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Let $M$ be a smooth manifold. I would like to understand why the moduli space of flat $U(1)$-connections modulo gauge equivalence is the torus $$ H^1(M;\mathbb{R})/H^1(M;\mathbb{Z}). $$ How should I see this?

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    Sorry, I'm not sure what you mean by the moduli space of flat $\text{U}(1)$-connections. Are these connections on the trivial bundle?2012-09-01
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    @QiaochuYuan The result holds for the moduli space of flat connections on any principal $\mathrm{U}(1)$-bundle, not just the trivial bundle. Maybe in a day or two I'll take the time to write up the details in an answer here.2012-09-02

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