Higher Lie theory
Formal Lie groupoids
Special and general types
Higher spin geometry
Critical string models
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
smooth Whitehead tower
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.
fivebrane 6-group string 2-group spin group special orthogonal group orthogonal group
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