1 Introduction

In his “vision talk” at the annual string theory conference in 2014, Greg Moore highlighted the following open question in string theory [42, Section 9]:

Perhaps we need to understand the nature of time itself better. [...] One natural way to approach that question would be to understand in what sense time itself is an emergent concept, and one natural way to make sense of such a notion is to understand how pseudo-Riemannian geometry can emerge from more fundamental and abstract notions such as categories of branes.

We are going to tell an origin story for spacetime, in which it emerges from the simplest kind of supermanifold: the superpoint, denoted \({\mathbb R}^{0|1}\). This is the supermanifold with no bosonic coordinates, and precisely one fermionic coordinate. From this minimal mathematical space, which has no Lorentz structure and no spinorial structure, we will give a systematic process to construct super Minkowski spacetimes up to dimension 11 and complete with their Lorentz structures and spinorial structures. Indeed, this is the same mathematical mechanism that makes, for instance, the M2-brane and then the M5-brane emerge from 11d spacetime. It is directly analogous to the D0-brane condensation by which 11d spacetime emerges out of the type IIA spacetime of dimension 10.

To make all this precise, first recall that the super p-branes of string theory and M-theory, in their incarnation as “fundamental branes” or “probe branes,” are mathematically embodied in terms of what are called “\(\kappa \)-symmetric Green–Schwarz-type functionals.” See Sorokin [50] for review and further pointers.

Not long after Green and Schwarz [32] discovered their celebrated action functional for the superstring, Henneaux and Mezincescu observed [33] that the previously somewhat mysterious term in the Green–Schwarz action, the one which ensures its \(\kappa \)-symmetry, is in a fact nothing but the WZW-type functional for super Minkowski spacetime regarded as a supergroup. This is mathematically noteworthy, because WZW-type functionals are a natural outgrowth of super Lie algebra cohomology [18, 27]. This suggests that the theory of super p-branes is to some crucial extent a topic purely in super Lie theory, hence amenable to mathematical precision and classification tools.

Indeed, Azcárraga and Townsend [17] later showed (following Achúcarro et al. [1]) that it is the \(\mathrm{Spin}(d-1,1)\)-invariant super Lie algebra cohomology of super Minkowski spacetime which classifies the Green–Schwarz superstring [32], the Green–Schwarz-type supermembrane [7], as well as all their double dimensional reductions [22] [29, Section 2], a fact now known as the “old brane scan” [19].Footnote 1

For example, for minimal spacetime supersymmetry there is, up to rescaling, a single non-trivial invariant \((p+2)\)-cocycle corresponding to a super p-brane in d dimensional spacetime, for just those pairs of (d, p) with \(d \le 11\) that are marked by an asterisk in Table 1.

Table 1 The old brane scan
Full size table

Here, the entry at \(d = 10\) and \(p = 1\) corresponds to the Green–Schwarz superstring, the entry at \(d = 10\) and \(p = 5\) to the NS5-brane, and the entry at \(d = 11\), \(p = 2\) to the M2-brane of M-theory fame [21, Chapter II]. Moving down and to the left on the table corresponds to double dimensional reduction [22] [29, Section 2].

This result is striking in its achievement and its failure: On the one hand, it is remarkable that the existence of super p-brane species may be reduced to a mathematical classification of super Lie algebra cohomology. But on the other hand, it is disconcerting that this classification misses so many p-brane species that are thought to exist: The M5-brane in \(d = 11\) and all the D-branes in \(d = 10\) are absent from the old brane scan, as are all their double dimensional reductions.Footnote 2

However, it turns out that this problem is not a shortcoming of super Lie theory as such, but only of the tacit restriction to ordinary super Lie algebras, as opposed to “higher” super Lie algebras, also called “super Lie n-algebras” or “super \(L_\infty \)-algebras” [27, 36].Footnote 3

One way to think of super Lie n-algebras is as the answer to the following question: Since, by a classical textbook fact, 2-cocycles on a super Lie algebra classify its central extensions in the category of super Lie algebras, what do higher-degree cocycles classify? The answer ([27, Prop. 3.5] based on [25, Theorem 3.1.13] and [3, Theorem 57]) is that higher-degree cocycles classify precisely higher central extensions, formed in the homotopy theory of super \(L_\infty \)-algebras. But in fact the Chevalley–Eilenberg algebras for the canonical models of these higher extensions are well known in parts of the supergravity literature, these are just the “free differential algebras”Footnote 4 or “FDA”s of D’Auria and Fré [16].

Hence, every entry in the “old brane scan,” since it corresponds to a cocycle, gives a super Lie n-algebraic extension of super Minkowski spacetime. Notably, the 3-cocycles for the superstring give rise to super Lie 2-algebras and the 4-cocycles for the supermembrane give rise to super Lie 3-algebras. These are superalgebraic analogs of the string Lie 2-algebra [3] [26, appendix] which controls the Green–Schwarz anomaly cancelation of the heterotic string [48], and hence, they are called the superstring Lie 2-algebra [36], to be denoted \(\mathfrak {string}\):

and the supermembrane Lie 3-algebra, denoted \(\mathfrak {m}2\mathfrak {brane}\):

A discussion of these structures as objects in higher Lie theory appears in Huerta’s thesis [36]. Note that \(\mathfrak {string}\) comes in several variants, denoted \(\mathfrak {string}_{\mathrm{IIA}}\), \(\mathfrak {string}_{\mathrm{IIB}}\), and \(\mathfrak {string}_{\mathrm{het}}\), corresponding to the type IIA, IIB, and heterotic variants of string theory. In their dual incarnation as “FDAs”, the \(\mathfrak {string}\) and \(\mathfrak {m}2\mathfrak {brane}\) algebras are the extended super Minkowski spacetimes considered by Chryssomalakos et al. [13]. We follow their idea and call extensions of super Minkowski spacetime to super Lie n-algebras extended super Minkowski spacetimes.

Now that each entry in the old brane scan is identified with a higher super Lie algebra in this way, something remarkable happens: new cocycles appear on these extended super Minkowski spacetimes, cocycles which do not show up on plain super Minkowski spacetime itself. (In homotopy theory, this is a familiar phenomenon: It is the hallmark of the construction of the “Whitehead tower” of a topological space.)

And indeed, it turns out that the new invariant cocycles thus found do correspond to the branes that were missing from the old brane scan [27]: On the super Lie 3-algebra \(\mathfrak {m}2\mathfrak {brane}\) there appears an invariant 7-cocycle, which corresponds to the M5-brane, on the super Lie 2-algebra \(\mathfrak {string}_{\mathrm{IIA}}\) there appears a sequence of \((p+2)\)-cocycles for \(p \in \{0,2,4,6,8\}\), corresponding to the type IIA D-branes, and on the superstring Lie 2-algebra \(\mathfrak {string}_{\mathrm{IIB}}\) there appears a sequence of \((p+2)\)-cocycles for \(p \in \{1,3,5,7,9\}\), corresponding to the type IIB D-branes. Under the identification of super Lie n-algebras with formal duals of “FDAs,” the algebra behind this statement is in fact an old result: For the M5-brane and the type IIA D-branes, this is due to Chryssomalakos et al. [13], while for the type IIB D-branes this is due to Sakaguchi [46, Section 2]. In fact, the 7-cocycle on the supermembrane Lie 3-algebra that corresponds to the M5-brane [6] was already discovered in the 1982 paper by D’Auria and Fré [16, Equations (3.27) and (3.28)].

Each of these cocycles gives a super Lie n-algebra extension. If we name these extensions by the super p-brane species whose WZW term is given by the cocycle, then we obtain the following diagram in the category of super \(L_\infty \)-algebras:

Hence, in the context of higher super Lie algebra, the “old brane scan” is completed to a tree of consecutive higher central extensions emanating out of the super Minkowski spacetimes, with one leaf for each brane species in string/M-theory and with one edge whenever one fundamental brane species may end on another, with its boundary sourcing a vector or tensor multiplet on the world volume of the other brane [27, Section 3]. This is the fundamental brane bouquet [27, Def. 3.9 and Section 4.5]. (The black branes and their more general intersection laws are obtained from this by passing to equivariant cohomology [37], but this will not concern us here.)

Interestingly, a fair bit of the story of string/M-theory is encoded in this purely super Lie-n-algebraic mathematical structure. This includes in particular the pertinent dualities: the KK-reduction between M-theory and type IIA theory, the HW-reduction between M-theory and heterotic string theory, the T-duality between type IIA and type IIB, the S-duality of type IIB, and the relation between type IIB and F-theory. All of these are reflected as equivalences of super Lie n-algebras obtained from the brane bouquet [28, 29, 37]. The diagram of super \(L_\infty \)-algebras that reflects these \(L_\infty \)-equivalences looks like a candidate to fill Polchinski’s famous schematic picture of M-theory [45, Figure 1] [53, Figure 4] with mathematical life:

Now note that not all of the super p-brane cocycles are of higher degree. One of them, the cocycle for the D0-brane, is an ordinary 2-cocycle. Accordingly, the extension that it classifies is an ordinary super Lie algebra extension. In fact, one finds that the D0-cocycle classifies the central extension of 10-dimensional type IIA super Minkowski spacetime to the 11-dimensional spacetime of M-theory. We can express these relationships by noting the following diagram of super Lie n-algebras is, in the sense of homotopy theory, a “homotopy pullback”:

This is the precise way to say that the D0-brane cocycle on \(\mathfrak {string}_{\mathrm{IIA}}\) comes from pulling back an ordinary 2-cocycle on \({\mathbb R}^{9,1|\mathbf{16 + {\overline{16}}}}\), which in turn is extended to \({\mathbb R}^{10,1|\mathbf{32}}\) by the same 2-cocycle. We may think of this as a super \(L_\infty \)-theoretic incarnation of the observation that D0-brane condensation in type IIA string theory leads to the growth of the 11th dimension of M-theory [27, Remark 4.6], as explained by Polchinski [45, Section 6].

This raises an evident question: Might there be a precise sense in which all dimensions of spacetime originate from the condensation of some kind of 0-branes in this way? Is the brane bouquet possibly rooted in the superpoint? Such that the ordinary super Minkowski spacetimes, not just extended super Minkowski spacetimes such as \(\mathfrak {string}\) and \(\mathfrak {m}2\mathfrak {brane}\), arise from a process of 0-brane condensation “from nothing”?

Since the brane bouquet proceeds at each stage by forming maximal invariant extensions, the mathematical version of this question is: Is there a sequence of maximal invariant central extensions that start at the superpoint and produce the super Minkowski spacetimes in which superstrings and supermembranes exist?

To appreciate the substance of this question, notice that it is clear that every super Minkowski spacetime is some central extension of a superpoint [13, Section 2.1]: the super 2 cocycle classifying this extension is just the superbracket that turns two supercharges into a translation generator. But there are many central extensions of superpoints that are nothing like super Minkowski spacetimes. The question is whether the simple principle of consecutively forming maximal invariant central extensions of super Lie algebras (as opposed to more general central extensions) discovers spacetime.

We shall prove that this is the case: This is our main result, Theorem 14. It says that in the following diagram of super Minkowski super Lie algebras, each diagonal morphism is singled out as being the maximal invariant central extension of the super Lie algebra that it points to:Footnote 5

Note that we do not specify by hand the groups under which these extensions are to be invariant. Instead, these groups are being discovered stagewise, along with the spacetimes. Namely, we say (Definition 7) that an extension \(\widehat{\mathfrak {g}} \rightarrow \mathfrak {g}\) is invariant if it is invariant with respect to the “simple external automorphisms” inside the automorphism group of \(\mathfrak {g}\) (Definition 1). This is a completely intrinsic concept of invariance.

We show that for \(\mathfrak {g}\) a super Minkowski spacetime, then this intrinsic group of simple external automorphisms is the spin group, the double cover of the connected Lorentz group in the corresponding dimension—this is Proposition 6, read at the Lie group level. This may essentially be folklore [24, p. 95], but it seems worthwhile to pinpoint this statement. For it says that as the extension process grows out of the superpoint, not only are the super Minkowski spacetimes being discovered as supertranslation supersymmetry groups, but also their Lorentzian metric structure is being discovered alongside.

Super Minkowski super Lie algebra

Simple external automorphisms

Induced Cartan geometry

Torsion-freeness

\({\mathbb {R}}^{d-1,1\vert N}\)

\(\mathrm {Spin}(d-1,1)\)

Supergravity

In \(d = 11\): Einstein’s equations

To highlight this, observe that with every pair (V, G) consisting of a super vector space V and a subgroup \(G \subset \mathrm {GL}(V)\) of its general linear supergroup, there is associated a type of geometry, namely the corresponding Cartan geometry: A (V, G)-geometry is a supermanifold with tangent spaces isomorphic to V and equipped with a reduction of the structure group of its super frame bundle from \(\mathrm {GL}(V)\) to G [41].

Now for the pairs \(({\mathbb {R}}^{d-1,1\vert N}, \mathrm {Spin}(d-1,1))\) that emerge out of the superpoint according to Proposition 6 and Theorem 14, this is what encodes a field configuration of d-dimensional N-supersymmetric supergravity: Supermanifolds locally modeled on \({\mathbb {R}}^{d-1,1\vert N}\) are precisely what underlie the superspace formulation of supergravity, and the reduction of its structure group to the \(\mathrm {Spin}(d-1,1)\)-cover of the connected Lorentz group \(\mathrm{SO}_0(d-1,1)\) is equivalently a choice of supervielbein field, which is a field configuration of supergravity.

Observe also that the mathematically most natural condition to demand from such a super Cartan geometry is that it be “torsion-free” [41]. In view of this, it is worthwhile to recall the remarkable theorem of Howe [35], based on Candiello and Lechner [12]: For \(d = 11\), the equations of motion of supergravity are implied by the torsion-freeness of the supervielbein.

In summary, Theorem 14 shows that the brane bouquet, and with it at least a fair chunk of the structure associated with the word “M-theory,” has its mathematical root in the superpoint, and Proposition 6 adds that as the superspacetimes grow out of the superpoint, they consecutively discover their relevant Lorentzian metric structure and spinorial structure, and finally their supergravity equations of motion.

2 Automorphisms of super Minkowski spacetimes

For our main result, Theorem 14, we need to know the automorphisms (Definition 18) of the “super Minkowski super Lie algebras” \({\mathbb {R}}^{d-1,1\vert N}\). We give the precise definition of \({\mathbb R}^{d-1,1|N}\) as Definition 22, but for the reader’s convenience, we quickly recall the idea. The super Minkowski super Lie algebra \({\mathbb R}^{d-1,1|N}\) is the version of Minkowski spacetime, \({\mathbb R}^{d-1,1}\), used when discussing supersymmetry. Unlike Minkowski spacetime, which is merely a vector space, super Minkowski is a super Lie algebra: It has an underlying vector space that is \({\mathbb Z}_2\)-graded, with an even and odd part:

$$\begin{aligned} {\mathbb R}^{d-1,1|N}_{\mathrm{even}} = {\mathbb R}^{d-1,1}, \quad {\mathbb R}^{d-1,1|N}_{\mathrm{odd}} = N. \end{aligned}$$

Here, \({\mathbb R}^{d-1,1}\) is ordinary Minkowski spacetime, while N is a spinor representation of \(\mathrm{Spin}(d-1,1)\). The Lie group \(\mathrm{Spin}(d-1,1)\) is the double cover of the connected Lorentz group, \(\mathrm{SO}_0(d-1,1)\), so it also acts on \({\mathbb R}^{d-1,1}\). The Lie bracket on this super Lie algebra is nonzero only on N and consists of a pairing turning spinors into vectors:

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

which is required to be an equivariant map between representations of \(\mathrm{Spin}(d-1,1)\).

Our key idea is that we can extract the Lorentz symmetries of \({\mathbb R}^{d-1,1\vert N}\) merely from its structure as a super Lie algebra, by looking at a particular piece of the automorphisms we call the “simple external automorphisms.” This result may be folklore (see Evans [24, p. 95]), but since we did not find a full account in the literature, we provide a proof here. After some simple lemmas, the result is Proposition 6. To begin, we define the “simple external automorphisms” of a super Lie algebra.

Definition 1

(External and internal automorphisms, admissible algebras) Let \(\mathfrak {g}\) be a super Lie algebra (Definition 15), and let \(\mathfrak {aut}(\mathfrak {g})\) be the ordinary Lie algebra of infinitesimal automorphisms of \(\mathfrak {g}\) which preserve the \({\mathbb Z}_2\)-grading (Proposition 19). We define the Lie algebra \(\mathfrak {int}(\mathfrak {g})\) of internal automorphisms of \(\mathfrak {g}\) as the Lie subalgebra of \(\mathfrak {aut}(\mathfrak {g})\) which acts trivially on the even part \(\mathfrak {g}_{\mathrm{even}}\). In other words, it is the Lie subalgebra of even derivations of \(\mathfrak {g}\) which vanish on \(\mathfrak {g}_{\mathrm {even}}\). This is clearly an ideal, so that the quotient

$$\begin{aligned} \mathfrak {ext}(\mathfrak {g}) := \mathrm {aut}(\mathfrak {g})/\mathfrak {int}(\mathfrak {g}) \end{aligned}$$

of all automorphisms by internal ones is again a Lie algebra, the Lie algebra of external automorphisms of \(\mathfrak {g}\). We thus have a short exact sequence:

$$\begin{aligned} 0 \rightarrow \mathfrak {int}(\mathfrak {g}) \rightarrow \mathfrak {aut}(\mathfrak {g}) \rightarrow \mathfrak {ext}(\mathfrak {g}) \rightarrow 0. \end{aligned}$$

We will say that \(\mathfrak {g}\) is admissible if this sequence splits and the external automorphism algebra \(\mathfrak {ext}(\mathfrak {g})\) is reductive. For an admissible algebra \(\mathfrak {g}\), we can thus view \(\mathfrak {ext}(\mathfrak {g})\) as a subalgebra of \(\mathfrak {aut}(\mathfrak {g})\). Moreover, because we demand \(\mathfrak {ext}(\mathfrak {g})\) be reductive, \(\mathfrak {ext}(\mathfrak {g})\) decomposes as a direct sum of its center and its maximal semisimple Lie subalgebra. We thus define the simple external automorphisms

$$\begin{aligned} \mathfrak {ext}_{\mathrm {simp}}(\mathfrak {g}) \hookrightarrow \mathfrak {ext}(\mathfrak {g}) \hookrightarrow \mathfrak {aut}(\mathfrak {g}) \end{aligned}$$

to be the semisimple part of \(\mathfrak {ext}(\mathfrak {g})\).

Example 2

The internal automorphisms (Definition 1) of the super Minkowski super Lie algebra \({\mathbb {R}}^{d-1,1\vert N}\) are the “R-symmetries” from the physics literature [30, p. 56].

Because the super Minkowski super Lie algebra \({\mathbb R}^{d-1,1|N}\) is built from \(\mathrm{Spin}(d-1,1)\) representations and \(\mathrm{Spin}(d-1,1)\)-equivariant maps, \(\mathrm{Spin}(d-1,1)\) acts on this super Lie algebra by automorphism. It thus acts on the full automorphism group \(\mathrm {Aut}({\mathbb R}^{d-1,1|N})\) by conjugation and on the Lie algebra \(\mathfrak {aut}({\mathbb R}^{d-1,1|N})\) by the adjoint action. These facts are key for our first lemma.

Lemma 3

Consider a super Minkowski super Lie algebra \({\mathbb {R}}^{d-1,1\vert N}\) (Definition 22) in any dimension \(d \ge 3\) and for any real spinor representation N of \(\mathrm{Spin}(d-1,1)\). Then, the automorphism Lie algebra \(\mathfrak {aut}({\mathbb {R}}^{d-1,1|N})\) (Proposition 19) is the graph of a surjective, \(\mathrm {Spin}(d-1,1)\)-equivariant Lie algebra homomorphism

$$\begin{aligned} K :\mathfrak {g}_s \longrightarrow \mathfrak {g}_v, \end{aligned}$$

where \(\mathfrak {g}_s \subseteq {\mathfrak {gl}}(N)\) and \(\mathfrak {g}_v \subseteq {\mathfrak {gl}}({\mathbb R}^{d-1,1})\) are the projections of \(\mathfrak {aut}({\mathbb R}^{d-1,1|N}) \subseteq {\mathfrak {gl}}(N) \oplus {\mathfrak {gl}}({\mathbb R}^{d-1,1})\) onto the summands. Here, K is equivariant with respect to the adjoint action of \(\mathrm{Spin}(d-1,1)\) restricted to \(\mathfrak {g}_s\) and \(\mathfrak {g}_v\).

In particular, the kernel of K is the internal automorphism algebra (Definition 1), also known as the R-symmetry algebra (Example 2):

$$\begin{aligned} \mathrm {ker}(K) \simeq \mathrm {int}({\mathbb {R}}^{d-1,1\vert N}). \end{aligned}$$

Proof

We will consider the corresponding inclusion at the level of groups

$$\begin{aligned} \mathrm {Aut}({\mathbb {R}}^{d-1,1\vert N}) \hookrightarrow \mathrm {GL}(N) \times \mathrm{GL}({\mathbb R}^{d-1,1}) \end{aligned}$$

with projections \(G_s \subseteq \mathrm{GL}(N)\) and \(G_v \subseteq \mathrm{GL}({\mathbb R}^{d-1,1})\). The result will then follow by differentiation.

Note that the spinor-to-vector pairing

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

is surjective, because it is a nonzero map of \(\mathrm {Spin}(d-1,1)\)-representations, and \({\mathbb R}^{d-1,1}\) is irreducible for \(d \ge 3\). Hence, for every vector \(v \in {\mathbb R}^{d-1,1}\), there is a pair of spinors \(\psi ,\phi \in N\) such that

$$\begin{aligned} v = [\psi ,\phi ]. \end{aligned}$$

It follows that for any automorphism \((f,g) \in \mathrm {Aut}({\mathbb {R}}^{d-1,1\vert N}) \subseteq G_s \times G_v\), g is uniquely determined by f because (f, g) is an automorphism:

$$\begin{aligned} g(v) = [f(\psi ), f(\phi )]. \end{aligned}$$

This determines a function \(k :G_s \rightarrow G_v\) sending f to g. It is surjective by construction of \(G_v\) and is a group homomorphism because its graph \(\mathrm {Aut}({\mathbb {R}}^{d-1,1\vert N})\) is a group. Finally, conjugating (f, g) by an element of \(\mathrm{Spin}(d-1,1)\), it is a quick calculation to check that k is \(\mathrm {Spin}(d-1,1)\)-equivariant, using the equivariance of the spinor-to-vector pairing \([-,-]\). \(\square \)

Lemma 4

Let \(\mathfrak {g}_s\) be as in Lemma 3. Then, \(\mathfrak {aut}({\mathbb R}^{d-1,1|N}) \simeq \mathfrak {g}_s\) as Lie algebras.

Proof

Because \(\mathfrak {aut}({\mathbb R}^{d-1,1|N})\) is the graph of the homomorphism \(K :\mathfrak {g}_s \rightarrow \mathfrak {g}_v\) from Lemma 3, it is isomorphic to the domain of this homomorphism, \(\mathfrak {g}_s\). \(\square \)

Lemma 5

Let N be a real spinor representation of \(\mathrm{Spin}(d-1,1)\) in some dimension \(d \ge 3\). Then, the Lie algebra \(\mathfrak {g}_v\) from Lemma 3 decomposes as a \(\mathrm {Spin}(d-1,1)\)-representation into the direct sum of the adjoint representation with the trivial representation:

$$\begin{aligned} \mathfrak {g}_v \;\simeq \; \mathfrak {so}(d-1,1) \oplus {\mathbb {R}}. \end{aligned}$$

Similarly, the Lie algebra \(\mathfrak {g}_s\) from Lemma 3 decomposes as a direct sum of exterior powers of the vector representation \({\mathbb R}^{d-1,1}\):

$$\begin{aligned} \mathfrak {g}_s \simeq \underset{i}{\oplus } \, \Lambda ^{n_i} {\mathbb R}^{d-1,1}. \end{aligned}$$

Proof

First assume that N is a Majorana spinor representation as in Example 25, and consider \(\mathfrak {g}_s\). Since the Majorana representation N is a real subrepresentation of a complex Dirac representation \({\mathbb {C}}^{\mathrm {dim}_{{\mathbb {R}}}(N)}\), there is a canonical \({\mathbb {R}}\)-linear inclusion

$$\begin{aligned} \mathrm {End}_{{\mathbb {R}}}(N) \hookrightarrow \mathrm {End}_{{\mathbb {C}}}({\mathbb {C}}^{\mathrm {dim}_{{\mathbb {R}}}(N)}). \end{aligned}$$

Therefore, it is sufficient to note that the space of endomorphisms of the Dirac representation over the complex numbers decomposes into a direct sum of exterior powers of the vector representation. This is indeed so, thanks to the inclusion:

$$\begin{aligned} \mathrm {End}_{{\mathbb {C}}}({\mathbb {C}}^{\mathrm {dim}_{{\mathbb {R}}}(N) }) \hookrightarrow \mathrm {Cl}({\mathbb {R}}^{d-1,1})\otimes {\mathbb {C}}. \end{aligned}$$

Explicitly, in terms of the Dirac Clifford basis of Example 25, the decomposition is given by the usual component formula:

$$\begin{aligned} \psi \otimes {\overline{\phi }} \; \mapsto \; {\overline{\phi }}\psi + \left( {\overline{\phi }} \Gamma _ a\psi \right) \Gamma ^a + \tfrac{1}{2} \left( {\overline{\phi }} \Gamma _{a b} \psi \right) \Gamma ^{a b} + \tfrac{1}{3!}\left( {\overline{\phi }} \Gamma _{a_1 a_2 a_3} \psi \right) \Gamma ^{a_1 a_2 a_3} + \cdots . \end{aligned}$$

Now consider \(\mathfrak {g}_v\). Recall that, by definition, the automorphism group of \({\mathbb {R}}^{d-1,1\vert N}\) is

and its Lie algebra is

As we noted above, \(\mathrm {Aut}({\mathbb {R}}^{d-1,1\vert N})\) always contains \(\mathrm {Spin}(d-1,1)\), acting canonically, since the spinor-to-vector pairing is \(\mathrm {Spin}(d-1,1)\)-equivariant. Another subgroup of automorphisms that exists generally is a copy of the multiplicative group of real numbers \({\mathbb {R}}^\times \) where \(t \in {\mathbb {R}}^\times \) acts on spinors \(\psi \) as rescaling by t and on vectors v as rescaling by \(t^2\):

$$\begin{aligned} \psi \mapsto t \psi , \quad v \mapsto t^2 v. \end{aligned}$$

The Lie algebra of this scaling action is the scaling derivations of Example 20. Hence, for all d and N we have the obvious Lie algebra inclusion

$$\begin{aligned} \mathfrak {so}(d-1,1) \oplus {\mathbb {R}} \hookrightarrow \mathfrak {aut}({\mathbb {R}}^{d-1,1\vert N}). \end{aligned}$$

This shows that there is an inclusion

$$\begin{aligned} \mathfrak {so}(d-1,1) \oplus {\mathbb {R}} \hookrightarrow \mathfrak {g}_v \hookrightarrow \mathfrak {gl}({\mathbb {R}}^{d}). \end{aligned}$$

Hence, it now only remains to see that there is no further summand in \(\mathfrak {g}_v\). But we know that there is at most one further summand in \(\mathfrak {gl}({\mathbb R}^{d-1,1})\), since this decomposes in the form

$$\begin{aligned} \mathfrak {gl}({\mathbb R}^{d-1,1}) \simeq \mathfrak {so}(d-1,1) \oplus {\mathbb {R}} \oplus \mathrm{Sym}^2_0({\mathbb R}^{d-1,1}), \end{aligned}$$

where \(\mathrm{Sym}^2_0({\mathbb R}^{d-1,1})\) denotes the space of traceless, symmetric \(d \times d\) matrices. It follows that the only further summand that could appear in \(\mathfrak {g}_v\) is \(\mathrm{Sym}^2_0({\mathbb R}^{d-1,1})\). But by Lemma 3, the homomorphism \(K :\mathfrak {g}_s \rightarrow \mathfrak {g}_v\) is surjective, so its image \(\mathfrak {g}_v\) must be a subset of the exterior powers appearing in \(\mathfrak {g}_s\). Since the symmetric traceless matrices and the exterior powers \(\Lambda ^\bullet {\mathbb {R}}^d\) are distinct irreducible representations of \(\mathrm{Spin}(d-1,1)\), we conclude \(\mathrm{Sym}^2_0({\mathbb R}^{d-1,1})\) is not a summand of \(\mathfrak {g}_v\).

This concludes the proof for the case that N is a Majorana representation. The argument for N symplectic Majorana (Example 25) is similar. Finally, a general real spin representation is a direct multiple of N or a sum of multiples of the two Weyl representations \(N \simeq N_- \oplus N_+\). We generalize to these cases in turn.

First, we consider nN, a direct multiple of N, for n some nonnegative integer. Since \(\mathrm{End}_{\mathbb R}(nN) \simeq n^2 \mathrm{End}_{\mathbb R}(N)\), the left-hand side is indeed a sum of exterior powers.

Next, if N decomposes as \(N_- \oplus N_+\), a general spin representation is of the form \(n_- N_- \oplus n_+ N_+\), for \(n_-\) and \(n_+\) nonnegative integers. We wish to show that

$$\begin{aligned} \mathrm{End}_{\mathbb R}(n_- N_- \oplus n_+ N_+)\simeq & {} n_-^2 \mathrm{End}_{\mathbb R}(N_-) \, \oplus \, n_- n_+ \mathrm{Hom}_{\mathbb R}(N_-, N_+) \\&\oplus \, n_+ n_- \mathrm{Hom}_{\mathbb R}(N_+, N_-) \, \oplus \, n_+^2 \mathrm{End}_{\mathbb R}(N_+) \end{aligned}$$

is a sum of exterior powers. Yet we have already shown that

$$\begin{aligned} \mathrm{End}_{\mathbb R}(N_- \oplus N_+) \simeq \mathrm{End}_{\mathbb R}(N_-) \, \oplus \, \mathrm{Hom}_{\mathbb R}(N_-, N_+) \, \oplus \, \mathrm{Hom}_{\mathbb R}(N_+, N_-) \, \oplus \, \mathrm{End}_{\mathbb R}(N_+) \end{aligned}$$

is a sum of exterior powers. Thus, every summand on the right-hand side is a sum of exterior powers, and it follows that \(\mathrm{End}_{\mathbb R}(n_-N_- \oplus n_+ N_+)\) is also. \(\square \)

Proposition 6

For any dimension \(d \ge 3\) and real spinor representation N of \(\mathrm{Spin}(d-1,1)\), the super Minkowski super Lie algebra \({\mathbb {R}}^{d-1,1\vert N}\) (Definition 22) is admissible (Definition 1). Moreover, the Lie algebra of external automorphisms (Definition 1) of \({\mathbb R}^{d-1,1|N}\) is the direct sum:

$$\begin{aligned} \mathfrak {ext}({\mathbb {R}}^{d-1,1\vert N}) \;\simeq \; \mathfrak {so}(d-1,1) \oplus {\mathbb {R}}, \end{aligned}$$

where \({\mathfrak {so}}(d-1,1)\) acts in the canonical way on \({\mathbb R}^{d-1,1|N}\) (Definition 22) and \({\mathbb R}\) acts by the scaling action from Example 20.

Proof

The admissibility of \({\mathbb R}^{d-1,1|N}\) will follow when we determine \(\mathfrak {ext}({\mathbb R}^{d-1,1|N})\) has the form claimed, since this form is reductive, and the action of \(\mathfrak {ext}({\mathbb R}^{d-1,1|N})\) on \({\mathbb R}^{d-1,1|N}\) described in the proposition gives the splitting. So we prove this form is correct. By Lemma 4, we have \(\mathfrak {aut}({\mathbb {R}}^{d-1,1\vert N}) \simeq \mathfrak {g}_s\), and by Lemma 5, we have a decomposition as \(\mathrm{Spin}(d-1,1)\)-representations

$$\begin{aligned} \mathfrak {aut}({\mathbb {R}}^{d-1,1\vert N}) \;\simeq \; \left( \mathfrak {so}(d-1,1) \;\oplus \; {\mathbb {R}} \right) \;\oplus \; \underset{= \mathfrak {int}({\mathbb {R}}^{d-1,1\vert N})}{\underbrace{\mathrm {ker}(K)}}, \end{aligned}$$

where the last summand is the algebra of internal automorphisms (Definition 1) and hence the R-symmetries (Example 2). Therefore, the claim follows by Definition 1. \(\square \)

3 The maximal invariant central extensions of the superpoint

With the results from the previous section in hand, we have a way of talking about the \({\mathfrak {so}}(d-1,1)\) symmetries of a super Minkowski super Lie algebra \({\mathbb R}^{d-1,1|N}\) purely in terms of its Lie bracket: It is the algebra of simple external automorphisms of \({\mathbb R}^{d-1,1|N}\), by Proposition 6. This allows us to begin with a super Lie algebra that lacks any apparent relation to spacetime, and discover spacetime symmetries via the automorphisms. We make repeated use of this in our construction of super Minkowski spacetimes by central extension of the superpoint, \({\mathbb R}^{0|1}\). This is our main result, Theorem 14.

To be precise, we compute consecutive “maximal invariant central extensions” of the superpoint. First, we state the definition of the extension process:

Definition 7

(maximal invariant central extensions) Let \(\mathfrak {g}\) be an admissible super Lie algebra (Definitions 1 and 15), let \(\mathfrak {h} \hookrightarrow \mathfrak {aut}(\mathfrak {g})\) be a subalgebra of its automorphism Lie algebra (Proposition 19) and let

be a central extension of \(\mathfrak {g}\) by a vector space V in even degree. Then, we say that \(\widehat{\mathfrak {g}}\) is

  1. 1.

    an \(\mathfrak {h}\)-invariant central extension if the even 2-cocycles that classify the extension, according to Example 17, are \(\mathfrak {h}\)-invariant 2-cocycles as in Definition 21;

  2. 2.

    an invariant central extension if it is \(\mathfrak {h}\)-invariant and \(\mathfrak {h} = \mathfrak {ext}_{\mathrm {simp}}(\mathfrak {g})\) is the semisimple part of its external automorphism Lie algebra (Definition 1);

  3. 3.

    a maximal \(\mathfrak {h}\)-invariant central extension if it is an \(\mathfrak {h}\)-invariant central extension such that the n-tuple of \(\mathfrak {h}\)-invariant even 2-cocycles that classifies it (according to Example 17) is a linear basis for the even \(\mathfrak {h}\)-invariant cohomology \(H_{\mathrm{even}}^2(\mathfrak {g},{\mathbb {R}})^{ \mathfrak {h} }\) (Definition 21).

When the central extension \({\widehat{\mathfrak {g}}}\) is both maximal and invariant, we say it is a maximal invariant central extension and distinguish it with the symbol \(\star \) on the projection map \({\widehat{\mathfrak {g}}} \rightarrow \mathfrak {g}\), like this:

We now begin to climb the tower of maximal invariant central extensions, beginning with the superpoint. But first we must note that our starting point is admissible.

Lemma 8

The superpoint \({\mathbb R}^{0|N}\) (Definition 24) is admissible (Definition 1) for any natural number \(N \in {\mathbb {N}}\).

Proof

All automorphisms of \({\mathbb R}^{0|N}\) are internal (Definition 1), so the external automorphisms are trivial. Hence, they are trivially reductive, and trivially a subalgebra of \(\mathfrak {aut}({\mathbb R}^{0|N})\), which is what it means to be admissible. \(\square \)

In our next proposition, we see spacetime appear by extending a superpoint.

Proposition 9

The maximal invariant central extension (Definition 7) of the superpoint \({\mathbb {R}}^{0\vert 2}\) (Definition 24) is the three-dimensional super Minkowski super Lie algebra \({\mathbb {R}}^{2,1\vert {\mathbf {2}} }\) as in Definition 22:

with \(N ={\mathbf {2}}\) the unique irreducible real spinor representation of \(\mathrm {Spin}(2,1)\) from Proposition 32.

Proof

Since \({\mathbb R}^{0|2}\) is concentrated in odd degree, the external automorphisms are trivial: \(\mathfrak {ext}({\mathbb {R}}^{0\vert 2}) = 0\). Thus, every central extension is invariant (Definition  7).

According to Example 17, the maximal central extension is the one induced by the all of the even super Lie algebra 2-cocycles on \({\mathbb {R}}^{0\vert 2}\). Since \({\mathbb {R}}^{0\vert 2}\) is concentrated in odd degree and has trivial Lie bracket, an even 2-cocycle in this case is given by a symmetric bilinear form on \({\mathbb {R}}^2\). There is a three-dimensional real vector space of these. This shows that the underlying supervector space of the maximal central extension is \({\mathbb {R}}^{3\vert 2}\). It remains to check that the Lie bracket is that of 3d super Minkowski.

If we let \(\{d \theta _1, d \theta _2\}\) denote the canonical basis of the dual space \({\mathbb {R}}^{{0\vert 2}*}\), then the space of even 2-cocycles is spanned by:

where the wedge product is symmetric between these odd elements. By the formula for the central extension from Example 17, this means that the super Lie bracket is given on the spinors and by

$$\begin{aligned} \left[ \psi , \phi \right] = \left( \begin{array}{l@{\quad }l} \psi _1 \phi _1 &{} \tfrac{1}{2}(\psi _1 \phi _2 + \phi _1 \psi _2)\\ \tfrac{1}{2}(\psi _1 \phi _2 + \phi _1 \psi _2) &{} \psi _2 \phi _2 \end{array} \right) \;\;\;=\;\;\; \tfrac{1}{2}\left( \psi \phi ^\dagger + \psi \phi ^\dagger \right) , \end{aligned}$$

Comparing this formula to Proposition 32, we see this is indeed the spinor-to-vector pairing for the real representation \({\mathbf {2}}\) of \(\mathrm {Spin}(2,1)\). \(\square \)

To deduce the maximal invariant central extensions of \({\mathbb {R}}^{2,1 \vert 2}\), we use the representation of spinors via the normed division algebras as a key tool. We give all the details in Proposition 32 of our appendix, but for the reader’s convenience, we quickly summarize the idea.

There are four real normed division algebras: the real numbers \({\mathbb R}\), the complex numbers \({\mathbb C}\), the quaternions \({\mathbb H}\), and the octonions \({\mathbb O}\). They have dimensions 1, 2, 4, and 8, respectively. It is a famous fact that the octonions \({\mathbb O}\) are not associative, while \({\mathbb R}, {\mathbb C}\) and \({\mathbb H}\) are. For \({\mathbb K}\) a normed division algebra of dimension k, we can construct spinors for spacetime of dimension \(k+2\). More precisely, we can cook up two irreducible, real spinor representations of the spin group \(\mathrm{Spin}(k+1,1)\), the double cover of the connected Lorentz group \(\mathrm{SO}_0(k+1,1)\). Both of these spinor representations are defined on the vector space \({\mathbb K}^2\), but they differ in the action of \(\mathrm{Spin}(k+1,1)\):

$$\begin{aligned} N_+ = {\mathbb K}^2, \quad N_- = {\mathbb K}^2 . \end{aligned}$$

We can also define Minkowski spacetime itself in terms of \({\mathbb K}\), as the space of \(2 \times 2\) hermitian matrices over \({\mathbb K}\):

$$\begin{aligned} {\mathbb R}^{k+1, 1} := \left\{ \begin{bmatrix} t + x&{\overline{y}} \\ y&t - x \end{bmatrix} : t, x \in {\mathbb R}, \, y \in {\mathbb K}\right\} , \end{aligned}$$

where \({\overline{y}} \in {\mathbb K}\) is denotes the conjugate of \(y \in {\mathbb K}\). For the more details, see Proposition 32.

Our next lemma relates the construction of spinors from \({\mathbb K}\) and from the ‘Cayley–Dickson double’, \({\mathbb K}_{\mathrm{dbl}}\) (Definition 26). Roughly, the Cayley–Dickson double takes a normed division algebra \({\mathbb K}\) of dimension k and gives the “next” normed division algebra \({\mathbb K}_{\mathrm{dbl}}\) of dimension 2k:

$$\begin{aligned} {\mathbb R}_{\mathrm{dbl}} = {\mathbb C}, \quad {\mathbb C}_{\mathrm{dbl}} = {\mathbb H}, \quad {\mathbb H}_{\mathrm{dbl}} = {\mathbb O}. \end{aligned}$$

This process breaks down for \({\mathbb K}= {\mathbb O}\), when \({\mathbb O}_{\mathrm{dbl}}\) fails to be a division algebra.

In any case, the Cayley–Dickson double contains the original algebra as a subalgebra, \({\mathbb K}\subseteq {\mathbb K}_{\mathrm{dbl}}\). This means that the \(2 \times 2\) hermitian matrices over \({\mathbb K}\) are a subset of those over \({\mathbb K}_{\mathrm{dbl}}\), and hence, there is an inclusion of spacetimes \({\mathbb R}^{k+1,1} \subseteq {\mathbb R}^{2k + 1, 1}\), and a corresponding inclusion of spin groups \(\mathrm{Spin}(k+1,1) \subseteq \mathrm{Spin}(2k+1,1)\). By restricting along this inclusion, spinor representations of \(\mathrm{Spin}(2k+1,1)\) become representations of \(\mathrm{Spin}(k+1,1)\). The next lemma tells us precisely which representations we obtain in this way.

Lemma 10

Let \({\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\}\) be an associative normed division algebra (Example 27) of dimension k, and let \({\mathbb {K}}_{\mathrm {dbl}} \in \{{\mathbb C}, {\mathbb H}, {\mathbb O}\}\) be its Cayley–Dickson double (Definition 26) of dimension 2k. Let \(N_+\) and \(N_-\) be the real spinor representations defined in terms of \({\mathbb K}\), and let \(N_{\mathrm{dbl}}\) denote either of the real spinor representations defined in terms of \({\mathbb K}_{\mathrm{dbl}}\), as in Proposition 32. Consider the inclusion of spin groups \(\mathrm{Spin}(k+1,1) \subseteq \mathrm{Spin}(2k + 1, 1)\) induced by the inclusion of normed division algebras \({\mathbb K}\subseteq {\mathbb K}_{\mathrm{dbl}}\). Restricting along this inclusion, the irreducible real \(\mathrm {Spin}( 2k+1,1 )\)-representation \(N_{\mathrm {dbl}}\) branches into the direct sum of the two irreducible real \(\mathrm {Spin}(k+1,1)\)-representations \(N_+, N_-\):

$$\begin{aligned} N_{\mathrm {dbl}} \simeq N_+ \oplus N_-. \end{aligned}$$

Proof

We will prove the result for \(N_{\mathrm{dbl}+}\), as the argument for \(N_{\mathrm{dbl}-}\) will be similar. By Proposition 32, the spin representation \(N_{\mathrm {dbl}+}\) is defined on the real vector space \({\mathbb {K}}_{\mathrm {dbl}}^2\). By Cayley–Dickson doubling (Definition 26), this is given in terms of \({\mathbb {K}}\) as the direct sum

$$\begin{aligned} {\mathbb {K}}_{\mathrm {dbl}}^2 \simeq {\mathbb {K}}^2 \oplus {\mathbb {K}}^2 \ell . \end{aligned}$$

This makes it immediate that the first summand \({\mathbb {K}}^2\) is \(N_+\) as a representation of \(\mathrm{Spin}(k+1,1)\). We need to show that the second summand is isomorphic to \(N_-\).

To that end observe, by the relations in the Cayley–Dickson construction (Definition 26), that for \(\psi \in {\mathbb {K}}^2\) and \(A \in \mathfrak {h}_2({\mathbb K})\) a \(2 \times 2\) hermitian matrix, we have the following identity:

$$\begin{aligned} \begin{aligned} A (\psi \ell )&= A (\ell {\overline{\psi }}) \\&= \ell ( {\overline{A}} \, {\overline{\psi }} ) \\&= \ell ({\overline{A}}_L {\overline{\psi }}) \\&= \ell ( \overline{A_R \psi } ) \\&= (A_R \psi ) \ell , \end{aligned} \end{aligned}$$

where \(A_L\) and \(A_R\) denote the right and left actions of the matrix A, respectively (Definition 28), and we have used Proposition 30 to relate left and right actions under conjugation.

Recall from Proposition 32 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:

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

It follows from our above calculation that the action of a generator \(AB \in \mathrm{Spin}(k+1,1)\) on the summand \({\mathbb {K}}^2 \ell \) is the composition of right actions on \({\mathbb {K}}^2\):

$$\begin{aligned} \begin{aligned} {\tilde{A}}_L B_L (\psi \ell )&= {\tilde{A}}( B (\psi \ell ) ) \\&= \left( {\tilde{A}}_R B_R (\psi ) \right) \ell . \end{aligned} \end{aligned}$$

Therefore, we are now reduced to showing that this action of \(\mathrm{Spin}(k+1,1)\) on \({\mathbb K}^2\):

$$\begin{aligned} \psi \mapsto {\tilde{A}}_R B_R (\psi )\quad \text{ for } \quad \psi \in {\mathbb K}^2 \end{aligned}$$

is isomorphic to the action of \(\mathrm{Spin}(k+1,1)\) on \(N_-\), which also has the underlying vector space \({\mathbb K}^2\):

$$\begin{aligned} \psi \mapsto A_L {\tilde{B}}_L (\psi ) \text{ for } \psi \in {\mathbb K}^2. \end{aligned}$$

We claim there is an isomorphism given by

$$\begin{aligned} F :\psi \mapsto J {\overline{\psi }}, \end{aligned}$$

where J is the matrix:

$$\begin{aligned} J := \left( \begin{array}{cc} 0 &{}\quad -\,1 \\ 1 &{}\quad 0 \end{array} \right) . \end{aligned}$$

A quick calculation shows that J satisfies the matrix identity:

$$\begin{aligned} J {\overline{A}} = - {\tilde{A}} J \end{aligned}$$

for any \(A \in \mathfrak {h}_2({\mathbb K})\) a \(2 \times 2\) hermitian matrix. We use this to show that F is indeed an isomorphism:

$$\begin{aligned} \begin{aligned} F( {\tilde{A}}_R B_R (\psi ) )&= J \, \overline{ {\tilde{A}}_R B_R (\psi ) } \\&= J \, \overline{{\tilde{A}}}_L \, {\overline{B}}_L ({\overline{\psi }} ) \\&= J \, \overline{{\tilde{A}}} ( {\overline{B}} \, {\overline{\psi }} ) \\&= A ( {\tilde{B}} J {\overline{\psi }} ) \\&= A_L {\tilde{B}}_L (F(\psi )). \end{aligned} \end{aligned}$$

\(\square \)

Next, for our spinor representation \(N_\pm \) constructed from a normed division algebra, we need to know certain invariants of the spin group action.

Lemma 11

Let \({\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}, {\mathbb {O}}\}\) be a normed division algebra (Example 27) of dimension k, and let \(N_{\pm }\) be the real \(\mathrm {Spin}(k+1,1)\) spinor representations from Proposition 32. Then,

  1. 1.

    \(\left( \mathrm {End}(N_{\pm })\right) ^{\mathrm {Spin}} \simeq \left\{ \begin{array}{ll} {\mathbb {K}} &{} \text{ if } \quad {\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\} \\ {\mathbb {R}} &{} \text{ if } \quad {\mathbb {K}} = {\mathbb {O}} \end{array} \right. \)

  2. 2.

    \(\left( \mathrm {Sym}^2(N_{\pm })\right) ^{\mathrm {Spin}} \simeq 0\)

where the superscript \(\mathrm{Spin}\) denotes the subspace left invariant by \(\mathrm{Spin}(k+1,1)\).

Proof

For part 1, the algebra of \(\mathrm{Spin}(k+1,1)\)-equivariant real linear endomorphisms of \(N_\pm \):

$$\begin{aligned} \mathrm{End}_{\mathrm{Spin}(k+1,1)}(N_\pm ) = \left( \mathrm{End}(N_\pm )\right) ^{\mathrm{Spin}} \end{aligned}$$

is called the commutant of \(N_\pm \). For an irreducible representation such as \(N_\pm \), Schur’s lemma tells us the commutant must be an associative division algebra. By the Frobenius theorem, the only associative real division algebras are \({\mathbb R},{\mathbb C}\) and \({\mathbb H}\). We must now determine which case occurs, but this is done by Varadarajan [52, Theorem 6.4.2].

For part 2, recall from Proposition 32 that we have an invariant pairing

$$\begin{aligned} \langle -, - \rangle :N_+ \otimes N_- \rightarrow {\mathbb R}. \end{aligned}$$

Thus, \(N_\pm \simeq N_\mp ^*\), and in particular, \(\mathrm{Sym}^2 N_\pm \simeq \mathrm{Sym}^2 N_\mp ^*\). But the latter is the space of symmetric pairings:

$$\begin{aligned} \mathrm{Sym}^2 N_\mp \rightarrow {\mathbb R}, \end{aligned}$$

which is a subspace of the space of all pairings on \(N_\mp \). The invariant elements of the space of all pairings are tabulated according to dimension and signature mod 8 by Varadarajan [52, Theorem 6.5.10]. In particular, for \({\mathbb K}= {\mathbb R}, {\mathbb C}\) where \(N_\mp = {\mathbb K}^2\) are the spinors in signature (2, 1) and (3, 1), respectively, the nonzero invariant pairings are antisymmetric, so \(\left( \mathrm{Sym}^2(N_\pm )\right) ^{\mathrm{Spin}} = 0\). For \({\mathbb K}= {\mathbb H}, {\mathbb O}\), where \(N_\mp = {\mathbb K}^2\) is the space of spinors in signature (5, 1) and (9, 1), respectively, \(N_\mp \) is not self-dual, so there are no nonzero invariant pairings, and again we conclude \(\left( \mathrm{Sym}^2(N_\pm )\right) ^{\mathrm{Spin}} = 0\). \(\square \)

Combining the previous two lemmas, we can prove a surprising relationship between the Cayley–Dickson double and maximal invariant central extension: They are in essence the same! More precisely, the maximal invariant central extension of \({\mathbb R}^{k+1,1|N_+ \oplus N_-}\), constructed from the normed division algebra \({\mathbb K}\), is given by \({\mathbb R}^{2k + 1, 1|N_{\mathrm{dbl}}}\), constructed from the Cayley–Dickson double \({\mathbb K}_{\mathrm{dbl}}\).

Proposition 12

Let \({\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}, {\mathbb {H}}\}\) be an associative normed division algebra (Example 27) of dimension k, and let \({\mathbb {K}}_{\mathrm {dbl}} \in \{{\mathbb C}, {\mathbb H}, {\mathbb O}\}\) be its Cayley–Dickson double (Definition 26) of dimension 2k. Then, the maximal invariant central extension of \({\mathbb {R}}^{k+1,1 \vert N_+ \oplus N_- }\), with \(N_{\pm }\) the irreducible real spinor representations constructed from \({\mathbb K}\) as in Proposition 32, is \({\mathbb {R}}^{2k +1,1 \vert N_{\mathrm {dbl}}}\):

for \(N_{\mathrm{dbl}}\) either of the irreducible real spinor representations induced by the Cayley–Dickson double \({\mathbb K}_\mathrm{dbl}\).

Proof

By Proposition 6, we need to compute the even \(\mathfrak {so}(k+1,1)\)-invariant cohomology of \({\mathbb {R}}^{k+1,1|N_+ \oplus N_-}\) in degree 2. Such even Lorentz-invariant 2-cocycles need to pair two spinors in \(N_+ \oplus N_-\): There are no even pairing between spinors in \(N_+ \oplus N_-\) and vectors in \({\mathbb R}^{k+1,1}\) and no antisymmetric Lorentz-invariant pairing between vectors in \({\mathbb R}^{k+1,1}\). Due to the simple nature of the Lie bracket on super Minkowski spacetime, this means that we need to compute the space of \(\mathfrak {so}(k+1,1)\)-invariant symmetric bilinear forms on \(N_+ \oplus N_-\), because every symmetric bilinear form on \(N_+ \oplus N_-\) is an even 2-cocycle.

We now apply Lemma 10 to produce these Lorentz-invariant 2-cocycles. Namely, let \(v \in {\mathbb {R}}^{2k+1,1 }\) be any vector in the orthogonal complement of \({\mathbb {R}}^{k+1,1}\). Then, the symmetric pairing

$$\begin{aligned} \begin{array}{ccc} N_{\mathrm {dbl}} \otimes N_{\mathrm {dbl}} &{} \rightarrow &{} {\mathbb {R}} \\ \psi \otimes \phi &{} \mapsto &{} \eta ( v,[\psi ,\phi ] ) \end{array} \end{aligned}$$

is clearly \(\mathrm {Spin}(k+1,1)\)-invariant, by the equivariance of the spinor pairing (Proposition 32) and the assumption on v. But by Lemma 10, \(N_{\mathrm {dbl}}\) is \(N_+ \oplus N_-\) as a \(\mathrm {Spin}(k+1,1)\)-representation. Therefore, this construction yields a k-dimensional space of Spin-invariant symmetric bilinear pairings on \(N_+ \oplus N_-\). Moreover, by the definition of the pairing above, it follows that the central extension classified by these pairings, regarded as 2-cocycles, is \({\mathbb {R}}^{2k+1,1\vert \mathrm {N}_{\mathrm {dbl}}}\).

To conclude the proof, it remains to show this invariant extension is maximal, hence that the dimension of the space of all invariant symmetric pairings on \(N_+ \oplus N_-\) is k. The space of all symmetric pairings, invariant or not, is:

$$\begin{aligned} \mathrm {Sym}^2(N_+ \oplus N_-) \;\simeq \; \mathrm {Sym}^2(N_+) \;\oplus \; N_+ \otimes N_- \;\oplus \; \mathrm {Sym}^2(N_-). \end{aligned}$$

So, we seek the invariant elements of the latter space. By Lemma 11, the invariant subspaces of \(\mathrm {Sym}^2(N_\pm )\) vanish. Therefore, the space of invariant 2-cocycles is the space of invariant elements in \(N_+ \otimes N_-\). By the spinor-to-scalar pairing from Proposition 32, the two spaces \(N_+\) and \(N_-\) are dual to each other as \(\mathrm {Spin(k+1,1)}\)-representations. Therefore, the invariant elements in \(N_+ \otimes N_-\) are equivalently the equivariant linear endomorphisms of \(N_+\):

$$\begin{aligned} N_+ \rightarrow N_+. \end{aligned}$$

By Lemma 11, this space of invariant endomorphisms is identified with \({\mathbb {K}}\)

$$\begin{aligned} \left( \mathrm {End}(N_+)\right) ^{\mathrm {Spin}} \simeq {\mathbb {K}}. \end{aligned}$$

Hence, the dimension of this space is k, which concludes the proof. \(\square \)

Proposition 13

The maximal invariant central extension (Definition 7) of the type IIA super Minkowski spacetime \({\mathbb {R}}^{9,1\vert \mathbf {16} \oplus \overline{{\mathbf {16}}}}\) is \({\mathbb {R}}^{10,1\vert {\mathbf {32}}}\):

Proof

By Proposition 6, we seek even \({\mathfrak {so}}(9,1)\)-invariant 2-cocycles. Since the extension in question is clearly \({\mathfrak {so}}(9,1)\)-invariant, it is sufficient to show that the space of all \({\mathfrak {so}}(9,1)\)-invariant 2-cocycles on \({\mathbb {R}}^{9,1\vert {\mathbf {16}} + \overline{{\mathbf {16}}}}\) is one dimensional. As in the proof of Proposition 12, that space is equivalently the space of \({\mathfrak {so}}(9,1)\)-invariant elements in

$$\begin{aligned} \mathrm {Sym}^2(N_+ \oplus N_-) \simeq \mathrm {Sym}^2(N_+) \;\oplus \; N_+ \otimes N_- \;\oplus \; \mathrm {Sym}^2(N_-). \end{aligned}$$

By Lemma 11, the invariants in \(\mathrm {Sym}^2(N_\pm )\) vanish and the space of invariants in \(N_+ \otimes N_-\) is one dimensional. \(\square \)

Putting together our results in this section, we prove our main theorem.

Theorem 14

The process that starts with the superpoint \({\mathbb {R}}^{0\vert 1}\) and then consecutively doubles the supersymmetries and forms the maximal invariant central extension according to Definition 7 discovers the super Minkowski super Lie algebras \({\mathbb {R}}^{d-1,1\vert N}\) from Definition 22 in dimensions \(d \in \{2,3,6,10,11\}\) for \(N = 1\) and \(N = 2\) supersymmetry: There is a diagram of super Lie algebras of the following form

where each single arrow denotes a maximal invariant central extension according to Definition 7 and where each double arrow denotes the two evident injections (Remark 23).

Proof

This is the joint statement of Propositions  9, 12, and 13. Here, we use in Proposition 12 that for \({\mathbb {K}} \in \{{\mathbb {R}}, {\mathbb {C}}\}\) a commutative division algebra, the two representations \(N_{\pm }\) from Proposition 32 are in fact isomorphic. \(\square \)

4 Outlook

In view of the brane bouquet [27], Theorem 14 is suggestive of phenomena still to be uncovered. Further corners of M-theory, currently less well understood, might be found by following the process of maximal invariant central extensions in other directions. Indeed, note that Theorem 14 only exhibits some maximal invariant central extensions. It does not claim that there are no further maximal central extensions.

For instance, the \(N=1\) superpoint \({\mathbb {R}}^{0 \vert 1}\) also has a maximal central extension, namely the superline \({\mathbb {R}}^{1\vert 1} = {\mathbb {R}}^{1,0\vert 1}\)

This follows immediately with the same argument as in Proposition 9.

The natural next question is, what is the bouquet of maximal central extensions emerging out of \({\mathbb {R}}^{0 \vert 3}\)? It is clear that the first step yields \({\mathbb {R}}^{6\vert 3}\), with the underlying even six-dimensional vector space canonically identified with the \(3 \times 3\) symmetric matrices with entries in the real numbers. Now if an analog of Proposition 12 continued to hold in this case, then the further consecutive maximal invariant extensions might involve the \(3 \times 3\) hermitian matrices with coefficients in the complex numbers \({\mathbb {C}}\), the quaternions \({\mathbb {H}}\), and finally the octonions \({\mathbb {O}}\). The last of these, denoted \(\mathfrak {h}_3({\mathbb O})\) for the \(3 \times 3\) hermitian matrices over \({\mathbb O}\), is the famous exceptional Jordan algebra. Just as \(\mathfrak {h}_2({\mathbb O})\), the \(2 \times 2\) hermitian matrices over \({\mathbb O}\), is isomorphic to Minkowski spacetime \({\mathbb {R}}^{9,1}\), so \(\mathfrak {h}_3({\mathbb O})\) is isomorphic to the 27-dimensional direct sum \({\mathbb {R}}^{9,1} \oplus {\mathbf {16}} \oplus {\mathbb {R}}\) consisting of 10d spacetime, one copy of the real 10d spinors and a scalar [2, Section 3.4]. This kind of data is naturally associated with heterotic M-theory, and grouping its spinors together with the vectors and the scalar to a 27-dimensional bosonic space is reminiscent of the speculations about bosonic M-theory [34]. Therefore, should the bouquet of maximal invariant extensions truly include \(\mathfrak {h}_3({\mathbb O})\), this might help to better understand the nature of the bosonic or heterotic corners of M-theory.

In a similar vein, we ought to ask how the tower of steps in Theorem 14 continues beyond dimension 11, and what the resulting structures mean.