In the paper "A Duality Web in 2 + 1 Dimensions and Condensed Matter Physics",
https://arxiv.org/abs/1606.01989
on page 34, it talks about the periods of a closed 2-form. Consider the following path integral,
∫DBexp{−i8π∫R4(ˉτ′F′+∧⋆F′+−τ′F′−∧⋆F′−)−12π∫R4F′∧dB}
where F′ is an arbitrary 2-form, F′±ab=12(F′ab±i2ϵabcdF′cd), and B is a U(1)-gauge field. It says that path-integral produces a delta functional δ[dF′].
My first question is that how the path-integral produces a delta-functional. Shouldn't there be an extra factor i in front of the second integral ∫F′∧dB, so that
I am thinking that the path-integral can be performed by extending the spacetime integral to a complex plane.
∫DBexp{−12π∫R4F′∧dB}
=∫DBexp{12π∫R4dF′∧B}→∫DBexp{12π∮C×R3dF′∧B},
where the last step means
∮C×R3dF′∧B=∫R×R3dF′∧B+∫iR×R3dF′∧iB,
where the integral over circles in upper and lower half plane at infinity vanishes. If the integrand dF′∧B has no singularity inside the contour, then one obtains the delta functional.
New Edition: I forgot that the path integral is integrating eiS.
It then says that the 2-form F′ is closed and has periods 2πZ, and so it is the field strength of some gauge field B′.
What's the definition of periods of a closed 2-form? Is that related with the fact that F′ belongs to the second cohomology class F′≡F′+dξ? Why is it 2πZ?
The delta functional shows that F′ belongs to the second de Rham cohomology, [F′]∈H2(R4,R).
Question: How do I show that it actually takes integral values?
From a physical aspect, that F′ belongs to integral cohomology means that it is the curvature of a non-trivial U(1)-bundle, and there exist magnetic monopoles with integer-valued magnetic charges. Is that correct? The delta functional δ[dF′] implies that F′ is closed. Why does it also implies that the bundle is non-trivial?