nLab super Minkowski spacetime

Idea

Super-Minkowski spacetime is a super spacetime which is an analog in supergeometry of ordinary Minkowski spacetime. It is a super Cartesian space whose odd coordinates form a real spin representation.

Definition

Ordinary $(d+1)$-dimensional Minkowski space may be understood as the quotient $Iso(\mathbb{R}^{d-1,1})/(Spin(d-1,1))$ of the Poincare group by the spin group cover of Lorentz group – the translation group.

Analogously, the for each real irreducible spin representation $N$ the $dim(N)$-extended supermanifold Minkowski superspace or super Minkowski space is the quotient of supergroups of the super Poincaré group by the corresponding spin group (a super Klein geometry).

The super-translation group. See there for more details.

Alternatively, regarded as a super Lie algebra this is the quotient of the super Poincaré Lie algebra by the relevant Lorentz Lie algebra.

Properties

Canonical coordinates

We briefly review some basics of the canonical coordinates and the super Lie algebra cohomology of the super Poincaré Lie algebra and super Minkowski space (see also at super Cartesian space and at signs in supergeometry).

By the general discussion at Chevalley-Eilenberg algebra, we may characterize the super Poincaré Lie algebra $\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ by its CE-algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ “of left-invariant 1-forms” on its group manifold.

Let $d \in \mathbb{N}$ and let $N$ be a real spin representation of $Spin(d-1,1)$. See at Majorana representation for details.

Definition

The Chevalley-Eilenberg algebra $CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}))$ is generated on

• elements $\{e^a\}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$

• and elements $\{\psi^\alpha\}$ of degree $(1,odd)$

where $a \in \{0,1, \cdots, d-1\}$ is a spacetime index, and where $\alpha$ is an index ranging over a basis of the chosen Majorana spinor representation $N$.

The CE-differential defined as follows

$$d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}$$

and

$$d_{CE} \, \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi \,.$$

(which is the differential for the semidirect product of the Poincaré Lie algebra acting on the given Majorana spinor representation)

and

$$d_{CE} \, e^{a } = \omega^a{}_b \wedge e^b + \overline{\psi} \wedge \Gamma^a \psi$$

where on the right we have the spinor-to-vector pairing in $N$ (def.).

This defines the super Poincaré super Lie algebra. After discarding the terms involving $\omega$ this becomes the CE algebra of the super translation algebra underlying super Minkowski spacetime.

In this way the super-Poincaré Lie algebra and its extensions is usefully discussed for instance in (D’Auria-Fré 82) and in (Azcárraga-Townsend 89, CAIB 99). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the coordinates on super Minkowski space.

Remark

The abstract generators in def. are identified with left invariant 1-forms on the super-translation group (= super Minkowski spacetime) as follows.

Let $N$ be a real spin representation and let $(x^a, \theta^\alpha)$ be the canonical coordinates on the supermanifold $\mathbb{R}^{d-1,1\vert N}$ underlying the super-Minkowski super translation group. Then the canonical super vielbein is the $\mathbb{R}^{d-1,1\vert N}$-valued super differential form with components

• $\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha$.

• $e^a \coloneqq \mathbf{d} x^a + \overline{\theta} \Gamma^a \mathbf{d} \theta$.

Notice that this then gives the above formula for the differential of the super-vielbein in def. as

$$\begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,.$$

Remark

The term $\frac{i}{2}\bar \psi \Gamma^a \psi$ is sometimes called the supertorsion of the super-vielbein $e$, because the defining equation

$$d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi$$

may be read as saying that $e$ is torsion-free except for that term. Notice that this term is the only one that appears when the differential is applied to “Lorentz scalars”, hence to object in $CE(\mathfrak{siso})$ which have “all indices contracted”. See also at torsion constraints in supergravity.

Notably we have

$$d \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} \right) \propto \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right) \,.$$

This remaining operation “$e \mapsto \Psi^2$” of the differential acting on Loretz scalars is sometimes denoted “$t_0$”, e.g. in (Bossard-Howe-Stelle 09, equation (8)).

This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of $(D,p)$ a Fierz identity implies that the term

$$\left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right)$$

vanishes identically, and hence in these dimensions the term

$$\overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p}$$

is a cocycle. See also the brane scan table below.

Super-Minkowski Lie group

We spell out the super-translation super Lie group-structure on the supermanifold $\mathbb{R}^{1,d\vert\mathbf{N}}$ underlying super Minkowski spacetime, hence equivalently of the quotient super Lie group of the super Poincaré group (the “supersymmetry” group) by its Lorentzian spin-subgroup:

(1)$$\mathbb{R}^{1,d\vert\mathbf{N}} \;\simeq\; Iso\big( \mathbb{R}^{1,d\vert\mathbf{N}} \big) \big/ Spin(1,d) \,.$$

Here

• $d \in \mathbb{N}$ is the spatial dimension (a natural number),

• $\mathbf{N} \in Rep_{\mathbb{R}}\big(Spin(1,d)\big)$ is a real spin representation equipped with a linear map

which is symmetric and $Spin(1,d)$-equivariant.

First, the super-Minkowski super Lie algebra structure on the super vector space

$$\mathbb{R}^{1,d\vert\mathbf{N}} \;\coloneqq\; \mathbb{R}^{1+d} \times \mathbb{R}^N_{odd}$$

is defined, dually, by the Chevalley-Eilenberg dgc-superalgebra with generators of $\mathbb{Z} \times \mathbb{Z}/2$ bidegree

generator	bidegree
$e^a$	$(1,evn)$
$\psi^\alpha$	$(1,odd)$

for $a \in \{0,1, \cdots, d\}$ indexing a linear basis of $\mathbb{R}^D$ and $\alpha \in \{1,\cdots, N\}$ indexing a linear basis of $\mathbf{N}$ by the differential equations

(2)$$\begin{array}{ccl} \mathrm{d}\, e^a &\coloneqq& \big( \overline{\psi} \,\Gamma^a\, \psi \big) \\ \mathrm{d}\, \psi &=& 0 \end{array}$$

The first differential is the linear dual of the archetypical super Lie bracket in the supersymmetry super Lie algebra which takes two odd elements to a spatial translation. The second differential is the linear dual of the fact that in the absence of rotational generators, no Lie bracket in the supersymmetry alegbra results in a non-vanishing odd element.

Next we regard $\mathbb{R}^{1,10\vert\mathbf{N}}$ not just as a super vector space but as a Cartesian supermanifold. As such it has canonical coordinate functions

generator	bidegree
$x^a$	$(0,evn)$
$\theta^\alpha$	$(0,odd)$

On this supermanifold, consider the super coframe field

$$(e,\psi) \;\colon\; T\mathbb{R}^{1,d\vert\mathbf{N}} \xrightarrow{\;} \mathbb{R}^{1,10\vert\mathbf{N}}$$

(where on the left we have the tangent bundle and on the right its typical fiber super vector space) given by

(3)$$\begin{array}{ccl} e^a &\coloneqq& \mathrm{d}x^a + \big(\overline{\theta} \,\Gamma^a \mathrm{d}\theta\big) \\ \psi &\coloneqq& \mathrm{d}\theta \end{array}$$

It is clear that this is a coframe field in that for all $x \in \mathbb{R}^{1,d\vert\mathbf{N}}$ it restricts to an isomorphism

$$T_{x}\mathbb{R}^{1,d\vert\mathbf{N}} \xrightarrow{\;\sim\;} \mathbb{R}^{1,d\vert\mathbf{N}}$$

and the peculiar second summand in the first line is chosen such that its de Rham differential has the same form as the differential in the Chevalley-Eilenberg algebra (2).

(Incidentally, a frame field linear dual to the coframe field (3) is

$$\begin{array}{ccl} D_a &\coloneqq& \partial_{x^a} \\ D_\alpha &\coloneqq& \partial_{\theta^\alpha} + \overline{\theta}\Gamma^a \partial_{v^a} \end{array} \;\;\;\;\;\;\; \text{such that} \;\;\;\; \begin{array}{ll} e^a(D_b) \,=\, \delta^a_b \,, & e^a(D_\alpha) \,=\, 0 \\ \psi^\alpha(D_a) \,=\, 0 \,, & \psi^\alpha(D_\beta) \,=\, \delta^\alpha_\beta \end{array}$$

which are the operators often stated right away in introductory texts on supersymmetry.)

This fact, that the Maurer-Cartan equations of a coframe field (3) coincide with the defining equations (2) of the Chevalley-Eilenberg algebra of a Lie algebra of course characterizes the left invariant 1-forms on a Lie group, and hence what remains to be done now is to construct a super Lie group-structure on the supermanifold $\mathbb{R}^{1,d\vert\mathbf{N}}$ with respect to which the coframe (3) is left invariant 1-form.

Recalling (from here) that a morphism of supermanifolds is dually given by a reverse algebra homomorphism between their function algebras, which in the present case are freely generated by the above coordinate functions, we denote the canonical coordinates on the Cartesian product $\mathbb{R}^{1,d\vert\mathbf{N}} \times \mathbb{R}^{1,d\vert\mathbf{N}}$ by $(x^a_{'}, \theta^\alpha_{'})$ for the first factor and $(x^a, \theta^\alpha)$ for the second, and declare a group product operation as follows:

(4)

(cf. CAIP99, (2.1) & (2.6))

Here the choice of notation for the coordinates on the left is adapted to thinking of this group operation equivalently as the left multiplication action of the group on itself, which makes the following computation nicely transparent.

Indeed, the induced left action of the super-group on its odd tangent bundle

is dually given by

and left-invariance of the coframe (2) means that it is fixed by this operation (so the differential $\mathrm{d}$ in the following computation is just that of the second factor, hence acting on unprimed coordinates only):

$$\begin{array}{ccl} \mathrm{act}^\ast e^a &=& \mathrm{act}^\ast \Big( \mathrm{d}x^a + \big(\overline{\theta} \,\Gamma^a\, \mathrm{d}\theta\big) \Big) \\ &=& \mathrm{d}\,\mathrm{act}^\ast x^a + \big(\overline{\mathrm{act}^\ast\theta} \,\Gamma^a\, \mathrm{d}\,\mathrm{act}^\ast\theta\big) \\ &=& \mathrm{d} \Big( x^a_{'} + x^a - \big( \overline{\theta}_{'} \,\Gamma^a\, \theta \big) \Big) + \big( (\overline{\theta}_{'} + \overline{\theta}) \,\Gamma^a\, \mathrm{d}( \theta_{'} + \theta ) \big) \\ &=& \mathrm{d}x^a - \big( \overline{\theta}_{'} \,\Gamma^a\, \mathrm{d}\theta \big) \,+\, \big( \overline{\theta}_{'} \,\Gamma^a\, \mathrm{d}\theta \big) \,+\, \big( \overline{\theta} \,\Gamma^a\, \mathrm{d}\theta \big) \\ &=& \mathrm{d}x^a \,+\, \big( \overline{\theta} \,\Gamma^a\, \mathrm{d}\theta \big) \\ &=& e^a \mathrlap{\,,} \end{array} \;\;\;\;\; \begin{array}{ccl} \mathrm{prd}^\ast \psi &=& \mathrm{prd}^\ast \mathrm{d}\theta \\ &=& \mathrm{d} \, \mathrm{prd}^\ast \theta \\ &=& \mathrm{d}\big( \theta_{'} + \theta \big) \\ &=& \mathrm{d}\theta \\ &=& \psi \mathrlap{\,.} \\ {} \end{array}$$

This shows that if (4) is the group product of a group object in SuperManifolds then the corresponding super Lie algebra is the super-Minkowski super translation Lie algebra and hence that this group object is the desired super-Minkowski super Lie group.

So, defining the remaining group object-operations as follows:

neutral element:

inverse elements:

we conclude by checking the group object-axioms:

For associativity we need to check that the following diagram commutes:

and indeed it does — the term $\big(\overline{\theta} \Gamma^a \theta\big)$ vanishes because the $\theta^\alpha$ anti-commute among themselves, while the pairing (1) is symmetric:

For unitality we need to check that the following diagram commutes:

and indeed it does:

And finally, for invertibility we need to check that the following diagram commutes:

and indeed it does:

$\Box$

Cohomology and super $p$-branes

As opposed to ordinary Minkowski space, the de Rham cohomology of left invariant forms of super-Minkowski space contains nontrivial exceptional cocycles (the brane scan). These serve as the WZW terms for the Green-Schwarz action functional (see there for more) of super-$p$-branes propagating on super-Minkowski space (FSS 13).

The corresponding $L_\infty$-extensions are extended superspacetime.

As a central extension of the superpoint

Regarded as a super Lie algebra, super Minkowski spacetime $\mathbb{R}^{d-1,1|N}$ has the single nontrivial super-Lie bracket given by the spinor bilinear pairing

$$\overline{(-)}\Gamma (-) \colon S \otimes S \longrightarrow V$$

discussed in detail at spin representation.

Notice that this means that if one regards the superpoint $\mathbb{R}^{0|dim(N)}$ as an abelian super Lie algebra?, then super Minkowski spacetime is the Lie algebra extension of that by this bilinear pairing regarded as a super-Lie algebra cocycle with coefficients in $\mathbb{R}^{d}$.

$$\array{ \mathbb{R}^{d} &\longrightarrow& \mathbb{R}^{d-1,1|N} \\ && \downarrow \\ && \mathbb{R}^{0|dim(N)} }$$

Related concepts

• super anti de Sitter spacetime

geometric context	gauge group	stabilizer subgroup	local model space	local geometry	global geometry	differential cohomology	first order formulation of gravity
differential geometry	Lie group/algebraic group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	Klein geometry	Cartan geometry	Cartan connection
examples	Euclidean group $Iso(d)$	rotation group $O(d)$	Cartesian space $\mathbb{R}^d$	Euclidean geometry	Riemannian geometry	affine connection	Euclidean gravity
	Poincaré group $Iso(d-1,1)$	Lorentz group $O(d-1,1)$	Minkowski spacetime $\mathbb{R}^{d-1,1}$	Lorentzian geometry	pseudo-Riemannian geometry	spin connection	Einstein gravity
	anti de Sitter group $O(d-1,2)$	$O(d-1,1)$	anti de Sitter spacetime $AdS^d$				AdS gravity
	de Sitter group $O(d,1)$	$O(d-1,1)$	de Sitter spacetime $dS^d$				deSitter gravity
	linear algebraic group	parabolic subgroup/Borel subgroup	flag variety	parabolic geometry
	conformal group $O(d,t+1)$	conformal parabolic subgroup	Möbius space $S^{d,t}$		conformal geometry	conformal connection	conformal gravity
supergeometry	super Lie group $G$	subgroup (monomorphism) $H \hookrightarrow G$	quotient (“coset space”) $G/H$	super Klein geometry	super Cartan geometry	Cartan superconnection
examples	super Poincaré group	spin group	super Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$	Lorentzian supergeometry	supergeometry	superconnection	supergravity
	super anti de Sitter group		super anti de Sitter spacetime
higher differential geometry	smooth 2-group $G$	2-monomorphism $H \to G$	homotopy quotient $G//H$	Klein 2-geometry	Cartan 2-geometry
	cohesive ∞-group	∞-monomorphism (i.e. any homomorphism) $H \to G$	homotopy quotient $G//H$ of ∞-action	higher Klein geometry	higher Cartan geometry	higher Cartan connection
examples			extended super Minkowski spacetime		extended supergeometry		higher supergravity: type II, heterotic, 11d

References

The $d = 4$, $N =2$ super Minkowski spacetime was originally introduced in

• Abdus Salam J.A. Strathdee, Supergauge Transformations, Nucl.Phys. B76 (1974) 477-482 (spire)

• Abdus Salam J.A. Strathdee, Physical Review D11, 1521-1535 (1975)

see at “superspace in physics”.

Further discussion includes:

• Super spacetimes and super Poincaré-group (pdf)

• Daniel Freed, lecture 6 of Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)

• Daniel Freed, Lecture 4 of Five lectures on supersymmetry

• Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction

• Leonardo Castellani, Riccardo D'Auria, Pietro Fre, page 370, part II section II.3.3 ofSupergravity and Superstrings - A Geometric Perspective_

Discussion of how super L-infinity algebra extensions of super Minkowski spacetime yield all the brane scan of string theory/M-theory is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The brane bouquet (2013)