I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a connection for the fibration - ie it's the same at all points. For the usual monopole connection $\omega$, what is the holonomy group Hol$_p(\nabla)$ of $\omega$ at a point $p$. The holonomy bundle will of course be a principal Hol$_p(\nabla)$-bundle over $S^2$, what is the total space of this bundle, how does Hol$_p(\nabla)$ act on it, what is the projection?
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asked Mar 11, 2011 at 19:40
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$\begingroup$ The second paragraph seems a little mixed up. The curve $c$ is in the base manifold $S^2$, correct? Parallel transport refers to a lifting of this curve up to the total space, so that the derivatives of the lift agree with the vector field $X$. In general, this lift of $c$ will no longer be a loop (think of the simplest example - a covering space). This is where non-trivial holonomy comes from. $\endgroup$– Dan RamrasCommented Mar 11, 2011 at 20:10
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$\begingroup$ I've removed the confused section. $\endgroup$– Jean DelinezCommented Mar 11, 2011 at 20:23
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The monopole bundle is a nontrivial circle over the 2-sphere with total space the 3-sphere. The structure group of this bundle is $U(1)$, so there are not that many choices for the holonomy group: it's either $U(1)$ or else the connection is flat, in which case, since the 2-sphere is simply connected, the holonomy group would be trivial. Now, the connection is not flat, so the holonomy is $U(1)$.
answered Mar 11, 2011 at 23:27
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$\begingroup$ So the holonomy bundle bundle of the monopole connection is the Hopf fibration? $\endgroup$ Commented Mar 12, 2011 at 16:17