super Minkowski spacetime


Riemannian geometry





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.


Ordinary (d+1)(d+1)-dimensional Minkowski space may be understood as the quotient Iso( d1,1)/(Spin(d1,1))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 NN the dim(N)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

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.


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 ℑ𝔰𝔬( d1,1|N)\mathfrak{Iso}(\mathbb{R}^{d-1,1|N}) by its CE-algebra CE(ℑ𝔰𝔬( d1,1|N))CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})) “of left-invariant 1-forms” on its group manifold.

Let dd \in \mathbb{N} and let NN be a real spin representation of Spin(d1,1)Spin(d-1,1). See at Majorana representation for details.


The Chevalley-Eilenberg algebra CE(ℑ𝔰𝔬( d1,1|N))CE(\mathfrak{Iso}(\mathbb{R}^{d-1,1|N})) is generated on

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

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

where a{0,1,,d1}a \in \{0,1, \cdots, d-1\} is a spacetime index, and where α\alpha is an index ranging over a basis of the chosen Majorana spinor representation NN.

The CE-differential defined as follows

d CEω ab=ω a bω bc d_{CE} \, \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}


d CEψ=14ω abΓ abψ. 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)


d CEe a=ω a be b+ψ¯Γ aψ 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 NN (def.).

This defines the super Poincaré super Lie algebra. After discarding the terms involving ω\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.


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

Let NN be a real spin representation and let (x a,θ α)(x^a, \theta^\alpha) be the canonical coordinates on the supermanifold d1,1|N\mathbb{R}^{d-1,1\vert N} underlying the super-Minkowski super translation group. Then the canonical super vielbein is the d1,1|N\mathbb{R}^{d-1,1\vert N}-valued super differential form with components

  • ψ αdθ α\psi^\alpha \coloneqq \mathbf{d} \theta^\alpha.

  • e adx a+θ¯Γ adθe^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. 1 as

de a =d(dx a+i2θ¯Γ adθ) =i2dθ¯Γ adθ =i2ψ¯Γ aψ. \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} \,.

The term i2ψ¯Γ aψ\frac{i}{2}\bar \psi \Gamma^a \psi is sometimes called the supertorsion of the super-vielbein ee, because the defining equation

d CEe aω a be b=i2ψ¯Γ aψ d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi

may be read as saying that ee 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(𝔰𝔦𝔰𝔬)CE(\mathfrak{siso}) which have “all indices contracted”. See also at torsion constraints in supergravity.

Notably we have

d(ψ¯Γ a 1a pψe a 1e a p)(ψ¯Γ a 1a pψe a 1e a p1)(Ψ¯Γ a pΨ). 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 “eΨ 2e \mapsto \Psi^2” of the differential acting on Loretz scalars is sometimes denoted “t 0t_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)(D,p) a Fierz identity implies that the term

(ψ¯Γ a 1a pψe a 1e a p1)(Ψ¯Γ a pΨ) \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

ψ¯Γ a 1a pψe a 1e a p \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.

Cohomology and super pp-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-pp-branes propagating on super-Minkowski space (FSS 13).

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

As a central extension of the superpoint

Regarded as a super Lie algebra, super Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1|N} has the single nontrivial super-Lie bracket given by the spinor bilinear pairing

()¯Γ():SSV \overline{(-)}\Gamma (-) \colon S \otimes S \longrightarrow V

discussed in detail at spin representation.

Notice that this means that if one regards the superpoint 0|dim(N)\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 d\mathbb{R}^{d}.

d d1,1|N 0|dim(N) \array{ \mathbb{R}^{d} &\longrightarrow& \mathbb{R}^{d-1,1|N} \\ && \downarrow \\ && \mathbb{R}^{0|dim(N)} }
geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


The d=4d = 4, N=2N =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:

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

Revised on February 11, 2017 11:43:12 by Urs Schreiber (