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Earlier we had been talking about elementary axiomatization in differential cohesion of the concept of order-$k$ integrable $G$-structures on some $V$-manifold $X$ (here). The resulting concept ought to naturally be a coalgebra for the Jet comonad, equivalently a bundle over $\Im X$, exhibiting the order-$k$ integrability condition manifestly as a differential equation. It’s roughly clear how this should proceed,but I don’t have a good universal construction yet.
So more in detail, given a group object $V$ (the local model space) and setting $GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V_e)$ for $\mathbb{D}^V_e \coloneqq V \underset{\Im V}{\times} \{e\}$ the infinitesimal disk around the neutral element (all for some fixed order of infinitesimality, i.e. $\Im = \Im_{(k)}$ here) and finally for $G\mathbf{Struc} \colon G \longrightarrow GL(V)$ some group homomorphism, then a $G$-structure
$\array{ X && \stackrel{\mathbf{g}}{\longrightarrow} && \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) }$being infinitesimally integrable is an equivalence between the restrictions of $\mathbf{g}$ and of $\mathbf{g}_{li}$ along all infinitesimal disk inclusions
$\array{ && \mathbb{D} \\ && \downarrow \\ && U \\ & \swarrow && \searrow \\ V && && X }$(for $U$ a $V$-cover which exhibits $X$ as a $V$-manifold), where
$\array{ V &&\stackrel{\mathbf{g}_{li}}{\longrightarrow} && \mathbf{B}G \\ & \searrow && \swarrow \\ && \mathbf{B}GL(V) }$is the canonical $G$-structure on $V$ induced from the framing of $V$ given by left translation, using that $V$ is a group. (For the present purpose it only matters that some such $\mathbf{g}_{li}$ exists, the question could be phrased with any other fixed background $G$-structure instead).
Now the question is, given such data, how to universally construct a bundle map
$Int \longrightarrow Jet( X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G )$such that infinitesimal integrability of a $G$-structure $\mathbf{c} \in \Gamma_X(X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G )$ is a factorization of the jet prolongation $Jet(\mathbf{c})$ through this map.
In words this is clear: a section of $Int$ over a point is to be a section of $X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G$ over an infinitesimal disk together with an equivalence of the corresponding $G$-structure over that infinitesimal disk with the given background $G$-structure.
Now how to formulate this internally by a diagrammatic universal construction?
I guess that by a “Jet-coalgebra” you mean what under usual circumstances is simply called a differential equation. Now, I don’t know much about $G$-structures. But I do know that back in the ’60s and ’70s, Spencer and co. have spent some effort studying precisely the equation you call $Int$ and the conditions under which it has at least formal and hopefully local solutions. I believe they called it the problem of the integrability of Lie structures. In fact, I believe that the impetus for Spencer to develop (or inspire other people to do so) his theory of over-determined PDEs was precisely to build the tools to attack this problem. Perhaps the following references might be helpful.
Not that I understand any of this stuff, but It seems that the actual equations $Int$ were constructed by Guillemin in the article The integrability problem for $G$-structures Trans. Amer. Math. Soc. 116 (1965), 544-560. From what I understand, these equations are natural with respect to the manifold $X$ and the group $G$, so that they are likely good candidates for a universal construction.
Also, some version of the state of the art on this topic was recorded in the (possibly unreadable) book by Kumpera & Spencer, LIE EQUATIONS VOLUME I: GENERAL THEORY (PUP, 1972).
Aha, there’s also a briefer summary of the Kumpera-Spencer monograph published as Systems of linear PDEs and deformations of pseudogroup structures (Univ. Montreal, 1974).
Hey Igor,
right, I know the traditional discussion of integrability of G-structures. What I am after here is the diagrammatic reformulation that will work also in more general stacky contexts where the traditional discussion fails.
There is the neat diagrammatic axiomatization of infinitesimally integrable $G$-stuctures on $V$-manifolds that I had recalled above in #1, where both $G$ and $V$ may be arbitrary smooth (or otherwise geometric) $\infty$-groups, hence where the “manifold” may be any “$V$-étale $\infty$-stack” or “higher derived $V$-scheme” or whatever one likes to call it.
While this already works well, I am thinking it should be beneficial to find an equivalent diagrammatic construction that spits out more manifestly an object in the slice over $\Im X$, hence equivalently a Jet comonad object.
I suppose what I should be doing is to look under the adjunction
$\frac{X \stackrel{}{ \longrightarrow} Jet(E)}{T_{inf} X \stackrel{\tilde {\mathbf{g}}}{\longrightarrow} E}$between sections of the jet bundle of some bundle $E$ over $X$ and bundle maps from the infinitesimal disk bundle of $X$ at those $\tilde {\mathbf{c}}$ that are equipped with a homotopy filling the diagram
$\array{ T_{inf} U & \longleftarrow & \mathbb{D}^V \times \flat U \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\tilde{\mathbf{g}}_{li}}} \\ T_{inf} X &\underset{\widetilde{j^\infty \mathbf{g}}}{\longrightarrow}& E }$where with $E = X \underset{\mathbf{B}GL(V)}{\times}\mathbf{B}G$ then $\tilde {\mathbf{g}}$ and $\tilde {\mathbf{g}}_{li}$ are the adjuncts of the corresponding maps in #1.
Then I am supposed to build the moduli object for this data, under adjoining back to sections of the jet bundle.
Here is a how it should work.
Write $\flat^{rel} X\coloneqq \flat X \underset{\Im X}{\times} X$ for the relative flat modality acting on objects $X$. This produces the collection of the infinitesimal disks around all global points of $X$. Assume that this has a right adjoint $\sharp^{rel}$, which is the case in the standard models.
Then observe that for $X$ a $V$-manifold exhibited by a $V$-cover
$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$(meaning that both maps here are local diffeomorphism/formally étale in that they are $\Im$-modal as objects in the slice, and in addition that the right map is 1-epi) we get a pasting diagram of the form
$\array{ && \flat^{rel} U \\ & &\downarrow& \searrow \\ \mathbf{B}G &\stackrel{\mathbf{g}_{li}}{\longleftarrow}& V && X \\ & \searrow & \downarrow^{\mathrlap{\tau_V}} & \swarrow_{\mathrlap{\tau_X}} \\ && \mathbf{B}GL(V) }$(where the right part is by the general properties of frame bundles of $V$-manifolds and where the left part is the $G$-structure on $V$ induced by the left-invariant framing of $V$) and hence a canonical morphism
$\flat^{rel} U \longrightarrow X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G$hence equivalently a map
$U \longrightarrow \sharp^{rel}\left(X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G\right) \,.$Hitting that map itself with $\sharp^{rel}$ and using idempotency of the modality finally produces a canonical morphism of the form
$\sharp^{rel}U \longrightarrow \sharp^{rel}\left(X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G\right) \,.$In words, this map sends every infinitesimal disk in the given $V$-cover to the $G$-structure on the restriction of the frame bundle of $X$ to the image of that infinitesimal disk in $X$ which is induced by identifying the infinitesimal disk with that in $V$ and using the left invariant $G$-structure there.
This is exactly the infinitesimal condition on $G$-structures which is to be exhibited as a differential equation to be imposed on the space of all $G$-structures on $X$.
So as a bundle over $\Im X$ that differential equation is the vertical composite in
$\array{ \mathbf{Int}G\mathbf{Struc}(X) &\longrightarrow& \sharp^{rel} U \\ \downarrow &(pb)& \downarrow \\ X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G &\longrightarrow& \sharp^{rel}\left(X \underset{\mathbf{B}GL(V)}{\times} \mathbf{B}G\right) \\ \downarrow \\ \Im X } \,.$Urs, how familiar are you with the article by Guillemin that I quoted earlier? I think he does give the differential equation that you are asking for, though of course not translated into the diagrammatic form that you might like.
Though, I’m out of my depth on this subject, let me give it a shot at makeing a short summary of the relevant result. I’ll do that in the next message.
I think he does give the differential equation that you are asking for, though of course not translated into the diagrammatic form that you might like.
Sure he gives the differential equations for order-$k$ integrability of traditional $G$-structures . But here I am not asking for that differential equation, I am taking this traditional concept as the starting point. What I am asking for is its synthetic formulation that makes it generalize.
What the construction in #5 gives is, I think, the accurate synthetic formulation of Guillemin’s definition on top of his page 550, first sentence.
So, Urs, the following might be well known to you, if you are well familiar with Guillemin’s paper. But it was worth for me to spell this out, at least to get a handle on what’s going on here. I still think that a differential equation defining local flatness is already given in that paper. It’s just a matter of extracting it and making it sufficiently explicit.
[Hmm, what I typed seems to be too long for a single comment, so I’m splitting it into parts.]
Definition of Spencer Cohomologies. Let $G\subset GL(V)$ be a sub-group and $\mathfrak{g} \subset \mathfrak{gl}(V)$ be the corresponding Lie sub-algebra, sitting respectively in the group $GL(V)$ of invertible linear operators and in the Lie algebra $\mathfrak{gl}(V)$ of linear operators on an $n$-dimensional vector space $V$. Now I need to define some vector spaces, the Spencer cohomology spaces $H^{i,p}(\mathfrak{g})$, canonically constructed from this inclusion. Let $C^{i,p}(V) = S^i(V^*) \otimes \Lambda^p(V^*)$, with the interpretation of differential $p$-forms on $V$ with homogeneous polynomial coefficients of degree $i$. Let $\delta\colon C^{i,p}(V) \to C^{i-1,p+1}(V)$ denote the usual de Rham differential acting on such forms (note that the the de Rham differential decreases the degree of the polynomial coefficients by $1$, preserving homogeneity, and increases the form degree by $1$, as indicated). Next, let me define $C^{i,p}(V;V) = V \otimes C^{i,p}(V)$ and extend the polynomial de Rham differential to $\delta \colon C^{i,p}(V;V) \to C^{i-1,p+1}(V;V)$, by acting trivially on the $V$ factor. Next, I define a bunch of subspaces $\mathfrak{g}^i \subset C^{i+1,0}(V;V) \cong V \otimes S^i(V^*)$, the $i$-symbol spaces of $\mathfrak{g}$. First, let $\mathfrak{g}^{-1} = V$, then $\mathfrak{g}^0 = \mathfrak{g} \subset \mathfrak{gl}(V) \cong V\otimes V^*$, and $\mathfrak{g}^i = \mathfrak{g} \otimes S^{i}(V^*) \cap V \otimes S^{i+1}(V^*)$ for all $i > 1$. Using the symbol spaces, let . These inclusions happend to respect the polynomial de Rham differential, $\delta\colon C^{i,p}(V,\mathfrak{g}) \to C^{i-1,p+1}(V,\mathfrak{g})$. Hence I have defined a bunch of cochain complexes:
$\array{ C^{0,0}(V;\mathfrak{g}) && C^{1,0}(V;\mathfrak{g}) && C^{2,0}(V;\mathfrak{g}) && C^{3,0}(V;\mathfrak{g}) && C^{4,0}(V;\mathfrak{g}) && \cdots \\ & \swarrow && \swarrow && \swarrow && \swarrow && \swarrow \\ C^{0,1}(V;\mathfrak{g}) && C^{1,1}(V;\mathfrak{g}) && C^{2,1}(V;\mathfrak{g}) && C^{3,1}(V;\mathfrak{g}) && C^{4,1}(V;\mathfrak{g}) && \cdots \\ & \swarrow && \swarrow && \swarrow && \swarrow && \swarrow \\ C^{0,2}(V;\mathfrak{g}) && C^{1,2}(V;\mathfrak{g}) && C^{2,2}(V;\mathfrak{g}) && C^{3,2}(V;\mathfrak{g}) && C^{4,2}(V;\mathfrak{g}) && \cdots \\ \vdots && \vdots && \vdots && \vdots && \vdots \\ & \swarrow && \swarrow && \swarrow && \swarrow && \swarrow \\ C^{0,n-1}(V;\mathfrak{g}) && C^{1,n-1}(V;\mathfrak{g}) && C^{2,n-1}(V;\mathfrak{g}) && C^{3,n-1}(V;\mathfrak{g}) && C^{4,n-1}(V;\mathfrak{g}) && \cdots \\ & \swarrow && \swarrow && \swarrow && \swarrow && \swarrow \\ C^{0,n}(V;\mathfrak{g}) && C^{1,n}(V;\mathfrak{g}) && C^{2,n}(V;\mathfrak{g}) && C^{3,n}(V;\mathfrak{g}) && C^{4,n}(V;\mathfrak{g}) && \cdots }$Denote the corresponding cohomology spaces by $H^{i,p}(V;\mathfrak{g}) = H(C^{i,p}(V;\mathfrak{g}), \delta)$. These are precisely finally the sought after Spencer cohomology spaces. Terminological note: Another name for the above complexes is (simple, or symbol or of the first kind) Spencer complexes. Another name for $\delta$ is the (simple, or symbol or of the first kind) Spencer differential. The important point is that Spencer cohomology spaces are functorially associated to the inclusion $\mathfrak{g} \subset \mathfrak{gl}(V)$.
Bundle of $G$-structure preserving jets. I’m not going to define in detail how a $G$-structure on an $n$-dimensional manifold $M$ is defined, nor what is the canonical flat $G$-structure on $V$, given $G \subset GL(V)$, nor what $k$-th order structure preserving jets are. (The main reason I’m not doing that is because I’m still fuzzy on some of the details.) I will just take for granted that we have a manifold $M$ with a $G$ structure on it and that for each integer $k\ge 0$, there is defined a subset $E^k \subset J^{k+1}_0(V,M)$, consisting of all jets at $0\in V$ of maps from $V$ to $M$ that are $k$-th order $G$-structure preserving at $0\in V$, with respect to the canonical flat $G$-structure on $V$ and the given $G$-structure on $M$. The points of $E^k$ look like $(0, x, jet\,data)$, where $0\in V$ and $x\in M$ are respectively the source and target of the jet. Let me also assume that everything is sufficiently regular for the projection $E^k \to M$, mapping $(0,x,jet\,data) \mapsto x$, defines a smooth bundle over $M$. Apparently, the inclusion $E^0 \subset J^1_0(V,M)$ is sufficient to identify the $G$ structure itself. Anyway, the point is that the bundle $E^k \to M$ is functorially associated to a given $G$-structure on $M$. Now, unless I’m mistaken, I believe that $E^k \to M$ can also be seen as a sub-bundle of $J^{k+1}(M,V) \to M$, whose fiber at $x\in M$ consist of jets of (locally defined in a neighborhood of $x$) of maps $M\to V$, sending $x\to 0$ and $k$-th order structure preserving.
$k$-th structure tensor. At the bottom of p.552, Guillemin defines the $k$-th structure tensor of the given $G$-structure on $M$ as a map $\mathbf{c}^k\colon E^k \to H^{k,2}(\mathfrak{g})$. And apparently this map $\mathbf{c}^k$ is also functorially associated to a given $G$-structure on $M$. Then, immediately after, at the top of p.553, there is a Corollary, that states: (roughly) the $G$-structure on $M$ is locally flat to order $k$ at every $x\in M$ iff $\mathbf{c}^k = 0$ everywhere on $E^k$. Let’s think about that for a bit. If the $G$-structure on $M$ were really locally flat in a neighborhood of some $x\in M$, then the locally defined map $M \to V$ establishing this equivalence would be $k$-th order structure preserving for any $k$. So $\mathbf{c}^k$ should vanish over a neighborhood of $x\in M$. On the other hand, if $\mathbf{c}^k$ does not vanish over any point $x\in M$, then the $G$-structure cannot be flat in any neighborhood of that point. Thus, we can conclude that the joint condition $\mathbf{c} = 0$, where $\mathbf{c} = (\mathbf{c}^0,\mathbf{c}^1,\mathbf{c}^2,\ldots)$, are necessary for flatness. (Whether this joint condition is sufficient for flatness apparently is a question of analysis and not just geometry. In the analytic category, yes it is sufficient. In the $C^\infty$ category, the question is much more subtle and I don’t know the full details.) This may seem like an infinite number of conditions, but one can actually show that $H^{k,2}(\mathfrak{g}) = 0$ for any $k \ge K$, for some finite $K$ that depends on the inclusion $\mathfrak{g} \subset \mathfrak{gl}(V)$. Thus, we need only consider the finite number of conditions $\mathbf{c} = (\mathbf{c}^0, \mathbf{c}^1, \ldots, \mathbf{c}^K)$.
Now, what is intuitively clear to me, is that $\mathbf{c}$ must be expressible as a differential operator of some kind. But I’m afraid that I can’t make that statement precise at the moment. However, once that statement can be formulated precisely, the differential operator $\mathbf{c}$ is functorially associated to a given $G$-structure on $M$ and the condition $\mathbf{c} = 0$ becomes the differential equation that I think you are looking for.
$G$-structures of finite type. Now, how high is this number $K$ so that $H^{k,2}(\mathfrak{g}) = 0$ for any $k\ge K$? I guess in general that’s a complicated question and requires a careful analysis of the Spencer complex. However, before even looking at the Spencer complex, one can notice that for some $\mathfrak{g} \subset \mathfrak{gl}(V)$, the symbol spaces $\mathfrak{g}^i = 0$ all vanish for sufficietly high $i$. Suppose that $\mathfrak{g}^{K-2} \ne 0$, but $\mathfrak{g}^{i} = 0$ for all $i \ge K-1$, then the $G$-structure is of type $K-1$. This automatically implies $H^{k,2}(\mathfrak{g}) = 0$ for al $k\ge K$. Thus, for $G$-structures of type $K-1$, the differential equation for local flatness is $\mathbf{c} = (\mathbf{c}^0, \mathbf{c}^1, \ldots, \mathbf{c}^K)$. Examples are discussed in Guillemin’s Sec.6.
Hey Igor,
good that you are energetic about typing up this material, that’s what the $n$Lab is for! You should copy that stuff over to the entry Spencer cohomology.
Regarding your boldface:
the condition $\mathbf{c}= 0$ becomes the differential equation that I think you are looking for.
Allow me to say again that I was not looking for that differential equation, but for a particular way of formalizing it by elementary means in a differentially cohesive $\infty$-topos. I already knew one way how to do it, recalled in #1, and what I was looking for here was another way of doing it, which I found in #5.
If you have the energy, you might enjoy looking at what is happening in #1 and #5, because it is pretty neat, if I may say that. Namely the thing is that once we are in a context of synthetic differential higher geometry where we may literally speak of order-$k$ infinitesimal neighbourhoods of points, then the condition that you wrote as $\mathbf{c} = 0$ has an immediate formalization by just taking the naive statement and interpreting it synthetically: for $X$ a manifold locally modeled on $V$ and for $V$ equipped with the “flat” $G$-structure, then order-$k$ infinitesimal integrability of a $G$-structure on $X$ means nothing but that after restriction of that $G$-structure along the inclusion $\mathbb{D}^X_x(k) \longrightarrow X$ of the order-$k$ infinitesimal disk $\mathbb{D}^X_x(k)$, around any $x \in X$, this restriction of the $G$-structure is equipped with an identification with the flat $G$-structure on $V$ restricted to $\mathbb{D}^V_e(e)$, relative to the identification of $\mathbb{D}^X_x(k)$ with $\mathbb{D}^V_e(k)$ that is given by $X$ being a $V$-manifold.
This is easily said synthetically, as recalled in #1. Now what I was after here, as I tried to say, is a formulation of this that exhibits the space of solutions of these conditions on $G$ structures as a single bundle (of stacks over the site of formal manifolds) over the de Rham space $\Im X$ of $X$, such that the space of bundle morphisms from $X \to \Im X$ to that bundle is equivalently the space of order-$k$ infinitesimally integrable $G$-structures on $X$. This, too, has an immediate synthetic formulation, though somehow I needed to think about it for a bit longer than might have been strictly necessary, this is what I gave in #5.
In any case, as said, if you are energetic about recording and typing up material on the traditional formulation of integrable $G$-structures, it would be good if, instead of writing this here in the $n$Forum thread where it will be lost, you’d move that material into the respective $n$Lab entries, where it will be more usefully archived.
Perhaps I am failing to understand the (possibly subtle) difference between what you are looking for (a “universal construction”) and what I understand the structure tensor $\mathbf{c}$ to be (a “natural differential operator”). After all, the fact that it is natural (or functorial) just means that it can be written down as a composition of a bunch of other natural maps, or that there exists a diagram of the form
$\array{ Jet(???) && \stackrel{\mathbf{c}}{\to} && H^{\bullet,2}(\mathfrak{g}) \\ & \searrow && \swarrow \\ && ({\cdots}) }$where $({\cdots})$ stands for some larger diagram consisting of more naturally constructed objects and natural maps between them.
Now, I don’t think I can really parse all that you’ve written in #1 and #5, but I don’t see anything like the structure tensor $\mathbf{c}$ (you do use the symbol $\mathbf{c}$, but I think it stands for something else).
I’m actually still debating whether understanding $G$-structures in a deeper way is useful to me. I might dedicate some more energy to that, but don’t know how much more. Anyway, I’m about to go offline for the rest of the day.
The structure tensor is, as you recalled, an explicit analytic realization of the differential condition “flat to order $k$ around a point $x \in X$”, which makes sense as long as $X$ is a smooth manifold and $G$ is a Lie group. What is discussed in #1 and #5 is an explicit but synthetic realization of the condition “flat to order $k$ around each point” which makes sense more generally. Namely the solution to this condition is given by exhibiting the homotopy in
$\array{ && \mathbb{D}^V(k) \\ & \swarrow && \searrow \\ V && \swArrow_{\mathrlap{\simeq}} && X \\ & \searrow && \swarrow \\ && \mathbf{B}G }$where the top morphisms include the abstract order-$k$ infinitesimal disk, and where the bottom morphisms exhibt the choice of $G$-structure, the whole diagram being over $\mathbf{B}GL(V)$ by the classifying maps of the frame bundles of $V$ and of $X$, respectively, which I am not showing here.
Given this situation in the case that $V = \mathbb{R}^n$, that $X$ is an $n$-dimensional manifold and that $G$ is a Lie group, then this is straightforwardly equivalent to Guillemin’s definition of order-$k$ flatness/itegrability on top of his page 550. What you recalled is his explicit analytic equivalent rewriting of this condition in terms of the vanishing of certain tensors.
While this is clearly useful, it has the disadvantage that it is practically intractable to see what replaces these explicit analytic tensor expressions when we pass to the more general situation where $V$ is allowed to be a higher group (one example appearing in practice is that $V$ is an extended super Minkowski spacetime), that $X$ is an orbifold or more generally an higher étale stack modeled on $V$, and that $G$ is a higher group (for instance the homotopy stabilizer group of a WZW term on $V$, which appears in practice when making $X$ the target space for a Green-Schwarz sigma model).
This is a general problem of higher differential geometry: beyond the very lowest degrees, it quickly becomes intractable to express any construction in a direct analytic fashion. If we summon enough energy we will be able to do so as long as everything is on the level of geometric homotopy 1-types or 2-types, but for the examples just mentioned one needs at least 6-types, and it is just out of the question that anyone writes down the relevant analytic data in this case, and even if an oracle gave it to us, it would be out of the question to then do anything with this mess.
There is however a good general solution to this problem that the analytic description of higher geometry is intractable. This is called the synthetic description. Here, instead of building and analyzing all structure in terms of point constituents, one synthesizes structures from the universal properties that characterize them. This is what I was after here for the case of the differential equation characterizing integrable $G$-structure.
Just quickly, before I hop on the train in 30 minutes… You are saying that you don’t need the definition of the structure tensor $\mathbf{c}$, because you have an alternative formulation of $k$-th order flatness using universal diagrams in a synthetic setting. OK. However, the main thrust of Guillemin’s article, the way I see it, is not the definition of $k$-th order flatness but the
Theorem: There exists a $K > 0$ such that $K$-th order flat $\implies$ $\infty$-flat (i.e., $k$-th order flat for any $k\ge K$).
Is that part of your formulation as well? Such a result is important because $\infty$-flat almost implies locally flat (flat in a neighborhood), which is what you really want (not just the definition of that property). That last implication actually does hold in the analytic category, while in the $C^\infty$ category it’s more complicated.
The main thrust of my question here was just the definition, that’s right, or rather the generalization of the definition to higher differential geometry. There are many aspects of order-$k$ integrable $G$-structures worth studying once one has a definition. For the purposes of studying higher gravity which I mentioned, one is interested in considering order-1 integrability (torsion freedom) but not higher order integrability (which here corresponds to vanishing of ever more pseudo-Riemannian curvature invariants). Specifically for 11-dimensional supergravity the order-1 integrability is already equivalent to the vacuum Einstein equations (Candiello-Lechner 93), and so here one interest in having the order-1 integrability as a differential equation defined as a bundle over the de Rham space is that it is already the correct covariant phase space of the theory.
Aha, that’s more clear now. Thanks!
In retrospect, I think I was unsettled by the term order-$k$ integrable, which I think I would phrase as order-$k$ flat. In my mind, integrability is the property the $G$-structure should have to have no obstructions to extending a witness of order-$k$ flatness to a witness of order-$(k+1)$ flatness. That’s what struck me as being conspicuously absent from your discussion above. But the terminology here is hugely confusing anyway, so I should have been more attentive. Having finally looked at the nLab pages on integrability of G-structures, which perhaps I should have done right away ;-), I see that the contents of the Guillemin article are not news to you. So thanks for putting up with my attempts to digest it!
That’s a good point about the terminology of either “integrable” or “flat”. I find the latter term problematic due to it invoking the common Riemannian concept of flatness, which is only a very special case. The alternative “integrable” is something in use, I don’t think that I made this up.
It’s unfortunate that no more suggestive term has become standard here. What one really means is more like “order-$k$ parallelizable” or “order-$k$ left invariant”, since it is about extending the framing given on coset model spaces $G/H$ by left translation along $G$ to manifolds locally modeled on $G/H$.
By the way, this fact which I mentioned (Candiello-Lechner 93), that the bosonic vacuum Einstein equations on an 11-dimensional spacetime $X$ are equivalent to the super-torsion freedom on an $N=1$ supermanifold extension of $X$ I find most curious. It seems to me this fact has been missed by the crowd of people who like to look for reformulations of Einstein gravity by more manifestly “topological”-looking constraints (e.g. BF- or CS-gravity or the like). It’s also interesting from a non-super point of view, since it indeed gives the ordinary bosonic Einstein equations, the supergeometry may be thought of just as a way to phrase that as an equivalent torsion constraint on an “auxiliary” super-extended space.
In any case, it makes me wonder that there should be a way to canonically construct a presymplectic 12-form current on the solution space to this super-torsion constraint. And hence I am wondering how generally the solution spaces to order-$k$ flatness/integrability differential equations might carry canonical presymplectic current forms. If this rings any bell with you, please let me know.
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