nLab Majorana spinor

Idea

A Majorana spin representation is essentially a real spin representation (see at spin representation – Real representations) but regarded as a complex spin representation equipped with real structure (recalled as def. below).

Accordingly a Majorana spinor or Majorana fermion is a spinor/fermion corresponding to such a representation under Wigner classification. None of the particles in the standard model of particle physics except possibly the neutrinos are Majorana fermions (for neutrinos this remains open). The relevance of Majorana representations is that these appear in supersymmetry, constituting for instance the odd-graded components of super-Minkowski spacetimes. See remark below.

The terminology Majorana spinor originates in and is standard in the physics literature, where it usually refers to the explicit expression of the reality condition in terms of chosen basis components. With standard conventions understood (see prop. below), then a complex spinor $\psi$ for $Spin(d-1,1)$, regarded as an element of $\mathbb{C}^{2^\nu}$ (with $d = 2 \nu, 2\nu+1$) is a Majorana spinor if it satisfies the condition

$$\psi^t C = \psi^\dagger \Gamma_0 \,,$$

where $(-)^T$ denotes forming transpose matrices, $(-)^\dagger = \overline{(-)}^T$ denotes forming hermitian adjoint and where $C$ is the charge conjugation matrix. This says that the Majorana conjugate (def. below) of $\psi$ (the left hand side) coincides with the “Dirac conjugate” (def. below) of $\psi$ (the right hand side). Equivalently this means that (e.g. Castellani-D’Auria-Fré, (II.7.22))

$$\psi = J(\psi) \coloneqq C \Gamma_0^T \psi^\ast \,,$$

where $J(-)$ is the given real structure (prop. below). See prop. below.

In some dimensions there are no complex spin representations with real structure, but there may be those with quaternionic structure. The corresponding physics jargon then is symplectic Majorana spinor.

Definition

Definition

Let $\rho \colon Spin(s,t) \longrightarrow GL_{\mathbb{C}}(V)$ be a unitary representation of a spin group. Then $\rho$ is called Majorana if it admits a real structure $J$ (def. ) and symplectic Majorana if it admits a quaternionic structure $J$ (def. ). An element $\psi \in V$ is called a Majorana spinor if $J(\psi) = \psi$.

In components

We work out in detail what def. comes down to in components (i.e. in terms of choices of linear bases), using standard notation and conventions from the physics literature (e.g. Castellani-D’Auria-Fré).

Conventions and Notation

In the following we use standard notation for operations on matrices with entries in the complex numbers (and of course these matrices may in particular be complex row/column vectors, which may in particular be single complex numbers):

• $(-)^\ast$ – componentwise complex conjugation;

• $(-)^T$ – transpose matrix

• $(-)^\dagger \coloneqq ((-)^\ast)^T = ((-)^T)^\ast$

• $A B$ for the matrix product of two matrices $A$ and $B$.

We will be discussing three different pairing operations on complex column vectors $\psi_1, \psi_2 \in \mathbb{C}^\nu$:

• $\psi_1^\dagger \psi_2$ – the standard hermitian form on $\mathbb{C}^\nu$, this will play a purely auxiliary role.

• $\langle \psi_1,\psi_2\rangle \coloneqq \overline{\psi}_1 \psi_2 \coloneqq \psi_1^\dagger \Gamma_0 \psi_2$ – the Dirac pairing, this will be the alternative hermitian form with respect to which the spin representation below is a unitary representation;

• $(\psi_1,\psi_2) \coloneqq \psi_1^T C \psi_2$ – the Majorana pairing (for $C$ the charge conjugation matrix), this turns out to coincide with the Dirac pairing above if $\psi_1$ is a Majorana spinor.

Then we use the following conventions on spacetime signature and the correspondig Clifford algebra:

Definition

We write $\mathbb{R}^{s,t}$ for the real vector space $\mathbb{R}^{s+t}$ of dimension $d = s + t$ equipped with the standard quadratic form $q$ of signature $(t,s)$ (“time”, “space”), i.e.

$$q(\vec x) \coloneqq (x^1)^2 + \cdots + (x^s)^2 - (x^{s+1})^2 - \cdots - (x^{s+t})^2 \,.$$

Hence the corresponding metric is

$$\eta = (\eta_{a b}) \coloneqq diag(\underset{t}{\underbrace{+1 , \cdots, +1}}, \underset{s}{\underbrace{-1, \cdots, -1}}) \,.$$

The real Clifford algebra $Cl(s,t)$ associated with this inner product space is the $\mathbb{R}$-algebra generated from elements $\{\Gamma_a\}_{0 = 1}^{s+t-1}$ subject to the relation

$$\Gamma_a \Gamma_b + \Gamma_b \Gamma_a = 2 \eta_{a b} \;\;\;\; \forall a,b \in \{0,1,\cdots, t+s-1\} \,.$$

For $n$-tuples $(a_i)_{i = 1}^n$ of indices we write

$$\Gamma_{a_1 \cdots a_n} \coloneqq \Gamma_{[a_1} \cdots \Gamma_{a_2]} \coloneqq \frac{1}{n!} \underset{\sigma}{\sum} (-1)^{\vert \sigma\vert} \Gamma_{a_{\sigma_1}} \cdots \Gamma_{a_{\sigma_n}}$$

for the skew-symmetrized product of Clifford generators with these indices. In partcular if all the $a_i$ are pairwise distinct, then this is simply the plain product of generators

$$\Gamma_{a_1 \cdots a_n} = \Gamma_{a_1} \cdots \Gamma_{a_n} \;\;\; \text{if} \; \underset{i,j}{\forall} (a_i \neq a_j) \,.$$

Indices are raised with $\eta^{-1} = (\eta^{a b})$ (which of course as a matrix coincides with $(\eta_{a b})$)

$$\Gamma^a \coloneqq \eta^{a b} \Gamma_b$$

Definition

The case $t = 1$ is that of Lorentzian signature.

In this case the single timelike Clifford genrator is $\Gamma_0$ and the remaining spatial Clifford generators are $\Gamma_1, \Gamma_2, \cdots, \Gamma_{d-1}$. So then

• $\Gamma^0 = \Gamma_0$ and $\Gamma_0^2 = + 1$;

• $\Gamma^a = - \Gamma_a$ and $\Gamma_a^2 = -1$ for $a \in \{1,\cdots, d-1\}$.

Dirac and Weyl representations

The following is a standard convention for the complex representation of the Clifford algebra for $\mathbb{R}^{s,1}$ (Castellani-D’Auria-Fré, (II.7.1)):

Proposition

(Dirac representation)

Let $t = 1$ (Lorentzian signature, def. ) and let

$$d = s + 1 \in \{ 2\nu, 2 \nu + 1 \} \;\;\;\; \text{for}\, \nu \in \mathbb{N}\,,\; d\geq 4 \,.$$

Then there is a choice of complex linear representation of the Clifford algebra $Cl(s,1)$ (def. ) on the complex vector space

$$V \coloneqq \mathbb{C}^{2^\nu}$$

such that

1. $\Gamma_{0}$ is hermitian

2. $\Gamma_{spatial}$ is anti-hermitian.

Moreover, the pairing

$$\langle -,-\rangle \coloneqq (-)^\dagger \Gamma_0 (-) \;\colon\; V \times V \longrightarrow \mathbb{C}$$

is a hermitian form (def. ) with respect to which the resulting representation of the spin group $\exp(\omega^{a b} \Gamma_{a b})$ is unitary:

$$\Gamma_0^{-1} \exp(\omega^{a b} \Gamma_{a b})^{\dagger} \Gamma_0 = \exp(\omega^{a b} \Gamma_{a b})^{-1} \,.$$

These representations are called the Dirac representations, their elements are called Dirac spinors.

Proof

In the case $d = 4$ consider the Pauli matrices $\{\sigma_{a}\}_{a = 0}^3$, defined by

$$\sigma_a x^a \coloneqq \left( \array{ x^0 + x^1 & x^2 + i x^3 \\ x^2 - i x^3 & x^0 - x^1 } \right) \,.$$

Then a Clifford representation as claimed is given by setting

$$\Gamma_0 \coloneqq \left( \array{ 0 & id \\ id & 0 } \right)$$

$$\Gamma_a \coloneqq \left( \array{ 0 & \sigma_a \\ -\sigma_a & 0 } \right) \,.$$

From $d = 4$ we proceed to higher dimension by induction, applying the following two steps:

odd dimensions

Suppose a Clifford representation $\{\gamma_a\}$ as claimed has been constructed in even dimension $d = 2 \nu$.

Then a Clifford representation in dimension $d = 2 \nu + 1$ is given by taking

$$\Gamma_a \coloneqq \left\{ \array{ \gamma_a & \vert \; a \leq d - 2 \\ \epsilon \gamma_0 \gamma_1 \cdots \gamma_{d-2} & \vert\; a = d-1 } \right.$$

where

$$\epsilon = \left\{ \array{ 1 & \vert \; \nu \, \text{odd} \\ i & \vert \; \nu \, \text{even} } \right. \,.$$

even dimensions

Suppose a Clifford representation $\{\gamma_a\}$ as claimed has been constructed in even dimension $d = 2 \nu$.

Then a corresponding representation in dimension $d+2$ is given by setting

$$\Gamma_{a \lt d} \coloneqq \left( \array{ 0 & \gamma_a \\ \gamma_a & 0 } \right) \;\;\,, \;\;\; \Gamma_{d} = \left( \array{ 0 & id \\ -id & 0 } \right) \;\;\,, \;\;\; \Gamma_{d+1} = \left( \array{ i \mathrm{id} & 0 \\ 0 & -i \mathrm{id} } \right) \,.$$

Finally regarding the statement that this gives a unitary representation:

That $\langle -,-\rangle \coloneqq (-)^\dagger \Gamma_0 (-)$ is a hermitian form follows since $\Gamma_0$ obtained by the above construction is a hermitian matrix.

Let $a,b \in \{1, \cdots, d-1\}$ be spacelike and distinct indices. Then by the above we have

$$\begin{aligned} \Gamma_0^{-1} (\Gamma_a \Gamma_b)^\dagger \Gamma_0 & = \Gamma_0^{-1} \Gamma_0 (\Gamma_b^\dagger \Gamma_a^\dagger) \\ & = (-\Gamma_b) (-\Gamma_a) \\ & = \Gamma_b \Gamma_a \\ & = - \Gamma_a \Gamma_b \end{aligned}$$

and

$$\begin{aligned} \Gamma_0^{-1} (\Gamma_0 \Gamma_a)^\dagger & = - \Gamma_0^{-1} \Gamma_0 \Gamma_a^\dagger \Gamma_0^\dagger \\ & = - (- \Gamma_a) (\Gamma_0) \\ & = \Gamma_a \Gamma_0 \\ & = - \Gamma_0 \Gamma_a \end{aligned} \,.$$

This means that the exponent of $\exp(\omega^{a b} \Gamma_a \Gamma_b)$ is an anti-hermitian matrix, hence that exponential is a unitary operator.

Definition

(Weyl representation)

Since by prop. the Dirac representations in dimensions $d = 2\nu$ and $d+1 = 2\nu+1$ have the same underlying complex vector space, the element

$$\Gamma_{d} \propto \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}$$

acts $Spin(d-1,1)$-invariantly on the representation space of the Dirac $Spin(d-1,1)$-representation for even $d$.

Moreover, since $\Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}$ squares to $\pm 1$, there is a choice of complex prefactor $c$ such that

$$\Gamma_{d+1} \coloneqq c \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}$$

squares to +1. This is called the chirality operator.

(The notation $\Gamma_{d+1}$ for this operator originates from times when only $d = 4$ was considered. Clearly this notation has its pitfalls when various $d$ are considered, but nevertheless it is commonly used this way e.g. Castellani-D’Auria-Fré, section (II.7.11) and top of p. 523).

Therefore this representation decomposes as a direct sum

$$V = V_+ \oplus V_-$$

of the eigenspaces $V_{\pm}$ of the chirality operator, respectively. These $V_{\pm}$ are called the two Weyl representations of $Spin(d-1,1)$. An element of these is called a chiral spinor (“left handed”, “right handed”, respectively).

Definition

For a Clifford algebra representation on $\mathbb{C}^\nu$ as in prop. , we write

$$\overline{(-)} \coloneqq (-)^\dagger \Gamma_0 \;\colon\; Mat_{\nu \times 1}(\mathbb{C}) \longrightarrow Mat(1 \times \nu)(\mathbb{C})$$

for the map from complex column vectors to complex row vectors which is hermitian congugation $(-)^\dagger = ((-)^\ast)^T$ followed by matrix multiplication with $\Gamma_0$ from the right.

This operation is called Dirac conjugation.

In terms of this the hermitian form from prop. (Dirac pairing) reads

$$\langle -,-\rangle = \overline{(-)}(-) \,.$$

Proposition

The operator adjoint $\overline{A}$ of a $\nu \times \nu$-matrix $A$ with respect to the Dirac pairing of def. , characterized by

$$\langle A (-), (-)\rangle = \langle - , \overline{A} -\rangle \;\;\;\text{and} \;\;\; \langle -, A -\rangle = \langle \overline{A} - , -\rangle$$

is given by

$$\overline{A} = \Gamma_0^{-1} A^\dagger \Gamma_0 \,.$$

All the Clifford generators from prop. are Dirac self-conjugate in that

$$\overline{\Gamma}_a = \Gamma_a \,.$$

Proof

For the first claim consider

$$\begin{aligned} \langle A \psi_1, \psi_2\rangle & = \psi_1^\dagger A^\dagger \Gamma_0 \psi_2 \\ & = \psi_1^\dagger \Gamma_0 (\Gamma_0^{-1} A^\dagger \Gamma_0) \psi_2 \\ & = \langle \psi_1, (\Gamma_0^{-1} A \Gamma_0)\psi_2\rangle \end{aligned} \,.$$

and

$$\begin{aligned} \langle \psi_1, A \psi_2\rangle & = \psi_1^\dagger \Gamma_0 A \psi_2 \\ & = \psi_1^\dagger \Gamma_0 A \Gamma_0^{-1} \Gamma_0 \psi_2 \\ & = ( (\Gamma_0^{-1})^\dagger A^\dagger (\Gamma_0)^\dagger \psi_1 )^\dagger \Gamma_0 \psi_2 \\ & = ( \Gamma_0^{-1} A^\dagger \Gamma_0 \psi_1 )^\dagger \Gamma_0 \psi_2 \\ &= \langle \overline{A} \psi_1, \psi_2\rangle \end{aligned} \,,$$

where we used that $\Gamma_0^{-1} = \Gamma_0$ (by def. ) and $\Gamma_0^\dagger = \Gamma_0$ (by prop. ).

Now for the second claim, use def. and prop. to find

$$\begin{aligned} \overline{\Gamma}_0 & = \Gamma_0^{-1}\Gamma_0^\dagger \Gamma_0 \\ & = \Gamma_0^{-1} \Gamma_0 \Gamma_0 \\ & = \Gamma_0 \end{aligned}$$

and

$$\begin{aligned} \overline{\Gamma}_{spatial} & = \Gamma_0^{-1} \Gamma_{spatial}^\dagger\Gamma_0 \\ &= - \Gamma_0^{-1} \Gamma_{spatial} \Gamma_0 \\ & = + \Gamma_0^{-1} \Gamma_0 \Gamma_{spatial} \\ &= \Gamma_{spatial} \end{aligned} \,.$$

Charge conjugation matrix

Proposition

Given the Clifford algebra representation of the form of prop. , consider the equation

$$C_{(\pm)} \Gamma_a = \pm \Gamma_a^T C_{(\pm)}$$

for $C_{(\pm)} \in Mat_{\nu \times n}(\mathbb{C})$.

In even dimensions $d = 2 \nu$ then both these equations have a solution, wheras in odd dimensions $d = 2 \nu + 1$ only one of them does (alternatingly, starting with $C_{(+)}$ in dimension 5). Either $C_{(\pm)}$ is called the charge conjugation matrix.

Moreover, all $C_{(\pm)}$ may be chosen to be real matrices

$$(C_{(\pm)})^\ast = C_{(\pm)}$$

and in addition they satisfy the following relations:

$d$
4	$C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$	$C_{(-)}^T = -C_{(+)}$; $C_{(-)}^2 = -1$
5	$C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$
6	$C_{(+)}^T = -C_{(+)}$; $C_{(+)}^2 = -1$	$C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$
7		$C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$
8	$C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$	$C_{(-)}^T = C_{(-)}$; $C_{(-)}^2 = 1$
9	$C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$
10	$C_{(+)}^T = C_{(+)}$; $C_{(+)}^2 = 1$	$C_{(-)}^T = -C_{(-)}$; $C_{(-)}^2 = -1$
11		$C_{(-)}^T = -C_{(-)}$; $C_{(-)}^2 = -1$

(This is for instance in Castellani-D’Auria-Fré, section (II.7.2), table (II.7.1), but beware that there $C_{(-)}$ in $d = 10, 11$ is claimed to be symmetric, while instead it is anti-symmetric as shown above, see van Proeyen 99, table 1, Laenen, table E.3).

Remark

Prop. implies that for all $C_{(\pm)}$ listed there then

$$C^{-1} = C^T \,.$$

This implies in all cases that

$$\Gamma_a C_{(\pm)}^T = \pm C_{(\pm)}^T \Gamma_a^T \,.$$

Majorana representations and Real structure

Proposition

For $d \in \{4,8,9,10,11\}$, let $V = \mathbb{C}^\nu$ as above. Write $\{\Gamma_a\}$ for a Dirac representation according to prop. , and write

$$C \coloneqq \left\{ \array{ C_{(-)} & \text{for}\; d = 4 \\ C_{(+)} & \text{for}\; d = 8 \\ C_{(+)} & \text{for}\; d = 9 \\ C_{(+)} or C_{(-)} & \text{for}\; d = 10 \\ C_{(-)} & \text{for}\; d = 11 } \right.$$

for the choice of charge conjugation matrix from prop. as shown. Then the function

$$J \colon V \longrightarrow V$$

given by

$$\psi \mapsto C \Gamma_0^T \psi^\ast$$

is a real structure (def. ) for the corresponding complex spin representation on $\mathbb{C}^\nu$.

Proof

The conjugate linearity of $J$ is clear, since $(-)^\ast$ is conjugate linear and matrix multiplication is complex linear.

To see that $J$ squares to +1 in the given dimensions: Applying it twice yields,

$$\begin{aligned} J^2 \psi &= C \Gamma_0^T (C \Gamma_0^T\psi^\ast)^\ast \\ & = C \Gamma_0^T C \Gamma_0^\dagger \psi \\ &= C \underset{= \pm C \Gamma_0}{\underbrace{\Gamma_0^T C}} \Gamma_0 \psi \\ & = \pm C_{(\pm)}^2 \Gamma_0^2 \psi \\ & = \pm C_{(\pm)}^2 \psi \end{aligned} \,,$$

where we used $\Gamma_0^\dagger = \Gamma_0$ from prop. , $C^\ast = \ast$ from prop. and then the defining equation of the charge conjugation matrix $\Gamma_a^T C_{(\pm)} = \pm C_{(\pm)} \Gamma_a$ (def. ), finally the defining relation $\Gamma_0^2 = +1$.

Hence this holds whenever there exists a choice $C_{(\pm)}$ for the charge conjugation matrix with $C_{(\pm)}^2 = \pm 1$. Comparison with the table from prop. shows that this is the case in $d = 4,8,9,10,11$.

Finally to see that $J$ is spin-invariant (in Castellani-D’Auria-Fré this is essentially (II.2.29)), it is sufficient to show for distinct indices $a,b$, that

$$J(\Gamma_a \Gamma_b \psi) = \Gamma_a \Gamma_b J(\psi) \,.$$

First let $a,b$ both be spatial. Then

$$\begin{aligned} J(\Gamma_a \Gamma_b \psi) & = C \Gamma_0^T \Gamma_a^\ast \Gamma_b^\ast \psi^\ast \\ & = C \Gamma_0^T (-\Gamma_a^T)(-\Gamma_b^T) \psi^\ast \\ & = C \Gamma_0^T \Gamma_a^T \Gamma_b^T \psi^\ast \\ & = C \Gamma_a^T \Gamma_b^T \Gamma_0^T \psi^\ast \\ & = \Gamma_a \Gamma_b C \Gamma_0^T \psi^\ast \\ & = \Gamma_a \Gamma_b J(\psi) \end{aligned} \,.$$

Here we first used that $\Gamma_{spatial}^\dagger = -\Gamma_{spatial}$ (prop. ), hence that $\Gamma_{spatial}^\ast = - \Gamma_{spatial}^T$ and then that $\Gamma_0$ anti-commutes with the spatial Clifford matrices, hence that $\Gamma_0^T$ anti-commutes the the transposeso fthe spatial Clifford matrices. Then we used the defining equation for the charge conjugation matrix, which says that passing it through a Gamma-matrix yields a transpose, up to a global sign. That global sign cancels since we pass through two Gamma matrices.

Finally, that the same conclusion holds for $\Gamma_{spatial} \Gamma_{spatial}$ replaced by $\Gamma_0 \Gamma_{spatial}$: The above reasoning applies with two extra signs picked up: one from the fact that $\Gamma_0$ commutes with itself, one from the fact that it is hermitian, by prop. . These two signs cancel:

$$\begin{aligned} J(\Gamma_0 \Gamma_a \psi) & = C \Gamma_0^T \Gamma_0^\ast \Gamma_a^\ast \psi^\ast \\ & = C \Gamma_0^T (+\Gamma_0^T)(-\Gamma_a^T) \psi^\ast \\ & = - C \Gamma_0^T \Gamma_0^T \Gamma_a^T \psi^\ast \\ & = + C \Gamma_0^T \Gamma_a^T \Gamma_0^T \psi^\ast \\ & = \Gamma_0 \Gamma_a \Gamma_0^T \psi^\ast \\ &= \Gamma_0 \Gamma_a J(\psi) \end{aligned} \,.$$

Definition

Prop. implies that given a Dirac representation (prop. ) $V$, then the real subspace $S \hookrightarrow V$ of real elements, i.e. elements $\psi$ with $J \psi = \psi$ according to prop. is a sub-representation. This is called the Majorana representation inside the Dirac representation (if it exists).

Proposition

If $C = C_{(\pm)}$ is the charge conjugation matrix according to prop. , then the real structure $J$ from prop. commutes or anti-commutes with the action of single Clifford generators, according to the subscript of $C = C_{(\pm)}$:

$$J(\Gamma_a(-)) = \pm \Gamma_a J(-) \,.$$

Proof

This is same kind of computation as in the proof prop. . First let $a$ be a spatial index, then we get

$$\begin{aligned} J(\Gamma_a \psi) & = C \Gamma_0^T \Gamma_a^\ast \psi^\ast \\ & = C \Gamma_0^T (-\Gamma_a^T) \psi^\ast \\ & = + C \Gamma_a^T \Gamma_0^T \psi^\ast \\ & = \epsilon C^T \Gamma_a^T \Gamma_0^T \\ & = \pm \epsilon \Gamma_a C^T \Gamma_0^T \psi^\ast \\ & = \pm \epsilon^2 \Gamma_a C \Gamma_0^T \psi^\ast \\ & = \pm \Gamma_a J(\psi) \end{aligned} \,,$$

where, by comparison with the table in prop. , $\epsilon$ is the sign in $C^T = \epsilon C$, which cancels out, and the remaining $\pm$ is the sign in $C = C_{(\pm)}$, due to remark .

For the timelike index we similarly get:

$$\begin{aligned} J(\Gamma_0 \psi) & = C \Gamma_0^T \Gamma_0^\ast \psi^\ast \\ & = + C \Gamma_0^T \Gamma_0^T \psi^\ast \\ & = \epsilon C^T \Gamma_0^T \Gamma_0^T \\ & = \pm \epsilon \Gamma_0 C^T \Gamma_0^T \psi^\ast \\ & = \pm \Gamma_0 C \Gamma_0^T \psi^\ast \\ & = \pm \Gamma_0 J(\psi) \end{aligned} \,.$$

We record some immediate consequences:

Proposition

The complex bilinear form

$$(-,-) \coloneqq \langle J(-),(-)\rangle$$

induced via the real structure $J$ of prop. from the hermitian form $\langle -,-\rangle$ of prop. is that represented by the charge conjugation matrix of prop.

$$(-,-) = (-)^T C (-) \,.$$

Proof

By direct unwinding of the various definitions and results from above:

$$\begin{aligned} \langle J(\psi_1),\psi_2 \rangle &= \langle C \Gamma_0^T\psi_1^\ast, \psi_2\rangle \\ & = (C \Gamma_0^T \psi_1^\ast)^\dagger \Gamma_0 \psi_2 \\ & = \psi_1^T C^\dagger \Gamma_0^\ast \Gamma_0 \psi_2 \\ & = \psi_1^T C \psi_2 \end{aligned} \,.$$

Definition

For a Clifford algebra representation on $\mathbb{C}^\nu$ as in prop. , then the map

$$(-)^T C \;\colon\; Mat_{\nu \times 1}(\mathbb{C}) \longrightarrow Mat_{1 \times \nu}(\mathbb{C})$$

(from complex column vectors to complex row vectors) which is given by transposition followed by matrix multiplication from the right by the charge conjugation matrix according to prop. is called the Majorana conjugation.

Proposition

In dimensions $d = 4,8,9,10,11$ a spinor $\psi \in \mathbb{C}^{2^\nu}$ is Majorana according to def. with respect to the real structure from prop. , precisely if

$$\psi = C \Gamma_0^T \psi^\ast$$

(as e.g. in Castellani-D’Auria-Fré, (II.7.22)),

This is equivalent to the condition that the Majorana conjugate (def. ) coincides with the Dirac conjugate (def. ) on $\psi$:

$$\psi^T C = \psi^\dagger \Gamma_0 \,,$$

which in turn is equivalent to the condition that

$$(\psi,-) = \langle \psi,-\rangle \,,$$

where on the left we have the complex bilinear form of prop. and on the right the hermitian form from prop. .

Proof

The first statement is immediate. The second follows by applying the transpose to the first equation, and using that $C^{-1} = C^T$ (from prop. ). Finally the last statement follows from this by prop. .

Pseudo-Majorana spinors and Symplectic structure

In $d = 5$, for example, the reality/Majorana condition

$$\psi = C \Gamma_0^T \psi^\ast$$

from prop. has no solution. But if we consider the direct sum of two copies of the complex spinor representation space, with elements denoted $\psi_1$ and $\psi_2$, then the following condition does have a solution

$$C \Gamma_0^T \psi_1^\ast = -\psi_2 \;\;\;\; C \Gamma_0^T \psi_2^\ast = +\psi_1$$

(e.g Castellani-D’Auria-Fré, II.8.41). Comparison with prop. and def. shows that this exhibits a quaternionic structure on the original complex spinor space, and hence a real structure on its direct sum double.

Majorana-Weyl spinors

Definition

In the even dimensions among those dimensions $d$ for which the Majorana projection operator (real structure) $J$ exists (prop. ) also the chirality projection operator $\Gamma_{d}$ exists (def. ). Then we may ask that a Dirac spinor $\psi$ is both Majorana, $J(\psi) = \psi$, as well as Weyl, $\Gamma_d \psi = \pm i \psi$. If this is the case, it is called a Majorana-Weyl spinor, and the sub-representation these form is a called a Majorana-Weyl representation.

Proposition

In Lorentzian signature (def. ) for $4 \leq d \leq 11$, then Majorana-Weyl spinors (def. ) exist precisely only in $d = 10$.

Proof

According to prop. Majorana spinors in the given range exist for $d \in \{4,8,9,10,11\}$. Hence the even dimensions among these are $d \in \{4,8,10\}$.

Majorana-Weyl spinors clearly exist precisely if the two relevant projection operators in these dimensions commute with each other, i.e. if

$$[J, \epsilon \Gamma_0 \cdots \Gamma_{d-1}] = 0$$

where

$$\epsilon = \left\{ \array{ 1 & \vert \; \nu \, \text{odd} \\ i & \vert \; \nu \, \text{even} } \right. \,.$$

with $d = 2\nu$ (from the proof of prop. ).

By prop. all the $\Gamma_a$ commute or all anti-commute with $J$. Since the product $\Gamma_0 \cdots \Gamma_{d-1}$ contains an even number of these, it commutes with $J$. It follows that $J$ commutes with $\Gamma_d$ precisely if it commutes with $\epsilon$. Now since $J$ is conjugate-linear, this is the case precisely if $\epsilon = 1$, hence precisely if $d = 2\nu$ with $\nu$ odd.

This is the case for $d = 10 = 2 \cdot 5$, but not for $d = 8 = 2 \cdot 4$ neither for $d = 4 = 2 \cdot 2$.

The spinor bilinear pairing to antisymmetric $p$-tensors

We now discuss, in the component expressions established above, the complex bilinear pairing operations that take a pair of Majorana spinors to a vector, and more generally to an antisymmetric rank $p$-tensor. These operations are all of the form

$$\psi \mapsto \epsilon \, \overline{\psi} \Gamma^{a_1 \cdots a_p} \psi \,,$$

where $\epsilon \in \mathbb{C}$ is some prefactor, constrained such as to make the whole expression be real, hence such as to make this the components of an element in $\wedge^p \mathbb{R}^{d-1,1}$.

For a $Spin(d-1,1)$ representation $V$ as in prop. , with real/Majorana structure as in prop. , write

$$S \hookrightarrow V$$

for the subspace of Majorana spinors, regarded as a real vector space.

Recall, by prop. , that on Majorana spinors the Majorana conjugate $(-)^T C$ coincides with the Dirac conjugate $\overline{(-)} \coloneqq (-)^\dagger \Gamma_0$. Therefore we write $\overline{(-)}$ in the following for the conjugation of Majorana spinors, unambiguously defined.

Definition

For a $Spin(d-1,1)$ representation $V$ as in prop. , with real/Majorana structure as in prop. , let

$$\overline{(-)}\Gamma (-) \;\colon\; S \times S \longrightarrow \mathbb{R}^{d-1,1}$$

be the function that takes Majorana spinors $\psi_1$, $\psi_2$ to the vector with components

$$\overline{\psi}_1\Gamma^a \psi_2 \coloneqq \psi_1^T C \Gamma^a \psi_2 \,.$$

Proposition

The pairing in def. is

1. symmetric:

   $$\overline{\psi}_1 \Gamma \psi_2 = \overline{\psi}_2 \Gamma \psi_1$$

2. component-wise real-valued (i.e. it indeed takes values in $\mathbb{R}^d \subset \mathbb{C}^d$);

3. $Spin(d-1,1)$-equivariant: for $g \in Spin(d-1,1)$ then

   $$\overline{g(-)}\Gamma(g(-)) = g(\overline{(-)}\Gamma(-)) \,.$$

Proof

Regarding the first point, we need to show that for all $a$ then $C \Gamma_a$ is a symmetric matrix. Indeed:

$$\begin{aligned} (C \Gamma_a)^T & = \Gamma_a^T C^T \\ & = \pm \Gamma_a^T C \\ & = \pm \pm C \Gamma_a \\ & = C \Gamma_a \end{aligned} \,,$$

where the first sign picked up is from $C^T = \pm C$, while the second is from $\Gamma_a^T C = \pm C \Gamma_a$ (according to prop. ). Imposing the condition in prop. one finds that these signs agree, and hence cancel out.

(In van Proeyen99 this is part of table 1, in (Castellani-D’Auria-Fré) this is implicit in equation (II.2.32a).)

With this the second point follows together with the relation $\psi^T C = \psi^\dagger \Gamma_0$ for Majorana spinors $\psi$ (prop. ) and using the conjugate-symmetry of the hermitian form $\langle -,-\rangle = (-)^\dagger \Gamma_0 (-)$ as well as the hermiticity of $\Gamma_0 \Gamma_a$ (both from prop. ):

$$\begin{aligned} (\overline{\psi}_1 \Gamma_a \psi_2)^\ast &= (\psi_1^T C \Gamma_a \psi_2)^\ast \\ & = (\psi_1^\dagger \Gamma_0 \Gamma^a \psi_2)^\ast \\ & = \psi_2^\dagger (\Gamma_0 \Gamma^a)^\dagger \psi_1 \\ & = \psi_2^\dagger \Gamma_0 \Gamma^a \psi_1 \\ & = \overline{\psi}_2 \Gamma_a \psi_1 \end{aligned} \,.$$

Regarding the third point: By prop. and prop. we get

$$\begin{aligned} (g(\psi_1), \Gamma_a g(\psi_2)) & = \langle g(\psi_1),\Gamma_a g(\psi_2)\rangle \\ & = \langle \psi_1, (\Gamma_0^{-1}g^\dagger\Gamma_0) \Gamma_a g \psi_2 \rangle \\ & = \langle \psi_1 (g^{-1} \Gamma_a g) \psi_2 \rangle \end{aligned} \,,$$

where we used that $\Gamma_0^{-1}(-)^\dagger \Gamma_0$ is the adjoint with respect to the hermitian form $\langle -,-\rangle = (-)^\dagger \Gamma_0 (-)$ (prop. ) and that $g$ is unitary with respect to this hermitian form by prop. .

(In (Castellani-D’Auria-Fré) this third statement implicit in equations (II.2.35)-(II.2.39).)

Remark

Proposition implies that adding a copy of $S$ to the Poincaré Lie algebra in odd degree, then the pairing of def. is a consistent extension of the Lie bracket of the latter to a super Lie algebra. This is the super Poincaré Lie algebra.

Definition

For a $Spin(d-1,1)$ representation $V$ as in prop. , with real/Majorana structure as in prop. , let

$$\overline{(-)}\Gamma\Gamma (-) \;\colon\; S \times S \longrightarrow \wedge^2 \mathbb{C}^d$$

be the function that takes Majorana spinors $\psi_1$, $\psi_2$ to the skew-symmetric rank 2-tensor with components

$$\overline{\psi}_1\Gamma^{a b} \psi_2 \coloneqq i \psi_1^T C \tfrac{1}{2}(\Gamma^a \Gamma^b - \Gamma^b \Gamma^a) \psi_2 \,,$$

Proposition

For $\psi_1 = \psi_2 = \psi$ then the pairing in prop. is real

$$\underset{a,b}{\forall} \;\;\;\; i \overline{\psi} \Gamma^{a b} \psi \in \mathbb{R} \subset \mathbb{C}$$

and $Spin(d-1,1)$-equivariant.

Proof

The equivariance follows exactly as in the proof of prop. .

The reality is checked by direct computation as follows:

$$\begin{aligned} (\overline{\psi}_1 \Gamma_a \Gamma_b \psi_2)^\ast & = (\psi_1^\dagger \Gamma_a \Gamma_b \psi_2)^\ast \\ & = \psi_2^\dagger (\Gamma_0 \Gamma_a \Gamma_b)^\dagger \psi_1 \\ & = -\langle \psi_2^\dagger \Gamma_0 \Gamma_a \Gamma_b \psi_1 \rangle \\ & = -\overline{\psi}_2 \Gamma_a \Gamma_b \psi_1 \end{aligned} \,,$$

where for the identification $(\Gamma_0 \Gamma_a \Gamma_b)^\dagger = - \Gamma_0 \Gamma_a \Gamma_b$ we used the properties in prop. .

Hence for $\psi_1 = \psi_2$ then

$$(\overline{\psi} \Gamma_a \Gamma_b \psi)^\ast = - \overline{\psi} \Gamma_a \Gamma_b \psi$$

and so this sign cancels against the sign in $i^\ast = -i$.

Generally:

Proposition

The following pairings are real and $Spin(d-1,1)$-equivariant:

$$\begin{aligned} & \overline{\psi} \Gamma_a \psi \\ i & \overline{\psi}\Gamma_{a_1 a_2} \psi \\ i & \overline{\psi} \Gamma_{a_1 a_2 a_3} \psi \\ & \overline{\psi} \Gamma_{a_1 \cdots a_4} \psi \\ & \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \\ i & \overline{\psi} \Gamma_{a_1 \cdots a_6} \psi \\ i & \overline{\psi} \Gamma_{a_1 \cdots a_7} \psi \\ & \vdots \end{aligned} \,.$$

Proof

The equivariance follows as in the proof of prop. .

Regarding reality:

Using that all the $\Gamma_a$ are hermitian with respect $\overline{(-)}(-)$ (prop. ) we have

$$\begin{aligned} \left( \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi \right)^\ast & = \overline{\psi} \Gamma_{a_p \cdots a_1} \psi \\ &= (-1)^{p(p-1)/2} \overline{\psi} \Gamma_{a_1 \cdots a_p} \psi \end{aligned} \,.$$

Supersymmetry: Super-Poincaré and super-Minkowski

Every real spin representation of $Spin(d-1,1)$ induces a super Lie algebra extension of the Poincaré Lie algebra $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ in that dimension, i.e. of the Lie algebra of the isometry group of the Minkowski spacetime (def. ) in that dimension.

Since we may recover a Minkowski spacetime from its Poincaré Lie algebra as the (vector space underlying the) coset of the Poincaré Lie algebra by the Lie algebra $\mathfrak{so}(d-1,1)$ of the spin group (the orthogonal Lie algebra in Lorentian signature)

$$\mathbb{R}^{d-1,1} \simeq \mathfrak{Iso}(\mathbb{R}^{d-1,1})/\mathfrak{so}(d-1,1)$$

(namely as the Lie algebra of translations along itself), every super Lie algebra extension of the Poincaré Lie algebra defines a super Lie algebra extension of Minkowski spacetime. These extensions are the super Minkowski spacetimes $\mathbb{R}^{d-1,1\vert N}$ of the following definition, and this justifies the following notation:

Definition

Let $d \in \mathbb{N}$ and let $N \in Rep(Spin(d-1,1))$ be a real spin representation, hence a direct sum of Majorana representations (def. ) and/or Majorana-Weyl representations (def. ) (or tensor product of two symplectic Majorana representations…).

We define the corresponding super Poincaré Lie algebra

$$\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$$

equivalently in terms of its Chevalley-Eilenberg algebra: $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ (a $\mathbb{N} \times \mathbb{Z}/2$-bigraded dg-algebra, see at signs in supergeometry).

This is generated on

• elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

• and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen Majorana spinor representation $N$.

The CE-differential is defined as follows

$$d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}$$

and

$$d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi \,.$$

(which so far is the differential for the semidirect product of the Poincaré Lie algebra acting on the given Majorana spinor representation)

and

$$d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \overline{\psi} \Gamma^a \psi$$

where on the right we have the spinor-to-vector pairing in $N$ (def. ).

That this is indeed a super Lie algebra follows from the fact that the Poincaré Lie algebra is a Lie algebra and the fact that the spinor-to-vector pairing is symmetric (which makes it qualify as the odd-odd component of a super-Lie algebra) and $Spin(d-1,1)$-equivariant (which is seen to be the super-Jacobi identity for it), all according to prop. .

This defines the super Poincaré super Lie algebra. After discarding the terms involving $\omega$ this becomes the CE algebra of the super translation algebra underlying super Minkowski spacetime

$$\mathbb{R}^{d-1,1\vert N} \,.$$

Examples

In dimensions 11, 10, and 9

We spell out some of the above constructions and properties for Majorana spinors in Lorentzian spacetimes (def. ) of dimensions 11, 10 and 9, and discuss some relations between these. These spinor structures are relevant for spinors in 11-dimensional supergravity and type II supergravity in 10d and 9d, as well as to the relation between these via Kaluza-Klein compactification and T-duality.

Proposition

Let $\{\gamma_a\}$ be any Dirac representation of $Spin(8,1)$ according to prop. . By the same logic as in the proof of prop. we get from this the Dirac representations in dimensions 9+1 and 10+1 by setting

$$\Gamma_{a \leq 8} \coloneqq \left( \array{ 0 & \gamma_a \\ \gamma_a & 0 } \right) \;\,,\;\; \Gamma_{9} \coloneqq \left( \array{ 0 & id \\ -id & 0 } \right) \;\,,\;\; \Gamma_{10} \coloneqq \left( \array{ i id & 0 \\ 0 & -i id } \right) \,.$$

Remark

By prop. the Dirac representation in $d = 11$ has complex dimension $2^{10/2} = 2^{5} = 32$. By prop. and prop. this representation carries a real structure and hence gives a real/Majorana spin representation $S \hookrightarrow \mathbb{C}^{32}$ of $Spin(10,1)$ of real dimension 32. This representation often just called “$\mathbf{32}$”. This way the corresponding super-Minkowski spacetime (remark ) is neatly written as

$$\mathbb{R}^{10,1\vert \mathbf{32}}$$

which thus serves to express both, the real dimension of the space of odd-graded coordinate functions at every point on it, as well as the way that the $Spin(10,1)$-cover of the Lorentz group $SO(10,1)$ acts on these. This is the local model space for super spacetimes in 11-dimensional supergravity.

As we regard $\mathbb{C}^{32}$ instead as the Dirac representation of $Spin(9,1)$ via def. , then it decomposes into to chiral halfs, each of complex dimension 16. This is the direct sum decomposition in terms of which the block decomposition of the above Clifford matrices is given.

Since in 10d the Weyl condition is compatible with the Majorana condition (by prop. ), the real Majorana representation $\mathbf{32}$ correspondingly decomposes as a direct sum two real representations of dimension 116 which often are denoted $\mathbf{16}$ and $\overline{\mathbf{16}}$. Hence as real/Majorana $Spin(9,1)$-representations there is a direct sum decomposition

$$\mathbf{32} \simeq \mathbf{16} \oplus \overline{\mathbf{16}} \,.$$

The corresponding super Minkowski spacetime (remark )

$$\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$$

is said to be of “type IIA”, since this is the local model space for superspacetimes in type IIA supergravity. This is as opposed to $\mathbb{R}^{9,1\vert \mathbf{16}\oplus \mathbf{16}}$, which is “type IIB” and in contrast to $\mathbb{R}^{9,1\vert \mathbf{16}}$ which is “heterotic” (the local model space for heterotic supergravity).

Now the Dirac-Weyl representation for $Spin(8,1)$ is of complex dimension $d = 2^{8/2} = 2^4 = 16$. By prop. and prop. this also admits real structure, and hence gives a Majorana representation fro $Spin(8,1)$, accordingly denoted $\mathbf{16}$. Notice that this is Majorana-Weyl.

We want to argue that both the $\mathbf{16}$ and the $\overline{\mathbf{16}}$ of $Spin(9,1)$ become isomorphic to the single $\mathbf{16}$ of $Spin(8,1)$ under forming the restricted representation along the inclusion $Spin(8,1)\hookrightarrow Spin(9,1)$ (the one fixed by the above choice of components).

For this it is sufficient to see that $\Gamma_9$, which as a complex linear map goes $\Gamma_9 \colon \mathbf{16} \longrightarrow \overline{\mathbf{16}}$ constitutes an isomorphism when regarded as a morphism in the category of representations of $Spin(8,1)$.

Clearly it is a linear isomorphism, so it is sufficient that it is a homomorphism of $Spin(8,1)$-representations at all. But that’s clear since by the Clifford algebra relations $\Gamma_9$ commutes with all $\Gamma_a \Gamma_b$ for $a,b \on \{0,\cdots, 8\}$.

Hence the branching rule for restricting the Weyl representation in 11d along the sequence of inclusions

$$Spin(8,1) \hookrightarrow Spin(9,1) \hookrightarrow Spin(10,1)$$

is

$$\underset{\in Rep(Spin(10,1))}\underbrace{\mathbf{32}} \mapsto \underset{\in Rep(Spin(9,1))}\underbrace{\mathbf{16} \oplus \overline{\mathbf{16}}} \mapsto \underset{\in Rep(Spin(8,1))}\underbrace{\mathbf{16} \oplus \mathbf{16}} \,.$$

If we write a Majorana spinor in $\mathbf{32}$ as $\vartheta \in \mathbb{C}^{32}$, decomposed as a $(1 \times 32)$-matrix as

$$\vartheta = \left( \array{ \psi_1 \\ \psi_2 } \right) \,.$$

and if we write for short

$$\psi_1 = \left( \array{ \psi_1 \\ 0 } \right) \,,\;\;\; \psi_2 = \left( \array{ 0 \\ \psi_2 } \right)$$

then this says that after restriction to $Spin(9,1)$-action then $\psi_1$ becomes a Majorana spinor in the $\mathbf{16}$, and $\psi_2$ a Majorana spinor in the $\overline{\mathbf{16}}$, and after further restriction to $Spin(8,1)$-action then either comes a Majorana spinor in one copy of $\mathbf{16}$.

The type IIA spinor-to-vector pairing is just that of 11d under this re-interpretation. We find:

Proposition

The type IIA spinor-to-vector pairing is given by

$$\begin{aligned} \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_a^{IIA} \left( \array{\psi_1 \\ \psi_2} \right) & = \left\{ \array{ \overline{\psi}_1 \gamma_a \psi_1 + \overline{\psi}_2 \gamma_a \psi_2 & \vert \; a \leq 8 \\ \overline{\psi}_1 \psi_1 - \overline{\psi}_2 \psi_2 & \vert \; a = 9 } \right. \end{aligned} \,.$$

Proof

Using that on Majorana spinors the Majorana conjugate coincides with the Dirac conjugate (prop. ) and applying prop. we compute:

$$\begin{aligned} \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_a^{IIA} \left( \array{\psi_1 \\ \psi_2} \right) &\coloneqq \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_a \left( \array{\psi_1 \\ \psi_2} \right) \\ & = \left(\array{\psi_1 \\ \psi_2}\right)^\dagger \Gamma_0 \Gamma_a \left( \array{\psi_1 \\ \psi_2} \right) \\ & = \left\{ \array{ \left(\array{\psi_1 \\ \psi_2}\right)^\dagger \left( \array{ \gamma_0 \gamma_a & 0 \\ 0 & \gamma_0 \gamma_a } \right) \left( \array{\psi_1 \\ \psi_2} \right) & \vert \; a\leq 8 \\ \left(\array{\psi_1 \\ \psi_2}\right)^\dagger \left( \array{ \gamma_0 & 0 \\ 0 & -\gamma_0 } \right) \left( \array{\psi_1 \\ \psi_2} \right) & \vert \; a = 9 } \right. \\ & = \left\{ \array{ \overline{\psi}_1 \gamma_a \psi_1 + \overline{\psi}_2 \gamma_a \psi_2 & \vert \; a \leq 8 \\ \overline{\psi}_1 \psi_1 - \overline{\psi}_2 \psi_2 & \vert \; a = 9 } \right. \end{aligned} \,.$$

Proposition

The type IIB spinor-to-vector pairing is

$$\begin{aligned} \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_a^{IIB} \left(\array{\psi_1 \\ \psi_2}\right) & = \left\{ \array{ \overline{\psi}_1 \gamma_a \psi_1 + \overline{\psi}_2 \gamma_a \psi_2 & \vert \; a \leq 8 \\ \overline{\psi}_1 \psi_1 + \overline{\psi}_2 \psi_2 & \vert \; a = 9 } \right. \end{aligned}$$

Proof

The type II pairing spinor-to-vector pairing is obtained from the type IIA pairing of prop. by replacing all bottom right matrix entries (those going $\overline{\mathbf{16}}\to \overline{\mathbf{16}}$ by the corresponding top left entries (those going $\mathbf{16} \to \mathbf{16}$ )). Notice that in fact all these block entries are the same, except for the one at $a = 9$, where they simply differ by a sign. This yields the claim.

Notice also the following relation between the different pairing in dimensions 11, 10 and 9:

Proposition

The $(9,10)$-component of the spinor-to-bivector pairing (def. ) in 11d equals the 9-component of the type IIB spinor-to-vector pairing

$$\begin{aligned} i \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_9 \Gamma_{10} \left(\array{\psi_1 \\ \psi_2}\right) & = \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_9^{IIB} \left(\array{\psi_1 \\ \psi_2}\right) \end{aligned}$$

Proof

Using prop. and prop. we compute:

$$\begin{aligned} i\, \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_9 \Gamma_{10} \left(\array{\psi_1 \\ \psi_2}\right) & = i \left(\array{\psi_1 \\ \psi_2}\right)^\dagger \Gamma_0\Gamma_9 \Gamma_{10} \left(\array{\psi_1 \\ \psi_2}\right) \\ & = i \, \left(\array{\psi_1 \\ \psi_2}\right)^\dagger \left( \array{ -i \gamma_0 & 0 \\ 0 & -i \gamma_0 } \right) \left(\array{\psi_1 \\ \psi_2}\right) \\ & = \overline{\psi}_1 \psi_1 + \overline{\psi}_2 \psi_2 \\ & = \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_9^{IIB} \left(\array{\psi_1 \\ \psi_2}\right) \end{aligned}$$

The following is an evident variant of the extensions considered in (CAIB 99, FSS 13).

Proposition

We have

1. The 11d $N = 1$ super-Minkowski spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}$ (def. ) is the central super Lie algebra extension of the 10d type IIA super-Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ by the 2-cocycle

   $$c_2 \coloneqq \overline{\psi} \wedge \Gamma_{10} \psi \;\;\; \in CE(\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}})$$

   (from def. ).

2. The 10d type IIA super-Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ is central super Lie algebra extension of th 9d $N = 2$ super-Minkowski spacetime by the 2-cocycle given by the type IIA spinor-to-vector pairing

   $$c_2^{IIA} \;\coloneqq\; \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \wedge \Gamma_9^{IIA} \left( \array{\psi_1 \\ \psi_2} \right) \;\;\;\in CE(\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}})$$

   (from prop. ).

3. The 10d type IIB super-Minkowski spacetime $\mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}}$ is central super Lie algebra extension of th 9d $N = 2$ super-Minkowski spacetime by the 2-cocycle given by the type IIB spinor-to-vector pairing

   $$c_2^{IIB} \;\coloneqq\; \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \wedge \Gamma_9^{IIB} \left( \array{\psi_1 \\ \psi_2} \right) \;\;\; \in CE(\mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}})$$

   (from prop. ).

In summary, we have the following diagram in the category of super L-infinity algebras

$$\array{ && && \mathbb{R}^{10,1\vert \mathbf{32}} \\ && && \downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16} + \mathbf{16}} && && \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} &\overset{c_2}{\longrightarrow}& B \mathbb{R} \\ & \searrow && \swarrow \\ && \mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}} \\ & {}^{\mathllap{c_2^{IIB}}}\swarrow && \searrow^{\mathrlap{c_2^{IIA}}} \\ B \mathbb{R} && && B \mathbb{R} } \,,$$

where $B\mathbb{R}$ denotes the line Lie 2-algebra, and where each “hook”

$$\array{ \widehat{\mathfrak{g}} \\ \downarrow \\ \mathfrak{g} &\overset{\omega_2}{\longrightarrow}& B\mathbb{R} }$$

is a homotopy fiber sequence (because homotopy fibers of super $L_\infty$-algebra cocycles are the corresponding extension that they classify, see at L-infinity algebra cohomology).

Proof

To see that the given 2-forms are indeed cocycles: they are trivially closed (by def. ), and so all that matters is that we have a well defined super-2-form in the first place. Since the $\psi^\alpha$ are in bidegree $(1,odd)$, they all commutes with each other (see at signs in supergeometry) and hece the condition is that the pairing is symmetric. This is the case by prop. .

Now to see the extensions. Notice that for $\mathfrak{g}$ any (super) Lie algebra (of finite dimension, for convenience), and for $\omega \in \wedge^2\mathfrak{g}^\ast$ a Lie algebra 2-cocycle on it, then the Lie algebra extension $\widehat{\mathfrak{g}}$ that this classifies is neatly characterized in terms of its dual Chevalley-Eilenberg algebra: that is simply the original CE algebra with one new generator $e$ (in degree $(1,even)$) adjoined, and with the differential of $e$ taking to be $\omega$:

$$CE(\widehat{\mathfrak{g}}) = (CE(\mathfrak{g}) \otimes \langle e\rangle), d e = \omega) \,.$$

Hence in the case of $\omega = c_2^{IIA}$ we identify the new generator with $e^9$ and see that the equation $d e^9 = c_2^{IIA}$ is precisely what distinguishes the CE-algebra of $\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}}$ from that of $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}$, by prop. and the fact that both spin representation have the same underlying space, by remark .

The other two cases are directly analogous.

Recall the following (e.g. from FSS 16 and references given there):

Definition

The cocycle for the higher WZW term of the Green-Schwarz sigma-model for the M2-brane is

$$\mu_{M2} \coloneqq i\,\overline{\vartheta} \wedge \Gamma_a \Gamma_b \vartheta \wedge e^a \wedge e^b \;\;\; \in CE(\mathbb{R}^{10,1\vert \mathbf{32}})$$

obtained from the spinor-to-bivector pairing of def. . (Here and in the following we are using the nation from remark .)

The cocycle for the WZW term of the Green-Schwarz sigma-model for the type IIA superstring is

$$\mu_{IIA} \coloneqq i\,\overline{\vartheta} \wedge \Gamma_a \Gamma_{10} \vartheta \wedge e^a = i\, \overline{\left( \array{\psi_1 \\ \psi_2} \right)} \Gamma_a \Gamma_{10} \left( \array{\psi_1 \\ \psi_2} \right) \;\;\; \in CE(\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}) \,,$$

i.e. this is the the $e^{10}$-component of $\mu_{M2}$ (“double dimensional reduction” FSS 16):

$$\mu_{IIA} = (\pi_{10})_\ast \mu_{M2} \,.$$

Proposition

The $e^9$-component of the cocycle for the IIA-superstring (def. ), regarded as an element in $CE(\mathbb{R}^{8,1}\vert \mathbf{16} + \mathbf{16})$, equals the 2-cocycle that defines the type IIB extension, according to prop. :

$$(\pi_9)_\ast \mu_{IIA} = c_2^{IIB} \,.$$

Proof

We have

$$\begin{aligned} (\pi_9)_\ast \mu_{IIA} & = i\, \overline{\left( \array{\psi_1 \\ \psi_2} \right)} \Gamma_9 \Gamma_{10} \left( \array{\psi_1 \\ \psi_2} \right) \\ & = \overline{\left(\array{\psi_1 \\ \psi_2}\right)} \Gamma_9^{IIB} \left(\array{\psi_1 \\ \psi_2}\right) \\ & = c_2^{IIB} \end{aligned}$$

where the first equality is by def. , the second is the statement of prop. , while the third is from prop. .

Properties of spinors in 11d

Consider the real irrep $\mathbf{32} \,\in\, Rep_{\mathbb{R}}\big(Spin(10,1)\big)$ from above.

First a couple of general statements about Clifford algebra:

Consider the standard shorthand

$$\Gamma_{a_1 \cdots a_p} \;\coloneqq\; \Gamma_{[a_1} \cdots \Gamma_{a_p]} \;:=\; \tfrac{1}{p!} \underset{ \sigma }{\sum} (-1)^{\vert\sigma\vert} \, \Gamma_{a_{\sigma(1)}} \cdots \Gamma_{a_{\sigma(p)}} \,.$$

Lemma

With the above conventions we have.

(1)$$\Gamma_{a_1 \cdots a_{11}} \;=\; \epsilon_{a_1 \cdots a_{11}} \cdot 1_{\mathbf{32}} \,.$$

Lemma

$$\Gamma^{a_j \cdots a_1} \, \Gamma_{b_1 \cdots b_k} \;=\; \sum_{l = 0}^{ min(j,k) } \pm l! \Big( { j \atop l } \Big) \Big( { k \atop l } \Big) \, \delta ^{[a_1 \cdots a_l} _{[b_1 \cdots b_l} \Gamma^{a_j \cdots a_{l+1}]} {}_{b_{l+1} \cdots b_k]} \,.$$

(e.g. Miemiec & Schnakenburg 2006, Prop. 2)

Proof

Observe that if the $a$-indices are not pairwise distinct or the $b$-indices are not pairwise distinct then both sides of the equation are zero.

Hence assume next that the indices are separately pairwise distinct, and consider their sets $A \coloneqq \{a_1, \cdots, a_j\}$, $B \coloneqq \{b_1, \cdots, b_k\}$ and their intersection $C := A \cap B$, with cardinality $\mathrm{card}(C) = l$. The idea is to recursively contract one pair $(\Gamma^c , \Gamma_{c})$ with $c \in C$ at a time. We claim that in the first step this can be written as

$$\Gamma^{a_j \cdots a_1} \Gamma_{b_1 \cdots b_k} \;=\; \frac{j k}{l} \, \Gamma^{[a_j \cdots a_2} \delta^{a_1]}_{[b_1} \Gamma_{b_2 \cdots b_k]} \,.$$

Namely, notice that for any tensor $X^{a_1 \cdots a_k}$ the expression $k X^{[a_k \cdots a_1]}$ is the signed sum over all ways of moving any one index to the far right, and similarly $l Y^{[b_1 \cdots b_l]}$ is the signed sum over all ways of moving any one index to the far left. In contracting all the indices that thus become coincident “in the middle” of our expression, we are contracting the one index that we set out to contract, but since we are doing this for all $c \in C$ we are overcounting by a factor of $l$.

In order to conveniently recurse on this expression, we just move the Kronecker-delta to the left to obtain

$$\Gamma^{a_j \cdots a_1} \Gamma_{b_1 \cdots b_k} \;=\; (-1)^{j-1} \frac{j k}{l} \, \delta^{[a_1}_{[b_1} \Gamma^{a_j \cdots a_2]} \Gamma_{b_2 \cdots b_k]} \,.$$

Now recursing, we arrive at

$$\Gamma^{a_j \cdots a_1} \Gamma_{b_1 \cdots b_k} \;=\; (-1)^{(j-1) \cdots (j-l)} \underbrace{ \frac{ j \cdots (j-l) \, k \cdots (k-l) }{l!} }_{ l! \Big( { k \atop l } \Big) \Big( { j \atop l } \Big) } \, \delta ^{[a_1 \cdots a_l} _{[b_1 \cdots b_l} \underbrace{ \Gamma^{a_j \cdots a_{l+1}]} \Gamma_{b_{l+1} \cdots b_{k}]} }_{ \Gamma ^{ a_j \cdots a_{l+1}] } {}_{ b_{j + l} \cdots b_k] } }$$

Under the brace on the far right we use that by assumption no further contraction is possible. With the subsitution under the brace made, the right hand side can just as well be summed over $l$, since it gives zero whenever $l \,\neq\, \mathrm{card}(C)$. This yields the claimed formula.

Proposition

Every $\mathbb{R}$-linear endomorphism on $\mathbf{32}$ $M \;\in\; \mathrm{End}_{\mathbb{R}}(\mathbf{32})$ may be expanded as:

$$M \;=\; \tfrac{1}{32} \sum_{p = 0}^5 \; \frac{ (-1)^{p(p-1)/2} }{ p! } \mathrm{Tr}\big( \phi \circ \Gamma_{a_1 \cdots a_p} \big) \Gamma^{a_1 \cdots a_p}$$

(e.g. Miemiec & Schnakenburg 2006 (2.61))

Proposition

(D’Auria & Fré 1982, (3.1-3)) The $Spin(10,1)$-irrep decomposition of the first few symmetric tensor powers of $\mathbf{32}$ is of the form:

(2)$$\begin{array}{rcl} \big( \mathbf{32} \otimes \mathbf{32} \big)_{\mathrm{symm}} &\simeq& \mathbf{11} \,\oplus\, \mathbf{55} \,\oplus\, \mathbf{462} \\ \big( \mathbf{32} \otimes \mathbf{32} \otimes \mathbf{32} \big)_{\mathrm{symm}} &\simeq& \mathbf{32} \,\oplus\, \mathbf{320} \,\oplus\, \mathbf{1408} \,\oplus\, \mathbf{4424} \\ \big( \mathbf{32} \otimes \mathbf{32} \otimes \mathbf{32} \otimes \mathbf{32} \big)_{\mathrm{symm}} &\simeq& \mathbf{1} \,\oplus\, \mathbf{165} \,\oplus\, \mathbf{330} \,\oplus\, \mathbf{462} \,\oplus\, \mathbf{65} \,\oplus\, \mathbf{429} \,\oplus\, \mathbf{1144} \,\oplus\, \mathbf{17160} \,\oplus\, \mathbf{32604} \,, \end{array}$$

where on the right each boldface summand is one irrep of that dimension.

Remark

(application in 11d supergravity)
In D=11 supergravity the gravitino-field is a super 1-form $\psi \in \Omega^1_{dR}(-;\mathbf{32}_1)$ of de Rham degree 1 with coefficients in a copy of $\mathbf{32}$ in odd degree. By the super-sign rule this means that these forms actuall commute with each other, hence that the $Spin(10,1)$-equivariant linear combinations of wedge products of these gravitino fields are $Spin(10,1)$-equivariant maps out of symmetric powers of $\mathbf{32}$ seen on the left of (2).

Now Schur's lemma implies that such cobinations can be non-zero only if they take values in the irreps appearing on the right of (2).

For example, the only $Spin(10,1)$-equiariant map $(\mathbf{32} \otimes \mathbf{32})_{sym} \to \mathbf{32}$ must be zero, which implies for instance that the gravitino-field strength $\rho \in \Omega^2_{dR}(-;\mathbf{32}_{odd})$, which on a local super-chart may be expanded as

$$\rho \;=\; \rho_{a b} e^a\, e^b + H_a \psi \, e^a + \underbrace{ \overline{\psi}\kappa\psi }_{ = 0} \,,$$

has to have vanishing last component $\kappa$, as show, since $\kappa$ has to be pointwise just such a $Spin(10,1)$-equivariant map $\big(\mathbf{32} \otimes \mathbf{32}\big)_{sym} \to \mathbf{32}$.

Proposition

The spinor pairing

(3)

(using the 4-component octonionic-spinor notation on the right)

is:

1. bi-linear

2. $Spin(10,1)$-equivariant

3. skew-symmetric.

Proposition

The followig quadratic forms on $\psi \in \mathbf{32}$ vanish:

$$\begin{array}{r} \overline{\psi} \psi \;=\;0 \\ \overline{\psi} \Gamma_{[a_1 a_2 a_3]} \psi \;=\; 0 \\ \overline{\psi} \Gamma_{[a_1 \cdots a_4]} \psi \;=\; 0 \\ \overline{\psi} \Gamma_{[a_1 \cdots a_7]} \psi \;=\; 0 \\ \overline{\psi} \Gamma_{[a_1 \cdots a_8]} \psi \;=\; 0 \\ \overline{\psi} \Gamma_{[a_1 \cdots a_{11}]} \psi \;=\; 0 \mathrlap{\,,} \end{array}$$

and so on.

Conversely and using (1), all non-trivial quadratic forms on $\mathbf{32}$ are linear combinations of the following ones:

(4)$$\begin{array}{ll} \big( \overline{\psi} \Gamma_a \psi \big) & \left( 11 \atop 1 \right) \;=\; 11 \\ \big( \overline{\psi} \Gamma_{a b} \psi \big) & \left( 11 \atop 2 \right) \;=\; 55 \\ \big( \overline{\psi} \Gamma_{a_1 \cdots a_5} \psi \big) & \left( 11 \atop 5 \right) \;=\; 462 \end{array}$$

and by (2) all of these are nontrivial.

Proof

With the skew-symmetry of the spinor pairing (3) we compute as follows:

$$\begin{array}{ll} \big( \overline{\psi} \Gamma_{[a_1 \cdots a_p]} \phi \big) & \;\coloneqq\; \mathrm{Re}\big( \psi^\dagger \Gamma_0 \Gamma_{[a_1 \cdots a_p]} \phi \big) \\ & \;=\; - \mathrm{Re}\Big( \phi^\dagger \big(\Gamma_{[a_1 \cdots a_p]}\big)^\dagger \Gamma_0 \psi \Big) \\ & \;=\; - \mathrm{Re}\Big( \phi^\dagger \Gamma_0 \Gamma^{-1}_0 \big(\Gamma_{[a_1 \cdots a_p]}\big)^\dagger \Gamma_0 \psi \Big) \\ & \;=\; - (-1)^{p + p(p-1)/2} \mathrm{Re}\Big( \phi^\dagger \Gamma_0 \Gamma_{[a_1 \cdots a_p]} \psi \Big) \\ & \;=\; -(-1)^{p(p+1)/2} \big( \overline{\phi} \,\Gamma_{[a_1 \cdots a_p]}\, \psi \big) \;. \end{array}$$

Proposition

(Fierz identities controlling D=11 supergravity)
The following quartic expressions in $\pin \in \mathbf{32}$ vanish:

(5)$$\begin{array}{r} \big( \overline{\psi} \Gamma_{a b} \psi \big) \big( \overline{\psi} \,\Gamma^{a}\, \psi \big) \;=\; 0 \\ \big( \overline{\psi} \Gamma_{a b_1\cdots b_4} \psi \big) \big( \overline{\psi} \,\Gamma^{a}\, \psi \big) \;+\; 3 \, \big( \overline{\psi} \,\Gamma_{b_1 b_2}\, \psi \big) \big( \overline{\psi} \,\Gamma_{b_3 b_4}\, \psi \big) \;=\; 0 \end{array}$$

Proof

On the first expression: This is the quartic diagonal of a $\mathrm{Spin}(1,10)$-equivariant map

$$\big( \mathbf{32} \,\otimes\, \mathbf{32} \,\otimes\, \mathbf{32} \,\otimes\, \mathbf{32} \big)_{\mathrm{symm}} \longrightarrow \mathbf{11} \,.$$

But by (2) the irrep summand $\mathbf{11}$ does not appear on the left, hence this map has to vanish by Schur's lemma (D’Auria & Fré 1982, (3.13)). For the second expression one needs a closer analysis (D’Auria & Fré 1982, (3.28a); also Naito, Osada & Kukui 1986, (2.27) and (2.28)), for more details see this example at geometry of physics – fundamental super p-branes.

Appendix

Review of unitary representations with real structure

For reference, we here collect some basics regarding unitary representations equipped with real structure.

All vector spaces in the following are taken to be finite dimensional vector spaces.

Definition

Let $V$ be a complex vector space. A real structure or quaternionic structure on $V$ is a real-linear map

$$\phi \;\colon\; V \longrightarrow V$$

such that

1. $\phi$ is conjugate linear ($\phi(\lambda v) = \overline{\lambda} \phi(v)$ for all $\lambda \in \mathbb{C}$, $v \in V$);

2. $\phi^2 = \left\{ \array{ +id & \text{for real structure} \\ -id & \text{for quaternionic structure} } \right.$

Remark

A real structure $\phi$, def. , on a complex vector space $V$ corresponds to a choice of complex linear isomorphism

$$V \simeq \mathbb{C} \otimes_{\mathbb{R}} V_+$$

of $V$ with the complexification of a real vector space $V_+$, namely the eigenspace of $\phi$ for eigenvalue +1, while $V_- \coloneqq i V_+$ is the eigenspace of eigenvalue -1.

A quaternionic structure, def. , o $V$ gives it the structure of a left module over the quaternions extending the underlying structure of a module over the complex numbers. Namely let

1. $I \coloneqq i(-) \colon V \to V$ be the operation of acting with $i \in \mathbb{C}$

2. $J \coloneqq \phi \colon V \to V$ be the given endomorphisms,

then the conjugate complex linearity of $\phi$ means that

$$J \circ I = - I \circ J$$

and hence with $J^2 = -1$ and $I^2 = -1$ this means that $I$, $J$ and $K \coloneqq I \circ J$ act like the imaginary quaternions.

Definition

Let $G$ be a Lie group, let $V$ be a complex vector space and let

$$\rho \;\colon\; G \longrightarrow GL_{\mathbb{C}}(V)$$

be a complex linear representation of $G$ on $V$, hence a group homomorphism form $G$ to the general linear group of $V$ over $\mathbb{C}$.

Then a real structure or quaternionic structure on $(V,\rho)$ is a real or complex structure, respectively, $\phi$ on $V$ (def. ) such that $\phi$ is $G$-invariant under $\rho$, i.e. such that for all $g \in G$ then

$$\phi \circ \rho(g) = \rho(g) \circ \phi \,.$$

We will be interested in complex finite dimensional vector spaces equipped with hermitian forms, i.e. finite-dimensional complex Hilbert spaces:

Definition

A hermitian form (or symmetric complex sesquilinear form) $\langle -,-\rangle$ on a complex vector space $V$ is a real bilinear form

$$\langle -,- \rangle \;\colon\; V \times V \longrightarrow \mathbb{C}$$

such that for all $v_1, v_2 \in V$ and $\lambda \in \mathbb{C}$ then

1. (sesquilinearity) $\langle v_1, \lambda v_2 \rangle = \lambda \langle v_1, v_2 \rangle$,

2. (conjugate symmetry) $\langle v_1, v_2\rangle^\ast = \langle v_2, v_1\rangle$.

3. (non-degeneracy) if $\langle v_1,-\rangle = 0$ then $v_1 = 0$.

A complex linear function $f \colon V \to V$ is unitary with respect to this hermitian form if it preserves it, in that

$$\langle f(-), f(-)\rangle = \langle -,-\rangle \,.$$

Write

$$U(V) \hookrightarrow GL_{\mathbb{C}}(V)$$

for the subgroup of unitary operators inside the general linear group.

A complex linear representation $\rho \colon G \longrightarrow GL_{\mathbb{C}}(V)$ of a Lie group on $V$ is called a unitary representation if it factors through this subgroup

$$\rho \;\colon\; G \longrightarrow U(V) \hookrightarrow GL_{\mathbb{C}}(V) \,.$$

The following proposition uses assumptions stronger than what we have in the application to Majorana spinors (compact Lie group, positive definite hermitian form) but it nevertheless helps to see the pattern.

Proposition

Let $V$ be a complex finite dimensional vector space, $\langle -,-\rangle$ some positive definite hermitian form on $V$, def. , let $G$ be a compact Lie group, and $\rho \colon G \to U(V)$ a unitary representation of $G$ on $V$. Then $\rho$ carries a real structure or quaternionc structure $\phi$ on $\rho$ (def. ) precisely if it carries a symmetric or anti-symmetric, respectively, non-degenerate complex-bilinear map

$$(-,-) \;\colon\; V \otimes_{\mathbb{C}} V \longrightarrow \mathbb{C} \,.$$

Explicitly:

Given a real/quaternionic structure $\phi$, then the corresponding symmetric/anti-symmetric complex bilinear form is

$$(-,-) \coloneqq \langle \phi(-), -\rangle \,.$$

Conversely, given $(-,-)$, first define $\tilde \phi$ by

$$(-,-) = \langle \tilde\phi(-),-\rangle \,,$$

and then $\phi \coloneqq \frac{1}{\vert \phi\vert} \phi$ is the corresponding real/quaternionic structure.

If $\tilde\phi = \phi$ then $(-,-)$ is called compatible with $\langle-,- \rangle$.

(e.g. Meinrenken 13, p. 81)

Related concepts

• charge conjugation matrix

• only vaguely related: Majorana zero modes in solid state physics

References

Named after Ettore Majorana.

The traditional component discussion in terms of a charge conjugation matrix is discussed for instance in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.7.3 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

• Antoine Van Proeyen, section 3 of Tools for supersymmetry (arXiv:hep-th/9910030)

• Eric Laenen, Classical Field Theory (web) chapter on Gamma matrices (pdf)

The relation to the concept of real structures on complex Spin-representations is highlighted in

• José Figueroa-O'Farrill, Majorana spinors [pdf, pdf]

See also

• Eckhard Meinrenken, Clifford algebras and Lie theory, Springer (2013)

• Theodor Bröcker, Tammo tom Dieck, Representations of Compact Lie Groups, Springer (1985)

• Wikipedia, Majorana fermion

The above discussion of cocycles on super-Minkowski spacetimes draws from

• C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces, Nuclear Physics B Volume 567, Issues 1–2, 14 February 2000, Pages 293–330 (arXiv:hep-th/9904137)

• Makoto Sakaguchi, section 2 of IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Rational sphere valued supercocycles in M-theory, (arXiv:1606.03206)