Classification of $SU(2)$-bundles versus the classification of $SO(3)$-bundles

Question

As explained in:

Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

principal $SU(2)$ bundles $P_{SU(2)}$ over a four-dimensional manifold $M$ are classified by their second Chern-class $c_{2}(P_{SU(2)})\in H^{4}(M,\mathbb{Z})$. Usually, when solving the Yang-Mills equations $\nabla F =0$, $\nabla\ast F =0$ in $P_{SU(2)}$, where $F$ is the curvature of the connection, one goes to a local patch in the base manifold $U$ and using a local section $s$ one works with the pullback $s^{\ast}A$ of the connection to $U$, and then one solves the equations locally in $U$. The pullback of the connection $s^{\ast}A$ is a one-form in $M$ with values in $\mathfrak{su}(2)$. But of course we have $\mathfrak{su}(2)\simeq so(3)$, so locally one could not distinguish if one is dealing with a $SU(2)$-bundle over $M$ or a $SO(3)$-bundle $P_{SO(3)}$ over $M$, at least at the level of locally solving the Yang-Mill equations for the connection. My question is then, how different is the topological classification of $SO(3)$ bundles over $M$ from the classification of $SU(2)$ bundles? In addition, it looks that one can use a local solution to the Yang-Mill equations of motions for a $SU(2)$-bundle in a $SO(3)$-bundle an vice-versa.

Thanks.

Answer 1

You are correct that locally, there is no difference between SU(2) and SO(3) bundles, but there are important global differences. In particular, SO(3) bundles may have non-trivial $w_2$, which obstructs their lifting to an SU(2) bundle. The Dold-Whitney theorem (Classification of oriented sphere bundles over a 4-complex. Ann. of Math. (2) 69 1959 667–677) says that the topological classification of SO(3) bundles is given exactly by $w_2$ and the first Pontrjagin class $p_1$. The only constraint on the two classes is that $p_1$ is the (mod 4 valued) Pontrjagin square of $w_2$; compare The Stiefel-Whitney and Pontryagin classes of SO(3)-bundles. There are also some subtle differences in the topology of the gauge group.

In mathematical gauge theory, as applied to 4-manifold topology, this is an important distinction. In many circumstances, one can prove that the moduli space of anti-self-dual SO(3) connections is compact, leading to somewhat simpler proofs of some of Donaldson's fundamental theorems. This was observed by Fintushel and Stern (SO(3)-connections and the topology of 4-manifolds, J. Differential Geom. 20 (1984), 523-539).