Higher spin geometry
Cohomology and Extensions
We frequently write as shorthand for .
For , the Lie group is the quotient
of the product of the spin group with the circle group by the common sub-group of order 2 and .
We have a short exact sequence
where is the canonical inclusion into the defining product .
As the homotopy fiber of the smooth
We discuss in the following that
the universal third integral Stiefel-Whitney class has an essentially unique lift from ∞Grpd Top to Smooth∞Grpd;
the smooth delooping is the homotopy fiber of , hence is the circle 2-bundle over classified by .
We have a homotopy pullback diagram
in Smooth∞Grpd, where
We present the sitation as usual in the projective model structure on simplicial presheaves over CartSp by ∞-anafunctors.
The first Chern class is given by the ∞-anafunctor
where denotes a presentation of a strict 2-group by a crossed module.
The second Stiefel-Whitney class is given by
Notice that the top horizontal morphism here is a fibration.
Therefore the homotopy pullback in question is (as discussed there) given by the ordinary pullback in
This pullback is , where
This is equivalent to
(notice the non-standard identification here, which is important below in prop. 3 for the identification of ) where now is the diagonal embedding of the subgroup
This in turn is equivalent to
which is def. 1.
Under the identificaton the “universal determinant line bundle map”
is given in components by
(where on the right we write the group structure additively).
By the proof of prop. 2 the -factor in arises from the identification . But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.
The third integral Stiefel-Whitney class
has an essentially unique lift through geometric realization Smooth∞Grpd ∞Grpd Top
where is simply given by the canonical subgroup embedding.
Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at smooth ∞-groupoid -- structures.
The ordinary Bockstein homomorphism is presented by the ∞-anafunctor
Accordingly we need to lift the canonical presentation of to a comparable -anafunctor. This is accomplished by
Here the top horizontal morphism is induced from the morphism of crossed modules that is given by the commuting diagram
Since is contractible, we have indeed under geometric realization an equivalence
where is the smoothly refined Bockstein homomorphism from prop. 4, is a fiber sequence.
The homotopy fiber of is . Thinking of this is one sees that the inclusion of this fiber is indeed .
Then consider the pasting diagram of homotopy pullbacks
The right square is a homotopy pullback by prop. 5. The left square is a homotopy pullback by prop. 2. The bottom composite is the smooth by prop 4.
This implies by claim by the pasting law.
There is a direct analogy between Spin, Spin^c and the metaplectic groups Mp and Mp^c (see there for more).