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Majorana spinor

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Representation theory

representation theory

geometric representation theory

Ingredients

Definitions

representation, 2-representation, ∞-representation

Geometric representation theory

Theorems

Representation theory

Higher spin geometry

Contents

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 ψ\psi for Spin(d1,1)Spin(d-1,1), regarded as an element of 2 ν\mathbb{C}^{2^\nu} (with d=2ν,2ν+1d = 2 \nu, 2\nu+1) is a Majorana spinor if it satisfies the condition

ψ tC=ψ Γ 0, \psi^t C = \psi^\dagger \Gamma_0 \,,

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

ψ=J(ψ)CΓ 0 Tψ *, \psi = J(\psi) \coloneqq C \Gamma_0^T \psi^\ast \,,

where J()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 ρ:Spin(s,t)GL (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 ψV\psi \in V is called a Majorana spinor if J(ψ)=ψJ(\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):

We will be discussing three different pairing operations on complex column vectors ψ 1,ψ 2 ν\psi_1, \psi_2 \in \mathbb{C}^\nu:

  • ψ 1 ψ 2\psi_1^\dagger \psi_2 – the standard hermitian form on ν\mathbb{C}^\nu, this will play a purely auxiliary role.

  • ψ 1,ψ 2ψ¯ 1ψ 2ψ 1 Γ 0ψ 2\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;

  • (ψ 1,ψ 2)ψ 1 TCψ 2(\psi_1,\psi_2) \coloneqq \psi_1^T C \psi_2 – the Majorana pairing (for CC the charge conjugation matrix), this turns out to coincide with the Dirac pairing above if ψ 1\psi_1 is a Majorana spinor.

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

Definition

We write s,t\mathbb{R}^{s,t} for the real vector space 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,,+1t,1,,1s). \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 \mathbb{R}-algebra generated from elements {Γ a} 0=1 s+t1\{\Gamma_a\}_{0 = 1}^{s+t-1} subject to the relation

Γ aΓ b+Γ bΓ a=2η aba,b{0,1,,t+s1}. \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 1a nΓ [a 1Γ a 2]1n!σ(1) |σ|Γ a σ 1Γ 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 1a n=Γ a 1Γ a nifi,j(a ia 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) \,.

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

Γ aη abΓ b \Gamma^a \coloneqq \eta^{a b} \Gamma_b
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 d1\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,,d1}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 s,1\mathbb{R}^{s,1} (Castellani-D’Auria-Fré, (II.7.1)):

Proposition

(Dirac representation)

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

d=s+1{2ν,2ν+1}forν,d4. 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 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 \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 2ix 3 x 0x 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 |ad2 ϵγ 0γ 1γ d2 |a=d1 \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,,d1}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 dΓ 0Γ 1Γ d1 \Gamma_{d} \propto \Gamma_0 \Gamma_1 \cdots \Gamma_{d-1}

acts Spin(d1,1)Spin(d-1,1)-invariantly on the representation space of the Dirac Spin(d1,1)Spin(d-1,1)-representation for even dd.

Moreover, since Γ 0Γ 1Γ d1\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+1cΓ 0Γ 1Γ d1 \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(d1,1)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 ν\mathbb{C}^\nu as in prop. , we write

()¯() Γ 0:Mat ν×1()Mat(1×ν)() \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} \,.

Charge conjugation matrix

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_{(\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:

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

(This is for instance in Castellani-D’Auria-Fré, section (II.7.2), table (II.7.1), but beware that there C ()C_{(-)} in d=10,11d = 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 (±)C_{(\pm)} listed there then

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

This implies in all cases that

Γ aC (±) T=±C (±) TΓ a T. \Gamma_a C_{(\pm)}^T = \pm C_{(\pm)}^T \Gamma_a^T \,.

Majorana representations and Real structure

Proposition

For d{4,8,9,10,11}d \in \{4,8,9,10,11\}, let V= 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:VV J \colon V \longrightarrow V

given by

ψCΓ 0 Tψ * \psi \mapsto C \Gamma_0^T \psi^\ast

is a real structure (def. ) for the corresponding complex spin representation on 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 SVS \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} \,.

We record some immediate consequences:

Proposition

The complex bilinear form

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

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

(,)=() TC(). (-,-) = (-)^T C (-) \,.
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 ν\mathbb{C}^\nu as in prop. , then the map

() TC:Mat ν×1()Mat 1×ν() (-)^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 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. .

Pseudo-Majorana spinors and Symplectic structure

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

ψ=CΓ 0 Tψ * \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 ψ 1\psi_1 and ψ 2\psi_2, then the following condition does have a solution

CΓ 0 Tψ 1 *=ψ 2CΓ 0 Tψ 2 *=+ψ 1 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 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 4d114 \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 0Γ d1]=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 0Γ d1\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=25d = 10 = 2 \cdot 5, but not for d=8=24d = 8 = 2 \cdot 4 neither for d=4=22d = 4 = 2 \cdot 2.

The spinor bilinear pairing to antisymmetric pp-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 pp-tensor. These operations are all of the form

ψϵψ¯Γ a 1a pψ, \psi \mapsto \epsilon \, \overline{\psi} \Gamma^{a_1 \cdots a_p} \psi \,,

where ϵ\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 p d1,1\wedge^p \mathbb{R}^{d-1,1}.

For a Spin(d1,1)Spin(d-1,1) representation VV as in prop. , with real/Majorana structure as in prop. , write

SV 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 () TC(-)^T C coincides with the Dirac conjugate ()¯() Γ 0\overline{(-)} \coloneqq (-)^\dagger \Gamma_0 . Therefore we write ()¯\overline{(-)} in the following for the conjugation of Majorana spinors, unambiguously defined.

Definition

For a Spin(d1,1)Spin(d-1,1) representation VV as in prop. , with real/Majorana structure as in prop. , let

()¯Γ():S×S d1,1 \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 2ψ 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 d d\mathbb{R}^d \subset \mathbb{C}^d);

  3. Spin(d1,1)Spin(d-1,1)-equivariant: for gSpin(d1,1)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).)

Remark

Proposition implies that adding a copy of SS 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(d1,1)Spin(d-1,1) representation VV as in prop. , with real/Majorana structure as in prop. , let

()¯ΓΓ():S×S 2 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 2iψ 1 TC12(Γ aΓ bΓ 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 \underset{a,b}{\forall} \;\;\;\; i \overline{\psi} \Gamma^{a b} \psi \in \mathbb{R} \subset \mathbb{C}

and Spin(d1,1)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.

Generally:

Proposition

The following pairings are real and Spin(d1,1)Spin(d-1,1)-equivariant:

ψ¯Γ aψ i ψ¯Γ a 1a 2ψ i ψ¯Γ a 1a 2a 3ψ ψ¯Γ a 1a 4ψ ψ¯Γ a 1a 5ψ i ψ¯Γ a 1a 6ψ i ψ¯Γ a 1a 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 1a pψ) * =ψ¯Γ a pa 1ψ =(1) p(p1)/2ψ¯Γ a 1a 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} \,.

Supersymmetry: Super-Poincaré and super-Minkowski

Every real spin representation of Spin(d1,1)Spin(d-1,1) induces a super Lie algebra extension of the Poincaré Lie algebra ℑ𝔰𝔬( d1,1)\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 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1) of the spin group (the orthogonal Lie algebra in Lorentian signature)

d1,1ℑ𝔰𝔬( d1,1)/𝔰𝔬(d1,1) \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 d1,1|N\mathbb{R}^{d-1,1\vert N} of the following definition, and this justifies the following notation:

Definition

Let dd \in \mathbb{N} and let NRep(Spin(d1,1))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( d1,1|N) \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})

equivalently in terms of its Chevalley-Eilenberg algebra: CE(ℑ𝔰𝔬( d1,1|N))CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})) (a ×/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,,d1}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 bω 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 be 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(d1,1)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

d1,1|N. \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 {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 a8(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) \,.
Remark

By prop. the Dirac representation in d=11d = 11 has complex dimension 2 10/2=2 5=322^{10/2} = 2^{5} = 32. By prop. and prop. this representation carries a real structure and hence gives a real/Majorana spin representation S 32S \hookrightarrow \mathbb{C}^{32} of Spin(10,1)Spin(10,1) of real dimension 32. This representation often just called “32\mathbf{32}”. This way the corresponding super-Minkowski spacetime (remark ) is neatly written as

10,1|32 \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)Spin(10,1)-cover of the Lorentz group SO(10,1)SO(10,1) acts on these. This is the local model space for super spacetimes in 11-dimensional supergravity.

As we regard 32\mathbb{C}^{32} instead as the Dirac representation of Spin(9,1)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 32\mathbf{32} correspondingly decomposes as a direct sum two real representations of dimension 116 which often are denoted 16\mathbf{16} and 16¯\overline{\mathbf{16}}. Hence as real/Majorana Spin(9,1)Spin(9,1)-representations there is a direct sum decomposition

321616¯. \mathbf{32} \simeq \mathbf{16} \oplus \overline{\mathbf{16}} \,.

The corresponding super Minkowski spacetime (remark )

9,1|16+16¯ \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 9,1|1616\mathbb{R}^{9,1\vert \mathbf{16}\oplus \mathbf{16}}, which is “type IIB” and in contrast to 9,1|16\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)Spin(8,1) is of complex dimension d=2 8/2=2 4=16d = 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)Spin(8,1), accordingly denoted 16\mathbf{16}. Notice that this is Majorana-Weyl.

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

For this it is sufficient to see that Γ 9\Gamma_9, which as a complex linear map goes Γ 9:1616¯\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)Spin(8,1).

Clearly it is a linear isomorphism, so it is sufficient that it is a homomorphism of Spin(8,1)Spin(8,1)-representations at all. But that’s clear since by the Clifford algebra relations Γ 9\Gamma_9 commutes with all Γ aΓ b\Gamma_a \Gamma_b for a,bon{0,,8}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)Spin(9,1)Spin(10,1) Spin(8,1) \hookrightarrow Spin(9,1) \hookrightarrow Spin(10,1)

is

32Rep(Spin(10,1))1616¯Rep(Spin(9,1))1616Rep(Spin(8,1)). \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 32\mathbf{32} as ϑ 32\vartheta \in \mathbb{C}^{32}, decomposed as a (1×32)(1 \times 32)-matrix as

ϑ=(ψ 1 ψ 2). \vartheta = \left( \array{ \psi_1 \\ \psi_2 } \right) \,.

and if we write for short

ψ 1=(ψ 1 0),ψ 2=(0 ψ 2) \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)Spin(9,1)-action then ψ 1\psi_1 becomes a Majorana spinor in the 16\mathbf{16}, and ψ 2\psi_2 a Majorana spinor in the 16¯\overline{\mathbf{16}}, and after further restriction to Spin(8,1)Spin(8,1)-action then either comes a Majorana spinor in one copy of 16\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

(ψ 1 ψ 2)¯Γ a IIA(ψ 1 ψ 2) ={ψ¯ 1γ aψ 1+ψ¯ 2γ aψ 2 |a8 ψ¯ 1ψ 1ψ¯ 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) |a8 (ψ 1 ψ 2) (γ 0 0 0 γ 0)(ψ 1 ψ 2) |a=9 ={ψ¯ 1γ aψ 1+ψ¯ 2γ aψ 2 |a8 ψ¯ 1ψ 1ψ¯ 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 |a8 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 1616\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.

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

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}

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

Proposition

We have

  1. The 11d N=1N = 1 super-Minkowski spacetime 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 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} by the 2-cocycle

    c 2ψ¯Γ 10ψCE( 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 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( 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 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( 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

10,1|32 9,1|16+16 9,1|16+16¯ c 2 B 8,1|16+16 c 2 IIB c 2 IIA B B, \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 BB\mathbb{R} denotes the line Lie 2-algebra, and where each “hook”

𝔤^ 𝔤 ω 2 B \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 8,1|16+16\mathbb{R}^{8,1\vert \mathbf{16}+ \mathbf{16}} from that of 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.

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

μ M2iϑ¯Γ aΓ bϑe ae bCE( 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 IIAiϑ¯Γ aΓ 10ϑe a=i(ψ 1 ψ 2)¯Γ aΓ 10(ψ 1 ψ 2)CE( 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 reductionFSS 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( 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. .

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

ϕ:VV \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\lambda \in \mathbb{C}, vVv \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.

Remark

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

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

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

A quaternionic structure, def. , o VV 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. Ii():VVI \coloneqq i(-) \colon V \to V be the operation of acting with ii \in \mathbb{C}

  2. Jϕ:VVJ \coloneqq \phi \colon V \to V be the given endomorphisms,

then the conjugate complex linearity of ϕ\phi means that

JI=IJ J \circ I = - I \circ J

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

Definition

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

ρ:GGL (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 \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 gGg \in G then

ϕρ(g)=ρ(g)ϕ. \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 VV is a real bilinear form

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

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

  1. (sesquilinearity) v 1,λv 2=λv 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:VVf \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 (V) U(V) \hookrightarrow GL_{\mathbb{C}}(V)

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

A complex linear representation ρ:GGL (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:GU(V)GL (V). \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 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:GU(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 V. (-,-) \;\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 ϕ1|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.

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

References

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

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

See also

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

Last revised on July 31, 2018 at 10:22:38. See the history of this page for a list of all contributions to it.