nLab super Klein geometry

Idea

The notion of super Klein geometry is essentially that of homogeneous space (coset space) $G/H$ in the context of supergeometry. It is the supergeometric counterpart of Klein geometry.

Super Klein geometries form the local models for super Cartan geometries.

Examples

• super Minkowski spacetimes are quotients of super Poincare groups by spin groups

  $$\mathbb{R}^{p,1\vert \mathbf{N}} \simeq Iso(\mathbb{R}^{p,1\vert \mathbf{N}})/Spin(p,1)$$

• similarly super anti de Sitter spacetimes are super cosets, this plays a central role in the AdS-CFT correspondence:

$d$	super anti de Sitter spacetime
4	$\;\;\;\;\frac{OSp(8\vert 4)}{Spin(3,1) \times SO(7)}\;\;\;\;$
5	$\;\;\;\;\frac{SU(2,2\vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;$
7	$\;\;\;\;\frac{OSp(6,2\vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;$

• The supersphere $S^{2|2}$ is the super coset space $UOSp(1|2)/U(1)$.

• The supersphere $S^{r-1|2s}$ is the super coset space $OSp(r|2s)/OSp(r-1|2s)$ of orthosymplectic groups (GJS 18).

Related concepts

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

• A. F. Kleppe, Chris Wainwright, Super coset space geometry, (arXiv:hep-th/0610039)

• A. F. Schunck, Chris Wainwright, A geometric approach to scalar field theories on the supersphere, (arXiv:hep-th/0409257)

• Constantin Candu, Vladimir Mitev, Volker Schomerus, Spectra of Coset Sigma Models, (arXiv:1308.5981)

On superspheres in relation to Brownian loop soup models:

• Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur, Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups, J. Phys. A: Math. Theor. 52 345001 (2019) [doi:10.1088/1751-8121/ab2aaa, arXiv:1810.07807]