Mathematical loop space literature

July 15, 2004

Posted by Urs Schreiber

Readers of my writings will have noticed that I mention the tem loop space from time to time.

Now, I have to admit that my understanding of loop space is very much that of a physicists, not that of a mathematician. My main tool to deal with the subtleties of the very concept of loop space is that I know how the exterior geometry on loop space is related to 2d superconformal field theory (by way of switching from Heisenberg to Schrödinger QFT picture), which is well understood.

Even though I have the suspicion that this, from the mathematical point of view rather unsophisticated, attitude is actually quite useful, my recent occupation with nonabelian connections on loop space indicates that I should try to get a more high-brow understanding of this space.

In particular, thinking about flat connections on loop space - and my claim is that a nonabelian 2-form on target space does correspond to a flat connection on loop space (which also follows from work by Girelli and Pfeiffer) - made me wonder about contractibility of paths in loop space.

A torus in target space is a closed path in loop space. This closed path can certainly be contracted to a point in loop space (this point corresponding to a constant function from the circle into target space) by shrinking the torus to a point.

But there are several closed paths in loop space that correspond to the same torus in target space, one corresponding to every foliation of that torus by circles.

All of these paths can be deformed into each other, but not all can be deformed into each other without using the constant path (in loop space). Namely the ‘horizontal’ and the ‘vertical’ slicings of the torus, corresponding to its 2 1-homotopy classes, can apparently never be continuously deformed into each other without using the ‘vanishing torus’.

Is that right?

See, that’s the level of understanding of global properties of loop space that I am currently sitting at…

For my own good I’ll list a couple of math books on loop space that certainly contain the answer to this and many other questions:

Jean-Luc Brylinski: Loop spaces, characteristic classes, and geometric quantization (1993)

John Frank Adams Infinite Loop Spaces (1978)

Hans Baues: Geometry of Loop Spaces and the Cobar Construction (1980)

L. Feher, A. Stipsicz: Topological Quantum Field Theories & Geometry of Loop Spaces
(1992)

- Sergeev, A. G. [Ed.] Loop spaces and groups of diffeomorphisms (1997)

Posted at July 15, 2004 3:00 PM UTC

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Re: Mathematical loop space literature

Hi Urs,
The basic mathematical fact is that
\pi_1(\Omega M)=\pi_2(M)

so if the second homotopy group of your space is trivial, then you can contract a loop in your loop space, otherwise not.

Here \Omega M is based loops.

For the basics of homotopy and cohomology, Bott and Tu “Differential Forms in Algebraic Topology” is hard to beat.

Posted by: Peter Woit on July 15, 2004 5:27 PM | Permalink | Reply to this

Re: Mathematical loop space literature

Hi Urs,
The basic mathematical fact is that
\pi_1(\Omega M)=\pi_2(M)

so if the second homotopy group of your space is trivial, then you can contract a loop in your loop space, otherwise not.

Here \Omega M is based loops.

For the basics of homotopy and cohomology, Bott and Tu “Differential Forms in Algebraic Topology” is hard to beat.

Posted by: Peter Woit on July 15, 2004 5:28 PM | Permalink | Reply to this

Re: Mathematical loop space literature

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Thanks for the reply.

Let’s assume that $\pi_2(\mathcal{M})$ is trivial, so that all these loop loops are contractible, as in my question.

Still, isn’t it strange (if correct) that for every torus in $\mathcal{M}$ there are, I think, always two loops in $\Omega\mathcal{M}$ that both map to this torus, but which can be taken into each other only by fully contracting both?

Let’s see, maybe I am formulating the question not well enough:

I am thinking of an equivalence relation for loops in loop space. Two loops in loop space shall be equivalent, if they correspond to the same surface in target space.

I could try to cook up some more mathy sounding formulation of that.

How about the following:

Let $\Gamma_1$ and $\Gamma_2$ be two loops in loop space, their elements $\gamma$ being loops in target space. Let

(1)

\mathrm{ev} : S^1\times \mathrm{Maps}(S^1\times \mathcal{M}) \rightarrow \mathcal{M}

be the trivial evaluation map. Then the equivalence relation I have in mind is

(2)

\Gamma_1 \sim \Gamma_2 \Leftrightarrow \bigcup_{\gamma \in \Gamma_1} \mathrm{ev}(S^1\times \gamma) = \bigcup_{\gamma \in \Gamma_2} \mathrm{ev}(S^1\times \gamma) \,.

This seems to be like an important equivalence relation for physical applications, where really only the surface in target space as such is of relevance, while its slicing into loops is essentially a choice of gauge (coordinates on the surface).

Now my question concerned the curious fact that the subspace of loop-loop space consisting of all loop loops which are equivalent $\sim$ to a given loop loop $\Gamma_0$ is

a) disconnected (I think)

and

b) such that every completion of it which makes it connected must include the ‘constant loop’ which maps all of $\mathrm{S}^1$ to a single point in $\mathcal{M}$ .

Is that true? Isn’t it somehow remarkable?

Posted by: Urs Schreiber on July 15, 2004 6:00 PM | Permalink | Reply to this

Re: Mathematical loop space literature

I should add that what does relate these two disconnected subsets of the full equivalence class is - T-duality! :-)

Posted by: Urs Schreiber on July 15, 2004 7:02 PM | Permalink | Reply to this

Re: Mathematical loop space literature

Let me see if I understand. You’re looking at the map from loops in loop space, call this LLM, to the subspace of M that is the image of the loop of loops.
This image can be 0, 1, 2 dimensional. If you fix the image to be a specific torus in M, you can ask what part of LLM maps to this torus.

I would guess this subspace of LLM would have lots of disconnected components, but don’t see what you mean about a completion and the constant loop. If you want to get from one component to another you can do it by going through subspaces corresponding to other images, but there are lots of ways of doing this.

Posted by: Peter Woit on July 15, 2004 7:16 PM | Permalink | Reply to this

Re: Mathematical loop space literature

Peter Woit wrote:

If you fix the image to be a specific torus in M, you can ask what part of LLM maps to this torus.

Yes, that’s what I am asking! Many thanks for your help!

I would guess this subspace of LLM would have lots of disconnected components

Hm, I would think there are precisely two, corresponding to the inequivalent 1-cycles of the torus. You can either slice it with 1-cycles of one type of the other. By continuously deforming these slices you can get all of those that correspond to one class, but not those of the other, I would think.

you can do it by going through subspaces corresponding to other images, but there are lots of ways of doing this.

This I don’t see yet. Seems to me that this can only work when you go through a degenerate point, where the image is 0 dimensional (maybe 1-dim, too), because outside and inside of the torus would have to be exchanged to go from one class to the other continuously. But maybe I am lacking visualization skill, which is unfortunately currently the only skill with which I am attacking this problem! ;-)

Posted by: Urs Schreiber on July 15, 2004 8:09 PM | Permalink | Reply to this

Re: Mathematical loop space literature

I would expect this to depend on what sort of space you are in. For instance, in S^3 I’m fairly sure that it’s possible to swap the “inside” and “outside” of any torus while deforming it such that at each step of the deformation you still have a torus, if that is what you had in mind here. In R^3 it is not.

Posted by: Matt Reece on July 15, 2004 9:36 PM | Permalink | Reply to this

Re: Mathematical loop space literature

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Thanks for the remark.

I there some keyword which I could look for in the loop space literature which would lead me to discussions of this point?

To maybe further clarify a bit:

What I am really concerned with is surface holonomy. I want to understand how connections in loop space $\Omega\mathcal{M}$ associate group elements with surfaces in target space $\mathcal{M}$ .

The most basic question seems to be: When does a connection on loop space assign a unique group element to some surface in target space?

This seems to be a nontrivial question to me, since many curves in loop space map to the same surface in target space.

But probably what I am really interested in are flat connections on loop space. Here the question is much simpler, since now certainly every contractible torus is uniquely associated trivial surface holonomy.

Still, I found it somewhat remarkable that, it seems to me, the equivalence class of loop loops accociated with this torus (in $\mathcal{M} = \mathrm{R}^D$ , say) is disconnected and that, unless I am confused, any path connecting them has to include a degenerate torus.

But I also think that what does relate these two components directly is what is known as T-duality, which effectively exchanges a foliation of the torus to an ‘orthogonal’ foliation, I think.

Maybe I should have a look at

Bunker, Turner, Willerton: Gerbes and homotopy quantum field theories, math.AT/0201116.

Posted by: Urs Schreiber on July 16, 2004 9:10 AM | Permalink | Reply to this

Re: Mathematical loop space literature

I’m probably out on deep water here, I’m definitely not an expert on loop spaces so please correct me if you think I’m wrong.
Some questions and comments though:

It’s not obvious to me which loop space is relevant here. Given the physical situation, it’s probably not based loop space (which also wouldn’t give toroidal images in M). Furthermore, I guess the loops are smooth and oriented, is that correct?

Regarding keeping a torus in M fixed and looking at the subspace of LLM (let’s call it C) with this image, I would also guess that this is disconnected with many components. The image-map from LLM into a fixed torus in M is a map between two tori. Assuming a continuous path in this subspace of LLM induces a homotopy of maps between tori, the number of components of C will be equal to the number of elements in the mapping class group of a torus.
The components of C correspond to slicings wrapping an arbitrary number of times around both cycles (I think).
As far as my understanding goes, however, this could depend substantially on the precise version of LM and LLM.

The two cases considered above, with the torus foliated by circles wrapping the A- and B-cycles respectively, are related by an S-transformation (or if it is a combination of S and T…don’t recall precisely) where S is one of the two generators of SL(2,Z) conventionally labelled S and T.
Now, what I don’t understand at all is why they should be related by T-duality. How do you see this?

A final comment: If what I wrote above is correct, then the equivalence of holonomies for all elements in C translates to modular invariance of the observable corresponding to the given torus in M.

Posted by: Jens Fjelstad on July 16, 2004 4:35 PM | Permalink | Reply to this

Re: Mathematical loop space literature

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Jens wrote:

It’s not obvious to me which loop space is relevant here. Given the physical situation, it’s probably not based loop space (which also wouldn’t give toroidal images in M). Furthermore, I guess the loops are smooth and oriented, is that correct?

Yes, if we are thinking of the configuration space of string then it is

- unbased

- parameterized

- consisting of functions from the circle into target space that have a Fourier decomposition

- oriented or unoriented depending on whether we want to do type II or type I strings.

The components of C correspond to slicings wrapping an arbitrary number of times around both cycles (I think).

Ah, ok. I was thinking about slicings which wrap one cycle once, the other not at all.

the number of components of C will be equal to the number of elements in the mapping class group of a torus.

Yes, I’d think so, too. So when restricting to the slices that I had in mind with 1 winding around one cycle and none around the other, the number of disconnected components should indeed be 2.

Now, what I don’t understand at all is why they should be related by T-duality. How do you see this?

T-duality exchanges $\partial_\tau X$ with $\partial_\sigma X$ .

Posted by: Urs Schreiber on July 16, 2004 5:13 PM | Permalink | Reply to this

Re: Mathematical loop space literature

Do you want to demand that the map from the torus T^2 in LLM=Maps(T^2,M) be one-to-one? If not, there should be lots of components because your loops can wrap multiple times around the non-contractible paths in your image T^2.

Posted by: Peter Woit on July 16, 2004 7:00 PM | Permalink | Reply to this

Re: Mathematical loop space literature

Yes, as I said in my reply to Jens I was thinking of the one-to-one case. But you are both right that there is no need to make this restriction.

Still, I think I was on the right track. There should be two disconnected components of 1-1 maps. This is just a special case of what Jens said about the mapping class group.

Ok, good. So what does this tell us about connections on loop space and how they are associated with unique surface holonomies?

One important question seems to be the following: If our surface in target space, that we whish to associate a surface holonomy with, is really a multiply-wrapped torus, should it get the same surface holonomy as the singly-wrapped torus?

(I am probably again using awkward wording. You’ll certainly find a better formulation. :-)

Probably not, I’d say. But if not, then we should really count all tori which occupy the same points in target space but differ in their wrapping numbers as different objects which a priori get different surface holonomies.

But that would be in contradiction to Jens’ proposal that the modular group plays a role. I am not sure about that. There should be no need to even mention a metric or conformal invariance as long as we are just talking about surface holonomies.

Here is one challenge that I think we are faced with:

Question: Is there any non-flat connection on loop space which assigns unique surface holonomy (in all these disconnected components of LLM) to a given torus in M?

If there is, how can these connections be characterized?

(This questions implicitly assumes that in general a connection on LM does not associated a unique surface holonomy, because there is no reason for it to agree on the disconnected components of LLM which correspond to the ‘same’ torus in target space.)

Posted by: Urs Schreiber on July 16, 2004 7:31 PM | Permalink | PGP Sig | Reply to this

Re: Mathematical loop space literature

I don’t think my comment has anything to do with “several-1” coverings of the torus (at least for unbased loops), they are all 1-1.

The reason the mapping class group should appear is as the group of homotopy types of homeomorphisms, no need to introduce metrics or conformal structures.

Regarding your question: As far as I understood, this was more or less the motivation for introducing 2-categories and 2-groups, i.e. to make holonomies well-defined, right? Unfortunately I don’t have a good understanding of how it should work.
For abelian gerbes I think this has been worked out in some papers, can’t remember the references though.

Posted by: Jens Fjelstad on July 19, 2004 11:17 AM | Permalink | Reply to this

Re: Mathematical loop space literature

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Jens wrote:

I don’t think my comment has anything to do with ‘several-1’ coverings of the torus (at least for unbased loops), they are all 1-1.

My apologies, I probably misunderstood you when you said

The components of C correspond to slicings wrapping an arbitrary number of times around both cycles (I think).

Now I realize that this means that we have slices which are not just simple circle clices, but slices which are like corkscrews winding around the torus, right?

Ok, then I get it. So there are indeed infinitely many disconnected components of LLM corresponding to the same target space torus. Sorry for being slow. Thanks for your help.

As far as I understood, this was more or less the motivation for introducing 2-categories and 2-groups, i.e. to make holonomies well-defined, right?

Yes, sure. As I said in the beginning of this entry, I am just trying to learn more about the mathematical results that are available. This thought about disconnected components in LLM was just meant to help me see that there are very strong constraints on any connection on LM to actually produce a unique surface holonomy in M. These strong constraints are trivially satisfied by flat connections on loop space. This is consistent with what Girelli and Pfeiffer have derived, which indicates that indeed only the flat connections have this property. I am wondering if this is a well known result, and where one could find it.

Posted by: Urs Schreiber on July 19, 2004 2:57 PM | Permalink | PGP Sig | Reply to this

Re: Mathematical loop space literature

“Now I realize that this means that we have slices which are not just simple circle clices, but slices which are like corkscrews winding around the torus, right?”

Yes, precisely.

Again, regarding your question, this is something I too would like to understand better. I don’t think it is well known. I have looked at (I dare not write “read”) all papers I have found dealing with these matters at some point, and I don’t recall such a statement anywhere before Girelli and Pfeiffer. But then again, my memory is very poor…

It seems obvious that a generic connection on loop space will have different holonomies in different components of C, also if the connection is flat.
So, where does the constraint for uniqueness appear? My guess is that it is a consequence of the fact that the interesting connections here all come from connections on the target, and are thus quite special. Do you agree?
Now, starting from a connection on loop space, what are the conditions on the connection that allows one to lift it to a connection on target?

Posted by: Jens Fjelstad on July 19, 2004 3:39 PM | Permalink | Reply to this

Re: Mathematical loop space literature

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Jens wrote:

Yes, precisely.

Good. Somehow I had a mental block here.

and I don’t recall such a statement anywhere before Girelli and Pfeiffer

I haven’t looked at any purely math papers on this stuff at all, I am only familiar with the math-ph stuff, like the papers by Hofman, Baez, Girreli&Pfeiffer. But since I figure that these people have some overview over the relevant math literature, and since they don’t mention any results on connections on loop space, I, too, get the impression that understanding of these is at least not very common.

It seems obvious that a generic connection on loop space will have different holonomies in different components of C, also if the connection is flat.

We know at least that if the connection is flat, and if $\pi_2(\mathcal{M})$ is trivial, then all flat connections will have trivial holonomy on all parts of $LL\mathcal{M}$ . So in this case the flat connections do solve the consistency condition.

So, where does the constraint for uniqueness appear?

Here I at the moment don’t fully understand what you are asking. The uniqueness constraint is necessary to have a consistent notion of surface holonomy. So we impose it by hand if we want to talk about surface holonoy, I’d say.

My guess is that it is a consequence of the fact that the interesting connections here all come from connections on the target, and are thus quite special. Do you agree?

Yes. I agee with that after having done the computations in my hep-th/0407122, which indicated to me that local gauge transformations on loop space will turn a loop space connection which ‘comes from’ a target space connection to another such connection if and only if it is flat. Stated this way, this sounds a little surprising, but actually when you look at the formulas (e.g. my (3.23) and the related text) then it seems to be pretty obvious. Plus, it agrees with what Girelli & Pfeiffer arrived at with completely different reasoning.

Now, starting from a connection on loop space, what are the conditions on the connection that allows one to lift it to a connection on target?

I think the condition is that the loop space connection be of the form $\oint_A (B)$ for some target space 1-form $A$ and 2-form $B$ .

Posted by: Urs Schreiber on July 19, 2004 9:24 PM | Permalink | PGP Sig | Reply to this

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July 15, 2004