Spin Foams and Noncommutative Geometry (video, Marcolli at Perimeter)

[marcus]

Marcolli has been a frequent collaborator with Alain Connes, and is now math prof at Caltech.
Since NC geometry has yielded the Standard Model particles, plus some predictions, there is interest in basing the NC matter on spinfoam quantum geometry
http://pirsa.org/11020110/
Spin Foams and Noncommutative Geometry
Speaker(s): Matilde Marcolli
Abstract: We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry. (Based on joint work with Domenic Denicola and Ahmad Zainy al-Yasry)
Date: 23/02/2011 - 4:00 pm

 

[marcus]

Before watching her "SF+NCG" talk it might be helpful to watch the talk she gave the day before. The first 36 minutes give an intuitive description of how NC geometry is set up and how it recovers the standard particle model (extended by righthand-neutrinos).

She is a remarkably good communicator. The talk gives much more understanding than you get in any of the online write-ups I have seen. The title of the talk is not descriptive of the first 36 minutes.

http://pirsa.org/11020109/
Cosmology and the Poisson summation formula
Speaker(s): Matilde Marcolli
Abstract: We show that, in a model of modified gravity based on the spectral action functional, there is a nontrivial coupling between cosmic topology and inflation, in the sense that the shape of the possible slow-roll inflation potentials obtained in the model from the nonperturbative form of the spectral action are sensitive not only to the geometry (flat or positively curved) of the universe, but also to the different possible non-simply connected topologies. We show this by explicitly computing the nonperturbative spectral action for some candidate cosmic topologies, spherical space forms and flat ones given by Bieberbach manifolds and showing that the resulting inflation potential differs from that of the sphere or flat torus by a multiplicative factor. We then show that, while the slow-roll parameters differ between the spherical and flat manifolds but do not distinguish different topologies within each class, the power spectra detect the different scalings of the slow-roll potential and therefore distinguish between the various topologies, both in the spherical and in the flat case. (Based on joint work with Elena Pierpaoli and Kevin Teh)
Date: 22/02/2011 - 2:00 pm
==================

The whole talk is 90 minutes. So in the second half she is going to talk about some new results (cosmology, inflation). But what is so valuable is that in the first half she is bringing nonspecialists up to speed on NC geometry.

 

[marcus]

The whole 1 and 1/2 hour talk on "Cosmology" by Marcolli was worth watching.
http://pirsa.org/11020109/
Also the whole Spinfoam+NCG talk I gave link to earlier.

She got a lot of questions, showing serious interest, from top people at Perimeter.

This is the approach to QG (and matter) which for me creates the strongest tension with LQG.

All the more so because, with the help of this talk http://pirsa.org/11020109/ it becomes much easier to understand what is going on with NC geometry.

One of the papers that Marcolli cited we noted in this thread:
"Merging NCG matter with Spinfoam LQG spacetime"
https://www.physicsforums.com/showthread.php?t=470478
The Holst action from the Spectral Action Principle by Pfäffle and Stephan
She also mentioned research by Christian Bär, whose name came up in that thread as well.

 

[humanino]

Thank you for posting those links Marcus, much appreciated. The idea of combining NCG and LQG is very tantalizing and it is refreshing to see that competent people attempt in this direction.

 

[marcus]

Thank you for posting those links Marcus, much appreciated. The idea of combining NCG and LQG is very tantalizing and it is refreshing to see that competent people attempt in this direction.

It's reassuring that someone else found this interesting (coming at it from your own direction, different from mine.) Thanks for the encouraging response. I find both videos cogent. She is a good communicator as well as a grade-A mathematician.

EDIT: to reply to Mitchell.

For me, the fantastic idea here is the "topologically enriched" spin network, where along with information about holonomies, you have information about monodromies which encode the topology of the background manifold.

Yes, that "topologically enriched" is a novel and intriguing idea! She's a creative mathematician. The Caltech department got a good deal.

As with the best ideas, I don't feel able to guess where this one could lead. It has to do with covering spaces of the hypersphere S3. Correct me if I am wrong but isn't it true that you can get all compact topological 3-manifolds as finite covers of that (and the analogous for 3-torus). So it suddenly seems very easy to have quantum uncertainty about topology. You just make cuts in the S3 to show where the covering space ramps up and later rejoins. I don't have time to check her paper on this or review the talk so I'm counting on you to clarify/amplify this as needed.

Beautiful idea, but it would make the whole QG game so much more complicated if the topology were allowed to waver while the geometry was flickering.

 

[mitchell porter]

For me, the fantastic idea here is the "topologically enriched" spin network, where along with information about holonomies, you have information about monodromies which encode the topology of the background manifold.

 

[marcus]

For me, the fantastic idea here is the "topologically enriched" spin network, where along with information about holonomies, you have information about monodromies which encode the topology of the background manifold.

I'm in strong agreement, but I can't give any reliable guess about how it will work out (LQG, Spinfoam, NCG, topology) in this nexus of ideas.
It's indicative other people are preparing for some crossover, who set up the school for young researchers getting into QG, e.g. with both Rovelli and Steinacker

QG school website with complete 2-week program:
http://www.fuw.edu.pl/~kostecki/school3/
==quote==
Carlo Rovelli - Spin foams
The most active field in the network in the last years has been spin foam models, starting with the development of the graviton propagator and the new models, to coherent state techniques and recent asymptotic results, the generalisation to arbitrary 2-complexes and cosmological applications. The lectures will present the current perspective on the construction of these models in terms of 2-complexes.

Harold Steinacker - Non-commutative geometry and matrix models
Non-commutative geometry is a natural extension of geometry in the context of quantum theories that potentially, may also include gravity.. NCG naturally occurs in particle physics, as shown by Alain Connes, and also appears naturally in the context of three-dimensional quantum gravity via Chern-Simons theory. It is also used as a technical tool in state sum models, particularly via quantum groups, which provide deformations of the usual spin network calculus which can be used to construct quantum gravity models. The lectures will cover the definition and construction of non-commutative spaces as well as the construction of QFTs on them. Another theme will be the relationship to matrix models.
==endquote==

some discussion here:
https://www.physicsforums.com/showthread.php?t=457381

 

[mitchell porter]

I'll describe what I immediately thought might be a use for the "topspin networks", although it's very off-topic for this thread, and I don't even know if my idea makes sense.

Nathan Berkovits studied a "zero-radius limit" of the AdS/CFT correspondence, that is, he studied string theory in the AdS₅ x S⁵ space, in the limit where the radial dimension of the AdS space shrinks to zero size. He constructs a direct correspondence between Feynman diagrams in the CFT (N=4 d=4 super-Yang-Mills, I'll call it SYM4) and string worldsheet histories in the zero-radius AdS space. Standard perturbative string theory represents an infinite-time string scattering process as a compact Riemann surface with special points corresponding to asymptotic string states. The compact surface is created by a conformal transformation of the worldsheet history, so that a string going off to infinity in some direction gets mapped onto a set of concentric circles on the Riemann surface converging on one of those special points, and then there's a "vertex operator" added at the point, which represents the asymptotic quantum state of the string. Berkovits then adds to this picture Wilson lines running between the special points. It seems that it's these Riemann surfaces decorated with Wilson lines which directly correspond to the CFT Feynman diagrams.

Meanwhile, it's generally believed that SYM4 completely describes IIB string theory on the AdS space above, including topological changes to the background space. And Lubos Motl once made a post pointing out that Berkovits's Wilson-line networks can be thought of as spin networks. So my thought was: could we make these topological fluctuations of the AdS space manifest, by placing the topspin networks of Marcolli et al on the string worldsheet? The immediately obvious problem with this is that the topspin networks encode 3-dimensional topological data, not 10-dimensional topological data.

While I'm on the subject of string theory, I may as well express a "string fundamentalist" interpretation of the significance of LQG, NCG as applied to physics, and any combination thereof. It would simply be that such theories represent truncations of string theory on a particular background. Again, it was Lubos Motl who I first saw express the idea that the noncommutative space in the NCG standard model must just be a truncation of a particular string compactification (including only massless excitations). Similarly, one possible approach to the problem of LQG dynamics would be to say that it will be well-defined only if it corresponds to some 4D solution of string theory; and the combination of LQG and NCG would exist as a sort of double limit.

I say all that just to state a particular hypothesis about where everything will end up. I don't especially believe it, and we are enormously far from being able to prove or disprove it. But it could certainly define a very large research program.

Going back to the question of what topspin networks will be good for in the long run, surely they will be mathematically consequential, to computations in combinatorial topology, maybe low-dimensional cohomology... But probably they need to be generalized, to objects other than graphs embedded in spaces with other than three dimensions. Marcolli et al already extend them to topspin foams in four dimensions. As for what people working in LQG will make of them, someone else will have to speculate.

 

[atyy]

I'll describe what I immediately thought might be a use for the "topspin networks", although it's very off-topic for this thread, and I don't even know if my idea makes sense.

Nathan Berkovits studied a "zero-radius limit" of the AdS/CFT correspondence, that is, he studied string theory in the AdS₅ x S⁵ space, in the limit where the radial dimension of the AdS space shrinks to zero size. He constructs a direct correspondence between Feynman diagrams in the CFT (N=4 d=4 super-Yang-Mills, I'll call it SYM4) and string worldsheet histories in the zero-radius AdS space. Standard perturbative string theory represents an infinite-time string scattering process as a compact Riemann surface with special points corresponding to asymptotic string states. The compact surface is created by a conformal transformation of the worldsheet history, so that a string going off to infinity in some direction gets mapped onto a set of concentric circles on the Riemann surface converging on one of those special points, and then there's a "vertex operator" added at the point, which represents the asymptotic quantum state of the string. Berkovits then adds to this picture Wilson lines running between the special points. It seems that it's these Riemann surfaces decorated with Wilson lines which directly correspond to the CFT Feynman diagrams.

Meanwhile, it's generally believed that SYM4 completely describes IIB string theory on the AdS space above, including topological changes to the background space. And Lubos Motl once made a post pointing out that Berkovits's Wilson-line networks can be thought of as spin networks. So my thought was: could we make these topological fluctuations of the AdS space manifest, by placing the topspin networks of Marcolli et al on the string worldsheet? The immediately obvious problem with this is that the topspin networks encode 3-dimensional topological data, not 10-dimensional topological data.

While I'm on the subject of string theory, I may as well express a "string fundamentalist" interpretation of the significance of LQG, NCG as applied to physics, and any combination thereof. It would simply be that such theories represent truncations of string theory on a particular background. Again, it was Lubos Motl who I first saw express the idea that the noncommutative space in the NCG standard model must just be a truncation of a particular string compactification (including only massless excitations). Similarly, one possible approach to the problem of LQG dynamics would be to say that it will be well-defined only if it corresponds to some 4D solution of string theory; and the combination of LQG and NCG would exist as a sort of double limit.

I say all that just to state a particular hypothesis about where everything will end up. I don't especially believe it, and we are enormously far from being able to prove or disprove it. But it could certainly define a very large research program.

Going back to the question of what topspin networks will be good for in the long run, surely they will be mathematically consequential, to computations in combinatorial topology, maybe low-dimensional cohomology... But probably they need to be generalized, to objects other than graphs embedded in spaces with other than three dimensions. Marcolli et al already extend them to topspin foams in four dimensions. As for what people working in LQG will make of them, someone else will have to speculate.

Hmm, you quoted Lubos twice up there, and he's clearly thinking about spin networks again here http://physics.stackexchange.com/questions/5517/dual-conformal-symmetry-and-spin-networks-in-abjm (I see you answered him!) maybe marcus can use this to show an increase in interest in LQG!

OK, to be a bit more serious, do "real" noncommutative field theories arise in strings, or can they all be made into "normal" field theories by a Seiberg-Witten transform?

 

[mitchell porter]

do "real" noncommutative field theories arise in strings, or can they all be made into "normal" field theories by a Seiberg-Witten transform?

I don't know how to prove that a generic noncommutative theory can't be rewritten as a "purely commutative" theory.

But let's just list some of the ways that space-time noncommutativity shows up in string theory:

There was the original example due to Douglas, Connes, and Schwarz, of M-theory compactified on a noncommutative torus, and the various examples involving noncommutative worldvolume theories on branes (often with a background flux included). I think most or all of these can be transformed.

Noncommutativity (the Moyal product) also appears in string field theory. (There are even versions of string field theory which are nonassociative.) The Moyal product also plays a role in Vasiliev theory (higher-spin gauge theory).