nLab super Klein geometry




Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



The notion of super Klein geometry is essentially that of homogeneous space (coset space) G/HG/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.


ddsuper anti de Sitter spacetime
4OSp(8|4)Spin(3,1)×SO(7)\;\;\;\;\frac{OSp(8\vert 4)}{Spin(3,1) \times SO(7)}\;\;\;\;
5SU(2,2|5)Spin(4,1)×SO(5)\;\;\;\;\frac{SU(2,2\vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;
7OSp(6,2|4)Spin(6,1)×SO(4)\;\;\;\;\frac{OSp(6,2\vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;
  • The supersphere S 2|2S^{2|2} is the super coset space UOSp(1|2)/U(1)UOSp(1|2)/U(1).

  • The supersphere S r1|2sS^{r-1|2s} is the super coset space OSp(r|2s)/OSp(r1|2s)OSp(r|2s)/OSp(r-1|2s) of orthosymplectic groups (GJS 18).

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d


  • 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)

  • Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur, Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups, (arXiv:1810.07807)

Last revised on April 30, 2019 at 09:43:16. See the history of this page for a list of all contributions to it.