mu}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt4{\cal A}_{\mathbb{Z}_{4},\mu})\varphi|^{2}+{\rm
V}(\varphi)+\dots.$ (3.6)

There are extra superconductivity-like term
$\phi^{2}\varphi^{\dagger}+(\varphi^{\dagger})^{2}\phi$ does not break the
$\mathbb{Z}_{4}$. Their $\mathbb{Z}_{4}$ or U(1) transformations are:

$\psi_{L}\to\psi_{L}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi}{4}},\;\;\psi_{R}\to\psi_{R}\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi}{4}},\;\;\phi\to\phi\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\frac{2\pi}{4}},\;\;\varphi\to\varphi.$ $\psi_{L}\to\psi_{L}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Theta},\;\;\psi_{R}\to\psi_{R}\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Theta},\;\;\phi\to\phi\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\Theta},\;\;\varphi\to\varphi\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt4\Theta}.$

In the $\langle\varphi\rangle\neq 0$ condensed Higgs phase as a discrete
$\mathbb{Z}_{4}$ gauge theory, which we can dualize the theory as a level-4 BF
theory [75]. The formulation starts from adding $\int[D\varphi]$ in the path
integral measure, and it ends with a 2-form ${\cal B}$ and 1-form gauge field
${\cal A}_{\mathbb{Z}_{4}}$

$\displaystyle\int[D{\cal B}][D{\cal A}_{\mathbb{Z}_{4}}]\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{4}{2\pi}\int_{M^{4}}{\cal B}\wedge\hskip
1.0pt\mathrm{d}{\cal A}_{\mathbb{Z}_{4}}+\dots).$ (3.7)

But more precisely, we really should formulate in terms of a
cohomology/cochain TQFT and taking care of the ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ structure, which we will do
in Sec. 3.2 (also in Sec. 5 of [9]).

  6. 6.

Yukawa-Higgs-Dirac term has $S_{\text{Yukawa-Higgs-Dirac}}=\int{\cal
L}_{\text{Yukawa-Higgs-Dirac}}\,\hskip
1.0pt\mathrm{d}^{4}x=\int({\psi}^{\dagger}_{L}\phi\psi_{R}+{\rm h.c.})\,\hskip
1.0pt\mathrm{d}^{4}x$. In this case, we pair the ${\psi}^{\dagger}_{L}$’s
$\bar{\bf 2}$ of SU(2) with the $\phi$’s ${\bf 2}$ of SU(2), and vice versa
pair ${\psi}_{L}$ with $\phi^{\dagger}$ to get an SU(2) singlet. The right-
handed ${\psi}_{R}$ here is (meant to be) an SU(2) singlet. This Yukawa-Higgs-
Dirac term at the kinetic level also preserves the $\mathbb{Z}_{4,X}$,
although the Higgs vacuum expectation value (vev) breaks the
$\mathbb{Z}_{4,X}$ dynamically.

  7. 7.

Yukawa-Higgs-Majorana term with Weyl fermion:
We can add Yukawa-Higgs-Majorana term for Weyl fermions. For example, for the
left-handed $\psi_{L}$, we can add a dimension-5 operator:
$S_{\text{Yukawa-Higgs-Majorana}}=\int-({\psi}^{\dagger}_{L}\phi(\phi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\psi_{L}^{*})+{\rm h.c.})\,\hskip
1.0pt\mathrm{d}^{4}x=\int-({\psi}^{\dagger}_{L}\phi(\phi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\psi_{L}^{*})+{(-\psi_{L}^{\rm T}}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\phi^{*})\phi^{\dagger}\psi_{L})\,\hskip
1.0pt\mathrm{d}^{4}x$.
Renormalizability is not an issue because we are concerned the effective field
theory. For the right-handed $\psi_{R}$, we can add a dimension-3 operator for
some Majorana mass coupling $M$:
$S_{\text{Yukawa-Higgs-Majorana}}=\int-M({\psi_{R}}^{\rm T}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2})\psi_{R}+{\rm h.c.})\,\hskip
1.0pt\mathrm{d}^{4}x=\int-M({\psi_{R}}^{\rm T}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2})\psi_{R}+\psi_{R}^{\dagger}(-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}){\psi_{R}^{*}})\,\hskip 1.0pt\mathrm{d}^{4}x$,
which breaks the lepton number conservation. However, in either cases, both
Yukawa-Higgs-Majorana terms above break the $\mathbb{Z}_{4,X}$ explicitly. So
they are not strongly favored if we pursue the $\mathbb{Z}_{4,X}$-preserving
theory at least at higher energy.

We have presented above the Standard Model coupled to a discrete $X$ gauge
field in the path integral (3.1), the action (3.2), and the Lagrangian (3.3).
Below we can quickly modify a few terms to obtain the $su(5)$ Grand
Unification coupled to a discrete $X$ gauge field.

##### The $su(5)$ Grand Unification Path Integral coupled to $X$

We have the $su(5)$ GUT path integral coupled to $X$:

$\displaystyle{\mathbf{Z}}_{{\rm GUT}}[{\cal
A}_{{\mathbb{Z}_{4}}}]\equiv\int[{\cal D}{\psi}][{\cal D}\bar{\psi}][{\cal
D}A][{\cal D}\phi]\dots\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\left.S_{\text{GUT}}[\psi,\bar{\psi},A,\phi,\dots,{\cal
A}_{\mathbb{Z}_{4}}]\right\rvert_{M^{4}}).$ (3.8)

We should write all $\bar{\bf 5}$ and ${\bf 10}$ of the SU(5) as the left-
handed Weyl fermions $\psi_{L}$, so there are 15 Weyl fermions $\psi_{L}$ per
generation. In Sec. 2.3, we may or may not introduce the right-handed
neutrinos here denoted as $\chi_{R}$. In the schematic way, we have the
action:

$S_{\text{GUT}}=-\int_{M^{4}}{\mathrm{Tr}}(F\wedge\star
F)+\frac{\theta}{8\pi^{2}}{\mathrm{Tr}}\Big{(}F\wedge
F\Big{)}+\int_{M^{4}}\Big{(}{\psi}^{\dagger}_{L}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\sigma}^{\mu}{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}})\psi_{L}+{\chi}^{\dagger}_{R}(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{\mu}{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}})\chi_{R}\\\ +|{D}_{\mu,A,{\cal
A}_{\mathbb{Z}_{4}}}\phi|^{2}-{\rm
U}(\phi)-({\psi}^{\dagger}_{L}\phi(\sigma^{2}{\psi_{L}^{\prime}}^{*})+{\rm
h.c.})+\dots\Big{)}\,\hskip 1.0pt\mathrm{d}^{4}x.$ (3.9)

We are left now only with a SU(5) gauge field whose 1-form connection written
as:

$\displaystyle g\,q_{{\bf R}}\,A=(g_{su(5)}\sum_{\mathrm{a}=1}^{24}\frac{{\rm
T}^{\mathrm{a}}}{2}A_{{su(5)},\mu}^{\mathrm{a}})\hskip
1.0pt\mathrm{d}x^{\mu}.$ (3.10)

We require the ${{\rm T}^{\mathrm{a}}}$ as the rank-5 and rank-10 Lie algebra
generator matrix representations for ${su(5)}$ to couple to $\bar{\bf 5}$ and
${\bf 10}$ of SU(5) respectively. Yukawa-Higgs pairs the appropriate
${\psi_{L}}$ and ${\psi_{L}^{\prime}}$ Weyl fermions. We may or may not
introduce the Majorana mass terms to $\chi_{R}$ in the $\dots$, while the
consequences are already discussed (which break the $\mathbb{Z}_{4,X}$
explicitly) in the Remark 7. The discussions about this path integral (3.8)
directly follow the above Remarks 1-7, so we should not repeat.

###  3.2 Topological Sector and TQFT Path Integral coupled to $X\equiv
5({\mathbf{B}-\mathbf{L}})-4Y$

The 15n Weyl fermion SM and $su(5)$ GUT path integrals, (3.1) and (3.8), are
_not_ gauge invariant under the $\mathbb{Z}_{4,X}$ gauge transformation.
$\bullet$ If the $\mathbb{Z}_{4,X}$ is only a background gauge field, this
only means the $\mathbb{Z}_{4,X}$ anomalous symmetry and the system has ’t
Hooft anomaly under the $\mathbb{Z}_{4,X}$ and the spacetime (Spin group)
transformations.
$\bullet$ If the $\mathbb{Z}_{4,X}$ is dynamically gauged and preserved at
high energy, then we must append a new sector to make the whole theory well-
defined.

In any case, follow Sec. 2.3, we now provide a path integral including
Topological Quantum Field Theories (TQFTs) to make the whole theory free from
the $\mathbb{Z}_{16}$ global anomaly (2.14):

${(-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(N_{\text{gen}}=3)}+n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}+\upnu_{\rm
4d}-\upnu_{\rm 5d})=0\mod 16.}$

  1. 1.

The 5d iTQFT partition function is given by (2.10):

$\displaystyle{\bf Z}_{{\text{5d-iTQFT}}}^{(\upnu_{\rm 5d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\equiv\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{16}\cdot\upnu_{\rm 5d}\cdot\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))\bigg{\rvert}_{M^{5}}),$ $\displaystyle\text{ with
}\upnu\in\mathbb{Z}_{16},\quad{\cal A}_{{\mathbb{Z}_{2}}}\equiv({\cal
A}_{{\mathbb{Z}_{4}}}\mod 2).$ (3.11)

  2. 2.

We propose the full gauge invariant path integral, invariant under the mixed
gauge-gravity transformation of ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ structure and free from its
$\mathbb{Z}_{16}$ global anomaly as follows:
The SM version employs (3.1) into:

$\displaystyle\boxed{{\bf Z}_{\begin{subarray}{c}\text{5d-iTQFT/}\\\
\text{4d-SM+TQFT}\end{subarray}}[{\cal A}_{{\mathbb{Z}_{4}}}]\equiv{\bf
Z}_{{\text{5d-iTQFT}}}^{(-\upnu_{\rm 5d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\bf Z}_{{\text{4d-TQFT}}}^{(\upnu_{\rm 4d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\mathbf{Z}}_{{\rm
SM}}^{(n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}})}[{\cal
A}_{{\mathbb{Z}_{4}}}].}$ (3.12)

The GUT version employs (3.8) into:

$\displaystyle\boxed{{\bf Z}_{\begin{subarray}{c}\text{5d-iTQFT/}\\\
\text{4d-GUT+TQFT}\end{subarray}}[{\cal A}_{{\mathbb{Z}_{4}}}]\equiv{\bf
Z}_{{\text{5d-iTQFT}}}^{(-\upnu_{\rm 5d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\bf Z}_{{\text{4d-TQFT}}}^{(\upnu_{\rm 4d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\mathbf{Z}}_{{\rm
GUT}}^{(n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}})}[{\cal
A}_{{\mathbb{Z}_{4}}}].}$ (3.13)

####  3.2.1 Symmetry extension $[\mathbb{Z}_{2}]\to{\rm
Spin}\times{\mathbb{Z}_{4,X}}\to{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$ and a 4d
$[\mathbb{Z}_{2}]$ gauge theory

Below we ask whether we can construct a fully gauge-invariant 5d-4d coupled
partition function preserving the ${{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}$ structure:

$\displaystyle{\bf Z}_{{\text{5d-iTQFT}}}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\bf Z}_{{\text{4d-TQFT}}}[{\cal
A}_{{\mathbb{Z}_{4}}}].$ (3.14)

Preserving the ${{\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}$ structure
means that under the spacetime background transformation and the ${\cal
A}_{{\mathbb{Z}_{4}}}$ background gauge transformation, the 5d-4d coupled
partition function is still fully gauge invariant.

First, we can rewrite the 5d iTQFT partition function (3.11) on a 5d manifold
$M^{5}$ into

${\bf Z}_{{\text{5d-iTQFT}}}^{(\upnu)}[{\cal
A}_{{\mathbb{Z}_{4}}}]=\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{16}\cdot\upnu\cdot\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))\bigg{\rvert}_{M^{5}})\\\ =\exp(\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{16}\cdot\upnu\cdot\Big{(}8\cdot\frac{p_{1}(TM)}{48}(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))+4\cdot\text{Arf}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{3}))+2\cdot{\tilde{\eta}}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{4}))+({\cal
A}_{{\mathbb{Z}_{2}}})^{5}\Big{)}\bigg{\rvert}_{M^{5}})\\\
=\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{16}\cdot\upnu\cdot\Big{(}8\cdot\frac{\sigma}{16}(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))+4\cdot\text{Arf}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{3}))+2\cdot{\tilde{\eta}}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{4}))+({\cal
A}_{{\mathbb{Z}_{2}}})^{5}\Big{)}\bigg{\rvert}_{M^{5}}),$ (3.15)

for a generic ${\upnu=-N_{\text{generation}}}\in\mathbb{Z}_{16}$.
$\bullet$ The $p_{1}(TM)$ is the first Pontryagin class of spacetime tangent
bundle $TM$ of the manifold $M$. Via the Hirzebruch signature theorem, we have
$\frac{1}{3}\int_{\Sigma^{4}}p_{1}(TM)=\sigma(\Sigma^{4})=\sigma$ on a
4-manifold $\Sigma^{4}$, where $\sigma$ is the signature of $\Sigma^{4}$. So
in (3.15), we evaluate the $\frac{1}{3}\int_{\Sigma^{4}}p_{1}(TM)=\sigma$ on
the Poincaré dual (PD) of $\Sigma^{4}$ manifold of the $({\cal
A}_{{\mathbb{Z}_{2}}})$ cohomology class within the $M^{5}$.
$\bullet$ The $\tilde{\eta}$ is a mod 2 index of 1d Dirac operator as a
cobordism invariant of $\Omega_{1}^{{\rm Spin}}=\mathbb{Z}_{2}$.
$\bullet$ The Arf invariant [41] is a mod 2 cobordism invariant of
$\Omega_{2}^{{\rm Spin}}=\mathbb{Z}_{2}$, whose realization is the 1+1d Kitaev
fermionic chain [42].
$\bullet$ The $({\cal A}_{{\mathbb{Z}_{2}}})^{5}$ is a mod 2 class purely
bosonic topological invariant, which corresponds to a 5d bosonic SPT phase
given by the group cohomology class data
$\operatorname{H}^{5}(\mathrm{B}\mathbb{Z}_{2},{\rm U}(1))=\mathbb{Z}_{2}$,
which is also one of the $\mathbb{Z}_{2}$ generators in $\Omega_{5}^{{\rm
SO}}(\mathrm{B}\mathbb{Z}_{2})$.

  1. 1.

When ${\upnu}$ is odd, such as ${\upnu}=1,3,5,7,\dots\in\mathbb{Z}_{16}$,
Ref. [46] suggested that the symmetry-extension method [51] cannot construct a
symmetry-gapped TQFT. Furthermore, Cordova-Ohmori [76] proves that a symmetry-
preserving gapped TQFT phase is impossible for this odd
${\upnu}\in\mathbb{Z}_{16}$ anomaly from $\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{16}$. The general
statement in [76] is that given an anomaly index $\upnu\in\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{16}$, we can at most
construct a fully symmetric gapped TQFT _if and only if_ $4\mid 2\upnu.$
Namely, 4 has to be a divisor of $2\upnu$. Apparently, the $4\mid 2\upnu$ is
true only when $\upnu$ is even.

Since ${\upnu=-N_{\text{generation}}}$, the case of ${\upnu}=1$ (for a single
generation) and ${\upnu}=3$ (for three generations) are particularly important
for the high energy physics phenomenology. This means that we are not able to
directly construct any 4d symmetric gapped TQFT that explicitly matches the
same $\mathbb{Z}_{16}$ anomaly for one right-handed neutrino (${\upnu}=1$) or
three right-handed neutrinos (${\upnu}=3$).

  2. 2.

When ${\upnu}$ is even, such as ${\upnu}=2,4,6,8,\dots\in\mathbb{Z}_{16}$,
Ref. [46, 38, 77, 78] suggested that the symmetry-extension method [51] can
trivialize the ’t Hooft anomaly. Furthermore, Cordova-Ohmori [76] shows that
there is no obstruction to construct a symmetry-preserving gapped TQFT phase
for any even ${\upnu_{\text{even}}}\in\mathbb{Z}_{16}$. We can verify the
claim by rewriting (3.15) in terms of the
$(\frac{\upnu_{\text{even}}}{2})\in\mathbb{Z}_{8}$ index:

$\Big{(}{\bf
Z}_{{\text{$5$d-iTQFT}}}\Big{)}^{\upnu_{\text{even}}}=\exp(\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{16}\cdot{\upnu_{\text{even}}}\cdot\eta(\text{PD}({\cal
A}_{{\mathbb{Z}_{2}}}))\bigg{\rvert}_{M^{5}})=\exp(\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{8}\cdot(\frac{{\upnu_{\text{even}}}}{2})\cdot\Big{(}\text{ABK}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{3}))\Big{)}\bigg{\rvert}_{M^{5}})\\\
=\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{8}\cdot(\frac{{\upnu_{\text{even}}}}{2})\cdot\Big{(}4\cdot\text{Arf}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{3}))+2\cdot{\tilde{\eta}}(\text{PD}(({\cal
A}_{{\mathbb{Z}_{2}}})^{4}))+({\cal
A}_{{\mathbb{Z}_{2}}})^{5}\Big{)}\bigg{\rvert}_{M^{5}}),$ (3.16)

with a 2d Arf-Brown-Kervaire (ABK) invariant. Notice that (3.16) can become
trivialized if we can trivialize the $({\cal A}_{{\mathbb{Z}_{2}}})^{2}$
factor. In fact, the $({\cal A}_{{\mathbb{Z}_{2}}})^{2}$ can be trivialized by
the symmetry extension [51]:

$0\to\mathbb{Z}_{2}\to{\mathbb{Z}_{4,X}}\to\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}\to
0.$

Namely, the 2-cocycle topological term $({\cal A}_{{\mathbb{Z}_{2}}})^{2}$ in
$\operatorname{H}^{2}(\mathrm{B}(\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}),{\rm
U}(1))$ becomes a coboundary once we lifting the
$\frac{\mathbb{Z}_{4,X}}{\mathbb{Z}_{2}^{F}}$-gauge field ${\cal
A}_{{\mathbb{Z}_{2}}}$ to a ${\mathbb{Z}_{4,X}}$ gauge field in
$\operatorname{H}^{2}(\mathrm{B}{\mathbb{Z}_{4,X}},{\rm U}(1))$. So this
suggests that the following symmetry extension for the spacetime-internal
symmetry161616See more discussions in Sec. 5 of [9], and in [78].

$\displaystyle 1\to[\mathbb{Z}_{2}]\to{\rm
Spin}\times{\mathbb{Z}_{4,X}}\to{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}\to 1.$ (3.17)

can fully trivialize any even ${\upnu_{\text{even}}}\in\mathbb{Z}_{16}$
cobordism invariant given in (3.16). The $[\mathbb{Z}_{2}]$ means that we can
gauge the _anomaly-free_ normal subgroup $[\mathbb{Z}_{2}]$ in the total group
${\rm Spin}\times{\mathbb{Z}_{4,X}}$. This symmetry extension (3.17) also
means that a 4d $[\mathbb{Z}_{2}]$ gauge theory preserves the ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$ symmetry while also
saturates the even ${\upnu_{\text{even}}}\in\mathbb{Z}_{16}$ anomaly. This 4d
$[\mathbb{Z}_{2}]$ gauge theory is the _anomalous_ symmetric gapped non-
invertible TQFT (with ’t Hooft anomaly of ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$-symmetry) desired in the
Scenario 2(2a).

Since a symmetric anomalous 4d TQFT only exists with even
${\upnu_{\text{even}}}\in\mathbb{Z}_{16}$, below we formulate the path
integral ${\bf Z}_{{\text{4d-TQFT}}}^{(\upnu_{\rm 4d}=2)}[{\cal
A}_{{\mathbb{Z}_{4}}}]$ of the root phase $\upnu_{\rm 4d}=2$. We generalize
the boundary TQFT construction in the Section 8 of [47]. With
${\upnu_{\text{even}}}=2\in\mathbb{Z}_{16}$, we have (3.14) with the input of
5d bulk iTQFT (3.16), then we can explicitly construct the partition function
on a 5d manifold $M^{5}$ with a 4d boundary $M^{4}\equiv\partial M^{5}$
as,171717We use the $\smile$ notation for the cup product between cohomology
classes, or between a cohomology class and a fermionic topological invariant
(paired via a Poincaré dual PD). We use the $\cup$ notation for the surgery
gluing the boundaries of two manifolds within relative homology classes. So
the $M_{1}\cup M_{2}$ means gluing the boundary $\partial
M_{1}=\overline{\partial M_{2}}$ such that the common orientation of
$\overline{\partial M_{2}}$ is the reverse of ${\partial M_{2}}$.

${\bf Z}_{{\text{5d-iTQFT}}}[{\cal A}_{{\mathbb{Z}_{4}}}]\cdot{\bf
Z}_{{\text{4d-TQFT}}}[{\cal
A}_{{\mathbb{Z}_{4}}}]=\sum_{c\in{\partial^{\prime}}^{-1}(\partial[\text{PD}({\cal
A}^{3})])}\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{8}\text{ABK}(c\cup\text{PD}({\cal
A}^{3}))}\quad\quad\quad\quad\quad\quad\\\
\cdot\frac{1}{2^{|\pi_{0}(M^{4})|}}\sum_{\begin{subarray}{c}a\in
C^{1}(M^{4},\mathbb{Z}_{2}),\\\ b\in
C^{2}(M^{4},\mathbb{Z}_{2})\end{subarray}}(-1)^{\int_{M^{4}}b(\delta a+{\cal
A}^{2})}\cdot\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{8}\text{ABK}(c\cup\text{PD}^{\prime}({\cal A}a))}.$ (3.18)

We write the mod 2 cohomology class $\mathbb{Z}_{2}$ gauge field as ${\cal
A}\equiv{\cal A}_{{\mathbb{Z}_{2}}}\equiv({\cal A}_{{\mathbb{Z}_{4}}}\mod
2)\in\operatorname{H}^{1}(M^{5},\mathbb{Z}_{2})$. Here come some remarks about
this 5d bulk iTQFT-4d boundary TQFT partition function (3.18):

  1. 1).

The $(\text{PD}({\cal A}^{3}))$ is a 2d manifold taking the Poincaré dual (PD)
of 3-cocycle ${\cal A}^{3}$ in the $M^{5}$; but the 2d manifold
$(\text{PD}({\cal A}^{3}))$ may touch the the 4d boundary $\partial
M^{5}=M^{4}$. The 1d boundary $\partial(\text{PD}({\cal A}^{3}))$ can be
regarded as the 1d intersection between the 2d $(\text{PD}({\cal A}^{3}))$ and
the $M^{4}$.

  2. 2).

More precisely, for a ${\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}$
manifold $M^{5}$ with a boundary, we have used the Poincaré-Lefschetz duality
for a manifold with boundaries:

$\displaystyle{\cal
A}^{2}\in\operatorname{H}^{3}(M^{5},\mathbb{Z}_{2})\xrightarrow{\cong}\operatorname{H}_{2}(M^{5},M^{4},\mathbb{Z}_{2})\ni\text{PD}({\cal
A}^{3}).$ (3.19)

  3. 3).

For any pair $({\rm S},{\rm S}^{\prime})$, where ${\rm S}^{\prime}$ is a
subspace of $S$, the short exact sequence of chain complexes

$\displaystyle 0\to C_{*}({\rm S}^{\prime})\to C_{*}({\rm S})\to C_{*}({\rm
S},{\rm S}^{\prime})\to 0$ (3.20)

with $C_{n}({\rm S},{\rm S}^{\prime})\equiv C_{n}({\rm S})/C_{n}({\rm
S}^{\prime})$, induces a long exact sequence of homology groups

$\displaystyle\cdots\to\operatorname{H}_{n}({\rm
S}^{\prime})\to\operatorname{H}_{n}({\rm S})\to\operatorname{H}_{n}({\rm
S},{\rm S}^{\prime})\xrightarrow{\partial}\operatorname{H}_{n-1}({\rm
S}^{\prime})\to\cdots.$ (3.21)

Here $\operatorname{H}_{n}({\rm S},{\rm S}^{\prime})$ is the relative homology
group, and $\partial$ is the boundary map.
$\bullet$ Take $({\rm S},{\rm S}^{\prime})=(\text{PD}({\cal
A}^{3}),\partial\text{PD}({\cal A}^{3}))$, we denote the boundary map by
$\partial$:

$\displaystyle\operatorname{H}_{2}(\text{PD}({\cal
A}^{3}),\partial\text{PD}({\cal
A}^{3}))\xrightarrow{\partial}\operatorname{H}_{1}(\partial\text{PD}({\cal
A}^{3})).$ (3.22)

Here $\text{PD}({\cal A}^{3})$ is not a closed 2-manifold, but which has a
boundary closed 1-manifold $\partial\text{PD}({\cal A}^{3})$.
$\bullet$ Take $({\rm S},{\rm S}^{\prime})=(M^{4}=\partial
M^{5},\partial\text{PD}({\cal A}^{3}))$, we denote another boundary map by
$\partial_{1}$:

$\displaystyle\operatorname{H}_{2}(M^{4},\partial\text{PD}({\cal
A}^{3}))\xrightarrow{\partial_{1}}\operatorname{H}_{1}(\partial\text{PD}({\cal
A}^{3})).$ (3.23)

Both $M^{4}=\partial M^{5}$ and $\partial\text{PD}({\cal A}^{3})$ are closed
manifolds, of 4d and 1d, respectively.

  4. 4).

Now, the $c$ is defined as a 2d surface living on the boundary $M^{4}$. The
${\partial_{1}}c$ uses the boundary map (3.23)’s $\partial_{1}$ of $c$ on the
$M^{4}$. We can compensate the 2-surface $\text{PD}({\cal A}^{3})$ potentially
with a 1-boundary, by gluing it with $c$ to make a closed 2-surface. To do so,
we require both $\text{PD}({\cal A}^{3})$ and $c$ share the same 1d boundary.

  5. 5).

The ${c\in{\partial_{1}}^{-1}(\partial[\text{PD}({\cal A}^{3})])}$ also means

$\displaystyle{{\partial_{1}}c=\partial[\text{PD}({\cal
A}^{3})]=[\partial\text{PD}({\cal A}^{3})]}.$ (3.24)

$\bullet$ The $[\text{PD}({\cal A}^{3})]$ means the fundamental class and the
relative homology class of the 2d manifold $\text{PD}({\cal A}^{3})$.
$\bullet$ The $\partial[\text{PD}({\cal A}^{3})]=[\partial\text{PD}({\cal
A}^{3})]$ means the boundary of the fundamental class (via the boundary
$\partial$ map in (3.22)) is equivalent to the fundamental class of the
boundary $\partial$ of $\text{PD}({\cal A}^{3})$. Beware that the two
$\partial$ operations in $\partial[\text{PD}({\cal
A}^{3})]=[\partial\text{PD}({\cal A}^{3})]$ have different meanings.
$\bullet$ Eqn. (3.24) exactly matches the requirement that the bulk 2-surface
$\text{PD}({\cal A}^{3})$ (living in $M^{5}$) and the boundary 2-surface $c$
(living on $M^{4}$) share the same 1d boundary.

  6. 6).

Note that when $M^{5}$ is a closed manifold with no boundary $M^{4}=\partial
M^{5}=\emptyset$ thus $c=\emptyset$, then the term

$\sum_{c\in{\partial_{1}}^{-1}(\partial[\text{PD}({\cal A}^{3})])}\hskip
1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{8}\text{ABK}(c\cup\text{PD}({\cal A}^{3}))}\text{ is equivalently
reduced to }\hskip 1.0pt\mathrm{e}^{\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{8}\text{ABK}(\text{PD}({\cal A}^{3}))}=\Big{(}{\bf
Z}_{{\text{$5$d-iTQFT}}}\Big{)}^{\upnu_{\text{even}}=2}.$

This is a satisfactory consistent check, consistent with the 5d bulk-only
iTQFT at ${\upnu_{\text{even}}}=2$ in (3.16).

  7. 7).

The $a\in C^{1}(M^{4},\mathbb{Z}_{2})$ means that $a$ is a 1-cochain, and the
$b\in C^{2}(M^{4},\mathbb{Z}_{2})$ means that $b$ is a 2-cochain. The factor
$(-1)^{\int_{M^{4}}b(\delta a+{\cal A}^{2})}=\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi{\int_{M^{4}}b(\delta a+{\cal A}^{2})})$ gives the weight of the 4d
$\mathbb{Z}_{2}$ gauge theory. The $b\delta a$ term is the level-2 BF theory
written in the mod 2 class.181818The continuum QFT version of this
$\mathbb{Z}_{2}$ gauge theory is $\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\int_{M^{4}}\frac{2}{2\pi}b\hskip
1.0pt\mathrm{d}a+\frac{1}{\pi^{2}}b{\cal A}^{2})})$, where the $b$ integration
over a closed 2-cycle, $\oiint b$, can be $n\pi$ with some integer
$n\in\mathbb{Z}$. The $a$ and ${\cal A}$ integration over a closed 1-cycle,
$\oint a$ and $\oint{\cal A}$, can be $n\pi$ with some integer
$n\in\mathbb{Z}$. The path integral sums over these distinct cochain classes.

The variation of $b$ gives the equation of motion $(\delta a+{\cal
A}^{2})=0\mod 2.$ In the path integral, we can integrate out $b$ to give the
same constraint $(\delta a+{\cal A}^{2})=0\mod 2.$ This is precisely the
trivialization of the second cohomology class,

$\displaystyle{\cal A}^{2}=({\cal A}_{{\mathbb{Z}_{2}}})^{2}=\delta a\mod 2$
(3.25)

so the 2-cocycle becomes a 2-coboundary which splits to a 1-cochain $a$. This
exactly matches the condition imposed by the symmetry extension (3.17):
$1\to[\mathbb{Z}_{2}]\to{\rm Spin}\times{\mathbb{Z}_{4,X}}\to{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}\to 1$, where the ${\cal
A}^{2}$ term in ${\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}$
becomes trivialized as a coboundary ${\cal A}^{2}=\delta a$ (so ${\cal
A}^{2}=0$ in terms of a cohomology or cocycle class) in ${\rm
Spin}\times{\mathbb{Z}_{4,X}}$.

  8. 8).

The $\frac{1}{2^{|\pi_{0}(M^{4})|}}$ factor mod out the gauge redundancy for
the boundary 4d $\mathbb{Z}_{2}$ gauge theory. The $\pi_{0}(M^{4})$ is the
zeroth homotopy group of $M^{4}$, namely the set of all path components of
$M^{4}$. Thus, we only sum over the gauge inequivalent classes in the path
integral.

  9. 9).

The ${\text{ABK}}(c\cup\text{PD}({\cal A}^{3}))$ is defined on a 2d manifold
with $\mathrm{Pin}^{-}$ structure. Recall the $M^{5}$ has the ${\rm
Spin}\times_{\mathbb{Z}_{2}}{\mathbb{Z}_{4}}$ structure. If $M^{5}$ is closed,
then there is a natural Smith map to induce the 2d $\mathrm{Pin}^{-}$
structure on the closed surface via $\text{PD}({\cal A}^{3})$. However, the
$M^{5}$ has a boundary $M^{4}$, so $\text{PD}({\cal A}^{3})$ may not be closed
— the previously constructed closed 2-surface $(c\cup\text{PD}({\cal A}^{3}))$
is meant to induce a 2d $\mathrm{Pin}^{-}$ structure.191919We do not yet know
whether it is always possible to induce a unique 2d $\mathrm{Pin}^{-}$
structure on $(c\cup\text{PD}({\cal A}^{3}))$ for any possible pair of data
$(M^{5},M^{4}=\partial M^{5})$ given any $\mathrm{Pin}^{-}$ $M^{5}$. However,
we claim that it is possible to find some suitable $M^{4}$ so that the 2d
$(c\cup\text{PD}({\cal A}^{3}))$ has $\mathrm{Pin}^{-}$ induced, thus in this
sense the ${\text{ABK}}(c\cup\text{PD}({\cal A}^{3}))$ is defined. For physics
purposes, it is enough that we can focus on studying the theory on these types
of $(M^{5},M^{4}=\partial M^{5})$.  Then we compute the ABK on this closed
2-surface $(c\cup\text{PD}({\cal A}^{3}))$.

  10. 10).

Let us explain the other term ${\text{ABK}(c\cup\text{PD}^{\prime}({\cal
A}a))}$ of the 4d boundary TQFT in (3.18). The $\text{PD}^{\prime}$ is the
Poincaré dual on $M^{4}=\partial M^{5}$. Since the fundamental classes of
$M^{5}$ and $M^{4}=\partial M^{5}$ are related by

$\displaystyle[M^{5}]\in\operatorname{H}_{5}(M^{5},M^{4},\mathbb{Z}_{2})\xrightarrow{\partial}\operatorname{H}_{4}(M^{4},\mathbb{Z}_{2})\ni[M^{4}]\equiv[\partial
M^{5}].$ (3.26)

Here we have the following relations:202020Let us clarify the notations:
$\partial$, $\partial^{\prime}$, and $\partial_{1}$. The boundary notation
$\partial$ may mean as (1) taking the boundary, or (2) in the boundary map of
relative homology class in (3.22). It should be also clear to the readers that
$\bullet$ the $\partial$ is associated with the operations on objects living
in the bulk $M^{5}$ or ending on the boundary $M^{4}$,
$\bullet$ while the $\partial^{\prime}$ is associated with the operations on
objects living on the boundary $M^{4}$ alone.
The $\partial_{1}$ is defined as another boundary map in (3.23).

PD $\displaystyle=$ $\displaystyle[M^{5}]\cap,$ (3.27)
$\displaystyle\text{PD}^{\prime}$ $\displaystyle=$
$\displaystyle[M^{4}]\cap=[\partial M^{5}]\cap=\partial[M^{5}]\cap,$ (3.28)
$\displaystyle{\partial^{\prime}}\text{PD}^{\prime}({\cal A}a)$
$\displaystyle=$ $\displaystyle{\text{PD}^{\prime}(\delta({\cal
A}a))=\text{PD}^{\prime}({\cal A}^{3})=\partial[\text{PD}({\cal
A}^{3})]=\partial_{1}c.}$ (3.29)

$\bullet$ The cap product $\cap$ here is to define PD homology class, such
that $\text{PD}({\cal A})=[M^{5}]\cap{\cal A}$.
$\bullet$ Here we use $[M^{4}]=[\partial M^{5}]=\partial[M^{5}]$: the
fundamental class of boundary of $M^{5}$ gives the boundary of fundamental
class.
$\bullet$ The (3.27)’s first equality
${\partial^{\prime}}\text{PD}^{\prime}({\cal
A}a)=\text{PD}^{\prime}(\delta({\cal A}a))$ uses the coboundary operator
$\delta$ on the cohomology class ${\cal A}a$.
$\bullet$ The (3.27)’s second equality $\text{PD}^{\prime}(\delta({\cal
A}a))=\text{PD}^{\prime}({\cal A}^{3})$ uses the condition $\delta{\cal A}=0$
and the trivialization condition (3.25): $\delta a={\cal A}^{2}$.
$\bullet$ The (3.27)’s third equality $\text{PD}^{\prime}({\cal
A}^{3})=\partial[\text{PD}({\cal A}^{3})]$, we use “ _the naturality of the
cap product_ ”. There are natural pushforward and pullback maps on homology
and cohomology, related by the projection formula, also known as “the
naturality of the cap product.”
$\bullet$ The (3.27)’s last equality $\partial[\text{PD}({\cal
A}^{3})]=\partial_{1}c$ is based on (3.24). Importantly, as a satisfactory
consistency check, this also shows that the 2-surface $c$ obeys:

the $c$ in ${\text{ABK}}(c\cup\text{PD}({\cal A}^{3}))$ is the same $c$ in
${\text{ABK}(c\cup\text{PD}^{\prime}({\cal A}a))}$. (3.30)

$\bullet$ As before, the union $(c\cup\text{PD}^{\prime}({\cal A}a))$ is a
closed 2-surface and we induce a $\mathrm{Pin}^{-}$ structure on this
2-surface. So we can compute the ABK on this closed 2-surface
$(c\cup\text{PD}^{\prime}({\cal A}a))$ on the $M^{4}$.212121Similar to
Footnote 19, we can find some suitable $M^{4}$ so that the 2d
$\mathrm{Pin}^{-}$ is induced, thus in this sense the
${\text{ABK}}(c\cup\text{PD}^{\prime}(Aa))$ is defined.

In summary, we have constructed the 4d $\mathbb{Z}_{2}$-gauge theory in (3.18)
preserving the $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}})$-structure, namely it is a
$({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}})$-symmetric TQFT but
with ${\upnu_{\text{even}}}=2\in\mathbb{Z}_{16}$ anomaly. This can be used to
compensate the anomaly $\upnu=-N_{\text{generation}}\mod 16$ with
$N_{\text{generation}}=2$, two generations of missing right-handed neutrinos.
We could not however directly construct the symmetric gapped TQFT for $\upnu$
is odd (thus symmetric TQFTs not possible for $N_{\text{generation}}=1$ or
$3$), due to the obstruction found in [46, 76].

###  3.3 General Principle

The discussions in Sec. 3.2.1 says that only the even integer
${\upnu_{4d,\text{even}}}\in\mathbb{Z}_{16}$, the $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}})$-symmetry-preserving TQFT
exists. This prompts us to improve the SM version (3.33) into:

$\displaystyle{{\bf Z}_{\begin{subarray}{c}\text{5d-iTQFT/}\\\
\text{4d-SM+TQFT}\end{subarray}}[{\cal A}_{{\mathbb{Z}_{4}}}]\equiv{\bf
Z}_{{\text{5d-iTQFT}}}^{(-\upnu_{\rm 5d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\bf Z}_{{\text{4d-TQFT}}}^{(\upnu_{\rm
4d,even})}[{\cal A}_{{\mathbb{Z}_{4}}}]\cdot{\mathbf{Z}}_{{\rm
SM}}^{(n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}})}[{\cal
A}_{{\mathbb{Z}_{4}}}].}$ (3.31)

The GUT version (3.13) should be adjusted into:

$\displaystyle{{\bf Z}_{\begin{subarray}{c}\text{5d-iTQFT/}\\\
\text{4d-GUT+TQFT}\end{subarray}}[{\cal A}_{{\mathbb{Z}_{4}}}]\equiv{\bf
Z}_{{\text{5d-iTQFT}}}^{(-\upnu_{\rm 5d})}[{\cal
A}_{{\mathbb{Z}_{4}}}]\cdot{\bf Z}_{{\text{4d-TQFT}}}^{(\upnu_{\rm
4d,even})}[{\cal A}_{{\mathbb{Z}_{4}}}]\cdot{\mathbf{Z}}_{{\rm
GUT}}^{(n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}})}[{\cal
A}_{{\mathbb{Z}_{4}}}].}$ (3.32)

Also the anomaly constraint (2.14) becomes:

${(-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}(N_{\text{gen}}=3)}+n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}+\upnu_{\rm
4d,even}-\upnu_{\rm 5d})=0\mod 16.}$

This implies that the existence of symmetry-preserving 4d TQFT sector
requires:

$\boxed{\text{$(n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}-\upnu_{\rm
5d})$ must be an even integer.}}$

Now we have derived a Ultra Unification path integral in (3.31) and (3.32),
including 4d SM (3.1), 4d GUT (3.8), 5d iTQFT (3.18) and 4d TQFT (3.15),
comprising many Scenarios and their linear combinations enlisted in Sec. 2.3:
1(1a), 1(1b), 1(1c), 2(2a), and 2(2b). Then we can dynamically gauge the
appropriate bulk-boundary global symmetries, promoting the theory to a bulk
gauge theory in Scenarios 2(2c) and 2(2d), or break some of the (global or
gauge) symmetries to 2(2e).

##### General Principle

In summary, we propose a general principle behind the Ultra Unification:
1\. We start with a QFT in general as an effective field theory (EFT) given
some full spacetime-internal global symmetry $G$, say in a D dimensional (Dd)
spacetime.
2\. We check the anomaly and cobordism constraint given by $G$ via computing
$\Omega^{\text{D+1}}_{G}\equiv\mathrm{TP}_{\text{D+1}}(G)$.
3\. We check the anomaly-matching of the Dd QFT/EFT constrained by cobordism
$\Omega^{\text{D+1}}_{G}\equiv\mathrm{TP}_{\text{D+1}}(G)$.
4\. If all anomalies are matched, then we do not require any new hidden sector
to define Dd QFT/EFT.
5\. If some anomalies are mismatched, then either
(1) we need to break some symmetry out of $G$, or
(2) we extend the symmetry $G$ to an appropriate $\tilde{G}$ to trivialize
nonperturbative global anomalies in
$\Omega^{\text{D+1}}_{\tilde{G}}\equiv\mathrm{TP}_{\text{D+1}}(\tilde{G})$, or
(3) we propose new hidden sectors appending to Dd QFT/EFT, with a schematic
path integral (say if we add Dd-TQFT and (D+1)d-iTQFT onto the original
theory):

$\displaystyle\boxed{{\bf Z}_{\begin{subarray}{c}\text{(D+1)d-iTQFT/}\\\
\text{Dd-QFT/EFT+TQFT}\end{subarray}}[{\cal A}]\equiv{\bf
Z}_{{\text{(D+1)d-iTQFT}}}[{\cal A}]\cdot{\bf Z}_{{\text{Dd-TQFT}}}[{\cal
A}]\cdot{\mathbf{Z}}_{\text{Dd-QFT/EFT+TQFT}}[{\cal A}].}$ (3.33)

###  3.4 Detect Topological Forces/Sectors and the Essence

##### Detect Topological Forces and Sectors: Neutrino Oscillations and Dark
Matter

By looking at Fig. 1, in the Higgs vacuum where our Standard Model effective
field theory (SM EFT) resides in, we have only detected the Strong,
Electromagnetic, and Weak in the subatomic physics. The GUT forces are weaker
than the Weak force, and the Topological Force is further weaker than the GUT
and Weak forces. So how could we experimentally detect Topological Forces?

Notice that the gravity is further weaker than all other forces. (So how could
we experimentally detect gravity?) But the gravity has accumulative effects
that only have the gravitational attractions. Without doubt, the gravity has
been detected by everyone and by all astrophysics and cosmology observations.
The gravity had been detected first in the human history among all the forces!

Similarly, although Topological Force is also weak (but stronger than the
gravity), Topological Force is infinite-range or long-range which does _not
decay in the long distance_ , and mediates between the linked
worldline/worldsheet/worldvolume trajectories of the charged (point-like or
extended) objects via fractional or categorical anyonic statistical
interactions. So in principle, _we may have already experienced Topological
Force in our daily life, in a previously scientifically unnoticed way._
222222In a metaphysics way sense, perhaps the Topological Force can be related
to the supernatural or religious phenomena. If so, the supernatural or
religious phenomena could be explained now in precise scientific,
mathematical, and physical ways (via the Beyond the Standard Model Topological
Force).
For example, if the $\mathbb{Z}_{4,X}$ is dynamically gauged, there is a
dynamical discrete gauge Wilson line connecting all SM fermions living in 4d
SM or GUT (e.g., quarks and electrons in our body). Namely, the SM fermions
can live at the open ends of the $\mathbb{Z}_{4,X}$ gauged Wilson line. Thus
there could be long-distance topological interactions and communications
between the $\mathbb{Z}_{4,X}$-gauge charged objects.
Moreover, the $\mathbb{Z}_{4,X}$ gauged Wilson line as a ${\cal
A}_{\mathbb{Z}_{4}}$ gauge field on the 4d theory can be leaked into the 5d
bulk theory as a ${\cal A}_{\mathbb{Z}_{2}}$ gauge field. In the 5d bulk, a
nontrivial link configuration can be charged under the other end of ${\cal
A}_{\mathbb{Z}_{2}}$ [9].
In any case, every phenomenon and every law of Nature should be explained by
mathematics and physics principles.

##### Neutrino Oscillations and Dark Matter

In fact, Ref. [9] had proposed that the Topological Force may cause (thus be
detected) by the phenomenon of neutrino oscillations. We can consider the
Majorana zero modes of the vortices in the 4d TQFT defects. The left-handed
neutrinos (confirmed in the experiments) are nearly gapless/ massless. When
the left-handed neutrinos traveling through the 4d TQFT defects, we may
observe nearly neutrino flavor oscillations interfering with the Majorana zero
modes trapped by the vortices in the 4d TQFT defects.

On the other hand, the gapped heavy excitations (point or extended objects) of
Topological Sector may be a significant contribution of Dark Matter [79].

##### The Essence of Ultra Unification

Finally, we come to the essence of Ultra Unification. What is unified after
all? We have united the Strong, Electromagnetic, Weak, GUT forces and
Topological forces into the same theory in the Ultra Unification QFT/TQFT path
integral (that this theory can also be coupled to curved spacetime geometry
and gravity, at least well-defined in a background non-dynamical way).

However, the Grand Unification [6, 7] united the three gauge interactions of
the SM into a single electronuclear force under a simple Lie group gauge
theory. Do we have the equivalent statement to also unite Strong,
Electromagnetic, Weak, GUT forces and Topological forces into a single force
at a high enough energy? We believe that the definite answer relies on
studying the details of analogous topological quantum phase transitions [10,
79], such as those explored in 4d [62, 80, 81, 82].

###  3.5 Summary of Ultra Unification in Drawings

Let us summarize what we have done in this work in drawings.
$\bullet$ Fig. 2: We have started from the Nature given Standard Model (SM)
quarks and leptons, and their quantum numbers, in three generations.
$\bullet$ Fig. 3: We have included gauge forces and (various) Higgs for SM and
Grand Unification (GUT).
$\bullet$ Fig. 4: After the essenthial check of the anomaly matching and
cobordism constraints, the detection of the $\mathbb{Z}_{16}$ global anomaly
for 15n Weyl fermion SM and GUT implies that we can choose (as one of many
options) to realized our 4d world living on an extra dimensional 5d invertible
TQFT (a 5d cobordism invariant). Here Fig. 4 may be understood as a 1-brane 4d
world with an extra large dimension.
$\bullet$ Fig. 5: Ultra Unification incorporates the SM, GUT, and Topological
forces into the same theory (that this theory can also be coupled to curved
spacetime geometry and gravity in a background non-dynamical way). (1) The
upper left has the 4d SM and GUT. (2) The upper right has the 4d non-
invertible TQFT. (3) The bottom has the 5d invertible TQFT (alternatively 5d
non-invertible TQFT if we dynamically gauge the full $\mathbb{Z}_{4,X}$). The
three sectors can communicate with each other mediated via the dynamical
$\mathbb{Z}_{4,X}$ gauge forces. Here Fig. 5 may be understood as a multi-
branes or 2-brane 4d world with an extra dimension. The issues of mirror
fermion doubling [83] on the mirror world, depending on the precise anomaly
matching on the mirror sector, may be fully trivially gapped (if anomaly-
free), or may be topologically ordered gapped with TQFT. These issues are
tackled in many recent (selective) works [84, 85, 86, 87, 88, 89, 90, 91, 13,
92, 93].
$\bullet$ Fig. 6: In Quantum Matter terminology, we show that SM and GUT
belong to a continuous gauge theory, (global/gauge) symmetry breaking mass,
and Ginzburg-Landau paradigm. In comparison, the new sectors that we introduce
include a discrete fermionic gauge theory, symmetry extension topological
mass, and modern issues on symmetry, topology, nonperturbative interactions
and short/long-range entanglements.

Figure 2: Standard Model, the $su(5)$ GUT and the $so(10)$ GUT quarks and
leptons. Figure 3: Includ gauge forces and (various) Higgs to the Standard
Model and GUT. Figure 4: Detection of the $\mathbb{Z}_{16}$ global anomaly for
15n Weyl fermion SM and GUT. It can be realized as adding an extra dimensional
5d invertible TQFT (a 5d cobordism invariant), where the 4d world lives on its
boundary.  Figure 5: Ultra Unification incorporates the Strong,
Electromagnetic, Weak, GUT, and Topological forces into the same theory in the
QFT/TQFT path integral (that this theory can also be coupled to curved
spacetime geometry and gravity in a background non-dynamical way). (1) The
upper left has the 4d SM and GUT. (2) The upper right has the 4d non-
invertible TQFT. (3) The bottom has the 5d invertible TQFT (alternatively 5d
non-invertible TQFT if we dynamically gauge the full $\mathbb{Z}_{4,X}$). The
three sectors can communicate with each other mediated via the dynamical
$\mathbb{Z}_{4,X}$ gauge forces.  Figure 6: Ultra Unification summarized in
terms of Quantum Matter terminology.

##  4 Acknowledgements and Bibliography

JW thanks the participants of Quantum Matter in Mathematics and Physics
program at Harvard University CMSA for the enlightening atmosphere. Part of
this work had been presented at Higher Structures and Field Theory at Erwin
Schrödinger Institute in Wien (August 4, 2020) [94], and at Harvard University
Particle Physics Lunch (November 30, 2020) and String Lunch (December 4,
2020). JW appreciates the mental support from Shing-Tung Yau.232323Instead of
writing or drawing an image of the author’s mental feelings, a piece of Johann
Sebastian Bach’s music “The Goldberg Variations, BWV 988 (1861)” (e.g.,
Variatio 25. a 2 Clav.: Adagio) may illuminate this well. This work is
supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical
Physics” and Center for Mathematical Sciences and Applications at Harvard
University.

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