Contents

Idea

Wu classes are a type of universal characteristic class in ${ℤ}_{2}$-cohomology that refine the Stiefel-Whitney classes.

Definition

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

${w}_{k}\in {H}^{k}\left(X,{ℤ}_{2}\right)$w_k \in H^k(X, \mathbb{Z}_2)

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

$\cup :{H}^{k}\left(X,{ℤ}_{2}\right)×{H}^{l}\left(X,{ℤ}_{2}\right)\to {H}^{k+1}\left(X,{ℤ}_{2}\right)$\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 ${ℤ}_{2}$-cohomology groups and write

${\mathrm{Sq}}^{k}\left(-\right):{H}^{l}\left(X,{ℤ}_{2}\right)\to {H}^{k+l}\left(X,{ℤ}_{2}\right)$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

${\nu }_{k}\in {H}^{k}\left(X,{ℤ}_{2}\right)$\nu_k \in H^k(X,\mathbb{Z}_2)

is defined to be the class that “represents” ${\mathrm{Sq}}^{k}\left(-\right)$ under the cup product, in the sense that for all $x\in {H}^{l}\left(X,{ℤ}_{2}\right)$ we have

${\mathrm{Sq}}^{k}\left(x\right)={\nu }_{k}\cup x\phantom{\rule{thinmathspace}{0ex}}.$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 ${\nu }_{k}$.

${w}_{k}=\mathrm{Sq}\left({\nu }_{k}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $TX:X\to B\mathrm{SO}\left(n\right)$ with spin structure $\stackrel{^}{T}X:X\to B\mathrm{Spin}\left(n\right)$. Then the following classes in integral cohomology of $X$, built from Pontryagin classes, coincide with Wu-classes under mod-2-reduction $ℤ\to {ℤ}_{2}$:

• ${\nu }_{4}=\frac{1}{2}{p}_{1}$

• ${\nu }_{8}=\frac{1}{8}\left(11{p}_{1}^{2}-20{p}_{2}\right)$

• ${\nu }_{12}=\frac{1}{16}\left(37{p}_{1}^{3}-100{p}_{1}{p}_{2}+80{p}_{3}\right)$.

This is discussed in Hopkins-Singer, page 101.

Corollary

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

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

$G\cup G={\mathrm{Sq}}^{4}\left(G\right)\phantom{\rule{thinmathspace}{0ex}}.$G \cup G = Sq^4(G) \,.

By the definition 1 of Wu classes, the image of this integral class in ${ℤ}_{2}$-coefficients equals the cup product with the Wu class

$G\cup G-G\cup \frac{1}{2}{p}_{1}={\mathrm{Sq}}^{4}\left(G\right)-G\cup {\nu }_{4}=0\mathrm{mod}2.\phantom{\rule{thinmathspace}{0ex}},$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

${G}_{4}=\frac{1}{2}\left(\frac{1}{2}{p}_{1}\right)+a\phantom{\rule{thinmathspace}{0ex}},$G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a \,,

for some $a\in {H}^{4}\left(X,ℤ\right)$, which makes direct sense as an equation in ${H}^{4}\left(X,ℤ\right)$ 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 $\stackrel{^}{a},\stackrel{^}{G}\in {\stackrel{^}{H}}^{4}\left(X\right)$ with

(1)$\stackrel{^}{G}=\frac{1}{2}\left(\frac{1}{2}{\stackrel{^}{p}}_{1}\right)+\stackrel{^}{a}\in {\stackrel{^}{H}}^{4}\left(X\right)\phantom{\rule{thinmathspace}{0ex}},$\hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,,

where the differential refinement $\frac{1}{2}{\stackrel{^}{p}}_{1}$ 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

$\stackrel{^}{G}↦\mathrm{exp}i{\int }_{X}\left(\stackrel{^}{G}\cup \stackrel{^}{G}-\left(\frac{1}{4}{\stackrel{^}{p}}_{1}{\right)}^{2}\right)\phantom{\rule{thinmathspace}{0ex}},$\hat G \mapsto \exp i \int_X ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,,

where

Using (1) this is

$\cdots =\mathrm{exp}i{\int }_{X}\left(\stackrel{^}{a}\cup \stackrel{^}{a}+\stackrel{^}{a}\cup \frac{1}{2}{\stackrel{^}{p}}_{1}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\stackrel{^}{G}↦\mathrm{exp}i{\int }_{X}\frac{1}{2}\left(\stackrel{^}{G}\cup \stackrel{^}{G}-\left(\frac{1}{4}{\stackrel{^}{p}}_{1}{\right)}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$\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.

References

The original reference is

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

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

This is based on or motivated from observations in

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.

Revised on June 1, 2012 07:32:57 by Urs Schreiber (212.236.23.114)