nLab
super Poincare Lie algebra

Idea

The super Poincaré Lie algebra is a super Lie algebra extension of a Poincaré Lie algebra.

The corresponding super Lie group is the super Euclidean group (except for the signature of the metric).

Definition

By the general discussion at Chevalley-Eilenberg algebra, we may characterize the super Poincaré Lie algebra $\mathrm{𝔰𝔦𝔰𝔬}(D-1,1)$ by its CE-algebra $\mathrm{CE}(\mathrm{𝔰𝔦𝔰𝔬}(D-1,1))$ “of left-invariant 1-forms” on its group manifold.

Definition

The Chevalley-Eilenberg algebra $\mathrm{CE}(\mathrm{𝔰𝔦𝔰𝔬}(d-1,1))$ is generated on

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

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

with the differential defined by

$$d_{\mathrm{CE}}{\omega }^{ab}={\omega }^a{}_b\wedge {\omega }^{bc}$$d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c}

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

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

Removing the terms involving $\omega$ here this is the super translation algebra.

Remark

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

Let $(x^a,{\theta }^{\alpha })$ be the canonical coordinates on the supermanifold ${ℝ}^{d\mid N}$ underlying the super translation group. Then the identification is

• ${\psi }^{\alpha }=d{\theta }^{\alpha }$.

• $e^a=d{x}^a+\frac{i}{2}\overline{\theta }{\Gamma }^{a}d\theta$.

This then gives the formula for the differential of the super-vielbein in def. 1 as

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

Properties

Lie algebra cohomology

The super Poincaré Lie algebra has, on top of the Lie algebra cocycles that it inherits from $\mathrm{𝔰𝔬}(n)$, a discrete number of exceptiona cocycles bilinear in the spinors, on the super translation algebra, that exist only in very special dimensions.

The following theorem has been stated at various placed in the physics literature (known there as the brane scan for $\kappa$-symmetry in Green-Schwarz action functionals for super-$p$-branes on super-Minkowski spacetime). A full proof is in Brandt 12-13. The following uses the notation in terms of division algebras (Baez-Huerta 10).

Theorem

• In dimensional $d=3,4,6,10$, $\mathrm{𝔰𝔦𝔰𝔬}(d-1,1)$ has a nontrivial 3-cocycle given by

  $$(\psi ,\varphi ,A)\mapsto g(\psi \cdot \varphi ,A)$$(\psi, \phi, A) \mapsto g(\psi \cdot \phi, A)

  for spinors $\psi ,\varphi \in 𝒮$ and vectors $A\in 𝒯$, and 0 otherwise.

• In dimensional $d=4,5,7,11$, $\mathrm{𝔰𝔦𝔰𝔬}(d-1,1)$ has a nontrivial 4-cocycle given by

  $$(\Psi ,\Phi ,𝒜,ℬ)\mapsto ⟨\Psi ,(𝒜ℬ-ℬ𝒜)\Phi ⟩$$(\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle

  for spinors $\Psi ,\Phi \in 𝒮$ and vectors $𝒜,ℬ\in 𝒱$, with the commutator taken in the Clifford algebra.

The 4-cocycle in $d=11$ is the one that induces the supergravity Lie 3-algebra.

Extensions

Super $L_{\mathrm{\infty }}$-algebra extensions

The super L-infinity algebra infinity-Lie algebra cohomology of the super Poincaré Lie algebra corresponding to the above cocycles involves

supergravity Lie 6-algebra $\to$ supergravity Lie 3-algebra $\to$ super-Poincaré Lie algebra

Polyvector extensions

The super-Poincaré Lie algebra has a class of super Lie algebra extensions called polyvector extensions , because they involve additional generators that transforn as skew-symmetric tensors. A complete classification is in (ACDP).

As automorphism Lie algebras of Lie $n$-superalgebras

At least some of the polyvector extensions of the super Poincaré Lie algebra arise as the automorphism super Lie algebras of the Lie n-algebra extensions classified by the cocycles discussed above.

For instance the automorphisms of the supergravity Lie 3-algebra gives the “M-theory Lie algebra”-extension of super-Poincaré in 11-dimensions. This is discussed here.

Related concepts

• super Poincaré group

• super translation algebra

• supersymmetry, Green-Schwarz action functional

References

Polyvector extensions

Some standard exposition is for instance in

• Super spacetimes and super Poincaré-group (pdf)

A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in

• D. V. Alekseevsky, V. Cortés, C. Devchand, A. Van Proeyen, Polyvector Super-Poincaré Algebras Commun.Math.Phys. 253 (2004) 385-422 (arXiv:hep-th/0311107)

See also

• C. Chrysso‌malakos, José de Azcárraga, J. M. Izquierdo and C. Pérez Bueno, The geometry of branes and extended superspaces (arXiv:hep-th/9904137)

Super Lie algebra cohomology

The super-Lie algebra cohomology of the super Poincare Lie algebra is discussed in

• José de Azcárraga and Paul Townsend, Superspace geometry and classification of supersymmetric extended objects, Phys. Rev. Lett. 62, 2579–2582 (1989)

and completely classified in

• Friedemann Brandt?, Supersymmetry algebra cohomology

  I: Definition and general structure J. Math. Phys.51:122302, 2010, (arXiv:0911.2118)

  II: Primitive elements in 2 and 3 dimensions, J. Math. Phys. 51 (2010) 112303 (arXiv:1004.2978)

  III: Primitive elements in four and five dimensions, J. Math. Phys. 52:052301, 2011 (arXiv:1005.2102)

  IV: Primitive elements in all dimensions from $D=4$ to $D=11$, J. Math. Phys. 54, 052302 (2013) (arXiv:1303.6211)

A classification of some special cases of signature/supersymmetry of this is also in the following (using a computer algebra system):

• Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries (arXiv:1011.4731)

• Michael Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries and of super Poincare Lie algebra (arXiv:1106.0335)

For applications of this classification see also at Green-Schwarz action functional and at brane scan.

An introduction to the exceptional fermionic cocycles on the super Poincaré Lie algebra, and their description using normed division algebras, are discussed here:

• John Baez, John Huerta, Division algebras and supersymmetry I (arXiv:0909.0551)

• John Baez, John Huerta, Division algebras and supersymmetry II (arXiv:1003.34360)

• John Huerta, Division Algebras, Supersymmetry and Higher Gauge Theory, (arXiv:1106.3385)

This subsumes some of the results in (Azcárraga-Townend)

A direct constructions of ordinary (Lie algebraic) extensions of the super Poincare Lie algebra by means of division algebras is in

• Jerzy Lukierski, Francesco Toppan, Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory (pdf)

For more on this see at division algebra and supersymmetry.