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at variational calculus I have started a section In terms of smooth spaces where I discuss a bit how for
$S \colon [\Sigma, X]_{\partial \Sigma} \to \mathbb{R}$a smooth “functional”, namely a smooth map of smooth spaces, its “functional derivative” is simply the plain de Rham differential of smooth functions on smooth spaces
$\mathbf{d}S \colon [\Sigma, X]_{\partial \Sigma} \stackrel{S}{\to} \mathbb{R} \stackrel{\mathbf{d}}{\to} \Omega^1 \,.$The notation can still be optimized. But I am running out of energy now. Has been a long day.
I made explicit at variational calculus the “mapping space with non-varying boundary configurations”, on which the variational caclulus is defined, as the pullback
$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,$Hm, I haven’t thought about this enough. This means for instance that for every $G$-principal bundle on the boundary configuration whose pullback to the bulk configuration space is equipped with a trivialization, there is a canonical flat $\mathfrak{g}$-valued differential form
$\omega \colon [\Sigma, X]_{\partial \Sigma} \to \flat_{dR}\mathbf{B}G$on the “configuration space with non-varying boundary configurations”, induced from the commuting diagram
$\array{ \flat [\partial \Sigma, X] &\stackrel{\flat \mathbf{c}}{\to}& \flat \mathbf{B}G \\ \downarrow && \downarrow \\ [\partial \Sigma,X] &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G \\ \uparrow && \uparrow \\ [\Sigma, X] &\to& * }$under the equivalence $\Omega^1_{flat}(-,\mathfrak{g}) \simeq \flat_{dR}\mathbf{B}G \coloneqq * \times_{\mathbf{B}G} \flat \mathbf{B}G$.
Hmm…
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