nLab Wu class

Idea

Wu classes are a type of universal characteristic class in $\mathbb{Z}_2$-cohomology that refine the Stiefel-Whitney classes.

Definition

For $X$ a topological space equipped with a class $E : X \to B SO(n)$ (a real vector bundle of some rank $n$), write

$$w_k \in H^k(X, \mathbb{Z}_2)$$

for the Stiefel-Whitney classes of $X$. Moreover, write

$$\cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)$$

for the cup product on $\mathbb{Z}_2$-cohomology groups and write

$$Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)$$

for the Steenrod square operations.

Definition

The Wu class

$$\nu_k \in H^k(X,\mathbb{Z}_2)$$

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

$$Sq^k(x) = \nu_k \cup x \,.$$

(e.g. Milnor-Stasheff 74, p. 131-133)

Remark

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.

Properties

Relation to Stiefel-Whitney classes

The total Stiefel-Whitney class $w$ is the total Steenrod square of the total Wu class $\nu$.

$$w = Sq(\nu) \,.$$

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

• $\nu_1 = w_1$;

• $\nu_2 = w_2 + w_1^2$

• $\nu_3 = w_1 w_2$

• $\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4$

• $\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3$

…

Relation to Pontryagin classes

Proposition

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).

Corollary

Suppose $X$ is 8 dimensional. Then, for $G \in H^4(X, \mathbb{Z})$ any integral 4-class, the expression

$$G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z})$$

is always even (divisible by 2).

Proof

By the basic properties of Steenrod squares, we have for the 4-class $G$ that

$$G \cup G = Sq^4(G) \,.$$

By the definition of Wu classes, the image of this integral class in $\mathbb{Z}_2$-coefficients equals the cup product with the Wu class

$$G \cup G - G \cup \frac{1}{2}p_1 = Sq^4(G) - G \cup \nu_4 = 0 \; mod \; 2. \,,$$

where the first step is by prop. .

Applications

To higher dimensional Chern-Simons theory

Remark

The relation 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

$$G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a \,,$$

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

(1)$$\hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,,$$

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

$$\hat G \mapsto \exp i \int_X ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,,$$

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

$$\cdots = \exp i \int_X \left( \hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1 \right) \,.$$

But by corollary this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this

$$\hat G \mapsto \exp i \int_X \frac{1}{2} ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,.$$

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.

Related concepts

• integral Wu structure, twisted Wu structure

• shifted C-field flux quantization

• Pontryagin class, Stiefel-Whitney class, one-loop anomaly polynomial I8

• Euler class

References

The original reference is

• Wen-Tsun Wu, On Pontrjagin classes: II Sientia Sinica 4 (1955) 455-490

See also around p. 228 of

• John Milnor, Jim Stasheff, Characteristic classes, Princeton University Press (1974)

and section 2 of

• Yanghyun Byun, On vanishing of characteristic numbers in Poincaré complexes, Transactions of the AMS, vol 348, number 8 (1996) (pdf)

and

• Robert Stong, Toshio Yoshida, Wu classes Proceedings of the American Mathematical Society Vol. 100, No. 2, (1987) (JSTOR)

Details are reviewed in appendix E of

• Mike Hopkins, Isadore Singer, Quadratic Functions in Geometry, Topology, and M-Theory

This is based on or motivated from observations in

• Edward Witten, Five-Brane Effective Action In M-Theory (arXiv:hep-th/9610234)

More discussion of Wu classes in this physical context is in

• Hisham Sati, Twisted topological structures related to M-branes II: Twisted $Wu$ and $Wu^c$ structures (arXiv:1109.4461)

which also summarizes many standard properties of Wu classes.