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added some content to variational bicomplex
added the definition (elegant version); expanded the list of commented references
I have been touching variational bicomplex and jet bundle, adding some more references and some further but scattered statements. Needs more work.
What is your “jet algebra functor” in down to Earth terms ? Do you have only $\infty$-jets in that language of adjoints or you have a variant of that abstract formulation for every level ? The questions of locality raise once you allow infinity jets for laws of motion. Is there a legitimate Euler-Lagrange formulation then ?
Just in passing: there are lecture notes ’Homological methods in mathematical physics’ from the Russian school available here.
Zoran, as it says at the beginning of the paragraph here, this is the construction of Beilinson-Drinfeld, just written in ideosyncratic notation
In the context of D-schemes this is (BeilinsonDrinfeld, 2.3.2). The abstract formulation as used here appears in (Lurie, prop. 0.9). See also (Paugam, section 2.3) for a review. There this is expressed dually in terms of algebras in D-modules. We indicate how the translation works
This does not seem to answer my questions. What about finite level, $k$-jets ? Is there a legitimate Euler-Lagrange formulation for infinite jets ?
This does not seem to answer my questions. W
Sorry, then I didn’t understand what you were asking. And maybe I still don’t. You write:
Is there a legitimate Euler-Lagrange formulation for infinite jets?
Not sure what you mean. Standard modern Euler-Lagrange variational theory works on the infinite jet bundle which, being a projective limit, is such that any function on it depends only on jets of finite order. See for instance here.
Right, this answers the second question!
What about the first question – what about the finite $k$-jets? (Can one get fixed finite level jet algebra functor as adjoint functor in the generality you had the infinite one ?)
Ah, now I see what you mean. Sorry for being slow.
I’d need to think about this, but what comes to mind is that the de Rham stack operation aka infinitesimal shape modality really comes as a filtration
$X \to \cdots \to \Im_{(3)}X \to \Im_{(2)}X \to \Im_{(1)}X \to \Im X \to ʃ X$where $\Im_{(k)}$ quotients out only higher order infinitesimals, retaining infinitesimals of order $\leq k$.
Presumeably the rank-$k$ jet bundle operation is the direct image of base change along $X \to \Im_{(k)}X$ in direct analogy to how the full jet bundle operation is the direct image base change along $X \to \Im X$. But I haven’t really thought this through. Good point.
Thanks (the filtration is on the modality level, wow), we can talk about it soon.
Yes, this simply comes from the filtration of morphisms of sites where you embed formal duals of rings with elements that are nilpotent of degree $k$ into formal duals of rings with elements that are nilpotent of degree $(k+1)$. All these inclusions are coreflective and left Kan extension turns this into adjoint quadruples between the toposes over these sites. The middle ones of the induced adjoint triples of modalities are those $\Im_{(k)}$.
I know how the filtration goes in the classical ring/scheme case (as well as for formal functions on analytic submanifold case; still, thanks for the details on middle adjunction); but I thought you were saying that there was an abstract nonsense derivation of filtration directly for any abstract infinitesimal shape modality from first principles.
(As analogy, at the level of Abelian categories (of qcoh sheaves on nc spaces) there are some properties of subcategories and adjunctions characterizing infinitesimal neighborhoods at finite and infinite level in Rosenberg’s work; some of the things worked for the jets there, though there is lack of certain representability properties).
I thought you were saying that there was an abstract nonsense derivation of filtration directly for any abstract infinitesimal shape modality from first principles.
Ah, no, this I wasn’t saying, and this isn’t true.
Ok, still interesting enough for thoughts :)
If I understood you correctly then, the $k$-jet and $\infty$-jet algebra functors with all the adjointness properties are then on the same footing from the abstract points of view (and then in the classical de Rham case there is an additional embedding of one into another).
Yes.
By the way, now that I thought of it: the $k$-jet bundle functor is base change reflection along $p \colon \Im X_{(k)}\to \Im X$.
The way this works is as follows: Given a bundle $E \to X$ regarded as an object in $\mathbf{H}_{/X} \to \mathbf{H}_{/\Im_{(k)}X}$ then to see what its image is under $p^\ast p_\ast$ for
$\mathbf{H}_{/\Im_{(k)}(X)} \stackrel{\overset{p_!}{\longrightarrow}}{\stackrel{\overset{p^\ast}{\longleftarrow}}{\underset{p_\ast}{\longrightarrow}}} \mathbf{H}_{/\Im(X)}$we test what its local sections are by mapping into it from $U \hookrightarrow X$ regarded as an object in $\mathbf{H}_{/X}\to \mathbf{H}_{\Im_{(k)}X}$.
Then by adjunction such maps $U \to p^\ast p_\ast E$ over $\Im_{(k)}X$ are maps from $p^\ast p_!U \to E$ over $\Im_{(k)}X$. But on the left this is the order-$k$ infinitesimal disk bundle $T_{(k)}U$, by the discussion at differential cohesion here.
But sections of $E$ from order $k$-infinitesimal disks aroun points of $U$ are precisely order-$k$ jets of $E$.
Thanks, looks right to me.
I have expanded a little on this in section 5.3.10 of dcct (pdf).
Thanks for the tip.
Some references seem missing in 5.5.7 (question marks).
Thanks. That’s a pointer to a section which exists, but not yet in the big file. I think for the moment I’ll just suppress the pointer to it. Thanks for the mentioning it.
I fixed an inaccurate statement about the symplectic form being degenerate on symmetries (that only holds for gauge symmetries, which had not been defined at that point). I also a more precise notion of a symmetry as an evolutionary vector field on the jet bundle (rather than just a vertical one).
Thanks, Igor. My fault, I suppose.
I also a more precise notion of a symmetry as an evolutionary vector field on the jet bundle (rather than just a vertical one).
I’m a bit confused about this point - maybe it’s just a question of terminology. If $v$ is an evolutionary vector field, i.e., a vertical generalized vector field on $E$, shouldn’t it then be $\operatorname{pr}v(L) = 0 \mod im d$, i.e., the prolongation of $v$ be used here? Or does your definition of an evolutionary vector field above describe what is in other places called “prolongation of an evolutionary vector field”? (Probably it does, since you define it as a vector field on jet space.)
Essentially yes. Since the vector field is already defined on $J^\infty E$, there is no need to prolong it. By a standard theorem, any such vector field is the prolongation of a “generalized vector field” on $E$, sometimes known as the “generator” (also “characteristic”) of the evolutionary vector field. But that way one first has to define what a “generalized vector field” is.
Thanks for the clarification, that’s what I thought. In my lecture notes I also tried to avoid introducing “generalized vector fields”, so I ended up defining an evolutionary vector field as a map $X: J^{\infty}E \to VE$, where $\nu: VE \to E$ is the vertical tangent bundle of $E$, such that $\nu \circ X = \pi_{\infty,0}$. This was a rather quick thought, but it should work like this.
Following your construction and defining an evolutionary vector field $Y$ as a particular vertical vector field on $J^{\infty}E$, my definition should simply be the characteristic and be given by $Xf = Y(f \circ \pi_{\infty,0})$ for all $f \in C^{\infty}(E)$.
have added to variational bicomplex the characterizations of the horizontal differential.
have created entries for Euler-Lagrange form, source form, Lepage form, and a stub for variational sequence.
Added reference to the longer version of a paper
Irina Kogan, Peter Olver, Invariant Euler-Lagrange Equations and the Invariant Variational Bicomplex, pdf
“This result will be based on combining two powerful ideas in the modern, geometric approach to differential equations and the variational calculus. The first is the variational bicomplex…The second ingredient in our method is Cartan’s moving frame theory”
It gives a list of “interesting further research directions” on p. 49.
Thanks! This one I had not seen yet.
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