Connection sectors, holonomy and presymplectic boundaries — Examples

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth principal-connection and gauge-action convention. Let $M_0={\mathbb{R}}_{x^0}\times U(1{)}_w\times {\mathbb{R}}_{x,y}^2$ have the product smooth structure and flat Lorentzian metric $g=-(d{x}^0{)}^2+R^{2}d{\varphi }^2+d{x}^2+d{y}^2$, where $x^0,x,y$ are length coordinates measured in metres, $R>0$ is a length, and $\varphi$ is dimensionless modulo $2\pi$. On each angular chart, $s=R\varphi$ is a length coordinate and $g=-(d{x}^0{)}^2+d{s}^2+d{x}^2+d{y}^2$, so all four coordinates in this chart are metre-valued. In the adopted Lorentzian length-coordinate convention, write $x^0=c{t}_{\mathrm{s}\mathrm{e}\mathrm{c}}$, where $t_{\mathrm{s}\mathrm{e}\mathrm{c}}$ is measured in seconds and $c>0$ is only a coordinate unit-conversion scale in $\mathrm{m}{\mathrm{s}}^{-1}$; no numerical measured value or physical role for $c$ is asserted. Give $M_0$ time orientation ${\mathrm{\partial }}_{x^0}$ and product orientation $d{x}^0\wedge R\nu \wedge dx\wedge dy$. Let $\nu =-i{w}^{-1}dw=d\varphi$ be the global real angular one-form on the base circle, normalized by ${\int }_{U(1)}\nu =2\pi$, and let $P=M_0\times U(1{)}_z$ have projection $\pi (x,z)=x$ and right action $(x,z)\cdot \zeta =(x,z\zeta )$. On the fibre let $\eta =-i{z}^{-1}dz$. For a dimensionless real parameter $\theta$, the normalized connection ${\omega }_{\theta }=\eta +\theta {\pi }^{\ast }\nu$ has zero curvature. For the fixed normalization ${\kappa }_0>0$ in webers, its physical local potential is $A_{\theta }={\kappa }_0\theta \nu$; in the length coordinate $s$ its component is ${\kappa }_0\theta \mathrm{/}R$ in $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$. For the positive base loop $\gamma (\varphi )=(0,e^{i\varphi },0,0)$, $\varphi \in [0,2\pi ]$, define principal holonomy as the fibre multiplier of the horizontal lift; it is $e^{-2\pi i\theta }$. The inverse Wilson-loop convention $W(\gamma ):=\exp (i{\int }_{\gamma }\theta \nu )$ gives $e^{2\pi i\theta }$. Under the pushforward action on horizontal distributions, this connection is gauge-equivalent to $\eta$ exactly when $\theta \in \mathbb{Z}$. The circle is an ideal topological factor of a smooth background; no monopole, charge quantization, or empirical claim is made.

Fixed-topology affine sector of normalized circle connections

Oriented Lorentzian fields, observers and coordinate conventions

Sphere winding, nonexact curvature and flat holonomy

The plus sign $e^{2\pi i\theta }$ is the inverse Wilson-loop convention used in the design. With the fixed total-space connection form $\eta +\theta \nu$, horizontal parallel transport has the inverse multiplier $e^{-2\pi i\theta }$. The two conventions agree on whether the phase is trivial and on the integer gauge-triviality criterion.