Fixed-topology affine sector of normalized circle connections

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the smooth tangent-bundle and principal-connection convention. Let $(M,g,N,o)$ be the smooth, Hausdorff, second-countable, oriented and time-oriented Lorentzian four-manifold used for normalized circle-connection data. Fix a smooth right principal $U(1)$ bundle $P\to M$, one countable trivializing cover $(U_i)$ with transition functions $g_{ij}$ in the convention $z_j=g_{ij}^{-1}{z}_i$, and one normalization constant ${\kappa }_0>0$ measured in webers. Let $(a_i)$ and $(a_i^{\mathrm{\prime }})$ be two smooth normalized connection representatives for this same atlas, each satisfying $a_j=a_i+\lambda (g_{ij})$ and $a_j^{\mathrm{\prime }}=a_i^{\mathrm{\prime }}+\lambda (g_{ij})$, where $\lambda (g):=-i{g}^{-1}dg$. In length-coordinate charts, the components of $a_i$ and $a_i^{\mathrm{\prime }}$ have units ${\mathrm{m}}^{-1}$; the physical potentials $A_i={\kappa }_0{a}_i$ and $A_i^{\mathrm{\prime }}={\kappa }_0{a}_i^{\mathrm{\prime }}$ have components $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$.

Then ${\alpha }_i:=a_i^{\mathrm{\prime }}-a_i$ is the restriction of a unique smooth global real one-form $\alpha$ on $M$. The normalized curvatures $f{\mathrm{\mid }}_{U_i}=d{a}_i$ and $f^{\mathrm{\prime }}{\mathrm{\mid }}_{U_i}=d{a}_i^{\mathrm{\prime }}$ are global, and $f^{\mathrm{\prime }}-f=d\alpha$. The physical curvatures $F={\kappa }_{0}f$ and $F^{\mathrm{\prime }}={\kappa }_0{f}^{\mathrm{\prime }}$ satisfy $F^{\mathrm{\prime }}-F={\kappa }_{0}d\alpha$; in length-coordinate charts their components have units $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. Consequently ${\int }_S{\iota }^{\ast }{F}^{\mathrm{\prime }}={\int }_S{\iota }^{\ast }F$ for every compact oriented smooth surface $S$ without boundary and smooth embedding $\iota :S\hookrightarrow M$. The representatives on a fixed bundle and atlas form an affine space over ${\mathrm{\Omega }}^1(M)$.

For a smooth local gauge map ${\mathrm{\Phi }}_h(x,z)=(x,h(x)z)$, the adopted pushforward action on horizontal distributions changes the local representative by $a_i\mapsto a_i-\lambda (h)$; this sign is part of the stated convention.

Local potentials and smooth circle-bundle connection data

Circle gluing, curvature and connection variations

Gauge transformations act on principal connections and carry curvature

The finite-chart diagram records the affine translation of a connection representative and the resulting curvature translation. It summarizes step 1.2 and step 2.1; the proof does not rely on the diagram.

\mathcal A(P) \arrow[r,"T_\alpha"] \arrow[d,"F_{\mathrm{phys}}"'] &
\mathcal A(P) \arrow[d,"F_{\mathrm{phys}}"] \\
\Omega^2(M) \arrow[r,"+\kappa_0\,d\alpha"'] & \Omega^2(M)

Here $\mathcal{A}(P)$ is the affine set of normalized connections on the fixed bundle and atlas, $T_{\alpha }$ adds the global one-form $\alpha$ to every local representative, and $F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ sends a connection to its global physical curvature. Step 2.1 proves that $F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(T_{\alpha }a)=F_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(a)+{\kappa }_{0}d\alpha$, which is exactly the commutativity asserted by the square. The proposition does not classify bundle sectors or claim that every flat difference is gauge trivial.