higher 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




tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)



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 (