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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeOct 16th 2012
• (edited Oct 16th 2012)

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 17th 2012
• (edited Oct 17th 2012)

I made explicit at variational calculus the “mapping space with non-varying boundary configurations”, on which the variational caclulus is defined, as the pullback

$\array{ [\Sigma,X]_{\partial \Sigma} &\to& \flat [\partial \Sigma, X] \\ \downarrow && \downarrow \\ [\Sigma,X] &\stackrel{(-)|_{\partial \Sigma}}{\to}& [\partial \Sigma,X] } \,,$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 17th 2012
• (edited Oct 17th 2012)

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…