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I suppose with all the entries on Witten genus and related what was missing was an entry titled Dirac-Ramond operator. So I created one and filled in a bare minimum.
I’m not sure of the ettiquete on this one. Is this the right page to refer to http://arxiv.org/abs/0809.3104 on?
I would say: definitely! Since these aren’t published (from a quick scan on MathSciNet — will they ever be, Andrew?), citing them on a standard reference such as the nLab is helpful dissemination.
Since these aren’t published (from a quick scan on MathSciNet — will they ever be, Andrew?)
Not through lack of trying on my part, but it does seem that this paper (and its companion) are doomed to wander the wastes of preprint land for ever.
And I thought you were being noble and shunning the evils of the publishing system… :-P I’d be interested to see who rejected this, in case I could make suggestions, but that’s perhaps a matter for private email discussion.
Hi Andrew,
that would be nice if you could write a brief review of the state of the art of building loop space Dirac operators! Preferably with a bit of comparative loopology, regarding the various proposals that are available, that would be great.
Konrad Waldorf tells me that he is having high hopes now that Melrose has a promising approach, based on Konrad’s fusion structures on smooth loop spaces. But I don’t know any details.
To me it seems unclear what one wants to achieve with Dirac operators on smooth loop spaces, given that Witten right at the beginning showed which operator one should look at, namely the 0-mode of the supercharge of a 2d SCFT, which is what is called the Dirac-Ramond operator in string theory.
That would also be my question regarding your article: you end the article with constructing something that probably deserves to be called a Dirac operator on loop space. But how is the Dirac-Stacey operator related to the Dirac-Ramond operator? Do you know? In particular, how is it related to the Witten genus? It seems you don’t consider the crucial second term in the Dirac-Ramond operator, that related to translation along the loop (or maybe I am missing it).
My understanding is (but if your review will set me straight on this I’d be grateful), that so far all attempts to relate an actual Dirac operator on smooth loop space back to the Dirac-Ramond operator in string theory didn’t work out. I certainly remember a few years back that Wurzbacher started his talks by highlighting this. For flat space one can say something, which however is not too interesting, and beyond that everything turned out to get stuck somewhere. That’s at least what he said, I have never since looked much deeper into it.
So to finally come to your question as to which would be the right entry to record a brief review of your article: I think that would be an entry titled
I started an Idea-section there, currently it reads as follows:
The traditional definition of a Dirac operator is formulated for operators acting on sections of spinor bundles over Riemannian manifolds, not however directly for bundles over infinite dimensional manifolds.
When the conceptual importance of the Dirac-Ramond operator in the superstring worldsheet 2d SCFT was realized (an operator in a super vertex operator algebra) via the relation in the large volume limit of its index (suitably regarded) to the Witten genus, then it seemed suggestive that it should be possible to regard the Dirac-Ramond operator as an actual Dirac operator on the infinite-dimensional smooth loop space of the underlying manifold, and regard the $S^1$-equivariant index of a Dirac operator in this sense.
The definition of Dirac operators on smooth loop spaces is technically tricky, but constructions do exist. It remains however unclear how these constructions relate to the Dirac-Ramond operator and a rigorous derivation of the Witten genus as an (equivariant) index of a Dirac operator along these lines seems to remain open.
that would be nice if you could write a brief review of the state of the art of building loop space Dirac operators
I’m afraid I can’t do that, Urs.
I am most definitely a frog so I don’t have a bird’s eye view of the state of the art. I can explain what I’ve done, but I don’t know why no-one else seems to be interested in it. I don’t even know if Wurzbacher has even heard of my work. I know that Melrose has heard of it as he was in the audience when I gave a talk on it, but I don’t know if he understood the point of the construction - at that conference then I had the impression that he didn’t. I had an email exchange with Chris Kottke not long ago which I initiated by asking what was wrong with my construction. I still don’t know the answer to that. He (and by extension Melrose) seemed pretty keen on these “very smooth” sections. I don’t know why they’re so important. Mathematically, I can see that they are a way to get round the fact that the obvious Clifford multiplication map doesn’t converge on sections of the cotangent bundle but then that’s precisely what my construction solves. So there must be some “philosophical” reason why they want those sections which I don’t understand. Again, I’m not a bird so I don’t have an overview.
In short, I’ve constructed a Dirac operator on loop spaces solving the main issue. The method is robust and should work for any modification of the Dirac operator that you want. To my knowledge, no-one else has even come close to the generality that I’ve achieved. I can also construct its topological index and show that this is the Witten genus, modulo removing the renormalisation (ie I don’t need to renormalise, which gets rid of that bizarre factor of 24 in the formula). The analytical index is still a long way off.
Thanks! This is just the kind of information I am after: what’s the technical issues being solved, and what’s the relation back to the string.
So maybe could you write a paragraph on what above you call the point of your construction?
Where do you have the discussion of the topological index being the Witten genus?
Hi Andrew,
I have looked through your article(s) a bit more. I certainly get the point. How about you submit this to the Publications of the nLab?
Of course I don’t know what your previous referees said, but I can tell you one impression that I am left with: one is left wondering what the upshot is.
First, I gather you are saying that all other existing proposals don’t work on the genuine loop space or are making some other re-definitions of the problem to be solved. Maybe this could be stated more explicitly, so that it is more clear which open problem your article solves that was not solved before. I am not doubting that it does, I am just saying that it would be good to state it more explicitly. The first line of the introduction might read “It has remained an open problem to construct an operator on sections of a xyz bundle over the uvw-space of smooth loops in a manifolds which satisfies abc. Previous proposals all fall short in either xyz or uvw or abc. Here we for the first time show that such an operator indeed exists.”
Second, you end up proving that some construction exists which you defined yourself. I mean, you end up saying something like “Corollary: loop space carries an $S^1$-equivariant Dirac operator”, where however “Dirac operator” here is not some generally agreed-upon concept, but is precisely what you define it is, namely that construction which is obvious only after one regards co-Riemannian structures.
I fully agree that your definition does look reasonable, but it seems to me for this to have an impact, it should be followed by at least some indication of why this is indeed an answer to the original question, some consistency check, some argument why this definition is good. Why is passing from Riemannian to co-Riemannian structure not a similar example of “redefining the problem” that you hint at is what all other authors did? I am not saying that it is, I just think it would be good to present more argument that indeed it is sensible.
Finally, whatever you know about the relation of your construction to the original observations by Witten would be good to state.
Submitting to Publications would seem a little like self-publishing.
One part of the story that’s probably worth knowing about - from the motivation side - is that I didn’t set out to construct anything at all resembling a Dirac operator. I set out to define a semi-infinite de Rham operator. They’re quite similar, but the Dirac operator turns out to be easier due to not worrying about issues of integrality. And even with the Dirac operator, I’m not really interested in it because the technical aspects were in showing that the co-Riemannian structure existed. That’s the bit that required actual work and the bit that I’d planned to develop further (only I switched to cohomology operations while waiting for the papers to get published). It just annoys me when people say “No one has constructed a Dirac operator”. I wouldn’t mind if they said “This isn’t what we want because of XYZ” (although I suspect that there cannot be an operator satisfying XYZ - the problem is that without an actual operator anyone is free to assume whatever properties they like about it and I strongly suspect that the accumulated assumptions are mutually exclusive). Even Konrad’s latest tome makes no mention of any of my work.
Having gotten that off my chest, I’ll turn to your specifics.
What is the open problem? To define any sort of operator on any sort of loop space. The only proposal that actually produces an operator before mine was Taubes’ (and Landweber’s extension) which works on an infinitesimal neighbourhood of the constant loops, and not even that but (as I understand it) works by exhibiting that as some colimit of finite dimensional manifolds. The idea behind this is that if index theory generalises then so should localisation and so we should be able to work in a neighbourhood of the fixed point set. But the number of “if”s there is quite large.
My concept of “Dirac operator” is “first order differential operator on sections of the spinor bundles whose symbol is Clifford multiplication” which seems fairly close to what it ought to be. Is that not reasonable?
As for the co-Riemannian/Riemannian swap, I did try to argue that this wasn’t “redefining the problem” but was rather a case of choosing how to extend from finite to infinite dimensions. Indeed, in one talk then as I laid out the standard definition of the Dirac operator in finite dimensions using the Riemannian structure then one person in the audience said that I ought to use the cotangent bundle instead.
Regarding the publication: after summer it will be different, and the refereeing will take time anyway, and if the refereeing is done publically as in the other cases, then there should be no problem.
Regarding Konrad Waldorf and Kottke-Melrose: I think they are still mainly into differential geometry and the description of smooth string structures as such, with the Dirac operator on smooth loop spaces serving as a motivation, but not being actually discussed yet by them. In fact I just removed the Kottke-Melrose reference from the entry on Dirac operators over smooth loop space, since it really does not say anything about these.
Regarding your construction: Let me just re-iterate that I am not doubting any of your steps, I am just trying to make you tell me more explicitly what you say your construction does in comparison to other available constructions.
And you did. Thanks!
I have now reworked a good bit further the References-section. Let me know if you feel this now reflects the situation decently. If not, tell me and I add more comments (in case you don’t want to edit there yourself).
(I should say that I looked at the articles by Brylinski and Landweber there, but not at those by Taubes and Léandre. I take it that you are saying that these, too, work just on formal loop space, and accordingly I have organized them in the list. But please check.)
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