# Cobordism and Deformation Class of the Standard Model

Juven Wang  Center of Mathematical Sciences and Applications, Harvard
University, MA 02138, USA  Zheyan Wan  Yau Mathematical Sciences Center,
Tsinghua University, Beijing 100084, China  Yi-Zhuang You  Department of
Physics, University of California, San Diego, CA 92093, USA

###### Abstract

’t Hooft anomalies of quantum field theories (QFTs) with an invertible global
symmetry $G$ (including spacetime and internal symmetries) in a $d$d spacetime
are known to be classified by a $d+1$d cobordism group TP${}_{d+1}(G)$, whose
group generator is a $d+1$d cobordism invariant written as an invertible
topological field theory (iTFT) with a partition function
${\mathbf{Z}}_{d+1}$. It has recently been proposed that the deformation class
of QFTs is specified by its symmetry $G$ and an iTFT ${\mathbf{Z}}_{d+1}$.
Seemingly different QFTs of the same deformation class can be deformed to each
other via quantum phase transitions. In this work, we ask which cobordism
class and deformation class control the 4d Standard Model (SM) of ungauged or
gauged $({\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1))/\mathbb{Z}_{q}$ group
for $q=1,2,3,6$ with a continuous or discrete baryon minus lepton
$\mathbf{B}-\mathbf{L}$ like symmetry. We show that the answer contains some
combination of 5d iTFTs: two $\mathbb{Z}$ classes associated with
$({\mathbf{B}-\mathbf{L}})^{3}$ and $({\mathbf{B}-\mathbf{L}})$-(gravity)2 4d
perturbative local anomalies, a $\mathbb{Z}_{16}$ class Atiyah-Patodi-Singer
$\eta$ invariant and a $\mathbb{Z}_{2}$ class Stiefel-Whitney $w_{2}w_{3}$
invariant associated with 4d nonperturbative global anomalies, and additional
$\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ global anomalies involving higher
symmetries whose charged objects are Wilson electric or ’t Hooft magnetic line
operators. Out of the multiple infinite $\mathbb{Z}$ classes of local
anomalies and 24576 classes of global anomalies, we pin down the deformation
class of the SM labeled by $(N_{f},n_{\nu_{R}},$ p${}^{\prime},q)$, the family
number, the total “right-handed sterile” neutrino number, the magnetic
monopole datum, and the mod $q$ relation. We show that Grand Unification such
as Georgi-Glashow $su(5)$, Pati-Salam $su(4)\times su(2)\times su(2)$, Barr’s
flipped $u(5)$, and the familiar or modified $so(n)$ models of Spin($n$) gauge
group, e.g., with $n=10,18$ can all reside in an appropriate SM deformation
class. We show that Ultra Unification, which replaces some of sterile
neutrinos with new exotic gapped/gapless sectors (e.g., topological or
conformal field theory) or gravitational sectors with topological origins via
cobordism constraints, also resides in an SM deformation class. Neighbor
quantum phases near SM or their phase transitions, and neighbor gapless
quantum critical regions naturally exhibit beyond SM phenomena.

##  I Introduction and Summary

Symmetries in quantum field theories (QFTs) in the strict sense mean that the
QFT is invariant under certain sets of transformations (i.e., symmetry
transformations). The symmetry invariance of QFT may be exhibited in many ways
depending on how the QFT is formulated: (1) in the path integral formulation
(say in $d$d for $d$-dimensional spacetime), the symmetry transformation on
quantum fields or spacetime coordinates leaves the path integral
${\mathbf{Z}}_{d}$ invariant, (2) in the Hamiltonian approach, the symmetry
transformation corresponds to a quantum operator $\hat{U}$ commuting with the
Hamiltonian $\hat{H}$, i.e., $[\hat{H},\hat{U}]=0$. Importantly, the symmetry
may only be invariant up to a global complex phase; for example, the symmetry
transformation can map the path integral ${\mathbf{Z}}_{d}\mapsto\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Theta}{\mathbf{Z}}_{d}$.
If $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Theta}$ cannot
be eliminated by $d$d local counter terms, this $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Theta}$ manifests a mild
violation of symmetry invariance. Such a symmetry is an _anomalous symmetry_
(i.e., symmetry with _’t Hooft anomaly_ ’t Hooft (1980)). In contrast, any
symmetry transformation without outputting any $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Theta}$ is an _anomaly-free symmetry_. Note that
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Theta}$ is
invertible (i.e., $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Theta}\cdot\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Theta}=1$) — the _anomaly inflow_ Callan and Harvey (1985); Witten and
Yonekura (2019) states that the $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Theta}$ not only characterizes a $d$d invertible
’t Hooft anomaly of a symmetry $G$, but also characterizes a one-higher
dimensional _invertible topological field theory_ (_iTFT_) with a partition
function ${\mathbf{Z}}_{d+1}$ of the symmetry $G$. The ${\mathbf{Z}}_{d+1}$
turns out to be a _cobordism invariant_ in $d+1$d (Witten and Yonekura (2019);
Kapustin (2014); Kapustin  _et al._ (2015); Wang  _et al._ (2015); Freed and
Hopkins (2016); Wan and Wang (2019) and references therein). These iTFTs
${\mathbf{Z}}_{d+1}$ form an abelian group: the group binary operation is the
stacking of iTFTs, the identity element is 1, the inverse is
${\mathbf{Z}}_{d+1}^{-1}$; they also obey closure, associativity, and
commutativity. By Freed and Hopkins (2016), this cobordism group is a specific
version $\mathrm{TP}_{d+1}(G)$ for smooth manifolds with $G$-structure.

_Deformation classes of QFTs_ : We can use the symmetry to organize the
quantum fields into the representations of the symmetry group, and give
powerful selection rules constraining the quantum dynamics. Anomalies are
invariant under continuous deformations of parameters of QFTs, including the
renormalization group flow ’t Hooft (1980) (Cordova  _et al._ (2020a, b) and
references therein). Given the importance of symmetries and their associated
’t Hooft anomalies, Seiberg Seiberg (2019) and others Wang and Wen (2020)
conjectured that any QFTs with the same symmetry $G$ and same anomaly
${\mathbf{Z}}_{d+1}$ can be deformed to each other by adding degrees of
freedom at short distances that preserve the same symmetry and maintain the
same anomaly. Namely, the whole system allows a large class of deformations
via all symmetric interactions between the original QFT and other new higher-
energy sectors of extra QFTs. This organization principle connects a large
class of QFTs together within the same data $(G,{\mathbf{Z}}_{d+1})$ via any
symmetric deformation (possibly with discontinuous or continuous quantum phase
transitions Sachdev (2011) between different phases). The _deformation class
of QFTs_ in $d$d Seiberg (2019), is indeed related to the _cobordism or
deformation class of iTFTs_ in one dimensional higher in $d+1$d Freed and
Hopkins (2016).

However, it is widely believed that there is no global symmetry in quantum
gravity (QG) (Harlow and Ooguri (2021) and references therein). Skeptical
readers may hold prejudice against this deformation class labeled by
$(G,{\mathbf{Z}}_{d+1})$, possibly not universal nor useful in QG. But several
reasons below suggest its importance even in QG. First, there could still be
various QFTs with global symmetries localized on a brane or obtained via
compactification from QG. Second, McNamara-Vafa McNamara and Vafa (2019)
suggests that if a $d$d effective theory is obtained via the compactification
from a quantum gravity in a total spacetime $D$d, and if the compact
${(D-d)}$d manifold is in a cobordism class ${\mathbf{Z}}_{D-d}$, it either
suggests the theory belongs to the Swampland in QG, or suggests the
${\mathbf{Z}}_{D-d}$ must be appropriately _trivialized_ to zero in the full
suitably interpreted QG cobordism group $\Omega_{D-d}^{\rm QG}=0$ (e.g., the
trivialization of cobordism class via introducing new defects to cancel each
other’s cobordism class). This original $d$d theory can be connected by a
finite-energy $(d-1)$d domain wall to nothing on the other side. So we may say
that Ref. [McNamara and Vafa, 2019] justifies the use of _deformation class of
quantum gravity_. Third, although there is no global symmetry in QG, there can
be either explicitly broken global symmetry or dynamically gauged symmetry
(i.e., global symmetry is dynamically gauged as gauge symmetry, which we shall
call Weyl’s _gauge principle_ Weyl (1929)) in QG. Therefore, for all the
reasons above, it is promising to explore the idea of cobordism and
deformation class further, in the context of both QFT Seiberg (2019) and QG
McNamara and Vafa (2019).

In this work, we shall focus on a specific question: _What is the deformation
class of the 4d Standard Model_ (SM) Glashow (1961); Salam and Ward (1964);
Salam (1968); Weinberg (1967)? Before attempting to solve this, we shall
sharpen the question further. We need to first specify the spacetime-internal
symmetries of SM, typically written as

$\displaystyle
G\equiv({\frac{{G_{\text{spacetime}}}\times{{G}_{\text{internal}}}}{{N_{\text{shared}}}}})\equiv{{G_{\text{spacetime}}}\times_{{N_{\text{shared}}}}{{G}_{\text{internal}}}}.$
(1)

The ${N_{\text{shared}}}$ is the shared common normal subgroup symmetry
between ${G_{\text{spacetime}}}$ and ${{G}_{\text{internal}}}$, e.g.
${N_{\text{shared}}}$ can be the fermion parity symmetry $\mathbb{Z}_{2}^{F}$,
which acts on fermions by $\psi\mapsto-\psi$. The Lie algebra of the internal
symmetry of SM is $su(3)\times su(2)\times u(1)$, but the global structure Lie
group $G_{\text{SM}_{q}}$ has four possible versions Aharony  _et al._ (2013);
Tong (2017):

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

We also include the baryon minus lepton ${\mathbf{B}-\mathbf{L}}$ vector
symmetry respected by the SM gauge structure and Yukawa-Higgs terms. We
consider both a continuous ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$ or ${\rm
U}(1)_{X}$, or a discrete order-4 finite abelian $\mathbb{Z}_{4,X}$ group;
there $X\equiv 5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}{\tilde{Y}}$ is Wilczek-
Zee’s chiral symmetry Wilczek and Zee (1979) with the electroweak hypercharge
${\tilde{Y}}$. We decide not to include extra discrete charge conjugation,
parity, or time-reversal (C, P, or T) symmetries as they are already
individually broken at the SM energy scale. The SM is a Lorentz invariant QFT
that obeys the spacetime rotation and boost symmetry
${G_{\text{spacetime}}}={\rm Spin}$ group (Spin(3,1) in Lorentz signature and
Spin(4) in Euclidean signature). Gathering the above data, we learn that two
physically pertinent versions of spacetime-internal symmetries of SM are:

$\displaystyle G^{{\rm U}(1)}_{{\rm SM}_{q}}\equiv{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)\times G_{{\rm SM}_{q}}\text{ and
}G^{\mathbb{Z}_{4,X}}_{{\rm SM}_{q}}\equiv{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm SM}_{q}}.$ (3)

Both versions of the symmetry (3) are compatible with the representation (rep)
of the 3 families of 15 Weyl fermions of quarks and leptons (below written as
left-handed 2-component Weyl spinors of the spacetime Spin group) confirmed by
experiments, possibly with or without the 16th Weyl fermion (the right-handed
neutrino sterile to SM gauge force):

$\displaystyle\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus n_{\nu_{j,R}}{\bar{\nu}_{j,R}}$
(4) $\displaystyle\sim(\overline{\bf 3},{\bf 1})_{2}\oplus({\bf 1},{\bf
2})_{-3}\oplus({\bf 3},{\bf 2})_{1}\oplus(\overline{\bf 3},{\bf
1})_{-4}\oplus({\bf 1},{\bf 1})_{6}\oplus n_{\nu_{j,R}}{({\bf 1},{\bf
1})_{0}}\text{ for each family of }su(3)\times su(2)\times u(1)_{\tilde{Y}}.$
(5)

We use $n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}}\in\\{0,1\\}$ to label
either the absence or presence of the 16th Weyl fermion for each of the three
families of electron $e$, muon $\mu$, or tauon $\tau$ types. In general, we
can consider $N_{f}$ families (typically $N_{f}=3$) and a total sterile
neutrino number $n_{\nu_{R}}\equiv\sum_{j}n_{\nu_{j,R}}$ that can be equal,
smaller, or larger than 3.

In Sec. II, we consider the 4d SM with internal symmetry $G_{\text{SM}_{q}}$
ungauged as a global symmetry group. Both versions of spacetime-internal
symmetries (3) are treated as global symmetries, so we study the global
symmetries and their ’t Hooft anomalies of the _ungauged Standard Model_ at
$d=4$. However, ’t Hooft anomaly of the ungauged theory is the obstruction of
gauging. Some _’t Hooft anomalies of global symmetries_ directly descend to
the _dynamical gauge anomalies of gauge symmetries_ , after gauging
$G_{\text{SM}_{q}}$. So we can also use the cobordism group
$\mathrm{TP}_{5}(G)$ to check the _anomaly cancellation consistency of the
gauge theory_.

In Sec. III, the internal symmetry $G_{\text{SM}_{q}}$ is thereafter
dynamically gauged and treated as a gauge group in the _gauged Standard Model_
with dynamical Yang-Mills gauge fields. We will see that the spacetime-
internal symmetry eq. (3) has to be modified to a new $G^{\prime}$ by
quotienting out the gauge group $[G_{\text{SM}_{q}}]$ and including extra
generalized _higher symmetry_ Gaiotto  _et al._ (2015) (which is also an
internal global symmetry but whose charged objects are extended 1d Wilson-’t
Hooft line operators). This $\mathrm{TP}_{5}(G^{\prime})$ instead will be used
to constrain the _quantum dynamics of the gauge theory_.

##  II Standard Model with Internal Symmetry Ungauged and ’t Hooft Anomalies

We aim to determine the deformation class $(G,{\mathbf{Z}}_{5})$ of ungauged
4d SM with spacetime-internal symmetries $G$ given in (3). Ref. [Freed, 2008;
Garcia-Etxebarria and Montero, 2019; Wang and Wen, 2020; Wan and Wang, 2019;
Davighi  _et al._ , 2020; Wan and Wang, 2020a; Wan  _et al._ , ”2021”]
consider systematic anomaly classifications include several SM-like or Grand
Unification (GUT)-like groups. The relevant cobordism groups that classify the
4d anomaly and 5d iTFT ${\mathbf{Z}}_{5}$ are Wan and Wang (2020a); Wan  _et
al._ (”2021”); Wang (2020a, b, 2021)

$\displaystyle\mathrm{TP}_{5}(G^{{\rm U}(1)}_{{\rm
SM}_{q}})=\mathrm{TP}_{5}({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)\times G_{{\rm SM}_{q}})$ $\displaystyle=$
$\displaystyle\mathrm{TP}_{5}({\rm Spin}^{c}\times G_{{\rm
SM}_{q}})=\mathbb{Z}^{11}.$ (6)
$\displaystyle\mathrm{TP}_{5}(G^{\mathbb{Z}_{4,X}}_{{\rm
SM}_{q}})=\mathrm{TP}_{5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm SM}_{q}})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\mathbb{Z}^{5}\times\mathbb{Z}_{2}\times\mathbb{Z}_{4}^{2}\times\mathbb{Z}_{16},&\quad
q=1,3.\\\
\mathbb{Z}^{5}\times\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{4}\times\mathbb{Z}_{16},&\quad
q=2,6.\end{array}\right.$ (9)

Here are some remarks:

$\bullet$ We turn on all possible _background fields_ (which are also called
_classical_ _non-dynamical_ fields, or _symmetry twists_ Wang  _et al._
(2015), that are _not_ summed over in the path integral) that couple to the
charged objects of QFT. Then we can perform all symmetry transformations of
QFT by changing the values of background fields to detect all invertible
${\mathbf{Z}}_{5}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Theta}$.

$\bullet$ The symmetry $G$ consists of _internal symmetry_ and _spacetime
symmetry_. Internal symmetry transformations act on charged objects that form
a representation (rep) of group $G_{\text{internal}}$. We couple the 0d
charged particle to a 1d background gauge field $A$. Spacetime symmetry
transformations act on the spacetime coordinates or metrics by spacetime
diffeomorphisms. The anomalies are detected by turning on (i) internal
symmetry’s background gauge fields, (ii) spacetime symmetry’s background
fields via diffeomorphism or via placing the theory on $G$-structured
manifolds, or (iii) mixed internal-spacetime background fields. These
anomalies are respectively called (i) _gauge anomaly_ , (ii) _gravitational
anomaly_ , or (iii) _mixed gauge-gravity_ or _gauge-diffeomorphism anomaly_.
These adjectives, _gauge_ or _gravity_ , characterize the nature of the
background fields of those anomalies.

$\bullet$ Cobordism group $\mathrm{TP}_{d+1}(G)$ not only contains all finite
subgroup $\mathbb{Z}_{\rm n}$ (the torsion part) of the standard bordism group
$\Omega_{d+1}^{G}$, but also contains the integer $\mathbb{Z}$ classes
descended from those $\mathbb{Z}$ (the free part) of the bordism group
$\Omega_{d+2}^{G}$.

$\bullet$ The $d$d _perturbative local anomalies_ Adler (1969); Bell and
Jackiw (1969); Alvarez-Gaume and Witten (1984) correspond to the torsion free
$\mathbb{Z}$ classes of $\mathrm{TP}_{d+1}(G)$. They are detected by
infinitesimal gauge/diffeomorphism transformations and captured by Feynman
diagram calculations.
The $d$d _nonperturbative global anomalies_ Witten (1982, 1985); Wang  _et
al._ (2019) correspond to the torsion $\mathbb{Z}_{\rm n}$ classes of
$\mathrm{TP}_{d+1}(G)$. They are detected via large gauge/diffeomorphism
transformations that _cannot_ be continuously deformed from the identity.
These adjectives, _perturbative or not_ , characterize the _local_ or _global_
nature of those anomalies.

Now we can obtain the deformation class ${\mathbf{Z}}_{5}$ of the ungauged SM
by checking which ’t Hooft anomaly persists:

  1. 1.

For SM with a ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$ vector symmetry, eq. (6)
shows $\mathbb{Z}^{11}$ classes of local anomalies captured by triangle
Feynman diagrams. Given the SM chiral fermion content (4), all of local
anomalies cancel to zero Weinberg (2013) except two $\mathbb{Z}$ classes Wang
(2020a): the cubic pure gauge ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}^{3}$ and
mixed gauge-gravity ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$-(gravity)2
anomalies. Their 5d iTFT evaluated on a 5d manifold ${M^{5}}$ is

$\displaystyle{\mathbf{Z}}_{5}^{{\rm U}(1)}\equiv\exp(\hskip
1.0pt\mathrm{i}\hskip
1.0pt(-N_{f}+n_{\nu_{R}})((\int_{M^{5}}{A}c_{1}^{2})+\frac{1}{48}{\rm
CS}_{3}^{T({\rm PD}(c_{1}))})),$ (10)

depending on the family number $N_{f}$ and total sterile neutrino number
$n_{\nu_{R}}$. Typically for $N_{f}=3$, we have

$-N_{f}+n_{\nu_{R}}\equiv-
N_{f}+\sum_{j}n_{\nu_{j,R}}=-3+n_{\nu_{e,R}}+n_{\nu_{\mu,R}}+n_{\nu_{\tau,R}}+\dots.$

Let us explain the components in (10). The $A$ is the ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}\equiv{\rm
Spin}^{c}$ gauge field, while $2A$ is the ordinary abelian gauge field, whose
first Chern class $c_{1}=\frac{\mathrm{d}(2A)}{2\pi}$ satisfies a spin-charge
relation $c_{1}=w_{2}(TM)\;{\rm mod}\;2$ Seiberg and Witten (2016) with the
second Stiefel-Whitney class $w_{2}(TM)$ of spacetime tangent bundle $TM$ of
the base manifold $M$. The gravitational Chern-Simons 3-form is ${\rm
CS}_{3}^{TM^{3}}\equiv\frac{1}{4\pi}\int_{M^{3}=\partial
M^{4}}{\mathrm{Tr}}(\omega\mathrm{d}\omega+\frac{2}{3}\omega^{3})=\frac{1}{4\pi}\int_{M^{4}}{\mathrm{Tr}}(R(\omega)\wedge
R(\omega))$ where $M^{3}$ is evaluated as a boundary $\partial M^{4}$, while
$\omega$ is the 1-connection of tangent bundle $TM$ and $R(\omega)$ is the
Riemann curvature 2-form of $\omega$. In eq. (10), we take ${M^{3}={\rm
PD}(c_{1})}$ to be a 3-manifold Poincaré dual (PD) to the degree-2 first Chern
class $c_{1}$ on the 5d $M^{5}$. On a closed oriented $M^{4}$, we further have
$\frac{1}{4\pi}\int_{M^{4}}{\mathrm{Tr}}(R(\omega)\wedge
R(\omega))={2\pi}\int_{M^{4}}p_{1}(TM)={2\pi}\cdot 3\sigma$, where $p_{1}$ is
the first Pontryagin class of $TM$ and the $\sigma$ is the $M^{4}$’s
signature. Overall, the 5d term (10) descends from a 6d anomaly polynomial on
$M^{6}$ at the U(1)-valued $\theta=2\pi$:

$\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta(-N_{f}+n_{\nu_{R}})((\int_{M^{6}}\frac{1}{2}c_{1}^{3})+\frac{1}{16}\sigma({\rm
PD}(c_{1})))).$

Since ${\rm U}(1)_{Y}$ is fully anomaly-free, our above result is applicable
to SM with a ${\rm U}(1)_{X\equiv
5({\mathbf{B}-\mathbf{L}})-\frac{2}{3}{\tilde{Y}}}$ chiral symmetry Wilczek
and Zee (1979); we still obtain the same deformation class (10) as
${\mathbf{Z}}_{5}^{{\rm U}(1)}$ when replacing ${\rm
U}(1)_{\mathbf{B}-\mathbf{L}}$ to ${\rm U}(1)_{X}$.

  2. 2.

For SM with a $\mathbb{Z}_{4,X}$ chiral symmetry, given the fermion content
(4), all $\mathbb{Z}^{5}$ classes of local anomalies in (9) cancel to zero.
All extra $\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$ classes of global anomalies in
(9) also cancel, but the $\mathbb{Z}_{16}$ mixed gauge-gravity global anomaly
can be nonzero Wang (2020a, 2021), whose corresponding 5d iTFT is

$\displaystyle{\mathbf{Z}}_{5}^{\mathbb{Z}_{4,X}}\equiv\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt(-N_{f}+n_{\nu_{R}})\frac{2\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{16}\eta_{4{\mathrm{d}}}(\text{PD}(A_{{\mathbb{Z}_{2,X}}}))).$ (11)

The background gauge field $A_{{\mathbb{Z}_{2,X}}}$ is the generator of the
cohomology group
$\operatorname{H}^{1}(\mathrm{B}{\mathbb{Z}_{2,X}},\mathbb{Z}_{2})$ with
${\mathbb{Z}_{2,X}}\equiv{\mathbb{Z}_{4,X}}/{\mathbb{Z}_{2}^{F}}$ of ${{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}}$-structure manifold $M$.
The classifying space $\mathrm{B}^{n}{\rm A}\equiv K({\rm A},n)$ is the
Eilenberg-MacLane space for an abelian group ${\rm A}$. Hereafter we use the
standard convention, all cohomology classes are pulled back to the manifold
$M$ along the maps given in the definition of cobordism groups, thus
$\operatorname{H}^{1}(\mathrm{B}{\mathbb{Z}_{2,X}},\mathbb{Z}_{2})$ can be
pulled back to $\operatorname{H}^{1}(M,\mathbb{Z}_{2})$. Here a 5d
$\mathbb{Z}_{16}$ class Atiyah-Patodi-Singer (APS Atiyah  _et al._ (1975)) eta
invariant
$\eta_{5{\mathrm{d}}}=\eta_{4{\mathrm{d}}}(\text{PD}(A_{{\mathbb{Z}_{2,X}}}))$
is written as the 4d eta invariant ${\eta}_{4{\mathrm{d}}}\in\mathbb{Z}_{16}$
on the Poincaré dual (PD) submanifold of $A_{{\mathbb{Z}_{2,X}}}$ (i.e., 4d
${{\rm Pin}^{+}}$-manifold), based on a Smith homomorphism for the bordism
group $\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}\cong\Omega_{4}^{{\rm
Pin}^{+}}=\mathbb{Z}_{16}$ Tachikawa and Yonekura (2018); Hsieh (2018); Guo
_et al._ (2020); Hason  _et al._ (2020). The $\mathbb{Z}_{16}$ class 4d eta
invariant ${\eta}_{4{\mathrm{d}}}$ is the topological invariant of the
interacting fermionic topological superconductor of condensed matter in three
spatial dimensions with an anti-unitary time-reversal symmetry
$\mathbb{Z}_{4}^{T}$ such that the time-reversal symmetry generator $T$
squares to the fermion parity operator, namely $T^{2}=(-1)^{F}$ (see a review
Senthil (2015); Guo  _et al._ (2018)). In contrast, in SM, we have the unitary
${\mathbf{B}-\mathbf{L}}$-like symmetry ${\mathbb{Z}_{4,X}}$ whose generator
$X$ squares to $X^{2}=(-1)^{F}$.

  3. 3.

Thanks to the Lie group embedding $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times G_{{\rm
SM}_{6}})\subset{\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm Spin}(10)\equiv
G^{{\rm Spin}(10)}$ Wan and Wang (2020a), we can embed the 16-Weyl-fermion SM6
(i.e., $n_{\nu_{e,R}},n_{\nu_{\mu,R}},n_{\nu_{\tau,R}}=1$) with rep 16 into a
Spin(10) chiral gauge theory, the $so(10)$ GUT. We can again ask the
deformation class $(G^{{\rm Spin}(10)},{\mathbf{Z}}_{5}^{{\rm Spin}(10)})$ of
the ungauged 4d model. Its cobordism group Wang and Wen (2020); Wan and Wang
(2019, 2020a) is a ${\rm p}\in\mathbb{Z}_{2}$ class for ${\rm Spin}(n)$ with
$n\geq 7$:

$\displaystyle\mathrm{TP}_{5}({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
Spin}(n\geq 7))$ $\displaystyle=$ $\displaystyle\mathbb{Z}_{2},$ (12)
$\displaystyle{\mathbf{Z}}_{5}^{{\rm Spin}(n)}\equiv{\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\;{\rm p}\int_{M^{5}}w_{2}w_{3})}={\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\;{\rm p}\int_{M^{5}}w_{2}(TM)w_{3}(TM))}$
$\displaystyle=$ $\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\;{\rm p}\int_{M^{5}}w_{2}(V_{{\rm SO}(n)})w_{3}(V_{{\rm SO}(n)})).$
(13)

Here $w_{j}(TM)$ or $w_{j}(V_{{\rm SO}(n)})$ is the $j$th Stiefel-Whitney
class of the spacetime tangent bundle $TM$, or the associated vector bundle of
the principal gauge bundle of ${{\rm SO}(n)}={\rm
Spin}(n)/\mathbb{Z}_{2}^{F}$. The only 5d iTFT represents a ${\rm
p}\in\mathbb{Z}_{2}$ class 4d mixed gauge-gravity global anomaly. The
conventional $so(10)$ GUT Fritzsch and Minkowski (1975) has no 4d anomaly thus
resides in a ${\rm p}=0$ trivial deformation class ${\mathbf{Z}}_{5}^{{\rm
Spin}(10)}=1$ Wang and Wen (2020); Wang  _et al._ (2019); Wan  _et al._
(2021). Ref. Wang and You (2022, 2021) construct a modified $so(10)$ GUT with
an extra discrete torsion class of 4d-5d Wess-Zumino-Witten term from the GUT-
Higgs field. The modified $so(10)$ GUT has a $w_{2}w_{3}$ anomaly, thus it
resides in a ${\rm p}=1$ deformation class. Since ${\rm Spin}(18)\supset{\rm
Spin}(10)$ contains the same $w_{2}w_{3}$ deformation class, one can work out
a similar story for a Spin(18) chiral gauge theory or $so(18)$ GUT Wilczek and
Zee (1982). In fact, within either ${\rm p}\in\mathbb{Z}_{2}$ deformation
class, Ref. Wang and You (2022, 2021) show that the SM can quantum-phase
transition to the GUT-like models of $su(5)$ Georgi and Glashow (1974),
$su(4)\times su(2)\times su(2)$ Pati and Salam (1974), or flipped $u(5)$ Barr
(1982) models. Especially for ${\rm p}=1$ deformation class, Ref. Wang and You
(2022, 2021) suggest a _gapless deconfined quantum critical region_ (analogous
to Senthil  _et al._ (2004)) with beyond-SM phenomena near the SM vacuum by
tuning the GUT-Higgs potential.

In summary, the deformation class $(G,{\mathbf{Z}}_{5})$ of ungauged SM with
${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$ or ${\rm U}(1)_{X}$ is $(G^{{\rm
U}(1)}_{{\rm SM}_{q}},{\mathbf{Z}}_{5}^{{\rm U}(1)})$, and that of ungauged SM
with ${\mathbf{Z}}_{5}^{\mathbb{Z}_{4,X}}$ is $(G^{\mathbb{Z}_{4,X}}_{{\rm
SM}_{q}},{\mathbf{Z}}_{5}^{\mathbb{Z}_{4,X}})$, for any $q=1,2,3,6$. For the
ungauged SM embeddable into a conventional or modified $so(n)$ GUT, we obtain
an extra deformation class data $(G^{{\rm Spin}(n)},{\mathbf{Z}}_{5}^{{\rm
Spin}(n)})$.

##  III Standard Model with Internal Symmetry Gauged and with Higher
Symmetries

Next we determine the deformation class of $G_{{\rm SM}_{q}}$-dynamically
gauged SM. Before solving that problem, let us recall the relationship between
the deformation classes of the ungauged SM (Sec. II) and the gauged SM:

$\bullet$ The vanishing ’t Hooft anomalies (thus not presented in the
deformation class ${\mathbf{Z}}_{5}$) for the symmetry that we will gauge in
the ungauged SM indeed one-to-one correspond to the _dynamical gauge-
diffeomorphism anomaly cancellation_ conditions for the gauged SM.

$\bullet$ If the gauged SM has only $G_{{\rm SM}_{q}}$ gauged, while keeping
the global symmetry ${\rm U}(1)_{{\mathbf{B}-\mathbf{L}}}$, ${\rm U}(1)_{X}$,
or ${\mathbb{Z}_{4,X}}$ ungauged, then this SM deformation class still
includes eq. (10), (11), and (13).

$\bullet$ If the gauged SM has not only $G_{{\rm SM}_{q}}$ but also
${\mathbf{B}-\mathbf{L}}$ or $X$ dynamically gauged, then the 5d iTFT also has
to be dynamically gauged to a new _noninvertible_ theory, which is either a
noninvertible gapped TFT or a gapless theory. In this scenario, we could no
longer study the 4d SM on its own. The 4d SM and 5d theories are strongly
correlated and coupled with each other through ${\mathbf{B}-\mathbf{L}}$ or
$X$ gauge forces mediating long-range entanglements.

$\bullet$ In a gauge theory, the internal symmetry also includes _higher
$n$-symmetries_ as generalized global symmetries Gaiotto  _et al._ (2015) that
act on $n$d charged objects (1d lines, 2d surfaces, etc.) by coupling the $n$d
objects to $n+1$d background gauge fields. We denote the higher $n$-symmetry
group $G$ as $G_{[n]}$. The $n+1$d gauge field lives in a higher classifying
space $\mathrm{B}^{n}G$ (namely, the Eilenberg-MacLane space) of the group
$G_{[n]}$. Once $G_{\text{SM}_{q}}$ is gauged, _kinematically_ , we may obtain
extra 1-form electric and magnetic symmetries, $G_{[1]}^{e}$ and
$G_{[1]}^{m}$, which are determined from the center of gauge group $Z(G_{{\rm
SM}_{q}})$, and the Pontryagin dual of the homotopy group $\pi_{1}(G_{{\rm
SM}_{q}})^{\vee}$. Moreover the 0d SM gauge charge fermionic particle (4)
explicitly breaks the $G_{[1]}^{e}$ from $Z(G_{{\rm SM}_{q}})$ to a subgroup
Wan  _et al._ (2020); Anber and Poppitz (2021); Wang and You (2021); Montero
_et al._ (to appear) that acts trivially on all particle representations:

$\displaystyle\begin{tabular}[]{|c | c |c| c| c |}\hline\cr&$Z(G_{{\rm SM}_{q}})$&$\pi_{1}(G_{{\rm SM}_{q}})^{\vee}$&1-form $e$ sym $G_{[1]}^{e}$&1-form $m$ sym $G_{[1]}^{m}$\\\ \hline\cr$G_{{\rm SM}_{q}}\equiv\frac{{{\rm SU}(3)}\times{{\rm SU}(2)}\times{\rm U}(1)_{\tilde{Y}}}{\mathbb{Z}_{q}}$&$\mathbb{Z}_{6/q}\times{\rm U}(1)$&${\rm U}(1)$&$\mathbb{Z}_{6/q,[1]}^{e}$&${{\rm U}(1)}_{[1]}^{m}$\\\ \hline\cr\end{tabular}.$ (16)

Thus, we have to modify (3), by removing $G_{{\rm SM}_{q}}$ and including
$G_{[1]}^{e}\times G_{[1]}^{m}$, to

$\displaystyle G^{\prime{\rm U}(1)}_{{\rm SM}_{q}}\equiv{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)\times\mathbb{Z}_{6/q,[1]}^{e}\times{\rm U}(1)_{[1]}^{m}\text{ and
}G^{\prime\mathbb{Z}_{4,X}}_{{\rm SM}_{q}}\equiv{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\mathbb{Z}_{6/q,[1]}^{e}\times{\rm
U}(1)_{[1]}^{m}.$ (17)

We denote ${q^{\prime}}\equiv 6/q$, where ${q}=1,2,3,6$ maps to
${q^{\prime}}=6,3,2,1$, and define ${q^{\prime}}\equiv{2^{n_{2}}\cdot
3^{n_{3}}}$ where ${q^{\prime}}=1,2,3,6$ corresponds to
$(n_{2},n_{3})=(0,0),(1,0),(0,1),(1,1)$. Below we examine what are additional
new terms in the deformation class of SM due to ’t Hooft anomalies of higher
symmetries of 1d line operators. All possible 4d invertible anomalies are
classified by the 5d cobordism groups $\mathrm{TP}_{5}$ of (17) Wan  _et al._
(”2021”):

$\displaystyle\mathrm{TP}_{5}(G^{\prime{\rm U}(1)}_{{\rm
SM}_{q}})={\mathrm{TP}_{5}({\rm Spin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)\times\mathrm{B}\mathbb{Z}_{q^{\prime}}\times\mathrm{B}{\rm
U}(1))=\mathrm{TP}_{5}({\rm
Spin}^{c}\times\mathrm{B}\mathbb{Z}_{q^{\prime}}\times\mathrm{B}{\rm
U}(1))=\mathbb{Z}^{2}\times\mathbb{Z}_{q^{\prime}}},$ (18) generated by
${A}c_{1}^{2}$, $\frac{1}{48}{\rm CS}_{3}^{T({\rm PD}(c_{1}))}$, and
$B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}(\mathrm{d}B^{m}_{{\rm U}(1),[1]}\;{\rm
mod}\;{q^{\prime}})$. (19)
$\displaystyle\mathrm{TP}_{5}(G^{\prime\mathbb{Z}_{4,X}}_{{\rm
SM}_{q}})=\mathrm{TP}_{5}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\mathrm{B}\mathbb{Z}_{q^{\prime}}\times\mathrm{B}{\rm
U}(1))=\mathbb{Z}_{16}\times(\mathbb{Z}_{4})^{n_{2}}\times\mathbb{Z}_{q^{\prime}},$
(20) generated by $\eta_{4{\mathrm{d}}}({\rm PD}({A_{\mathbb{Z}_{2,X}}}))$,
$\eta^{\prime}_{1{\mathrm{d}}}({\rm PD}({\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}})))$, and
$B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}(\mathrm{d}B^{m}_{{\rm U}(1),[1]}\;{\rm
mod}\;{q^{\prime}})$. (21)

Below we first explain the 5d iTFTs of (18) and (20), and then determine the
deformation class of the gauged SM:

  1. 1.

The $\mathbb{Z}^{2}$ and $\mathbb{Z}_{16}$ classes give the same 5d iTFTs
already appeared in Sec. II. So the deformation class of ungauged SM in eq.
(10) and (11) still remains in that of gauged SM.

  2. 2.

$B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}$ and $B^{m}_{{\rm U}(1),[1]}$ are the
background fields of 1-form electric and magnetic symmetries respectively,
namely ${\mathbb{Z}_{q^{\prime},[1]}^{e}}$ and ${{\rm U}(1)}_{[1]}^{m}$ in
(16). Again, all cohomology classes are pulled back to the manifold $M$. Since
the maps $M\to\mathrm{B}^{2}\mathbb{Z}_{q^{\prime}}$ and
$M\to\mathrm{B}^{2}{\rm U}(1)$ are given in the definitions of the cobordism
groups, the cohomology classes
$B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}\in\operatorname{H}^{2}(\mathrm{B}^{2}\mathbb{Z}_{q^{\prime}},\mathbb{Z}_{q^{\prime}})$
and $B^{m}_{{\rm U}(1),[1]}\in\operatorname{H}^{2}(\mathrm{B}^{2}{\rm
U}(1),{\rm U}(1))$ are pulled back to
$\operatorname{H}^{2}(M,\mathbb{Z}_{q^{\prime}})$ and
$\operatorname{H}^{2}(M,{\rm U}(1))$ respectively.

  3. 3.

$\eta^{\prime}_{1{\mathrm{d}}}({\rm PD}({\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}})))$: $\eta^{\prime}_{1{\mathrm{d}}}$ is a
generator of the cobordism group $\mathrm{TP}_{1}({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{X})=\mathbb{Z}_{4}$. The
Pontryagin square ${\cal P}(B^{e}_{\mathbb{Z}_{2,[1]}})\equiv
B^{e}_{\mathbb{Z}_{2,[1]}}\smile
B^{e}_{\mathbb{Z}_{2,[1]}}+B^{e}_{\mathbb{Z}_{2,[1]}}\smile_{1}\delta
B^{e}_{\mathbb{Z}_{2,[1]}}$ is a cohomology class in
$\operatorname{H}^{4}(M,\mathbb{Z}_{4})$, while $B^{e}_{\mathbb{Z}_{2,[1]}}$
is a cohomology class in $\operatorname{H}^{2}(M,\mathbb{Z}_{2})$, the
$\smile_{j}$ is the $j$-th cup product between cochains (we make the $0$-th
cup product $\smile$ implicit intentionally throughout the article), and
$\delta$ is a coboundary operator. Since $M$ is $\mathbb{Z}$-orientable hence
$\mathbb{Z}_{4}$-orientable, the Poincaré dual ${\rm PD}({\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}}))$ is represented by a 1d ${\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}$-submanifold of $M$, thus
$\eta^{\prime}_{1{\mathrm{d}}}({\rm PD}({\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}})))$ is well-defined. Whether the gauged SM has
this mixed anomaly between the chiral $\mathbb{Z}_{4,X}$ and the
${\mathbb{Z}^{e}_{q^{\prime},[1]}}$ symmetries (only at $q^{\prime}=2,6$ thus
$q=3,1$) can be computed via the techniques in Shimizu and Yonekura (2018);
Tanizaki  _et al._ (2018); Cordova and Dumitrescu (2018); Wan and Wang
(2020b).

Ref. [Cordova and Dumitrescu, 2018] finds that two adjoint Weyl fermions of
SU(2) gauge theory with ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{8}\times{\mathbb{Z}_{2,[1]}^{e}}$
symmetry has an anomaly index $1\in\mathbb{Z}_{4}$ class of $A^{\prime}{\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}})$ due to fractional SO(3) instantons, where
$A^{\prime}\in\operatorname{H}^{1}(\mathrm{B}({\frac{\mathbb{Z}_{8}}{\mathbb{Z}_{2}^{F}}}),\mathbb{Z}_{4})$
is pulled back to $\operatorname{H}^{1}(M,\mathbb{Z}_{4})$. In comparison, we
consider that each family of SM has four SU(2) fundamental Weyl fermions as
four SU(2) doublets 2 gauged under ${\rm SU}(2)\times{\rm U}(1)$ gauge theory
for $G_{{\rm SM}_{q}}$ at $q=1,3$ with ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times{\mathbb{Z}_{2,[1]}^{e}}$
symmetry. But this theory does not carry an anomaly index in the
$\mathbb{Z}_{4}$ class of $\eta^{\prime}_{1{\mathrm{d}}}({\rm PD}({\cal
P}(B^{e}_{\mathbb{Z}_{2,[1]}})))$, because the ${\rm U}(1)_{X}$ symmetry is
_not_ broken down to $\mathbb{Z}_{4,X}$ due to SU(2) instantons. So any
fractional ${\rm U}(2)$ instantons cannot induce any further mixed anomaly
between the $\mathbb{Z}_{4,X}$-${\mathbb{Z}^{e}_{q^{\prime},[1]}}$ symmetry.

  4. 4.

$B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}(\mathrm{d}B^{m}_{{\rm U}(1),[1]}\;{\rm
mod}\;{q^{\prime}})$: This 5d term represents a mixed anomaly between the
electric 1-form $\mathbb{Z}_{q^{\prime}}$ symmetry and the magnetic 1-form
${\rm U}(1)$ symmetry that we will detect in the 4d SM. To define this 5d
$B\mathrm{d}B^{\prime}$-like term mathematically precisely, we first define a
generator $\tilde{\tau}_{3}\equiv\mathrm{d}B^{m}_{{\rm
U}(1),[1]}\in\operatorname{H}^{3}(\mathrm{B}^{2}{\rm U}(1),\mathbb{Z})$
represented by the identity map $\mathrm{B}^{2}{\rm
U}(1)=\mathrm{B}^{3}\mathbb{Z}\to\mathrm{B}^{3}\mathbb{Z}=\mathrm{B}^{2}{\rm
U}(1)$. The $\tilde{\tau}_{3}$ is pulled back to
$\operatorname{H}^{3}(M,\mathbb{Z})$ along the map $M\to\mathrm{B}^{2}{\rm
U}(1)$ given in the definition of the cobordism group. Here we regard
$B^{m}_{{\rm U}(1),[1]}$ as the generator of
$\operatorname{H}^{2}(\mathrm{B}^{2}{\rm U}(1),{\rm U}(1))=\mathbb{Z}$. Also,
we regard $\mathrm{d}$ as the boundary map in the long exact sequence
$\cdots\to\operatorname{H}^{2}(-,{\rm
U}(1))\xrightarrow{\mathrm{d}}\operatorname{H}^{3}(-,\mathbb{Z})\to\cdots$
induced from the short exact sequence $1\to\mathbb{Z}\to{\mathbb{R}}\to{\rm
U}(1)\to 1$. Note that $\mathrm{d}$ is not the coboundary map $\delta$ which
sends a cochain to a coboundary. The factorization
$\mathbb{Z}_{q^{\prime}}=(\mathbb{Z}_{3})^{n_{3}}\times(\mathbb{Z}_{2})^{n_{2}}$
implies $B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}(\mathrm{d}B^{m}_{{\rm
U}(1),[1]}\;{\rm
mod}\;{q^{\prime}})=-2n_{3}B^{e}_{\mathbb{Z}_{3,[1]}}(\mathrm{d}B^{m}_{{\rm
U}(1),[1]}\;{\rm
mod}\;3)+3n_{2}B^{e}_{\mathbb{Z}_{2,[1]}}(\mathrm{d}B^{m}_{{\rm
U}(1),[1]}\;{\rm mod}\;2)$. We check that, for $G_{{\rm SM}_{q}}$ with Weyl
fermion content in (4), there is indeed a 4d mixed higher anomaly of 1-form
symmetries between $\mathbb{Z}_{6/q,[1]}^{e}$ and ${\rm U}(1)_{[1]}^{m}$. For
any family, the gauged SM contains the anomaly index ${\rm
p}^{\prime}=1\in\mathbb{Z}_{q^{\prime}}$ of the 5d iTFT. However, if the ${\rm
U}(1)_{\tilde{Y}}$ magnetic monopole exists, kinematically it can break ${\rm
U}(1)_{[1]}^{m}$ down to a discrete subgroup or to none (thus ${\rm
p}^{\prime}=0$) at the higher-energy monopole scale; nonetheless ${\rm
U}(1)_{[1]}^{m}$ can still re-emerge at the lower-energy SM scale. In short,
the value ${\rm p}^{\prime}\in\mathbb{Z}_{q^{\prime}}$ depends on both the
${\rm U}(1)_{\tilde{Y}}$ magnetic monopole and the energy scale.

  5. 5.

For a discrete ${\mathbf{B}-\mathbf{L}}$ vector or $X$ chiral symmetry, we so
far focus on $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}\times\dots)$ for the SM. But
we could have other alternatives, e.g., $({\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2{\rm m}}\times\dots)$ for ${\rm
m}=2,3,4,\dots$ for a larger discrete $\mathbb{Z}_{2{\rm m},X}$ symmetry Hsieh
(2018); Guo  _et al._ (2020). We could also include the discrete
${\mathbf{B}+\mathbf{L}}$ vector symmetry: the ${\rm
U}(1)_{\mathbf{B}+\mathbf{L}}$ is explicitly broken down, to
$\mathbb{Z}_{2N_{f},\mathbf{B}+\mathbf{L}}$ by the Adler-Bell-Jackiw anomaly
Adler (1969); Bell and Jackiw (1969) due to SU(2) instantons Belavin  _et al._
(1975), or to $\mathbb{Z}_{2N_{f}\\#,\mathbf{B}+\mathbf{L}}$ with an extra
$q$-dependent factor denoted as $\\#$ due to fractional instantons Anber and
Poppitz (2021). But we have excluded ${\mathbf{B}+\mathbf{L}}$ in our work
because it is not a good symmetry for the $su(5)$ or other GUTs. We leave
other scenarios for future directions Wang  _et al._ (2022).

  6. 6.

In summary, out of the multiple $\mathbb{Z}$ classes of local anomalies and
$\mathbb{Z}_{n}$ classes of global anomalies (counted as a total finite group
of order $2\cdot 4^{2}\cdot 16\cdot 2\cdot 4\cdot 6=2^{13}\cdot 3=24576$ from
(6), (9), (12), (18), and (20)), we determine the deformation class of
ungauged or gauged SM, labeled by the family number $N_{f}$, total sterile
neutrino number $n_{\nu_{R}}$, and magnetic monopole datum ${\rm p}^{\prime}$,
the mod $q\equiv 6/{q^{\prime}}$ of $G_{{\rm SM}_{q}}$, written as a 5d iTFT:

${\mathbf{Z}}_{5}\equiv\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-N_{f}+n_{\nu_{R}})(\dots)+\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\rm
p}^{\prime}}{{q^{\prime}}}\int_{M^{5}}B^{e}_{\mathbb{Z}_{q^{\prime},[1]}}(\mathrm{d}B^{m}_{{\rm
U}(1),[1]}\;{\rm mod}\;{q^{\prime}})).\quad\quad$ (22)

Here the $(\dots)$ are the $\mathbb{Z}^{2}$ or $\mathbb{Z}_{16}$ class iTFT
already detected in eq. (10) and (11). For the ungauged SM, if we embed the SM
into the modified $so(10)$ GUT Wang and You (2022, 2021), we could add eq.
(13) into (22). But again, once the ${\mathbf{B}-\mathbf{L}}$, $X$, or
Spin(10) are dynamically gauged, the 4d SM and the 5d becoming _noninvertible_
theories are correlated via mediating ${\mathbf{B}-\mathbf{L}}$, $X$, or
Spin(10) gauge forces.

The last term of (22) with ${\rm p}^{\prime}/q^{\prime}$ dependence imposes a
dynamical constraint on the 4d quantum gauge theory of the gauged SM: the high
energy theory or low energy fate of this gauged SM must match the same ’t
Hooft anomaly. For example, the $B^{e}\mathrm{d}B^{m}$ term suggests that
either ${\mathbb{Z}_{q^{\prime},[1]}^{e}}$ or ${{\rm U}(1)}_{[1]}^{m}$ or both
are spontaneously broken Gaiotto  _et al._ (2015), or there is no symmetric
gapped trivial vacuum. The $\mathbb{Z}_{16}$ class 4d anomaly of (11) and (22)
imposes another dynamical constraint in 4d: there is no symmetric gapped phase
for the odd classes in $\mathbb{Z}_{16}$, but there exists symmetric gapped
phases (namely gapped without $\mathbb{Z}_{4,X}$ chiral symmetry breaking)
with low energy TQFTs for the even classes in $\mathbb{Z}_{16}$ Wang  _et al._
(2018); Hsieh (2018); Cordova and Ohmori (2019, 2020).

  7. 7.

Ultra Unification (UU) Wang (2020a, b, 2021) replaces some of “right-handed
sterile neutrinos” (sterile only to $G_{{\rm SM}_{q}}$ gauge force, but not
sterile to $X$ or certain GUT forces) by new exotic gapped/gapless sectors
(e.g., topological or conformal field theory) or by gravitational sectors with
topological origins, via nonperturbative global anomaly cancellation and
cobordism constraints. Namely $(-N_{f}+n_{\nu_{R}}+\dots)=0$ in
$\mathbb{Z}_{16}$ for ${\rm Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$
symmetry, where $\dots$ is the anomaly index of the new gapped/gapless sectors
from 4d or 5d. It can be generalized to ${\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2^{m},X}$ symmetry with the
$\mathbb{Z}_{2^{m+2}}$ global anomaly constraint for $m\geq 2$ Hsieh (2018);
Guo  _et al._ (2020). If we remove the new gapped/gapless sectors out of UU,
the UU also resides in an SM deformation class. There is a rich interplay
between SM, GUT, and UU, where _topological quantum phase transitions_ occur
between the 15 vs 16 Weyl fermion models. The new sectors Wang (2020a, b,
2021); Wang and You (2022, 2021) (gapped or gapless, including quantum phases
or phase transitions, and quantum critical regions), neighbor to the SM vacuum
in the quantum phase diagram landscape and in the deformation class, naturally
exhibit beyond SM phenomena and thus may provide Dark Matter candidates.
Gapless quantum criticality also challenges the spacetime concept near the
highly entangled critical regions, which may affect the energy density or
pressure of spacetime, thus may relate to Dark Energy. The deformation between
neutrinos and exotic gapped/gapless sectors may shed new light on the quantum
version of the equivalence principle on the inertia mass (of the quantum gauge
theory origin) and gravitational mass (of the gravitational origin).

_Acknowledgments_ — JW thanks Mohamed Anber, Yuta Hamada, Miguel Montero,
Cumrun Vafa, and Edward Witten for conversations. We thank Alek Bedroya,
Cameron Krulewski, and Leon Liu for feedback on the manuscript. JW is
supported by Harvard University CMSA. ZW is supported by the Shuimu Tsinghua
Scholar Program. YZY is supported by a startup fund at UCSD.

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