# What Non-commutative Geometry Is and Can Do

1. May 26, 2005

Staff Emeritus
I decided to start this thread to tempt Kneemo and Kea to come and post on the title subject. If they want to copy some prior posts here that's fine. My idea is that it become link-rich like Marcus's Rovelli thread.

{Added} I didn't intend tf or this thread to compete with Kea's third road thread. This one was to be specifically about non-comm geom and its effect on other discplines like LQG and CDT, and Strings. As a start which non-commutative geometries are really non-associative, and how does that relate to the non-associative lattices of quantum algebra?

Last edited: May 26, 2005
2. May 26, 2005

### marcus

If you want links, this is a very readable article which can give a lot of intuition about NCG

http://arxiv.org/gr-qc/9906059

Noncommutative Geometry for Pedestrians
Lecture given at the International School of Gravitation, Erice, 28 pages

"A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature."

Last edited: May 26, 2005
3. May 26, 2005

### marcus

Here is a link to all the Alain Connes postings on arxiv

http://arxiv.org/find/hep-th/1/au:+Connes_A/0/1/0/all/0/1

Here is an encyclopedia article by Thomas Schucker about attempts to
derive the Standard Model of particle physics from NCG

http://arxiv.org/abs/hep-th/0310145
Noncommutative geometry and the standard model
Thomas Schucker (Marseille)
11 pages, 4 figures, contribution to Encyclopedia of Mathematical Physics, J.-P. Francoise, G. Naber & Tsou Sheung Tsun (eds.), Elsevier Science

"The aim of this contribution is to explain how Connes derives the standard model of electromagnetic, weak and strong forces from noncommutative geometry. The reader is supposed to be aware of two other derivations in fundamental physics: the derivation of the Balmer-Rydberg formula for the spectrum of the hydrogen atom from quantum mechanics and Einstein's derivation of gravity from Riemannian geometry."

4. May 26, 2005

### marcus

Alain Connes addresses background independence in this paper

http://arxiv.org/abs/math.QA/0505475
Background independent geometry and Hopf cyclic cohomology
Alain Connes, Henri Moscovici
50 pages
Quantum Algebra; Operator Algebras

"This is primarily a survey of the way in which Hopf cyclic cohomology has emerged and evolved, in close relationship with the application of the noncommutative local index formula to transverse index theory on foliations. Being Diff-invariant, the geometric framework that allowed us to treat the 'space of leaves' of a general foliation provides a 'background independent' set-up for geometry that could be of relevance to the handling of the the background independence problem in quantum gravity. With this potential association in mind, we have added some new material, which complements the original paper and is also meant to facilitate its understanding. Section 2 gives a detailed description of the Hopf algebra that controls the 'affine' transverse geometry of codimension $n$ foliations, and Section 5 treats the relative version of Hopf cyclic cohomology in full generality, including the case of Hopf pairs with noncompact isotropy."

I just checked the "third way" thread and saw that the most recent post there is from three weeks ago, on 3 May.
This Connes paper posted on arxiv just this week, on 24 May, so it will not already have been discussed on that other thread. I shall post it here, therefore, without fear of repetition.

It is nice to know that Connes is thinking about the needs of quantum gravity (BTW see his strong plug for Rovelli's Quantum Gravity book at Cambridge Press! Basically Connes wrote the jacket blurb for LQG.)

but the actual mathematical support proffered here is rather tentative and iffy. See connes page 6:

"The precise construction of D, to be recalled below, involves the choice of a connection on the frame bundle but this choice does not affect the principal symbol of D and thus plays an innocent role which does not alter the fundamental Diff+(M)-invariance of the spectral triple. More precisely, we have shown in [8] that it does define in full generality a spectral triple on the crossed product of PM by Diff+(M). It is worth mentioning at this point that this construction, besides allowing to handle arbitrary foliations, could be of relevance in handling the basic problem of background independence, which is inherent to any attempt at a quantization of the theory of gravitation."

If it is frame bundles and diffeomorphisms Diff+(M) on a manifold M, well there we are. Manifolds again.
He suggests what he says applies to any attempt at a quantization of the theory of gravitation. We need to be careful though, not every attempt at QG is immersed in the context of differentiable manifolds.

well, have to go. maybe someone would like to make something of this paper, or others I've linked to here

Last edited: May 26, 2005
5. May 26, 2005

### Mike2

And what are the prerequisites to enable a reading of this paper? And how many on PF have that background?

6. May 26, 2005

### marcus

And how many have the time? And think it relates to their main interests?
We will have to see, Mike
The two K kids, kneemo and Kea, are the ones that have been making big claims for NonComGeom lately and seem very enthusiatic (evangelistic?) about it. I think selfAdjoint set the thread up in the expectation that they would discuss NCG, and "what it can do", for us.

My personal hope is that one or the other, Kea or kneemo, will make good on their assertion that the triangulations approach to quantum gravity (CDT) which interests me is logically contained in NCG.

7. May 26, 2005

### Kea

Oh, dear. More to do! But a very good idea, selfAdjoint. Must go, now. Back later.

Kea

8. May 26, 2005

### kneemo

Great thread and excellent references so far!

The flavor of NCG I've been discussing in relation to M-theory is the kind mentioned in Noncommutative geometry and the standard model on page 6.

The 0-dimensional approach is central to the classic BFSS paper:
M Theory As A Matrix Model: A Conjecture
Authors: T. Banks, W. Fischler, S.H. Shenker, L. Susskind
hep-th/9610043

Two months after the BFSS paper, the IKKT paper emerged.

A Large-N Reduced Model as Superstring
Authors: N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya
hep-th/9612115

The IKKT paper argued that space-time is generated dynamically as follows:
Influenced by the BFSS and IKKT papers, Lee Smolin introduced his own matrix model, as a candidate for unification of LQG and String theory:

The exceptional Jordan algebra and the matrix string
Authors: Lee Smolin
hep-th/0104050

A fellow researcher of H. Kawai (of IKKT), Y. Ohwashi, generalized Smolin's 'exceptional cubic matrix model' to include E6 symmetry.

E6 Matrix Model
Authors: Yuhi Ohwashi
hep-th/0110106

The spectral triple formalism (Connes' NCG) for matrix models eventually became explicit in the K-matrix model:

D-branes, Matrix Theory and K-homology
Authors: T. Asakawa, S. Sugimoto, S. Terashima
hep-th/0108085

Thus the evolution of string matrix models did not begin with the rigor demanded by Connes. It took years to find that spectral triples were behind the matrix model approach. So noncommutative geometry, by Connes' standards, only emerged after a closer hindsight analysis of the matrix model constructions.

Matrix models provide 0-dimensional spaces with differential calculi based on derivations of matrix algebras. Smolin was able to treat LQG and string theory in a 0-dimensional fashion, so the remaining challenge is CDT.

Conveniently, AJL have already discussed 3d Lorentzian triangulations arising from the ABAB-matrix model.

Renormalization of 3d quantum gravity from matrix models
Authors: J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)
hep-th/0307263

Hence, since all matrix models are really noncommutative geometry, the CDT/NCG connection is established by AJL themselves.

Regards,

Mike

9. May 26, 2005

### marcus

Sounds like you MIGHT be on to something. But 3D is not real gravity. Can you demonstrate a connection for 4D?

You still owe us some explanation. What is an ABAB-matrix model? Why is it part of the theory usually known as NCG? At PF we dont only throw esoteric terms around, we try to balance it with some down-to-earth explanation. Are you up for that?

BTW I am still skeptical (as is my right to be) about a connection of 4D Causal DT with any kind of matrix model. That was back in 2003 and I have not heard anything since then. But if you can explain the connection of ABAB-matrix to NCG then you have gone most of the way to satisfying what i asked. Please give it a shot.

BTW kneemo, congratulations on a very nice list of NCG online articles, with links.

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10. May 27, 2005

### Chronos

This is a fascinating subject. I'm a little surprised no one has referenced any works by B.G. Sidharth. A recent, and intriguing entry:
http://arxiv.org/abs/physics/0502106
I am curious if anyone has an opinion. He seems a bit 'out there' at times, but the ideas are captivating.

11. May 27, 2005

Staff Emeritus
I went back to AJL's ref 15, where the cited the symmetic ABAB model and found this paper:

http://arxiv.org/PS_cache/hep-th/pdf/9808/9808043.pdf

It discusses the solvability of the model and also shows the connection to networks. OK? Seems to me you could have gone that far yourself before sinking to lecturing your peers on good PF behavior.

12. May 27, 2005

### marcus

SelfAdjoint, I have a feeling that kneemo is not going to offer a clear reason why the first four articles he mentioned belong in this thread about What Is NCG. Could you please explain why "Matrix" or "M-theory" is an essential part of NCG? It would be nice if in this thread, at least right at the start, we could focus on some essentials of core-NCG and not start indulging in free-association.

Here are the first four articles:
-----THE NEXT ARTICLE GIVES SOME HINT OF A CONNECTION. But I still have the feeling of being passed a bad check. Are not "spectral triples" something that can come up in a variety of contexts? (I ask as a non-expert, correct me if I am wrong.) So is it legitimate to simply equate "spectral triple formalism" with "Connes' NCG? IS THAT ALL THAT NCG IS ABOUT, spectral triples?

In that case any time a spectral triple comes to light in some mathematical research then the research must automatically be part of NCG. :uhh:

So did NCG only "emerge" when some connection with M-theory was noticed?

It is extremely unclear to me what is supposed to be a part of what.

It looks to me that we have a picture of M-theory with delusions of grandeur. Sighing like an Alexander, because it has no more worlds to gobble.
The answer to the question "What is Noncommutative Geometry?" is "It is a part of M-theory". Or any other branch of mathematics. Whatever branch it is, if you are asked to describe what it is and what it is good for, you are supposed to say "It is a part of M-theory."

Thomas Swift and Moliere might have had great times with today's theorists.

I am still hoping for a focused presentation of What is NCG. And an answer to questions about its actual logical connection to other theories. I must reject the claim that NCG and something that interests me, CDT, are connected simply because both are imagined to be parts of a grandiose
M-theory Empire

13. May 27, 2005

### marcus

What is an ABAB matrix model? Is it part of Alain Connes Noncommutative Geometry? I am asking for a simple explanation that as many of us as possible can understand.

Judging from this article you just cited, ABAB matrix is not a part of NCG. I dont see any connection. But since i am not familiar with the context, I am relying on you to draw the connection with NCG.

http://arxiv.org/hep-th/9808043
Two-Matrix model with ABAB interaction
V.A. Kazakov, P. Zinn-Justin
24 pages, 5 figures (1 color figure)
Nucl.Phys. B546 (1999) 647-668

"Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitean matrices A and B with the action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4) -{\beta\over 2} \tr(AB)^2. This model can be mapped onto a special case of the 8-vertex model on dynamical planar graphs. The solution is parametrized in terms of elliptic functions. A phase transition is found: the critical point is a conformal field theory with central charge c=1 coupled to 2D quantum gravity."

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14. May 27, 2005

### marcus

selfAdjoint pointed us to this paper, which must be germane to this NCG thread:
Two-Matrix model with ABAB interaction

this paper was cited by "Triangulations" authors Ambjorn et al in a 1998 paper.

No one is explaining or telling us what is going on, it is supposed to be obvious. So I will try do supply the necessary explanation (even tho M-stuff is way out of my range of interest)

this article, I would say, is NOT an NonComGeom article at all, curiously it seems totally out of the orbit of Alain Connes. But it IS at least on the fringes of M-theory. Its references include some mainstream 1990 M-theory papers

[21] E. Brezin and V. Kazakov, Phys. Lett. B236 (1990), 144.
[22] M. Douglas and S. Shenker, Nucl. Phys. B335 (1990), 635.
[23] D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990), 127.

here, even non-M people like myself can recognize Mike Douglas, Steve Shenker, David Gross. On page 17, in their conclusions they say:

"If such a representation exists it could allow us to analyse the critical behaviour for all genera of random graphs in the double scaling limit, like for the more standard matrix models [21], [22], [23]."

So what can we make of this? One thing to remember is Ambjorn has been a string theorist. Just like Renate Loll has done LQG research, Ambjorn has actually done a fair amount of String/M. So when the two collaborate on "Triangulations" research they comprise collectively a fairly broad experience of alternatives. Renate may also have done some string as well, I dont happen to know offhand.

So is not too surprising that Ambjorn et al would write this----which kneemo introduced to this thread:
Renormalization of 3d quantum gravity from matrix models
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)
http://arxiv.org/hep-th/0307263

1. I dont see how this is connected to the thread topic NCG, but let's ignore that question and go with it some. (If anyone can explain clearly how the matrix models in these two papers are actually NCG, I would be pleased to know.)

3. Not sure the cited paper, about 2+1 dimensional gravity, is conclusive in any sense

have to go, back soon

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15. May 27, 2005

### marcus

Renormalization of 3d quantum gravity from matrix models
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)
http://arxiv.org/hep-th/0307263

To expand on this and try to better understand its significance, let's look at Ambjorn's papers
http://arxiv.org/find/hep-th/1/au:+Ambjorn_J/0/1/0/all/0/1

Ambjorn has written 109 papers going back to 1991
a considerable number of them, especially before 2001, seem to be MATRIX papers

http://arxiv.org/find/grp_physics/1/AND+au:+Ambjorn+abs:+AND+Matrix+Model/0/1/0/all/0/1

this search (Ambjorn AND matrix model) gets 33 of his papers.
of these, 27 are from 2001 and earlier.

here is another great list, search (Ambjorn AND string)! one gets 22 papers
http://arxiv.org/find/grp_physics/1/AND+au:+Ambjorn+abs:+string/0/1/0/all/0/1
the list includes one where AMBJORN DISCUSSES RELATIONS OF QG AND STRING AND NCG HIMSELF
http://arxiv.org/hep-lat/0201012
Strings, quantum gravity and non-commutative geometry on the lattice
J. Ambjorn
Lattice(plenary)
Nucl.Phys.Proc.Suppl. 106 (2002) 62-70

Finally, here is a combined search: Ambjorn AND (string OR (matrix AND model))

http://arxiv.org/find/grp_physics/1/AND+au:+Ambjorn+abs:+OR+string+AND+matrix+model/0/1/0/all/0/1

It gets 50 papers, of which 10 are since 2001.

I think I may say, with some satisfaction, that Ambjorn qualifies as a working string/M theorist. I would say that he is a successful string research professional (judging by publication) but that is not all he is.

Ambjorn's trajectory is interesting, at least to me. It seems to be taking him out of differential geometry manifold-land and into new territory. Maybe other string and Matrix model can follow like paths as it becomes time to move on.

here, for comparision, are Lolls 48 papers going back to 1993.
http://arxiv.org/find/grp_physics/1/au:+Loll/0/1/0/all/0/1
I see she has published with Jose Mourao and Abhay Ashtekar (LQG people) but does not seem to have continued with mainstream LQG past 1997.

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16. May 27, 2005

### marcus

It looks to me as if a little bitty corner of the Triangulations approach is tenuously connected to Matrix Model

and some aspect of Matrix Model is connected (somehow i dont understand) to Noncommutative.

But I have no assurance that Triangulations connects to the PART of the M-theory M-pire that is connected to NCG.
So I still have to regard the claim of CDT being subsumed by NCG (Kea says gobbled up or devoured, somesuch expression) as empty handwaving.
It would be delightful if true but I see no evidence that it's a fact.

So I must sink or stoop to the level of requesting some more careful reasoned clarification. I would like to see some things defined, please.
I would like to see page references with the same mathematical object defined in a NCG context and a CDT context.

17. May 27, 2005

Staff Emeritus
The paper I cited was for a description of the ABAB model, it was not intended to make a connection to CDT. It rather shows a connection to certain dynamic planar networks. Suggestive, though.. The connection to CDT, or rather to "Lorentzian dynamic triangulations in three dimensions", is made in the AJL paper. itself! THEY claim the abitlity of the ABAB model to count states of changing triangulations can be applied to make it describe their dynamic triangulations in two space and one time dimension Lorentzian spacetime.

The basic idea of a matrix model is that you start with one or more random NXN hermitian matrices as your basic variables, with the purpose of describing some kind of combinatorial system, and require that the matrices satisfy some given partition function, given as an integral of the basic matrix interactions you are going to consider to apply on the configurations. The integral will have parameters multiplying different terms, and it all superficially resembles the kind of thing you see in a Lagrangean. You then expand this integral perturbatively using Feynmann diagrams. The parameters will come into these a la coupling coefficients. The partition function will then express the counts of the combinatorial system, generating the given partition function, and in the case of the ABAB model the system is the set of planar graphs with vertices swapping around. By your perturbative solution you are able to describe such a configuration space within the perturbative limits.

Because the ABAB model, with the quartic (AB)^2 in its partition function, is so fully adapted to two dimensional link problems (there is another paper I found where it is applied to knots), it seems unlikely that it could be applied to four dimensional Lorentzian physics - triangular or otherwise. Devising a matrix model to do that, and providing a solution of that model, would be a challenging research problem.

Last edited: May 27, 2005
18. May 27, 2005

### marcus

I have tried to track this down, but cant: in their most recent papers they seem to be suggest that there would be no point to connecting up with Matrix----they now have a different idea for renormalization. (the changing dimensionality at small scale)---not sure about this.

been some time since i read it, have to go back and refresh my memory.

thanks mucho for the clarification!

PS: I cant find the quote I was looking for. I could find this in the recent
hep-th/0505113 "Spectral Dimension..." paper:

"We conclude that quantum gravity may be 'self-renormalizing' at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction....

...This suggests a picture of physics at the Planck scale which is radically different from frequently invoked scenarios of fundamental discreteness: through the dynamical generation of a scale-dependent dimensionality, nonperturbative quantum gravity provides an effective ultraviolet cut-off through dynamical dimensional reduction."

but I can't find where they say that they are looking for this to make a Matrix-related renormalization gambit unnecessary. Maybe this came up in conversation, or I am misremembering.

Here is a relevant quote from page 24 of the even more recent
hep-th/0505154
"...A dynamically generated scale-dependent dimension with this behaviour is truly exciting news, because it signals the existence of an effective ultraviolet cut-off for theories of gravity with and without matter coupling, brought about by the (highly non-perturbative) behaviour of the quantum-geometric degrees of freedom on the very smallest scale."

but can find no mention in these recent papers of the ABAB Matrix involvement they discussed back in 2003

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19. May 27, 2005

### marcus

https://www.physicsforums.com/showpost.php?p=574119&postcount=15
https://www.physicsforums.com/showpost.php?p=581338&postcount=70
these two posts highlight, for me, why we need this thread on what NonCommutative Geometry is and what it can do. they illustrate what seems to be a widespread belief that one thing NCG is, or does, is other branches of mathematics.

The underlined statements here have neither been retracted, so far, or substantiated.

Some of this may give the false impression that i am someone who dislikes modern abstract mathematics. that is wrong. but it is of little importance.

What seems important to me is that I cant "consume" pizza without getting my teeth into it. Noncommutative cannot "consume" Triangulations without a rigorous logical connection. So I am waiting to hear the statement justified that
"NCG easily consumes CDT."

kneemo predicts that people doing CDT (the authors Ambjorn et al) will come to realize that they are actually doing NCG, in other words that Triangulations research is subsumed by NonComGeom. this requires demonstrating a logical connection as well.

I would be delighted to hear of a rigorous connection by which NCG somehow contains CDT, and am very far from resisting this, or being made unhappy by it. But I dont want to be passed a bad check on this. I would like a real connection but would very much not like a phony or handwaving one.

So I hope this thread will get us straightened out on the topic of NCG and what other mathematics can be derived from it.

20. May 27, 2005

### Kea

I'm hoping to have a better understanding of NCG after attending an instructional program in July. People like Marcolli will be there. Some of the lecture notes are already online:

Very Basic Noncommutative Geometry
Masoud Khalkhali
http://arxiv.org/abs/math.KT/0408416

Warning: it's not as basic as the title suggests!