Lecture note on Clifford algebra

Abstract

This lecture note surveys the gamma matrices in general dimensions with arbitrary signatures, the study of which is essential to understand the supersymmetry in the corresponding spacetime. The contents supplement the lecture presented by the author at Modave Summer School in Mathematical Physics, Belgium, june, 2005.

Full text

Vol.:(0123456789)
Journal of the Korean Physical Society
https://doi.org/10.1007/s40042-022-00493-z
Vol.:(0123456789)
1 3
REVIEW - PARTICLES ANDNUCLEI
Lecture note onClifford algebra
Jeong‑HyuckPark1
Received: 7 March 2022 / Accepted: 5 April 2022
© The Korean Physical Society 2022
Abstract
This lecture note surveys the gamma matrices in general dimensions with arbitrary signatures, the study of which is essential
to understand the supersymmetry in the corresponding spacetime. The contents supplement the lecture presented by the
author at Modave Summer School in Mathematical Physics, Belgium, June, 2005.
Keywords Clifford algebra· Gamma matrix· Lecture note
1 Preliminary
Where do we see Clifford algebra?
• Dirac equation, for sure.
• Supersymmetry algebra.
• Non-anti-commutative superspace.
• Division algebra,
ℝ,ℂ,ℍ,𝕆
.
• Atiyah–Drinfeld–Hitchin–Manin construction of Yang–
Mills instantons,
F=±∗F
.
The gamma matrices in the Euclidean two-dimensions pro-
vide the fermionic oscillators:
where f=
1
2
(𝛾1+i𝛾2
)
,
̄
f=
1
2
(𝛾1−i𝛾2
)
. Consequently, the
irreducible representation is given uniquely by
2×2
matri-
ces acting on two dimensional spinors,
�+⟩
and
�−⟩
:
Higher dimensional gamma matrices are then constructed
by the direct products of them.
2 Gamma matrix
We start with the following Theorem on linear algebra.
Theorem
Any matrix,M, satisfying
M2=𝜆2≠0
,
𝜆∈𝐂
is diago-
nalizable, and furthermore, if there is another invertible
matrix, N, which anti-commutes with M,
{N,M}=0
, then
M is
2n×2n
matrix of the form:
In particular,
tr M=0
. See Sect.A for our proof.
2.1 In even dimensions
In even
d=t+s
dimensions, with metric1
gamma matrices,
𝛾𝜇
, satisfy the Clifford algebra:
With2
(1.1)
f2=0,
̄
f2=0, {f,
̄
f}=1,
(1.2)
f=

−

+

=

00
10

,
̄
f=

+

−

=

01
00
.
(2.1)
M
=S
(
𝜆0
0−𝜆
)
S−1
.
(2.2)
 =
diag
(+ +
⋯
+

t
−
⋯
−

s
),
(2.3)
𝛾𝜇𝛾𝜈+𝛾𝜈𝛾𝜇=2𝜂𝜇𝜈 .
(2.4)
𝛾𝜇
1
𝜇
2⋯
𝜇
m
=𝛾[𝜇
1
𝛾𝜇
2⋯
𝛾𝜇
m
],
Online ISSN 1976-8524
Print ISSN 0374-4884
* Jeong-Hyuck Park
park@sogang.ac.kr
1 Department ofPhysics, Sogang University, 35 Baekbeom-ro,
Mapo-gu, Seoul04107, Korea
1 Note that throughout the lecture note we adopt the field theorists’
convention rather than string theorists, such that the time directions
have the positive signature. The conversion is straightforward:
2 “[]” means the standard anti-symmetrization with “strength one”:

J.-H.Park
Vol.:(0123456789)
1 3
we define
ΓM,M=1, 2, …2d
by assigning numbers to inde-
pendent
𝛾𝜇
1
𝜇
2
…𝜇m
, e.g.,imposing
𝜇1<𝜇
2<
⋯
<𝜇
m
:
Then,
{ΓM}∕
Z
2
forms a group:
where L is a function of M, N and
ΩMN =±1
does not
depend on the specific choice of representation of the
gamma matrices.
Theorem(2.1) implies
which shows the linear independence of
{ΓM}
, so that any
gamma matrix should not be smaller than
2
d
∕
2
×2
d
∕2
.
In two-dimensions, one can take the Pauli sigma matrices,
𝜎1,𝜎2
as gamma matrices with a possible factor,i, depending
on the signature. In general, one can construct
d+2
dimen-
sional gamma matrices from d dimensional gamma matrices
by taking tensor products as
Thus, the smallest size of irreducible representations is
2d∕2×2d∕2
and
{ΓM}
forms a basis of
2d∕2×2d∕2
matrices.
By induction on the dimensions, from Eq.(2.8), we may
require gamma matrices to satisfy the hermiticity condition:
With this choice of gamma matrices we define
𝛾(d+1)
as
satisfying
For two sets of irreducible gamma matrices,
𝛾𝜇
,𝛾′𝜇
which
are
2n×2n,2n�×2n�
, respectively, we consider a matrix:
where T, is an arbitrary
2n�×2n
matrix.
This matrix satisfies for any N from Eq. (2.6):
(2.5)
ΓM=(1, 𝛾𝜇,𝛾𝜇𝜈 ,…,𝛾𝜇
1
𝜇
2⋯
𝜇
m
,…,𝛾12…d).
(2.6)
ΓMΓN=Ω
MN ΓL,ΩMN =±1,
(2.7)
1
2n
tr (ΓMΓN)=Ω
MN 𝛿MN
,
(2.8)
(𝛾𝜇⊗𝜎
1,1⊗𝜎
2,1⊗𝜎
3): up to a factor i.
(2.9)
𝛾
𝜇†=𝛾𝜇=
⎧
⎪
⎨
⎪
⎩
+𝛾𝜇for time-like 𝜇
−𝛾𝜇for space-like 𝜇
.
(2.10)
𝛾
(d+1)=
√
(−1)
t−s
2𝛾1𝛾2⋯𝛾d
,
(2.11)
𝛾(d+1)
=(𝛾
(d+1)
)
−1
=𝛾
(d+1)† ,
{𝛾
𝜇
,𝛾
(d+1)
}=0.
(2.12)
S
=
∑
M
Γ�MT(ΓM)−1
,
(2.13)
Γ�NS=SΓN
.
By Schur’s Lemmas, it should be either
S=0
or
n=n�, det S≠0
. Furthermore, S is unique up to constant,
although T is arbitrary. This implies the uniqueness of the
irreducible
2d∕2×2d∕2
gamma matrices in even d dimen-
sions, up to the similarity transformations. These similar-
ity transformations are also unique up to constant. Conse-
quently there exist similarity transformations which relate
𝛾𝜇
to
𝛾𝜇†
,𝛾𝜇∗
,𝛾𝜇T
, since the latter form also representations
of the Clifford algebra. By combining
𝛾(d+1)
with the simi-
larity transformations, from Eq. (2.11), we may acquire the
opposite sign,
−𝛾𝜇†
,−𝛾𝜇∗
,−𝛾𝜇T
as well.
Explicitly we define3
satisfying
If we write
then from
one can normalize
B±
to satisfy[2, 3]
where the unitarity follows from
and the positive definiteness of
B†
±
B
±
. The calculation of
𝜀±
is essentially counting the dimensions of symmetric and
anti-symmetric matrices[2, 3]4.
(2.14)
A
=
√
(−1)
t(t−1)
2
𝛾
1
𝛾
2⋯
𝛾
t
,
(2.15)
A=A−1=A†,
(2.16)
𝛾𝜇†= (−1)t+1A𝛾𝜇A−1.
(2.17)
±
𝛾
𝜇∗
=B
±
𝛾
𝜇
B
−1
±,
(2.18)
𝛾𝜇
=(𝛾
𝜇∗
)
∗
=B
∗
±
B
±
𝛾
𝜇
(B
∗
±
B
±
)
−1,
(2.19)
B
∗
±
B
±
=𝜀
±
1, 𝜀
±
= (−1)
1
8(s−t)(s−t±2)
,
(2.20)
B†
±
B
±
=
1,
(2.21)
BT
±
=𝜀
±
B
±,
(2.22)
𝛾𝜇
=𝛾
†
𝜇
= (±B
−1
±
𝛾
∗
𝜇
B±)
†
=±B
†
±
𝛾
𝜇∗
(B
†
±
)
−1
=B
†
±
B±𝛾
𝜇
(B
†
±
B±)
−1,
3 Alternatively, one can construct
C±
explicitly out of the gamma
matrices in a certain representation[1]:
4 From (2.24) we have
(
C
±
𝛾
𝜇
1
𝜇
2⋯
𝜇
n
)T=
𝜒
n±
C
±
𝛾
𝜇
1
𝜇
2⋯
𝜇n
,
𝜒n±∶=
𝜀
±(±
1
)
t+n
(−
1
)
n+
1
2(t+n)(t+n−1
)
(2.29). Thus, one can obtain the
dimension of the symmetric
2
d∕2
×2
d∕
2
matrices as
From this one can obtain the value of
𝜀±
(2.19).
2
d∕2−1(2d∕2+1)=
d
∑
n=0
1
2(1+𝜒n±)d!
n!(d−n)!
.

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
What is worthy of notice is the case
𝜀±=+1
. As we see
later in (4.4), (4.5), if
𝜀+=+1
, the gamma matrices can be
chosen to real, i.e.,
B+=1
, while if
𝜀−=+1
, the gamma
matrices can be chosen to pure imaginary, i.e.,
B−
=1
. Espe-
cially when the gamma matrices are real we say they are in
the Majorana representation.
The charge conjugation matrix,
C±
, given by
satisfies5 from the properties of A and
B±
:
The sign factors
𝜀±
and
𝜁
are related by
Equations (2.24,2.26) imply
𝛾(d+1)
satisfies
where
{
A
+
,A
−
}={A,
𝛾(d+1)
A
}
.
In stead of Eq. (2.8) one can construct
d+2
dimensional
gamma matrices from d dimensional gamma matrices by
taking tensor products as
Therefore, the gamma matrices in even dimensions can be
chosen to have the “off-block diagonal” form:
(2.23)
C±
=B
T
±
A
,
(2.24)
C±
𝛾
𝜇
C
−1
±
=𝜁𝛾
𝜇T
,𝜁= ±(−1)
t+1,
(2.25)
C†
±
C
±
=
1,
(2.26)
C
T
±
= (−1)
1
8d(d−𝜁2)C
±
=𝜀
±
(±1)t(−1)
1
2t(t−1)C
±,
(2.27)
𝜁
t(−1)
1
2t(t−1)AT=B
±
AB−1
±
=C
±
AC−1
±.
(2.28)
𝜀±
=
𝜁
t(−1)
1
2t(t−1)+
1
8d(d−𝜁2)
.
(2.29)
(
C±𝛾𝜇1𝜇2⋯𝜇n)T=𝜁n(−1)
1
8d(d−𝜁2)+
1
2n(n−1)C±𝛾𝜇1𝜇2⋯𝜇n
=
𝜀±
(±1)t+n(−1)n+1
2(t+n)(t+n−1)C
±𝛾
𝜇1𝜇2⋯𝜇n
.
(2.30)
𝛾(d+1)†
= (−1)
t
A±𝛾
(d+1)
A
−1
±=𝛾
(d+1),
𝛾
(d+1)∗ = (−1)
t−s
2B±𝛾(d+1)B−1
±,
𝛾
(d+1)T= (−1)
t+s
2C
±
𝛾(d+1)C−1
±
,
(2.31)
(𝛾𝜇⊗𝜎
1,𝛾(d+1)⊗𝜎
1,1⊗𝜎
2): up to a factor
i
.
where the
2
d
2
−1
×2
d
2
−
1
matrices,
𝜎𝜇
,̃𝜎 𝜇
satisfy
In this choice of gamma matrices, from Eq. (2.30),
A±,B±,C±
are either “block diagonal” or “off-block diag-
onal” depending on whether
t
,
t−s
2
,
t+s
2
are even or odd,
respectively.
In particular, in the case of odd t, we write from Eqs.
(2.14,2.15) A as
and in the case of odd
t+s
2
we write from Eq. (2.26)
C±
as
where
a,
̃
a,c,
̃
c
satisfy from Eqs. (2.16,2.24):
If both of t and
t+s
2
are odd then from Eq. (2.27):
2.2 In Odd dimensions
The gamma matrices in odd
d+1=t+s
dimensions are
constructed by combining a set of even d dimensional
gamma matrices with either
±𝛾(d+1)
or
±i𝛾(d+1)
depending
on the signature of even d dimensions. This way of construc-
tion is general, since
𝛾(d+1)
serves the role of
𝛾d+1
:
and such a matrix is unique in irreducible representations
up to sign.
However, contrary to the even dimensional Clifford algebra,
in odd dimensions two different choices of the signs in
𝛾d+1
bring
(2.32)
𝛾
𝜇=
(
0𝜎
𝜇
̃𝜎 𝜇0
)
,𝛾(d+1)=
(
10
0−1
),
(2.33)
𝜎𝜇̃𝜎 𝜈+𝜎𝜈̃𝜎 𝜇=2𝜂𝜇𝜈 ,
(2.34)
𝜎𝜇†
=̃𝜎
𝜇.
(2.35)
A
=
(
0a
̃
a0
)
,a=
√
(−1)
t(t−1)
2𝜎1̃𝜎 2⋯𝜎t=̃
a†=̃
a−1
,
(2.36)
C
±=
(
0c
±̃
c0
)
,c=𝜀+(−1)
t(t−1)
2̃
cT= (c†)−1
,
(2.37)
𝜎
𝜇
†
=̃
a𝜎𝜇̃
a,̃𝜎 𝜇
†
=ã𝜎 𝜇a,
𝜎
𝜇T
= (−1)
t+1
̃
c𝜎
𝜇
c
−1
,̃𝜎
𝜇T
= (−1)
t+1
c̃𝜎
𝜇
̃
c
−1
.
(2.38)
a
T
= (−1)
t−1
2
̃
cac
−1
,̃
a
T
= (−1)
t−1
2
c̃
ã
c
−1
.
(2.39)
−
𝛾𝜇=𝛾
d+1
𝛾𝜇(𝛾
d+1
)
−1
, for 𝜇=1, 2, …,d
,
(𝛾
d+1
)
2
=±1,
5 Essentially all the properties of the charge conjugation matrix,
C±
depends only on d and
𝜁
. However, it is useful here to have expression
in terms of the signature to dicuss the Majorana supersymmetry later.

J.-H.Park
Vol.:(0123456789)
1 3
two irreducible representations for the Clifford algebra, which
can not be mapped to each other6 by similarity transformations:
If there were a similarity transformation between these two,
it should have been identity up to constant because of the
uniqueness of the similarity transformation in even dimen-
sions. Clearly this would be a contradiction due to the pres-
ence of the two opposite signs in
𝛾d+1
.
In general, one can put7

2
d
∕2×2
d
∕2
gamma matrices in odd
d+1
dimensions,
𝛾𝜇,𝜇=1, 2, …,d+1
, induce the following basis of
2
d
∕2×2
d
∕2
matrices,
̃
ΓM
:
From Eq. (2.41)
Here, contrary to the even dimensional case,
̃
ΩMN
depends
on each particular choice of the representations due to the
arbitrary sign factor in
𝛾d+1
. This is why Eq. (2.13) does not
hold in odd dimensions. Therefore, it is not peculiar that
not all of
±𝛾𝜇†
,±𝛾𝜇∗
,±𝛾𝜇T
are related to
𝛾𝜇
by similarity
transformations. In fact, if it were true, say for
±𝛾𝜇∗
, then the
similarity transformation should have been
B±
(2.17) by the
uniqueness of the similarity transformations in even dimen-
sions, but this would be a contradiction to Eq. (2.30), where
the sign does not alternate under the change of
B+
↔
B−
.
Thus, in odd dimensions, only the half of
±𝛾𝜇†
,±𝛾𝜇∗
,±𝛾𝜇T
are related to
𝛾𝜇
by similarity transformations and hence
from Eq. (2.30) there exist three similarity transformations,
A,B,C, such that
(2.40)
𝛾𝜇=(𝛾1,𝛾2,…,𝛾d+1)and 𝛾�𝜇=(𝛾1,𝛾2,
⋯
,𝛾d,−𝛾d+1).
(2.41)
𝛾
d+1=
⎧
⎪
⎨
⎪
⎩
±𝛾12⋯dfor t−s
≡1 mod 4 ,
±i𝛾12⋯dfor t−s≡
3 mod 4 .
(2.42)
̃
Γ
M
=(1, 𝛾
𝜇
,𝛾
𝜇𝜈
,
⋯
,𝛾𝜇
1
𝜇
2⋯
𝜇
d∕2
),M=1, 2, …2
d
.
(2.43)
̃
ΓM
̃
ΓN=
̃
ΩMN
̃
ΓL,
̃
Ω
MN =
⎧
⎪
⎨
⎪
⎩
±1 for t−s≡1 mod 4 ,
±1, ±iFor t−s≡
3 mod 4 .
A,B,C
are all unitary and satisfy
In particular, A is given by Eq. (2.14).
2.3 Lorentz transformations
Lorentz transformations,L can be represented by the follow-
ing action on gamma matrices in a standard way:
where L and
L
are given by
For even d, if a
2d∕2×2d∕2
matrix,
M𝜇
1
𝜇
2⋯
𝜇n
, is totally anti-
symmetric over the n spacetime indices:
and transforms covariantly under Lorentz transformations in
d or
d+1
dimensions as
then for
0≤n≤max(d∕2, 2)
, the general forms of
M𝜇
1
𝜇
2⋯
𝜇n
are
where c is a constant.
To show this, one may first expand
M𝜇
1
𝜇
2⋯
𝜇n
in terms
of
𝛾𝜈1𝜈2
⋯
𝜈m
,𝛾
(d+1)
𝛾
𝜈1𝜈2
⋯
𝜈m
or
𝛾𝜈1𝜈2
⋯
𝜈m
depending on the
(2.44)
(−1)t+1𝛾𝜇†=A𝛾𝜇A−1,
(2.45)
(−1)
t−s−1
2
𝛾
𝜇∗
=B𝛾
𝜇
B
−1
,
(2.46)
(−1)
t+s−1
2
𝛾
𝜇T
=C𝛾
𝜇
C
−1
.
(2.47)
A=A−1=A†,C=BTA,
(2.48)
B
∗
B=𝜀1= (−1)
1
8(t−s+1)(t−s−1)
1,
(2.49)
B
T
=𝜀B,C
T
=𝜀(−1)
ts
2
C= (−1)
1
8(t+s+1)(t+s−1)
C,
(2.50)
(−1)
ts
2
A
T
=BAB
−1
=CAC
−1
.
(2.51)
L−1
𝛾
𝜇
L=L
𝜇
𝜈
𝛾
𝜈,
(2.52)
L
=ew𝜇𝜈M𝜇𝜈 ,L=e
1
2w𝜇𝜈𝛾𝜇𝜈
,
(M𝜇𝜈)𝜆
𝜌
=𝜂𝜇𝜆𝛿𝜈
𝜌
−𝜂𝜈𝜆𝛿𝜇
𝜌.
(2.53)
M𝜇
1
𝜇
2⋯
𝜇
n
=M[𝜇
1
𝜇
2⋯
𝜇
n
],
(2.54)
L
−1M𝜇1𝜇2⋯𝜇nL=
n
∏
i=1
L𝜇i
𝜈i
M𝜈1𝜈2⋯𝜈n
,
(2.55)
M
𝜇1𝜇2⋯𝜇n=
⎧
⎪
⎨
⎪
⎩
(1+c𝛾(d+1))𝛾𝜇1𝜇2⋯𝜇nIn even ddimensions ,
𝛾𝜇1𝜇2⋯𝜇nIn odd d+
1 dimensions ,
6 Nevertheless, this can be cured by the following transforma-
tion. Under x
𝜇=(
x
1,
x
2,…,
x
d+1)
→x
�𝜇=(
x
1,
x
2,…,−
x
d+1)
, we
transform the Dirac field
𝜓(x)
as
𝜓(
x
)
→
𝜓�(
x
�)=𝜓(
x
),
to get
̄𝜓 (
x
)𝛾
⋅
𝜕𝜓(
x
)
→
̄𝜓 �(
x
�)𝛾�
⋅
𝜕�𝜓�(
x
�)= ̄𝜓 (
x
)𝛾
⋅
𝜕𝜓(
x
).
Hence, those
two representations are equivalent describing the same physical sys-
tem.
7 Our results (2.41–2.50) do not depend on the choice of the sig-
nature in d dimensions, i.e., they hold for either increasing the time
dimensions,
d=(t−1)+s
or the space dimensions,
d=t+(s−1)
:

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
dimensions,d or
d+1
, with
0≤m≤d∕2
. Then Eq. (2.54)
implies that the coefficients of them, say
T𝜇
1
𝜇
2
…𝜇
m+
n
, are Lor-
entz invariant tensors satisfying
Finally, one can recall the well known fact[4] that the gen-
eral forms of Lorentz invariant tensors are multi-products of
the metric,
𝜂𝜇𝜈
, and the totally antisymmetric tensor,
𝜖𝜇
1
𝜇
2
…
,
which verifies Eq.(2.55).
2.4 Crucial identities forsuper Yang–Mills
The following identities are crucial to show the existence of
the non-Abelian super Yang–Mills in THREE, FOUR, SIX
and TEN dimensions.
(i) The following identity holds only in THREE or
FOUR dimensions with arbitrary signature:
To verify the identity in even dimensions we con-
tract
(
𝛾
𝜇
C
−1
)
𝛼𝛽
(𝛾
𝜇
)
𝛾𝛿
with
(C𝛾𝜈
1
𝜈
2⋯
𝜈
n
)𝛽𝛼
and take
cyclic permutations of
𝛼,𝛽,𝛾
to get
This equation must be satisfied for all
0≤n≤d
,
which is valid only in
d=4, 𝜁=−1
. Similar analysis
can be done for the
d+1
odd dimensions by adding
(
𝛾
(d+1)
C
−1
)
𝛼𝛽
(𝛾
(d+1)
C
−1
)
𝛾𝛿
term into Eq.(2.57). We
get
Only in
d=2
and hence three dimensions, this equa-
tion is satisfied for all
0≤n≤d
.
(ii) The following identity holds only in TWO, FOUR or
SIX dimensions with arbitrary signature:
To verify this identity we take d dimensional sigma
matrices from
f=d−2
dimensional gamma matri-
ces as in Eq.(2.31):
to get
(2.56)
m+n
∏
i=1
L𝜇i
𝜈i
T𝜈1𝜈2⋯𝜈m+n=T𝜇1𝜇2…𝜇m+
n
(2.57)
0
=(𝛾
𝜇
C
−1
)
𝛼𝛽
(𝛾
𝜇
C
−1
)
𝛾𝛿
+cyclic permutations of 𝛼,𝛽,𝛾
(2.58)
0
=2d∕2𝛿n
1
+(d−2n)(𝜁+𝜁n(−1)
1
2n(n−1))(−1)n+
1
8d(d−𝜁2
)
(2.59)
0
=2
d∕2
(𝛿
n
1+𝛿
n
d)+(d−2n+1)
(𝜁+𝜁
n
(−1)
1
2n(n−1)
)(−1)
n+1
8d(d−𝜁2)
,𝜁= (−1)
d∕
2
(2.60)
0=(𝜎𝜇)𝛼𝛽 (𝜎𝜇)𝛾𝛿 +(𝜎𝜇)𝛾𝛽 (𝜎𝜇)𝛼𝛿
(2.61)
𝜎𝜇=(𝛾𝜇,𝛾(f+1),i)
Again this expression is valid for any signature, (t,s).
Now we contract this equation with
(
𝛾
𝜈
1
𝜈
2⋯
𝜈
nC
−1
+
)
𝛽𝛿
.
From Eqs. (2.24,2.30) in the case of odd t, we get
To satisfy Eq. (2.60) this expression must be
anti-symmetric over
𝛼↔𝛾
for any
0≤n≤f
.
Thus from Eq. (2.29) we must require
0= (−1)
n
(f−2n)+(−1)
f
2+n
−1
for all n satisfying
(−1)
1
8f(f−2)+
1
2n(n−1)
=1
. This condition is satisfied
only in
f=0, 2, 4
and hence
d=2, 4, 6
(
f=6
case
is excluded by choosing
n=6
and
f≥8
cases are
excluded by choosing either
n=0
or
n=3
).
(iii) The following identity holds only in TWO or TEN
dimensions with arbitrary signature:
3 Spinors
3.1 Weyl spinor
In any even d dimensions, Weyl spinor,
𝜓
, satisfies
and so
̄𝜓 =𝜓†A
satisfies from Eq.(2.30):
3.2 Majorana Spinor
By definition Majorana spinor satisfies
depending on the dimensions, even or odd. This is possible
only if
𝜀±,𝜀=1
and so from Eqs. (2.19,2.48):
where
𝜂
is the sign factor,
±1
, occurring in Eq.(2.17) or
Eq.(2.45)8.
(2.62)
(
𝜎
𝜇
)
𝛼𝛽
(𝜎
𝜇
)
𝛾𝛿
=(𝛾
𝜇
)
𝛼𝛽
(𝛾
𝜇
)
𝛾𝛿
+(𝛾
(f+1)
)
𝛼𝛽
(𝛾
(f+1)
)
𝛾𝛿
−𝛿
𝛼𝛽
𝛿
𝛾𝛿
(2.63)
(
(−1)n(f−2n) + (−1)
f
2+n−1
)
(𝛾𝜈1𝜈2⋯𝜈nC−1
+)
𝛼𝛾
(2.64)
0
=(𝜎
𝜇
c
−1
)
𝛼𝛽
(𝜎
𝜇
c
−1
)
𝛾𝛿
+cyclic permutations of 𝛼,𝛽,𝛾
(3.1)
𝛾(d+1)𝜓=𝜓
(3.2)
̄𝜓 𝛾
(d+1)= (−1)t̄𝜓 𝛾 (d+1)C−1
±
̄𝜓 T= (−1)
t−s
2C−1
±
̄𝜓 T
(3.3)
̄𝜓
=
𝜓T
C
±
or
̄𝜓
=
𝜓TC
(3.4)
𝜂=+1∶t−s=0, 1, 2 mod 8
𝜂=−1∶t−s=0, 6, 7 mod 8
8 In [2],
𝜂=−1
case is called Majorana and
𝜂=+1
case is called
pseudo-Majorana.

J.-H.Park
Vol.:(0123456789)
1 3
3.3 Majorana–Weyl spinor
Majorana–Weyl spinor satisfies both of the two conditions
above:
Majorana–Weyl Spinor exists only if
4 Majorana Representation and
SO (8)
Fact 1 Consider a finite dimensional vector space,
V
with the
unitary and symmetric matrix,
B=BT
,
BB†=1
. For every
�v⟩∈V
if
B�v⟩∗∈V
then there exists an orthonormal “semi-
real ” basis,
V={�l⟩,l=1, 2,
⋯
}
, such that
B�l⟩∗=�l⟩
.
Proof Start with an arbitrary orthonormal basis,
{�vl⟩,l=1, 2,
⋯
}
and let
�1⟩∝�v1⟩+B�v1⟩∗
. After
the normalization,
⟨1�1⟩=1
, we can take a new ortho-
normal basis,
{�1⟩,�2�⟩,�3�⟩,
⋯
}
. Now we assume that
{�1⟩,�2⟩,
⋯
�k−1⟩,�k�⟩,�(k+1)�⟩,
⋯
}
is an orthonormal
basis, such that
B�j⟩∗=�j⟩
for
1≤j≤k−1
. To construct the
kth such a vector,
�k⟩
we set
�k⟩∝�k�⟩+B�k�⟩∗
with the nor-
malization. We check this is orthogonal to
�j⟩,1≤j≤k−1
:
In this way one can construct the desired basis.
In the spacetime which admits Majorana spinor from
Eq.(3.4):
more explicitly in the even dimensions having
𝜀+=1
(or
𝜀−
=1
), where
B+
(or
B
−
) is symmetric and also in the odd
dimensions of
𝜀=1
, where B is symmetric, from Fact1
above we can choose an “semi-real ” orthonormal basis,
such that
B−1
𝜂�
l
⟩∗
=
�
l
⟩
(here it is
B−1
𝜂
that plays the role of
B
in Fact 1). In the basis, we write the gamma matrices:
From
𝜂𝛾
𝜇∗
=B
𝜂
𝛾
𝜇
B
−1
𝜂
and the property of the semi-real
basis,
B−1
𝜂�
l
⟩∗
=
�
l
⟩
, we get
Since
R𝜇
is also a representation of the gamma matrix:
(3.5)
𝛾(d+1)𝜓
=
𝜓 ̄𝜓
=
𝜓T
C
±
(3.6)
𝜂=+1∶
t
−
s
=0 mod 8
𝜂=−1∶t−s=0 mod 8
(4.1)

j

k
�
+B

k
�∗
=0+

k

j

=
0.
(4.2)
𝜂=+1∶t−s=0, 1, 2 mod 8
𝜂=−1∶t−s=0, 6, 7 mod 8 ,
(4.3)
𝛾
𝜇=

R𝜇
lm
l

m
.
(4.4)
(
R
𝜇
lm)∗
=𝜂R
𝜇
lm .
adopting the true real basis, we conclude that there exists
a Majorana representation, where the gamma matrices are
real,
𝜂=+
or pure imaginary,
𝜂=−
in any spacetime admit-
ting Majorana spinors.
Furthermore, from Eq.(2.30), in the even dimension of
t−s≡0
mod 8,
𝜀±=1
and
𝛾(d+1)∗ =B𝛾(d+1)B−1
(here we
omit the subscript index ± or
𝜂
for simplicity.). The action,
�v⟩
→
B†�v⟩∗
preserves the chirality, and from the Fact 1
above we can choose an orthonormal semi-real basis for
the chiral and anti-chiral spinor spaces,
V=V++V−
,
V±={�
l
±⟩}
, such that
With the semi-real basis
and the gamma matrices are in the Majorana representation:
From Eq.(4.6) any two sets of semi-real basis, say
{�
l
±⟩}
and
{�
̃
l
±⟩}
are connected by an
O((2d∕2−1))
transformation:
If we define
then
�
̃
l
±⟩
=Λ
±�
l
±⟩
and from the definition of the semi-real
basis:
We write
Thus, for
Λ±
, such that the infinity sum converges, we have
This gives a strong constraint when we express
M±
by the
gamma matrix products. For the Euclidean eight dimensions
(4.5)
R𝜇R𝜈+R𝜈R𝜇=2𝜂𝜇𝜈 ,
(4.6)
⟨l
±
�m
±
⟩=
𝛿lm
,⟨l
±
�m
∓
⟩=0,
𝛾
(d+1)�l
±
⟩=±�l
±
⟩,B†�l
±
⟩∗=�l
±
⟩
.
(4.7)
𝛾
(d+1)=
(
10
0−1
),
(4.8)
𝛾
𝜇=
(
0r
𝜇
rT
𝜇
0
)
,r𝜇∈O(2d∕2−1),r𝜇r𝜈T+r𝜈r𝜇T=2𝛿𝜇𝜈
.
(4.9)
̃
l±

=

m
Λ±ml

m±

,

m
Λ±lmΛ±nm =𝛿ln
.
(4.10)
Λ
±=

l,m
Λ±lm

l±

m±
,
(4.11)
Λ±
=B†Λ∗
±
B=Λ
±
P
±
=P
±
Λ
±
,Λ
±
Λ
†
±
=P
±.
(4.12)
Λ
±=eM±,M±
≡
∞
∑
n=1
(−1)n+11
n(Λ±−P±)n=ln Λ±
.
(4.13)
M±
=−M
†
±
=B
†
M
∗
±
B=M
±
P
±
=P
±
M
±.

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
only the
SO (8)
generators for the spinors survive in the
expansion:
Namely, we find an isomorphism between the two
SO (8)
’s,
one for the semi-real vectors and the other for the spinors in
the conventional sense. Alternatively this can be seen from
where the each block diagonal is a generator of
SO (D)
,
while the dimension of the chiral space is
2d∕2−1
. Only in
d=8
both coincide leading to the “
so (8)
triolity” among
sov(8)
,
soc(8)
and
sōc(8)
.
Fact 2 Relation to octonions.
In Euclidean eight dimensions, the
16 ×16
gamma
matrices can be taken of the off-block diagonal form:
where the
8×8
real matrices,
ra
,
1≤a≤8
, give the multi-
plication of the octonions,
oa
:
Fact 3 Consider an arbitrary real self-dual or anti-self-dual
four form in
D=8
:
Using the
SO (8)
rotations one can transform the four form
into the canonical form, where the non-vanishing compo-
nents are
T±
1234
,T
±
1256
,T
±
1278
,T
±
1357
,T
±
1368
,T
±
1458
,T
±
1467
and
their dual counter parts only.
Proof We start with the seven linearly independent traceless
Hermitian matrices:
As they commute with each other, there exists a basis
V±={�l±⟩}
diagonalizing the seven quantities:
(4.14)
M
±=
1
2
wab𝛾ab P±
.
(4.15)
𝛾
ab =
(
r
[a
r
b]T
0
0r[aTrb]
),
(4.16)
𝛾
a=
(
0ra
rT
a
0
)
,rarT
b+rbrT
a=2𝛿ab
,
(4.17)
oaob=(ra)b
coc.
(4.18)
T±
abcd
=±
1
4!
𝜖abcdefghT±efgh
.
(4.19)
E
±1=𝛾
2341
P±,E±2=𝛾
2561
P±,E±3=𝛾
2781
P±,E±4=𝛾
1357
P±
,
E±5
=
𝛾
3681P
±
,E
±6
=
𝛾
4581P
±
,E
±7
=
𝛾
4671P
±
.
(4.20)
E
±r=

l
𝜆rl

l±

l±

,(𝜆rl)2=
1.
Furthermore, since
C�
l
±⟩∗
is also an eigenvector of the same
eigenvalues, from the Fact 1 we can impose the semi-reality
condition without loss of generality,
C�
l
±⟩∗=�
l
±⟩
.
Now, for the self-dual four form, we let
Since
T±
is Hermitian and
C(T±)∗C†=T±
, one can diago-
nalize
T±
with a semi-real basis:
For the two semi-real basis above, we define a transforma-
tion matrix:
Then, since
T±
is traceless,
O±
T±O
†
±
can be written in terms
of
E±i
’s. Finally the fact
O±
gives a spinorial
SO (8)
rotation
completes our proof.
Some useful formulae are
For an arbitrary self-dual or anti-self-dual four form tensor
in
D=8
, from
we obtain an identity
5 Superalgebra
5.1 Graded Lie algebra
Supersymmetry algebra is a
̂
Z2
graded Lie algebra,
𝐠={Ta}
,
which is an algebra with commutation and anti-commutation
relations[5, 6]:
where
Cc
ab
is the structure constant and
with
#a
, the
̂
Z2
grading of
Ta
,
(4.21)
T±
=
1
4
T
±
abcd
𝛾abcd
.
(4.22)
T
±=

l
𝜆l

̃
l±

̃
l±

,C

̃
l±

∗=

̃
l±
.
(4.23)
O±
=
�
l
±⟩⟨
̃
l
±�.
(4.24)
±P
±
=E
±1
E
±2
E
±3
=E
±1
E
±4
E
±5
=E
±1
E
±6
E
±7
=E
±2
E
±4
E
±
6
=E
±2
E
±5
E
±7
=E
±3
E
±4
E
±7
=E
±3
E
±5
E
±6
.
(4.25)
T
±
acdeT±bcde =(
1
4!)2𝜖acdefghi𝜖bcdejklm T±fghiT±
jklm
=1
4
𝛿abT±
cdef
T±cdef −T±
acde
T±bcde
,
(4.26)
T
±
acde
T±bcde =
1
8
𝛿abT±
cdef
T±cdef
.
(5.1)
[
T
a
,T
b}=
C
c
ab
T
c
(5.2)
[
T
a
,T
b
}=T
a
T
b
− (−1)
#a#b
T
b
T
a

J.-H.Park
Vol.:(0123456789)
1 3
The generalized Jacobi identity is
which implies
For a graded Lie algebra, we consider
where
za
is a superspace coordinate component which has
the same bosonic or fermionic property as
Ta
and hence
zaTa
is bosonic.
In the general case of non-commuting objects, say A and B,
the Baker–Campbell–Haussdorff formula gives
where
Cn(A,B)
involves n commutators. The first three of
these are
Since for the graded algebra
the Baker–Campbell–Haussdorff formula(5.7) implies that
g(z) forms a group, the graded Lie group. Hence we may
define a function on superspace,
fa(w,z)
, by
Since
g(0)=e
, the identity, we have
f(0, z)=z,f(w,0)=w
and further we assume that f(w,z) has a Taylor expansion in
the neighbourhood of
w=z=0
.
Associativity of the group multiplication requires f(w,z)
to satisfy
5.2 Left andright invariant derivatives
For a graded Lie group, left and right invariant derivatives,
La,Ra
are defined by
(5.3)
#
a=
{
0 for bosonic a
1 for fermionic
a
(5.4)
[Ta,[Tb,Tc}} − (−1)#a#b[Tb,[Ta,Tc}} = [[Ta,Tb},Tc}
(5.5)
(−
1)
#a#c
C
d
ab
C
e
dc
+ (−1)
#b#a
C
d
bc
C
e
da
+ (−1)
#c#b
C
d
ca
C
e
db
=
0
(5.6)
g(z)= exp (zaTa)
(5.7)
e
AeB=exp
(∞
∑
n=0
Cn(A,B)
)
(5.8)
C
0
(A,B)=A+B
C
1(A,B)= 1
2[A,B]
C
2(A,B)= 1
12
[[A,B],B]+ 1
12
[A,[A,B
]]
(5.9)
[
z
a
T
a
,z
b
T
b
]=z
b
z
a
[T
a
,T
b
}=z
b
z
a
C
c
ab
T
c
(5.10)
g(w)g(z)=g(f(w,z))
(5.11)
f(f(u,w),z)=f(u,f(w,z))
Explicitly, we have
where
𝜕
b=
𝜕
𝜕z
b.
It is easy to see that
La
is invariant under left action,
g(z)
→
hg(z)
, and
Ra
is invariant under right action,
g(z)
→
g(z)h
.From Eqs. (5.12,5.13), we get
and from Eqs. (5.12,5.13) we can also easily show
Thus,
La(z)
,
Ra(z)
form representations of the graded Lie
algebra separately. For the supersymmetry algebra, the left
invariant derivatives become covariant derivatives, while
the right invariant derivatives become the generators of the
supersymmetry algebra acting on superfields.
5.3 Superspace andsupermatrices
In general a superspace may be denoted by
𝐑p|q
, where p,
q are the number of real commuting (bosonic) and anti-
commuting (fermionic) variables, respectively. A super-
matrix which takes
𝐑p|q→𝐑p|q
may be represented by a
(p+q)×(p+q)
matrix, M, of the form:
where a,d are
p×p
,
q×q
matrices of Grassmanian even
or bosonic variables and b,c are
p×q
,
q×p
matrices of
Grassmanian odd or fermionic variables, respectively.
The inverse of M can be expressed as
where we may write
(5.12)
Lag(z)=g(z)Ta
(5.13)
Rag(z)=−Tag(z)
(5.14)
L
a=La
b(z)𝜕bLa
b(z)= 𝜕fb(z,u)
𝜕ua
|
|
|
|
|u=0
(5.15)
R
a=Ra
b(z)𝜕bRa
b(z)=−𝜕fb(u,z)
𝜕ua
|
|
|
|
|u=0
(5.16)
[L
a,
L
b
}=Cc
abLc
(5.17)
[Ra,Rb}=Cc
abRc
(5.18)
[La,Rb}=0
(5.19)
M
=
(
ab
cd
)
(5.20)
M
−1=
(
(a−bd
−1
c)
−1
−a
−1
b(d−ca
−1
b)
−1
−d−1c(a−bd−1c)−1(d−ca−1b)−1
)

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
Note that due to the fermionic property of b,c, the power
series terminates at
n≤pq +1
.
The supertrace and the superdeterminant of M are
defined as
The last equality comes from
which may be shown using
and observing
From Eq.(5.23) we note that
sdet M≠0
implies the exist-
ence of
M−1
. Thus the set of supermatrices for
sdet M≠0
forms the supergroup,
Gl (p|q)
. If
sdet M=1
then
M∈Sl (p|q)
.
The supertrace and the superdeterminant have the
properties:
We may define the transpose of the supermatrix, M, either as
or as
where
at
,bt
,ct
,dt
are the ordinary transposes of a,b,c,d,
respectively.We note that
(5.21)
(
a−bd−1c)−1=a−1+
∞
∑
n=1
(a−1bd−1c)na−
1
(5.22)
str M=tr a−tr d
(5.23)
sdet M=det(a−bd−1c)∕ det d=det a∕det(d−ca−1b)
(5.24)
det(1−a−1bd−1c)= det −1(1−d−1ca−1b)
(5.25)
det
(1−a)= exp
(
−
∞
∑
n=1
1
n
tr an
)
(5.26)
tr (a−1bd−1c)n=−tr (d−1ca−1b)n
(5.27)
str (M1M2)= str (M2M1)
(5.28)
sdet (M1M2)= sdet M1sdet M2
(5.29)
M
t
=
(
a
t
c
t
−btdt
)
(5.30)
M
t�
=
(
a
t
−c
t
btdt
)
(5.31)
(
M
1
M
2
)t=Mt
2
Mt
1
(M
1
M
2
)t
�
=Mt
�
2
Mt
�
1
(5.32)
(Mt)t�=(Mt�)t=M
6 Super Yang–Mills
6.1
(3+1)D
N=1
super Yang–Mills
In four-dimensional Minkowskian spacetime of the metric,
𝜂= diag(− + ++)
, the
4×4
gamma matrices satisfy with
𝜇=0, 1, 2, 3
:
The Majorana spinor,
𝜓
satisfies then
The four-dimensional super Yang–Mills Lagrangian reads
The supersymmetry transformations are
6.2
(5+1)D
(1,0) super Yang–Mills
In six-dimensional Minkowskian spacetime of the metric,
𝜂= diag(− + + + ++)
, the
8×8
gamma matrices satisfy with
M=0, 1, 2, 3, 4, 5
:
The gamma “seven” is given by
Γ(7)=Γ
012345
to satisfy
Γ(7)=Γ
(7)† =Γ
(7)−1
and
where
𝜖012345 =+1
.
The
su (2)
Majorana–Weyl spinor,
𝜓i
,
i=1, 2
, satisfies
then
(6.1)
Γ𝜇†
=Γ
𝜇=−AΓ
𝜇
A
†
,A=Γ
t
=−A
†
,
Γ
𝜇∗=+BΓ𝜇B†,BT=B,B†=B−1,
Γ
𝜇T
=−CΓ
𝜇
C
†
,C=−C
T
=BΓ
t
,C
†
=C
−1
.
(6.2)
̄𝜓 =𝜓†Γt=𝜓TC
⟺
𝜓∗=B𝜓.
(6.3)
L
4D= tr
(
−1
4F𝜇𝜈F𝜇𝜈 −i1
2̄𝜓 Γ𝜇D𝜇𝜓
).
(6.4)
𝛿
A
𝜇
=ī𝜀 Γ
𝜇
𝜓=−ī𝜓 Γ
𝜇
𝜀,𝛿𝜓 =−
1
2
F
𝜇𝜈
Γ𝜇𝜈𝜀
.
(6.5)
ΓM†
=Γ
M=AΓ
M
A
†
,A∶= Γ
12345
=A
†
=A
−1
,
Γ
MT =CΓMC†,CT=−C,C†=C−1
,
Γ
M∗
=BΓ
M
B
†
,B=CA =−B
T
,B
†
=B
−1
.
(6.6)
Γ
LMN =
1
6
𝜖LMNPQR ΓPQRΓ(7)
,
(6.7)
Γ(7)
𝜓i=+𝜓i,̄𝜓
i
Γ
(7)
=−̄𝜓
i
∶chiral ,
̄𝜓
i=(𝜓
i
)†A=𝜖ij(𝜓
j
)TC∶
su(2) Majorana ,

J.-H.Park
Vol.:(0123456789)
1 3
where
𝜖ij
is the usual
2×2
skew-symmetric unimodular
matrix. It is worth to note that
̄𝜓 i
Γ
M
1
M
2
…M
2n
𝜌i
=
0
and
where
𝜓i
,
𝜌i
are two arbitrary Lie algebra valued
su (2)
Majo-
rana–Weyl spinors.
The six-dimensional super Yang–Mills Lagrangian reads
where all the fields are in the adjoint representation of the
gauge group, such that, with the Hermitian Lie algebra val-
ued gauge field,
AM
:
From (6.8) the action is real valued.
The supersymmetry transformations are given by with a
su (2)
Majorana–Weyl supersymmetry parameter,
𝜀i
:
so that, in particular,
𝛿 ̄𝜓
i=+
1
2
F
MN
̄𝜀 iΓMN . The crucial Fierz
identity for the supersymmetry invariance is with the chiral
projection matrix,
P
∶=
1
2
(1+Γ
(7)
)
:
which ensures the vanishing of the terms cubic in
𝜓i
:
The equations of motion are
6.3 6D super Yang–Mills inthespacetime
ofarbitrary signature
With
we have
We introduce a pair of Weyl spinors of the same chirality:
and define the charge conjugate spinor by
(6.8)
tr(
ī𝜓 iΓM1M2…M2n+1𝜌i)=[tr(ī𝜓 iΓM1M2…M2n+1𝜌i)]
†
= −(−1)ntr(i
̄𝜌
iΓM1M2…M2n+1
𝜓i
)
,
(6.9)
L
6D= tr
(
−1
4
FLM FLM −i1
2
̄𝜓 iΓLDL𝜓i
),
(6.10)
DL𝜓i=𝜕L𝜓i−i[AL,𝜓i],FLM =𝜕LAM−𝜕MAL−i[AL,AM].
(6.11)
𝛿
AM=+ī𝜀 iΓM𝜓i=−ī𝜓 iΓM𝜀i,𝛿𝜓i=−
1
2
FMN ΓMN 𝜀i
,
(6.12)
(
Γ
L
P
)𝛼𝛽(
ΓLP
)𝛾𝛿
+
(
Γ
L
P
)𝛾𝛽(
ΓLP
)𝛼𝛿
=
0,
(6.13)
tr(
̄𝜓
i
Γ
L
[𝛿A
L
,𝜓
i
]
)
= tr
(
̄𝜓
i
Γ
L
[ī𝜀
j
Γ
L
𝜓
j
,𝜓
i
]
)
=
0.
(6.14)
DL
F
LM +̄𝜓 iΓM𝜓i=0, ΓM
D
M𝜓i=0.
(6.15)
(
ΓM
)T
=±C
±
ΓMC−1
±
,CT
±
=∓C
±,
(6.16)
(
C
±
ΓM
)T
=−C
±
ΓM
.
(6.17)
(
𝜓
1
,𝜓
2)
,Γ(7)𝜓
i
=s𝜓
i
,s2=
1,
The super Yang–Mills Lagrangian reads
and the supersymmetry transformations are given by
so that, in particular,
𝛿 ̄𝜓
i
c
=+
1
2
FMN ̄𝜀 i
c
Γ
MN
. The Lagrangian
transforms as, from (2.60):
Only if
B∗
±B±=−1
, as in the Minkowskian signature, one
can impose the pseudo-Majorana condition:
6.4
(9+1)D
SYM, its reduction, and4D
superconformal symmetry
• Conventions for
(9+1)D
gamma matrices Spacetime
signature :
𝜂=diag (−++
⋯
+)
, mostly plus signature.
32 ×32
Gamma matrices:
(i) Hermitian conjugate:
(ii) Complex conjugate
(iii) Transpose
Let the spinorial indices be located as
(6.18)
̄𝜓 i
c
∶= 𝜖
−1ij
𝜓
T
j
C
±.
(6.19)
L
6D= tr
(
1
4
FMN FMN +1
2
̄𝜓 i
cΓMDM𝜓i
),
(6.20)
𝛿
AM=̄𝜀
i
cΓM𝜓i=−̄𝜓
i
cΓM𝜀i
,
𝛿𝜓
i=−
1
2
FMN ΓMN 𝜀i,
(6.21)
𝛿
L6D=𝜕Mtr
(
FMN 𝛿AN−1
2
̄𝜓 i
cΓM𝛿𝜓i
).
(6.22)
̄𝜓 i
c
=̄𝜓
i
D
∶= (𝜓
i
)
†
A
.
(6.23)
(Γ
M)†=Γ
M= −Γ0ΓMΓ0=AΓMA
†
,
A
=Γ
12⋯9=A†=A−1,
(
A
Γ
M1M2⋯Mn
)
†
= (−1)
1
2n(n−1)A
Γ
M1M2⋯Mn
,
(6.24)
(Γ
M)∗=±B±ΓMB
†
±
,
B±
=BT
±
=(B†
±
)−1,
(6.25)
(Γ
M)T=±C±ΓMC
†
±,
C
±=BT
±A=±CT
±=(C†
±)−1,
(
C
+Γ
M1M2⋯Mn
)
T
= (−1)
1
2n(n−1)C
+Γ
M1M2⋯Mn
.

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
Define
The crucial identity for the super Yang–Mills action is
where
Γ
±=
1
2
(1±Γ
(10)
)
is either the chiral or the anti-
chiral projector, and
𝛼,𝛽,𝛾
are symmetrized. Note also
the symmetric property,
(
C
+
Γ
M
Γ
±
)
T
=C
+
Γ
M
Γ
±
.
For spinors, we set
Majorana–Weyl Spinor,
𝜓
, satisfies
or equivalently
Hence, for the fermionic Majorana–Weyl spinors:
and9
We can further set
Namely,
(𝛾M
,̃𝛾 N)
are the real
16 ×16
matrices appear-
ing in the off block-diagonal parts of the
32 ×32
gamma
matrices, satisfying10
(6.26)
(ΓM
)
𝛼
𝛽
,(A)
𝛼
𝛽
,(B
±
)
𝛼𝛽
=(B
±
)
𝛽𝛼
,(C
±
)
𝛼𝛽
= ±(C
±
)
𝛽𝛼 .
(6.27)
Γ(10)
=Γ
012
⋯
9
= (Γ
(10)
)
†
= (Γ
(10)
)
−1
=−C
†
+
(Γ
(10)
)
T
C
+.
(6.28)
(
C
+
Γ
M
Γ
±
)
(𝛼𝛽
(C
+
Γ
M
Γ
±
)
𝛾)𝛿
=
0
(6.29)
̄𝜓 =𝜓†
A
.
(6.30)
Γ(10)𝜓=+𝜓∶Weyl condition ,
(6.31)
𝜓∗=B+𝜓∶Majorana condition ,
(6.32)
̄𝜓
Γ
(10)
=−̄𝜓 ∶
opposite chirality ,
̄𝜓
=
𝜓
TC
+
.
(6.33)
̄𝜓 1ΓM
1
M
2⋯
M
2n
𝜓2=0,
(6.34)
̄𝜓
1Γ
M
1
M
2⋯
M
2n+1𝜓2= (−1)
n+1
̄𝜓 2Γ
M
1
M
2⋯
M
2n+1𝜓1
= −( ̄𝜓
1
ΓM1M2⋯M2n+1𝜓
2
)†∶
imaginary .
(6.35)
lll
ΓM=
(
0̃𝛾
M
𝛾M0
)
,
𝛾M̃𝛾 N+𝛾Ñ𝛾 M=2𝜂MN
,
𝜂=diag (−+++
⋯
+) .
• Lagrangian Let the gauge group be
su (N)
or
u(N)
.
Lie algebra valued fields:
Field strength and the covariant derivative are
Bianchi identity reads
The gauge symmetry is given by, for
g†=g−1
:
The Lagrangian of 10D super Yang–Mills theory reads
where
Ψ≡(𝜓0)T
and
𝜓𝛼
is a sixteen component spinor
and
̄𝜓 ∶= 𝜓T̃𝛾 0
.
Under arbitrary infinitesimal transformations,
𝛿AM
,
𝛿Ψ
:
• Summary of supersymmetry in
D≤10
. The ordinary
supersymmetry and kinetic supersymmetry are given by
so that
where
𝜉+
and
𝜉�
+
are constant Majornana–Weyl spinors
corresponding to the ordinary and kinetic supersymmetry
parameters.
+
denotes the chirality. The above is the sym-
metry of the
(9+1)D
and also any dimensionally reduced
super Yang–Mills action. In four-dimensions of either
Minkowskian or Euclidean signature, the supersymmetry
gets enhanced to the superconformal symmetry as
(6.36)
(
𝛾
M
)
∗
=𝛾
M
,(𝛾
M
)
T
=̃𝛾
0
𝛾
M
̃𝛾
0
=̃𝛾 M
,
̃𝛾
0
𝛾
1
̃𝛾
2⋯
𝛾
9
=+1, 𝛾
0
̃𝛾
1
𝛾
2⋯
̃𝛾
9
=−1.
(6.37)
AM
=A
p
M
T
p
,Ψ=Ψ
p
T
p
,(T
p
)
†
=T
p.
(6.38)
FMN =𝜕MAN−𝜕NAM−i[AM,AN],DMΨ=𝜕MΨ−i[AM,Ψ] .
(6.39)
DLFMN +DMFNL +DNFLM =0.
(6.40)
AM
→gA
M
g
−1+
ig
𝜕M
g
−1
,F
MN
→gF
MN
g
−1
,
Ψ
→g
Ψ
g
−1.
(6.41)
L
= tr
[
−1
4FMN FMN −i1
2
̄
ΨΓMDMΨ
]
= tr
[
−1
4FMN FMN −i1
2̄𝜓 𝛾 MDM𝜓
],
(6.42)
𝛿
L=tr

DLF
LM
+
̄
ΨΓ
M
Ψ

𝛿AM−i
̄
ΨΓ
M
DM𝛿Ψ

+𝜕Ntr

FMN 𝛿AM−i1
2𝛿̄
ΨΓNΨ

.
(6.43)
𝛿
AM=i
̄
ΨΓM𝜉+=−ī
𝜉+ΓMΨ,𝛿Ψ=
1
2
FMN ΓMN 𝜉++𝜉�
+
1N×N
,
(6.44)
𝛿̄
Ψ=−
1
2
̄
𝜉+FMN Γ
MN
+
̄
𝜉
�
+
1N×N
,
9 When the spinor is Lie algebra valued, Eq.(6.34) does not hold in
general:
10 From
̃𝛾 M=(𝛾M)−1
it also follows that
̃𝛾 M𝛾N+̃𝛾 N𝛾M=
2
𝜂MN
. One
may further impose the symmetric property,
(𝛾M)T=𝛾M
, but it is not
necessary in our paper:

J.-H.Park
Vol.:(0123456789)
1 3
where m is for the four-dimensions and a is for the rest.
𝜉−
is a constant Majornana–Weyl spinor of the opposite
chirality corresponding to the special superconformal
symmetry parameter, and
In any case, the conserved supercurrent is of the uni-
versal form:
In Appendix C, we present the derivation.
• Superconformal symmetry in 4D of arbitrary signa-
ture. The 32 supersymmetries in 4D super Yang–Mills
which consist of ordinary supersymmetry and special
superconformal symmetry read
where
𝜉
is a 32 component Majorana spinor:
The chiral decomposition of the spinor gives the
ordinary supersymmetry and special superconformal
symmetry,11
The 32 component Majorana supercurrent is of the form:
The supercharge is given by
(6.45)
ll
𝛿AM=i
̄
ΨΓME(x)=−i
̄
E(x)ΓMΨ,
𝛿
Ψ= 1
2
FMN ΓMN E(x)−2ΦaΓa𝜉−+𝜉�
+
1N×N
,
(6.46)
E(x)=xmΓm𝜉−+𝜉+.
(6.47)
JM
=−itr
(̄
ΨΓ
M
𝛿Ψ
)
=+itr
(
𝛿
̄
ΨΓ
M
Ψ
).
(6.48)
𝛿
AM=i
̄
ΨΓM(1+xmΓm)𝜉=−i
̄
𝜉(1+xmΓm)ΓMΨ,
𝛿
Ψ= 1
2(1+Γ
(10))[1
2FMN ΓMN (1+xmΓm)−2ΦaΓa]𝜉,
𝛿
̄
Ψ= ̄
𝜉
[
−1
2
(1+xmΓm)FMN ΓMN −2ΦaΓa
]
1
2
(1−Γ
(10))
,
(6.49)
𝜉∗=B+𝜉.
(6.50)
𝜉
=𝜉++𝜉−,𝜉±=
1
2
(1±Γ
(10))𝜉
.
(6.51)
J
M=+ī
Q
M
𝜉=−ī
𝜉QM,
Q
M= tr[(1
2(1+xmΓm)FKLΓKL +2ΦaΓa)ΓMΨ],
̄
Q
M= tr
[
̄
ΨΓM
(
−1
2FKLΓKL (1+xmΓm)+2ΦaΓa
)]
=(QM)†A=(QM)TC+
.
(6.52)
Q
=
∫
d3xQ0
.
Acknowledgements This work is supported by Basic Science Research
Program through the National Research Foundation of Korea Grants,
NRF-2016R1D1A1B01015196 and NRF-2020R1A6A1A03047877
(Center for Quantum Space Time).
Appendix
Proof oftheTheorem(2.1)
Theorem(2.1)
Any
N×N
matrix, M, satisfying
M2
=
𝜆2
1
N×N
,
𝜆≠0
, is
diagonalizable.
Proof Suppose for some K,
1≤K≤N
, we have found a
basis:
such that
From
M2=𝜆21N×N
:
and hence
The assumption holds for
K=1
surely. To construct
eK+1
we first consider an eigenvector of the
(N−K)×(N−K)
matrix, P:
and set
(A.1)
{ea,vr∶1≤a≤K,1≤r≤N−K}
(A.2)
Me
a
=𝜆
a
e
a
, for 1 ≤a≤K,
Mvr
=Ps
r
v
s
+ha
r
e
a
, for K+1
≤
r,s
≤
N
.
(A.3)
𝜆2
a=𝜆
2
,
𝜆
2v
r
=(P2)s
r
v
s
+
[
(hP)a
r
+𝜆
a
ha
r]
e
a,
(A.4)
P2
=𝜆
2
1(N−K)×(N−K),
(
hP)a
r
+
𝜆a
ha
r
=
0.
(A.5)
Pr
s
c
s
=𝜆
K+1
c
r
,𝜆
2
K+1
=𝜆
2,
(A.6)
v=crv
r
,ha=ha
r
cr,
Mv
=𝜆
K+1
v+hae
a
.
11 Note also
E
(x)=
1
2
(1+Γ
(10))(1+xmΓm)𝜉
.

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
Consequently
so that
We construct
eK+1
, with K unknown coefficients,
da
, as
From
we determine
From (A.8) and
𝜆2
K+1
=𝜆
2
a
=𝜆
2≠0
, we have
This completes our proof.
If we set a
N×N
invertible matrix, S, by
then
Gamma matrices in4,6,10,12 dimensions
Our conventions are such that
Four dimensions
In Minkowskian four dimension of the metric,
̂𝜂 = diag(− + ++)
, the gamma matrices satisfy
(A.7)
(𝜆K+1+𝜆a)ha=0∶not asum ,
(A.8)
ha=0 if 𝜆K+1+𝜆a≠0.
(A.9)
eK+1=v+daea.
(A.10)
MeK+1
=𝜆
K+1
e
K+1
+
[
h
a
+(𝜆
a
−𝜆
K+1
)d
a]
e
a,
(A.11)
d
a=
⎧
⎪
⎨
⎪
⎩
ha
𝜆K+1−𝜆a
if 𝜆K+1
≠
𝜆a
,
any number if 𝜆
K+1
=𝜆
a.
(A.12)
MeK+1=𝜆K+1eK+1.
(A.13)
(
S)
b
a
=(e
a
)
b
,Me
a
=
𝜆a
e
a
,1
≤
a,b
≤
N
,
(A.14)
S−1MS = diag(𝜆1,𝜆2,…,𝜆N).
(B.1)
̂𝛾 m∶m=0, 1, 2, 3 for 1 +3D,
𝛾
𝜇∶𝜇=1, 2, …, 6 for 2 +4D,
𝛾
a∶a=7, 8, …, 12 for 0 +6D,
Γ
M∶M=0, 1, 2, 3, 7, …, 12 for 1 +9D,
𝚪
𝐌
∶𝐌=1, 2, …, 12 for 2 +10D.
where
m,n=0, 1, 2, 3
. The chiral matrix reads
The three pairs of unitary matrices,
̂
A
±
,
̂
B
±
,
̂
C
±
, relate the
hermitain conjugate, complex conjugate, and the transpose
of the gamma matrices:
Especially in Minkowskian four dimensions, they can be
chosen further to satisfy
Four tosix dimensions
Using the four dimensional gamma matrices above, one
can construct the six dimensional gamma matrices in the
off-block diagonal form:
With the relevant choice of the metric:
we require
̄𝜌 𝜇
=(𝜌
𝜇
)
†
and set
Explicitly with (B.3), (B.5)
Note
(B.2)
̂𝛾 m̂𝛾 n+̂𝛾 n̂𝛾 m=2̂𝜂 mn ,(̂𝛾 m)†=̂𝛾 m,
(B.3)
̂𝛾 (5)=−î𝛾 0123 =(̂𝛾 (5))−1=(̂𝛾 (5))†.
(B.4)
±(
̂𝛾 m)
†
=
̂
A±̂𝛾 m
̂
A
†
±,
̂
A
†
±
̂
A±=
1,
±(
̂𝛾 m)∗=̂
B±̂𝛾 m̂
B†
±,̂
B†
±̂
B±=
1,
±(̂𝛾
m)T=̂
C
±̂𝛾
m̂
C†
±
,̂
C†
±
̂
C
±
=
1.
(B.5)
̂
A
+=−i𝛾123 ,
̂
A−=−̂𝛾 0,
̂
A−=
̂
A+̂𝛾 (5)
,
̂
B
∗
±̂
B±=±1, ̂
BT
±=±
̂
B±,̂
B−=̂
B+̂𝛾 (5)
,
̂
C±
=̂
BT
+
̂
A
±
=̂
BT
±
̂
A
+
,̂
CT
±
=−̂
C
±
,̂
C
−
=̂
C
+
̂𝛾 (5)
.
(B.6)
𝛾
𝜇=
(
0𝜌
𝜇
̄𝜌 𝜇0
)
,𝜇=1, 2, …,6, 𝜌𝜇̄𝜌 𝜈+𝜌𝜈̄𝜌 𝜇=2𝜂𝜇𝜈
.
(B.7)
𝜂= diag(− − + + ++) ,
(B.8)
𝛾1
=U

−i𝜏2⊗1

U
†
,𝛾
m+2
=U

𝜏1⊗ ̂𝛾
m
U
†,
𝛾
6=U

𝜏1⊗ ̂𝛾 (5)

U†,U=

̂
C+0
01

.
(B.9)
𝜌
1=−
̂
C+,𝜌m
+
2=
̂
C+̂𝛾 m,𝜌6=
̂
C−,
̄𝜌
1=+
̂
C−1
+
,̄𝜌 m+2=̂𝛾 m̂
C−1
+
,̄𝜌 6=̂
C−1
−.

J.-H.Park
Vol.:(0123456789)
1 3
and especially the anti-symmetric property of the
4×4
matrices:
The spinorial indices,
𝛼,𝛽=1, 2, 3, 4
, denote the fundamen-
tal representation of
su (2, 2)
. It follows that
{𝜌𝜇}
and
{̄𝜌 𝜇}
separately form bases for the anti-symmetric
4×4
matrices
with the completeness relation:
On the other hand, Eq.(B.8) implies that12
so each of the sets
𝜌[𝜇̄𝜌 𝜈𝜌𝜆]≡𝜌𝜇𝜈𝜆
or
̄𝜌 [𝜇𝜌𝜈̄𝜌 𝜆]≡̄𝜌 𝜇𝜈𝜆
has
only 10 independent components and forms a basis for sym-
metric
4×4
matrices:
Finally,
{
𝜌𝜇𝜈
≡
1
2
(𝜌𝜇̄𝜌 𝜈−𝜌𝜈̄𝜌 𝜇
)}
or
{
̄𝜌 𝜇𝜈
≡
1
2
(̄𝜌 𝜇𝜌𝜈−̄𝜌 𝜈𝜌𝜇
)}
forms an orthonormal basis for the general
4×4
traceless
matrices:
satisfying
Six dimensions
The result above can be straightforwardly generalized to
other signatures in six dimensions. In Euclidean six dimen-
sions, gamma matrices satisfy
where we set a,b run from 7 to 12, instead of 1–6, as the
latter have been reserved for
so (2, 4)
. With the choice:
(B.10)
𝛾
(7)=i𝛾1𝛾2…𝛾6=
(
10
0−1
),
(B.11)
(
𝜌𝜇)𝛼𝛽 = −(𝜌𝜇)𝛽𝛼 ,(̄𝜌 𝜇)𝛼𝛽 =−
1
2
𝜖𝛼𝛽𝛾𝛿 (𝜌𝜇)𝛾𝛿
.
(B.12)
tr(
𝜌
𝜇
̄𝜌
𝜈
)=4𝛿
𝜇
𝜈
,(𝜌
𝜇
)
𝛼𝛽
(̄𝜌
𝜇
)
𝛾𝛿
=2(𝛿
𝛼
𝛿
𝛿
𝛽
𝛾
−𝛿
𝛽
𝛿
𝛿
𝛼
𝛾
)
.
(B.13)
𝜌
[𝜇̄𝜌 𝜈𝜌𝜆]=+i
1
6
𝜖𝜇𝜈𝜆𝜎 𝜏𝜅 𝜌[𝜎̄𝜌 𝜏𝜌𝜅],̄𝜌 [𝜇𝜌𝜈̄𝜌 𝜆]=−i
1
6
𝜖𝜇𝜈𝜆𝜎 𝜏𝜅 ̄𝜌 [𝜎𝜌𝜏̄𝜌 𝜅]
,
(B.14)
tr(
𝜌𝜇𝜈𝜆 ̄𝜌 𝜎𝜏𝜅 )=−i4𝜖𝜇𝜈 𝜆
𝜎𝜏𝜅 −24𝛿
[𝜇
𝜎𝛿𝜈
𝜏𝛿𝜆]
𝜅
,
(
𝜌𝜇𝜈𝜆)
𝛼𝛽
(̄𝜌
𝜇𝜈𝜆
)𝛾𝛿 =−24(𝛿
𝛼
𝛾𝛿
𝛽
𝛿+𝛿
𝛽
𝛾𝛿
𝛼
𝛿)
.
(B.15)
tr(
𝜌
𝜇𝜈
𝜌𝜆𝜅 )=4(𝛿
𝜇
𝜅𝛿
𝜈
𝜆−𝛿
𝜈
𝜅𝛿
𝜇
𝜆),
−1
8(𝜌𝜇𝜈)𝛼
𝛽(𝜌𝜇𝜈)𝛾
𝛿+1
4𝛿𝛼
𝛽𝛿𝛾
𝛿
=𝛿
𝛼
𝛿𝛿
𝛾
𝛽,
(B.16)
(̄𝜌 𝜇𝜈)𝛼
𝛽= −(𝜌𝜇𝜈)𝛽
𝛼
.
(B.17)
𝛾a𝛾b+𝛾b𝛾a=2𝛿ab ,
the six dimensional gamma matrices are in the block diago-
nal form:
satisfying the hermiticity conditions:
We can further set all the
4×4
matrices,
𝜌a
,̄𝜌 a
to be
anti-symmetric[7]:
which makes the relation,
su (4)≡so (6)
, manifest. That is,
the indices,
̇𝛼 ,̇
𝛽=1, 2, 3, 4
, denote the fundamental repre-
sentation of
su (4)
.
Note that precisely the same equations as (B.12)–(B.16)
hold for the
so (6)
gamma matrices,
{𝜌a
,̄𝜌 b}
after replacing
𝜇,𝜈
,
𝛼,𝛽
by a,b,
̇𝛼 ,̇
𝛽
, etc.
Ten dimensions again
Using the four and six dimensional gamma matrices above,
we write the ten dimensional gamma matrices:
In the above choice of gamma matrices, we have from (6.27),
(B.3), (B.18):
and
Majorana spinor is now of the form:
(B.18)
𝛾
(7)=i𝛾7𝛾8⋯𝛾12 =
(
10
0−1
),
(B.19)
𝛾
a
=
(
0𝜌
a
̄𝜌 a0
),
(B.20)
̄𝜌 a=(𝜌a)†
.
(B.21)
(
𝜌a)̇𝛼 ̇
𝛽= −(𝜌a)̇
𝛽̇𝛼 ,(̄𝜌 a)̇𝛼
̇
𝛽=−
1
2
𝜖̇𝛼
̇
𝛽̇𝛾
̇
𝛿(𝜌a)̇𝛾 ̇
𝛿
,
(B.22)
Γm
=̂𝛾
m
⊗𝛾
(7)
for m=0, 1, 2, 3
Γ
a
=1⊗𝛾
a
for a=7, 8, 9, 10, 11, 12 .
(B.23)
Γ(10)=̂𝛾 (5)⊗𝛾
(7),
(B.24)
A
=
̂
A+⊗1, B±=C±A,
B
+=̂
B−⊗(0+1
−10
),B−=̂
B+⊗(0+1
+10
)
,
C
+=̂
C−⊗
(
0−1
+10
)
,C−=̂
C+⊗
(
0+1
+10
).
(B.25)
Ψ=
B−1
+Ψ∗=
⎛
⎜
⎜
⎝
𝜓𝛼
+̇𝛼
𝜓𝛼 ̇𝛼
−
⎞
⎟
⎟
⎠
,(𝜓†
+)𝛼̇𝛼 =(
̂
B−)𝛼𝛽𝜓𝛽 ̇𝛼
−
,
12 We put
𝜖123456 =1
and “[]” denotes the standard anti-symmetriza-
tion with “strength one”:

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
where
𝛼
is the
so (1, 3)
spinor index and ± denote the
so (6)
chirality.
Further to have 10 dimensional Majorana–Weyl spinor,
imposing the chirality condition,
Γ(10)Ψ=Ψ
, we also have
For the later convenience, we define
𝜓𝛼 ̇𝛼
,
̄𝜓 𝛼 ̇𝛼
by
The Majorana condition is equivalent to
Twelve dimensions
To make the
SO (2, 4)× SO (6)
isometry of AdS
5
×S
5
geometry manifest, it is convenient to employ the twelve
dimensional gamma matrices of spacetime signature,
(--++++++++++), and write them in terms of two sets
of six dimensional gamma matrices,
{𝛾𝜇}
,
{𝛾a}
, which we
reviewed above:
In the above choice of gamma matrices, the twelve dimen-
sional charge conjugation matrices,
𝐂±
, are given by
while the complex conjugate matrices,
𝐀±
, read
satisfying
In particular, for
𝜇=1, 2, …,6
, we have
(B.26)
̂𝛾 (5)𝜓±
=±
𝜓±.
(B.27)
𝜓𝛼 ̇𝛼
=i(
̂
C
+
)
𝛼𝛽
𝜓
𝛽
+̇𝛼
,̄𝜓 𝛼 ̇𝛼 =𝜓𝛼 ̇𝛼
−.
(B.28)
̄𝜓
𝛼 ̇𝛼 =A𝛼
𝛽
(𝜓†)𝛽 ̇𝛼 ,A=i
̂
A
−
=A†=A−1
.
(B.29)
𝚪𝜇
=𝛾
𝜇
⊗𝛾
(7)
for 𝜇=1, 2, 3, 4, 5, 6
𝚪
a
=1⊗𝛾
a
for a=7, 8, 9, 10, 11, 12 .
(B.30)
±(𝚪
𝐌
)
T
=𝐂±𝚪
𝐌
𝐂
−1
±,𝐌=1, 2, …
, 12,
𝐂
±=
(
01
±10
)
⊗
(
01
∓10
)
,
(B.31)
𝐀
±=
(
A
t
0
0∓A
)
⊗
(
10
0±1
)
,
A
=−i
̄𝜌 12
=−i
̂𝛾
0=î
A
−
=A†=A−1
,
(B.32)
±(
𝚪
𝐌
)
†
=𝐀
±
𝚪
𝐌
𝐀
−1
±.
(B.33)
(
𝜌
𝜇
)
†
=−Ā𝜌
𝜇
A
t
=̄𝜌
𝜇
,(̄𝜌
𝜇
)
†
=−A
t
𝜌
𝜇
A=𝜌
𝜇.
Now if we define the twelve dimensional chirality operator as
then
In 2+10 dimensions it is possible to impose the Majorana-
Weyl condition on spinors to have sixteen independent
complex components which coincides with the number of
supercharges in the
AdS5
×S
5
superalgebra,
su (2, 2|4)
. Up
to the redefinition of the spinor by a phase factor, there are
essentially two choices for the Majorana-Weyl condition
depending on the chirality:
Our choice will be the plus sign so that the
2+10
dimen-
sional Weyl spinor carries the same chiral indices for
su (2, 2)
and
su (4)
, i.e.,
𝚿=(𝜓𝛼 ̇𝛼 ,̄𝜓 𝛼 ̇𝛼 )T
, while the Majo-
rana condition relates them as
̄𝜓 𝛼 ̇𝛼
=A
𝛼
𝛽
(𝜓
†
)
𝛽 ̇𝛼
which is
identical to (B.28). Hence, the Majorana-Weyl spinor in
2+10
dimensions can be identified as the Majorana spinor
in
1+9
dimensions.
Looking forthegeneral odd symmetry
With a Majorana–Weyl spinor,
E
,
ΔΨ
, which may depend on
xM
, we focus on the following transformations:
so that
Note that
ΔΨ
is Lie algebra valued, while
E
is not.
From
and the identity (6.28), we note that the second term in
(6.42) vanishes
We also get, using the Bianchi identity (6.39):
(B.34)
𝚪(𝟏𝟑)≡𝛾(7)⊗𝛾
(7),
(B.35)
{
𝚪
(13)
,𝚪
𝐌
}=0, 𝐂
−
=𝐂
+
𝚪
(13)
,𝐀
−
=𝐀
+
𝚪
(13).
(B.36)
𝚿
=±𝚪
(
13
)
𝚿, and
̄
𝚿=𝚿
†
𝐀
+
=𝚿T𝐂
+.
(C.1)
𝛿
AM=ī
ΨΓME=−ī
EΓMΨ,𝛿Ψ=
1
2
FMN ΓMN E+Δ
Ψ
,
(C.2)
𝛿
̄
Ψ=−
1
2
̄
EFMN ΓMN +ΔΨ
.
(C.3)
Ψ
p
𝛼
Ψq
𝛽
Ψr
𝛾
tr(TpTqTr)=Ψ
p
𝛾
Ψq
𝛼
Ψr
𝛽
tr(TpTqTr)
=Ψ
p𝛽Ψq𝛾Ψr𝛼tr(T
p
T
q
T
r
)
,
(C.4)
tr(̄
ΨΓ
M
Ψ
̄
ΨΓ
M
E
)
=
0.
(C.5)
̄
ΨΓ
MDM𝛿Ψ=
1
2DLFMN
̄
Ψ(ΓLMN +2𝜂LM ΓN)E+
1
2
̄
ΨΓLΓMN 𝜕LEFMN +
̄
ΨΓLDLΔ
Ψ
=−iDMFMN 𝛿AN+1
2
̄
ΨΓLΓMN 𝜕LEFMN +̄
ΨΓLDLΔΨ.

J.-H.Park
Vol.:(0123456789)
1 3
Thus, semi-finally, we obtain
We first note that constant
E
, and constant
ΔΨ
which is
central in the Lie algebra lead to the ordinary and kinetic
supersymmetries:
Henceforth, keeping the dimensional reduction either to
Minkowskian d-dimensions,
0≤m≤d−1
,
d≤a≤9
, or
Euclidean d-dimensions,
1≤m≤d
,
a=0
,
d+1≤a≤9
,
we set
Aa=Φ
a
, “
𝜕a≡0
”, and look for some possibilities
of more general symmetries.
Since
we first require
or equivalently
It follows after multiplying
Γnm
without m,n summing:
Equations (C.9),(C.10),(C.11) are trivial when
d=0, 1
. For
d≥2
, summing over
m≠n
in (C.11) we get
Hence, for
d=2, 3
,
d≥5
• For
d=3
,
d≥5
, we easily conclude
𝜕mE=0
, i.e., con-
stant parameter,
E
.
• When
d=2
, we get
so that
Let
𝜎≠𝜏
be the two different spacetime indices in
d=2
case. Eq.(C.9) is simply equivalent to
(C.6)
𝛿
L=−itr
[
1
2FMN
̄
ΨΓLΓMN 𝜕LE+̄
ΨΓLDLΔΨ
]
+𝜕Ntr
[
FMN 𝛿AM+i1
2
̄
ΨΓN𝛿Ψ
]
.
(C.7)
E,ΔΨ∶constant and ΔΨ∝1N×N.
(C.8)
FMN
Γ
L
Γ
MN
𝜕
L
E=
(
F
mn
Γ
l
Γ
mn
+2D
m
Φ
b
Γ
l
Γ
mb
+D
a
Φ
b
Γ
l
Γ
ab)
𝜕
l
E
,
(C.9)
Γl
Γ
mn𝜕l
E=
0,
(C.10)
ΓmnΓl𝜕l
E
=2Γm𝜕n
E
−2Γn𝜕m
E
.
(C.11)
Γl𝜕l
E
=2Γm𝜕m
E
+2Γn𝜕n
E
∶no sum for m≠n.
(C.12)
(d−1)(d−4)Γl𝜕l
E
=0.
(C.13)
Γm𝜕mE= −Γn𝜕nE∶no sum and m≠n.
(C.14)
𝜕mE= −Γmn𝜕nE∶for d=2,
(C.15)
𝜕m𝜕mE=0.
(C.16)
(𝜕𝜎+Γ
𝜎
𝜏𝜕𝜏)E=0.
This can be solved easily in the diagonal basis of
Γ𝜎
𝜏
.
In the Minkowskian two-dimensions, as
Γ0
1
is hermitian,
the solution is given by the left and right modes,
𝜎±𝜏
.
On the other hand, in the Euclidean two-dimensions,
Γ1
2
is anti-hermitian and the solution involves holomorphic
functions,
𝜎±i𝜏
.
• For
d=4
we have for any m:
From
𝜕[m𝜕n]E=0
we get an essentially same relation
as (C.13):
Hence,
𝜉−
is a constant spinor, and
where
𝜉+,𝜉−
are constant Majorana-Weyl spinors of
the opposite chiralities, corresponding to the ordinary
supersymmetry and special superconformal symmetry,
respectively.
Provided the above solutions for (C.9), we are ready for the
full analysis.
1. When
d=0
: IKKT matrix model. Equation (C.8)
becomes trivial, and we naturally require
We need to find the algebraic solution for
ΔΨ
in terms of
the Lie algebra valued fields,
Φa
,
d≤a≤9
. Clearly, the
kinetic supersymmetry transformation, i.e.,
ΔΨ∝1N×N
,
satisfies the above equation. In fact, we can show that
this is the most general solution.
Proof We consider the special case,
Φa=0
,
d≤a≤7
.
Eq.(C.20) gives
Multiplying
Φ8
and taking the
u(N)
trace we get
Since the commutator,
[Φ8,Φ9]
, can be arbitrary except
1N×N
,
we conclude that
ΔΨ∝1N×N
. This completes our proof.
Therefore, when
d=0
,
E
and
ΔΨ
are simply constant
Majorana–Weyl spinors corresponding to the ordinary and
the kinetic supersymmetries.
2. When
d=1
: BFSS matrix model. Equations(C.9)
is trivial, and with the coordinate,
𝜏
for
d=1
, From
Eq.(C.6) we require
(C.17)
𝜕
mE=Γ
m𝜉−,𝜉−=
1
4
Γl𝜕lE
.
(C.18)
Γm𝜕m𝜉−= −Γn𝜕n𝜉−∶no sum and m≠n.
(C.19)
E=xmΓm𝜉−+𝜉+,
(C.20)
Γa[Φa,ΔΨ]=0.
(C.21)
[Φ8,Γ8ΔΨ]+[Φ
9,Γ9ΔΨ]=0.
(C.22)
tr(
[Φ
8
,Φ
9
]Δ
Ψ)
=
0.

Lecture note onClifford algebra
Vol.:(0123456789)
1 3
The only possible algebraic solutions are (C.7)
corresponding to the ordinary and the kinetic
supersymmetries.
3. When
d=2
. From (C.6) we require, using (C.14),
(C.15),
We conclude again that the only possible algebraic solu-
tions are (C.7) corresponding to the ordinary and the
kinetic supersymmetries.
4. When
d=3
,
d≥5
. Since
E
is constant, the only pos-
sible algebraic solutions are (C.7) corresponding to the
ordinary and the kinetic supersymmetries.
5. When
d=4
. From (C.6) we require, using (C.19),
(C.23)
ll
0=
1
2FMN ΓLΓMN 𝜕LE+Γ
LDLΔ
Ψ
=Γ
𝜏D𝜏(ΔΨ+Φ
aΓ𝜏a𝜕𝜏E)
+Γ
bDb(ΔΨ−1
2ΦaΓ𝜏a𝜕𝜏E)
−Φ
a
Γa𝜕𝜏𝜕
𝜏
E.
(C.24)
0
=
1
2FMN ΓLΓMN 𝜕LE+Γ
LDLΔΨ
=Γ
m
DmΔΨ+Γ
a
DaΔΨ+2(D
𝜏
Φa−D
𝜎
ΦaΓ𝜎
𝜏
)Γ
a
𝜕𝜏
E
.
(C.25)
0
=
1
2FMN ΓLΓMN 𝜕LE+Γ
LDLΔ
Ψ
=Γ
LD
L
(Δ
Ψ
+2Φ
a
Γa
𝜉−
).
Thus, the algebraic solution reads
References
1. J. Strathdee, Extended Poincaré supersymmetry. Int. J. Mod. Phys.
A 2, 273 (1987)
2. T. Kugo, P. Townsend, Supersymmetry and the division algebras.
Nucl. Phys. B 221, 357 (1983)
3. J. Scherk, F. Gliozzi, D. Olive, Supersymmetry, supergravity theo-
ries and the dual spinor model. Nucl. Phys. B 122, 253 (1977)
4. H. Weyl, The classical groups (Princeton University Press, Prince-
ton, 1946)
5. B. Dewitt, Supermanifolds (Cambridge University Press, Cam-
bridge, 1984)
6. J.F. Cornwell, Group theory in physics, vol. III (Academic Press,
Cambridge, 1989)
7. Jeong-Hyuck. Park, Superconformal symmetry in six-dimensions
and its reduction to four-dimensions. Nucl. Phys. B 539, 599–642
(1999)
Publisher's Note Springer Nature remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
(C.26)
ΔΨ+2ΦaΓa𝜉−∝1N×N.