A Hodge decomposition of a 1-form on a torus

[Mastermind01]

TL;DR Summary

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.

 

[mathwonk]

I don't quite understand the language of that paper but can say a few things that may help some. In using differential 1-forms we are interested in integrating them over paths. In the plane, if the form is "closed" i.e. d of it is zero, then the integral is path independent, so the integral over a closed path, or loop, is zero. But that is becaue each loop in the plane is the boundary of a piece of surface and we can then use Stokes or Greens theorem to replace the path integral by the double integral, over the surface, of d of the form, which is zero by assumption.

But on a closed surface like a torus, or doughnut, there are two closed paths that do not bound a piece of surface, and the integral of even a closed form over those may not be zero. e.g. on the parallelogram with opposite sides identified, to make a torus, the coordinate function z differs by a constant at any two identified points, hence its differential dz, is a well defined holomorphic, hence closed, 1-form on the torus.

This dz is the 1-form in the article you link, called there omega. For some reason he says you can take omega equal to 1, which makes no sense to me, rather it is clearly dz.

Now homology theory is the analysis of closed loops on a surface which essentially do not bound, and hence over which closed forms may have non zero integral. I.e. the homology space of a torus has dimension 2, and those two non bounding loops are a basis.

deRham cohomology is the dual theory, namely the deRham cohomology spce of the torus is the 2 dimensional space of all closed one forms, modulo exact one forms. Hodge theory says that we can dispense with equivalence classes of forms if we consider only harmonic forms, i.e. that each equivalence class of closed mod exact forms contains exactly one harmonic form. In the case of a Riemann surface, in fact we can split the (always even dimensional) (first) deRham cohomology space into a direct sum of holomorphic and antiholomorphic one-forms. So in the case of a torus, every deRham cohomology class in H^1, is represented by something locally of form f(z)dz + g(z)dzbar.

 

[Mastermind01]

Thanks, this at least gives me something to go on. I realize I should have attached the previous paragraph, which defines $A$ and $\chi$. I'll have a look at deRham cohomology to see if I get something.