nLab super Poincaré 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

Let $d \in \mathbb{N}$ and consider Minkowski spacetime $\mathbb{R} ^{d-1,1}$ of dimension $d$. For $Spin(d-1,1)$ the corresponding spin group, let

$$S \in Rep(Spin(d-1,1))$$

be a real spin representation (Majorana spinor), which has the property (def., prop.) that there exists a linear map

$$\Gamma \;\colon\; S \otimes S \longrightarrow \mathbb{R}^{d-1,1}$$

which is

1. symmetric:

2. $Spin(V)$ equivariant (a homomorphism of spin representations).

For a classification of spin representations with this property see at spin representations the sections real irreducible spin representations in Lorentz signature and super Poincaré brackets. For explicit construction in components see at Majorana spinor the section The spinor pairing to vectors.

Definition

The super Poincaré Lie algebra $\mathfrak{siso}_S(d-1,1)$ of $d$-dimensional Minkowski spacetime with respect to the spin representation $S$ with symmetric and $Spin(V)$-equivariant pairing $\Gamma \colon S \otimes S \to \mathbb{R}^{d-1,1}$ is the super Lie algebra extension of the Poincaré Lie algebra by $\Pi S$ (the vector space underlying $S$ taken in odd degree)

$$\Pi S \longrightarrow \mathfrak{siso}_S(d-1,1) \longrightarrow \mathfrak{iso}(d-1,1) \simeq \mathbb{R}^{d-1,1} \ltimes \mathfrak{so}(d-1,1) \,,$$

where the Lie bracket of elements in $\mathfrak{so}(d-1,1)$ with those in $S$ is the given action, the Lie bracket of elements of $\mathbb{R}^{d-1,1}$ with those on $S$ is trivial, and the Lie bracket of two elements $s_1, s_2 \in S$ is given by $\Gamma$:

$$[s_1,s_2] \coloneqq \Gamma(s_1,s_2) \,.$$

Remark

It is precisely the symmetry and $Spin(V)$-equivariant assumption on $\Gamma$ that makes this a well defined super Lie algebra: the symmetry corresponds to the graded skew-symmetry of the Lie bracket on elements in $S$, which are regarded as odd, and the $Spin(V)$-equivariance yields the nontrivial Jacobi identity for $o \in \mathfrak{so}(d-1,1)$ and $s_1, s_2 \in S$:

$$\Gamma([o,s_1], s_2) + \Gamma(s_1, [o,s_2]) = [o, \Gamma(s_1,s_2)] \,.$$

Remark

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

Write $\{\omega_a{}^b\}_{a,b}$ for the canonical basis of the special orthogonal matrix Lie algebra $\mathfrak{so}(D-1,1)$ and write $\{\psi_\alpha\}_\alpha$ for a corresponding basis of the spin representation $S$.

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

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

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

with the differential defined by

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

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

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

Remark

Removing all terms involving $\omega$ here yields the Chevalley-Eilenberg algebra of the super translation algebra $\mathbb{R}^{D;N}$.

Remark

The abstract generators in def. 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 $\mathbb{R}^{d|N}$ 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. as

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

Properties

Lie algebra cohomology

The super Poincaré Lie algebra has, on top of the Lie algebra cocycles that it inherits from $\mathfrak{so}(n)$, 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 $d = 3,4,6, 10$, $\mathfrak{siso}(d-1,1)$ has a nontrivial 3-cocycle given by

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

  for spinors $\psi, \phi \in \mathcal{S}$ and vectors $A \in \mathcal{T}$, and 0 otherwise.

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

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

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

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

All these cocycles are controled by the relevant Fierz identities.

Extensions

Super $L_\infty$-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

Extended super Poincaré Lie algebra – Polyvector extensions

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

For instance the “M-theory Lie algebra” is a polyvector extension of the super Poincaré Lie algebra $\mathfrak{siso}_{N=1}(10,1)$ by polyvectors of rank $p = 2$ and $p=5$ (the M2-brane and the M5-brane in the brane scan), see below Polyvector extensions as automorphism Lie algebras.

As current algebras of the GS super $p$-branes

The polyvector extensions arise as the super Lie algebras of conserved currents of the Green-Schwarz super p-brane sigma-models (AGIT 89).

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 (FSS 13). This is also discussed at supergravity Lie 3-algebra – Polyvector extensions.

Contractions

One may consider breaking super-Poincaré invariance by passage to non-relativistic or ultrarelativistic limits, formally understood as taking the speed of light to $\infty$ or to 0, respectively, referred to as Galilean of Carrollian limits, respectively. This is achieved via the process of contraction, as described in İnönü & Wigner 1953.

The corresponding super Lie algebras are defined as follows (see e.g. Section 2 of Koutrolikos & Najafizadeh 2023). We denote the generators of the $4d$ $N=1$ super-Poincaré algebra as $\{ J_{\mu\nu}, P_{\mu}, Q_{\alpha}, \bar{Q}_{\dot \alpha} \}$.

The super-Carroll algebra is defined by the rescaled generators

$$\Big\{K^C _i \coloneqq c J_{0i} , P^C _0 \coloneqq c P_0, J_{ij} ^C \coloneqq J_{ij} , P_{i} ^C \coloneqq P_{i}, Q^C _{\alpha} \coloneqq \sqrt{c} Q_{\alpha} , \bar{Q}^C _{\dot \alpha} \coloneqq \bar{Q} _{\dot \alpha} \Big\}$$

satisfying the commutation relations inherited from the super-Poincaré algebra, with the exceptions that now the commutators

$$\begin{array}{lcl} [K^{^{(\rm{C})}}_i ,J^{C}_{jk} ] &=& - i \delta_{i[j} K^{C}_{k]} \\ [J^{C}_{ij} ,J^{C}_{kl} ] &=& i \big( \delta_{[i|k|} J^{C}_{j]l} - \delta_{[i|l|} J^{C}_{j]k} \big) \\ [K^{C}_i ,P^{C}_j ] &=& -i \delta_{ij} P^{C}_0 \\ [J^{C}_{ij} ,P^{C}_k ] &=& i \delta_{[i|k|}P^{C}_{j]} \\ [J^{C}_{ij}, Q^{C}_{\alpha} ] &=& (\sigma_{ij} )_{\alpha} ^{\beta} Q^{C}_{\beta} \\ [J^{C}_{ij}, \bar{Q}^{C}_{\dot\alpha} ] &=& (\bar{\sigma }_{ij})_{\dot\alpha}^{\dot\beta} \bar{Q}^{C}_{\dot\beta} \\ \{Q^{C}_{\alpha},\bar{Q}^{C}_{\dot\alpha}\} &=& -(\sigma^0 )_{\alpha\dot\alpha} P^{C}_0 \,. \end{array}$$

The super-Galilean algebra is defined with generators

$$\Big\{ K^{G}_i = \frac{1}{c} J_{0i} , P^{G}_0 = c P_0 , Q^{G}_{\alpha} = Q_{\alpha}, J^{G}_{ij} = J_{ij}, P^{G}_{i} = P_{i}, \bar{Q}^{G}_{\dot\alpha} = Q_{\dot\alpha} \Big\}$$

and the following non-zero commutators:

$$\begin{array}{lcl} [K^{G}_i ,J^{G}_{jk} ] &=& -i \delta_{i[j} K^{G}_{k]} \\ [J^{G}_{ij}, J^{G}_{kl} ] &=& i(\delta_{[i|k|} J^{G}_{j]l} - \delta_{[i|l|}J^{G}_{j]k} ) \\ [K^{G}_i ,P^{G}_0 ] &=& - i P^{G}_i \\ [J^{G}_{ij}, P^{G}_k ] &=& i \delta_{[i|k|} P^{G}_{j]} \\ [J^{G}_{ij},Q^{G}_{\alpha} ] &=& (\sigma_{ij} )_{\alpha}^{\beta} Q^{G}_{\beta} \\ [J^{G}_{ij}, \bar{Q}^{G}_{\dot\alpha} ] &=& (\bar{\sigma}_{ij})_{\dot\alpha}^{\dot\beta} \bar{Q}^{G}_{\dot\beta} \\ \{Q^{G}_{\alpha},\bar{Q}^{G}_{\dot\alpha}\} &=& -(\sigma^i)_{\alpha\dot\alpha} P^{G}_i \,. \end{array}$$

See Bergshoeff et al. 2023 and the references therein for more.

Related concepts

• super Minkowski spacetime, signs in supergeometry

• super Poincaré group, supersymmetry

• supermultiplet, BPS state

• division algebra and supersymmetry

• R-symmetry

• super translation algebra

• Green-Schwarz action functional, brane scan

• 4d supergravity Lie 2-algebra, supergravity Lie 3-algebra, supergravity Lie 6-algebra

References

Introducing the super Poincaré Lie algebra (“supersymmetry”):

• Yuri Golfand, Evgeny Likhtman,_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: Victor Ginzburg et al. (eds.) I. E. Tamm Memorial Volume Problems of Theoretical Physics, (Nauka, Moscow 1972), page 37,

  translated and reprinted in: Mikhail Shifman (ed.) The Many Faces of the Superworld pp. 44-53, World Scientific (2000) (doi:10.1142/9789812793850_0006)

General

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2.1 of Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)

The seminal classification result of simple supersymmetry algebras is due to

• Werner Nahm, Supersymmetries and their Representations, Nucl.Phys. B135 (1978) 149 (spire)

Lecture notes include

• 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, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

See also

• C. Chrysso‌malakos, José de Azcárraga, José M. Izquierdo, C. Pérez Bueno, The geometry of branes and extended superspaces, Nucl. Phys. B 567 (2000) 293-330 [arXiv:hep-th/9904137, doi:10.1016/S0550-3213(99)00512-X]

for discussion in the view of the brane scan and The brane bouquet of super-$p$-brane Green-Schwarz sigma-models.

Polyvector extensions

The Polyvector extensions of $\mathfrak{Iso}(\mathbb{R}^{10,1|32})$ (the “M-theory super Lie algebra”) were first considered in

• Riccardo D'Auria, Pietro Fré Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140

• Jan-Willem van Holten, Antoine Van Proeyen, $N=1$ supersymmetry algebras in $d=2,3,4 \,mod\, 8$ J.Phys. A15, 3763 (1982).

Polyvector extensions were found as the algebra of conserved currents of the Green-Schwarz super p-branes in

• José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys.Rev.Lett. 63 (1989) 2443 (spire)

reviewed in section 8.8. of

• José de Azcárraga, José M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics , Cambridge monographs of mathematical physics, (1995)

and specifically for super-D-branes this discussion is in

• Hanno Hammer, Topological Extensions of Noether Charge Algebras carried by D-p-branes, Nucl.Phys. B521 (1998) 503-546 (arXiv:hep-th/9711009)

The role of polyvector extended supersymmetry algebras in supergravity and string theory is further highlighted in

• Paul Townsend, $p$-Brane Democracy (arXiv:hep-th/9507048)

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

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

Super Lie algebra cohomology

Discussion of the super-Lie algebra cohomology of the super Poincare Lie algebra goes back to work on Green-Schwarz sigma models in

• José de Azcárraga, Paul Townsend, Superspace geometry and classification of supersymmetric extended objects, Phys. Rev. Lett. 62 (1989) 2579-2582 [doi:10.1103/PhysRevLett.62.2579, spire:284635]

A rigorous classification of these cocycles was later given 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):

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

• Mikhail Movshev, Albert Schwarz, Renjun Xu, Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 (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)

Discussion of the corresponding super L-∞ algebra L-∞ extensions in the context of Green-Schwarz action functionals and ∞-Wess-Zumino-Witten theory is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields

A direct constructions of ordinary (Lie algebraic) extensions of the super Poincaré 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.

On the notion of contraction used for non-Lorentzian limits:

• Erdal İnönü, Eugene Wigner. On the Contraction of Groups and Their Representations. Proceedings of the National Academy of Sciences 39, no. 6 (1953): 510-524. (doi).

Discussion of the Carrollian- and Galilean limits:

• Konstantinos Koutrolikos, Mojtaba Najafizadeh. Super-Carrollian and super-Galilean field theories. Physical Review D 108, no. 12 (2023): 125014. (doi).

• Eric Bergshoeff, José Figueroa-O'Farrill, and Joaquim Gomis. A non-lorentzian primer. SciPost Physics Lecture Notes (2023): 069. (doi).