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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:[Σ,X]∂Σ→ℝa 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
dS:[Σ,X]∂ΣS→ℝd→Ω1.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
[Σ,X]∂Σ→♭[∂Σ,X]↓↓[Σ,X](−)|∂Σ→[∂Σ,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 𝔤-valued differential form
ω:[Σ,X]∂Σ→♭dRBGon the “configuration space with non-varying boundary configurations”, induced from the commuting diagram
♭[∂Σ,X]♭c→♭BG↓↓[∂Σ,X]c→BG↑↑[Σ,X]→*under the equivalence Ω1flat(−,𝔤)≃♭dRBG≔*×BG♭BG.
Hmm…
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