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Picking up from G+:
I see three main points in the comments you made:
a) on the organization of dcct and how to navigate interdependency of results there
b) on what the result and point of higher geometric quantization is
c) whether n-fold correspondences are really a good way to speak of Lagrangian correspondences and similar in higher codimension.
Is that roughly right?
Right.
Regarding a) I am the first to admit that there is much room for improving that document. I am working on it slowly but surely.
But regarding what seemed to be an implicit complaint of yours that the propositions in the section 1 are not being referred to in the rest of the text I’d like to say: section 1 is the Introduction, meant to put stuff in context and down to earth, and it depends on the rest, not the other way around. The section “geometry of physics” that we are looking at now could also move to the last sections “Applications”, but I thought it would be good to move this material to the introduction.
Not really a complaint, just a remark. I did have the impression that you would need the de Donder-Weyl-Hamilton stuff to set up higher pre-quantum geometry and then proceed to apply higher geometric quantization. However, doing the obvious thing, searching for the numbers of the relevant propositions and theorems in the full text did not lead to much. So I was just looking for pointers to better navigate the text.
Regarding b), remember that a key point is that it is an illusion that the n-plectic forms aka symplectic current densities in covariant phase space technology are globally defined, since that is only true before passing to the reduced phase space. The point of the section on “kinetic action functionals” and related is to point out that in general there is no way around having the (n-)symplectic form be the curvature of a (higher) line bundle. This is the higher prequantization.
And the mystery in the literature that I had referred to is the question of what a Poisson algebra of observable is for n-plectic forms, hence for local currents. We talked about that at length in Trento (and maybe I have to apologize for not having gotten back to this, but somehow I am still absorbed with surviving and now yet back to full research mode). The higher quantization that I am discussing solves this as it gives the higher quantomorphisms groups and so forth. That’s in dcct, but the standalone article that discusses it is “Higher geometric prequantum theory”.
First, no need to apologize. It will be some time before I can get back to the details of our previous discussion myself. Second, I myself must apologize because it might be some time before I can pick up this thread of higher geometric quantization, even though I was the one who brought it up. I find the examples that you use to illustrate this method (Chern-Simions, AKSZ, Poisson -model, WZW model) somewhat removed from my experience. Hence it’s tough for me to find a point of entry into the (somewhat) lengthy relevant material in dcct or this other stand alone paper. What I’d like to get to grips with is the claim, which I believe you’ve made before, that there are some Hamiltonian (or de Donder-Weyl-Hamiltonian) aspects of local classical field theory that are necessary to make higher geometric quantization work. Since I don’t understand this quantization method, it’s hard for me to see how true that is and what these aspects are precisely. The example that would help would be the Klein-Gordon field on a globally hyperbolic space (heck, even just Minkowski space!) treated via higher geometric quantization. There, I at least know what to expect from conventional quantization (performed in any number of standard ways).
Finally regarding c), the correspondences: I may have to have a closer look at your construction, but generally the point of the n-fold corespondences is to connect to the axiomatics of functorial (pre-)quantum field theory. For this one makes crucial use of facts about n-fold correspodences forming a symmetric monoidal (infinity,n)-category with all duals. This is key to the actual goal of constructing full local TQFTs, and from there one is to proceed to work out what these n-fold correspondences come down to in examples, or conversely how to build them from explicit data.
The construction that I outlined (Lagrangian subspaces (of solutions) of symplectic spaces of boundary data) is more flexible than Lagrangian correspondences, which are included as a special case. So, my intuition suggests that anything that correspondences can do, this alternative can do as well. I’d need to learn more about these categorical properties to say precisely how, though.
P.S.: To bystanders, no hate related to the title, please. :-) The pun is purely syntactic and I don’t actually have any idea about the semantics that it touches on.
Regarding b): right so I am not talking much about taking a non-topological theory at face value and applying higher geometric quantization to it. Instead I am talking about regarding it as the boundary theory of a topological theory and applying higher geometric boundary quantization to that toplogical theory. For the generic quantum mechanical system regarded non-covariantly, that cobounding theory is the 2d Poisson-Chern-Simons theory and indeed its 2-plectic geometric boundary quantization yields the ordinary quantization that you expect to see.
Incidentally, this is the topic of the third session of my Paris lectures tomorrow. I’ll be busy preparing notes for that today.
This is why there is all that discussion of higher CS-type field theories in the notes, which you say you have a hard time connecting to. The general strategy is based on the observation that the system where traditional geometric quantization really came to flourish is 3dCS, and that this did end up being at the same time the quantization of the 2d boson with values in a compact group manifold (say in a torus, to get close to the Klein-Gordon field).
Regarding b): right so I am not talking much about taking a non-topological theory at face value and applying higher geometric quantization to it. Instead I am talking about regarding it as the boundary theory of a topological theory and applying higher geometric boundary quantization to that toplogical theory. For the generic quantum mechanical system regarded non-covariantly, that cobounding theory is the 2d Poisson-Chern-Simons theory and indeed its 2-plectic geometric boundary quantization yields the ordinary quantization that you expect to see.
OK, but this really is what I have trouble connecting to. What is the holographic -dimensional theory whose higher geometric quantization produces a fullly extended QFT (or TQFT) such that it assigns the ordinary quantization of Klein-Gordon theory to codimension- surfaces? And if such a thing is available, why bother with the holographic part at all? By that, I only mean that it should be possible to treat the -theory as auxiliary in the construction and, eliminating references to it, end up with a stand-alone quantization of the Klein-Gordan field (possibly fully extended itself). Would such a construction be what you would call a “higher geometric quantization” of Klein-Gordon theory? If such a thing were a available, it shouldn’t be too hard to compare it to standard methods to see where the similarities and differences are.
In my, rather limited I must admit, understanding, the closest thing that is currently available in your notes is the application of the 2d Poisson -model to the particular case of the Klein-Gordon field treated as infinite dimensional mechanical system. However, this approach totally throws out any any (spatial) locality or (spacetime) covariance properties of the theory. Is this a fair assessment or am I missing something?
P.S.: Good luck with the lectures!
Yes, that’s exactly what I was referring to in #2 when saying
For the generic quantum mechanical system regarded non-covariantly,
That’s what we currently have: quantization of 1d quantum evolution via 2-plectic quantization of 2dPCS theory, quantization of covariant 2d CFT via 3dCS theory. The pattern is clear, but somebody needs to write out more details in higher dimension. I wrote a grant proposal proposing to do so, now it needs somebody to fund that. Should they ask you to be my referee, you’ll have a chance to say that these are open questions that you are wondering about, too! :-)
Haha, I doubt that my name would come across the desk of any funding agency. But if it does, you’ll have my full support!
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