∞-Lie theory (higher geometry)
superalgebra and (synthetic ) supergeometry
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).
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
be a real spin representation (Majorana spinor), which has the property (def., prop.) that there exists a linear map
which is
symmetric:
$Spin(V)$ 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.
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)
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 $\Gamma$:
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 $s_1, s_2 \in S$:
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
Removing all terms involving $\omega$ here yields the Chevalley-Eilenberg algebra of the super translation algebra $\mathbb{R}^{D;N}$.
The abstract generators in def. 2 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. 2 as
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
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
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.
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
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.
The polyvector extensions arise as the super Lie algebras of conserved currents of the Green-Schwarz super p-brane sigma-models (AGIT 89).
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.
The seminal classification result of simple supersymmetry algebras is due to
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
for discussion in the view of the brane scan and The brane bouquet of super-$p$-brane Green-Schwarz sigma-models.
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
reviewed in section 8.8. of
and specifically for super-D-branes this discussion is in
The role of polyvector extended supersymmetry algebras in supergravity and string theory is further highlighted in
A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in
Discussion of the super-Lie algebra cohomology of the super Poincare Lie algebra goes back to work on Green-Schwarz sigma models in
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):
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 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)
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
A direct constructions of ordinary (Lie algebraic) extensions of the super Poincaré Lie algebra by means of division algebras is in
For more on this see at division algebra and supersymmetry.