nLab
higher Klein geometry

Context

Geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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.

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 \,.
Proof

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

Remark

That fiber sequence continues to the left as

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

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 :

Definition

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

Definition

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.

Remark

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.

Remarks

Examples

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 \,,

where

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.

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group SO(d)SO(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz groupMinkowski space d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 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

Revised on December 17, 2014 20:49:20 by Urs Schreiber (127.0.0.1)