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February 15, 2005

Project: 2-NCG

Posted by Urs Schreiber

In this entry here I want to develop some notes on categorification of Not necessarily Commutative Geometry by means of spectral triples in the sense of Connes, and, eventually and if possible, relations of this to superstrings, loop space geometry and maybe generalized cohomology.

This entry will be work in progress for a while since I plan to develop these notes as I go along.

The original reason for developing these notes in the public is actually the following: I usually type all my notes on my notebook computer. Currently I am visiting John Baez and his group at UC Riverside. It seems that the screen of my notebook computer did not survive the security checks at the airport, unfortunately, so I am left without my crucial thinking tool. This weblog must now serve as a substitute.

But maybe it is an interesting experiment in its own right. I’d be grateful for anyone interested in accompanying me on my quest for seeing more of the big picture of which ordinary NCG, superstrings, loop spaces, Dirac operators, 2-bundles, etc. are jigsaw pieces. Collaboration on papers not excluded.

I had been thinking and talking about related ideas for a long time already. But it was not before last Sunday when I was sipping hot chocololate at Starbucks late at night that something occurred to me which suddenly made all the pieces fall in place.

The crucial catalyst was John Baez’s remark that what I was talking about seemed to call for the 2-Hilbert spaces, 2 *-algebras and the categorification of the Gelfand-Naimark theorem he discussed in [1]. Meditating over that paper I realized that it should be possible to start with a bosonic 2- *-algebra and form its abstract differential 2-calculus by throwing in a formal d-functor. By an analogous procudure one can obtain nice ordinary spectral triples, as I have discussed together with Eric Forgy in [2].

Therefore I propose that a 2-spectral triple should be a triple


consisting of an 2-algebra A, a 2-Hilbert space H on which this algebra acts functorially and on which a 2-Dirac operator D is represented.

My first step shall be to develop in detail a simple but interesting example for such a 2-spectral triple, namely one describing discrete superstrings. Depending on how that works out there are many directions to follow and things to work out.

Last modified: March 20

Here is an outline of some ideas:

Consider the ordinary commutative case first. The algebra of sunctions from the manifold to the complex numbers obviously must become a category of functors from some 2-space to some target.

For simplicity, assume the 2-space to be a graph category Q for the moment. In HDA2 it has been suggested that the correct target is Vect. However, (Vect) Q=modA (where A is the path algebra of the graph) is more like a categorification of functions taking values in the natural numbers. In order to fix that throw in multiplicative inverses to obtain the category bimodA of A bimodules with the product being the tensor product. This is still lacking addtive inverses. In order to get these we need to know how to decompose objects. Given any object B in an abelian category we can consider any exact sequence ABC as a decomposition of B in A and C. This suggests to let the objects be chain complexes in bimod A. Hence we are led to take the derived category D(bimodA) as the correct categorification of the algebra of complex functions.

In the uncategorified case we use the algebra of functions to construct vector bundles as suitable modules of this algebra. The wave function of the system itself is a section of such a bundle.

Hence we should be looking for 2-modules of the 2-algebra D(bimodA). The simplest one is just D(modA). This suggests to interpret elements of D(modA) as states of the categorified mechanics.

Indeed, these objects are known to describe D-brane states with strings stretching between them. More here.

Posted at February 15, 2005 8:10 PM UTC

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Re: Project: 2-NCG


How was the talk? :)

I have a small problem with Section 1.2. Basically, the order in which things happens is a little off.

What came first? The chicken or the egg?

I mean, the inner product or the adjoint? :)

I spent a good fraction of my graduate research trying to find a nice inner product from which I could determine the adjoint. It seems like a natural enough thing to do, but then you came along and said, “Hey, why don’t we just define an adjoint first?” That one simple observation is what made everything fall in place.

Once you have an adjoint, you can construct the inner product in an obvious way (which is also outlined in one of Baez’ papers).

Once you have any adjoint, you construct more general inner products by deforming the original adjoint, not the inner product directly. This places more emphasis on the algebra.

I know it is not a big deal and both chronologies are probably mathematically equivalent, but in terms of “naturalness”, I think cooking up adjoints first is a better way to approach new avenues.

Just a thought…

Best regards,

Posted by: Eric on February 16, 2005 7:21 PM | Permalink | Reply to this


Yes, we have this kind of backwards discussion in our notes. I believe though that at least for sufficiently nice graphs it is somewhat more direct to define the inner product first.

Posted by: Urs on February 17, 2005 4:05 AM | Permalink | Reply to this

Re: Adjoint

True. I guess when I get a new toy, I like to play with it until I get bored :)

Still, if you find yourself a nice graded differential algebra with an obvious involution (with appropriate degree) and you are trying to find an inner product to go along with it, just let your involution be an adjoint for some base inner product and just deform things from there.

As if I can teach you something :) I will never stop trying though!

Posted by: Eric on February 17, 2005 1:06 PM | Permalink | Reply to this

Re: Project: 2-NCG

So does this blog simply consist of Urs research or will there be postings about the work of other people

Posted by: craig on February 16, 2005 9:20 PM | Permalink | Reply to this

Re: Project: 2-NCG

Hi craig,

As has been stated before, discussions on any topic related to string theory are more than welcome. The only reason it seems to be Urs’ playground is that no one else is taking advantage of this neat forum. To start a new topic thread requires one of the admins though.


Posted by: Eric on February 16, 2005 9:25 PM | Permalink | Reply to this


It’s a weblog, not a discussion forum per se. And, yes, only one of the authors (6 currently) can add new entries.

I 'spose that the other authors have not been terribly prolific posters here. (Two have weblogs of their own, so that’s part of the reason.) But nothing prevents them from posting more.

And I do consider adding more authors, when suitable candidates present themselves.

(the Administrator)

Posted by: Jacques Distler on February 16, 2005 10:43 PM | Permalink | PGP Sig | Reply to this

Re: Authors

I am hoping both that there will eventually be more people posting here about a larger variety of topics than I am able to come up with, and also that it is OK with everybody that meanwhile I post here the stuff I do. Hopefully these two hopes are not mutually exclusive. Social dynamics of public forums is a delicate issue.

In any case, I am very grateful to Jacques for providing this weblog. If I am getting on anyone’s nerves with what I am writing about please let me know.

Posted by: Urs on February 17, 2005 4:13 AM | Permalink | Reply to this

Re: Authors

I certainly enjoy reading your `notes’ and comments by other readers. Variety will not necessarily make this blog better, and since some of the more prolific authors have their own blogs, they would either end up duplicating their articles, or spreading their comments over different blogs.

What you are doing is commendable and useful, I hope you don’t lose heart over occasional comments from detractors.

Posted by: Amitabha on February 18, 2005 6:22 AM | Permalink | Reply to this

Re: Authors

Thanks for the nice words.

Posted by: Urs on February 19, 2005 4:40 AM | Permalink | Reply to this

Re: Project: 2-NCG

Maybe one should point out that already 1-NCG depends on a background metric. Already the inner product assumes that there is a volume form; you can define an invariant inner product of half-forms (and of n/2-forms in even dimension n), but not of functions. Also, a Dirac operator always assumes that there is a metric. This is obvious since the square of the Dirac operator is the Laplacian. The place where the metric is introduced is in the adjoint of the exterior derivative, -*d*, since the Hodge star depends on the metric.

Mathematically, there is of course nothing wrong with assuming a metric, since any Riemannian manifold can be equipped with one, and in the beginning you do the same thing as Connes. However, if this is some approach to quantum gravity, then you do introduce a background structure which is not present in general relativity. Your construction is background dependent if you use the Dirac operator prior to quantization, e.g. to polarize the Hilbert space. It seems to me that you, and Connes, will not get a unique quantum theory, but rather a family of quantum theories parametrized by the reference metric and Dirac operator used to polarize the Hilbert space, and it is by no means obvious that these theories are unitarily equivalent even if they exist.

It is possible of course that your construction is independent of the choice of background, even though you have to introduce one. However, many people believe that background independence is a crucial property of gravity, and one which will survive quantization. It is unclear to me whether a mathematician like Connes appreciates this.

Posted by: Thomas Larsson on February 17, 2005 5:52 AM | Permalink | Reply to this

Re: Project: 2-NCG

Can you describe a graph category in 20 words or less? Objects are nodes and morphisms are edges? Do we need it to be a directed graph?

Sorry for the silly questions, but I’d like to understand your construction. When you say “discrete string geometry”, what I think you really mean is “discrete loop space geometry”? Is that right? If so, this is what I’ve been waiting for! :)


Posted by: Eric on February 18, 2005 5:24 AM | Permalink | Reply to this

Re: Project: 2-NCG

Whatever happened to the idea of starting with a directed graph Γ and then constructing the product graph Γ×Γ?

This gives a painfully obvious way to do discrete 2-calculus. With nodes of Γ denoted e i and directed edges of Γ denoted e ij=e ide j, then a 2-node of Γ×Γ is simply of the form

(1)e ie j.

with 2-edges

(2)e ij,kl=(e ie j)d(e ke l).

Maybe I am talking nonsense. Sorry! :)


Posted by: Eric on February 18, 2005 5:42 AM | Permalink | Reply to this

Re: Project: 2-NCG

One more thing before I pass out for the night…

If it wasn’t painfully obvious already, we can think of the 2-node e ie j as an open string connecting the 1-nodes e i and e j. The 2-edge e ij,kl describes the evolution of 2-node (i.e. open string) e ie j to the 2-node (i.e. open string) e ke l. Causality of the 2-graph is inheritted from that of the 1-graph Γ.

Posted by: Eric on February 18, 2005 5:48 AM | Permalink | Reply to this

Re: Project: 2-NCG

I can’t resist one more comment…

Γ n is polymer space. If we closed the polymer and let n, we should get loop space (I think).

Posted by: Eric on February 18, 2005 5:53 AM | Permalink | Reply to this

Re: Project: 2-NCG

Yes! Note that this is pretty much what happens in that discrete toy example of that 2-NCG program automotically!. I’ll spell that out in more detail as soon as possible.

Posted by: Urs on February 19, 2005 4:45 AM | Permalink | Reply to this

Re: Project: 2-NCG

Yes! Note that this is pretty much what happens in that discrete toy example of that 2-NCG program automotically!. I’ll spell that out in more detail as soon as possible.

I’m looking forward to it :)

There seems to be at least one significant difference though.

so δ γ(dδ γ) is a discrete differential 1-form on discrete path space in precisely the same sense as δ x(dδ x) turned out to be a discrete differential 1-form on ordinary discrete space [2].

In the quoted line above, we are limiting ourselves to 2-nodes that are actually 1-edges in the orginal graph Γ.

If you look at Γ×Γ, then we can have 2-edges

(1)e i,j(de k,l)=(e ie j)[d(e ke l)],

where e ij and e kl need not be edges in Γ. This latter scenario makes more sense to me because the edges of Γ are probably going to turn out to be lightlike. If that is the case, then dealing with light-like paths seems a little restrictive.

Just a thought…


Note on notation: This is a little confusing so I’ll say a quick word on notation. The symbol “e ij” denotes an edge in Γ from e i to e j. The symbol “e i,j” denotes e ie j, and does not require an edge connecting e i and e j in Γ. Maybe I should choose different letters :)

Posted by: Eric on February 20, 2005 4:14 PM | Permalink | Reply to this

Re: Project: 2-NCG

At this point I really mean ‘discrete path space geometry’ since the morphism spaces on which I want to do differential calculus will consist of discrete paths (paths of edges).

Yes, we have thought about similar things before. But now I see a nice way to really do it.

Posted by: Urs on February 19, 2005 4:43 AM | Permalink | Reply to this

Re: Project: 2-NCG

Hi Urs :)

I think I understand. Just a sanity check…

A point in loop space corresponds to a loop in some base space.

A point in discrete loop space would then correspond to a discrete loop in some discrete base space.

Ok so far?

I have never heard you talk about “path space” before, but I am assuming a point in path space corresponds to an directed path in some base space.

Is that correct? If so, I was trying to get you interested in that way back when we were trying to write our “loop space for dummies” paper, but you did not seem the least bit interest. Hmmph! :)

If I haven’t strayed too far from reality so far, then a point in discrete path space should correspond to a directed path in some discrete base space. Right?

If so, what do you plan to use for your discrete base space? Those kinds of things don’t just grow on trees (no pun intended) you know! I have a suggestion for what you should use. There is a nice, little known, paper on the archive by one very questionable author and one very good one that you might like to take a look at :)

Gotta run!


PS: I got a nice email from Professor Baez. He said we should publish that paper.

PPS: If we do ever finish that paper, I can add some new material on discrete spheres. That might be nice to dispel the initial impression some people might get that the paper is only valid for discrete Euclidean/Minkowski space.

Posted by: Eric on February 19, 2005 1:41 PM | Permalink | Reply to this

Re: Project: 2-NCG

Just a sanity check…

Yes, this sounds good.

I hope to find the time to continue working out my notes on this, but for the moment let me just indicate some more details here in the comment section:

The plan of action is this: For starters, pick a hypercubical graph. Call that an example of a discrete 2-space.

Let a 2-function from this 2-space be a functor from the associated graph category to the category of finite dimensional Hilbert spaces.

This just means that such a 2-function is something

- which assigns natural numbers n(v) to vertices v in the graph

- which assigns linear maps

(1)A γ: n(v 1 ) n(v 2 )

to any edge

(2)v 1 γv 2

- and which assigns the obvious composition of such maps to compositions of elementary edges.

Of course these maps are just n(v 1 )×n(v 2 ) matrices with complex entries.

There is a full-blown theory of 2-Hilbert spaces behind this but actually what matters right now is that the set of all such 2-functions forms an algebra if we define the addition and multiplication of two such 2-maps edgewise as the direct sum and direct product, respectivbely, of these matrices A.

So for example consider the case where our graph consist just of two vertices and the single edge

(3)v 1 γv 2 .

2-functions f i maps this to

(4)A γ i: n i(v 1 ) n i(v 2 ).

The sum f 1 +f 2 is then the 2-function which assigns

(5)A γ 1 A γ 2 : (n 1 (v 1 )+n 2 (v 1 )) (n 1 (v 2 )+n 2 (v 2 )).

to γ, while the product f 1 f 2 assigns

(6)A γ 1 A γ 2 : (n 1 (v 1 )n 2 (v 1 )) (n 1 (v 2 )n 2 (v 2 )).

So you see how all these 2-functions form an algebra. Forget about all the categorification business for a moment. Given an algebra, we can form abstract differential calculi over it by throwing in an abstract d which we demand to be nilpotent and graded Leibnitz.

So there is a differential calculus with elements of the form

(7)f 1 df 2

and the like. There is the universal such calculus which is just the free algebra over all these generators. That’s not too interesting. It becomes interesting once we specify certain rules for when such an element is nonvanishing.

In the very simple example where the graph has only a single edge this is not very illuminating. But suppose the graph is actually consisting of four vertices forming a square. Let f 1 be a 2-function supported on the top edge of that square and f 2 one supported on the bottom edge. Then f 1 df 2 describes a discrete differential 1-form on path space which says that there is a ‘2-edge’ between the upper and the lower edge, i.e. a plaquette (surface element).

Since every 2-function involves an ordinary function on vertices (assigning the dimension of a Hilbert space to each vertex) the above induces a differential calculus on these ordinary functions. This is just the one that we have studied in our notes! Hence such a plaquette interpolates between two ordinary discrete differential 1-forms as in our notes.

At the level of discrete hypercubical graphs pretty much the only technical difference to what we did before is that now the underlying algebra of the f i is highly noncommutative by itself. This makes it non-obvious to figure out if there is any set of f i that canonically and naturally should be regarded as characterizing the elementary paths.

Posted by: Urs on February 22, 2005 4:52 PM | Permalink | Reply to this

Re: Project: 2-NCG

Hi Urs,

What is the difference between that and what we already did with group algebra-valued forms?


Posted by: Eric on February 22, 2005 8:11 PM | Permalink | Reply to this

Re: Project: 2-NCG

What is the difference between that and what we already did with group algebra-valued forms?

Yes, it’s very very similar. That is the big insight: A 2-algebra is essentially an ordinary algebra of functors (2-functions) and given that we can turn the ordinary crank.

There are some subtleties, thoug. First, the d which we define on 2-functions should really be a functor itself. There is not just a set of 2-functions, but a category of them. And d should preserve the morphism structure.

That category of 2-functions has 2-functions as objects and natural transformations between 2-functions as morphisms. Recalling that a 2-functions is much like a holonomy it does not come as a surprise that a natural transformation between two 2-functions is much like a gauge transformation.

So for instance given that graph with only one edge γ the two 2-functions F and G with

(1)F(xγy)=A γ


(2)G(xγy)=U xA γU y 1

would be connected by a morphism FUG. Hence for d to be a functor we should have that the discrete path space 1-forms dF and dG are also connected by a morphism


I think this is easily accomplished, but it needs to be taken care of.

I wish I had more time working this out! Right now I am busy finishing a paper with John Baez, Danny Stevenson and Alissa Crans. And starting tomorrow I will be on vacation until May 7. Right after that I’ll be in Berlin on a conference. So probably I won’t be able to do much more on my 2-NCG project until March 10. But if you are still around then I’d be happy to get some things done!

Posted by: Urs on February 23, 2005 7:58 PM | Permalink | Reply to this

Re: Project: 2-NCG

Ah, yes. I might have accidentally stumbled on some of these issues when I was trying to modify our group algebra valued stuff to account for parallel transport. For example, if you have 2 G-valued edges

(1)g ije ij


(2)g kle kl

and we multiply them, we should get something like

(3)[g ije ij][g kle kl]=g ijg klije ije kl.

If you know what I mean :)

Hopefully I will still be around in March. I am being sucked into the abyss and you are the only thing keeping me from falling in! :)

Posted by: Eric on February 23, 2005 8:32 PM | Permalink | Reply to this

Re: Project: 2-NCG

Maybe that’s related. It is important to note, though, that with the product of 2-functions defined as I mentioned in a previous comment, you’ll never really multiply these group elements but instead take their tensor product. This sort of makes sense, since the group elements should really be multiplied only when their respective paths are composed, which is not what happens when 2-functions are multiplied.

I have some ideas on how to give that curious algebra of 2-functuions a sensible physical underpinning in terms of string boundary states. But I am still not completely sure if maybe in the end we want the 2-functions to take values in something else than Hilb for that.

Posted by: Urs on February 23, 2005 8:43 PM | Permalink | Reply to this

Re: Project: 2-NCG

Hi Urs,

I’m sure you’ve explained it before and I should just review all these entries, but the motivation isn’t very clear to me why you want to deal with functors right out of the gate.

It feels kind of like topological field theory (which I know almost nothing about), which is a functor


It seems like you want to do something similar with graphs, i.e.

(2)Z:2 Graph2 Hilb

and are hoping something interesting comes out. However, before you can study a functor


I think it makes sense to first understand A and B :)

It almost seems like there is work to be done to understand 2-Graphs and there is work to be done to understand 2-Hilb. If one of these is understood, a functor might be useful in helping to understand the other. Is that the motivation?

Sorry for the naive questions :)


Posted by: Eric on February 23, 2005 9:16 PM | Permalink | Reply to this

Re: Project: 2-NCG

but the motivation isn’t very clear to me why you want to deal with functors right out of the gate.

The work on 2-bundles convinced me that by categorifying point stuff we should get string stuff. The categorification of the set of complex-valued functions on some space just happens to be the category of functors from some 2-space into some target category like Vect.

And it seems that this indeed shows up in string theory in a way pretty much as I have envisioned, but naturally much more sophisticated than I had guessed. I’ll try to post more about all that soon.

You are right, field theories can also be formulated as functors from a cobordism category to Hilb. That’s a different angle of attack though, which is not what I currently have in mind.

Posted by: Urs Schreiber on March 11, 2005 4:51 PM | Permalink | PGP Sig | Reply to this

Re: Project: 2-NCG

Hi Urs,

This is going to sound out of the blue, but hey, what do you expect from me? :)

Do you ever think that maybe all this abstract mathematics is not how nature really operates? I think that physicists come to a turning point early in their careers where they need to decide on a philosophy. Are you going to try to develop theories that might describe some kind of phenomological aspect of nature, or are you going to really try to understand the true nature of the universe at the most fundamental level.

Are you pursuing this because you truly believe that nature operates according to the rules of gerbes and n-categories? It’s kind of a silly philosophical question, I suppose, but what is it that drives you?


Posted by: Eric on March 12, 2005 3:16 PM | Permalink | Reply to this

Re: Project: 2-NCG

Well, I guess I am trying to investigate the idea that looks coolest and most promising while trying to make sanity checks all along and learning more about what other ideas are out there and then recursively modify things. When something does not work it should be dropped. Right now it seems that the categorical stuff does work nicely in ‘formal hep’, so why not.

At Riverside they had this running gag, which I am not sure about how much it was meant to mock a certain guest from Germany:

Q: ‘How to have many good ideas?’

A: ‘Have many ideas.’


Posted by: Urs Schreiber on March 13, 2005 8:19 PM | Permalink | PGP Sig | Reply to this

Re: Project: 2-NCG

Thanks for all the comments. I had tried to expand on my existing notes the last days, giving more details on some of the things that Eric mentioned, but when I tried to save my changes the server responded with an error message.

Right now some calculations on loop groups, 2-groups and the string group is assigned highest priority here at Riverside, so I’ll have to delay the 2-NCG project for a moment.

Posted by: Urs on February 18, 2005 5:59 PM | Permalink | Reply to this

Re: Project: 2-NCG

Are you saying that you would prefer to hang out with John Baez, Toby Bartels, and Derek Wise than explain what a graph category is to us! Sheesh! :)

You could always try to clone yourself :)

As always (you should know since I say it enough), my silly questions are not urgent and will be here for whenever, if ever, you feel like responding. Have fun!!


PS: You have to at least go the beach once! Even if it means dragging the gang with you and discussing n-categories in the sand :)

Posted by: Eric on February 18, 2005 6:44 PM | Permalink | Reply to this

Re: Project: 2-NCG

We were planning to go hiking in some nearby desert on the week end, but it looks like it will – rain! Hey, why am I flying all these miles from rainy Germany to sunny California!?

By ‘graph category’ I simply mean the category you get from any graph by taking all the vertices as objects and all edges and paths of edges as morphisms.

More later! 2-many 2-things 2-do!

Posted by: Urs on February 18, 2005 8:24 PM | Permalink | Reply to this

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