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In Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces, a special case of what they do is to prove that Unitary Yang-Mills connections over a R.S $M$ are in bijective correspondence with unitary representations of a central extension of the fundamental group. To this end, given a representation, they construct a unitary bundle as follows (and therein lies my confusion): Take a degree one $U(1)$ bundle $Q$. Now, pull this bundle back to the universal cover $\tilde{M}$. The new bundle is a $U(1) \times \pi_1(M)$ bundle over $M$. Now apparently, one can "lift the representation of the central extension and thus form a unitary bundle" (with a connection). Why can one do this?

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    $\begingroup$ I think Tom Baird discusses this in some detail in his thesis. I suggest asking him for a copy, or asking him your question directly. $\endgroup$
    – Dan Ramras
    Commented Jul 10, 2011 at 3:30

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