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Given (not-necessarily linear) fibre bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ with structure sheaf denoted $\mathcal{O}_M$, a (not-necessarily-linear) differential operator taking sections of $E$ to sections of $F$ can be defined as a bundle map $D: J^\infty E \to F$ covering the identity on $M$, where $J^\infty E$ is the limit of finite-dimensional jet bundles with some appropriate infinite-dimensional smooth manifold structure. A finite-order differential operator is one which factors through some finite-order jet bundle.

If $E$ and $F$ are indeed vector bundles, we can characterize linear differential operators by requiring that the bundle map $J^\infty E \to F$ is linear.

However, there is an alternative inductive definition of linear differential operators, which seems to work for any space (or topos) $M$ equipped with a sheaf of commutative algebras $\mathcal{O}_M$ over a sheaf of commutative rings $K$ .

For $\mathcal{O}_M$-modules $E$ and $F$, define the sheaf of $0$th order linear differential operators to be $\mathrm{Diff}^0(E,F)=\mathcal{Hom}_{\mathcal{O}_M}(E,F)$. The sheaf $\mathrm{Diff}^{k+1}(E,F)$ of differential operators of order at most $k+1$ is the subsheaf of $\mathcal{Hom}_K(E,F)$ consisting of sections $D$ whose commutators $[D,f]$ with every section $f$ of $\mathcal{O}_M$ lie in $\mathrm{Diff}^{k}(E,F)$. Then the sheaf of differential operators $\mathrm{Diff}(E,F)$ is defined to be the colimit of these modules.

Does the notion of a differential operator of infinite order in this second definition agree with that given via jet bundles?

It seems to me that this is not the case, since every differential operator in $\mathrm{Diff}(E,F)$ will be locally of finite order, whereas a differential operator of infinite order defined as a map out of $J^\infty(E) \to F$ might not be of finite order locally (as far as I can tell; I don't know any infinite-dimensional manifold theory). For instance, on a compact smooth manifold, by construction $\mathrm{Diff}(E,F)$ should just consist of finite-dimensional operators. If indeed my suspicion is right, that leads me to my second question:

Are there any interesting infinite-order differential operators, especially those that are not locally of finite order?

I can think of one natural infinite-order differential operator which is interesting from a more abstract point of view: the comultiplication $J^\infty \to J^\infty J^\infty$ of the jet comonad. I suspect there might be some infinite-order differential operators of more practical interest to physicists as well, though perhaps these might involve dealing with infinite-dimensional manifolds.

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    $\begingroup$ Is the shift (by 1) operator an interesting infinite-order differential operator? $\endgroup$ – M.G. Jul 5 '17 at 14:59
  • $\begingroup$ @July: Differential operators are local, whereas the shift operator is clearly nonlocal. $\endgroup$ – Dmitri Pavlov Jul 5 '17 at 16:00
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    $\begingroup$ In the linear case all differential operators have locally finite order, this is precisely the content of Peetre's theorem. Nonlinear analogs of Peetre's theorem have been developed since then, and an answer to the question can probably be found in one of these papers, though I don't have time to look. The Wikipedia page en.wikipedia.org/wiki/Peetre_theorem has some references. $\endgroup$ – Dmitri Pavlov Jul 5 '17 at 16:02
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    $\begingroup$ @DmitriPavlov: I guess my question is if $\mathrm{exp}(d/dx)$ is considered a differential operator of infinite order. $\endgroup$ – M.G. Jul 5 '17 at 16:05
  • $\begingroup$ Just a remark about the comonad map $\Delta\colon J^\infty \to J^\infty J^\infty$. The reason that it is of "infinite order" is that its codomain is itself infinite dimensional. When looking at any of the projections $J^\infty \xrightarrow{\Delta} J^\infty J^\infty \to J^k J^l$ (or "components" of $\Delta$), it is necessarily of locally finite order, as required by the conclusion of Peetre's theorem. $\endgroup$ – Igor Khavkine Jul 6 '17 at 12:02
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I think your answer to your first question is correct. If I remember correctly, the definition of the sheaf of differential operators $\mathrm{Diff}(E,F)$ is exactly the one usually used in the theory of D-modules and I think that the only operators of infinite order arise from global sections.

To your second question -- big natural class of interesting examples comes from splitting operators. The story starts with unfolding of differential equations. This is a process of adding new variables for derivatives of fields. For overdetermined systems of PDEs this process ends after finite number of steps and one obtains a vector bundle of finite rank. But for other systems this leads to infinite number of new variables and then the mapping "fields $\to$ unfolded form" is a differential operator of infinite order. This mapping is called splitting operator. Think of it as a formal power series that respects the symmetries of your original problem. In favourable cases the systems of PDEs are reinterpreted as parallel sections of a connection on infinite-dimensional bundle (basically on $J^\infty E$). These parallel sections appear disguised as "covariant constancy equation" in the classification of conformally invariant equations in Unfolded form of conformal equations in M dimensions and $\mathfrak{o}(M+2)$-modules which deals only with the flat space. I've managed to extend some of these cases to general conformal manifolds in Yamabe operator via BGG sequences.

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    $\begingroup$ Thanks for your reply! Could you explain a bit what you mean when you say that connections are infinite order differential operators? I thought that connections are always first order operators. $\endgroup$ – ಠ_ಠ Jul 6 '17 at 5:01
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    $\begingroup$ Oops, sorry. I don't know what I was thinking yesterday. I've clarified my answer. The short version is that connections on infinite-dimensional bundles can be intimately tied with PDEs on finite-dimensional bundles and that this correspondence is given by differential operators of infinite order. $\endgroup$ – Vít Tuček Jul 6 '17 at 8:50
  • $\begingroup$ Ah OK, that seems reasonable. I'm also working in parabolic geometry at the moment, and trying to learn about BGG sequences. Do you happen to know of a good place to learn this stuff? Unfortunately it's not really covered in the big text by Cap & Slovak. At the moment I'm reading this paper from 2000 by Cap, Slovak, and Soucek, but it's a bit terse in some parts. $\endgroup$ – ಠ_ಠ Jul 6 '17 at 9:01
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    $\begingroup$ The place to start is article by Calderbank and Diemer arxiv.org/abs/math/0001158 $\endgroup$ – Vít Tuček Jul 6 '17 at 9:13
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    $\begingroup$ What exactly are you working on if I may ask? $\endgroup$ – Vít Tuček Jul 6 '17 at 9:38

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