group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Wu classes are a type of universal characteristic class in $\mathbb{Z}_2$-cohomology that refine the Stiefel-Whitney classes.
For $X$ a topological space equipped with a class $E : X \to B SO(n)$ (a real vector bundle of some rank $n$), write
for the Stiefel-Whitney classes of $X$. Moreover, write
for the cup product on $\mathbb{Z}_2$-cohomology groups and write
for the Steenrod square operations.
The Wu class
is defined to be the class that “represents” $Sq^k(-)$ under the cup product, in the sense that for all $x \in H^{n-k}(X, \mathbb{Z}_2)$ where $n$ is the dimension of $X$, we have
The total Stiefel-Whitney class $w$ is the Steenrod square of the total Wu class $\nu$.
Solving this for the components of $\nu$ in terms of the components of $w$, one finds the first few Wu classes as polynomials in the Stiefel-Whitney classes as follows
$\nu_1 = w_1$;
$\nu_2 = w_2 + w_1^2$
$\nu_3 = w_1 w_2$
$\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4$
$\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3$
…
Let $X$ be an oriented manifold $T X : X \to B SO(n)$ with spin structure $\hat T X : X \to B Spin(n)$. Then the following classes in integral cohomology of $X$, built from Pontryagin classes, coincide with Wu-classes under mod-2-reduction $\mathbb{Z} \to \mathbb{Z}_2$:
$\nu_4 = \frac{1}{2} p_1$
$\nu_8 = \frac{1}{8}(11 p_1^2 - 20 p_2)$
$\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3)$.
(all products are cup products).
This is discussed in Hopkins-Singer, page 101.
Suppose $X$ is 8 dimensional. Then, for $G \in H^4(X, \mathbb{Z})$ 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 $G$ that
By the definition 1 of Wu classes, the image of this integral class in $\mathbb{Z}_2$-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).
Specifically, in this context $G$ is the 4-class of the circle 3-bundle underlying the supergravity C-field, subject to the quantization condition
for some $a \in H^4(X, \mathbb{Z})$, which makes direct sense as an equation in $H^4(X, \mathbb{Z})$ if the spin structure on $X$ happens to be such $\frac{1}{2}p_1$ 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 $\frac{1}{2}p_1$ of $X$ 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 $\hat a, \hat G \in \hat H^4(X)$ with
where the differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ 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 $X$ contains a higher Chern-Simons term which up to prefactors is of the form
where
the cup product now is the differential Beilinson-Deligne cup product refinement of the integral cup product;
the symbol $\exp(i \int_X (-))$ 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
In (Witten 1996) it is discussed that the space of states of this “fractional” functional over a 6-dimensional $\Sigma$ is the space of conformal blocks of the self-dual higher gauge theory on the M5-brane.
The original reference is
See also around p. 228 of
and section 2 of
and
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.