$\tilde{G}=\mathbb{Z}_{4}^{T}$-symmetry-extended gapped boundary [24]. Gauging
the normal subgroup $K=\mathbb{Z}_{2}$ results in a (2+1)D discrete $K$-gauge
theory with both electric and magnetic gauge charges are Kramers doublet with
$T^{2}=-1$.

$\bullet$ The (2+1)D surface state of (3+1)D $k\mathcal{P}(B_{2})$ higher-SPT
state with $k\in\mathbb{Z}_{4}$: This higher-SPT state is protected by a
1-form symmetry (denoted as $G=\mathrm{SO}\times\mathbb{Z}_{2,[1]}$) which
couples to a 2-form background field $B_{2}$. The $\mathcal{P}(B_{2})$ means
the Pontryagin square of $B_{2}$. The symmetry-extension construction can
obtain a gapped phase for an even $k$ via extending to
$\tilde{G}=\mathrm{Spin}\times\mathbb{Z}_{2,[1]}$ (although gauging
$K=\mathbb{Z}_{2}$ results in $G$ SSB), but the symmetry-extension
trivialization is proven to not exist for an odd $k$ [160]. Later Ref. 176
proves a no-go theorem that the symmetry-preserving TQFT also does not exist
for an odd $k$. This means the (2+1)D surface state must be either symmetric
gapless or symmetry-breaking for an odd $k$.

  4. 4.

$d=4$, a (3+1)D anomalous theory and a (4+1)D bulk:

$\bullet$ The (3+1)D boundary of a (4+1)D $\mathbb{Z}_{2}$-SPT state allows a
symmetric gapless, symmetry-breaking, or symmetric gapped surface topological
order boundary. The $G=\mathbb{Z}_{2}$ can be extended to give a
$\tilde{G}=\mathbb{Z}_{4}$-symmetry-extended gapped boundary [24]. Gauging the
normal subgroup $K=\mathbb{Z}_{2}$ results in a (3+1)D discrete $K$-gauge
theory with both electric and magnetic gauge charges carries a fractional $G$
charge.

$\bullet$ The (3+1)D boundary of a (4+1)D $w_{3}(TM)B_{2}^{e}$ higher-SPT
state is protected by a 1-form electric symmetry (denoted as
$G=\mathbb{Z}_{2,[1]}^{e}$, coupled to a 2-form $B_{2}^{e}$ field), while the
$w_{j}(TM)$ is the $j$th Stiefel-Whitney class of the tangent bundle $TM$ of
spacetime manifold $M$. The corresponding anomaly occurs as a part of the
anomaly of (3+1)D SU(2) Yang-Mills theory coupled to two Weyl fermions in the
adjoint representation of SU(2) (below called this theory as the adjoint QCD4,
see [177, 178, 179] and [160]). The $G=\mathbb{Z}_{2,[1]}^{e}$ can be extended
to give a $\tilde{G}=\mathbb{Z}_{2}^{F}\times\mathbb{Z}_{2,[1]}^{e}$-symmetry-
extended gapped boundary [160]. Gauging the normal subgroup $K=\mathbb{Z}_{2}$
results in a (3+1)D discrete $K$-gauge theory such that its electric gauge
charge has fermionic statistics.

$\bullet$ The (3+1)D boundary of a (4+1)D $kA\mathcal{P}(B_{2}^{e})$ higher-
SPT state: Again this is part of the anomaly of the adjoint QCD4 [178]. The
higher-SPT is protected by a $\mathbb{Z}_{8}$-axial symmetry (coupled to a
1-form $A$ field, with its fourth power of the symmetry generator equals to
$(-1)^{F}$) and a 1-form electric symmetry $\mathbb{Z}_{2,[1]}^{e}$ (coupled
to a 2-form $B_{2}^{e}$ field), which can be denoted as a spacetime-internal
symmetry
$G=\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{8}\times\mathbb{Z}_{2,[1]}^{e}$.
There is a $k\in\mathbb{Z}_{4}$ class. The even $k$ class can be trivialized
by a $\mathbb{Z}_{2}$-extension to
$\tilde{G}=\mathrm{Spin}\times\mathbb{Z}_{8}\times\mathbb{Z}_{2,[1]}^{e}$. The
odd $k$ class cannot be trivialized by any symmetry extension [160]. Later
Ref. 176 proves a no-go theorem that the symmetry-preserving TQFT also does
not exist. The above results [160, 176] turn out to rule out certain UV-IR
duality proposal of the adjoint QCD4 hypothesized in [179].

$\bullet$ The (3+1)D boundary of a (4+1)D
$\frac{1}{2}\tilde{w}_{1}(TM)\mathcal{P}(B_{2}^{e})$ higher-SPT state: The
boundary turns out to associate with the anomaly of the (3+1)D SU(2) Yang-
Mills gauge theory with a ${\theta=\pi}$ $F\wedge F$ topological term (denoted
as $\mathrm{SU}(2)_{\theta=\pi}$ YM) [180, 181, 182], while the
$\tilde{w}_{1}(TM)$ is the twisted first Stiefel-Whitney class of $TM$ such
that $\frac{1}{2}\tilde{w}_{1}(TM)\mathcal{P}(B_{2}^{e})$ is a mod 2 class.
The $\mathrm{SU}(2)_{\theta=\pi}$ YM kinematically at UV has time-reversal
$\mathbb{Z}_{2}^{T}$ and 1-form $\mathbb{Z}_{2,[1]}^{e}$ symmetries. The
$G=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2,[1]}^{e}$ can be extended to
$\tilde{G}=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{4,[1]}^{e}$ to trivialize the
anomaly, thus the $\tilde{G}$-symmetric extended gapped phase can be
constructed [181, 182]. However, upon gauging $K=\mathbb{Z}_{2,[1]}$, this
induces the $G$ spontaneous symmetry breaking (SSB) [182]. Later Ref. 176
proves a no-go theorem that the symmetry-preserving TQFT also does not exist.
The above results together demand that the IR fate of
$\mathrm{SU}(2)_{\theta=\pi}$ YM must be either symmetric gapless or symmetry-
breaking only.

$\bullet$ The (3+1)D boundary of a (4+1)D $k\eta_{\rm 4d}(\mathrm{PD}(A))$ SPT
state: This is a $k\in\mathbb{Z}_{16}$ class of (3+1)D anomaly and (4+1)D SPT
state, protected by a unitary $\mathbb{Z}_{4,X}$ symmetry such that
$X^{2}=(-1)^{F}$; in terms of a spacetime-internal symmetry
$G=\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$. The $\eta_{\rm
4d}(\mathrm{PD}(A))$ is a 4d Atiyah-Patodi-Singer eta invariant $\eta_{\rm
4d}$ evaluated on the 4-manifold Poincaré dual (PD) to the
$(\mathbb{Z}_{4,X}/\mathbb{Z}_{2}^{F})$-gauge field $A$. The $\eta_{\rm
4d}(\mathrm{PD}(A))$ is a cobordism invariant of the bordism group
$\Omega_{5}^{\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}}=\mathbb{Z}_{16}$,
see [183, 184] and [57]. It turns out that $k=(-N_{f})$ of such SPT state
captures a global anomaly of (3+1)D $15N_{f}$-Weyl-fermion Standard Model (SM)
where $N_{f}$ is the number of families of quarks and leptons [183, 185]. If
$k$ is odd, Ref. 186 proves an obstruction, so the symmetry-gapped TQFT is not
possible to saturate this odd $k$ anomaly. If $k$ is even, Ref. 184, 187, 188
show that two layers of symmetry extensions can construct the
$\tilde{G}=\mathrm{Spin}\times{\mathbb{Z}_{8}}$-symmetry extended gapped
phase: the first layer
$1\to\mathbb{Z}_{2}\to\mathrm{Spin}\times{\mathbb{Z}_{4}}\to\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}{\mathbb{Z}_{4,X}}\to
1$ and the second layer
$1\to\mathbb{Z}_{2}\to\mathrm{Spin}\times{\mathbb{Z}_{8}}\to\mathrm{Spin}\times{\mathbb{Z}_{4}}\to
1$. These constructions may have applications to beyond the SM physics [187,
189, 188].

##  V Features and Applications

###  V.1 Green’s Function Zeros

Given different mass generation mechanisms discussed above, a key question is
how to diagnose the SMG mechanism: if a mass gap is observed in an interacting
fermionic system, how do we know the mass gap is opened up through the SMG
mechanism, rather than a more conventional symmetry-breaking mechanism, say by
condensing a fermion-bilinear mass. Over the years it was gradually realized
that one of the characteristic features of the SMG phase is that, the fermion
Green’s function
$G(\mathrm{i}\omega)=-\langle\psi(\mathrm{i}\omega)\psi^{\dagger}(\mathrm{i}\omega)\rangle$
in the Matsubara frequency space should have a zero at $\omega\rightarrow 0$,
i.e. $\det G(\mathrm{i}\omega\rightarrow 0)=0$. This applies to the SMG phase
either in the bulk or on the boundary of a fermionic system.

There are various ways to argue the necessary existence of this Green’s
function zero in a system with SMG. For the Green’s function defined in the
bulk, one general argument for the existence of zero is based on the
topological number associated with the free fermionic SPT state [190, 191,
120]. For a fermionic system with translation symmetry, the topological number
can be defined in the Matsubara frequency and momentum space using the
Euclidean spacetime full fermion Green’s function [192, 193, 194, 195, 196]:

$n\sim\int\mathrm{d}\omega\mathrm{d}^{{d-1}}k\ \mathrm{tr}[B(G^{-1}\partial
G)\wedge(G^{-1}\partial G)\cdots],$ (59)

where $G$ is the matrix of the full fermion Green’s function, and $B$ is
certain matrix in the flavor space. For a Chern insulator at even spatial
dimension, $B$ is an identity matrix, and $n$ is equivalent to the Chern
number. One can prove mathematically that the number $n$ must be a quantized
integer, and it can only change discontinuously when the Green’s function has
singularity.

Figure 6: Schematic phase diagram between a trivial phase and a (fake)
fermionic SPT phase that can be trivialized by interaction. Massless bulk
fermions along the critical line in the weak-coupling limit undergoes the SMG
as the interaction $g$ exceeds a critical value $g_{c}$. Although the two
phases (technically one phase) are smoothly connected through the strong-
coupling regime, the topological number $n$ must still jump across the missing
“phase boundary”, along which the Green’s function must have a zero.

The singularity of $G$ happens at two types of “transitions”. The first type
of transition is a physical transition where $\det
G^{-1}(\mathrm{i}\omega=0)=0$ vanishing to zero, i.e. the fermions become
gapless. This corresponds to the quantum critical line between the fermionic
SPT and trivial phases in the weak-coupling regime, as shown in Fig. 6.
However, note that in the definition of the topological number $n$ in Eq.
(59), $G^{-1}$ and $G$ appear on an equal footing, hence $n$ can also change
when $\det G(\mathrm{i}\omega=0)=0$, i.e. when the Green’s function has a
zero. Hence when the free fermion SPT phase is trivialized by interactions,
although there is no unavoidable phase transition between the SPT and the
trivial phase, the topological number $n$ still has to change discontinuously
somewhere in the phase diagram, as shown in Fig. 6. Since there is no real
physical transition, the number $n$ has to change through the zero of the full
interacting fermion’s Green’s function. This must hold throughout the SMG
phase (line) regardless of the interaction strength $g$ (as long as
$g>g_{c}$).

Computing the full fermion’s Green’s function for a given interacting
Hamiltonian is generally a challenging task. To argue that the Green’s
function zero is a general feature of the SMG state independent of microscopic
details, one relies on the _topological defect condensation_ construction for
the SMG state, introduced in Sec. IV.1. Starting with the SMG of eight
Majorana zero modes in (0+1)D, the fermion Green’s function can be explicitly
computed [197, 136]:

$G_{ab}(\mathrm{i}\omega)\sim\frac{\mathrm{i}\omega\delta_{ab}}{(\mathrm{i}\omega)^{2}-m^{2}},$
(60)

where $m$ is an effective mass proportional to the strength of the fermion
interaction. Obviously, in (0+1)D, $G_{ab}(\mathrm{i}\omega)$ approaches zero
as $\mathrm{i}\omega\rightarrow 0$. Now one can construct a higher dimensional
SMG state by decorating every topological point defect of the Yukawa field
with eight interacting Majorana fermions in its SMG state, and then
proliferating the point defects to put the Yukawa field in a disordered phase,
following the strategy of the decorated domain wall construction [198]. The
fermion Green’s function can be evaluated in the spacetime by patching the
(0+1)D Green’s functions along the world line of the topological defect, and
then path integrating all possible world line configurations. Following this
approach, Ref. 199 was able to show that the fermion Green’s function takes
the general form of

$G_{ab}(k)\sim\frac{k_{\mu}\gamma^{\mu}\delta_{ab}}{k_{\mu}k^{\mu}-m^{2}},$
(61)

which universally exhibits a zero $G(k_{\mu}=0)=0$ in zero momentum-energy
limit. Before the non-perturbative prove by Ref. 199, Eq. (61) was first
obtained in Ref. 197 by a perturbative calculation, and later argued in Ref.
99, 35, 154 using the fluctuating bilinear mass picture.

###  V.2 Deconfined Quantum Criticality

While the Green’s function zero provides a key diagnosis of the SMG phase,
what about the diagnosis for the SMG transition? The theoretical framework of
fermion fractionalization indicates that if the SMG transition is direct and
continuous, it should be a deconfined quantum critical point (DQCP) [35],
where the physical fermions $\psi$ fractionalize into _deconfined_ bosonic
$\upeta$ and fermionic $\uppsi$ partons _at and only at_ the critical point.

The concept of DQCP [200, 201, 202] was originally introduced to describe the
direct continuous transition between the antiferromagnetic phase and the
valence bond solid phase in (2+1)D quantum spin models. The two phases break
distinct symmetries (spin-rotation and lattice-rotation symmetries
respectively) and cannot be connected by a single quantum critical point
without fine-tuning in the Landau-Ginzburg-Wilson paradigm. The DQCP provides
an explanation for this exotic quantum critical point by fractionalizing the
physical spin into deconfined spinons (partons) at and only at the critical
point. The continuous SMG transition is also an exotic quantum critical point
that requires the DQCP description. However, Unlike the conventional (bosonic)
DQCP that fractionalizes bosonic degrees of freedoms, the continuous SMG
transition is a _fermionic_ version of DQCP, as it fractionalizes fermionic
degrees of freedoms.

The SMG critical point can be viewed as the intersection among four phases:

  1. (i)

the massless fermion phase ($\langle\upeta\rangle\neq
0,\langle\upphi\rangle=0$),

  2. (ii)

the SMG phase ($\langle\upeta\rangle=0,\langle\upphi\rangle\neq 0$),

  3. (iii)

the spontaneous symmetry breaking (SSB) massive phase
($\langle\upeta\rangle\neq 0,\langle\upphi\rangle\neq 0$),

  4. (iv)

the fermionic parton QCD phase (if stable)
($\langle\upeta\rangle=0,\langle\upphi\rangle=0$).

The four phases corresponds to the four different states of the bosonic parton
$\upeta$ and the parton Higgs $\upphi$ fields, as summarized in Fig. 5. Fig.
7(a) shows a schematic phase diagram. The DQCP emerges at the origin of the
phase diagram, where both $\upeta$ and $\upphi$ are critical. It seems that a
direct transition between the massless fermion phase to the SMG phase requires
fine tuning through the DQCP. However, if the fermionic parton QCD theory is
dynamically unstable, the QCD phase will shrink to a single transition line
between the massless fermion and the SMG phases, such that a direct continuous
SMG transition can persist without fine tuning.

Figure 7: (a) Schematic phase diagram determined by whether $\upeta$ and
$\upphi$ are condensed or not. The phase transitions belongs to the following
universality classes: Gross-Neveu (GN), Wilson-Fisher (WF), Higgs, or
confinement (Conf.). The DQCP emerges at the intersection all phases. (b,c)
Flow of coupling constant $g$ and the scaling dimensions for physical fermions
$\Delta_{\psi}$ and/or the interaction term $\Delta_{\text{int}}$ in (b) the
(1+1)D 3-4-5-0 model, (c) higher dimensional models.

Strictly speaking, the DQCP picture for the SMG transition is only valid and
necessary in (2+1)D or higher dimensions. The SMG in lower dimensions (i.e.
(0+1)D and (1+1)D) are exceptional and can be understood without involving
fermion fractionalization or DQCP.

  * •

In the (0+1)D Fidkowski-Kitaev model, the four-fermion interaction will
immediately opens the gap, which can be understood by solving the quantum
mechanical problem in Eq. (1) exactly. There is no notion of phase transition
and quantum criticality in (0+1)D, not to mention DQCP.

  * •

In (1+1)D, take the 3-4-5-0 model for example, the SMG can be understood
within the Luttinger liquid framework as a BKT transition. The six-fermion SMG
interaction in Eq. (11) has a bare scaling dimension
$\Delta_{\text{int}}|_{g=0}=5$ in the free fermion limit, which is
perturbatively irrelevant. However, with a non-perturbative (finite)
interaction strength, under the RG flow, the Luttinger parameters (as exact
marginal parameters) will be modified by the interaction, leading to the
decrease of the scaling dimension of the interaction term. When the scaling
dimension drops below 2 (which is the spacetime dimension), the SMG
interaction will become relevant, driving the system into the SMG phase.[34,
105] So the SMG transition is triggered right at
$\Delta_{\text{int}}|_{g=g_{c}}=2$, see Fig. 7(b).

  * •

In higher dimensions ((2+1)D and above), interactions are always
perturbatively irrelevant at the free fermion fixed point, such that an
infinitesimal interaction $g$ will not immediately drive the SMG transition.
Therefore, the transition generally requires a finite critical interaction
strength $g_{c}$. The critical point ($g=g_{c}$) is expected to be an unstable
fixed point under RG, which either flows to the free fermion fixed point
($g=0$) or the SMG (gapped phase) fixed point ($g=\infty$), as illustrated in
Fig. 7(c).

One implication of fermion fractionalization at the DQCP is that the physical
fermion scaling dimension will generally be larger at the SMG transition
compared to the free fermion fixed point [35]. Massless free fermion
(Dirac/Weyl/Majorana) in $d$-dimensional spacetime has the scaling dimension
$\Delta_{\psi}|_{g=0}=(d-1)/2$. At the SMG transition point ($g=g_{c}$), the
fermion fractionalizes to the critical bosonic parton $\upeta$ and the gapless
fermionic parton $\uppsi$. In the large-$N$ limit, the scaling dimensions of
partons are $\Delta_{\upeta}=d/2-1$ and $\Delta_{\uppsi}=(d-1)/2$, such that
the leading order estimation of the physical fermion scaling dimension becomes
$\Delta_{\psi}|_{g=g_{c}}=\Delta_{\upeta}+\Delta_{\uppsi}=d-3/2$. For $d>2$,
the large-$N$ estimation implies

$\Delta_{\psi}|_{g=g_{c}}>\Delta_{\psi}|_{g=0},$ (62)

i.e. the fermion scaling dimension will jump to a higher value right at the
SMG transition, as shown in Fig. 7(c). For $d=2$, the above naive dimension
counting seems to indicate $\Delta_{\psi}|_{g=g_{c}}=\Delta_{\psi}|_{g=0}$
(which is not correct, unless $g_{c}=0$). More careful Luttinger liquid RG
analysis [105] shows that the fermion scaling dimension increases continuously
with the interaction until the transition happens, as shown in Fig. 7(b).
Thus, the statement $\Delta_{\psi}|_{g=g_{c}}>\Delta_{\psi}|_{g=0}$ still
holds for the (1+1)D model.

###  V.3 Symmetric Mass Generation of the Standard Model

One important motivation driving the study of SMG is to seek for the lattice
regularization of chiral gauge theories, in particular the Standard Model of
particle physics. The goal is to gap the fermion doublers (mirror fermions)
without affecting the original chiral fermions [36, 41, 93, 95, 97]. Possible
routes to gap the Standard Model with either 15$N_{f}$ or 16$N_{f}$ Weyl
fermions via the SMG are reviewed in the following, where $N_{f}$ stands for
the family (or generation) number and can be taken to be $N_{f}=3$.

####  V.3.1 Symmetry Extension of the 15$N_{f}$\- or 16$N_{f}$-Weyl-Fermion
Standard Model

The (3+1)D Standard Model (SM) has a chiral internal symmetry group of the Lie
algebra $\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}$,
which could correspond to either of the four versions of Lie groups for
$q=1,2,3,6$:

$G_{\mathrm{SM}_{q}}\equiv\frac{\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)_{Y}}{\mathbb{Z}_{q}},$
(63)

which are all compatible with all known particle representation data to date.
The following discussion is applicable to any of the four versions of SM for
$q=1,2,3,6$. For any specific version, the SM contains $N_{f}=3$ families
(generations) of matter fermions. In each family, there can be either 15 or 16
left-handed Weyl fermions. The corresponding SM phases will be denoted as
15$N_{f}$-SM and 16$N_{f}$-SM respectively. The first 15 Weyl fermions
transform in the following representations of
$\mathfrak{su}(3)\times\mathfrak{su}(2)\times\mathfrak{u}(1)_{Y}$

$\displaystyle\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}$ (64) $\displaystyle=(\overline{\bf
3},{\bf 1})_{2}\oplus({\bf 1},{\bf 2})_{-3}\oplus({\bf 3},{\bf
2})_{1}\oplus(\overline{\bf 3},{\bf 1})_{-4}\oplus({\bf 1},{\bf 1})_{6}\quad$
(65)

in each family. Here $u_{R}$ and $d_{R}$ are up and down types of right-handed
quarks. The $q_{L}$ is the $\mathfrak{su}(2)$ doublet of up and down types of
left-handed quarks. The $e_{R}$ is the right-handed electron. The $l_{L}$ is
the $\mathfrak{su}(2)$ doublet of neutrino and electron types of left-handed
leptons. The 16th Weyl fermion corresponds to the sterile neutrino
$\bar{\nu}_{R}=({\bf 1},{\bf 1})_{0}$, which can be appended to Eq. (64) in
any of the families.

For simplicity, all SM fermions are assumed to be massless, without including
the electroweak Higgs and its symmetry breaking. To apply SMG to 15$N_{f}$-SM
or 16$N_{f}$-SM, one shall confirm that the two necessary conditions in Sec.
II.5 are satisfied. For the second condition in Sec. II.5, it is true in the
SM. For the first condition, several recent works had checked the cobordism
group classification of anomalies in the SM [183, 203, 185, 187]. Given the
spacetime-internal symmetry $G=\mathrm{Spin}\times G_{\mathrm{SM}_{q}}$, the
anomaly index $\nu\in\text{TP}_{5}(G)$ belongs to the cobordism group

$\displaystyle\mathrm{TP}_{5}(\mathrm{Spin}\times
G_{\mathrm{SM}_{q}})=\left\\{\begin{array}[]{ll}\mathbb{Z}^{5}\times\mathbb{Z}_{2},&q=1,3,\\\
\mathbb{Z}^{5},&q=2,6.\end{array}\right.$ (68)

As checked in Ref. 203, 185, 187, The perturbative local anomalies
($\mathbb{Z}$ classes) and nonperturbative global anomalies ($\mathbb{Z}_{n}$
classes) all vanish ($\nu=0$) for both the 15$N_{f}$-SM and 16$N_{f}$-SM, such
that the SMG is possible in either cases. However, if an additional continuous
baryon minus lepton symmetry $\mathrm{U}(1)_{\bf B-L}$ is to be preserved, the
spacetime-internal symmetry is enlarged to
$G=\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)_{\bf B-L}\times
G_{\mathrm{SM}_{q}}=\mathrm{Spin}^{c}\times G_{\mathrm{SM}_{q}}$, then the
anomaly index $\nu\in\text{TP}_{5}(G)$ belongs to a different cobordism group

$\mathrm{TP}_{5}(\mathrm{Spin}^{c}\times
G_{\mathrm{SM}_{q}})=\mathbb{Z}^{11},\quad q=1,2,3,6,$ (69)

which only vanishes ($\nu=0$) for $16N_{f}$ Weyl fermions [203, 187].
Therefore, the SMG preserving an additional $\mathrm{U}(1)_{\bf B-L}$ only
works for the $16N_{f}$-SM.

Razamat and Tong (Ref. 98 and Ref. 34) showed that both the 15$N_{f}$-SM and
16$N_{f}$-SM can be embedded into a left-right (LR) model with 27 Weyl
fermions (denoted as the 27$N_{f}$-LR) preserving the $G_{\mathrm{SM}_{q}}$
symmetry. The key idea is to bring additional fermions down from high-energy
that are vector-like under $G_{\mathrm{SM}_{q}}$ to mix with the low-energy SM
chiral fermions in each family:

$\displaystyle\begin{array}[]{| l | c | ccc | c}\cline{1-5}\cr(\overline{\bf 3},{\bf 1})_{2}&({\bf 1},{\bf 2})_{-3}&\lx@intercol\hfil({\bf 3},{\bf 2})_{1}\hfil\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil(\overline{\bf 3},{\bf 1})_{-4}\hfil\lx@intercol\vrule\lx@intercol&({\bf 1},{\bf 1})_{6}&\\\ \cline{3-6}\cr(\overline{\bf 3},{\bf 1})_{2}&({\bf 1},{\bf 2})_{-3}&&&&\lx@intercol\hfil{({\bf 1},{\bf 1})_{0}}\hfil\lx@intercol\vrule\lx@intercol\\\ \cline{1-2}\cr({\bf 3},{\bf 1})_{-2}&({\bf 1},{\bf 2})_{+3}&&&&\lx@intercol\hfil{({\bf 1},{\bf 1})_{0}}\hfil\lx@intercol\vrule\lx@intercol\\\ \cline{1-2}\cr\cline{6-6}\cr\end{array}.$ (73)

The first row of Eq. (73) corresponds to the original 15 left-handed Weyl
fermions in Eq. (64), and the second and third rows of Eq. (73) correspond to
6 left-handed and 6 right-handed additional Weyl fermions in total in a
vector-like theory, which add up to $27$ Weyl fermions per family. In (73),
all right-handed fermions are complex conjugated to be written as their anti-
particles, so all fermions are written in the left-handed versions. The
sterile neutrino $\bar{\nu}_{R}$ (the 16th Weyl fermion) corresponds to one of
the $(\mathbf{1},\mathbf{1})_{0}$ representation in Eq. (73).

Hereafter for the left-handed and right-handed notations, we always use the
italic font $L$ and $R$ to denote that of spacetime symmetry (Spin group),
while use the text font L and R for that of internal symmetry.

The 27$N_{f}$-LR model has an enlarged symmetry $G_{\mathrm{LR}_{q,p}}$ with
$q,p\in\\{1,2,3,6\\}$, totally sixteen versions,

$\begin{split}&G_{\mathrm{LR}_{q,p}}\equiv\frac{G_{\mathrm{SM}_{q}}\times\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{R}}}{\mathbb{Z}_{p}}\\\
&=\frac{\mathrm{SU}(3)\times\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{R}}}{\mathbb{Z}_{q}\times\mathbb{Z}_{p}},\end{split}$
(74)

where $\mathrm{SU}(2)\times\mathrm{U}(1)_{Y}$ in $G_{\mathrm{SM}_{q}}$ is
renamed to $\mathrm{SU}(2)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{L}}$ in
$G_{\mathrm{LR}_{q,p}}$. The 27 Weyl fermions can be organized by the
$\mathrm{SU}(2)_{\mathrm{R}}$ irreducible representations, as indicated by the
framed boxes in Eq. (73), or written as the representation of
${\mathfrak{su}(3)\times\mathfrak{su}(2)_{\rm L}\times\mathfrak{su}(2)_{\rm
R}\times\mathfrak{u}(1)_{\rm L}\times\mathfrak{u}(1)_{\rm R}}$:

$\begin{split}&\big{(}\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus\bar{\nu}_{R}\big{)}\oplus{d}^{\prime}_{L}\oplus\bar{l}^{\prime}_{R}=\\\
&\big{(}(\overline{\bf 3},{\bf 1},{\bf 2})_{2,-1}\oplus({\bf 1},{\bf 2},{\bf
2})_{-3,3}\oplus({\bf 3},{\bf 2},{\bf 1})_{1,-2}\oplus(\overline{\bf 3},{\bf
1},{\bf 1})_{-4,2}\\\ &\oplus({\bf 1},{\bf 1},{\bf 1})_{6,-6}\oplus{({\bf
1},{\bf 1},{\bf 2})_{0,-3}}\big{)}\oplus({\bf 3},{\bf 1},{\bf
1})_{-2,4}\oplus({\bf 1},{\bf 2},{\bf 1})_{3,0}.\end{split}$ (75)

This essentially promotes $\bar{d}_{R},l_{L},\bar{\nu}_{R}$ to
$\mathrm{SU}(2)_{\mathrm{R}}$ doublets and introduces new
$\mathrm{SU}(2)_{\mathrm{R}}$ singlet fermions
${d}^{\prime}_{L},\bar{l}^{\prime}_{R}$.

Figure 8: A schematic phase diagram among the SMG, the 27$N_{f}$-LR, the
15$N_{f}$-SM or the 16$N_{f}$-SM phases. The SMG phase is separated from the
27$N_{f}$-LR phase by the SMG transition. The 15$N_{f}$-SM and 16$N_{f}$-SM
phases are separated by the Ultra Unification (UU) transition [187, 188, 189],
and they can be both obtain from the 27$N_{f}$-LR model phase by different
Higgs transitions.

The 27$N_{f}$-LR phase can be viewed as an intermediate quantum phase between
the 15$N_{f}$-SM or 16$N_{f}$-SM and the SMG phases, as shown in Fig. 8. The
transition from the 27$N_{f}$-LR phase to the 16$N_{f}$-SM phase can be driven
by condensing a scalar Higgs field
$h_{\mathrm{R}}=(\mathbf{1},\mathbf{1},\mathbf{2})_{0,-3}$ that couples to the
Weyl fermions by the following Higgs term

$\big{(}\epsilon_{R}h_{\mathrm{R}}(\bar{d}_{R}d^{\prime}_{L}+\bar{l}^{\prime}_{R}l_{L})+(h_{\mathrm{R}}^{\dagger}\bar{\nu}_{R})(h_{\mathrm{R}}^{\dagger}\bar{\nu}_{R})\big{)}+\text{h.c.},$
(76)

where $\epsilon_{R}$ is an anti-symmetric tensor in the
$\mathrm{SU}(2)_{\mathrm{R}}$ doublet subspace. The $\epsilon_{L}$ and
$\epsilon$ tensor for the $\mathrm{SU}(2)_{\mathrm{L}}$ and Lorentz
$\mathfrak{su}(2)$ subspace are omitted. All Lagrangian terms become a scalar
in a trivial singlet representation of both spacetime and internal symmetries.
For each family, the Higgs condensation $\langle h_{\mathrm{R}}\rangle\neq 0$
(only its _upper_ $\mathrm{SU}(2)_{\mathrm{R}}$ doublet component is nonzero
and condenses) lifts 11 Weyl fermions with a mass gap (the _lower_ half of
both $\mathrm{SU}(2)_{\mathrm{R}}$ doublets of $\bar{d}_{R}$ and $l_{L}$ get
mass, while the dimension-5 term
$(h_{\mathrm{R}}^{\dagger}\bar{\nu}_{R})(h_{\mathrm{R}}^{\dagger}\bar{\nu}_{R})$
gives Majorana mass to only the _upper_ half of the
$\mathrm{SU}(2)_{\mathrm{R}}$ doublet $\bar{\nu}_{R}$), leaves 16 Weyl
fermions at low energy, and breaks the symmetry from $G_{\mathrm{LR}_{q,p}}$
down to $G_{\mathrm{SM}_{q}}\times\mathrm{U}(1)_{\bf B-L}$.

To further lift the sterile neutrino, an additional scalar Higgs field
$h^{\prime}_{\mathrm{R}}=(\mathbf{1},\mathbf{1},\mathbf{1})_{0,6}$ should be
introduced (with $\mathrm{U}(1)_{\bf B-L}$ charge $-1$), such that the Higgs
term becomes

$\big{(}\epsilon_{R}h_{\mathrm{R}}(\bar{d}_{R}d^{\prime}_{L}+\bar{l}^{\prime}_{R}l_{L})+h^{\prime}_{\mathrm{R}}(\epsilon_{R}\bar{\nu}_{R}{\nu}_{R})\big{)}+\text{h.c.}$
(77)

In each family, the Higgs condensation $\langle h_{\mathrm{R}}\rangle,\langle
h^{\prime}_{\mathrm{R}}\rangle\neq 0$ leaves 15 Weyl fermions at low energy
(the $h^{\prime}_{\mathrm{R}}(\epsilon_{R}\bar{\nu}_{R}{\nu}_{R})$ gives a
Dirac mass to both upper and lower components of the
$\mathrm{SU}(2)_{\mathrm{R}}$ doublet $\bar{\nu}_{R}$), breaks the symmetry
from $G_{\mathrm{LR}_{q,p}}$ down to $G_{\mathrm{SM}_{q}}$ with no
$\mathrm{U}(1)_{\bf B-L}$, and drives the transition from the 27$N_{f}$-LR
phase to the 15$N_{f}$-SM phase.

Embedding the 15$N_{f}$-SM into the 27$N_{f}$-LR amounts to the following
symmetry extension

$\frac{\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{R}}}{{\mathbb{Z}_{\gcd(p,2)}}}\to\mathrm{Spin}\times
G_{\mathrm{LR}_{q,p}}\to\frac{\mathrm{Spin}\times
G_{\mathrm{SM}_{q}}}{{\mathbb{Z}_{\gcd(p,3)}}},$ (78)

and embedding the 16$N_{f}$-SM into the 27$N_{f}$-LR amounts to the following
symmetry extension

$\mathrm{SU}(2)^{\prime}_{\mathrm{R}}\to\mathrm{Spin}\times
G_{\mathrm{LR}_{q,p}}\to\frac{\mathrm{Spin}^{c}\times
G_{\mathrm{SM}_{q}}}{\mathbb{Z}_{\gcd(p,3)}},$ (79)

where
$\mathrm{SU}(2)^{\prime}_{\mathrm{R}}=\frac{\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{R}}}{\mathbb{Z}_{\gcd(p,2)}\times\mathrm{U}(1)_{\bf
B-L}}$.

####  V.3.2 Symmetric Mass Generation in a 27$N_{f}$-Weyl-Fermion Left-Right
Model

The SMG in the 27$N_{f}$-LR model can be achieved by both the parton-Higgs
mechanism in Sec. IV.2.1 and the s-confinement mechanism in Sec. IV.2.2.

  1. 1.

The s-confinement mechanism: According to Razamat and Tong [98], one first
supersymmetrizes the fermions in Eq. (73) to their corresponding
$\mathcal{N}=1$ chiral multiplets as $\bar{d}_{R}\oplus{l}_{L}\oplus
q_{L}\oplus\bar{u}_{R}\oplus\bar{e}_{R}\oplus\bar{\nu}_{R}\oplus{d}^{\prime}_{L}\oplus\bar{l}^{\prime}_{R}\to{\rm
D}\oplus{\rm L}\oplus{\rm Q}\oplus{\rm U}\oplus{\rm E}\oplus{\rm N}\oplus{\rm
D}^{\prime}\oplus{\rm L}^{\prime}$. Then gauge the $\mathrm{SU}(2)_{\rm R}$
symmetry by turning on a dynamical $\mathrm{SU}(2)_{\rm R}$ gauge field that
couples to the $\mathfrak{su}(2)_{\rm R}$ doublet: ${\rm D},{\rm L},{\rm N}$.
A _dangerously irrelevant_ superpotential $\mathcal{W}_{\rm UV}$ at UV

$\displaystyle\mathcal{W}_{\rm UV}$ $\displaystyle=$ $\displaystyle{\rm L}{\rm
L}{\rm E}+{\rm D}{\rm D}{\rm U}+{\rm L}{\rm D}{\rm Q}+{\rm L}{\rm N}{\rm
L}^{\prime}+{\rm D}{\rm N}{\rm D}^{\prime}+{\rm h.c.}\quad\;$ (80)

becomes a $G_{\mathrm{SM}_{q}}$-symmetric _relevant_ deformation that pairs
the 15 mesons ($\tilde{\rm E},\tilde{\rm U},\tilde{\rm Q},\tilde{\rm
L},\tilde{\rm D}$) formed by ${\rm D},{\rm L},{\rm N}$ via s-confinement) with
the remained 15 superfields (namely ${\rm Q},{\rm U},{\rm E},{\rm
D}^{\prime},{\rm L}^{\prime}$) in a quadratic manner $\mathcal{W}_{\rm IR}$ at
IR, consequently gapping out all fields as SMG:

$\displaystyle\mathcal{W}_{\rm IR}$ $\displaystyle=$ $\displaystyle\tilde{\rm
E}{\rm E}+\tilde{\rm U}{\rm U}+\tilde{\rm Q}{\rm Q}+\tilde{\rm L}{\rm
L}^{\prime}+\tilde{\rm D}{\rm D}^{\prime}+{\rm h.c.}$ (81)

Here the $\mathfrak{su}(3)$ color and $\mathfrak{su}(2)_{\mathrm{L}}$ flavor
indices are suppressed, with the understanding that they should be contracted
properly to make the Lagrangian a singlet. As $\mathcal{W}_{\rm IR}$ flows
strong, all fermions are gapped from low-energy, resulting in the SMG phase.
When there are multiple families, independent $\mathrm{SU}(2)_{\mathrm{R}}$
gauge fields are introduced in each family, such that the total gauge group is
$\mathrm{SU}(2)_{\rm R_{1}}\times\mathrm{SU}(2)_{\rm
R_{2}}\times\cdots\times\mathrm{SU}(2)_{\mathrm{R}_{N_{f}}}$. This guarantees
that the s-confinement can induce the fully gapped SMG phase in each family
independently.

  2. 2.

The parton-Higgs mechanism: Tong [34] shows that $N_{f}$ families of 27 Weyl-
fermion model can be fully gapped by preserving _not only_ the SM internal
symmetry group $G_{\mathrm{SM}_{q}}$ for $q=1,2,3,6$, _but also_ an additional
continuous baryon minus lepton symmetry $\mathrm{U}(1)_{\bf B-L}$.

The parton-Higgs mechanism introduces the scalar Higgs fields ${\phi}={({\bf
1},{\bf 2},{\bf 2})_{-3,3}}$. Ref. 34 suggests to fully gap (the 27 Weyl
fermions per family) to achieve the SMG by adding

$\displaystyle\big{(}\phi({{\bar{d}_{R}}}{{q_{L}}}+{{\bar{\nu}_{R}}}{{\bar{l}^{\prime}_{R}}}+{{\bar{e}_{R}}}{{l_{L}}})+\phi^{2}{{{d^{\prime}_{L}}}{{\bar{u}_{R}}}}\big{)}+{\rm
h.c.}$ (82)

when the condensation $\langle\phi\rangle\neq 0$ occurs. The internal symmetry
breaking pattern of this SMG deformation is

$\displaystyle
G_{\mathrm{SM}_{q}}\times_{\mathbb{Z}_{\gcd(p,3)}}K=G_{\mathrm{SM}_{q}}\times_{\mathbb{Z}_{\gcd(p,3)}}\frac{\mathrm{SU}(2)_{\mathrm{R}}\times\mathrm{U}(1)_{\mathrm{R}}}{{\mathbb{Z}_{\gcd(p,2)}}}$
(83)
$\displaystyle\mapsto\frac{\mathrm{SU}(3)\times\mathrm{SU}(2)_{\text{diagonal}}\times\mathrm{U}(1)_{\text{diagonal}}}{\mathbb{Z}_{q}}\times\mathrm{U}(1)_{\bf
B-L},\quad\quad$ (84)

which leaves a continuous baryon minus lepton number symmetry preserved.

##  VI Summary and Outlook

We have reviewed the topic of symmetric mass generation (SMG) which has
attracted considerable ongoing interests from both the condensed matter and
high energy physics communities in recent years. The SMG transition is beyond
the classic Landau-Ginzburg-Wilson paradigm, which potentially involves
fractionalization of physical fermions at and only at the critical point. We
have reviewed various aspects related to SMG, including interacting
topological insulators (TI), topological superconductors (TSC) [204, 205], SPT
states [206, 207], anomalies, lattice regularization of chiral gauge theories,
and the current status of numerical efforts from both the lattice gauge theory
and condensed matter communities. We have also discussed theoretical
understanding related to SMG, such as the s-confinement mechanism, the
connection between SMG and DQCP, etc.

Various numerical works have suggested that the SMG could be a continuous
transition, and the critical exponents have been measured in some of the
numerical works. These works pose a challenge to further analytical
understanding of SMG. Unlike the standard Higgs-Yukawa types of theories where
a large-$N$ or small-$\epsilon$ expansion can be applied, theoretically we do
not yet have a controlled theory where we can perform a reliable analytical
calculation for the critical exponents for SMG, and compare with the numerical
simulation. If the SMG is indeed a continuous transition and corresponds to a
certain type of conformal field theories, then the conformal Bootstrap method
can also provide very helpful insights into the nature of the transition. All
these require further efforts from different disciplines of theoretical
physics.

The problem of the nonperturbative regularization of chiral fermion and chiral
gauge theories also potentially bring the high-energy lattice, mathematical
physics, quantum information, numerical simulation, and condensed matter
communities to work closer together. What else can stimulate different
communities to work together other than tackling a profound mysterious
problem? Looking back at science history, the phenomena of the anomaly-inflow
[5, 6] and domain wall fermion [76, 78] (as the precursors of the TI/TSC/SPT
states) had an old tradition rooted in the high-energy theory and lattice
community in the mid 1980s and mid 1990s. Closely related phenomena like
integer and fractional quantum Hall states are studied in the condensed matter
community already in the early 1980s. Yet the acceleration came much later
until the concrete materialization discovery of TI/TSC in 2005 (in both
theories and experiments [204, 205]) ignites the serious interest in the
classification of the interacting many-body quantum systems of SPT states in
the early 2010s [59, 60, 61, 206, 207]. This generates crossing-disincline
interests in the classification of anomalies, cobordism classes, and the TQFTs
in quantum matter, high-energy string theory, and mathematical physics [41,
42, 43, 48, 44, 4]. Look back and ponder this history, some curious minds
might ask: Why were the discovery of SMG and classification of TI/TSC/SPT
states not made in the high-energy lattice community even earlier, although
the lattice gauge theory pioneers like Wilson and Kogut already had earlier
achievements on the related lattice topics in the 1970s [208]? Perhaps the
inputs of the condensed matter material experiments are crucial, also perhaps
the crossing-boundary interdisciplinary ideas are much more welcomed in the
recent condensed matter developments. As Preskill summarized: Combing more
ideas together is better than each of the isolated ideas [209]. Hopefully
combing the ideas of (1) domain wall fermions [76], (2) gapping the mirror
fermions [36], and (3) the anomaly-free SMG [41, 93, 95, 97], already gives
the ample insights to fully solve the lattice definition of chiral fermion,
chiral gauge theory, and chiral Standard Model problems.

###### Acknowledgements.

The authors are listed in alphabetical order by a standard convention. We
would like to acknowledge Cenke Xu, who initiated this review project,
contributed to the writing of Secs. II.1, V.1, VI, and suggested numerous
improvements. We would also like to thank David Tong for patiently explaining
a few points of his works to us, and John McGreevy for reading through our
draft and providing many helpful suggestions. JW thanks Yuta Hamada for
discussions. JW is supported by Center for Mathematical Sciences and
Applications at Harvard University and NSF Grant DMS-1607871 “Analysis,
Geometry and Mathematical Physics.” YZY is supported by a startup fund at
UCSD.

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