higher Klein geometry



Differential geometry

differential geometry

synthetic differential geometry








Higher Klein geometry is the generalization of Klein geometry from traditional (differential) geometry to higher geometry:

where Klein geometry is about (Lie) groups and their quotients, higher Klein geometry is about (smooth) ∞-groups and their ∞-quotients.

The way that the generalization proceeds is clear after the following observation.


Let GG be a discrete group and HGH \hookrightarrow G a subgroup. Write BG\mathbf{B}G and BH\mathbf{B}H for the corresponding delooping groupoids with a single object. Then the action groupoid G//HG//H is the homotopy fiber of the inclusion functor

BHBG \mathbf{B}H \to \mathbf{B}G

in the (2,1)-category Grpd: we have a fiber sequence

G//HBHBG G//H \to \mathbf{B}H \to \mathbf{B}G

that exhibits G//HG//H as the GG-principal bundle over BH\mathbf{B}H which is classified by the cocycle BHBG\mathbf{B}H \to \mathbf{B}G.

Moreover, the decategorification of the action groupoid (its 0-truncation) is the ordinary quotient

τ 0(G//H)=G/H. \tau_0 (G//H) = G/H \,.

This should all be explained in detail at action groupoid.

The fact that a quotient is given by a homotopy fiber is a special case of the general theorem discussed at ∞-colimits with values in ∞Grpd


That fiber sequence continues to the left as

HGG//HBHBG. H \to G \to G//H \to \mathbf{B}H \to \mathbf{B}G \,.

The above statement remains true verbatim if discrete groups are generalized to Lie groups – or other cohesive groups – if only we pass from the (2,1)-topos Grpd of discrete groupoids to the (2,1)-topos SmoothGrpd of smooth groupoids .

This follows with the discussion at smooth ∞-groupoid -- structures.

Since the quotient G/HG/H is what is called a Klein geometry and since by the above observations we have analogs of these quotients for higher cohesive groups, there is then an evident definition of higher Klein geometry :


Let H\mathbf{H} be a choice of cohesive structure. For instance choose


An \infty-Klein geometry in H\mathbf{H} is a fiber sequence in H\mathbf{H}

G//HBHiBG G//H \to \mathbf{B}H \stackrel{i}{\to} \mathbf{B}G

for ii any morphism between two connected objects, as indicated, hence Ωi:HG\Omega i : H \to G any morphism of ∞-group objects.


For XX an object equipped with a GG-action and f:YXf : Y \to X any morphism, the higher Klein geometry induced by “the shape YY in XX” is given by taking i:HGi : H \to G be the stabilizer ∞-group Stab(f)GStab(f) \to G of ff in XX. See there at Examples – Stabilizers of shapes / Klein geometry.



Higher super Poincaré Klein geometry

Let H=\mathbf{H} = SuperSmooth∞Grpd be the context for synthetic higher supergeometry.

Write 𝔰𝔲𝔤𝔯𝔞 11\mathfrak{sugra}_11 for the super L-∞ algebra called the supergravity Lie 6-algebra. This has a sub-super L L_\infty-algebra of the form

B(𝔰𝔬(10,1)bb 5)B𝔰𝔲𝔤𝔯𝔞 11, \mathbf{B}(\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}) \hookrightarrow \mathbf{B}\mathfrak{sugra}_11 \,,


The quotient

𝔰𝔲𝔤𝔯𝔞 11/((𝔰𝔬(10,1)bb 5)) \mathfrak{sugra}_11 / ((\mathfrak{so}(10,1) \oplus b \mathbb{R} \oplus b^{5}\mathbb{R}))

is the super translation Lie algebra in 11-dimensions.

This higher Klein geometry is the local model for the higher Cartan geometry that describes 11-dimensional supergravity. See D'Auria-Fre formulation of supergravity for more on this.

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


That there ought to be a systematic study of higher Klein geometry and higher Cartan geometry has been amplified by David Corfield since 2006.

Such a formalization is offered in

For more on this see at higher Cartan geometry and Higher Cartan Geometry.

Revised on August 19, 2015 07:39:19 by David Corfield (