# C-R-T Fractionalization, Fermions, and Mod 8 Periodicity

Zheyan Wan<EMAIL_ADDRESS>Yanqi Lake Beijing Institute of Mathematical
Sciences and Applications, Beijing 101408, China  Juven Wang
<EMAIL_ADDRESS>http://sns.ias.edu/~juven/ Center of Mathematical
Sciences and Applications, Harvard University, MA 02138, USA  Shing-Tung Yau
<EMAIL_ADDRESS>Yau Mathematical Sciences Center, Tsinghua University,
Beijing 100084, China  Yanqi Lake Beijing Institute of Mathematical Sciences
and Applications, Beijing 101408, China  Yi-Zhuang You<EMAIL_ADDRESS>Department of Physics, University of California San Diego, CA 92093, USA

###### Abstract

Charge conjugation (C), mirror reflection (R), time reversal (T), and fermion
parity $(-1)^{\rm F}$ are basic discrete spacetime and internal symmetries of
the Dirac fermions. In this article, we determine the group, called the C-R-T
fractionalization, which is a group extension of $\mathbb{Z}_{2}^{\rm
C}\times\mathbb{Z}_{2}^{\rm R}\times\mathbb{Z}_{2}^{\rm T}$ by the fermion
parity $\mathbb{Z}_{2}^{\rm F}$, and its extension class in all spacetime
dimensions $d$, for a single-particle fermion theory. For Dirac fermions, with
the canonical CRT symmetry $\mathbb{Z}_{2}^{\rm CRT}$, the C-R-T
fractionalization has two possibilities that only depend on spacetime
dimensions $d$ modulo 8, which are order-16 nonabelian groups, including the
famous Pauli group. For Majorana fermions, we determine the R-T
fractionalization in all spacetime dimensions $d=0,1,2,3,4\mod 8$, which is an
order-8 abelian or nonabelian group. For Weyl fermions, we determine the C or
T fractionalization in all even spacetime dimensions $d$, which is an order-4
abelian group. For Majorana-Weyl fermions, we only have an order-2
$\mathbb{Z}_{2}^{\rm F}$ group. We discuss how the Dirac and Majorana mass
terms break the symmetries C, R, or T. We study the domain wall dimensional
reduction of the fermions and their C-R-T fractionalization: from $d$-dim
Dirac to $(d-1)$-dim Dirac or Weyl; and from $d$-dim Majorana to $(d-1)$-dim
Majorana or Majorana-Weyl.

###### Contents

  1. I Introduction and summary
  2. II Clifford algebras and representations
    1. II.1 Clifford algebras
    2. II.2 The groups Pin and Spin
    3. II.3 Representations
    4. II.4 C, R, and T as bundle maps
  3. III Dirac fermions
  4. IV Majorana, Weyl, and Majorana-Weyl fermions
  5. V Mass term
  6. VI Domain wall dimensional reduction
  7. VII Conclusion
  8. VIII Acknowledgments
  9. A Chiral representation in any $d$ (Weyl representation in even $d$)
  10. B Majorana representation
  11. C Charge-like and isospin-like U(1) symmetries

##  I Introduction and summary

The study of symmetries and their breaking patterns is a cornerstone of modern
theoretical physics, providing deep insights into the laws that govern the
behavior of fundamental particles and fields. Among the symmetries of
particular interest are charge conjugation (C), mirror reflection (R), and
time reversal (T), each representing a fundamental operation that can be
applied to physical systems Schwinger (1951); Pauli (1955, 1957); Luders
(1954, 1957); Streater and Wightman (1989). Charge conjugation inverts the
charge of particles, mirror reflection flips one of the spatial coordinates,
and time reversal switches the direction of time’s flow. These discrete
symmetries, and their conservation or violation, have profound implications
for our understanding of the universe at its most fundamental level.

The time-reversal T and mirror reflection R are discrete spacetime symmetries
that are not part of the proper orthochronous continuous Lorentz symmetry
group ${\rm SO}^{+}(d-1,1)$. As passive transformations on the spacetime
coordinates $(t,x)=(t,x_{1},\dots,x_{d-1})$,

$\displaystyle{\rm T}(t,x){\rm T}^{-1}$ $\displaystyle=$
$\displaystyle(-t,x),{}$ $\displaystyle{\rm R}_{j}(t,x){\rm R}_{j}^{-1}$
$\displaystyle=$
$\displaystyle(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),$ (1)

where T flips the time coordinate and Rj flips the spatial coordinate $x_{j}$.

However, the charge conjugation C is not a spacetime symmetry but an internal
symmetry. It can only manifest itself under an active transformation on a
particle or field, such as a complex-valued Lorentz scalar boson $\phi$ (which
is a function of the spacetime coordinates). The C flips between particles and
anti-particles, or more generally, between energetic excitations and anti-
excitations

$\displaystyle{\rm C}(\text{excitations}){\rm C}^{-1}=(\text{anti-
excitations})$ (2)

which involves the complex conjugate (denoted *). The active transformations
act on this Lorentz scalar boson $\phi$ as

$\displaystyle{\rm C}\phi{\rm C}^{-1}$ $\displaystyle=$
$\displaystyle\phi^{*},{}$ $\displaystyle{\rm R}_{j}\phi{\rm R}_{j}^{-1}$
$\displaystyle=$
$\displaystyle\phi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),{}$
$\displaystyle{\rm T}\phi{\rm T}^{-1}$ $\displaystyle=$
$\displaystyle\phi(-t,x).$ (3)

In this scalar boson example, C, Rj, and T are $\mathbb{Z}_{2}$ involutions
and commutative with each other. Thus the C-R-T symmetry form a direct product
group $G_{\phi}=\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$. However, this is not the case in
general. We find that all these discrete C, R, and T symmetries can form a
nonabelian finite group for Dirac fermions in all spacetime dimensions.

In recent times, the concept of symmetry fractionalization has gained
traction, offering a nuanced view of how symmetries behave in certain quantum
systems. By Wang (2022), the symmetry fractionalization means the following:
the matter field is not in the linear representation of the original symmetry
group $G$, but in the projective representation of $G$ and in the linear
representation of the extended group $\tilde{G}$. Namely, there is a group
extension $1\to N\to\tilde{G}\to G\to 1$ where $N$ is the normal subgroup and
$G$ is the quotient group of the total group $\tilde{G}$, so $\tilde{G}/N=G$.
A well-known example is the gapped 1+1d isospin-1 Haldane chain with $G={\rm
SO}(3)$ symmetry Affleck (1989), whose 0+1d boundary can have a two-fold
degenerate isospin-1/2 doublet with $\tilde{G}={\rm SU}(2)$ symmetry and
$N=\mathbb{Z}_{2}$. This doublet is in a projective representation of $G={\rm
SO}(3)$, also in a linear representation of $\tilde{G}={\rm SU}(2)$.

In this article, we will find the analogous C-R-T symmetry fractionalization.
In contrast to a scalar boson’s $G_{\phi}=\mathbb{Z}_{2}^{{\rm
C}}\times\mathbb{Z}_{2}^{{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$, we
discover an order 16 nonabelian $\tilde{G}_{\psi}$ for a Dirac fermion in all
spacetime dimensions. The C-Rj-T fractionalization $\tilde{G}_{\psi}$ for a
Dirac fermion in any spacetime dimension $d$ is a group extension

$\displaystyle 1\to\mathbb{Z}_{2}^{{\rm F}}\to\tilde{G}_{\psi}\to
G_{\phi}=\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}\to 1.$ (4)

This group extension is classified by its extension class in
$\operatorname{H}^{2}(\mathrm{B}(\mathbb{Z}_{2}^{{\rm
C}}\times\mathbb{Z}_{2}^{{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm
T}}),\mathbb{Z}_{2}^{{\rm F}})=\mathbb{Z}_{2}^{6}$. We find that only 8 of the
64 kinds of group extensions are realized as the C-Rj-T fractionalization with
the canonical CRT for a Dirac fermion.

The fermion parity $\mathbb{Z}_{2}^{{\rm F}}$ generated by $(-1)^{{\rm
F}}:\psi\mapsto-\psi$ plays a crucial role in the group extension (4). The
Dirac fermion $\psi$ is in a projective representation of $G_{\phi}$, also in
an (anti-)linear representation of $\tilde{G}_{\psi}$. (It is anti-linear
because $\tilde{G}_{\psi}$ contains the anti-linear time-reversal symmetry.)

In Wang (2022), the second author studied the C-P-T fractionalization in 3+1d
and 1+1d. In this article, we consider the mirror reflection R instead of the
parity P. The reasons are as follows:

  * •

The P is a composition of several mirror reflections, so R is more basic. In
odd dimensional spacetime, ${\rm P}\subset{\rm SO}(d-1)\subset{\rm
SO}^{+}(d-1,1)$, so P is not an independent discrete symmetry. We should
replace P with R. For example, the CPT theorem Schwinger (1951); Pauli (1955,
1957); Luders (1954, 1957); Streater and Wightman (1989) should be called the
CRT theorem Witten (2016); Freed and Hopkins (2021) in any general spacetime
dimension.

  * •

The CRT theorem asserts that the symmetry $H$ of any relativistic QFT extends
to a double cover $G$ of $H$, elements of $G\setminus H$ act antilinearly, and
CRT is an element of $G\setminus H$ Freed and Hopkins (2021). So there is a
natural CRT symmetry for any relativistic QFT but it is not unique. However,
there is a unique (up to rotations) canonical CRT which is obtained by
breaking all the internal global symmetries, it is unbreakable Hason  _et al._
(2020). The domain wall dimensional reduction requires that the combined
symmetry CRT is canonical Hason  _et al._ (2020); Córdova  _et al._ (2020);
Wang  _et al._ (2020), so we will only consider the case when CRT is
canonical. We observe that CPT can not be canonical for $d=0\mod 4$. For
example, the CPT of the C-P-T fractionalization in 3+1d in Wang (2022) is not
canonical. So we consider R instead of P.

This article aims to dissect the intricate concept of C-R-T fractionalization,
a group extension that has profound implications in various spacetime
dimensions and for different types of fermionic fields. Our results are
multifaceted and encompass a range of scenarios:

  * •

We determine the C-R-T fractionalization with the canonical CRT and its
extension class for Dirac fermions in any spacetime dimension, and we find an
8-periodicity. See Table 10 and Table 11.

  * •

We determine the R-T fractionalization for Majorana fermions in any spacetime
dimension $d=0,1,2,3,4\mod 8$. See Table 12.

  * •

We determine the C or T fractionalization for Weyl fermions in any even
spacetime dimension. See Table 13.

  * •

We study how the Dirac and Majorana mass terms break the symmetries C, R, or
T. See Table 15 and Table 16.

  * •

We study the domain wall dimensional reduction of the C-R-T fractionalization
and R-T fractionalization. See (VI) and (VI). We also study the domain wall
dimensional reduction of fermions. See Fig. 4 and Table 17. These results are
consistent.

In this article, we mainly consider the Lorentz signature version of spacetime
symmetry, such as ${\rm SO}^{+}(d-1,1)$ and ${\rm Spin}(d-1,1)$. However, the
readers may wonder what is the relation between the Lorentz signature and the
Euclidean signature of our theory. By Freed and Hopkins (2021), the unitarity
of Lorentz is equivalent to the reflection positivity of Euclidean under Wick
rotation. The (anti)-unitary C, R, and T symmetries in Lorentz become
reflection-positive C and R symmetries in Euclidean under Wick rotation.

In Sec. II, we introduce the mathematical definitions of Dirac fermions, Weyl
fermions, Majorana fermions, Majorana-Weyl fermions, charge conjugation,
mirror reflection, and time reversal. In Sec. III, we determine the C-R-T
fractionalization with the canonical CRT and its extension class for Dirac
fermions in any spacetime dimension, and we find an 8-periodicity. In Sec. IV,
we determine the R-T fractionalization for Majorana fermions in any spacetime
dimension $d=0,1,2,3,4\mod 8$. We also determine the C or T fractionalization
for Weyl fermions in any even spacetime dimension. In Sec. V, we discuss how
the Dirac and Majorana mass terms break the symmetries C, R, or T. In Sec. VI,
we study the domain wall dimensional reduction of the C-R-T fractionalization,
R-T fractionalization, and fermions. In Sec. VII, we summarize the results in
this article.

##  II Clifford algebras and representations

A fermion is a particle that follows Fermi–Dirac statistics. There are many
varieties of fermions, with the four most common types being:

  * •

Dirac fermions,

  * •

Weyl fermions,

  * •

Majorana fermions, and

  * •

Majorana-Weyl fermions.

Mathematically, Dirac fermions, Weyl fermions, Majorana fermions, and
Majorana-Weyl fermions are defined in Definitions II.4 (or II.6), II.5, and
II.8. (these definitions are consistent with those given in physics textbooks
Peskin and Schroeder (1995); Weinberg (2005); Zee (2003); Srednicki (2007);
Polchinski (2007)).

More precisely, Dirac fermions are defined as maps from the Minkowski
spacetime ${\mathbb{R}}^{d-1,1}$ to the complex spin representation of the
Spin group ${\rm Spin}_{d-1,1}$ or maps from the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$ to an irreducible complex representation of the
Clifford algebra $\mathrm{Cl}_{d-1,1}$. The irreducible complex representation
of the Clifford algebra $\mathrm{Cl}_{d-1,1}$ restricts to the complex spin
representation of the Spin group ${\rm Spin}_{d-1,1}$. For even $d$, the
complex spin representation of the Spin group ${\rm Spin}_{d-1,1}$ is the
direct sum of two complex semi-spin representations and maps from the
Minkowski spacetime ${\mathbb{R}}^{d-1,1}$ to the two complex semi-spin
representations are defined as left and right Weyl fermions respectively.
Majorana fermions are defined as Dirac fermions with the trivial charge
conjugation symmetry. Majorana-Weyl fermions are defined as fermions which are
both Majorana and Weyl. In the language of fiber bundle theory, Dirac
fermions, Weyl fermions, Majorana fermions, and Majorana-Weyl fermions are
sections of certain trivial fiber bundles over the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$.

In order to give the mathematical definitions of Dirac fermions, Weyl
fermions, Majorana fermions, and Majorana-Weyl fermions, we first introduce
some mathematical notions, such as Clifford algebras, Pin and Spin groups,
their representations, and bundle maps. The main reference is Lawson and
Michelsohn (1989).

###  II.1 Clifford algebras

Let $V$ be a real vector space of dimension $n$ with a nondegenerate quadratic
form $Q$ of signature $(p,q)$, $p+q=n$. The Clifford algebra
$\mathrm{Cl}_{p,q}(V)$ is the algebra generated by $V$ with the relation

$\displaystyle v^{2}=-Q(v)1,\;\;\;\forall v\in V.$ (5)

This is equivalent to

$\displaystyle vw+wv=-2f_{Q}(v,w)1,\;\;\;\forall v,w\in V$ (6)

where

$\displaystyle f_{Q}(v,w)=\frac{1}{2}(Q(v+w)-Q(v)-Q(w))$ (7)

is the associated bilinear form of $Q$.

If $e_{1},\dots,e_{n}$ is an orthogonal basis for $V$ with $e_{i}^{2}=1$ for
$1\leq i\leq p$, $e_{i}^{2}=-1$ for $p+1\leq i\leq n$, then
$e_{i}e_{j}=-e_{j}e_{i}$, $i\neq j$. The elements $e_{I}=e_{i_{1}}\cdots
e_{i_{k}}$, $I=\\{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n\\}$ form a basis for
$\mathrm{Cl}_{p,q}(V)$. So $\dim\mathrm{Cl}_{p,q}(V)=2^{n}$. Note that as
vector spaces, $\mathrm{Cl}_{p,q}(V)$ is isomorphic to the exterior algebra
generated by $V$, but the multiplications are different, e.g. $e_{i}e_{i}=\pm
1\neq 0=e_{i}\wedge e_{i}$.

We define an algebra homomorphism
$\alpha(e_{I})=(-1)^{k}e_{I}=(-1)^{|I|}e_{I}$ and extend linearly to
$\mathrm{Cl}_{p,q}(V)$. Then $\mathrm{Cl}_{p,q}(V)$ is a
$\mathbb{Z}_{2}$-graded algebra

$\displaystyle\mathrm{Cl}_{p,q}(V)=\mathrm{Cl}_{p,q}^{0}(V)\oplus\mathrm{Cl}_{p,q}^{1}(V)$
(8)

where
$\mathrm{Cl}_{p,q}^{i}(V)=\\{x\in\mathrm{Cl}_{p,q}(V)|\alpha(x)=(-1)^{i}x\\}$
and

$\displaystyle\mathrm{Cl}_{p,q}^{i}(V)\cdot\mathrm{Cl}_{p,q}^{j}(V)\subseteq\mathrm{Cl}_{p,q}^{i+j}(V).$
(9)

We write $\mathrm{Cl}_{p,q}\equiv\mathrm{Cl}_{p,q}(V)$ where
$V={\mathbb{R}}^{p+q}$.

###  II.2 The groups Pin and Spin

We now consider the multiplicative group of units in the Clifford algebra,
which is defined to be the subset

$\displaystyle\mathrm{Cl}_{p,q}^{*}(V):=\\{x\in\mathrm{Cl}_{p,q}(V)\mid\exists
x^{-1}\text{ with }x^{-1}x=xx^{-1}=1\\}.$ (10)

The group of units always acts naturally as automorphisms of the algebra. That
is, there is a homomorphism

$\displaystyle\mathrm{Ad}:\mathrm{Cl}_{p,q}^{*}(V)\to\mathrm{Aut}(\mathrm{Cl}_{p,q}(V))$
(11)

called the adjoint representation, which is given by

$\displaystyle\mathrm{Ad}_{x}(y):=xyx^{-1}.$ (12)

######  Proposition II.1.

((Lawson and Michelsohn, 1989, Proposition I.2.2)) Let $v\in
V\subset\mathrm{Cl}_{p,q}(V)$ be an element with $Q(v)\neq 0$. Then
$\mathrm{Ad}_{v}(V)=V$. In fact, for all $w\in V$, the following equation
holds:

$\displaystyle-\mathrm{Ad}_{v}(w)=w-2\frac{f_{Q}(v,w)}{Q(v)}v.$ (13)

This leads us naturally to consider the subgroup of elements
$x\in\mathrm{Cl}_{p,q}^{*}(V)$ such that $\mathrm{Ad}_{x}(V)=V$. By
Proposition II.1, this group contains all elements $v\in V$ with $Q(v)\neq 0$.
Furthermore, we see from (13) that whenever $Q(v)\neq 0$, the transformation
$\mathrm{Ad}_{v}$ preserves the quadratic form $Q$. That is,
$(\mathrm{Ad}_{v}^{*}Q)(w)=Q(\mathrm{Ad}_{v}(w))=Q(w)$ for all $w\in V$.
Therefore, we define $P_{p,q}(V)$ to be the subgroup of
$\mathrm{Cl}_{p,q}^{*}(V)$ generated by the elements $v\in V$ with $Q(v)\neq
0$, and observe that there is a homomorphism

$\displaystyle P_{p,q}(V)\xrightarrow{\mathrm{Ad}}{\rm O}_{p,q}(V)$ (14)

where ${\rm O}_{p,q}(V)$ is the group of invertible linear transformations of
$V$ that preserve the quadratic form $Q$.

######  Definition II.2.

((Lawson and Michelsohn, 1989, Definition I.2.3)) Let ${\rm Pin}_{p,q}(V)$ be
the set of elements of $\mathrm{Cl}_{p,q}(V)$ which can be written in the form
$v_{1}v_{2}\cdots v_{k}$ where $v_{i}\in V$ with $Q(v_{i})=\pm 1$. Those
elements $v_{1}v_{2}\cdots v_{k}\in{\rm Pin}_{p,q}(V)$ for which $k$ is even
form ${\rm Spin}_{p,q}(V)$. Note that ${\rm Spin}_{p,q}(V)={\rm
Pin}_{p,q}(V)\cap\mathrm{Cl}_{p,q}^{0}(V)$.

We write ${\rm Pin}_{p,q}\equiv{\rm Pin}_{p,q}(V)$ and ${\rm
Spin}_{p,q}\equiv{\rm Spin}_{p,q}(V)$ where $V={\mathbb{R}}^{p+q}$.

###  II.3 Representations

Mathematically, a representation of a group $G$ is a vector space $V$ together
with a homomorphism $G\to{\rm GL}(V)$ where ${\rm GL}(V)$ is the group of
invertible linear transformations of $V$.

By (Lawson and Michelsohn, 1989, Theorem I.3.7 and Table II),

$\displaystyle{\rm
Spin}_{d-1,1}\subset\mathrm{Cl}_{d-1,1}^{0}\cong\mathrm{Cl}_{d-2,1}\cong\left\\{\begin{array}[]{cccccccc}{\mathbb{C}}(2^{\frac{d}{2}-1})&d=0\mod
8\\\ {\mathbb{R}}(2^{\frac{d-1}{2}})&d=1\mod 8\\\
{\mathbb{R}}(2^{\frac{d}{2}-1})\oplus{\mathbb{R}}(2^{\frac{d}{2}-1})&d=2\mod
8\\\ {\mathbb{R}}(2^{\frac{d-1}{2}})&d=3\mod 8\\\
{\mathbb{C}}(2^{\frac{d}{2}-1})&d=4\mod 8\\\
\mathbb{H}(2^{\frac{d-3}{2}})&d=5\mod 8\\\
\mathbb{H}(2^{\frac{d}{2}-2})\oplus\mathbb{H}(2^{\frac{d}{2}-2})&d=6\mod 8\\\
\mathbb{H}(2^{\frac{d-3}{2}})&d=7\mod 8\end{array}\right.$ (23)

where $A(n)$ is the $n\times n$ matrix algebra over the algebra $A$ for
$A={\mathbb{R}},{\mathbb{C}},\mathbb{H}$.

By the proof of (Lawson and Michelsohn, 1989, Theorem I.3.7), the isomorphism
$f:\mathrm{Cl}_{d-2,1}\to\mathrm{Cl}_{d-1,1}^{0}$ is given by
$f(e_{\mu})=e_{d-1}e_{\mu}$ for $\mu=0,1,\dots,d-2$ and extending it linearly
where $e_{\mu}$ ($\mu=0,1,\dots,d-1$) form an orthogonal basis of
$V={\mathbb{R}}^{d}$ with $e_{0}^{2}=1$ and $e_{j}^{2}=-1$
($j=1,2,\dots,d-1$).

Any complex representation of $\mathrm{Cl}_{p,q}$ automatically extends to a
representation of
$\mathrm{Cl}_{p,q}\otimes_{{\mathbb{R}}}{\mathbb{C}}\cong\mathbb{C}\mathrm{l}_{p+q}$,
the complexification of the real Clifford algebra $\mathrm{Cl}_{p,q}$.

By (23) or the classification of Clifford algebras (Lawson and Michelsohn,
1989, Theorem I.5.8),

$\displaystyle\mathrm{Cl}_{d-2,1}\otimes_{{\mathbb{R}}}{\mathbb{C}}\cong\mathbb{C}\mathrm{l}_{d-1}\cong\left\\{\begin{array}[]{ll}{\mathbb{C}}(2^{\frac{d-1}{2}}),&\text{for
odd }d,\\\
{\mathbb{C}}(2^{\frac{d}{2}-1})\oplus{\mathbb{C}}(2^{\frac{d}{2}-1}),&\text{for
even }d.\end{array}\right.$ (26)

######  Definition II.3.

  1. 1.

When $d$ is even, $\mathrm{Cl}_{d-2,1}$ has two irreducible complex
representations. The complex semi-spin representation $S_{\pm}$ of ${\rm
Spin}_{d-1,1}$ is the homomorphism

$\displaystyle\Delta_{d-1,1,\pm}^{{\mathbb{C}}}:{\rm Spin}_{d-1,1}\to{\rm
GL}_{{\mathbb{C}}}(S_{\pm})$ (27)

given by restricting an irreducible complex representation
$\mathrm{Cl}_{d-2,1}\to\operatorname{Hom}_{{\mathbb{C}}}(S_{\pm},S_{\pm})$ to
${\rm Spin}_{d-1,1}\subset\mathrm{Cl}_{d-1,1}^{0}\cong\mathrm{Cl}_{d-2,1}$.
The complex spin representation $S$ of ${\rm Spin}_{d-1,1}$ is the direct sum
of the two complex semi-spin representations $S_{+}$ and $S_{-}$. Namely,

$\displaystyle\Delta_{d-1,1}^{{\mathbb{C}}}=\Delta_{d-1,1,+}^{{\mathbb{C}}}\oplus\Delta_{d-1,1,-}^{{\mathbb{C}}}.$
(28)

  2. 2.

When $d$ is odd, $\mathrm{Cl}_{d-2,1}$ has only one irreducible complex
representation. The complex spin representation $S$ of ${\rm Spin}_{d-1,1}$ is
the homomorphism

$\displaystyle\Delta_{d-1,1}^{{\mathbb{C}}}:{\rm Spin}_{d-1,1}\to{\rm
GL}_{{\mathbb{C}}}(S)$ (29)

given by restricting the irreducible complex representation
$\mathrm{Cl}_{d-2,1}\to\operatorname{Hom}_{{\mathbb{C}}}(S,S)$ to ${\rm
Spin}_{d-1,1}\subset\mathrm{Cl}_{d-1,1}^{0}\cong\mathrm{Cl}_{d-2,1}$.

Mathematically, a fiber bundle Nakahara (2003) is a quintuple $(E,\pi,M,F,G)$
satisfying the local triviality condition where $E$ is the total space, $M$ is
the base space, $\pi:E\to M$ is the projection, $F$ is the fiber, and $G$ is
the structure group. A section of a fiber bundle $(E,\pi,M,F,G)$ is a map
$s:M\to E$ such that $\pi\circ s=\text{id}_{M}$. Since the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$ is contractible, any fiber bundle over the Minkowski
spacetime is trivial Nakahara (2003), namely the product of the Minkowski
spacetime and the fiber. A section of a trivial fiber bundle can be regarded
as a map from the base space to the fiber. We can construct the complex
(semi-)spin bundle over the Minkowski spacetime ${\mathbb{R}}^{d-1,1}$ whose
structure group is ${\rm Spin}_{d-1,1}$ and fiber is the complex (semi-)spin
representation of ${\rm Spin}_{d-1,1}$.

######  Definition II.4.

The Dirac fermions are sections of the complex spin bundle over the Minkowski
spacetime ${\mathbb{R}}^{d-1,1}$.

We will give another equivalent definition for Dirac fermions later, see
Definition II.6.

######  Definition II.5.

For even $d$, the complex spin representation of ${\rm Spin}_{d-1,1}$ is a
direct sum of two complex semi-spin representations, the left and right Weyl
fermions are sections of the two complex semi-spin bundles over the Minkowski
spacetime ${\mathbb{R}}^{d-1,1}$ respectively. Namely, left and right Weyl
fermions $\psi_{L}$ and $\psi_{R}$ are the components of the Dirac fermion
$\psi=\left(\begin{array}[]{cc}\psi_{L}\\\ \psi_{R}\end{array}\right)$.

The Dirac Lagrangian ${\cal L}=\bar{\psi}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}-m)\psi$ where
$\bar{\psi}\equiv\psi^{\dagger}\Gamma^{0}$, $\bar{\psi}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{\mu}\partial_{\mu}\psi$ is the kinetic
term, and $\bar{\psi}m\psi$ is the mass term. Free Dirac fermions with mass
$m$ satisfy the Dirac equation

$\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}-m)\psi=0.$ (30)

It involves the Gamma matrices which represent the real Clifford algebra
$\mathrm{Cl}_{d-1,1}$ with

$\displaystyle\\{\Gamma^{\mu},\Gamma^{\nu}\\}=2\eta^{\mu\nu}.$ (31)

Here $\\{,\\}$ means the anticommutator. We consider the Lorentz signature
$\eta^{\mu\nu}={\rm diag}(+,-,-,\dots,-)$. Let $d$ denote the total spacetime
dimension. We shall use $\mu=0,1,2,\dots,d-1$ to denote the spacetime index
and $j=1,2,\dots,d-1$ to denote the space index.

Dirac fermions can also be regarded as maps from the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$ to an irreducible complex representation of
$\mathrm{Cl}_{d-1,1}$. We have the following alternative definition for Dirac
fermions.

######  Definition II.6.

The Dirac fermions are sections of the Clifford module bundle111A Clifford
module bundle is a vector bundle whose fibers are Clifford modules, the
representations of Clifford algebras. over the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$ with fiber an irreducible complex representation of
$\mathrm{Cl}_{d-1,1}$. For even $d$, $\mathrm{Cl}_{d-1,1}$ has a unique
irreducible complex representation. For odd $d$, $\mathrm{Cl}_{d-1,1}$ has two
non-isomorphic irreducible complex representations. We choose one of them such
that the image of the volume element in the representation is $-1$, see Lemma
II.7.

The following lemma is well-known, but we still include its proof here.

######  Lemma II.7.

For odd $d=2k+3$, the two non-isomorphic irreducible complex representations
of $\mathrm{Cl}_{d-1,1}$ can be distinguished by the image of the volume
element $\omega=\hskip 1.0pt\mathrm{i}\hskip 1.0pt^{k+1}e_{0}e_{1}\cdots
e_{2k+2}$ in the representation where $e_{\mu}$ ($\mu=0,1,\dots,2k+2$) are
generators of $\mathrm{Cl}_{d-1,1}$ with $e_{0}^{2}=1$ and $e_{j}^{2}=-1$ for
$j=1,\dots,2k+2$. The image of the volume element in the representation is
either 1 or $-1$. If
$(S,\Delta:\mathrm{Cl}_{d-1,1}\to\text{End}_{{\mathbb{C}}}(S))$ is one of the
two irreducible complex representations, then the other one is
$(S,\Delta^{\prime}:\mathrm{Cl}_{d-1,1}\to\text{End}_{{\mathbb{C}}}(S))$ where
$\Delta^{\prime}(e_{\mu})=\Delta(-e_{\mu})$.

###### Proof.

By the relations in the Clifford algebra, $\omega^{2}=1$ and $\omega$ lies in
the center of $\mathrm{Cl}_{d-1,1}$. Let $\Gamma=\Delta(\omega)$ be the image
of $\omega$ in the irreducible complex representation
$(S,\Delta:\mathrm{Cl}_{d-1,1}\to\text{End}_{{\mathbb{C}}}(S))$, then
$\Gamma^{2}=1$ and $S$ decomposes as the direct sum $S=S_{+}\oplus S_{-}$
where $S_{\pm}=\ker(\Gamma\mp 1)$. Since $\omega$ lies in the center of
$\mathrm{Cl}_{d-1,1}$, $S_{\pm}$ are $\mathrm{Cl}_{d-1,1}$-invariant subspaces
of $S$. By the irreducibility of $S$, $S=S_{+}$ or $S=S_{-}$. Therefore,
$\Gamma=\pm 1$. If $\Delta(\omega)=1$, then $\Delta^{\prime}(\omega)=-1$.
Therefore, $(S,\Delta)$ and $(S,\Delta^{\prime})$ are not isomorphic. By (34),
these are the only two irreducible representations (up to isomorphism). ∎

For both $d=2k+2$ and $d=2k+3$, we can choose the Gamma matrices as complex
$2^{k+1}\times 2^{k+1}$ matrices (see App. A).  Since

$\displaystyle\mathrm{Cl}_{d-1,1}\otimes_{{\mathbb{R}}}{\mathbb{C}}\cong\mathbb{C}\mathrm{l}_{d}\cong\left\\{\begin{array}[]{ll}{\mathbb{C}}(2^{\frac{d}{2}}),&\text{for
even }d,\\\
{\mathbb{C}}(2^{\frac{d-1}{2}})\oplus{\mathbb{C}}(2^{\frac{d-1}{2}}),&\text{for
odd }d,\end{array}\right.$ (34)

this amounts to choosing the Gamma matrices in only one direct summand of
${\mathbb{C}}(2^{\frac{d-1}{2}})\oplus{\mathbb{C}}(2^{\frac{d-1}{2}})$ for odd
$d$. Namely, we choose an irreducible complex representation $S^{\prime}$ of
$\mathrm{Cl}_{d-1,1}$:

$\displaystyle{\Delta^{\prime}}^{{\mathbb{C}}}_{d-1,1}:\mathrm{Cl}_{d-1,1}\to\text{End}_{{\mathbb{C}}}(S^{\prime}).$
(35)

The restriction of $S^{\prime}$ to ${\rm Spin}_{d-1,1}$ is

$\displaystyle S^{\prime}|_{{\rm
Spin}_{d-1,1}}=\left\\{\begin{array}[]{cc}S=S_{+}\oplus S_{-},&d\text{
even},\\\ S,&d\text{ odd}\end{array}\right.$ (38)

where $S_{\pm}$ are the complex semi-spin representations of ${\rm
Spin}_{d-1,1}$ and $S$ is the complex spin representation of ${\rm
Spin}_{d-1,1}$. Namely, we have

(43)

Therefore, Definitions II.4 and II.6 are indeed equivalent.

###  II.4 C, R, and T as bundle maps

The symmetries of the Dirac equation (the transformations that preserve the
equation) include charge conjugation C, mirror reflection ${\rm R}_{j}$, time-
reversal T, and fermion parity $(-1)^{{\rm F}}$.

If we regard Dirac fermions as sections of a Clifford module bundle $M\times
S^{\prime}$ over the Minkowski spacetime $M={\mathbb{R}}^{d-1,1}$ where
$S^{\prime}$ is the irreducible complex representation of
$\mathrm{Cl}_{d-1,1}$ (see Definition II.6), then the charge conjugation sends
the Clifford module bundle to its complex conjugate, and the mirror reflection
and time-reversal send the Clifford module bundle to itself. Mathematically,
they are bundle maps between Clifford module bundles.222Let
$E\xrightarrow{\pi}M$ and $E^{\prime}\xrightarrow{\pi^{\prime}}M^{\prime}$ be
fiber bundles. A map $\overline{f}:E\to E^{\prime}$ is called a bundle map if
it maps each fiber $\pi^{-1}(p)$ of $E$ onto $\pi^{\prime-1}(q)$ of
$E^{\prime}$. Then $\overline{f}$ naturally induces a map $f:M\to M^{\prime}$
such that $f(p)=q$. More precisely, we have

(48) (53)

and

(58)

where $E=M\times S^{\prime}$ is the Clifford module bundle over the Minkowski
spacetime $M={\mathbb{R}}^{d-1,1}$, $S^{\prime}$ is the irreducible complex
representation of $\mathrm{Cl}_{d-1,1}$,
$E^{\prime}=M\times\overline{S^{\prime}}$ is the complex conjugate of the
Clifford module bundle $E$, and $\overline{S^{\prime}}$ is the complex
conjugate of the irreducible complex representation
$S^{\prime}$,333Mathematically, if $A$ is an algebra and
$\rho:A\to\text{End}(V)$ is a representation of it over the complex vector
space $V$, then the complex conjugate representation $\overline{\rho}$ is
defined over the complex conjugate vector space $\overline{V}$ as follows:
$\overline{\rho}(a)$ is the complex conjugate of $\rho(a)$ for all $a$ in $A$.
. We should consider $S^{\prime}$ and $\overline{S^{\prime}}$ simultaneously
and regard C, Rj, and T as bundle automorphisms of $E\oplus E^{\prime}$ (the
Whitney sum of $E$ and $E^{\prime}$). Namely,

(63) (68)

and

(73)

For a Dirac fermion $\psi$, the symmetry transformations are (see (III) and
(III))

$\displaystyle{\rm C}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)\mapsto\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)^{{\rm C}}=\left(\begin{array}[]{cc}0&M_{{\rm C}}\\\
M_{{\rm C}}^{*}&0\end{array}\right)\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right){}$ (82) $\displaystyle{\rm R}_{j}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)\mapsto\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)^{{\rm R}_{j}}=\left(\begin{array}[]{cc}M_{{\rm
R}_{j}}&0\\\ 0&M_{{\rm
R}_{j}}^{*}\end{array}\right)\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}){}$
(91) $\displaystyle{\rm T}$ $\displaystyle:$
$\displaystyle\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)\mapsto\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)^{{\rm T}}=\left(\begin{array}[]{cc}M_{{\rm T}}&0\\\
0&M_{{\rm T}}^{*}\end{array}\right)\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)(-t,x){}$ (100) $\displaystyle(-1)^{{\rm F}}$
$\displaystyle:$ $\displaystyle\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right)\mapsto-\left(\begin{array}[]{cc}\psi\\\
\psi^{*}\end{array}\right).$ (105)

Here $\psi^{*}$ is the complex conjugate of $\psi$ but it does take the
adjoint of the components of the field operator. (The complex conjugate of a
field operator is not defined in any representation-independent way. The right
concept here is adjoint, not complex conjugate.) Here $M^{*}$ is the complex
conjugate of the matrix $M$, see footnote 3, $M_{{\rm C}}$ is an invertible
matrix in $\text{End}_{{\mathbb{C}}}(\overline{S^{\prime}})$, $M_{{\rm
R}_{j}}$ and $M_{{\rm T}}$ are invertible matrices in
$\text{End}_{{\mathbb{C}}}(S^{\prime})$, hence $M_{{\rm C}}^{*}$, $M_{{\rm
R}_{j}}$, and $M_{{\rm T}}$ are in $\text{End}_{{\mathbb{C}}}(S^{\prime})$.

The matrices $M_{{\rm C}}$, $M_{{\rm R}_{j}}$, and $M_{{\rm T}}$ satisfy (see
(113), (2), and (3))

$\displaystyle M_{{\rm C}}\Gamma^{\mu*}M_{{\rm C}}^{-1}$ $\displaystyle=$
$\displaystyle-\Gamma^{\mu},\;\;\;\mu=0,1,\dots,d-1,{}$ $\displaystyle M_{{\rm
R}_{j}}\Gamma^{0}M_{{\rm R}_{j}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{0},{}$ $\displaystyle M_{{\rm R}_{j}}\Gamma^{j}M_{{\rm
R}_{j}}^{-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{j},{}$ $\displaystyle
M_{{\rm R}_{j}}\Gamma^{i}M_{{\rm R}_{j}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{i},\;\;\;i=1,\dots,j-1,j+1,\dots,d-1,{}$ $\displaystyle
M_{{\rm T}}\Gamma^{0*}M_{{\rm T}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{0},{}$ $\displaystyle M_{{\rm T}}\Gamma^{j*}M_{{\rm
T}}^{-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{j},\;\;\;j=1,\dots,d-1.$
(106)

######  Definition II.8.

Majorana fermions are the Dirac fermions $\psi$ with the trivial C symmetry,
namely $\psi^{{\rm C}}=\psi$. Majorana-Weyl fermions are the fermions which
are both Majorana and Weyl.

We will see later that Majorana fermions only exist for $d=0,1,2,3,4\mod 8$
and Majorana-Weyl fermions only exist for $d=2\mod 8$. They are actually
sections of a real (semi-)spin bundle over the Minkowski spacetime
${\mathbb{R}}^{d-1,1}$, namely maps from ${\mathbb{R}}^{d-1,1}$ to a real
(semi-)spin representation of ${\rm Spin}_{d-1,1}$. By (23), the real spin
representation of ${\rm Spin}_{d-1,1}$ only exists for $d=0,1,2,3,4\mod
8$444For $d=5,6,7\mod 8$, there exists only the pseudo-real spin
representation of ${\rm Spin}_{d-1,1}$ and the corresponding fermions are
symplectic Majorana fermions Stone (2022) that we will not study in this
article. and it can be decomposed as the direct sum of two real semi-spin
representations of ${\rm Spin}_{d-1,1}$ only for $d=2\mod 8$.

##  III Dirac fermions

In this section, we determine the C-R-T fractionalization with the canonical
CRT and its extension class for Dirac fermions in any spacetime dimension, and
we find an 8-periodicity.

Our main result is the following theorem.

######  Theorem III.1.

  1. 1.

For a Dirac fermion and each $d$, there are two possibilities of the C-Rj-T
fractionalization with the canonical CRT.

  2. 2.

For a Dirac fermion, the two possibilities of the C-Rj-T fractionalization
with the canonical CRT depend only on the remainder of the spacetime dimension
$d$ modulo 8.

By Hason  _et al._ (2020), the canonical CRT satisfies the following
properties:

  1. 1.

Any unitary internal symmetry U commutes with the canonical CRT:

$\displaystyle{\rm U}\cdot{\rm C}{\rm R}_{j}{\rm T}={\rm C}{\rm R}_{j}{\rm
T}\cdot{\rm U}.$ (107)

  2. 2.

Any time reversal symmetry T commutes with the canonical CRT up to the fermion
parity:

$\displaystyle{\rm T}\cdot{\rm C}{\rm R}_{j}{\rm T}=(-1)^{{\rm F}}{\rm C}{\rm
R}_{j}{\rm T}\cdot{\rm T}.$ (108)

  3. 3.

The canonical CRT is of order 2:

$\displaystyle({\rm C}{\rm R}_{j}{\rm T})^{2}=1.$ (109)

We check that the CRT for our two possibilities of the C-Rj-T
fractionalization satisfies (109), (108), and (107) for ${\rm U}={\rm C}$ (see
the discussion around (170)) and the internal charge-like and isospin-like
U(1) symmetries (see App. C), but we do not check (107) for other U.

To prove Theorem III.1, we prove the following lemmas.

######  Lemma III.2.

The order of elements of $\tilde{G}_{\psi}$ in the group extension (4) divides
4.

###### Proof.

The elements of $\tilde{G}_{\psi}$ in the group extension (4) are expressed as
pairs $(a,b)$ where $a\in\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$ and $b\in\mathbb{Z}_{2}^{{\rm F}}$.
Then $(a,b)^{2}=(a+a,b+b+f(a,a))=(0,f(a,a))$ and $(a,b)^{4}=(0,0)$ where $f$
is a 2-cocycle on $\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$ valued in $\mathbb{Z}_{2}^{{\rm F}}$. ∎

######  Lemma III.3.

The matrices of C, Rj, and T for a Dirac fermion are uniquely determined up to
a phase factor in U(1).

###### Proof.

Let $M_{{\rm C}}$, $M_{{\rm R}_{j}}$, and $M_{{\rm T}}$ be the matrices of C,
Rj, and T, then $M_{{\rm C}}^{*}$, $M_{{\rm R}_{j}}$, and $M_{{\rm T}}$ are
invertible matrices in $\text{End}_{{\mathbb{C}}}(S^{\prime})$. The matrices
of C, Rj, and T act on Gamma matrices by conjugation, see (II.4). Note that
$\Gamma^{\mu}$ is either real or imaginary, so (II.4) implies that $M_{{\rm
C}}^{*}$, $M_{{\rm R}_{j}}$, and $M_{{\rm T}}$ commute with some Gamma
matrices while anti-commute with other Gamma matrices.

If $M_{{\rm C}}$ and $M_{{\rm C}}^{\prime}$ are two solutions to (II.4), then
$M_{{\rm C}}^{\prime}M_{{\rm C}}^{-1}$ commutes with all Gamma matrices, so
does $M_{{\rm C}}^{\prime*}{M_{{\rm C}}^{*}}^{-1}$. Since $S^{\prime}$ is an
irreducible complex representation of $\mathrm{Cl}_{d-1,1}$, by Schur’s lemma,
$M_{{\rm C}}^{\prime}M_{{\rm C}}^{-1}=\eta_{{\rm C}}\cdot I$ where $\eta_{{\rm
C}}\in{\mathbb{C}}$ and $I$ is the identity matrix. Since C, Rj, and T
preserve the norm of $\psi$, we have $\eta_{{\rm C}}\in{\rm U}(1)$. Similarly
for Rj and T.

∎

The proof of Theorem III.1 consists of the following steps:

  * •

To determine the C-Rj-T fractionalization, we only need to determine the
squares of C, Rj, T, CRj, RjT, and CT, hence we obtain a group presentation of
the C-Rj-T fractionalization.

  * •

We observe that the order of the elements in the group $\tilde{G}_{\psi}$
divides 4, see Lemma III.2.

  * •

By Schur’s lemma, we can prove that the matrices of C, Rj, and T for a Dirac
fermion are unique up to a phase factor in U(1), see Lemma III.3. Moreover, we
can determine the matrices of C, Rj, and T for a Dirac fermion up to a phase
factor, see the discussion below.

  * •

Based on the composition rules (III) and (III), we can prove that among the
squares of C, Rj, T, CRj, RjT, and CT, only the squares of Rj and CT depend on
the phase factors. More precisely,

$\displaystyle{\rm C}^{2}\psi{\rm C}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm C}}M_{{\rm C}}^{*}\psi,{}$ $\displaystyle{\rm R}_{j}^{2}\psi{\rm
R}_{j}^{-2}$ $\displaystyle=$ $\displaystyle M_{{\rm R}_{j}}^{2}\psi,{}$
$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{*}M_{{\rm T}}\psi,{}$ $\displaystyle({\rm C}{\rm
R}_{j})^{2}\psi({\rm C}{\rm R}_{j})^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm R}_{j}}M_{{\rm C}}M_{{\rm R}_{j}}^{*}M_{{\rm C}}^{*}\psi,{}$
$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{*}M_{{\rm
T}}M_{{\rm R}_{j}}\psi,{}$ $\displaystyle({\rm C}{\rm T})^{2}\psi({\rm C}{\rm
T})^{-2}$ $\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm C}})^{*}M_{{\rm
T}}^{*}M_{{\rm C}}^{*}\psi$ (110)

where $M_{{\rm C}}$, $M_{{\rm R}_{j}}$, and $M_{{\rm T}}$ are the matrices of
C, Rj, and T respectively. In particular, the phase factors of Rj and CT can
only be $\pm 1$ or $\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt$ since their order
divides 4.

  * •

Since the order of the canonical CRT is 2, see (109), we can determine the
product of the phase factors of C, Rj, and T if we consider the canonical CRT.

  * •

In summary, there are two possibilities of the C-Rj-T fractionalization with
the canonical CRT corresponding to the two choices of the phase factor of Rj:
$\pm 1$ and $\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt$.

Now we determine the matrices of C, Rj, and T for a Dirac fermion up to a
phase factor.

  1. 1.

Charge conjugation

The Dirac equation interacting with an external electromagnetic field is

$\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}-e\Gamma^{\mu}A_{\mu}-m)\psi=0$ (111)

and hence the charge-conjugated field equation is

$\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}+e\Gamma^{\mu}A_{\mu}-m)\psi^{{\rm C}}=0.$
(112)

We find that if $\psi$ satisfies (111), then $\psi^{{\rm C}}=M_{{\rm
C}}\psi^{*}$ satisfies (112) provided that

$\displaystyle M_{{\rm C}}\Gamma^{\mu*}M_{{\rm C}}^{-1}=-\Gamma^{\mu}.$ (113)

First, we consider $d=2k+2$. Note that only
$\Gamma^{2},\Gamma^{4},\dots,\Gamma^{2k}$ are imaginary, this implies that
$M_{{\rm C}}$ commutes with $\Gamma^{2},\Gamma^{4},\dots,\Gamma^{2k}$,
anticommutes with $\Gamma^{0},\Gamma^{1},\Gamma^{3},\dots,\Gamma^{2k+1}$.
Therefore,

$\displaystyle M_{{\rm
C}}=\left\\{\begin{array}[]{ll}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}&k\text{
odd,}\\\ \Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}&k\text{
even}\end{array}\right.$ (116)

is a solution to (113).

Next, we consider $d=2k+3$. Recall that $\Gamma^{2k+2}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt^{k+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$. So
if $M_{{\rm C}}\Gamma^{\mu*}M_{{\rm C}}^{-1}=-\Gamma^{\mu}$ for
$\mu=0,1,\dots,2k+1$, then $M_{{\rm C}}\Gamma^{2k+2*}M_{{\rm
C}}^{-1}=(-1)^{k+1}\Gamma^{2k+2}.$ Let
$\Gamma=\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$.

For even $k$, we can choose $M_{{\rm C}}$ as we did for $d=2k+2$ such that
(113) is satisfied. So $\psi^{{\rm C}}=M_{{\rm
C}}\psi^{*}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$.

For odd $k$, if we choose $M_{{\rm C}}$ as we did for $d=2k+2$, then (113) can
not be satisfied, while $(\Gamma M_{{\rm C}})\Gamma^{\mu*}(\Gamma M_{{\rm
C}})^{-1}=\Gamma^{\mu}$ for $\mu=0,1,\dots,2k+2$. Therefore, the mass term
breaks the C symmetry in this case. For the massless case, $\psi^{{\rm
C}}=\Gamma M_{{\rm
C}}\psi^{*}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$.

  2. 2.

Mirror reflection

The mirror reflected Dirac equation is

$\displaystyle(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{0}\partial_{0}+\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{1}\partial_{1}+\cdots+\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{j-1}\partial_{j-1}-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{j}\partial_{j}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j+1}\partial_{j+1}+\cdots+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}\partial_{d-1}-m)\cdot{}$ $\displaystyle\psi^{{\rm
R}_{j}}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})=0.$ (117)

Let $\psi^{{\rm R}_{j}}(t,x)=M_{{\rm
R}_{j}}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$, then by
comparing with (30), we get

$\displaystyle M_{{\rm R}_{j}}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\partial_{0}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\partial_{1}+\cdots+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}\partial_{d-1}-m)M_{{\rm R}_{j}}^{-1}{}$ (118)
$\displaystyle=$ $\displaystyle\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\partial_{0}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\partial_{1}+\cdots+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j-1}\partial_{j-1}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j}\partial_{j}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j+1}\partial_{j+1}+\cdots+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}\partial_{d-1}-m.$

Since ${\rm R}_{j}$ is unitary, we get

$\displaystyle M_{{\rm R}_{j}}\Gamma^{0}M_{{\rm R}_{j}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{0},{}$ $\displaystyle M_{{\rm R}_{j}}\Gamma^{j}M_{{\rm
R}_{j}}^{-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{j},{}$ $\displaystyle
M_{{\rm R}_{j}}\Gamma^{i}M_{{\rm R}_{j}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{i},\;\;\;i=1,\dots,j-1,j+1,\dots,d-1.$ (119)

For $d=2k+2$, we find that $M_{{\rm
R}_{j}}=\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$ is a
solution to (2) and $\psi^{{\rm
R}_{j}}(t,x)=\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$
where $\hat{\Gamma}^{j}$ means that $\Gamma^{j}$ is omitted.

For $d=2k+3$, recall that $\Gamma^{2k+2}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt^{k+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$. If $j<2k+2$ and $M_{{\rm
R}_{j}}\Gamma^{0}M_{{\rm R}_{j}}^{-1}=\Gamma^{0}$, $M_{{\rm
R}_{j}}\Gamma^{j}M_{{\rm R}_{j}}^{-1}=-\Gamma^{j}$, and $M_{{\rm
R}_{j}}\Gamma^{i}M_{{\rm R}_{j}}^{-1}=\Gamma^{i}$ for
$i=1,\dots,j-1,j+1,\dots,2k+1$, then $M_{{\rm R}_{j}}\Gamma^{2k+2}M_{{\rm
R}_{j}}^{-1}=-\Gamma^{2k+2}$. So (2) can not be satisfied. Therefore, the mass
term breaks the ${\rm R}_{j}$ symmetry in odd dimensional spacetime. For the
massless case, $M_{{\rm R}_{j}}=\Gamma^{j}$ and $\psi^{{\rm
R}_{j}}(t,x)=\Gamma^{j}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$.
If $j=2k+2$ and $M_{{\rm R}_{2k+2}}\Gamma^{0}M_{{\rm
R}_{2k+2}}^{-1}=\Gamma^{0}$, $M_{{\rm R}_{2k+2}}\Gamma^{i}M_{{\rm
R}_{2k+2}}^{-1}=\Gamma^{i}$ for $i=1,\dots,2k+1$, then $M_{{\rm
R}_{2k+2}}\Gamma^{2k+2}M_{{\rm R}_{2k+2}}^{-1}=\Gamma^{2k+2}$. So (2) can not
be satisfied. Therefore, the mass term breaks the ${\rm R}_{2k+2}$ symmetry in
odd dimensional spacetime. For the massless case, $M_{{\rm
R}_{2k+2}}=\Gamma^{2k+2}$ and $\psi^{{\rm
R}_{2k+2}}(t,x)=\Gamma^{2k+2}\psi(t,x_{1},\dots,x_{2k+1},-x_{2k+2})$.

  3. 3.

Time-reversal

Since T is anti-unitary, ${\rm T}\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\rm
T}^{-1}=-\hskip 1.0pt\mathrm{i}\hskip 1.0pt$. The time-reversed Dirac equation
is

$\displaystyle(-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\partial_{0}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j}\partial_{j}-m)^{*}\psi^{{\rm T}}(-t,x)=0.$ (120)

Let $\psi^{{\rm T}}(t,x)=M_{{\rm T}}\psi(-t,x)$, then by comparing with (30),
we get

$\displaystyle M_{{\rm T}}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\partial_{0}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j}\partial_{j}-m)^{*}M_{{\rm T}}^{-1}=-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{0}\partial_{0}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j}\partial_{j}-m.$ (121)

Then we get

$\displaystyle M_{{\rm T}}\Gamma^{0*}M_{{\rm T}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{0},{}$ $\displaystyle M_{{\rm T}}\Gamma^{j*}M_{{\rm
T}}^{-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{j}.$ (122)

First, we consider $d=2k+2$. Note that only
$\Gamma^{2},\Gamma^{4},\dots,\Gamma^{2k}$ are imaginary, this implies that
$M_{{\rm T}}$ commutes with $\Gamma^{0},\Gamma^{2},\dots,\Gamma^{2k}$,
anticommutes with $\Gamma^{1},\Gamma^{3},\dots,\Gamma^{2k+1}$. Therefore,

$\displaystyle M_{{\rm
T}}=\left\\{\begin{array}[]{ll}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}&k\text{
odd,}\\\ \Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}&k\text{
even}\end{array}\right.$ (125)

is a solution to (3).

Next, we consider $d=2k+3$. Recall that $\Gamma^{2k+2}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt^{k+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$. So
if $M_{{\rm T}}\Gamma^{0*}M_{{\rm T}}^{-1}=\Gamma^{0}$ and $M_{{\rm
T}}\Gamma^{j*}M_{{\rm T}}^{-1}=-\Gamma^{j}$ for $j=1,2,\dots,2k+1$, then
$M_{{\rm T}}\Gamma^{2k+2*}M_{{\rm T}}^{-1}=(-1)^{k}\Gamma^{2k+2}$. Recall that
$\Gamma=\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$.

For odd $k$, we can choose $M_{{\rm T}}$ as we did for $d=2k+2$ such that (3)
is satisfied. So $\psi^{{\rm T}}(t,x)=M_{{\rm
T}}\psi(-t,x)=\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$.

For even $k$, if we choose $M_{{\rm T}}$ as we did for $d=2k+2$, then (3) can
not be satisfied, while $(\Gamma M_{{\rm T}})\Gamma^{0*}(\Gamma M_{{\rm
T}})^{-1}=-\Gamma^{0}$ and $(\Gamma M_{{\rm T}})\Gamma^{j*}(\Gamma M_{{\rm
T}})^{-1}=\Gamma^{j}$ for $j=1,2,\dots,2k+2$. Therefore, the mass term breaks
the T symmetry in this case. For the massless case, $\psi^{{\rm
T}}(t,x)=\Gamma M_{{\rm
T}}\psi(-t,x)=\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$.

We summarize the ${\rm C}$, ${\rm R}_{j}$, ${\rm T}$ transformations on a
Dirac fermion in $d$d for all $d$ in Table 1555Similar results were obtained
in Shimizu (1985) but P instead of R was considered and only one choice of the
phase factors was specified there..

Dimension | $M_{{\rm C}}\psi^{*}$ | $M_{{\rm R}_{j}}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$ | $M_{{\rm T}}\psi(-t,x)$
---|---|---|---
$d=2k+2$ ($k$ odd) | $\eta_{{\rm C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}$ | $\eta_{{\rm R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$ | $\eta_{{\rm T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$
$d=2k+2$ ($k$ even) | $\eta_{{\rm C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$ | $\eta_{{\rm R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$ | $\eta_{{\rm T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\psi(-t,x)$
$d=2k+3$ | $\eta_{{\rm C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$ | $\eta_{{\rm R}_{j}}\Gamma^{j}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$ | $\eta_{{\rm T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$
Table 1: The ${\rm C}$, ${\rm R}_{j}$, ${\rm T}$ transformations on a Dirac
fermion in $d$d for all $d$. Here the phase factors $\eta_{{\rm C}}$,
$\eta_{{\rm R}_{j}}$, and $\eta_{{\rm T}}$ are to be determined.

We find that the Dirac mass term breaks the C symmetry for $d=2k+3$ ($k$ odd),
it breaks the T symmetry for $d=2k+3$ ($k$ even), and it breaks the Rj
symmetry for $d=2k+3$ and $j=1,2,\dots,2k+2$, see Table 15.

Now we determine the group structure of the C-Rj-T fractionalization for Dirac
fermions. We need to determine the squares of C, Rj, T, CRj, RjT, and CT.
Because C and Rj are linear and unitary, while T is anti-linear and anti-
unitary, the composition obeys the following rule:

$\displaystyle{\rm C}(z\psi){\rm C}^{-1}$ $\displaystyle=$ $\displaystyle
zM_{{\rm C}}\psi^{*},{}$ $\displaystyle{\rm R}_{j}(z\psi){\rm R}_{j}^{-1}$
$\displaystyle=$ $\displaystyle zM_{{\rm
R}_{j}}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),{}$
$\displaystyle{\rm T}(z\psi){\rm T}^{-1}$ $\displaystyle=$ $\displaystyle
z^{*}M_{{\rm T}}\psi(-t,x)$ (126)

where $z$ is a complex number and $M_{{\rm C}}$, $M_{{\rm R}_{j}}$, and
$M_{{\rm T}}$ are the matrices of C, Rj, and T transformations respectively.

In general, for a matrix $M$,

$\displaystyle{\rm C}(M\psi){\rm C}^{-1}$ $\displaystyle=$ $\displaystyle
MM_{{\rm C}}\psi^{*},{}$ $\displaystyle{\rm R}_{j}(M\psi){\rm R}_{j}^{-1}$
$\displaystyle=$ $\displaystyle MM_{{\rm
R}_{j}}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),{}$
$\displaystyle{\rm T}(M\psi){\rm T}^{-1}$ $\displaystyle=$ $\displaystyle
M^{*}M_{{\rm T}}\psi(-t,x).$ (127)

If $M$ is a scalar matrix, then these rules are reduced to eq. (III).

Moreover, since $\psi^{*}$ actually means the adjoint of the components of
$\psi$ and ${\rm C}\psi^{\dagger}{\rm C}^{-1}=({\rm C}\psi{\rm
C}^{-1})^{\dagger}$, we have ${\rm C}\psi^{*}{\rm C}^{-1}=({\rm C}\psi{\rm
C}^{-1})^{*}=M_{{\rm C}}^{*}\psi$. Since C is linear and unitary, we get ${\rm
C}(M\psi^{*}){\rm C}^{-1}=MM_{{\rm C}}^{*}\psi$. Similarly for Rj and T.
Therefore,

$\displaystyle{\rm C}(M\psi^{*}){\rm C}^{-1}$ $\displaystyle=$ $\displaystyle
MM_{{\rm C}}^{*}\psi,{}$ $\displaystyle{\rm R}_{j}(M\psi^{*}){\rm R}_{j}^{-1}$
$\displaystyle=$ $\displaystyle MM_{{\rm
R}_{j}}^{*}\psi^{*}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}){}$
$\displaystyle{\rm T}(M\psi^{*}){\rm T}^{-1}$ $\displaystyle=$ $\displaystyle
M^{*}M_{{\rm T}}^{*}\psi^{*}(-t,x).$ (128)

The full matrices of C, Rj, and T in
$\text{End}_{{\mathbb{C}}}(S^{\prime}\oplus\overline{S^{\prime}})$ are
$\left(\begin{array}[]{cc}0&M_{{\rm C}}\\\ M_{{\rm
C}}^{*}&0\end{array}\right)$, $\left(\begin{array}[]{cc}M_{{\rm R}_{j}}&0\\\
0&M_{{\rm R}_{j}}^{*}\end{array}\right)$, and
$\left(\begin{array}[]{cc}M_{{\rm T}}&0\\\ 0&M_{{\rm
T}}^{*}\end{array}\right)$ respectively. Based on these rules, we do the
following computation.

For $d=2k+2$ ($k$ odd), ${\rm C}\psi{\rm C}^{-1}=\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}$, then

$\displaystyle{\rm C}^{2}\psi{\rm C}^{-2}$ $\displaystyle=$
$\displaystyle\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}(\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k})^{*}\psi{}$
(131) $\displaystyle=$
$\displaystyle(-1)^{k}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi{}$
$\displaystyle=$
$\displaystyle(-1)^{k}(-1)^{k-1}(-1)(-1)^{k-2}(-1)\cdots(-1)\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{\frac{k(k-1)}{2}}\psi{}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if }k=4l,4l+1,\\\
-\psi&\text{if }k=4l+2,4l+3.\end{array}\right.$

For $d=2k+2$ ($k$ even) and $d=2k+3$, ${\rm C}\psi{\rm C}^{-1}=\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$, then

$\displaystyle{\rm C}^{2}\psi{\rm C}^{-2}$ $\displaystyle=$
$\displaystyle\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}(\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi{}$
(134) $\displaystyle=$
$\displaystyle(-1)^{k+1}(-1)^{k}(-1)(-1)^{k-1}(-1)\cdots(-1)\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{\frac{k(k+1)}{2}}\psi{}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if }k=4l,4l+3,\\\
-\psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

Note that the phase factor of C does not affect the order of C. The square of
C in $d$d for all $d$ is summarized in Table 2.

Dimension |  | ${\rm C}^{2}\psi{\rm C}^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $-\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $-\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $-\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $-\psi$
Table 2: The square of C in $d$d for all $d$.

For $d=2k+2$, ${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
then

$\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$
$\displaystyle\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x){}$
(135) $\displaystyle=$ $\displaystyle(-1)^{k}\eta_{{\rm R}_{j}}^{2}\psi(t,x).$

For $d=2k+3$, ${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{j}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
then

$\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$
$\displaystyle\eta_{{\rm R}_{j}}\Gamma^{j}\eta_{{\rm
R}_{j}}\Gamma^{j}\psi(t,x)=-\eta_{{\rm R}_{j}}^{2}\psi(t,x).$ (136)

The square of ${\rm R}_{j}$ in $d$d for all $d$ is summarized in Table 3.

Dimension |  | ${\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $\eta_{{\rm R}_{j}}^{2}\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $-\eta_{{\rm R}_{j}}^{2}\psi$
Table 3: The square of ${\rm R}_{j}$ in $d$d for all $d$.

For $d=2k+2$ ($k$ even), ${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\psi(-t,x)$, then

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k})^{*}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\psi{}$
(139) $\displaystyle=$
$\displaystyle(-1)^{k}(-1)^{k}(-1)^{k-1}(-1)(-1)^{k-2}(-1)\cdots(-1)\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{\frac{k(k+1)}{2}}\psi{}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if }k=4l,4l+3,\\\
-\psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

For $d=2k+2$ ($k$ odd) and $d=2k+3$, ${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$, then

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$
$\displaystyle(\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi{}$
(142) $\displaystyle=$
$\displaystyle(-1)^{k}(-1)(-1)^{k-1}(-1)\cdots(-1)\psi{}$ $\displaystyle=$
$\displaystyle(-1)^{\frac{(k+1)(k+2)}{2}}\psi{}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-\psi&\text{if }k=4l,4l+1,\\\
\psi&\text{if }k=4l+2,4l+3.\end{array}\right.$

Note that the phase factor of T does not affect the order of T. The square of
T in $d$d for all $d$ is summarized in Table 4.

Dimension |  | ${\rm T}^{2}\psi{\rm T}^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $-\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $-\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $-\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $-\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $\psi$
Table 4: The square of T in $d$d for all $d$.

For $d=2k+2$ ($k$ odd), ${\rm C}\psi{\rm C}^{-1}=\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}$,

${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),$

then

$\displaystyle{\rm C}{\rm R}_{j}\psi({\rm C}{\rm R}_{j})^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(143)

Thus

$\displaystyle({\rm C}{\rm R}_{j})^{2}\psi({\rm C}{\rm R}_{j})^{-2}$
$\displaystyle=$
$\displaystyle\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}(\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{*}(\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k})^{*}\psi{}$
(146) $\displaystyle=$ $\displaystyle(-1)^{\frac{k(k-1)}{2}}\psi{}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if
}k=4l,4l+1,\\\ -\psi&\text{if }k=4l+2,4l+3.\end{array}\right.$

For $d=2k+2$ ($k$ even), ${\rm C}\psi{\rm C}^{-1}=\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$,

${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}),$

then

$\displaystyle{\rm C}{\rm R}_{j}\psi({\rm C}{\rm R}_{j})^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(147)

Thus

$\displaystyle({\rm C}{\rm R}_{j})^{2}\psi({\rm C}{\rm R}_{j})^{-2}$
$\displaystyle=$
$\displaystyle\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}(\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{*}(\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi{}$
(150) $\displaystyle=$ $\displaystyle-(-1)^{\frac{k(k+1)}{2}}\psi{}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-\psi&\text{if
}k=4l,4l+3,\\\ \psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

For $d=2k+3$, ${\rm C}\psi{\rm C}^{-1}=\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$, ${\rm
R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{j}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
then

$\displaystyle{\rm C}{\rm R}_{j}\psi({\rm C}{\rm R}_{j})^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{j}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(151)

Thus

$\displaystyle({\rm C}{\rm R}_{j})^{2}\psi({\rm C}{\rm R}_{j})^{-2}$
$\displaystyle=$
$\displaystyle\Gamma^{j}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}(\Gamma^{j})^{*}(\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi{}$
(154) $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if
}k=4l,4l+1,\\\ -\psi&\text{if }k=4l+2,4l+3.\end{array}\right.$

Note that the phase factors of C and Rj do not affect the order of CRj. The
square of CRj in $d$d for all $d$ is summarized in Table 5.

Dimension |  | $({\rm C}{\rm R}_{j})^{2}\psi({\rm C}{\rm R}_{j})^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $-\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $-\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $-\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $-\psi$
Table 5: The square of CRj in $d$d for all $d$.

For $d=2k+2$ ($k$ even), ${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\psi(-t,x)$, then

$\displaystyle({\rm R}_{j}{\rm T})\psi({\rm R}_{j}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(-t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(155)

Thus

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{*}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi{}$
(158) $\displaystyle=$ $\displaystyle(-1)^{\frac{k(k+1)}{2}}\psi{}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if
}k=4l,4l+3,\\\ -\psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

For $d=2k+2$ ($k$ odd), ${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$, then

$\displaystyle({\rm R}_{j}{\rm T})\psi({\rm R}_{j}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
R}_{j}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi(-t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(159)

Thus

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$
$\displaystyle(\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{*}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}\psi{}$
(162) $\displaystyle=$ $\displaystyle-(-1)^{\frac{(k+1)(k+2)}{2}}\psi{}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-\psi&\text{if
}k=4l,4l+3,\\\ \psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

For $d=2k+3$, ${\rm R}_{j}\psi{\rm R}_{j}^{-1}=\eta_{{\rm
R}_{j}}\Gamma^{j}\psi(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$,
${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$, then

$\displaystyle({\rm R}_{j}{\rm T})\psi({\rm R}_{j}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
R}_{j}}\Gamma^{j}\psi(-t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1}).$
(163)

Thus

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$
$\displaystyle(\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\Gamma^{j})^{*}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\Gamma^{j}\psi{}$
(166) $\displaystyle=$ $\displaystyle(-1)^{\frac{k(k+1)}{2}}\psi{}$
$\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\psi&\text{if
}k=4l,4l+3,\\\ -\psi&\text{if }k=4l+1,4l+2.\end{array}\right.$

Note that the phase factors of Rj and T do not affect the order of RjT. The
square of RjT in $d$d for all $d$ is summarized in Table 6.

Dimension |  | $({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $-\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $-\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $-\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $-\psi$
Table 6: The square of RjT in $d$d for all $d$.

For $d=2k+2$ ($k$ odd), ${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$ and ${\rm C}\psi{\rm
C}^{-1}=\eta_{{\rm C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}$.
Therefore, $({\rm C}{\rm T})\psi({\rm C}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}\psi^{*}(-t,x)$. Thus

$\displaystyle({\rm C}{\rm T})^{2}\psi({\rm C}{\rm T})^{-2}$ $\displaystyle=$
$\displaystyle{\rm C}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k})^{*}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi^{*}(t,x){\rm C}^{-1}{}$
(167) $\displaystyle=$ $\displaystyle(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k})^{*}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
C}}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k})^{*}\psi(t,x){}$ $\displaystyle=$
$\displaystyle(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}(-1)^{k+1}\psi{}$
$\displaystyle=$ $\displaystyle(\eta_{{\rm T}}^{*}\eta_{{\rm
C}}^{*})^{2}\psi.$

For $d=2k+2$ ($k$ even), ${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\psi(-t,x)$ and ${\rm C}\psi{\rm
C}^{-1}=\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$. Therefore,
$({\rm C}{\rm T})\psi({\rm C}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}(-t,x)$. Thus

$\displaystyle({\rm C}{\rm T})^{2}\psi({\rm C}{\rm T})^{-2}$ $\displaystyle=$
$\displaystyle{\rm C}(\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k})^{*}\psi^{*}(t,x){\rm C}^{-1}{}$
(168) $\displaystyle=$ $\displaystyle(\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
T}}\Gamma^{0}\Gamma^{2}\cdots\Gamma^{2k})^{*}(\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi(t,x){}$
$\displaystyle=$ $\displaystyle(\eta_{{\rm T}}^{*}\eta_{{\rm
C}}^{*})^{2}(-1)^{k+1}\psi{}$ $\displaystyle=$ $\displaystyle-(\eta_{{\rm
T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi.$

For $d=2k+3$, ${\rm T}\psi{\rm T}^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi(-t,x)$ and ${\rm C}\psi{\rm
C}^{-1}=\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}$. Therefore,
$({\rm C}{\rm T})\psi({\rm C}{\rm T})^{-1}=\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\psi^{*}(-t,x)$. Thus

$\displaystyle({\rm C}{\rm T})^{2}\psi({\rm C}{\rm T})^{-2}$ $\displaystyle=$
$\displaystyle{\rm C}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi^{*}(t,x){\rm C}^{-1}{}$
(169) $\displaystyle=$ $\displaystyle(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
T}}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}(\eta_{{\rm
C}}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1})^{*}\psi(t,x){}$
$\displaystyle=$ $\displaystyle(\eta_{{\rm T}}^{*}\eta_{{\rm
C}}^{*})^{2}\psi.$

The square of CT in $d$d for all $d$ is summarized in Table 7.

Dimension |  | $({\rm C}{\rm T})^{2}\psi({\rm C}{\rm T})^{-2}$
---|---|---
$d\equiv 0\mod 8$ | $d=2k+2,\;k\equiv 3\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 1\mod 8$ | $d=2k+3,\;k\equiv 3\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 2\mod 8$ | $d=2k+2,\;k\equiv 0\mod 4$ | $-(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 3\mod 8$ | $d=2k+3,\;k\equiv 0\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 4\mod 8$ | $d=2k+2,\;k\equiv 1\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 5\mod 8$ | $d=2k+3,\;k\equiv 1\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 6\mod 8$ | $d=2k+2,\;k\equiv 2\mod 4$ | $-(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
$d\equiv 7\mod 8$ | $d=2k+3,\;k\equiv 2\mod 4$ | $(\eta_{{\rm T}}^{*}\eta_{{\rm C}}^{*})^{2}\psi$
Table 7: The square of CT in $d$d for all $d$.

Now we consider the canonical CRT Hason  _et al._ (2020). By (109), we have
$({\rm C}{\rm R}_{j}{\rm T})^{2}=1$. Note that

$\displaystyle({\rm C}{\rm R}_{j}{\rm T})^{2}\psi({\rm C}{\rm R}_{j}{\rm
T})^{-2}=(M_{{\rm T}}M_{{\rm R}_{j}}M_{{\rm C}})^{*}M_{{\rm T}}^{*}M_{{\rm
R}_{j}}^{*}M_{{\rm C}}^{*}\psi.$ (170)

By Table 1, $M_{{\rm T}}M_{{\rm R}_{j}}M_{{\rm C}}=\eta_{{\rm T}}\eta_{{\rm
R}_{j}}\eta_{{\rm C}}\Gamma^{0}\Gamma^{j}$, we get $(\eta_{{\rm T}}\eta_{{\rm
R}_{j}}\eta_{{\rm C}})^{2}=1$ and $\eta_{{\rm T}}\eta_{{\rm R}_{j}}\eta_{{\rm
C}}=\pm 1$.

By (107), we have ${\rm C}\cdot{\rm C}{\rm R}_{j}{\rm T}={\rm C}{\rm
R}_{j}{\rm T}\cdot{\rm C}$, hence ${\rm C}{\rm R}_{j}{\rm T}={\rm R}_{j}{\rm
T}{\rm C}$ and ${\rm C}{\rm C}{\rm R}_{j}{\rm T}{\rm R}_{j}{\rm T}={\rm C}{\rm
R}_{j}{\rm T}{\rm C}{\rm R}_{j}{\rm T}$. By (109), we have ${\rm C}^{2}({\rm
R}_{j}{\rm T})^{2}=1$. This is consistent with Table 2 and Table 6.
Conversely, ${\rm C}^{2}({\rm R}_{j}{\rm T})^{2}=1$ and (109) imply (107) for
${\rm U}={\rm C}$.

By (108), we have ${\rm T}\cdot{\rm C}{\rm R}_{j}{\rm T}=(-1)^{{\rm F}}{\rm
C}{\rm R}_{j}{\rm T}\cdot{\rm T}$, hence ${\rm C}{\rm R}_{j}{\rm T}=(-1)^{{\rm
F}}{\rm T}{\rm C}{\rm R}_{j}$ and ${\rm T}{\rm T}{\rm C}{\rm R}_{j}{\rm C}{\rm
R}_{j}={\rm T}(-1)^{{\rm F}}{\rm C}{\rm R}_{j}{\rm T}{\rm C}{\rm
R}_{j}=(-1)^{{\rm F}}{\rm C}{\rm R}_{j}{\rm T}{\rm C}{\rm R}_{j}{\rm T}$. By
(109), we have ${\rm T}^{2}({\rm C}{\rm R}_{j})^{2}=(-1)^{{\rm F}}$. This is
consistent with Table 4 and Table 5. Conversely, ${\rm T}^{2}({\rm C}{\rm
R}_{j})^{2}=(-1)^{{\rm F}}$ and (109) imply (108) because $({\rm T}{\rm C}{\rm
R}_{j})^{2}={\rm T}({\rm C}{\rm R}_{j}{\rm T})^{2}{\rm T}^{-1}=1$ as long as
$({\rm C}{\rm R}_{j}{\rm T})^{2}=1$.

Combining Table 2, Table 3, Table 4, Table 5, Table 6, and Table 7, we get
Table 8 and Table 9. The CRT for our two possibilities satisfies (109), ${\rm
C}^{2}({\rm R}_{j}{\rm T})^{2}=1$, and ${\rm T}^{2}({\rm C}{\rm
R}_{j})^{2}=(-1)^{{\rm F}}$, hence also satisfies (108) and (107) for ${\rm
U}={\rm C}$. We will check in App. C that it also satisfies (107) for the
internal charge-like and isospin-like U(1) symmetries. Thus the CRT is
canonical.

Dimension | ${\rm C}^{2}$ | ${\rm R}_{j}^{2}$ | ${\rm T}^{2}$ | $({\rm C}{\rm R}_{j})^{2}$ | $({\rm R}_{j}{\rm T})^{2}$ | $({\rm C}{\rm T})^{2}$
---|---|---|---|---|---|---
$d\equiv 0\mod 8$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1
$d\equiv 1\mod 8$ | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1 | 1
$d\equiv 2\mod 8$ | 1 | 1 | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$
$d\equiv 3\mod 8$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | 1 | 1
$d\equiv 4\mod 8$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | 1 | 1
$d\equiv 5\mod 8$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1
$d\equiv 6\mod 8$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$
$d\equiv 7\mod 8$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1
Table 8: The first possibility for the squares of C, Rj, T, CRj, RjT, and CT with the canonical CRT in $d$d for all $d$.  Dimension | ${\rm C}^{2}$ | ${\rm R}_{j}^{2}$ | ${\rm T}^{2}$ | $({\rm C}{\rm R}_{j})^{2}$ | $({\rm R}_{j}{\rm T})^{2}$ | $({\rm C}{\rm T})^{2}$
---|---|---|---|---|---|---
$d\equiv 0\mod 8$ | $(-1)^{{\rm F}}$ | 1 | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$
$d\equiv 1\mod 8$ | 1 | 1 | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$
$d\equiv 2\mod 8$ | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1 | 1
$d\equiv 3\mod 8$ | 1 | 1 | $(-1)^{{\rm F}}$ | 1 | 1 | $(-1)^{{\rm F}}$
$d\equiv 4\mod 8$ | 1 | 1 | $(-1)^{{\rm F}}$ | 1 | 1 | $(-1)^{{\rm F}}$
$d\equiv 5\mod 8$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$
$d\equiv 6\mod 8$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | 1 | $(-1)^{{\rm F}}$ | 1
$d\equiv 7\mod 8$ | $(-1)^{{\rm F}}$ | 1 | 1 | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$ | $(-1)^{{\rm F}}$
Table 9: The second possibility for the squares of C, Rj, T, CRj, RjT, and CT
with the canonical CRT in $d$d for all $d$.

The extension class of the group extension (4) is represented by a 2-cocycle
$f$ on $\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$ valued in $\mathbb{Z}_{2}^{{\rm F}}$.
Note that for $a\in\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm
R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}$ and $b\in\mathbb{Z}_{2}^{{\rm F}}$,
$(a,b)^{2}=(a+a,b+b+f(a,a))=(0,f(a,a))$. We can determine the extension class
of the C-Rj-T fractionalization from the squares of C, Rj, T, CRj, RjT, and
CT.

We summarize the two possibilities of the C-Rj-T fractionalization with the
canonical CRT and its extension class for a Dirac fermion in Table 10 and
Table 11. We find that the C-Rj-T fractionalizations for a Dirac fermion only
depend on the remainder of the spacetime dimension $d$ modulo 8. Hence we have
proved the second part of Theorem III.1. Note that the case $d=1$ is special,
there is only one possibility because there is no Rj for $d=1$.

Dimension | Group presentation | Group | Extension class
---|---|---|---
$d\equiv 0\mod 8$ |  ${\rm C}^{2}={\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm T}^{2}=({\rm C}{\rm T})^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm C},{\rm T}}\times\mathbb{Z}_{4}^{{\rm R}_{j}{\rm T}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm R}_{j}})^{2}+a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
$d=1$ | ${\rm C}^{2}={\rm T}^{2}=({\rm C}{\rm T})^{2}=1$ | $\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm T}}\times\mathbb{Z}_{2}^{{\rm F}}$ | $0$
$d\equiv 1\mod 8$ ($d>1$) |  ${\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm R}_{j},{\rm C}{\rm T}}\times\mathbb{Z}_{2}^{{\rm C}}$ | $(a^{{\rm R}_{j}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 2\mod 8$ |  $({\rm C}{\rm R}_{j})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm R}_{j}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm T},{\rm C}}\times\mathbb{Z}_{2}^{{\rm R}_{j}{\rm T}}$ | $a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
$d\equiv 3\mod 8$ |  ${\rm R}_{j}^{2}={\rm T}^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm C}{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm R}_{j}{\rm T}}$ | $(a^{{\rm R}_{j}}+a^{{\rm T}})(a^{{\rm C}}+a^{{\rm R}_{j}}+a^{{\rm T}})$
$d\equiv 4\mod 8$ |  ${\rm R}_{j}^{2}={\rm T}^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm C}{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm R}_{j}{\rm T}}$ | $(a^{{\rm R}_{j}}+a^{{\rm T}})(a^{{\rm C}}+a^{{\rm R}_{j}}+a^{{\rm T}})$
$d\equiv 5\mod 8$ |  ${\rm C}^{2}={\rm R}_{j}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=(-1)^{{\rm F}}$, $({\rm C}{\rm R}_{j})^{2}=({\rm C}{\rm T})^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm C}{\rm R}_{j}}\times\mathbb{Z}_{4}^{{\rm C}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm R}_{j}})^{2}+(a^{{\rm T}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 6\mod 8$ |  ${\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=1$, ${\rm C}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm C}{\rm R}_{j}{\rm T}}\times\mathbb{Z}_{4}^{{\rm R}_{j}{\rm T}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm T}})^{2}+a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
$d\equiv 7\mod 8$ |  ${\rm C}^{2}={\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm T}^{2}=({\rm C}{\rm T})^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm C},{\rm T}}\times\mathbb{Z}_{4}^{{\rm R}_{j}{\rm T}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm R}_{j}})^{2}+a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
Table 10: The first possibility for the structure of the C-Rj-T fractionalization with the canonical CRT for a Dirac fermion and its extension class in $d$d for all $d$. Here ${\mathbb{D}}_{8}^{{\rm F},x,y}\equiv\langle x,y|x^{4}=y^{2}=1,yxy=x^{-1},x^{2}=(-1)^{{\rm F}}\rangle$, and $\frac{{\mathbb{D}}_{8}^{{\rm F},x,y}\times\mathbb{Z}_{4}^{z{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}\equiv\langle x,y,z|x^{4}=y^{2}=z^{4}=1,yxy=x^{-1},zx=xz,zy=yz,x^{2}=z^{2}=(-1)^{{\rm F}}\rangle$. Here $a^{X}$ ($X={\rm C}$, ${\rm R}_{j}$, and ${\rm T}$) are the three generators of $\operatorname{H}^{1}(\mathrm{B}(\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}),\mathbb{Z}_{2}^{{\rm F}})=\mathbb{Z}_{2}^{3}$ respectively. Dimension | Group presentation | Group | Extension class
---|---|---|---
$d\equiv 0\mod 8$ |  ${\rm C}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm R}_{j}^{2}={\rm T}^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm R}_{j},{\rm T}}\times\mathbb{Z}_{4}^{{\rm C}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d=1$ | ${\rm C}^{2}={\rm T}^{2}=({\rm C}{\rm T})^{2}=1$ | $\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm T}}\times\mathbb{Z}_{2}^{{\rm F}}$ | $0$
$d\equiv 1\mod 8$ ($d>1$) |  $({\rm C}{\rm R}_{j})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm R}_{j}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm R}_{j},{\rm C}{\rm R}_{j}{\rm T}}\times\mathbb{Z}_{2}^{{\rm R}_{j}{\rm T}}$ | $a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
$d\equiv 2\mod 8$ |  ${\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm R}_{j},{\rm C}{\rm R}_{j}{\rm T}}\times\mathbb{Z}_{2}^{{\rm C}}$ | $(a^{{\rm R}_{j}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 3\mod 8$ |  ${\rm T}^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm C}}$ | $(a^{{\rm T}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 4\mod 8$ |  ${\rm T}^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm C}^{2}={\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm C}}$ | $(a^{{\rm T}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 5\mod 8$ |  ${\rm C}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm R}_{j}^{2}=({\rm C}{\rm R}_{j})^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm C}{\rm R}_{j}{\rm T}}\times\mathbb{Z}_{4}^{{\rm R}_{j}{\rm T}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm T}})^{2}+a^{{\rm C}}a^{{\rm R}_{j}}+a^{{\rm C}}a^{{\rm T}}$
$d\equiv 6\mod 8$ |  $({\rm C}{\rm R}_{j})^{2}=({\rm C}{\rm T})^{2}=1$, ${\rm C}^{2}={\rm R}_{j}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=(-1)^{{\rm F}}$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm R}_{j},{\rm C}{\rm R}_{j}{\rm T}}\times\mathbb{Z}_{4}^{{\rm C}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+(a^{{\rm R}_{j}})^{2}+(a^{{\rm T}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
$d\equiv 7\mod 8$ |  ${\rm C}^{2}=({\rm C}{\rm R}_{j})^{2}=({\rm R}_{j}{\rm T})^{2}=({\rm C}{\rm T})^{2}=(-1)^{{\rm F}}$, ${\rm R}_{j}^{2}={\rm T}^{2}=1$ | $\frac{{\mathbb{D}}_{8}^{{\rm F},{\rm C}{\rm R}_{j},{\rm T}}\times\mathbb{Z}_{4}^{{\rm C}{\rm F}}}{\mathbb{Z}_{2}^{{\rm F}}}$ | $(a^{{\rm C}})^{2}+a^{{\rm R}_{j}}a^{{\rm T}}$
Table 11: The second possibility for the structure of the C-Rj-T
fractionalization with the canonical CRT for a Dirac fermion and its extension
class in $d$d for all $d$. Here ${\mathbb{D}}_{8}^{{\rm F},x,y}\equiv\langle
x,y|x^{4}=y^{2}=1,yxy=x^{-1},x^{2}=(-1)^{{\rm F}}\rangle$, and
$\frac{{\mathbb{D}}_{8}^{{\rm F},x,y}\times\mathbb{Z}_{4}^{z{\rm
F}}}{\mathbb{Z}_{2}^{{\rm F}}}\equiv\langle
x,y,z|x^{4}=y^{2}=z^{4}=1,yxy=x^{-1},zx=xz,zy=yz,x^{2}=z^{2}=(-1)^{{\rm
F}}\rangle$. Here $a^{X}$ ($X={\rm C}$, ${\rm R}_{j}$, and ${\rm T}$) are the
three generators of $\operatorname{H}^{1}(\mathrm{B}(\mathbb{Z}_{2}^{{\rm
C}}\times\mathbb{Z}_{2}^{{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm
T}}),\mathbb{Z}_{2}^{{\rm F}})=\mathbb{Z}_{2}^{3}$ respectively.

Let $G_{{\rm D},i}(d\mod 8)$ $(d>1)$ denote the $i$-th possibility of the
C-Rj-T fractionalization with the canonical CRT for a Dirac fermion. We will
use this notation in Sec. VI. We find that

  * •

$G_{{\rm D},i}(0\mod 8)=G_{{\rm D},i}(7\mod 8)$ for $i=1,2$,

  * •

$G_{{\rm D},i}(3\mod 8)=G_{{\rm D},i}(4\mod 8)$ for $i=1,2$,

  * •

$G_{{\rm D},1}(1\mod 8)=G_{{\rm D},2}(2\mod 8)$, $G_{{\rm D},1}(2\mod
8)=G_{{\rm D},2}(1\mod 8)$,

  * •

$G_{{\rm D},1}(5\mod 8)=G_{{\rm D},2}(6\mod 8)$, $G_{{\rm D},1}(6\mod
8)=G_{{\rm D},2}(5\mod 8)$.

##  IV Majorana, Weyl, and Majorana-Weyl fermions

In this section, we determine the R-T fractionalization for Majorana fermions
in any spacetime dimension $d=0,1,2,3,4\mod 8$. We also determine the C or T
fractionalization for Weyl fermions in any even spacetime dimension.

Majorana fermions are defined as Dirac fermions with the trivial C symmetry.
Majorana fermions exist only for $d=0,1,2,3,4\mod 8$ Polchinski (2007); Stone
(2022). For $d=0,1\mod 8$, there exist only massless Majorana fermions Stone
(2022). First, we reproduce these results in terms of the chiral
representation (see App. A).

To determine the Rj-T fractionalization for Majorana fermions, we then
construct imaginary Gamma matrices for the massive case (for $d=2,3,4\mod 8$)
and real Gamma matrices for the massless case (for $d=0,1,2\mod 8$) (see App.
B), then we can choose the matrix of C to be the identity matrix. Since the
matrices of Rj and T for a Majorana fermion have to be real, we can determine
the matrices of Rj and T for a Majorana fermion. Moreover, we can determine
the squares of Rj, T, and RjT, hence we obtain a group presentation of the
Rj-T fractionalization.

For Majorana fermions, the C symmetry is trivial. First, we consider the
massless Majorana fermions. The massless Dirac equation is

$\displaystyle\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}\psi=0.$ (171)

The charge-conjugated massless Dirac equation is

$\displaystyle\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}\psi^{{\rm C}}=0.$ (172)

Since $\psi^{{\rm C}}=M_{{\rm C}}\psi^{*}$, we get

$\displaystyle M_{{\rm C}}\Gamma^{\mu}=-\Gamma^{\mu*}M_{{\rm C}}$ (173)

or

$\displaystyle M_{{\rm C}}\Gamma^{\mu}=\Gamma^{\mu*}M_{{\rm C}}$ (174)

for $\mu=0,1,\dots,d-1$. Only (173) is valid if we consider the massive
Majorana fermions.

First, we choose the Gamma matrices as (239). For $d=2k+2$, we get $M_{{\rm
C}}=\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}$ (it satisfies (173) for odd $k$ and
(174) for even $k$) or $M_{{\rm
C}}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$ (it satisfies (173) for
even $k$ and (174) for odd $k$). For $d=2k+3$, since $\Gamma^{2k+2}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt^{k+1}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{2k+1}$, if
(173) holds for $\mu=0,1,\dots,2k+1$, then (173) holds for $\mu=2k+2$ only if
$k$ is even. If (174) holds for $\mu=0,1,\dots,2k+1$, then (174) holds for
$\mu=2k+2$ only if $k$ is odd. We get $M_{{\rm
C}}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$ for $d=2k+3$ and all
$k$ (it satisfies (173) for even $k$ and (174) for odd $k$). The Majorana
condition is $\psi^{{\rm C}}=\psi=M_{{\rm C}}\psi^{*}=M_{{\rm C}}M_{{\rm
C}}^{*}\psi$, hence $M_{{\rm C}}M_{{\rm C}}^{*}=1$.

For $d=2k+2$, if $M_{{\rm C}}=\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}$, then
$M_{{\rm C}}M_{{\rm C}}^{*}=1$ implies that $(-1)^{\frac{k(k-1)}{2}}=1$ and
$k=0,1\mod 4$ where $M_{{\rm C}}$ satisfies (174) for $k=0\mod 4$ and (173)
for $k=1\mod 4$. If $M_{{\rm
C}}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$, then $M_{{\rm
C}}M_{{\rm C}}^{*}=1$ implies that $(-1)^{\frac{k(k+1)}{2}}=1$ and $k=0,3\mod
4$ where $M_{{\rm C}}$ satisfies (173) for $k=0\mod 4$ and (174) for $k=3\mod
4$. For $d=2k+3$, $M_{{\rm
C}}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$. $M_{{\rm C}}M_{{\rm
C}}^{*}=1$ implies that $(-1)^{\frac{k(k+1)}{2}}=1$ and $k=0,3\mod 4$ where
$M_{{\rm C}}$ satisfies (173) for $k=0\mod 4$ and (174) for $k=3\mod 4$.

In summary, the Majorana condition implies that $d=0,1,2,3,4\mod 8$. This
agrees with Table B.1 of Polchinski (2007). Here $d=0\mod 8$ requires (174),
$d=1\mod 8$ requires (174), $d=2\mod 8$ requires either (173) or (174),
$d=3\mod 8$ requires (173), and $d=4\mod 8$ requires (173). Therefore, there
exist only massless Majorana fermions for $d=0,1\mod 8$. This also agrees with
the results in Stone (2022).

We can construct imaginary Gamma matrices for $d=2,3,4\mod 8$, namely
$\Gamma^{\mu*}=-\Gamma^{\mu}$. Then we can choose the matrix of the charge
conjugation to be the identity matrix since $M_{{\rm C}}\Gamma^{\mu*}M_{{\rm
C}}^{-1}=-\Gamma^{\mu}$. We can construct real Gamma matrices for $d=0,1,2\mod
8$, namely $\Gamma^{\mu*}=\Gamma^{\mu}$. In the massless case, we can choose
the matrix of the charge conjugation to be the identity matrix since $M_{{\rm
C}}\Gamma^{\mu*}M_{{\rm C}}^{-1}=\Gamma^{\mu}$.

Therefore, for $d=0,1,2,3,4\mod 8$, the Majorana condition becomes
$\psi^{*}=\psi$. In the massive case, the matrices of Rj and T should satisfy
(II.4). If the Gamma matrices are imaginary ($d=2,3,4\mod 8$), then (II.4)
implies that

$\displaystyle M_{{\rm T}}\Gamma^{0}M_{{\rm T}}^{-1}$ $\displaystyle=$
$\displaystyle-\Gamma^{0},{}$ $\displaystyle M_{{\rm T}}\Gamma^{j}M_{{\rm
T}}^{-1}$ $\displaystyle=$ $\displaystyle\Gamma^{j},\;\;\;j=1,\dots,d-1.$
(175)

If the Gamma matrices are real ($d=0,1,2\mod 8$), then (II.4) implies that

$\displaystyle M_{{\rm T}}\Gamma^{0}M_{{\rm T}}^{-1}$ $\displaystyle=$
$\displaystyle\Gamma^{0},{}$ $\displaystyle M_{{\rm T}}\Gamma^{j}M_{{\rm
T}}^{-1}$ $\displaystyle=$ $\displaystyle-\Gamma^{j},\;\;\;j=1,\dots,d-1.$
(176)

For Majorana fermions, the matrices of Rj and T should be real. For
$d=2k+2=4\mod 8$, we can choose imaginary Gamma matrices (see App. B), and we
have $M_{{\rm T}}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1}$ and $M_{{\rm R}_{j}}=\hskip
1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$. In this
case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (177) $\displaystyle=$ $\displaystyle(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=-\psi,$ $\displaystyle{\rm
R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$ $\displaystyle M_{{\rm
R}_{j}}^{2}\psi{}$ (178) $\displaystyle=$ $\displaystyle(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k+1}\psi=\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (179)
$\displaystyle=$ $\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle\psi.$

For $d=2k+3=3\mod 8$, we can choose imaginary Gamma matrices (see App. B), and
(II.4) has no solutions for the matrices of Rj and T. In the massless case, we
have $M_{{\rm T}}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{0}$ and $M_{{\rm
R}_{j}}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{j}$. In this case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (180) $\displaystyle=$ $\displaystyle(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{0})^{2}\psi{}$ $\displaystyle=$
$\displaystyle-\psi,$ $\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$
$\displaystyle=$ $\displaystyle M_{{\rm R}_{j}}^{2}\psi{}$ (181)
$\displaystyle=$ $\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{j})^{2}\psi{}$ $\displaystyle=$ $\displaystyle\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (182)
$\displaystyle=$ $\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{j})^{2}\psi{}$
$\displaystyle=$ $\displaystyle\psi.$

For $d=2k+2=2\mod 8$, the Majorana fermions can be massive or massless and we
can choose imaginary or real Gamma matrices (see App. B). We have $M_{{\rm
T}}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1}$
and $M_{{\rm R}_{j}}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$ (if the
Gamma matrices are imaginary) or $M_{{\rm T}}=\Gamma^{0}$ and $M_{{\rm
R}_{j}}=\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$ (if the
Gamma matrices are real). In the first case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (183) $\displaystyle=$ $\displaystyle(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=\psi,$ $\displaystyle{\rm
R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$ $\displaystyle M_{{\rm
R}_{j}}^{2}\psi{}$ (184) $\displaystyle=$ $\displaystyle(\hskip
1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k+1}\psi=-\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (185)
$\displaystyle=$ $\displaystyle(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{2k+1}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle\psi.$

In the second case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (186) $\displaystyle=$
$\displaystyle(\Gamma^{0})^{2}\psi{}$ $\displaystyle=$ $\displaystyle\psi,$
$\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$
$\displaystyle M_{{\rm R}_{j}}^{2}\psi{}$ (187) $\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (188)
$\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=\psi.$

For $d=2k+3=1\mod 8$, the Majorana fermions have to be massless and we can
choose real Gamma matrices (see App. B). We have $M_{{\rm T}}=\Gamma^{0}$ and
(II.4) has no solutions for the matrix of Rj. In the massless case, we have
$M_{{\rm R}_{j}}=\Gamma^{j}$. In this case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (189) $\displaystyle=$
$\displaystyle(\Gamma^{0})^{2}\psi{}$ $\displaystyle=$ $\displaystyle\psi,$
$\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$
$\displaystyle M_{{\rm R}_{j}}^{2}\psi{}$ (190) $\displaystyle=$
$\displaystyle(\Gamma^{j})^{2}\psi{}$ $\displaystyle=$ $\displaystyle-\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (191)
$\displaystyle=$ $\displaystyle(\Gamma^{0}\Gamma^{j})^{2}\psi{}$
$\displaystyle=$ $\displaystyle\psi.$

For $d=2k+2=0\mod 8$, the Majorana fermions have to be massless and we can
choose real Gamma matrices (see App. B). We have $M_{{\rm T}}=\Gamma^{0}$ and
$M_{{\rm
R}_{j}}=\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$. In
this case,

$\displaystyle{\rm T}^{2}\psi{\rm T}^{-2}$ $\displaystyle=$ $\displaystyle
M_{{\rm T}}^{2}\psi{}$ (192) $\displaystyle=$
$\displaystyle(\Gamma^{0})^{2}\psi{}$ $\displaystyle=$ $\displaystyle\psi,$
$\displaystyle{\rm R}_{j}^{2}\psi{\rm R}_{j}^{-2}$ $\displaystyle=$
$\displaystyle M_{{\rm R}_{j}}^{2}\psi{}$ (193) $\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=-\psi,$

and

$\displaystyle({\rm R}_{j}{\rm T})^{2}\psi({\rm R}_{j}{\rm T})^{-2}$
$\displaystyle=$ $\displaystyle(M_{{\rm T}}M_{{\rm R}_{j}})^{2}\psi{}$ (194)
$\displaystyle=$
$\displaystyle(\Gamma^{0}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1})^{2}\psi{}$
$\displaystyle=$ $\displaystyle(-1)^{k}\psi=-\psi.$

Note that the case $d=1$ is special. In this case, there is only the T
symmetry and a single Majorana fermion can not have a mass. In the massless
case, $M_{{\rm T}}=1$ and ${\rm T}^{2}=1$.

Therefore, a single Majorana fermion can not have a mass for $d=0,1\mod 8$.
The Majorana mass term breaks the ${\rm R}_{j}$ and T symmetry for $d=3\mod
8$. See Table 16.

We summarize the Rj-T fractionalization for a Majorana fermion in Table 12.
Note that the case $d=1$ is special because there is no Rj for $d=1$.

Dimension | Group presentation | Group
---|---|---
$d\equiv 0\mod 8$ |  ${\rm T}^{2}=1$, ${\rm R}_{j}^{2}=({\rm R}_{j}{\rm T})^{2}=(-1)^{{\rm F}}$ | $\mathbb{Z}_{4}^{{\rm R}_{j}{\rm F}}\times\mathbb{Z}_{2}^{{\rm T}}$
$d=1$ | ${\rm T}^{2}=1$ | $\mathbb{Z}_{2}^{{\rm T}}\times\mathbb{Z}_{2}^{{\rm F}}$
$d\equiv 1\mod 8$ ($d>1$) |  ${\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=1$, ${\rm R}_{j}^{2}=(-1)^{{\rm F}}$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm R}_{j},{\rm T}}$
$d\equiv 2\mod 8$ | $\begin{array}[]{cc}{\rm R}_{j}^{2}=(-1)^{{\rm F}},{\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=1\\\ \text{or }{\rm R}_{j}^{2}={\rm T}^{2}=({\rm R}_{j}{\rm T})^{2}=1\end{array}$ | $\begin{array}[]{cc}{\mathbb{D}}_{8}^{{\rm F},{\rm R}_{j},{\rm T}}\\\ \text{or }\mathbb{Z}_{2}^{{\rm R}_{j}}\times\mathbb{Z}_{2}^{{\rm T}}\times\mathbb{Z}_{2}^{{\rm F}}\end{array}$
$d\equiv 3\mod 8$ |  ${\rm T}^{2}=(-1)^{{\rm F}}$, ${\rm R}_{j}^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm R}_{j}}$
$d\equiv 4\mod 8$ |  ${\rm T}^{2}=(-1)^{{\rm F}}$, ${\rm R}_{j}^{2}=({\rm R}_{j}{\rm T})^{2}=1$ | ${\mathbb{D}}_{8}^{{\rm F},{\rm T},{\rm R}_{j}}$
Table 12: The structure of the Rj-T fractionalization for a Majorana fermion
in $d$d for all $d=0,1,2,3,4\mod 8$. Here ${\mathbb{D}}_{8}^{{\rm
F},x,y}\equiv\langle x,y|x^{4}=y^{2}=1,yxy=x^{-1},x^{2}=(-1)^{{\rm
F}}\rangle$. For $d=2\mod 8$, there are two choices because Majorana fermions
can be massive or massless in $d$d. The first choice is for the massive case,
and the second choice is for the massless case.

Let $G_{\rm M}(d\mod 8)(d>1)$ denote the Rj-T fractionalization for a Majorana
fermion and $G_{M,i}(2\mod 8)$ denote the $i$-th choice of the Rj-T
fractionalization for a Majorana fermion. We will use this notation in Sec.
VI. We find that

  * •

$G_{\rm M,1}(2\mod 8)=G_{\rm M}(1\mod 8)$.

  * •

$G_{\rm M}(3\mod 8)=G_{\rm M}(4\mod 8)$.

  * •

$G_{\rm M}(d\mod 8)$ is a subgroup of $G_{{\rm D},1}(d\mod 8)$ for $d=0,1\mod
8$.

  * •

$G_{\rm M}(d\mod 8)$ is a subgroup of $G_{{\rm D},2}(d\mod 8)$ for $d=3,4\mod
8$.

  * •

$G_{\rm M,1}(2\mod 8)$ is a subgroup of $G_{{\rm D},2}(2\mod 8)$ and $G_{\rm
M,2}(2\mod 8)$ is a subgroup of $G_{{\rm D},1}(2\mod 8)$.

For Weyl fermions, the spacetime dimension $d$ has to be even because of
Definition II.5. For $d=2k+2$, the Dirac fermion $\psi$ decomposes as
$\psi=\left(\begin{array}[]{cc}\psi_{L}\\\ \psi_{R}\end{array}\right)$ where
$\psi_{L}$ and $\psi_{R}$ are left and right Weyl fermions. Since $M_{{\rm
R}_{j}}=\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{j}\cdots\Gamma^{2k+1}$ is not
block diagonal in the chiral representation (see App. A),

$\displaystyle\psi_{L}^{{\rm
R}_{j}}=(-1)^{k}\gamma^{1}\cdots\hat{\gamma}^{j}\cdots\gamma^{2k+1}\psi_{R}(t,x_{1},\dots,x_{j-1},-x_{j},x_{j+1},\dots,x_{d-1})$
(195)

(see (239)), the left Weyl fermion $\psi_{L}$ does not have the ${\rm R}_{j}$
symmetry. $\psi_{L}$ has the C or T symmetry if and only if $M_{{\rm C}}$ (or
$M_{{\rm T}}$) is block diagonal in the chiral representation (see App. A).
Note that the phase factors of C and T do not affect their orders, hence we
ignore the phase factors of C and T in the following discussion.

For odd $k$, $d=0\mod 4$, $M_{{\rm C}}=\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}$
is not block diagonal in the chiral representation (see App. A), while
$M_{{\rm T}}=\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$ is block diagonal in the
chiral representation (see App. A), hence $\psi_{L}$ has the T symmetry but
not the C symmetry. In this case,

$\displaystyle\psi_{L}^{{\rm
T}}=(-1)^{\frac{k+1}{2}}\gamma^{1}\gamma^{3}\cdots\gamma^{2k+1}\psi_{L}(-t,x)$
(196)

(see (239)) and ${\rm T}^{2}=(-1)^{\frac{k(k+1)}{2}}$.

For even $k$, $d=2\mod 4$, $M_{{\rm
C}}=\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{2k+1}$ is block diagonal in
the chiral representation (see App. A), while $M_{{\rm
T}}=\Gamma^{0}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{2k}$ is not block diagonal in
the chiral representation (see App. A), hence $\psi_{L}$ has the C symmetry
but not the T symmetry. In this case,

$\displaystyle\psi_{L}^{{\rm
C}}=(-1)^{\frac{k}{2}+1}\gamma^{1}\gamma^{3}\cdots\gamma^{2k+1}\psi_{L}^{*}$
(197)

(see (239)) and ${\rm C}^{2}=(-1)^{\frac{k(k+1)}{2}}$.

We summarize the C or T fractionalization for a Weyl fermion in Table 13.

Dimension | Group presentation | Group
---|---|---
$d=0\mod 8$ | ${\rm T}^{2}=1$ | $\mathbb{Z}_{2}^{{\rm T}}\times\mathbb{Z}_{2}^{{\rm F}}$
$d=2\mod 8$ | ${\rm C}^{2}=1$ | $\mathbb{Z}_{2}^{{\rm C}}\times\mathbb{Z}_{2}^{{\rm F}}$
$d=4\mod 8$ | ${\rm T}^{2}=(-1)^{{\rm F}}$ | $\mathbb{Z}_{4}^{{\rm T}{\rm F}}$
$d=6\mod 8$ | ${\rm C}^{2}=(-1)^{{\rm F}}$ | $\mathbb{Z}_{4}^{{\rm C}{\rm F}}$
Table 13: The C or T fractionalization for a Weyl fermion in $d$d for all even
$d$.

Let $G_{\rm W}(d\mod 8)$ denote the C or T fractionalization for a Weyl
fermion. We will use this notation in Sec. VI. We find that the $G_{\rm
W}(d\mod 8)$ is a subgroup of $G_{{\rm D},i}(d\mod 8)$ ($i=1,2$) for all even
$d$.

For Majorana-Weyl fermions, since the Majorana-Weyl fermion is Majorana, it
has the trivial C symmetry and $d=0,1,2,3,4\mod 8$. Since the Majorana-Weyl
fermion is also Weyl, $d$ is even. By the above discussion, Weyl fermion does
not have the C symmetry unless $d=2\mod 4$. But Majorana-Weyl fermion has a
trivial C symmetry, thus there is only a trivial C symmetry but no Rj or T
symmetries, and $d=2\mod 8$. Let $G_{\rm MW}(2\mod 8)=\mathbb{Z}_{2}^{{\rm
F}}$ denote the trivial C fractionalization for a Majorana-Weyl fermion. We
will use this notation in Sec. VI.

We summarize the symmetries of Dirac, Majorana, Weyl, and Majorana-Weyl
fermions in even spacetime dimension $d$ in Table 14.

| C | ${\rm R}_{j}$ | T | C${\rm R}_{j}$T
---|---|---|---|---
Dirac | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Majorana | trivial | $\checkmark$ | $\checkmark$ | $\checkmark$
Weyl ($d=2\mod 4$) | $\checkmark$ | ✗ | ✗ | $\checkmark$
Weyl ($d=0\mod 4$) | ✗ | ✗ | $\checkmark$ | $\checkmark$
Majorana-Weyl ($d=2\mod 8$) | trivial | ✗ | ✗ | $\checkmark$
Table 14: The symmetries of Dirac, Majorana, Weyl, and Majorana-Weyl fermions
in even spacetime dimension $d$. Here $\checkmark$ means the symmetry is
preserved. Here trivial means that the symmetry acts trivially. The C acts on
Majorana fermion $\psi$ as $\psi^{\rm C}=\psi$. Here ✗ means the symmetry is
violated.

##  V Mass term

In this section, we discuss how the Dirac and Majorana mass terms break the
symmetries C, R, or T.

Based on the discussion in Sec. III, we find that the Dirac mass term breaks
the C symmetry for $d=2k+3$ ($k$ odd), it breaks the T symmetry for $d=2k+3$
($k$ even), and it breaks the Rj symmetry for $d=2k+3$ and $j=1,2,\dots,2k+2$,
see Table 15.

Dirac mass term | C | Rj | T
---|---|---|---
$d=2k+2$ ($k\in\mathbb{Z}_{\geq 0}$) or $d=0,2\mod 4$ | $\checkmark$ | $\checkmark$ | $\checkmark$
$d=2k+3$ ($k$ odd) or $d=1\mod 4$ | ✗ | ✗ | $\checkmark$
$d=2k+3$ ($k$ even) or $d=3\mod 4$ | $\checkmark$ | ✗ | ✗
Table 15: The Dirac mass term (for a single particle field) breaks the
symmetries C, Rj, and T or not. Here $\checkmark$ means the Dirac mass term
preserves the symmetry. Here ✗ means the Dirac mass term breaks the symmetry.

Based on the discussion in Sec. IV, we find that a single Majorana fermion can
not have a mass for $d=0,1\mod 8$. The Majorana mass term breaks the ${\rm
R}_{j}$ and T symmetry for $d=3\mod 8$. See Table 16.

Majorana mass term | Rj | T
---|---|---
$d=0,1\mod 8$ (massless) |  |
$d=2,4\mod 8$ | $\checkmark$ | $\checkmark$
$d=3\mod 8$ | ✗ | ✗
Table 16: The Majorana mass term (for a single particle field) breaks the
symmetries Rj and T or not. Here $\checkmark$ means the Majorana mass term
preserves the symmetry. Here ✗ means the Majorana mass term breaks the
symmetry. For a single particle Majorana fermion, the dimension $d=0,1\mod 8$
forbids Majorana mass thus Majorana fermion is massless.

Dirac mass and Majorana mass break the same kind of symmetry in the same
pattern. Majorana has trivial C symmetry, but Dirac has nontrivial C symmetry.
Dirac mass can break C symmetry in $d=1\mod 4$, thus the corresponding
Majorana mass breaks C symmetry which contradicts the premise Majorana
condition — this implies that Majorana fermion must be massless in $d=1\mod
4$. Dirac mass exists in all dimensions. Majorana fermion exists in
$d=0,1,2,3,4\mod 8$, while Majorana mass exists in $d=2,3,4\mod 8$.

##  VI Domain wall dimensional reduction

In this section, we study the domain wall dimensional reduction of the C-R-T
fractionalization and R-T fractionalization. We also study the domain wall
dimensional reduction of fermions and arrive at a consistent story.

The mass profile of a fermion in $d$d is shown in Fig. 1 where the domain wall
that separates the two domains $m>0$ and $m<0$ is $m=0,x_{d-1}=0$.

$x_{d-1}$$m$$d$d$(d-1)$d$m=0,x_{d-1}=0$ Figure 1: Mass profile. The $d$d
fermion obeys the Lagrangian $\bar{\psi}^{d}(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{\mu}\partial_{\mu}-m(x))\psi^{d}$, then there is an effective
massless $(d-1)$d domain wall fermion theory $\psi^{d-1}$ at $x_{d-1}=0$, with
its time and spatial coordinates $(t,x_{1},x_{2},\dots,x_{d-2})$, which is
obtained by the projection ${\rm P}_{\pm}=\frac{1\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}}{2}$, $\psi^{d-1}_{\pm}={\rm
P}_{\pm}\psi^{d}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}}{2}\psi^{d}$ where ${\rm P}_{\pm}^{2}={\rm P}_{\pm}$, ${\rm
P}_{+}{\rm P}_{-}={\rm P}_{-}{\rm P}_{+}=0$, and $\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}{\rm P}_{\pm}=\pm{\rm P}_{\pm}$. For odd $d$,
$\Gamma^{d-1}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt{\rm diag}(-I,I)$ in the
chiral representation (see App. A). The Gamma matrices $\Gamma^{\mu}$ in
$(d-1)$d are the same as those in $d$d for odd $d$ and $\mu=0,1,\dots,d-2$.
Therefore, for odd $d$, the projection maps a Dirac (Majorana) fermion in $d$d
to a Weyl (Majorana-Weyl) fermion in $(d-1)$d. For even $d$, the projection
maps a Dirac (Majorana) fermion in $d$d to a Dirac (Majorana) fermion in
$(d-1)$d. See also Fig. 4. $x_{d-1}=0$${\rm U}$${\rm U}$${\rm C}{\rm
R}_{d-1}{\rm T}$ Figure 2: Domain wall dimensional reduction. Suppose ${\rm
U}$ is (spontaneously or explicitly) broken by mass, ${\rm U}$ switches the
two degenerate ground states, and assume that ${\rm U}$ does not flip the
spatial coordinate $x_{d-1}$ (which excludes ${\rm R}_{d-1}$, but ${\rm U}$
can be ${\rm C}$, ${\rm R}_{i}$ for $i\neq d-1$, and ${\rm T}$, or other
internal symmetries), we need to combine it with the space-orientation-
reversing symmetry ${\rm C}{\rm R}_{d-1}{\rm T}$ to obtain a new symmetry on
the domain wall Hason  _et al._ (2020); Córdova  _et al._ (2020); Wang  _et
al._ (2020). However, some of the domain wall symmetries may not be directly
induced from ${\rm U}\cdot({\rm C}{\rm R}_{d-1}{\rm T})$, the domain wall
symmetries may be induced from the ${\rm U}={\rm R}_{d-1}$ itself or other
symmetries.

We assume that the fermion parity $(-1)^{{\rm F}}$ is not spontaneously
broken, if the C, R, and T symmetries are spontaneously broken
$\mathbb{Z}_{4}$ symmetries (they are $\mathbb{Z}_{2}$ or $\mathbb{Z}_{4}$
symmetries in our assumption, see Lemma III.2), then there are still only two
degenerate ground states. We consider the mass profile (see Fig. 1) which can
be spontaneously or explicitly generated, and see which symmetry is broken by
mass.

If $d$ is odd, then we limit ourselves to consider only the broken symmetry
${\rm U}$ by mass, see Fig. 2. First, we consider Dirac fermions. By Table 15,
the Dirac mass term breaks the C and R symmetries for $d=1\mod 4$ and breaks
the R and T symmetries for $d=3\mod 4$. For $d=1\mod 4$, we consider ${\rm
U}={\rm C}$ or ${\rm R}$. For $d=3\mod 4$, we consider ${\rm U}={\rm T}$ or
${\rm R}$. Next, we consider Majorana fermions. By Table 16, the Majorana mass
term breaks the R and T symmetries for $d=3\mod 8$. In this case, we consider
${\rm U}={\rm T}$ or ${\rm R}$.

If ${\rm U}={\rm C}$ is unitary, then it becomes anti-unitary ${\rm
T}^{\prime}={\rm U}\cdot{\rm C}{\rm R}_{d-1}{\rm T}={\rm R}_{d-1}{\rm T}$ up
to $(-1)^{{\rm F}}$ on the domain wall.

If ${\rm U}={\rm T}$ is anti-unitary, then it becomes unitary ${\rm
C}^{\prime}={\rm U}\cdot{\rm C}{\rm R}_{d-1}{\rm T}={\rm C}{\rm R}_{d-1}$ up
to $(-1)^{{\rm F}}$ on the domain wall.

If ${\rm U}={\rm R}_{d-1}$ is unitary, then it switches the two degenerate
ground states and flips the spatial coordinate $x_{d-1}$, hence it preserves
the mass profile, so it becomes an internal unitary symmetry $X$ on the domain
wall. By Table 1, the matrix of ${\rm R}_{d-1}$ is $\eta_{{\rm
R}_{d-1}}\Gamma^{d-1}=\eta_{{\rm R}_{d-1}}\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\rm diag}(-I,I)$ in the chiral representation where $I$ is an identity
matrix (see App. A), the matrix of $X$ is $\mp\eta_{{\rm R}_{d-1}}\hskip
1.0pt\mathrm{i}\hskip 1.0ptI$ (which generates a unitary $\mathbb{Z}_{4,X}$
symmetry if $\eta_{{\rm R}_{d-1}}=\pm 1$ or $\mathbb{Z}_{2}^{{\rm F}}$
symmetry or trivial symmetry if $\eta_{{\rm R}_{d-1}}=\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt$) under the projection ${\rm
P}_{\pm}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}}{2}$.

If ${\rm U}={\rm R}_{i}$ ($i\neq d-1$) is unitary, then it becomes anti-
unitary ${\rm U}\cdot{\rm C}{\rm R}_{d-1}{\rm T}={\rm R}_{i}{\rm C}{\rm
R}_{d-1}{\rm T}$ on the domain wall. Since the matrix of ${\rm R}_{i}{\rm
C}{\rm R}_{d-1}{\rm T}$ is $\eta\Gamma^{i}\Gamma^{0}\Gamma^{d-1}$ which is
block diagonal in the chiral representation (see App. A), ${\rm R}_{i}$
becomes an unbroken anti-unitary symmetry ${\rm C}{\rm R}_{i}{\rm T}\cdot X$
up to $(-1)^{{\rm F}}$ on the domain wall where ${\rm C}{\rm R}_{i}{\rm T}$ is
induced from the ${\rm C}{\rm R}_{i}{\rm T}$ in the bulk and it is unbroken,
and $X$ is induced from ${\rm R}_{d-1}$ in the bulk.

Therefore, for odd $d$, the domain wall dimensional reduction of the
symmetries C, R, and T obeys the following rule (up to $(-1)^{{\rm F}}$):

  1. 1.

For Dirac fermions and $d=1\mod 4$,

$\displaystyle d\text{-}\dim$ $\displaystyle\xrightarrow{\rm DW}$
$\displaystyle(d-1)\text{-}\dim{}$ $\displaystyle{\rm R}_{d-1}$
$\displaystyle\xrightarrow{{\rm P}_{\pm}}$ $\displaystyle X{}$
$\displaystyle{\rm R}_{i}$ $\displaystyle\xrightarrow{\cdot{\rm C}{\rm
R}_{d-1}{\rm T}}$ $\displaystyle{\rm C}{\rm R}_{i}{\rm T}\cdot X\;\;\;(i\neq
d-1){}$ $\displaystyle{\rm C}$ $\displaystyle\xrightarrow{\cdot{\rm C}{\rm
R}_{d-1}{\rm T}}$ $\displaystyle{\rm T}^{\prime}=X{\rm T}{}$
$\displaystyle{\rm T}$ $\displaystyle\xrightarrow{{\rm P}_{\pm}}$
$\displaystyle{\rm T}.$ (198)

The $(d-1)$d domain wall (DW) Weyl fermion has the corresponding symmetry such
that ${\rm C}_{\rm DW}$ and ${\rm R}_{\rm DW}$ are broken, ${\rm T}_{\rm
DW}={\rm T}$ or ${\rm T}^{\prime}=X{\rm T}$, and ${\rm C}_{\rm DW}{\rm R}_{\rm
DW}{\rm T}_{\rm DW}$ is unbroken, while there is also an extra internal
unitary $X$ symmetry (which generates $\mathbb{Z}_{4,X}$ or
$\mathbb{Z}_{2}^{{\rm F}}$ or 1). By Table 1, the matrices of T for Dirac
fermions in $d=1\mod 4$ and $d-1=0\mod 4$ are the same and they are block
diagonal in the chiral representation (see App. A). So the matrix of T for
Dirac fermions in $d=1\mod 4$ becomes the matrix of T for Weyl fermions in
$d-1=0\mod 4$ under the projection ${\rm P}_{\pm}=\frac{1\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}}{2}$.

  2. 2.

For Dirac fermions and $d=3\mod 4$,

$\displaystyle d\text{-}\dim$ $\displaystyle\xrightarrow{\rm DW}$
$\displaystyle(d-1)\text{-}\dim{}$ $\displaystyle{\rm R}_{d-1}$
$\displaystyle\xrightarrow{{\rm P}_{\pm}}$ $\displaystyle X{}$
$\displaystyle{\rm R}_{i}$ $\displaystyle\xrightarrow{\cdot{\rm C}{\rm
R}_{d-1}{\rm T}}$ $\displaystyle{\rm C}{\rm R}_{i}{\rm T}\cdot X\;\;\;(i\neq
d-1){}$ $\displaystyle{\rm C}$ $\displaystyle\xrightarrow{{\rm P}_{\pm}}$
$\displaystyle{\rm C}{}$ $\displaystyle{\rm T}$
$\displaystyle\xrightarrow{\cdot{\rm C}{\rm R}_{d-1}{\rm T}}$
$\displaystyle{\rm C}^{\prime}={\rm C}X.$ (199)

The $(d-1)$d domain wall (DW) Weyl fermion has the corresponding symmetry such
that ${\rm T}_{\rm DW}$ and ${\rm R}_{\rm DW}$ are broken, ${\rm C}_{\rm
DW}={\rm C}$ or ${\rm C}^{\prime}={\rm C}X$, and ${\rm C}_{\rm DW}{\rm R}_{\rm
DW}{\rm T}_{\rm DW}$ is unbroken, while there is also an extra internal
unitary $X$ symmetry (which generates $\mathbb{Z}_{4,X}$ or
$\mathbb{Z}_{2}^{{\rm F}}$ or 1). By Table 1, the matrices of C for Dirac
fermions in $d=3\mod 4$ and $d-1=2\mod 4$ are the same and they are block
diagonal in the chiral representation (see App. A). So the matrix of C for
Dirac fermions in $d=3\mod 4$ becomes the matrix of C for Weyl fermions in
$d-1=2\mod 4$ under the projection ${\rm P}_{\pm}=\frac{1\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}}{2}$.

  3. 3.

For Majorana fermions and $d=3\mod 8$, the matrix of ${\rm R}_{d-1}$ is
$\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}$ where $\Gamma^{d-1}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{3}\otimes I$ in the Majorana representation
(see App. B) and $I$ is an identity matrix. The matrix of $X$ is $\pm I$
(which generates a unitary $\mathbb{Z}_{2}^{{\rm F}}$ symmetry or trivial
symmetry) under the projection ${\rm P}_{\pm}=\frac{1\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{d-1}}{2}$. So $X=1$ up to $(-1)^{{\rm F}}$.

$\displaystyle d\text{-}\dim$ $\displaystyle\xrightarrow{\rm DW}$
$\displaystyle(d-1)\text{-}\dim{}$ $\displaystyle{\rm R}_{d-1}$
$\displaystyle\xrightarrow{{\rm P}_{\pm}}$ $\displaystyle X=1{}$
$\displaystyle{\rm R}_{i}$ $\displaystyle\xrightarrow{\cdot{\rm C}{\rm
R}_{d-1}{\rm T}}$ $\displaystyle{\rm C}{\rm R}_{i}{\rm T}\cdot X\;\;\;(i\neq
d-1){}$ $\displaystyle{\rm C}=1$ $\displaystyle\xrightarrow{{\rm P}_{\pm}}$
$\displaystyle{\rm C}=1{}$ $\displaystyle{\rm T}$
$\displaystyle\xrightarrow{\cdot{\rm C}{\rm R}_{d-1}{\rm T}}$
$\displaystyle{\rm C}^{\prime}={\rm C}X=1.$ (200)

The $(d-1)$d domain wall (DW) Majorana-Weyl fermion has the corresponding
symmetry such that ${\rm T}_{\rm DW}$ and ${\rm R}_{\rm DW}$ are broken, ${\rm
C}_{\rm DW}$ is trivial, and ${\rm C}_{\rm DW}{\rm R}_{\rm DW}{\rm T}_{\rm
DW}$ is unbroken. Here ${\rm C}=1$ because the C symmetry of a Majorana
fermion is trivial.

By comparing the results in Sec. III and Sec. IV, we conclude that for odd
$d$,

$\displaystyle d\text{-}\dim$ $\displaystyle\xrightarrow{\rm DW}$
$\displaystyle(d-1)\text{-}\dim{}$ $\displaystyle G_{{\rm D},i}(d\mod 8)$
$\displaystyle\to$ $\displaystyle G_{\rm W}(d-1\mod 8){}$ $\displaystyle
G_{\rm M}(3\mod 8)$ $\displaystyle\to$ $\displaystyle G_{\rm MW}(2\mod 8).$
(201)

Here $G_{{\rm D},i}(d\mod 8)$ $(i=1,2)$ are the two possibilities for the
C-R-T fractionalization with the canonical CRT for Dirac fermions in spacetime
dimensions $d$, $G_{\rm W}(d\mod 8)$ is the C or T fractionalization for Weyl
fermions in even spacetime dimensions $d$, $G_{\rm M}(d\mod 8)$ is the R-T
fractionalization for Majorana fermions in spacetime dimensions
$d=0,1,2,3,4\mod 8$, and $G_{\rm MW}(2\mod 8)=\mathbb{Z}_{2}^{{\rm F}}$ is the
trivial C fractionalization for Majorana-Weyl fermions in spacetime dimensions
$d=2\mod 8$. There is no domain wall dimensional reduction from $G_{\rm
M}(1\mod 8)$ because a Majorana fermion in $d=1\mod 8$ has to be massless
while the fermion in the bulk has to be massive.

If $d$ is even, the mass term does not break C, R, and T symmetries. First, we
consider Dirac fermions. Since $\Gamma^{d-1}$ is not block diagonal in the
chiral representation (see App. A), we replace $\Gamma^{d-1}$ with
${\Gamma^{d-1}}^{\prime}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt^{\frac{d}{2}}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{d-1}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt{\rm diag}(-I,I)$ which is block diagonal to do the
projection. Note that ${\Gamma^{d-1}}^{\prime}$ is $\Gamma^{d}$ in $(d+1)$d
chiral representation (see App. A). However, ${\Gamma^{d-1}}^{\prime}$ is
imaginary while $\Gamma^{d-1}$ is real. So the conditions (II.4) on the
matrices of C, R, and T imply that

$\displaystyle M_{{\rm C}}$ $\displaystyle=$ $\displaystyle\eta_{{\rm
C}}\left\\{\begin{array}[]{ll}\Gamma^{0}\Gamma^{1}\Gamma^{3}\cdots\Gamma^{d-3}&d=0\mod
4\\\ \Gamma^{2}\Gamma^{4}\cdots\Gamma^{d-2}{\Gamma^{d-1}}^{\prime}&d=2\mod
4\end{array}\right.{}$ (204) $\displaystyle M_{{\rm T}}$ $\displaystyle=$
$\displaystyle\eta_{{\rm
T}}\left\\{\begin{array}[]{ll}\Gamma^{0}\Gamma^{2}\Gamma^{4}\cdots\Gamma^{d-2}{\Gamma^{d-1}}^{\prime}&d=0\mod
4\\\ \Gamma^{1}\Gamma^{3}\cdots\Gamma^{d-3}&d=2\mod 4\end{array}\right.{}$
(207) $\displaystyle M_{{\rm R}_{i}}$ $\displaystyle=$
$\displaystyle\eta_{{\rm
R}_{i}}\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{i}\cdots{\Gamma^{d-1}}^{\prime}.$
(208)

These matrices (except $M_{{\rm R}_{d-1}}$) are block diagonal in the chiral
representation (see App. A). Under the projection ${\rm
P}_{\pm}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\Gamma^{d-1}}^{\prime}}{2}$, these matrices become

$\displaystyle M_{{\rm C}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\eta_{{\rm
C}}\left\\{\begin{array}[]{ll}\pm(-1)^{\frac{d}{4}}\gamma^{1}\gamma^{3}\cdots\gamma^{d-3}&d=0\mod
4\\\ \mp\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-1)^{\frac{d-2}{4}}\gamma^{2}\gamma^{4}\cdots\gamma^{d-2}&d=2\mod
4\end{array}\right.{}$ (211) $\displaystyle M_{{\rm T}}^{d-1,\pm}$
$\displaystyle=$ $\displaystyle\eta_{{\rm
T}}\left\\{\begin{array}[]{ll}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-1)^{\frac{d}{4}}\gamma^{2}\gamma^{4}\cdots\gamma^{d-2}&d=0\mod 4\\\
(-1)^{\frac{d-2}{4}}\gamma^{1}\gamma^{3}\cdots\gamma^{d-3}&d=2\mod
4\end{array}\right.{}$ (214) $\displaystyle M_{{\rm R}_{i}}^{d-1,\pm}$
$\displaystyle=$ $\displaystyle\eta_{{\rm R}_{i}}\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-1)^{\frac{d}{2}}\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2}\;\;\;(i\neq
d-1).$ (215)

The symmetry ${\rm R}_{d-1}$ flips the spatial coordinate $x_{d-1}$, it is
possible to be reduced to an internal unitary symmetry $X$ on the domain wall.
The matrix of ${\rm R}_{d-1}$ is $\eta_{{\rm
R}_{d-1}}\Gamma^{0}\Gamma^{1}\cdots\Gamma^{d-2}$ which is not block diagonal
in the chiral representation (see App. A), so $X$ is broken on the domain
wall.

Therefore, for Dirac fermions and even $d$, the domain wall dimensional
reduction of the symmetries C, R, and T obeys the following rule:

$\displaystyle d\text{-}\dim$ $\displaystyle\xrightarrow{\rm DW}$
$\displaystyle(d-1)\text{-}\dim{}$ $\displaystyle{\rm R}_{d-1}$
$\displaystyle\xrightarrow{{\rm P}_{\pm}}$ $\displaystyle 1{}$
$\displaystyle{\rm R}_{i}$ $\displaystyle\xrightarrow{{\rm P}_{\pm}}$
$\displaystyle{\rm R}_{i}\;\;\;(i\neq d-1){}$ $\displaystyle{\rm C}$
$\displaystyle\xrightarrow{{\rm P}_{\pm}}$ $\displaystyle{\rm C}{}$
$\displaystyle{\rm T}$ $\displaystyle\xrightarrow{{\rm P}_{\pm}}$
$\displaystyle{\rm T}.$ (216)

The $(d-1)$d domain wall (DW) Dirac fermion has the corresponding symmetry
such that ${\rm T}_{\rm DW}={\rm T}$, ${\rm R}_{\rm DW}={\rm R}$, and ${\rm
C}_{\rm DW}={\rm C}$. Namely, the C, R, and T symmetries on the domain wall
are induced from the C, R, and T symmetries in the bulk.

Based on the composition rules (III) and $\eqref{eq:composition-matrix2}$, we
do the following computation.

  1. 1.

For Dirac fermions and $d=0\mod 4$,

$\displaystyle{\rm C}_{\rm DW}^{2}\psi_{\pm}^{d-1}{\rm C}_{\rm DW}^{-2}$
$\displaystyle=$
$\displaystyle(\gamma^{1}\gamma^{3}\cdots\gamma^{d-3})^{2}\psi_{\pm}^{d-1}{}$
(217) $\displaystyle=$ $\displaystyle(-1)^{\frac{d}{4}-1}\psi_{\pm}^{d-1}.$
$\displaystyle{\rm T}_{\rm DW}^{2}\psi_{\pm}^{d-1}{\rm T}_{\rm DW}^{-2}$
$\displaystyle=$
$\displaystyle(\gamma^{2}\gamma^{4}\cdots\gamma^{d-2})^{2}(-1)^{\frac{d}{2}-1}\psi_{\pm}^{d-1}{}$
(218) $\displaystyle=$ $\displaystyle(-1)^{\frac{d}{4}}\psi_{\pm}^{d-1}.$
$\displaystyle{\rm R}_{{\rm DW},i}^{2}\psi_{\pm}^{d-1}{\rm R}_{{\rm
DW},i}^{-2}$ $\displaystyle=$ $\displaystyle\eta_{{\rm
R}_{i}}^{2}(-1)(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})^{2}\psi_{\pm}^{d-1}{}$
(219) $\displaystyle=$ $\displaystyle\eta_{{\rm
R}_{i}}^{2}(-1)(-1)^{\frac{(d-4)(d-3)}{2}}\psi_{\pm}^{d-1}{}$ $\displaystyle=$
$\displaystyle-\eta_{{\rm R}_{i}}^{2}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm C}_{\rm DW}{\rm R}_{{\rm DW},i})^{2}\psi_{\pm}^{d-1}({\rm
C}_{\rm DW}{\rm R}_{{\rm DW},i})^{-2}$ $\displaystyle=$
$\displaystyle(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2}\gamma^{1}\gamma^{3}\cdots\gamma^{d-3})(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2}\gamma^{1}\gamma^{3}\cdots\gamma^{d-3})^{*}\psi_{\pm}^{d-1}{}$
(220) $\displaystyle=$ $\displaystyle(-1)^{\frac{d}{4}-1}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm R}_{{\rm DW},i}{\rm T}_{\rm DW})^{2}\psi_{\pm}^{d-1}({\rm
R}_{{\rm DW},i}{\rm T}_{\rm DW})^{-2}$ $\displaystyle=$
$\displaystyle(\gamma^{2}\gamma^{4}\cdots\gamma^{d-2}\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})^{*}(\gamma^{2}\gamma^{4}\cdots\gamma^{d-2}\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})\psi_{\pm}^{d-1}{}$
(221) $\displaystyle=$ $\displaystyle(-1)^{\frac{d}{4}-1}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm C}_{\rm DW}{\rm T}_{\rm DW})^{2}\psi_{\pm}^{d-1}({\rm
C}_{\rm DW}{\rm T}_{\rm DW})^{-2}$ $\displaystyle=$ $\displaystyle(\eta_{{\rm
C}}^{*}\eta_{{\rm
T}}^{*})^{2}(-1)(\gamma^{1}\gamma^{2}\cdots\gamma^{d-2})^{2}\psi_{\pm}^{d-1}{}$
(222) $\displaystyle=$ $\displaystyle(\eta_{{\rm C}}^{*}\eta_{{\rm
T}}^{*})^{2}(-1)(-1)^{\frac{(d-3)(d-2)}{2}}\psi_{\pm}^{d-1}{}$
$\displaystyle=$ $\displaystyle(\eta_{{\rm C}}^{*}\eta_{{\rm
T}}^{*})^{2}\psi_{\pm}^{d-1}.$

Recall that $\eta_{{\rm R}_{i}}=\pm 1$ and $\eta_{{\rm C}}\eta_{{\rm T}}=\pm
1$ in the first possibility of the C-R-T fractionalization with the canonical
CRT for Dirac fermions, while $\eta_{{\rm R}_{i}}=\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt$ and $\eta_{{\rm C}}\eta_{{\rm T}}=\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt$ in the second possibility of the C-R-T
fractionalization with the canonical CRT for Dirac fermions. Therefore, the
first (second) possibility of the C-R-T fractionalization with the canonical
CRT for Dirac fermions in $d=0\mod 4$ is reduced to the first (second)
possibility of the C-R-T fractionalization with the canonical CRT for Dirac
fermions in $d-1=3\mod 4$, see Table 10 and Table 11.

  2. 2.

For Dirac fermions and $d=2\mod 4$,

$\displaystyle{\rm C}_{\rm DW}^{2}\psi_{\pm}^{d-1}{\rm C}_{\rm DW}^{-2}$
$\displaystyle=$
$\displaystyle(\gamma^{2}\gamma^{4}\cdots\gamma^{d-2})^{2}(-1)^{\frac{d}{2}-1}\psi_{\pm}^{d-1}{}$
(223) $\displaystyle=$ $\displaystyle(-1)^{\frac{d-2}{4}}\psi_{\pm}^{d-1}.$
$\displaystyle{\rm T}_{\rm DW}^{2}\psi_{\pm}^{d-1}{\rm T}_{\rm DW}^{-2}$
$\displaystyle=$
$\displaystyle(\gamma^{1}\gamma^{3}\cdots\gamma^{d-3})^{2}\psi_{\pm}^{d-1}{}$
(224) $\displaystyle=$ $\displaystyle(-1)^{\frac{d-2}{4}}\psi_{\pm}^{d-1}.$
$\displaystyle{\rm R}_{{\rm DW},i}^{2}\psi_{\pm}^{d-1}{\rm R}_{{\rm
DW},i}^{-2}$ $\displaystyle=$ $\displaystyle\eta_{{\rm
R}_{i}}^{2}(-1)(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})^{2}\psi_{\pm}^{d-1}{}$
(225) $\displaystyle=$ $\displaystyle\eta_{{\rm
R}_{i}}^{2}(-1)(-1)^{\frac{(d-4)(d-3)}{2}}\psi_{\pm}^{d-1}{}$ $\displaystyle=$
$\displaystyle\eta_{{\rm R}_{i}}^{2}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm C}_{\rm DW}{\rm R}_{{\rm DW},i})^{2}\psi_{\pm}^{d-1}({\rm
C}_{\rm DW}{\rm R}_{{\rm DW},i})^{-2}$ $\displaystyle=$
$\displaystyle(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2}\gamma^{2}\gamma^{4}\cdots\gamma^{d-2})(\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2}\gamma^{2}\gamma^{4}\cdots\gamma^{d-2})^{*}\psi_{\pm}^{d-1}{}$
(226) $\displaystyle=$ $\displaystyle(-1)^{\frac{d-2}{4}-1}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm R}_{{\rm DW},i}{\rm T}_{\rm DW})^{2}\psi_{\pm}^{d-1}({\rm
R}_{{\rm DW},i}{\rm T}_{\rm DW})^{-2}$ $\displaystyle=$
$\displaystyle(\gamma^{1}\gamma^{3}\cdots\gamma^{d-3}\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})^{*}(\gamma^{1}\gamma^{3}\cdots\gamma^{d-3}\gamma^{1}\cdots\hat{\gamma}^{i}\cdots\gamma^{d-2})\psi_{\pm}^{d-1}{}$
(227) $\displaystyle=$ $\displaystyle(-1)^{\frac{d-2}{4}}\psi_{\pm}^{d-1}.$

$\displaystyle({\rm C}_{\rm DW}{\rm T}_{\rm DW})^{2}\psi_{\pm}^{d-1}({\rm
C}_{\rm DW}{\rm T}_{\rm DW})^{-2}$ $\displaystyle=$ $\displaystyle(\eta_{{\rm
C}}^{*}\eta_{{\rm
T}}^{*})^{2}(-1)(\gamma^{1}\gamma^{2}\cdots\gamma^{d-2})^{2}\psi_{\pm}^{d-1}{}$
(228) $\displaystyle=$ $\displaystyle(\eta_{{\rm C}}^{*}\eta_{{\rm
T}}^{*})^{2}(-1)(-1)^{\frac{(d-3)(d-2)}{2}}\psi_{\pm}^{d-1}{}$
$\displaystyle=$ $\displaystyle-(\eta_{{\rm C}}^{*}\eta_{{\rm
T}}^{*})^{2}\psi_{\pm}^{d-1}.$

Recall that $\eta_{{\rm R}_{i}}=\pm 1$ and $\eta_{{\rm C}}\eta_{{\rm T}}=\pm
1$ in the first possibility of the C-R-T fractionalization with the canonical
CRT for Dirac fermions, while $\eta_{{\rm R}_{i}}=\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt$ and $\eta_{{\rm C}}\eta_{{\rm T}}=\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt$ in the second possibility of the C-R-T
fractionalization with the canonical CRT for Dirac fermions. Therefore, the
first (second) possibility of the C-R-T fractionalization with the canonical
CRT for Dirac fermions in $d=2\mod 4$ is reduced to the second (first)
possibility of the C-R-T fractionalization with the canonical CRT for Dirac
fermions in $d-1=1\mod 4$, see Table 10 and Table 11.

Next, we consider Majorana fermions and $d=2,4\mod 8$.

  1. 1.

For Majorana fermions and $d=4\mod 8$, we choose imaginary Gamma matrices.
Note that $\Gamma^{d-1}$ is block diagonal in the Majorana representation (see
App. B). By the results in Sec. IV, we have $M_{{\rm T}}=\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{1}\Gamma^{2}\cdots\Gamma^{d-1}$ and
$M_{{\rm R}_{i}}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{i}\cdots\Gamma^{d-1}$. These
matrices (except $M_{{\rm R}_{d-1}}$) are block diagonal in the Majorana
representation (see App. B). For example, we consider the case $d=4$. Under
the projection ${\rm P}_{\pm}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{d-1}}{2}$, these matrices become (for $d=4$)

$\displaystyle M_{{\rm T}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2},{}$ $\displaystyle M_{{\rm
R}_{1}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\sigma^{1},{}$
$\displaystyle M_{{\rm R}_{2}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp\sigma^{3}.$ (229)

By comparison with the Majorana representation for $d-1=3$, we find that (up
to $\pm 1$)

$\displaystyle M_{{\rm T}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Gamma^{0}_{d-1},{}$ $\displaystyle M_{{\rm
R}_{i}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{i}_{d-1}\;\;\;(i\neq d-1).$ (230)

These are exactly the matrices of R and T for $d-1=3$. Therefore, the R-T
fractionalization for Majorana fermions in $d=4\mod 8$ is reduced to the R-T
fractionalization for Majorana fermions in $d-1=3\mod 8$.

  2. 2.

For Majorana fermions and $d=2\mod 8$, we can choose imaginary or real Gamma
matrices. However, we choose imaginary Gamma matrices because the Majorana
fermion in the bulk has to be massive. Since $\Gamma^{d-1}$ is not block
diagonal in the Majorana representation (see App. B), we replace
$\Gamma^{d-1}$ with ${\Gamma^{d-1}}^{\prime}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{3}\otimes I$ which is block diagonal to do the projection where
$I$ is an identity matrix. Note that ${\Gamma^{d-1}}^{\prime}$ is $\Gamma^{d}$
in $(d+1)$d Majorana representation (see App. B). By the results in Sec. IV,
we have $M_{{\rm T}}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{1}\Gamma^{2}\cdots{\Gamma^{d-1}}^{\prime}$ and $M_{{\rm
R}_{i}}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\Gamma^{0}\Gamma^{1}\cdots\hat{\Gamma}^{i}\cdots{\Gamma^{d-1}}^{\prime}$
if the Gamma matrices are imaginary. These matrices (except $M_{{\rm
R}_{d-1}}$) are block diagonal in the Majorana representation (see App. B).
For example, we consider the case $d=10$. Under the projection ${\rm
P}_{\pm}=\frac{1\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt{\Gamma^{d-1}}^{\prime}}{2}$, these matrices become (for $d=10$)

$\displaystyle M_{{\rm T}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp\sigma^{2}\otimes\sigma^{1}\otimes I\otimes\sigma^{2},{}$
$\displaystyle M_{{\rm R}_{1}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp\sigma^{1}\otimes I\otimes I\otimes\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2},{}$ $\displaystyle M_{{\rm
R}_{2}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\pm\sigma^{3}\otimes
I\otimes I\otimes\hskip 1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2},{}$
$\displaystyle M_{{\rm R}_{3}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}\otimes\sigma^{2}\otimes I\otimes\sigma^{2},{}$ $\displaystyle
M_{{\rm R}_{4}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\pm\hskip
1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}\otimes\sigma^{3}\otimes\sigma^{2}\otimes\sigma^{2},{}$
$\displaystyle M_{{\rm R}_{5}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}\otimes\sigma^{3}\otimes\sigma^{1}\otimes I,{}$ $\displaystyle
M_{{\rm R}_{6}}^{d-1,\pm}$ $\displaystyle=$ $\displaystyle\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\otimes\sigma^{3}\otimes\sigma^{3}\otimes
I,{}$ $\displaystyle M_{{\rm R}_{7}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\mp I\otimes I\otimes\hskip 1.0pt\mathrm{i}\hskip
1.0pt\sigma^{2}\otimes\sigma^{1},{}$ $\displaystyle M_{{\rm R}_{8}}^{d-1,\pm}$
$\displaystyle=$ $\displaystyle\pm I\otimes I\otimes\hskip
1.0pt\mathrm{i}\hskip 1.0pt\sigma^{2}\otimes\sigma^{3}.$ (231)

By comparison with the Majorana representation for $d-1=9$, we find that (up
to $\pm 1$)

$\displaystyle M_{{\rm T}}^{d-1,\pm}$ $\displaystyle=$
$\displaystyle\Gamma^{0}_{d-1},{}$ $\displaystyle M_{{\rm R}_{i}}^{d-1,\pm}$
$\displaystyle=$ $\displaystyle\Gamma^{i}_{d-1}\;\;\;(i\neq d-1).$ (232)

These are exactly the matrices of R and T for $d-1=9$. Therefore, the first
choice of the R-T fractionalization for Majorana fermions in $d=2\mod 8$ is
reduced to the R-T fractionalization for Majorana fermions in $d-1=1\mod 8$.