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started TCFT
eventually we should better reflect at the relevant nLab entries that Costello’s classification result of 2d TCFTs – proving the corresponding conjecture by Kontsevich – was the precurser of the proof of the full cobordism hypothesis: Hopkins and Lurie tried to generalize Costello’s proof (they had not actually heard of Baez-Dolan back then).
the line “This means in particular that when two Riemann surfaces are homologous as chains in the moduli space of Riemann surfaces, then the TCFT will send them to two equivalent morphisms of chain complexes between the in- and the output states.” does not convince me: two Riemann surfaces are just two points of the moduli space, and for fixed genus and number of states moduli spaces are connected, so everything would reduce to say that surfaces with the same topology give the same morphism. I rather think here one is referring to families of Riemann surfaces: these are maps form the base of the family to the moduli space and so can be thought of as singular chains in the moduli space (e.g. by fixing a triangulation of the base).
so everything would reduce to say that surfaces with the same topology give the same morphism.
Yes, but more precisely: equivalent morphisms. That’s why it is topological conformal field theory. But the point is that the equivalences between these morphisms encode more information: if for instance there are two non-homologous 1-chains connecting two 0-chains=(sums of)cobordisms, then this will yield to non-equivalent equivalences of the corresponding morphisms.
oh, I see!
so, we are talking of a functor from a $\Pi_\infty(moduli)$-enriched category to $Ch_\bullet$? mmm.. indeed it should be true that the datum of such a fuctor is the same thing as a morphism from a $Chains(moduli)$-enriched category to $Ch_\bullet$..
added the content of #3 above to TCFT
so, we are talking of a functor from a $\Pi_\infty(moduli)$-enriched category to $Ch_\bullet$? mmm..
That’s exactly the idea, yes! I tried to say this in the entry, likely my discussion needs to be improved.
More precisely:
a “topological CFT” proper would be, as you say, a functor from the $\Pi_\infty(moduli)$-enriched category of conformal cobordisms to some other $(\infty,1)$-category.
But this is not actually considered in the literature. Instead what is considered is the slight simplification, where we stabilize this setup by sending all hom- $\infty$-groupoids to their homology chain complexes.
Have a look at the articles by Costello that I linked to.
I should admit the following:
While I have some idea how one connects the formal definition of a TCFT to the path-integral heuristics that are usually used to characterize the A-model and the B-model, that idea is vague.
Does anyone have literature on how to make this more precise?
Ages back, I had a small conversation with Eric Sharpe on that point here. He said effectively that it is so very obvious that it is almost noth worth writing it out. But I wold still see it written out in as much detail as possible.
So why don’t you invite Eric to join creating a little piece of nlab ?
that’s funny, it’s the kind of ubiquitous statement I am used to find in mathematically oriented string theory papers and I’m never able to understand :) you can also find it, for instance, in Witten’s Chern-Simons Gauge Theory As A String Theory, where differential forms on moduli spaces of curves appear in an “obvious” way..
I love this place :)
Ah, now I remember, this is in Costello’s work, too:
See his TCFTs and gauge theory.
Section 2.2 on page 3 surveys his statements concerning differential forms on moduli spaces and Witten’s article “CS theory as a string theory”.
So Costello in that article does not explicitly construct a TCFT from a worldsheet action functional. Rather, he constructs one connected to its effective background actional functional:
He defines differential forms on the moduli space of worldsheets in an adhoc fashion given the input datum of a “CY manifold” or a generalization thereot, and then shows two things:
that these define a 2d TCFT
that the string perturbation series of this TCFT produces an effective background theory which comes form a Chern-Simons type action functional.
So this supposedly makes Witten’s construction from “CS theory as a string theory” precise. Since Witten in that article argues the connection to A-model and B-model, we get something like a closed circle of ideas. Or at least a closed ellipsoid of ideas ;-)
I added to TCFT a section Worldsheet and effective background theories with a quick brief summary of my understanding of the global story of Witten-Costello. Quite impressive, actually. I hadn’t been fully aware of the scope of this before.
note that there’s a remarkable difference between Witten’s and Costello’s construction. namely, in Witten’s picture, one has differential forms on moduli spaces on the nose, while in Costello the construction heavily (not to say completely) relies on the ribbon graph cellularization of the moduli space. The same happens, for intance, in Kontsevich’s proof of Witten’s conjecture in Intersection theory on the moduli space of curves and the matrix Airy function. Somehow, Witten’s argument says since this space is a moduli space for conformal structures, then this is a differential form on it, while Costello says since this space has a cellular decomposition indexed by ribbon graphs, then this is a differential form on it. so, in a sense, they are of a completely different nature.
the same for Sharpe on TCFT: in his point of view everything is clear and does not rely on particular combinatorial descriptions of the spaces involved.
I seem to remember that Costello somewhere has a remark on how forms on the cell decomposition give forms on the moduli space. Maybe I am hallucinating, but I thought I saw this yesterday somewhere,
Don’t have time to check right now, though. Gotta run.
no, no hallucination: the cell decomposition is a cell decomposition of the moduli space :)
what I was remarking is that constructions involving the cell decomposition invole a particular combinatorics of a space that happnes to be the homeomorphic to the moduli space of Riemann surfaces, while string theorists arguments rely directly on the fact that that space is a moduli space. in this sense the two constructions are of a different nature. the final statement is clearly the same, it is the path to the statement to be different (and I feel that the string theorists point of view which uses the “nature” of the spce rather than its “shape”, is the deeper one, so I’d like to understand that)
Kontsevich’s proof of Witten conjecture can be done for different uniformizations; various analytic strengthenings of toplogical picture are crucial (and not spelled out in the original paper). Especially interesting is the story behind the Hilbert uniformization. There is an arxiv paper on this, by a former student from Bonn.
of a space that happnes to be the homeomorphic to the moduli space of Riemann surfaces
Youneed more than homeomorphic!
yes, of course, you need smoothness on the interior of cells and regularity when you go from a cell to the next one, but this are (however subtle and fundamental to make a rigorous mathematical costruction) technicalities I skipped over. what I’m trying to stress here is that the original string theory reasoning deals directly with the idea of moduli space and does not care of how this space can be “concretely” described.
I’m not saying here that the naive picture works as it is, and that it is not a wonderful piece of math giving the rigorous notion of moduli space of stable maps, virtual fundamental class, and all that. but at a certain point the original ideas which were so neat go lost into the technicalities. and my feeling is that everything is so subtle and difficult because we are using a non-adequate language. there’s somewhere on the Lab a similar statement I love about triangulated categories: it claims that most of the subtleties of triangulated categories depend on the fact they try to capture at a 1-categorical level something that is really higher categorical. I feel that reducing sigma-model generated ideas to combinatorial constructions made up with “known pieces” of maths we are loosing a lot. I’ll write more on this later.
Domenico,
yes, I am aware that the cell decomposition that we are talking about is one of the moduli space that we are taking about! :-)
ok, maybe now I see our misunderstanding: when I said forms on the cells and forms on the moduli were not the same thing that could have seemed a problem of gluing forms :)
my position is: I want to believe, but please tell me what to believe! :)
when I was a first year student, or maybe before that I could have heard or read it somewhere, I was told that every polynomial equation over $\mathbb{C}$ had a zero, that I would have seen a proof of this at my third year, but that we would used this from the very beginning. something like a deal with the devil: belive $\mathbb{C}$ is algebraically closed, and I’ll show you the power of spectral theory! or something like that :) looking at that back, I now see I had become quite expert in a whole series of computations which were ultimately based on an assumption I had no idea of how to prove, I was just accepting it to be true, and building from that.
at a certain point I lost this ability of trusting the assumptions of some theory without seeing how they are rigorously proven, so I now figure myself as a first year student which meets the fundamental theorem of algebra and starts yelling Show me a proof, show me a proof! and when his linear algebra instructor starts to describe a proof, goes on yelling No, I don’t want to hear about compactness, or holomorphic functions, or fundamental groups! Polynomials, I only know polynomials, I want polynomials!
so I feel very stupid :(
what I’d like to have given is a Credo of basic sigma-model manipulations, something like: i believe everything can be integrated; I believe every integral can be localized, etc. so to become able in doing all the manipulations, and only when I’d be extremely confident with manipulating objects I could start thinking of the language to speak of them. it’s like having a language A with 1000 words and a language B with 100000 words, and wanting to translate a poem from language B to language A using a dictionary. it simply does not works. what works is learning language B, understanding the poem and then expand language A in such a way that the poem can be transferred.
ok, too much nonsense for today, better go and have a look at Mirror Manifolds And Topological Field Theory :)
Domenico,
yes, I think I understand what you want. And I meant to say that I believed I had seen a statement that asserted that what you want can be provided, but that maybe I hallucinated that statement. I’ll try to dig it out again. Possibly I was hallucinating after all.
Meanwhile, I added to the entry TCFT the (re)definitions and (re)formulations by Lurie from his TFT article. He actually shows that TCFT is a misnomer: the fact that we are talking about the moduli space of conformal surfaces is just an artefact of how that is parameterized. Inm reality we are just dealing with an $(\infty,1)$-TFT.
(BTW, I was in a seminar all day and now on the train. Therefore all a bit telegraohic…)
thanks, I’ve expanded the paragraph on Lurie’s approach in TCFT a bit. this reminds me of something I was trying to say here from post #210 on. there I was speaking in terms of Frobenius algebra objects rather than Calabi-Yau, but at a suitable general level the two terms are synonyms. rather, what I was missing is that I was looking only at the closed sector, while Lurie correctly looks at the whole open-closed theory 9which can be read from the open sector by the Kan extensione open —> open-closed; then eventually one looks at the closed sector inside open-closed).
Myself, I have no time for this at the moment, but just for the record and for later: this discussion at MO is relevant, and we should eventually put the facts mentioned there into the entry (preferably in a bit more digestible form).
Kevin,
thanks for that! All your help is very much appreciated here.
I am a bit busy with other things right now, as usual, but I would very much enjoy seeing the cluster of entries revolving around TCFT be develop further, eventually.
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