Majorana spinor in nLab

Context

Representation theory

Higher spin geometry

• Idea
• Definition
• In components
• Conventions and Notation
• Dirac and Weyl representations
• Charge conjugation matrix
• Majorana representations and Real structure
• Pseudo-Majorana spinors and Symplectic structure
• Majorana-Weyl spinors
• The spinor bilinear pairing to antisymmetric pp-tensors
• Supersymmetry: Super-Poincaré and super-Minkowski
• Examples
• In dimensions 11, 10, and 9
• Properties of spinors in 11d
• Appendix
• Review of unitary representations with real structure
• Related concepts
• References

Definition

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

Definition

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

q(x→)≔(x 1) 2+⋯+(x s) 2−(x s+1) 2−⋯−(x s+t) 2. 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

η=(η ab)≔diag(+1,⋯,+1⏟t,−1,⋯,−1⏟s). \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)Cl(s,t) associated with this inner product space is the ℝ\mathbb{R}-algebra generated from elements {Γ a} 0=1 s+t−1\{\Gamma_a\}_{0 = 1}^{s+t-1} subject to the relation

Γ aΓ b+Γ bΓ a=2η ab∀a,b∈{0,1,⋯,t+s−1}. \Gamma_a \Gamma_b + \Gamma_b \Gamma_a = 2 \eta_{a b} \;\;\;\; \forall a,b \in \{0,1,\cdots, t+s-1\} \,.

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

Γ a 1⋯a n≔Γ [a 1⋯Γ a 2]≔1n!∑σ(−1) |σ|Γ a σ 1⋯Γ a σ n \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 ia_i are pairwise distinct, then this is simply the plain product of generators

Γ a 1⋯a n=Γ a 1⋯Γ a nif∀i,j(a i≠a j). \Gamma_{a_1 \cdots a_n} = \Gamma_{a_1} \cdots \Gamma_{a_n} \;\;\; \text{if} \; \underset{i,j}{\forall} (a_i \neq a_j) \,.

Definition

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

In this case the single timelike Clifford genrator is Γ 0\Gamma_0 and the remaining spatial Clifford generators are Γ 1,Γ 2,⋯,Γ d−1\Gamma_1, \Gamma_2, \cdots, \Gamma_{d-1}. So then

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

• Γ a=−Γ a\Gamma^a = - \Gamma_a and Γ a 2=−1\Gamma_a^2 = -1 for a∈{1,⋯,d−1}a \in \{1,\cdots, d-1\}.

Proposition

(Dirac representation)

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

d=s+1∈{2ν,2ν+1}forν∈ℕ,d≥4. 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)Cl(s,1) (def. ) on the complex vector space

V≔ℂ 2 ν V \coloneqq \mathbb{C}^{2^\nu}

such that

1. Γ 0\Gamma_{0} is hermitian

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

Moreover, the pairing

⟨−,−⟩≔(−) †Γ 0(−):V×V⟶ℂ \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(ω abΓ ab)\exp(\omega^{a b} \Gamma_{a b}) is unitary:

Γ 0 −1exp(ω abΓ ab) †Γ 0=exp(ω abΓ ab) −1. \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=4d = 4 consider the Pauli matrices {σ a} a=0 3\{\sigma_{a}\}_{a = 0}^3, defined by

σ ax a≔(x 0+x 1 x 2+ix 3 x 2−ix 3 x 0−x 1). \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

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

Γ a≔(0 σ a −σ a 0). \Gamma_a \coloneqq \left( \array{ 0 & \sigma_a \\ -\sigma_a & 0 } \right) \,.

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

odd dimensions

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

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

Γ a≔{γ a |a≤d−2 ϵγ 0γ 1⋯γ d−2 |a=d−1 \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

ϵ={1 |νodd i |νeven. \epsilon = \left\{ \array{ 1 & \vert \; \nu \, \text{odd} \\ i & \vert \; \nu \, \text{even} } \right. \,.

even dimensions

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

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

Γ a<d≔(0 γ a γ a 0),Γ d=(0 id −id 0),Γ d+1=(iid 0 0 −iid). \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 ⟨−,−⟩≔(−) †Γ 0(−)\langle -,-\rangle \coloneqq (-)^\dagger \Gamma_0 (-) is a hermitian form follows since Γ 0\Gamma_0 obtained by the above construction is a hermitian matrix.

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

Γ 0 −1(Γ aΓ b) †Γ 0 =Γ 0 −1Γ 0(Γ b †Γ a †) =(−Γ b)(−Γ a) =Γ bΓ a =−Γ aΓ b \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

Γ 0 −1(Γ 0Γ a) † =−Γ 0 −1Γ 0Γ a †Γ 0 † =−(−Γ a)(Γ 0) =Γ aΓ 0 =−Γ 0Γ a. \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(ω abΓ aΓ b)\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=2νd = 2\nu and d+1=2ν+1d+1 = 2\nu+1 have the same underlying complex vector space, the element

Γ d∝Γ 0Γ 1⋯Γ d−1 \Gamma_{d} \propto \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}

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

Moreover, since Γ 0Γ 1⋯Γ d−1\Gamma_0 \Gamma_1 \cdots \Gamma_{d-1} squares to ±1\pm 1, there is a choice of complex prefactor cc such that

Γ d+1≔cΓ 0Γ 1⋯Γ d−1 \Gamma_{d+1} \coloneqq c \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}

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

(The notation Γ d+1\Gamma_{d+1} for this operator originates from times when only d=4d = 4 was considered. Clearly this notation has its pitfalls when various dd 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 +⊕V − V = V_+ \oplus V_-

of the eigenspaces V ±V_{\pm} of the chirality operator, respectively. These V ±V_{\pm} are called the two Weyl representations of Spin(d−1,1)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 ℂ ν\mathbb{C}^\nu as in prop. , we write

(−)¯≔(−) †Γ 0:Mat ν×1(ℂ)⟶Mat(1×ν)(ℂ) \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 (−) †=((−) *) T(-)^\dagger = ((-)^\ast)^T followed by matrix multiplication with Γ 0\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 A¯\overline{A} of a ν×ν\nu \times \nu-matrix AA with respect to the Dirac pairing of def. , characterized by

⟨A(−),(−)⟩=⟨−,A¯−⟩and⟨−,A−⟩=⟨A¯−,−⟩ \langle A (-), (-)\rangle = \langle - , \overline{A} -\rangle \;\;\;\text{and} \;\;\; \langle -, A -\rangle = \langle \overline{A} - , -\rangle

is given by

A¯=Γ 0 −1A †Γ 0. \overline{A} = \Gamma_0^{-1} A^\dagger \Gamma_0 \,.

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

Γ¯ a=Γ a. \overline{\Gamma}_a = \Gamma_a \,.

Proof

For the first claim consider

⟨Aψ 1,ψ 2⟩ =ψ 1 †A †Γ 0ψ 2 =ψ 1 †Γ 0(Γ 0 −1A †Γ 0)ψ 2 =⟨ψ 1,(Γ 0 −1AΓ 0)ψ 2⟩. \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

⟨ψ 1,Aψ 2⟩ =ψ 1 †Γ 0Aψ 2 =ψ 1 †Γ 0AΓ 0 −1Γ 0ψ 2 =((Γ 0 −1) †A †(Γ 0) †ψ 1) †Γ 0ψ 2 =(Γ 0 −1A †Γ 0ψ 1) †Γ 0ψ 2 =⟨A¯ψ 1,ψ 2⟩, \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 Γ 0 −1=Γ 0\Gamma_0^{-1} = \Gamma_0 (by def. ) and Γ 0 †=Γ 0\Gamma_0^\dagger = \Gamma_0 (by prop. ).

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

Γ¯ 0 =Γ 0 −1Γ 0 †Γ 0 =Γ 0 −1Γ 0Γ 0 =Γ 0 \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

Γ¯ spatial =Γ 0 −1Γ spatial †Γ 0 =−Γ 0 −1Γ spatialΓ 0 =+Γ 0 −1Γ 0Γ spatial =Γ spatial. \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} \,.

Proposition

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

C (±)Γ a=±Γ a TC (±) C_{(\pm)} \Gamma_a = \pm \Gamma_a^T C_{(\pm)}

for C (±)∈Mat ν×n(ℂ)C_{(\pm)} \in Mat_{\nu \times n}(\mathbb{C}).

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

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

(C (±)) *=C (±) (C_{(\pm)})^\ast = C_{(\pm)}

and in addition they satisfy the following relations:

Proposition

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

C≔{C (−) ford=4 C (+) ford=8 C (+) ford=9 C (+)orC (−) ford=10 C (−) ford=11 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:V⟶V J \colon V \longrightarrow V

given by

ψ↦CΓ 0 Tψ * \psi \mapsto C \Gamma_0^T \psi^\ast

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

Proof

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

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

J 2ψ =CΓ 0 T(CΓ 0 Tψ *) * =CΓ 0 TCΓ 0 †ψ =CΓ 0 TC⏟=±CΓ 0Γ 0ψ =±C (±) 2Γ 0 2ψ =±C (±) 2ψ, \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 Γ 0 †=Γ 0\Gamma_0^\dagger = \Gamma_0 from prop. , C *=*C^\ast = \ast from prop. and then the defining equation of the charge conjugation matrix Γ a TC (±)=±C (±)Γ a\Gamma_a^T C_{(\pm)} = \pm C_{(\pm)} \Gamma_a (def. ), finally the defining relation Γ 0 2=+1\Gamma_0^2 = +1.

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

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

J(Γ aΓ bψ)=Γ aΓ bJ(ψ). J(\Gamma_a \Gamma_b \psi) = \Gamma_a \Gamma_b J(\psi) \,.

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

J(Γ aΓ bψ) =CΓ 0 TΓ a *Γ b *ψ * =CΓ 0 T(−Γ a T)(−Γ b T)ψ * =CΓ 0 TΓ a TΓ b Tψ * =CΓ a TΓ b TΓ 0 Tψ * =Γ aΓ bCΓ 0 Tψ * =Γ aΓ bJ(ψ). \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 Γ spatial †=−Γ spatial\Gamma_{spatial}^\dagger = -\Gamma_{spatial} (prop. ), hence that Γ spatial *=−Γ spatial T\Gamma_{spatial}^\ast = - \Gamma_{spatial}^T and then that Γ 0\Gamma_0 anti-commutes with the spatial Clifford matrices, hence that Γ 0 T\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 Γ spatialΓ spatial\Gamma_{spatial} \Gamma_{spatial} replaced by Γ 0Γ spatial\Gamma_0 \Gamma_{spatial}: The above reasoning applies with two extra signs picked up: one from the fact that Γ 0\Gamma_0 commutes with itself, one from the fact that it is hermitian, by prop. . These two signs cancel:

J(Γ 0Γ aψ) =CΓ 0 TΓ 0 *Γ a *ψ * =CΓ 0 T(+Γ 0 T)(−Γ a T)ψ * =−CΓ 0 TΓ 0 TΓ a Tψ * =+CΓ 0 TΓ a TΓ 0 Tψ * =Γ 0Γ aΓ 0 Tψ * =Γ 0Γ aJ(ψ). \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. ) VV, then the real subspace S↪VS \hookrightarrow V of real elements, i.e. elements ψ\psi with Jψ=ψJ \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 (±)C = C_{(\pm)} is the charge conjugation matrix according to prop. , then the real structure JJ from prop. commutes or anti-commutes with the action of single Clifford generators, according to the subscript of C=C (±)C = C_{(\pm)}:

J(Γ a(−))=±Γ aJ(−). J(\Gamma_a(-)) = \pm \Gamma_a J(-) \,.

Proof

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

J(Γ aψ) =CΓ 0 TΓ a *ψ * =CΓ 0 T(−Γ a T)ψ * =+CΓ a TΓ 0 Tψ * =ϵC TΓ a TΓ 0 T =±ϵΓ aC TΓ 0 Tψ * =±ϵ 2Γ aCΓ 0 Tψ * =±Γ aJ(ψ), \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. , ϵ\epsilon is the sign in C T=ϵCC^T = \epsilon C, which cancels out, and the remaining ±\pm is the sign in C=C (±)C = C_{(\pm)}, due to remark .

For the timelike index we similarly get:

J(Γ 0ψ) =CΓ 0 TΓ 0 *ψ * =+CΓ 0 TΓ 0 Tψ * =ϵC TΓ 0 TΓ 0 T =±ϵΓ 0C TΓ 0 Tψ * =±Γ 0CΓ 0 Tψ * =±Γ 0J(ψ). \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} \,.

Proof

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

⟨J(ψ 1),ψ 2⟩ =⟨CΓ 0 Tψ 1 *,ψ 2⟩ =(CΓ 0 Tψ 1 *) †Γ 0ψ 2 =ψ 1 TC †Γ 0 *Γ 0ψ 2 =ψ 1 TCψ 2. \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 ℂ ν\mathbb{C}^\nu as in prop. , then the map

(−) TC:Mat ν×1(ℂ)⟶Mat 1×ν(ℂ) (-)^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=4,8,9,10,11d = 4,8,9,10,11 a spinor ψ∈ℂ 2 ν\psi \in \mathbb{C}^{2^\nu} is Majorana according to def. with respect to the real structure from prop. , precisely if

ψ=CΓ 0 Tψ * \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 ψ\psi:

ψ TC=ψ †Γ 0, \psi^T C = \psi^\dagger \Gamma_0 \,,

which in turn is equivalent to the condition that

(ψ,−)=⟨ψ,−⟩, (\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 TC^{-1} = C^T (from prop. ). Finally the last statement follows from this by prop. .

Definition

In the even dimensions among those dimensions dd for which the Majorana projection operator (real structure) JJ exists (prop. ) also the chirality projection operator Γ d\Gamma_{d} exists (def. ). Then we may ask that a Dirac spinor ψ\psi is both Majorana, J(ψ)=ψJ(\psi) = \psi, as well as Weyl, Γ dψ=±iψ\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≤d≤114 \leq d \leq 11, then Majorana-Weyl spinors (def. ) exist precisely only in d=10d = 10.

Proof

According to prop. Majorana spinors in the given range exist for d∈{4,8,9,10,11}d \in \{4,8,9,10,11\}. Hence the even dimensions among these are d∈{4,8,10}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,ϵΓ 0⋯Γ d−1]=0 [J, \epsilon \Gamma_0 \cdots \Gamma_{d-1}] = 0

where

ϵ={1 |νodd i |νeven. \epsilon = \left\{ \array{ 1 & \vert \; \nu \, \text{odd} \\ i & \vert \; \nu \, \text{even} } \right. \,.

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

By prop. all the Γ a\Gamma_a commute or all anti-commute with JJ. Since the product Γ 0⋯Γ d−1\Gamma_0 \cdots \Gamma_{d-1} contains an even number of these, it commutes with JJ. It follows that JJ commutes with Γ d\Gamma_d precisely if it commutes with ϵ\epsilon. Now since JJ is conjugate-linear, this is the case precisely if ϵ=1\epsilon = 1, hence precisely if d=2νd = 2\nu with ν\nu odd.

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

Definition

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

(−)¯Γ(−):S×S⟶ℝ d−1,1 \overline{(-)}\Gamma (-) \;\colon\; S \times S \longrightarrow \mathbb{R}^{d-1,1}

be the function that takes Majorana spinors ψ 1\psi_1, ψ 2\psi_2 to the vector with components

ψ¯ 1Γ aψ 2≔ψ 1 TCΓ aψ 2. \overline{\psi}_1\Gamma^a \psi_2 \coloneqq \psi_1^T C \Gamma^a \psi_2 \,.

Proposition

The pairing in def. is

1. symmetric:

   ψ¯ 1Γψ 2=ψ¯ 2Γψ 1 \overline{\psi}_1 \Gamma \psi_2 = \overline{\psi}_2 \Gamma \psi_1

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

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

   g(−)¯Γ(g(−))=g((−)¯Γ(−)). \overline{g(-)}\Gamma(g(-)) = g(\overline{(-)}\Gamma(-)) \,.

Proof

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

(CΓ a) T =Γ a TC T =±Γ a TC =±±CΓ a =CΓ a, \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=±CC^T = \pm C, while the second is from Γ a TC=±CΓ a\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 ψ TC=ψ †Γ 0\psi^T C = \psi^\dagger \Gamma_0 for Majorana spinors ψ\psi (prop. ) and using the conjugate-symmetry of the hermitian form ⟨−,−⟩=(−) †Γ 0(−)\langle -,-\rangle = (-)^\dagger \Gamma_0 (-) as well as the hermiticity of Γ 0Γ a\Gamma_0 \Gamma_a (both from prop. ):

(ψ¯ 1Γ aψ 2) * =(ψ 1 TCΓ aψ 2) * =(ψ 1 †Γ 0Γ aψ 2) * =ψ 2 †(Γ 0Γ a) †ψ 1 =ψ 2 †Γ 0Γ aψ 1 =ψ¯ 2Γ aψ 1. \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

(g(ψ 1),Γ ag(ψ 2)) =⟨g(ψ 1),Γ ag(ψ 2)⟩ =⟨ψ 1,(Γ 0 −1g †Γ 0)Γ agψ 2⟩ =⟨ψ 1(g −1Γ ag)ψ 2⟩, \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 Γ 0 −1(−) †Γ 0\Gamma_0^{-1}(-)^\dagger \Gamma_0 is the adjoint with respect to the hermitian form ⟨−,−⟩=(−) †Γ 0(−)\langle -,-\rangle = (-)^\dagger \Gamma_0 (-) (prop. ) and that gg 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).)

Definition

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

(−)¯ΓΓ(−):S×S⟶∧ 2ℂ d \overline{(-)}\Gamma\Gamma (-) \;\colon\; S \times S \longrightarrow \wedge^2 \mathbb{C}^d

be the function that takes Majorana spinors ψ 1\psi_1, ψ 2\psi_2 to the skew-symmetric rank 2-tensor with components

ψ¯ 1Γ abψ 2≔iψ 1 TC12(Γ aΓ b−Γ bΓ a)ψ 2, \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 ψ 1=ψ 2=ψ\psi_1 = \psi_2 = \psi then the pairing in prop. is real

∀a,biψ¯Γ abψ∈ℝ⊂ℂ \underset{a,b}{\forall} \;\;\;\; i \overline{\psi} \Gamma^{a b} \psi \in \mathbb{R} \subset \mathbb{C}

and Spin(d−1,1)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:

(ψ¯ 1Γ aΓ bψ 2) * =(ψ 1 †Γ aΓ bψ 2) * =ψ 2 †(Γ 0Γ aΓ b) †ψ 1 =−⟨ψ 2 †Γ 0Γ aΓ bψ 1⟩ =−ψ¯ 2Γ aΓ bψ 1, \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 (Γ 0Γ aΓ b) †=−Γ 0Γ aΓ b(\Gamma_0 \Gamma_a \Gamma_b)^\dagger = - \Gamma_0 \Gamma_a \Gamma_b we used the properties in prop. .

Hence for ψ 1=ψ 2\psi_1 = \psi_2 then

(ψ¯Γ aΓ bψ) *=−ψ¯Γ aΓ bψ (\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 *=−ii^\ast = -i.

Proposition

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

ψ¯Γ aψ i ψ¯Γ a 1a 2ψ i ψ¯Γ a 1a 2a 3ψ ψ¯Γ a 1⋯a 4ψ ψ¯Γ a 1⋯a 5ψ i ψ¯Γ a 1⋯a 6ψ i ψ¯Γ a 1⋯a 7ψ ⋮. \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 Γ a\Gamma_a are hermitian with respect (−)¯(−)\overline{(-)}(-) (prop. ) we have

(ψ¯Γ a 1⋯a pψ) * =ψ¯Γ a p⋯a 1ψ =(−1) p(p−1)/2ψ¯Γ a 1⋯a pψ. \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} \,.

Definition

Let d∈ℕd \in \mathbb{N} and let N∈Rep(Spin(d−1,1))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

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

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

This is generated on

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

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

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

The CE-differential is defined as follows

d CEω ab=ω a b∧ω bc d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}

and

d CEψ=14ω abΓ abψ. 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 CEe a=ω a b∧e b+ψ¯Γ aψ 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 NN (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)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 ω\omega this becomes the CE algebra of the super translation algebra underlying super Minkowski spacetime

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

Proposition

Let {γ a}\{\gamma_a\} be any Dirac representation of Spin(8,1)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

Γ a≤8≔(0 γ a γ a 0),Γ 9≔(0 id −id 0),Γ 10≔(iid 0 0 −iid). \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) \,.

Proposition

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

(ψ 1 ψ 2)¯Γ a IIA(ψ 1 ψ 2) ={ψ¯ 1γ aψ 1+ψ¯ 2γ aψ 2 |a≤8 ψ¯ 1ψ 1−ψ¯ 2ψ 2 |a=9. \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:

(ψ 1 ψ 2)¯Γ a IIA(ψ 1 ψ 2) ≔(ψ 1 ψ 2)¯Γ a(ψ 1 ψ 2) =(ψ 1 ψ 2) †Γ 0Γ a(ψ 1 ψ 2) ={(ψ 1 ψ 2) †(γ 0γ a 0 0 γ 0γ a)(ψ 1 ψ 2) |a≤8 (ψ 1 ψ 2) †(γ 0 0 0 −γ 0)(ψ 1 ψ 2) |a=9 ={ψ¯ 1γ aψ 1+ψ¯ 2γ aψ 2 |a≤8 ψ¯ 1ψ 1−ψ¯ 2ψ 2 |a=9. \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

(ψ 1 ψ 2)¯Γ a IIB(ψ 1 ψ 2) ={ψ¯ 1γ aψ 1+ψ¯ 2γ aψ 2 |a≤8 ψ¯ 1ψ 1+ψ¯ 2ψ 2 |a=9 \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 16¯→16¯\overline{\mathbf{16}}\to \overline{\mathbf{16}} by the corresponding top left entries (those going 16→16\mathbf{16} \to \mathbf{16} )). Notice that in fact all these block entries are the same, except for the one at a=9a = 9, where they simply differ by a sign. This yields the claim.

Proposition

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

i(ψ 1 ψ 2)¯Γ 9Γ 10(ψ 1 ψ 2) =(ψ 1 ψ 2)¯Γ 9 IIB(ψ 1 ψ 2) \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:

i(ψ 1 ψ 2)¯Γ 9Γ 10(ψ 1 ψ 2) =i(ψ 1 ψ 2) †Γ 0Γ 9Γ 10(ψ 1 ψ 2) =i(ψ 1 ψ 2) †(−iγ 0 0 0 −iγ 0)(ψ 1 ψ 2) =ψ¯ 1ψ 1+ψ¯ 2ψ 2 =(ψ 1 ψ 2)¯Γ 9 IIB(ψ 1 ψ 2) \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}

Proposition

We have

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

   c 2≔ψ¯∧Γ 10ψ∈CE(ℝ 9,1|16+16¯) 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 ℝ 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} is central super Lie algebra extension of th 9d N=2N = 2 super-Minkowski spacetime by the 2-cocycle given by the type IIA spinor-to-vector pairing

   c 2 IIA≔(ψ 1 ψ 2)¯∧Γ 9 IIA(ψ 1 ψ 2)∈CE(ℝ 8,1|16+16) 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 ℝ 9,1|16+16\mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}} is central super Lie algebra extension of th 9d N=2N = 2 super-Minkowski spacetime by the 2-cocycle given by the type IIB spinor-to-vector pairing

   c 2 IIB≔(ψ 1 ψ 2)¯∧Γ 9 IIB(ψ 1 ψ 2)∈CE(ℝ 8,1|16+16) 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

ℝ 10,1|32 ↓ ℝ 9,1|16+16 ℝ 9,1|16+16¯ ⟶c 2 Bℝ ↘ ↙ ℝ 8,1|16+16 c 2 IIB↙ ↘ c 2 IIA Bℝ Bℝ, \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 BℝB\mathbb{R} denotes the line Lie 2-algebra, and where each “hook”

𝔤^ ↓ 𝔤 ⟶ω 2 Bℝ \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 ∞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 ψ α\psi^\alpha are in bidegree (1,odd)(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 𝔤\mathfrak{g} any (super) Lie algebra (of finite dimension, for convenience), and for ω∈∧ 2𝔤 *\omega \in \wedge^2\mathfrak{g}^\ast a Lie algebra 2-cocycle on it, then the Lie algebra extension 𝔤^\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 ee (in degree (1,even)(1,even)) adjoined, and with the differential of ee taking to be ω\omega:

CE(𝔤^)=(CE(𝔤)⊗⟨e⟩),de=ω). CE(\widehat{\mathfrak{g}}) = (CE(\mathfrak{g}) \otimes \langle e\rangle), d e = \omega) \,.

Hence in the case of ω=c 2 IIA\omega = c_2^{IIA} we identify the new generator with e 9e^9 and see that the equation de 9=c 2 IIAd e^9 = c_2^{IIA} is precisely what distinguishes the CE-algebra of ℝ 8,1|16+16\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}} from that of ℝ 9,1|16+16¯\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.

Definition

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

μ M2≔iϑ¯∧Γ aΓ bϑ∧e a∧e b∈CE(ℝ 10,1|32) \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

μ IIA≔iϑ¯∧Γ aΓ 10ϑ∧e a=i(ψ 1 ψ 2)¯Γ aΓ 10(ψ 1 ψ 2)∈CE(ℝ 9,1|16+16¯), \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 10e^{10}-component of μ M2\mu_{M2} (“double dimensional reduction” FSS 16):

μ IIA=(π 10) *μ M2. \mu_{IIA} = (\pi_{10})_\ast \mu_{M2} \,.

Proposition

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

(π 9) *μ IIA=c 2 IIB. (\pi_9)_\ast \mu_{IIA} = c_2^{IIB} \,.

Proof

We have

(π 9) *μ IIA =i(ψ 1 ψ 2)¯Γ 9Γ 10(ψ 1 ψ 2) =(ψ 1 ψ 2)¯Γ 9 IIB(ψ 1 ψ 2) =c 2 IIB \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. .

Lemma

With the above conventions we have.

(1)Γ a 1⋯a 11=ϵ a 1⋯a 11⋅1 32. \Gamma_{a_1 \cdots a_{11}} \;=\; \epsilon_{a_1 \cdots a_{11}} \cdot 1_{\mathbf{32}} \,.

Lemma

Γ a j⋯a 1Γ b 1⋯b k=∑ l=0 min(j,k)±l!(jl)(kl)δ [b 1⋯b l [a 1⋯a lΓ a j⋯a l+1] b l+1⋯b k]. \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]} \,.

Proof

Observe that if the aa-indices are not pairwise distinct or the bb-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≔{a 1,⋯,a j}A \coloneqq \{a_1, \cdots, a_j\}, B≔{b 1,⋯,b k}B \coloneqq \{b_1, \cdots, b_k\} and their intersection C:=A∩BC := A \cap B, with cardinality card(C)=l\mathrm{card}(C) = l. The idea is to recursively contract one pair (Γ c,Γ c)(\Gamma^c , \Gamma_{c}) with c∈Cc \in C at a time. We claim that in the first step this can be written as

Γ a j⋯a 1Γ b 1⋯b k=jklΓ [a j⋯a 2δ [b 1 a 1]Γ b 2⋯b k]. \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⋯a kX^{a_1 \cdots a_k} the expression kX [a k⋯a 1]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 lY [b 1⋯b l]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∈Cc \in C we are overcounting by a factor of ll.

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

Γ a j⋯a 1Γ b 1⋯b k=(−1) j−1jklδ [b 1 [a 1Γ a j⋯a 2]Γ b 2⋯b k]. \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

Γ a j⋯a 1Γ b 1⋯b k=(−1) (j−1)⋯(j−l)j⋯(j−l)k⋯(k−l)l!⏟ l!(kl)(jl)δ [b 1⋯b l [a 1⋯a lΓ a j⋯a l+1]Γ b l+1⋯b k]⏟ Γ a j⋯a l+1] b j+l⋯b k] \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 ll, since it gives zero whenever l≠card(C)l \,\neq\, \mathrm{card}(C). This yields the claimed formula.

Proposition

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

M=132∑ p=0 5(−1) p(p−1)/2p!Tr(ϕ∘Γ a 1⋯a p)Γ a 1⋯a p 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}

Proposition

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

(2)(32⊗32) symm ≃ 11⊕55⊕462 (32⊗32⊗32) symm ≃ 32⊕320⊕1408⊕4424 (32⊗32⊗32⊗32) symm ≃ 1⊕165⊕330⊕462⊕65⊕429⊕1144⊕17160⊕32604, \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.

Proposition

The spinor pairing

(3)

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

is:

1. bi-linear

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

3. skew-symmetric.

Proposition

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

ψ¯ψ=0 ψ¯Γ [a 1a 2a 3]ψ=0 ψ¯Γ [a 1⋯a 4]ψ=0 ψ¯Γ [a 1⋯a 7]ψ=0 ψ¯Γ [a 1⋯a 8]ψ=0 ψ¯Γ [a 1⋯a 11]ψ=0, \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.

Proof

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

(ψ¯Γ [a 1⋯a p]ϕ) ≔Re(ψ †Γ 0Γ [a 1⋯a p]ϕ) =−Re(ϕ †(Γ [a 1⋯a p]) †Γ 0ψ) =−Re(ϕ †Γ 0Γ 0 −1(Γ [a 1⋯a p]) †Γ 0ψ) =−(−1) p+p(p−1)/2Re(ϕ †Γ 0Γ [a 1⋯a p]ψ) =−(−1) p(p+1)/2(ϕ¯Γ [a 1⋯a p]ψ). \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∈32\pin \in \mathbf{32} vanish:

(5)(ψ¯Γ abψ)(ψ¯Γ aψ)=0 (ψ¯Γ ab 1⋯b 4ψ)(ψ¯Γ aψ)+3(ψ¯Γ b 1b 2ψ)(ψ¯Γ b 3b 4ψ)=0 \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 Spin(1,10)\mathrm{Spin}(1,10)-equivariant map

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

But by (2) the irrep summand 11\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.

Definition

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

ϕ:V⟶V \phi \;\colon\; V \longrightarrow V

such that

1. ϕ\phi is conjugate linear (ϕ(λv)=λ¯ϕ(v)\phi(\lambda v) = \overline{\lambda} \phi(v) for all λ∈ℂ\lambda \in \mathbb{C}, v∈Vv \in V);

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

Definition

Let GG be a Lie group, let VV be a complex vector space and let

ρ:G⟶GL ℂ(V) \rho \;\colon\; G \longrightarrow GL_{\mathbb{C}}(V)

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

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

ϕ∘ρ(g)=ρ(g)∘ϕ. \phi \circ \rho(g) = \rho(g) \circ \phi \,.

Definition

A hermitian form (or symmetric complex sesquilinear form) ⟨−,−⟩\langle -,-\rangle on a complex vector space VV is a real bilinear form

⟨−,−⟩:V×V⟶ℂ \langle -,- \rangle \;\colon\; V \times V \longrightarrow \mathbb{C}

such that for all v 1,v 2∈Vv_1, v_2 \in V and λ∈ℂ\lambda \in \mathbb{C} then

1. (sesquilinearity) ⟨v 1,λv 2⟩=λ⟨v 1,v 2⟩\langle v_1, \lambda v_2 \rangle = \lambda \langle v_1, v_2 \rangle ,

2. (conjugate symmetry) ⟨v 1,v 2⟩ *=⟨v 2,v 1⟩\langle v_1, v_2\rangle^\ast = \langle v_2, v_1\rangle .

3. (non-degeneracy) if ⟨v 1,−⟩=0\langle v_1,-\rangle = 0 then v 1=0v_1 = 0.

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

⟨f(−),f(−)⟩=⟨−,−⟩. \langle f(-), f(-)\rangle = \langle -,-\rangle \,.

Write

U(V)↪GL ℂ(V) U(V) \hookrightarrow GL_{\mathbb{C}}(V)

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

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

ρ:G⟶U(V)↪GL ℂ(V). \rho \;\colon\; G \longrightarrow U(V) \hookrightarrow GL_{\mathbb{C}}(V) \,.

Proposition

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

(−,−):V⊗ ℂV⟶ℂ. (-,-) \;\colon\; V \otimes_{\mathbb{C}} V \longrightarrow \mathbb{C} \,.

Explicitly:

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

(−,−)≔⟨ϕ(−),−⟩. (-,-) \coloneqq \langle \phi(-), -\rangle \,.

Conversely, given (−,−)(-,-), first define ϕ˜\tilde \phi by

(−,−)=⟨ϕ˜(−),−⟩, (-,-) = \langle \tilde\phi(-),-\rangle \,,

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

If ϕ˜=ϕ\tilde\phi = \phi then (−,−)(-,-) is called compatible with ⟨−,−⟩\langle-,- \rangle.