In Euclidean signature: $\mathbb{Z}_{2}^{F_{+}}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{T_{\rm E}}$ and
$\mathbb{Z}_{2}^{F_{-}}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{4}^{{T_{\rm E}}F_{-}}$.

     * •

In Lorentzian signature:
$\mathbb{Z}_{2}^{F_{+}}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{4}^{TF_{+}}$ and
$\mathbb{Z}_{2}^{F_{-}}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F_{-}}$.

     * •

$\mathbb{Z}_{2}^{F_{\pm}}\rtimes\Big{(}{\rm
SO}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}\Big{)}=\mathrm{Pin}^{\pm}(d)$.

  3. (3).

Another interpretation of Table 4’s involves a $\mathcal{T}^{2}=-1$ boson $B$
and a $\mathcal{T}^{2}=-1$ fermion $F_{+}$:

$\mathbb{Z}_{2}^{F_{+}}\to\frac{\mathbb{Z}_{4}^{TB}\times\mathbb{Z}_{4}^{TF_{+}}}{\mathbb{Z}_{2}^{T}}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F_{-}},$

by including SO$(d)$ which becomes Table 5’s

$\displaystyle\mathbb{Z}_{2}^{B}\to{\rm
E}\mathrm{Pin}^{BF_{+}}(d)\equiv\Big{(}(\mathbb{Z}_{2}^{F_{+}}\rtimes{\rm
SO}(d))\times\mathbb{Z}_{2}^{B}\Big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}\to\mathrm{Pin}^{-}(d).$ (6.3)

Note that ${\rm
E}\mathrm{Pin}^{BF_{+}}(d)\equiv\Big{(}(\mathbb{Z}_{2}^{F_{+}}\rtimes{\rm
SO}(d))\times\mathbb{Z}_{2}^{B}\Big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm E}}$ has
the properties _specifically_ here:

     * •

$\mathrm{Pin}^{+}(d)=(\mathbb{Z}_{2}^{F_{+}}\rtimes{\rm
SO}(d))\rtimes\mathbb{Z}_{2}^{T_{\rm E}}={\rm
Spin}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}=\frac{{\rm
Spin}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm E}F}}{\mathbb{Z}_{2}^{F}}$.

     * •

In Euclidean signature: $\mathbb{Z}_{2}^{B}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{4}^{{T_{\rm E}}B}$ and
$\mathbb{Z}_{2}^{F_{+}}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{2}^{T_{\rm E}}\times\mathbb{Z}_{2}^{F_{+}}$.

     * •

In Lorentzian signature:
$\mathbb{Z}_{2}^{B}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{4}^{TB}$ and
$\mathbb{Z}_{2}^{F_{+}}\rtimes\mathbb{Z}_{2}^{T}=\mathbb{Z}_{4}^{TF_{+}}$.

     * •

$\big{(}{\rm
SO}(d)\times\mathbb{Z}_{2}^{B}\big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm
E}}=\mathbb{Z}_{2}^{B}\rtimes{\rm O}(d)={\rm
SO}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm E}B}={\rm E}(d).$

Ref. [48] finds that the three interpretations and expressions eq (6.1),
(6.2), and (6.3) are equivalent by relabeling the group elements, so we can
generally write all of them as the same as the ${\rm E}\mathrm{Pin}$
extension:

$\displaystyle\mathbb{Z}_{2}\to{\rm E}\mathrm{Pin}(d)\to\mathrm{Pin}^{-}(d).$
(6.4)

Our claim is that the 1+1d $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
fermionic SPTs classified by $\Omega_{2}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{8}$
with $\nu_{\rm BDI}\in\mathbb{Z}_{8}$ class 2d topological invariants over a
2-manifold ${M^{2}}$ with $\mathrm{Pin}^{-}$ structure

$\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi\nu_{\rm
BDI}}{8}\text{ABK})|_{M^{2}}$ (6.5)

can be trivialized at $\nu_{\rm BDI}=4$ by an extension from
$\mathrm{Pin}^{-}$ to ${\rm E}\mathrm{Pin}$. The reason is that 4ABK (4 layers
of Kitaev fermionic chain) equals to the topological invariant of the first
Stiefel-Whitney classes $w_{1}^{2}\equiv w_{1}(TM)^{2}$ (1 layer of Haldane
spin chain) [3].202020Here we denote $w_{j}\equiv w_{j}(TM)$ are the $j$-th
Stiefel-Whitney (SW) classes of the tangent bundle $TM$ of the spacetime
manifold $M$. For the SW class for the vector bundle $V_{G}$ associated with
certain group $G$ (where $G$ can be gauge group or global symmetry group), we
denote it as $w_{j}(V_{G})$ or simply $w_{j}(G)$. All the product notations
between cohomology classes are cup products, such as
$w_{i}w_{j}:=w_{i}(TM)w_{j}(TM)=w_{i}(TM)\cup w_{j}(TM)$.  Namely, the two
topological invariants are related

$\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
4}{8}\text{ABK})|_{M^{2}}\simeq\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi(w_{1})^{2})|_{M^{2}}$ (6.6)

up to a trivially gapped fermionic sector specifying spin structures.

Let us show the eq (6.4)’s EPin can trivialize the eq (6.6). First, the ${\rm
E}\mathrm{Pin}$ structure requires both $\mathrm{Pin}^{+}$ and
$\mathrm{Pin}^{-}$ to hold, while $\mathrm{Pin}^{+}$ requires $w_{2}=0$ and
$\mathrm{Pin}^{-}$ requires $w_{2}+w_{1}^{2}=0$, so ${\rm E}\mathrm{Pin}$
requires $w_{2}=w_{1}^{2}=0$, but all can have nontrivial $w_{1}$ to be on
unorientable manifolds. (See Ref. [48] and Table 6 for a summary of the bundle
constraints for various symmetry groups and corresponding spacetime
structures.) Second, eq (6.6) also implies $\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi(w_{1})^{2})=\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi w_{2})$ in 2d
$\mathrm{Pin}^{-}$, while both expressions become trivial as 1 in ${\rm
E}\mathrm{Pin}$. Thus EPin as a symmetry extension of Pin- trivializes eq
(6.4) for $\nu_{\rm BDI}=4$.

We can also write down how the symmetry operator S of EPin$(d)$ group acts on
the state vectors of Hilbert space

$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F_{+}}\oplus\mathcal{H}_{F_{-}},$

spanned by three sectors, the bosonic sector $|B\rangle$ with $T^{2}=+1$, the
fermionic sector $|F_{+}\rangle$ with $T^{2}=-1$, and the fermionic sector
$|F_{-}\rangle$ with $T^{2}=+1$. Naively one looks for the larger symmetry
including ${\rm O}(d)$, $\mathrm{Pin}^{+}(d)$, and $\mathrm{Pin}^{-}(d)$ for
three sectors in the continuum, in a more general manifold on $M^{d+1}$. But
for quantum mechanics, it is more practical to define the Hilbert space on a
flat space on a $\mathbb{R}^{d-1}$ with a continuous time $t$ on
$\mathbb{R}^{1}$. Then we find the operator S acts on state vectors living on
the $\mathbb{R}^{d-1}$ generating the following groups for each sector:

$\displaystyle\begin{pmatrix}{\rm O}(d-1)\times\mathbb{Z}_{2}^{T}&0&0\\\
0&\mathrm{Pin}^{+}(d-1)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}&0\\\
0&0&\mathrm{Pin}^{-}(d-1)\times\mathbb{Z}_{2}^{T}\end{pmatrix}\text{ acts on
}\begin{pmatrix}[l]|B\rangle\\\ |F_{+}\rangle\\\ |F_{-}\rangle\end{pmatrix}.$
(6.7)

Note that $\mathbb{Z}_{2}^{F_{+}}$ and $\mathbb{Z}_{2}^{F_{-}}$ have the two
generators

$\begin{pmatrix}1&0&0\\\ 0&-1&0\\\ 0&0&1\end{pmatrix}\text{ and
}\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-1\end{pmatrix}.\text{ act on
}\begin{pmatrix}[l]|B\rangle\\\ |F_{+}\rangle\\\ |F_{-}\rangle\end{pmatrix},$

each sits at the $\mathbb{Z}_{2}$ center of Spin group within Pin+ and Pin-
groups for each. If we choose $|B\rangle$ with $T^{2}=-1$, we must change eq
(6.7)’s ${\rm O}(d-1)\times\mathbb{Z}_{2}^{T}$ to ${\rm
E}(d-1)\times_{\mathbb{Z}_{2}^{B}}\mathbb{Z}_{4}^{TB}$. This eq (6.7) is our
interpretation of $\mathrm{EPin}(d)$ acting on the assumed Hilbert space.

In summary, eq (6.4) on $\mathbb{Z}_{2}\to{\rm
E}\mathrm{Pin}(d)\to\mathrm{Pin}^{-}(d)$ gives the mathematical way to
understand the unwinding of 4 layers of Kitaev chains via the symmetry
extension
$\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
demonstrated in Ref. [3]’s Sec. 4.

###  6.2 1+1d: Lift ${\rm O}(d)$ to ${\rm E}(d)$ for $\nu_{\rm
CII}=1\in\mathbb{Z}_{2}$ or even $\nu_{\rm AIII}\in\mathbb{Z}_{4}$ class

In fact, the previous extension method in Sec. 6.1 also works for 1+1d SPTs
with the same topological invariants $\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi(w_{1})^{2})$ but with other symmetry groups of Table 4, such as CII
and AIII classes. In the discrete version, we have
$\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\to\mathbb{Z}_{2}^{T}$, converting to a
continuum language becomes $\mathbb{Z}_{2}\to{\rm E}(d)\to{\rm O}(d)$.

For example, Sec. 3.5’s CII class symmetry extension eq (3.59) and eq (3.63),
and AIII even $\nu_{\rm AIII}\in\mathbb{Z}_{4}$ class studied in Ref. [3]’s
Sec. 4, can be understood by including the spacetime symmetry in the continuum
as the following extensions $N\to\tilde{G}_{\text{Tot}}\to G_{\text{Tot}}$:

$\displaystyle\begin{array}[]{rcccccccc}\text{CII: }&\mathbb{Z}_{2}&\to&{\rm
E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}{\rm
SU}(2)^{F}&\to&\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm SU}(2)^{F}.\\\
&\mathbb{Z}_{2}&\to&{\rm E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}&\to&\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}\frac{({{\rm
U}}(1)\rtimes{\mathbb{Z}}_{4}^{C})}{{\mathbb{Z}}_{2}^{F}}.\\\ \text{AIII:
}&\mathbb{Z}_{2}&\to&{\rm E}\mathrm{Pin}\times_{\mathbb{Z}_{2}^{F}}{\rm
U}(1)^{F}&\to&\mathrm{Pin}^{-}\times_{\mathbb{Z}_{2}^{F}}{\rm U}(1)^{F}.\\\
\text{In pinciple: }&\mathbb{Z}_{2}&\to&{\rm E}(d)&\to&{\rm O}(d).\\\
\end{array}$ (6.12)

Here $d=2$. The above extension works because the $\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi(w_{1})^{2})$ is a nontrivial topological
invariant in $\Omega_{2}^{{\rm O}}$ but becomes trivial by pulling it back to
$\Omega_{2}^{{\rm E}}$ due to the constraint $(w_{1})^{2}=0$ for ${{\rm
E}}(d)$ structure, see Table 6. More generally, the cocycle $(w_{1})^{2}$
becomes a coboundary, thus $(w_{1})^{2}=\delta\beta_{1}$, which splits into
one lower dimensional 1-cochain $\beta_{1}$ under the pullback
trivialization.212121More generally in a multiplicative notation, in terms of
2-cocycle with U(1) coefficient: $\omega_{2}=(-1)^{\int(w_{1})^{2}}$, we mean
the $\omega_{2}=\delta\alpha_{1}$ can split into one lower dimensional
1-cochain $\alpha_{1}$ under the pullback trivialization.

###  6.3 1+1d supersymmetry extension: Lift $\mathrm{Pin}^{-}$ to
$\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ for even $\nu_{\rm BDI}\in\mathbb{Z}_{8}$
class

We have given a discrete group interpretation of $\mathbb{D}_{8}^{F,T}$ of
Table 4 and the extension:

$\mathbb{Z}_{2}^{B}\to\big{(}\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}\big{)}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}.$

Now we will like to include the continuous ${\rm SO}(d)$ sector in order to
generalize our formulation to the continuum QFT. Note that
$\mathbb{D}_{8}^{F,T}\mathrm{Pin}(d)$ contains the ingredients

$\displaystyle\left\\{\begin{array}[]{l}\mathbb{Z}_{2}^{F}\rtimes{\rm
SO}(d)={\rm Spin}(d),\\\ {\rm SO}(d)\rtimes\mathbb{Z}_{4}^{TB}={\rm E}(d),\\\
\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}.\end{array}\right.$
(6.16)

However, there is a _caveat_ : The short exact sequence

$1\to\mathbb{Z}_{2}\to\tilde{G}_{\text{Tot}}\to\mathrm{Pin}^{-}(d)\to 1$

is classified by
$\tilde{G}_{\text{Tot}}=\mathbb{Z}_{2}\rtimes_{\rho,\varphi}\mathrm{Pin}^{-}(d)$
controlled by no $\rho\in G_{\text{Tot}}\to\text{Aut}(\mathbb{Z}_{2})=0$, but
only by a 2-cocycle
$\varphi\in\operatorname{H}^{2}(\mathrm{B}\mathrm{Pin}^{-}(d),\mathbb{Z}_{2})=\mathbb{Z}_{2}$:

  * •

The trivial $\varphi$ gives
$\tilde{G}_{\text{Tot}}=\mathbb{Z}_{2}\times\mathrm{Pin}^{-}(d)$.

  * •

The nontrivial $\varphi=w_{1}^{2}$ gives the
$\tilde{G}_{\text{Tot}}=\mathrm{EPin}(d)$.

So naively $\tilde{G}_{\text{Tot}}$ has only two solutions,
$\mathbb{Z}_{2}\times\mathrm{Pin}^{-}(d)$ and $\mathrm{EPin}$, there is _no_
$\mathbb{D}_{8}^{F,T}\mathrm{Pin}$. How can we make sense the extension from
$\mathrm{Pin}^{-}(d)$ to $\mathbb{D}_{8}^{F,T}\mathrm{Pin}(d)$?222222We should
clarify that our $\mathbb{D}_{8}^{F,T}\mathrm{Pin}(d)$ is distinct from the
$\mathrm{DPin}(d)$ studied in [49, 50]. The
$\mathbb{D}_{8}^{F,T}\mathrm{Pin}(d)$ contains a
$\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}$, where
fermion parity is not at the $\mathbb{D}_{8}^{F,T}$’s center. The
$\mathrm{DPin}(d)$ contains a different $\mathbb{D}_{8}$ which is
$\mathbb{D}_{8}=(\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{F_{-}})\rtimes_{\rho,0}\mathbb{Z}_{2}^{T}$
where $\rho$ is a nontrivial $\mathbb{Z}_{2}^{T}$ action on ${\rm
Aut}(\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{F_{-}})$ with two kinds of
fermion parity $\mathbb{Z}_{2}^{F_{+}}\times\mathbb{Z}_{2}^{F_{-}}$ at the
$\mathbb{D}_{8}$’s center. In fact, in this article, we will not pursue a
Euclidean spacetime realization of $\mathbb{D}_{8}^{F,T}\mathrm{Pin}(d)$.
Since we do not specify the full $d$d spacetime rotational part, we will
denote it as $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ from now on. We shall only
discuss the $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ symmetry generator acts on the
states in the Hilbert space.

The answer comes to the subtle fact that the time-reversal $\mathcal{T}$ in
$\mathbb{D}_{8}^{F,T}$ has to switch between bosonic and fermionic sectors,
thus a _supersymmetry extension_ comes in. We can write down how the symmetry
operator S of $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ group acts on the state
vectors of Hilbert space

$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F},$

spanned by two sectors, the bosonic sector $|B\rangle$ with $T^{2}=-1$, the
fermionic sector $|F\rangle$ with $T^{2}=-1$. Again it is practical to define
the Hilbert space on a flat space on a $\mathbb{R}^{d-1}$ with a continuous
time $t$ on $\mathbb{R}^{1}$. Then we find the S acts on state vectors living
on the $\mathbb{R}^{d-1}$ generating the following groups for each
sector:232323 We represent the complex and Majorana fermionic operators acting
on 2-dimensional Hilbert space $\begin{pmatrix}[l]|B\rangle\\\
|F\rangle\end{pmatrix}$ as:  $\displaystyle\psi\equiv\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\equiv\sigma_{-}\equiv\frac{(\sigma_{x}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sigma_{y})}{2},\quad\psi^{\dagger}\equiv\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\equiv\sigma_{+}\equiv\frac{(\sigma_{x}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma_{y})}{2},\quad F\equiv
N_{F}\equiv\psi^{\dagger}\psi\equiv\sigma_{+}\sigma_{-}\equiv\begin{pmatrix}0&0\\\
0&1\end{pmatrix}.\quad\quad$ (6.17)
$\displaystyle{\gamma}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=\sigma_{x},\quad\bar{\gamma}=\begin{pmatrix}0&-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\\\ \hskip 1.0pt\mathrm{i}\hskip
1.0pt&0\end{pmatrix}=\sigma_{y},\quad P_{F}=(-1)^{F}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\bar{\gamma}{\gamma}=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\gamma}\bar{\gamma}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}=\sigma_{z}.$
(6.18) The following matrix representations of higher-dimensional Hilbert
spaces are the direct parallel story of 2-dimensional Hilbert space.

  1. 1.

The unitary $\mathbb{Z}_{2}^{F}$ generator is

$\displaystyle P_{F}=(-1)^{F}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}=\sigma_{z}\text{, which acts on
}\begin{pmatrix}[l]|B\rangle\\\ |F\rangle\end{pmatrix}.$ (6.19)

We omit a square identity matrix $\mathds{1}$ that acts on the whole Hilbert
space of $|B\rangle$ and $|F\rangle$ (fermionic and bosonic sectors have equal
Hilbert space dimensions), so precisely $P_{F}=\sigma_{z}\otimes\mathds{1}$.

  2. 2.

The anti-unitary $\mathbb{Z}_{4}^{T}$ generator, with complex conjugation $K$,
is

$\displaystyle\mathcal{T}=\begin{pmatrix}0&-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\\\ \hskip 1.0pt\mathrm{i}\hskip 1.0pt&0\end{pmatrix}K=\sigma_{y}K\text{,
which acts on }\begin{pmatrix}[l]|B\rangle\\\ |F\rangle\end{pmatrix},$ (6.20)

or precisely $\mathcal{T}=(\sigma_{y}\otimes\mathds{1})K$. The $\mathcal{T}$
switches between $|B\rangle$ and $|F\rangle$.

  3. 3.

The
$\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{4}^{TB}\rtimes\mathbb{Z}_{2}^{F_{-}}$
is generated by
$\\{\mathcal{T},P_{F}|\mathcal{T}^{4}=P_{F}^{2}=1,P_{F}\mathcal{T}P_{F}=\mathcal{T}^{3}=-\mathcal{T}\\}$.
Note that $\mathcal{T}^{2}=-1$ for all bosonic and fermionic states.

  4. 4.

The Spin group has elements $\rm{S}_{\rm{rot}}$ acting on the Hilbert space
associated with a $\mathbb{R}^{d-1}$ space:

$\displaystyle\rm{S}_{\rm{rot}}=\begin{pmatrix}{\rm SO}(d-1)&0\\\ 0&{\rm
Spin}(d-1)\end{pmatrix}\text{ acts on }\begin{pmatrix}[l]|B\rangle\\\
|F\rangle\end{pmatrix}.$ (6.21)

The $P_{F}$ is the $\mathbb{Z}_{2}$ center of Spin group.

Combining the group elements $P_{F}$, $\mathcal{T}$, and $\rm{S}_{\rm{rot}}$
together, we generate the full structure of $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$
that acts on the assume Hilbert space
$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F}$ associated with the
$\mathbb{R}^{d-1}$ space. We name such an extension as a _supersymmetry
extension_ when we require some of the group elements (here $\mathcal{T}$)
permutes bosonic and fermionic sectors. Formally, we propose that by pulling
back $\mathrm{Pin}^{-}$ to $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ via the
_supersymmetry extension_ , we can trivialize the even
$\nu\in\Omega_{2}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{8}$ whose cobordism invariant
eq (6.5) can be written as:

$\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
2}{8}\text{ABK})|_{M^{2}}\simeq\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{2\pi}{4}\eta^{\prime}\mathrm{PD}(w_{1}(TM)))|_{M^{2}}.$ (6.22)

The $\eta^{\prime}$ is the 1d cobordism invariant of $\Omega_{1}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{4}$, which is a
$\mathbb{Z}_{4}$ extended version of the $\tilde{\eta}$ (which is the 1d
invariant of $\Omega_{1}^{{\rm Spin}}=\mathbb{Z}_{2}$). See more discussions
about cobordism invariants in Sec. 7.1.

###  6.4 1+1d supersymmetry extension: Lift $\mathrm{Pin}^{+}$ to
$\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ for odd $\nu_{\rm DIII}\in\mathbb{Z}_{2}$
class

We have given a discrete group interpretation of $\mathbb{M}_{16}^{F,T}$ of
Table 4 via an extension:

$\mathbb{Z}_{4}\to\big{(}\mathbb{M}_{16}^{F,T}\equiv\mathbb{Z}_{4}^{TF}\rtimes\mathbb{Z}_{4}\big{)}\to\mathbb{Z}_{4}^{TF}.$

Similar to Sec. 6.3, we write down how the symmetry operator S of
$\mathbb{M}_{16}^{F,T}\mathrm{Pin}(d)$ group acts on the state vectors of
Hilbert space

$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F},$

spanned by two sectors, the bosonic sector $|B\rangle$ and the fermionic
sector $|F\rangle$ both with $T^{4}=-1$ and $T^{8}=+1$. Again it is practical
to define the Hilbert space on a flat space on a $\mathbb{R}^{d-1}$ with a
continuous time $t$ on $\mathbb{R}^{1}$. Then we find the S acts on state
vectors living on the $\mathbb{R}^{d-1}$ generating the following groups for
each sector:

  1. 1.

The unitary $\mathbb{Z}_{2}^{F}$ generator is the same as eq (6.19).

  2. 2.

The anti-unitary $\mathbb{Z}_{8}^{T}$ generator, with complex conjugation $K$,
is

$\displaystyle\mathcal{T}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1+\hskip
1.0pt\mathrm{i}\hskip 1.0pt\\\ 1-\hskip 1.0pt\mathrm{i}\hskip
1.0pt&0\end{pmatrix}K=(\frac{\sigma_{x}-\sigma_{y}}{\sqrt{2}})K\text{, which
acts on }\begin{pmatrix}[l]|B\rangle\\\ |F\rangle\end{pmatrix},$ (6.23)

or precisely
$\mathcal{T}=((\frac{\sigma_{x}-\sigma_{y}}{\sqrt{2}})\otimes\mathds{1})K$.
The $\mathcal{T}$ switches between $|B\rangle$ and $|F\rangle$.

  3. 3.

The $\mathbb{M}_{16}^{F,T}$ is generated by
$\\{\mathcal{T},P_{F}|\mathcal{T}^{8}=P_{F}^{2}=1,P_{F}\mathcal{T}P_{F}=\mathcal{T}^{5}=-\mathcal{T}\\}$.
Note that $\mathcal{T}^{2}=-\hskip 1.0pt\mathrm{i}\hskip 1.0ptP_{F}=-\hskip
1.0pt\mathrm{i}\hskip 1.0pt(\sigma_{z}\otimes\mathds{1})$, while
$\mathcal{T}^{4}=-1$ for all bosonic and fermionic states.

  4. 4.

The Spin group has elements $\rm{S}_{\rm{rot}}$ the same as eq (6.21), with
$P_{F}$ its $\mathbb{Z}_{2}$ center.

Combining the group elements $P_{F}$, $\mathcal{T}$, and $\rm{S}_{\rm{rot}}$
together, we generate the full structure of
$\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ that acts on the assume Hilbert space
$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F}$ associated with the
$\mathbb{R}^{d-1}$ space. Again such an extension is a _supersymmetry
extension_ when we require some group element (here $\mathcal{T}$) permutes
bosonic and fermionic sectors. Formally, we propose that we can trivialize the
odd $\nu\in\Omega_{2}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{2}$ class whose cobordism
invariant is the same as in eq (6.22), by pulling back $\mathrm{Pin}^{+}$ to
$\mathbb{M}_{16}^{F,T}\mathrm{Pin}$ via a _supersymmetry extension_.

##  7 Conclusion and Higher Dimensional Generalization

Let us discuss the higher dimensional generalization of the _fermionic
symmetry extension_ and _supersymmetry extension_ :

Earlier we have studied $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
($\mathrm{Pin}^{-}$) and $\mathbb{Z}_{4}^{TF}$ ($\mathrm{Pin}^{+}$) and their
_fermionic symmetry extension_ and _supersymmetry (SUSY) extension_ via a
finite group extension. In fact, there are maps known as Smith homomorphisms
in any dimension relating the following SPT classes and symmetry groups (see
[23, 51, 52] and also [26, 48]):

$(n+3){\rm{d}}\;\text{unitary
}\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}\Rightarrow(n+2){\rm{d}}\;\text{anti-
unitary }\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}\Rightarrow\\\
(n+1){\rm{d}}\;\text{unitary }\mathbb{Z}_{4}^{F}\Rightarrow
n{\rm{d}}\;\text{anti-unitary }\mathbb{Z}_{4}^{TF}\Rightarrow\dots.\quad$
(7.1)

By an adding SO rotational symmetry in terms of continuous groups, eq (7.1)
becomes

$\displaystyle(n+3){\rm{d}}\;{\rm
Spin}\times\mathbb{Z}_{2}\Rightarrow(n+2){\rm{d}}\;\mathrm{Pin}^{-}\Rightarrow(n+1){\rm{d}}\;{\rm
Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}\Rightarrow
n{\rm{d}}\;\mathrm{Pin}^{+}\Rightarrow\dots.\quad$ (7.2)

Smith homomorphism maps include the maps between SPT classifications of these
symmetries [23, 51, 52, 48]:

$\displaystyle\begin{array}[]{rrrl}\Omega_{8}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{32}\times\mathbb{Z}_{2}&\to&\Omega_{7}^{{\rm
Spin}\times{\mathbb{Z}_{2}}}=\mathbb{Z}_{16},&\\\ \Omega_{7}^{{\rm
Spin}\times{\mathbb{Z}_{2}}}=\mathbb{Z}_{16}&\to&\Omega_{6}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{16},&\\\
\Omega_{6}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{16}&\to&\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{16},&\\\
\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{16}&\to&\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16},&\\\
\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}&\to&\Omega_{3}^{{\rm
Spin}\times{\mathbb{Z}_{2}}}=\mathbb{Z}_{8},&\\\ \Omega_{3}^{{\rm
Spin}\times{\mathbb{Z}_{2}}}=\mathbb{Z}_{8}&\to&\Omega_{2}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{8},&\\\
\Omega_{2}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{8}&\to&\Omega_{1}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{4},&\\\
\Omega_{1}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{4}&\to&\Omega_{0}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{2}.&\end{array}$
(7.11)

In fact, as found in Ref. [48] and [53], the $\mathbb{Z}_{2}$ extensions can
trivialize many SPT classes — thus this symmetry extension can help to
construct the symmetry-extended gapped boundaries (lifting all boundary zero
modes):

$\displaystyle 1\to\mathbb{Z}_{2}\to\mathrm{EPin}(d)\to\mathrm{Pin}^{\pm}\to
1.$ (7.12) $\displaystyle 1\to\mathbb{Z}_{2}\to{\rm
Spin}(d)\times\mathbb{Z}_{4}\to\begin{matrix}{\rm
Spin}(d)\times\mathbb{Z}_{2}\\\ \text{ or }\\\ \frac{{\rm
Spin}(d)\times\mathbb{Z}_{4}}{\mathbb{Z}_{2}}\end{matrix}\to 1.$ (7.13)

In the following subsections, we make some comments about these extensions and
the generalization to supersymmetry extensions, for various dimensions.

###  7.1 1+1d

For $d=2$ or $1+1$d: In the bulk part of our work, we had focused on $d=2$ or
$1+1$d cases. We have $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ SPTs
classified by $\Omega_{2}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{8}$, which involves
3d SPT cobordism invariant with $\nu\in\mathbb{Z}_{8}$ [23]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{8}\int_{M^{2}}\mathrm{ABK}).$ (7.14)

$\bullet$ When $\nu=4$, we can show that the four layers of Kitaev fermionic
chain ($\nu=4$) has SPT invariants $4\mathrm{ABK}$ relates to the bosonic
Haldane spin chain $w_{1}(TM)^{2}$ up to a trivial gapped fermionic sector,

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{8}\int_{M^{2}}\mathrm{ABK})=\exp({\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}\int_{M^{2}}w_{1}(TM)^{2}).$

such as in $\mathrm{Pin}^{-}$ which becomes trivialized
$w_{1}(TM)^{2}=\delta\mu$ as a coboundary in EPin, via the group extension
$1\to\mathbb{Z}_{2}\to\mathrm{EPin}(d)\to\mathrm{Pin}^{-}\to 1$, see the
result of [48], Sec. 6.1, and Sec. 6.2.
$\bullet$ When $\nu=2$, we can show that $2\mathrm{ABK}$ is related to
$w_{1}(TM){\eta^{\prime}}\equiv{\eta^{\prime}}\;\mathrm{PD}(w_{1}(TM))$ via

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{8}\int_{M^{2}}\mathrm{ABK})=\exp(\frac{\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{2}}{\eta^{\prime}}\;\mathrm{PD}(w_{1}(TM))).$

The ${\eta^{\prime}}$ is the 1d cobordism invariant of $\Omega_{1}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{4}$, a
$\mathbb{Z}_{4}$ enhancement of the $\tilde{\eta}$. The $\tilde{\eta}$ is the
1d cobordism invariant of $\Omega_{1}^{{\rm Spin}}=\mathbb{Z}_{2}$ which
assigns different spin structures with different values:

$\begin{cases}\text{For the periodic (Ramond) boundary condition,
$\tilde{\eta}=1$.}\\\ \text{For the anti-periodic (Neveu-Schwarz) boundary
condition, $\tilde{\eta}=0$.}\end{cases}$

The ${\eta^{\prime}}\;\mathrm{PD}(w_{1}(TM))$ is also a $\mathbb{Z}_{4}$ class
like ${\eta^{\prime}}$. In Sec. 3.2, we have shown this 2 ABK invariant can be
trivialized by a _SUSY extension_ involving a discrete $\mathbb{D}_{8}^{F,T}$
that we introduce (or we denote it as a $\mathbb{D}_{8}^{F,T}\mathrm{Pin}$ in
the continuum).
$\bullet$ When $\nu=1$, our method does not apply. But Dijkgraaf-Witten [54]
suggests that 2d topological gravity can lift up a single Majorana zero mode
and thus trivialize the Kitaev chain. Other related 10 Cartan symmetry classes
and 2d topological gravity are explored subsequently by Stanford-Witten [55].

###  7.2 2+1d

For $d=3$ or $2+1$d: we have $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ SPTs
classified by $\Omega_{3}^{{\rm Spin}\times{\mathbb{Z}_{2}}}=\mathbb{Z}_{8}$,
which involves 3d SPT cobordism invariant with $\nu\in\mathbb{Z}_{8}$ [23,
56]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{8}\int_{M^{3}}\mathrm{ABK}(\mathrm{PD}(A))),$ (7.15)

where ABK is the Arf-Brown-Kervaire (ABK) invariant and $A$ is a
$\mathbb{Z}_{2}$ gauge field so it is a mod 2 cohomology class
$A\in\operatorname{H}^{1}(M,\mathbb{Z}_{2})$.
$\bullet$ When $\nu=4$, Ref. [4, 48] shows that the
$4\mathrm{ABK}(\mathrm{PD}(A))$ is related to $A^{3}$. This is the same
generator as a 2+1d bosonic SPTs with $\mathbb{Z}_{2}$ symmetry given by
$\operatorname{H}^{3}(\mathrm{B}\mathbb{Z}_{2},{\rm U}(1))=\mathbb{Z}_{2}$,
namely up to a trivial gapped fermionic sector

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{8}\int_{M^{3}}\mathrm{ABK}(\mathrm{PD}(A)))=\exp({\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}\int_{M^{3}}A^{3}).$

$\bullet$ When $\nu=2$, the $2\mathrm{ABK}(\mathrm{PD}(A))$ is related to
$A^{2}{\eta}^{\prime}\equiv{\eta}^{\prime}(\mathrm{PD}(A^{2}))$. Namely,

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{8}\int_{M^{3}}\mathrm{ABK}(\mathrm{PD}(A)))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{3}}{\eta}^{\prime}(\mathrm{PD}(A^{2}))).$

Again, the ${\eta^{\prime}}$ is the 1d cobordism invariant of
$\Omega_{1}^{{\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{4}$.
By pulling back ${\rm Spin}\times{\mathbb{Z}_{2}}$ to ${\rm
Spin}\times{\mathbb{Z}_{4}}$, we can trivialize $A^{2}$ by a $\mathbb{Z}_{2}$
extension, thus we also trivialize all the even classes $\nu=4,2$ at once [48,
53].
$\bullet$ When $\nu=1$ or odd, it will be interesting to check whether our
_SUSY extension_ method applies.

###  7.3 3+1d

For $d=4$ or $3+1$d: We have $\mathbb{Z}_{4}^{TF}$ SPTs classified by
$\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$, which involves 4d SPT
cobordism invariant with $\nu\in\mathbb{Z}_{16}$ [18, 23]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{16}\int_{M^{4}}\eta),$ (7.16)

where $\eta$ is the famous eta invariant.
$\bullet$ When $\nu=8$, the $8\eta$ is related to $w_{1}(TM)^{4}$ up to
trivial gapped fermions, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt8}{16}\int_{M^{4}}\eta)=\exp(\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int_{M^{4}}w_{1}(TM)^{4}).$

$\bullet$ When $\nu=4$, the $4\eta$ is related to
$w_{1}(TM)^{3}{\eta}^{\prime}\equiv{\eta}^{\prime}(\mathrm{PD}(w_{1}(TM)^{3})$,
namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{16}\int_{M^{4}}\eta)=\exp(\frac{\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{4}}{\eta}^{\prime}(\mathrm{PD}(w_{1}(TM)^{3})).$

$\bullet$ When $\nu=2$, the $2\eta$ is related to
$w_{1}(TM)^{2}\mathrm{ABK}\equiv\mathrm{ABK}(\mathrm{PD}(w_{1}(TM)^{2})$,
namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{16}\int_{M^{4}}\eta)=\exp(\frac{\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt}{4}\int_{M^{4}}{\mathrm{ABK}}(\mathrm{PD}(w_{1}(TM)^{2})).$

By pulling back $\mathrm{Pin}^{+}$ to $\mathrm{EPin}$, we can trivialize
$w_{1}(TM)^{2}$ by a $\mathbb{Z}_{2}$ extension, thus we also trivialize all
the even classes $\nu=8,4,2$ at once [48, 53].
$\bullet$ When $\nu=1$ or odd, it will be interesting to check whether our
_SUSY extension_ method applies.

###  7.4 4+1d

For $d=5$ or $4+1$d: We have $\mathbb{Z}_{4}^{F}$ SPTs classified by
$\Omega_{5}^{{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}=\mathbb{Z}_{16}$, which involves
5d SPT cobordism invariant with $\nu\in\mathbb{Z}_{16}$ [51, 26, 48]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{16}\int_{M^{5}}\eta(\mathrm{PD}(A))).$ (7.17)

$\bullet$ When $\nu=8$, the $8\eta(\mathrm{PD}(A)))$ is related to $A^{5}$ up
to trivial gapped fermions, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt8}{16}\int_{M^{5}}\eta(\mathrm{PD}(A))))=\exp(\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\int_{M^{5}}A^{5}).$

$\bullet$ When $\nu=4$, the $4\eta(\mathrm{PD}(A)))$ is related to
$A^{4}{\eta}^{\prime}\equiv{\eta}^{\prime}(\mathrm{PD}(A^{4}))$, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{16}\int_{M^{5}}\eta(\mathrm{PD}(A))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{5}}{\eta}^{\prime}(\mathrm{PD}(A^{4}))).$

$\bullet$ When $\nu=2$, the $2\eta(\mathrm{PD}(A)))$ is related to
$A^{3}\mathrm{Arf}\equiv\mathrm{ABK}(\mathrm{PD}(A^{3}))$, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{16}\int_{M^{5}}\eta(\mathrm{PD}(A))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{4}\int_{M^{5}}\mathrm{ABK}(\mathrm{PD}(A^{3}))).$

By pulling back ${{\rm Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}$ to ${{\rm
Spin}\times\mathbb{Z}_{4}}$, we can trivialize $A^{3}$ by a $\mathbb{Z}_{2}$
extension, thus we also trivialize all the even classes $\nu=8,4,2$ at once
[48, 53].
$\bullet$ When $\nu=1$ or odd, it will be interesting to check whether our
SUSY extension method applies.

These results of 4d $\mathbb{Z}_{16}$ ’t Hooft anomalies have applications to
the 3+1d Beyond the Standard Model physics and non-perturbative interactions
[57, 58, 59, 60]. Our work suggests that the $\mathbb{Z}_{2}$ symmetry
extension [48, 53] and possible _SUSY extensions_ can help to compensate the
$\nu\in\mathbb{Z}_{16}$ ’t Hooft anomaly via:

$\begin{cases}\text{a 3+1d symmetry-extended TQFT by trivializing the
$\nu\in\mathbb{Z}_{16}$ anomaly via the symmetry-extension
\cite[cite]{[\@@bibref{}{WangWenWitten_SymmetricGapped_PRX2018}{}{}]}.}\\\
\text{a 3+1d symmetric anomalous gapped TQFT with the $\nu\in\mathbb{Z}_{16}$
't Hooft anomaly.}\end{cases}$

The 3+1d symmetric anomalous gapped TQFT with the $\nu\in\mathbb{Z}_{16}$ ’t
Hooft anomaly may only require a $\mathbb{Z}_{2}$ gauge group if the symmetry
extension only requires a $\mathbb{Z}_{2}$-extension.

If there is no ’t Hooft anomaly, namely $\nu=0\mod{16}$, then we can in
principle have a 3+1d symmetric gap boundary preserving the ${{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathbb{Z}_{4}}$ symmetry with a trivial order (no
topological order, no TQFT, no symmetry-breaking order, etc.). See the related
work on the 3+1d physics along this research direction [61, 62, 63, 64, 65,
66].

###  7.5 5+1d

For $d=6$ or $5+1$d: We have $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ SPTs
classified by $\Omega_{6}^{\mathrm{Pin}^{-}}=\mathbb{Z}_{16}$, which involves
6d SPT invariant with $\nu\in\mathbb{Z}_{16}$ [53]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{16}\int_{M^{6}}\eta(\mathrm{PD}(w_{1}(TM)^{2}))).$ (7.18)

$\bullet$ When $\nu=8$, the $8\eta(\mathrm{PD}(w_{1}(TM)^{2})))$ is related to
$w_{1}(TM)^{6}$ up to trivial gapped fermions, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt8}{16}\int_{M^{6}}\eta(\mathrm{PD}(w_{1}(TM)^{2}))))=\exp(\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\int_{M^{6}}w_{1}(TM)^{6}).$

$\bullet$ When $\nu=4$, the $4\eta(\mathrm{PD}(w_{1}(TM)^{2})))$ is related to
$w_{1}(TM)^{5}{\eta}^{\prime}\equiv{\eta}^{\prime}(\mathrm{PD}(w_{1}(TM)^{6}))$,
namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{16}\int_{M^{6}}\eta(\mathrm{PD}(w_{1}(TM)^{2}))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{6}}{\eta}^{\prime}(\mathrm{PD}(w_{1}(TM)^{5}))).$

$\bullet$ When $\nu=2$, the $2\eta(\mathrm{PD}(w_{1}(TM)^{2})))$ is related to
$w_{1}(TM)^{4}\mathrm{ABK}\equiv\mathrm{ABK}(\mathrm{PD}(w_{1}(TM)^{4}))$,
namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{16}\int_{M^{6}}\eta(\mathrm{PD}(w_{1}(TM)^{2}))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{4}\int_{M^{6}}\mathrm{ABK}(\mathrm{PD}(w_{1}(TM)^{4}))).$

$\bullet$ When $\nu=1$, we have the $\eta(\mathrm{PD}(w_{1}(TM)^{2})))$.
However, by pulling back $\mathrm{Pin}^{-}$ to $\mathrm{EPin}$, we can
trivialize $w_{1}(TM)^{2}$ by a $\mathbb{Z}_{2}$ extension, thus we also
trivialize all these classes $\nu=8,4,2,1$ and any $\nu\in\mathbb{Z}_{16}$ at
once [48, 53].

###  7.6 6+1d

For $d=7$ or $6+1$d: We have $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ SPTs
classified by $\Omega_{7}^{{\rm Spin}\times\mathbb{Z}_{2}}=\mathbb{Z}_{16}$,
which involves 7d SPT invariant with $\nu\in\mathbb{Z}_{16}$ [53]:

$\displaystyle\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\nu}{16}\int_{M^{7}}\eta(\mathrm{PD}(A^{3}))).$ (7.19)

$\bullet$ When $\nu=8$, the $8\eta(\mathrm{PD}(A^{3})))$ is related to $A^{7}$
up to trivial gapped fermions, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt8}{16}\int_{M^{7}}\eta(\mathrm{PD}(A^{3}))))=\exp(\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt\int_{M^{7}}A^{7}).$

$\bullet$ When $\nu=4$, the $4\eta(\mathrm{PD}(A^{3})))$ is related to
$A^{6}\tilde{\eta}\equiv\tilde{\eta}(\mathrm{PD}(A^{6}))$, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt4}{16}\int_{M^{7}}\eta(\mathrm{PD}(A^{3}))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip
1.0pt}{2}\int_{M^{7}}{\eta}^{\prime}(\mathrm{PD}(A^{6}))).$

$\bullet$ When $\nu=2$, the $2\eta(\mathrm{PD}(A^{3})))$ is related to
$A^{5}\mathrm{ABK}\equiv\mathrm{ABK}(\mathrm{PD}(A^{5}))$, namely

$\exp(\frac{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt2}{16}\int_{M^{7}}\eta(\mathrm{PD}(A^{3}))))=\exp(\frac{\pi\hskip
1.0pt\mathrm{i}\hskip 1.0pt}{4}\int_{M^{7}}\mathrm{ABK}(\mathrm{PD}(A^{5}))).$

$\bullet$ When $\nu=1$, we have the $\eta(\mathrm{PD}(A^{3}))$. However, by
pulling back ${{\rm Spin}\times\mathbb{Z}_{2}}$ to ${{\rm
Spin}\times\mathbb{Z}_{4}}$, we can trivialize $A^{3}$ by a $\mathbb{Z}_{2}$
extension, thus we also trivialize all these classes $\nu=8,4,2,1$ and any
$\nu\in\mathbb{Z}_{16}$ at once [48, 53].

###  7.7 Smooth (DIFF) manifolds to topological (TOP) manifolds

Sec. 7.3 shows that the following extension can trivialize the cobordism
invariant of $\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$ when $\nu$ is
even ($2,4,6,8,10,12,14,16$) $\in\mathbb{Z}_{16}$:

$\displaystyle\mathbb{Z}_{2}\to\mathrm{EPin}\to{\mathrm{Pin}^{+}},$ (7.20)

because we can trivialize the $w_{1}^{2}(TM)$ in $\mathrm{EPin}$. The EPin is
introduced in Ref. [48] and reviewed in Sec. 6.1. Is there any way we can
trivialize when the $\nu$ is odd ($1,3,5,7,9,11,13,15$) $\in\mathbb{Z}_{16}$?

$\bullet$ One known way is by the construction of Ref. [67], which shows that
the 3d ${\rm SO}(3)_{3,\pm}$ Chern-Simons (CS) theory can saturate the ’Hooft
anomaly of $\nu=\pm 3\in\mathbb{Z}_{16}$. The $\pm$ in 3d ${\rm
SO}(3)_{3,\pm}$ CS means two version of time-reversal symmetry assignment.

$\bullet$ Here we provide another possibility by looking into more general
categories of manifolds. The bordism group
$\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$ classifies the 4d smooth and
differentiable (DIFF) manifolds with ${\mathrm{Pin}^{+}}$ structures. However,
Ref. [68] looks at the bordism group

$\displaystyle\Omega_{4}^{{\rm
Top}\mathrm{Pin}^{+}}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}$ (7.21)

which classifies the 4d non-triangulable topological (Top) manifolds also with
${\mathrm{Pin}^{+}}$ structures. It turns out that what survives as
$\mathbb{Z}_{8}$ in $\Omega_{4}^{{\rm Top}\mathrm{Pin}^{+}}$ is the even $\nu$
class in $\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$. Which means all the
odd $\nu$ class in $\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$ is
“trivialized” in the more general category of topological manifolds in
$\Omega_{4}^{{\rm Top}\mathrm{Pin}^{+}}$.242424The remained $\mathbb{Z}_{2}$
in $\Omega_{4}^{{\rm Top}\mathrm{Pin}^{+}}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}$
is given by the nontrivial Kirby-Siebenmann class
$\kappa\in\operatorname{H}^{4}(M,\mathbb{Z}_{2})$, whose manifold generator is
the Freedman’s E8 manifold. The E8 manifold is a unique compact, simply
connected 4d topological manifold whose intersection form is the positive-
definite even unimodular rank-8 matrix of the E8 lattice. Th E8 manifold does
not have a smooth DIFF or piecewise linear PL structure.

Since EPin trivializes the even $\nu\in\mathbb{Z}_{16}$ of
$\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$, while the ${{\rm
Top}\mathrm{Pin}^{+}}$ trivializes odd $\nu\in\mathbb{Z}_{16}$ of
$\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$, therefore we expect that the
combined categories of topological manifolds with the extended EPin structure,
which we name ${{\rm Top}\mathrm{EPin}}$, can trivialize all
$\nu\in\mathbb{Z}_{16}$ of $\Omega_{4}^{\mathrm{Pin}^{+}}=\mathbb{Z}_{16}$.
For future work, it will be illuminating to study the bordism group
$\Omega_{4}^{{\rm Top}\mathrm{EPin}}$ further.

###  7.8 Intrinsic fermionic gapless topological phase protected by
supersymmetry

Recent work [69] points out the relation between (1) the boundary of $D+1$
dimensional SPTs whose ’t Hooft anomaly can be trivialized by symmetry
extension, and (2) the intrinsic gapless $D$ dimensional SPTs living in one
lower dimension.

Ref. [69] focus on the bosonic SPT example, here we can propose a novel
intrinsically fermionic “gapless” SPTs protected by supersymmetry.

Take the 0+1d boundary of the 1+1d 2-layer of Kitaev chains, we had found that

$\begin{cases}\text{the bulk $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
symmetry can be extended to the boundary $\mathbb{D}_{8}^{F,T}$ with 't Hooft
anomaly trivialized.}\\\ \text{the bulk $\mathbb{Z}_{4}^{TF}$ symmetry can be
extended to the boundary $\mathbb{M}_{16}^{F,T}$ with 't Hooft anomaly
trivialized.}\end{cases}$

So there should be a 0+1d intrinsic fermionic “gapless” SPTs protected by
supersymmetry $\mathbb{D}_{8}^{F,T}$ or $\mathbb{M}_{16}^{F,T}$, living in
0+1d (without the requirement of an extra dimension 1+1d). “Gapless” in 0+1d
only means degenerate zero energy modes. How to find such a 0+1d intrinsic
fermionic “gapless” SPTs with degenerate zero energy modes? We can consider
$N=0\mod 8$-layer of Kitaev chains, such as the 8-layer of 1+1d Kitaev chains.
The 8-layer of 1+1d Kitaev chains can be gapped in the bulk with a trivial
order without any SPTs. Thus, the 0+1d boundary of 1+1d 8-layer Kitaev chains
must be intrinsically 0+1d.

Say we consider the Hilbert space of 8 Majorana fermion operators:
$\gamma_{1},\gamma_{1^{\prime}},\gamma_{2},\gamma_{2^{\prime}},\gamma_{3},\gamma_{3^{\prime}},\gamma_{4},\gamma_{4^{\prime}}$
(on the boundary of 8-layer of Kitaev chains). The 8 Majorana fermions have at
most $\mathrm{GSD}=2^{4}=16$ ground states. But we can turn on the 0+1d time-
reversal symmetric Hamiltonian:

  1. (1).

Consider the 0+1d Hamiltonian

$\displaystyle H^{\prime}_{BB}$ $\displaystyle=$
$\displaystyle(\gamma_{1}\gamma_{1^{\prime}}\gamma_{2}\gamma_{2^{\prime}})+(\gamma_{3}\gamma_{3^{\prime}}\gamma_{4}\gamma_{4^{\prime}})+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{B})$
(7.22) $\displaystyle=$
$\displaystyle(-P_{F,1}\;P_{F,2})+(-P_{F,3}\;P_{F,4})+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{B})\equiv
H^{\prime}_{B,11^{\prime}22^{\prime}}+H^{\prime}_{B,33^{\prime}44^{\prime}}+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{B}).\quad$
(7.23)

The $P_{F,a}=-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\gamma_{a}\gamma_{a^{\prime}}$
is the fermion parity from the $a$-th Majorana pairs, for $a=1,2,3,4$. The
$H^{\prime}_{BB}$ leaves us a single bosonic ground state ($\mathrm{GSD}=1$)
with a total fermion parity $P_{F}=P_{F,1}P_{F,2}P_{F,3}P_{F,4}=+1$. The
reasoning is the following.

     * •

The first interaction term
$H^{\prime}_{B,11^{\prime}22^{\prime}}\equiv(\gamma_{1}\gamma_{1^{\prime}}\gamma_{2}\gamma_{2^{\prime}})$
in $H^{\prime}_{BB}$ energetically selects the bosonic ground state sector in
$P_{F,1}\;P_{F,2}=+1$ out of the $\mathrm{GSD}=2^{2}=4$ from the first four
Majorana $\gamma_{1},\gamma_{1^{\prime}},\gamma_{2},\gamma_{2^{\prime}}$.
There are two bosonic sectors: $P_{F,1}=P_{F,2}=+1$ and $P_{F,1}=P_{F,2}=-1$.
In short, we gap the other two fermionic sectors out of $\mathrm{GSD}=4$ to
leave with only bosonic $\mathrm{GSD}=2$ remained for the first four Majorana
modes.

     * •

The second interaction term
$H^{\prime}_{B,33^{\prime}44^{\prime}}\equiv(\gamma_{3}\gamma_{3^{\prime}}\gamma_{4}\gamma_{4^{\prime}})$
in $H^{\prime}_{BB}$ also selects the bosonic ground state sector in
$P_{F,3}\;P_{F,4}=+1$ out of the $\mathrm{GSD}=2^{2}=4$ from the second four
Majorana $\gamma_{3},\gamma_{3^{\prime}},\gamma_{4},\gamma_{4^{\prime}}$. We
gap the other two fermionic sectors out of $\mathrm{GSD}=4$ to leave with only
bosonic $\mathrm{GSD}=2$ remained for the last four Majorana modes.

     * •

So far the first two interaction terms
$H^{\prime}_{B,11^{\prime}22^{\prime}}+H^{\prime}_{B,33^{\prime}44^{\prime}}$
in $H^{\prime}_{BB}$ energetically select 4 bosonic ground states out of the
original 16 states. Out of the bosonic $\mathrm{GSD}=4$, the first four
Majorana modes form a bosonic doublet, while the last Majorana modes form
another bosonic doublet. Each bosonic doublet is the same degree of freedom as
a $J=\frac{1}{2}$ spin (as a representation of SU(2)) at the one end of
Haldane spin-1 chain. We can write these two SU(2) bosonic spin-$\frac{1}{2}$
operators as, $\vec{S}^{\alpha}_{B}$ and $\vec{S}^{\beta}_{B}$, respectively.
Since the two bosonic doublets can form a spin-0 singlet and a spin-1 triplet
(via $\frac{1}{2}_{B}\otimes\frac{1}{2}_{B}=0_{B}\oplus 1_{B}$), we can gap
the spin-1 triplet via the third interaction term, the Heisenberg-type
interaction $(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{B})$. Thus turning on
the full $H^{\prime}_{BB}$, we end up with only a bosonic sector of spin-0
singlet with $\mathrm{GSD}=1$. Let us call this single bosonic ground state
sector $|B_{0}\rangle$.

     * •

In fact the $H^{\prime}_{B}$ of eq (7.23) can be rewritten as 14 terms of
four-Majorana interactions similar to Ref. [32], firstly known by Fidkowski-
Kitaev.

  2. (2).

We can also tune the 0+1d Hamiltonian via different interactions:

$\displaystyle H^{\prime}_{FF}$ $\displaystyle=$
$\displaystyle(-\gamma_{1}\gamma_{1^{\prime}}\gamma_{2}\gamma_{2^{\prime}})+(-\gamma_{3}\gamma_{3^{\prime}}\gamma_{4}\gamma_{4^{\prime}})+(\vec{S}^{\alpha}_{F}\cdot\vec{S}^{\beta}_{F})$
(7.24) $\displaystyle=$
$\displaystyle(P_{F,1}\;P_{F,2})+(P_{F,3}\;P_{F,4})+(\vec{S}^{\alpha}_{F}\cdot\vec{S}^{\beta}_{F})\equiv
H^{\prime}_{F,11^{\prime}22^{\prime}}+H^{\prime}_{F,33^{\prime}44^{\prime}}+(\vec{S}^{\alpha}_{F}\cdot\vec{S}^{\beta}_{F}).$
(7.25)

The $H^{\prime}_{F,11^{\prime}22^{\prime}}$ gives two fermionic ground states
with $(P_{F,1}\;P_{F,2})=-1$, and the $H^{\prime}_{F,33^{\prime}44^{\prime}}$
gives another two fermionic ground states with $(P_{F,3}\;P_{F,4})=-1$. Here
we write these two SU(2) fermionic spin-$\frac{1}{2}$ operators as,
$\vec{S}^{\alpha}_{F}$ and $\vec{S}^{\beta}_{F}$, respectively. However, the
full fermion parity of each of these $\mathrm{GSD}=4$ is still
$P_{F}=P_{F,1}P_{F,2}P_{F,3}P_{F,4}=+1$ thus bosonic. We can further gap the
spin-1 triplet (3 ground states) via the spin-1 interaction in
$\frac{1}{2}_{F}\otimes\frac{1}{2}_{F}=0_{B}\oplus 1_{B}$, which can be
constructed out of two fermionic spin-$\frac{1}{2}$ doublets as
$(\vec{S}^{\alpha}_{F}\cdot\vec{S}^{\beta}_{F})$. Thus turning on the full
$H^{\prime}_{FF}$, we still end up with only a bosonic sector of spin-0
singlet with $\mathrm{GSD}=1$.

  3. (3).

An interesting question is whether we can tune Hamiltonian ending up with only
a fermionic sector with $\mathrm{GSD}=1$? Consider the 0+1d Hamiltonian

$\displaystyle H^{\prime}_{BF}$ $\displaystyle=$
$\displaystyle(\gamma_{1}\gamma_{1^{\prime}}\gamma_{2}\gamma_{2^{\prime}})+(-\gamma_{3}\gamma_{3^{\prime}}\gamma_{4}\gamma_{4^{\prime}})+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{B})$
(7.26) $\displaystyle=$
$\displaystyle(-P_{F,1}\;P_{F,2})+(+P_{F,3}\;P_{F,4})+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{F})\equiv
H^{\prime}_{B,11^{\prime}22^{\prime}}+H^{\prime}_{F,33^{\prime}44^{\prime}}+(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{F}).\quad$
(7.27)

The $H^{\prime}_{B,11^{\prime}22^{\prime}}$ gives two bosonic ground states
with $(P_{F,1}\;P_{F,2})=+1$, and the $H^{\prime}_{F,33^{\prime}44^{\prime}}$
gives another two fermionic ground states with $(P_{F,3}\;P_{F,4})=-1$.
However, the full fermion parity of each of these $\mathrm{GSD}=4$ is
$P_{F}=P_{F,1}P_{F,2}P_{F,3}P_{F,4}=-1$ fermionic. We further gap the spin-1
triplet (3 ground states) via the spin-1 interaction
$\frac{1}{2}_{B}\otimes\frac{1}{2}_{F}=0_{F}\oplus 1_{F}$ constructed out of
one bosonic and one fermionic spin-$\frac{1}{2}$ doublet as
$(\vec{S}^{\alpha}_{B}\cdot\vec{S}^{\beta}_{F})$. Thus turning on the full
$H^{\prime}_{BF}$, we end up with only a fermionic sector of spin-0 singlet
with $\mathrm{GSD}=1$. Let us call this single fermionic ground state sector
$|F_{0}\rangle$.

Similarly, we can also define $H^{\prime}_{FB}$ analogous to eq (7.27), which
also ends up with only a fermionic sector of spin-0 singlet with
$\mathrm{GSD}=1$.

The combined Hamiltonian above, by tuning the coupling appropriately

$\displaystyle
H^{\prime}=g_{BB}\;H^{\prime}_{BB}+g_{FF}\;H^{\prime}_{FF}+g_{BF}\;H^{\prime}_{BF}+g_{FB}\;H^{\prime}_{FB}+\dots$
(7.28)

can energetically select ground states out of the original 16 ground states.
With certain appropriate couplings, when the bosonic and fermionic states,
$|B_{0}\rangle$ and $|F_{0}\rangle$, are both favored as ground states, we
have $\mathrm{GSD}=2$. We can imagine a process when the $|B_{0}\rangle$ is
favored on one side of coupling choice, while the $|F_{0}\rangle$ is favored
on the other side of coupling choice (both have $\mathrm{GSD}=1$); while at
some critical coupling, we have the $\mathrm{GSD}=2$ formed by a 2-dimensional
Hilbert space
$\mathcal{H}=\mathcal{H}_{B}\oplus\mathcal{H}_{F}=\\{|B_{0}\rangle,|F_{0}\rangle\\}$.
Although we are studying the boundary of 8-layer Kitaev chains, we find with
appropriate interactions, we land on the same 2-dimensional Hilbert space
$\mathcal{H}=\\{|B_{0}\rangle,|F_{0}\rangle\\}$ for 2-layer of Kitaev chain
studied in Sec. 3 and Sec. 5!

As mentioned, the boundary of 8-layer Kitaev chains is intrinsically 0+1d.
Thus, this theory at the critical coupling with a Hilbert space
$\mathcal{H}=\\{|B_{0}\rangle,|F_{0}\rangle\\}$, can be regarded as _a
critical “gapless” intrinsically 0+1d fermionic SPTs protected by
supersymmetry_ (here gapless only means the degenerate ground states in 0+1d)!
Depends on the assignment of time-reversal $\mathcal{T}$ symmetry shown in
Sec. 3 , the supersymmetry here can be $\mathbb{D}^{F,T}_{8}$ or
$\mathbb{M}^{F,T}_{8}$.

We leave the generalization of this understanding on _a critical “gapless”
intrinsically fermionic SPTs protected by supersymmetry_ in higher-dimensions
for future work.

###  7.9 Final comments and future directions

We conclude by enlisting some additional comments and future directions:

  1. 1.

_All intrinsically fermionic 1+1d SPT phases have supersymmetric 0+1d
boundaries_ :
$\bullet$ In the present work, especially in Sec. 3 and Sec. 5, we have shown
by examples that when the number of layers of 1+1d Kitaev chains is $8N+2$ or
$8N+6$ (namely the layer number is $2\mod 4$), then the 0+1d boundary theory
exhibits at least an ${\cal N}=2$ supersymmetric quantum mechanics. In the
sense that the boundary energy spectrum always have bosonic energy eigenstate
$|\mu,+\rangle$ and fermionic energy eigenstate $|\mu,-\rangle$ paired
together for every eigenenergy $E_{\mu}$.

$\bullet$ In a companion work [36], we prove that “all intrinsically fermionic
1+1d SPT phases have supersymmetric 0+1d boundaries.” Another way to rephrase
this result in a mathematically precise statement is the following:

Consider a 2d fermionic bordism group $\Omega_{2}^{G_{f,\text{total}}}$ with a
total group ${G_{f,\text{total}}}$ obtained from the extension of a normal
subgroup of fermionic internal $G_{f}$ global symmetry, and the
$d$-dimensional spacetime rotational symmetry (say ${\rm O}(d)/{\rm SO}(d)$,
etc.)252525The Spin/Pin group can be obtained via the
$\mathbb{Z}_{2}^{F}$-extension of O and SO groups (see Appendix B.1).:

$\displaystyle 1\to G_{f}\to G_{f,\text{total}}\to{\rm O}(d)\to 1,\quad\text{
with time-reversal symmetry}.$ (7.29) or $\displaystyle 1\to G_{f}\to
G_{f,\text{total}}\to{\rm SO}(d)\to 1,\quad\text{ without time-reversal
symmetry}.$ (7.30)

The fermionic internal $G_{f}$ contains the fermion parity
$\mathbb{Z}_{2}^{F}$ and the bosonic internal $G_{b}$, via a central
extension:

$\displaystyle 1\to\mathbb{Z}_{2}^{F}\to G_{f}\to G_{b}\to 1.$ (7.31)

The intrinsically fermionic SPTs, corresponds to the cobordism invariants
(representing some invertible TQFTs in field theory) of
$\Omega_{2}^{G_{f,\text{total}}}$, but those do _not overlap_ with cobordism
invariants of fermionic invertible topological order’s bordism group
$\Omega_{2}^{{\rm Spin}}(pt)$, _nor_ overlap with those of bosonic SPT’s
bordism group $\Omega_{2}^{G_{b,\text{total}}}$. Here the
${G_{b,\text{total}}}$ is obtained from:

$\displaystyle 1\to G_{b}\to G_{b,\text{total}}\to{\rm O}(d)\to 1,\quad\text{
with time-reversal symmetry}.$ (7.32) or $\displaystyle 1\to G_{b}\to
G_{b,\text{total}}\to{\rm SO}(d)\to 1,\quad\text{ without time-reversal
symmetry}.$ (7.33)

We can prove that the 0+1d boundary of these 1+1d cobordism invariants
(namely, the invertible TQFTs) of intrinsically fermionic SPTs must be
supersymmetric [36].

  2. 2.

_Boundary supersymmetry and the symmetry extension_ :
Various previous works pointed out the fractionalized boundary symmetry of
fermionic SPTs may be related to supersymmetry [70, 71, 72, 34, 73,
74].262626Naively Ref. [34] results look similar to ours, such as Gu’s
superalgebra mentioned in his equation (39). However, we can tell the
_differences_ of our result and Gu’s Ref. [34] by comparing the commutativity
of time reversal $\mathcal{T}$ and fermion parity $P_{f}$. Ref. [34] shows the
commutative $\mathcal{T}P_{f}=+P_{f}\mathcal{T}$ (which has no SUSY, in our
definition). Our work shows the anti-commutative
$\mathcal{T}P_{f}=-P_{f}\mathcal{T}$ which implies the SUSY quantum mechanics.

In the present work, we point out that on the boundary of intrinsic fermionic
SPTs, the non-commutative properties between fermion parity $P_{f}$ and other
symmetry generator (such as time reversal $\mathcal{T}$) implies the boundary
supersymmetry. (In a companion work [36], we present a precise proof for this
statement.) We also provide a framework based on a generalization of symmetry
extension in terms of group extension [4], to the case of an extended total
group involving supersymmetry — which we name it a supersymmetry extension.

  3. 3.

_Quantum criticality and emergent supersymmetry_ : In Sec. 7.8, we point out a
potential relation between the 0+1d boundary supersymmetric degenerate ground
states can be regarded as a 0+1d intrinsically gapless fermionic SPTs (here
gapless means degenerate in 0+1d), indeed also as a 0+1d critical theory on
the boundary of the 1+1d trivial class of SPTs (by tuning the 0+1d boundary
Hamiltonian). This result may be generalizable to other higher dimensions,
thus connecting to the literature [70, 71, 72, 73, 74] of quantum criticality
and supersymmetry on the boundary of fermionic SPTs. Hopefully our approach
can provide a more systematic and mathematically precise framework.

  4. 4.

_’t Hooft anomalies on the boundary of worldsheet or worldvolume in string
theory_ : In a different language setting. we can also apply our results to
the fermionic SPTs living on the 1+1d worldsheet of string theory. The 0+1d
boundary of worldsheet may have various ’t Hooft anomalies related to the
reflection, time-reversal $\mathcal{T}$, and other symmetries. These ’t Hooft
anomalies impose the constraints on the consistency of worldsheet and string
theories [5, 49, 50, 75, 76]. The constraints are tightly connected to the
string landscape vs swampland program, where the _cobordism conjecture_ plays
an important role to restrict the consistency of quantum gravity [77].

  5. 5.

_Higher-symmetry extension and fermionic symmetry extension construction of
symmetric gapped boundaries of SPTs_ : The present work focus on the ordinary
0-form global symmetry acting on bosonic and fermionic Hilbert spaces. We can
also attempt to include higher-form generalized global symmetries [78]. In
fact the symmetry extension method had be generalized to a higher-symmetry
extension [6, 79, 80, 81], also include fermionic symmetry extension [82, 26,
83] — these methods are helpful to construct symmetric gapped boundaries of
SPTs, as explained in [4]. By including the supersymmetry extension, we hope
to also construct more general boundary theories of fermionic SPTs in higher
dimensions.

##  8 Acknowledgments

AP acknowledges useful discussions with Subhro Bhattacharjee, Ashvin
Vishwanath, Nat Tantivasadakarn, J.P. Ang and Tarun Grover. JW thanks
conversations with Jun Hou Fung, Enno Keßler, Jacob McNamara, Miguel Montero,
Pavel Putrov, Ryan Thorngren, Cumrun Vafa, and Yizhuang You, and especially
thanks Zheyan Wan and Yunqin Zheng for the previous collaboration and the
enlightening discussions, in particularly on Ref. [48]’s Section 6. AP
receives funding from the Simon’s foundation through the ICTS-Simons
postdoctoral fellowship. JW was initially supported by NSF Grant PHY-1606531
and Institute for Advanced Study when this work started in 2018. JW is also
later supported by NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical
Physics” and Center for Mathematical Sciences and Applications at Harvard
University.

##  Appendix A Tantivasadakarn-Vishwanath model and the basis change

Here, we give details of how the model in section 4 was arrived at. We start
by writing down the model constructed by Tantivasadakarn and Vishwanath
in[35]. They start with the Hilbert space of one fermion on odd sites
($\gamma,\bar{\gamma}$), one fermion on odd sites ($\eta,\bar{\eta}$) and one
qubit on even sites and the following trivial Hamiltonian

$H_{0}=-\sum_{\text{even}~{}j}\sigma^{x}_{j}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{even}~{}j}\bar{\eta}_{j}\eta_{j}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{odd}~{}j}\bar{\gamma}_{j}\gamma_{j}.$ (A.1)

The SPT Hamiltonian is obtained from $H_{0}$ by a finite depth quantum circuit
$W$. Up to an inconsequential difference in minus signs, we can write $W$
considered by the authors of [35] as a two-layer FDUC:

$\displaystyle W$ $\displaystyle=W_{2}W_{1}.$ (A.2) $\displaystyle W_{1}$
$\displaystyle=\prod_{\text{even}~{}j}\left[|0\rangle\langle
0|_{j}+|1\rangle\langle 1|_{j}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\bar{\eta}_{j}\xi_{jj+1}\gamma_{j+1}\right].$ (A.3) $\displaystyle W_{2}$
$\displaystyle=\prod_{\text{even}~{}j}\left[|0\rangle\langle
0|_{j}+|1\rangle\langle 1|_{j}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma_{j-1}\bar{\eta}_{j}\right].$ (A.4)

Where $|0/1\rangle$ are the eigenstates of $\sigma^{z}$ with eigenvalues
$(-1)^{0/1}$. The $\xi_{jj+1}=\pm 1$ are $\mathbb{Z}_{2}$ valued background
fields that denote the spin structure to be Ramond (periodic) or Neveu-Schwarz
(anti-periodic) depending on whether $\prod_{j}\xi_{jj+1}=+1$ or
$\prod_{j}\xi_{jj+1}=-1$ respectively. The time reversal symmetry
$\mathcal{T}$ is

$\mathcal{T}=\prod_{\text{even}~{}j}\sigma^{x}_{j}~{}\mathcal{K}.$ (A.5)

It can easily be checked that $\mathcal{T}$ does not commute with $W_{1}$ or
$W_{2}$. However, depending on the spin structure, $\mathcal{T}$ commutes
(Neveu-Schwarz) or anti-commutes (Ramond) with $W$

$\mathcal{T}W\mathcal{T}^{-1}=-\left(\prod_{j}\xi_{jj+1}\right)W.$ (A.6)

Regardless of spin structure, the SPT Hamiltonian $H=WH_{0}W^{\dagger}$ shown
below commutes with $\mathcal{T}$.

$H=-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sum_{\text{odd}~{}j}\sigma^{z}_{j-1}\bar{\gamma}_{j}\gamma_{j}\sigma^{z}_{j+1}+\sum_{\text{odd}~{}j}\xi_{j+1j+2}\gamma_{j}(\sigma^{z}_{j+1}\sigma^{x}_{j+1})\gamma_{j+2}\sigma^{z}_{j+3}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sum_{\text{even}~{}j}\bar{\eta}_{j}\eta_{j}.$
(A.7)

Notice that the term $\sum_{\text{even}~{}j}\bar{\eta}_{j}\eta_{j}$ remains
unchanged from its trivial form and only plays an intermediary role in
defining the FDUC, $W$. It is ignored in the final form by the authors of
[35]. However, we shall retain it. To get the form of the Hamiltonian and
symmetry presented in the main text, we redefine the Hilbert space by
considering both the fermions to live on the sites and qubits to live on the
links with the following relabeling

$(\bar{\gamma}_{2k+1},\gamma_{2k+1})\mapsto(\bar{\gamma}_{\downarrow,k},\gamma_{\downarrow,k}),~{}~{}~{}(\bar{\eta}_{2k+2},\eta_{2k+2})\mapsto(\bar{\gamma}_{\uparrow,k},\gamma_{\uparrow,k}),~{}~{}\vec{\sigma}_{2k+2}\mapsto\vec{\sigma}_{k,k+1}.$
(A.8)

Finally, by performing a change-of-basis using $W_{2}$, we get the form of the
Hamiltonian and symmetry operators used in the main text.

$\displaystyle W_{2}HW_{2}^{\dagger}$ $\displaystyle=-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sum_{j}\left[\sigma^{z}_{j,j+1}\left(\bar{\gamma}_{\uparrow,j}\gamma_{\uparrow,j}+\bar{\gamma}_{\downarrow,j+1}\gamma_{\downarrow,j+1}\right)+{\sigma^{x}_{j,j+1}\bar{\gamma}_{\uparrow,j}\gamma_{\downarrow,j+1}}\right].$
(A.9) $\displaystyle W_{2}\mathcal{T}W_{2}^{\dagger}$
$\displaystyle=\prod_{j}\left(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\downarrow,j}\bar{\gamma}_{\uparrow,j}\right)\sigma^{x}_{j,j+1}~{}\mathcal{K}.$
(A.10)

##  Appendix B Appendices for Section 6

###  B.1 Group extension: Formal setup

Below we illuminate further the group extension used throughout our work:

$\displaystyle 1\to N\to\tilde{G}\to G\to 1.$ (B.1)

We will later include also the continuous spacetime symmetry ${\rm SO}(d)$ in
the continuum limit272727By the continuous spacetime symmetry, we mean the
Lorentz symmetry ${\rm SO}(d-1,1)$ or Euclidean rotational symmetry ${\rm
SO}(d)$ in the continuum limit. In the condensed matter system with a
discretized lattice, the spacetime symmetry, such as continuous rotations, is
not usually manifested. We may regard this continuous spacetime symmetry as an
emergent symmetry at an intermediate energy scale of some effective field
theories, _below_ the ultraviolet (UV) high-energy lattice cutoff, but _above_
the infrared (IR) low-energy field theory. In this work, we mainly do $d=2$ or
1+1d, however we will also discuss general $d$ in Sec. 7.  in terms of

$\displaystyle 1\to N\to\tilde{G}_{\text{Tot}}\to G_{\text{Tot}}\to 1,$ (B.2)

where $\tilde{G}_{\text{Tot}}$ and $G_{\text{Tot}}$ can be written in terms of
the spacetime-internal symmetry group form of
$\frac{G_{\text{spacetime}}\ltimes{G}_{\text{internal}}}{N_{\text{shared}}}$.
In Table 4, Table 5, and discussions below, we shorthand the short exact
sequences in eq (B.1) and eq (B.2) as $N\to\tilde{G}\to G$ and
$N\to\tilde{G}_{\text{Tot}}\to G_{\text{Tot}}$ for simplicity.

To precisely specify an extended group

$\displaystyle\tilde{G}=N\rtimes_{\rho,\varphi}G$ (B.3)

in of eq (B.1), we need the following data

  * •

The normal subgroup $N$ and the quotient group $Q$.

  * •

$\rho:G\to{\rm Aut}(N)$. If and only if a nontrivial $\rho$ implies a non-
central extension eq (B.1).

  * •

A 2-cocycle $\varphi\in\operatorname{H}^{2}(\mathrm{B}G,N)$ in the cohomology
group with the classifying space $\mathrm{B}G$ mapping to values in $N$
(usually defined for an Abelian group) is solved from the cocycle condition,
in the additive form,

$0=\delta\varphi(g_{1},g_{2},g_{3})=\big{(}\rho(g_{1}).\varphi(g_{2},g_{3})\big{)}-\varphi(g_{1}g_{2},g_{3})+\varphi(g_{1},g_{2}g_{3})-\varphi(g_{1},g_{2})$

or in the multiplicative form
$\delta\varphi(g_{1},g_{2},g_{3})=\frac{\big{(}\rho(g_{1}).\varphi(g_{2},g_{3})\big{)}\varphi(g_{1},g_{2}g_{3})}{\varphi(g_{1}g_{2},g_{3})\varphi(g_{1},g_{2})}$.

With two group elements in $\tilde{G}$ specified by a doublet $(n_{1},g_{1})$
and $(n_{2},g_{2})\in(N,G)$, we have the group law

$(n_{1},g_{1})(n_{2},g_{2})=(n_{1}+\rho(g_{1}).n_{2}+\varphi(g_{1},g_{2}),\;g_{1}\cdot
g_{2}).$

Here we use the addition $+$ for the group operation in $N$, and the product
$\cdot$ for the group operation in $G$. Here the 2-cocycle
$\varphi(g_{1},g_{2})$ specifies how much the group extension is twisted. We
illustrate the precise $\tilde{G}=N\rtimes_{\rho,\varphi}G$ for the examples
of Table 4 in a footnote.282828Here let us illustrate the mathematically
precise $\tilde{G}=N\rtimes_{\rho,\varphi}G$ for the examples in Table 4:  1.
$\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}=\mathbb{Z}_{2}\rtimes_{0,\varphi}(\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F})$
has no $\rho$ but a nontrivial
$\varphi\in\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{2}^{T},\mathbb{Z}_{2})=\mathbb{Z}_{2}$.
We may denote this generator $\varphi=w_{T}^{2}$ with $w_{T}$ can be regarded
as a time-reversal background field as a generator from
$\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2}^{T},\mathbb{Z}_{2})=\mathbb{Z}_{2}$.
2.
$\mathbb{D}_{8}^{F,T}=\mathbb{Z}_{2}\rtimes_{0,\varphi}(\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F})$
has no $\rho$ but a nontrivial
$\varphi=w_{T}A_{F}\in\operatorname{H}^{2}(\mathrm{B}(\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}),\mathbb{Z}_{2})=\mathbb{Z}_{2}^{3}$
with $A_{F}$ can be regarded as a fermion-parity background field as a
generator from
$\operatorname{H}^{1}(\mathrm{B}\mathbb{Z}_{2}^{F},\mathbb{Z}_{2})=\mathbb{Z}_{2}$.
But for
$\mathbb{D}_{8}^{F,T}=\mathbb{Z}_{4}^{T}\rtimes_{\rho,0}\mathbb{Z}_{2}^{F}$,
it has a nontrivial
$\rho:\mathbb{Z}_{2}\to\text{Aut}(\mathbb{Z}_{4})=\mathbb{Z}_{2}$ thus not a
central extension, but no $\phi$. 3.
$\mathbb{M}_{16}^{F,T}=\mathbb{Z}_{4}\rtimes_{0,\varphi}\mathbb{Z}_{4}^{TF}$
has no $\rho:\mathbb{Z}_{4}\to\text{Aut}(\mathbb{Z}_{4})=\mathbb{Z}_{2}$ and a
nontrivial odd class
$\varphi\in\operatorname{H}^{2}(\mathrm{B}\mathbb{Z}_{4}^{TF},\mathbb{Z}_{4})=\mathbb{Z}_{4}$.
4. Other groups in Table 4 obtained by adding extra U(1) and SU(2) in group
extension twisted by ${(\rho,\varphi)}$ are similar in the spirit as the
previous examples. We can also use the information below to specify the group
laws for $\tilde{g}_{1}=(n_{1},g_{1})\in\tilde{G}$ and
$\tilde{g}_{2}=(n_{2},g_{2})\in\tilde{G}$, with $n_{i}\in N$ and $g_{i}\in G$
as
$\tilde{g}_{1}\tilde{g}_{2}=(n_{1},g_{1})(n_{2},g_{2})=(n_{1}+\rho(g_{1}).n_{2}+\varphi(g_{1},g_{2}),g_{1}+g_{2})$.

###  B.2 Bosonic, Fermionic, Time Reversal, Spacetime Symmetries and Group
Extensions

We start from elaborating the familiar groups: the continuous rotation ${\rm
SO}(d)$,292929The major part of our paper focus on $d=2$ or 1+1d, but now we
keep the general $d$ notation in order to generalize our approach to higher
dimensions in Sec. 7. the reflection $\mathbb{Z}_{2}^{T_{\rm E}}$ or time-
reversal $\mathbb{Z}_{2}^{T}$, and the fermion parity $\mathbb{Z}_{2}^{F}$
(See useful discussions in [84, 25]). Then we can write down the short exact
sequences relating the familiar groups to ${\rm O}(d)$, ${\rm Spin}(d)$, and
$\mathrm{Pin}^{\pm}(d)$, by symmetry group extension along the horizontal and
vertical directions of:

---
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}^{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}^{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
Spin}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Pin}^{\pm}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2}^{T_{\rm
E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
SO}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\rm
O}(d)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\det}=\pm
1}$$\textstyle{\mathbb{Z}_{2}^{T_{\rm
E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$$\textstyle{1}$
(B.4)

Some remarks about these symmetry groups:

  1. 1.

The ${\rm SO}(d)$ is a spatial rotation group. The general linear group
$\mathrm{GL}(d,\mathbb{R})$ is the group of the rank-$d$ invertible matrix
with all the entry with $\mathbb{R}$ coefficients as group elements, together
with the operation of ordinary matrix multiplication. The special linear group
$\mathrm{SL}(d,\mathbb{R})$ is the subgroup of $\mathrm{GL}(d,\mathbb{R})$,
then

$\displaystyle{\rm SO}(d)=\\{g_{{\rm
SO}(d)}\in{\mathrm{SL}(d,\mathbb{R})}\;\;|\;\;g_{{\rm SO}(d)}g_{{\rm
SO}(d)}^{\rm T}=g_{{\rm SO}(d)}^{\rm T}g_{{\rm SO}(d)}=1\\},$ (B.5)

with the transpose operation ${\rm T}$ and its determinant $\det(g_{{\rm
SO}(d)})=1$.

  2. 2.

The ${\rm Spin}(d)=\mathbb{Z}_{2}^{F}\rtimes{\rm SO}(d)$ is from the spatial
rotation ${\rm SO}(d)$ extended/graded by the fermion parity
$\mathbb{Z}_{2}^{F}$. For $d>2$, ${\rm Spin}(d)$ is double cover of ${\rm
SO}(d)$ and simply-connected, thus the (unique) universal cover of ${\rm
SO}(d)$. So its first homotopy group is $\pi_{1}({\rm Spin}(d))=0$. The
centers of ${\rm Spin}(d)$ and ${\rm SO}(d)$ for $d\geq 2$ are:

$\displaystyle Z({\rm Spin}(d))=\begin{cases}\mathbb{Z}_{2}^{2},&d=0\mod 4\\\
\mathbb{Z}_{2},&d=1\mod 4\\\ \mathbb{Z}_{4},&d=2\mod 4\\\
\mathbb{Z}_{2},&d=3\mod 4\end{cases}.\quad Z({\rm
SO}(d))=\begin{cases}\mathbb{Z}_{2},&d=0\mod 2\\\ 1,&d=1\mod 2\end{cases}.$
(B.6)

  3. 3.

The ${\rm O}(d)={\rm SO}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}$ is

$\displaystyle{\rm O}(d)=\\{g_{{\rm
O}(d)}\in{\mathrm{GL}(d,\mathbb{R})}\;\;|\;\;g_{{\rm O}(d)}g_{{\rm O}(d)}^{\rm
T}=g_{{\rm O}(d)}^{\rm T}g_{{\rm O}(d)}=1\\}$ (B.7)

from the time reversal $\mathbb{Z}_{2}^{T_{\rm E}}$ extended by the spatial
rotation ${\rm SO}(d)$:

$\displaystyle{\rm O}(d)\cong\begin{cases}{\rm
SO}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}},&d\in~{}\text{\text{even}}\\\ {\rm
SO}(d)\times\mathbb{Z}_{2}^{T_{\rm E}},&d\in~{}\text{\text{odd}}\end{cases}.$
(B.8)

When $d$ is odd, the $\mathbb{Z}_{2}^{T_{\rm E}}=\\{+1,-1\\}$ along the
diagonal element, exactly matches the determinant map $\det(g_{{\rm
O}(d)})=\pm 1$. The center of ${\rm O}(d)$ for $d\geq 2$ is: $Z({\rm
O}(d))=\mathbb{Z}_{2}^{T_{\rm E}}.$

  4. 4.

The $\mathrm{Pin}^{\pm}(d)=\mathbb{Z}_{2}^{F}\rtimes{\rm O}(d)$ are double
covers of ${\rm O}(d)$. For physics purpose, we need to interpret the
spacetime symmetry in the Lorentzian signature (i.e., Minkowski signature).
Let us clarify the notations of groups in Euclidean signature (time-reversal
$\mathcal{T}_{\rm E}$) and Lorentzian signature (time-reversal
$\mathcal{T}$):303030Follow eq (3.32), for
$1\to\mathbb{Z}_{2}^{F}\to\tilde{G}\to\mathbb{Z}_{2}^{T}\to 1$, we have two
possible extended $\tilde{G}=\mathbb{Z}_{4}^{TF}$ or
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$. We can also replace $T$ to
$T_{\rm E}$ so $\tilde{G}=\mathbb{Z}_{4}^{T_{\rm E}F}$ or
$\mathbb{Z}_{2}^{T_{\rm E}}\times\mathbb{Z}_{2}^{F}$.

$\displaystyle\mathrm{Pin}^{+}(d)$ $\displaystyle\cong$ $\displaystyle{\rm
Spin}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}.~{}~{}\quad\quad\mathcal{T}_{\rm
E}^{2}=+1\text{ or }\mathcal{T}^{2}=(-1)^{F}.~{}~{}$
$\displaystyle\mathbb{Z}_{2}^{T_{\rm
E}}\times\mathbb{Z}_{2}^{F}\quad\quad\quad\text{ or
}\quad\mathbb{Z}_{4}^{TF}\text{ for discrete subgroups}.{}$
$\displaystyle\mathrm{Pin}^{-}(d)$ $\displaystyle\cong$
$\displaystyle\frac{{\rm Spin}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm
E}F}}{\mathbb{Z}_{2}^{F}}.~{}~{}\quad\quad\mathcal{T}_{\rm
E}^{2}=(-1)^{F}\text{ or }\mathcal{T}^{2}=+1.~{}~{}$
$\displaystyle\mathbb{Z}_{4}^{T_{\rm E}F}\quad\quad\quad\quad\quad\text{ or
}\quad\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}\text{ for discrete
subgroups}.{}$

In Euclidean signature, the $\mathbb{Z}_{2}^{{T_{\rm E}}}$ is generated by a
reflection $\mathcal{T}_{\rm E}$. The reflection $\mathcal{T}_{\rm E}\in{\rm
O}(d)$ has $\mathcal{T}_{\rm E}^{2}=+1$. The pullback from $\mathcal{T}_{\rm
E}\in{\rm O}(d)$ to $\mathcal{T}_{\rm E}\in\mathrm{Pin}^{\pm}(d)$ via the
$\mathbb{Z}_{2}^{F}$ extension shows that $\mathcal{T}_{\rm E}^{2}=+1$ for
$\mathrm{Pin}^{+}(d)$ and $\mathcal{T}_{\rm E}^{2}=(-1)^{F}$ for
$\mathrm{Pin}^{-}(d)$.

In Lorentzian signature, the reflection $\mathcal{T}_{\rm E}$ becomes the
time-reversal $\mathcal{T}$ by a Wick rotation [23]. The Wick rotation
introduces the imaginary $\hskip 1.0pt\mathrm{i}\hskip 1.0pt$ sign to the
Dirac gamma matrix $\gamma_{\rm E}\to\gamma_{0}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\gamma_{\rm E}$ of the temporal component in order to fit the Lorentzian
version of Clifford algebra $(\gamma_{0})^{2}=-(\gamma_{\rm E})^{2}$, so
$\mathcal{T}^{2}=(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\mathcal{T}_{\rm
E})^{2}=-\mathcal{T}_{\rm E}^{2}$, which suggests that
$\mathcal{T}^{2}=(-1)^{F}$ for $\mathrm{Pin}^{+}(d)$ and $\mathcal{T}^{2}=1$
for $\mathrm{Pin}^{-}(d)$.

The centers of $\mathrm{Pin}^{+}(d)$ and $\mathrm{Pin}^{-}(d)$ for $d\geq 2$
are:

$\displaystyle Z(\mathrm{Pin}^{+}(d))=\begin{cases}\mathbb{Z}_{2},&d=0\mod
4\\\ \mathbb{Z}_{2}^{2},&d=1\mod 4\\\ \mathbb{Z}_{2},&d=2\mod 4\\\
\mathbb{Z}_{4},&d=3\mod 4\end{cases}.\quad
Z(\mathrm{Pin}^{-}(d))=\begin{cases}\mathbb{Z}_{2},&d=0\mod 4\\\
\mathbb{Z}_{4},&d=1\mod 4\\\ \mathbb{Z}_{2},&d=2\mod 4\\\
\mathbb{Z}_{2}^{2},&d=3\mod 4\end{cases}.$ (B.11)

  5. 5.

When $d$ is an odd integer, because of ${\rm O}(d)={\rm
SO}(d)\times\mathbb{Z}_{2}^{T_{\rm E}}$, we can simplify eq (4) and eq (4), in
the Euclidean or Lorentzian signatures (with time coordinates ${T_{\rm E}}$
and ${T}$ respectively), to:

$\displaystyle\mathrm{Pin}^{+}(d)\cong\begin{cases}{\rm
Spin}(d)\times\mathbb{Z}_{2}^{T_{\rm E}},&d=+1~{}(\text{mod}~{}4)\\\ ({\rm
Spin}(d)\times\mathbb{Z}_{4}^{T_{\rm
E}F})/\mathbb{Z}_{2}^{F},&d=+3~{}(\text{mod}~{}4)\end{cases}~{}.$ (B.12)
$\displaystyle\mathrm{Pin}^{-}(d)\cong\begin{cases}({\rm
Spin}(d)\times\mathbb{Z}_{4}^{T_{\rm
E}F})/\mathbb{Z}_{2}^{F},&d=+1~{}(\text{mod}~{}4)\\\ {\rm
Spin}(d)\times\mathbb{Z}_{2}^{T_{\rm
E}},&d=+3~{}(\text{mod}~{}4)\end{cases}~{}.$ (B.13)

These forms match the center of $\mathrm{Pin}^{\pm}$ groups in eq (B.11).
Furthermore, we can still find in these expressions, $\mathrm{Pin}^{+}(d)$ and
$\mathrm{Pin}^{-}(d)$ contain the discrete subgroups $\mathbb{Z}_{2}^{T_{\rm
E}}\times\mathbb{Z}_{2}^{F}$ and $\mathbb{Z}_{4}^{T_{\rm E}F}$ respectively.

  6. 6.

The ${\rm E}(d)$ is defined in [24, 48, 81] which is a subgroup of ${\rm
O}(d)\times\mathbb{Z}_{4}$, described by

$\displaystyle{\rm E}(d)=\\{(M,j)\in({\rm O}(d),\mathbb{Z}_{4})\;|\;\det
M=j^{2}\\}$ (B.14)

in our interpretations, we take $\mathbb{Z}_{4}=\mathbb{Z}_{4}^{TB}$ which is
from $1\to\mathbb{Z}_{2}^{B}\to\mathbb{Z}_{4}^{TB}\to\mathbb{Z}_{2}^{T}\to
1$.313131For bosonic case, we can identify the Lorentz and Euclidean time
reversal symmetries directly (without extra $-1$ sign), thus
$\mathbb{Z}_{4}^{TB}=\mathbb{Z}_{4}^{T_{{\rm E}}B}$ and
$\mathbb{Z}_{2}^{T}=\mathbb{Z}_{2}^{T_{{\rm E}}}$. The ${\rm E}(d)$ sits at
the following extensions: $1\to\mathbb{Z}_{2}^{B}\to{\rm E}(d)\to{\rm O}(d)\to
1$ and $1\to{\rm SO}(d)\to{\rm E}(d)\to\mathbb{Z}_{4}^{TB}\to 1$ [81].

For the later convenience, we will specify two species of fermions $F_{-}$,
$F_{+}$, and their boson bound states $B=F_{+}F_{-}$; their time-reversal
$\mathcal{T}$ and spacetime symmetries in the continuum correspond to:

$\displaystyle\text{Fermion }F_{-}$ $\displaystyle:$
$\displaystyle\mathcal{T}^{2}=+1,\;\mathcal{T}_{\rm
E}^{2}=-1,\quad(\mathbb{Z}_{2}^{F_{-}}\rtimes{\rm
SO}(d))\rtimes\mathbb{Z}_{2}^{T_{\rm E}}={\rm
Spin}(d)\rtimes\mathbb{Z}_{2}^{T_{\rm E}}=\mathrm{Pin}^{-}(d).\quad\quad\quad$
(B.15) $\displaystyle\text{Fermion }F_{+}$ $\displaystyle:$
$\displaystyle\mathcal{T}^{2}=-1,\;\mathcal{T}_{\rm
E}^{2}=+1,\quad(\mathbb{Z}_{2}^{F_{+}}\rtimes{\rm
SO}(d))\rtimes\mathbb{Z}_{2}^{T_{\rm E}}={\frac{{\rm
Spin}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm
E}F}}{\mathbb{Z}_{2}^{F}}}=\mathrm{Pin}^{+}(d).\quad\quad\quad$ (B.16)
$\displaystyle\text{Boson }B$ $\displaystyle:$
$\displaystyle{\mathcal{T}^{2}=-1,\;\mathcal{T}_{\rm
E}^{2}=-1,\quad\big{(}{\rm
SO}(d)\times\mathbb{Z}_{2}^{B}\big{)}\rtimes\mathbb{Z}_{2}^{T_{\rm E}}={{\rm
SO}(d)\rtimes\mathbb{Z}_{4}^{T_{\rm E}B}}={\rm E}(d).}$ (B.17)

## References

  * [1] X. Chen, Z.-C. Gu, Z.-X. Liu and X.-G. Wen, _Symmetry protected topological orders and the group cohomology of their symmetry group_ , _Phys. Rev. B_ 87 155114 (2013 Apr).
  * [2] T. Senthil, _Symmetry-Protected Topological Phases of Quantum Matter_ , _Annual Review of Condensed Matter Physics_ 6 299–324 (2015 Mar.), [arXiv:1405.4015].
  * [3] A. Prakash, J. Wang and T.-C. Wei, _Unwinding Short-Range Entanglement_ , _Phys. Rev. B_ 98 125108 (2018), [arXiv:1804.11236].
  * [4] J. Wang, X.-G. Wen and E. Witten, _Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions_ , _Phys. Rev. X_ 8 031048 (2018), [arXiv:1705.06728].
  * [5] E. Witten, _The ”Parity” Anomaly On An Unorientable Manifold_ , _Phys. Rev. B_ 94 195150 (2016), [arXiv:1605.02391].
  * [6] Y. Tachikawa, _On gauging finite subgroups_ , _SciPost Phys._ 8 015 (2020), [arXiv:1712.09542].
  * [7] Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, _Comments on Abelian Higgs Models and Persistent Order_ , _SciPost Phys._ 6 3 (2019).
  * [8] A. Vishwanath and T. Senthil, _Physics of three-dimensional bosonic topological insulators: Surface-deconfined criticality and quantized magnetoelectric effect_ , _Phys. Rev. X_ 3 011016 (2013 Feb).
  * [9] L. Fidkowski, X. Chen and A. Vishwanath, _Non-abelian topological order on the surface of a 3d topological superconductor from an exactly solved model_ , _Phys. Rev. X_ 3 041016 (2013 Nov).
  * [10] I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki, _Rigorous results on valence-bond ground states in antiferromagnets_ , _Phys. Rev. Lett._ 59 799–802 (1987 Aug).
  * [11] E. H. Lieb, T. Schultz and D. Mattis, _Two soluble models of an antiferromagnetic chain_ , _Annals Phys._ 16 407–466 (1961).
  * [12] A. Prakash, _An elementary proof of 1d lsm theorems_ , arXiv:2002.11176.
  * [13] S. Sachdev and J. Ye, _Gapless spin fluid ground state in a random, quantum Heisenberg magnet_ , _Phys. Rev. Lett._ 70 3339 (1993), [arXiv:cond-mat/9212030].
  * [14] Z. Wan and J. Wang, _Higher anomalies, higher symmetries, and cobordisms I: classification of higher-symmetry-protected topological states and their boundary fermionic/bosonic anomalies via a generalized cobordism theory_ , _Ann. Math. Sci. Appl._ 4 107–311 (2019), [arXiv:1812.11967].
  * [15] A. P. Schnyder, S. Ryu, A. Furusaki and A. W. W. Ludwig, _Classification of topological insulators and superconductors_ , _AIP Conference Proceedings_ 1134 10–21 (2009), [arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.3149481].
  * [16] A. Kitaev, _Periodic table for topological insulators and superconductors_ , _AIP Conference Proceedings_ 1134 22–30 (2009), [arXiv:https://aip.scitation.org/doi/pdf/10.1063/1.3149495].
  * [17] Z.-C. Gu and X.-G. Wen, _Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory_, _Phys. Rev. B_ 90 115141 (2014), [arXiv:1201.2648].
  * [18] A. Kitaev, _Homotopy-theoretic approach to SPT phases in action: Z(16) classification of three-dimensional superconductors, in Symmetry and Topology in Quantum Matter Workshop, Institute for Pure and Applied Mathematics, University of California, Los Angeles, California.http://www.ipam.ucla.edu/abstract/?tid=12389_, .
  * [19] A. Kapustin and R. Thorngren, _Fermionic SPT phases in higher dimensions and bosonization_ , _JHEP_ 10 080 (2017), [arXiv:1701.08264].
  * [20] Q.-R. Wang and Z.-C. Gu, _Towards a Complete Classification of Symmetry-Protected Topological Phases for Interacting Fermions in Three Dimensions and a General Group Supercohomology Theory_ , _Phys. Rev. X_ 8 011055 (2018), [arXiv:1703.10937].
  * [21] D. Gaiotto and T. Johnson-Freyd, _Symmetry Protected Topological phases and Generalized Cohomology_ , _JHEP_ 05 007 (2019), [arXiv:1712.07950].
  * [22] Q.-R. Wang and Z.-C. Gu, _Construction and classification of symmetry protected topological phases in interacting fermion systems_ , arXiv:1811.00536.
  * [23] A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, _Fermionic Symmetry Protected Topological Phases and Cobordisms_ , _JHEP_ 12 052 (2015), [arXiv:1406.7329].
  * [24] D. S. Freed and M. J. Hopkins, _Reflection positivity and invertible topological phases_ , _ArXiv e-prints_ (2016 Apr.), [arXiv:1604.06527].
  * [25] M. Guo, P. Putrov and J. Wang, _Time reversal, SU(N) Yang-Mills and cobordisms: Interacting topological superconductors/insulators and quantum spin liquids in 3 + 1 D_ , _Annals of Physics_ 394 244–293 (2018 Jul), [arXiv:1711.11587].
  * [26] M. Guo, K. Ohmori, P. Putrov, Z. Wan and J. Wang, _Fermionic Finite-Group Gauge Theories and Interacting Symmetric/Crystalline Orders via Cobordisms_ , _Communications in Mathematical Physics_ 1073?1154 (2018 Jan), [arXiv:1812.11959].
  * [27] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder and S. Ryu, _Classification of topological quantum matter with symmetries_ , _Rev. Mod. Phys._ 88 035005 (2016 Aug).
  * [28] J. C. Wang, Z.-C. Gu and X.-G. Wen, _Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond_ , _Phys. Rev. Lett._ 114 031601 (2015), [arXiv:1405.7689].
  * [29] X.-G. Wen, _Zoo of quantum-topological phases of matter_ , _Rev. Mod. Phys._ 89 041004 (2017), [arXiv:1610.03911].
  * [30] A. Y. Kitaev, _Unpaired majorana fermions in quantum wires_ , _Physics-Uspekhi_ 44 131–136 (2001 oct).
  * [31] L. Fidkowski and A. Kitaev, _Effects of interactions on the topological classification of free fermion systems_ , _Phys. Rev. B_ 81 134509 (2010 Apr).
  * [32] L. Fidkowski and A. Kitaev, _Topological phases of fermions in one dimension_ , _Phys. Rev. B_ 83 075103 (2011 Feb).
  * [33] “M16.” https://groupprops.subwiki.org/wiki/M16.
  * [34] Z.-C. Gu, _Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing_ , _Phys. Rev. Res._ 2 033290 (2020), [arXiv:1308.2488].
  * [35] N. Tantivasadakarn and A. Vishwanath, _Full commuting projector hamiltonians of interacting symmetry-protected topological phases of fermions_ , _Phys. Rev. B_ 98 165104 (2018 Oct).
  * [36] A. Prakash and J. Wang, _Boundary supersymmetry of 1+1 d fermionic spt phases_ , arXiv:2011.12320.
  * [37] E. Witten, _Introduction to Supersymmetry_.  Springer US, Boston, MA, 1983, 10.1007/978-1-4613-3655-6_7.
  * [38] F. Cooper, A. Khare and U. Sukhatme, _Supersymmetry and quantum mechanics_ , _Physics Reports_ 251 267–385 (1995 Jan).
  * [39] E. Witten, _Dynamical breaking of supersymmetry_ , _Nuclear Physics B_ 188 513 – 554 (1981).
  * [40] E. Witten, _Constraints on supersymmetry breaking_ , _Nuclear Physics B_ 202 253 – 316 (1982).
  * [41] E. Witten, _Supersymmetry and Morse theory_ , _J. Diff. Geom._ 17 661–692 (1982).
  * [42] J. Behrends and B. Béri, _Supersymmetry in the Standard Sachdev-Ye-Kitaev Model_ , _Phys. Rev. Lett._ 124 236804 (2020), [arXiv:1908.00995].
  * [43] A. Kapustin, A. Turzillo and M. You, _Spin topological field theory and fermionic matrix product states_ , _Phys. Rev. B_ 98 125101 (2018 Sep).
  * [44] Y.-Z. You, A. W. W. Ludwig and C. Xu, _Sachdev-ye-kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states_ , _Phys. Rev. B_ 95 115150 (2017 Mar).
  * [45] A. Kitaev, _A simple model of quantum holography (part 1 and 2)_ , _online.kitp.ucsb.eduhttps://online.kitp.ucsb.edu/online/entangled15/kitaev/ https://online.kitp.ucsb.edu/online/entangled15/kitaev2/_ (2015).
  * [46] J. Maldacena and D. Stanford, _Remarks on the Sachdev-Ye-Kitaev model_ , _Phys. Rev. D_ 94 106002 (2016), [arXiv:1604.07818].
  * [47] A. Kitaev and S. J. Suh, _The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual_ , _JHEP_ 05 183 (2018), [arXiv:1711.08467].
  * [48] Z. Wan, J. Wang and Y. Zheng, _Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory_ , _Ann. Math. Sci. Appl._ 5 (2020 (to appear)), [arXiv:1912.13504].
  * [49] J. Kaidi, J. Parra-Martinez and Y. Tachikawa, _GSO projections via SPT phases_ , arXiv:1908.04805.
  * [50] J. Kaidi, J. Parra-Martinez and Y. Tachikawa, _Topological Superconductors on Superstring Worldsheets_ , arXiv:1911.11780.
  * [51] Y. Tachikawa and K. Yonekura, _Why are fractional charges of orientifolds compatible with Dirac quantization?_ , _arXiv e-prints_ arXiv:1805.02772 (2018 May), [arXiv:1805.02772].
  * [52] I. Hason, Z. Komargodski and R. Thorngren, _Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism_ , _SciPost Phys._ 8 062 (2020), [arXiv:1910.14039].
  * [53] P. Putrov, R. Thorngren, Z. Wan and J. Wang, _(in preparation)_ , _to appear_.
  * [54] R. Dijkgraaf and E. Witten, _Developments in Topological Gravity_ , _Int. J. Mod. Phys. A_ 33 1830029 (2018), [arXiv:1804.03275].
  * [55] D. Stanford and E. Witten, _JT Gravity and the Ensembles of Random Matrix Theory_ , arXiv:1907.03363.
  * [56] P. Putrov, J. Wang and S.-T. Yau, _Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions_ , _Annals Phys._ 384 254–287 (2017), [arXiv:1612.09298].
  * [57] I. Garcia-Etxebarria and M. Montero, _Dai-Freed anomalies in particle physics_ , _JHEP_ 08 003 (2019), [arXiv:1808.00009].
  * [58] J. Wang, _Anomaly and Cobordism Constraints Beyond Standard Model: Topological Force_ , arXiv:2006.16996.
  * [59] J. Wang, _Anomaly and Cobordism Constraints Beyond Grand Unification: Energy Hierarchy_ , arXiv:2008.06499.
  * [60] S. S. Razamat and D. Tong, _Gapped Chiral Fermions_ , arXiv:2009.05037.
  * [61] X.-G. Wen, _A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model_ , _Chin. Phys. Lett._ 30 111101 (2013), [arXiv:1305.1045].
  * [62] Y. You, Y. BenTov and C. Xu, _Interacting Topological Superconductors and possible Origin of $16n$ Chiral Fermions in the Standard Model_, arXiv:1402.4151.
  * [63] Y.-Z. You and C. Xu, _Interacting topological insulator and emergent grand unified theory_ , _Phys. Rev. B_ 91 125147 (2015 Mar.), [arXiv:1412.4784].
  * [64] Y. Kikukawa, _On the gauge invariant path-integral measure for the overlap Weyl fermions in $\underline{16}$ of SO(10)_, _PTEP_ 2019 113B03 (2019), [arXiv:1710.11618].
  * [65] J. Wang and X.-G. Wen, _Nonperturbative definition of the standard models_ , _Phys. Rev. Res._ 2 023356 (2020), [arXiv:1809.11171].
  * [66] S. Catterall, _Chiral Lattice Theories From Staggered Fermions_ , arXiv:2010.02290.
  * [67] L. Fidkowski, X. Chen and A. Vishwanath, _Non-Abelian Topological Order on the Surface of a 3D Topological Superconductor from an Exactly Solved Model_ , _Phys. Rev. X_ 3 041016 (2013), [arXiv:1305.5851].
  * [68] R. C. Kirby and L. R. Taylor, _Pin structures on low-dimensional manifolds_ , _London Mathematical Society Lecture Note Series_ 2 177–242 (1990).
  * [69] R. Thorngren, A. Vishwanath and R. Verresen, _Intrinsically Gapless Topological Phases_ , arXiv:2008.06638.
  * [70] T. Grover and A. Vishwanath, _Quantum Criticality in Topological Insulators and Superconductors: Emergence of Strongly Coupled Majoranas and Supersymmetry_ , arXiv:1206.1332.
  * [71] P. Ponte and S.-S. Lee, _Emergence of supersymmetry on the surface of three dimensional topological insulators_ , _New J. Phys._ 16 013044 (2014), [arXiv:1206.2340].
  * [72] T. Grover, D. Sheng and A. Vishwanath, _Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase_ , _Science_ 344 280–283 (2014), [arXiv:1301.7449].
  * [73] N. Zerf, C.-H. Lin and J. Maciejko, _Superconducting quantum criticality of topological surface states at three loops_ , _Phys. Rev. B_ 94 205106 (2016), [arXiv:1605.09423].
  * [74] J. Yu, R. Roiban, S.-K. Jian and C.-X. Liu, _Finite-Scale Emergence of 2+1D Supersymmetry at First-Order Quantum Phase Transition_ , _Phys. Rev. B_ 100 075153 (2019), [arXiv:1902.07407].
  * [75] M. Montero and C. Vafa, _Cobordism Conjecture, Anomalies, and the String Lamppost Principle_ , arXiv:2008.11729.
  * [76] J. Kaidi, _Stable Vacua for Tachyonic Strings_ , arXiv:2010.10521.
  * [77] J. McNamara and C. Vafa, _Cobordism Classes and the Swampland_ , arXiv:1909.10355.
  * [78] D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 172 (2015), [arXiv:1412.5148].
  * [79] Z. Wan and J. Wang, _Adjoint QCD 4, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension_, _Phys. Rev._ D99 065013 (2019), [arXiv:1812.11955].
  * [80] Z. Wan, J. Wang and Y. Zheng, _New higher anomalies, SU(N) Yang–Mills gauge theory and $\mathbb{CP}^{\mathrm{N}-1}$ sigma model_, _Annals Phys._ 414 168074 (2020), [arXiv:1812.11968].
  * [81] Z. Wan, J. Wang and Y. Zheng, _Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory_ , _Phys. Rev._ D100 085012 (2019), [arXiv:1904.00994].
  * [82] J. Wang, K. Ohmori, P. Putrov, Y. Zheng, Z. Wan, M. Guo et al., _Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions_ , _PTEP_ 2018 053A01 (2018), [arXiv:1801.05416].
  * [83] R. Kobayashi, K. Ohmori and Y. Tachikawa, _On gapped boundaries for SPT phases beyond group cohomology_ , _JHEP_ 11 131 (2019), [arXiv:1905.05391].
  * [84] C. Cordova, P.-S. Hsin and N. Seiberg, _Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups_ , _SciPost Phys._ 4 021 (2018), [arXiv:1711.10008].