Classification of manifolds and smoothness structures

[torsten]

The whole discussion was confusing, one spoke about gravity, Tom spoke about Dirac operator (as the self-adjoint operator) and all the others spoke about the general NCG.
My post was a direct reaction to Toms. I wanted to clarify that in Connes model using the Dirac operator (which is Connes version of the standard model, no other model was proposed by him) he assumed a Riemannian spin manifold.
I'm deeply sorry for the confusion. Next time I will retain myself in posting such things.

 

[marcus]

The whole discussion was confusing, one spoke about gravity, Tom spoke about Dirac operator (as the self-adjoint operator) and all the others spoke about the general NCG.
My post was a direct reaction to Toms. I wanted to clarify that in Connes model using the Dirac operator (which is Connes version of the standard model, no other model was proposed by him) he assumed a Riemannian spin manifold.
I'm deeply sorry for the confusion. Next time I will retain myself in posting such things.

No problem, Torsten! I'm sorry for whatever confusion I contributed by not being clear. You surely know a lot more about non-manifold geometry than I do. I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?

They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.

 

[torsten]

I'm very excited by the Suijlekom-Marcolli paper which introduces the concept of a "gauge network", have you looked at the paper?

They are aiming at the Connes standard model without any commutative part of the algebra! All the spectral triples are purely non-manifold and also they are finite dimensional i.e. in A,H,D the algebra A is finite dimensional and so is the hilbertspace.

I remembered to read the paper in January but now I read it again. In the paper there was a clear motivation. Everything looks very interesting. But I'm not surprised that the authors obtained the Higgs sector. A spin network with a NCG at every vertex...
One thing troubles me: the spin network was usually constructed to be related to the triangulation of the 3-manifold. I know of only one approach to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.

 

[marcus]

... to simplify a triangulatio of a 3-manifold: use a 2-complex (a so-called spine of a 3-manifold). It is known that every 3-manifold can be constructed by using spines but nothing less (including no 1-complex like a spin network).
But I have to think more carefully about it.

Is a "spine" in this case the same as the 2-skeleton?
I think I know what you mean. The 2-skeleton is a 2-complex that simplifies the triangulation of a 3 -manifold by discarding the 3-cells and just leaving the vertices edges faces.

Torsten, I decided it would be more "on-topic" to reply to you in the "Dynamics of NCG" thread that PhysicsMonkey started as an offshoot of this one. So I quoted and replied to your post here:
https://www.physicsforums.com/showthread.php?p=4414879#post4414879

 

[julian]

You can't classify 3 manifolds with 1d spin networks but what about q-deformed spin networks? Here vertices are replaced with discs and links with ribbons which can have twists in them...isn't this related to Dehn surgery and Kirby calculus or something? What is the additional information in q-deformed spin network states? Just the twists?

 

[torsten]

You can't classify 3 manifolds with 1d spin networks but what about q-deformed spin networks? Here vertices are replaced with discs and links with ribbons which can have twists in them...isn't this related to Dehn surgery and Kirby calculus or something? What is the additional information in q-deformed spin network states? Just the twists?

Hi Julian, it is not directly but related to Kirby calculus etc. q-deformed spin networks are more related to skein theory (as far as I can remember Kauffman proved that there is a strong relation). Skeins are related to knot theory and the knot polynomials. The relation to Kirby calculus was proved by Lickorish, he obtained 3-manifold invariants (of Witten type).

 

[torsten]

Sorry I forgot something to write:

Hi Julian, it is not directly but related to Kirby calculus. The q-deformed spin networks are more related to skein theory (as far as I can remember Kauffman proved that there is a strong relation, see the paper in Int. J. Mod. Phys. A 5 (1990) 93). Skeins are related to knot theory and the knot polynomials. The relation to Kirby calculus was proved by Lickorish, he obtained 3-manifold invariants (of Witten type), see J. Knot Theory and Its Ramifications, 2 (1993) 171.
In both cases the quantum group SU_q(2) was used.
Thanks fro asking!
Torsten

 

[MTd2]

It seems some big shot guys from string theory uploaded a paper on exotic smoothness but did not cite Torsten. I feel really sad because I consider him a friend. It's also sad that they mentioned an expression contained in the title which is a famouls book on exotic smoothness, without citing it "wild world of 4 manifolds".

http://arxiv.org/pdf/1306.4320v1.pdf

 

[Haelfix]

yes, of course

yes, why not?

I know; nevertheless it may be interesting to take it seriously.

The interesting thing is that if you define the counting AND sum over ALL dimensions (!) then the sum is naturally peaked at dim=4 due to the uncountably many smoothness structures for non-compact 4-manifolds.

So if you go over the paper I linked too where they compute 2+1 pure Ads gravity with toric boundary conditions, they actually explicitly compute the sum over manifolds (and topological structures). What's fascinating is the choices they make on how to partition the sum.

B/c of the AdS/CFT correspondance they are able to bootstrap an answer and it seems to be self consistent on both sides of the duality. That's really cool, and as far as I know, has only occurred for Chern-Simmons theory exactly.

Anyway, what peaked my interest was that they did indeed include nontrivial diffeomorphisms in the sum (so called modular transformations), that they did arrange the sum as an expansion around classical solutions followed by loop corrections (where the coefficients need not be small) and that they explicitly did not include singular configurations of the geometry. That they then compute an answer that seems to agree with the CFT side is quite nontrivial and interesting, and at least motivates that these might be the correct choices for 4d.