On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), that I translate by "The subfactors : an enriched Galois theory".
I don't find a video of the same conference in English, nevertheless, for people interested, Vaughan gave a series of lectures (in English) introducing to the subfactors theory (see here).

The key point is that most of the tools and objects for studying a field extension $\mathbb{K} \subset \mathbb{L}$ exist naturally for studying a subfactor $N \subset M$: for example the Galois groups $G=Gal_{\mathbb{K}}(\mathbb{L})$ or $Gal_{N}(M)$.
The reason why it's an enriched Galois theory is that on one hand $\mathbb{K} = \mathbb{L}^{G}$ whereas on the other hand, the case $N=M^{G}$ is very particular, and given by the group subfactor $R^G \subset R$, with $R$ the hyperfinite II$_1$ factor (for simplifying we suppose $G$ to be a finite group). In general, a subfactor is related to an object called the standard invariant, with a "quantum flavour" (even more general than just quantum groups). There are several approaches (see here): paragroups, $\lambda$-lattices, planar algebras...

Now, I'm completely ignorant about motives, I just know it should be (also) an enriched Galois theory:

<< One of the hopes of the theory of motives is to provide a framework generalizing Galois theory, allowing the definition of motivic Galois groups. These generalize usual Galois groups for systems of several multi-variable polynomials. It is no more finite groups, but algebraic groups >>.

The motive is one of the famous (and for most people, mysterious) notions introduced by Alexander Grothendieck for his need in algebraic geometry, the other one being the topos. But the notion of topos is (in some sense) a categorical generalization of the notion of topological space, as the notion of $C^{\star}$-algebra is (in some sense) the quantum generalization. In this paper (p402) Pierre Cartier wrote about a link between works of Alexander on topos and works of Alain Connes on $C^{\star}$-algebras, via works of Joseph Tapia. Nowadays, Alain works on topos, his last course at the "Collège de France" is "The epicyclic site". He develops fruitful links with its noncommutative geometry (see this interview on 5th February 2014 at the IHES). See also the post Grothendieck and Non-commutative Geometry?.

See also From toposes to non-commutative geometry through the study of internal Hilbert spaces PhD-dissertation (defended on sept. 25, 2014) by Simon Henry (directed by Alain Connes).

Question: Is there a link between subfactors and motives (as enriched Galois theories), as there is this link between operators algebras and topos (as enriched topological spaces) ?

In the same paper Pierre wrote (p405):

<< Grothendieck's broken dream was to develop a theory of motives, which would in particular unify Galois theory and topology >>.

It really looks like to what the theory of subfactors do at the quantum level: it's a quantum Galois theory and a subfactor can be related to quantum topologies, i.e. topological quantum field theories (tqft).
See the book: (2+1-dim.) Topological quantum field theories from subfactors (Kodiyalam-Sunder, 2000), the paper: Subfactors (planar algebras) and 1+1-dim. TQFTs (Kodiyalam-Vishwambhar-Sunder, 2005)

See the book: Noncommutative Geometry, Quantum Field Theory and Motives (Connes-Marcolli 2008) and the other book: Feynman motives (Marcolli, 2010).

Here is the translation (in English) of the transcription of (a bit cryptic) exchange (in French) between Vaughan Jones and the physicist Edouard Brézin (on 27th March 2007, see here: 1:10:15 – 1:11:30) :

Edouard : << Naive question, the Feynman diagrams, at the limit, where the gauge group is infinite dimensional, become planar. Is it possible that your planar algebras are restrictions of more simple algebras ? >>
Vaughan: << Ah, this is a very good question, I know, there are others things we want to put in the inner disks of a planar tangle, these are random matrices, as we say ; and it's the case... so this story of Feynman diagrams when the number of indices go to infinity, we only see planar things at the limit... Well, I didn't speak about this, because justly I'm working on this, it's my current obsession, and I don't know what answer, I would love answer..., but there is certainly a link because we see the same structures in the random matrices and in the planar algebras, but we don't know how. So this disappearance of higher genre things are still a bit mysterious for the planar algebras...
So this was not at all a naive question ! >>

Do the motives encode categorical symmetries of topos, as the subfactors encode quantum symmetries of operator algebras ?

Is there a dictionary between these two theories ?
Can the subfactors theory help the development of the motives theory (and vice versa)?

For example, the (finite depth finite index irreducible) subfactors admit a categorical equivalent formulation (of Tannaka-Krein flavour): the fusion categories (together with an algebra object).
The fusion ring is its Grothendieck ring.

Perhaps these theories are two complementary aspects of a more general theory, generalizing for example the notion of Galois groups for system of several multi-variable noncommutative polynomials.

See my new post on physics.stackexchange: TQFTs and Feynman motives

  • $\begingroup$ One of the basic ideas for motives is that they shouldn't be closey related with any particular topos. $\endgroup$ – Mikhail Bondarko Mar 27 '14 at 20:34
  • $\begingroup$ @MikhailBondarko: as for the subfactors $(N \subset M)$: the quantum symmetry encoded in its planar algebra (or standard invariant) is essentially independent of the choice of the factors $N$ and $M$ (a factor is a center-less von Neumann algebra). For example for $G$ a finite group, the subfactor $(M^G \subset M)$ encodes the same symmetry if $M$ is the hyperfinite II$_1$ factor $R$ or the factor $L(\mathbb{F}_{\infty})$. $\endgroup$ – Sebastien Palcoux Mar 27 '14 at 21:22
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    $\begingroup$ There is a stone-soup flavor here, since you are asking for a speculative connection between two fields, without any concrete principles that would motivate such a belief. For example, the theory of motives can be motivated by the very concrete observation that different cohomology functors for algebraic varieties yield the same Betti numbers, suggesting that there is some universal abelian category (the category of motives) through which those functors factor. I don't see anything like that in your question. $\endgroup$ – S. Carnahan Mar 28 '14 at 19:56
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    $\begingroup$ There is no renormalization in a tqft! $\endgroup$ – S. Carnahan Mar 28 '14 at 20:29
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    $\begingroup$ Before you make conjectures about deep connections between areas of mathematics, you should at least familiarize yourself with basic properties, instead of just putting words together. To answer your question, renormalization is concerned with how the interactions in a field theory change with respect to length scale or energy scale. In a topological field theory, there is no length scale (you see, it gives topological invariants), so the lack of renormalization is a basic property. This breaks your indirect link, by the way. $\endgroup$ – S. Carnahan Mar 28 '14 at 20:47