Consider a Yang-Mills U(1) gauge theory in d=2+1 flat dimensions.
It is known that in d=2+1 the field strength F=dA can be written in terms of a dual scalar field a, in the following way:
F=⋆da.
(The star is the hodge dual of forms)
My question is why a is compact, i.e. its target space is a circle (which is, a is invariant under a∼a+2π
In all the reviews of 3d gauge dynamics this fact is simply stated, with no proof at all.
To me, it is not obvius.
Thank you very much.