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 $E:X\to B\mathrm{SO}(n)$ (a real vector bundle of some rank $n$), write

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

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

$$\cup :H^k(X,{\mathbb{Z}}_2)\times H^l(X,{\mathbb{Z}}_2)\to H^{k+1}(X,{\mathbb{Z}}_2)$$\cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2) \to H^{k+1}(X, \mathbb{Z}_2)

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

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

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

is defined to be the class that “represents” ${\mathrm{Sq}}^k(-)$ under the cup product, in the sense that for all $x\in H^l(X,{\mathbb{Z}}_2)$ we have

$${\mathrm{Sq}}^k(x)={\nu }_k\cup x.$$Sq^k(x) = \nu_k \cup x \,.

Properties

Relation to Stiefel-Whitney classes

The Stiefel-Whitney class $w_k$ is the Steenrod square of the Wu class ${\nu }_k$.

$$w_k=\mathrm{Sq}({\nu }_k).$$w_k = Sq(\nu_k) \,.

One finds the first few Wu classes as polynomials in the Stiefel-Whitney classes as follows

• $v_1=w_1$;

• $v_2=w_2+w_1^2$

• $v_3=w_1{w}_2$

• $v_4=w_4+w_3{w}_1+w_2^2+w_1^4$

• $v_5=w_4{w}_1+w_3{w}_1^2+w_2^2{w}_1+w_2{w}_1^3$

…

Relation to Pontryagin classes

Proposition

Let $X$ be an oriented manifold $TX:X\to B\mathrm{SO}(n)$ with spin structure $\hat{T}X:X\to B\mathrm{Spin}(n)$. Then the following classes in integral cohomology of $X$, built from Pontryagin classes, coincide with Wu-classes under mod-2-reduction $\mathbb{Z}\to {\mathbb{Z}}_2$:

• ${\nu }_4=\frac{1}{2}{p}_1$

• ${\nu }_8=\frac{1}{8}(11p_1^2-20p_2)$

• ${\nu }_{12}=\frac{1}{16}(37p_1^3-100p_1{p}_2+80p_3)$.

(all products are cup products).

This is discussed in Hopkins-Singer, page 101.

Corollary

For $G\in H^4(X,\mathbb{Z})$ any integral 4-class, the expression

$$G\cup G-G\cup \frac{1}{2}{p}_1\in H^4(X,\mathbb{Z})$$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

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

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

$$G\cup G-G\cup \frac{1}{2}{p}_1={\mathrm{Sq}}^4(G)-G\cup {\nu }_4=0\mathrm{mod}2.,$$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. 1.

Applications

To higher dimensional Chern-Simons theory

Remark

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

$$G_4=\frac{1}{2}(\frac{1}{2}{p}_1)+a,$$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)$$\hat{G}=\frac{1}{2}(\frac{1}{2}{\hat{p}}_1)+\hat{a}\in {\hat{H}}^4(X),$$\hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,,

where the differential refinement $\frac{1}{2}{\hat{p}}_1$ 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

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

$$\cdots =\exp i{\int }_X(\hat{a}\cup \hat{a}+\hat{a}\cup \frac{1}{2}{\hat{p}}_1).$$\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 1 this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this

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

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 $\mathrm{Wu}$ and ${\mathrm{Wu}}^c$ structures (arXiv:1109.4461)

which also summarizes many standard properties of Wu classes.