M-theory from the superpoint

Abstract

The “brane scan” classifies consistent Green–Schwarz strings and membranes in terms of the invariant cocycles on super Minkowski spacetimes. The “brane bouquet” generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete fundamental brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super Minkowski spacetimes themselves arise as maximal invariant central extensions. Here, we prove that they do. Starting from the simplest possible super Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive “maximal invariant central extensions” and show that it discovers the super Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet.

A Background

For reference, we briefly recall some definitions and facts that we use in the main text.

1.1 A.1 Super Lie algebra cohomology

We recall the definition of super Lie algebras and their cohomology. All our vector spaces and algebras are over ${\mathbb {R}}$, and they are all finite dimensional. We write even for $0 \in {\mathbb Z}_2$ and odd for $1 \in {\mathbb Z}_2$.

Definition 15

The tensor category of supervector spaces is the category of ${\mathbb Z}_2$-graded vector spaces and grade-preserving linear maps, equipped with the unique non-trivial braiding, $\tau ^{\mathrm{super}}$. For any two supervector space V and W, $\tau ^{\mathrm{super}}$ is the isomorphism

$$\begin{aligned} \begin{array}{rcl} \tau ^{\mathrm{super}} :V \otimes W &{} \rightarrow &{} W \otimes V \\ v_1 \otimes v_2 &{} \mapsto &{} (-1)^{\sigma _1 \sigma _2} \, v_2 \otimes v_1 , \end{array} \end{aligned}$$

for elements $v_1 \in V$, $v_2 \in W$ of homogeneous degree $\sigma _i \in {\mathbb Z}_2$.

A super Lie algebra is a Lie algebra internal to supervector spaces. That is, it is a supervector space

$$\begin{aligned} \mathfrak {g} = \mathfrak {g}_{\mathrm {even}} \oplus \mathfrak {g}_{\mathrm {odd}} \end{aligned}$$

equipped with a bilinear map, called the Lie bracket:

$$\begin{aligned}{}[-,-] :\mathfrak {g} \otimes \mathfrak {g} \longrightarrow \mathfrak {g} \end{aligned}$$

which is graded skew symmetric:

$$\begin{aligned}{}[v_1, v_2] = -(-1)^{\sigma _1 \sigma _2} [v_2, v_1] \end{aligned}$$

and which satisfies the graded Jacobi identity:

$$\begin{aligned}{}[v_1, [v_2,v_3]] = [[v_1, v_2], v_3] + (-1)^{ \sigma _1 \sigma _2 } [v_2, [v_1, v_3]]. \end{aligned}$$

A homomorphism of super Lie algebras $\mathfrak {g}_1 \longrightarrow \mathfrak {g}_2$ is a linear map preserving the ${\mathbb Z}_2$-grading and the bracket.

Definition 16

(super Lie algebra cohomology) Let V be a finite-dimensional supervector space. Then, the super Grassmann algebra $\Lambda ^\bullet V^*$ is the ${\mathbb {Z}} \times {\mathbb Z}_2$-bigraded-commutative associative algebra freely generated by $V^*$ in degree $1 \in {\mathbb Z}$. That is to say, it is generated by the elements in $V^*_{\mathrm {even}}$ regarded as being in bidegree $(1,\mathrm {even})$, and the elements in $V^*_{\mathrm {odd}}$ regarded as being in bidegree $(1,\mathrm {odd})$, subject to the relation that for $\alpha _i$ two elements of homogeneous bidegree $(n_i, \sigma _i)$, then

$$\begin{aligned} \alpha _1 \wedge \alpha _2 \;=\; (-1)^{n_1 n_2} (-1)^{\sigma _1 \sigma _2} \, \alpha _2 \wedge \alpha _1. \end{aligned}$$

In particular, this relation says that elements of bidegree $(1,\mathrm{even})$ anticommute with each other, those of bidegree $(1,\mathrm{odd})$ commute with each other, while an element of bidegree $(1,\mathrm{even})$ anticommutes with an element of bidegree $(1,\mathrm{odd})$.

Now let $(\mathfrak {g}, [-,-])$ be a finite-dimensional super Lie algebra. Then, its Chevalley–Eilenberg algebra $\mathrm {CE}(\mathfrak {g})$ is the super Grassmann algebra $\Lambda ^\bullet \mathfrak {g}^*$ equipped with the differential $d_{\mathrm{CE}}$ defined as follows. On the generators $\mathfrak {g}^*$, $d_{\mathrm{CE}}$ acts as the linear dual of the Lie bracket:

$$\begin{aligned}{}[-,-]^* :\mathfrak {g}^*\rightarrow \Lambda ^2 \mathfrak {g}^*. \end{aligned}$$

The action of $d_{\mathrm{CE}}$ on generators is then extended to all of $\Lambda ^\bullet \mathfrak {g}^*$ as a derivation, bigraded of bidegree $(1,\mathrm{even})$. This makes $\mathrm{CE}(\mathfrak {g})$ into a differential graded algebra. A calculation shows $d_{\mathrm{CE}}^2 = 0$, so $\mathrm{CE}(\mathfrak {g})$ is also a cochain complex.

For $p \in {\mathbb {N}}$, we say that a $(p+2)$-cocycle on $\mathfrak {g}$ with coefficients in ${\mathbb {R}}$ is a $d_{\mathrm {CE}}$-closed element in $\Lambda ^{p+2} \mathfrak {g}^*$. We say that cocycle is even if its degree in ${\mathbb Z}_2$ is even, and odd if it is odd. The super Lie algebra cohomology of $\mathfrak {g}$ with coefficients in ${\mathbb {R}}$ is the cohomology of its Chevalley–Eilenberg algebra, regarded as a cochain complex:

$$\begin{aligned} H^\bullet (\mathfrak {g}, {\mathbb {R}}) := H^\bullet (\mathrm {CE}(\mathfrak {g})). \end{aligned}$$

The ${\mathbb Z}_2$-grading on $\mathrm{CE}(\mathfrak {g})$ makes $H^p({\mathfrak {g}},{\mathbb {R}})$ into a supervector space for each p. We will be interested in its even part, $H_{\mathrm {even}}^p({\mathfrak {g}},{\mathbb {R}})$

Example 17

Let $\mathfrak {g}$ be a finite-dimensional super Lie algebra, and let $\omega \in \Lambda ^2 \mathfrak {g}^*$ be an even 2-cocycle as in Definition 16. Then, there is a new super Lie algebra $\widehat{\mathfrak {g}}$ whose underlying supervector space is

$$\begin{aligned} \widehat{\mathfrak {g}} := \underset{\mathrm {even}}{\underbrace{\mathfrak {g}_{\mathrm {even}} \oplus {\mathbb {R}}}} \oplus \underset{\mathrm {odd}}{\underbrace{\mathfrak {g}_{\mathrm {odd}}}} \end{aligned}$$

and with super Lie bracket given by

$$\begin{aligned}{}[ (x_1,c_1), (x_2,c_2) ] \;=\; ([x_1,x_2], \omega (x_1, x_2)). \end{aligned}$$

We have a short exact sequence giving ${\widehat{\mathfrak {g}}}$ as a central extension of $\mathfrak {g}$:

$$\begin{aligned} 0 \rightarrow {\mathbb R}\rightarrow {\widehat{\mathfrak {g}}} \rightarrow \mathfrak {g}\rightarrow 0 . \end{aligned}$$

In the paper, we will often write this short exact sequence as follows; in the style, an algebraic topologist might use to write down a fibration:

Just as for ordinary Lie algebras, this construction establishes a natural equivalence between central extensions of $\mathfrak {g}$ by ${\mathbb {R}}$ (in even degree) and even super Lie algebra 2-cocycles on $\mathfrak {g}$.

More generally, a central extension in even degree is by some vector space $V \simeq {\mathbb {R}}^n$

which is equivalently the result of extending by n even 2-cocycles, one after the other, in any order.

We will be interested not in the full super Lie algebra cohomology, but in the invariant cohomology with respect to some action:

Definition 18

For $\mathfrak {g}$ a super Lie algebra (Definition 15), its automorphism group is the Lie subgroup

$$\begin{aligned} \mathrm {Aut}(\mathfrak {g}) \hookrightarrow \mathrm {GL}(\mathfrak {g}_{\mathrm {even}}) \times \mathrm {GL}(\mathfrak {g}_{\mathrm {odd}}) \end{aligned}$$

of the group of degree-preserving linear isomorphisms on the underlying vector space, consisting of those elements which are super Lie algebra isomorphisms.

Proposition 19

For $\mathfrak {g}$ a super Lie algebra, the Lie algebra of its automorphism Lie group (Definition 18)

$$\begin{aligned} \mathfrak {aut}(\mathfrak {g}) \end{aligned}$$

is called the automorphism Lie algebra of $\mathfrak {g}$. It is the Lie algebra of those linear maps $\Delta :\mathfrak {g} \rightarrow \mathfrak {g}$ which preserve the degree and satisfy the derivation property:

$$\begin{aligned} \Delta [X,Y] = [\Delta X, Y] + [X, \Delta Y] \end{aligned}$$

for all $X, Y\in \mathfrak {g}$. The Lie bracket on $\mathfrak {aut}(\mathfrak {g})$ is the commutator:

$$\begin{aligned}{}[\Delta _1, \Delta _2] := \Delta _1 \Delta _2 - \Delta _2 \Delta _1. \end{aligned}$$

We caution the reader that, even though $\mathfrak {g}$ is a super Lie algebra, its automorphism algebra $\mathfrak {aut}(\mathfrak {g})$ is merely a Lie algebra. This is because we want elements of $\mathfrak {aut}(\mathfrak {g})$ to preserve the degree on $\mathfrak {g}$.

Example 20

The super Minkowski super Lie algebras ${\mathbb {R}}^{d-1,1\vert N}$ from Definition 22 all carry an automorphism action of the abelian Lie algebra ${\mathbb {R}}$ which is spanned by the scaling derivation that acts on vectors $v \in {\mathbb {R}}^{d-1,1}$ by

$$\begin{aligned} v \mapsto 2v \end{aligned}$$

and on spinors $\psi \in N$ by

$$\begin{aligned} \psi \mapsto \psi . \end{aligned}$$

Definition 21

Let $\mathfrak {g}$ be a super Lie algebra (Definition 15). Clearly, every automorphism of $\mathfrak {g}$ will induce an automorphism of the Chevalley–Eilenberg algebra $\mathrm{CE}(\mathfrak {g})$ (Definition 16). Explicitly, this works as follows. Let $\Delta \in \mathfrak {aut}(\mathfrak {g})$ be an infinitesimal automorphism (Proposition 19). The induced automorphism $\Delta _\mathrm{CE}:\mathrm{CE}(\mathfrak {g}) \rightarrow \mathrm{CE}(\mathfrak {g})$ acts on the generators $\mathfrak {g}^*$ of $\mathrm{CE}(\mathfrak {g})$ as the linear dual $\Delta ^*$:

$$\begin{aligned} \Delta _{\mathrm {CE}} :\mathfrak {g}^*\overset{\Delta ^*}{\longrightarrow } \mathfrak {g}^*. \end{aligned}$$

This is then extended to all of $\mathrm{CE}(\mathfrak {g})$ as a derivation of bidegree $(0,\mathrm{even})$. The fact that $\Delta _\mathrm{CE}$ commutes with $d_\mathrm{CE}$ is equivalent to the fact that $\Delta$ is a derivation of $\mathfrak {g}$.

Now let $\mathfrak {h} \hookrightarrow \mathfrak {aut}(\mathfrak {g})$ be a Lie subalgebra of its automorphism Lie algebra. The elements of $\mathrm {CE}(\mathfrak {g})$ which are annihilated by $\Delta _{\mathrm {CE}}$ for all $\Delta \in \mathfrak {h}$ form a differential graded subalgebra of $\mathrm{CE}(\mathfrak {g})$:

$$\begin{aligned} \mathrm {CE}(\mathfrak {g})^{\mathfrak {h}} \hookrightarrow \mathrm {CE}(\mathfrak {g}). \end{aligned}$$

We say an $\mathfrak {h}$-invariant $(p+2)$-cocycle on $\mathfrak {g}$ is an element in $\mathrm {CE}(\mathfrak {g})^{\mathfrak {h}}$ which is $d_{\mathrm {CE}}$-closed and the $\mathfrak {h}$-invariant cohomology of $\mathfrak {g}$ with coefficients in ${\mathbb {R}}$ is the cochain cohomology of this subcomplex:

$$\begin{aligned} H^\bullet (\mathfrak {g}, {\mathbb {R}})^{\mathfrak {h}} := H^\bullet (\mathrm {CE}(\mathfrak {g})^{\mathfrak {h}}). \end{aligned}$$

We define even and odd invariant cocycles as before. The vector space $H^p(\mathfrak {g}, {\mathbb R})^\mathfrak {h}$ is ${\mathbb Z}_2$-graded for each p, and our focus will be on its even part, $H^p_\mathrm{even}(\mathfrak {g},{\mathbb R})^\mathfrak {h}$.

1.2 A.2 Super Minkowski spacetimes

We recall the definition of “super Minkowski super Lie algebras” (Definition 22) as well as their construction, on the one hand via Majorana or symplectic Majorana spinors (Example 25) and on the other hand via the four normed division algebras (Proposition 32). We freely use basic facts about spinors, as may be found in the book of Lawson and Michelsohn [40].

Definition 22

(super Minkowski Lie algebras) Let $d \in {\mathbb {N}}$ (spacetime dimension) and let N be a real spinor representation of $\mathrm{Spin}(d-1,1)$, the double cover of the connected Lorentz group $\mathrm{SO}_0(d-1,1)$. Then, d-dimensional N-supersymmetric super Minkowski spacetime ${\mathbb {R}}^{d-1,1\vert N}$ is the super Lie algebra (Definition 15) whose underlying supervector space is

$$\begin{aligned} {\mathbb {R}}^{d-1,1\vert N} := \underset{\mathrm {even}}{\underbrace{{\mathbb {R}}^{d-1,1}}} \oplus \underset{\mathrm {odd}}{\underbrace{N}}. \end{aligned}$$

The Lie bracket is nonzero only on N and is a choice of symmetric, bilinear, $\mathrm{Spin}(d-1,1)$-equivariant map:

$$\begin{aligned}{}[-,-] :N \otimes N \longrightarrow {\mathbb {R}}^{d-1,1} . \end{aligned}$$

Such a map is always available in spacetime signature $(d-1,1)$ [30], though there may be more than one choice [52].

There is a canonical action of $\mathrm {Spin}(d-1,1)$ on ${\mathbb {R}}^{d-1,1\vert N}$ by Lie algebra automorphisms, and the corresponding semidirect product Lie algebra is the super Poincaré super Lie algebra

$$\begin{aligned} \mathfrak {iso}({\mathbb {R}}^{d-1,1\vert N}) = {\mathbb {R}}^{d-1,1\vert N} \rtimes \mathfrak {so}(d-1,1). \end{aligned}$$

It is also called the supersymmetry algebra.

Remark 23

(number of supersymmetries) In the physics literature, the choice of real spin representation in Definition 22 is often referred to as the “number of supersymmetries.” While this is imprecise, it fits well with the convention of labeling irreducible representations by their dimension in boldface. For example, when $d = 10$ there are two irreducible real spinor representations, both of real dimension 16, but of opposite chirality, and hence traditionally denoted ${\mathbf {16}}$ and $\overline{{\mathbf {16}}}$. Hence, we may speak of $N = {\mathbf {16}}$ supersymmetry (also called $N = 1$, type I or heterotic) and $N = {\mathbf {16}} \oplus \overline{{\mathbf {16}}}$ supersymmetry (also called $N = (1,1)$ or type IIA) and $N = {\mathbf {16}} \oplus {\mathbf {16}}$ supersymmetry (also called $N = (2,0)$ or type IIB).

In Sect. 3, the generalization of the last of these cases plays a central role, where for any given real spin representation N we pass to the doubled supersymmetry $N \oplus N$. Observe that the two canonical linear injections $N \rightarrow N \oplus N$ into the direct sum induce two super Lie algebra homomorphisms

The following degenerate variation of super Minkowski spacetime will play a key role:

Definition 24

(Superpoint) A superpoint is the super Lie algebra

$$\begin{aligned} {\mathbb {R}}^{0|N}, \end{aligned}$$

which has zero Lie bracket, and whose underlying supervector space is concentrated in odd degree, where it is of dimension N.

We will use two different ways of constructing real spinor representation, and hence super Minkowski spacetimes: via “Majorana” or “symplectic Majorana” spinors (Example 25) and via real normed division algebras (Proposition 32).

Example 25

(Majorana representations) For $d = 2\nu$ or $2 \nu +1$, there exists a complex representation of the Clifford algebra $\mathrm {Cl}({\mathbb {R}}^{d-1,1})\otimes {\mathbb {C}}$, hence of the spin group $\mathrm {Spin}(d-1,1)$ on ${\mathbb {C}}^{2^\nu }$ such that

• 1. all skew-symmetrized products of $p \ge 1$ Clifford elements $\Gamma _{a_1 \cdots a_p}$ are traceless;

• 2. $\Gamma _0^\dagger = \Gamma _0$ and $\Gamma _i^\dagger = - \Gamma _i$, for $1 \le i \le d - 1$.

This is the Dirac representation, a complex representation of $\mathrm{Spin}(d-1,1)$. For $d = 2\nu$, this is the direct sum of two subrepresentations on ${\mathbb {C}}^{2^{\nu }-1}$, the Weyl representations.

For $d \in \{1,2,3,4,8,9,10,11\}$, there exists a real structure J on the complex Dirac representation, restricting to the Weyl representations for $d = 2$ or $d = 10$. This is a $\mathrm {Spin}(d-1,1)$-equivariant antilinear endomorphism $J :S \rightarrow S$ which squares to the identity: $J^2 = +1$. It carves out a real representation called the Majorana representation $N := \mathrm {Eig}(J,+1)$, the eigenspace of J of eigenvalue +1, whose elements are called the Majorana spinors. In this case, the Dirac conjugation $\psi \mapsto \psi ^\dagger \Gamma _0$ on elements $\psi \in {\mathbb {C}}^{2^\nu }$ restricts to N and is called the Majorana conjugation. We write it as simply $\psi \mapsto {\overline{\psi }}$. In terms of this matrix representation, then the spinor bilinear pairing that appears in Definition 22 is given by the following matrix product expression:

$$\begin{aligned}{}[\psi ,\phi ] \;=\; \left( {\overline{\psi }} \Gamma ^a \phi \right) _{a = 0}^{d-1}. \end{aligned}$$

Similarly, for $d \in \{5,6,7\}$ there exists a quaternionic structure on the Dirac representation. This is a $\mathrm{Spin}(d-1,1)$-equivariant antilinear endomorphism ${\tilde{J}}$ which squares to minus the identity, ${\tilde{J}}^2 = -1$. It follows that

$$\begin{aligned} J := \left( \begin{array}{c@{\quad }c} 0 &{} -\,{\tilde{J}} \\ {\tilde{J}} &{} 0 \end{array} \right) \end{aligned}$$

is a real structure on the direct sum of the Dirac representation with itself. Hence as before $N := \mathrm {Eig}(J,+1)$ is a real subrepresentation, called the symplectic Majorana representation. The spinor-to-vector bilinear pairing for symplectic Majorana spinors is similar to the case of Majorana spinors.

Definition 26

(Cayley–Dickson double [2, Section 2.2]) Let ${\mathbb {K}}$ be a real $*$-algebra. This is a real, not necessarily associative algebra ${\mathbb K}$ equipped with a conjugation $\overline{(-)} :{\mathbb {K}} \rightarrow {\mathbb {K}}$, satisfying:

$$\begin{aligned} \overline{a + b} = {\overline{a}} + {\overline{b}}, \quad {\overline{ab}} = {\overline{b}} \, {\overline{a}}, \quad \overline{{\overline{a}}} = a , \end{aligned}$$

for any $a, b \in {\mathbb K}$. Then, the Cayley–Dickson double ${\mathbb {K}}_{\mathrm{dbl}}$ of ${\mathbb {K}}$ is the real $*$-algebra obtained from ${\mathbb {K}}$ by adjoining one element $\ell$ such that $\ell ^2 = -1$ and such that the following relations hold, for all $a, b \in {\mathbb {K}}$:

$$\begin{aligned} a(\ell b) = \ell ({\overline{a}} b),\;\;\;\; (a \ell ) b = (a {\overline{b}}) \ell ,\;\;\;\; (\ell a)(b \ell ) = -\, \overline{(a b)}. \end{aligned}$$

Finally, the conjugation $\overline{(-)}$ on ${\mathbb {K}}_{\mathrm {dbl}}$ acts on elements of ${\mathbb K}$ by the conjugation on ${\mathbb K}$, and sends the new generator $\ell$ to $-\ell$.

Example 27

Consider ${\mathbb {R}}$ the real numbers regarded as a $*$-algebra with trivial conjugation ${\overline{a}} = a$. Then, its Cayley–Dickson double (Definition 26) is the complex numbers ${\mathbb {C}}$ with the usual conjugation, the Cayley–Dickson double of ${\mathbb C}$ is the quaternions ${\mathbb {H}}$, and the Cayley–Dickson double of ${\mathbb H}$ is the octonions ${\mathbb {O}}$.

By a classical result of Hurwitz, these four algebras are the only normed division algebras over the real numbers, as reviewed by Baez [2].

In the next proposition and elsewhere in the text, we will use $n \times n$ matrices over ${\mathbb K}$ to describe real linear operators on ${\mathbb K}^n$. We will write ${\mathbb K}[n]$ for the set of all $n \times n$ matrices with entries in ${\mathbb K}$. For any such matrix, there are two natural ways for it to induce a linear operator, one using left multiplication in ${\mathbb K}$ and the other right multiplication.

Definition 28

(Matrices over${\mathbb K}$ as linear operators) Let ${\mathbb K}$ be a normed division algebra. Any element of $a \in {\mathbb K}$ induces a linear endomorphism on ${\mathbb K}$ by left or right multiplication, which we will denote by $a_L$ or $a_R$, respectively:

$$\begin{aligned} \begin{array}{rcl} a_L :{\mathbb K}&{} \rightarrow &{} {\mathbb K}\\ x &{} \mapsto &{} ax \end{array} , \quad \begin{array}{rcl} a_R :{\mathbb K}&{} \rightarrow &{} {\mathbb K}\\ x &{} \mapsto &{} xa . \end{array} \end{aligned}$$

More generally, any $n \times n$ matrix $A \in {\mathbb K}[n]$ induces a linear endomorphism on ${\mathbb K}^n$ via either left multiplication or right multiplication:

$$\begin{aligned} \begin{array}{rcl} A_L :{\mathbb K}^n &{} \rightarrow &{} {\mathbb K}^n \\ x &{} \mapsto &{} \sum a_{ij} x_j \end{array} , \quad \begin{array}{rcl} A_R :{\mathbb K}^n &{} \rightarrow &{} {\mathbb K}^n \\ x &{} \mapsto &{} \sum x_j a_{ij} \end{array} \end{aligned}$$

where we are using the subscript $x_j$ to denote the jth coordinate of $x \in {\mathbb K}^n$, and $A = (a_{ij})$. In other words, $A_L$ and $A_R$ are the linear maps on ${\mathbb K}^n$ with components $\left( (a_{ij})_L \right)$ and $\left( (a_{ij})_R \right)$, respectively. We say that $A_L$ is the left action of A and $A_R$ is the right action of A. We caution that because ${\mathbb K}$ is non-associative, $(AB)_L \ne A_L B_L$ in general, and because ${\mathbb K}$ is non-associative and non-commutative, $(AB)_R \ne A_R B_R$ in general.

Remark 29

The left action of a matrix A by $A_L$ is just the usual matrix multiplication, so we will sometimes write:

$$\begin{aligned} A_L x = Ax . \end{aligned}$$

The utility of defining the linear transformation $A_L$ is that the composition of linear transformations is associative, so we do not need to worry about the non-associativity of ${\mathbb K}$ when we compose them. For example:

$$\begin{aligned} A_L B_L C_L x = A(B(Cx)) . \end{aligned}$$

Since ${\mathbb K}$ comes with a conjugation, we can define the conjugate of any matrix in ${\mathbb K}[n]$ by taking the conjugate of each entry, and the conjugate of any element of ${\mathbb K}^n$ by taking the conjugate of each coordinate. It is then an elementary calculation to show that the action of a matrix A by $A_L$ and $A_R$ is related by conjugation:

Proposition 30

Let $A \in {\mathbb K}[n]$ be an $n \times n$ matrix over the normed division algebra ${\mathbb K}$ (as defined in Example 27). Then,

$$\begin{aligned} \overline{A_L x} = ({\overline{A}})_R \, {\overline{x}} \text{ and } \overline{A_R x} = ({\overline{A}})_L \, {\overline{x}} \end{aligned}$$

for all $x \in {\mathbb K}^n$.

The next definition is straightforward, but is central to realizing spin representations via normed division algebras.

Definition 31

[49] For $A \in \mathfrak {h}_2({\mathbb K})$, a hermitian matrix with coefficients in one of the four real normed division algebras from Example 27. Then, its trace reversal is

$$\begin{aligned} {\widetilde{A}} := A - \mathrm {tr}(A)\cdot {\mathbf {1}}. \end{aligned}$$

Proposition 32

[4] Let ${\mathbb K}\in \{{\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}, {\mathbb {O}}\}$ be one of the normed division algebras as in Example 27. Write $\mathfrak {h}_2({\mathbb K})$ for the real vector space of $2 \times 2$ hermitian matrices with coefficients in ${\mathbb K}$, and k for the dimension of ${\mathbb K}$.

Then:

• 1. There is an isomorphism of inner product spaces (“forming Pauli matrices over ${\mathbb {K}}$”)

  $$\begin{aligned} ({\mathbb {R}}^{k+1,1}, \eta ) {\mathop {\longrightarrow }\limits ^{\simeq }} \left( \mathfrak {h}_2({\mathbb K}), -\mathrm {det} \right) \end{aligned}$$

  identifying ${\mathbb {R}}^{k+1,1}$ equipped with its Minkowski inner product

  $$\begin{aligned} \eta (A,B) := -A^0 B^0 + A^1 B^1 + \cdots + A^{k+1} B^{k+1}, \text{ for } A, B \in {\mathbb R}^{k+1,1} \end{aligned}$$

  with the space of hermitian matrices equipped with the negative of the determinant operation.

• 2. Let $N_+$ and $N_-$ both denote the vector space ${\mathbb K}^2$. Then, $N_+ \oplus N_-$ is a module of the Clifford algebra ${\mathcal {C}\ell }(k+1,1)$, with the action of a vector in $A \in {\mathbb R}^{k+1,1}$ given by

  $$\begin{aligned} \Gamma (A) (\psi , \phi ) = (\tilde{A}_L \phi , A_L \psi ) \end{aligned}$$

  for any element $(\psi , \phi ) \in N_+ \oplus N_-$, where we are using the identification of vectors with $2 \times 2$ hermitian matrices. Here, $\widetilde{(-)}$ is the trace reversal operation from Definition 31, and $(-)_L$ denotes the linear map given by left multiplication as in Definition 28

• 3. Realizing the spin group $\mathrm{Spin}(k+1,1)$ inside the Clifford algebra ${\mathcal {C}\ell }(k+1,1)$ by the standard construction, this induces irreducible representations $\rho _\pm$ of $\mathrm{Spin}(k+1,1)$ on $N_\pm$. Explicitly, recall that $\mathrm{Spin}(k+1,1)$ is the subgroup of the Clifford algebra generated by products of pairs of unit vectors of the same sign:

  $$\begin{aligned} \mathrm{Spin}(k+1, 1) = \langle AB \in {\mathcal {C}\ell }(k+1,1) \, : \, A, B \in {\mathbb R}^{k+1,1}, \, \eta (A,A) = \eta (B,B) = \pm 1 \rangle . \end{aligned}$$

  Then, restricting the Clifford action to these elements, a generator AB of $\mathrm{Spin}(k+1,1)$ acts as

  $$\begin{aligned} \rho _+(AB) = \tilde{A}_L B_L \text{ on } N_+ \end{aligned}$$

  and as

  $$\begin{aligned} \rho _-(AB) = A_L \tilde{B}_L \text{ on } N_-, \end{aligned}$$

  where again $\widetilde{(-)}$ is the trace reversal operation from Definition 31 and where $(-)_L$ denotes the linear map given by left multiplication as in Definition 28. For ${\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}\}$, then these two representations are in fact isomorphic and are the Majorana representation of $\mathrm {Spin}(2,1)$ and $\mathrm{Spin}(3,1)$, respectively, while for ${\mathbb {K}} \in \{ {\mathbb {H}}, {\mathbb {O}} \}$ they are the two non-isomorphic symplectic Majorana representations of $\mathrm {Spin}(5,1)$ and Majorana–Weyl representations of $\mathrm {Spin}(9,1)$, respectively.

• 4. Under the above identifications, the symmetric bilinear $\mathrm{Spin}(k+1,1)$-equivariant spinor-to-vector pairings are given by

  $$\begin{aligned} \begin{array}{rcl} [-,-] :N_+ \otimes N_+ &{} \rightarrow &{} {\mathbb R}^{k+1,1} \\ \psi \otimes \phi &{} \mapsto &{} \tfrac{1}{2}\left( \widetilde{ \psi \phi ^\dagger + \phi \psi ^\dagger } \right) \, \end{array} \end{aligned}$$

  and

  $$\begin{aligned} \begin{array}{rcl} [-,-] :N_- \otimes N_- &{} \rightarrow &{} {\mathbb R}^{k+1,1} \\ \psi \otimes \phi &{} \mapsto &{} \tfrac{1}{2}\left( \psi \phi ^\dagger + \phi \psi ^\dagger \right) \, \end{array} \end{aligned}$$

• 5. There is a bilinear symmetric, non-degenerate and $\mathrm {Spin}(k+1,1)$-invariant spinor-to-scalar pairing given by

  $$\begin{aligned} \begin{array}{rcl} \langle -, -\rangle :N_\pm \otimes N_{\mp } &{} \rightarrow &{} {\mathbb R}\\ \psi \otimes \phi &{} \mapsto &{} \mathrm {Re}(\psi ^\dagger \phi ). \end{array} \end{aligned}$$