## March 23, 2007

### History of Understanding Bundles with Connection using Parallel Transport around Loops

#### Posted by Urs Schreiber

In the process of finishing a paper, I was today busy collecting some background history literature on the development of the idea that a principal $G$-bundle with connection may equivalently be encoded in its parallel transport around based loops.

Apparently one of the oldest occurrences of the idea that a bundle $P \to X$ with connection may be reconstructed from its holonomies for any fixed base point in the connected base space $X$ appears in

S. Kobayashi,
Comptes Rendus, 238, 443-444. (1954).
(full text)

A more detailed and more general discussion, has then been given in

Milnor, J.
Annals of Mathematics, 63, 272-284.(1956).

A detailed discussion of the differentiable case appears is

C. Teleman
Généralisation du groupe fondamental
Annales Scientifiques de l’école Normale Supérieure 3, 77, 195-234. (1960)
(full text)

C. Teleman
Annali di Matematica, Pura ed Applicata, LXII, 379-412. (1963).

This history is recollected in the introduction of

J. W. Barrett
Holonomy and path structures in general relativity and Yang-Mills theory
Int. J. of Theor. Phys., Iiol. 30, No. 9, 1991
(pdf)

who himself gives a proof.

Barret implicitly uses the diffeological – or Chen – smooth structure on the space of paths (except that he restricts attention to closed paths).

In

Jerzy Lewandowski
Group of loops, holonomy maps, path bundle and path connection
Class. Quantum Grav. 10 (1993) 879-904
(pdf)

the statement of the equivalence is attributed to

J. Anandan
Holonomy groups in gavity and gauge fields Pmc. Conf Differential Geometric Methods in Plystcs (fieste 1981)
ed G Denardo and H D Doebner (Singapore: World Scientific)

Therein it is shown that smoothness of the parallel transport is a necessary condition for it to come from a smooth bundle with connection. Barrett also shows that this is sufficient.

Lewandowski adds to this a formulation of an equivalence of bundles with connections and the subset of loops around which the corresponding parallel transport is trivial.

He also comments on the relation to generalized connections (those functors from paths to the group which are required neither to be smooth nor, in fact, continuous) in section 3. (I mentioned this issue here.)

Around the same time appeared

A. Caetano, R.F. Picken
An axiomatic definition of holonomy
Int. Journ. Math. 5 (1994) 835
(scanned version).

These authors introduced the idea of sitting instants of paths and noticed that the most elegant way to (re)state the maximal equivalence relation on paths which is respected by parallel transport is in terms of thin homotopy, i.e. smooth homotopy between paths with differential of less than maximal rank.

Barrett originally had something very similar but slightly different. With Caetano and Picken’s relation, the space of thin homotopy classes of paths in $X$ becomes an honest groupoid $P_1(X)$ internal to smooth spaces.

There would be more to say. But I have to run now, unfortunately (same problem every Friday…)

I am grateful to Christian Fleischhack and to Laurent Guillopé for help with tracking down some of the above links.

Posted at March 23, 2007 6:30 PM UTC

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### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Surely this story should start with Riemann and the early days of complex analysis.

Posted by: Changbao on March 23, 2007 10:04 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Perhaps we should all take the advice Arnold gives on page 17 of his first Toronto lecture:

To learn the state-of-the-art in a domain new to me, I usually start with the German Encyclopaedia of Mathematical Sciences edited by Klein and published around 1925. It contains an enormous amount of information. Then there are papers in the Jahrbuch which was published before mathematical Reviews and Zentralblatt had been organized - it is full of information. Then, I usually consult the collected works of Felix Klein and Poincaré. In Klein’s Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert there’s a lot of information on whatever has happened in 19th century and before.

Posted by: David Corfield on March 24, 2007 9:36 AM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Perhaps we should all take the advice Arnold gives […]

Okay, thanks for forwarding this advice. I have not looked at either of these yet.

For the pre-Barret literture on the issue here I had relied on the introduction of Barrett’s thesis.

The date of the first occurence of the (uproven, at this point) statement of the fact that a bundle is reconstructible from its parallel transport around based loops he gives as 1954.

But he also emphasizes in that introduction that this statement has appeared over and over again, more or less independently, in a plethora of papers (with this phenomenon being most pronounced in the physics literature).

And in fact the proof is, in broad outline, quite simple. (So I could imagine that many people might maybe have found the statement to be “obvious”.) There are some subtleties in the technical details, though.

Posted by: urs on March 25, 2007 4:37 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Thanks for adding the full text to the Kobayashi reference; had never seen it before. Interesting to see the quotient of the loop space he uses.

The comment preceding the Milnor reference is a little misleading since the emphasis, as I recall, is on the CW structure. And I don’t see any mention of a connection??

On the other hand, in the case of fibrations (locally homotopy trivial maps p: E –> B), higher homotopies
take the palce of quotienting.

Posted by: jim stasheff on March 3, 2010 8:37 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Thanks for adding the full text to the Kobayashi reference; had never seen it before. Interesting to see the quotient of the loop space he uses.

I’m puzzled by the Milnor reference since there is no mention of connection there and getting a universal bundle * with connection* has even more `technical subtleties’.

Posted by: jim stasheff on March 3, 2010 8:45 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Surely this story should start with Riemann and the early days of complex analysis.

One has to start somewhere. Here I am interested in who was the first to

a) conjecture

b) state

c) attempt to prove

d) prove

that a bundle with connection may be reconstructed from its parallel transport around based loops.

Who do you think would that be?

Posted by: urs on March 25, 2007 3:03 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

I guess if you want the oldest paper that you can cite in a proof, maybe you’ve found it. But the history of understanding monodromy – or anything – goes back farther than conjectures, statements, and proofs about bundles and connections. The notion of what we would call a flat connection on a Riemann surface was pretty explicit by the time Hilbert came up with his 21st problem. You should probably start looking even earlier, at something like analytic continuation.

Posted by: Changbao on March 29, 2007 7:00 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

If we take parallel transport as the fundamental,
then we have the easier statement:

that a principal bundle may be reconstructed from its parallel transport around based loops.

Posted by: jim stasheff on March 3, 2010 8:46 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

This is all from memory: Milnor did the PL case, Lashof topological, and I think I was the first to do the smooth case, though the statement but not proof appeared earlier in Anandan. I can’t remember what Teleman did, perhaps someone should check. If all you are interested in is flat bundles then it probably goes back very much further.

Posted by: John Barrett on October 4, 2007 6:13 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

It would be good if someone would do a proper history here. Notice one could/should first do a history of understanding bundles THEN with connection.
Btw, I think parallel transport should come first, especially in the topological and even homotopy theoretic cases. Dold-Lashof
did some work on the case of principal fibrations with monoid rather than group as I dimly recall and I did the case of parallel transport for homotopy fibrations;
The corresponding map based loops on B into self homotopy equivalences of F is an A_\infty map.

Posted by: jim stasheff on October 6, 2007 2:09 AM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

Just for the record, I should recall that I started this thread here when Konrad Waldorf and I were looking for references to cite in our article Parallel Transport and Functors.

I think we do list the major predecessors, but apparently, as has been noticed here, the very early history is long and diverse. I’d still be interested in compiling a good historical account, though.

a history of understanding bundles THEN with connection.

Certainly this would be interesting, too. On the other hand, what I was originally mostly interested here is that idea of describing bundles with connection entirely in terms of their parallel transport, i.e. without even necessarily having the total space of a bundle manifestly present a priori. So this is not about starting first with a bundle and then putting a connection on it, but about the idea of regarding the connection as the fundamental entity and the total space of a bundle as a derived concept.

Posted by: Urs Schreiber on October 6, 2007 10:16 AM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

If you don’t mind, I have another question about the history of category theory. This paper from 11 years ago by Michael Farber defines the categories of Hilbert representations, of von Neumann algebras, and of Hilbertian representations.

Is this the first paper to define such concepts? It looks to me as if the answer is “yes”.

Posted by: Charlie Stromeyer Jr on October 6, 2007 2:03 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

If you don’t yet have the bundle, then your connection must be defined only locally.

OK but parallel transport allows you to start globally, using just the paths in the base

Posted by: jim stasheff on October 6, 2007 2:29 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

If you don’t yet have the bundle, then your connection must be defined only locally.

Well, that depends. The description which I am after here conceives the globally defined connection without making the total space of the possibly nontrivial bundle that it is defined on manifest.

The original statement of this kind of result, whose history the above entry was concerned with, asserted that a $G$-bundle with connection on a connected space can be reconstructed, up to isomorphism, from the information encoded in a group homomorphism from the group of thin homotopy classes of based loops in base space to the structure group.

So, a bundle with connection “is” an element in

$\mathrm{Hom}(\Omega_\star^\mathrm{thin}(X),G)$

and this doesn’t mention the total space of the bundle, while at the same time encoding all the information necessary to describe a nontrivial bundle with globally defined connection on it.

When the base space is not conected, one needs to pass from based loops to paths. Same for higher bundles. This way we arrive at the description of bundles with connection as parallel transport functors which send paths in base space to certain morphisms.

In this kind of description, the bundle and its total space is a secondary concept. It can be reconstructed from the information encoded in the parallel transport functor, but the latter exists, globally, in its own right.

One reason for being interested in this point of view of “bundles with connection as transport functors without total space” is that it seems to be this point of view which lends itself most seamlessly to categorification and quantization – two crucial edges of the cube.

Posted by: Urs Schreiber on October 6, 2007 3:59 PM | Permalink | Reply to this

### Re: History of Understanding Bundles with Connection using Parallel Transport around Loops

I have now added some comments concerning the relevance of describing bundles with connection entirely in terms of their holonomy and/or their parallel transport in String- and Chern-Simons $n$-Transport.

You can find this by

- then following the link to section “Parallel $n$-transport”

- then following the link to the subsection “History and comparison with Cheeger-Simons differential characters”.

As the headline suggests, there I point out that another famous example for this kind of description of $n$-bundles with connection in which the total space does not manifestly appear, while only the holonomy map does, is the definition of Cheeger-Simons differential characters (currently slides 135).

In fact, as I amplify in the comments after that slide, one way to think of parallel $n$-transport is as the nonabelian generalization of Cheeger-Simons differential characters.

Posted by: Urs Schreiber on October 10, 2007 6:17 PM | Permalink | Reply to this