# nLab spin representation

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Representation theory

representation theory

geometric representation theory

spin geometry

string geometry

# Contents

## Idea

A representation of the spin group.

## Definition

###### Definition

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

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

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

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

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

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

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

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

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

### Real and quaternionic representations

One may ask in which dimensions $d$ 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 $S$ there is a non-vanishing symmetric and $Spin(d-1,1)$-invariant bilinear form $S \otimes S \longrightarrow \mathbb{R}^d$, projectively unique if $S$ 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 $S$. This is “supersymmetry” in physics.

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

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

#### Real irreducible spin representations in Lorentz signature

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

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

| 1 | $\mathbb{Z}_2$ | $S$ real | 1 | $V \simeq (S^\ast)^{\otimes}^2$ | | | 2 | $\mathbb{R}^{\gt 0} \times \mathbb{Z}_2$ | $S^+, S^-$ real | 1 | $V \simeq ({S^+}^\ast)^{\otimes^2} \oplus ({S^-}^\ast)^{\otimes 2}$ | | | 3 | $SL(2,\mathbb{R})$ | $S$ real | 2 | $V \simeq Sym^2 S^\ast$ | | | 4 | $SL(2,\mathbb{C})$ | $S_{\mathbb{C}} \simeq S' \oplus S''$ | 4 | $V_{\mathbb{C}} \simeq {S'}^\ast \oplus {S''}^\ast$ | | | 5 | $Sp(1,1)$ | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 8 | $\wedge^2 S_0^\ast \simeq \mathbb{C} \oplus V_{\mathbb{C}}$ | | | 6 | $SL(2,\mathbb{H})$ | $S^\pm_{\mathbb{C}} \simeq S_0^\pm \otimes_{\mathbb{C}} W$ | 8 | $V_{\mathbb{C}} \simeq \wedge^2 {S_0^+}^\ast \simeq (\wedge^2 {S_0^-}^\ast)^\ast$ | | | 7 | | $S_{\mathbb{C}} \simeq S_0 \otimes_{\mathbb{C}} W$ | 16 | $\wedge^2 S_0^\ast \simeq V_{\mathbb{C}} \oplus \wedge^2 V_{\mathbb{C}}$ | | | 8 | | $S_{\mathbb{C}} \simeq S^\prime \oplus S^{\prime\prime}$ | 16 | ${S'}^\ast {S''}^\ast \simeq V_{\mathbb{C}} \oplus \wedge^3 V_{\mathbb{C}}$ | | | 9 | | $S$ real | 16 | $Sym^2 S^\ast \simeq \mathbb{R} \oplus V \wedge^4 V$ | | | 10 | | $S^+ , S^-$ real | 16 | $Sym^2(S^\pm)^\ast \simeq V \oplus \wedge_\pm^5 V$ | type II supergravity | | 11 | | $S$ real | 32 | $Sym^2 S^\ast \simeq V \oplus \wedge^2 V \oplus \wedge^5 V$ | 11-dimensional supergravity |

Here $W$ 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

$\Gamma \;\colon\; S^\ast \otimes S^\ast \longrightarrow \mathbb{R}^{d-1,1}$

which is

1. symmetric;

2. $Spin(V)$-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.

### Spinor bilinear forms

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

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

$S_1 \otimes S_2 \longrightarrow \wedge^p V \,.$

These appear notably in the following applications:

• for $p = 0$ symmetric bilinears $(-,-) \;\colon\; S \otimes S \longrightarrow k$ define a metric on the space of spinors, also known as a charge conjugation matrix. This appears for instance in the Lagrangian for a spinor field $\psi$, which is of the form $\psi \mapsto (\psi, D \psi)$, for $D$ a Dirac operator;

• for $p = 1$ symmetric bilinear $Spin(V)$-homomorphisms $\Gamma \;\colon\; S \otimes S \longrightarrow V$ constitutes the odd Lie bracket in a super Poincaré Lie algebra extension of the a Poincaré Lie algebra by $S$.

• for $p \geq 2$ higher spin bilinears $S \otimes S \longrightarrow \wedge^p V$ appear in further polyvector extensions.

#### $p = 0$ – spinor metric

We discuss spinor bilinear pairings to scalars.

##### Over the complex numbers
###### Proposition

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

$C \;\colon\; S \otimes S \longrightarrow \mathbb{C}$

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

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

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

$d \, mod\, 8$C
0symmetric
1symmetric
2$S^\pm$ dual to each other
3skew-symmetric
4skew-symmetric
5skew-symmetric
6$S^\pm$ 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 $C^T$ satisfies

$C^T = - \epsilon C$

with $\epsilon \in \{-1,1\}$ given in dimension $d$ by the following table

$d \, mod \, 8$$C$
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 $V$ be a quadratic vector space, def. 1 over the real numbers of dimension $d$ with Loentzian signature. Then there exists, up to rescaling, a unique $Spin(V)$-invariant bilinear form

$C \;\colon\; S \otimes S \longrightarrow \mathbb{R}$

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

$d \, mod \, 8$$C$
0symmetric
1symmetric
2$S^{\pm}$ dual to each other
3skew symmetric
4skew symmetric
5symmetric
6$S^{\pm}$ dual to each other
7symmetric

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

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

We discuss spinor bilinear pairings to vectors.

##### Over the complex numbers
###### Proposition

Let $V$ be a quadratic vector space, def. 1 over the complex numbers of dimension $d$.

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

$\Gamma \;\colon\; S\otimes S \longrightarrow \mathbb{C}$

for odd $d$ and $S$ the unique irreducible representation, and

$\Gamma \;\colon\; S^{\pm} \otimes S^{\mp} \longrightarrow \mathbb{C}$

for even $d$ and $S^\pm$ the two inequivalent irreducible representations.

This is (Varadarajan 04, theorem 6.6.3).

##### Over the real numbers (for Majorana spinors)
###### Proposition

Let $V$ be a quadratic vector space, def. 1 over the real numbers of dimension $d$.

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

$d \,mod \, 8$
0$S^\pm \otimes S^\mp \to V$
1$S \otimes S \to V$
2$S^\pm \otimes S^\pm \to V$
3$S \otimes S \to V$
4$S^\pm \otimes S^\mp \to V$
5$S \otimes S \to V$
6$S^\pm \otimes S^\pm \to V$
7$S \otimes S \to V$

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 $\Gamma^a = \{(\Gamma^a)^\alpha{}_\beta\}$ 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 $\phi = (\phi^\alpha)$ 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

$\overline{\phi} \coloneqq \phi^T C$

hence

$\overline{\phi}_\alpha \coloneqq \phi^\beta C_{\beta \alpha} \,.$

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

\begin{aligned} C(\phi,\psi) &= \phi^\alpha C_{\alpha \beta} \,\psi^\beta \\ & = \overline{\phi}_\alpha \psi^\alpha \\ & = \overline{\phi} \psi \end{aligned}

and

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

$N = (N_+, N_-)$

to indicate that there are $N_+$ copies of the irreducible $Spin(V)$-representation of one chirality, and $N_-$ 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 $N_+$ and $N_-$ 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 $d = 4$ and $d = 8$ in Lorentzian signature (see the above table) it is the complex representations $\mathbf{2} \oplus \mathbf{2}'$ and $\mathbf{16} \oplus \mathbf{16}'$, respectively, which carry a real structure. Hence the real representation underlying this parameterizes $N = 1$ supersymmetry in terms Majorana spinors, even though its complexification would be $N = (1,1)$.

See for instance (Freed 99, p. 53).

## 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

Revised on October 15, 2013 19:49:26 by Urs Schreiber (80.237.234.132)