for the Stiefel-Whitney classes of . Moreover, write
for the Steenrod square operations.
The Wu class
is defined to be the class that “represents” under the cup product, in the sense that for all where is the dimension of , we have
In other words this says that the lifts of Wu classes to integral cohomology (integral Wu structures) are characteristic elements of the intersection product on integral cohomology, inducing quadratic refinements.
Solving this for the components of in terms of the components of , one finds the first few Wu classes as polynomials in the Stiefel-Whitney classes as follows
(all products are cup products).
This is discussed in Hopkins-Singer, page 101.
Suppose is 8 dimensional. Then, for any integral 4-class, the expression
is always even (divisible by 2).
By the basic properties of Steenrod squares, we have for the 4-class that
By the definition 1 of Wu classes, the image of this integral class in -coefficients equals the cup product with the Wu class
where the first step is by prop. 1.
The relation 1 plays a central role in the definition of the 7-dimensional Chern-Simons theory which is dual to the self-dual higher gauge theory on the M5-brane. In this context it was first pointed out in (Witten 1996) and later elaborated on in (Hopkins-Singer).
for some , which makes direct sense as an equation in if the spin structure on happens to be such is further divisible by 2, and can be made sense of more generally in terms of twisted cohomology (which was suggested in (Witten 1996) and made precise sense of in (Hopkins-Singer) ).
For simplicity, assume that of is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to ordinary differential cohomology and so we consider differential cocycles with
where the differential refinement is discussed in detail at differential string structure.
Now, after dimensional reduction on a 4-sphere, the action functional of 11-dimensional supergravity on the remaining 7-dimensional contains a higher Chern-Simons term which up to prefactors is of the form
the cup product now is the differential Beilinson-Deligne cup product refinement of the integral cup product;
the symbol denotes fiber integration in ordinary differential cohomology.
Using (1) this is
But by corollary 1 this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this
The original reference is
See also around p. 228 of
and section 2 of
Details are reviewed in appendix E of
This is based on or motivated from observations in
More discussion of Wu classes in this physical context is in
which also summarizes many standard properties of Wu classes.