How Space-Times Emerge from the Superpoint

TL;DR

This paper explains how super Minkowski space-times, essential to string and M-theory, can be derived from the superpoint using higher structures and rational homotopy theory, revealing a foundational emergence process.

Contribution

It introduces a novel framework for deriving super Minkowski space-times from the superpoint through central extensions within higher structures.

Findings

Super Minkowski space-times emerge from the superpoint via central extensions.

The approach uses rational homotopy theory and higher structures.

Connections are made to the brane bouquet and string/membrane theories.

Abstract

We describe how the super Minkowski space-times relevant to string theory and M-theory, complete with their Lorentz metrics and spin structures, emerge from a much more elementary object: the superpoint. In the sense of higher structures, this comes from treating the superpoint as an object in a flavor of rational homotopy theory, and repeatedly constructing central extensions. We will fit this story into the larger picture of the brane bouquet of Fiorenza-Sati-Schreiber: string theories and membrane theories emerge from super Minkowski space-times in precisely the same way as the super Minkowski space-times themselves emerge from the superpoint. This note is adapted from a talk I gave at the Durham symposium Higher Structures in M-Theory.

Equations

$$f(\theta )=f(0)+f^{\prime }(0)\theta .$$

$$M^{9,1}⇝N^{10,1}.$$

$${\text{mathbbm}R}^{9,1}⇝{\text{mathbbm}R}^{10,1}.$$

$${\text{mathbbm}R}^{9,1|16+\overline{16}}⇝{\text{mathbbm}R}^{10,1|32}.$$

$${\text{mathbbm}R}_{\mathrm{even}}^{d-1,1|\mathbf{N}}={\text{mathbbm}R}^{d-1,1},{\text{mathbbm}R}_{\mathrm{odd}}^{d-1,1|\mathbf{N}}=\mathbf{N}.$$

$$[-,-]:{\text{mathbbm}R}^{d-1,1|\mathbf{N}}\otimes {\text{mathbbm}R}^{d-1,1|\mathbf{N}}\to {\text{mathbbm}R}^{d-1,1|\mathbf{N}}.$$

$$[-,-]:\mathbf{N}\otimes \mathbf{N}\to {\text{mathbbm}R}^{d-1,1}.$$

$${\text{mathbbm}R}^{9,1|16+\overline{16}}⇝{\text{mathbbm}R}^{10,1|32}.$$

$$0⟶\text{mathbbm}R⟶\tilde{\mathfrak{g}}⟶\mathfrak{g}⟶0$$

$$\omega :{\Lambda }^2\mathfrak{g}\to \text{mathbbm}R$$

$$\omega ([X,Y],Z)\pm \omega ([Y,Z],X)\pm \omega ([Z,X],Y)=0$$

$${\mathfrak{g}}_{\omega }=\mathfrak{g}\oplus \text{mathbbm}Rc$$

$$[X,Y{]}_{\omega }=[X,Y]+\omega (X,Y)c\text{mbox}forX,Y\in \mathfrak{g}.$$

$$0⟶\text{mathbbm}R⟶{\mathfrak{g}}_{\omega }⟶\mathfrak{g}⟶0$$

$$\omega =\mathrm{d}{\theta }^{\alpha }\wedge {\Gamma }_{\alpha \beta }^{01\cdots 9}\mathrm{d}{\theta }^{\beta },$$

$${\text{mathbbm}R}^{10,1|32}⟶{\text{mathbbm}R}^{9,1|16+\overline{16}}.$$

$${\text{mathbbm}R}_{\mathrm{even}}^{0|1}=0,{\text{mathbbm}R}_{\mathrm{odd}}^{0|1}=\text{mathbbm}R.$$

$$\mathrm{d}\theta \wedge \mathrm{d}\theta .$$

$${\text{mathbbm}R}^{1|1}⟶{\text{mathbbm}R}^{0|1}.$$

$${\mathrm{Aut}}_0({\text{mathbbm}R}^{d-1,1|\mathbf{N}})\cong {\text{mathbbm}R}^{+}\times \mathrm{Spin}(d-1,1)\times \text{mbox}R-group$$

$$\text{mathbbm}{R}^{0|2}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{d}{\theta }_1\wedge \mathrm{d}{\theta }_1,\mathrm{d}{\theta }_1\wedge \mathrm{d}{\theta }_2,\mathrm{d}{\theta }_2\wedge \mathrm{d}{\theta }_2.$$

$${\text{mathbbm}R}^{3|2}⟶{\text{mathbbm}R}^{0|2}.$$

$${\mathrm{Aut}}_0({\text{mathbbm}R}^{3|2})={\text{mathbbm}R}^{+}\times \mathrm{Spin}(2,1).$$

$$2\otimes 2\to \text{mathbbm}R$$

$$\text{mathbbm}{R}^{2,1|2+2}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\text{mathbbm}R}^{3,1|4}⟶{\text{mathbbm}R}^{2,1|2+2}$$

$${\mathrm{Aut}}_0({\text{mathbbm}R}^{3,1|4})={\text{mathbbm}R}^{+}\times \mathrm{Spin}(3,1)\times \mathrm{U}(1).$$

$$\text{mathbbm}{R}^{10,1|32}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathfrak{m}5\mathfrak{brane}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

How Space-Times Emerge from the Superpoint

missing

John Huerta 111Corresponding author e-mail: jhuerta@math.tecnico.ulisboa.pt aa CAMGSD, Instituto Superior Técnico, Av. Ravisco Pais, 1049-001 Lisboa, Portugal

\shortabstract

1 Introduction

The superpoint, denoted ${\mathbbm{R}}^{0|1}$, is the space with a single odd coordinate $\theta$. Because it is odd, its square vanishes, $\theta^{2}=0$, and a power series expansion terminates at first order:

[TABLE]

For this reason, we regard $\theta$ as infinitesimal. Geometrically, the space ${\mathbbm{R}}^{0|1}$ consists of a single point with an infinitesimal neighborhood around it.

We can probe this straightforward space with the tools of homotopy theory. When we do this, we discover something remarkable: all the super Minkowski spacetimes of importance to string theory and M-theory, including their metrics and spin structures, can be constructed as extensions of the superpoint. From these space-times, using the brane bouquet of Fiorenza–Sati–Schreiber, we then find the strings and branes themselves.

This note is a gentle introduction to these ideas. It is based on a talk I gave at the Durham symposium Higher Structures and M-theory in August 2018, and that talk was about work with Schreiber [1]. Although our results concern M-theory, a part of physics, our techniques are pure mathematics. Let us nonetheless begin with the physical motivation.

2 M-theory

In the 1990s, the string theory community realized they had to study objects of dimension larger than 1, called branes. Witten christened this topic M-theory [2], where the M arguably stands for ‘membrane’ [3]. The idea of M-theory, not yet fully realized today, is that it should be single physical theory having the five superstring theories in 10d as limits, and its classical limit should be 11d supergravity. This idea is often pictured schematically as in Figure 1.

The idea that we can have an 11d theory and 10d theory on equal footing shows off how the concept of ‘dimension’ is flexible in M-theory. While M-theory is posited to have these various limits, in practice we do not even know its degrees of freedom. What we glimpse of M-theory comes in fact from taking certain limits of string theories. Most directly, we can take a certain limit of 10d type IIA string theory to obtain 11d physics, as pictured in Figure 2.

This suggests there must be some mathematical process that turns the 10d space-time of string theory into the 11d space-time of M-theory:

[TABLE]

Infinitesimally, at tangent spaces, this process turns 10d Minkowski space-time into 11d:

[TABLE]

But string theory and M-theory are both supersymmetric, so really this should be between super Minkowski space-times:

[TABLE]

In this last case, there is a natural mathematical choice for this process: it is called central extension. To understand it, we need to understand super Minkowski space-time.

3 Super Minkowski space-time

Super Minkowski space-time ${\mathbbm{R}}^{d-1,1|{\bf N}}$ is the supersymmetric version of Minkowski space-time, ${\mathbbm{R}}^{d-1,1}$, which itself is just ${\mathbbm{R}}^{d}$ with the indefinite metric $\eta(u,v)=-u^{0}v^{0}+u^{1}v^{1}+\cdots+u^{d-1}v^{d-1}$. Mathematically, ${\mathbbm{R}}^{d-1,1|{\bf N}}$ is a super Lie algebra. This is a Lie algebra in super vector spaces, so ${\mathbbm{R}}^{d-1,1|{\bf N}}$ has an underlying super vector space:

[TABLE]

The even part is ordinary Minkowski space-time ${\mathbbm{R}}^{d-1,1}$. The odd part, ${\bf N}$, is a new ingredient: it is a spinor representation.

On this super vector space, we have a bracket:

[TABLE]

This bracket satisfies the axioms of a Lie algebra, up to some signs. The bracket, and indeed the whole theory of super Minkowski space-time, is governed by the representation theory of the spin group, ${\rm Spin}(d-1,1)$. This Lie group is the double cover of the connected Lorentz group, ${\rm SO}_{0}(d-1,1)$.

The spin group ${\rm Spin}(d-1,1)$ has more representations than ${\rm SO}_{0}(d-1,1)$. In particular, it has ‘spinor representations’: there is a well-known inclusion ${\rm Spin}(d-1,1)\hookrightarrow{\rm C\ell}(d-1,1)$ of the spin group into the Clifford algebra. In fact, the Clifford algebra is ${\mathbbm{Z}}_{2}$-graded, and the spin group lands in the even subalgebra ${\rm Spin}(d-1,1)\hookrightarrow{\rm C\ell}_{0}(d-1,1)$. Modules of ${\rm C\ell}_{0}(d-1,1)$ thus become representations of ${\rm Spin}(d-1,1)$. These are spinor representations. Our ${\bf N}$ is one of these.

In detail:

1. i)

${\rm Spin}(d-1,1)$ acts on ${\mathbbm{R}}^{d-1,1}$ by linear transformations preserving the metric; 2. ii)

${\bf N}$ is a real spinor representation of ${\rm Spin}(d-1,1)$; 3. iii)

The bracket $[-,-]\colon{\mathbbm{R}}^{d-1,1|{\bf N}}\otimes{\mathbbm{R}}^{d-1,1|{\bf N}}\to{\mathbbm{R}}^{d-1,1|{\bf N}}$ is equivariant with respect to the action of ${\rm Spin}(d-1,1)$; 4. iv)

In fact, the only nonzero part of the bracket is the spinor-to-vector pairing:

[TABLE]

Physicists write the bracket as $[Q_{\alpha},Q_{\beta}]=-2\Gamma^{\mu}_{\alpha\beta}P_{\mu}$, where $\{Q_{\alpha}\}$ is a basis for ${\bf N}$, $\{P_{\mu}\}$ is a basis for ${\mathbbm{R}}^{d-1,1}$, and $\Gamma$ is the gamma matrix for our spinor representation. When ${\bf N}$ is irreducible, the bracket is the unique equivariant map up to rescaling. Otherwise it involves a choice.

Because super Minkowski space-times are super Lie algebras, we can consider their central extensions. This will provide our sought after mathematical process going from 10d to 11d:

[TABLE]

Here’s how it works: for any super Lie algebra $\mathfrak{g}$, a central extension is a short exact sequence of super Lie algebras:

[TABLE]

such that ${\mathbbm{R}}$ lands in the center of $\tilde{\mathfrak{g}}$. Mathematically, central extensions are classified by super Lie algebra cohomology, and this allows us to describe them very concretely. Specifically, 2-cocycle on $\mathfrak{g}$ is skew-symmetric map:

[TABLE]

satisfying the cocycle condition:

[TABLE]

where the signs depend on whether $X,Y,Z$ are even or odd elements of $\mathfrak{g}$. Given one of these 2-cocycles, we can define a central extension $\mathfrak{g}_{\omega}$ to be the super Lie algebra obtained from $\mathfrak{g}$ by including one extra generator $c$, even and central:

[TABLE]

and modifying the bracket with the 2-cocycle:

[TABLE]

Since $c$ is central, this defines the bracket on all of $\mathfrak{g}_{\omega}$. We thus get a central extension:

[TABLE]

where ${\mathbbm{R}}$ is included as ${\mathbbm{R}}c$, and $\mathfrak{g}_{\omega}\longrightarrow\mathfrak{g}$ is the map setting $c$ to zero. Every central extension is isomorphic to one of this form: thus, 2-cocycles give us a central extensions, and vice versa. In what follows, we will often denote a central extension by the homomorphism $\mathfrak{g}_{\omega}\longrightarrow\mathfrak{g}$ which sets $c$ to zero.

The cocycle condition may look mysterious, but it is exactly what we need to guarantee that the Lie bracket on $\mathfrak{g}_{\omega}$ satisfies the Jacobi identity. It also has a beautiful geometric interpretation: if $G$ is a super Lie group with super Lie algebra $\mathfrak{g}$, then $\omega$ defines a 2-form on $G$ by left-translation. The cocycle condition then holds if and only if this form is closed: ${\rm d}\omega=0$.

Thus, all we need to extend from 10d to 11d is a 2-cocycle. Here is one, written as a 2-form:

[TABLE]

where $\Gamma^{01\cdots 9}=\Gamma^{0}\Gamma^{1}\cdots\Gamma^{9}$ is the product of all the gamma matrices, and $\{\theta^{\alpha}\}$ are odd coordinates on ${\mathbbm{R}}^{9,1|{\bf 16}+\overline{\bf 16}}$. This 2-form is indeed left-invariant on the super Lie group corresponding to ${\mathbbm{R}}^{9,1|{\bf 16}+\overline{\bf 16}}$, and $d\omega=0$ by the naive calculation. Hence, $\omega$ is in fact a 2-cocycle. Finally, centrally extending ${\mathbbm{R}}^{9,1|{\bf 16}+\overline{\bf 16}}$ by this 2-cocycle does indeed yield ${\mathbbm{R}}^{10,1|{\bf 32}}$, by some Clifford algebraic yoga. We thus have the central extension:

[TABLE]

This example raises a few questions. First, why should we use the 2-cocycle $\omega$? There could be others on${\mathbbm{R}}^{9,1|{\bf 16}+\overline{\bf 16}}$. What singles out $\omega$? The answer is an invariance condition: $\omega$ is invariant under the action of ${\rm Spin}(9,1)$. Next, can we account for more dimensions in space-time by central extension? Taking this to extremes, can we realize all the space-times we care about by centrally extending the superpoint, ${\mathbbm{R}}^{0|1}$? Indeed we can; this is our main result.

To begin, let us define the superpoint more precisely: the superpoint ${\mathbbm{R}}^{0|1}$ is the super vector space with vanishing even degree and ${\mathbbm{R}}$ in odd degree:

[TABLE]

It is crucial to note that ${\mathbbm{R}}^{0|1}$ has no Lie bracket, no metric, and no spin structure. We will discover all of these by central extension.

Despite this lack of structure, the superpoint has a 2-cocycle:

[TABLE]

This is nonzero precisely because $\theta$ is odd. Centrally extending by this cocycle, we get ${\mathbbm{R}}^{1|1}$, the worldline of the super particle:

[TABLE]

That is already something, but we can go a lot further.

4 The dimensional ladder

Let us play a game with two moves, starting with the superpoint:

1. i)

extend by all nontrivial 2-cocycles subject to a suitable invariance condition; 2. ii)

if no 2-cocycles are available, double the number of odd dimensions.

We need to spell out that invariance condition. We want to say that our 2-cocycles need to be Lorentz-invariant, or more precisely, invariant under the spin group. But we cannot, because there is no metric as yet. Fortunately, the symmetries of the metric turn out to be encoded in the Lie bracket:

Proposition 1** (H.–Schreiber, folklore).**

For a super Minkowski space-time ${\mathbbm{R}}^{d-1,1|{\bf N}}$, its connected automorphism group is, up to cover:

[TABLE]

where the R-group acts trivially on ${\mathbbm{R}}^{d-1,1}$.

The R-group is known, in the physics literature, as the R-symmetries of ${\mathbbm{R}}^{d-1,1|{\bf N}}$. We can generalize this to any super Lie algebra: an R-symmetry of $\mathfrak{g}$ is an automorphism that acts trivially on $\mathfrak{g}_{\rm even}$. This theorem was probably folklore among physicists, but not finding a proof in the literature, we went ahead and proved it. Its significance is that it allows us to get our hands on the ${\rm Spin}(d-1,1)$ symmetries, in the terms of the super Lie algebra structure alone, without mentioning the metric. Specifically: a 2-cocycle on a super Lie algebra $\mathfrak{g}$ is called invariant if it is invariant under the quotient of $\mathrm{Aut}_{0}(\mathfrak{g})$ by rescalings and R-symmetries. With this definition, when $\mathfrak{g}$ is a super Minkowski space-time, a 2-cocycle is invariant precisely when it is invariant under ${\rm Spin}(d-1,1)$.

Let us begin. First, we will double the number of odd dimensions of ${\mathbbm{R}}^{0|1}$, yielding ${\mathbbm{R}}^{0|2}$. We will write this operation as follows:

[TABLE]

Now, ${\mathbbm{R}}^{0|2}$ has two odd generators, $\theta_{1}$ and $\theta_{2}$, and there are three 2-cocycles:

[TABLE]

Because ${\mathbbm{R}}^{0|2}$ has no even part, any automorphism must be an R-symmetry. Hence, all of these 2-cocycles are invariant under the maximal subgroup containing no nontrivial R-symmetries. Extending by all three we get:

[TABLE]

At this point, something remarkable happens: a metric appears,

[TABLE]

We did not put it in, but by looking at the automorphisms of the algebra, the three even generators in ${\mathbbm{R}}^{3|2}$ transform under ${\rm Spin}(2,1)$ as vectors, and the two odd generators as spinors.

Thanks to this metric, we can look for ${\rm Spin}(2,1)$-invariant 2-cocycles on ${\mathbbm{R}}^{2,1|2}$. There are none, because the only ${\rm Spin}(2,1)$-invariant map:

[TABLE]

is antisymmetric.

Since we are out of 2-cocycles, let us double the number of odd dimensions again:

[TABLE]

There is precisely one ${\rm Spin}(2,1)$-invariant 2-cocycle, and extending by this gives:

[TABLE]

Again, the metric is not a choice:

[TABLE]

Here, ${\rm U}(1)$ is the R-group. There are no further ${\rm Spin}(3,1)$-invariant 2-cocycles.

We can keep going in exactly this way, up to dimension 11. Two notable phenomena occur. First, we sometimes encounter several 2-cocycles after doubling the number of spinors: in dimensions 4, there are two 2-cocycles, so we jump directly to dimension 6. In dimension 6, after doubling, there are four 2-cocycles, so we jump directly to dimension 10. Moreover, in dimensions 6 and 10, there are two distinct spinor representations, so there are two ways to double, a type IIA, where we include both kinds of spinor, and a type IIB, where we just include one kind. In summary, we have the following collection of doublings and central extensions that we display in Theorem 2.

Theorem 2** (H.–Schreiber).**

[TABLE]

5 The brane bouquet

In the last section, we saw what we could do with invariant 2-cocycles. Modulo a suitable equivalence relation, 2-cocycles form a group, $H^{2}(\mathfrak{g})$, the second cohomology group of the super Lie algebra $\mathfrak{g}$. There are cohomology groups in higher degree, $H^{p}(\mathfrak{g})$. Can we fit these into our story? What is the significance of $p$-cocycles for $p\geq 3$?

There are two remarkable answers to this question, one coming from physics, and the other from mathematics:

• Physics: Invariant $(p+2)$-cocycles on ${\mathbbm{R}}^{d-1,1|{\bf N}}$ correspond to Green–Schwarz $p$-branes [5].
• Mathematics: Central extensions by $(p+2)$-cocycles on the super Lie algebra $\mathfrak{g}$ yield ‘super $L_{\infty}$-algebras’ [6].

A super $L_{\infty}$-algebra $\mathfrak{g}$ is like a Lie algebra, defined on a chain complex of super vector spaces:

[TABLE]

But the Jacobi identity does not hold:

[TABLE]

Instead, it holds up to coherent homotopy: we get infinitely many identities like this:

[TABLE]

This says the Jacobi identity holds up to a chain homotopy, given by a trilinear bracket:

[TABLE]

satisfying its own Jacobi-like identity up to a 4-linear bracket…and so on, forever.

The key insight of the brane bouquet due to Fiorenza–Sati–Schreiber [7] is that we can combine these two strands, one from physics and one from mathematics: we can centrally extend by the higher degree cocycles classifying the Green–Schwarz $p$-branes to obtain super $L_{\infty}$-algebras. Then we can look for additional invariant cocycles on those $L_{\infty}$-algebras. Lo and behold, these new cocycles turn out to correspond to additional branes, also very important in physics: D-branes and the M5-branes. We can then centrally extend by these cocycles, and continue our hunt for invariant cocycles, which should correspond to new branes.

Thus, by including higher degree cocycles, we get the brane bouquet, growing out of the superpoint as shown in Figure 3. There, we have named the super $L_{\infty}$-algebras after the physical objects to which their cocycles correspond.

In this note, I have recounted what we know so far. But I have not claimed to be exhaustive: there may be more cocycles, and thus more extensions, waiting to be found. A full computation of the brane bouquet has not been done. There may be many more surprises waiting for us inside the humble superpoint.