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I'm reading this paper and at the beginning of the second section, he states many results that aren't clear to me.

Consider a principal $SO(3)$-bundle $P\rightarrow R^2\times \Sigma$, where $\Sigma$ is a Riemann surface.

If $\mathcal{w}_2(P)=0$, then $P$ is covered by a principal $SU(2)$-bundle to which we may associate a rank-2 vector bundle $V$, with $\mathcal{c}_1(V)=0$.

My questions:

1.1) What does he mean by saying "$P$ is covered by another bundle"?

1.2) Why $\mathcal{w}_2(P)=0$ implies the existence of such a cover?

If $\mathcal{w}_2(P)\neq 0$, then there is a principal $U(2)$ bundle $\hat P$ to which $P$ is associeted via homomorphism $U(2)/Z(U(2))\simeq SO(3)$. Associeted to $\hat P$ is a rank-2 vector bundle $V$ with $\mathcal{c}_1(V)$ is odd. Fixing a connection $A_0$ on $\wedge^2V$, we find that a connection $A$ on $\hat P$ lifts to one on $P$, whose curvature is $F(A)+\frac{1}{2}F(A_0)1$.

More questions:

2.1) Again, why $\mathcal{w}_2(P)=0$ implies the existence of such a cover?

2.2) How can I find this connection $A$?

Sorry for all these questions, but I'm pretty new in this subject and I have no idea of where can I find these results. So, any explanation and reference are very welcome.

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    $\begingroup$ The keywords you're looking for are "spin structure": en.wikipedia.org/wiki/Spin_structure $\endgroup$
    – Oliver Nash
    Commented Jun 25, 2019 at 12:09
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    $\begingroup$ $SU(2)$ is a $2$-fold cover of $SO(3)$. In particular, there is a quotient map $q : SU(2) \to SO(3)$. $P$ is covered by a principal $SU(2)$-bundle $Q$ if the bundle projection $Q \to R^2 \times \Sigma$ factors through $P$ and $Q$ is an $SU(2)$-bundle in such a way that the $SU(2)$-multiplication is compatible with the $SO(3)$ multiplication on $P$. $\endgroup$
    – Ulrich Pennig
    Commented Jun 25, 2019 at 12:29
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    $\begingroup$ The Lie algebras of $SU(2)$ and $SO(3)$ are canonically isomorphic, so local connection forms for both bundles are essentially valued in the same Lie algebra, and the conjugation action of the structure group that appears in the formula on overlaps is the same for the original transition functions and the lifted ones. $\endgroup$
    – David Roberts ♦
    Commented Jun 25, 2019 at 12:40

1 Answer 1

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This is an answer only to the question 1.2 (1.1 is already answered in a comment by Ulrich Pennig).

Consider the adjoint representation of $SU(2)$ on its Lie algebra. Noting that the real dimension of $SU(2)$ is 3, this gives a homomorphism $SU(2)\to SO(3)$. One can prove that it is surjective with the kernel isomorphic to $Z/2$. Thus we get a fibration sequence $$Z/2 \to SU(2) \to SO(3) \to BZ/2\to BSU(2)\to BSO(3) \to K(Z/2,2).$$ Suppose now $P$ is a principal $SO(3)$ bundle over $X$, classified by a map $f:X\to BSO(3)$. Then $w_2(P)$ is nothing but the composition $X\stackrel{f}{\to}BSO(3)\to K(Z/2,2)$. Thus its vanishing is equivalent to existence of $f$ to $BSO(3)$. Take the corresponding $SO(3)$-principal bundle, and you get the desired cover.

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  • $\begingroup$ Thanks for your answer! Could you please clarify what do you mean by $BZ/2$ and $K(Z/2)$? $\endgroup$
    – Leonardo Schultz
    Commented Jun 26, 2019 at 7:56
  • $\begingroup$ Sorry, that was a typo, I meant $K(Z/2.2)$ (Eilenberg-Maclane space with $\pi _2=Z/2$. $BG$ is the classifying space for the group $G$. $\endgroup$
    – user43326
    Commented Jun 26, 2019 at 13:21

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