Last summer at the Hausdorff institute in Bonn # we had started working on the following; now it is finally converging to something. Maybe somebody is interested in having a look.
Hisham Sati, U. S., Jim Stasheff,
Twisted differential String- and Fivebrane-Structures
(pdf, Lab)
TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1935
“leat to any natural” should be read “leads to any natural description, or dual, to what was”
Thanks for the references. I’ll have a look. Good to see from the Lee-Park abstract that apparently finally it is recognized that the membrane 3-algebra is the old Nambu algebra. We talked about that at Lie 3-algebra on the membrane(?).
I have to say that am still not sure what an -Lie algebra that is not an example of an -algebras, i.e. without grading and without the brackets being of the right homogeneous degree in the grading would “mean”, abstractly. Things that can be integrated to Lie -groupoids are not of this form, it seems. But I may be missing something.
Daniel de Franca asked about the relation, if any, between our discussion of the Green-Schwarz mechanism in terms of -algebra connections and the use of 3-Lie algebras that is currently so popular over at hep-th.
I observe two things:
from what I have seen, the current hep-th discussion of 3-algebras revolves around the following two aspects:
1) the Nambu-Goto worldvolume action of the fundamental -brane, and accordingly its Polyakov form, are naturally expressed in terms of the -ary Nambu Bracket on the embedding fields induced by the skew-symmetric degree differential operator on the worldvolume;
2) specifically for another generic way to produce a 3-Lie algebra is in terms of a Chern-Simons term, namely using the trinary bracket of a String Lie-2-algebra (e.g. section 6.4, p. 47 here) (yes, since in these circles an ordinary Lie algebra is called a 2-Lie algebra it is the String Lie 2-algebra which corresponds to what is called a 3-Lie algebra).
Here my remarks on both points:
the first point should really be well known and established knowledge. My impression is that it has been fogotton by the community and is now being redicsovered. I may be wrong, but Lee and Park rederive this on p. 3 of their article Three-algebra for supermembrane and two-algebra for superstring which Daniel mentioned and point to their summer 2008 article for details. But this old observation was the basis for ideas about covariant membrane matrix models back in the late 1990s. I don’t know to whom it shoulod first be addressed. I just remember that I review it as a simple well-known fact in my dimploma ( MaS) thesis around p. 176.
Now concerning the second point, the construction of “3-algebras” in terms of structure constants of ordinary semisimple Lie algebras:
this is the point where there is a sort of contact to what we are talking about in the above article, and in particular to what we are talking about concering -connections.
Namely a popular class of examples for “3-algebras” is really the String-Lie 2-algebra in disguise, built using the canonical 3-cocycle on a semisimple Lie algebra with invariant bilinear pairing (Killing form) . One can see this for instance on p. 2 of Sato’s Covariant formulation of M-theory (as well as in other articles which we discussed recetly in this context): if you disregard the generator there and take the to be in degree 1 and the to be in degree 2, then this is precisely the String Lie 2-algebra, with the trinary bracket being its associator! This is nicely described in detail in the original Baez-Crans article, see in particular section 6 or see the – for the present purpose more amplified – review of these points in section 2 of From Loop groups to 2-groups.
Now, indeed, this String Lie 2-algebra does and is known to control the Chern-Simons coupling of the membrane.
From the point of view of generalized Poisson-Sigma-models this was amplified by Kontsevich et al. in their seminal AKSZ formalism.
From the point of view of generalization of the WZW term of the String, this was the punchline of section 9 of our discussion of -connections.
As we discussed there and amplify in the article above again: the Chern-Simons background field can be understood as arising as the obstruction to lifting the ordinary background -bundle to a -2-bundle, controlled by the String Lie 2-algebra, and the form of the Chern-Simons term is directly induced from that.
Indeed, as Daniel notices, this is the mechanism that is at work and indeed spelled out in the discussion surrounding the diagram on p. 72 of the above document, which in turn is a slight extension of the corresponding discussion surrounding the diagram on page p. 79 of -connections.
Urs wrote:
1) the Nambu-Goto worldvolume action of the fundamental k-brane, and accordingly its Polyakov form, are naturally expressed in terms of the (k+1)-ary Nambu Bracket on the embedding fields induced by the skew-symmetric degree (k+1) differential operator on the worldvolume;
Is this action the only one known for the k-brane? if not, any reason it should be preferred?
Any reason physics needs k-brackets with only one k in any specific problem?
Any reason physics needs k-brackets with only one in any specific problem?
Actually, it would surprise me if this is really the case.
It doesn’t seem that the current “3-algebra mini-revolution”, whatever it is, has been conclusive in this respect.
Is [the Nambu-Goto] action the only one known for the k-brane? if not, any reason it should be preferred?
All kinetic actions for branes seem to be variants and generalizations of the Nambu-Goto action.
The NG action functional itself is certainly singled out by its conceptual simplicity: it simply computes the (proper) volume of the brane’s worldvolume as measured by the background metric.
For general D-branes, the action is the “DBI-action” (Dirac-Born-Infeld): this is the like the NG action but where the metric is modified by a gauge field.
Urs wrote:
From the point of view of generalized Poisson-Sigma-models this was amplified by Kontsevich et al. in their seminal AKSZ formalism.
The link is to Roytenberg’s article of which the abstract neatly emphasizes that AKSZ casts things in terms of graded manifolds. In the recent arXiv:0903.0995 Finite dimensional AKSZ-BV theories
by Francesco Bonechi, Pavel Mnev, Maxim Zabzine, in section 7, it is pointed out that the mechanism has a purely homological version quite in keeping with the original BV (Batalin and Vilkovisky) treatment.
In the recent arXiv:0903.0995 Finite dimensional AKSZ-BV theories
by Francesco Bonechi, Pavel Mnev, Maxim Zabzine, in section 7, it is pointed out that the mechanism has a purely homological version quite in keeping with the original BV (Batalin and Vilkovisky) treatment.
Thanks for the link! Will have to read this. For the moment I don’t have the leisure but just collected the reference at a preliminary entry Lab: AKSZ formalism.
Eventually I should be able to figure out in more detail than I have so far how AKSZ follows from the general idea of -models after passing from -groupoids to their -algebroids. At the moment I am lacking some details here, unfortunately.
I am wondering though if David Ben-Zvi in the context of his work on -models can see the AKSZ picture arise more manifestly….
While I try to digest your comments, take a look at this Lubos Motl’s post. He wrote a nice only about Matsuo Sato’s first article. There might be insightful things there.
OK,
so after a few months I digests some of the meat, but I have some really basic concept questions about string and 5 brane structure. Well, the question is very simple:
Are these string and 5-brane structure related to M2 and M5 branes in any way?
I’m sorry if the question is obvious, but it is just that these articles seems to be all all about 10 dimension, but not in 11, like M-Theory.
so after a few months I digests some of the meat,
Hey, that’s great. Thanks for coming back to this here.
but I have some really basic concept questions about string and 5 brane structure. Well, the question is very simple:
Are these string and 5-brane structure related to M2 and M5 branes in any way?
This is an excellent question, I think.
I don’t have the full answer. I do have some kind of idea, though. I will now pronounce this idea, but I want it to be understood that this is tentative and that I don’t speak for my co-authors in as far as this is wrong, while I am possibly just parroting what I learned from osmosis as far as this is right (maybe not very far).
So, here is the picture I can provide:
recall the situation one level down:
there is a bulk space that carries an abelian [[gerbe with connection]]/[[differential 2-cocycle]] – the Kalb-Ramond field.
On certain subspaces (the [[D-branes]]) this cocycle serves as the [[twist]] of a twisted differential 1-cocycle (the Chan-Paton field).
It is this twisted 1-cocycle (twisted bundle with connection) that the boundary of the -brane (the fundamental string) couples to.
Now we boost this scenario up one categorical dimension. It seems to yield the following story.
there is a bulk space that carries an abelian 2-grbe/differential 3-cocycle – the supergravity -field.
On certain subspaces – called Hořava-Witten 9-branes – this cocycle serves as the twist of a twisted differential 2-cocycle on (the [[Green-Schwarz twisted]] Kalb-Ramond field).
It is this twisted 2-cocycle (twisted gerbe with connection) that the boundary of the -brane (this boundary is the -brane) couples to.
I guess you see the pattern that this suggests.
Urs, regarding Chan-Paton factors, how do string theorists know whether to choose the full symmetry group as, say, U(N), SO(N) or USp(N)?
What might happen if as a general starting principle we assume that N=infinity? Thanks.
What might happen if as a general starting principle we assume that N=infinity?
There might be several answers to this, some of them maybe even good answers.
Explicitly not attempting anything even close to a comprehensive answer, let me just mention this:
it’s an oversimplification to say that the “Chan-Paton field” on coinciding D-branes is a (twisted) -cocycle, at least in the susystring theory, which I’ll take for granted we are talking about.
There the physics tale has it that only the net number of branes and anti-branes matters, which translates mathematically to the fact that its not differential refinements of that control the structure of the field, but of the K-theory spectrum , where the “” here, as you may know, is some version of “”.
So the “Chan-Paton field”, if we do call it that way, is not really an element in nonabelian differential twisted -cohomology for any , but rather in twisted differential K-cohomology, which in some rough sense is a ““-version.
Hmmm. I think I disagree with several of the things you are trying to categorify.
there is a bulk space X 10 that carries an abelian [[gerbe with connection]]/[[differential 2-cocycle]] – the Kalb-Ramond field.
In the bosonic string, the gauge equivalence class of the Kalb-Ramond field is an element of . In the Type-II string, the Kalb-Ramond field is a more subtle object. It is a twisting of differential K-theory.
The equivalence class of such a twisting is an element of a generalized differential cohomology group that Dan Freed, Greg Moore and I call . This abelian group has, as a subgroup, the aforementioned . More precisely, there’s an exact sequence
(Actually, this is a little fancy. In the ordinary (non-orientifold) Type-II case, this sequence is partially split, and . For orientifolds, the corresponding sequence is not even partially split.)
On certain subspaces (the [[D-branes]]) this cocycle serves as the [[twist]] of a twisted differential 1-cocycle (the Chan-Paton field).
It’s a twist of K-theory, not of “differential 1-cocycles”.
On certain subspaces – called Hořava-Witten 9-branes – this cocycle serves as the twist of a twisted differential 2-cocycle on (the [[Green-Schwarz twisted]] Kalb-Ramond field).
I don’t see that at all. The C-field doesn’t “twist” the B-field on the boundary. Rather, it “is” the B-field.
Also, note that, in contrast to the Type-II case, the gauge equivalence class of the heterotic Kalb-Ramond field is not a differential cocycle (of any kind). Rather, it’s a non-flat trivialization of something you would call a twisted string structure.
Note that the Hořava-Witten setup, that you are talking about here, crucially depends on a choice of orientation-reversing involution of . The subspace, , is not just any-old subspace, it’s the fixed-point set of this involution.
So, if you want to see this a some categorified version of the situation one dimension down, you should start with the orientifold story there.
Thanks for the reply, Jacques. Much appreciated.
I see two issues addressed here, which I’ll try to reply to
- 1st ist that, indeed, I am talking here for the moment about a slightly simple minded version where I assume the Chan-Paton field not as a cocycle in full (twisted) K-theory, but just as a (twisted) -bundle representing a class in there. I am working on getting the full correct K-theory picture under control, so bear with me hear while I gloss over K-theoritic subtleties that you rightly point out. I gather that in addition to the orientifold there is another -grading in the game here is the one that is missing in our hep-th/0512283 as compared to your Orientifold precis and which comes from susy/K-grading essentially. I’ll catch up on this in little while, promised.
- 2nd is that I am thinkind that what Dan Freed calls “non-flat trivialization” is the abelian part of a twist in nonabelian differential cohomology. So I am being a bit unwise here in not entirely following Dan Freed’s terminology, which I realize is risky (for me), but I’ll try to provide some motivation for it now.
The C-field doesn’t “twist” the B-field on the boundary. Rather, it “is” the B-field.
Also, note that, in contrast to the Type-II case, the gauge equivalence class of the heterotic Kalb-Ramond field is not a differential cocycle (of any kind). Rather, it’s a non-flat trivialization of something you would call a twisted string structure.
So this “non-flat trivialization” (a term that, I gather, was introduced in Dan Freed’s “Dirac charge quantization”) is what I would call a twisted cocycle.
The C-field on the heterotic boundary is a differential refinement of the first fractional Pontryagin class of the Spin bundle (minus the corresponding degree 4 class of the gauge bundle). If that would trivialize we had a (differential) String structure given by a String-2-bundle with connection.
If it doesn’t trivialize, we still get a twisted String-2-bundle with connection. This has an abelian component, which is, I think, what Dan Freed calls a “non-flat trivialization” in abelian differential cohomology.
More concretely (ignoring the gauge bundle contribution for simplicity for the moment) the C-field is a differential refinement of a cocycle with values in , which is the obstruction under
to lifting (the differential refinement of) our Spin bundle through .
That String-bundle will not exist when the C-class is non-trivial, but a “twisted” version will exist, namely a (differential refinement of) a cocycle whose projection is the prescribed -field.
The differential refinement of this is locally yields differential forms with values in the corresponding Lie 3-algebra, which is of the form with structure such that differential form data with values in it is
such that
Forgetting the nonabelian cocycle with local differential form involved here this reproduces the “non-flat trivialization” of the differential = the Chern-Simons 2-gerbe in .
All this is exactly analogous to the story one dimension down, if you allow me for the time being the slight oversimplification, as I mentioned, that I regard the Chan-Paton field not as a cocycle in proper, but just as a (twisted, in turn) -bundle.
The KR gerbe is the differential refinement of a cocycle in , i.e. in . Under
its restriction to the brane is the obstruction to lifting the -bundle with connection on the brane to a -bundle with connection (which I dare to call here a nonabelian differential 1-cocycle, even though more properly this should be regarded properly in K-theory, yes).
The lift through won’t exist, but again a twisted lift will exist, which we can realize as a cocycle with values in whose projection under is the prescribed restriction of our KR-class to the brane.
Such cocycles are precisely what people call twisted -bundles or gerbe modules. Taking the differential refinement and emplyoing -algebra valued forms as before, we find the familiar twisted Bianchi identities relating the bundle’s curvature with the gerbe’s curvature , i.e. again the cortrect “non-flat trivialization”. In view of the result of section 6 of Twisted K-theory and K-theory of bundle gerbes we find that these twisted -cocycles indeed represent the expected twisted -classes here, even in the somewhat simple-minded approach.
Hi Urs,
I don’t really understand what you are saying, I have to study more, because now, from what you say, I have even more basic questions, since I am completely confused:
I heard that M2 and M5 branes are the fundamental objects of M-Theory. Shouldn’t you look for “self consistent” sigma models based on these n-particles?
By self consistent I mean, more or less, a QFT on the n-particle that made it possible to find the critical dimension of the target space without resorting to string theory. For example, finding that the M2 and M5 branes does not lead to ghosts in D=11, without thinking about superstrings. But finding them upon compactification, of course.
I am LOST.
Shouldn’t you look for “self consistent” sigma models based on these -particles?
Somebody should, if maybe not me. And somebody did, if maybe not completely exhaustively.
Various quantum consistency conditions on the “higher gauge fields” that appear in these higher supergravity theories have been worked out, notably due to anomaly cancellation in 10 and 11 dimensions. Various local formulas for the twisted Bianchi identities are known. Some things about in which cohomology theories these local expressions live when they are expanded to global structures that live on the entire spacetime are known.
But several aspects are not fully understood yet and in any case the goal is to draw a more complete and more precise picture of what the gauge field background in a consistent geometric string/M-background are.
This is what Dan Freed, Jacques Distler and Greg Moore have been working on, and about which they have now put first results to the arXiv: to say precisely what a String-background is, with all the right global cohomological conditions.
And this is, if I may say so, the kind of thing that me and my co-authors are interested in here, too, if maybe from a slightly different angle. The idea to take the existing physical literature, the existing known anomaly cancellation mechanisms etc, and see what that now actually means in terms of coherent global structures.
The aspect that I am feeling I’d like to and could add to the existing insights is that of a bit of nonabelian differential cohomology. For it turns out that various of the twists of the abelian gauge fields that one sees in these higher supergravity gauge theories are naturally thought of as intrinsically living not in abelian differential generalized cohomology, but in nonabelian version. For instance the claim is that the famous twisted Bianchi identity that one sees in the Green-Schawarz mechanism, with all its ingreditent including the data of the nonabelian connection forms on the underlying Spin and U-bundles is naturally interpreted as the connection data on a “twisted String-2bundle”, a nonabelian gadget, quite analog to the “twisted vector bundles” one dimension down.
(These twisted String-gerbesm by the way, had first been considered by Aschiere and Jurčo in their article of M5 brane anomalies.)
“the goal is to draw a more complete and more precise picture of what the gauge field background in a consistent geometric string/M-background are.”
And one of the unknown parts, that you are trying to figure out it is “that various of the twists of the abelian gauge fields that one sees in these higher supergravity gauge theories are naturally thought of as intrinsically living not in abelian differential generalized cohomology, but in nonabelian version.”
Frow what you’ve seen, can you tell me a guess what kind of sigma model yields all these backgrounds? Could it be a non supersymmetric model?
“Could it be a non supersymmetric model?”
Let me explain why this question is not random. According to article on wikipedia: “Quantum group symmetry, present in some two-dimensional integrable quantum field theories like the sine-Gordon model, exploits a similar loophole.”
Then, I looked for Quantum Groups
In Drinfeld’s approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0.
Then, I looked for Hopf algebra
In botton there is a link for anyon algebra, which is a stub. Nevertheless it made me look for anyons
So, we have there:
“Here the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO(2,1) which do not arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). These representations are called anyons.“
It looks like a string thinkened in one dimension. Notice that we are talking about operations dealing with fermions and bosons, but instead of exchanging them, like in SUSY, one is generalizing.
This is why I asked if, say, M2 might not be supersymmetric on high energies, but an anyon.
What do you think?
Frow what you’ve seen, can you tell me a guess what kind of sigma model yields all these backgrounds? Could it be a non supersymmetric model?
Daniel, remind me where in the discussion we are.
I take it that you know that everything 10-dimensional we talked about here comes from certain 2d supersymmetric conformal field theory -models with or supersymmetry and central charge . Right?
The name of the game is: given that parts of the space of the SCFTs may be coarsely parameterized by certain differential geometric data on a 10-dimensional manifold, what is the fine-grained description of these geometric parts of the space of these SCFTs. That’s the question of understanding the differential cohomological natural of string backgrounds.
Now, at some point you started asking about the M2 brane. That’s a somewhat more speculative idea where one imagines that all these 2d SCFTs points above are really boundary theories of certain 3d QFTs of sorts.
Comparatively little is known about these 3d theories. There used to be some industry on studying these super-membrane sigma models back in the late 80s early ninties. If I recall correctly that got washed away when the meme spread that applying these and those tricks and limits, these QFTs would look like “matrix mechanis” and all problems of this world would be solved by saying “matrix mechanics”.
Then nothing much happned for a long time. Then somebody said “membrane 3-algebra” and again a storm broke loose. I can’t really tell you, though, what is seen when the dust of that particular storm settled.
Anyway, give me and the rest of us a few more coordinates on context of your questions (maybe make them simpler, more concrete, less speculative) and chances are that one or two good answer will materialize.
“Anyway, give me and the rest of us a few more coordinates on context of your questions (maybe make them simpler, more concrete, less speculative) and chances are that one or two good answer will materialize.”
p.6 of David Gross’s opening lecture of strings 2009
What is that 3D Chern Simons doing in the M-Thoery, in the top of the pic?
p.6 of David Gross’s opening lecture of strings 2009
What is that 3D Chern Simons doing in the M-Thoery, in the top of the pic?
That’s a good question!
Years back, when I certainly knew less what I was talking about than now, I made an uneductaed guess about that right at the beginning of this blog entry here.
I guess the general idea is that one notices that the relation by which the WZW model sits on the boundary of CS theory shouldn’t be entirely unrelated to the way the heterotic background theory sits on the boundary of the Hořon-Witten 11-d background.
And indeed, if you think of the -current algebra bit in the CFT of the heterotic string, which looks like one chiral half of an WZW model, and when you notice that due to quantization constraints on the supergravity -field, whose integral over the membrane is its gauge kinetic term, is actually required to be a Chern-Simons 3-form, then this begins to look a little less than entirely speculative.
But I am not really sure. I remeber years back Aaron Bergman made some useful reply to me on this in some newsgroup, somewhere, but I lost it.
That happens when you don’t archive insights into the Lab. You just lose them! :-)
Thanks Urs, I typed “E8 WZW” on google and found this.
I guess it explains what you wanted to say, doesn’t it? There is even the discussion with Aaron there!
I guess it explains what you wanted to say, doesn’t it? There is even the discussion with Aaron there!
Yes, thanks!
The only thing more useful than a well kept Lab is an army of attentive blog commenters. :-)
” attentive blog commenters”
I guess this is why I cannot forget “Branes and Quantization”.
Since we are talking about WZW and Gerbes, would you mind reading this article some this article, please?
I guess I already pointed out this article to you, but right now I really don’t know anyone better than you to take a look at this. See the blog link on the trackback.
Question from the peanut gallery…
What is “nonabelian” about nonabelian cohomology?
I’m shaky even on cohomology, but when I hear “(non)abelian” I think of products and the product that comes to mind when I think of cohomology is cup product.
Given cochains and , then we get cohomology classes and . The cup product of cohomology classes is “(skew) abelian”
However, we know that cup product is not only defined at the level of cohomology, but we can define cup product at the cochain level (before passing to cohomology). In this case, we have
I would be tempted to call this “nonabelian”, but as we know from our paper, this is not a bug, but rather a feature. The failure of cup product to be (skew) abelian at the cochain level gives all kinds of wonderful stuff.
Is that at all related to nonabelian cohomology? I’ve read the nLab entries, but they are frankly way over my head :)
It would be kind of neat if nonabelian cohomology were related to doing cohomology-type stuff, but with cochains instead of cohomology classes (if you know what I mean).
Conversely, it would be kind of neat if what we were doing was actually nonabelian cohomology in our paper.
Hi Eric, there are two earlier ways to derive non-abelian cochains from classical cochains. The first was due to P. Dedecker and the second due to L. Breen. Please see Section 2 on page 2 of this paper by L.M. Ionescu for further explanation:
http://www.intlpress.com/HHA/v6/n1/a5/v6n1a5.pdf
I quite agree - non-ableian cohomology is a catchy name (even grant worthy? cf. quantum groups) but almost lacking in clear meaning
as for the (co) chain level, that’s what derived cats are all about
not to mention my life’s work
Interesting. Then again, everything you point to is interesting. I should make the rest of my life’s work trying to understand your life’s work :)
I don’t know the first thing about nonabelian cohomology or derived categories, so this question is probably meaningless, but is there a relation between the two? Looking at the reference Charlie provided (thanks for that by the way!) I see they mention “derived functors”.
Note: In asking this question I’m basically trying to rephrase my original question above.
non-abelian cohomology is a catchy name (even grant worthy? cf. quantum groups) but almost lacking in clear meaning
Actually, it has a very precise meaning. Some researchers like Breen built large parts of their career on studying it.
I’ll reply to all of this more extensively in a little while when I am done with what I need to do.
Meanwhile there is, you have guessed it, the Lab, which has introductions, expositions, detailed definitions, examples and commented link lists to references and other resources on all things cohomology and nonabelian cohomology.
Start at [[cohomology]] and follow links from there to your heart’s content.
To answer Eric’s question in a more pedestrian fashion, you’re thinking about the ring structure on the cohomology (in particular, the rule for multiplying cohomology classes).
Most cohomology theories don’t have a ring structure. They’re just (graded) abelian groups.
Now, for any cohomology theory, there’s a notion of taking the same theory, “with coefficients in an abelian group, .” The result is, again, a graded abelian group (which may or may not have a ring structure).
What people mean by “nonabelian cohomology” is something along the lines of taking cohomology “with coefficients in an nonabelian group, ”.
Depending on what cohomology theory you started with, there is often a good theory of , but the status of the higher “nonabelian” cohomology groups is pretty murky (at least, to me), and — like Jim — I’m not sure there’s a good theory.
In any case, I find it kinda amusing to call any sort of nonabelian cohomology a “simple minded” stand-in for K-theory (which is a perfectly nice ‘abelian’ cohomology theory, with a ring structure 'n everything).
Thanks for that very clear explanation.
So if nonabelian cohomology is (related to) cohomology with nonabelian group coefficients and derived categories are (related to) cohomology-type stuff performed on (co)chains, what do you call cohomology-type stuff performed on (co)chains with nonabelian group coefficients? Nonabelian derived categories?
I guess I’m wondering where our toy model fits into the big picture (if it does). There we had the graded differential structure so we could have looked at cohomology, but stuck with (co)chains, which makes me think we might have been doing something along the lines of derived categories (if a toned down toy version), but then in the lattice gauge field theory application, we had nonabelian group coefficients. So we were working with a nonabelian graded algebra of cochains on a finitary cell complex with nonabelian group coefficients (see Section 5, Page 67).
PS: You guys probably know my motivation, but just to reiterate, I come at this from the perspective of an “engineer” interested in numerical modeling. To me, spacetime is a cell complex and fields are cochains. My PhD was in computational electromagnetics, which in some company I might describe as applied abelian lattice gauge field theory. My goal is “numerical modeling done right”.
Then I have the secret motivation that “numerical modeling done right” may serve as a fundamental model for describing nature, but that is another story.
Eric, I got stuck trying to answer your question above because sometimes derived categories may be missing cokernels.
Also, in his PhD thesis, Ionescu shows that there is a correspondence between nonabelian group cohomology and the cohomology of a functor, but I have not read that far yet.
I’ll just drop a very brief remark on the following, happy to expand on it in a more constructive fashion later, I don’t really have the time to post here.
I find it kinda amusing to call any sort of nonabelian cohomology a “simple minded” stand-in for K-theory
The reason is that doing Grothendieck group completion after decategorification is easier than doing it before, which is what one needs to do to “do it right” homotopically.
The non-simple minded correct -theory -stack on for smooth K-theory to use is of the kind that Toën-Vezzosi describe on page 82 here.
That’s arguable a little less simple minded than working with the -stack and group completing after taking cohomology classes.
What people mean by “nonabelian cohomology” is something along the lines of taking cohomology “with coefficients in an nonabelian group, ”.
Depending on what cohomology theory you started with, there is often a good theory of , but the status of the higher “nonabelian” cohomology groups is pretty murky […]
That's because you really want to have an -groupoid to define .
Now, the -fold delooping of an -tuply groupal set is an -groupoid, and an -tuply groupal set for is simply an abelian group, so that works. A -tuply groupal set is simply a group, so you can do with that. (And a -tuply groupal set is simply a pointed set, so you can do with that.)
On the other hand, the automorphism -group (for a group) is a -group, which can be delooped once to form a -groupoid, so you can get that way … but it's inconsistent with the previous way when happens to be abelian. (That is, the map differs from the map , even when you further deloop and follow this with , except for degenerate cases of .)
To tie in with David’s comment, while there is no pointed space with for a nonabelian group, there is a space with for a -groupoid. (Note capitalisation; is an -tuply groupal set for a pointed space, while is an -groupoid for a space. For the (pointed) Eilenberg–Mac Lane space , is equivalent to the -fold delooping of , which in turn is .)
I am assuming throughout the validity of the homotopy hypothesis and the delooping hypothesis. Although these have been proved for some definitions (perhaps ones that make them trivial), I'm not sure that they have been proved for anything that covers all of the above.
I’m not sure that they have been proved for anything that covers all of the above.
But you can embed everything you just said into Kan complexes for which everything works.
I should add that one thing that makes nonabelian cohomology murky to those used only to abelian cohomology is that (for an abelian group) is a nice family of cohomology groups, which in fact might even interact (say through the cup product) in certain situations. (In terms of the above, this comes from interpreting as an -tuply groupal set and delooping times to get an -groupoid.)
So while makes sense for an -groupoid, that is only one cohomology object for any given . Maybe you can make a family out of an arbitrary group by applying an operation for iterations; I'm not sure. Certainly this doesn't match the previous family when happens to be abelian. Even the (yet more) general viewpoint of cohomology as giving spaces of maps (as mentioned in David's comment op cit and elsewhere) doesn't give us this family that we're used to; you have to use a family of spaces to reproduce abelian cohomology.
Nonabelian cohomology (as Urs has explained several times) is a fairly straightforward generalization of cohomology (once all the dust settles) in which the coefficient abelian groups are replaced with homotopy types of spaces. Namely there’s a construction H(X,A) where X is the space we’re measuring, A are the coefficients, and H just means homotopy classes of maps - though we can easily make X be something like a variety, stack, or any other kind of geometric object (more below). Usual cohomology H^n(X,Z) is the special case A=K(Z,n), the Eilenberg-MacLane space.
Lifting from cohomology groups to cochains means replacing the homotopy classes of maps by the SPACE of maps from X to A (of which the former is just pi_0).
The word “nonabelian” I find highly misleading: all of the nonabelian nature of homotopy types has to do with pi_1, which is where we find arbitrary nonabelian groups. pi_0 is just a set and the higher pi_1 are abelian! of course there is interesting “nonabelian” nature to the way spaces are built out of these homotopy groups, but at the end of the day higher nonabelian cohomology gets more and more abelian as we look higher and higher up. There isn’t a space with pi_2=G for a nonabelian group G, so saying H^2(X,G) doesn’t make sense in this way (there are other ways of making sense of it, as say the 2-category of G-gerbes, but if you try to get a space or set from this in the end you’ll be seeing a cohomology theory of the kind we discussed above, so something like the center of G in pi_2 anyway.. so there isn’t a set or space which you could call honestly nonabelian H^2.. happy to be explained otherwise).
In another direction, we can replace spaces by sheaves of spaces over X, and get nonabelian cohomology with twisted coefficients.
(ie we replace maps to A by sections of an A-bundle over X). Or we can replace A by a spectrum, like the K-theory spectrum, or a bundle of spectra over X, to get (twisted) generalized ABELIAN cohomology theories — spectra are the “abelianization” of spaces..
At the end of the day though all of these theories are just looking at (derived) global sections of some sheaf (of spaces or spectra) over X.. which to me sounds a lot less fanciful then what the word nonabelian cohomology initially connotes.. but can equally easily be formulated for any notion of space X (as was Grothendieck’s vision for algebraic geometry).
Thanks, David.
The word “nonabelian” I find highly misleading:
Yes, it’s bad. Also because the more general term should have less adjectives then the special one.
If we could rewrite history, we’d speak of just cohomology and abelian cohomology. Well we are rewriting. On the Lab…
all of the nonabelian nature of homotopy types has to do with pi_1
and its action on the rest, and the rest of the Postnikov information, yes.
And this is precisely the data that is secretly involved when people speak about various “twisted” cohomologies.
So in my last Oberwolfach talk I offered the audience the following principle to think about this (full text here):
Principle
Higher nonabelian cohomology disguises as twisted higher abelian cohomology;
conversely: twisted higher abelian cohomology is really nonabelian cohomology
So the claim is that when one is looking at twisted cohomology one can guess it to some extent. The picture clarifies in nonabelian cohomology.
Why don’t you invent a new word for “nonabelian cohomology” and then “cohomology” is an abelian version of whatever you call “nonabelian cohomology”? :)
Or, as your comment hinted at, just “Play Bourbaki” and call nonabelian cohomology simply “cohomology” and what was formerly known as cohomology becomes “abelian cohomology”.
Is is possible to clarify the relation between
Principle
Higher nonabelian cohomology disguises as twisted higher abelian cohomology;
conversely: twisted higher abelian cohomology is really nonabelian cohomology
and the more usual notion of twisted abelian cohomology, noted by David, above, namely
replace maps to A by sections of an A-bundle over X
(where is a spectrum).
Is is possible to clarify the relation between
Principle
Higher nonabelian cohomology disguises as twisted higher abelian cohomology;
conversely: twisted higher abelian cohomology is really nonabelian cohomology
and the more usual notion of twisted abelian cohomology, noted by David, above, namely
replace maps to by sections of an -bundle over
(where is a spectrum).
Yes, in more detail the principle above implies in particular that a situation of a pair consisting of an abelian thing acted on by a nonabelian thing can be equivalently thought of as one single nonabelian thing: the homotopy quotient of by , a concrete realization of which is the Borel construction . This is not a spectrum anymore, but is a space.
Once we place ourselves in a context where both and live, this is best thought of as a game with fibration sequences.
So for instance the fact that some acts on some is witnessed by the existence of a left-long fibration sequence
where in language used among nonabelian cohomologists, as it were, we have
is the action
and
is the action groupoid.
For our general nonsense purposes we don’t need to care here what exactly is, as long as the action on it induces this sequence, but in concrete realizations that’ll be of interest. So we can just as well assume that , which will be useful in the following.
So we look at the remaining fibration sequence, which sits by definition in a homotopy pullback square
The universal property of this homotopy pullback says precisely that:
- the obstruction to lifting a (“nonabelian” or “twisted”) -cocycle to an -cocycle is its image in first -cohomology under the above horizontal map. That’s the twist.
Read the other way round it says:
-cocycles are precisely those -twisted -cocycles whose twist vanishes.
More formally, but without adding any genuine new information, since cohomology is just connected components of the Hom in our context, we know that for any we have a fibration sequence
So if we fix the twist in (which you’d want to write as ) to some then the -twisted -cohomology is precisely that bit of that sits in the homotopy fiber over .
Therefore we may say that the -twisted cohomology is the homotopy pullback in
To see that this does indeed reproduce the description in terms of sections of associated bundles,
look at the long fibration sequence one step down the row, where it reads
and exhibts as the bundle with fiber -associated to the universal -bundle.
For the given -cocycle the corresponding associated bundle with fiber over is the further homotopy pullback in
And again it is precisely the universal property of the homotopy pullback that asserts that sections of this bundle are in bijection, up to homotopy, with those maps whose projection to reproduces the prescribed twist.
Toën writes:
The expression non-abelian cohomology can be quite ambiguous. One could try to make it clearer by the following observation. Abelian cohomology is certainly dual to (abelian) homology. On the other side, homology is nothing else than the abelianization of homotopy, or abelian homotopy. Therefore, non-abelian cohomology should really be understood as dual to homotopy.
Hey, why don’t we cancel the current use of cohomotopy and use it to rename non-abelian cohomology?
I should of course say extend rather than cancel the current use of cohomotopy, which seems rather limited at present for a potentially weighty term.
David C quotes Betrand Toën:
Therefore, non-abelian cohomology should really be understood as dual to homotopy.
Yes, we keep talking about that. Maybe we still haven’t presented that with the necessary impact on the Lab. I have now added a short section “relation to homotopy” to [[cohomology]].
We should do a more mirror-symmetric discussion at homotopy, eventually.
Hey, why don’t we cancel the current use of cohomotopy and use it to rename non-abelian cohomology?
I should of course say extend rather than cancel the current use of cohomotopy, which seems rather limited at present for a potentially weighty term.
Yes, somehow the historically grown terms are mixed up, really.
I agree that logically speaking homotopy and cohomotopy (for cohomology) would be the good dual pair of words.
But I see personal trouble ahead if in my everyday business I start saying cohomotopy for cohomology. Even more trouble, that is…
I was actually going to suggest using “cohomotopy” as well (without realizing it was actually justified). I like that. I think you can avoid any political ruckus by always clarifying that cohomology is “abelian cohomotopy”.
I think you can simply start using “cohomotopy” instead of (the loaded?) “nonabelian cohomology” and continue to use “cohomology” for abelian cohomotopy, but then you have the opportunity to explain the relation whenever the issue comes up.
This is basically what I said a few comments back about making up a new word, but it seems the natural “new” word would be “cohomomotopy”.
So can we go through line by line the [[cohomology]] entry and dualize?
E.g.
the objects are the cocycles on with values in ,
becomes
the objects are the cycles on with (co)-values in ?
And “classes of special cases of cohomologies with their own entries include” becomes “classes of special cases of homotopies with their own entries include”: group homotopy; generalized (Eilenberg-Steenrod) homotopy; abelian cosheaf homotopy; etc.?
So can we go through line by line the [[cohomology]] entry and dualize?
In principle, yes, I think so.
E.g.
the objects are the cocycles on with values in A,
becomes
the objects are the cycles on with (co)-values in ?
Yes, only that we may have to beware that we might be using terminology in a non-standard way. This cycle language is usually reserved for homology.
and “classes of special cases of cohomologies with their own entries include” becomes “classes of special cases of homotopies with their own entries include”: group homotopy; generalized (Eilenberg-Steenrod) homotopy; abelian cosheaf homotopy; etc.?
Yes, only that nobody seems to have considered these terms before. So we should make clear what’s going on.
Maybe best would be to start with a sub-entry on the special case of generalized homotopy discussed in Baues’ “Algebraic Homotopy” book (section II.6, starting on page 115), that Tim Porter pointed out.
OK. Made a small start here, but as noted there am wondering whether entry should just be about stable generalized homotopy.
I have been gossiping a bit with David on this so let me add in some extra snippets here. First some points that I have mentioned in my gossips, but will recall here.
The dual of Cech cohomology is not Cech homology. By dual I mean Alexander Spanier (AS) dual, so slightly different (but not so much perhaps)! What is the AS-dual of Cech cohomology? It is Steenrod Sitnikov homology. For this you form up the prosimplicial set of nerves of open covers of your space, form its homotopy limit and then take homology.
This is significant for several reasons but most interestingly for its links with -algebras and other parts of functional analysis. Back in the 1970s Brown, Douglas and Fillmore solved a well known problem of von Neumann’s. The obstruction to something being true was the non-vanishing of a Steenrod-Sitnikov K-homology group. They took this a lot further in subsequent work and eventually it was noted that the duality between compact spaces and (commutative) -algebras led to there being viable homotopy theory on the algebra side. This is very closely related to algebraic K-theory, but I am not really competent to give that connection here (without looking it up … and where on earth did I leave that paper!)
This whole area perhaps suggests that the homotopy as dual of cohomology idea may be important for links with other areas as well as being fun in itself.
The idea that everything in a homotopy type is somehow abelian in dimensions greater than 1 is false. What is correct to say is that the rather weak invariants that are the higher homotopy groups are Abelian.
If you look at a homotopy 2-type then it can be modelled in various ways e.g. a crossed module, 2-group, or other variants of the same idea. These are NOT Abelian objects. The kernel of the crossed module will be the second homotopy group of that homotopy type and that is Abelian.
When you go to 3-types.. even more non-Abelian (NOT LESS) with very pretty models Conduché’s 2-crossed modules, Loday’s cat-groups (highly non-Abelian), crossed squares (Loday, Ellis-Steiner, etc.), and Gray groupoids, etc.
A crossed module is a natural generalisation of a pair consisting of a group and a normal subgroup. A crossed square is a natural generalisation of a group and two normal subgroups. Nothing Abelian in sight!
The space for -gerbes is the classifying space of the automorphism crossed module of the group . Its second homotopy group is the centre of so is Abelian.
To control the cohomology, one should expect to have to use nicely structured coefficients (n-types woul give a neat structure for instance). The cohomology is, as has already been said, really the SPACE or its homotopy type not the . The observations of that homotopy type will be done using a whole range of different `probes’ such as the homotopy groups.
Tim - I’m not sure I see any contradictions between what I said and what you write. When I wrote that things get more abelian higher up I was referring to looking at more and more connective spaces (or covers of a fixed space), not higher and higher truncations of a given type. In any case I think we all have a reasonable intuition for what a topological space (or homotopy type) is, and wanted to emphasize that’s all there is to the coefficients of nonabelian cohomology (independent of the complexity level of the algebraic models for homotopy types one uses).
David, I think I see more clearly what you are meaning but I still think there is a danger (not a contradiction!) in what you say. The invariants of a homotopy type get more subtle as you go up in dimension. The Whitehead product structure and other primary operations give the first layer of this.
You say:the homotopy is “all there is to the coefficients of nonabelian cohomology”, and that we have a reasonable intuition of what a homotopy type is. (Yes an (infinity,1)-groupoid, or a Kan complex or …), but what is perhaps `dangerous’ is that there seems to be an implicit meaning for `cohomology’ with an echo of `group’ hoped for. Why not cohomology homotopy types, cohomology crossed module, cohomology Gray groupoid (up to equivalence) and following Grothendieck’s idea or dream a suitable Galois theory in each suitable situation.
You mention increasing connectivity.. why go down that road. It does lead into an Abelian wilderness, not the rich rain forest jungle of coconnectity. (Aside : Lurking in the undergrowth was an untamed Whitehead product, waiting to devour any unwary cocycle that strayed that way.)
The truncation idea is just as logical as a way to get information out of the original space and does seem to correspond to classification of geometrically significant objects.
I agree completely.. maybe it’s better to talk about “nonlinear cohomology” or “unstable cohomology” to signify moving from spectra to spaces, since the word “nonabelian” puts undue emphasis on groups..
David, I think nonabelian is better than having emphasis on changing little bit in topological, selfdual situation. There are many other algebraic structures where homotopy theory and model categories do not apply, but either homology or cohomology makes sense. Quillen model categories have symmetric (self-dual) axioms, while one can have derived functors in situations which are less symmetric. For example, could one explain sheaves in Q-categories as promoted by Rosenberg via homotopy theory ? I think not, because one does not have exactness properties which hold for localizations yielding to topoi or localizations coming from homotopy. Thus I would prefer historical “nonabelian” because there are extensions which do not fit into the unstable description you just quoted. E.g. puzzling nonabelian cohomology for Hopf algebras, as proposed by Majid, see
[[bialgebra cocycles]]. Could you get at least that one in full generality using “unstable homotopy” of any kind ?
Zoran said
There are many other algebraic structures where homotopy theory and model categories do not apply, but either homology or cohomology makes sense.
So you disagree with this idea outlined by Mark Hovey in problem 1 here?
The essential idea is: whenever someone uses the word homology, there ought to be a model category around.
Looking further down Hovey’s list, has anyone made any progress on 8?
My general theory is that the category of model categories is not itself a model category, but a 2-model category. Weak equivalences of model categories are Quillen equivalences, and weak equivalences of Quillen functors are natural weak equivalences. Define a 2-model category and show the 2-category of model categories is one.
Yes, David, I disagree with Hovey’s metaconjecture in his Problem 1. He might be right, though, with stronger assumption: if one has homology and cohomology simultaneously then yes, I think he is right, a model category is responsible for it (or a slight modification thereof, look e.g. at the axioms for model stacks of Durov). But there are also fundamentally less symmetric situations allowing just for one of the two. I also know from few short conversations that Kontsevich has his own reasons to believe in assymetric story at the fundamental level.
As far as “2-model categories”, I was thinking on that speculative possibility; and then again if there are 2-analogoues of weak equivalences one should do bicategory of fractions in the sense of Moerdijk-Pronk to get 2-coherent improvement of homotopy category. I also think as discussed with Urs yesterday that a 3-step rather than 2-step weak factorization systems should be devised as a building block for the axiomatics.
A thought. We know that in some -topos is the cohomology of with coefficients in .
If the 2-category of model categories turned out to be a 2-model category as Hovey suggests, might we expect that to form a -topos, and if so, would we think of in as a sort of cohomology of ?
If the 2-category of model categories turned out to be a 2-model category as Hovey suggests, might we expect that to form a -topos, and if so, would we think of in as a sort of cohomology of ?
I’d think something slightly simpler would be the right thing to consider here, but of course this will depend on what you want “2-cohomology” to be, really.
But if your “2-cohomology” is supposed to be given by homotopy classes of maps so that you just identify cocycles that are equivalent as morphisms and you don’t want to identify two cocycles if there is just an arbitrary morphism from one to the other, possibly non-invertible, then you would be content with computing the in the maximal -category inside the given -topos.
So for instance it ought to be true that the collection of -categories forms an -category called . This should be the archetypical -topos.
But that -category is maybe hard to get hold of. On the other hand, we are not really that much interested in the non-invertible 2-morphisms in , then the maximal -category inside – let’s call it for definiteness – will be sufficient for all purposes.
And this is easy to get under control. That’s one reason why -categories are so useful: they do reflect on themselves straightforwardly.
Namely the -category of all -categories is, when regarded as an -groupoid-enriched category, simply the subcategory of the -enriched category of simplicial sets on those that are quasi-categories, where in each hom-simplicial set we pick the maximal Kan complex.
So a natural proposal for the notion “cohomology of an -category with coefficients in the -category ” would be
If we do want to think of cohomology in such a higher sense, it might maybe be best to think of the cohomology defined by as that of directed space. That should give the right intuition, I suppose.
Is there any candidate known structure that looks like it might secretly better thought of as “2-cohomology” in this sense than as what it is traditionally thought of?
The comment in -category of all -categories says
…for many purposes it is quite sufficient to regard only invertible transformations between -functors…
Do we know any purposes where this is insufficient? We know of cases where it’s insufficient to consider the mere 1-category of 1-categories.
I wrote:
…for many purposes it is quite sufficient to regard only invertible transformations between -functors…
David asked:
Do we know any purposes where this is insufficient?
Oh, sure, for many other purposes it is insufficient. Say if you need to talk about lax instead of homotopy-pullbacks.
Take your favorite example from higher classifiers:
For an -group, its delooping and a cocycle, the principal -bundle classified by it is the homotopy pullback
of the point in the ambient -category.
But take instead a cocycle to the non-groupoidal object , then the total bundle classified by this (in the usual sense) is the lax pullback
in the ambient -category (where the right vertical map picks the canonical 1-dimensional vector space).
Crucially here the 2-cell filling this square is not invertible: instead: its component on a vector in the fiber over is the non-invertible map that that sends to .
In the same lax pullback manner works the Grothendieck construction that sends a functor to the lax pullback
If one doesn’t have a notion of lax pullback, one has to fake this by instead choosing a suitable replacement pullback diagram whose ordinare pullback already computes the desired lax pullback. For the Grothendieck construction that’s the 2-category of pointed categories in
Similarly I’d expect that if Jacob Lurie had not restricted himself to to the -category of -categories but had used the full -category, he would have been able to classify Cartesian fibrations simply as lax pullbacks
Without this notion he used instead the universal fibration of -categories and realized this as just a homotopy pullback
David also wrote:
We know of cases where it’s insufficient to consider the mere 1-category of 1-categories.
Yes. On the other hand it is quite often quite sufficient to consider the -category of categories, obtained by discarding the non-invertible 2-morphisms.
Note that there is a proper article [[-category]].
Pointing out how nonabelian the algebraic gadgets (-crossed complexes) are that model homotopy types gives a vivid impression, but maybe it is important to emphasize that there is an intrinsic non-abelianness here.
Simply: not every loop-space (= -group) has an abelian product (is symmetric monoidal) . That’s why we talk about nonabelian cohomology.
This intrinsic non-abelianness is there, even if all the input in its algebraic description is abelian.
Consider the 2-group which is the coefficient object for orientifold gerbes.
Its crossed module description consists entirely of abelian groups. Yet, as a monoidal groupoid (= monoidal space) it is non-abelian.
The reason of course is, as Tim emphasizes, the Postnikov data: acts on and the semidirect product group is not abelian.
Urs wrote:
not every loop-space G (= ∞-group) has an abelian product (is symmetric monoidal)
IN fact, very very few do. those that do are products of Eilenberg-Mac Lane spaces
I wrote:
not every loop-space (= -group) has an abelian product (is symmetric monoidal)
Jim remarked:
IN fact, very very few do. those that do are products of Eilenberg-Mac Lane spaces
So here you are think of strict group structures (topological groups). In my comment above I was implicitly taking “symmetric monoidal” in the more general higher categorical sense, as in symmetric monoidal -groupoid. That’s just another way of talking about connective spectra, of course.
So to avoid revisionist language, I should have said above: not every loop space is a spectrum. :-)
The point of rephrasing this in terms of the revisionist monoidal category language was to make that point about what nonabelian cohomology means.
Has anyone else noticed that the examples at the end of
Atiyah - Segal: Twisted k-theory and cohomology implicitly contrast the twisted cohomology of their space Y x CP(2) with the cohomology of the twisted spaces i.e. bundles??
I have just posted
ncatlab.org/nlab/show/History+of+cohomology+with+local+coefficients
whihc is very preliminary but hopefully everyone with additional or improved info will jump in - thanks
Finally, this has made it to a form that made it to the arXiv:
H. Sati, U.S., J. Stasheff, Differential twisted String and Fivebrane Structures (arXiv)
In parts this finally happened because Hisham exerted unnegotiable time pressure on us.
In parts it worked because we switched at some point from working on a LaTeX document with a disheartening maze of internal comments and replies to working with the Lab and developing the various building blocks there
This happened mostly on my personal web, where the overview material at schreiber:theory of differential nonabelian cohomology is essentially section 2.4 now, but for the other parts of section 2 it happened on Lab:cohomology and Lab:twisted cohomology.
Since, by its nature, the article on the arXiv just indicates a theory and toolset that shall appear formally developed elsewhere, the Lab material also had the pleasant side effect of keeping in view all the background material that isn’t spelled out in the article for brevity, but which the article does build on. It seems to me that without that we would still be fighting over whether to split in two articles or not.
Congratulations! This is a perfect example of my light-hearted but sincere comment here.
Yeps, well done. I printed it out and I felt I was at least almost able to understand section 1! Could you perhaps give an explicit equation reference for the intimidated reader in the following statement?
… and the explicit derivation of the twisted Bianchi identities of $L_\infty$ algebra connections corresponding to the Green-Schwarz mechanism and its magnetic dual is in section 5.
Could you perhaps give an explicit equation reference for the intimidated reader in the following statement?
Thanks for the question!
I’ll do something better: I point you to the entry
curvature of -Lie algebroid valued differential forms
(which was greatly expanded a few minutes ago, so be sure to look at it, you haven’t see that entry yet!)
that explains the notion (subsection twisted Bianchi identities) and discusses the examples (subsection examples).
When you have looked at that, open the article towards the end at these big page-filling diagrams. You’ll recognize the diagrammatic form of the twisted Bianchi identities there from the entries above, and then right after that the corresponding equations.
Explicitly, in the present arXiv version v1 twisted Bianchi identities are displayed in (5.14), boxed equation on p. 47, repeated in (5.25) below that and then (5.29).
I beg to differ on several counts. Urs describes well how _he_ works. Note that either the neat feature at n-Lab of the Tex command or the plain source code need significant massaging to make it into a proper article.
Re: Twisted Differential String- and Fivebrane-Structures
Hi Urs,
Does this new article, specificaly what is written in the tables written on section 5.2, leat to any natural seen in the recently formulated Covariant Matrix Theoy part I part II? It uses 3-algebras to derive with practicity other string theories. There are 2 similar articles that foolows a close approach: 1, 2(uses Fillipov algebra). These also attempts to generlize the descriptions of M-Branes.