The n-Category Café

Van Est??
as in the famous theorems?
if so, how did his name end up here?

how did his name end up here?

To honor a great Utrechtian and to indicate the good tradtion of study of geometry that the seminar hopefully continues. And because the name provides 3.5 letters for the word quest .

I’m even more curious about the nonassociative string theory backgrounds, but I gather that these have received rather little attention.

Have you seen Atiyah’s lecture (last on the page) – From Quantum Physics to Number Theory? It seems to cover similar ground to the one mentioned here.

There’s the idea that to bring gravity within quantum field theory the octonions need to be involved via G2-manifolds, and that while cyclic cohomology is fine for noncommutative associative algebras, there needs to be an analogue for nonassociative algebras.

I haven’t seen Atiyah’s lecture and I’m not sure it would give many clues about what he’s really thinking, but Conan Leung told me something like: Atiyah and others have developed a theory of flop transitions for $G_2$ manifolds, which should play a role in M-theory vaguely analogous to the usual flop transitions for Calabi-Yau manifolds in string theory, but with triality entering in a crucial way.

As far as I can tell, none of this has any known direct connection to nonassociative tori, which supposedly show up when you do torus compactifications of M-theory and the 3-form field has nontrivial flux over the torus fibers.

(By the way, I don’t understand either of these subjects in anything but the most superficial way.)

non-associative referring to your beloved octonions
OR
homotopy associative
OR
???
non-associative strings - that wasn’t an active link
could you provide one?

jim

I’m guessing that Urs is trying to get a conversation started on the subject matter of the talks

That’s right. And now I need to recover from the shock of having received more comments on a single entry than in total on the last dozen.

and that maybe he’d be happy for a question from anyone,

Yes! Maybe we can get back to the point where everyone feels like asking lots of questions about our blog entries. Where would I be had I never asked about the Leinster measure, for instance. And now after lots of posts about $\sigma$-models we discover that Tom never asked what these were!

to what extent have noncommutative string theory backgrounds been applied to particle physics phenomenology? In other words, to what extent have they been studied as ‘quasi-realistic’ models of physics? What are the best candidates, if any?

Alain Connes described (see here) a spectral nc Kaluza-Klein background where the compactified fiber is shrunk to an actual classical point, retaining only noncommutatively visible dimension to a total of 10 mod 8. Being given by a spectral triple, this is the target for a superparticle. Maybe it’s the point particle limit of some noncommutative superstring $\sigma$-model. The KO dimensions at least do match.

A $\sigma$-model is a theory of physics where you have a manifold $M$ standing for ‘spacetime’, another manifold $X$, and a function

$$F : M \to X$$

which is called a ‘field’. Then there will be some differential equations that $F$ needs to satisfy.

A $\sigma$-model is a kind of ‘field theory’: a theory where the world is described by some function on spacetime that wiggles around according to some equation. Physicists love field theories. You may know about the electromagnetic field, which at one level of sophistication can be thought of as a pair of vector fields, namely functions

$$E: M \to \mathbb{R}^3$$

$$B : M \to \mathbb{R}^3$$

where $E$ obviously stands for ‘electric’ and $B$ obviously stands for, umm, ‘magnetic’. At a slightly higher level of sophistication we lump them into a single field:

$$F : M \to \mathbb{R}^6$$

This obeys a differential equation called ‘Maxwell’s equations’. That’s the most famous and useful field theory of all.

Building on this idea, physicists went ahead and studied all sorts of ‘fields’. At first these always took values in a vector space. The $\sigma$-model idea was that they could also take values in some other sort of manifold: like a sphere, or whatever you want.

They’re called ‘$\sigma$-models’ for an absolutely idiotic reason. Back in the 1960s people were interested in a hypothetical particle called the $\sigma$ particle. A famous physicist named Gell-Mann (the guy who invented ‘quarks’) came up with a theory of them. It was called ‘the $\sigma$-model’. It was an old-fashioned field theory where the field took values in a vector space. You see, particles in nature are really fields. (Yeah, it’s confusing.)

Then someone came up with a modified version of the $\sigma$-model where the field took values in some other manifold. They called this ‘the nonlinear $\sigma$-model’.

Then all field theories where the fields took values in a manifold other than a vector space got called ‘nonlinear $\sigma$-models’.

Then people got tired of saying ‘nonlinear’ and called them simply ‘$\sigma$-models’.

By now people have generalized the heck out of this idea. If you look in the $n$Lab you’ll find talk of $\sigma$-models where $X$, instead of being a mere manifold, is (say) a symplectic Lie $n$-algebroid, or an $\infty$-stack. But somewhere, deep at the bottom of the history of this mathematics, is a poor little particle that might or might not even exist.

Tom wrote:

what is T-duality?

There is a small booklet about T-Duality (for mathematicians):

• Jonathan Rosenberg: Topology, $C^*$-algebras, and string duality (ZMATH).

(I’ve added it to the references of topological T-duality).

Tom asked:

what is a sigma-model a model of?

Quoting from sigma model – Terminology and history:

In physics one tends to speak of a model if one specifies a particular quantum field theory for describing a particular situation, for instance by specifying a Lagrangian or local action functional on some configuration space. This is traditionally not meant in the mathematical sense of model of some theory . But in light of progress of mathematically formalizing quantum field theory (see FQFT and AQFT), it can with hindsight be interpreted in this way:

a $\sigma$-model is supposed to be a type of model for the theory called quantum field theory . This sounds like a tautology, but much effort in mathematical physics is devoted to eventually making this a precise statement. In special cases and toy examples this has been achieved, but for the examples that seem to be directly relevant for the phenomenological description of the observed world, lots of clues still seem to be missing.