Closed Maxwell current and gauge boundary term

Let $g$ be a fixed oriented Lorentzian metric of signature $(-,+,+,+)$ on a smooth four-manifold, and let $\alpha ,\beta \in {\mathrm{\Omega }}^1(M)$ be smooth normalized connection variations satisfying $d{\star }_{g}d\alpha =d{\star }_{g}d\beta =0$. For ${\omega }_g(\alpha ,\beta )=\alpha \wedge {\star }_{g}d\beta -\beta \wedge {\star }_{g}d\alpha$, one has $d{\omega }_g(\alpha ,\beta )=0$. For every smooth real function $\chi$ on $M$, replacing $\alpha$ by $\alpha +d\chi$ changes the current by $d(\chi {\star }_{g}d\beta )$. Hence, for every finite smooth singular three-chain $c$, the integrated change is ${\int }_{\mathrm{\partial }c}\chi {\star }_{g}d\beta$; it vanishes when $c$ is a cycle.

For a noncompact oriented spacelike hypersurface ${\iota }_{\mathrm{\Sigma }}:\mathrm{\Sigma }\hookrightarrow M$, suppose both ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )$ and ${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha +d\chi ,\beta )$ are defined by the absolute-integrability convention of the preceding presymplectic-current definition. Write ${\iota }_{\mathrm{\Sigma }}^{\ast }({\omega }_g(\alpha +d\chi ,\beta )-{\omega }_g(\alpha ,\beta ))=u{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$. For every compact Borel exhaustion $K_n\subseteq K_{n+1}$ with ${\bigcup }_n{K}_n=\mathrm{\Sigma }$, one has ${\int }_{K_n}ud{\mu }_{\gamma }\to {\int }_{\mathrm{\Sigma }}ud{\mu }_{\gamma }$. If the exhaustion has finite smooth singular three-chains $c_n$ supported in $K_n$ satisfying ${\int }_{c_n}\tau ={\int }_{K_n}{u}_{\tau }d{\mu }_{\gamma }$ for every smooth three-form $\tau =u_{\tau }{\mathrm{v}\mathrm{o}\mathrm{l}}_{\gamma }$ on $\mathrm{\Sigma }$, using the induced metric and density fixed in that definition, then

${\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha +d\chi ,\beta )-{\mathrm{\Omega }}_{\mathrm{\Sigma }}(\alpha ,\beta )={\lim }_{n\to \mathrm{\infty }}{\int }_{\mathrm{\partial }{c}_n}\chi {\star }_{g}d\beta \mathrm{.}$

The limit exists and is independent of the compact exhaustion and its smooth-chain representatives. For a fixed $\beta$, the pairing shift vanishes exactly when this limiting boundary flux is zero. Given any linear subspace $\mathcal{V}$ of smooth real one-form solutions $\beta$ of $d{\star }_{g}d\beta =0$ for which both pairings are defined, call the exact gauge direction $d\chi$ degenerate on $\mathcal{V}$ when its pairing shift vanishes for every $\beta \in \mathcal{V}$. Thus degeneracy on $\mathcal{V}$ holds exactly when this limit vanishes for every $\beta \in \mathcal{V}$.

Fixed-background Maxwell presymplectic current

Lorentzian exterior calculus and Hodge star

Stokes theorem for smooth singular chains

The calculation treats an exact gauge variation with a globally defined real parameter $\chi$. A circle-valued large gauge transformation need not admit such a global phase; this identity alone therefore does not declare every flat connection difference to be presymplectically degenerate.