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I am currently writing an article as follows:
Classical field theory via higher differential geometry
Abstract We discuss here how the refined formulation of classical mechanics/classical field theory (Hamiltonian mechanics, Lagrangian mechanics) that systematically takes all global effects properly into account – such as notably non-perturbative effects, classical anomalies and the definition of and the descent to reduced phase spaces – naturally is a formulation in “higher differential geometry”. This is the context where smooth manifolds are allowed to be generalized first to smooth orbifolds and then further to Lie groupoids, then further to smooth groupoids and smooth moduli stacks and finally to smooth higher groupoids forming a higher topos for higher differential geometry. We introduce and explain this higher differential geometry as we go along. At the same time as we go along, we explain how the classical concepts of classical mechanics all follow naturally from just the abstract theory of “correspondences in higher slice toposes”.
This text is meant to serve the triple purpose of being an exposition of classical mechanics for homotopy type theorists, being an exposition of geometric homotopy theory for physicists, and finally to serve as the canonical example for and seamlessley lead over to the formulation of a local prequantum field theory which supports a localized quantization to local quantum field theory in the sense of the cobordism hypothesis.
This started out as a motivational subsection of Local prequantum field theory (schreiber) and as the nLab page prequantized Lagrangian correspondences, but for various reasons it seems worthwhile to have this as a standalone exposition and as a pdf file.
I am still working on it. Section 1 and two have already most of the content which they are supposed to have, need more polishing, but should be readable. Section 3 is currently just piecemeal, to be ignored for the moment.
Should we expect this formulation to tell us things about most features of classical mechanics. I mean what about say integrable systems?
I jotted down some typos
From Sept 18 version
smootjh; equivalrntly; such morphism are; This mans that; togther with.
From the Sept 16 version, but you may have removed them.
notably non-pertrubative; a local preqauntum; namley the formulation; In partiucular; the resulung; Lagragian; from the the notion; then motivate (needs s); betwee them; Euler-Lagrane; corresponding funcitons; def. 2.11 (X_1, \omega_1); two Lagrangian correspondence; with is a smooth space; in which set up everything; differentially cohesive [23]). full stop inside bracket; By constuction.
Thanks for the spotted typos! Yes, I had caught some of them meanwhile, but by far not all. Fixed now.
Concerning integrable systems etc.: so taken at face value what the discussion in section 2 gives is a translation/dictionary between tradtional symplectic formalism and a natural formulation that has pre-quantization/global effects automatically built in. I think of the perspective of section 2, if you allow, as being to the perspective of Arnold’s book as that is to the formulation of classical mechanics before him.
It is very unlikely that there are genuinely undiscovered new phenomena in classical mechanics, but I think the translation here helps a good bit to make transparent those phenomena which are notorious for being subtle and are actually mostly ignored: global reduced phase spaces, classical anomalies and similar global effects.
Specifically concerning integrable systems: here the dictionary says that an $n$-dimension system is integrable if in addition to its Hamiltonian time-flow one finds further $n$ commuting 1-parameter subgroups of $\mathbf{Aut}_{\mathbf{B}U(1)_{conn}}(prequantum\;bundle)$. For whatever that’s worth.
I think the point where something more than a new polished toolbox appears is section 3, which is meant to clarify the “multisymplectic” lore a little.
But in general, as the introduction hopefully says clearly enough, the goal of this particular set of notes is expressedly not to do a lot of new stuff, but to provide an introduction for how to chart the known stuff nicely. That’s why these notes are being being split off from Local prequantum field theory (schreiber).
I have now content that should be readable in the first part of section 3, including
3.1 – Local field theory Lagrangians and $n$-plectic smooth spaces
3.2 – Local observables, conserved currents and higher Poisson bracket homotopy Lie algebras
3.3 – Field equations of motion, higher Maurer-Cartan elements, and higher Lie integration
Section 3.3 currently ends with the observation that the Hamilton-de Donder-Weyl field equations on an $n$-plectic extended phase space $(X,\omega)$ characterize equivalently the $L_\infty$-homomorphisms
$\mathbb{R}^n \longrightarrow \mathfrak{pois}(X,\omega)$into the Lie $n$-algebra of local observables/of currents, hence that they characterize the Maurer-Cartan elements in $\wedge^\bullet(\mathbb{R}^n)\otimes \mathfrak{pois}(X,\omega)$.
Since $\mathfrak{pois}(X,\omega)$ Lie integrates to the automorphism $n$-group of a prequantum $n$-bundle in the higher cohesive slice topos over $\mathbf{B}^n U(1)_{conn}$, this are equivalently something like infinitesimal approximations to $n$-functors of the form
$Bord_n^{Riem} \longrightarrow \mathbf{H}_{/\mathbf{B}^n U(1)_{conn}} \,.$That will be in section 3.4, I guess, to be written still.
Specifically concerning integrable systems… For whatever that’s worth.
I was wondering why Ben-Zvi wrote on geometric representation theory
Geometric representation theory has close and profound connections to many fields of mathematics, which we expect to play a significant role in the program. Perhaps the most significant are to number theory, via the theory of automorphic forms, L-functions and modularity. Much current activity in the field is motivated either directly by problems in number theory or by more tractable geometric analogues thereof. Another major influence on the subject comes from physics, in particular gauge theory, integrable systems, and recently topological string theory
but I see the answer is at his page
The characteristic feature of integrable mechanical systems is the presence of tori in the phase space, along which the flows are linearized. A crucial development in the study of infinite-dimensional integrable systems, or soliton equations, was the discovery that the phase spaces often contain many such tori that are in fact Jacobian varieties, associated with algebraic curves known as spectral curves. Thus one can start from an algebraic curve with some extra structure, and construct a special “algebro-geometric” solution to a soliton equation. (See [SW].) In my thesis [BF1] a simple, direct and general link is established between integrable systems in algebraic geometry (the study of spectral curves and their Jacobians) and soliton equations (usually expressed as flows on spaces of flat connections). It is derived from an abstract group theoretic construction, when applied to loop groups.
I have now typed what I’ll use as talk slides here from slide 69 on.
Typo
“Physical concept and mathmatical formalization”, p. 5
“can - and traditionally is - also be applied” p. 33, awkward. Maybe “can also be - and traditionally is - applied” or “can also be - and traditionally also is - applied”
(while we’re there, why ” if only we allow Y to be a smooth space more general than a finite-dimensional manifold”, when you explicitly say in example 2.2. that F is “an infinite dimensional Frechet manifold”)? Or did you mean you already made this generalisation in example 2.2?
Section 3.1, p. 33, have you explained what $X$ and $\Sigma$ are, or should they be the same? But $X$ then reappears as a jet bundle.
eqipped; are are the canonical coordinates; p. 33
Thanks, David!
Right now I am in the awkward situation that I would like to fix these typos, but my computer still hasn’t recovered from being watered. I am hoping that once when all fluid remnants inside it have evaporated, it will come back, but it still hasn’t. (Though the characters that it produces when I hit keys keep changing over time, so something is still happening inside it…)
And bad luck continues: The institute computers here don’t have LaTeX installed. Next: The web-application “WriteLaTeX” effectively freezes when I hand it my lengthy tex-file, so that doesn’t work either.
So I need to see what I can do about it…
But thanks again so much for consistently providing so much feedback! I really appreciate it.
So I just come out of my talk. This seems to have worked well, now that I had more than 15 minutes… Maybe I make a post about it later.
Is anything interesting likely to happen when passing from classical mechanics to statistical mechanics in the framework of HoTT?
So this is something I definitely need to explore further. The thing is that the “motivic quantization” in homotopy type theory already works by a mix of “Euclidean”/”Wick rotated” QFT (statistical mechanics) and genuine QFT.
I am not claiming that I fully understand what the big story is. I just observe that for the quantizations that we do understand, such as discussed in master thesis Nuiten (schreiber), the formalism gives both, automatically.
For instance a mechanical system is quantized by realizing it as the boundary field theory of the Poisson sigma model and the corresponding motivic quantization effectively computes the Wick-rotated partition function of an auxiliary Dirac operator/supersymmetric quantum mechanics, which then by holography turns out to be the genuine quantization of the boundary mechanical system.
See also remark 1 here which is some kind of shadow of this effect.
under T-duality
We had a tad more of a mention at T-fold. Here is now a start for: double field theory.
Not sure what you have in mind. I find this a strange statement because DFT is sort of by definition about T-duality equivariant field theory, so what would it even mean to “expand beyond”? Also, the literature so far discusses only a rather naive and restricted version of what one imagines T-folds should actually be, so I think on the contrary that my statement “field theory on T-folds” is already beyond what is actually in the literature. But maybe you could point me to what you are thinking of.
I know that Thomas Nikolaus has some non-naive ideas about how to go about general T-folds. based on the “T-duality 2-group” which I called $\Omega A$ in remark 3 at T-Duality and Differential K-Theory.
I’ll add
Is this a form of higher Cartan geometry?
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