∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
The fivebrane 6-group is a smooth version of the topological space that appears in the second step of the Whitehead tower of the orthogonal group.
It is a lift of this through the geometric realization functor ?LieGrpd? ∞Grpd.
One step below the fivebrane 6-group in the Whitehead tower is the string Lie 2-group.
For the time being see the discussions at
and the Motivation section at
for more background.
In the (∞,1)-topos ?LieGrpd? we have a smooth refinement of the second fractional Pontryagin class
defined on the delooping of the string Lie 2-group. Strictly speaking, we need , since for low , is not 6-connected. See orthogonal group for a table of the relevant homotopy groups.
The delooping of the fivebrane 6-group is the principal ∞-bundle classified by this in , that is the homotopy fiber
Along the lines of the description at Lie integration and string 2-group, in a canonical model for the morphism is given by a morphism out of a resolution of that is built in degree from smooth -simplices in the Lie group . This morphism assigns to a 7-simplex the integral
of the degree 7 Lie algebra cocycle of the special orthogonal Lie algebra which is normalized such that its pullback to (..explain…) is the deRham image of the generator in integral cohomology there.
More in detail, a resolution of is given by the coskeleton
where the subobjects are those consisting of 3-simplices in with 2-faces labeled in such that the integral of over the 3-simplex in is the signed product of these labels.
(…)
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Whitehead tower of orthogonal group | orientation | spin group | string group | fivebrane group | ninebrane group | |||||||||||||
higher versions | special orthogonal group | spin group | string 2-group | fivebrane 6-group | ninebrane 10-group | |||||||||||||
homotopy groups of stable orthogonal group | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||
stable homotopy groups of spheres | 0 | 0 | 0 | |||||||||||||||
image of J-homomorphism | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
The topological fivebrane group with its interpretation in dual heterotic string theory was discussed in
and the smooth fivebrane 6-group was indicated. The latter is discussed in more detail in section 4.1 of
Jesse Wolfson says he has shown the existence of a presentation of the smooth 6-group by a locally Kan and degreewise finite-dimensional simplicial smooth manifold.
Last revised on May 27, 2022 at 02:58:10. See the history of this page for a list of all contributions to it.