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- How to go about decomposing a 1-form on a torus.

I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be written as the sum of an exact, co-exact and harmonic form. But, I've only ever seen it as a one line result without proof in a mathematical methods book. I don't know anything about homologies either. If someone could at least point me towards a resource where I could learn how to derive the result I would be thankful. I have attached the section of the paper I'm having difficulty with.