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spin representation

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Idea

A representation of the spin group.

Definition

Definition

A quadratic vector space (V,,)(V, \langle -,-\rangle) is a vector space VV over finite dimension over a field kk of characteristic 0, and equipped with a symmetric bilinear form ,:VVk\langle -,-\rangle \colon V \otimes V \to k.

Conventions as in (Varadarajan 04, section 5.3).

Properties

Complex representations

Complex representations of the spin group follow a mod-2 Bott periodicity.

In even d=2nd = 2n there are two inequivalent complex-linear irreducible representations of Spin(d1,1)Spin(d-1,1), each of complex dimension 2 d/212^{d/2-1}, called the two chiral representations, or the two Weyl spinor representations.

For instance for d=10d = 10 one often writes these as 16\mathbf{16} and 16\mathbf{16}'.

The direct sum of the two chiral representation is called the Dirac spinor representation, for instance 16+16\mathbf{16} + \mathbf{16}'.

In odd d=2n+1d = 2n+1 there is a single complex irreducible representation of complex dimension 2 (d1)/22^{(d-1)/2}. For instance for d=11d = 11 one often writes this as 32\mathbf{32}. This is called the Dirac spinor representation in this odd dimension.

For d=2nd = 2n, if {Γ 1,,Γ n}\{\Gamma^1, \cdots, \Gamma^n\} denote the generators of the Clifford algebra Cl d1,1Cl_{d-1,1} then there is the chirality operator

Γ d+1Γ 1Γ 2Γ d \Gamma^{d+1} \coloneqq \Gamma^1 \cdot \Gamma^2 \cdots \Gamma^d

on the Dirac representation, whose eigenspaces induce its decomposition into the two chiral summands.

The unique irreducible Dirac representation in the odd dimension d+1d+1 is, as a complex vector space, the sum of the two chiral representations in dimension dd, with the Clifford algebra represented by Γ 1\Gamma^1 through Γ d\Gamma^d acting diagonally on the two chiral representations, and the chirality operator Γ d+1\Gamma^{d+1} in dimension dd acting on their sum, now being the representation of the (d+1)(d+1)st Clifford algebra generator.

Real and quaternionic representations

One may ask in which dimensions dd the above complex representations admit a real structure or a quaternionic structure.

Real spinor representations are also called Majorana representations, and an element of a real/Majorana spin representation is also called a Majorana spinor. On a Majorana representation SS there is a non-vanishing symmetric and Spin(d1,1)Spin(d-1,1)-invariant bilinear form SS dS \otimes S \longrightarrow \mathbb{R}^d, projectively unique if SS is irreducible. This serves as the odd-odd Lie bracket in the super Lie algebra called the super Poincaré Lie algebra extension of the ordinary Poincaré Lie algebra induced by SS. This is “supersymmetry” in physics.

The above irreducible complex representations admit a real structure for d=1,2,3mod8d = 1,2,3 \, mod \, 8. Therefore in dimension d=2mod8d = 2 \, mod \, 8 there exist Majorana-Weyl spinor representations.

The above irreducible complex representations admit a quaternionic structure for d=5,6,7mod8d = 5,6,7 \, mod \, 8.

Real irreducible spin representations in Lorentz signature

Let VV be a quadratic vector space, def. 1, over the real numbers with bilinear form or Lorentzian signature, hence V= d1,1V = \mathbb{R}^{d-1,1} is Minkowski spacetime of some dimension dd.

The following table lists the irreducible real representations of Spin(V)Spin(V) (Freed 99, page 48).

ddSpin(d1,1)Spin(d-1,1)minimal real spin representation SSdim Sdim_{\mathbb{R}} S\;\;VV in terms of S *S^\astsupergravity
1 2\mathbb{Z}_2SS real1V(S *) 2V \simeq (S^\ast)^{\otimes}^2
2 >0× 2\mathbb{R}^{\gt 0} \times \mathbb{Z}_2S +,S S^+, S^- real1V(S + *) 2(S *) 2V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}
3SL(2,)SL(2,\mathbb{R})SS real2VSym 2S *V \simeq Sym^2 S^\ast
4SL(2,)SL(2,\mathbb{C})S SSS_{\mathbb{C}} \simeq S' \oplus S''4V S *S *V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast
5Sp(1,1)Sp(1,1)S S 0 WS_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W8 2S 0 *V \wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}
6SL(2,)SL(2,\mathbb{H})S ±S 0 ± WS^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W8V 2S 0 + *( 2S 0 *) *V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast
7S S 0 WS_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W16 2S 0 *V 2V \wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}
8S S S S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}16S *S *V 3V {S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}
9SS real16Sym 2S *V 4VSym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V
10S +,S S^+ , S^- real16Sym 2(S ±) *V ± 5VSym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 Vtype II supergravity
11SS real32Sym 2S *V 2V 5VSym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V11-dimensional supergravity

Here WW is the 2-dimensional complex vector space on which the quaternions naturally act.

Remark

The last column implies that in each dimension there exists a linear map

Γ:S *S * d1,1 \Gamma \;\colon\; S^\ast \otimes S^\ast \longrightarrow \mathbb{R}^{d-1,1}

which is

  1. symmetric;

  2. Spin(V)Spin(V)-equivariant.

This allows to form the super Poincaré Lie algebra in each of these cases. See there and see Spinor bilinear forms below for more.

Quaternionic irreducible spin representations in Lorentz signature

Spinor bilinear forms

Let (V,,)(V, \langle -,-\rangle) be a quadratic vector space, def. 1. For pp \in \mathbb{R} write pV\wedge^p V for its ppth skew-symmetrized tensor power, regarded naturally as a representation of the spin group Spin(V)Spin(V).

For S 1,S 2Rep(Spin(V))S_1, S_2 \in Rep(Spin(V)) two irreducible representations of Spin(V)Spin(V), we discuss here homomorphisms of representations (hence kk-linear maps respecting the Spin(V)Spin(V)-action) of the form

S 1S 2 pV. S_1 \otimes S_2 \longrightarrow \wedge^p V \,.

These appear notably in the following applications:

p=0p = 0 – spinor metric

We discuss spinor bilinear pairings to scalars.

Over the complex numbers
Proposition

Let VV be a quadratic vector space, def. 1 over the complex numbers of dimension dd. Then there exists in dimensions d2,6mod8d \neq 2,6 \; mod \, 8, up to rescaling, a unique Spin(V)Spin(V)-invariant bilinear form

C:SS C \;\colon\; S \otimes S \longrightarrow \mathbb{C}

on a complex irreducible representation SS of Spin(V)Spin(V), or in dimension 2 and 6 a bilinear pairing

C:S +S C \;\colon\; S^+ \otimes S^- \longrightarrow \mathbb{C}

which is non-degenerate and whose symmetry is given by the following table:

dmod8d \, mod\, 8C
0symmetric
1symmetric
2S ±S^\pm dual to each other
3skew-symmetric
4skew-symmetric
5skew-symmetric
6S ±S^\pm dual to each other
7symmetric

This appears for instance as (Varadarajan 04, theorem 6.5.7).

Remark

The matrix representation of the bilinear form in prop. 1 is known in the physics literature as the charge conjugation matrix. In matrix calculus the symmetry property means that the transpose matrix C TC^T satisfies

C T=ϵC C^T = - \epsilon C

with ϵ{1,1}\epsilon \in \{-1,1\} given in dimension dd by the following table

dmod8d \, mod \, 8CC
0-1
1-1
2either
3+1
4+1
5+1
6either
7-1

For instance (van Proeyen 99, table 1).

Over the real numbers (for Majorana spinors)
Proposition

Let VV be a quadratic vector space, def. 1 over the real numbers of dimension dd with Loentzian signature. Then there exists, up to rescaling, a unique Spin(V)Spin(V)-invariant bilinear form

C:SS C \;\colon\; S \otimes S \longrightarrow \mathbb{R}

on a real irreducible representation SS of Spin(V)Spin(V), and its symmetry is given by the following table

dmod8d \, mod \, 8CC
0symmetric
1symmetric
2S ±S^{\pm} dual to each other
3skew symmetric
4skew symmetric
5symmetric
6S ±S^{\pm} dual to each other
7symmetric

This appears for instance as (Freed 99, around (3.4), Varadarajan 04, theorem 6.5.10).

p=1p = 1 – super Poincaré bracket (supersymmetry)

We discuss spinor bilinear pairings to vectors.

Over the complex numbers
Proposition

Let VV be a quadratic vector space, def. 1 over the complex numbers of dimension dd.

Then there exists unique Spin(V)Spin(V)-representation morphisms

Γ:SS \Gamma \;\colon\; S\otimes S \longrightarrow \mathbb{C}

for odd dd and SS the unique irreducible representation, and

Γ:S ±S \Gamma \;\colon\; S^{\pm} \otimes S^{\mp} \longrightarrow \mathbb{C}

for even dd and S ±S^\pm the two inequivalent irreducible representations.

This is (Varadarajan 04, theorem 6.6.3).

Over the real numbers (for Majorana spinors)
Proposition

Let VV be a quadratic vector space, def. 1 over the real numbers of dimension dd.

Then there exists unique Spin(V)Spin(V)-representation morphisms

dmod8d \,mod \, 8
0S ±S VS^\pm \otimes S^\mp \to V
1SSVS \otimes S \to V
2S ±S ±VS^\pm \otimes S^\pm \to V
3SSVS \otimes S \to V
4S ±S VS^\pm \otimes S^\mp \to V
5SSVS \otimes S \to V
6S ±S ±VS^\pm \otimes S^\pm \to V
7SSVS \otimes S \to V

This is (Varadarajan 04, theorem 6.5.10).

For more see (Varadarajan 04, section 6.7).

Pairing to a vector in terms of the charge conjugation matrix
Remark

In terms of a matrix representation with respect to a chosen basis as in remark 2 the pairing of prop. 4 is given by the matrices Γ a={(Γ a) α β}\Gamma^a = \{(\Gamma^a)^\alpha{}_\beta\} that represent the Clifford algebra by raising and lowering indices with the charge conjugation matrix of remark 2 (e.g Freed 99 (3.5)).

In such a notation if ϕ=(ϕ α)\phi = (\phi^\alpha) denotes the component-vector of a spinor, then the result of “lowering its index” is given by acting with the metric in form of the charge conjugation matrix. The result is traditionally denoted

ϕ¯ϕ TC \overline{\phi} \coloneqq \phi^T C

hence

ϕ¯ αϕ βC βα. \overline{\phi}_\alpha \coloneqq \phi^\beta C_{\beta \alpha} \,.

This yields the component formula for the pairings to scalars and to vectors which is traditional in the physics literature as follows:

C(ϕ,ψ) =ϕ αC αβψ β =ϕ¯ αψ α =ϕ¯ψ \begin{aligned} C(\phi,\psi) &= \phi^\alpha C_{\alpha \beta} \,\psi^\beta \\ & = \overline{\phi}_\alpha \psi^\alpha \\ & = \overline{\phi} \psi \end{aligned}

and

Γ a(ϕ,ψ) =ϕ αΓ a αβψ β =ϕ αC ακΓ aκ βψ β =ϕ TCΓ aψ ϕ¯Γ aψ. \begin{aligned} \Gamma^a(\phi, \psi) &= \phi^\alpha \Gamma^a{}_{\alpha \beta} \psi^\beta \\ &= \phi^\alpha C_{\alpha \kappa} \Gamma^{a \kappa}{}_\beta \psi^\beta \\ & = \phi^T C \Gamma^a \psi \\ & \coloneqq \overline{\phi} \Gamma^a \psi \end{aligned} \,.

(Recall that all this is here for Majorana spinors, as in the previous prop. 4.)

This yields the component expressions for the bilinear pairings as familiar from the physics supersymmetry literature, for instance (Polchinski 01, (B.2.1), (B.5.1))

Counting numbers of supersymmetries

A spinor bilinear pairing to a vector Γ:SSV\Gamma \;\colon\; S \otimes S \to V as above serves as the odd-odd bracket in a super Poincaré Lie algebra extension of VV. Since this is also called a “supersymmetrysuper Lie algebra, with the spinors being the supersymmetry generators, the decomposition of SS into minimal/irreducible representations is also called the number of supersymmetries. This is traditionally denoted by a capital NN and in even dimensions and over the complex numbers it is traditional to write

N=(N +,N ) N = (N_+, N_-)

to indicate that there are N +N_+ copies of the irreducible Spin(V)Spin(V)-representation of one chirality, and N N_- of those of the other chirality (i.e. left and right handed Weyl spinors).

This counting however is more subtle over the real numbers (Majorana spinors) and the notation in this case (which happens to be the more important case) is not entirely consistent through the literature.

There is no issue in those dimensions in which the complex Weyl representation already admits a real structure itself, hence when there are Majorana-Weyl spinors. In this case one just counts them with N +N_+ and N N_- as in the case over the complex numbers.

However, in some dimensions it is only the direct sum of two Weyl spinor representations which carries a real structure. For instance for d=4d = 4 and d=8d = 8 in Lorentzian signature (see the above table) it is the complex representations 22\mathbf{2} \oplus \mathbf{2}' and 1616\mathbf{16} \oplus \mathbf{16}', respectively, which carry a real structure. Hence the real representation underlying this parameterizes N=1N = 1 supersymmetry in terms Majorana spinors, even though its complexification would be N=(1,1)N = (1,1). See for instance (Freed 99, p. 53).

Similarly in dimensions 5,6 and 7 mod 8, the minimal real representation is obatained from tensoring the complex spinors with the complex 2-dimensional canonical quaternionic representation WW (as in the above table). These are also called symplectic Majorana representations. For instance in in 6d one typically speaks of the 6d (2,0)-superconformal QFT to refer to that with a single “symplectic Majorana-Weyl” supersymmetry (e.g. Figueroa-OFarrill, p. 9), which might therefore be counted as (1,0)(1,0) real supersymmetric, but which involves two complex irreps and is hence often denoted counted as (2,0)(2,0).

References

Chapter I.5 of

For Lorentzian signature and with an eye towards supersymmetry in QFT, see for mathematical accounts

and for the traditional component notation used in physics see

For good math/physics discussion with special emphasis on the symplectic Majorana spinors and their role in the 6d (2,0)-superconformal QFT see

Revised on September 1, 2014 10:39:21 by Urs Schreiber (185.37.147.14)