## October 31, 2006

### Markl on Natural Differential Operators

#### Posted by Urs Schreiber

Just heard a talk by Martin Markl on

He explained

• a way to make precise the idea that certain differential operators (like the Lie derivative, or the covariant derivative) are more natural than others,
• that all natural differential operators of a certain “type” arise as the 0th cohomology of a complex of graphs,
• where the graphs appearing here are like string diagrams representing the action of linear operators on tensor powers of vector spaces.

Given a space like $\mathbb{R}^n$ with coordinate functions $\{X^i\}$, we can form plenty of differential operators on the space of functions on this space by writing expressions in local coordinates like

(1)$X^i X^j \frac{\partial}{\partial X^k}$

or

(2)$\left( f^i \frac{\partial g^j}{\partial X^i} - g^i \frac{\partial f^j}{\partial X^i} \right) \frac{\partial}{\partial X^j} \,.$

The second one is “natural” (it comes from the Lie derivative), the first one isn’t.

What exactly does this mean?

Let $\mathrm{Man}_n$ be the category of $n$-dimensional smooth manifolds with morphisms being open embeddings. Let $\mathrm{Fib}_n$ be the category of fiber bundles over $\mathrm{Man}_n$.

We say a kind of bundle is natural if we can functorially associate it to manifolds:

Definition: A natural bundle is a functor $\mathbf{F} : \mathrm{Man}_n \to \mathrm{Fib}_n$ such that $F(M)$ is a bundle over $M$ and such that for any open submanifold $M' \subset M$ we have $F(M') = F(M)|_{M'}$.

In 1977 Palais and Terng proved a theorem which characterized natural bundles as precisely those fiber bundles that are associated by $\mathrm{GL}^k(n)$ to a $k$-frame bundle.

The bundle of $k$-frames over $M$

(3)$F_n^k(M)$

is defined to be the bundle of $k$-jets of local coordinate systems. This is like the ordinary bundle of frames plus higher derivatives of that. In particular, $F^1_n(M) = F_n(M)$ is the ordinary frame bundle of $M$.

Similarly, $\mathrm{GL}^k(n)$ is the group of $k$-jets of local diffeomorphisms of $M$.

So:

Theorem: For each natural bundle $\mathbf{F}$ there is a natural number $k \geq 1$ and a $\mathrm{GL}^k(n)$-space $F$ such that

(4)$\mathbf{F}(M) = F_n^k(M) \; \times_{\mathrm{GL}^k(n)} \; F \,.$

For instance, the tangent bundle is natural and we have

(5)$T(M) := T M = F_n^1(M) \times_{\mathrm{GL}(n)}\; \mathbb{R}^n \,.$

Now,

Definition:A natural differential operator is any morphism $D : \mathbf{F}^{(l)} \to \mathbf{G} \,,$ where $\mathbf{F}$ and $\mathbf{G}$ are natural bundles and $\mathbf{F}^{(l)}$ is the natural bundle obatined by taking $l$-jets of $\mathbf{F}$.

For instance, the Lie derivative takes two copies of the tangent bundle to the tangent bundle, depending on the first derivative of the vector fields involved. Hence it is a natural operator of the form

(6)$T^{(1)} \oplus T^{(1)} \to T \,.$

The nice thing is that such natural differential operators can be understood in terms of equivariant maps:

Theorem: Let $\mathbf{F}$ and $\mathbf{G}$ be natural bundles with typical fibers $F$ and $G$, respectively, according to the above theorem. Then we have a bijection between natural differential operators $D : \mathbf{F}^{(l)} \to \mathbf{G}$ and $\mathrm{GL}^{k+l}(n)$-equivariant maps $U : F^{(l)} \to G \,.$

$U$ is the local formula for the differential operator. For a reason unknown to the speaker and his audience, this theorem is known as IT-reduction.

Next, we can decompose $F^{(l)}$ as well as $G$ into reps of the ordinary general linear group $\mathrm{GL}(n)$. The basic invariant theorem for such reps then tells us that all natural differential operators must be obtainable by doing the familiar index contraction on linear maps, roughly.

But we don’t want to think in terms of index contraction, but instead in terms of plumbing. A linear map is represented by a graph with a single vertex, with a couple of incoming and a couple of outgoing edges. Index contraction is conneting outgoing with incoming edges between maps.

The main message is that one can define a differential $\delta$ on graphs, which acts locally - in the sense that its action is completely specified by the action on graphs containing a single vertex - such that the natural differential operators come from precisely those linear maps $U$ such that

(7)$\delta U = 0 \,.$

Markl indicated that there is something much more profound going on in the background, involving operads. But that’s where the talk ended.

Posted at October 31, 2006 8:48 PM UTC

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### Re: Markl on Natural Differential Operators

a way to make precise the idea that certain differential operators (like the Lie derivative, or the covariant derivative) are more natural than others

A morphism is natural if it commutes with diffeomorphisms. Why is that not precise enough?

Naturality depends on your symmetry group, though. If you consider symplectic or contact manifolds, say, a morphism is natural if it commutes with symplectomorphisms or contact transformations, respectively.

I think that natural morphisms wrt simple Lie algebras have been classified by Russian mathematicians, locally; perhaps some multi-linear cases remain. For natural morphisms wrt simple infinite-dimensional Lie superalgebras see math.RT/0202193. Extra credit if you can actually decipher this paper - I failed, despite knowing the result in a different formalism.

### Re: Markl on Natural Differential Operators

A morphism is natural if it commutes with diffeomorphisms. Why is that not precise enough?

It is indeed precise enough.

I think the point of the exercise was to find another (equivalent) precise definition, such that one obtains from it a constructive method of enumerating all natural differential operators.

This aspect is probably more pronounced in Markl’s abstract, than in my summary of his talk. (Partly because he ran over time before coming to his main points.)

The claim is apparently that studying the 0th cohomology of these graph complexes is helpful.

Posted by: urs on November 1, 2006 6:04 PM | Permalink | Reply to this

### Re: Markl on Natural Differential Operators

Dear Friends

I am pleased by your interest in my talk. Right now, I am writing things down to spell them up, so I believe something more definite will be available in a month or so.

All started by my attempt to understand S. Merkulov’s idea of “PROP profiles” and see if and how one might apply his approach to differential geometry (whereas his methods live in formal geometry).

It turned out that classification of natural operators in lot of interesting cases boils down to calculating the cohomology of graph complexes. These graph complexes are in fact isomorphic to subspaces of stable elements in certain Chevalley-Eilenberg complexes, so, formaly speaking, the claim is that a certain Chevalley-Eilenberg cohomology is the cohomology of the orresponding graph complex. Instances of this phenomenon were probably observed first by M. Kontsevich.

But what makes it EXCITING is that there are powerful methods developed during the “rennaisance of operads” which give a deep understanding of these graph complexes. In fact, a miracle is already the fact that the graph complexes arising in this context are of the type studied by operad people.

I have a couple of examples that are based on very difficult and apparently unknown calculations with graph complexes, which lead me to believe that also the results implied for natural operators are unknown. This was also corroborated by some differential geometers I talked to.

And yes, I know Fuchs and the results of his school.

Another way to interpret the results is that they give an exact meaning to the “abstract tensor calculus.” When we studied differential geometry in kindergarden, many of us, instead of writing hundreds of indices, draw simple pictures. My attempt then tries to put this kindergarden approach on solid footing. Which, by the way, means that all textbooks on differential geometry ought to be rewritten to simplify the lives of readers.

As I said, I am working on a paper, so in a month or so, I will explain everything in detail.

Regards, Martin

Posted by: Martin Markl on November 3, 2006 10:07 AM | Permalink | Reply to this

### Re: Markl on Natural Differential Operators

A preprint containing the above mentioned resuls is now available as math.DG/0612183.

Martin

Posted by: Martin Markl on December 11, 2006 11:15 AM | Permalink | Reply to this
Read the post Codescent and the van Kampen Theorem
Weblog: The n-Category Café
Excerpt: On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
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