nLab extended super Minkowski spacetime

Idea

As opposed to ordinary Minkowski spacetime, super Minkowski spacetime has, when regarded as the super translation Lie algebra, exceptional cocycles. The super L-infinity algebra extensions classified by these serve as extended super-spacetimes which are target spaces for super $p$-branes.

Notably the supergravity Lie 3-algebra is the extended Minkowski spacetime on which superspacetimes of 11-dimensional supergravity are modeled.

Related concepts

• extended supersymmetry

• M-theory supersymmetry algebra

• type II supersymmetry algebra

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

Discussion in the language of the D'Auria-Fre formulation of supergravity (“FDA”s) and the brane scan is in

• 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]

• Makoto Sakaguchi, IIB-Branes and New Spacetime Superalgebras, JHEP 0004 (2000) 019 (arXiv:hep-th/9909143)

• José de Azcárraga, J. M. Izquierdo, Superalgebra cohomology, the geometry of extended superspaces and superbranes, Nucl.Phys.B567:293-330,2000 (arXiv:hep-th/0105125)

and particularly with emphasis of the extended super Minkowski spacetimes as targets for the Green-Schwarz super-p brane sigma models with gauge fields on the worldvolume (tensor multiplets) in

• José de Azcárraga, Superbranes, $D=11$ CJS supergravity and enlarged superspace coordinates/fields correspondence, AIP Conf.Proc.767:243-267,2005 (arXiv:hep-th/0501198)

Discussion that makes the super L-infinity algebra homotopy theory underlying this manifest is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics. Volume 12, Issue 02 (2015) 1550018 (35 pages) 2013 (arXiv:1308.5264)