Real and complex normalization

A neutral real field uses $\beta =0$. For the charged field,

Thus $⟨{\Phi }^{†}\Phi ⟩$ has the same scalar covariance normalization as one real Wick square, whereas the complex stress tensor has twice the corresponding real differential expression. The real components need not be separate globally trivial bundles.

The antiparticle sector uses the conjugate bundle and conjugate extension. A real sign bundle at $\beta =1/2$ is not what is called one charged complex field here.

For

a periodic gauge may be chosen locally with $eA=\beta \mathrm{d}\phi$, up to an integer gauge shift. The twisted parallel representation satisfies

For a flux $\mathcal{F}$, our convention is $\beta =\mathrm{frac}(e\mathcal{F}/2\pi )$.