The n-Category Café

Is there a reason why 1d or 3d field theories can’t be regarded as generalized manifolds?

No, they can be.

For 1d this is well known. For 3d almost nothing is really known, but it should work analogously:

A 1-dimensional Riemannian QFT with a single binary interaction is essentially the same thing as a spectral triple (Kontsevich-Soibelman speak of graph field theory , see the $n$Café entry here):

the Hilbert space assigned to a point is the Hilbert space of the spectral triple, the Hamiltonian of the 1d QFT is the Laplace operator of the spectral triple, or rather the Dirac operator for a supersymmetric 1d QFT.

In other words, quantum mechanics is a way to talk about the Riemannian geometry of the target space in which a quantum particle propagates in terms of the energy spectrum of the particle (can you hear the shape of spacetime?). I guess this is how Connes came up with the idea of spectral geometry (usually known, misleadingly, as (Connes-style) _ noncommutative geometry_ ) in the first place, possibly inspired by Witten’s early observations on how supersymmetric quantum mechanics encodes interesting geometry on target space. I once wrote a series of $n$Café entries explaining how Connes tries to realize our observed world as the effective target space of a 1d QFT (a quantum superparticle) in this fashion: here.

When one goes from 1d to 2d, spectral triples become 2-spectral triples, usually known as 2d CFTs: there are now two Laplace operators or Dirac operators, the “left moving” and the “right moving” one, known as the Virasoro generators if we are speaking of a 2d conformal CFT. The premise of string theory is to take this seriously and investigate the hypothesis that the world around us is not actually the 1-spectral geometry that Connes considers, but an actual 2-spectral geometry. (And it is maybe amusing to note that both Connes as well as string theory find that these spectral geometries have to have KO-dimension 10, for consistency.)

Liang Kong has a nice set of notes with an exposition of this idea for a course he taught. He calls it “stringy algebraic geometry”. We are just in a middle of an email conversation where I was arguing that he should rather call it “string spectral geometry”. The plan is that his notes will make it to a contribution in our book.

One can take the “point particle limit” of a 2dQFT describing a string propagating inside such a spectral geometry, in which it goes over to look just like a point particle. Accordingly there is a limit that takes a “2-spectral triple” (a 2d CFT) to an ordinary spectral triple. This has been originally studied at a mathematical level by Roggenkamp-Wendland and by Kontsevich-Soibelman (see the above link). Yan Soibelman has now written it up all nicely in his contribution to our book. Not much longer and I can show this around.

For 3d the guess is that the story continues analogously, becoming ever richer. That’s the hypothesis, essentially, underlying the idea that “the super-membrane is the fundamental degree of freedom of M-theory”: based on the above well-understood story for particles and strings, and given a host of hints that it makes sense to consider these 2d CFTs in turn as string-limits (called “double dimensional reduction”) of 3d QFTs of membranes, the natural guess is that for suitable 3d QFTs one can understand these as describing the dynamics of quantum membranes in the effective spectral geometry encoded by the QFT.

So I think the answer is: dimension 2 here is special (only) in that it is the highest dimension where we can make progress with understanding the situation. There are hints that also d = 6 should be tractable, being the EM-dual of $d = 2$ for 10-dimensional targets

I don’t understand this stuff nearly as well as I’d like, but my impression is that ultimately, the modern relation between arithmetic geometry and conformal field theory is just a modern outgrowth of the old analogy between number fields and function fields of Riemann surfaces. Now that string theorists have become experts at doing quantum field theory on Riemann surfaces, a bunch of their ideas should generalize to number fields.

Dear Urs,

there are videos of his lectures:

http://www.mpim-bonn.mpg.de/node/3372/abstracts

http://www.mpim-bonn.mpg.de/node/3372/program

Best wishes and regards

Jenny

there are videos of his lectures:

Ah, thanks for the links.

I have collected them now on the $n$Lab at Three Roles of Quantum Field Theory .

Have just watched the first one, where Segal surveys all the ingredients from classical physics. Nice to see that 1. he discusses the canonical presymplectic structure on covariant phase space; 2. he amplifies the observation that of the algebra structure on the quantum algebra of observables really only the “symplectic” structure (notably its semilattice of isotropic subspaces) should matter.

Two simple but important points that are often not emphasized.