higher geometry / derived geometry
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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 be a discrete group and a subgroup. Write and for the corresponding delooping groupoids with a single object. Then the action groupoid is the homotopy fiber of the inclusion functor
in the (2,1)-category Grpd: we have a fiber sequence
that exhibits as the -principal bundle over which is classified by the cocycle .
Moreover, the decategorification of the action groupoid (its 0-truncation) is the ordinary quotient
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
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 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 be a choice of cohesive structure. For instance choose
Disc∞Grpd for discrete higher Klein geometry (no actual geometric structure);
ETop∞Grpd for continuous higher Klein geometry (with topological structure);
Smooth∞Grpd for higher Klein geometry based on differential geometry;
SuperSmooth∞Grpd for the supergeometry version of higher Klein geometry
and so on.
An -Klein geometry in is a fiber sequence in
for any morphism between two connected objects, as indicated, hence any morphism of ∞-group objects.
For an object equipped with a -action and any morphism, the higher Klein geometry induced by “the shape in ” is given by taking be the stabilizer ∞-group of in . See there at Examples – Stabilizers of shapes / Klein geometry.
By the discussion at looping and delooping, and using that a cohesive (∞,1)-topos has homotopy dimension 0 it follows that every connected object indeed is the delooping of an ∞-group object.
The above says that is the principal ∞-bundle over that is classified by the cocycle .
Continuing this fiber sequence further to the left yields the long fiber sequence
This exhibits indeed as the fiber of .
Let SuperSmooth∞Grpd be the context for synthetic higher supergeometry.
Write for the super L-∞ algebra called the supergravity Lie 6-algebra. This has a sub-super -algebra of the form
where
is the special orthogonal Lie algebra;
is the line Lie n-algebra.
The quotient
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.
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.
Last revised on May 10, 2024 at 16:12:51. See the history of this page for a list of all contributions to it.